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+Consensus based optimization with memory effects:
+random selection and applications
+Giacomo Borghi∗
+Sara Grassi†
+Lorenzo Pareschi†
+February 1, 2023
+Abstract
+In this work we extend the class of Consensus-Based Optimization (CBO) metaheuris-
+tic methods by considering memory effects and a random selection strategy. The proposed
+algorithm iteratively updates a population of particles according to a consensus dynamics
+inspired by social interactions among individuals. The consensus point is computed taking
+into account the past positions of all particles. While sharing features with the popular Parti-
+cle Swarm Optimization (PSO) method, the exploratory behavior is fundamentally different
+and allows better control over the convergence of the particle system. We discuss some im-
+plementation aspects which lead to an increased efficiency while preserving the success rate
+in the optimization process. In particular, we show how employing a random selection strat-
+egy to discard particles during the computation improves the overall performance. Several
+benchmark problems and applications to image segmentation and Neural Networks training
+are used to validate and test the proposed method. A theoretical analysis allows to recover
+convergence guarantees under mild assumptions of the objective function. This is done by
+first approximating the particles evolution with a continuous-in-time dynamics, and then by
+taking the mean-field limit of such dynamics. Convergence to a global minimizer is finally
+proved at the mean-field level.
+Keywords: consensus-based optimization, stochastic particle methods, memory effects, ran-
+dom selection, machine learning, mean-field limit
+Contents
+1
+Introduction
+2
+2
+Consensus-based optimization with memory effects
+4
+2.1
+Particles update rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.2
+Random selection strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.3
+Comparison with CBO and PSO . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+∗RWTH
+Aachen
+University,
+Institute
+for
+Geometry
+and
+Applied
+Mathematics,
+Aachen,
+Germany
+([email protected])
+†University of Ferrara, Department of Mathematics and Computer Science & Center for Modelling Computing
+and Statistics, Ferrara, Italy ([email protected], [email protected])
+1
+arXiv:2301.13242v1  [math.OC]  30 Jan 2023
+
+3
+Numerical results
+7
+3.1
+Tests on benchmark problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.2
+Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+3.2.1
+Image segmentation
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+3.2.2
+Approximating functions with NN . . . . . . . . . . . . . . . . . . . . . .
+16
+3.2.3
+Application on MNIST dataset . . . . . . . . . . . . . . . . . . . . . . . .
+17
+4
+Theoretical analysis
+19
+4.1
+Mean-field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+19
+4.2
+Convergence in mean-field law . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+4.3
+Random selection analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+5
+Conclusions
+25
+A Proofs
+25
+A.1 Notation and auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+A.2 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+A.3 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+A.4 Proof of Proposition 4.2 and Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . .
+30
+1
+Introduction
+Meta-heuristic algorithms are recognized as trustworthy, easy to understand and to adapt op-
+timization methods which have been widely applied to a several fields such as Machine Learn-
+ing [28], path planning [29] and image processing [45], to name a few. Starting form a set of
+possible solutions, a meta-heuristic algorithm typically updates such set iteratively by combining
+deterministic and stochastic choices, often inspired by natural phenomena. Exploration of the
+search space and exploitation of the current knowledge are the two fundamental mechanisms
+driving the algorithm iteration [46]. Examples of established meta-heuristic algorithms are given
+by Genetic Algorithm (GA) [17,42], Simulated Annealing (SA) [25], Particle Swarm Optimiza-
+tion (PSO) [24] and Differential Evolution (DE) [40]. We refer to [21] for a complete literature
+review.
+Consensus-Based Optimization (CBO) is a class of gradient-free meta-heuristic algorithms
+inspired by consensus dynamics among individuals. After its introduction [34] it has gained
+popularity among the mathematical community due to its robust mathematical framework [3,9,
+16,19]. In CBO algorithms, a population of particles concentrates around a consensus point given
+by a weighted average of the particles position. In the computation of such consensus point, more
+importance is given to those particles attaining relatively low values of the objective function.
+The exploration mechanism is introduced by randomly perturbing the particles positions at each
+iteration. Particles which are close to the consensus point are subject to small perturbations,
+while those that are far from it display a more exploratory behavior.
+In this work, following the recent analysis in [14], we study a Consensus-Based Optimization
+algorithm with Memory Effects (CBO-ME) where the consensus point is computed among the
+whole history of the particles positions and not just on the positions of the current iteration, as
+2
+
+in the original CBO method. This is done by keeping track of the best position found so far by
+each particle, and computing the consensus point among these “personal” bests. While sharing
+common elements with PSO, such as convergence to a promising point and the presence of
+personal bests, CBO-ME differs in the way the exploration mechanism is implemented. Indeed,
+in CBO-ME, as in CBO algorithms, the stochastic behavior is given by adding Gaussian noise to
+the particles dynamics and can be tuned independently on the exploitation mechanisms, leading
+to a better control over the particles convergence. Therefore, while in classical PSO methods it
+is the balance between local best and global best that governs the optimization strategy, in CBO
+methods it is the balance between exploration and exploitation mechanisms that determines the
+choice of parameters. We recall that a generalization of PSO methods that allows leveraging
+the same flexibility in searching the global minimum as in CBO algorithms has been recently
+presented in [14].
+Many real-life problems, especially those regarding Machine Learning, require to optimize a
+large number of parameters. Therefore, it essential to design fast algorithm to save computa-
+tional time and memory. This is a major weakness of swarm-based methods, which require a set
+of particles to minimize the problem, unlike gradient-based methods that can work on a single
+particle trajectory. For methods based on a collection of particles, existing algorithms can be
+improved by discarding particles whenever the system has a prominent exploitative behavior.
+This is sometimes referred as “natural selection strategy” in the DE literature [27,40] and aims
+to discard the non-promising solutions. Inspired by particle simulations techniques where it
+is important to preserve the particles distribution, we examine a “random selection strategy”
+where particles are discarded randomly based on the local consensus achieved. We will discuss
+such implementation aspects by testing CBO-ME against high-dimensional learning problems
+and theoretically analyze the impact of the random selection strategy on the system. In partic-
+ular, we prove that if the full particle system is expected to converge towards a solution to (2.1),
+so will the reduce one, provided a sufficient number of particles remains active. Note that, such
+analysis can be generalized to other particle dynamics and may be of independent interest.
+Owing to the convergence analysis of CBO algorithms [3, 9, 10, 19] and recent analysis of
+PSO [14, 20] we are able to prove convergence of the algorithm under mild assumption on the
+objective function. This is done by first approximating the algorithm with a continuous-in-
+time dynamics and secondly by giving a probabilistic description to the particles system. By
+assuming propagation of chaos [41], particles are considered to behave independently according
+to the same law. This allows to reduce the possible large system of equations to a single partial
+differential equation: the so-called mean-field model. Such model is then analyzed to recover
+convergence guarantees under precise assumption on the objective function. Developed in the
+field of statistical physics, this approach has shown be fruitful in studying particle-based meta-
+heuristic algorithms [9,10,20]. We note that convergence in mean-field law was recently proved
+in [37] in an independent work.
+The rest of the paper is organized as follows.
+Section 2 is devoted to the introduction
+of the CBO-ME algorithm with random selection and comparison with CBO methods without
+memory effects and PSO. In Section 3 validate the proposed methods against several benchmark
+problems and two Machine Learning tasks. Theoretical convergence guarantees and analysis of
+the random selection strategy are summarized in Section 4. Some final remarks are given in
+Section 5. Technical details of the theoretical analysis are given in Appendix A.
+3
+
+2
+Consensus-based optimization with memory effects
+In this section, we present the Consensus-Based Optimization algorithm with Memory Effects
+(CBO-ME) to solve problems of the form
+x∗ ∈ argmin
+x∈Rd F(x) ,
+(2.1)
+where Rd, d ∈ N is the, possibly large, search domain for the continuous function F ∈ C(Rd, R).
+We will do so by highlighting similarities and differences between classical CBO methods and
+PSO algorithms.
+2.1
+Particles update rule
+At each iteration step k and for every particle i = 1, . . . , N, we store its position xk
+i and its best
+position found so far yk
+i = argmin{F(xk
+1), . . . , F(xk
+N)}. The best positions are used to compute
+a consensus point
+¯yα,k =
+N
+�
+i=1
+ωk
+i yk
+i
+with
+ωk
+i =
+e−αF(yk
+i )
+�N
+j=1 e−αF(yk
+j )
+(2.2)
+which approximate the global best solution ¯yα,k among all particles and all times for α > 1.
+Indeed, thanks to the choice of the weights ωk
+i , we have that
+¯yα,k
+−→
+¯y∞,k := argmin{F(yk
+1), . . . , F(yk
+N)}
+(2.3)
+as α → ∞, provided that there is only one global best position among {yk
+1, . . . , yk
+N}. Such ap-
+proximation was first introduced for CBO methods [34] as it leads to more amenable theoretical
+analysis, but it also allows more flexibility. Indeed, relatively small values of α are typically
+used at the beginning of the computation to promote exploration. Large values of α, on the
+other hand, lead to better exploitation of the computed solutions and to higher accuracy. We
+note that the weights used in (2.2) correspond in statistical mechanics to the Boltzmann-Gibbs
+distribution associated with the energy F. In this context, α plays the role of the inverse of the
+system temperature T and the limit α → ∞ corresponds to T → 0.
+Once the consensus point ¯yα,k is computed, the particle positions are then updated according
+to the law
+xk+1
+i
+= xk
+i + λ
+�
+¯yα,k − xk
+i
+�
++ σ
+�
+¯yα,k − xk
+i
+�
+⊗ θk
+i
+(2.4)
+with θk
+i ∈ Rd randomly sampled from the normal distribution (θk
+i ∼ N(0, Id)) and where ⊗ is
+the component-wise product.
+The update rule is characterized by a deterministic component of strength λ ∈ (0, 1) promot-
+ing concentration around the consensus point ¯yα,k and a stochastic component of strength σ > 0
+promoting exploration of the search space. As the latter depends on the difference (¯yα,k − xk
+i ),
+the random behavior is stronger for particles which are far form the consensus point, whereas
+it is weaker for those that are close to it. Also, such exploration resemble an anisotropic diffu-
+sive behavior exploring every coordinate direction at a different rate. This approach was first
+proposed in [4] in the context of CBO methods and has been proved to suffer less from the
+curse of dimensionality with the respect to the originally proposed isotropic diffusion given by
+σ∥¯yα,k − xk
+i ∥2θk
+i with θk
+i being again a normally distributed d-dimensional vector [4].
+4
+
+2.2
+Random selection strategy
+When the particle system concentrates around the consensus point, showing a mostly exploita-
+tive behavior, we employ a particle selection strategy. Discarding particles introduces additional
+stochasticity to the system, while reducing the computational cost. Following the approach sug-
+gested in [7], we check the evolution of the system variance to decide how many particles to
+(eventually) discard.
+For a given set of particles z = {zi}i∈J, the system variance is given by
+var(z) := 1
+|J|
+�
+j∈J
+∥zj − m(z)∥2
+2
+with
+m(z) := 1
+|J|
+�
+i∈J
+zi ,
+(2.5)
+where |J| indicates the cardinality of I, that is, the number of particles in this context.
+Let Ik ⊆ {1, . . . , N} be the set of active particles at step k and Nk = |Ik|.
+To decide
+how many particles to select, we compare the variance of the particle system before the position
+update (2.4), xk = {xk}i∈Ik and after it, ˜xk+1 = {xk+1
+i
+}i∈Ik. Then, the number Nk+1 of particles
+we select for the next iteration is given by
+˜Nk+1 =
+�
+Nk
+�
+1 + µ var(˜xk+1) − var(xk+1)
+var(xk+1)
+��
+Nk+1 = min
+�
+max
+� ˜Nk+1, Nmin
+�
+, Nk
+�
+(2.6)
+⌊z⌋ being the integer part of a number z and Nmin ∈ N the smallest amount of particles we allow
+to have. Then, a subset Ik+1 ⊂ Ik, |Ik+1| = Nk+1, of particles is randomly selected to continue
+the computation. The parameter µ ∈ [0, 1] regulates the mechanism: for µ = 0 there is no
+particle discarding, while for µ = 1 the maximum number of particles is discarded if the variance
+is decreasing. As we will see in Section 3, this random selection mechanism dramatically reduces
+the computational time without affecting the algorithm performance. We will also theoretically
+analyze this aspect in Section 4.3, where we show that convergence properties are preserved.
+As stopping criterion, we keep a counter n on how many times ∥¯yα,k+1 − ¯yα,k∥2 is smaller
+than a certain tolerance δstall. If this happens for more than a given nstall number of times in a
+row, we assume the particles system found a solution and stop the computation. A maximum
+number of iteration kmax representing the computational budget is also given. The proposed
+CBO-ME is summarized in Algorithm 1.
+Remark 2.1. In the meta-heuristic literature, particles are usually discarded depending on their
+objective value, in a way that particles with high values are more likely to be discarded [27,40].
+The proposed strategy does not add a further heuristic strategy but simply cut down the algorithm
+complexity. Also, the convergence properties are in this way expected to be preserved. We note
+that, on the other hand, there is no straightforward way to generate particles and, at the same
+time, preserve the particle system distribution.
+2.3
+Comparison with CBO and PSO
+What distinguishes CBO-ME from plain CBO, see e.g [4,34], is clearly the introduction of the
+best positions {yk
+i }N
+i=1 and the fact that the consensus point is calculated among them and not
+5
+
+Algorithm 1: Consensus-Based Optimization with Memory Effects (CBO-ME)
+Input: F, N0, kmax, λ, σ, α, nstall and δstall;
+1 Inizialize N0 particle positions xi
+0, i = 1, . . . , N;
+2 y0
+i ← x0
+i for all i = 1, . . . , Nk;
+3 Compute yα,0 according to (2.2);
+4 k ← 0, n ← 0;
+5 while k < kmax and n < nstall do
+6
+for i = 1 to Nk do
+7
+θk
+i ∼ N(0, Id);
+8
+Compute xk+1
+i
+according to (2.4);
+9
+if F(xk+1
+i
+) < F(yk
+i ) then
+10
+yk+1
+i
+← xk+1
+i
+;
+11
+else
+12
+yk+1
+i
+← yk
+i ;
+13
+end
+14
+end
+15
+Compute ¯yα,k+1 according to (2.2);
+16
+if ∥¯yα,k+1 − ¯yα,k∥2 < δstall then
+17
+n ← n + 1;
+18
+else
+19
+n ← 0;
+20
+end
+21
+Compute Nk+1 according to (2.6);
+22
+if Nk+1 < Nk then
+23
+Randomly discard Nk+1 − Nk particles;
+24
+k ← k + 1;
+25 end
+26 return ¯yα,k, F(¯yα,k)
+just among the particle positions {xk
+i }N
+i=1 at that given time k. Indeed, the classical CBO update
+rule without memory effects (and with anisotropic diffusion and projection step) is given by
+xk+1
+i
+= xk
+i + λ
+�
+¯xα,k − xk
+i
+�
++ σ
+�
+¯xα,k − xk
+i
+�
+⊗ θk
+i
+(2.7)
+where ¯xα,k is defined consistently with (2.2) (by substituting yk
+i with xk
+i ). As we will see in the
+numerical tests, the use of memory effects improves the algorithm performance.
+Since alignment towards personal bests yk
+i and towards the global best ¯y∞,k are also the
+fundamental building blocks of PSO algorithms, we highlight now the main differences and
+similarities between PSO and CBO-ME. For completeness, we recall the canonical PSO method,
+see e.g. [36], using the notation of (2.4) for easier comparison
+�
+xk+1
+i
+= xk
+i + vk+1
+i
+vk+1
+i
+= wvk
+i + C1
+�
+yk
+i − xk
+i
+�
+⊗ ˆθk
+i,1 + C2
+�
+¯y∞,k − xk
+i
+�
+⊗ ˆθk
+i,2
+(2.8)
+6
+
+where vk
+i are the particles velocities, w, C1, C2 > 0 are the algorithm parameters and θk
+i,1, θk
+i,2
+are uniformly sampled from [0, 1]d (ˆθk
+i,1, ˆθk
+i,2) ∼ Unif([0, 1]d). Several variants and improvements
+have been proposed starting from the above dynamics, but a complete review is beyond the
+scope of this paper and we refer to the recent survey [47] for more references.
+We are interested in highlighting the main differences between (2.4) and (2.8) regarding
+the stochastic components: in CBO-ME deterministic and stochastic steps are de-coupled and
+tuned by two different parameters (λ and σ), while in PSO they are coupled. Indeed, in (2.8),
+deterministic and stochastic components are both controlled by the same parameter: C1 in
+the case of personal best dynamics and C2 for the global best one.
+By splitting the term
+C2
+�
+¯y∞,k − xk
+i
+� ˆθk
+i,2 into a deterministic step and a zero-mean term we obtain
+C2
+�
+¯y∞,k − xk
+i
+�
+⊗ ˆθk
+i,2 = C2
+2
+�
+¯y∞,k − xk
+i
+�
++ C2
+2
+�
+¯y∞,k − xk
+i
+�
+⊗ θk
+i,2
+(2.9)
+with θk
+i,2 = 2ˆθk
+i,2 − 1, θk
+i,2 ∼ Unif([−1, 1]d). Suggested in [14], such rewriting highlights how
+increasing the alignment strength towards the global best (by increasing C2) necessary increases
+the stochasticity of the system as well. In (2.4) and (2.7), on the other hand, one is allowed to
+tune the exploration and exploitation behaviors separately, by either changing parameters λ or
+σ.
+Clearly, CBO-ME also differs from PSO due to its first-order dynamics. Having the aim of
+resembling birds flocking, the first PSO algorithm [24] was proposed as a second-order dynamics.
+The inertia weight w, introduced later in [39], became an essential parameter to prevent early
+convergence of the swarm and to increase the global exploration behavior, especially at the
+beginning of the computation, see e.g. [31, 39] and reviews [18, 36, 47] for more references. We
+note that several other strategies have proposed to improve PSO exploration behavior, see,
+for example, [50]. As already mentioned, in CBO methods convergence and exploration are
+de-coupled and can be tuned separately. Therefore, to keep the algorithm more amenable to
+theoretical analysis, we consider a simpler first-order dynamics. We note that a CBO dynamics
+with inertia mechanism was proposed in [5].
+Similarly, we found the contribution given by the personal best alignment non-essential and
+difficult to tune. Thus, the lack of alignment towards personal best in (2.4). Replacing alignment
+towards personal best with gaussian noise was also suggested in [48] where authors proposed the
+Accelerated PSO (APSO) algorithm. Further studied in [11,49], APSO also allows to de-couple
+the stochastic component from the deterministic one and the noise is heuristically tuned to
+decrease during the computation (as in Simulated Annealing [25]). In CBO methods, the noise
+strength automatically adapts as it depends on the distance from the consensus point, which
+is also different for every particle. For completeness, we note that many other variants of PSO
+have been proposed to include the explorative behavior, see e.g. Chaotic PSO [30].
+3
+Numerical results
+Having discussed the fundamental features of the CBO dynamics with memory effects, we now
+validate Algorithm 1 and compare its performance with plain CBO and PSO algorithms. We will
+test the methods against several benchmark optimization problems and analyze the impact of the
+7
+
+Name
+Objective function F(x)
+Search space
+x∗
+F(x∗)
+Ackley
+−20 exp
+�
+−0.2
+�
+1
+d
+�d
+i=1 (xi)2
+�
+− exp
+�
+1
+d
+�d
+i=1 cos (2π(xi))
+�
++ 20 + e
+[−32, 32]d
+(0, . . . , 0)
+0
+Griewank
+1 + �d
+i=1
+(xi)2
+4000 − �d
+i=1 cos
+� xi
+i
+�
+[−600, 600]d
+(0, . . . , 0)
+0
+Rastrigin
+10d + �d
+i=1
+�
+(xi)2 − 10 cos (2π(xi))
+�
+[−5.12, 5.12]d
+(0, . . . , 0)
+0
+Rosenbrock
+1 − cos
+�
+2π
+��d
+i=1 (xi)2
+�
++ 0.1
+��d
+i=1 (xi)2
+[−5, 10]d
+(1, . . . , 1)
+0
+Salomon
+1 − cos
+�
+2π
+��d
+i=1 (xi)2
+�
++ 0.1
+��d
+i=1 (xi)2
+[−100, 100]d
+(0, . . . , 0)
+0
+Schwefel 2.20
+�d
+i=1 |xi|
+[−100, 100]d
+(0, . . . , 0)
+0
+XSY random
+�d
+i=1 ηi|xi|i,
+ηi ∼ Unif([0, 1])
+[−5, 5]d
+(0, . . . , 0)
+0
+XSY 4
+��d
+i=1 sin2(xi) − e − �d
+i=1(xi)2�
+e − �d
+i=1 sin2 √
+|xi|
+[−10, 10]d
+(0, . . . , 0)
+−1
+Table 1: Considered benchmark test functions for global optimization. For each function,
+the corresponding search space and global solution is given.
+random selection technique on the convergence speed. We also employ 1 to solve problems arising
+form applications, such as image segmentation and training of a machine learning architectures
+for function approximation and image recognition.
+3.1
+Tests on benchmark problems
+We test the proposed algorithm against different optimization problems, by considering 8 bench-
+mark objective functions, see e.g. [22], which we report in Table 1 for completeness. The search
+space dimension is set to d = 20.
+As in plain CBO methods, we expect the most important parameters are those governing
+the balance between the exploitative behavior (λ in this case) and the explorative one (σ). In
+particular, we are interested in the algorithm performance as we change the ratio between λ
+and σ. Therefore, in the first experiment we fix λ = 0.01, while considering different values of
+σ. The parameter α is adapted during the computation: starting form α0 = 10, it increases
+according to the law
+α = α0 · k · log2(k) .
+(3.1)
+Fig. 1 shows the accuracy and the objective value reached for σ ∈ [0, 2] after kmax = 104
+algorithm iterations with N = 200 particles, with no random selection. The optimal value for σ
+is clearly problem-dependent, but we note that the optimal values for the problems considered
+all fall within a relative small range (underlined in gray in Fig. 1).
+From Fig.
+1 we infer that a good value for all benchmark problems considered is given
+by σ = 0.8. Using this value, we now compare CBO-ME, with plain CBO and the standard
+PSO (with and without alignment towards personal best) for different population sizes N =
+50, 100, 200. We keep the random selection mechanism off by setting µ = 0 and use the same
+8
+
+0
+0.5
+1
+1.5
+2
+<
+10!10
+10!5
+100
+105
+1010
+ky,;k ! x$k1
+Rastrigin
+Ackley
+Griewank
+Rosenbrock
+Salomon
+Schwefel
+XSY 4
+XSY random
+(a) ∥¯yα,k − x∗∥∞
+0
+0.5
+1
+1.5
+2
+<
+10!10
+10!5
+100
+105
+1010
+F(7y,;k)
+Rastrigin
+Ackley
+Griewank
+Rosenbrock
+Salomon
+Schwefel
+XSY 4
+XSY random
+(b) F(¯yα,k)
+Figure 1: Optimization on benchmark functions using CBO-ME. Behavior of the expec-
+tation error and fitness value for different values of σ. Here λ = 0.01 and α is adaptive,
+with α0 = 10. The particle population is N = 200. Grey bands (of values [0.70, 1.05] for
+the error and [0.65, 1] for the fitness) show the range in which the minima of the different
+benchmark functions fall. The dotted line marks the visually estimate pseudo-optimal value
+σ = 0.8. Results averaged on 250 runs, are obtained with kmax = 104 iterations and without
+stopping criterion.
+previously chosen parameters when memory effects are used. For plain CBO, without memory
+effects, we set σ = 0.71 ≈
+√
+2/2. Concerning PSO, we use the solver provided by the MATLAB
+Global Optimisation Toolbox (particleswarm), changing the maximum number of iterations
+and the stall condition to the one used for CBO methods, to make the results comparable.
+The remaining parameters are kept as described in the relative documentation [33]. We set
+kmax = 104, δstall = 10−4 and consider a run successful when either
+∥¯yα,k − x∗∥∞ < 0.1
+or
+|F(¯yα,k) − F(x∗)| < 0.01 .
+(3.2)
+Table 2 reports success rate, final error given by ∥¯yα,k −x∗∥∞, mean objective function value
+and total number of iterations, averaged over 250 runs. In addition to the classic PSO method,
+where the acceleration coefficients are chosen to be equal C1 = C2 = 1.49, Table 2 also shows
+the results when only the alignment towards global best is considered in PSO (C1 = 0).
+While CBO already manages to find the global minimizer in most of the problems considered,
+we note that it fails when Rastrigin, Rosenbrock or XSY random functions are optimized. CBO-
+ME, on the other hand, is able to solve the optimization problem correctly even in these cases
+if the population size N is large enough. CBO seems to achieve greater accuracy in some cases,
+such as with Schwefel 2.20 and Salomon objectives, at the cost of more iterations. Standard
+PSO in many cases fails to solve the problem, see e.g. Rastrigin, Salomon or XSY 4 functions.
+PSO success rate is also lower among all problems, with the exception of the Schwefel 2.20
+benchmark problem. Considering only global adjustment seems to show advantages with respect
+to the classical PSO method, except in the case of Ackley where setting C1 = 0 decreases the
+success rate or, in the case of XSY 4, Salomon or Rastrigin, where convergence is not achieved
+even for C1 = 0 Consensus methods, however, seem to perform better in terms of both success
+9
+
+(a) Error: ∥¯yα,k − x∗∥∞
+(b) Fitness Value: F(¯yα,k)
+Figure 2: Optimization of Ackley function for different values of the random selection
+parameter µ, where the initial particle population is N 0 = 104. We report error (on the
+left) and fitness values (on the right) as the number of function evaluations increases.
+Parameters are set as λ = 0.01, σ = 0.8, α adaptive starting from α0 = 10 and following
+the law α = α0 · k · log2(k). Results are averaged over 250 runs.
+rate and speed up. In addition, for most problems, the population size N seems not to play a
+significant role in the algorithms performance. This further motivates the introduction of the
+random selection strategy described in the Section 2.1 in order to save computational costs.
+In the third experiment, we test the proposed random selection mechanism (2.6) for different
+values of the parameter µ. We recall that with µ = 0 we have no particles removal, while as
+µ increases, more particles are likely to be discarded when the system variance decreases. The
+initial population is set to N0 = 200, while the minimum number of particles to Nmin = 10.
+Results are reported in Tables 3 and 4 in terms of: success rate, error, objective value, weighted
+number of iterations, given by
+witer =
+kend
+�
+k=1
+Nk
+N0
+(3.3)
+and percentage of Computational Time Saved (CTS). Results show that relative large values
+of µ allow to reach fast convergence without affecting the algorithm performance. The values of
+µ considered in Table 4 as different from those in Table 3 as in our experiments, the Rastrigin
+problem allows for larger values of µ, while the Rosenbrock one seems to be more sensitive to
+the selection mechanism with respect to the other objectives. In both cases, a suitable value of
+µ reduces the computational time with almost no impact in terms of accuracy.
+Fig.s 2 and 3 show error and fitness value as a function of the number of fitness evaluation
+during the algorithm computation, for the Ackley and Rastrigin problem respectively. Several
+values of µ are considered to display how the random selection mechanism affects the convergence
+speed.
+Initial particle population is set to N0 = 104 and particles evolve for kmax = 104
+iterations. We note how convergence speed increases as µ increases.
+10
+
+CBO (σ =
+√
+2/2)
+CBO-ME (σ = 0.8)
+PSO
+PSO (C1 = 0)
+N = 50
+N = 100
+N = 200
+N = 50
+N = 100
+N = 200
+N = 50
+N = 100
+N = 200
+N = 50
+N = 100
+N = 200
+Ackley
+Rate
+99.3%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+14.6%
+38.6%
+53.3%
+4.0%
+16.0%
+39.3%
+Error
+4.11e-06
+2.03e-06
+2.55e-06
+2.39e-06
+1.73e-06
+1.74e-06
+6.97e-09
+8.56e-11
+2.16e-12
+2.30e-08
+1.90e-10
+8.96e-13
+Favg
+1.18e-04
+5.81e-05
+7.30e-05
+1.54e-04
+4.96e-05
+4.99e-05
+6.24e-09
+7.65e-11
+1.94e-12
+2.06e-08
+1.70e-10
+8.01e-13
+Iterations
+954.3
+778.2
+678.1
+997.7
+724.3
+626.9
+493.8
+420.6
+391.4
+502.0
+436.0
+390.1
+Griewank
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+46.0%
+48.6%
+55.3%
+50.0%
+58.7%
+78.0%
+Error
+2.20e-02
+2.21e-02
+2.24e-02
+2.13e-02
+2.16e-02
+2.25e-02
+7.34e-02
+1.56e-02
+9.45e-03
+1.17e-01
+1.10e-01
+8.96e-02
+Favg
+5.26e-02
+5.31e-02
+5.47e-02
+4.95e-02
+5.15e-02
+5.82e-02
+3.23e-03
+4.11e-03
+3.78-03
+3.73e-03
+3.71e-03
+2.90e-03
+Iterations
+927.5
+777.0
+682.7
+891.9
+735.0
+635.4
+436.0
+394.5
+374.5
+427.2
+370.2
+345.5
+Rastrigin
+Rate
+9.3%
+27.3%
+60.7%
+26.0%
+68.7%
+89.3%
+0.0%
+0.0%
+0.0%
+0.0%
+0.0%
+0.0%
+Error
+1.28e-04
+1.83e-04
+2.34e-04
+9.73e-05
+1.27e-04
+1.76e-04
+-
+-
+-
+-
+-
+-
+Favg
+4.51e-06
+9.03e-06
+1.46e-05
+2.54e-06
+4.31e-06
+8.28e-06
+-
+-
+-
+-
+-
+-
+Iterations
+1083.0
+933.7
+819.8
+1007.6
+922.5
+769.9
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+Rosenbrock
+Rate
+65.3%
+86.7%
+97.3%
+72.7%
+98.0%
+100.0%
+9.3%
+22.6%
+36.6%
+46.7%
+60.7%
+76.7%
+Error
+1.81e-02
+2.44e-02
+1.48e-02
+3.62e-02
+4.04e-02
+1.78e-02
+6.19e-04
+2.56e-04
+1.67e-04
+4.44e-02
+4.45e-02
+4.46e-02
+Favg
+6.13e-03
+7.57e-03
+2.40e-03
+1.26e-02
+1.42e-02
+2.65e-03
+3.80e-02
+3.76e-02
+2.56e-02
+2.56e-03
+8.95e-04
+3.71e-04
+Iterations
+5772.0
+5440.3
+5439.2
+5955.7
+4977.3
+4275.9
+4830.0
+3322.4
+2892.7
+5886.4
+3419.1
+2164.2
+Schwefel 2.20
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+Error
+5.79e-06
+8.23e-07
+2.44e-07
+8.42e-06
+1.03e-06
+2.76e-07
+8.34e-10
+1.97e-12
+4.58e-14
+1.68e-07
+3.41e-10
+8.03e-14
+Favg
+1.04e-03
+2.15e-04
+8.36e-05
+1.50e-03
+3.12e-04
+9.37e-05
+1.94e-09
+6.36e-12
+1.52e-13
+2.44e-07
+6.48e-10
+2.46e-13
+Iterations
+814.7
+691.5
+619.2
+670.8
+547.2
+467.7
+484.3
+428.0
+401.7
+593.3
+457.8
+410.3
+Salomon
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+100.0%
+0.0%
+0.0%
+0.0%
+0.0%
+0.0%
+0.0%
+Error
+3.12e-02
+2.14e-02
+1.87e-02
+5.28e-02
+4.49e-02
+3.91e-02
+-
+-
+-
+-
+-
+-
+Favg
+3.14e-01
+2.15e-01
+1.88e-01
+2.44e-01
+1.86e-01
+1.91e-01
+-
+-
+-
+-
+-
+-
+Iterations
+10000.0
+10000.0
+10000.0
+8886.4
+9296.2
+2456.5
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+XSY random
+Rate
+55.3%
+84.7%
+92.0%
+100.0%
+100.0%
+100.0%
+1.2%
+11.7%
+21.0%
+100.0%
+100.0%
+100.0%
+Error
+2.64e-02
+1.62e-02
+9.80e-03
+3.06e-02
+1.86e-02
+1.15e-02
+2.25e-01
+9.56e-02
+8.42e-02
+6.23e-02
+5.12e-02
+2.34e-02
+Favg
+6.95e-08
+3.54e-08
+2.13e-08
+2.21e-06
+4.85e-08
+3.17e-08
+3.35e-04
+2.28e-04
+1.34e-04
+8.22e-04
+4.11e-04
+3.45e-04
+Iterations
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+XSY 4
+Rate
+22.0%
+87.3%
+98.7%
+23.3%
+86.7%
+100.0%
+0.0%
+0.0%
+0.0%
+0.0%
+0.0%
+0.0%
+Error
+8.07e-01
+7.48e-01
+7.16e-01
+8.44e-01
+7.35e-01
+6.95e-01
+-
+-
+-
+-
+-
+-
+Favg
+4.79e-07
+3.78e-07
+3.46e-07
+1.58e-06
+8.56e-07
+5.43e-07
+-
+-
+-
+-
+-
+-
+Iterations
+10000.0
+10000.0
+10000.0
+9677.5
+9128.4
+8943.2
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+10000.0
+Table 2: Comparison between classical CBO, CBO-ME and standard PSO with and with-
+out alignment towards personal best on benchmark problems. The solver particleswarm
+available in the MATLAB Global Optimisation Toolbox was used for the results concerning
+the PSO method. Optimal choice of parameters, different for each method, are used for the
+CBO algorithms. Same stopping criterion and definition of success, see (3.2), were used.
+Performance metric considered: success rate (see (3.2)), error (∥¯yα,k−x∗∥∞), fitness value
+F(¯yα,k) and number of iterations. Results are averaged over 250 runs.
+11
+
+µ = 0
+µ = 0.05
+µ = 0.1
+µ = 0.2
+Ackley
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+Error
+1.84e-06
+2.16e-06
+6.54e-06
+1.34e-05
+Favg
+7.30e-05
+6.17e-05
+1.87e-04
+3.95e-04
+witer
+674.2
+505.2
+357.2
+182.1
+CTS
+-
+31.1%
+51.6 %
+73.8%
+Griewank
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+Error
+2.35e-02
+2.22e-02
+2.32e-02
+2.28e-02
+Favg
+5.82e-02
+5.20e-02
+5.72e-02
+5.70e-02
+witer
+635.4
+395.9
+204.6
+184.6
+CTS
+-
+31.8%
+58.2%
+73.3%
+Schwefel 2.20
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+Error
+2.76e-07
+9.08e-07
+8.21e-07
+2.73e-08
+Favg
+9.37e-05
+2.93e-05
+1.58e-05
+3.74e-05
+witer
+467.7
+359.8
+318.7
+172.4
+CTS
+-
+24.4%
+32.9%
+64.1%
+Salomon
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+Error
+4.11e-02
+3.35e-02
+2.74e-02
+1.75e-02
+Favg
+4.34e-01
+4.43e-01
+4.07e-01
+3.26e-01
+witer
+2456.5
+1595.8
+1289.2
+913.0
+CTS
+-
+36.7%
+49.1%
+63.7%
+XSY random
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+Error
+1.50e-02
+8.62e-02
+8.89e-02
+9.08e-02
+Favg
+5.97e-07
+1.75e-05
+5.48e-05
+1.06e-04
+witer
+10000.0
+2642.3
+1755.7
+1123.7
+CTS
+-
+73.6%
+82.4%
+88.7%
+XSY 4
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+Error
+5.30e-01
+3.78e-01
+1.35e-01
+1.37e-01
+Favg
+1.17e-05
+6.28e-06
+3.41e-06
+3.55e-06
+witer
+8943.2
+3910.4
+1890.2
+1060.1
+CTS
+-
+46.9%
+68.1%
+79.4%
+Table 3: CBO-ME algorithm with random selection of particles tested against different
+benchmark functions with different values of µ, which regulates the random selection mech-
+anism. The system is initialized with N0 = 200 particles and σ = 0.8. Performance metric
+considered: success rate (see (3.2)), error (∥¯yα,k − x∗∥∞), fitness value F(¯yα,k), weighted
+iteration (3.3), and Computational Time Saved (CTS). Results are averaged over 250 runs.
+µ = 0
+µ = 0.1
+µ = 0.2
+µ = 0.5
+Rastrigin
+Rate
+100.0%
+100.0%
+100.0%
+100.0%
+Error
+9.14e-05
+7.12e-05
+3.77e-05
+1.24e-05
+Favg
+2.23e-06
+2.19e-06
+1.98e-06
+1.27e-06
+witer
+1161.1
+719.6
+256.5
+111.2
+CTS
+-
+38.1%
+77.9%
+90.4%
+µ = 0
+µ = 0.01
+µ = 0.02
+µ = 0.05
+Rosenbrock
+Rate
+100.0%
+100.0%
+99.4%
+99.0%
+Error
+2.55e-02
+2.23e-02
+1.66e-02
+1.341e-02
+Favg
+4.20e-03
+5.23e-03
+4.10e-03
+4.24e-03
+witer
+3172.3
+852.9
+347.8
+82.5
+CTS
+-
+73.1%
+89.1%
+97.4%
+Table 4: CBO-ME algorithm with particle reduction tested against Rastrigin and Rosen-
+brock functions with an higher diffusion parameter σ = 1.1 and for different values of µ ,
+which regulates the random selection mechanism. The system is initialized with N0 = 200
+particles. Performance metric considered: success rate (see (3.2)), error (∥¯yα,k − x∗∥∞),
+fitness value F(¯yα,k), weighted iteration (3.3), and Computational Time Saved (CTS).
+12
+
+(a) Error: ∥¯yα,k − x∗∥∞
+(b) Fitness Value: F(¯yα,k)
+Figure 3: Optimization of Rastigin function for different values of the random selection
+parameter µ where the initial particle population is N0 = 104. We report error (on the
+left) and fitness values (on the right) as the number of function evaluations increases.
+Parameters are set as λ = 0.01, σ = 1.1, α adaptive starting from α0 = 10 and following
+the law α = α0 · k · log2(k). Results are averaged over 250 runs.
+3.2
+Applications
+In this section, we propose some applications of the proposed optimization algorithm. First
+we consider a image segmentation problem using multi-thresholding, then we use the CBO-
+ME to train a Neural Network (NN) architecture to approximate functions and perform image
+classification on MNIST database of handwritten digits.
+3.2.1
+Image segmentation
+To perform image segmentation, we use a threshold detection technique, namely, the multidimen-
+sional Otsu algorithm [32,44] in order to compare the results to similar optimization algorithm,
+such as the Modified PSO in [43].
+In the Otsu algorithm, every pixel of the image is assigned to one of the possible L grayscale
+values. We denote with ηi the number of pixel with gray level i, 1 ≤ i ≤ L and Npix = �L
+i=1 ηi
+the total number of pixels [32]. Then, the image is divided into object C0 with gray-level [1, . . . , l]
+and background C1 with gray-level [l + 1, . . . , L] by inserting a threshold l. The probabilities of
+class occurrence and the class mean level for the object, respectively, are given by
+ω0(l) =
+l
+�
+i=1
+pi,
+pi =
+ηi
+Npix
+µ0(l) =
+l
+�
+i=1
+ipi
+ω0(k) .
+For the background, the class occurrence probabilities and the class mean level are given by
+ω1(l) =
+L
+�
+i=l+1
+pi,
+pi =
+ηi
+Npix
+13
+
+µ1(l) =
+L
+�
+i=l+1
+ipi
+ω1(k) .
+As in [32], the best threshold l∗ is obtained when the variance formula
+f(l) = ω0(l) ω1(l) (µ0(l) − µ1(l))2
+(3.4)
+between object group and background reaches its maximum value, i.e. l∗ = argmaxlf(l). The
+problem is then reduced to a threshold problem, which we can solve with optimization methods.
+Since segmentation is a trivial one-dimensional problem, we consider an extension of Otsu’s
+technique to the multidimensional case [44] to test capabilities of method. Assuming we want to
+optimize the choice of d thresholds, we require d + 1 classes of different gray-scales (C0, . . . , Cd)
+with relative probabilities of occurrence classes defined as
+ω0(l1) =
+l1
+�
+i=1
+pi , . . . , ωd(ld) =
+L
+�
+i=ld+1
+pi,
+pi =
+ηi
+Npix
+and classes mean levels
+µ0(l1) =
+�l1
+i=1 ipi
+ω0
+, . . . , µd(ld) =
+�L
+i=ld+1 ipi
+ωd
+,
+The optimal thresholds (ˆl1, . . . , ˆld) are those that satisfy ˆl1 < · · · < ˆld and maximise
+f(l1, . . . ld) =
+d
+�
+i=1
+ωi(li)µ2
+i (li)
+(3.5)
+For the experiment, we chose d = 5 thresholds and compare the segmentation performed by
+Otsu’s method, solved with both standard PSO and CBO-ME, with segmentation obtained by
+dividing the greyscale into d + 1 uniformly spaced intervals. For PSO, we use to the default
+parameters in the particleswarm function in the MATLAB Global Optimisation Toolbox, while
+for CBO-ME we used optimal parameters found in Section 3.1 and exploit the random selection
+technique to speed up the algorithm.
+We report the results on two sample images, Fig.s 4 and 5. We fix kmax = 103 and average
+results over 250 runs. As in [2], we evaluate multi-thresholding segmentation through the Peak
+Signal to Noise Ratio (PSNR) computed as:
+PSNR = 20 · log10
+�
+255
+RMSE
+�
+where RMSE is the Root Mean-Squared Error, defined as
+RMSE =
+�
+�
+�
+�
+1
+Npix
+Nrow
+�
+i=1
+Ncol
+�
+j=1
+[I(i, j) − S(i, j)]2
+where Npix = Nrow · Ncol, I is the original image and S is the associated segmented image.
+The higher the value of PSNR is, the greater the similarity between the clustered image and
+the original image is. From Fig.s 4,5, we note that the most accurate segmentation on details
+is obtained by the CBO-ME method. This is quantitatively confirmed by the PSNR values
+reported in Table 5.
+14
+
+(a) Original
+(b) Standard segmentation
+(c) Otsu seg. (PSO)
+(d) Otsu seg. (CBO-ME)
+Figure 4: Image segmentation of darkhair woman image (256 × 256 pixels) with standard
+segmentation and Otsu segmentation solved respectively by PSO (c) and by CBO-ME (d);
+results are averaged over 250 runs, with an initial population of N0 = 103 particles.
+(a) Original
+(b) Standard segmentation
+(c) Otsu seg. (PSO)
+(d) Otsu seg. (CBO-ME)
+Figure 5: Image segmentation of lake image (256×256 pixels) with standard segmentation
+and Otsu segmentation solved respectively by PSO (c) and by CBO-ME (d); results are
+averaged over 250 runs, with an initial population of N0 = 103 particles.
+cameraman
+lake
+lena
+peppers
+woman darkhair
+Standard segmentation
+22.83
+21.72
+24.35
+27.24
+25.33
+Otsu segmentation
+(PSO)
+34.62
+32.33
+38.19
+38.03
+37.14
+Otsu segmentation
+(CBO-ME)
+37.22
+35.44
+38.72
+38.28
+39.57
+Table 5: PSNR values to evaluating the advantages of the method in optimising threshold
+values in 5 sample images known in literature. For these results, we compared the Otsu
+segmentation solved by the proposed CBO-ME method with the classical PSO method with
+equispaced thresholding segmentation. Experiments are performed with d = 5 thresholds.
+15
+
+-1
+-0.5
+0
+0.5
+1
+-2
+-1
+0
+1
+2
+(a) 2000 epochs
+-1
+-0.5
+0
+0.5
+1
+-2
+-1
+0
+1
+2
+(b) 3000 epochs
+-1
+-0.5
+0
+0.5
+1
+-2
+-1
+0
+1
+2
+(c) 5000 epochs
+-1
+-0.5
+0
+0.5
+1
+-2
+-1
+0
+1
+2
+(d) 8000 epochs
+Figure 6: Approximating smooth function u1 (3.8) using a network with n = 50 and m = 3.
+The learning rate is λ = 0.2 and we initially use N0 = 500 particles. The others parameters
+are set as λ = 1, σ = 0.8 and α adaptive starting from α0 = 10.
+3.2.2
+Approximating functions with NN
+In this section, we use the proposed CBO-ME algorithm to train a NN architecture into approx-
+imating a function u : I → R, I ⊂ R with low regularity. As in [5], we use a fully-connected NN
+with m layers
+f(x; θ) = (Lm ◦ . . . L2 ◦ L1)(x)
+(3.6)
+where each layer is given by
+Li = σ(W ix + bi)
+with σ(x) = 1/(1+exp(−x)) being the sigmoid function. We use internal layers of dimension n,
+so W 1 ∈ Rn×1, b1 ∈ R, W m ∈ R1×n, bm ∈ Rd and W i ∈ Rn×n for all i = 2, . . . , m − 1. In (3.6),
+all DNN parameters are collected in θ = {W i, bi}m
+i=1.
+As loss function which need to be minimized, we consider the L2-norm between the target
+function u and its NN approximation f(· ; θ)
+F(θ) := ∥f(· ; θ) − u∥L2(I) .
+(3.7)
+Again, similarly to [5], we test the method against the following two functions:
+u1(x) = sin(2πx) + sin(8πx2)
+(3.8)
+16
+
+-1
+-0.5
+0
+0.5
+1
+-2
+-1
+0
+1
+2
+(a) 2000 epochs
+-1
+-0.5
+0
+0.5
+1
+-2
+-1
+0
+1
+2
+(b) 3000 epochs
+-1
+-0.5
+0
+0.5
+1
+-2
+-1
+0
+1
+2
+(c) 5000 epochs
+-1
+-0.5
+0
+0.5
+1
+-2
+-1
+0
+1
+2
+(d) 8000 epochs
+Figure 7:
+Approximating non-smooth u2 (3.9) function using a network with n = 50,
+m = 3. The learning rate is λ = 0.2 and we use initially N0 = 500 particles. The others
+parameters are set as λ = 1, σ = 0.8 and α adaptive starting from α0 = 10.
