aainstitutetext: Rudjer Boskovic Institute, Division of Theoretical Physics,
Bijenička 54, 10000 Zagreb, Croatiabbinstitutetext: Kavli IPMU (WPI), UTIAS,
The University of Tokyo, Kashiwa, Chiba 277-8583, Japanccinstitutetext: Asia
Pacific Center for Theoretical Physics, Pohang, 37673, Koreaddinstitutetext:
Quantum Theory Center ($\hbar$QTC) at IMADA and D-IAS, Southern Denmark Univ.,
Campusvej 55, 5230 Odense M, Denmarkeeinstitutetext: Dipartimento di Fisica
“E. Pancini" and INFN sezione di NapoliUniversità di Napoli Federico II, via
Cintia, 80126 Napoli, Italyffinstitutetext: Scuola Superiore Meridionale,
Largo S. Marcellino, 10, 80138 Napoli NA, Italygginstitutetext: School of
Physics, The University of Electronic Science and Technology of China, 88
Tian-run Road, Chengdu, China

# Infinite order results
for
Charged sectors of the Standard Model

Oleg Antipin b Jahmall Bersini c Pantelis Panopoulos d,e,f Francesco Sannino g
Zhi-Wei Wang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We determine anomalous dimensions of a family of fixed hypercharge operators
in the Standard Model featuring the general Cabibbo-Kobayashi-Maskawa
structure. The results are obtained at infinite orders in the couplings and to
leading and subleading orders in the charge. The computed anomalous dimensions
are shown to agree with the maximum known order in perturbation theory. We
further show that the large hypercharge sector of the Standard Model is
characterised by a non-Abelian vector condensation phase.

###### Keywords:

Standard Model, Gauge Symmetries, Large Charge

##  1 Introduction

The Standard Model (SM) of particle interactions is currently the most
successful theory of Nature. Therefore, testing it via direct comparison with
experiments constantly requires reducing theoretical uncertainty by resorting
to higher order computations. Additionally, certain sectors of the SM have not
yet been explored enough both theoretically and experimentally. These include
for example the physics stemming from families of composite operators
featuring multiple Higgses.

To inch forward in this direction, in this work we show how to renormalize the
family of lowest-lying Higgs operators with fixed hypercharge $Q$ by
determining their anomalous dimensions to infinite orders in the SM couplings
and leading and subleading orders in $Q$. This will be achieved by exploiting
and extending a bag of tools related to the semiclassical approach known as
the large charge expansion Hellerman:2015nra ; Monin:2016jmo ; Badel:2019oxl ;
Badel:2019khk ; Orlando:2019hte ; Orlando:2019skh ; Arias-Tamargo:2019xld ;
Gaume:2020bmp ; Orlando:2020yii ; Antipin:2020abu ; Antipin:2020rdw ;
Antipin:2022dsm ; Jack:2020wvs ; Jack:2021lja ; Cuomo:2022kio ; Cuomo:2020rgt
; Dondi:2022zna ; Antipin:2022naw ; Hellerman:2021qzz ; Antipin:2022hfe ;
Hellerman:2023myh ; Caetano:2023zwe ; Banerjee:2017fcx ; Jafferis:2017zna .
The leading order in $Q$ is given in (5) and at the next-to-leading in (6.2),
(6.2), (6.3), and (6.3). For the reader’s benefit and to keep the paper self-
contained we review the approach in the next section.

The analysis is semiclassical in nature Hellerman:2015nra ; Monin:2016jmo ;
Badel:2019oxl and determines the aforementioned anomalous dimensions via a
saddle-point expansion around a nontrivial vacuum. In the case of the SM, the
relevant classical trajectory is spatially homogeneous but anisotropic and
defines a vector condensation phase characterized by a nonvanishing
expectation value for both the Higgs field and the electroweak gauge bosons.
This is in contrast with most previous applications of the semiclassical large
charge approach where the ground state was an isotropic superfluid
Hellerman:2015nra ; Monin:2016jmo ; Badel:2019oxl ; Badel:2019khk ;
Gaume:2020bmp ; Antipin:2020abu ; Antipin:2020rdw ; Antipin:2022naw ;
Jack:2020wvs ; Jack:2021lja .

The Standard Model Lagrangian, notation and conventions are presented in Sec.
3. The identification of the family of Higgs composite operators with fixed
hypercharge is discussed in Sec. 4 while the leading and subleading
corrections in the charge $Q$ are determined respectively in Sec. 5 and Sec.
6. We show in Sec. 7 that our anomalous dimensions reproduce perturbative
results. The work further features several appendices providing supplementary
details of the computations including the explicit expression for the six-loop
anomalous dimensions.