+u2(x) =
+�
+�
+�
+�
+�
+1
+if x < − 7
+8, − 1
+8 < x < 1
+8, x > 7
+8
+−1
+if
+3
+8 < x < 5
+8, − 5
+8 < x < − 3
+8,
+0
+otherwise .
+(3.9)
+We note that u1 is smooth, while u2 is discontinuous. Parameters of the CBO-ME algorithm
+have been set to λ = 0.01, σ = 0.8, as in the previous sections. Parameter α is adapted during
+the computation as in 3.1 and random selection mechanism is used. We employ m = 3 layers
+with internal dimension n = 50. Results are displayed in Fig.s 6 and 7. We note that smooth
+function u1 is well-approximated already after 5000 epochs, while convergence is slower for the
+discontinuous step function u2.
+3.2.3
+Application on MNIST dataset
+We now employ the proposed algorithm to train a NN architecture to solve a image classification
+tasks. We will consider the MNIST dataset [26] composed of handwritten digits in grayscale with
+28 × 28 pixels. For better comparability with CBO methods without memory effects, we closely
+follow the experiment settings used in the literature [4,10,37], which we summarize below.
+We consider a 1-layer NN where input images x ∈ R28×28 are first vectorized x �→ vec(x) ∈
+R728 and then processed through a fully-connected layer with parameters θ = {W, b}, with
+17
+
+10
+20
+30
+40
+50
+Epochs
+0.4
+0.6
+0.8
+1
+Accuracy on test data
+CBO-ME
+CBO
+10
+20
+30
+40
+50
+Epochs
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+Loss
+CBO-ME
+CBO
+Figure 8:
+Performance during training of shallow NN (3.10) on image classification
+(MNIST dataset) with CBO-ME optimizer and plain CBO without memory effects [10].
+Training is performed by Algorithm 1 with N = 100 particles and no particle selection.
+Cross-entropy loss function (3.11) and adaptive parameters strategy (3.12) were used in
+the training.
+W ∈ R10×728, b ∈ R10. That is, the network is given by
+fSNN(x; θ) = softmax (ReLU (Wvec(x) + b) ) ,
+(3.10)
+where ReLU(z) = max{z, 0} (component-wise) and softmax(z) = (ez1, . . . , ezn)/(�
+i ezi) are
+the commonly activation functions.
+During the training, batch regularization is performed
+after ReLU is applied in order to speed up convergence. Given a training set {(xm, ℓm)}M
+m=1,
+xm ∈ R28×28, ℓm ∈ {0, 1}10 made of M image-label tuples we train the model by minimizing the
+categorical cross-entropy loss
+F(θ) = 1
+M
+M
+�
+m=1
+�
+−
+10
+�
+i=1
+ℓm
+i log(fi(xm, θ))
+�
+.
+(3.11)
+We employ a population on N = 100 particles throughout the entire computation, initially
+sampled from the standard normal distribution N(0, Id). Following the mini-batch approach
+suggested in [4], the consensus points ¯yα,k is computed only among a random subset of nN = 10
+particles, but all particles are updated at each step. The training data is divided in batches of
+nF = 60 images. The drift parameter is set to λ = 0.01, while σ and α are adapted during the
+computation after each epoch as
+σepoch = σ0/ log2 (epoch + 2)
+αepoch+1 = 2 · αepoch+1
+(3.12)
+starting form σ0 =
+√
+0.04, and α0 = 50,
+Fig.
+8 shows the results in terms of loss function (3.11) over the test data set and the
+accuracy reached in the classification task. While challenging state-of-the-art training methods
+18
+
+is beyond the scope of the experiment, we note how high-dimensional data optimization tasks
+can be solved with as little as N = 100 particles by the proposed method, obtaining results
+comparable with the literature on CBO methods [4, 10, 37]. Also, we remark that parameters
+have not been tuned extensively.
+4
+Theoretical analysis
+A strength of CBO algorithms lays on the possibility of theoretically analyze the particle system
+by relying on a mean-field approximation of the dynamics. We will illustrate in this section how
+to derive such approximation and present the main theoretical result regarding the convergence
+of the particle system towards a solution to (2.1), in case of no selection mechanism. Next,
+we will study the impact of the random selection strategy on the convergence properties of the
+algorithm. Technical details are left to Appendix A.
+4.1
+Mean-field approximation
+First, we note that a simple update rule for the personal bests yk
+i is given by
+yk+1
+i
+= yk
+i + 1
+2
+�
+xk+1
+i
+− yk
+i
+�
+S(xk+1
+i
+, yk
+i ) ,
+with
+S(x, y) = 1 + sign (F(y) − F(x)) .
+(4.1)
+As in [14], we approximate it for β ≫ 1 as
+yk+1
+i
+= yk
+i + ν
+2
+�
+xk+1
+i
+− yk
+i
+�
+Sβ(xk+1
+i
+, yk
+i ) ,
+(4.2)
+with Sβ(x, y) being a continuous approximation of S(x, y) as β → ∞. By choosing ν = 1 we
+get (4.1) with the only difference of having Sβ instead of S. As for ¯yα with respect to ¯y∞, this
+is needed to make the update rule easier to handle mathematically, but it does have an impact
+on the performance for large values of β.
+With the aim of deriving a continuous-in-time reformulation of (2.4) and (4.2), we introduce
+a single parameter ∆t > 0 which controls the step length of all involved update mechanisms.
+By performing the rescaling
+λ ← λ∆t ,
+σ ← σ
+√
+∆t ,
+ν ← ν∆t
+to get the update rules
+�
+xk+1
+i
+= xk
+i + λ∆t
+�
+¯yα,k − xk
+i
+�
++ σ
+√
+∆t
+�
+¯yα,k − xk
+i
+�
+⊗ θk
+i
+yk+1
+i
+= yk
+i + (ν∆t/2)
+�
+xk+1
+i
+− yk
+i
+�
+Sβ(xk+1
+i
+, yk
+i )
+(4.3)
+which differ form the original formulation (2.4), (4.1) only due to the use of Sβ instead of S.
+As already noted in [14], the iterative process (4.3) corresponds to an Euler-Maruyama
+scheme applied to a system of Stochastic Differential Equations (SDEs). Indeed, (4.3) corre-
+sponds to a discretization of the system
+�
+dXi
+t
+= λ
+�
+¯yα(ρN
+t ) − Xi
+t
+�
+dt + σ
+�
+¯yα(ρN
+t ) − Xi
+t
+�
+⊗ dBi
+t
+dY i
+t
+= ν(Xi
+t − Y i
+t )Sβ(Xi
+t, Y i
+t ) dt
+(4.4)
+19
+
+where, for convenience, we underlined above the dependence of the consensus point on the
+empirical distribution ρN
+t = �
+i δY i
+t (δy being the Dirac measure at y ∈ Rd) by using
+¯yα(ρ) :=
+�
+ye−αF(y)dρ(y)
+�
+e−αF(y)dρ(y) ,
+(4.5)
+defined for any Borel probability measure ρ over Rd (ρ ∈ P(Rd)). In this way, we generalized
+the definition introduced in (2.2) to any ρ ∈ P(Rd), provided the above integrals exists. In (4.4),
+the random component of the dynamics is now described by N independent Wiener processes
+(Bi
+t)t>0. As before, we supplement the system with initial conditions Xi
+0 ∼ ρ0, Y i
+0 = Xi
+0 for some
+ρ0 ∈ P(Rd).
+The continuous-in-time description (4.4) already simplifies the analytical analysis of the
+optimization algorithm, but still pays the price of a possible large number O(N) of equations.
+This issue is typically addressed by assuming that for large populations N, the particles become
+indistinguishable from one another and start behaving, in some sense, as a unique system.
+More precisely, let F N(t) ∈ P(R(2d)N) denote the joint probability distribution of N tuples
+(Xi
+t, Y i
+t ). We assume propagation of chaos [41] for large N ≫ 1, that is, we assume that the
+joint probability distribution decomposes as F N(t) = f(t)⊗N for some f(t) ∈ P(R2d). System
+(4.4) becomes independent on the index i and hence every particle moves according to the
+mono-particle process
+d ¯Xt = λ(¯yα(¯ρt) − ¯Xt) dt + σ (¯yα(¯ρt) �� ¯Xt) ⊗ d ¯Bt
+d ¯Yt = ν( ¯Xt − ¯Yt)Sβ( ¯Xt, ¯Yt) dt
+(4.6)
+where ¯ρt = Law( ¯Yt).
+Assume ( ¯Xt, ¯Yt) are initially distributed according to f0 = ρ⊗2
+0 , by applying Itˆo formula we
+have that f(t) = Law( ¯Xi
+t, ¯Y i
+t ) satisfies
+∂tf + ∇x · (λ(¯yα(¯ρ) − x)f) + ∇y ·
+�
+ν(x − y)Sβ(x, y)f
+�
+=
+d
+�
+ℓ=1
+∂2
+xℓ
+�
+σ(¯yα(¯ρ) − x)2
+ℓf
+�
+(4.7)
+and initial data limt→0 f(t) = f0 in a weak sense.
+Dynamics (4.6), or, equivalently, (4.7),
+corresponds to the mean-field approximation of the particle system (4.4) as N → ∞. We remark
+that the above derivation has only been possible thanks to the approximations S ≈ Sβ and
+¯y∞ ≈ ¯yα for large α and β. Well-posedness of the system is also granted by such approximations
+(proof details are given in Appendix A.2).
+Proposition 4.1 (well-posedness of (4.6)). There exists a unique process ( ¯X, ¯Y ) ∈ C([0, T], Rd),
+T > 0 satisfying (4.4) with initial conditions ( ¯X0, ¯Y0) with ¯X0 ∼ ρ0 ∈ P4(Rd) and ¯Y0 = ¯X0.
+Being mathematically tractable, we show next that the mean-field dynamics converges to a
+global solution to (2.1) if F, Sβ.
+20
+
+4.2
+Convergence in mean-field law
+We start by enunciating the necessary assumptions to the convergence result.
+Assumption 4.1 (Assumptions on F). The objective function F ∈ C(Rd, R), satisfies:
+A1
+there exists some constant LF > 0 such that
+|F(x) − F(x′)| ≤ LF
+�
+∥x∥2 + ∥x′∥2
+�
+∥x − x′∥2,
+∀ x, x′ ∈ Rd ;
+A2
+there exists uniquely x∗ ∈ Rd solution to (2.1);
+A3
+there exist η, R0 > 0 and γ ∈ (2, ∞) such that
+F(x) − inf F ≥ η ∥x − x∗∥γ
+∞
+∀x ∈ Rd , ∥x − x∗∥∞ ≤ R0
+F(x) − inf F ≥ η Rγ
+0
+∀x ∈ Rd , ∥x − x∗∥∞ > R0 .
+A4
+F is convex in a (possibly small) neighborhood {x ∈ Rd : ∥x − x∗∥∞ ≤ R1} of x∗ for
+some R1 < R0.
+A5
+There exists cg, R2 > 0 such that
+F(x) − inf F ≥ cg∥x − x∗∥2
+2
+∀x ∈ Rd , ∥x − x∗∥2 > R2 .
+Assumption 4.2 (Assumptions on Sβ). The function Sβ ∈ C(R2d, [0, 2]), with β > 0
+A6
+has the following structure
+Sβ(x, y) = 2ψ (β(F(y) − F(x))) ,
+(4.8)
+with ψ ∈ C1(R, [0, 1]) being an increasing function with Lipschitz constant Lψ = 1.
+A7
+The value Sβ(x, y) is positive only when x is strictly better than y in terms of objective
+value F:
+Sβ(x, y)
+�
+≥ 0
+if
+F(x) < F(y)
+= 0
+else .
+Assuming uniqueness of global minimum is a typical assumption for analysis of CBO methods
+[9,10] and it is due to the definition of the consensus point ¯yα (or ¯xα in the case without memory
+mechanism). Indeed, in presence of two global minima, ¯yα may be placed between them, no
+matter how large α is. Assumption A2 ensure to avoid such situations. Furthermore, A3 also
+allows to give quantitative estimates on the difference between the global minimum and eventual
+local minima. In the literature, such property is known as conditioning [12]. Requirements A4
+and A7 ensure that if a personal best yk
+i enters such small neighborhood where F is convex, it
+will not leave it for the rest of the computation. Condition A5 (quadratic growth at infinity) is
+needed for the well-posedness of the mean-field mono-particle process (4.6), see also [3]. For an
+intuition of A3 and A4 we refer to Figure 9, where the Rastrigin function is considered.
+21
+
+x$ ! R0
+x$
+x$ + R0
+0
+20
+40
+objective
+lower bound (A3)
+convex area (A4)
+Figure 9: Assumptions 4.1 illustrated for Rastrigin function. For example, such objective
+function satisfies A3 with η = 1, γ = 1.8, R0 = 1.42 and A4 with R1 = 0.25.
+Theorem 4.1 (Convergence in mean-field law). Assume F satisfies A1–A5, Sβ satisfies A6,
+A7 for some β > 0 fixed. Let ( ¯Xt, ¯Yt)t≥0 be a solution to (4.6) for t ∈ [0, T], with initial data
+¯X0 ∼ ρ0 ∈ P4(Rd), Y0 = X0 such that x∗ ∈ supp(ρ0) .
+Fix an accuracy ε > 0. If 2λ > σ2, there exists a time T ∗ such that the expected ℓ2-error
+satisfies
+E
+�
+∥ ¯XT ∗ − x∗∥2
+2
+�
+≤ ε
+(4.9)
+provided T, α > 0 are large enough.
+We refer to Appendix A for a proof.
+Remark 4.1. The mean-field mono-particle process (4.6) aims to approximate the algorithm
+iterative dynamics (4.3) for small time steps ∆t ≪ 1 and large particle populations N ≫ 1.
+Therefore, convergence of the algorithm dynamics towards the global solution x∗ can be proven
+by coupling Theorem 4.1 with error estimates of such approximation.
+For instance, assuming that all considered dynamics take place on a bounded set D ensures
+that the error introduced by the continuous-in-time particle system will be of order ∆t thanks to
+classical results on Euler-Maruyama schemes [35]. Likewise, considering a bounded dynamics
+allows to prove that the error introduced by the mean-field approximation is of order N−1 (see
+e.g. [8, Theorem 3.1], [9, Proposition 16]). Let {(xk
+i , yk
+i )}N
+i=1 be given by (4.3), {(Xi
+t, Y i
+t )}N
+i=1 be
+a solution (4.4) and {( ¯Xi
+t, ¯Y i
+t )}N
+i=1 be N-copies of a solution to (4.6). Altogether, one obtains
+the following error decomposition for K∆t = T ∗
+E
+�
+1
+N
+N
+�
+i=1
+∥xK
+i − x∗∥2
+2
+�
+≤ C
+�
+E
+�
+1
+N
+N
+�
+i=1
+∥xK
+i − Xi
+T ∗∥2
+2
+�
++ E
+�
+1
+N
+N
+�
+i=1
+∥Xi
+T ∗ − ¯Xi
+T ∗∥2
+2
+�
++ E
+�
+1
+N
+N
+�
+i=1
+∥ ¯Xi
+T ∗ − x∗∥2
+2
+� �
+≤ CEM∆t + CMFAN−1 + ε
+22
+
+where C, CEM, CMFA are positive constant independent on N, ∆t.
+4.3
+Random selection analysis
+In this section, we analytically investigate the impact of randomly discarding particles during
+the computation. We are particularly interested in tracking the distance between a particle
+system {xk
+i , xk
+j }N0
+i=1 evolving according to (4.3) where no particles are discarded, and a second
+system {ˆxk
+i , ˆyk
+i }Ik, |Ik| = Nk where Nk − Nk+1 particles are discarded after update rule (4.3).
+Clearly, we have that Nk+1 ≤ Nk and Ik+1 ⊆ Ik ⊆ I0 = {1, . . . , N0} for all k. Similarly to the
+analysis carried out in [15,16], we restrict to the simpler dynamics where, at every step k, the
+random variables θk
+i and ˆθk
+i used to generate such systems are the same for all particles:
+θk
+i = ˆθk
+j = θk ∼ N(0, Id)
+for all
+i ∈ Ik, j ∈ I0.
+(4.10)
+To compare particle systems with a different number of particles, we rely on their represen-
+tation as empirical probability measures and the notion of 2-Wasserstein distance. For {ˆxk
+i }i∈Ik
+and {xk
+i }N0
+i=1 we consider, respectively, the following probability measures
+ρk
+Nk := 1
+Nk
+�
+i∈Ik
+δˆxk
+i
+and
+ρk
+N0 := 1
+N0
+�
+i∈I0
+δxk
+i .
+(4.11)
+Informally, the 2-Wasserstein distance W2(ρk
+Nk, ρk
+N0) quantifies the minimal effort needed to
+move the mass from distribution ρk
+Nk into ρk
+N0 (or vice versa) [38]. Let wij denote the amount
+of mass leaving particle xk
+i and going into ˆxk
+i : the cost of such movement is assumed to be given
+by wij∥xk
+i − ˆxk
+j ∥2
+2. Therefore, if we indicate the set of all admissible couplings between the two
+discrete probability measures as
+Γ(ρk
+Nk, ρk
+N0) =
+�
+�
+�w ∈ RN0×Nk :
+Nk
+�
+j=1
+wij = 1
+N0
+,
+N0
+�
+i=1
+wij = 1
+Nk
+, wij ≥ 0, ∀ i, j
+�
+�
+� ,
+(4.12)
+the 2- Wasserstein distance is defined as
+W2(ρk
+Nk, ρk
+N0) :=
+min
+w∈Γ(ρk
+Nk,ρk
+N0)
+�
+��
+i,j
+wij∥xk
+i − ˆxk
+j ∥2
+2
+�
+�
+1
+2
+(4.13)
+see, for instance, [38, Section 6.4.1].
+Before providing estimates on (4.12), let us present a more general result on the impact that
+the random selection strategy has on an arbitrary particle distribution.
+Proposition 4.2 (Stability of random selection procedure). Let z = {zi}i∈I, |I| = N be an
+ensemble of particles and {zi}j∈Isel with Isel ⊆ I, |I| = Nsel a random sub-set of such ensemble.
+Consider the associated empirical distributions µN and µNsel (defined consistently to (4.11)), it
+holds
+E
+�
+W 2
+2 (µN, µNsel)
+�
+≤ 2 var(z) N − Nsel
+N − 1
+(4.14)
+where the expectation is taken with respect to the random selection of Isel.
+23
+
+The proof is provided Appendix A.4. We note how the system variance var(z) enters the
+error estimate due to the randomness of the selection, similar to the Law of Large Number error
+for random variables. In particular, the smaller the particles variance is, the closer the reduced
+particle system will be to the original distribution. This justifies the choice of Nk+1 proposed
+in Section 2.2 where we are allowed to discard particles only if the system shows a contractive
+behavior, see (2.6).
+By iteratively applying Proposition 4.2 and by using suitable stability estimates of dynamics
+(4.3), we are able to bound the error introduced by the random selection procedure as follows.
+Proof details are a given in Appendix A.4.
+Theorem 4.2. Let {xk
+i , yk
+i }N0
+i=1 be constructed according to (4.3) were particles are not discarded,
+and {ˆxk
+i , ˆyk
+i }Ik, |Ik| = Nk where Nk−Nk+1 particles are discarded after update rule (4.3). Assume
+(4.10) is satisfied and consider the probability measures (4.11). If {xk
+i , yk
+i }N0
+i=1, {ˆxk
+i , ˆyk
+i }i∈Ik ⊂
+BM(0) at all step k for some M > 0, it holds
+E
+�
+W 2
+2
+�
+ρk
+Nk, ρk
+N0
+��
+≤ C
+max
+h=1,...,k var
+�
+˜zh� N0 − Nk
+Nk − 1
+(4.15)
+where C = C(∆t, λ, σ, ν, β, α, k, LF, M) and ˜zh = {(ˆxh
+i , ˆyh
+i )}i∈Ih−1 describes the particle system
+just before the random selection procedure at step h ≤ k. The expectation is taken with respect
+to the sampling of {θh}k
+h=1 and with respect to the selection procedure.
+We can directly apply the above result to relate the expected ℓ2-errors of the two particle
+system, which we define as
+Err(k) = E
+�
+� 1
+N0
+�
+i∈I0
+∥xk
+i − x∗∥2
+2
+�
+� ,
+Err(k) = E
+�
+� 1
+Nk
+�
+i∈Ik
+∥ˆxk
+i − x∗∥2
+2
+�
+� ,
+that is, the discrete counterpart of the mean-field error E[∥ ¯Xi
+t − x∗∥2
+2] studied in Theorem 4.1.
+By definition of the Wasserstein-2 distance, we have
+Err(k) = E
+�
+W 2
+2 (ρk
+N0, δx∗)
+�
+for any solution x∗ to (2.1), and the same holds of Errsel(k). We then apply inequality
+W 2
+2 (ρk
+Nk, δx∗) ≤ 2
+�
+W 2
+2 (ρk
+Nk, ρk
+N0) + W 2
+2 (ρk
+N0, δx∗)
+�
+to obtain the following estimate.
+Corollary 4.1. Under the assumptions of Theorem 4.2, at all steps k, it holds
+Errsel(k) ≤ 2
+�
+Err(k) + C
+max
+h=1,...,k var(˜zh) N0 − Nk
+Nk − 1
+�
+.
+(4.16)
+Before concluding the section, let us report some remarks concerning the theoretical results
+just presented.
+24
+
+Remark 4.2.
+• Proof of Theorem 4.2 can be adapted to any other particle system with random selection,
+provided that the update rule is stable with respect to the 2-Wasserstein distance. In the
+proposed method, such stability was proved thanks to the approximation of the global best
+¯y∞,k with ¯yα,k for α ≫ 1 (see (2.2)) and S(x, y) with Sβ(x, y) for β ≫ 1 in the personal
+best update (4.2).
+• Quantitative estimates on the variance decay can be used, if available, to improve the error
+bound in Theorem 4.2, see also proof in Appendix A.4.
+• The error introduced by a sub-sampling technique in a Monte Carlo integral approximation
+is expected to be of order
+2 var(z)
+�
+1
+N − 1 −
+1
+Nsel − 1
+�
+= 2 var(z)
+N − Nsel
+(N − 1)(Nsel − 1) ,
+(4.17)
+see e.g. [23]. Therefore, an additional factor of order 1/(Nsel − 1) seems to be missing
+in Proposition 4.2. We remark, though, that Proposition 4.2 does not concern the Monte
+Carlo approximation of an integral quantity, but rather consider the 2-Wasserstein distance
+between discrete measures. Numerical simulations suggest that estimates of order (4.17)
+do not hold on in this case, see Fig.10.
+5
+Conclusions
+In this work, we studied a Consensus-Based Optimization algorithm with Memory Effects (CBO-
+ME) and random selection for single objective optimization problems of the form (2.1). While
+sharing common features with Particle Swarm Optimization (PSO) methods, CBO-ME differs
+on the way the particle system explore the search space. Its structure provides greater flexi-
+bility in balancing the exploration and exploitation processes. In particular, we implemented
+and analytically investigates a random selection strategy which allows to reduce the algorithm
+computational complexity, without affecting convergence properties and overall accuracy. This
+analysis is entirely general and, in perspective, applicable to other particle swarm-based opti-
+mization methods as well. The convergence analysis to the global minimum is carried out by
+relying on a mean-field approximation of the particle system and error estimates are given un-
+der mild assumptions on the objective function. We compared CBO-ME against CBO without
+memory effects and PSO against several benchmark problem and showed how the introduction
+of memory effects and random selection improves the algorithm performance. Applications to
+image segmentation and machine learning problems are finally reported.
+A
+Proofs
+A.1
+Notation and auxiliary lemmas
+We will use the following notation. For any a ∈ R, |a| indicates the absolute value. For a given
+vector b ∈ Rd, ∥b∥p indicates its p-norm, p ∈ [1, ∞]; (b)ℓ its ℓ-th component; while diag(b) ∈ Rd×d
+25
+
+0
+20
+40
+60
+80
+100
+# particle selected
+!
+Nsel
+"
+0
+0.5
+1
+1.5
+2
+N = 100; d = 3
+squared Wasserstein dist.
+estimate (4.18)
+estimate (4.21)
+0
+20
+40
+60
+80
+100
+# particle selected
+!
+Nsel
+"
+0
+1
+2
+3
+4
+5
+6
+7
+N = 100; d = 10
+squared Wasserstein dist.
+estimate (4.18)
+estimate (4.21)
+Figure 10: Numerical validation of Proposition 4.2 with different dimensions d = 3, 10.
+N = 100 points are randomly, uniformly sampled over [0, 1]d to construct the empirical
+distribution µN and Nsel ∈ [2, N − 1] are discarded to obtain µNsel. The experiment is
+repeated 500 times for all Nsel to obtain an approximation of E
+�
+W 2
+2 (µN, µNsel)
+�
+(blue line).
+In red, estimate provided by Proposition 4.2 (RHS of (4.14)), in yellow the one given
+equation (4.17). Wasserstein distances are computed with the ot.emd function provided by
+the Python Optimal Transport library [6].
+is the diagonal matrix with elements of b on the main diagonal. Let a, b ∈ Rd, ⟨a, b⟩ denotes
+the scalar product in Rd. For a given closed convex set A ⊂ Rd, N(A, x), T (A, x) denote the
+normal and the tangential cone at x ∈ A respectively. The ball or radius r centered at x ∈ Rd
+is indicated with Br(x) = {x ∈ Rd | ∥x∥2 ≤ r}. All considered stochastic processes are assumed
+to take their realizations over the common probability space (Ω, ¯F, P). P(Rd) is the set of Borel
+probability measures over Rd and Pq(Rd) = {µ ∈ P(Rd) |
+�
+∥x∥q
+2dµ < ∞} which we equip with
+the Wasserstein distance Wq, q ≥ 1, see [38]. For a random variable X, X ∼ µ, µ ∈ P(Rd)
+indicates a sampling procedure such that P(X ∈ A) = µ(A) for any Borel set A ⊂ Rd. With
+Unif(A) ∈ P(Rd) we denote the uniform probability measure over a bounded Borel set A.
+Throughout the computations, C will denote an arbitrary positive constant, whose value may
+vary from line to line. Dependence on relevant parameters or variables, will be underlined.
+Lemma A.1 ( [3, Lemma 3.2]). Let F satisfy Assumption 4.1 (in particular the locally Lipschitz
+assumption A1) and ρ1, ρ2 ∈ P4(Rd) with
+�
+∥x∥4
+2dρ1 ,
+�
+∥x∥4
+2dρ2 ≤ M .
+Then, the following stability estimate holds
+∥¯yα(ρ1) − ¯yα(ρ2)∥2 ≤ C W2(ρ1, ρ2)
+for a constant C = C(α, LF, M).
+26
+
+Lemma A.2. Under Assumptions A1 and A6, for any x1, x2, y1, y2 ∈ BM(0) and β > 0, it
+holds
+∥(x1 − y1)Sβ(x1, y1) − (x2 − y2)Sβ(x2, y2)∥2 ≤ C (∥x1 − y1∥2 + ∥x2 − y2∥2)
+where C = C(β, LF, M).
+Proof. Thanks to the Lipschitz continuity of ψ, F and the choice of ψ (Assumptions A1 and
+A6), it holds
+|Sβ(x1, y1) − Sβ(x2, y2)| = |2ψ (β(F(y1) − F(x1)) − 2ψ(β(F(y2) − F(x2)) |
+≤ 2β |F(y1) − F(x1) − F(y2) + F(x2)|
+≤ 2βLF (∥x1 − x2∥2 + ∥y1 − y2∥2) .
+Next, we have
+∥(x1 − y1)Sβ(x1, y1) − (x2 − y2)Sβ(x2, y2)∥2 ≤ ∥(x1 − y1)Sβ(x1, y1) − (x2 − y2)Sβ(x1, y1)∥2
++ (x2 − y2)Sβ(x1, y1) − (x2 − y2)Sβ(x2, y2)∥2
+≤ ∥(x1 − x2 + y2 − y1)Sβ(x1, y1)∥2
++ ∥(x2 − y2)
+�
+Sβ(x1, y1) − Sβ(x2, y2)
+�
+∥2
+≤ 2 (∥x1 − x2∥2 + ∥y1 − y2∥2)
++ 2M|Sβ(x1, y1) − Sβ(x2, y2)|
+≤ C (∥x1 − x2∥2 + ∥y1 − y2∥2)
+with C = C(β, LF, M), where we used the first estimate to conclude.
+A.2
+Proof of Proposition 4.1
+Proof of Proposition 4.1. The proof is based on the Leray–Schauder fixed point theorem [13,
+Chapter 11], and we follow closely the proof steps of [3].
+Step 1. For any ξ ∈ C([0, T], Rd) there exists a unique process ( ˆXt, ˆYt) ∈ C([0, T], Rd)
+satisfying
+d ˆXt = λ(ξ(t) − ˆXt) dt + σ(ξ(t) − ˆXt) ⊗ d ˆBt
+d ˆYt = ν( ˆXt − ˆYt)Sβ( ˆXt, ˆYt) dt
+with Law( ˆX0) = Law( ˆY0) = ρ0 ∈ Rd, by the Lipschitz continuity of the coefficients.
+As a
+consequence, we have that f(t) := Law( ˆXt, ˆYt) satisfies
+d
+dt
+�
+φ df(t) =
+� �
+−λ⟨∇xφ, ξ(t) − x⟩ +
+�
+ℓ=1
+∂2φ
+∂x2
+ℓ
+(ξt) − y)2
+ℓ − νSβ⟨∇yφ, y − x⟩
+�
+df(t)
+for all φ ∈ C2
+b (R2d). Therefore, let ¯ρ(t) = Law( ˆYt), we can set T ξ := ¯yα(¯ρ(·)) ∈ C([0, T], Rd) to
+define
+T : C([0, T], Rd) → C([0, T], Rd).
+27
+
+Step 2. We prove now compactness of T . Thanks to ρ0 ∈ P4(Rd) and standard results for
+SDEs (see [1, Chapter 7]) we have boundedness of the forth moments
+E
+�
+∥ ˆXt∥4
+2 + ∥ ˆYt∥4
+2
+�
+≤ c1
+�
+1 + E[∥ ˆX0∥4
+2 + ∥ ˆY0∥4
+2]ec2t�
+for some c1, c2 > 0. Therefore, we can apply Lemma A.1 to obtain for any 0 < s < t < T,
+∥¯yα(¯ρ(t)) − ¯yα(¯ρ(s))∥2 ≤ CW2 (¯ρ(t), ¯ρ(s)) ≤ ˜C|t − s|1/2
+for some constants C, ˜C > 0, from which H¨older continuity of t �→ ¯yα(¯ρ(t) follows. Therefore,
+by
+T (C([0, T], Rd)) ⊂ C0, 1
+2 ([0, T], Rd) �→ C([0, T], Rd)
+we get compactness of T .
+Step 3. Consider ξ ∈ C([0, T], Rd) satisfying ξ = τT ξ, for τ ∈ [0, 1]. Thanks to [3][Lemma
+3.3] and boundedness of second moments, we obtain compactness of the set
+{ξ ∈ C([0, T], Rd) : ξ = τT ξ, τ ∈ [0, 1]}
+and by Leray–Schauder fixed point theorem there exists a fixed point for the mapping T and
+hence a solution to (4.6).
+Step 4. Assume now there exist two solutions, ( ¯X1
+t , ¯Y 1
+t ) and ( ¯X2
+t , ¯Y 2
+t ) to (4.6) with same
+Brownian process ¯Bt and initial conditions. Let ¯ρℓ = Law( ¯Y ℓ
+t ), ℓ = 1, 2, we have
+∥ ¯X1
+t − ¯X2
+t ∥2
+2 =
+� t
+0
+� ¯X1
+s − ¯X2
+s , ¯yα(¯ρ1(s)) − ¯yα(¯ρ2(s)) − ¯X1
+s + ¯X2
+s
+�
+dt
++
+� t
+0
+�
+diag
+�
+¯yα(¯ρ1(s)) − ¯X1
+s
+�
+− diag
+�
+¯yα(¯ρ2(s)) − ¯X2
+s
+��
+d ¯Bs .
+(A.1)
+We note that all terms can be estimated by means of W 2
+2 (¯ρ1(s), ¯ρ2(s)) and ∥ ¯X1
+s − ¯X2
+s ∥2
+2. Similarly,
+∥ ¯Y 1
+t − ¯Y 2
+t ∥2
+2 can be bounded in terms ∥ ¯X1
+s − ¯X2
+s ∥2
+2 thanks to the Lipschitz continuity of Sβ and
+Lemma A.2. Therefore, for some constant C > 0
+∥ ¯X1
+t − ¯X2
+t ∥2
+2 + ∥ ¯Y 1
+t − ¯Y 2
+t ∥2
+2 ≤ C
+� t
+0
+�
+∥ ¯X1
+s − ¯X2
+s ∥2
+2 + ∥ ¯Y 1
+s − ¯Y 2
+s ∥2
+2 + W 2
+2 (¯ρ1(s), ¯ρ2(s))
+�
+ds
+from which, together with (A.1), follows for some ˜C > 0
+E
+�
+∥ ¯X1
+t − ¯X2
+t ∥2
+2 + ∥ ¯Y 1
+t − ¯Y 2
+t ∥2
+2
+�
+≤ E
+�
+∥ ¯X1
+0 − ¯X2
+0∥2
+2 + ∥ ¯Y 1
+0 − ¯Y 2
+0 ∥2
+2
+�
+e
+˜C t
+by Gr¨onwall’s inequality. Since E
+�
+∥ ¯X1
+0 − ¯X2
+0∥2
+2 + ∥ ¯Y 1
+0 − ¯Y 2
+0 ∥2
+2
+�
+= 0, we proved uniqueness.
+A.3
+Proof of Theorem 4.1
+Having proved there exists a solution ( ¯Xt, ¯Yt)t∈[0,T] to the mean-field process (4.6) we are here
+interested in studying the expected ℓ2-error given by
+E∥ ¯Xt − x∗∥2
+2
+where x∗ is the unique solution to the minimization problem (2.1), see Assumption 4.1. We do
+so by means of the following quantitative version of the Laplace principle.
+28
+
+Proposition A.1 (quantitative Laplace principle [10, Proposition 1]). Let ρ ∈ P(Rd) be such
+that x∗ ∈ supp(ρ) and fix α > 0. For any r > 0, define Fr = supx∈B∗r F(x) − F(x∗) with
+B∗
+r := {x | ∥x − x∗∥∞ ≤ r} .
+Then, under Assumption 4.1, for any r ∈ (0, R0] and q > 0 such that q + Fr ≤ F∞ = ηRγ
+0,
+it holds
+∥yα(ρ) − x∗∥2 ≤
+√
+d(q + Fr)γ
+η
++
+√
+d exp(−αq)
+ρ(B∗r)
+�
+∥x − x∗∥2 dρ(x).
+(A.2)
+We remark that RHS of (A.2) can be made arbitrary small by taking large values of α and
+small values of q, r. To apply Proposition A.1 to all ¯ρ(t) = Law( ¯Yt), we need though to provide
+lower bounds on ¯ρ(t)(B∗
+r) for any small radius r and times t ∈ [0, T].
+Lemma A.3. Let ¯ρ(t) = Law( ¯Yt), with ¯Yt evolving according to (4.6) and limt→0 ¯ρ(t) = ρ0 with
+x∗ ∈ supp(ρ0). Under Assumptions 4.1 and 4.2 , it holds ¯ρ(t)(B∗
+r) ≥ mr > 0, for all t ∈ [0, T]
+and for all r ≤ R0.
+Proof. Let δ = η min{R1, r}γ, we start by proving that the mass in the set
+Lδ = {x ∈ Rd | F(x) ≤ inf F + δ}
+is non-decreasing. We note that for this choice of δ, Lδ is convex due to Assumption 4.1. Consider
+now (Ω, ¯F, P) to be the common probability space over which the considered processes take
+their realization and define Ωδ = {ω : ¯Y0(ω) ∈ Lδ}. By Assumption 4.2, Sβ( ¯Xt(ω), ¯Yt(ω)) = 0
+whenever ¯Xt(ω) /∈ Lδ. Therefore, it holds
+�
+( ¯Xt(ω) − ¯Yt(ω))Sβ( ¯Xt(ω), ¯Yt(ω)) , n( ¯Yt(ω))
+� �
+= 0
+if ¯Xt(ω) /∈ Lδ
+≤ 0
+if ¯Xt(ω) ∈ Lδ
+for
+¯Yt(ω) ∈ ∂Lδ
+for any n( ¯Yt(ω)) ∈ N(Lδ, x) from which follows that ¯Yt(ω) solves
+¯Yt(ω) = ¯Y0(ω) +
+� t
+0
+ΠT (Lδ, ¯Ys(ω))
+�
+( ¯Xs(ω) − ¯Ys(ω))Sβ( ¯Xs(ω), ¯Ys(ω))
+�
+ds
+for all ω ∈ Ωδ. As a consequence, if ¯Y0(ω) ∈ Lδ, ¯Yt(ω) ∈ Lδ for all t ≥ 0 and so
+¯ρ(t)(B∗
+r) = P( ¯Yt ∈ Lδ) ≥ P( ¯Y0 ∈ Lδ) =: mr
+for all t ≥ 0. We conclude by noting that mr > 0 since x∗ ∈ supp(ρ0).
+Next, we study the evolution of the error E∥ ¯Xt − x∗∥2
+2 and, in particular, we try to bound it
+in terms of ∥¯yα(¯ρ(s)) − x∗∥2 and E∥ ¯Xt − x∗∥2 itself for s ∈ [0, t].
+29
+
+Proposition A.2.
+[10, Lemma 1] Let ( ¯Xt, ¯Yt) ∈ C([0, T], R2d) be the solution to (4.6) with
+initial datum ¯X0 ∼ ρ0, ¯Y0 = ¯X0 for some time horizon T > 0. For all t ∈ [0, T], it holds
+E∥ ¯Xt − x∗∥2
+2 ≤
+� t
+0
+�
+− (2λ − σ2)E∥ ¯Xs − x∗∥2
+2 +
+√
+2(λ + σ2)E∥ ¯Xs − x∗∥2∥¯yα(¯ρ(s)) − x∗∥2
++ σ2
+2 ∥¯yα(¯ρ(s)) − x∗∥2
+2
+�
+ds
+(A.3)
+where ¯ρ(t) = Law( ¯Yt).
+Proof of Theorem 4.1. The above result, together with Lemma A.3, leads to the convergence
+in mean-field law of the dynamics towards the solution to (2.1). The proof can be carried out
+exactly as in [10, Theorem 12].
+A.4
+Proof of Proposition 4.2 and Theorem 4.2
+We start by collecting a preliminary result.
+Lemma A.4. Let {xk
+1,i, yk
+1,i}N1
+i=1 and {xk
+2,j, yk
+2,j}N2
+j=1 be two particle populations generated through
+update rules (4.3) with θk
+1,i = θk
+2,j = θk for all i, j and k ∈ Z+. At any iteration step k and for
+any couple of indexes (i, j), it holds
+E
+�
+∥xk+1
+1,i
+− xk+1
+2,j ∥2
+2 + ∥yk+1
+1,i
+− yk+1
+2,j ∥2
+2
+�
+≤
+CE
+�
+∥xk
+1,i − xk
+2,j∥2
+2 + ∥yk
+1,i − yk
+2,j∥2
+2 + ∥¯yα(¯ρk
+1) − ¯yα(¯ρk
+2)∥2
+2
+�
+where C = C(∆t, λ, σ, ν, β) is a positive constant and ¯ρk
+1, ¯ρk
+2 are the empiricial distributions
+associated with {yk
+1,i}N1
+i=1 and {yk
+2,j}N2
+j=1 respectively.
+Proof. For all k ∈ Z+ and i, j
+E∥xk+1
+1,i
+− xk+1
+2,j ∥2
+2 ≤ E
+���xk
+1,i + λ∆t
+�
+¯yα(¯ρk
+1) − xk
+1,i
+�
++ σ
+√
+∆t
+�
+¯yα(¯ρk
+1) − xk
+1,i
+�
+⊗ θk
+1,i
+−
+�
+xk
+2,j + λ∆t
+�
+¯yα(¯ρk
+2) − xk
+2,j
+�
++ σ
+√
+∆t
+�
+¯yα(¯ρk
+2) − xk
+2,j
+�
+⊗ θk
+2,j
+� ���
+2
+2
+≤ 2E
+���
+�
+1 − λ∆t − σ
+√
+∆t diag(θk)
+�
+(xk
+1,i − xk
+2,j)
+���
+2
+2
++ 2E
+���
+�
+λ∆t + σ
+√
+∆t diag(θk)
+� �
+¯yα(¯ρk
+1) − ¯yα(¯ρk
+2)
+����
+2
+2
+≤ 2(1 + σ2∆t)E∥xk
+1,i − xk
+2,j∥2
+2
++ 2(λ2∆t2 + σ2∆t)E∥¯yα(¯ρk
+1) − ¯yα(¯ρk
+2)∥2
+2 ,
+(A.4)
+where we also used that E[(θk)2
+ℓ] = 1 for all ℓ = 1, . . . , d. We now bound ∥yk+1
+1,i
+− yk+1
+2,j ∥2
+2 as
+E∥yk+1
+1,i
+− yk+1
+2,j ∥2
+2 ≤ E
+���yk
+1,i + (ν∆t/2)
+�
+xk+1
+i,1
+− yk
+1,i
+�
+Sβ(xk+1
+1,i , yk
+1,i)
+30
+
+−
+�
+yk
+2,j + (ν∆t/2)
+�
+xk+1
+2,j − yk
+2,j
+�
+Sβ(xk+1
+2,j , yk
+2,j)
+� ���
+2
+2
+≤ CE
+�
+∥xk+1
+i,1
+− xk+1
+j,2 ∥2
+2 + ∥yk
+i,1 − yk
+j,2∥2
+2
+�
+(A.5)
+where we used Lemma A.2 and C = C(∆t, β, ν). By combining (A.4) and (A.5) we get the
+desired estimate.
+Next, we show how the particle update rule (4.3) is stable with respect to the 2-Wasserstein
+distance.
+Proposition A.3 (Stability of update rule (4.3)). Let {xk
+1,i, yk
+1,i}N1
+i=1, {xk
+2,j, yk
+2,j}N2
+j=1 ⊂ BM(0),
+for some M > 0, be two particle populations generated through the update rules (4.3) with
+θk
+1,i = θk
+2,j = θk for all i, j and k ∈ Z+. Let µk
+1, µk
+2 ∈ P(R2d) the empirical probability measures
+defined as
+µk
+1 := 1
+N1
+N1
+�
+i=1
+δ(xk
+1,i,yk
+1,i) ,
+µk
+2 := 1
+N2
+N2
+�
+j=1
+δ(xk
+2,j,yk
+2,j) ,
+it holds
+E
+�
+W 2
+2 (µk+1
+1
+, µk+1
+2
+)
+�
+≤ C1 E
+�
+W 2
+2 (µk
+1, µk
+2)
+�
+,
+where C1 = C1(∆, λ, σ, ν, α, β, LF, M) is positive constant.
+Proof. Let Eθk[·] denote the expectation taken with respect to the sampling of θk only and
+w ∈ RN1×N2 be the optimal coupling between µk
+1, µk
+2, see (4.12) and (4.13). Being w a sub-
+optimal coupling for µk+1
+1
+, µk+1
+2
+, it holds
+Eθk[W 2
+2 (µk+1
+1
+, µk+1
+2
+)] ≤ Eθk
+�
+i,j
+wij
+�
+∥xk+1
+1,i
+− xk+1
+2,j ∥2
+2 + ∥yk+1
+1,i
+− yk+1
+2,j ∥2
+2
+�
+≤ C
+�
+i,j
+wij
+�
+∥xk
+1,i − xk
+2,j∥2
+2 + ∥yk
+1,i − yk
+2,j∥2
+2
+�
++ ∥¯yα(¯ρk
+1) − ¯yα(¯ρk
+2)∥2
+2
+where we used the linearity of the expectation, estimates given by Lemma A.4 and, to take the
+last term out of the sum, the fact that �
+ij wij = 1.
+To estimate the distance between the two consensus points, we use Lemma A.1 and note
+that the coupling w is sub-optimal for ¯ρk
+1, ¯ρk
+2. By Lemma A.1, it follows
+∥¯yα(¯ρk
+1) − ¯yα(¯ρk
+2)∥2
+2 ≤ CW 2
+2 (¯ρk
+1, ¯ρk
+2) ≤ C
+�
+i,j
+wij∥yk
+1,i − yk
+2,j∥2 .
+Therefore,
+Eθk[W 2
+2 (µk+1
+1
+, µk+1
+2
+)] ≤ C1
+�
+i,j
+wij
+�
+∥xk
+1,i − xk
+2,j∥2
+2 + ∥yk
+1,i − yk
+2,j∥2
+2
+�
+= C1 W 2
+2 (µk
+1, µk
+2) ,
+thanks to the optimality of w, with C1 = C1(∆, λ, σ, ν, α, β, LF, M) being a positive constant.
+One can conclude by taking the expectation of the above inequality with respect to the remaining
+sampling processes.
+31
+
+We now quantify the impact of the particle discarding step.
+Proof of Proposition 4.2. For notational simplicity, let us introduce zi = (xi, yi) ∈ R2d.
+As
+in (4.13), the 2-Wasserstein distance is given by an optimal coupling between the full particle
+system {zi}i∈I and the reduced one {zj}j∈Isel. We consider the following transportation of mass
+from µN to µNsel: if particle i has not been discarded, all its mass remains in xi, otherwise the
+mass is uniformly distributed among the selected particles to generate an admissible coupling
+w ∈ RN×Nsel. This means that w is given by
+wij =
+�
+�
+�
+�
+�
+1/N
+if j = i, i ∈ Isel
+1/(N · Nsel)
+if i ∈ I \ Isel, j ∈ Isel
+0
+else .
+(A.6)
+We note that such coupling w satisfies the coupling conditions
+�
+j∈Isel
+wij = 1
+N
+�
+i∈I
+wij =
+1
+Nsel
+,
+∀ i ∈ I, j ∈ Isel
+(A.7)
+and that this choice will be in general sub-optimal. Therefore, it holds
+W 2
+2 (µN, µNsel) ≤
+�
+i∈I, j∈Isel
+wij∥zi − zj∥2
+2
+= 1
+N
+�
+i∈Isel
+∥zi − zi∥2
+2 +
+1
+N · Nsel
+�
+i∈I\Isel, j∈Isel
+∥zi − zj∥2
+2
+=
+1
+N · Nsel
+�
+i,j∈I
+∥zi − zj∥2
+2 1i∈I\Isel 1j∈Isel
+where 1i∈I = 1 if i ∈ I and zero otherwise.
+Now, the probability of having i ∈ I \ Isel is given by (N − Nsel)/N, while the probability of
+having j ∈ Isel (condition i ∈ I \ Isel) is given by Nsel/(N − 1). Hence, we have
+E
+�
+1i∈I\Isel 1j∈Isel
+�
+= P [i ∈ I \ Isel, j ∈ Isel] = (N − Nsel)Nsel
+N(N − 1)
+from which follows
+E
+�
+W 2
+2 (µN, µNsel)
+�
+≤
+1
+N · Nsel
+�
+i,j∈I
+∥zi − zj∥2
+2 E
+�
+1i∈I\Isel 1j∈Isel
+�
+=
+1
+N · Nsel
+· (N − Nsel)Nsel
+N(N − 1)
+�
+i,j∈I
+∥zi − zj∥2
+2 .
+The desired estimates can finally be obtained by noting that the variance can be computed as
+var(z) = 1/(2N2) �
+i,j∈I ∥zi − zj∥2
+2, see definition (2.5).
+32
+
+Finally, we are ready to provide a proof of Theorem 4.2.
+Proof of Theorem 4.2. Let {(xk
+i , yk
+i )}i∈Ik, |Ik| = Nk be the sequence of particles generated by
+iteration (4.3) where additionally Nk+1 − Nk particles are discarded after each step k ≥ 0. We
+denote with µk
+Nk ∈ P(R2d) the empirical measure associated with such particle system given by
+µk
+Nk = 1
+Nk
+�
+i∈Ik
+δ(xk
+i ,yk
+i ) .
+We also introduce the measures µk
+N0, k ≥ 0 corresponding to a particle system generated with
+the same initial conditions µ0
+N0 but where no particle reduction occurs. Consistently, we define
+µh
+Nk, h > k to represent the particle system generated starting from µk
+Nk, after h − k iterations,
+with no random selection. The relation between such measures is summarized in the following
+diagram
+µ0
+N0
+→
+µ1
+N0
+→
+µ2
+N0
+→
+. . .
+→
+µk
+N0
+...
+���
+µ1
+N1
+→
+µ2
+N1
+→
+. . .
+→
+µk
+N1
+...
+���
+µ2
+N2
+→
+. . .
+→
+µk
+N2
+...
+...
+...
+µk
+Nk
+...
+(A.8)
+where → indicates an iteration step (4.3) while ��� a particle reduction procedure. Therefore,
+we are interested in studying the distance between the main diagonal of such diagram µk
+Nk, cor-
+responding to the system with particle reduction, and the first line µk
+N0 where particle reduction
+is never performed.
+We note that the 2-Wasserstein distance between subsequent rows can be estimated thanks
+to Proposition A.3 and Proposition 4.2. Let ˜zh+1 denote the set of particles associated with
+the probability measure µh+1
+Nh , that is, the particle systems before the selection procedure (up-
+per diagonal elements in scheme (A.8)). By first applying Proposition A.3 and, subsequently,
+Proposition 4.2 to ˜zh+1, we obtain that for some constant C > 0
+E
+�
+W 2
+2 (µk
+Nk, µk
+N0)
+�
+≤ C
+k−1
+�
+h=0
+E
+�
+W 2
+2
+�
+µk
+Nh, µk
+Nℓ+1
+��
+≤ C
+k−1
+�
+h=0
+Ck−h+1
+1
+E
+�
+W 2
+2
+�
+µh+1
+Nh , µh+1
+Nh+1
+��
+≤ 2 C
+k−1
+�
+h=0
+Ck−h+1
+1
+var
+�
+˜zh+1� Nh − Nh+1
+Nh − 1
+33
+
+≤ C2
+max
+h=1,...,k var
+�
+˜zh�
+1
+Nk − 1
+k−1
+�
+h=0
+Nh − Nh+1
+= C2
+max
+h=1,...,k var
+�
+˜zh� N0 − Nk
+Nk − 1
+with C2 = C2(∆t, λ, σ, ν, β, α, k, M). Finally, the desired estimate follows after noting that
+W 2
+2 (ρk
+Nk, ρk
+N0) ≤ W 2
+2 (µk
+Nk, µk
+N0)
+since ∥xk
+i − xk
+j ∥2
+2 ≤ ∥(xk
+i , yk
+i ) − (xk
+j , yk
+j )∥2
+2 for all couples of particles (i, j).
+Acknowledgments
+This work has been written within the activities of GNCS group of INdAM (National Institute
+of High Mathematics). L.P. acknowledges the partial support of MIUR-PRIN Project 2017,
+No. 2017KKJP4X “Innovative numerical methods for evolutionary partial differential equations
+and applications”. The work of G.B. is funded by the Deutsche Forschungsgemeinschaft (DFG,
+German Research Foundation) through 320021702/GRK2326 “Energy, Entropy, and Dissipative
+Dynamics (EDDy)” and SFB 1481 “Sparsity and Singular Structures”. S.G. acknowledges the
+support of the ESF PhD Grant “Mathematical and statistical methods for machine learning in
+biomedical and socio-sanitary applications”.
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+

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+arXiv:2301.04415v1  [hep-th]  11 Jan 2023
+Charges for Hypertranslations and Hyperrotations
+Chethan Krishnana and Jude Pereirab
+aCentre for High Energy Physics, Indian Institute of Science,
+C.V. Raman Road, Bangalore 560012, India.
+Email: [email protected]
+b Department of Physics, Arizona State University,
+Tempe, Arizona 85287-1504, USA.
+Email:
+[email protected]
+Hypertranslations and hyperrotations are asymptotic symmetries of flat space, on top of the
+familiar supertranslations and superrotations. They were discovered in arXiv:2205.01422 by working
+in the Special Double Null (SDN) gauge, where I + and I − are approached along null directions. It
+was observed there that while the hair degrees of freedom associated to these diffeomorphisms show
+up in the covariant surfaces charges, the diffeomorphisms themselves do not. This made their status
+intermediate in some ways between global symmetries and trivial gauge transformations, making
+interpretation ambiguous.
+In this paper, we revisit the fall-offs considered in arXiv:2205.01422
+which were strictly subleading to Minkowski in conventional double null coordinates. We identify
+a new class of fall-offs where this assumption is relaxed, but whose charges nonetheless remain
+finite. Remarkably, the leading behavior is still Riemann flat, indicating that these are soft modes.
+With this more refined definition of asymptotic flatness, we show that leading hypertranslations
+and leading hyperrotations explicitly show up in the charges.
+This makes them genuine global
+symmetries of asymptotically flat Einstein gravity in the SDN gauge.
+We write down the new
+algebra of asymptotic Killing vectors that subsumes the BMS algebra.
+Introduction:
+When the cosmological constant Λ is
+negative, it is widely believed that the radial direction of
+the resulting anti-de Sitter (AdS) spacetime is holograph-
+ically emergent [1]. When Λ is positive, less is known,
+but there are many suggestions in the literature that the
+timelike direction of de Sitter (dS) space may have a holo-
+graphic origin [2]. These observations make one suspect
+that it may be useful to view the holographic direction
+for flat space, which has Λ = 0, as a null coordinate.
+Historically however, the null boundary of flat space is
+typically approached along a spacelike direction, eg., in
+the famous Bondi gauge [3].
+With quite different motivations, various aspects of flat
+space were explored from a holographic perspective in [4–
+6]. Along the way, it was realized that a natural gauge
+for asymptotically flat space is the Special Double Null
+(SDN) gauge [7], defined by
+guu = 0, gvv = 0, guA = gvA.
+(1)
+Here u and v are null coordinates and I + and I − are
+covered (generically) by two separate patches around v →
+∞ and u → −∞ respectively. The holographic directions
+are v and −u in these patches.
+The notion of a double null coordinate system has been
+explored in various contexts in the literature before, see
+eg.
+[8, 9].
+But usually in these settings, not enough
+constraints are imposed to fix all the coordinate free-
+dom; they are therefore not genuine gauge choices. In
+fact in the context of mathematical relativity, the form
+of the double null metric that is sometimes written down
+(see eg., eqn (70) of [10]) does not fall into the gauge we
+have presented above. This reflects a difference in phi-
+losophy. General relativists are interested in I + as the
+eventual location of gravitational waves from localized
+objects. But if one is interested in graviton scattering,
+as perhaps necessary in quantum gravity, we need access
+to both I + and I −. Our gauge has a natural u ↔ −v
+symmetry which relates I + and I −. This manifests an
+asymptotic CPT invariance [7], which is believed to be a
+symmetry of quantum gravity in flat space [11].
+In [12] we considered the most general asymptotic sym-
+metry algebra in SDN gauge with fall-offs which are
+power laws in the respective null coordinate. Minkowski
+space in double null coordinates can be obtained by writ-
+ing u = t − r and v = t + r:
+ds2 = −du dv + 2
+�v − u
+2
+�2
+γz¯zdzd¯z
+(2)
+which has guA
+=
+gvA
+=
+0.
+This suggested that
+one should allow fall-offs where guA and gvA are at
+most O(v−1) at I +.
+The result was found to con-
+tain new classes of non-trivial asymptotic diffeomor-
+
+2
+phisms on top of the BMS symmetries [13]. These were
+named hypertranslations, subleading hypertranslations
+and subleading hyperrotations1. The algebra of asymp-
+totic Killing vectors that extends the BMS algebra was
+identified and the covariant surface charges [14, 15] were
+computed.
+It was noted that these charges had non-
+trivial dependence on the corresponding “hair” (the met-
+ric parameters affected by these asymptotic diffeomor-
+phisms). But at the same time, they did not contain the
+new asymptotic diffeomorphisms themselves, and there-
+fore the interpretation of these charges was ambiguous.
+Typically for global symmetries that emerge from an
+asymptotic symmetry calculation, both the diffeomor-
+phisms and the associated hair parameters appear in the
+charge expression. On the other hand, for trivial diffeo-
+morphisms, neither the diffeomorphisms nor the parame-
+ters associated to them appear in the charges. This made
+the status of hypertranslations and hyperrotations in-
+termediate between global symmetries and trivial gauge
+transformations, making them challenging to interpret.
+Part of the problem here is that because we are working
+with null directions, the formalism that is most suited
+for our purposes is the covariant phase space approach of
+Wald and followers [14, 15], while a more Hamiltonian-
+like formalism is perhaps more suited for interpretational
+purposes.
+In this paper, we will bypass this problem by identify-
+ing a new set of fall-offs which are not strictly subleading
+to (2), but for which the charges are still finite. These
+fall-offs are presented in (3) and also in more detail in the
+Supplementary Material. In particular, our fall-offs will
+allow guA = O(v0) = gvA. A key feature of these fall-offs
+is that they can change the metric at an order more lead-
+ing than (2), and yet remarkably, we are able to show that
+their charges remain finite. In particular, a striking fact
+that we note is that demanding Riemann flatness allows
+these terms. This allows us to adopt the philosophy that
+there is nothing too sacred about the specific form in ex-
+pression (2), it is the demand of Riemann flatness that
+should be respected in deciding the leading behavior. We
+will find that Riemann flatness still leaves the possibility
+that these modes can be functions of the angular coordi-
+nates (z, ¯z). We will eventually identify these as related
+to the hyperrotation hair. This should be compared to
+the familiar fact that purely angle dependent shear modes
+1In [12], the latter were simply called hyperrotations. But in
+the present paper we will find more leading counterparts to these
+AKVs which are more naturally called (leading) hyperrotations.
+Therefore the ones noted in [12] will be referred to as subleading
+hyperrotations in this paper.
+in Bondi gauge are soft hair associated to supertrans-
+lations, and turning them on can still leave the metric
+Riemann flat [16]. Similarly in SDN gauge, turning on
+supertranslation hair or hypertranslation hair, leaves the
+metric Riemann flat. But both in Bondi gauge as well as
+in SDN gauge, the supertranslation and hypertranslation
+soft modes were subleading to the corresponding conven-
+tional form of the Minkowski metric. The new feature
+of hyperrotation hair here is that it is more leading than
+(2) while remaining Riemann flat. Riemann flatness in
+SDN gauge has many remarkable properties, which will
+be discussed in detail elsewhere [17].
+Once we adopt these relaxed fall-offs the nature of the
+calculation is parallel to that in [12], even though techni-
+cally more involved due to the increased number of metric
+functions that we start with. The result of this exercise is
+that we find that (a) the charges are still finite, (b) there
+is a new set of Diff(S2) transformations (the leading hy-
+perrotations) that appear before subleading hyperrota-
+tions but are subleading to superrotations, (c) both the
+hair parameters as well as the diffeomorphisms associ-
+ated to the leading hypertranslations and leading hyper-
+rotations appear in the charge expressions, on top of the
+conventional BMS quantities, (d) demanding Riemann
+flatness still allows soft hair associated to these diffeo-
+morphisms to appear in the metric, and (e) the algebra
+of the asymptotic symmetries is enhanced with respect
+to both the BMS algebra as well as the BBMS algebra of
+[12].
+The next section contains the main results of this pa-
+per. To avoid repetition, we will only emphasize aspects
+of the discussion that are distinct from those in [12]. In
+particular, we will simply present the final algebra with-
+out presenting the details of the derivation – the approach
+is identical to that in [12], even though technically more
+involved.
+Results: We will work with SDN gauge discussed in [7].
+The fall-offs are presented in great detail in the Supple-
+mentary Material in terms of functions appearing in the
+metric. Here we will write the fall-offs as
+guv = −2 + O
+�
+v−1�
+(3a)
+gAB = 4γAB v−2 + O
+�
+v−3�
+(3b)
+guA = gvA = O
+�
+v−2�
+(3c)
+Even though technically this is a small change from our
+previous paper, we emphasize that this is a pretty sub-
+stantive departure from experience in other gauges. We
+are demanding that the metric be distinct from the con-
+ventional form Minkowski metric (2), already at leading
+order.
+There are three reasons why we believe this is
+
+3
+reasonable. Firstly, the charges remain finite. Secondly,
+demanding Riemann flatness does not force these terms
+to be zero. Thirdly, with this choice, we get a perfectly
+conventional structure for the leading hypertranslation
+and hyperrotation charges.
+The asymptotic Killing vector conditions take the
+form:
+Lξguv = O
+�
+v−1�
+(4a)
+LξguA = O
+�
+v−2�
+(4b)
+LξgvA = O
+�
+v−2�
+(4c)
+LξgAB = O
+�
+v−3�
+(4d)
+These and the exact Killing conditions (21), lead to the
+solutions:
+ξu = f +
+ξu
+(1)
+v
++
+ξu
+(2)
+v2 +
+ξu
+(3)
+v3 + O
+�
+v−4�
+(5a)
+ξv = −ψ
+2 v + ξv
+(0) +
+ξv
+(1)
+v
++
+ξv
+(2)
+v2 + O
+�
+v−3�
+(5b)
+ξA = Y A +
+ξA
+(1)
+v
++
+ξA
+(2)
+v2 +
+ξA
+(3)
+v3 + O
+�
+v−4�
+(5c)
+where
+f = ξu
+(0) = ψ(z, ¯z) u/2 + T (z, ¯z),
+with ψ(z, ¯z) = DAY A
+(6a)
+ξu
+(1) = αA
+2 ∂Af
+(6b)
+ξu
+(2) = 1
+2
+�
+αA
+3 ∂Af + αA
+2 ∂Aξu
+(1)
+�
+(6c)
+ξu
+(3) = 1
+3
+�
+αA
+4 ∂Af + αA
+3 ∂Aξu
+(1) + αA
+2 ∂Aξu
+(2)
+�
+(6d)
+Here T (z, ¯z) denotes supertranslations, and Y z(z), Y ¯z(¯z)
+denote superrotations.
+On top of the BMS diffeo-
+morphisms, the ξv
+(0), ξv
+(1), ξA
+(1) and ξA
+(2) are also de-
+termined by the exact and asymptotic Killing condi-
+tions. The independent functions contained in them are
+hypertranslations φ(z, ¯z), sub-leading hypertranslations
+τ(z, ¯z), hyperrotations XA(z, ¯z) and sub-leading hyper-
+rotations ZA(z, ¯z) respectively. They are related to the
+ξv
+(0), ξv
+(1), ξA
+(1), ξA
+(2) via:
+ξA
+(1) = XA − 2 DAf
+(7a)
+ξv
+(0) = φ + T + △γT − 1
+4aA
+2 DAψ − 1
+2DAXA
+(7b)
+ξv
+(1) = ˜τ + 1
+2A A
+2 DAψ
+(7c)
+ξA
+(2) = ˜ZA + C AB DBψ + A A
+2 ψ − u XA + 2 u DAξv
+(0)
+− u2 DAψ − L1 DAψ
+We have introduced ˜τ and ˜ZA for convenience which are
+related to the sub-leading hypertranslations τ and sub-
+leading hyperrotations ZA via
+˜τ = τ − 1
+4 aA
+3 DAψ
++
+�
+D¯zcz¯z − Dzczz + γz¯zDzD¯zaz
+2 − γz¯zD2
+¯za¯z
+2
+�
+DzT
++
+�
+Dzcz¯z − D¯zc¯z¯z + γz¯zD¯zDza¯z
+2 − γz¯zD2
+zaz
+2
+�
+D¯zT
++ aA
+2 DAξv
+(0) + aA
+2 DAT + 1
+4aA
+2 DA
+�
+aB
+2 DBψ
+�
+(8)
+˜Zz = Zz + czz DzT + cz¯z D¯zT + T Dzczz − T D¯zcz¯z
++ az
+2 ξv
+(0) − 1
+2XBDBaz
+2 + 1
+2aB
+2 DBXz − γz¯zD¯za¯z
+2D¯zT
+− γz¯zD¯zaz
+2DzT + 1
+4az
+2a¯z
+2D¯zψ − γz¯za¯z
+2D2
+¯zT + 1
+4
+�
+az
+2
+�2Dzψ
+− 1
+2az
+2∆γT − γz¯zT DzD¯zaz
+2 + γz¯zT D2
+¯za¯z
+2
+(9)
+˜Z ¯z = Z ¯z + c¯z¯z D¯zT + cz¯z DzT + T D¯zc¯z¯z − T Dzcz¯z
++ a¯z
+2 ξv
+(0) − 1
+2XBDBa¯z
+2 + 1
+2aB
+2 DBX ¯z − γz¯zDzaz
+2DzT
+− γz¯zDza¯z
+2D¯zT + 1
+4a¯z
+2az
+2Dzψ − γz¯zaz
+2D2
+zT + 1
+4
+�
+a¯z
+2
+�2D¯zψ
+− 1
+2a¯z
+2∆γT − γz¯zT D¯zDza¯z
+2 + γz¯zT D2
+zaz
+2
+(10)
+These expressions are significantly more complicated
+than those in [12], so let us pause to explain some of the
+details.
+The integration “constants” in the shear are2
+introduced via
+CAB(u, z, ¯z) = cAB(z, ¯z) +
+� u
+−∞
+du′NAB(u′, z, ¯z)(11)
+with NAB ≡ ∂uCAB, being the SDN news tensor. Sim-
+ilarly, we have defined the integration “constant” in αA
+2
+as
+αA
+2 (u, z, ¯z) = aA
+2 (z, ¯z) +
+� u
+−∞
+du′βA
+2 (u′, z, ¯z)
+(12)
+where βA
+2 ≡ ∂uαA
+2 . See [7, 18] for a discussion on integrals
+of this type that are defined from I +
+− to u. On shell (ie.,
+when Einstein equations hold), we have Nz¯z = 0 and
+βA
+2 = 0, so we will have
+Cz¯z(u, z, ¯z) = cz¯z(z, ¯z)
+αA
+2 (u, z, ¯z) = aA
+2 (z, ¯z)
+(13)
+In addition to this, the Einstein constraints also require
+2The notation here is slightly different from that in [12].
+
+4
+that λ1 = 0. For ξA
+(2), combining all the relevant equa-
+tions, we can write [18]
+∂uξA
+(2) = CAB DBψ − 2 u DAψ + 2 DAξv
+(0) + αA
+2 ψ
+− λ1DAψ − XA
+=⇒ ξA
+(2) = C AB DBψ − u2 DAψ + 2 u DAξv
+(0) + A A
+2 ψ
+− L1DAψ − u XA + ˜ZA(z, ¯z)
+(14)
+The u-independence of ψ, ξv
+(0) and XA has been used in
+writing the integrated version in the second step. Also
+C AB, A A
+2
+and L2 have been defined via
+∂uC AB = CAB(u, z, ¯z)
+(15)
+∂uA A
+2 = αA
+2 (u, z, ¯z)
+(16)
+∂uL1 = λ1(u, z, ¯z)
+(17)
+As in [12], ˜ZA(z, ¯z) is taken as the u-independent piece in
+ξA
+(2). The shift is done on ˜ZA via (9)-(10) and the result
+is what we call sub-leading hyperrotations ZA.
+The rest of the notation follows that of [12]. As empha-
+sized there, the idea in (7) is to do certain shifts so that
+the structure of the diffeomorphisms is cleanest.
+This
+“diagonalizes” the algebra of diffeomorphisms. The phi-
+losophy here is identical, even though the expressions are
+more complicated.
+The hair associated to the various diffeomorphisms are
+therefore as follows: supertranslations T (z, ¯z) are associ-
+ated to the u-independent shifts in Czz and C¯z¯z, hyper-
+translations φ(z, ¯z) are associated to the u-independent
+shifts of Cz¯z, subleading hypertranslations τ(z, ¯z) are as-
+sociated to u-independent shifts of λ2, hyperrotations
+XA(z, ¯z) are associated to u-independent shifts of α A
+2
+and sub-leading hyperrotations ZA(z, ¯z) are associated
+to u-independent shifts of α A
+3 . As in Bondi gauge, we
+also have superrotations Y z(z), Y ¯z(¯z).
+The shifts in-
+volved in the definitions of φ, τ, XA and ZA are detailed
+in the Supplementary Material. As in [12], supertrans-
+lations and leading-&-subleading hypertranslations are
+diffeomorphisms of u and v respectively. Leading hyper-
+rotations were not present in [12], but both leading and
+subleading hyperrotations are subleading to superrota-
+tions on the sphere.
+We will define the “Beyond BBMS” algebra b2-bms4
+as the asymptotic symmetry algebra of the nine non-
+trivial diffeomorphisms – supertranslations, superrota-
+tions, hypertranslations & subleading hypertranslations,
+and hyperrotations & subleading hyperrotations. Follow-
+ing [12, 13], we define the bracket
+��Y , �T, �φ, �τ, �
+X, �Z
+�
+=
+�
+(Y1, T1, φ1, τ1, X1, Z1), (Y2, T2, τ2, φ2, X2, Z2)
+�
+(18)
+The notation is the natural generalization of that in [12,
+13] and the reader should consult those papers for the
+detailed definitions. The new algebra is defined via �Y ,
+�T, �φ, �τ, �
+X and �Z given by the following expressions:
+�Y A = Y B
+1 ∂BY A
+2 − Y B
+2 ∂BY A
+1
+(19a)
+�T = Y A
+1 ∂AT2 − Y A
+2 ∂AT1 + 1
+2 (T1 ψ2 − T2 ψ1).
+(19b)
+�φ = 1
+2(ψ1φ2 − ψ2φ1) +
+�
+Y A
+1 ∂Aφ2 − Y A
+2 ∂Aφ1
+�
+(19c)
+�τ = (ψ1τ2 − ψ2τ1) +
+�
+Y A
+1 ∂Aτ2 − Y A
+2 ∂Aτ1
+�
+(19d)
+�
+XA = 1
+2
+�
+ψ1XA
+2 − ψ2XA
+1
+�
++
+�
+Y B
+1 ∂BXA
+2 − Y B
+2 ∂BXA
+1
+�
++
+�
+XB
+1 ∂BY A
+2 − XB
+2 ∂BY A
+1
+�
+(19e)
+�ZA =
+�
+ψ1ZA
+2 − ψ2ZA
+1
+�
++
+�
+Y B
+1 ∂BZA
+2 − Y B
+2 ∂BZA
+1
+�
++
+�
+ZB
+1 ∂BY A
+2 − ZB
+2 ∂BY A
+1
+�
+(19f)
+This is what we call the b2-bms4 algebra. The fact that
+these nine non-trivial diffeomorphisms form a closed al-
+gebra is checked by the same procedure as outlined in
+[12]. The calculations are straightforward but lengthier
+variations of those there. In order to identify the capped
+quantities, we need to consider the Barnich-Troessaert
+bracket [ξ1, ξ2]M of two AKVs ξ1 and ξ2 [12, 13]. The
+structure is parallel to that presented in [12], with a no-
+
+5
+table difference in the A-component which takes the form
+[ξ1, ξ2]A
+M = �Y A +
+�ξA
+(1)
+v
++
+�ξA
+(2)
+v2 + O
+�
+v−3�
+.
+(20)
+In computing all four components of the Barnich-
+Troessaert bracket, we need �ξv
+(0), �ξv
+(1), �ξA
+(1) and �ξA
+(2), which
+are defined as in (7) but with Y A, T, φ, τ, XA, ZA re-
+placed by their capped versions, defined in (19).
+Equations (19) define the b2-bms4 algebra. Setting the
+hyperrotations XA to zero results in the BBMS algebra
+of [12], and setting φ, τ and ZA as well to zero results in
+the familiar BMS algebra [13].
+Discussion: In this paper, we observed that demand-
+ing finite covariant surface charges in Einstein gravity
+allows fall-offs that are not necessarily subleading to (2).
+Turning on the soft modes associated to supertransla-
+tions and leading hypertranslations/hyperrotations takes
+us beyond (2) even though the metric is still Riemann
+flat. We exploited this fact to work with fall-offs that
+allowed these modes, to show that the covariant surface
+charges contain these diffeomorphisms as well as the as-
+sociated soft hair. This places them on an equal footing
+with conventional global symmetries (eg. supertransla-
+tions), resolving some of the ambiguities pointed out in
+[12].
+Of course, these results open up further questions. Our
+work strongly suggests that the charges associated to hy-
+pertranslations should be interpreted as soft, so it would
+be interesting to connect these results to soft theorems
+(perhaps to the subsubleading soft graviton theorem of
+[20]?) and also to new memory effects. Some of these
+questions are currently under investigation. Hypertrans-
+lations have many similarities to supertranslations, but
+there are also crucial distinctions. The lowest modes of
+supertranslations are simply the action of Poincare trans-
+lations on the boundary (u, z, ¯z). Hypertranslations on
+the other hand are truly distinct from bulk translations –
+we have already subtracted out the supertranslations in
+our shifted diffeomorphisms, when defining hypertransla-
+tions. It should be clear from (5) that the interpretation
+of hypertranslations is more like a bulk diffeomorphism
+at infinity (note that infinity is along the null direction
+v in SDN gauge). It is more naturally compared to ξr
+than ξu in Bondi gauge.
+A related interesting feature of hypertranslations and
+their associated hair is that they can be spherically sym-
+metric.
+This raises a subtlety in the usual statement
+of Birkhoff’s theorem, which will be discussed in an up-
+coming work.
+Note that while supertranslations allow
+soft hair on Schwarzschild, the only spherically symmet-
+ric supertranslation is a time translation, so this subtlety
+does not arise for Schwarzschild in Bondi gauge. It is also
+important to emphasize that hypertranslations should be
+distinguished from the shifts in v at the past boundary
+I −. The latter are simply supertranslations, but now
+acting in the past. What we mean by hypertranslations
+are shifts in v at I +. There is no obvious connection
+between the two (other than the future-past matching at
+i0 that was discussed in [7]) because these coordinates
+live in different charts.
+What about subleading hypertranslations and sublead-
+ing hyperrotations? They do not show up in the charges
+even with the new fall-offs, but their associated hair was
+present both in [12] as well as here. So their interpre-
+tation remains ambiguous. It is natural to consider the
+sub-algebra obtained by setting the subleading hyper-
+translations/hyperrotations to zero.
+This would mean
+that we are working with supertranslations, superrota-
+tions, leading hypertranslations and leading hyperrota-
+tions. This is a natural generalization of the conventional
+BMS algebra in the SDN gauge; it is clearly of interest to
+study it more closely. One could also consider the even
+simpler generalization of BMS, obtained by adding only
+the leading hypertranslations and suppressing the lead-
+ing hyperrotations. This algebra has the advantage that
+we are not turning on diffeomorphisms on the sphere,
+but only the Virasoro (super)rotations.
+While it may
+be difficult to conclusively argue for such a choice from
+a purely asymptotic symmetry perspective, it is natural
+from a celestial holography perspective [19]. This is the
+algebra of supertranslations, (leading) hypertranslations
+and superrotations.
+ACKNOWLEDGMENTS
+We thank Sudip Ghosh and Sarthak Talukdar for dis-
+cussions.
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+Witten,
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+Sitter
+space
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+BMS/CFT
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+Supplementary material
+REFINED FALL-OFFS
+In this section, we will present the falloffs in some detail. Our emphasis will be on the distinctions from those
+presented in [12]. We start with a quick review of the notation: in d + 1 dimensions, the SDN gauge [7] is defined by
+eqn (1). We will restrict ourselves to 3+1 dimensions here. The exact Killing vector equations are
+Lξguu = 0, Lξgvv = 0, LξguA = LξgvA
+(21)
+and we will write the general metric in this gauge as
+ds2 = −eλdu dv +
+�v − u
+2
+�2
+ΩAB(dxA − αAdu − αAdv)(dxB − αBdu − αBdv)
+(22)
+
+7
+In [12], we presented a set of fall-offs in terms of the functions in this ansatz, which the reader should consult. The
+fall-offs we consider in this paper are distinct in the following functions:
+λ(u, v, z, ¯z) = λ1(u, z, ¯z)
+v
++ λ2(u, z, ¯z)
+v2
++ λ3(u, z, ¯z)
+v3
++ λ4(u, z, ¯z)
+v4
++ O
+�
+v−5�
+(23a)
+αz(u, v, z, ¯z) = αz
+2(u, z, ¯z)
+v2
++ αz
+3(u, z, ¯z)
+v3
++ αz
+4(u, z, ¯z)
+v4
++ αz
+5(u, z, ¯z)
+v5
++ O
+�
+v−6�
+(23b)
+α¯z(u, v, z, ¯z) = α¯z
+2(u, z, ¯z)
+v2
++ α¯z
+3(u, z, ¯z)
+v3
++ α¯z
+4(u, z, ¯z)
+v4
++ α¯z
+5(u, z, ¯z)
+v5
++ O
+�
+v−6�
+(23c)
+In terms of the metric, this results in the fall-offs:
+guu = gvv = O
+�
+v−2�
+(24a)
+guv = −1
+2 + O
+�
+v−1�
+(24b)
+gAB = 1
+4 γAB v2 + O(v)
+(24c)
+guA = gvA = O
+�
+v0�
+(24d)
+Compared to the discussion in [12], we also allow αA
+2 as the O(1/v2) term in the αA fall-off. Just like Cz¯z, αA
+2 also
+turns out to be u-independent once we demand Einstein equations. Hence it is an integration “constant” in Einstein
+constraints in the language of [7, 12, 18].
+Demanding Ricci (or Riemann) flatness forces λ1 to be zero and αA
+2 to be functions only of the angles. We have
+kept them general in the discussions of the AKVs because they can be defined on arbitrary backgrounds, without
+worrying about the equations satisfied by those backgrounds. But one can in principle start a-priori with fall-offs
+(23) where λ1 is set to zero and αA
+2 are functions only of z and ¯z. Some of the expressions we have presented will
+simplify somewhat in that case, but the main results do not change.
+DIFFEOMORPHISM SHIFTS
+As in [12] we will define the various diffeomorphisms after a suitable shift in the fall-off coefficient of ξ. This is more
+elaborate in the present case, and we discuss them in detail below. The philosophy behind these shifts was discussed
+in [12].
+Hyperrotations: The simplest case arises for the leading hyperrotations XA(z, ¯z), so we start with them. From
+the exact Lie derivative conditions, we obtain the following constraint on ξA
+1 ,
+∂uξA
+1 = −DAψ
+(25)
+which on integrating both sides becomes
+ξA
+1 = ˜XA − u DAψ
+(26)
+The metric function corresponding to leading hyperrotations is αA
+2 . Under the action of AKVs, the transformation
+of αA
+2 can be obtained by evaluating δξguA = LξguA at O(v−2) as follows
+δαA
+2 =
+�
+f∂u + LY + ψ
+2
+�
+αA
+2 + ˜XA − u DAψ + 2 DAf
+(27)
+where
+LY αA
+2 = Y B ∂BαA
+2 − αB
+2 ∂BY A.
+(28)
+
+8
+is the Lie derivative of αA
+2 with respect to Y A. Recalling that on-shell αA
+2 = aA
+2 (z, ¯z) and substituting f = ψ(z, ¯z) u/2+
+T (z, ¯z), we obtain
+δaA
+2 =
+�
+LY + ψ
+2
+�
+aA
+2 + ˜XA + 2 DAT
+(29)
+Next we would like to interpret XA(z, ¯z) as the diffeomorphism that causes αA
+2 to be turned on if it was initially zero.
+This immediately suggests the following shift
+˜XA = XA − 2 DAT
+(30)
+Substituting this in (26) and using f =
+�
+ψ/2
+�
+u + T yields
+ξA
+1 = XA − 2 DAf
+(31)
+Hypertranslations: In the case of the leading hypertranslations φ(z, ¯z), the shift is of the form
+ξv
+(0) = φ + T + △γT − 1
+4aA
+2 DAψ − 1
+2DAXA
+(32)
+This reduces to the form presented in [12] when the hyperrotations and their hair are set to zero. The change in Cz¯z
+can be computed by evaluating δξgz¯z = Lξgz¯z at O(v−3). The result is
+δCz¯z =
+�
+f ∂u + LY − 1
+2 ψ
+�
+Cz¯z − 4 ∂z∂¯zf + 2 γz¯z
+�
+ξv
+(0) − f − u
+2 ψ + 1
+2DAXA + 1
+4αA
+2 DAψ
+�
+(33)
+Here LY is the Lie derivative of Cz¯z with respect to Y A defined as in [12]:
+LY Cz¯z = Y A∂ACz¯z +
+�
+∂AY A�
+Cz¯z
+(34)
+On-shell we have Cz¯z = cz¯z(z, ¯z) and αA
+2 = aA
+2 (z, ¯z). Using these and substituting f = ψ(z, ¯z) u/2 + T (z, ¯z), we obtain
+δcz¯z =
+�
+LY − 1
+2 ψ
+�
+cz¯z + 2γz¯z
+�
+ξv
+(0) − T − ∆γT + 1
+2DAXA + 1
+4aA
+2 DAψ
+�
+(35)
+It is clear that ξv
+(0) mixes with supertranslations, superrotations and leading hyperrotations. We wish to remove
+this mixing, so that we can interpret φ(z, ¯z) as the diffeomorphism that causes cz¯z to be turned on if it was initially
+zero. From this it follows that the shift is ξv
+(0) = φ + T + △γT − 1
+4aA
+2 DAψ − 1
+2DAXA, as we presented above. This
+defines hypertranslations, φ(z, ¯z). Note that in deriving the algebra for hypertranslations, we have made use of the
+identity
+δξξv
+(0) = −1
+4
+�
+δaA
+2
+�
+DAψ
+(36)
+where we have demanded that δξφ = 0 and δξXA = 0. This shifted definition above of the hypertranslations ensures
+the vanishing of the hatted �φ on the left hand side of algebra, when φ1 and φ2 are zero. As we pointed out in [12], this
+feature can be viewed as one of the motivations behind doing the shifts. This generalizes to the other diffeomorphisms
+as well.
+Subleading Hyperrotations: Now we turn to the case of subleading hyperrotations ZA(z, ¯z) and the correspond-
+ing metric functions αA
+3 . The same procedure as in [12] now yields
+δαz
+3 =
+�
+f ∂u + LY + ψ
+�
+αz
+3 + 2 ξz
+(2) + 4 u Dzf − 2 CzB DBf − 2 αz
+2 ξv
+(0) + XBDBαz
+2 − αB
+2 DBXz
++ 2 γz¯zD¯zα¯z
+2D¯zf + 2 γz¯zD¯zαz
+2DzT − 1
+2αz
+2α¯z
+2D¯zψ + 2 γz¯zα¯z
+2D2
+¯zT + 2u γz¯zα¯z
+2D2
+¯zψ
+− u γz¯zDzαz
+2D¯zψ − 1
+2
+�
+αz
+2
+�2Dzψ − 2u αz
+2ψ + αz
+2∆γT + 2 λ1Dzf + ∂uαz
+2
+�
+αA
+2 DAf
+�
+(37)
+
+9
+where the Lie derivative is defined as
+LY αA
+3 = Y B ∂BαA
+3 − αB
+3 ∂BY A.
+(38)
+Note that in obtaining the above equation, we have used (27) along with
+δλ1 =
+�
+f ∂u + LY + 1
+2ψ
+�
+λ1 + ∂uαA
+2 DAf
+(39)
+which has been obtained by evaluating δξguv = Lξguv at O(v−1) where LY λ1 = Y A∂Aλ1 is the Lie derivative of λ1
+with respect to Y A. On-shell, we have [18]
+∂uαz
+3 = −2 DzCzz + 2 D¯zcz¯z + 2 γz¯zDzD¯zaz
+2 − 2 γz¯zD2
+¯za¯z
+2
+=⇒ αz
+3(u, z, ¯z) = −2 DzC zz + u
+�
+2 D¯zcz¯z + 2 γz¯zDzD¯zaz
+2 − 2 γz¯zD2
+¯za¯z
+2
+�
++ az
+3(z, ¯z)
+(40)
+and a similar equation for α¯z
+3(u, z, ¯z). Recalling that on-shell λ1 = 0, substituting (13), (40), (14) and (6a) into (37)
+and extracting the u-independent terms, we find
+δaz
+3 =
+�
+LY + ψ
+�
+az
+3 + 2 ˜Zz − 2 czz DzT − 2 cz¯z D¯zT − 2 T Dzczz + 2 T D¯zcz¯z − 2 az
+2 ξv
+(0) + XB DBaz
+2
+− aB
+2 DBXz + 2 γz¯zD¯za¯z
+2 D¯zT + 2 γz¯zD¯zaz
+2 DzT − 1
+2az
+2a¯z
+2 D¯zψ + 2 γz¯za¯z
+2 D2
+¯zT − 1
+2
+�
+az
+2
+�2 Dzψ
++ az
+2 ∆γT + 2 γz¯zT DzD¯zaz
+2 − 2 γz¯zT D2
+¯za¯z
+2
+(41)
+As in [12], the inhomogeneous part of the variation gives the shift:
+˜Zz = Zz + czz DzT + cz¯z D¯zT + T Dzczz − T D¯zcz¯z + az
+2 ξv
+(0) − 1
+2XBDBaz
+2
++ 1
+2aB
+2 DBXz − γz¯zD¯za¯z
+2D¯zT − γz¯zD¯zaz
+2DzT + 1
+4az
+2a¯z
+2D¯zψ − γz¯za¯z
+2D2
+¯zT + 1
+4
+�
+az
+2
+�2Dzψ
+− 1
+2az
+2∆γT − γz¯zT DzD¯zaz
+2 + γz¯zT D2
+¯za¯z
+2
+(42)
+For completeness, we also present the result for α¯z
+3(u, z, ¯z), which gives an analogous shift for the ¯z-component of the
+subleading hyperrotations:
+˜Z ¯z = Z ¯z + c¯z¯z D¯zT + cz¯z DzT + T D¯zc¯z¯z − T Dzcz¯z + a¯z
+2 ξv
+(0) − 1
+2XBDBa¯z
+2
++ 1
+2aB
+2 DBX ¯z − γz¯zDzaz
+2DzT − γz¯zDza¯z
+2D¯zT + 1
+4a¯z
+2az
+2Dzψ − γz¯zaz
+2D2
+zT + 1
+4
+�
+a¯z
+2
+�2D¯zψ
+− 1
+2a¯z
+2∆γT − γz¯zT D¯zDza¯z
+2 + γz¯zT D2
+zaz
+2
+(43)
+The point of the shifts is that after doing them, the ZA’s are the independent diffeomorphisms. So it is natural to
+demand
+δξZA = 0.
+(44)
+This leads to
+δξ ˜Zz =
+�
+δczz�
+DzT +
+�
+δcz¯z�
+D¯zT + T
+�
+Dzδczz�
+− T
+�
+D¯zδcz¯z�
++
+�
+δaz
+2
+�
+ξv
+(0) + az
+2
+�
+δξξv
+(0)
+�
+− 1
+2XB�
+DBδaz
+2
+�
++ 1
+2
+�
+δaB
+2
+�
+DBXz − γz¯z�
+D¯zδa¯z
+2
+�
+D¯zT − γz¯z�
+D¯zδaz
+2
+�
+DzT + 1
+4a¯z
+2
+�
+δaz
+2
+�
+D¯zψ + 1
+4az
+2
+�
+δa¯z
+2
+�
+D¯zψ − γz¯z�
+δa¯z
+2
+�
+D2
+¯zT
++ 1
+2az
+2
+�
+δaz
+2
+�
+Dzψ − 1
+2
+�
+δaz
+2
+�
+∆γT − γz¯zT
+�
+DzD¯zδaz
+2
+�
++ γz¯zT
+�
+D2
+¯zδa¯z
+2
+�
+(45)
+with a similar expression for δξ ˜Z ¯z. When computing the algebra for the shifted subleading hyperrotations ZA, these
+expressions come in handy for cancelling out certain unpleasant pieces, and leading to the simple form of our final
+
+10
+algebra (19).
+Subleading Hypertranslations: Following the same procedure as in [12], we find
+δλ2 =
+�
+f∂u + LY + ψ
+�
+λ2 − 1
+4 αA
+3 DAψ + 1
+2 ∂uαA
+3 DAf − ξv
+(1) + αA
+2 DAξv
+(0) + αA
+2 DAT + 1
+4αA
+2 DA
+�
+αB
+2 DBψ
+�
+− λ1 ξv
+(0) + ξA
+(1)DAλ1 + ∂uλ1 αA
+2 DAf + 1
+2∂u
+�
+αA
+2 DAαB
+2
+�
+DBf + αA
+2 ∂uαB
+2 DADBf
+(46)
+with LY λ2 = Y A∂Aλ2. By demanding the Einstein equations as in [12], we can write
+λ2 = λ0
+2(z, ¯z) + u λ1
+2(z, ¯z) + Λ2(u, z, ¯z)
+(47)
+where the form of Λ2(u, z, ¯z) will not be important in what follows. This leads to
+δλ0
+2 =
+�
+ψ + LY
+�
+λ0
+2 + T λ1
+2 − ˜τ − 1
+4 aA
+3 DAψ +
+�
+D¯zcz¯z − Dzczz + γz¯zDzD¯zaz
+2 − γz¯zD2
+¯za¯z
+2
+�
+DzT
++
+�
+Dzcz¯z − D¯zc¯z¯z + γz¯zD¯zDza¯z
+2 − γz¯zD2
+zaz
+2
+�
+D¯zT + aA
+2 DAξv
+(0) + aA
+2 DAT + 1
+4aA
+2 DA
+�
+aB
+2 DBψ
+�
+(48)
+The inhomogeneous part of this is the independent subleading hypertranslation, which takes the form
+˜τ = τ − 1
+4 aA
+3 DAψ +
+�
+D¯zcz¯z − Dzczz + γz¯zDzD¯zaz
+2 − γz¯zD2
+¯za¯z
+2
+�
+DzT
++
+�
+Dzcz¯z − D¯zc¯z¯z + γz¯zD¯zDza¯z
+2 − γz¯zD2
+zaz
+2
+�
+D¯zT + aA
+2 DAξv
+(0) + aA
+2 DAT + 1
+4aA
+2 DA
+�
+aB
+2 DBψ
+�
+(49)
+This results in the modified algebra we presented earlier.