##  2 Brief review of the semiclassical approach

The aim of this section is to provide a general overview of the strategy we
used to compute the anomalous dimension of fixed charge operators in the SM.
In particular, we will briefly describe the semiclassical framework developed
in Badel:2019oxl ; Badel:2019khk ; Antipin:2020abu ; Antipin:2020rdw ;
Antipin:2022naw ; Antipin:2022hfe .

Consider a quantum field theory invariant under the action of a certain global
symmetry group $\mathcal{G}$ in $D$ dimensions. We collectively denote the
couplings of the theory as $\kappa_{I}$. Any global symmetry implies the
existence of a conserved current $\mathcal{J}^{\mu}$ and a conserved charge
$Q$ given by

$Q=\int d^{D-1}x\,\mathcal{J}^{0}(x)\,,\qquad\frac{dQ}{dt}=0\,.$ (1)

The method uses an expansion in one over the charge to determine quantities
such as the anomalous dimension of certain fixed charge operators
$\mathcal{O}_{Q}$. To this extent, one exploits the fact that quantum systems
can be well approximated by classical dynamics in the presence of large
quantum numbers. In general, it is possible to fix up to rank$(\mathcal{G})$
charges, $Q_{i}$ which we can rescale as $Q_{i}=Qq_{i}$ in order to take $1/Q$
to be our small expansion parameter. The $\\{q_{i}\\}$ are a set of parameters
of order unity that define the charge configuration and uniquely specify the
irreducible representation according to which $\mathcal{O}_{Q}$ transforms. We
first engineer a perturbative fixed point of the renormalization group (RG)
flow where the theory is conformal and the anomalous dimensions are physical.
We then make use of standard conformal field theory (CFT) tools to set up the
$1/Q$ expansion and lastly, we rewrite our results for the anomalous
dimensions as perturbative series in the running coupling constants.
Crucially, in the latter form, the results remain valid outside the fixed
point, i.e. for the real-world SM, and are shown to match the outcome of
standard perturbation theory.

One starts by devising a perturbative fixed point of the Wilson-Fisher type by
tuning the mass parameters to $0$ and moving infinitesimally away from the
upper critical dimension of the theory. In the SM, this amounts to setting the
Higgs mass to $0$ and considering the theory in $D=4-\epsilon$ dimensions with
$\epsilon\ll 1$. Generically the zeros of the beta functions are complex but,
as has been shown in Antipin:2020rdw , we can still resort to the CFT toolbox.
These allow us to determine the anomalous dimension of fixed-charge operators
as a perturbative series in $\epsilon$. One can then use the fixed point
values of the couplings $\kappa_{I}^{*}=\kappa_{I}^{*}(\epsilon)$ to express
the results as a power series in the renormalized couplings $\kappa_{I}$. The
obtained results are valid away from the fixed point as confirmed
diagrammatically to high-loop orders in e.g. Jack:2020wvs ; Jack:2021lja ;
Bednyakov:2022guj ; Antipin:2022hfe .

We can now exploit the power of state-operator correspondence Cardy:1984rp by
first using Weyl invariance to map the theory onto a cylinder
$\mathbb{R}\times S^{D-1}$. Concretely, if we use polar coordinates
$(r,\Omega^{D-1})$ on $\mathbb{R}^{D}$ and we parameterize $\mathbb{R}\times
S^{D-1}$ by $(\tau,\Omega^{D-1})$, the map reads $r=Re^{\tau/R}$ with $R$ the
radius of $S^{D-1}$. Note that Weyl invariance requires the introduction of a
mass term for the scalar fields of the form $m^{2}H^{\dagger}H$ where
$m^{2}=(\frac{D-2}{2R})^{2}$. The latter arises from the coupling of the
scalar fields to the Ricci scalar of $S^{D-1}$. For the fermions and gauge
bosons, there are no extra correction terms when coupled to curved backgrounds
and we just replace simple derivatives with the covariant ones. Hence, the
free Dirac action becomes
$S=\int_{\mathcal{M}}d^{D}x\,\sqrt{-g}\,\,\bar{\psi}i\not{\nabla}_{\mathcal{M}}\psi\,,$
where
$\nabla_{\mu}\psi\equiv\partial_{\mu}\psi+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\,\psi$
and $\not{\nabla}_{\mathcal{M}}\equiv\gamma^{\mu}\nabla_{\mu}$ is the Dirac
operator on the manifold $\mathcal{M}=\mathbb{R}\times S^{D-1}$ with
$\gamma^{\mu}$ 111The infinitesimal Weyl transformations for the scalar,
fermion, and the vierbein are given respectively by
$\delta\phi=\frac{D-2}{2}\sigma(x)\phi\,$,
$\delta\psi=\frac{(D-1)}{2}\sigma(x)\psi(x)$ and $\delta
e^{a}_{\mu}=-\sigma(x)e^{a}_{\mu}$. .