+COVARIANT SURFACE CHARGES
+We will compute the covariant surface charges of [14] as in [12], see also [21]. For the set up in the present paper
+putting all the ingredients together leads to a potentially divergent term
+/δQξ[h; g] =
+1
+16πG lim
+v→∞
+�
+d2Ω
+�1
+8
+�
+ψ DAδαA
+2 − 2 YA δαA
+2 − ψ γABδCAB − δαA
+2 DAψ − YA ∂uδαA
+3
+�
+v + O
+�
+v0��
+(50)
+This should be compared to eqn. (54) of [12]. Substituting
+∂uδαz
+3 = 2 D¯zδCz¯z − 2 DzδCzz + 2 γz¯zDzD¯zδαz
+2 − 2 γz¯zD¯zD¯zδα¯z
+2
+(51a)
+∂uδα¯z
+3 = 2 DzδCz¯z − 2 D¯zδC ¯z¯z + 2 γz¯zDzD¯zδα¯z
+2 − 2 γz¯zDzDzδαz
+2
+(51b)
+we obtain
+/δQξ[h; g] =
+1
+16πG lim
+v→∞
+�
+d2Ω
+�1
+4γz¯z �
+Y z�
+D¯zδCzz − DzδCz¯z
+�
++ Y ¯z�
+DzδC¯z¯z − D¯zδCz¯z
+�
+− ψ δCz¯z
+�
++ 1
+4
+�
+Y z DzDzδαz
+2 − δαz
+2 Dzψ + 1
+2Dz
+�
+ψ δαz
+2
+�
+− Y ¯z DzD¯zδαz
+2 − γz¯zY ¯z δαz
+2
++ Y ¯z D¯zD¯zδα¯z
+2 − δα¯z
+2 D¯zψ + 1
+2D¯z
+�
+ψ δα¯z
+2
+�
+− Y z D¯zDzδα¯z
+2 − γz¯zY z δα¯z
+2
+�
+v + O
+�
+v0��
+(52)
+
+11
+We will establish finiteness of the charges by showing that the O
+�
+v
+�
+term vanishes. The terms on the first line in the
+parenthesis at O
+�
+v
+�
+in the above expression can be rewritten as
+Y z�
+D¯zδCzz − DzδCz¯z
+�
++ Y ¯z�
+DzδC¯z¯z − D¯zδCz¯z
+�
+− ψ δCz¯z
+= Y z D¯zδCzz + Y ¯z DzδC¯z¯z − Y zDzδCz¯z − Y ¯zD¯zδCz¯z −
+�
+DzY z + D¯zY ¯z�
+δCz¯z
+= D¯z
+�
+Y z δCzz
+�
++ Dz
+�
+Y ¯z δC¯z¯z
+�
+− Dz
+�
+Y z δCz¯z
+�
+− D¯z
+�
+Y ¯z δCz¯z
+�
+= Dz
+�
+Y ¯z δC¯z¯z − Y z δCz¯z
+�
++ D¯z
+�
+Y z δCzz − Y ¯z δCz¯z
+�
+(53)
+Similarly, the terms on the second line in the parenthesis at O
+�
+v
+�
+can be rewritten as
+Y z DzDzδαz
+2 − δαz
+2 Dzψ + 1
+2Dz
+�
+ψ δαz
+2
+�
+− Y ¯z DzD¯zδαz
+2 − γz¯zY ¯z δαz
+2
+= Y z DzDzδαz
+2 − δαz
+2 DzDzY z − δαz
+2 DzD¯zY ¯z + 1
+2Dz
+�
+ψ δαz
+2
+�
+− Y ¯z DzD¯zδαz
+2 − γz¯zY ¯z δαz
+2
+=
+�
+Y z DzDzδαz
+2 + Dzδαz
+2 DzY z�
+− Dz
+�
+δαz
+2 DzY z�
+− δαz
+2 DzD¯zY ¯z + 1
+2Dz
+�
+ψ δαz
+2
+�
+− Y ¯z DzD¯zδαz
+2 − γz¯zY ¯z δαz
+2
+= Dz(Y zDzδαz
+2) − Dz
+�
+δαz
+2DzY z�
+−
+�
+δαz
+2 D¯zDzY ¯z − γz¯z δαz
+2 Y ¯z�
++ 1
+2Dz
+�
+ψ δαz
+2
+�
++
+�
+DzY ¯z D¯zδαz
+2
+− Dz
+�
+Y ¯zD¯zδαz
+2
+��
+− γz¯zY ¯z δαz
+2
+= Dz(Y zDzδαz
+2) − Dz
+�
+δαz
+2DzY z�
++ 1
+2Dz
+�
+ψ δαz
+2
+�
+− Dz
+�
+Y ¯zD¯zδαz
+2
+�
+(54)
+Note that in writing down the third equality in the above expression, we have commuted the covariant derivatives
+acting on Y ¯z using the definition of the Riemann tensor. That is, we have evaluated [DA, DB]Y C = RC
+DABY D to
+obtain DzD¯zY ¯z − D¯zDzY ¯z = −γz¯zY ¯z. To simplify and obtain the final expression, we have made use of the fact
+that Y z and Y ¯z are holomorphic functions of z and ¯z respectively. A similar procedure can be implemented for δα¯z
+2
+to rewrite the terms on the third line in the parenthesis at O
+�
+v
+�
+as follows
+Y ¯z D¯zD¯zδα¯z
+2 − δα¯z
+2 D¯zψ + 1
+2D¯z
+�
+ψ δα¯z
+2
+�
+− Y z D¯zDzδα¯z
+2 − γz¯zY z δα¯z
+2
+= D¯z(Y ¯zD¯zδα¯z
+2) − D¯z
+�
+δα¯z
+2D¯zY ¯z�
++ 1
+2D¯z
+�
+ψ δα¯z
+2
+�
+− D¯z
+�
+Y zDzδα¯z
+2
+�
+(55)
+After integration over the 2-sphere, the “total” derivative terms disappear. The vanishing of O
+�
+v
+�
+terms guarantees
+that the surface charges remain finite in the limit v → ∞. This is one of our key results in this paper.
+Due to the vanishing of the O
+�
+v
+�
+terms, only the O
+�
+v0�
+terms remain in the v → ∞ limit. These constitute our
+charge expression and they can be evaluated to be
+/δQξ[h; g] =
+1
+16πG
+�
+d2Ω
+�
+u YA δαA
+2 − 3
+8YA δαA
+3 − 1
+4Y AαB
+2 δCAB − f
+2 γz¯z δCz¯z − 1
+4γAB ξA
+(1) δαB
+2 − ψ
+8 γAB αA
+2 δαB
+2
+− ψ
+4 δλ2 + u ψ
+2 γz¯zδCz¯z − 3
+8ψ γz¯zDz¯z − 1
+4Y A δαB
+2 CAB + 3
+16ψ δCAB CAB + ψ γAC δCAB DCαB
+2 + f
+4 DA�A
+2
+− 1
+4ξv
+(0) DAδαA
+2 − u
+4 ψ DAδαA
+2 + 1
+4δαA
+2 DAξv
+(0) + u
+4 δαA
+2 DAψ − 1
+8δαA
+3 DAψ + ψ
+8 γz¯zCz¯z DAδαA
+2
++ 1
+4γz¯z ψ αA
+2 DAδCz¯z − 1
+4γz¯zδCz¯z αA
+2 DAψ − 1
+8γz¯zCz¯z δαA
+2 DAψ + 1
+8γz¯z ψ δαA
+2 DACz¯z − 1
+4δαA
+2 DAf
++ ψ
+8 DAδαA
+3 + u ψ
+4 γz¯zδCz¯z − ψ
+8 γz¯zδDz¯z − 1
+8Y A δCAB ∂uαB
+3 + u
+2 YA ∂uδαA
+3 + f
+2 γz¯z ∂uδDz¯z − f
+8 N AB δCAB
+− 1
+8γAB ξA
+(1) ∂uδαB
+3 − ψ
+16 γAB αA
+2 ∂uδαB
+3 − 1
+8Y A CAB ∂uδαB
+3 − 1
+8YA ∂uδαA
+4 − f
+4 CAB δNAB
+�
+(56)
+
+12
+Further on, rearranging the terms and simplifying the above expression gives
+/δQξ[h; g] =
+1
+16πG
+�
+d2Ω
+�
+YA
+�
+u δαA
+2 − 3
+8δαA
+3 + u
+2 ∂uδαA
+3 − 1
+8∂uδαA
+4 − 1
+4δ
+�
+CA
+B αB
+2
+�
+− 1
+8δ
+�
+CA
+B ∂uαB
+3
+��
++ ψ
+�
+− u
+2 DAδαA
+2 + 1
+4DAδαA
+3 + 3
+4u γz¯z δCz¯z − 1
+2 γz¯z δDz¯z − 1
+4δλ2 − 1
+8γAB αA
+2 δαB
+2
++ 3
+16CAB δCAB + DAαB
+2 δCAB + 1
+4γz¯z αA
+2 DAδCz¯z + 1
+4δ
+�
+DA(γz¯z Cz¯z αA
+2 )
+�
+− 1
+16γAB αA
+2 ∂uδαB
+3
+�
++ f
+�1
+2DAδαA
+2 − 1
+2γz¯z δCz¯z + 1
+2γz¯z ∂uδDz¯z − 1
+8N AB δCAB − 1
+4 CAB δNAB
+�
++ ξA
+(1)
+�
+− 1
+4γAB δαB
+2 − 1
+8γAB ∂uδαB
+3
+�
++ ξv
+(0)
+�
+− 1
+2DAδαA
+2
+��
+(57)
+Next we can substitute in the shifts that we obtained earlier, namely
+ξA
+(1) = XA − 2 DAf
+(58a)
+ξv
+(0) = φ + f + ∆γf + u
+2 ψ − 1
+2DAXA − 1
+4αA
+2 DAψ
+(58b)
+to write down the final form of the charge expression as follows:
+/δQξ[h; g] =
+1
+16πG
+�
+d2Ω
+�
+YA
+�
+u δαA
+2 − 3
+8δαA
+3 + u
+2 ∂uδαA
+3 − 1
+8∂uδαA
+4 − 1
+4δ
+�
+CA
+B αB
+2
+�
+− 1
+8δ
+�
+CA
+B ∂uαB
+3
+��
++ ψ
+�
+− 3
+4u DAδαA
+2 + 1
+4DAδαA
+3 + 3
+4u γz¯z δCz¯z − 1
+2 γz¯z δDz¯z − 1
+4δλ2 − 1
+8γAB αA
+2 δαB
+2 + 3
+16CAB δCAB
++ DAαB
+2 δCAB + 1
+4γz¯z αA
+2 DAδCz¯z + 1
+4δ
+�
+DA(γz¯z Cz¯z αA
+2 )
+�
+− 1
+16γAB αA
+2 ∂uδαB
+3 − 1
+8DA
+�
+αA
+2 DBδαB
+2
+��
++ f
+�
+− 1
+2DAδαA
+2 − 1
+2γz¯z δCz¯z + 1
+2γz¯z ∂uδDz¯z − 1
+4∂uDAδαA
+3 − 1
+8N AB δCAB − 1
+4 CAB δNAB
+�
+− 1
+2∆γf DAδαA
+2 + XA
+�
+− 1
+4δαA
+2 − 1
+8∂uδαA
+3 − 1
+4DADBδαB
+2
+�
++ φ
+�
+− 1
+2DAδαA
+2
+��
+(59)
+This can be expanded further by substituting the Einstein constraints on the metric parameters
+∂uαz
+3 = 2 D¯zCz¯z − 2 DzCzz + 2 γz¯zDzD¯zαz
+2 − 2 γz¯zD¯zD¯zα¯z
+2
+(60a)
+∂uα¯z
+3 = 2 DzCz¯z − 2 D¯zC ¯z¯z + 2 γz¯zDzD¯zα¯z
+2 − 2 γz¯zDzDzαz
+2
+(60b)
+but we will not do so here.
+The key observation we take away from the final form of the charges is that both leading hypertranslations and
+leading hyperrotations show up in these charges. This should be contrasted to our previous paper [12] where only the
+metric parameters corresponding to these diffeomorphisms showed up, and not the diffeomorphisms themselves. We
+will investigate the physical significance of hypertranslations and their connections to new memory effects in follow
+up work.
+

6NE4T4oBgHgl3EQfcAzU/content/tmp_files/2301.05080v1.pdf.txt

@@ -0,0 +1,1161 @@
+Non-linear correlation analysis in financial markets using
+hierarchical clustering
+J. E. Salgado-Hern´andez and Manan Vyas
+Instituto de Ciencias F´ısicas, Universidad Nacional
+Aut´onoma de M´exico, 62210 Cuernavaca, M´exico
+1
+arXiv:2301.05080v1  [q-fin.ST]  12 Jan 2023
+
+Abstract
+Distance correlation coefficient (DCC) can be used to identify new associations and correlations
+between multiple variables. The distance correlation coefficient applies to variables of any dimen-
+sion, can be used to determine smaller sets of variables that provide equivalent information, is zero
+only when variables are independent, and is capable of detecting nonlinear associations that are
+undetectable by the classical Pearson correlation coefficient (PCC). Hence, DCC provides more
+information than the PCC. We analyze numerous pairs of stocks in S&P500 database with the
+distance correlation coefficient and provide an overview of stochastic evolution of financial market
+states based on these correlation measures obtained using agglomerative clustering.
+I.
+INTRODUCTION
+Correlation coefficient is a number which is used to describe dependence between random
+observations. Most popular correlation coefficient is the Pearson one which is defined on the
+interval [−1, 1] [1]. For random variables X and Y , with finite and positive variances, Pearson
+correlation coefficient (PCC) is defined as PCC(X, Y ) = cov(X, Y )/
+�
+var(X) var(Y ). If
+Pearson correlation coefficient between two random variables is zero, it does not necessarily
+mean that the variables are independent. Distance correlation coefficient does not suffer
+from this drawback.
+The distance correlation coefficient (DCC) is a product-moment correlation and a gener-
+alized form of bivariate measures of dependency [2]. It is a very useful and unexplored area
+for statistical inference. The range of the distance correlation is 0 ≤ DCC ≤ 1 [3]. For two
+real random variables X and Y with finite variances, distance correlation coefficient is de-
+fined as DCC(X, Y ) = dcov(X, Y )/
+�
+dcov(X, X) dcov(Y, Y ). Here, the distance covariance
+dcov is defined in the following way. Let (X, Y ), (X′, Y ′) and (X′′, Y ′′) be i.i.d. copies, then
+dcov2(X, Y ) = E(|X − X′||Y − Y ′|) + E(|X − X′|)E(|Y − Y ′|) − 2E(|X − X′||Y − Y ′′|) .
+Thus, DCC is the correlation between the dot products which the ”double centered” (it
+is the operation of converting the distances to the scalar products while placing the origin
+at the geometric center) matrices are comprised of. It is important to mention that the
+definition of distance correlation coefficient can be extended to variables with finite first
+moments only and lack of DCC defines independence.
+2
+
+As both PCC and DCC quantify strength of dependence, is important to understand
+how large the differences between these two measures can possibly be. A natural question is
+how large the DCC can be for variables for which PCC is zero, since uncorrelatedness only
+means the lack of linear dependence. Importantly, nonlinear or nonmonotone dependence
+may exist. The fact that PCC requires finite second moments while DCC requires finite
+first moments implies that PCC is more sensitive to the tails of the distribution. Although
+methods based on ranks (Spearman rank correlation) can be applied in some problems, these
+methods are effective only for testing linear or monotone types of dependence. Importantly,
+uncorrelatedness (PCC = 0) is too weak to imply a central limit theorem which requires
+independence (DCC = 0) necessarily [4–7].
+We have used stocks listed under S&P 500 for the time period August 2000 to August
+2022 and focus on financial market crisis that occurred in the years 2008 (subprime mortgage
+crisis), 2010 (European debt crisis), 2011 (August 2011 stock market fall), 2015 (Great fall
+of China), 2020 (COVID-19 recession) and 2022 (ongoing Russo-Ukrainian war) along with
+bubble periods of 2002 (stock market downturn of 2002) and 2007 (Chinese stock bubble).
+In order to point out the differences of using DCC, we will also focus on epochs for which
+PCC ≈ 0.
+We analyze the Pearson and Distance correlation matrices and their moments along with
+eigenvalue distributions and participation ratios distribution. Participation ratios quantify
+the number of components that participate significantly in each eigenvector [8, 9]. We show
+that there are strong correlations in all these three measures at the times of crisis. Using
+correlation matrices to represent market states [10–12], we employ agglomerative clustering
+[13] to identify correlation matrices that act similarly and compare the clustering results for
+the selected stocks using PCC and DCC.
+II.
+DATA SET
+We use the 5552 daily closing prices of N = 370 stocks listed under S&P 500 for the
+time period August 2000 to August 2022 downloaded freely from Yahoo finance webpage
+[14]. The selected stocks are the ones that have been continuously traded for the chosen
+time period. Using the daily closing prices Pi(t), with index i representing a given stock and
+time t = 1, 2, . . . , T. daily returns ri(t) = [Pi(t) − Pi−1(t)]/Pi−1(t) are calculated. Here T is
+3
+
+Sector
+Ticker Stocks Weight
+Communication Services
+TS
+11
+0.03
+Consumer Discretionary
+CD
+38
+0.10
+Consumer Staples
+CS
+27
+0.07
+Energy
+EN
+18
+0.05
+Financials
+FI
+49
+0.13
+Health Care
+HC
+51
+0.14
+Industrials
+IN
+54
+0.15
+Information Technology
+IT
+49
+0.13
+Materials
+MA
+21
+0.06
+Real Estate
+RE
+25
+0.07
+Utilities
+UT
+27
+0.07
+TABLE I. Distribution of the constituent sectors of selected stocks of financial market S&P 500.
+total number of the trading days present in the considered time horizon. We then have 5551
+daily returns and use these to compute the equal-time cross-correlation matrices based on
+PCC and DCC. Table I gives the distribution of the sectors.
+The disadvantage of working with long financial time series is the loss of information over
+short periods of time, it is convenient to divide it into short time series (epochs). Computing
+returns and dealing with epochs guarantees (weakly) stationary time series. With this, one
+can study the evolution over time, for example, of the average correlations. This helps focus
+on details in a given particular time interval as financial market is a dynamic entity.
+First, we analyze the distribution of correlation matrix elements, eigenvalues and partici-
+pation ratios obtained using both PCC and DCC for all the 138 time epochs (non-overlapping
+epochs of size 40 days each). We show that there are strong correlations in all these three
+measures at the times of crisis. In other words, there is collective motion during crashes.
+III.
+CORRELATIONS AND SPECTRAL ANALYSIS
+To begin with, we plot the correlation matrices obtained using PCC and DCC in Fig.
+1. As expected, one loses the details due to long time averaging. We plot both PCC and
+4
+
+FIG. 1. Correlation matrices for the total time horizon considered. Left panel shows the PCC
+matrix and the right panel shows the DCC matrix. The minimum values for PCC and DCC are
+0.003 and 0.06 respectively. Similarly, the average PCC and DCC are 0.347 and 0.34.
+DCC correlation matrices on the same scale as there are no negative correlations in the PCC
+matrix computed for the total time horizon. Sectorial correlations are stronger for PCC in
+comparison to DCC.
+As one can not see any specific structures in the plots for the correlation matrices for
+the complete time horizon, we study the distribution of correlation matrix elements for each
+epoch as shown in Fig. 2. DCC and PCC both show a clear shift towards higher values
+of correlation during the crisis periods of interest (2002, 2008, 2010, 2011, 2020 and 2022).
+Also, DCC shows the peaks of distributions at lower values of correlation for the non-crisis
+periods, unlike PCC. Notably, DCC ≥ 0.2 for the time horizon considered implying that
+there are non-monotonic correlations present in financial markets at all times.
+Next, we analyze the time evolution of distribution of eigenvalues of correlation matrices
+as shown in Fig. 3; note that the plot is logarithmic. All the correlation matrices are singular
+and thus, we have a delta peak at zero eigenvalues in addition to bulk distribution (which
+follows random matrix theory predictions) and outliers that represent correlations [9, 15–
+19]. The largest eigenvalue, which is linearly correlated with average correlations, attains
+very large values in crisis periods as seen from distribution of eigenvalues for both PCC and
+DCC. Around end of 2016, the gap between the bulk eigenvalue distribution and outliers
+for PCC is little. One can clearly see a comparatively broad bulk eigenvalue distribution
+5
+
+PCC; August-2000 -- August-2022
+TS
+1.0
+CD
+CS
+ 0.8
+EN
+FI
+ 0.6
+HC
+ 0.4
+IN
+IT
+ 0.2
+MA
+RE
+UT
+8
+Z
+5DCC; August-2000 -- August-2022
+TS
+1.0
+CD
+CS
+0.8
+EN
+FI
+一0.6
+HC
+0.4
+IN
+JI
+0.2
+MA
+RE
+UT
+ 0.0
+Z
+5-1
+-0.5
+ 0
+ 0.5
+ 1 2000-10-02
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+ 0
+ 1
+ 2
+ 3
+ 4
+Cij
+P(Cij)
+ 0
+ 1
+ 2
+ 3
+ 4
+Pearson Correlation
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+2000-10-02
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+ 0
+ 2
+ 4
+ 6
+ 8
+Cij
+P(Cij)
+ 0
+ 2
+ 4
+ 6
+ 8
+Distance Correlation
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+Time (YYYY/MM/DD)
+-1
+-0.5
+ 0
+ 0.5
+ 1
+Cij
+ 0
+ 1
+ 2
+ 3
+ 4
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+Time (YYYY/MM/DD)
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+Cij
+ 0
+ 2
+ 4
+ 6
+ 8
+FIG. 2. Time evolution of the distribution of correlation matrix elements P(Cij). Left panel shows
+the P(Cij) for PCC and the right panel shows P(Cij) for the DCC. Bottom panel shows the 2D
+projection of the corresponding figures in the top panel.
+for DCC beyond 2019. This feature is also seen in the plot for PCC, however it is equally
+broad for 2001 when the largest eigenvalue is < 100. Like the distribution of correlation
+matrix elements in Fig. 1, eigenvalue distributions for DCC and PCC both show a clear
+shift towards higher values of outliers during the crisis periods of interest (2002, 2008, 2010,
+2011, 2015, 2020 and 2022).
+We use participation ratio (PR) to quantify the number of components that participate
+significantly in each eigenvector νi,
+PRν =
+� N
+�
+i=1
+|νi|4
+�−1
+.
+(1)
+PR gives the number of elements of an eigenvector that are different from zero that contribute
+significantly to the value of the eigenvector and thus, takes values between 1 (only one
+6
+
+2003-11-10 2007-02-05 2010-04-09 2013-06-13 2016-10-12 2019-12-172022-08-30
+Time (YYYY/MM/DD)
+ 0
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+Eigenvalues
+10-6
+10-5
+10-4
+10-3
+10-2
+2003-11-10 2007-02-05 2010-04-09 2013-06-13 2016-10-12 2019-12-172022-08-30
+Time (YYYY/MM/DD)
+ 0
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+Eigenvalues
+10-6
+10-5
+10-4
+10-3
+10-2
+FIG. 3. Time evolution of the distribution of eigenvalues. Left panel shows the eigenvalue distri-
+bution for PCC and the right panel shows for the DCC.
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+Time (YYYY/MM/DD)
+ 0
+ 50
+ 100
+ 150
+ 200
+Participation ratios
+ 0
+ 0.005
+ 0.01
+ 0.015
+ 0.02
+ 0.025
+ 0.03
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+Time (YYYY/MM/DD)
+ 0
+ 50
+ 100
+ 150
+ 200
+Participation ratios
+ 0
+ 0.005
+ 0.01
+ 0.015
+ 0.02
+ 0.025
+ 0.03
+ 0.035
+FIG. 4. Time evolution of the distribution of participation ratios. Left panel shows the participation
+ratios distribution for PCC and the right panel shows for the DCC. The horizontal line shows the
+expectation value obtained from random matrix theory.
+component) and N (all components contributing equally). The expectation value of PR for
+a Gaussian Orthogonal Ensemble (classical random matrix ensemble) has the limiting value
+of ⟨PR⟩ ≈ N/3 [8, 20]. We show the time evolution of distribution of PR for PCC and
+DCC in Fig. 4. The horizontal line in the plots gives the average PR value estimated using
+Gaussian Orthogonal Ensemble. As seen from the plots, the average PR for PCC is ≈ 160
+while that for DCC is ≈ 110. The distribution of PR in case of PCC shows a slight upward
+shift during crisis years of 2008, 2010 and 2011 while we see a slight downward shift in case
+of DCC during the crisis years 2002, 2008, 2010, 2011 and 2020. The lesser the average
+correlation, prominent is the downward shift in the distribution of PR in case of DCC.
+Next we analyze the scatter plots between various moments [21] corresponding to PCC
+and DCC and the results are presented in Fig. 5. Note that each point corresponds to an
+7
+
+epoch and we represent the bubble and crisis periods of interest as solid circles. As seen in
+Fig. 1, the crisis periods appear at higher values of mean correlations µ for both PCC and
+DCC. For PCC, the crisis periods of 2008, 2010, 2011, 2015 and 2020 appear with largest µ
+while the bubble periods of 2002 and 2007 alongwith the ongoing Russo-Ukrainian war have
+relatively lower values of µ. Skewness is negative for all the crisis periods and the bubble
+periods implying that the distribution has a longer left tail and bulk is concentrated towards
+the right side. Kurtosis for the crisis periods of 2008, 2010, 2011, 2015 and 2020 is positive
+implying the distributions are leptokurtic while distributions are platykurtic for the bubble
+periods of 2002 and 2007, and the ongoing Russo-Ukrainian war. Emax reflects a similar
+behavior as average correlations µ and PREmax is also maximum for crisis periods of 2008,
+2010, 2011, 2015 and 2020. In summary, PCC distinguishes the bubble periods of 2002 and
+2007, and the ongoing Russo-Ukrainian war from the crisis periods of 2008, 2010, 2011, 2015
+and 2020 depending on kurtosis of the distribution of correlation matrix elements.
+Similarly, in case of DCC: for µ < 0.5, σ increases with increasing µ and for µ > 0.5, σ
+decreases with increasing µ. The crisis periods of 2008, 2010, 2011, 2015 and 2020 appear
+with largest µ while the bubble periods of 2002 and 2007, and the ongoing Russo-Ukrainian
+war have relatively lower values of µ. Skewness is negative for the crisis periods of 2008,
+2010, 2011, 2015 and 2020 implying that the distribution has a longer left tail and bulk
+is concentrated towards the right side, while distribution has a longer right tail for the
+bubble periods of 2002 and 2007, and distribution is symmetric for the ongoing Russo-
+Ukrainian war. Kurtosis for the crisis periods of 2010, 2011 and 2020 is positive implying
+the distributions are leptokurtic while distributions are platykurtic for the bubble periods
+of 2002 and 2007, crisis periods of 2008 and 2015, and the ongoing Russo-Ukrainian war.
+Emax reflects a similar behavior as average correlations µ and PREmax is constant around the
+maximum value for all the epochs. In summary, DCC distinguishes the bubble periods from
+the crisis periods depending on skewness of the distribution of correlation matrix elements.
+Also, DCC distinguishes the bubble periods of 2002 and 2007, the crisis periods of 2008 and
+2015, and the ongoing Russo-Ukrainian war from the crisis periods of 2010, 2011 and 2020
+depending on kurtosis of the distribution of correlation matrix elements.
+8
+
+FIG. 5. Scatter plots corresponding to PCC (top panel) and DCC (bottom panel) between (a)
+mean correlation µ and standard deviation σ, (b) skewness γ1 and σ, (c) excess kurtosis γ2 and σ,
+(d) largest eigenvalue Emax and σ, (e) PR for the largest eigenvalues PREmax and σ, and (f) PR
+for the largest eigenvalue PREmax and largest eigenvalues Emax.
+9
+
+Pearsoncorrelation
+0.8F
+(a) 
+2
+(b) 
+(c)
+1
+3
+≤ 0.4
+0
+Y2
+0
+-1
+0
+0
+0.1 0.2 0.3 0.4
+00.1 0.2 0.3 0.4
+0
+0.1 0.2 0.3 0.4
+a
+a
+300
+400
+400
+(d)
+(e)
+200
+i300
+300
+xeu
+100
+200
+008
+200
+98
+(f) 
+0
+100
+100
+0
+0.1 0.2 0.3 0.4
+0
+0.1 0.2 0.3 0.4
+0
+100
+200
+300
+a
+E
+maxDistance correlation
+0.8
+N
+(a)
+(b) 
+(c)
+1
+≤. 0.4
+0
+2
+0
+-1
+0
+0
+0.1
+0.2
+0
+0.1
+0.2
+0
+0.1
+0.2
+300
+400
+400
+200
+300
+300
+max
+E
+100
+200
+200
+(d) 
+(e)
+().
+0
+100
+0.1
+0.2
+0.1
+0.2
+100
+0
+100
+200
+300
+a
+a2000-10-02
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+2000-10-02
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+0
+50
+100
+150
+200
+250
+2000-10-02
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+2000-10-02
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+0
+20
+40
+60
+80
+100
+120
+140
+160
+FIG. 6. Euclidean distance matrix obtained using Eq. (2) for PCC (left panel) and DCC (right
+panel).
+IV.
+AGGLOMERATIVE CLUSTERING
+In this section, we compare the clustering results for the selected stocks using PCC and
+DCC. We employ agglomerative clustering that creates clusters by successively merging
+epochs starting with singleton clusters. Using the linkage criterion in each iteration, the
+clusters are joined together until obtaining a single cluster [13].
+Dendrograms give the
+representation of this hierarchy. Choosing the threshold value then decides the number of
+clusters that will be obtained. We cluster similar correlation matrices into these optimized
+n number of “market states”.
+This is a variance-minimizing approach tackled with an
+agglomerative hierarchical approach.
+Dendrograms obtained for the PCC and DCC are
+given in Appendix V.
+In order to implement this algorithm, we need to compute the distance matrix ξ based
+on correlation coefficients C’s,
+ξ(ti, tj) = dE|C(ti) − C(tj)| ,
+(2)
+with dE representing the Euclidean norm and indices i, j = 1, 2, , . . . , 138 representing dif-
+ferent epochs. Figure 6 gives the Euclidean matrices for PCC and DCC respectively. Note
+that the crash periods of 2008, 2010, 2011 and 2020 are visible in these. Once the algorithm
+was trained with its respective distance matrix, the average correlation coefficients PCC and
+DCC were used as inputs to be able to group them into n = 5 clusters that were considered
+10
+
+FIG. 7. Average correlation matrices for each market state obtained using agglomerative clustering
+for PCC [(a)-(e)] and DCC [(f)-(j)]. The average correlation coefficients (from left to right) are
+PCC: 0.12, 0.22, 0.37, 0.52, and 0.65; DCC: 0.35, 0.41, 0.46, 0.54, and 0.66, respectively.
+adequate; see Figs. 10 and 11 for corresponding dendrograms.
+The average correlation matrices of each market states corresponding to both (a) PCC
+and (b) DCC are shown in Fig. 7. The correlation structures vary for each market state
+corresponding to PCC and DCC. The average correlation coefficients (from left to right) are
+(a) PCC: 0.12, 0.22, 0.37, 0.52, and 0.65, (b) DCC: 0.35, 0.41, 0.46, 0.54, and 0.66. The
+number of matrices that are grouped together in each of the market states (from left to right)
+are (a) PCC: 9, 49, 66, 7, and 7 and (b) DCC: 51, 23, 47, 10, and 7. The market states with
+highest correlation coefficient are 7 for both PCC and DCC. For PCC, the market state
+with highest average correlation includes the crash periods of 2008, 2010, 2015 and 2022
+with two matrices not belonging to crash periods. For DCC, the market state with highest
+average correlation includes the crash periods of 2008, 2010, 2011 and 2020. For PCC, the
+market state with second highest average correlation includes epochs in the vicinity of the
+crash periods of 2008, 2010, 2011 and 2020 and for DCC, the market state with second
+highest average correlation includes epochs in the vicinity of the crash periods of 2015 and
+2022. The bubble periods of years 2002 and 2007 are included in the market state with third
+highest average correlation for both PCC and DCC. There are two epochs for which PCC
+11
+
+(a)PCC,mar-(b)PCC,mar-(c)PCC,mar-(
+(d) PCC, mar- (e) PCC, mar-
+ket state1
+ket state 2
+ket state 3
+ket state 4
+ketstate5
+(f) DCC,mar-(g)DCC,mar-(h)DCC,mar-
+(i) DCC, mar-
+G) DCC,
+mar-
+ket state 1
+ket state 2
+ket state 3
+ket state 4
+ket state52000-10-02
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+1
+2
+3
+4
+5
+States
+2000-10-02
+2003-11-10
+2007-02-05
+2010-04-09
+2013-06-13
+2016-10-12
+2019-12-17
+2022-08-30
+1
+2
+3
+4
+5
+States
+FIG. 8. Dynamical evolution of financial market in time: PCC (top panel) and DCC (bottom
+panel).
+The market states 1, 2, . . . , 5 obtained using agglomerative clustering are arranged in
+increasing order of average correlation coefficients for both PCC and DCC.
+≈ 0 and these epochs are in the market state corresponding to the lowest average correlation
+coefficient both for PCC and DCC. Note that this market state has respectively 9 and 51
+matrices in the cluster for PCC and DCC.
+Dynamical evolution of the financial market can be studied by the transitions between
+these market states. The financial market can remain in a particular market state, can
+jump to another market state and bounce back or evolve to another market state. In Fig.
+8 the results of the temporal evolution of the market are shown based on both PCC and
+DCC and Fig. 9 shows the corresponding transition matrices. For each market state, the
+average correlation coefficients are ordered in ascending order. Transitions are counted when
+changing epoch, either from one market state to another or if it remained in the same market
+state. Most of the values stay close to the diagonal, this means that the transitions occur in
+12
+
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+3
+5
+1
+0
+0
+4
+26
+16
+1
+2
+1
+16
+42
+4
+2
+0
+1
+5
+1
+0
+0
+1
+2
+1
+3
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+31
+10
+7
+1
+2
+9
+5
+8
+1
+0
+8
+8
+26
+3
+1
+1
+0
+6
+2
+1
+1
+0
+0
+3
+3
+FIG. 9. Transition matrices corresponding to PCC (left panel) and DCC (right panel) showing
+transition between the five market states obtained using agglomerative clustering.
+small jumps towards the closest market states or continue in itself and transitions between
+states with low average correlation and high average correlations are avoided [12, 22].
+In case of PCC, the state with lowest average correlation (1) never connects to state with
+highest (5) or second highest (4) average correlation coefficient. There is a transition from
+state state 2 to 5 and state 2 to 5 which are indirect transitions as they are in the sequence
+1 → 2 → 5 and 2 → 3 → 5 and these correspond to the crash periods of 2020 and 2011
+respectively. Similarly, for DCC, the state with second lowest average correlation (2) never
+connects to state with highest (5) average correlation coefficient. However, there are two
+transitions between 5 and 1 and one transition from 1 to 5. These correspond to the crash
+periods of 2010, 2020 and 2011 respectively. There is also a transition between 1 and 4 that
+corresponds to the crash period of 2011. This is an indirect one as first transition happens
+between 1 and 3 and then to 4.
+V.
+CONCLUSIONS
+We analyzed correlations in S&P 500 market data for the time period August 2000 to
+August 2022 using both PCC and DCC. Notably, DCC ≥ 0.2 for the time horizon considered
+implying that there are non-monotonic correlations present in financial markets at all times.
+Eigenvalue distributions for DCC and PCC both show a clear shift towards higher values of
+13
+
+outliers during the crisis periods of interest (2002, 2008, 2010, 2011, 2015, 2020 and 2022).
+The distribution of PR in case of PCC shows a slight upward shift during crisis years of
+2008, 2010 and 2011 while we see a slight downward shift in case of DCC during the bubble
+period of 2002 and crisis years 2008, 2010, 2011 and 2020. The lesser the average correlation,
+prominent is the downward shift in the distribution of PR in case of DCC.
+PCC distinguishes the bubble periods of 2002 and 2007, and the ongoing Russo-Ukrainian
+war from the crisis periods of 2008, 2010, 2011, 2015 and 2020 depending on kurtosis of the
+distribution of correlation matrix elements.
+DCC distinguishes the bubble periods from
+the crisis periods depending on skewness of the distribution of correlation matrix elements.
+Also, DCC distinguishes the bubble periods of 2002 and 2007, the crisis periods of 2008 and
+2015, and the ongoing Russo-Ukrainian war from the crisis periods of 2010, 2011 and 2020
+depending on kurtosis of the distribution of correlation matrix elements.
+Going further, we compare the clustering results for correlation matrices obtained for
+the selected stocks using PCC and DCC. We employ agglomerative clustering that uses
+Euclidean distances and minimizes the sum of squared differences within all clusters. We
+obtain five market states corresponding to both PCC and DCC. The crisis periods are in
+market states with largest and second largest average correlation coefficients. Bubble periods
+are in the market state with third largest average correlation coefficient. The two epochs for
+PCC ≈ 0 are in the market state with smallest average correlation coefficient; note that this
+market state has respectively 9 and 51 matrices in the cluster for PCC and DCC. We also
+compare the transitions between these market states for both PCC and DCC. In summary,
+results for clustering depend upon the linear (PCC) and non-linear (DCC) nature of the
+correlation coefficient employed. Preliminary results on financial markets can be viewed in
+a bachelor thesis [23].
+ACKNOWLEDGMENTS
+Authors thank Harinder Pal for many useful discussions on clustering algorithms and
+help with many figures. Authors acknowledge financial support from CONACYT project
+14
+
+Fronteras 10872.
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+A Study on Stock and Currency Market”, in A. S. Chakrabarti et al. (eds.), Network Theory
+and Agent-Based Modeling in Economics and Finance (2019) pp. 331-352.
+[19] H. K. Pharasi, K. Sharma, A. Chakraborti, and T. H. Seligman, ”Complex market dynamics
+in the light of random matrix theory”, in New Perspectives and Challenges in Econophysics
+15
+
+and Sociophysics, edited by F. Abergel, B. K. Chakrabarti, A. Chakraborti, N. Deo, and K.
+Sharma (Springer International Publishing, Cham, 2019) pp. 13–34.
+[20] V. K. B. Kota, Embedded Random Matrix Ensembles in Quantum Physics (Springer, Heidel-
+berg, 2014).
+[21] A. Stuart and J. K. Ord, Kendall’s Advanced Theory of Statistics : Distribution Theory
+(Oxford University Press, New York, 1987).
+[22] A. J. Heckens and T. Guhr, J. Stat. Mech. 2022, 043401 (2022)
+[23] J. E. Salgado-Hern´andez, (Licenciatura thesis, UNAM) Correlaci´on y agrupaciones de series
+de tiempo financieras (2023).
+16
+
+APPENDIX A: DENDROGRAMS OBTAINED USING PCC AND DCC
+114
+73
+136
+93
+62
+53
+61
+60
+70
+69
+50
+51
+68
+122
+22
+44
+28
+35
+75
+82
+87
+17
+24
+31
+36
+54
+55
+56
+63
+48
+46
+47
+13
+15
+16
+11
+14
+74
+76
+77
+79
+80
+81
+84
+90
+91
+49
+134
+130
+125
+133
+126
+99
+123
+83
+92
+95
+58
+65
+72
+94
+100
+135
+137
+110
+118
+12
+88
+89
+57
+40
+45
+42
+43
+67
+52
+78
+124
+96
+109
+120
+0
+107
+1
+2
+101
+112
+105
+108
+5
+7
+8
+3
+39
+33
+38
+127
+131
+121
+106
+103
+104
+111
+102
+115
+113
+117
+85
+97
+132
+128
+129
+6
+119
+98
+116
+59
+64
+66
+34
+26
+29
+19
+20
+23
+27
+25
+18
+21
+32
+30
+41
+71
+86
+37
+4
+9
+10
+Epochs
+0
+500
+1000
+1500
+2000
+2500
+3000
+Euclidean distance
+Dendrogram (PCC)
+FIG. 10. Dendrogram obtained for PCC using agglomerative clustering.
+17
+
+50
+69
+70
+51
+60
+68
+122
+97
+98
+75
+87
+31
+6
+17
+24
+35
+85
+119
+113
+117
+76
+84
+80
+82
+36
+11
+22
+77
+28
+44
+0
+1
+107
+3
+8
+2
+5
+120
+108
+105
+112
+7
+9
+37
+39
+21
+33
+38
+71
+115
+102
+106
+101
+86
+103
+104
+30
+32
+29
+25
+26
+18
+34
+41
+4
+10
+19
+20
+23
+131
+128
+129
+127
+132
+116
+111
+121
+27
+64
+59
+66
+93
+53
+61
+123
+114
+136
+62
+73
+52
+67
+99
+110
+118
+88
+109
+12
+65
+72
+89
+94
+95
+45
+42
+43
+96
+57
+78
+49
+47
+48
+124
+126
+134
+135
+137
+100
+90
+92
+81
+83
+74
+79
+56
+63
+58
+54
+55
+91
+40
+46
+130
+125
+133
+13
+15
+14
+16
+Epochs
+0
+500
+1000
+1500
+2000
+Euclidean Distance
+Dendrogram (DCC)
+FIG. 11. Dendrogram obtained for DCC using agglomerative clustering.
+18
+

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+ 
+ 
+1 
+How Effective are COVID-19 Vaccine Health Messages in Reducing Vaccine Skepticism? 
+Heterogeneity in Messages’ Effectiveness by Just-World Beliefs 
+ 
+ 
+ 
+Juliane Wiese, corresponding author  
+ 
+ 
+ 
+  Nattavudh Powdthavee 
+ 
+Warwick Business School 
+ 
+ 
+ 
+          Nanyang Technological University 
+University of Warwick 
+ 
+ 
+ 
+ 
+ 
+         50 Nanyang Avenue 
+Scarman Road  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 639798 Singapore 
+Coventry CV4 7AL 
+ 
+ 
+ 
+ 
+ 
+ 
+          
+United Kingdom 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+    
+ 
+ORCID ID : 0000-0002-4314-5934   
+ 
+          ORCID ID: 0000-0002-9345-4882 
+ 
+[email protected] 
++33 6 68 88 18 27 
+ 
+ 
+Declarations of interest: none. 
+ 
+Abstract 
+ 
+To end the COVID-19 pandemic, policymakers have relied on various public health messages to 
+boost vaccine take-up rates amongst people across wide political spectra, backgrounds, and 
+worldviews. However, much less is understood about whether these messages affect different 
+people in the same way. One source of heterogeneity is the belief in a just world (BJW), which is 
+the belief that in general, good things happen to good people, and bad things happen to bad people. 
+This study investigates the effectiveness of two common messages of the COVID-19 pandemic: 
+vaccinate to protect yourself and vaccinate to protect others in your community. We then examine 
+whether BJW moderates the effectiveness of these messages. We hypothesize that just-world 
+believers react negatively to the prosocial pro-vaccine message, as it charges individuals with the 
+responsibility to care for others around them. Using an unvaccinated sample of UK residents before 
+vaccines were made widely available (N=526), we demonstrate that the individual-focused 
+message significantly reduces overall vaccine skepticism, and that this effect is more robust for 
+
+ 
+ 
+2 
+individuals with a low BJW, whereas the community-focused message does not.  Our findings 
+highlight the importance of individual differences in the reception of public health messages to 
+reduce COVID-19 vaccine skepticism.  
+ 
+Keywords: vaccine skepticism; health messages; justice beliefs; individual differences; COVID-
+19 
+ 
+ 
+
+ 
+ 
+3 
+1. Introduction 
+ 
+Before the vaccine rollout in the UK, 28% of the British population, particularly those in Black 
+and South Asian minority ethnic groups, were skeptical about getting vaccinated (Robertson et al., 
+2021). To maximize vaccine take-up, governments have been delivering simple messages that 
+emphasize people’s responsibility to themselves and the community. For example, the National 
+Health Services in the UK urges the public to “join the millions already vaccinated, to protect 
+yourself and others” (NHS UK, 2021). These foci, given their central role in public health 
+messaging during the COVID-19 pandemic so far, have shaped the two themes of messages 
+examined in this study: individual and community responsibilities.  
+ 
+Despite the extensive literature on the framing approaches of public health messages around 
+vaccines (e.g., Gallagher & Updegraff, 2012; McPhee et al., 2003; Kelly & Kornik, 2016), the 
+overall effectiveness of COVID-19 vaccine messages on individual or community responsibility 
+is currently imperfectly understood. While recent evidence suggests that individual-focused 
+messages more effectively increase vaccine uptake and support for mandates than community-
+focused messages, these effects are heterogeneous across individualistic and communitarian 
+worldviews (Yuan & Chu, 2022). Furthermore, we do not know which underlying beliefs about 
+the vaccine are best addressed by these messages. Nevertheless, they continue to be used by 
+public health officials worldwide. 
+ 
+In contexts of extreme urgency, who are the types of people who might respond poorly to these 
+messages and experience stronger vaccine skepticism? We build our investigation around the 
+strong theoretical link between belief in a just world (BJW) and vaccine skepticism. Just-world 
+
+ 
+ 
+4 
+believers conceive a universal justice structure which holds that both normatively and positively 
+speaking, good things tend to happen to good people and vice versa (Furnham, 2003). This 
+adaptive function (Dalbert, 2009), manifesting at varying levels of intensity and therefore 
+influencing a large portion of the population (White et al., 2019), allows individuals to 
+rationalize negative consequences in the world as justified, predictable, and manageable. Doing 
+so promotes well-being and a sense of stability in the world (Correia et al., 2009; Jiang et al., 
+2016). In the context of the COVID-19 pandemic, where an unprecedented public health 
+emergency and sweeping government regulations significantly reduced individual freedoms, 
+just-world believers struggled to make sense of such undeserved restrictions. This sense of 
+unfairness fosters a resistance against the government-promoted solution to the problem: 
+specifically, a vaccine that has been developed in record speed. Suggestive evidence of this link 
+between just-world believers and anti-vaxxers is demonstrated by their numerous shared 
+psychological traits, including conspiracy thinking (Nestik et al., 2020; Jolley & Douglas, 2014) 
+and individualistic attitudes (Wenzel et al., 2017; Motta et al., 2021). Government-sponsored 
+pro-vaccine messages, particularly ones that focus on the responsibility we hold to our 
+communities, are therefore likely to threaten the just-world believers’ worldview, as their 
+personal role in the pandemic is limited, and others’ health outcomes are independent of their 
+own decision to get vaccinated. Their worldview threatened, just-world believers defensively 
+dismiss the message that threatens their BJW, and deny the existence of a problem in the first 
+place (Furnham, 2003). 
+ 
+This study makes two main contributions to the literature. First, we experimentally investigate the 
+effectiveness of two commonly used pro-vaccine messages. Second, we examine whether BJW 
+
+ 
+ 
+5 
+moderates the effectiveness of each message. Given policymakers’ priority to increase COVID-19 
+vaccine uptake, understanding individual differences in the messages’ effectiveness by BJW is 
+critical to understanding the potential threats to their overall effectiveness on the entire population. 
+ 
+2. Existing literature and hypotheses 
+Before the vaccine rollout, researchers’ main concern was whether the COVID-19 vaccines safely 
+reduce illness and transmissibility. Having established this (Katella, 2021; Pritchard et al., 2021), 
+vaccine uptake has emerged as a more enduring challenge for public health officials. A nationally 
+representative survey of 316 Americans shows that demonstrating its efficacy and  emphasizing 
+the costs of the pandemic encourages vaccine uptake (Pogue et al., 2020). However, their survey 
+did not engage with messages that focus on the simple facts that give value to the vaccine: that it 
+protects its recipients and their community. These facts have been central to policymakers’ 
+messaging during the COVID-19 pandemic, and there continues to be little empirical investigation 
+into their effectiveness in shifting public perception around the vaccine’s effectiveness. 
+ 
+The decision to vaccinate weighs the benefits against the risks of vaccination, which could range 
+from fears of side effects and needles to mistrust of healthcare authorities. Previous research 
+demonstrates the importance of highlighting vaccines’ protective benefits, as doing so can crowd 
+out concerns about risks (Porter et al., 2018). Similarly, a COVID-19 vaccine message highlighting 
+the vaccine’s protective benefits to the individual has been shown to increase intended vaccine 
+uptake (Yuan & Chu, 2022). Our work examines how such an individualistic message can drive 
+the underlying beliefs around the vaccine’s protective function to its recipients. 
+ 
+
+ 
+ 
+6 
+In addition, researchers have found prosocial vaccine messages to have a positive impact on 
+vaccination rates (Betsch et al., 2017; Betsch & Böhm, 2018; McPhee et al., 2003). For example, 
+messages that emphasize the benefits of an avian flu vaccine to others significantly increase 
+vaccination intentions, compared to messages which emphasize its benefits to the individual (Kelly 
+& Hornick, 2016). While these findings link the community-oriented message to increased 
+vaccination intentions, they do not examine how such a message impacts beliefs around 
+transmission rates, which is the mechanism that connects the prosocial messages with increased 
+vaccine uptake. We aim to show experimentally that prosocial messages increase confidence in 
+the underlying belief that the vaccines reduce transmission.  
+     
+Based on this evidence, we predict that the individual message will more effectively decrease 
+overall skepticism than the community message, and that this effect is driven by the fact that the 
+individual message shifts the underlying belief that the vaccine protects its recipients. The 
+prosocial messages will more moderately increase confidence that the vaccine reduces 
+transmission.  
+ 
+Despite the predicted overall success of the two messages, the question remains around 
+heterogeneous effects, specifically around moral worldviews that play a role in the decision to 
+vaccinate. While Devereux et al. (2021) discover a link between stronger BJW and a greater 
+likelihood to adhere to COVID-19 measures, such as social distancing, these measures come at 
+essentially zero risk, resulting in a very different cost-benefit analysis. In contrast, accepting a 
+vaccine requires accepting the risk of potential negative side-effects, and might therefore have a 
+different relationship with BJW.  
+
+ 
+ 
+7 
+ 
+Demographic factors (Peretti-Watel et al., 2020; Khubchandani et al., 2021), psychological traits 
+(Browne et al., 2015; Jolley & Douglas, 2014), and beliefs about vaccine safety (Karlsson et al., 
+2021) predict vaccine attitudes. However, studies that examine how such traits, like BJW, interfere 
+with public health messages are scarce. While recent evidence has shown that people with more 
+individualistic, rather than communitarian, values respond more favorably to individual-centered 
+COVID-19 vaccine messages (Yuan & Chu, 2022), it remains unclear how such worldviews 
+moderate individuals’ understanding of the many ways in which the vaccine protects the public. 
+Furthermore, rather than simply capturing individualistic or community-oriented worldviews, 
+BJW contains a deeper moral around one’s deservingness of one’s place in the world, telling us 
+more about the reasoning behind an individual’s action (or inaction).   
+ 
+While people who see public health as a moral issue tend to consider prosocial (vs. self-centered) 
+social distancing messages more persuasive (Luttrell & Petty, 2020), BJW is not an altruistic moral 
+belief system. Instead, it holds individuals responsible for their own fate. BJW inherently commits 
+fundamental attribution error, in which individuals place more weight on dispositional, as opposed 
+to environmental or situational, factors (Ross, 1977). By further emphasising societal 
+responsibility as a motive to get vaccinated, public health officials transfer the responsibility for a 
+COVID patient’s health onto the community’s vaccination decision-making. This clashes with the 
+tendency of just-world believers to blame patients for their own misfortunes and to separate the 
+consequences of their own actions from the outcomes of others (Lerner & Simmons, 1966; Lucas 
+et al., 2009). Therefore, by asking people to take responsibility for others’ health and safety during 
+the COVID-19 pandemic, policymakers inevitably challenge the justice structure of the world in 
+
+ 
+ 
+8 
+which individuals are responsible for their own fate. In response, just-world believers might 
+discredit the vaccine altogether. We therefore hypothesise that for individuals with a strong BJW, 
+the prosocial messages are less effective at reducing vaccine skepticism.  
+ 
+3. Method 
+3.1 Data 
+In this pre-registered experiment (tinyurl.com/bxv23), 600 UK-based Prolific (www.prolific.co) 
+users aged between 18 and 49 joined a longitudinal online study on attitudes towards COVID-19 
+and vaccination. At the time, the UK general public under 50 years of age was not yet eligible to 
+receive a COVID-19 vaccine. Just over a quarter of the UK population had received its first dose, 
+and only 1% of the population had received both doses (Vaccinations in United Kingdom, 30 April, 
+2021).  
+ 
+Part one of the study (𝑇!) took place on 24 February 2021, and part two (𝑇") on 1 March 2021. We 
+collected data at two points in time to reduce the likelihood that (i) participants suspect the study 
+purpose and bias their responses, and (ii) participants’ responses to vaccine skepticism questions 
+are biased by exposure to questions around justice beliefs (Zizzo, 2010). Participants gave 
+informed consent and were compensated £0.25 at 𝑇! and £1.00 at 𝑇".      
+ 
+527 participants (88%) remained at 𝑇" and were randomised evenly across Control, Individual-
+Treatment, and Prosocial-Treatment (N = 172, 181, and 174, respectively). Only one participant 
+failed all three attention checks and was removed from the sample, resulting in 526 participants 
+with complete longitudinal data. This sample size (i) allowed sufficient power for a reasonable 
+
+ 
+ 
+9 
+minimal detectable effect size and (ii) is slightly larger than what was used in a similar research 
+design studying BJW and climate change messaging (Feinberg & Willer, 2011). Of the final 
+sample of 526 individuals, 70% were females, 87% were ethnically White, and 59% have an annual 
+income of £30,000 or over. The mean age was 31. Balance checks confirm that our sample was 
+balanced on observable characteristics across all groups; see Table A.1 in the appendix.     
+ 
+3.2 Measures and procedure 
+3.2.1 BJW scales 
+Because vaccination evokes concepts of justice both for the individual and for society, participants 
+completed the general BJW scale, six questions about the justice structure in the world in general 
+(Dalbert et al., 1987), and the personal BJW scale, seven questions which posit that the world is 
+just for me personally but not for others (Dalbert, 1999) at 𝑇!. The two scales have a correlation 
+coefficient of 0.52. To attain a linear combination of BJW factors, we conducted a separate factor 
+analysis on each scale, yielding two distinct factors (a = 0.78 for general BJW and a = 0.88 for 
+personal BJW), and then conducted a factor analysis on these factors, resulting in a combined BJW 
+factor (a = 0.68); the factor analysis results are in Table A.2. The resulting combined BJW factor 
+was standardized to a mean of 0 and a standard deviation of 1. It was transformed into a dummy 
+variable which marks above- or below-median strength of BJW. This allows us to investigate the 
+differential effects of the treatments on vaccine skepticism by the strength of BJW. 
+ 
+3.2.2 Vaccine skepticism 
+
+ 
+ 
+10 
+At 𝑇! and 𝑇", participants completed four questions on COVID-19 vaccine skepticism, with 
+possible answers ranging from 0 (not at all certain/likely) to 100 (extremely certain/likely). The 
+precise wording of the questions was:  
+• “How certain are you that the COVID-19 vaccines are a useful tool in fighting the 
+pandemic?”  
+• “How likely are you to accept the COVID-19 vaccine when offered?” 
+• “How certain are you that the COVID-19 vaccine reduces transmission between 
+individuals?”  
+• 
+“How certain are you that the COVID-19 vaccine would prevent you personally from 
+getting very ill due to COVID-19?”  
+For simplicity, we reversed the responses so that higher values represent higher levels of vaccine 
+skepticism in each of the four outcomes. The baseline mean responses are 16.8, 13.0, 29.1, and 
+22.4, respectively, which suggest that at 𝑇!, the study population was relatively prepared to take 
+the vaccine but was more skeptical of its illness and transmission prevention. These outcomes are 
+moderately correlated, with correlations ranging from 0.53 to 0.76. To circumvent the multiple 
+comparisons problem, we also derived an overall skepticism outcome by conducting a factor 
+analysis on the four reversed individual skepticism variables for both outcomes at 𝑇! (a = 0.87) 
+and 𝑇" (a = 0.89); see Tables A.3 and A.4 in the appendix for the estimates. All skepticism 
+variables were standardized to have a mean of 0 and a standard deviation of 1 and were included 
+in analysis. 
+ 
+At 𝑇", 5 days after 𝑇!, participants were randomised into one of three groups: control (no article), 
+individual, and community responsibility treatment. In both treatments, participants were asked to 
+
+ 
+ 
+11 
+read a news-style article. The articles, Figure A.3.1 in the appendix, are identical in the first 
+paragraphs, which discuss the context of the pandemic and vaccine development at the time of 
+writing. They deviate towards the end by treatment group. The individual responsibility article 
+explains that the vaccine reduces the risk of severe COVID-19 illness to vaccine recipients, and 
+the prosocial article explains that to combat the virus, individuals must accept the vaccine to reduce 
+community transmission.  
+ 
+3.2.3 Attention and manipulation check 
+Participants in the treatment groups were asked two fact-based questions from the article, as well 
+as whether taking the recommended steps during the pandemic will mainly protect them, or mainly 
+protect others, from COVID-19 illness. Amongst the final sample of participants who passed all 
+three attention checks, we find a significant difference between the two treatments on the 
+manipulation-check item, t(352) = 13.64, p = 0.000 for indicating that the vaccine protects 
+yourself, and t(352) = -13.07, p = 0.000 for indicating that the vaccine protects others. 
+ 
+3.2.4 Sociodemographic controls 
+Participants also completed a post-experiment questionnaire, which elicited their ethnicity, 
+education level, region, income, political views, optimism, risk attitudes, COVID-19 history, and 
+adhesion to government guidelines. Age and gender were collected automatically by Prolific. 
+ 
+Figure A.1 shows the procedural flow of the experiment and consort diagram, and Figures A.2 and 
+A.3 present screenshots of the materials used. 
+ 
+
+ 
+ 
+12 
+3.3 Analysis 
+We conduct all analyses of vaccine skepticism using Ordinary Least Squares (OLS) regression 
+with robust standard errors clustered on the participant-level. Our primary analysis examines the 
+treatment effects on the overall vaccine skepticism factor. We regress Equation (1) and present the 
+results in column 3 of Table 1: 
+∆𝑆#$ = 𝑎 + 𝛽"𝑇# + 𝑋#
+%𝛾 + 𝛽&𝐵𝐽𝑊# + 𝛽'(𝑇# × 𝐵𝐽𝑊#) + 𝑒,   (1) 
+ 
+where 𝑖 = 1, … , 𝑁; 𝑡 = 1, … ,2. ∆𝑆#$ represents the change in the overall vaccine skepticism factor 
+from t=0 to t=1, where a higher value represents greater vaccine skepticism; Ti represents the 
+treatment condition (control, individual, or community message) and 𝛽" is the effect of this 
+condition on skepticism;	𝑋#
+% represents the matrix of covariates, including a standardized optimism 
+factor (a=0.8134), age, age-squared, gender dummy, £30,000+ annual income (vs. below £30,000 
+annual income) dummy, London (vs. non-London) dummy, undergraduate education (vs. non-
+undergraduate education) dummy, white (vs. non-white) dummy, Labour party (vs. non-Labour) 
+dummy; 𝛽& is the effect of holding a strong (vs. weak) BJW;	𝛽' represents the interaction of 
+treatment and BJW, i.e. the differential effect of the treatment when participants have either a 
+stronger or a weaker BJW; and e is the error term. Columns 1 and 2 model the parsimonious 
+specifications of Eq. (1), with covariates excluding and including BJW, respectively. 
+ 
+Table 2 models the effects of the interaction between treatment and BJW on each of the four 
+skepticism outcomes. Their forms are identical to Eq. (1), with the exception that the outcome 
+variable is replaced by each of the four vaccine skepticism subscales, standardized to mean of 0 
+and standard deviation of 1. 
+
+ 
+ 
+13 
+∆𝑆()$
+; = 𝑎 + 𝛽"𝑇# + 𝑋#
+%𝛾 + 𝛽&𝐵𝐽𝑊# + 𝛽'(𝑇# × 𝐵𝐽𝑊#) + 𝑒,   (2) 
+where 𝑖 = 1, … , 𝑁; 𝑗 = 1, … ,4; 𝑡 = 1, … ,2. Here, ∆𝑆()
+; =	𝑆()$
+; −	𝑆()$*"
+? , where the notation j 
+represents different domains of beliefs, e.g., 𝑆"# represents the belief that the vaccine is not useful; 
+𝑆&# represents the likelihood of not accepting the vaccine; 𝑆'# represents the belief that the vaccine 
+will not reduce transmission; and 𝑆+# represents the belief that the vaccine will not prevent serious 
+illness. The rest of the specification is identical to Eq. (1). 
+ 
+Note that we deviate from the pre-registered document in two respects. First, we include an overall 
+skepticism factor as an outcome variable in our primary analysis, circumventing the multiple 
+comparisons problem in our primary analysis. Second, we run OLS regressions with standard 
+errors clustered at the participant level as the primary analysis rather than using analysis of 
+variance (ANOVA). This change is made due to the inclusion of continuous independent variables 
+in the regression. 
+ 
+4. Results 
+4.1 Message effectiveness 
+We begin by examining the within-person changes in vaccine skepticism by treatment group. As 
+predicted, Figure 1 shows that the individual message significantly reduces overall skepticism by 
+0.04 standard deviation, compared to the control group which increases overall skepticism by 0.07 
+standard deviation (Wilcoxon signed-rank test, p = 0.030). There is weaker evidence that the 
+community message also reduces overall skepticism, which decreased by 0.02 standard deviation 
+(Wilcoxon signed-rank test, p = 0.103). Figure 1 thus provides raw data evidence that individual-
+focused public health message is most effective at reducing overall vaccine skepticism. 
+
+ 
+ 
+14 
+[Figure 1 here] 
+To understand this result more thoroughly, Table 1 estimates regression equations that adjust for 
+other covariates, i.e., Eq.1. We find the regression results to be consistent with Figure 1’s findings. 
+The individual-focused message decreases overall skepticism more robustly than the community 
+message, b = -0.11, [95% C.I.: -0.20, -0.02], p = 0.014, versus b = -0.09, [95% C.I.: - 0.19, 0.01], 
+p = 0.083, respectively. 
+[Table 1 here] 
+4.2 BJW as a moderator of pro-vaccine message impacts  
+To formally test for the heterogeneous effect of public health messages by BJW, Tables 1 and 2  
+include the interaction terms between treatment and a high BJW dummy. Column 3 of Table 1 
+shows that for people with a low BJW, the individual message is extremely effective at lowering 
+their overall skepticism factor, b = - 0.19, [95% C.I.: - 0.32, -0.06], p = 0.004. As discussed 
+earlier, columns 1 and 2 of Table 1 demonstrate a greater effectiveness of the individual message 
+on average. The results of column 3 suggest that the effectiveness of this individualistic message 
+is more robust for people with a low BJW, whereas we see no such differential effect for the 
+collective message. Figures 2 and 3 show this distinction visually, with the predictive margins 
+plots of the control and individual treatment overlapping (Figure 2), and the predictive margins 
+plots of the control and collective treatment (Figure 3) not overlapping. When examining the 
+interaction regressions for each sub-scale of vaccine skepticism (Table 2), we find that the strong 
+effect of the individual treatment on overall skepticism for people with a low BJW is driven by a 
+reduction in skepticism around the belief that the vaccine will not prevent illness, b = - 0.32, 
+[95% C.I.: - 0.50, -0.14], p < 0.001. This suggests that people with a low BJW, i.e. those who do 
+not believe that there is a justice system which ensures that overall good things happen to good 
+
+ 
+ 
+15 
+people and bad things happen to bad people, are extremely reactive to the individualistic message. 
+It increases their confidence in the vaccine being able to protect them from serious illness. In other 
+words, receiving the individualistic message, which accurately highlights that receiving the 
+vaccine can prevent serious illness, correctly updates the beliefs around this issue for those with a 
+low BJW, but not for those with a strong BJW. This suggests that for someone with a strong BJW, 
+the belief in this just world order overpowers the belief in the science of the vaccine, as perhaps 
+the deservingness of a person to fall ill would govern their likelihood of sickness moreso than the 
+vaccine’s protective properties. 
+[Figures 2 and 3 here] 
+Furthermore, we do not find evidence that people with a strong BJW react particularly poorly to 
+the community message, b = 0.03, [95% C.I.: - 0.16, 0.22], p = 0.778. This suggests that a 
+message which urges the public to take care of its community does not come into strong conflict 
+with believers of a just world who may not feel responsible for the pandemic. This lack of 
+resistance is consistent with just world believers’ willingness to engage in other COVID-19 
+preventative measures (Devereux et al., 2021).  
+[Table 2 here] 
+ 
+5. Discussions 
+Our findings that the individual and community messages concerning the COVID-19 vaccine can 
+shift beliefs around the vaccine’s various protective functions demonstrates an unsurprising link 
+between the presentation of fact and its influence on a corresponding attitude. Nevertheless, in 
+their desperate attempts to convince the public to get vaccinated, policymakers have sometimes 
+turned to extreme measures, such as million-dollar lotteries, rifle giveaways, and free beer and 
+
+ 
+ 
+16 
+donuts (Lewis 2021). However, while policymakers may have expected a clear increase in uptake, 
+emerging evidence suggests that there is limited evidence in favor of these creative incentivizing 
+strategies (Walkey et al., 2022; Acharya & Dhakal, 2021), perhaps due to newfound suspicion of 
+such gimmicky programs. Instead, policymakers should provide truthful information about the 
+capacities of the COVID-19 vaccine, relying on existing evidence that these strategies effectively 
+lower vaccine skepticism (Pennycook et al., 2020; Yuan & Chu, 2022).  
+ 
+Our messages do not easily shift the belief that the vaccine reduces transmission of the virus. This 
+is especially important as new evidence emerges around the limited effectiveness of the vaccines 
+against mutations of the coronavirus and in preventing transmission. Early studies suggest that the 
+COVID-19 vaccines may not be as effective in preventing transmission as previously thought 
+(Reuters, 2021). While policymakers should highlight the protective benefits of the vaccine, they 
+must be cautious in not overstating the vaccine’s effectiveness around transmission. Doing so 
+could give vaccinated individuals a false sense of security, and ultimately reduced trust in public 
+health authorities, resulting in less social distancing and respect for COVID-19 guidelines. As new 
+scientific evidence about the vaccine emerges, officials must update their messaging content 
+accordingly. 
+ 
+The literature shows that prosocial messages play an important role in motivating COVID-19 
+preventative actions, like signing up for contact-tracing apps (Jordan et al., 2020). In contrast, 
+vaccine skepticism responds differently. Consistent with previous findings (Yuan & Chu, 2022), 
+we show that individual responsibility messages work as well, and sometimes better, than the 
+community messages in reducing vaccine skepticism, depending on the dimension of skepticism 
+
+ 
+ 
+17 
+in question. This discrepancy between non-vaccine COVID-19 prevention and vaccine messages 
+could be because general preventative measures are perceived to be less risky than taking the 
+vaccine. Riskier behaviors require more self-gain, which explains why the individual message is 
+more successful.  
+ 
+Furthermore, the pro-vaccine messages used in this experiment affect different domains of vaccine 
+skepticism differently. More specifically, they do not convince the population that the vaccine is 
+useful to ending the pandemic, nor do they influence vaccination intentions. In the urgent 
+pandemic context, while attitudes matter, vaccination behaviors are even more critical. Alternative 
+strategies to motivate behavior must not be overlooked or confounded with strategies that target 
+attitudes in future research.  
+ 
+When further examining heterogeneous treatment affects by intensity of BJW, we find that the 
+overall success of the individual message is more robust among individuals with a low BJW, 
+compared to those with a high BJW. The individual message, which focusses on the primary effect 
+of the vaccine, may speak more particularly to people with a weak BJW because they see the world 
+in a more factual, cartesian way. Someone with a strong BJW, on the other hand, may consider 
+competing justice-related reasonings for the spread of or protection against COVID. The same is 
+not true of the effects of the community message. Individuals with a strong BJW were found to be 
+unmoved by the community message, possibly because this prosocial message sets an expectation 
+that challenges the distribution of responsibility in a just world, as previously discussed. While 
+individuals who see public health as a moral issue are more persuaded by other-focused (rather 
+than self-focused) social distancing messages (Luttrell & Petty, 2020), BJW is not a worldview 
+
+ 
+ 
+18 
+based on altruistic morals. Rather, where others may fall ill due to COVID-19, strong believers of 
+a just world would blame the patients for their own misfortune, rather than assuming responsibility 
+over the pandemic via mass collective vaccination.  
+ 
+Our results suggest that evidence-based messages (e.g.: the vaccine will protect you) have 
+heterogeneous effects according to worldview. This heterogeneity replicates the findings of Yuan 
+& Chu, who recently demonstrate that the individual-centered COVID-19 vaccine message is more 
+impactful than a community-centered one, largely due to people whose worldview aligns with a 
+more individualistic outlook (2022). Our studies differ in that we examine BJW, rather than 
+individualism/communitarianism, and our sample was based in the UK, rather than the US. 
+However, broadly speaking, the results confirm one another’s findings, which is that the 
+individual-centered message works best overall, but that this effect is driven largely by people with 
+a worldview that places themselves, the individual, independent of a larger community or justice 
+structure, at the center. Authorities ought to take into consideration the extent to which their 
+vaccine messaging can have heterogeneous effects according to the worldviews of their 
+population, especially as they encourage vaccine take-up amongst people with more extreme 
+worldviews.  
+ 
+6. Conclusions   
+Simple messages that promote the COVID-19 vaccine effectively reduce vaccine skepticism of 
+the corresponding beliefs around the vaccine’s effectiveness. This reassuringly highlights the 
+importance for policymakers to focus the information of their vaccination campaigns on the 
+specific concerns of the public. The differences we find in effectiveness by psychological outlook 
+
+ 
+ 
+19 
+are important for policymakers to consider, especially as the remaining unvaccinated likely hold 
+more extreme world views. Messages that work well for people with low-level BJW evidently 
+work less well for those with a more extreme worldview, suggesting that policymakers must 
+reconsider how to motivate those harder-to-reach populations to get vaccinated. Custom messages 
+that directly target people with such views could be an interesting line of research to follow.  
+ 
+This research is not without limitations. First, the data is restricted to a specific age-group in the 
+United Kingdom and therefore has not been tested in other contexts, where just-world beliefs and 
+vaccine skepticism differ. For example, in the United States, conservatism links with both BJW 
+(Furnham, 2003) and COVID-19 vaccine skepticism (Latkin et al., 2021), suggesting that BJW 
+might be negatively correlated with pro-vaccine attitudes. Second, the sample in our study is not 
+quota matched to the U.K. population, nor was it obtained using probability sampling. Hence, the 
+results cannot be considered nationally representative, and there is likely a degree of selection bias 
+amongst users of Prolific. Third, our dataset does not capture whether participants ultimately took 
+up the vaccination, as it only captures attitudes and intentions. As previously discussed, behaviors 
+in this context are more important than attitudes, and would be valuable to follow up on. 
+ 
+The authors declare no conflicts of interest. 
+ 
+ 
+
+ 
+ 
+20 
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+
+ 
+ 
+25 
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+
+ 
+ 
+26 
+ 
+1 
+
+How Effective are COVID-19 Vaccine Health Messages in Reducing Vaccine Skepticism? 
+Heterogeneity in Messages’ Effectiveness by Just-World Beliefs 
+ 
+Tables and Figures 
+ 
+ 
+
+ 
+Figure 1. Proportions of overall skepticism changes across control, individual, and community messages.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Change in standardized vaccine skepticism
+Change in skepticism (SD) from T=0 to T=1
+5
+.05
+0
+Overall skepticism
+Control
+Individual
+Community
+95% CI
+n = 526Table 1: The effects of public health messages on overall vaccine skepticism factor 
+outcome: OLS regressions 
+ 
+ 
+ 
+ 
+  
+(1) 
+(2) 
+(3) 
+ 
+∆	Skepticism 
+factor (std) 
+∆	Skepticism 
+factor (std) 
+∆	Skepticism 
+factor (std) 
+  
+  
+  
+  
+Individual 
+-0.113** 
+-0.113** 
+-0.189*** 
+ 
+(0.0453) 
+(0.0457) 
+(0.0657) 
+Community 
+-0.0872 
+-0.0872 
+-0.101 
+ 
+(0.0502) 
+(0.0502) 
+(0.0639) 
+High BJW 
+ 
+0.000134 
+-0.0569 
+ 
+ 
+(0.0434) 
+(0.0658) 
+Individual x High BJW 
+ 
+ 
+0.147 
+ 
+ 
+ 
+(0.0911) 
+Community x High BJW 
+ 
+ 
+0.0274 
+ 
+ 
+ 
+(0.0969) 
+Optimism (std) 
+-0.00429 
+-0.00432 
+-0.00569 
+ 
+(0.0201) 
+(0.0215) 
+(0.0215) 
+Age 
+0.00640 
+0.00640 
+0.00625 
+ 
+(0.0190) 
+(0.0191) 
+(0.0188) 
+Age squared 
+-8.56e-05 
+-8.56e-05 
+-8.20e-05 
+ 
+(0.000293) 
+(0.000294) 
+(0.000290) 
+Female 
+-0.000917 
+-0.000910 
+0.00305 
+ 
+(0.0438) 
+(0.0440) 
+(0.0436) 
+£ 30k+ 
+-0.0179 
+-0.0179 
+-0.0153 
+ 
+(0.0410) 
+(0.0414) 
+(0.0411) 
+London 
+-0.0313 
+-0.0313 
+-0.0292 
+ 
+(0.0635) 
+(0.0636) 
+(0.0633) 
+University+ 
+0.0467 
+0.0467 
+0.0431 
+ 
+(0.0430) 
+(0.0434) 
+(0.0429) 
+White 
+0.0210 
+0.0210 
+0.0254 
+ 
+(0.0703) 
+(0.0705) 
+(0.0708) 
+Labour 
+-0.00769 
+-0.00768 
+-0.00469 
+ 
+(0.0375) 
+(0.0372) 
+(0.0378) 
+Constant 
+-0.0700 
+-0.0701 
+-0.0499 
+ 
+(0.303) 
+(0.307) 
+(0.300) 
+Cluster individuals 
+526 
+526 
+526 
+R-squared 
+0.024 
+0.024 
+0.029 
+Note: *** p<0.001, ** p<0.05. Robust standard errors clustered at the individual level and are in parentheses. 
+Dependent variables represent the change from #! to #"	and are standardized to have a mean of 0 and a standard 
+deviation of 1. 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+Table 2: The effects of public health messages on individual skepticism outcomes: OLS 
+regressions with BJW interactions 
+ 
+  
+(1) 
+(2) 
+(3) 
+(4) 
+ 
+∆	Vaccine not 
+useful (std) 
+∆	Not accept 
+vaccine (std) 
+∆	Not reduce 
+transmission (std) 
+∆	Not prevent 
+illness (std) 
+  
+  
+  
+  
+  
+Individual 
+-0.166 
+-0.0212 
+-0.0458 
+-0.323*** 
+ 
+(0.102) 
+(0.0466) 
+(0.107) 
+(0.0917) 
+Community 
+-0.101 
+0.0175 
+-0.0904 
+-0.139 
+ 
+(0.0951) 
+(0.0545) 
+(0.110) 
+(0.0972) 
+High BJW 
+-0.0605 
+-0.00896 
+0.0244 
+-0.103 
+ 
+(0.0951) 
+(0.0543) 
+(0.116) 
+(0.113) 
+Individual x High BJW 
+0.147 
+0.0565 
+0.0416 
+0.214 
+ 
+(0.135) 
+(0.0790) 
+(0.158) 
+(0.146) 
+Community x High BJW 
+0.0651 
+-0.0288 
+-0.189 
+0.0998 
+ 
+(0.134) 
+(0.0827) 
+(0.170) 
+(0.155) 
+Optimism (std) 
+-0.0281 
+-4.72e-05 
+0.00949 
+0.0194 
+ 
+(0.0253) 
+(0.0188) 
+(0.0378) 
+(0.0344) 
+Age 
+-0.00199 
+0.0131 
+0.0512 
+-0.00812 
+ 
+(0.0276) 
+(0.0156) 
+(0.0311) 
+(0.0281) 
+Age squared 
+5.24e-05 
+-0.000208 
+-0.000845 
+0.000158 
+ 
+(0.000424) 
+(0.000227) 
+(0.000481) 
+(0.000433) 
+Female 
+0.0213 
+-0.0378 
+0.0621 
+-0.0174 
+ 
+(0.0616) 
+(0.0397) 
+(0.0760) 
+(0.0696) 
+£ 30k+ 
+-0.00347 
+0.0365 
+-0.0707 
+-0.0745 
+ 
+(0.0606) 
+(0.0381) 
+(0.0706) 
+(0.0662) 
+London 
+-0.00687 
+-0.0415 
+0.00830 
+-0.0467 
+ 
+(0.0989) 
+(0.0448) 
+(0.0875) 
+(0.0791) 
+University+ 
+0.0975 
+-0.0117 
+-0.0588 
+0.0427 
+ 
+(0.0601) 
+(0.0326) 
+(0.0678) 
+(0.0667) 
+White 
+0.0705 
+-0.0778 
+0.122 
+-0.0191 
+ 
+(0.108) 
+(0.0591) 
+(0.118) 
+(0.115) 
+Labour 
+-0.0578 
+0.0123 
+-0.0130 
+0.0311 
+ 
+(0.0557) 
+(0.0321) 
+(0.0687) 
+(0.0605) 
+Constant 
+-0.0239 
+-0.114 
+-0.716 
+0.284 
+ 
+(0.458) 
+(0.259) 
+(0.498) 
+(0.444) 
+Cluster individuals 
+526 
+526 
+526 
+526 
+R-squared 
+0.023 
+0.021 
+0.032 
+0.034 
+Note: *** p<0.001, ** p<0.05. Robust standard errors clustered at the individual level and are in parentheses. 
+Dependent variables represent the change from #! to #"	and are standardized to have a mean of 0 and a standard 
+deviation of 1. 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2: Predictive margins of the individual treatment and control group, over the 
+standardized BJW factor 
+
+ 
+ 
+
+Predictive Margins of treat with 95% Cls
+2
+Linear Prediction
+0
+2
+-4
+-2
+0
+2
+4
+Standardized BJW Factor
+Control
+IndividualFigure 3: Predictive margins of the community treatment and control group, over the 
+standardized BJW factor 
+ 
+ 
+ 
+ 
+ 
+
+Predictive Margins of treat with 95% Cls
+Linear Prediction
+2
+0
+2
+-4
+-2
+0
+2
+4
+Standardized BJW Factor
+Control
+Community 1 
+How Effective are COVID-19 Vaccine Health Messages in Reducing Vaccine Skepticism? 
+Heterogeneity in Messages’ Effectiveness by Just-World Beliefs 
+ 
+Appendix 
+ 
+ 
+
+ 2 
+Table A.1: Balance checks on all observable characteristics amongst the final analysis sample. 
+  
+  
+Control 
+(0) 
+Individual 
+(1) 
+Community 
+(2) 
+(0) vs. (1), 
+p-value 
+(0) vs. (2), 
+p-value 
+(1) vs. (2), 
+p-value 
+𝑇!(baseline) 
+Not vaccine useful 
+17.0 
+16.2 
+17.3 
+0.704 
+0.904 
+0.616 
+  
+(1.5) 
+(1.4) 
+(1.6) 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+Not accept vaccine 
+12.5 
+12.6 
+13.9 
+0.988 
+0.628 
+0.625 
+  
+(1.9) 
+(1.7) 
+(2.0) 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+Not reduce transmission 
+29.9 
+28.7 
+29.0 
+0.645 
+0.670 
+0.983 
+  
+(1.9) 
+(1.9) 
+(2.0) 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+Not prevent illness 
+22.0 
+22.3 
+22.8 
+0.895 
+0.741 
+0.839 
+ 
+(1.8) 
+(1.7) 
+(1.8) 
+ 
+ 
+ 
+  
+172 
+180 
+174 
+  
+  
+  
+𝑇" (endline) 
+Not vaccine useful 
+15.6 
+13.5 
+14.5 
+0.253 
+0.615 
+0.566 
+  
+(1.4) 
+(1.2) 
+(1.5) 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+Not accept vaccine 
+11.6 
+12.0 
+13.0 
+0.892 
+0.602 
+0.684 
+  
+(1.8) 
+(1.7) 
+(1.8) 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+Not reduce transmission 
+30.7 
+29.0 
+25.0 
+0.548 
+0.039 
+0.124 
+  
+(2.0) 
+(1.9) 
+(1.8) 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+Not prevent illness 
+22.1 
+17.8 
+20.8 
+0.054 
+0.581 
+0.187 
+ 
+(1.7) 
+(1.5) 
+(1.7) 
+ 
+ 
+ 
+  
+172 
+180 
+174 
+  
+  
+  
+ 
+Quartile 1 BJW 
+0.3 
+0.2 
+0.3 
+0.626 
+0.671 
+0.358 
+ 
+ 
+(0.0) 
+(0.0) 
+(0.0) 
+ 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+ 
+Quartile 4 BJW 
+0.2 
+0.3 
+0.3 
+0.529 
+0.352 
+0.756 
+ 
+ 
+(0.0) 
+(0.0) 
+(0.0) 
+ 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+ 
+Optimism factor (Std) 
+0.0 
+0.0 
+-0.0 
+0.928 
+0.779 
+0.850 
+ 
+ 
+(0.1) 
+(0.1) 
+(0.1) 
+ 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+ 
+Age 
+30.4 
+31.0 
+31.5 
+0.479 
+0.231 
+0.600 
+ 
+  
+(0.7) 
+(0.6) 
+(0.7) 
+ 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+ 
+Female 
+0.7 
+0.8 
+0.7 
+0.309 
+0.459 
+0.781 
+ 
+  
+(0.0) 
+(0.0) 
+(0.0) 
+ 
+ 
+ 
+ 
+ 
+166 
+171 
+170 
+ 
+ 
+ 
+ 
+£30,000+ 
+0.7 
+0.7 
+0.8 
+0.987 
+0.455 
+0.423 
+ 
+  
+(0.1) 
+(0.0) 
+(0.1) 
+ 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+ 
+London 
+0.1 
+0.1 
+0.2 
+0.652 
+0.248 
+0.472 
+ 
+  
+(0.0) 
+(0.0) 
+(0.0) 
+ 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+ 
+Undergraduate+ 
+0.6 
+0.6 
+0.6 
+0.778 
+0.808 
+0.597 
+ 
+  
+(0.0) 
+(0.0) 
+(0.0) 
+ 
+ 
+ 
+ 
+ 
+171 
+179 
+174 
+ 
+ 
+ 
+ 
+White 
+0.9 
+0.9 
+0.9 
+0.525 
+0.724 
+0.779 
+ 
+  
+(0.0) 
+(0.0) 
+(0.0) 
+ 
+ 
+ 
+ 
+ 
+172 
+180 
+174 
+ 
+ 
+ 
+ 
+Labour party 
+0.3 
+0.4 
+0.4 
+0.825 
+0.438 
+0.578 
+
+ 3 
+ 
+ 
+(0.0) 
+(0.0) 
+(0.0) 
+ 
+ 
+ 
+  
+  
+169 
+172 
+172 
+  
+  
+  
+Note: standard deviations in parenthesis, sample size of respondents in italics. 
+ 
+ 
+ 
+ 
+
+ 4 
+Figure A.1: Experimental process and consort diagram 
+ 
+ 
+ 
+ 
+ 
+
+T0
+600participants
+Completed BJW scale and baseline
+skepticism outcomes
+T1
+600 participants
+Invited to return for part 2
+Control
+Individual treatmentCollective treatment
+172 participants
+181 participants
+174 participants
+Read article about
+Read article about
+individual benefits
+communitybenefits
+to vaccination
+tovaccination
+Passed attention
+Passed attention
+checks
+checks
+180 participants
+174 participants
+526 participants
+Completed endline skepticism outcomes
+and demographic questions 5 
+Figure A.2: Survey design: questions at 𝑇!. 
+ 
+ 
+ 
+ 
+
+Please read each statement carefully and indicate the extent to which you personally
+agree or disagree with it.
+Very
+Very
+strongly
+Slightly
+Slightly
+strongly
+disagree
+Disagree
+disagree
+agree
+Agree
+agree
+I think basically the
+0
+0
+0
+0
+0
+0
+world is a justplace.
+I believe that, by and
+large, people get what
+0
+0
+0
+0
+0
+0
+they deserve.
+Iam confidentthat
+justice always prevails
+0
+0
+0
+0
+0
+over injustice.
+I am convinced that in
+the long run, people
+0
+will be compensated
+for injustices.
+I firmly believe that
+injustices inallareas
+of life (e.g.
+professional, family,
+0
+0
+0
+politics) are the
+exception rather than
+the rule.
+I think people try to be
+fair when making
+0
+0
+0
+important decisions. 6 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Please read each statement carefully and indicate the extent to which you personally
+agree ordisagree with it.
+Very
+Very
+strongly
+Slightly
+Slightly
+strongly
+disagree
+Disagree
+disagree
+agree
+Agree
+agree
+I believe that, by and
+large, I deserve what
+0
+0
+0
+0
+0
+0
+happens to me.
+I am usually treated
+0
+0
+0
+0
+0
+0
+fairly.
+I believe that I usually
+0
+0
+0
+0
+0
+0
+get what I deserve.
+Overall, events in my
+0
+0
+0
+0
+0
+0
+life are just.
+In my life injustice is
+theexceptionrather
+0
+0
+0
+0
+0
+0
+than the rule.
+I believe that most of
+0
+0
+0
+0
+0
+0
+the things that happen
+in my life are fair.
+I think that important
+decisionsthatare
+0
+made concerning me
+are usually just. 7 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+How certainareyouthattheCOViD-19vaccinesare ausefultool infightingthepandemic?
+Not at all certain
+Extremelycertain
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+How likelyareyouto accept the CovID-19vaccinewhen offered?
+Not at all likely
+Extremely likely
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+HowcertainareyouthattheCOViD-19vaccinereducestransmissionbetweenindividuals?
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+How certainareyouthattheCoviD-19vaccinewouldpreventyoupersonallyfromgettingveryill dueto
+COVID-19?
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100 8 
+Figure A.3.1: Survey design: treatment messages at 𝑇". Control participants were asked to 
+respond to the same four skepticism outcomes shown in Figure A.2. Individual (left) and 
+community (right) messages participants were first asked to read the following fictitious news 
+articles and were then prompted to respond to the four skepticism outcomes.  
+ 
+ 
+ 
+ 
+ 
+
+Belowisanewsstorysimilartoothernewsstoriesyoumighthavereadbefore.Please
+readthestoryandrespondtothequestionsthatfollow.
+BOsTON --"Coronavirus disease (COVID-19)is a highly contagious illness, caused bythe
+transmission of the SARS-CoV-2 virus. First identified in December 2019, the virus has
+causedapandemicthatresulted inshutdownsall aroundtheglobe.It wasfirst widely
+haswreakedhavocontheglobe.Claimingmillionsoflives,thispandemichascreateda
+cleardemarcationintime:pre-covid,andpost-covid,"says ProfessorArthurMichali,a
+publichealthexpertfromaleadingresearchuniversity."Beforethispandemic,Tcouldhave
+attendedaconferenceinTokyo oneday,ledaresearchcollaboration inGenevathenext,
+andarrivedback inBostonthethirdday.Thiskindoftravel issimplynolongerpossible
+under currentcircumstances,andit's likelythatthis sort ofbehaviourcontributedtothe
+rapidspreadofthediseaseworldwide."
+ProfessorMichali,whohaswonnumerousawardsforhis researchoverthelasttwo
+decades,ispartoftheCOViD-19EmergencyCommitteeattheWorldHealthOrganisation.
+Amongstothertopics,thiscommitteeisworkingtobetterunderstandthevarious
+responses and interventions that can help curb the spread of the disease.
+Michali is co-authoring a forthcoming pamphlet, entitled"The COViD-19Vaccine:what
+circulatingandforthcomingvaccines.ThepamphletdescribesCOviD-19as"adangerous
+disease,particularlyforthe elderlyand clinicallyvulnerable,astheyare more likelyto
+suffersevere,andpossiblyfatal,respiratoryillness.Nevertheless,anyone,regardlessof
+ageormedical background,isatriskof sufferingaharshillness.Michaliwishesto
+emphasisethatthebestthingyoucandotoprotectyourselffromthisdiseaseisto
+takeupthevaccinewhenyouareofferedit."Some ofthevaccinesonthemarketare
+boasting95%efficacyrates.Thismeansthatreceivingthevaccinedramaticallyreduces
+yourriskofdevelopingseriousCOViD-19symptomsifyouareexposedtotheviruslater
+downtheline."Althoughexpertsarecontinuingtoemphasisetheimportanceofsocial
+distancingandwearingmasks,thesemeasuresare notperfect,andthereremainsariskof
+inadvertentlycatchingthediseasethatcouldleaveyoubed-riddenforweeks,even
+months.Receivingthe vaccine isthesingle most important stepan individualcantaketo
+protect him orherself fromthe virus.Michali reflects intheconcludingthoughts of the
+pamphlet,"thereisnotmuchthatwecancontrolintimeslikethese,butyouneedto
+do whatyoucantoprotectyourself inthese uncertain times.Takingup thevaccine
+whenoffered isthebestactionyoucantaketokeepyourself safe!"Importantly,
+Michali wants individuals to rememberthat it is their personal responsibility to keep
+themselvesprotected.Below isanewsstorysimilartoothernewsstoriesyoumighthavereadbefore.Please
+readthestoryand respondtothequestionsthatfollow.
+BOsTON--“Coronavirus disease (COVID-19)is a highly contagious illness, caused bythe
+transmission oftheSARS-CoV-2 virus.First identified inDecember2019,the virushas
+caused apandemic that resulted in shutdowns all aroundtheglobe.Itwasfirst widely
+spreadbetweenhumansatawholesaleseafoodmarket inWuhan,China."Thisdisease
+haswreakedhavocontheglobe.Claimingmillionsof lives,thispandemichascreateda
+cleardemarcation intime:pre-covid,and post-covid,says ProfessorArthurMichali,a
+publichealthexpertfromaleadingresearchuniversity."Beforethispandemic,Icouldhave
+attendedaconference inTokyooneday,ledaresearchcollaboration inGenevathenext,
+andarrivedbackinBostonthethirdday.Thiskindoftravelissimplynolongerpossible
+undercurrentcircumstances,andit's likelythatthissortof behaviourcontributedtothe
+rapidspreadofthediseaseworldwide."
+ProfessorMichali,whohaswonnumerousawardsforhisresearchoverthelasttwo
+decades, is part of the COviD-19 Emergency Committee at the World Health Organisation.
+Amongstothertopics,thiscommitteeisworkingtobetterunderstandthevarious
+Michali is co-authoring a forthcoming pamphlet, entitled “"The COviD-19 Vaccine: what
+circulatingandforthcomingvaccines.ThepamphletdescribesCOviD-19as"adangerous
+disease,particularly forthe elderly and clinically vulnerable,as they are more likely to
+suffersevere,andpossiblyfatal,respiratoryillness.Nevertheless,anyone,regardlessof
+ageormedical background,is atrisk ofsufferingaharsh illness."Michali wishesto
+emphasisethatthebestthingyoucandotoprotectothersfromthisdiseaseisto
+takeupthevaccinewhenyouareoffered it."Someofthevaccinesonthemarketare
+boasting95%efficacyrates.Thismeansthatreceivingthevaccinedramaticallyreduces
+yourriskofdevelopingseriousCOviD-19symptomsifyouareexposedtotheviruslater
+downtheline.CommunitytransmissionhasbeenshowntobelowerwhensevereCOviD
+19symptomsdonotpresent,soyouareprotectingyourneighbours,parents,
+grandparents,andfriendsbyreceivingthevaccine."Althoughexpertsarecontinuingto
+emphasisetheimportanceofsocialdistancingandwearingmasks,thesemeasuresare
+notperfect,astheviruscanstillspreadbetweenpeople.Theworryisnotsomuchabout
+individual cases, but rather, it is about reducing transmission in communities, as it is that
+type oftransmissionthat will preventus from everseeing anendtothispandemic.
+thecommunityfromthevirus.Michali reflectsintheconcludingthoughtsofthepamphlet,
+"thereisnotmuchthatwe can control intimes likethese,butweneedtotake
+collectiveactiontofightthispandemic!Takingupthevaccinewhenofferedisthe
+bestactionyoucantakeforyourfamily,friends,andforyourcommunity!
+Importantly,Michaliwantspeopletorememberthatitistheirresponsibilitytokeeppeople
+intheircommunity,especiallythosewhoarevulnerabletothedisease,protected 9 
+Figure A.3.2: Survey design : manipulation check at 𝑇". 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+According to the article, where was the COVID-19 virus first widely spread?