Next, we use the state-operator correspondence Cardy:1984rp to relate the
scaling dimension $\Delta_{\mathcal{O}}$ of an operator $\mathcal{O}$ to the
energy $E_{\mathcal{O}}$ of the corresponding state $\mathcal{O}(0)\ket{0}$ on
the cylinder as $E_{\mathcal{O}}=\Delta_{\mathcal{O}}/R$. In particular, the
scaling dimension $\Delta_{Q,\\{q_{i}\\}}$ of the lowest-lying operators
carrying a given charge configuration $\left(Q,\\{q_{i}\\}\right)$ is given by
the ground state energy $E_{Q,\\{q_{i}\\}}$ of the theory on the cylinder in
the presence of a finite charge density. We can obtain the latter by
considering an arbitrary state $|{Q,\\{q_{i}\\}}\rangle$ with charges
$Q_{i}=Qq_{i}$ and compute the expectation value of the evolution operator
$e^{-HT}$ with $T$ the time interval and $H$ the Hamiltonian. It reads

$\langle
Q,\\{q_{i}\\}|e^{-HT}|Q,\\{q_{i}\\}\rangle=\mathcal{Z}^{-1}\int\mathcal{D}\Phi\,e^{-S_{\text{fixed}}[\Phi]}\,,$
(2)

where $S_{\text{fixed}}$ is the charge fixed action, $\Phi$ represents
collectively the field content of the theory, and $\mathcal{Z}$ is the
partition function of the theory. Taking the $T\to\infty$ limit the ground
state saturates the above matrix element as

$\langle Q,\\{q_{i}\\}|e^{-HT}|Q,\\{q_{i}\\}\rangle\stackrel{{\scriptstyle
T\to\infty}}{{\sim}}e^{-E_{Q,\\{q_{i}\\}}T}=e^{-\frac{\Delta_{Q,\\{q_{i}\\}}T}{R}}\,,$
(3)

thereby yielding $\Delta_{Q,\\{q_{i}\\}}$.

Having mapped the computation of the anomalous dimensions to the evaluation of
the ground state energy of a finite density QFT on the cylinder, we now
emphasize the advantages of the approach. The matrix element (3) can now be
computed via a semiclassical expansion equivalent to consider the following
double-scaling limit 222More precisely, since at the fixed point all the
couplings are functions of $\epsilon$, the double scaling limit is
$Q\to\infty\,,\quad\epsilon\to 0\,,\quad Q\epsilon=(\text{fixed})\,.$ (4)

$Q\to\infty\,,\quad\kappa_{I}\to 0\,,\quad Q\kappa_{I}=(\text{fixed})\,.$ (5)

Accordingly, the semiclassical expansion for the scaling dimension of the
lowest-lying operator with charge $Q$ takes the form

$E_{Q,\\{q_{i}\\}}R=\Delta_{Q,\\{q_{i}\\}}=\sum_{j=-1}\frac{1}{Q^{j}}\Delta_{j}\left(Q\kappa_{I}^{*},\\{q_{i}\\}\right)\,.$
(6)

The leading order $\Delta_{-1}$ is simply obtained by solving the equations of
motion (EOMs) and plugging the solution into the action to get the classical
energy. The subleading contribution $\Delta_{0}$ is given by the one-loop
grand potential on the cylinder which is a Gaussian path integral over the
fluctuations around the classical solutions. By varying the charge
configuration $\\{q_{i}\\}$ one accesses different operators transforming
according to different irreducible representations of $\mathcal{G}$.
Noticeably, each $\Delta_{j}$ in eq.(6) is the resummation of an infinite
series of Feynman diagrams. Concretely, $\Delta_{-1}$ resums the terms with
the leading power of $Q$ at every loop order, $\Delta_{0}$ resums the next-to-
leading powers, and so on. In fact, the conventional perturbative series for
the anomalous dimensions can be written as

$\Delta_{Q,\\{q_{i}\\}}=Q\left(\frac{D-2}{2}\right)+\sum_{l=1}P_{Q}^{(l-\text{loop})}\qquad\text{where}\qquad
P_{Q}^{(l-\text{loop})}=\sum_{k=0}^{l}C_{kl}Q^{l+1-k}\,.$ (7)

By comparing to eq.(6) one can see that the coefficients $C_{kl}$ appear in
the small ’t Hooft-like coupling $Q\kappa_{I}^{*}$ expansion of
$\Delta_{k-1}$. At any loop order, there are $l+1$ coefficients $C_{kl}$ to
compute. Two of them can be read off from the expressions for $\Delta_{-1}$
and $\Delta_{0}$. This opens the intriguing possibility of fixing the
remaining $l-1$ unknown coefficients by matching eq.(7) to the perturbative
results for $l-1$ different values of $Q$ if they are known in the literature.