+Geneva,Switzerland
+Boston, USA
+Wuhan, China
+Tokyo, Japan
+Accordingtothearticle,whatisProfessorArthurMichali currentlyworkingon?
+A strategy to liaise with journalists and media about COVID-19
+Apamphletto informthe average citizenaboutthe currentandforthcoming COviD-19vaccines
+Atravel itineraryfromTokyotoGenevato Boston
+Asociologicalstudyonthespreadof COviD-19
+According to the article, whom will you primarily protect by taking up a CoOVID-19 vaccine?
+Yourself
+Healthworkersinothercountries
+Peoplewho have justdiedofCOvID-19-related illness
+Others in your community 10 
+Figure A.3.3: Survey design: demographic questions at 𝑇".  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Areyou generally aperson who tries to avoid taking risks orare youfully prepared to take
+risks?
+Won't take risks
+Ready to take risks
+0
+1
+2
+3
+4
+5
+6
+7
+8
+6
+10
+HaveyoubeendiagnosedwithCovID-19atanypoint?
+Howfrequentlydoyoufollowgovernmentguidelinesonfacecoveringswheninshops?
+I neverwearafacecoveringbecauseIamexemptfromwearingone
+I neverwearafacecoveringand I amnotexemptfromwearingone.
+Most of the time I do not wear a face covering.
+HalfofthetimeIwearafacecovering,halfI donot.
+Most ofthe time I wearaface covering.
+Ialwayswearafacecovering
+How confident are you thatface coverings area useful tool in fighting the pandemic?
+Not at all confident
+Extremelyconfident
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100What isyourethnicity?
+What is the highest level of education that you have completed?
+Inwhichregiondoyoucurrentlyreside?
+What is youryearlyhousehold incomebeforetax?
+V
+Whichpolitical party do you consideryourself to beclosest to?
+Please indicateyourattitudesto eachof thefollowingstatements.
+I neither
+I disagree a
+I disagree a
+agree nor
+Iagree a
+lot
+little
+disagree
+little
+I agree a lot
+In uncertain times,
+usuallyexpectthe
+0
+0
+0
+0
+0
+best.
+I'm always optimistic
+0
+0
+0
+0
+0
+aboutmy future.
+Overall,Iexpectmore
+good things to happen
+0
+0
+0
+0
+0
+to me than bad. 11 
+Table A.2: Factor analysis on the personal and general BJW factors, which produce the 
+combined BJW factor.  
+ 
+Factor analysis/correlation 
+ 
+ 
+Factor 
+Eigenvalue 
+Difference 
+Proportion 
+Cumulative 
+Factor1 
+ 0.79                         1.04 
+ 1.46 
+1.46 
+ 
+Factor loadings (pattern matrix) and unique variances 
+Variable 
+Factor1 
+Uniqueness         
+General BJW 
+0.63 
+0.61 
+Personal BJW 
+0.63 
+0.61 
+ 
+ 
+Scoring coefficients 
+Variable 
+Factor1 
+General BJW 
+0.41 
+Personal BJW 
+0.41 
+ 
+Cronbach’s alpha 
+a 
+0.69 
+ 
+ 
+Table A.3: Factor analysis on the skepticism outcomes at 𝑇!. 
+Factor analysis/correlation 
+ 
+ 
+Factor 
+Eigenvalue 
+Difference 
+Proportion 
+Cumulative 
+Factor1 
+2.57 
+2.60 
+1.10 
+1.10 
+ 
+Factor loadings (pattern matrix) and unique variances 
+Variable 
+Factor1 
+Uniqueness         
+Vaccine Useful 
+0.86 
+0.26 
+Accept Vaccine 
+0.83 
+0.31 
+Reduce Transmission 
+0.64 
+0.59 
+Prevent Illness 
+0.85 
+0.28 
+ 
+ 
+Scoring coefficients 
+Variable 
+Factor1 
+Vaccine Useful 
+0.35 
+Accept Vaccine 
+0.28 
+Reduce Transmission 
+0.12 
+Prevent Illness 
+0.31 
+ 
+ 
+
+ 12 
+Cronbach’s alpha 
+a 
+0.88 
+ 
+ 
+Table A.4: Factor analysis on the skepticism outcomes at 𝑇". 
+ 
+Factor analysis/correlation 
+ 
+ 
+Factor 
+Eigenvalue 
+Difference 
+Proportion 
+Cumulative 
+Factor1 
+2.69 
+2.70 
+1.09 
+1.09 
+ 
+Factor loadings (pattern matrix) and unique variances 
+Variable 
+Factor1 
+Uniqueness         
+Vaccine Useful 
+0.88 
+ 0.22 
+Accept Vaccine 
+0.82 
+ 0.33 
+Reduce Transmission 
+0.70  
+ 0.51 
+Prevent Illness 
+0.87 
+0.25 
+ 
+ 
+Scoring coefficients 
+Variable 
+Factor1 
+Vaccine Useful 
+0.37 
+Accept Vaccine 
+ 0.23 
+Reduce Transmission 
+0.13 
+Prevent Illness 
+0.32 
+ 
+ 
+Cronbach’s alpha 
+a 
+0.89 
+ 
+ 
+ 
+

89AzT4oBgHgl3EQfFPox/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
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8tFAT4oBgHgl3EQfpB3g/content/tmp_files/2301.08638v1.pdf.txt

@@ -0,0 +1,589 @@
+arXiv:2301.08638v1  [hep-th]  20 Jan 2023
+Enlarging the symmetry of pure R2 gravity, BRST invariance
+and its spontaneaous breaking
+Ariel Edery∗
+Department of Physics and Astronomy, Bishop’s University, 2600 College Street,
+Sherbrooke, Qu´ebec, Canada, J1M 1Z7.
+Abstract
+Pure R2 gravity was considered originally to possess only global scale symmetry. It was
+later shown to have the larger restricted Weyl symmetry where it is invariant under the
+Weyl transformation gµν → Ω2(x) gµν when the conformal factor Ω(x) obeys the harmonic
+condition □Ω(x) = 0. Restricted Weyl symmetry has an analog in gauge theory. Under a
+gauge transformation Aµ → Aµ + 1
+e∂µf(x), the gauge-fixing term (∂µAµ)2 has a residual
+gauge symmetry when □f = 0. In this paper, we consider scenarios where the symmetry
+of pure R2 gravity can be enlarged even further. In one scenario, we add a massless scalar
+field to the pure R2 gravity action and show that the action becomes on-shell Weyl invari-
+ant when the equations of motion are obeyed. We then enlarge the symmetry to a BRST
+symmetry where no on-shell or restricted Weyl condition is required. The BRST trans-
+formations here are not associated with gauge transformations (such as diffeomorphisms)
+but with Weyl (local scale) transformations where the conformal factor consists of a prod-
+uct of Grassmann variables. BRST invariance in this context is a generalization of Weyl
+invariance that is valid in the presence of the Weyl-breaking R2 term. In contrast to the
+BRST invariance of gauge theories like QCD, it is not preserved after quantization since
+renormalization introduces a scale (leading to the well-known Weyl (conformal) anomaly).
+We show that the spontaneous breaking of the BRST symmetry yields an Einstein action;
+this still has a symmetry which is also anomalous. This is in accord with previous work
+that shows that there is conformal anomaly matching between the unbroken and broken
+phases when conformal symmetry is spontaneously broken.
+∗[email protected]
+1
+
+1
+Introduction
+Pure R2 gravity (R2 alone with no additional R term) is unique among quadratic gravity the-
+ories as it is unitary and moreover has been shown to be conformally equivalent to Einstein
+gravity with non-zero cosmological constant and massless scalar field [1–5] (though in a Pala-
+tini formalism one can avoid having a massless scalar [6]). It has been known for a long time
+that it is invariant under the global scale transformation gµν → λ2 gµν where λ is a constant. It
+was later discovered to possess a larger symmetry than global scale symmetry called restricted
+Weyl symmetry [7] where it is invariant under the transformation gµν → Ω2(x) gµν when the
+conformal factor Ω(x) obeys the harmonic condition □Ω = gµν∇µ∇νΩ = 0. The conformal
+factor Ω(x) is therefore not limited to being a constant. The aforementioned equivalence be-
+tween pure R2 gravity and Einstein gravity with cosmological constant was then interpreted
+in a new light: it occurs when the restricted Weyl symmetry is spontaneously broken [3,5]. In
+the broken sector, the Ricci scalar of the background (vacuum) spacetime has R ̸= 0 which
+excludes a flat background. This is why the equivalence requires a non-zero cosmological con-
+stant on the the Einstein side. The unbroken sector which has an R = 0 vacuum (background)
+has no relation to Einstein gravity. In fact, it has been shown that a linearization of pure R2
+gravity about Minkowski spacetime does not yield gravitons but only a propagating scalar [4];
+simply put, pure R2 gravity does not gravitate about a flat background [4]. However, it was
+later shown that if one includes a non-minimally coupled scalar field in the restricted Weyl-
+invariant action and the field acquires a non-zero VEV, then the theory can gravitate about
+flat spacetime [5,8]. Various aspects of restricted Weyl symmetry, it spontaneous breaking as
+well as its role in critical gravity were then explored further in [5–7,9–11]
+Restricted Weyl symmetry has an analog in gauge theory. The gauge-fixing term (∂µAµ)2
+is invariant under the gauge transformation Aµ → Aµ + 1
+e∂µf(x) only when the arbitrary
+smooth function f(x) obeys the condition □f = 0 where □ here represents the flat space
+d’Alembertian. Therefore, the gauge-fixing term has a residual gauge symmetry when □f = 0
+is satisfied [12]. This is the analog to the restricted Weyl symmetry of pure R2 gravity when
+the conformal factor Ω(x) satisfies □Ω = 0. As we will see, this analogy is fruitful as it provides
+a bridge to the BRST symmetry of pure R2 gravity. Recent work on the BRST invariance of
+other gravitational theories can be found in [13,14,16].
+In this paper, we consider scenarios where the symmetry of pure R2 gravity is enlarged further.
+We show that when a massless scalar field is added to pure R2 gravity, the action becomes Weyl
+invariant when the equations of motion are satisfied. No separate external condition is required
+to be imposed on the conformal factor Ω(x) as this occurs naturally via the equations of motion.
+One passes from restricted Weyl invariance to on-shell Weyl invariance. One can then enlarge
+the symmetry further to include BRST symmetry. In analogy with the BRST invariance in
+gauge theories in the presence of a gauge-fixing term, we establish BRST invariance in the
+2
+
+presence of the Weyl-breaking pure R2 gravity term. The BRST transformations here are
+not associated with gauge transformations (such as diffeomorphisms) but are a generalization
+of Weyl (local scale) transformations where the conformal factor is composed of Grassmann
+variables. Therefore, in contrast to the BRST invariance in gauge theory, it is anomalous since
+renormalization introduces a scale (leading to the well-known Weyl (conformal) anomaly). We
+show that the spontaneous breaking of the BRST symmetry yields an Einstein action with its
+own symmetry that is also anomalous. This is in agreement with previous work where it was
+shown that when conformal symmetry is spontaneously broken there is conformal anomaly
+matching in the unbroken and and broken phases [18,19].
+The paper is organized as follows. In section 2, we obtain the on-shell Weyl invariance of pure
+R2 gravity when a massless scalar field is included in the action. In section 3, we obtain the
+BRST invariance of pure R2 gravity. In section 4, we show that the spontaneous breaking
+of the BRST symmetry yields an Einstein action and that there is a quantum anomaly in
+both the unbroken and broken sectors. We conclude with section 5 where we summarize our
+results, provide further physical insights and discuss directions for future work. We relegate
+to Appendix A some technical details on the symmetry of the Einstein action.
+2
+Pure R2 gravity plus a massless scalar: from restricted to
+on-shell Weyl invariance
+The action of pure R2 gravity is given by
+S =
+� √−g d4x α R2
+(1)
+where R is the Ricci scalar and α a dimensionless constant. This action is restricted Weyl
+invariant i.e. it is invariant under the Weyl transformation gµν → Ω2(x) gµν if the conformal
+factor Ω(x) obeys the condition □Ω(x) = 0. This invariance stems from the fact that R →
+R/Ω2 when □Ω(x) = 0. As already mentioned, this implies that pure R2 gravity has a greater
+symmetry than global scale symmetry (where Ω(x) would have to be a constant).
+We now show that pure R2 gravity can be Weyl-invariant on-shell when a minimally coupled
+real massless scalar field is added to the action. Here, the condition □Ω(x) = 0 is not imposed
+as an external condition but satisfied automatically by the equations of motion. The action of
+pure R2 gravity with a minimally coupled real massless scalar field φ is given by
+Sa =
+� √−g d4x
+�
+α R2 − 1
+2 gµν ∂µφ ∂νφ
+�
+(2)
+where φ(x) is a real scalar field. Under the Weyl transformation gµν → e−2 ǫ φ gµν, where ǫ is
+3
+
+a real constant, the Ricci scalar transforms as
+R → R e2ǫφ − 6 e3ǫ φ □(e−ǫ φ)
+(3)
+and √−g → e−4 ǫ φ √−g so that action (2) transforms to
+Sb =
+� √−g d4x
+�
+α
+�
+R2 − 12 R eǫ φ □(e−ǫ φ) + 36 e2ǫ φ (□(e−ǫ φ))2�
+− 1
+2 e−2 ǫ φ gµν ∂µφ ∂νφ
+�
+.
+(4)
+The equations of motion yield □(e−ǫ φ) = 0. Therefore, when the equations of motion are
+satisfied, the above action reduces to
+Sc =
+� √−g d4x
+�
+α R2 − 1
+2 gµν ∂µψ ∂νψ
+�
+(5)
+where ψ is a real massless scalar field (related to the old scalar φ via ψ = e−ǫ φ/ǫ). Note that
+the equation of motion for ψ is □ψ = 0 which is equivalent to □(e−ǫ φ) = 0 and consistent with
+what we previously obtained. We therefore recover pure R2 gravity with a minimally coupled
+real massless scalar field ψ. What happened here is that the restricted Weyl condition □ Ω = 0
+with Ω = e−ǫ φ did not have to be imposed as a separate condition because it was satisfied
+automatically by the equations of motion. In short, pure R2 gravity became Weyl invariant
+on-shell in the presence of a massless scalar field. It passed from restricted Weyl invariance to
+on-shell Weyl invariance.
+3
+BRST invariance of pure R2 gravity
+Before discussing BRST invariance in the case of pure R2 gravity, let us first recall how BRST
+invariance works in gauge theories in Minkowski spacetime. For illustrative purposes, we will
+consider the case of scalar QED. The Abelian version of the Faddeev-Popov Lagrangian is then
+given by [12]
+L = −1
+4 F 2
+µν − (Dµφ∗
+a)(Dµφa) − m2 φ∗
+a φa − 1
+2 ξ (∂µ Aµ)2 + ¯c □c
+(6)
+where c(x) and ¯c(x) are independent Grassmann-valued fields, φa are a set of complex scalar
+fields and Dµ is the usual covariant derivative. The gauge fixing term,
+1
+2 ξ(∂µ Aµ)2 breaks the
+gauge symmetry since it is not invariant under the transformation Aµ → Aµ + 1
+e ∂µf(x) where
+f(x) is an arbitrary function. However, it has a residual symmetry: it is invariant if f(x)
+obeys the condition □f = 0. As previously mentioned, this residual symmetry is the analog of
+restricted Weyl symmetry in pure R2 gravity.
+4
+
+The equation of motion for c(x) is □c = 0. Consider the gauge transformation with f(x) =
+θ c(x) for arbitrary Grassmann number θ. Then, if the equation of motion for c is satisfied,
+the scalar QED Lagrangian (6) is invariant under the following transformations
+Aµ → Aµ + 1
+e θ ∂µc(x)
+φa(x) → eiθ c(x) φa(x) = φa(x) + iθ c(x)φa(x) .
+(7)
+In other words, the equation □f = θ □c = 0 is automatically satisfied on-shell and does not
+have to be imposed as a separate condition. This is similar to what we saw in the previous
+section for pure R2 gravity which was invariant under gµν → Ω2 gµν with Ω = e−ǫφ when the
+equations of motion were satisfied.
+If the equation of motion for c is not used, the only term in the Lagrangian (6) which is not
+invariant under the transformation (7) is (∂µAµ)2 which transforms as
+(∂µAµ)2 → (∂µAµ)2 + 2
+e(∂µAµ)(θ□c)
+(8)
+where we used the fact that θ2 = 0 since θ is Grassmann. Now, if under (7) we also have ¯c
+transforming as
+¯c(x) → ¯c(x) − θ
+e ξ (∂µAµ)
+(9)
+then the scalar QED Lagrangian (6) is invariant without having to use the equation of motion
+for c. This is BRST invariance. The crucial point is that under the BRST transformations
+given by (7) and (9), the Lagrangian is invariant despite the presence of the gauge-fixing term
+(∂µAµ)2.
+We now turn to pure R2 gravity. Consider the action
+S =
+�
+d4x√−g (α R2 + ¯c □c)
+(10)
+where again c(x) and ¯c(x) are independent Grassmann-valued fields. This action is not Weyl-
+invariant i.e. it is not invariant under the transformation gµν → Ω2(x)gµν where Ω(x) is an
+arbitrary smooth function. Consider now the Weyl transformation
+gµν → e2 θ c(x)gµν = (1 + 2 θ c) gµν
+(11)
+where θ is again an arbitrary Grassmann number. Under this transformation we have
+√−g α R2 → √−g (α R2 − 12 α R θ □c )
+(12)
+5
+
+where the following transformations were used: √−g → (1+4 θ c) √−g and R → (1−2 θ c) R−
+6 θ □c. Again, we used that θ2 = 0. Under the transformation (11), □c transforms as
+□c → (1 − 2 θ c) □c
+(13)
+where gµν∂µc ∂νc = 0 was used (this stems from the fact that gµν is symmetric and c is
+Grassmann). The equation of motion for c is □c = 0 and we see from (12) that √−g α R2 is
+Weyl invariant on-shell. However, we can dispense with the on-shell condition if we also allow
+¯c to transform as
+¯c → (1 − 2 θ c) ¯c + 12 α R θ .
+(14)
+We then obtain
+√−g ¯c □c → √−g (¯c □c + 12 α R θ □c) .
+(15)
+The last term on the right hand side of (15) above cancels precisely the last term on the right
+hand side of (12). Therefore, the action (10) is invariant under the combined transformations
+of (11) and (14) (which we refer to to as BRST transformations).
+This is the BRST invariance
+of pure R2 gravity. Note that BRST invariance does not require any on-shell or restricted Weyl
+condition. It is a generalization of Weyl (conformal) invariance that is valid in the presence of
+the Weyl-breaking R2 term.
+Let us now take a closer look at what is common and what is different between the BRST
+invariance of pure R2 gravity and the BRST invariance in the gauge theories of particle physics
+(for concreteness and simplicity, we will consider scalar QED again but the main points apply
+also to QCD). The BRST invariance in scalar QED can be viewed as a generalization of gauge
+invariance in the presence of the gauge-fixing (and hence gauge-breaking) term (∂µ Aµ)2. The
+are two points in common between the scalar QED and R2 cases. First, the Ricci scalar R under
+a Weyl transformation and the term ∂µ Aµ under a gauge transformation both pick up an extra
+□Φ(x) term (where Φ(x) represents either a conformal factor Ω(x) in a Weyl transformation
+or a function f(x) in a gauge transformation). Recall that in a BRST transformation, Φ(x) is
+a product of a Grassmann number θ with a Grassmann field (the product yields a commuting
+(bosonic) quantity). The second point in common is that R and ∂µ Aµ are both squared. The
+squaring yields a (□Φ(x))2 term which is zero since θ2 = 0. The squaring still leaves one
+extra □Φ(x) term and this is cancelled out in both cases via the transformation property of a
+Grassmann field. These two common points render the analogy between the two cases quite
+strong. However, there is one important difference. In scalar QED (and in QCD) , the BRST
+transformations are associated with gauge transformations. The BRST invariance of pure R2
+gravity that we are considering here is not associated with gauge transformations (such as
+diffeomorphisms) but with Weyl (local scale) transformations. We will see that this difference
+plays an important role when the theory is quantized.
+6
+
+4
+Spontaneous breaking of BRST symmetry
+We now show that the BRST-invariant action
+S =
+�
+d4x√−g (αR2 + ¯c □c)
+(16)
+is conformally equivalent to an action that involves the Einstein-Hilbert term; this will involve
+the spontaneous breaking of BRST symmetry. The starting point is to introduce a auxiliary
+field σ(x) to rewrite the above action into the equivalent form
+S1 =
+�
+d4x√−g (−α(b σ + R)2 + αR2 + ¯c □c)
+�
+d4x√−g (−α b2 σ2 − 2 α b R σ + ¯c □c)
+(17)
+where b is a real non-zero constant with dimensions of mass squared and σ(x) is dimensionless.
+Action (17) is equivalent to the original action (16) since adding the squared term in the first
+line of (17) does not alter anything (classically, the equations of motion are unaffected and
+quantum mechanically, the path integral over σ is a Gaussian which yields a constant). The
+equivalent action (17) is also BRST invariant; it is invariant under the following transforma-
+tions:
+gµν → (1 + 2 θ c) gµν
+;
+¯c → (1 − 2 θ c) ¯c − 12 θ α b σ
+;
+σ → (1 − 2θ c) σ
+(18)
+where θ is again a Grassmann number. Note that the BRST invariance requires the auxiliary
+field σ to transform besides the fields gµν and ¯c. We now perform the following conformal
+(Weyl) transformation:
+gµν → σ−1 gµν
+¯c → σ ¯c
+(19)
+which leads to √−g → σ−2 √−g and R → σ R − 6 σ3/2□(σ−1/2). Under the above conformal
+transformation, action (17) becomes
+S2 =
+�
+d4x√−g (−α b2 − 2 α b R + 3α b
+σ2 ∂µσ ∂µσ + ¯c □c − 1
+σ ¯c ∂µc ∂µσ) .
+(20)
+The above action is no longer invariant under the BRST transformations given by (18). The
+BRST symmetry has been spontaneously broken. The factor σ−1 appearing in the confor-
+mal transformation (19) is valid only for non-zero σ so that the VEV (vacuum expectation
+value) of the field σ must be non-zero. The VEV is therefore not invariant under the BRST
+transformation σ → (1 − 2θ c) σ leading to the spontaneous breaking of the BRST symmetry.
+7
+
+We can identify −2 α b R as an Einstein-Hilbert term if we equate −2 α b with
+1
+16π G where G
+is Newton’s constant. The constant term −α b2 can then be associated with a cosmological
+constant Λ = −b/4. Note that though −2 α b is positive, the constant b can be either positive
+or negative (but not zero). This implies that the cosmological constant can be either positive
+corresponding to a de Sitter (dS) background or negative corresponding to an anti-de Sitter
+(AdS) background but it cannot be identically zero. We can then express (20) as the following
+Einstein action,
+SE =
+�
+d4x√−g
+�
+1
+16π G(R − 2 Λ) + 3α b
+σ2 ∂µσ ∂µσ + ¯c □c − 1
+σ ¯c ∂µc ∂µσ)
+�
+.
+(21)
+We have left the constant 3 α b in the action for simplicity but it is not an independent constant;
+it is equal to
+−3
+32 πG. We therefore obtain an Einstein-Hilbert action with non-zero cosmological
+constant, a kinetic term for the scalar σ (which we will express in canonical form later) and
+an interaction term.
+Recall that σ is non-zero so that divisions by σ pose no issue.
+It is
+well-known that in spontaneously broken theories, the vacuum breaks the symmetry but it is
+not actually broken in the Lagrangian but manifested or realized in a different way [12]. It
+can be directly verified (see Appendix A) that the Einstein action (21) is invariant under the
+following transformations:
+σ → (1 − 2θ c) σ , gµν → gµν and ¯c → ¯c − 12 θ α b .
+(22)
+The BRST symmetry of action (17) manifests itself in the Einstein action (21) via its symmetry
+under the above transformations (22). We now show how transformation (22) stems from the
+BRST transformations (18). In the Einstein action and transformation (22) label the metric
+and the barred Grassmann field with a subscript E i.e.
+gµνE and ¯cE.
+In action (17) and
+transformation (18) we leave gµν and ¯c as is. Then the conformal transformation (19) yields
+gµνE = σ gµν and ¯cE = σ−1 ¯c. Under the BRST transformations (18) we obtain gµνE = σ gµν →
+(1 − 2 θ c) σ (1 + 2 θ c) gµν = σ gµν = gµνE and ¯cE = σ−1 ¯c → (1 + 2 θ c) σ−1�
+(1 − 2 θ c) ¯c −
+12 θ α b σ
+�
+= σ−1¯c − 12 θ α b = ¯cE − 12 θ α b. We have therefore obtained the transformations
+gµνE → gµνE and ¯cE → ¯cE − 12 θ α b which correspond to those in (22). Note that we used
+σ → (1 − 2θ c) σ in (18) to derive this, so the transformation of σ is also part of (22).
+We can define a real massless scalar field ψ(x) =
+√
+−3α b ln σ(x) so that the kinetic term for
+σ is expressed in canonical form. The Einstein action (21) expressed in terms of the field ψ is
+S =
+�
+d4x√−g
+�
+1
+16π G(R − 2 Λ) − ∂µψ ∂µψ + ¯c □c −
+1
+√
+−3α b ¯c ∂µc ∂µψ
+�
+.
+(23)
+The massless scalar field ψ corresponds to the Nambu-Goldstone boson of the broken sector.
+Under transformation (22), the field ψ transforms as a shift ψ → ψ−
+√
+−3α b 2 θ c (whereas ¯c →
+¯c−12 θ α b and gµν → gµν). The above action (23) is invariant under those transformations (see
+8
+
+Appendix A). This is in accord with what we expect from spontaneously broken theories: the
+original symmetry in the Lagrangian manifests itself in the broken sector as a shift symmetry
+of the Goldstone bosons [12].
+4.1
+Quantum anomaly
+We saw that the action (17) is BRST invariant under the following transformations:
+gµν → (1 + 2 θ c) gµν , ¯c → (1 − 2 θ c) ¯c − 12 θ α b σ , σ → (1 − 2θ c) σ. Each transformation
+involves a Weyl transformation where the conformal factor is expressed in terms of of a prod-
+uct of two Grassmann variables The BRST symmetry is therefore a generalization of Weyl
+(conformal) symmetry. After quantization, renormalization introduces a scale which breaks
+the BRST symmetry since it automatically breaks Weyl symmetry (leading to the well-known
+Weyl (conformal) anomaly). So the BRST symmetry of pure R2 gravity is anomalous. This is
+in contrast to the BRST invariance of gauge theories like QCD which have no anomaly.
+After the BRST symmetry is spontaneously broken and we obtain the Einstein action (21),
+we saw that the BRST symmetry manifests itself now in the Einstein action as a symmetry
+under the transformations (22). This symmetry is also anomalous since the transformation of
+the field σ is a Weyl transformation and renormalization breaks this symmetry (leading again
+to the Weyl (conformal anomaly). Another way to see this is to note that the only fields that
+transform in (22) are ¯c and σ. The transformation for ¯c is simply a constant shift so that its
+path integral measure D¯c is invariant. However, σ undergoes a Weyl transformation and this
+introduces a non-trivial Jacobian J (i.e. J ̸= 1) to the measure Dσ. Since the measure is not
+invariant, this implies there is an anomaly [17]. So the symmetry in the unbroken phase and
+its associated symmetry in the broken phase are both anomalous. Our finding here is in accord
+with previous work that shows that when the Weyl or conformal symmetry is spontaneously
+broken there is conformal anomaly matching between the unbroken and broken phases [18,19].
+5
+Conclusion
+In the last six years or so, we have kept discovering new aspects of pure R2 gravity. A non-
+exhaustive list includes its unitarity among quadratic gravity theories [4], its conformal equiv-
+alence to Einstein gravity with non-zero cosmological constant and massless scalar field [1–5],
+its restricted Weyl symmetry [7,10,11], its spontaneous symmetry breaking to Einstein grav-
+ity [3,5] and the lack of a propagating graviton when the theory is linearized about a Minkowski
+background [4] (where there is no Einstein equivalence since the cosmological constant is zero).
+In this paper, we have gained further insights into this theory. We saw that pure R2 gravity
+has an analog with the gauge-fixing term (∂µAµ)2 in gauge theory. R2 is not invariant under
+9
+
+the Weyl transformation gµν → Ω2(x) gµν just like (∂µ Aµ)2 is not invariant under the gauge
+transformation Aµ → Aµ + 1
+e ∂µf(x). However, each have a residual symmetry (when □Ω = 0
+is satisfied in the gravity case and □f = 0 is satisfied in the gauge case). This analogy opened
+the door towards enlarging the symmetry of pure R2 gravity to include BRST symmetry.
+We first showed that when a massless scalars field was included in the pure R2 action, the
+condition □Ω = 0 could be met automatically when the equations of motion were satisfied
+i.e. we went from restricted Weyl to on-shell Weyl invariance. Finally, we obtained the BRST
+invariance of pure R2 gravity where no restricted Weyl or on-shell condition is required. The
+BRST transformations involve Weyl transformations where the conformal factor is composed of
+products of Grassmann variables (the conformal factor itself is commutative). The important
+point is that the BRST invariance exists despite the Weyl-breaking R2 term.
+There is one important difference between the BRST symmetry in gauge theories like QCD
+and the BRST symmetry that we have considered here for pure R2 gravity. Gauge invari-
+ance in particle physics is preserved after quantization. The BRST invariance of QCD is a
+generalization of gauge invariance so that it is also preserved after quantization; there is no
+anomaly. In contrast to gauge symmetry, global scale or Weyl (local scale) symmetry is broken
+after quantization since renormalization introduces a scale. The BRST symmetry of pure R2
+gravity is a generalization of Weyl (conformal) symmetry so that it is also broken after quan-
+tization leading to the well-known Weyl (conformal) anomaly. After the spontaneous breaking
+of the BRST symnmetry, we obtained an Einstein action. We showed that this action has
+its own symmetry and that it is also anomalous. This is in accord with previous work that
+shows that when the Weyl (conformal) symmetry is spontaneously broken there is conformal
+anomaly matching between the unbroken and broken sectors [18,19].
+The focus of this paper was pure R2 gravity because of its many special and attractive fea-
+tures that we previously mentioned. All other quadratic gravity theories (like Weyl-squared,
+Riemann-squared, etc.), apart from boundary terms, can be expressed as a linear combina-
+tion of R2 and RµνRµν. The latter term, the square of the Ricci tensor, appears in quantum
+corrections to General Relativity (GR) and even though it does not constitute a valid UV
+completion of GR due to its non-unitarity (yields a massive spin two ghost [4, 20]), it still
+makes a well-known calculable short-range correction to the Newtonian potential [12,21]. Like
+R2, the term RµνRµν is not Weyl-invariant so it would be of interest to see if it can be BRST
+invariant. It is not in the form of a scalar squared like (∂µAµ)2 or R2, so one may be inclined
+to think that the BRST formalism would not apply here. However, like pure R2, it was shown
+in [7] that RµνRµν is restricted Weyl invariant (up to a boundary term). This suggests that
+the procedure used to establish the BRST invariance of pure R2 gravity might in the end also
+work for this quadratic theory. It would therefore be worthwhile and interesting to investigate
+this further.
+10
+
+Acknowledgments
+A.E. acknowledges support from a discovery grant of the National Science and Engineering
+Research Council of Canada (NSERC).
+A
+Symmetry of Einsten Action
+In this appendix we show that the Einstein action (21) given by
+SE =
+�
+d4x√−g
+�
+1
+16π G(R − 2 Λ) + 3α b
+σ2 ∂µσ ∂µσ + ¯c □c − 1
+σ ¯c ∂µc ∂µσ)
+�
+(A.1)
+is invariant under the transformations (22) given by
+σ → (1 − 2θ c) σ , gµν → gµν and ¯c → ¯c − 12 θ α b .
+(A.2)
+Under the above transformation, the metric gµν does not change so that √−g as well as the
+term √−g
+1
+16π G(R − 2 Λ) does not change.
+The other terms in the above Einstein action
+transform as
+3α b
+σ2 ∂µσ ∂µσ → 3α b
+σ2 ∂µσ ∂µσ − 12 θ α b
+σ
+∂µc ∂µσ
+− 1
+σ ¯c ∂µc ∂µσ) → − 1
+σ ¯c ∂µc ∂µσ + 12 θ α b
+σ
+∂µc ∂µσ
+¯c □c → ¯c □c − 12 θ α b □c
+(A.3)
+where we used that θ2 = 0 (since θ is a Grassmann number) and that gµν ∂µc ∂νc = 0 since gµν
+is symmetric and c(x) and its derivatives are Grassmann fields. We see that the extra term
+− 12 θ α b
+σ
+∂µc ∂µσ in the first line of (A.3) is canceled exactly by the extra term in the second
+line. The extra term in the third line, −12 θ α b □c, where −12 θ α b is a constant, does not
+cancel out with any other extra term in (A.3). However, √−g □c is a total derivative that
+yields an inconsequential boundary term in the action. We have therefore shown that action
+(A.1) is invariant under transformations (A.2).
+We saw in section 4 that the Einstein action (A.1) can be expressed in terms of a real massless
+scalar field ψ(x) =
+√
+−3α b ln σ(x) as action (23):
+S =
+�
+d4x√−g
+�
+1
+16π G(R − 2 Λ) − ∂µψ ∂µψ + ¯c □c −
+1
+√
+−3α b ¯c ∂µc ∂µψ
+�
+(A.4)
+where ψ was identified as the Nambu-Goldstone boson of the broken sector. We stated in
+section 4 that the action (A.4) was invariant under the following transformations:
+ψ → ψ −
+√
+−3α b 2 θ c , ¯c → ¯c − 12 θ α b and gµν → gµν .
+(A.5)
+11
+
+We now verify this statement. Under (A.5) the last three terms in action (A.4) transform as:
+− ∂µψ ∂µψ → −∂µψ ∂µψ + 4 θ
+√
+−3 α b ∂µψ ∂µc
+−
+1
+√
+−3α b ¯c ∂µc ∂µψ → −
+1
+√
+−3α b ¯c ∂µc ∂µψ − 4 θ
+√
+−3 α b ∂µψ ∂µc
+¯c □c → ¯c □c − 12 θ α b □c .
+(A.6)
+We see that the extra term +4 θ
+√
+−3 α b ∂µψ ∂µc in the first line above is cancelled by the
+extra term on the second line which is equal to its negative.
+The only extra term that is
+not cancelled is the term −12 θ α b □c appearing in the last line. However, √−g □c is a total
+derivative which yields a boundary term with no physical consequence. We have therefore
+verified that the Einstein action (A.4) is indeed invariant under the transformations (A.5).
+References
+[1] C. Kounnas, D. L¨ust and N. Toumbas, R2 inflation from scale invariant supergravity and
+anomaly free superstrings with fluxes, Fortsch. Phys. 63, 12 (2015) [arXiv:1409.7076].
+[2] A. Kehagias, C. Kounnas, D. L¨ust and A. Riotto, Black Hole Solutions in R2 Gravity,
+JHEP 05, 143 (2015)[arXiv:1502.04192].
+[3] A. Edery and Y. Nakayama, Generating Einstein gravity, cosmological constant and
+Higgs mass from restricted Weyl invariance, Mod. Phys. Lett. A 30, 1550152 (2015)
+[arXiv:1502.05932].
+[4] L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. L¨ust and A. Riotto, Aspects of Quadratic
+Gravity, Fortsch. Phys. 64, 176 (2016) [arXiv:1505.07657].
+[5] A. Edery and Y. Nakayama, Gravitating magnetic monopole via the spontaneous symmetry
+breaking of pure R2 gravity, Phys. Rev. D 98 (2018) 064011 [arXiv:1807.07004].
+[6] A. Edery and Y. Nakayama, Palatini formulation of pure R2 gravity yields Einstein gravity
+with no massless scalar, Phys. Rev. D 99, 124018 (2019) [arXiv:1902.07876].
+[7] A. Edery and Y. Nakayama, Restricted Weyl invariance in four-dimensional curved space-
+time, Phys. Rev. D 90, 043007 (2014) [arXiv:1406.0060].
+[8] A. Edery, Pure R2 gravity can gravitate about a flat background, J. Phys. Conf. Ser. 1956,
+012005 (2021).
+[9] A. Edery and Y. Nakayama, Critical gravity from four dimensional scale invariant gravity,
+JHEP 11, 169 (2019) [arXiv:1908.08778].
+12
+
+[10] I. Oda, Restricted Weyl symmetry, Phys. Rev. D 102, 045008 (2020)[arXiv:2005.04771].
+[11] I. Oda, Restricted Weyl Symmetry and Spontaneous Symmetry Breakdown of Conformal
+Symmetry, Mod. Phys. Lett. A 36, 2150203 (2021) [arXiv:2104.04694].
+[12] M. Schwartz, Quantum Field Theory and the Standard Model, (Cambridge University
+Press, Cambridge, UK, 2014).
+[13] I. Oda and P. Saake, BRST formalism of Weyl conformal gravity, Phys. Rev. D 106,
+106007 (2022) [arXiv:2209.14533].
+[14] D. Prinz, The BRST Double Complex for the Coupling of Gravity to Gauge Theories,
+[arXiv:2206.00780].
+[15] T. Kugo, R. Nakayama and N.Ohta, Covariant BRST quantization of unimodular grav-
+ity: Formulation with antisymmetric tensor ghosts, Phys. Rev. D 105, 086006 (2022),
+[arXiv:2202.03626].
+[16] L. Berezhiani, G. Dvali and O. Sakhelashvili, de Sitter space as a BRST invariant coherent
+state of gravitons Phys. Rev. D 105, 025022 (2022) [arXiv:2111.12022].
+[17] K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies, (Oxford University
+Press, Oxford, UK, 2004).
+[18] A. Schwimmer and S. Theisen, Spontaneous Breaking of Conformal Invariance and Trace
+Anomaly Matching,Nucl. Phys. B 847, 590 (2011) [arXiv:1011.0696.
+[19] A. Cabo-Bizet, E. Gava and K. S. Narain, Holography and Conformal Anomaly Matching,
+JHEP 11,044 (2013) [arXiv:1307.3784].
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+953 (1977).
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+72, 2996 (1994).
+13
+

8tFAT4oBgHgl3EQfpB3g/content/tmp_files/load_file.txt

@@ -0,0 +1,355 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf,len=354
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='08638v1  [hep-th]  20 Jan 2023  Enlarging the symmetry of pure R2 gravity, BRST invariance  and its spontaneaous breaking  Ariel Edery∗  Department of Physics and Astronomy, Bishop’s University, 2600 College Street,  Sherbrooke, Qu´ebec, Canada, J1M 1Z7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  Abstract  Pure R2 gravity was considered originally to possess only global scale symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' It was  later shown to have the larger restricted Weyl symmetry where it is invariant under the  Weyl transformation gµν → Ω2(x) gµν when the conformal factor Ω(x) obeys the harmonic  condition □Ω(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Restricted Weyl symmetry has an analog in gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Under a  gauge transformation Aµ → Aµ + 1  e∂µf(x), the gauge-fixing term (∂µAµ)2 has a residual  gauge symmetry when □f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In this paper, we consider scenarios where the symmetry  of pure R2 gravity can be enlarged even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In one scenario, we add a massless scalar  field to the pure R2 gravity action and show that the action becomes on-shell Weyl invari-  ant when the equations of motion are obeyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We then enlarge the symmetry to a BRST  symmetry where no on-shell or restricted Weyl condition is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The BRST trans-  formations here are not associated with gauge transformations (such as diffeomorphisms)  but with Weyl (local scale) transformations where the conformal factor consists of a prod-  uct of Grassmann variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' BRST invariance in this context is a generalization of Weyl  invariance that is valid in the presence of the Weyl-breaking R2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In contrast to the  BRST invariance of gauge theories like QCD, it is not preserved after quantization since  renormalization introduces a scale (leading to the well-known Weyl (conformal) anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  We show that the spontaneous breaking of the BRST symmetry yields an Einstein action;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  this still has a symmetry which is also anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is in accord with previous work  that shows that there is conformal anomaly matching between the unbroken and broken  phases when conformal symmetry is spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  ∗aedery@ubishops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='ca  1  1  Introduction  Pure R2 gravity (R2 alone with no additional R term) is unique among quadratic gravity the-  ories as it is unitary and moreover has been shown to be conformally equivalent to Einstein  gravity with non-zero cosmological constant and massless scalar field [1–5] (though in a Pala-  tini formalism one can avoid having a massless scalar [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' It has been known for a long time  that it is invariant under the global scale transformation gµν → λ2 gµν where λ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' It  was later discovered to possess a larger symmetry than global scale symmetry called restricted  Weyl symmetry [7] where it is invariant under the transformation gµν → Ω2(x) gµν when the  conformal factor Ω(x) obeys the harmonic condition □Ω = gµν∇µ∇νΩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The conformal  factor Ω(x) is therefore not limited to being a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The aforementioned equivalence be-  tween pure R2 gravity and Einstein gravity with cosmological constant was then interpreted  in a new light: it occurs when the restricted Weyl symmetry is spontaneously broken [3,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In  the broken sector, the Ricci scalar of the background (vacuum) spacetime has R ̸= 0 which  excludes a flat background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is why the equivalence requires a non-zero cosmological con-  stant on the the Einstein side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The unbroken sector which has an R = 0 vacuum (background)  has no relation to Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In fact, it has been shown that a linearization of pure R2  gravity about Minkowski spacetime does not yield gravitons but only a propagating scalar [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  simply put, pure R2 gravity does not gravitate about a flat background [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, it was  later shown that if one includes a non-minimally coupled scalar field in the restricted Weyl-  invariant action and the field acquires a non-zero VEV, then the theory can gravitate about  flat spacetime [5,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Various aspects of restricted Weyl symmetry, it spontaneous breaking as  well as its role in critical gravity were then explored further in [5–7,9–11]  Restricted Weyl symmetry has an analog in gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The gauge-fixing term (∂µAµ)2  is invariant under the gauge transformation Aµ → Aµ + 1  e∂µf(x) only when the arbitrary  smooth function f(x) obeys the condition □f = 0 where □ here represents the flat space  d’Alembertian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Therefore, the gauge-fixing term has a residual gauge symmetry when □f = 0  is satisfied [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is the analog to the restricted Weyl symmetry of pure R2 gravity when  the conformal factor Ω(x) satisfies □Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' As we will see, this analogy is fruitful as it provides  a bridge to the BRST symmetry of pure R2 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Recent work on the BRST invariance of  other gravitational theories can be found in [13,14,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  In this paper, we consider scenarios where the symmetry of pure R2 gravity is enlarged further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  We show that when a massless scalar field is added to pure R2 gravity, the action becomes Weyl  invariant when the equations of motion are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' No separate external condition is required  to be imposed on the conformal factor Ω(x) as this occurs naturally via the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  One passes from restricted Weyl invariance to on-shell Weyl invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' One can then enlarge  the symmetry further to include BRST symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In analogy with the BRST invariance in  gauge theories in the presence of a gauge-fixing term, we establish BRST invariance in the  2  presence of the Weyl-breaking pure R2 gravity term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The BRST transformations here are  not associated with gauge transformations (such as diffeomorphisms) but are a generalization  of Weyl (local scale) transformations where the conformal factor is composed of Grassmann  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Therefore, in contrast to the BRST invariance in gauge theory, it is anomalous since  renormalization introduces a scale (leading to the well-known Weyl (conformal) anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We  show that the spontaneous breaking of the BRST symmetry yields an Einstein action with its  own symmetry that is also anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is in agreement with previous work where it was  shown that when conformal symmetry is spontaneously broken there is conformal anomaly  matching in the unbroken and and broken phases [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In section 2, we obtain the on-shell Weyl invariance of pure  R2 gravity when a massless scalar field is included in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In section 3, we obtain the  BRST invariance of pure R2 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In section 4, we show that the spontaneous breaking  of the BRST symmetry yields an Einstein action and that there is a quantum anomaly in  both the unbroken and broken sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We conclude with section 5 where we summarize our  results, provide further physical insights and discuss directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We relegate  to Appendix A some technical details on the symmetry of the Einstein action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  2  Pure R2 gravity plus a massless scalar: from restricted to  on-shell Weyl invariance  The action of pure R2 gravity is given by  S =  � √−g d4x α R2  (1)  where R is the Ricci scalar and α a dimensionless constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This action is restricted Weyl  invariant i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' it is invariant under the Weyl transformation gµν → Ω2(x) gµν if the conformal  factor Ω(x) obeys the condition □Ω(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This invariance stems from the fact that R →  R/Ω2 when □Ω(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' As already mentioned, this implies that pure R2 gravity has a greater  symmetry than global scale symmetry (where Ω(x) would have to be a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  We now show that pure R2 gravity can be Weyl-invariant on-shell when a minimally coupled  real massless scalar field is added to the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Here, the condition □Ω(x) = 0 is not imposed  as an external condition but satisfied automatically by the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The action of  pure R2 gravity with a minimally coupled real massless scalar field φ is given by  Sa =  � √−g d4x  �  α R2 − 1  2 gµν ∂µφ ∂νφ  �  (2)  where φ(x) is a real scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Under the Weyl transformation gµν → e−2 ǫ φ gµν, where ǫ is  3  a real constant, the Ricci scalar transforms as  R → R e2ǫφ − 6 e3ǫ φ □(e−ǫ φ)  (3)  and √−g → e−4 ǫ φ √−g so that action (2) transforms to  Sb =  � √−g d4x  �  α  �  R2 − 12 R eǫ φ □(e−ǫ φ) + 36 e2ǫ φ (□(e−ǫ φ))2�  − 1  2 e−2 ǫ φ gµν ∂µφ ∂νφ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (4)  The equations of motion yield □(e−ǫ φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Therefore, when the equations of motion are  satisfied, the above action reduces to  Sc =  � √−g d4x  �  α R2 − 1  2 gµν ∂µψ ∂νψ  �  (5)  where ψ is a real massless scalar field (related to the old scalar φ via ψ = e−ǫ φ/ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Note that  the equation of motion for ψ is □ψ = 0 which is equivalent to □(e−ǫ φ) = 0 and consistent with  what we previously obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We therefore recover pure R2 gravity with a minimally coupled  real massless scalar field ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' What happened here is that the restricted Weyl condition □ Ω = 0  with Ω = e−ǫ φ did not have to be imposed as a separate condition because it was satisfied  automatically by the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In short, pure R2 gravity became Weyl invariant  on-shell in the presence of a massless scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' It passed from restricted Weyl invariance to  on-shell Weyl invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  3  BRST invariance of pure R2 gravity  Before discussing BRST invariance in the case of pure R2 gravity, let us first recall how BRST  invariance works in gauge theories in Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' For illustrative purposes, we will  consider the case of scalar QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The Abelian version of the Faddeev-Popov Lagrangian is then  given by [12]  L = −1  4 F 2  µν − (Dµφ∗  a)(Dµφa) − m2 φ∗  a φa − 1  2 ξ (∂µ Aµ)2 + ¯c □c  (6)  where c(x) and ¯c(x) are independent Grassmann-valued fields, φa are a set of complex scalar  fields and Dµ is the usual covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The gauge fixing term,  1  2 ξ(∂µ Aµ)2 breaks the  gauge symmetry since it is not invariant under the transformation Aµ → Aµ + 1  e ∂µf(x) where  f(x) is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, it has a residual symmetry: it is invariant if f(x)  obeys the condition □f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' As previously mentioned, this residual symmetry is the analog of  restricted Weyl symmetry in pure R2 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  4  The equation of motion for c(x) is □c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Consider the gauge transformation with f(x) =  θ c(x) for arbitrary Grassmann number θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Then, if the equation of motion for c is satisfied,  the scalar QED Lagrangian (6) is invariant under the following transformations  Aµ → Aµ + 1  e θ ∂µc(x)  φa(x) → eiθ c(x) φa(x) = φa(x) + iθ c(x)φa(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (7)  In other words, the equation □f = θ □c = 0 is automatically satisfied on-shell and does not  have to be imposed as a separate condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is similar to what we saw in the previous  section for pure R2 gravity which was invariant under gµν → Ω2 gµν with Ω = e−ǫφ when the  equations of motion were satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  If the equation of motion for c is not used, the only term in the Lagrangian (6) which is not  invariant under the transformation (7) is (∂µAµ)2 which transforms as  (∂µAµ)2 → (∂µAµ)2 + 2  e(∂µAµ)(θ□c)  (8)  where we used the fact that θ2 = 0 since θ is Grassmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Now, if under (7) we also have ¯c  transforming as  ¯c(x) → ¯c(x) − θ  e ξ (∂µAµ)  (9)  then the scalar QED Lagrangian (6) is invariant without having to use the equation of motion  for c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is BRST invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The crucial point is that under the BRST transformations  given by (7) and (9), the Lagrangian is invariant despite the presence of the gauge-fixing term  (∂µAµ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  We now turn to pure R2 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Consider the action  S =  �  d4x√−g (α R2 + ¯c □c)  (10)  where again c(x) and ¯c(x) are independent Grassmann-valued fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This action is not Weyl-  invariant i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' it is not invariant under the transformation gµν → Ω2(x)gµν where Ω(x) is an  arbitrary smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Consider now the Weyl transformation  gµν → e2 θ c(x)gµν = (1 + 2 θ c) gµν  (11)  where θ is again an arbitrary Grassmann number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Under this transformation we have  √−g α R2 → √−g (α R2 − 12 α R θ □c )  (12)  5  where the following transformations were used: √−g → (1+4 θ c) √−g and R → (1−2 θ c) R−  6 θ □c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Again, we used that θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Under the transformation (11), □c transforms as  □c → (1 − 2 θ c) □c  (13)  where gµν∂µc ∂νc = 0 was used (this stems from the fact that gµν is symmetric and c is  Grassmann).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The equation of motion for c is □c = 0 and we see from (12) that √−g α R2 is  Weyl invariant on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, we can dispense with the on-shell condition if we also allow  ¯c to transform as  ¯c → (1 − 2 θ c) ¯c + 12 α R θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (14)  We then obtain  √−g ¯c □c → √−g (¯c □c + 12 α R θ □c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (15)  The last term on the right hand side of (15) above cancels precisely the last term on the right  hand side of (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Therefore, the action (10) is invariant under the combined transformations  of (11) and (14) (which we refer to to as BRST transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  This is the BRST invariance  of pure R2 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Note that BRST invariance does not require any on-shell or restricted Weyl  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' It is a generalization of Weyl (conformal) invariance that is valid in the presence of  the Weyl-breaking R2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  Let us now take a closer look at what is common and what is different between the BRST  invariance of pure R2 gravity and the BRST invariance in the gauge theories of particle physics  (for concreteness and simplicity, we will consider scalar QED again but the main points apply  also to QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The BRST invariance in scalar QED can be viewed as a generalization of gauge  invariance in the presence of the gauge-fixing (and hence gauge-breaking) term (∂µ Aµ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The  are two points in common between the scalar QED and R2 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' First, the Ricci scalar R under  a Weyl transformation and the term ∂µ Aµ under a gauge transformation both pick up an extra  □Φ(x) term (where Φ(x) represents either a conformal factor Ω(x) in a Weyl transformation  or a function f(x) in a gauge transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Recall that in a BRST transformation, Φ(x) is  a product of a Grassmann number θ with a Grassmann field (the product yields a commuting  (bosonic) quantity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The second point in common is that R and ∂µ Aµ are both squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The  squaring yields a (□Φ(x))2 term which is zero since θ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The squaring still leaves one  extra □Φ(x) term and this is cancelled out in both cases via the transformation property of a  Grassmann field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' These two common points render the analogy between the two cases quite  strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, there is one important difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In scalar QED (and in QCD) , the BRST  transformations are associated with gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The BRST invariance of pure R2  gravity that we are considering here is not associated with gauge transformations (such as  diffeomorphisms) but with Weyl (local scale) transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We will see that this difference  plays an important role when the theory is quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  6  4  Spontaneous breaking of BRST symmetry  We now show that the BRST-invariant action  S =  �  d4x√−g (αR2 + ¯c □c)  (16)  is conformally equivalent to an action that involves the Einstein-Hilbert term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' this will involve  the spontaneous breaking of BRST symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The starting point is to introduce a auxiliary  field σ(x) to rewrite the above action into the equivalent form  S1 =  �  d4x√−g (−α(b σ + R)2 + αR2 + ¯c □c)  �  d4x√−g (−α b2 σ2 − 2 α b R σ + ¯c □c)  (17)  where b is a real non-zero constant with dimensions of mass squared and σ(x) is dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  Action (17) is equivalent to the original action (16) since adding the squared term in the first  line of (17) does not alter anything (classically, the equations of motion are unaffected and  quantum mechanically, the path integral over σ is a Gaussian which yields a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The  equivalent action (17) is also BRST invariant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' it is invariant under the following transforma-  tions:  gµν → (1 + 2 θ c) gµν  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  ¯c → (1 − 2 θ c) ¯c − 12 θ α b σ  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  σ → (1 − 2θ c) σ  (18)  where θ is again a Grassmann number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Note that the BRST invariance requires the auxiliary  field σ to transform besides the fields gµν and ¯c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We now perform the following conformal  (Weyl) transformation:  gµν → σ−1 gµν  ¯c → σ ¯c  (19)  which leads to √−g → σ−2 √−g and R → σ R − 6 σ3/2□(σ−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Under the above conformal  transformation, action (17) becomes  S2 =  �  d4x√−g (−α b2 − 2 α b R + 3α b  σ2 ∂µσ ∂µσ + ¯c □c − 1  σ ¯c ∂µc ∂µσ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (20)  The above action is no longer invariant under the BRST transformations given by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The  BRST symmetry has been spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The factor σ−1 appearing in the confor-  mal transformation (19) is valid only for non-zero σ so that the VEV (vacuum expectation  value) of the field σ must be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The VEV is therefore not invariant under the BRST  transformation σ → (1 − 2θ c) σ leading to the spontaneous breaking of the BRST symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  7  We can identify −2 α b R as an Einstein-Hilbert term if we equate −2 α b with  1  16π G where G  is Newton’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The constant term −α b2 can then be associated with a cosmological  constant Λ = −b/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Note that though −2 α b is positive, the constant b can be either positive  or negative (but not zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This implies that the cosmological constant can be either positive  corresponding to a de Sitter (dS) background or negative corresponding to an anti-de Sitter  (AdS) background but it cannot be identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We can then express (20) as the following  Einstein action,  SE =  �  d4x√−g  �  1  16π G(R − 2 Λ) + 3α b  σ2 ∂µσ ∂µσ + ¯c □c − 1  σ ¯c ∂µc ∂µσ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (21)  We have left the constant 3 α b in the action for simplicity but it is not an independent constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  it is equal to  −3  32 πG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We therefore obtain an Einstein-Hilbert action with non-zero cosmological  constant, a kinetic term for the scalar σ (which we will express in canonical form later) and  an interaction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  Recall that σ is non-zero so that divisions by σ pose no issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  It is  well-known that in spontaneously broken theories, the vacuum breaks the symmetry but it is  not actually broken in the Lagrangian but manifested or realized in a different way [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' It  can be directly verified (see Appendix A) that the Einstein action (21) is invariant under the  following transformations:  σ → (1 − 2θ c) σ , gµν → gµν and ¯c → ¯c − 12 θ α b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (22)  The BRST symmetry of action (17) manifests itself in the Einstein action (21) via its symmetry  under the above transformations (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We now show how transformation (22) stems from the  BRST transformations (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In the Einstein action and transformation (22) label the metric  and the barred Grassmann field with a subscript E i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  gµνE and ¯cE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  In action (17) and  transformation (18) we leave gµν and ¯c as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Then the conformal transformation (19) yields  gµνE = σ gµν and ¯cE = σ−1 ¯c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Under the BRST transformations (18) we obtain gµνE = σ gµν →  (1 − 2 θ c) σ (1 + 2 θ c) gµν = σ gµν = gµνE and ¯cE = σ−1 ¯c → (1 + 2 θ c) σ−1�  (1 − 2 θ c) ¯c −  12 θ α b σ  �  = σ−1¯c − 12 θ α b = ¯cE − 12 θ α b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We have therefore obtained the transformations  gµνE → gµνE and ¯cE → ¯cE − 12 θ α b which correspond to those in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Note that we used  σ → (1 − 2θ c) σ in (18) to derive this, so the transformation of σ is also part of (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  We can define a real massless scalar field ψ(x) =  √  −3α b ln σ(x) so that the kinetic term for  σ is expressed in canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The Einstein action (21) expressed in terms of the field ψ is  S =  �  d4x√−g  �  1  16π G(R − 2 Λ) − ∂µψ ∂µψ + ¯c □c −  1  √  −3α b ¯c ∂µc ∂µψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (23)  The massless scalar field ψ corresponds to the Nambu-Goldstone boson of the broken sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  Under transformation (22), the field ψ transforms as a shift ψ → ψ−  √  −3α b 2 θ c (whereas ¯c →  ¯c−12 θ α b and gµν → gµν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The above action (23) is invariant under those transformations (see  8  Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is in accord with what we expect from spontaneously broken theories: the  original symmetry in the Lagrangian manifests itself in the broken sector as a shift symmetry  of the Goldstone bosons [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='1  Quantum anomaly  We saw that the action (17) is BRST invariant under the following transformations:  gµν → (1 + 2 θ c) gµν , ¯c → (1 − 2 θ c) ¯c − 12 θ α b σ , σ → (1 − 2θ c) σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Each transformation  involves a Weyl transformation where the conformal factor is expressed in terms of of a prod-  uct of two Grassmann variables The BRST symmetry is therefore a generalization of Weyl  (conformal) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' After quantization, renormalization introduces a scale which breaks  the BRST symmetry since it automatically breaks Weyl symmetry (leading to the well-known  Weyl (conformal) anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' So the BRST symmetry of pure R2 gravity is anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is  in contrast to the BRST invariance of gauge theories like QCD which have no anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  After the BRST symmetry is spontaneously broken and we obtain the Einstein action (21),  we saw that the BRST symmetry manifests itself now in the Einstein action as a symmetry  under the transformations (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This symmetry is also anomalous since the transformation of  the field σ is a Weyl transformation and renormalization breaks this symmetry (leading again  to the Weyl (conformal anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Another way to see this is to note that the only fields that  transform in (22) are ¯c and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The transformation for ¯c is simply a constant shift so that its  path integral measure D¯c is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, σ undergoes a Weyl transformation and this  introduces a non-trivial Jacobian J (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' J ̸= 1) to the measure Dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Since the measure is not  invariant, this implies there is an anomaly [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' So the symmetry in the unbroken phase and  its associated symmetry in the broken phase are both anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Our finding here is in accord  with previous work that shows that when the Weyl or conformal symmetry is spontaneously  broken there is conformal anomaly matching between the unbroken and broken phases [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  5  Conclusion  In the last six years or so, we have kept discovering new aspects of pure R2 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' A non-  exhaustive list includes its unitarity among quadratic gravity theories [4], its conformal equiv-  alence to Einstein gravity with non-zero cosmological constant and massless scalar field [1–5],  its restricted Weyl symmetry [7,10,11], its spontaneous symmetry breaking to Einstein grav-  ity [3,5] and the lack of a propagating graviton when the theory is linearized about a Minkowski  background [4] (where there is no Einstein equivalence since the cosmological constant is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  In this paper, we have gained further insights into this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We saw that pure R2 gravity  has an analog with the gauge-fixing term (∂µAµ)2 in gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' R2 is not invariant under  9  the Weyl transformation gµν → Ω2(x) gµν just like (∂µ Aµ)2 is not invariant under the gauge  transformation Aµ → Aµ + 1  e ∂µf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, each have a residual symmetry (when □Ω = 0  is satisfied in the gravity case and □f = 0 is satisfied in the gauge case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This analogy opened  the door towards enlarging the symmetry of pure R2 gravity to include BRST symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  We first showed that when a massless scalars field was included in the pure R2 action, the  condition □Ω = 0 could be met automatically when the equations of motion were satisfied  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' we went from restricted Weyl to on-shell Weyl invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Finally, we obtained the BRST  invariance of pure R2 gravity where no restricted Weyl or on-shell condition is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The  BRST transformations involve Weyl transformations where the conformal factor is composed of  products of Grassmann variables (the conformal factor itself is commutative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The important  point is that the BRST invariance exists despite the Weyl-breaking R2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  There is one important difference between the BRST symmetry in gauge theories like QCD  and the BRST symmetry that we have considered here for pure R2 gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Gauge invari-  ance in particle physics is preserved after quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The BRST invariance of QCD is a  generalization of gauge invariance so that it is also preserved after quantization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' there is no  anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' In contrast to gauge symmetry, global scale or Weyl (local scale) symmetry is broken  after quantization since renormalization introduces a scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The BRST symmetry of pure R2  gravity is a generalization of Weyl (conformal) symmetry so that it is also broken after quan-  tization leading to the well-known Weyl (conformal) anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' After the spontaneous breaking  of the BRST symnmetry, we obtained an Einstein action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We showed that this action has  its own symmetry and that it is also anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This is in accord with previous work that  shows that when the Weyl (conformal) symmetry is spontaneously broken there is conformal  anomaly matching between the unbroken and broken sectors [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  The focus of this paper was pure R2 gravity because of its many special and attractive fea-  tures that we previously mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' All other quadratic gravity theories (like Weyl-squared,  Riemann-squared, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' ), apart from boundary terms, can be expressed as a linear combina-  tion of R2 and RµνRµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The latter term, the square of the Ricci tensor, appears in quantum  corrections to General Relativity (GR) and even though it does not constitute a valid UV  completion of GR due to its non-unitarity (yields a massive spin two ghost [4, 20]), it still  makes a well-known calculable short-range correction to the Newtonian potential [12,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Like  R2, the term RµνRµν is not Weyl-invariant so it would be of interest to see if it can be BRST  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' It is not in the form of a scalar squared like (∂µAµ)2 or R2, so one may be inclined  to think that the BRST formalism would not apply here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, like pure R2, it was shown  in [7] that RµνRµν is restricted Weyl invariant (up to a boundary term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' This suggests that  the procedure used to establish the BRST invariance of pure R2 gravity might in the end also  work for this quadratic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' It would therefore be worthwhile and interesting to investigate  this further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  10  Acknowledgments  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' acknowledges support from a discovery grant of the National Science and Engineering  Research Council of Canada (NSERC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  A  Symmetry of Einsten Action  In this appendix we show that the Einstein action (21) given by  SE =  �  d4x√−g  �  1  16π G(R − 2 Λ) + 3α b  σ2 ∂µσ ∂µσ + ¯c □c − 1  σ ¯c ∂µc ∂µσ)  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='1)  is invariant under the transformations (22) given by  σ → (1 − 2θ c) σ , gµν → gµν and ¯c → ¯c − 12 θ α b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='2)  Under the above transformation, the metric gµν does not change so that √−g as well as the  term √−g  1  16π G(R − 2 Λ) does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  The other terms in the above Einstein action  transform as  3α b  σ2 ∂µσ ∂µσ → 3α b  σ2 ∂µσ ∂µσ − 12 θ α b  σ  ∂µc ∂µσ  − 1  σ ¯c ∂µc ∂µσ) → − 1  σ ¯c ∂µc ∂µσ + 12 θ α b  σ  ∂µc ∂µσ  ¯c □c → ¯c □c − 12 θ α b □c  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='3)  where we used that θ2 = 0 (since θ is a Grassmann number) and that gµν ∂µc ∂νc = 0 since gµν  is symmetric and c(x) and its derivatives are Grassmann fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We see that the extra term  − 12 θ α b  σ  ∂µc ∂µσ in the first line of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='3) is canceled exactly by the extra term in the second  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' The extra term in the third line, −12 θ α b □c, where −12 θ α b is a constant, does not  cancel out with any other extra term in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, √−g □c is a total derivative that  yields an inconsequential boundary term in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We have therefore shown that action  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='1) is invariant under transformations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  We saw in section 4 that the Einstein action (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='1) can be expressed in terms of a real massless  scalar field ψ(x) =  √  −3α b ln σ(x) as action (23):  S =  �  d4x√−g  �  1  16π G(R − 2 Λ) − ∂µψ ∂µψ + ¯c □c −  1  √  −3α b ¯c ∂µc ∂µψ  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='4)  where ψ was identified as the Nambu-Goldstone boson of the broken sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We stated in  section 4 that the action (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='4) was invariant under the following transformations:  ψ → ψ −  √  −3α b 2 θ c , ¯c → ¯c − 12 θ α b and gµν → gµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='5)  11  We now verify this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Under (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='5) the last three terms in action (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='4) transform as:  − ∂µψ ∂µψ → −∂µψ ∂µψ + 4 θ  √  −3 α b ∂µψ ∂µc  −  1  √  −3α b ¯c ∂µc ∂µψ → −  1  √  −3α b ¯c ∂µc ∂µψ − 4 θ  √  −3 α b ∂µψ ∂µc  ¯c □c → ¯c □c − 12 θ α b □c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='6)  We see that the extra term +4 θ  √  −3 α b ∂µψ ∂µc in the first line above is cancelled by the  extra term on the second line which is equal to its negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  The only extra term that is  not cancelled is the term −12 θ α b □c appearing in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' However, √−g □c is a total  derivative which yields a boundary term with no physical consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' We have therefore  verified that the Einstein action (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='4) is indeed invariant under the transformations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' D 16,  953 (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Donoghue, Leading quantum correction to the Newtonian potential, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  72, 2996 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}
+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFAT4oBgHgl3EQfpB3g/content/2301.08638v1.pdf'}

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+Wasserstein Gradient Flows of the Discrepancy
+with Distance Kernel on the Line⋆
+Johannes Hertrich, Robert Beinert, Manuel Gräf, and Gabriele Steidl
+TU Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany
+{hertrich,beinert, graef,steidl}@math.tu-berlin.de
+https://tu.berlin/imageanalysis/
+Abstract. This paper provides results on Wasserstein gradient flows between measures on
+the real line. Utilizing the isometric embedding of the Wasserstein space P2(R) into the Hilbert
+space L2((0, 1)), Wasserstein gradient flows of functionals on P2(R) can be characterized as
+subgradient flows of associated functionals on L2((0, 1)). For the maximum mean discrepancy
+functional Fν := D2
+K(·, ν) with the non-smooth negative distance kernel K(x, y) = −|x − y|,
+we deduce a formula for the associated functional. This functional appears to be convex,
+and we show that Fν is convex along (generalized) geodesics. For the Dirac measure ν = δq,
+q ∈ R as end point of the flow, this enables us to determine the Wasserstein gradient flows
+analytically. Various examples of Wasserstein gradient flows are given for illustration.
+Keywords: Maximum Mean Discrepancy · Wasserstein gradient flows · Riesz kernel.
+1
+Introduction
+Gradient flows provide a powerful tool for computing the minimizers of modeling functionals in
+certain applications. In particular, gradient flows on the Wasserstein space are an interesting field
+of research that combines optimization with (stochastic) dynamical systems and differential geom-
+etry. For a good overview on the theory, we refer to the books of Ambrosio, Gigli and Savaré [3],
+and Santambrogio [31]. Besides Wasserstein gradient flows of the Kullback–Leibler (KL) functional
+KL(·, ν) and the associated Fokker–Planck equation related to the overdamped Langevin dynamics,
+which were extensively examined in the literature, see, e.g., [19,26,28], flows of maximum mean
+discrepancy (MMD) functionals Fν := D2
+K(·, ν) became popular in machine learning [4] and image
+processing [14]. On the other hand, MMDs were used as loss functions in generative adversarial
+networks [6,13,22]. Wasserstein gradient flows of MMDs are not restricted to absolutely continuous
+measures and have a rich structure depending on the kernel. So the authors of [4] showed that for
+smooth kernels K, particle flows are indeed Wasserstein gradient flows meaning that Wasserstein
+flows starting at an empirical measure remain empirical measures and coincide with usual gradi-
+ent descent flows in Rd. The situation changes for non-smooth kernels like the negative distance,
+where empirical measures can become absolutely continuous ones and conversely, i.e. particles may
+explode. The concrete behavior of the flow depends also on the dimension, see [11,12,17,18]. The
+crucial part is the treatment of the so-called interaction energy within the discrepancy, which is
+repulsive and responsible for the proper spread of the measure. This nicely links to another field of
+mathematics, namely potential theory [21,30].
+⋆ Supported by the German Research Foundation (DFG) [grant numbers STE571/14-1, STE 571/16-1]
+and the Federal Ministry of Education and Research (BMBF, Germany) [grant number 13N15754].
+arXiv:2301.04441v1  [math.OC]  11 Jan 2023
+
+2
+J. Hertrich et al.
+In this paper, we are just concerned with Wasserstein gradient flows on the real line. Optimal
+transport techniques that reduce the original transport to those on the line were successfully used
+in several applications [1,5,9,10,20,27]. When working on R, we can exploit quantile functions of
+measures to embed the Wasserstein space P2(R) into the Hilbert space of (equivalence classes) of
+square integrable functions L2((0, 1)). Then, instead of dealing with functionals on P2(R), we can
+just work with associated functionals which are uniquely defined on a cone of L2((0, 1)). If the asso-
+ciated functional is convex, we will see that the original one is convex along (generalized) geodesics,
+which is a crucial property for the uniqueness of the Wasserstein gradient flow. Furthermore, we
+can characterize Wasserstein gradient flows using regular subdifferentials in L2((0, 1)). Note that
+the special case of Wasserstein gradient flows of the interaction energy was already considered in
+[7]. We will have a special look at the Wasserstein gradient flow of Fδq := D2
+K(·, δq) for the negative
+distance kernel, i.e. flows ending in δq. We will deduce an analytic formula for this flow and provide
+several examples to illustrate its behavior.
+Outline of the paper. In Section 2, we recall the basic notation on Wasserstein gradient flows in d
+dimensions. Then, in Section 3, we show how these flows can be simpler treated as gradient descent
+flows of an associated function on the Hilbert space L2((0, 1)). MMDs are introduced in Section 4.
+Then, in Section 5, we restrict our attention again to the real line and show how the associated
+functional looks for the MMD with negative distance kernel. In particular, this functional is convex.
+For the Dirac measure ν = δq, q ∈ R, we give an explicit formula for the Wasserstein gradient flow
+of the MMD functional. Examples illustrating the behavior of the Wasserstein flows are provided
+in Section 6. Finally, conclusions are drawn in Section 7.
+2
+Wasserstein Gradient Flows
+Let M(Rd) denote the space of σ-additive, signed measures and P(Rd) the set of probability
+measures. For µ ∈ M(Rd) and measurable T : Rd → Rn, the push-forward of µ via T is given by
+T#µ := µ ◦ T −1. We consider the Wasserstein space P2(Rd) := {µ ∈ P(Rd): �
+Rd ∥x∥2
+2 dµ(x) < ∞}
+equipped with the Wasserstein distance W2 : P2(Rd) × P2(Rd) → [0, ∞),
+W 2
+2 (µ, ν) :=
+min
+π∈Γ (µ,ν)
+�
+Rd×Rd ∥x − y∥2
+2 dπ(x, y),
+µ, ν ∈ P2(Rd),
+(1)
+where Γ(µ, ν) := {π ∈ P2(Rd × Rd) : (π1)#π = µ, (π2)#π = ν} and πi(x) := xi, i = 1, 2 for
+x = (x1, x2). The set of optimal transport plans π realizing the minimum in (1) is denoted by
+Γ opt(µ, ν). A curve γ : I → P2(Rd) on an interval I ⊂ R, is called a geodesic if there exists a
+constant C ≥ 0 such that
+W2(γ(t1), γ(t2)) = C|t2 − t1|,
+for all t1, t2 ∈ I.
+The Wasserstein space is a geodesic space, meaning that any two measures µ, ν ∈ P2(Rd) can be
+connected by a geodesic. The regular tangent space at µ ∈ P2(Rd) is given by
+TµP2(Rd) :=
+�
+λ(T − Id) : (Id, T)#µ ∈ Γ opt(µ, T#µ), λ > 0
+�L2,µ.
+Here L2,µ denotes the Bochner space of (equivalence classes of) functions ξ : Rd → Rd with
+finite ∥ξ∥2
+L2,µ :=
+�
+Rd ∥ξ(x)∥2
+2 dµ(x) < ∞. Note that TµP2(Rd) is not a “classical” tangent space, in
+
+Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line
+3
+particular it is an infinite dimensional subspace of L2,µ if µ is absolutely continuous and just Rd
+if µ = δx, x ∈ Rd. In particular, this means that the Wasserstein space has only a “manifold-like”
+structure.
+For λ ∈ R, a function F : P2(Rd) → (−∞, +∞] is called λ-convex along geodesics if, for every
+µ, ν ∈ dom F := {µ ∈ P2(Rd) : F(µ) < ∞}, there exists at least one geodesic γ : [0, 1] → P2(Rd)
+between µ and ν such that
+F(γ(t)) ≤ (1 − t) F(µ) + t F(ν) − λ
+2 t(1 − t) W 2
+2 (µ, ν),
+t ∈ [0, 1].
+In the case λ = 0, we just speak about convex functions. For a proper and lower semi-continuous
+(lsc) function F : P2(Rd) → (−∞, ∞] and µ ∈ P2(Rd), the reduced Fréchet subdifferential at µ is
+defined as
+∂F(µ) :=
+�
+ξ ∈ L2,µ : F(ν) − F(µ) ≥
+inf
+π∈Γ opt(µ,ν)
+�
+R2d
+⟨ξ(x), y − x⟩ dπ(x, y) + o(W2(µ, ν)) ∀ν ∈ P2(Rd)
+�
+. (2)
+A curve γ : I → P2(Rd) is absolutely continuous, if there exists a Borel velocity field vt : Rd → Rd
+with
+�
+I ∥vt∥L2,γ(t) dt < +∞ such that
+∂tγ(t) + ∇x · (vt γ(t)) = 0
+(3)
+on I × Rd in the distributive sense, i.e., for all ϕ ∈ C∞
+c (I × Rd) it holds
+�
+I
+�
+Rd ∂tϕ(t, x) + vt(x) · ∇x ϕ(t, x) dγ(t) dt = 0.
+A locally absolutely continuous curve γ : (0, +∞) → P2(Rd) with velocity field vt ∈ Tγ(t)P2(Rd) is
+called a Wasserstein gradient flow with respect to F : P2(Rd) → (−∞, +∞] if
+vt ∈ −∂F(γ(t)),
+for a.e. t > 0.
+(4)
+3
+Wasserstein Gradient Flows on the Line
+Now we restrict our attention to d = 1, i.e., we work on the real line. We will see that the above
+notation simplifies since there is an isometric embedding of P2(R) into L2((0, 1)). To this end, we
+consider the cumulative distribution function Rµ : R → [0, 1] of µ ∈ P2(R), which is defined by
+Rµ(x) := µ((−∞, x]), x ∈ R. It is non-decreasing and right-continuous with limx→−∞ Rµ(x) = 0 as
+well as limx→∞ Rµ(x) = 1. The quantile function Qµ : (0, 1) → R is the generalized inverse of Rµ
+given by
+Qµ(p) := min{x ∈ R: Rµ(x) ≥ p},
+p ∈ (0, 1).
+It is non-decreasing and left-continuous. The quantile functions form a convex cone C((0, 1)) :=
+{Q ∈ L2((0, 1)) : Q nondecreasing} in L2((0, 1)). Note that both the distribution and quantile
+functions are continuous except for at most countably many jumps. For a good overview see [29,
+§ 1.1]. By the following theorem, the mapping µ �→ Qµ is an isometric embedding of P2(R) into
+L2((0, 1)).
+Theorem 1 ([32, Thm 2.18]). For µ, ν ∈ P2(R), the quantile function Qµ ∈ C((0, 1)) satisfies
+µ = (Qµ)#λ(0,1) and
+W 2
+2 (µ, ν) =
+� 1
+0
+|Qµ(s) − Qν(s)|2ds.
+
+4
+J. Hertrich et al.
+Next we will see that instead of working with functionals F : P2(R) → (−∞, +∞], we can just
+deal with associated functionals F: L2((0, 1)) → (−∞, ∞] fulfilling F(Qµ) := F(µ). Note that F
+is defined in this way only on C((0, 1)), and there exist several continuous extensions to the whole
+linear space L2((0, 1)). Instead of the extended Fréchet subdifferential (2), we will use the regular
+subdifferential in L2((0, 1)) defined by
+∂G(f) :=
+�
+h ∈ L2((0, 1)) : G(g) ≥ G(f) + ⟨h, g − f⟩ + o(∥g − f∥L2) ∀g ∈ L2((0, 1))
+�
+.
+The following theorem characterizes Wasserstein gradient flows by this regular subdifferential and
+states a convexity relation between F : P2(R) → (−∞, +∞] and the associated functional F.
+Theorem 2. i) Let γ : (0, ∞) → P2(R) be a locally absolutely continuous curve and F: L2((0, 1)) →
+(−∞, ∞] such that the pointwise derivative ∂tQγ(t) exists and fulfills the L2 subgradient equation
+∂tQγ(t) ∈ −∂F(Qγ(t)),
+for almost every t ∈ (0, +∞).
+Then γ is a Wasserstein gradient flow with respect to the functional F : P2(R) → (−∞, +∞] defined
+by F(µ) := F(Qµ).
+ii) If F : C((0, 1)) → (−∞, ∞] is convex, then F(µ) := F(Qµ) is convex along geodesics.
+Proof. i) Since γ is (locally) absolute continuous, the velocity field vt from (3) fulfills by [3,
+Prop 8.4.6] for almost every t ∈ (0, ∞) the relation
+0 = lim
+h→0
+W2(γ(t + h), (Id + hvt)#γ(t))
+|h|
+= lim
+h→0
+W2((Qγ(t+h))#λ(0,1),
+�
+Qγ(t) + h(vt ◦ Qγ(t))
+�
+#λ(0,1))
+|h|
+= lim
+h→0
+���Qγ(t+h) − Qγ(t)
+h
+− vt ◦ Qγ(t)
+���
+L2 = ∥∂tQγ(t) − vt ◦ Qγ(t)∥L2.
+Thus, by assumption, vt ◦ Qγ(t) ∈ −∂F(Qγ(t)) a.e. In particular, for any µ ∈ P2(R), we obtain
+0 ≤ F(Qµ) − F(Qγ(t)) +
+� 1
+0
+vt(Qγ(t)(s)) (Qµ(s) − Qγ(t)(s)) ds + o(∥Qµ − Qγ(t)∥L2)
+= F(µ) − F(γ(t)) +
+�
+R×R
+vt(x) (y − x) dπ(x, y) + o
+�
+W2(µ, γ(t))
+�
+,
+where π := (Qγ(t), Qµ)#λ(0,1). Since π the unique optimal transport plan between γ(t) and µ, this
+yields by (2) that vt ∈ −∂F(γ(t)) showing the assertion by (4).
+ii) Let F: L2((0, 1)) → R be convex. For any geodesic γ : [0, 1] → P2(R), since µ �→ Qµ is an
+isometry, the curve t �→ Qγ(t) is a geodesic in L2((0, 1)) too. Since L2((0, 1)) is a linear space, the
+convexity of F: L2((0, 1)) → R yields that t �→ F(Qγ(t)) = F(γ(t)) is convex. Thus, F is convex
+along γ.
+⊓⊔
+Remark 1. If F : P2(R) → (−∞, +∞] is proper, lsc, coercive and λ-convex along so-called general-
+ized geodesics, then the Wasserstein gradient flow starting at any µ0 ∈ dom F is uniquely determined
+and is the uniform limit of the miminizing movement scheme of Jordan, Kinderlehrer and Otto [19]
+when the time step size goes to zero, see [3, Thm 11.2.1]. In R, but not in higher dimensions,
+λ-convex functions along geodesics fulfill also the stronger property that they are λ-convex along
+generalized geodesics, see [18].
+
+Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line
+5
+4
+Discrepancies
+We consider symmetric and conditionally positive definite kernels K : Rd × Rd → R of order one,
+i.e., for any n ∈ N, any pairwise different points x1, . . . , xn ∈ Rd and any a1, . . . , an ∈ R with
+�n
+i=1 ai = 0 the relation �n
+i,j=1 aiajK(xi, xj) ≥ 0 is satisfied. Typical examples are Riesz kernels
+K(x, y) := −∥x − y∥r,
+r ∈ (0, 2),
+where we have strict inequality except for all aj, j = 1, . . . , n being zero. The maximum mean
+discrepancy (MMD) D2
+K : P(Rd) × P(Rd) → R between two measures µ, ν ∈ P(Rd) is defined by
+D2
+K(µ, ν) := EK(µ − ν)
+with the so-called K-energy on signed measures
+EK(σ) := 1
+2
+�
+Rd
+�
+Rd K(x, y) dσ(x)dσ(y),
+σ ∈ M(Rd).
+The relation between discrepancies and Wasserstein distances is discussed in [15,24]. For fixed
+ν ∈ P(Rd), the MMD can be decomposed as
+Fν(µ) = D2
+K(µ, ν) = EK(µ) + VK,ν(µ) + EK(ν)
+� �� �
+const.
+with the interaction energy on probability measures
+EK(µ) = 1
+2
+�
+Rd
+�
+Rd K(x, y) dµ(x)dµ(y),
+µ ∈ P2(Rd)
+and the potential energy of µ with respect to the potential of ν,
+VK,ν(µ) :=
+�
+Rd VK,ν(y)dµ(x),
+VK,ν(x) := −
+�
+Rd K(x, y)dν(y).
+In dimensions d ≥ 2 neither EK nor D2
+K with the Riesz kernel are λ-convex along geodesics, see
+[18], so that certain properties of Wasserstein gradient flows do not apply. We will see that this is
+different on the real line.
+5
+MMD Flows on the Line
+In the rest of this paper, we restrict our attention to d = 1 and negative distance K(x, y) =
+−|x − y|, i.e. to Riesz kernels with r = 1. For fixed ν ∈ P2(R), we consider the MMD functional
+Fν := D2
+K(·, ν). Note that the unique minimizer of this functional is given by µ = ν.
+Lemma 1. Let Fν := D2
+K(·, ν) with the negative distance kernel. Then the convex functional
+Fν : L2((0, 1)) → R defined by
+Fν(f) :=
+� 1
+0
+�
+(1 − 2s)(f(s) + Qν(s)) +
+� 1
+0
+|f(s) − Qν(t)| dt
+�
+ds.
+(5)
+fulfills Fν(Qµ) = Fν(µ) for all µ ∈ P2(R). In particular, Fν is convex along (generalized) geodesics
+and there exists a unique Wasserstein gradient flow.
+
+6
+J. Hertrich et al.
+Proof. We reformulate Fν as
+Fν(µ) = −1
+2
+�
+R×R
+|x − y|(dµ(x) − dν(x))(dµ(y) − dν(y))
+= −1
+2
+� 1
+0
+� 1
+0
+|Qµ(s) − Qµ(t)| − 2|Qµ(s) − Qν(t)| + |Qν(s) − Qν(t)| ds dt
+=
+� 1
+0
+� 1
+t
+Qµ(t) − Qµ(s) + Qν(t) − Qν(s) ds dt +
+� 1
+0
+� 1
+0
+|Qµ(s) − Qν(t)| ds dt
+=
+� 1
+0
+� 1
+t
+Qµ(t) + Qν(t) ds dt −
+� 1
+0
+� s
+0
+Qµ(s) + Qν(s) dt ds +
+� 1
+0
+� 1
+0
+|Qµ(s) − Qν(t)| ds dt
+=
+� 1
+0
+�
+(1 − 2s)(Qµ(s) + Qν(s)) +
+� 1
+0
+|Qµ(s) − Qν(t)| dt
+�
+ds,
+which yields the first claim. The second one follows by Theorem 2ii) and Remark 1.
+⊓⊔
+Note that the lemma cannot immediately be generalized to Riesz kernels with r = (1, 2).
+Finally, we derive for the special choice ν = δq in D2
+K(·, ν) an analytic formula for its Wasserstein
+gradient flow.
+Proposition 1. Let Fδq := D2
+K(·, δq) with the negative distance kernel. Then the unique Wasser-
+stein gradient flow of Fδq starting at µ0 = γ(0) ∈ P2(R) is γ(t) = (gt)#λ(0,1), where the function
+gt : (0, 1) → R is given by
+gt(s) :=
+�
+�
+�
+�
+�
+�
+�
+min{Qµ0(s) + 2st, q},
+Qµ0(s) < q,
+q,
+Qµ0(s) = q,
+max{Qµ0(s) + 2st − 2t, q},
+Qµ0(s) > q.
+(6)
+Proof. First, note that gt ∈ C((0, 1)) such that it holds gt = Qγ(t). Since Qδq ≡ q, the subdifferential
+of Fδq in (5) at gt consists of all functions
+h(s) =
+�
+�
+�
+�
+�
+�
+�
+−2s,
+Qµ0(s) < q and t < q−Qµ0(s)
+2s
+,
+2 − 2s,
+Qµ0(s) > q and t < Qµ0(s)−q
+2−2s
+,
+1 − 2s + n(s),
+otherwise,
+with −1 ≤ n(s) ≤ 1 for s ∈ (0, 1). On the other hand, the pointwise derivative of gt in (6) can be
+written as
+∂tgt(s) =
+�
+�
+�
+�
+�
+�
+�
+2s,
+Qµ0(s) < q and t < q−Qµ0(s)
+2s
+,
+2s − 2,
+Qµ0(s) > q and t < Qµ0(s)−q
+2−2s
+,
+0,
+otherwise,
+such that we obtain ∂tQγ(t) = ∂tgt ∈ −∂Fν(gt) = −∂Fν(Qγ(t)). Thus, by Lemma 1 and Theorem 2,
+we obtain that γ is a Wasserstein gradient flow. It is unique since Fν is convex along geodesics by
+Theorem 2.ii, Lemma 1 and Remark 1.
+⊓⊔
+
+Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line
+7
+6
+Intuitive Examples
+Finally, we provide some intuitive examples of Wasserstein gradient flows of Fν := D2
+K(·, ν) with
+the negative distance kernel.
+6.1
+Flow between Dirac Measures
+We consider the flow of Fδ0 starting at the initial measure γ(0) = µ0 := δ−1. Due to Qδ0 ≡ 0,
+Proposition 1 yields the gradient flow γ(t) := (Qt)#λ(0,1) given by
+γ(t) =
+�
+�
+�
+�
+�
+�
+�
+δ−1,
+t = 0,
+1
+2tλ[−1,−1+2t],
+0 ≤ t ≤ 1
+2,
+1
+2tλ[−1,0] +
+�
+1 − 1
+2t
+�
+δ0,
+1
+2 < t.
+For t ∈ (0, 1
+2], the initial Dirac measure becomes a uniform measure with increasing support, and
+for t ∈ ( 1
+2, 1) it is the convex combination of a uniform measure and δ0. A visualization of the flow
+is given in Figure 1.
+□
+Fig. 1: Visualization of the Wasserstein gradient flow of Fδ0 from δ−1 to δ0. At various times t, the
+absolute continuous part is visualized by its density in blue (area equals mass) and the atomic part
+by the red dotted vertical line (height equals mass). The atomic part at the end point x = 0 starts
+to grow at time t = 1
+2, where the support of the density touches this point for the first time.
+6.2
+Flow on Restricted Sets
+Next, we are interested in the Wasserstein gradient flows on the subsets Si, i = 1, 2, given by
+(i) S1 := {δx : x ∈ R},
+(ii) S2 := {µm,σ =
+1
+2
+√
+3σλ[m−
+√
+3σ,m+
+√
+3σ] : m ∈ R, σ ∈ R≥0}.
+Note that S2 is a special instance of sets of scaled and translated measures µ ∈ P2(R) defined by
+{Ta,b#µ : a ∈ R≥0, b ∈ R}, where Ta,b(x) := ax + b. As mentioned in [16] the Wasserstein distance
+between measures µ1, µ2 from such sets has been already known to Fréchet:
+W 2
+2 (µ1, µ2) = |m1 − m2|2 + |σ1 − σ2|2,
+
+t= 0.00
+t= 0.25
+t= 0.29
+t= 0.33
+t=0.40
+t= 0.50
+t= 0.67
+t= 1.00
+t= 2.00
+t= 8
+2.0
+1.5
+1.0 -
+0.5 
+0.0
+1
+0
+0
+0
+0
+0
+0
+08
+J. Hertrich et al.
+where mi and σi are the mean value and standard deviation of µi, i = 1, 2. This provides an
+isometric embedding of R×R≥0 into P2(R). The boundary of S2 is the set of Dirac measures S1 and
+is isometric to R. The sets are convex in the sense that for µ, ν ∈ Si all geodesics γ : [0, 1] → P(R)
+with γ(0) = µ and γ(1) = ν are in Si, i ∈ {1, 2}. For i = 1, 2, we consider
+Fi,ν(µ) :=
+�
+Fν
+µ ∈ Si,
++∞
+otherwise.
+Due to the convexity of Fν along geodesics and the convexity of the sets Si, we obtain that the
+functions Fi,ν are convex along geodesics.
+Flows of F1,ν We use the notation fx ≡ x for the constant function on (0, 1) with value x. It is
+straightforward to check that the function F: L2((0, 1)) → (−∞, ∞] given by
+F(f) =
+�
+F(x),
+if f = fx for some x ∈ R,
++∞,
+otherwise,
+with
+F(x) :=
+�
+R
+|x − y| dν(y) − 1
+2
+�
+R×R
+|y − z| dν(y)dν(z)
+fulfills F(Qµ) = F1,ν(µ). In the following, we aim to find x: [0, ∞) → R satisfying
+˙x(t) = −∂F(x(t)).
+Since the set {Qµ : µ ∈ S1} is a one-dimensional linear subspace of L2((0, 1)) spanned by the
+constant one-function f1, this yields fx(t) ∈ −∂F(fx(t)) such that the Wasserstein gradient flow is
+by Theorem 2 given by γ(t) = (fx(t))#λ(0,1) = δx(t).
+In the special case ν = δq for some q ∈ R, we have
+F(x) = |x − q|,
+∂F(x) =
+�
+�
+�
+�
+�
+�
+�
+{−1},
+x < q,
+[−1, 1],
+x = q,
+{1},
+x > q.
+Therefore, the Wasserstein gradient flow for x(0) = x0 ̸= 0 is given by
+γ(t) = δx(t),
+with
+x(t) =
+�
+x0 + t,
+x0 < q,
+x0 − t,
+x0 > q. ,
+0 ≤ t < |x0 − q|
+and γ(t) = δq for t ≥ |x0 − q|.
+For ν = 1
+2λ[−1,1] the gradient flow starting at x0 ∈ [−1, 1] is
+x(t) = x0e−t,
+t ≥ 0,
+and converges to the midpoint of the interval for t → ∞. If it starts at x0 ∈ R \ [−1, 1] the gradient
+flow is
+x(t) =
+�
+x0 + t,
+x0 < −1,
+x0 − t,
+x0 > 1.
+,
+0 ≤ t ≤ min |x0 − 1|, |x0 + 1|,
+where it reaches the nearest interval end point in finite time. In Figure 2, we plotted the x(t) for
+different initial values x(0).
+
+Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line
+9
+Fig. 2: Wasserstein gradient flow of F1,ν for ν = δ0 (left) and ν = 1
+2λ[−1,1] (right) from various
+initial points δx, x ∈ [−2, 2]. The support of the right measure ν is depicted by the blue region. The
+examples show that gradient flows may reach the optimal points in finite or infinite time.
+Flows of F2,ν We observe that Qµm,σ = fm,σ, where fm,σ(x) = m + 2
+√
+3σ(x − 1
+2). By Lemma 1
+we obtain that the function F: L2((0, 1)) → (−∞, ∞] given by
+F(f) =
+�
+F(m, σ),
+if f = fm,σ for (m, σ) ∈ R × R≥0,
++∞,
+otherwise,
+fulfills F(Qµ) = F2,ν(µ), where
+F(m, σ) :=
+�
+(0,1)
+(1 − 2s)(fm,σ(s) + Qν(s))ds +
+�
+(0,1)2 |fm,σ(s) − Qν(t)|dtds,
+The set {fm,σ : m, σ ∈ R} is a two dimensional linear subspace of L2((0, 1)) with orthonormal basis
+{f1,0, f0,1}. We aim to compute m: [0, ∞) → R and σ: [0, ∞) → R≥0 with
+( ˙m(t), ˙σ(t)) = −∂F(m(t), σ(t)),
+t ∈ I ⊂ R,
+(7)
+because this yields fm(t),σ(t) ∈ −∂F(fm(t),σ(t)) such that γ(t) = (fm(t),σ(t))#λ(0,1) = µm,σ is by
+Theorem 2 the Wasserstein gradient flow.
+In the following, we consider the special case ν = δ0 = µ0,0. Then, the function F reduces to
+F(m, σ) =
+�
+R
+(1 − 2s)(m + 2
+√
+3σ(s − 1
+2)) + |m + 2
+√
+3σ(s − 1
+2)|ds
+= − σ
+√
+3 +
+�
+�
+�
+|m|,
+if |m| ≥
+√
+3σ,
+m2+3σ2
+2
+√
+3σ2
+if |m| <
+√
+3σ,
+and the subdifferential is given by
+∂F(m, σ) =
+�
+�
+�
+sgn(m) × {− 1
+√
+3},
+if |m| ≥
+√
+3σ,
+{(
+m
+√
+3σ2 , −m2
+√
+3σ3 −
+1
+√
+3)},
+if |m| <
+√
+3σ,
+sgn(m) =
+�
+{ |m|
+m },
+if m ̸= 0,
+[−1, 1],
+if m = 0.
+We observe that F is differentiable for σ > 0. Thus, for any initial intial value (m(0), σ(0)) =
+(m0, σ0), we can compute the trajectory (m(t), σ(t)) solving (7) using an ODE solver. In Figure 3
+
+2.0
+1.5
+1.0
+0.5
+X
+0.0
+-0.5
+1.0
+1.5
+-2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+t2.0
+1.5
+1.0
+0.5
+X
+0.0
+-0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+t10
+J. Hertrich et al.
+(left), we plotted the level sets of the function F(m, σ) as well as the solution trajectory (m(t), σ(t))
+for different initial values (m(0), σ(0)). For (m(0), σ(0)) = (−1, 0), the resulting flow is illustrated
+in Figure 3, right.
+Fig. 3: Wasserstein gradient flow F2,δ0 from (m(0), σ(0)) to δ0 (left) and from δ−1 to δ0 (right). In
+contrast Figure 1 it is a uniform measure for all t ∈ (0, 1).
+Flows for a Smooth Kernel For smooth, positive definite kernels K the MMD functional Fν :=
+D2
+K(·, ν) is in general not convex and leads to a more complex energy landscape than for the
+negative distance kernel. This may lead to problems for optimization algorithms. To illustrate this
+observation, we let ν := λ[−1,1] and compare the energy landscape of the restricted functional F2,ν
+for K(x, y) := −|x − y| and the kernel
+˜K(x, y) :=
+�
+(1 − 1
+2|x − y|)2(|x − y| + 1),
+|x − y| ≤ 2,
+0,
+else.
+(8)
+In contrast to the negative distance kernel K, the kernel ˜K is positive definite (without restrictions
+on the ai), cf. [33], and has a Lipschitz continuous gradient. The two energy landscapes of F2,ν are
+visualized in Figure 4. The non-convexity of Fν for ˜K is readily seen by the presence of a saddle
+point for F2,ν at µ = δ0 (equivalently to (m, σ) = (0, 0) in the mσ-plane). Note that any Wasserstein
+gradient flow of Fν starting at a Dirac measure δx converges to this saddle point µ = δ0.
+7
+Conclusions
+We provided insight into Wasserstein gradient flows of MMD functionals with negative distance
+kernels and characterized in particular flows ending in a Dirac measure. We have seen that such flows
+are not simple particle flows, e.g. starting in another Dirac measure the flow becomes immediately
+uniformly distributed and after a certain time a mixture of a uniform and a Dirac measure. In
+our future work, we want to extend our considerations to empirical measures and incorporate
+
+2.00
+1.75
+1.50
+1.25
+b 1.00
+0.75
+0.50
+0.25
+0.00
+-1.00
+-0.75
+-0.50
+-0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+mt= 0.00
+t= 0.25
+t= 0.50
+t= 0.75
+t= 1.00
+t= 1.25
+t= 1.50
+t= 1.75
+t= 2.00
+t = 2.25
+2.0
+1.5
+1.0
+0.5
+0.0
+1
+0
+0
+0
+0
+0
+1
+0Wasserstein Gradient Flows of the Discrepancy with Distance Kernel on the Line
+11
+Fig. 4: Visualization of the energy landscapes of F2,λ[−1,1] for the convex negative distance kernel
+(left) and the non-convex, smooth kernel given in (8). The red dot is the global minimizer λ[−1,1]
+(left and right) and the blue point (right) is the saddle point δ0. The black lines depict selected
+gradient flows.
+deep learning techniques as in [2]. Also the treatment of other functionals which incorporate an
+interaction energy part appears to be interesting. Further, we may combine univariate techniques
+with multivariate settings using Radon transform like techniques as in [8,23,25].
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+

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+Investigating fission dynamics of neutron shell closed nuclei 210Po, 212Rn and 213Fr
+within a stochastic dynamical approach
+Divya Arora, P. Sugathan,∗ and A. Chatterjee
+Inter-University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi 110067, India
+Dissipative dynamics of nuclear fission is a well confirmed phenomenon described either by a
+Kramers-modified statistical model or by a dynamical model employing the Langevin equation.
+Though dynamical models as well as statistical models incorporating fission delay are found to
+explain the measured fission observables in many studies, it nonetheless shows conflicting results
+for shell closed nuclei in the mass region 200.
+Analysis of recent data for neutron shell closed
+nuclei in excitation energy range 40−80 MeV failed to arrive at a satisfactory description of the
+data and attributed the mismatch to shell effects and/or entrance channel effects, without reaching
+a definite conclusion. In the present work we show that a well established stochastic dynamical
+code simultaneously reproduces the available data of pre-scission neutron multiplicities, fission and
+evaporation residue excitation functions for neutron shell closed nuclei 210Po and 212Rn and their
+isotopes 206Po and 214,216Rn without the need for including any extra shell or entrance channel
+effects. The calculations are performed by using a phenomenological universal friction form factor
+with no ad-hoc adjustment of model parameters. However, we note significant deviation, beyond
+experimental errors, in some cases of Fr isotopes.
+I.
+INTRODUCTION
+Fission of atomic nuclei is considered to be one of the
+most complex physical phenomena in nuclear physics. It
+involves rapid re-arrangement of nuclear matter with a
+delicate interplay between the macroscopic bulk matter
+and the microscopic quantal properties [1, 2]. Though
+properties of fission have been studied exhaustively, many
+aspects of the dynamics are still not well-understood. For
+instance, discrepancies are reported between the mea-
+sured fission observables and the predictions of the clas-
+sical theory based on the standard Bohr-Wheeler statis-
+tical model of fission [3].
+Fission hindrance, enhanced
+pre-scission particle and giant dipole resonance (GDR)
+γ-ray multiplicities observed in hot nuclei suggested the
+effects of nuclear dissipation slowing down the fission pro-
+cess [4–10]. To account for frictional effects, Kramers dif-
+fusion model formalism with modified fission width [11],
+referred to as Kramers-modified statistical model was in-
+cluded in the standard statistical theory.
+Although nature and strength of the nuclear dissipa-
+tion have been studied quite extensively, a simultaneous
+description of the experimental observables, namely, pre-
+scission neutron multiplicities (νpre), fission excitation
+functions and evaporation residue (ER) cross-sections
+still remains challenging.
+Additionally, the dissipation
+coefficient is treated as an adjustable free parameter in
+the statistical model analysis. The pre-fission lifetime (or
+the dissipation strength), the level density parameter at
+ground state and saddle point deformation and fission
+barrier are empirically fitted to explain the νpre and/or
+fission and ER cross-section data [6, 12–15]. As a result,
+the conclusions reached are often system dependent and
+are inadequate to provide a consistent description of the
+∗ [email protected]
+fission process.
+Inadequate modelling of fission in statistical model can
+drastically influence the understanding of the fission phe-
+nomenon [16, 17].
+This is especially observed in mass
+(A) ≈ 200 region that is explored here, to understand the
+role of N=126 neutron shell closure in the fissioning com-
+pound nucleus (CN). An anomalous increase in the exper-
+imental fission fragment angular anisotropy was reported
+for 210Po (N=126) as compared to 206Po (non-shell closed
+nuclei) across an excitation energy range (Eex) ≈ 40−60
+MeV and was conjectured to be a manifestation of shell
+effects at the unconditional saddle [18]. Further, a con-
+siderable amount of saddle shell correction was invoked
+to describe the experimental νpre data for 210Po nuclei
+[19]. However, a re-investigation of the experimental ex-
+citation functions and νpre data of 210Po ruled out any
+significant shell influence on the saddle [20] after corre-
+lated tuning of statistical-model parameters and inclu-
+sion of fission delay.
+Another interesting aspect is the contradictory inter-
+pretation for correlation between neutron shell structure
+and nuclear dissipation strength that was required to re-
+produce the measured ER and νpre excitation functions
+in N=126 shell closed nuclei, namely 212Rn and 213Fr.
+The theoretical analysis of νpre data of 212Rn [21] and
+213Fr [22] reported a low dissipation strength at Eex ≈
+50 MeV which was attributed to the influence of neu-
+tron shell closure. On the contrary, no discernible shell
+influence was reported from ER cross-section studies of
+212Rn and its isotope [23], though moderate nuclear dis-
+sipation was required to describe the data. It must be
+noted that the magnitude of dissipation invoked to ex-
+plain the experimental ER cross-sections varied within
+Rn isotopes [23, 24], which is again found to be different
+for the description of the νpre data [21]. Interestingly,
+in case of Fr nuclei, the finite-range liquid drop model
+fission barrier was scaled down, particularly for 213Fr to
+fit measured ER cross-section [15]. This reduction of the
+arXiv:2301.13461v1  [nucl-th]  31 Jan 2023
+
+2
+fission barrier is in disagreement with the predictions for
+the shell closed nuclei [25]. One notable observation is the
+reported interpretation of reduced survival probability of
+213Fr nucleus due to neutron shell which is in contrast
+to the isotopic trend reported for Rn isotopes. Further,
+the fission cross-section of 213Fr is reported to exhibit
+no extra stability from N=126 shell closure [26]. In the
+statistical model approach followed in these works, no at-
+tempts were made to extract a global prescription of the
+parameters, rather, a case specific adjustment of dissipa-
+tion strength was involved. The influence of neutron shell
+structures on the potential energy surface and hence fis-
+sion observables are still quite ambiguous. Apart from
+just shell influence, entrance channels effects are also
+probed in a couple of recent publications to understand
+the experimental νpre data for 213Fr nuclei [27, 28]. These
+studies reportedly observed a deviation in the measured
+data from the predictions of entrance channel model for
+16O- and 19F-induced reactions.
+These studies substantiate the view that no consistent
+picture has emerged from recent independent analysis
+of each fission observable for neutron shell closed nuclei
+210Po, 212Rn and 213Fr and their isotopes.
+Inadequa-
+cies of standard statistical model interpretations have
+been addressed by employing Kramers-modified fission
+width taking into account shape-dependent level den-
+sity, temperature-dependent fission transition points, ori-
+entation (K state) degree of freedom and temperature-
+independent reduced dissipation coefficient [16, 17]. At-
+tempts for restraining the statistical-model parameters
+have also been reported [29], but a consistent description
+of experimental data for all three observables, namely
+νpre, fission and ER excitation functions for shell closed
+nuclei still could not be achieved. Recent developments
+in multi-dimensional stochastic approach are fairly suc-
+cessful in describing the fission characteristics of excited
+nuclei [30–35]. However, a simultaneous description of
+the experimental data and a systematic study for shell
+closed nuclei has not been attempted yet and is further
+required.
+In this paper, we show that the dynamical model based
+on 1D Langevin equation coupled with a statistical ap-
+proach [36] can simultaneously reproduce νpre, fission
+and ER cross-section data of shell closed nuclei over a
+range of excitation energies (Eex ≈ 40−80 MeV) of the
+measurements. The present calculations are performed
+without adjusting any of the model parameters, thus pro-
+vides a unified framework for a simultaneous study of
+these fission observables for nuclei in A ≈ 200 region.
+We re-investigated the available experimental data for
+the neutron shell closed nuclei 210Po, 212Rn and 213Fr,
+and their non-shell closed isotopes 206Po, 214,216Rn and
+215,217Fr.
+It is observed that a universal deformation-
+dependent reduced friction parameter is able to describe
+the fission observables simultaneously at all measured en-
+ergies irrespective of the shell structure of the nuclei.
+II.
+THEORETICAL MODEL DESCRIPTION
+A combined dynamical and statistical model code [37]
+is utilized to compute the fission observables of nuclei
+under study.
+The detailed description of the theoreti-
+cal aspects of the model can be found in Refs. [36, 38].
+The dynamical part of the model is carried out with a
+1D Langevin equation of motion governed by a driving
+potential that is determined by free energy F(q, T), as
+employed in recent Refs.
+[39–44].
+The free energy as
+derived from the Fermi gas model is related to the defor-
+mation dependent level density parameter a(q, A) as F(q,
+T) = V(q) - a(q, A)T 2 where T is the nuclear temper-
+ature, q is the dimensionless deformation coordinate de-
+fined as the ratio of half the distance between the center
+of masses of future fission fragments to the radius of CN
+and V (q) is the nuclear potential energy obtained from
+the finite-range liquid drop model [45, 46]. Fr¨obrich [47]
+and Lestone et al. [17] have emphasized on using nuclear
+entropy given by, S(q, A, Etot) = 2
+�
+a(q, A)[Etot − V (q)]
+in determining the driving force and therefore, it is em-
+ployed as a crucial quantity in the model. The nuclear
+driving force K = - dV (q)
+dq
++ da(q)
+dq T 2, not only consists of
+a conservative force but also contain a thermodynami-
+cal correction that enters the dynamics via. level density
+parameter a(q, A). The deformation dependent level den-
+sity parameter used in constructing the entropy has the
+form [48]:
+a(q, A) = ˜a1A + ˜a2A2/3Bs(q)
+(1)
+where A is the mass number of the CN and ˜a1 = 0.073
+MeV−1 and ˜a2 = 0.095 MeV−1 are taken from Ref. [49].
+Bs(q) is the dimensionless functional of the surface en-
+ergy [34, 38, 43, 50], expressed as the ratio of surface
+energy of the composite system to that of a sphere.
+The over-damped Langevin equation which describes
+the fission process in the dynamical part of the model
+thus, has the form [36]:
+dq
+dt =
+T
+Mβ(q)
+�∂S(q)
+∂q
+�
+Etot
++
+�
+T
+Mβ(q)Γ(t)
+(2)
+where Etot is the total energy of the composite system
+that remains conserved and Γ(t) is a Markovian stochas-
+tic variable with a normal distribution. The reduced dis-
+sipation coefficient β(q) = γ/M (as employed in litera-
+ture, see e.g., Refs. [16, 29, 42, 44] (and Refs. therein))
+is the ratio of friction coefficient γ to the inertia param-
+eter M calculated with Werner-Wheeler approximation
+of an incompressible irrotational fluid [51]. The present
+model employs ”funny−hills” parameters {c,h,α} [52]
+for describing the shape of the fissioning nuclei.
+Tak-
+ing into account only symmetric fission, the mass asym-
+metry parameter of the shape evolution is set to α=0
+[36, 38, 50]. The dimensionless fission coordinate (q) is
+given by q(c,h)= ( 3c
+8 )(1+ 2
+15[2h+ (c−1)
+2
+]c3), where c and h
+
+3
+defines the elongation and neck degree of freedom of the
+fissioning nucleus, respectively [36, 43, 53, 54].
+Following the fission dynamics through full Langevin
+dynamical calculation is quite time consuming. Similar
+to previous Langevin studies [31, 36, 39–43], a compu-
+tationally less intensive approach is adopted in present
+study where the dynamical stage is coupled with a sta-
+tistical model. In the present calculations, the emission
+of light particles from ground state to scission config-
+uration along the Langevin trajectories is treated as a
+discrete process.
+The evaporation of pre-scission light
+particles from ground state of Langevin trajectories to
+the scission point is coupled to the fission mode by a
+Monte Carlo procedure. The decay width for light parti-
+cle evaporation at each Langevin time step is calculated
+with the formalism as suggested by Fr¨obrich et al. [36]
+and later incorporated in Refs. [34, 40–43]. The emission
+width of a particle of kind ν (n,p,α) is given by [55]:
+Γν = (2sν + 1)
+mν
+π2ℏ2ρc(Eex)
+×
+� (Eex−Bν)
+0
+dϵνρR(Eex − Bν − ϵν)ϵνσinv(ϵν)
+(3)
+where sν is the spin of emitted particle ν, and mν is
+its reduced mass with respect to the residual nucleus.
+The level densities of the compound and residual nuclei
+are denoted by ρc(Eex) and ρR(Eex − Bν − ϵν). Bν is
+the liquid-drop binding energy, ϵ is the kinetic energy
+of the emitted particle and σinv(ϵν) is the inverse cross
+sections [55]. The decay width for light particle emission
+is calculated at each Langevin time step τ [43, 53, 54].
+When a stationary flux over the barrier is reached af-
+ter a sufficiently long delay time, the decay of the CN
+is then modelled by an adequately modified statistical
+model [38, 56, 57]. To have continuity when switching
+from dynamical to statistical branch, an entropy depen-
+dent fission width is incorporated in the latter. While en-
+tering the statistical branch, the particle emission width
+Γν is re-calculated and the fission width Γf = ℏRf [36]
+is calculated with fission rate (Rf) given by,
+Rf =
+Tgs
+�
+|S
+′′
+gs|S
+′′
+sd
+2πMβgs
+exp[S(qgs) − S(qsd)]
+× 2(1+erf[(qsc − qsd)
+�
+S
+′′
+sd/2])−1
+(4)
+Here erf(x) = (2/√π)
+� x
+0 dt exp(−t2) is the error func-
+tion and βgs is ground state dissipation coefficient. The
+saddle-point (qsd) and the ground-state positions (qgs)
+are defined by the entropy and not, as in the conventional
+approach, by the potential energy. The standard Monte
+Carlo cascade procedure was used to select the kind of
+decay with weights Γi/Γtot (i=fission,n,p,d,α) and Γtot =
+�
+i Γi. Pre-scission particle multiplicities are calculated
+by counting the number of corresponding evaporated par-
+ticle events registered in the dynamical and statistical
+branch of the model.
+The Langevin equation is started from a ground state
+configuration with a temperature corresponding to the
+initial excitation energy.
+The fusion cross-section can
+be determined from the partial cross section dσ(l)
+dl
+which
+represent the contribution of angular momenta l to the
+total fusion cross-section.
+Each Langevin trajectory is
+started with an orbital angular momentum which is sam-
+pled from a fusion spin distribution that reads as [34, 36]:
+dσ(l)
+dl
+= 2π
+k2
+2l + 1
+1 + exp (l−lc)
+δl
+(5)
+The final results are weighted over all relevant waves, that
+is, the spin distribution is used as an angular momen-
+tum weight function with which the Langevin calcula-
+tions for fission are started. As shown in recent Langevin
+studies, [34, 39–44], the spin distribution is calculated
+with the surface friction model [58].
+This calculation
+also fixes the fusion cross-section thus guaranteeing the
+correct normalization of fission and evaporation residue
+cross-sections within the accuracy of the surface friction
+model. The parameters lc and δl are the critical angular
+momentum for fusion and diffuseness, respectively.
+The fission observables that will be discussed in sub-
+sequent sections are calculated in the model as follows.
+The pre-scission neutron multiplicity is the number of
+neutrons emitted by the CN till it reaches the scission
+configuration. The fission probability (Pf) is given by
+the ratio of fissioned trajectories to total trajectories.
+The CN survival probability (1-Pf) is given by number
+of trajectories leading to ER formation divided by total
+trajectories and the fission (ER) cross-section is given by
+the product of fission (survival) probability and fusion
+cross-section.
+III.
+RESULTS AND DISCUSSION
+In the present study, pre-scission neutron multiplic-
+ities, fission and ER excitation functions for 206,210Po,
+212,214,216Rn and 213,215,217Fr compound nuclei are com-
+puted and compared with available experimental data
+wherein 210Po, 212Rn and 213Fr are N=126 neutron shell
+closed nuclei. The table I shows important parameters
+for the reactions studied in this work. The dynamical cal-
+culations are performed with a universal frictional form
+of Refs. [36, 47, 57] without adjusting any of the model
+parameters with a consistent prescription of the dissipa-
+tion coefficient. To account for sufficient statistics, 107
+Langevin trajectories are considered in the model calcu-
+lations.
+Fig.
+1 shows the results of dynamical calculations
+compared with the experimental data of νpre, fission
+and ER cross-sections for 206Po formed via.
+12C+194Pt
+[18, 19, 59] reaction and 210Po formed through two dif-
+ferent entrance channel reactions, namely
+12C+198Pt
+[18, 19, 60] and 18O+192Os [5, 60, 61], spanning a wide
+range of excitation energy. The excitation energies shown
+
+4
+TABLE I. Important parameters of reactions studied
+CN
+fissility
+Sn
+Bf(l=0)
+Reaction
+Mass excess (MeV) α/αBG
+(MeV)
+(MeV)
+target(proj)
+CN
+206Po
+0.717
+7.99
+10.51
+12C+194Pt
+-34.79(0)
+-18.83
+1.043
+210Po
+0.711
+7.38
+11.22
+12C+198Pt
+-29.93(0)
+-16.33
+1.050
+18O+192Os -35.89(-0.78) -16.33
+0.982
+212Rn
+0.732
+7.83
+8.88
+18O+194Pt -34.79(-0.78)
+-9.26
+0.970
+214Rn
+0.729
+7.54
+9.19
+16O+198Pt -29.93(-4.74)
+-4.77
+0.996
+216Rn
+0.727
+7.25
+9.49
+18O+198Pt -29.93(-0.78)
+0.70
+0.977
+213Fr
+0.743
+8.06
+7.83
+16O+197Au -31.16(-4.74)
+-4.01
+0.987
+19F+194Pt
+-34.79(-1.49)
+-4.01
+0.954
+215Fr
+0.740
+7.76
+8.13
+19F+196Pt -32.67 (-1.49) -0.07
+0.958
+217Fr
+0.737
+7.47
+8.42
+19F+198Pt
+-29.93(-1.49)
+5.00
+0.961
+here are with respect to the liquid drop ground state CN
+mass and experimental mass of projectile and target [62].
+Our calculations are restricted to excitation energies at
+and above 40 MeV where the present macroscopic model
+is valid. We emphasize that the microscopic shell correc-
+tions are not accounted for in the present calculations,
+as we are dealing with hot nuclei where shell effects are
+expected to be negligible at high excitation energies that
+are populated in heavy-ion reactions. The results of cal-
+culations using only the statistical model (dashed line)
+are also shown in Fig. 1. These calculations are made
+with the same code with Langevin dynamics turned off.
+The statistical model calculations under-predict the mea-
+sured νpre data as shown in panels (a) to (c), even more
+so as excitation energy increases. The dynamical model
+calculations using universal reduced friction coefficient
+are in excellent agreement with the measured data of
+νpre (panels (a) to (c)), fission cross-sections σfiss (pan-
+els (d) to (f)) and ER cross-sections σER (panels (g) to
+(i)) for the neutron shell closed nuclei 210Po as well as
+its isotope 206Po. The measured data of 210Po formed
+through two different entrance channels agree well with
+the theory in a broad range of excitation energies up to
+80 MeV. The model calculations describe the available
+experimental data for 206,210Po simultaneously at these
+excitation energies without any microscopic corrections
+included in the model. These observations are at vari-
+ance with the statistical model analysis of 12C+194Pt and
+12C+198Pt reactions that reported a significant shell cor-
+rection at the saddle deformation to describe the angular
+anisotropy and νpre data [18, 19]. A recent 4D Langevin
+dynamical study [63] that was carried on 206Po and 210Po
+populated from reaction 12C+198Pt, reported a reason-
+able description of the measured data for these reactions
+without invoking any extra shell corrections at the saddle
+state; shown as open triangles in panels (a), (c), (d) and
+(f) of Fig. 1. A better agreement of the measured data is
+observed for 12C+198Pt reaction in comparison to its 4D
+Langevin calculations [63], particularly at low excitation
+energies as shown in panels (a) and (d) of Fig. 1. The
+overestimation of νpre and fission cross-section of 210Po
+in Ref.
+[63] was attributed to the remnant of ground
+state shells and hence, a consequence of not using a pure
+macroscopic potential energy surface as suggested in Ref.
+[64]. Nonetheless, the predictions of multi-dimensional
+Langevin model for νpre data of 206Po by Karpov et al.
+[30] are also found to be in reasonable agreement with the
+results of the present analysis. Moreover, the measured
+mass distribution of fragments in the fission of 206,210Po
+[65, 66] reaffirms the absence of any shell corrections on
+the potential energy surface at the saddle point.
+Figs.
+2 and 3 display the comparison between ex-
+perimental data and theoretical calculations of νpre, fis-
+sion, ER and fusion cross-sections for N=126 shell closed
+nuclei viz.
+212Rn [21, 23, 24, 67] formed through re-
+action 18O+194Pt and 213Fr formed through reactions
+19F+194Pt [15, 22, 26, 68] and 16O+197Au [5, 6], and
+their non-shell closed isotopes 214,216Rn populated via.
+reactions 16,18O+198Pt [21, 23, 24, 67] and 215,217Fr pop-
+ulated via. reactions 19F+196,198Pt [15, 22, 26, 68]. The
+model calculations describe the νpre and fission excita-
+tion functions for 212Rn and its isotopes 214,216Rn quite
+successfully. In reactions forming 213,215,217Fr nuclei, the
+same parameter set is able to account for the experi-
+mental fission excitation functions but not νpre. A re-
+cent work [26] using an extended version of statistical-
+model employing collective enhancement of level density
+also reported an under-estimation of νpre data for same
+reactions when fitted simultaneously with fission cross-
+section. In the present work, the disagreement between
+experimental νpre and theory is prominent above ≈50
+MeV excitation energy and it increases with rise in exci-
+tation energy. Considering that νpre of other studied nu-
+clei are well reproduced by the model, it is unclear why
+
+5
+0
+1
+2
+3
+4
+νpre
+(a)
+12C+198Pt −→ 210Po
+(b)
+18O+192Os −→ 210Po
+(c)
+12C+194Pt −→ 206Po
+100
+101
+102
+103
+σfiss(mb)
+(d)
+(e)
+(f)
+40
+60
+80
+100
+100
+101
+102
+103
+σER(mb)
+(g)
+40
+60
+80
+100
+(h)
+40
+60
+(i)
+Eex (MeV)
+FIG. 1. (Colour online) Measured and calculated pre-scission neutron multiplicities (νpre), fission cross-sections (σfiss) and evap-
+oration residue cross-sections (σER) as a function of excitation energy for the reactions 12C+198Pt, 18O+192Os and 12C+194Pt.
+The continuous line (red) denote calculated results with a universal frictional form factor and dashed line (black) represent
+statistical model calculations. The symbols in the legend represent different experimental data sets, for νpre: (filled squares)
+Ref. [19], (filled circles) Ref. [5] and (open square) Ref. [59]; for σfission and σER: (filled diamonds) Ref. [18], (filled hexagons)
+Ref. [61] and (open diamonds) Ref. [60]. The open triangles represent results of νpre and σfission from 4D Langevin calculations
+of Ref. [63].
+the same frictional form fails, particularly for reactions
+forming Fr nuclei. It is to be noted that, an energy de-
+pendent dissipation was used in Ref.[21, 22] to describe
+the νpre data for these reactions.
+We also attempted
+similar approach by employing a temperature-dependent
+friction (TDF) in the stochastic calculations [69] (with-
+out changing any other parameter). This frictional form
+factor is deformation dependent, unlike the ones used in
+Refs. [21, 22, 70]. The maximum of β(q) in TDF corre-
+sponds to the ground state, that tends to decrease with
+increasing deformation with its minimum near the sad-
+dle configuration and is followed by an increase in the
+dissipation strength when approaching the scission. The
+dissipation coefficient assumes a higher value with in-
+creasing temperature of the CN. It is observed that a
+better agreement of νpre data is achieved for reactions
+19F+194,196,198Pt and 16O+197Au after invoking temper-
+ature dependence of the dissipation. The same frictional
+form, however, is found to over-predict the measured νpre
+data of other studied nuclei and hence is not shown here.
+Deviation in ER excitation functions are also to be
+noted for 212Rn and 213,215,217Fr nuclei wherein the cal-
+culated ER cross-sections underpredict the experimental
+data for these nuclei at high excitation energies. The case
+
+6
+0
+2
+4
+νpre
+(a)
+18O+198Pt −→ 216Rn
+(b)
+16O+198Pt −→ 214Rn
+(c)
+18O+194Pt −→ 212Rn
+101
+102
+103
+σfiss(mb)
+(d)
+(e)
+(f)
+101
+102
+103
+σER(mb)
+(g)
+(h)
+(i)
+40
+60
+80
+101
+102
+103
+σfus(mb)
+(j)
+40
+60
+80
+(k)
+40
+60
+80
+(l)
+Eex (MeV)
+FIG. 2. (Colour online) Measured and calculated pre-scission neutron multiplicities (νpre), fission cross-sections (σfiss), evapora-
+tion residue cross-sections (σER) and fusion cross-sections (σfus) as a function of excitation energy for the reactions 18O+198Pt,
+16O+198Pt, 18O+194Pt. The continuous (red) and dashed (black) lines have the same meaning as in Fig. 1. The calculations
+of fusion cross-section are independent of the frictional form and are represented by dotted line (brown). The symbols in the
+legend represent different experimental data sets, for νpre: (filled squares) Ref. [21]; for σfiss: (filled diamonds) Ref. [67] and
+(open diamonds) Ref. [23]; for σER: (filled circles) Ref. [24] and (filled hexagons) Ref. [23] and for σfus: (filled triangles) Refs.
+[23, 24].
+of Rn isotopes is of particular interest as the ER cross-
+section data for 214,216Rn [24] agrees fairly well with the
+model calculations at all measured energies but differ for
+212Rn [23] except at the lowest energy. For 213,215,217Fr
+nuclei, the measured ER cross-sections of Ref. [15] differ
+above excitation energy ≈ 55 MeV and the deviation is
+prominent for 213,215Fr. It is quite interesting to note that
+the ER measurement by a different group [68] for same
+reactions forming 213,217Fr at Eex ≤ 55 MeV follows the
+trend of the model predictions quite successfully. Unfor-
+tunately, Ref. [68] has reported only three data points.
+Moreover, the ER cross-section data of 215Fr formed in
+reaction 18O+197Au [71] is reproduced reasonably well
+with results of 19F+196Pt particularly, above 50 MeV ex-
+citation energy (displayed as open pentagons in panel (j)
+of Fig. 3). The present dynamical calculations assume
+
+7
+0
+2
+4
+6
+νpre
+(a)
+19F+198Pt → 217Fr
+(b)
+19F+196Pt → 215Fr
+(c)
+19F+194Pt → 213Fr
+(d)
+16O+197Au → 213Fr
+101
+102
+103
+σfiss(mb)
+(e)
+(f)
+(g)
+(h)
+101
+102
+103
+σER(mb)
+(i)
+(j)
+(k)
+(l)
+50
+75
+100
+101
+102
+103
+σfus(mb)
+(m)
+50
+75
+100
+(n)
+50
+75
+100
+(o)
+50
+100
+(p)
+Eex (MeV)
+FIG. 3. (Colour online) Measured and calculated pre-scission neutron multiplicities (νpre), fission cross-sections (σfiss), evapora-
+tion residue cross-sections (σER) and fusion cross-sections (σfus) as a function of excitation energy for the reactions 19F+198Pt,
+19F+196Pt, 19F+194Pt and 16O+197Au. The continuous (red), dashed (black) and dotted (brown) lines have the same meaning
+as in Figs. 1 and 2. The dash-dotted line (magenta) represent calculated results with temperature-dependent friction. The
+symbols in the legend represent different experimental data sets, for νpre: (filled squares) Ref. [22] and (partially filled squares)
+Ref. [5] ; for σfiss: (filled diamonds) Ref. [26],(partially filled diamonds) Ref. [6] and (open diamonds) Ref. [68]; for σER:
+(filled circles) Ref. [15], (partially filled circles) Ref. [6] and (open circles) Ref. [68] and for σfus: (filled triangles) Refs.
+[15, 26, 68] and (open triangles) Refs. [6]. The open pentagons denote σER for 215Fr nuclei formed via 18O+197Au Ref. [71].
+decay from an equilibrated CN and any entrance channel
+effects are not included. It takes account of only the dif-
+ferent angular momenta that are populated in different
+entrance channels. Taking into consideration the insignif-
+icant difference in angular momenta between two en-
+trance channels forming 215Fr, the observed deviation in
+ER cross-section for 19F-induced reaction is quite unex-
+pected. These observations further necessitated the need
+to confront the deviations in describing ER cross-sections
+by comparing the measured fusion cross-sections for Rn
+and Fr nuclei with the model. It is revealed that the cal-
+culated fusion cross-sections are in good agreement with
+the measured fusion data, augmenting the validity of the
+present calculations. Furthermore, the under-prediction
+of ER cross-sections indicates the need for a strong dis-
+sipation in the pre-saddle region [72].
+However, 3D
+
+8
+Langevin dynamical calculations [31] reported a reduc-
+tion in the wall friction coefficient to reproduce the mass
+and kinetic energy distribution of fission fragments, and
+their influence on νpre for 215Fr nucleus. The strength
+of the reduction coefficient, ks = 0.25 − 0.5 indicates
+a weak dissipation in the initial stages of the fissioning
+nucleus. The experimental analysis of fission fragment
+nuclear-charge distributions and fission cross-sections of
+Fr, Rn isotopes and their neighbouring nuclei also re-
+ported a pre-saddle dissipation strength of magnitude
+(4.5 ± 0.5) × 1021 s−1 [73] and 2 × 1021 s−1 [74], respec-
+tively. The more recent microscopic study of energy de-
+pendent dissipation using time-dependent Hartree-Fock
++ BCS method [75] also observed a strength of deforma-
+tion dependent friction coefficient, ranging from 1 to 6
+× 1021 s−1 in heavy nuclei. The strength of these fric-
+tional parameterizations are quite in agreement with the
+dissipation form factor employed in the present calcula-
+tions.
+These observations affirm a weak dissipation in
+the pre-saddle region; so, the observed enhancement of
+ER cross-sections in Fr nuclei populated via. 19F-induced
+reactions is not well-understood from the perspective of
+dissipation strength alone.
+In fact, a satisfactory de-
+scription of the excitation functions including ER cross-
+sections for reactions 12C+194Pt, 12C+198Pt, 18O+192Os
+and 16,18O+198Pt and survival probabilities for a range
+of fissilities [36] is observed within the framework of this
+1D Langevin dynamics with a universal friction param-
+eter. However, it is also important to bear in mind the
+possible bias coming from experimental uncertainty. It is
+striking that the observed deviations are pronounced in
+ER cross-section data where measurements are reported
+to have large uncertainty in ER separator transmission
+efficiency [15, 23]. It would be highly desirable to have
+additional ER measurements to rule out any possible ex-
+perimental bias in the interpretation of ER data.
+It must be noted that, the entrance channel dynam-
+ics of the fusion stage might also play a role influenc-
+ing neutron emission at the formation stage [14]. It is
+known that interplay of CN excitation energy, angular
+momentum and fission barrier play crucial role in fission
+process [28]. Present study do not take into account any
+entrance channel dynamics influencing the fusion stage.
+The model only considers the entrance channel depen-
+dent ’l’ distribution calculated within the surface friction
+model [58]. In Fig. 4 we show the calculated fission bar-
+rier height Bf(l) for three compound systems and mean
+angular momentum < l > calculated from ’l’ distribu-
+tion for different entrance channels forming same CN.
+The variation of Bf is plotted as a function of ’l’ in Fig.
+4(a) and variation of < l > of the compound systems is
+plotted as a function of Eex in Fig 4(b). From Fig. 4,
+it is clear that, the difference in angular momenta be-
+tween two entrance channels forming same CN at similar
+Eex is not very significant to cause any ’l’ induced ef-
+fects on measured fission observable. This is evident in
+the νpre data for 210Po formed in reactions 12C+198Pt
+and 18O+192Os which are well described in the present
+work (see Fig. 1) without invoking any entrance channel
+effects in the model.
+Recent studies investigating en-
+trance channel dynamics [27, 28] reported disagreement
+between experimental νpre and predictions of entrance
+channel model for 213Fr nuclei formed via.
+16O+197Au
+and 19F+194Pt reactions. These studies were, however,
+not extended to other isotopes of Fr, namely 215,217Fr
+that also show similar discrepancy as reported in the
+present study.
+The current 1D Langevin analysis provides a simul-
+taneous description of the experimental data for neutron
+magic nuclei 210Po without invoking any saddle shell cor-
+rections or a nuclear dissipation strength dependent on
+system/observable under study. In order to understand
+qualitatively that consideration of saddle shell correc-
+tions are not required to explain νpre data, we consider
+the nature of neutron emission during the fission process.
+It is to be noted that these neutrons are emitted from dy-
+namical trajectories that originated from compact config-
+uration till scission point is reached. The prompt and
+beta-delayed neutron emissions from fission fragments
+are not taken into consideration.
+As recent publica-
+tions have advocated for the inclusion of shell correc-
+tions on the saddle configuration to describe the angular
+anisotropy and νpre data at moderate excitation energies
+[12, 18, 19], we have attempted to find the distribution
+of pre-scission neutrons as it evolves from ground state
+to scission point. The model calculated potential energy
+V(q) and distribution of percentage yield of pre-scission
+neutrons are plotted as a function of the deformation co-
+ordinate (q) for these nuclei at 50 MeV excitation energy
+and shown in Fig. 5. It is evident that more than 90% of
+the neutron emission occurs at an early stage of fission
+before the saddle deformation (q ≈ 0.8) [38] is reached.
+The mean of the distribution corresponds to νpre emis-
+sion close to the ground state configuration. In-fact, a
+multi-dimensional Langevin study of 215Fr by Nadtochy
+et al. [31] have also pointed out that an appreciable part
+of pre-scission neutrons are emitted at an early stage of
+fission before saddle is reached. As most of the neutrons
+are emitted close to the ground state configuration, it is
+unlikely to be influenced by any shell corrections applied
+at the saddle.
+Though the present code uses classical 1D approach to
+describe fission observables, the main objective of this
+work is to have a simultaneous description of experi-
+mental data without any parameter adjustment thus,
+removing some of the reported ambiguities.
+A com-
+parison between νpre calculated with 1D model and re-
+cent macroscopic multi-dimensional models is displayed
+in Fig. 6. It can be seen that the νpre values predicted
+by different models are very similar and also reproduce
+the measurements quite well for reactions spanning a
+wide range of fissility parameter Z2/A.
+Additionally,
+the multi-dimensional calculations [34, 50, 76] also use
+the formalisms adopted from Refs. [36, 69] such as the
+parameterization of surface friction model and weakest
+coordinate dependence of the level-density parameter as
+
+9
+0
+20
+40
+60
+80
+ℓ (¯h)
+0
+2
+4
+6
+8
+10
+12
+14
+Bf (MeV)
+(a)
+210Po
+212Rn
+213Fr
+30
+40
+50
+60
+70
+80
+90
+10
+15
+20
+25
+30
+35
+40
+45
+< ℓ > (¯h)
+(b)
+12C+198Pt
+18O+192Os
+19F+194Pt
+16O+197Au
+Eex (MeV)
+FIG. 4. (Colour online) (a) The angular momentum ’l’ de-
+pendent fission barrier height Bf(l) for three CN 210Po,212Rn
+and 213Fr and (b) Variation of mean angular momentum < l >
+with compound nucleus excitation energy for 210Po,and 213Fr
+populated by different entrance channels.
+employed in the present work.
+Hence, the qualitative
+nature of the observed features presented here is not ex-
+pected to be different with multi-dimensional approach.
+As the present framework is found to provide realistic
+values close to measured data, we believe that the 1D
+approach still can be a potential tool to study a wider
+systematics which can be accomplished within minimum
+0
+10
+20
+30
+40
+V (MeV)
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+0
+1
+2
+3
+4
+5
+6
+7
+8
+d<νpre>/dq (%)
+qneck
+qsadd
+210Po
+212Rn
+213Fr
+deformation coordinate (q)
+FIG. 5.
+(Colour online) Potential energy distribution as a
+function of nuclear deformation coordinate (q) for three fis-
+sioning nuclei 210Po, 212Rn and 213Fr (top panel) and distri-
+bution of percentage yield of evaporated pre-scission neutrons
+as a function of (q) for three CN at 50 MeV excitation energy
+(bottom panel). The deformation coordinate (q) assumes a
+value of 0.6 (qneck) when the neck of the fissioning nucleus
+starts to develop and q=0.8 (qsadd) at the saddle state con-
+figuration.
+computational resources.
+It must be remarked here that, even though present
+analysis provides a reasonable reproduction of the exper-
+imental data without invoking any shell corrections at
+high excitation energies, it shall not be concluded from
+this work that shell effects are not relevant in the analy-
+sis. As present investigation consider only the first chance
+fission at Eex ∼ 40 MeV and above where shell effects are
+expected to be washed out, no indication for the need of
+including shell corrections was found. However, for the
+case when the CN is populated at low excitation energies
+or reaches low excitation energy due to neutron emission
+as a consequence of competition between neutron evapo-
+ration and fission (multi-chance fission), the microscopic
+effects are required to be taken into consideration. Re-
+cent microscopic study of dissipation within Hartree-Fock
++ BCS framework [75] have shown a strong dependence
+of dissipation on deformation and initial excitation ener-
+gies of the hot nuclei. Possible influence of microscopic
+temperature dependence of fission barrier height and its
+curvature were also emphasized in some recent studies
+of fully microscopic description of fission process [77, 78].
+
+10
+30
+35
+40
+Z2/A
+0
+1
+2
+3
+4
+5
+6
+νpre
+162Yb
+206Po
+210Po
+215Fr
+244Cm
+264Rf
+216Ra
+248Cf
+Expt. data
+Present
+Multi-dimensional model
+FIG. 6.
+(Colour online) Comparison of measured pre-
+scission neutron multiplicities (νpre) with the results of the
+1D model (present work) and multi-dimensional models. The
+filled triangles (blue) denote experimental data [6, 14, 59, 80–
+83], the present dynamical model calculations are represented
+by filled circles (orange) and the filled squares (green) de-
+note the results of multi-dimensional dynamical calculations
+[28, 30, 34, 84].
+A microscopic framework based on the finite-temperature
+Skyrme-HartreeFock+BCS approach [79] was adopted to
+demonstrate the essential role of energy dependent fission
+barriers by studying the experimental fission probability
+of 210Po. It would be quite interesting to extend the in-
+vestigation of Fr nuclei within such a microscopic frame-
+work.
+IV.
+SUMMARY AND CONCLUSION
+In the present work we report a systematic study on
+the fission dynamics of N=126 shell closed nuclei in mass
+region 200 with a simultaneous description of three fis-
+sion observables. The present work highlights the limited
+reliability of the conclusions drawn from the recent statis-
+tical model analysis of shell closed nuclei, namely 210Po,
+212Rn and 213Fr at excitation energies 40 MeV and above,
+that advocated for extra shell effects at saddle configu-
+ration even after their inclusion in the level density for-
+mulation. Earlier analyses of νpre and ER cross-sections
+were based on different assumptions and case dependent
+parameter adjustments, without reaching a definite con-
+clusion.
+On the basis of present analysis we conclude
+that, without many of those assumptions and parameter
+adjustments, a well established combined dynamical and
+statistical model can simultaneously reproduce the avail-
+able data of νpre, fission and evaporation residue excita-
+tion functions (also fusion cross-sections in certain cases)
+for neutron shell closed nuclei, viz.
+210Po, 212Rn and
+their non-shell closed isotopes 206Po and 214,216Rn with-
+out the need of including any extra shell effects. There
+appears to be no discernible influence of N=126 neutron
+shell structure on these measured fission observables in
+the medium excitation energy range. The present work
+also points to a relatively smaller role of entrance channel
+effects in the studied systems.
+However, we find a significant mismatch between mea-
+sured νpre data and its model predictions for Fr nuclei
+formed in reactions 19F+194,196,198Pt and 16O+197Au,
+despite a reasonable description of fission and fusion
+cross-sections.
+The νpre data in Fr nuclei could only
+be reproduced after invoking a temperature dependent
+frictional form.
+The difficulty in completely reproduc-
+ing some specific measurements of Fr nuclei still remains
+not well-understood and additional measurements are de-
+sired. Although the present work is limited to the study
+of three fission observables, it would also be interesting
+to extend the systematic study using recent microscopic
+theory within Hartree-Fock + BCS framework.
+V.
+ACKNOWLEDGMENTS
+We are thankful to K. S. Golda and N. Saneesh for
+fruitful discussions. One of the authors (D.A.) acknowl-
+edges the financial support in the form of research fel-
+lowship received from the University Grants Commission
+(UGC).
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