In the opposite regime where $Q\kappa_{I}^{*}$ is taken to be large
$\Delta_{Q}$ assumes the following general form

$\displaystyle\Delta_{Q}=Q^{\frac{D}{D-1}}\left[\alpha_{1}+\alpha_{2}Q^{\frac{-2}{D-1}}+\alpha_{3}Q^{\frac{-4}{D-1}}+\ldots\right]+Q^{0}\left[\beta_{0}+\beta_{1}Q^{\frac{-2}{D-1}}+\ldots\right]+\mathcal{O}\left(Q^{-\frac{D}{D-1}}\right)\,.$
(8)

The above structure can be also derived by considering an EFT description for
the large charge sectors of generic interacting scalar CFT and is, therefore,
insensitive to the microscopic dynamics. However, the latter is encapsulated
in the parameters entering eq.(8) which are related to the Wilson coefficient
of the large charge EFT. At the same time, in odd dimensions, the coefficient
$\beta_{0}$ is universal and can be predicted within the EFT framework
Hellerman:2015nra . Analogously, in even dimensions $\log(Q)$ terms with
universal coefficients appear Cuomo:2020rgt . The known field theories whose
large charge sector is not described by the large charge EFT and do not
satisfy eq.(8) are non-interacting QFTs, supersymmetric theories where the
lowest-lying state is related to a BPS operator, and certain fermionic models
Dondi:2022zna .

Finally, in Antipin:2022hfe , the approach has been generalized to $U(1)$
gauge theories in the concrete framework of the Abelian Higgs model in
$D=4-\epsilon$ dimensions. In parallel, the construction of an on-shell
conserved charge in Abelian gauge theories has been recently elucidated in
Aoki:2022ugd . There are two differences with respect to the Abelian global
symmetry case. First, one needs to add a neutralizing charge background to
avoid long-range electric fields producing infrared divergences. Being non-
dynamical such a background only affects the EOM for the $U(1)$ gauge field as
follows

$\partial_{\mu}F^{\mu\nu}=J_{\text{tot}}^{\nu}=J_{\text{Matter}}^{\nu}-J_{\text{Background}}^{\nu}=0\,,$
(9)

where $F^{\mu\nu}$ and $J^{\nu}$ are the electromagnetic field strength and
current, respectively. Secondly, while in the global symmetry case,
$\Delta_{Q}$ corresponds to the scaling dimension of a certain local operator
333For instance, in the $\lambda(\bar{\phi}\phi)^{2}$ theory at its weakly
coupled Wilson-Fisher fixed point in $D=4-\epsilon$, $\Delta_{Q}$ is the
conformal dimension of the $\phi^{Q}$ operator Badel:2019oxl ., when the
$U(1)$ symmetry is gauged $\Delta_{Q}$ is related to a dressed two-point
function Antipin:2022hfe . For $Q=1$, the latter defines a non-local order
parameter for the continuous phase transitions described by the model e.g. for
the three-dimensional superconducting phase transition in the case of the
Abelian Higgs model.

##  3 Standard Model, notation and conventions

We are now ready to investigate the fixed hypercharge sector of the SM by
starting with a concise review of the model Lagrangian. The latter is
invariant under the gauge $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ symmetry
and can be decomposed as follows

$\mathcal{L}_{SM}=\mathcal{L}_{\text{kin}}+\mathcal{L}_{H}+\mathcal{L}_{\text{Yukawa}}\,.$
(10)

The gauge-fermion part of the Lagrangian is

$\displaystyle\mathcal{L}_{\text{kin}}=$ $\displaystyle\
i\sum_{i=1,2,3}\left(\bar{\mathcal{Q}}_{i}^{L}\not{D}\mathcal{Q}_{i}^{L}+\bar{u}_{i}^{R}\not{D}u_{i}^{R}+\bar{d}_{i}^{R}\not{D}d_{i}^{R}+\bar{L}_{i}^{L}\not{D}L_{i}^{L}+\bar{l}_{i}^{R}\not{D}l_{i}^{R}\right)$
$\displaystyle-\frac{1}{4}G_{\mu\nu}^{A}G^{\mu\nu\
A}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}W_{\mu\nu}^{a}W^{a\,\mu\nu}\,\,\,,$
(11)

where the covariant derivative is succinctly written as

$D_{\mu}=\partial_{\mu}-ig^{{}^{\prime}}\frac{Y}{2}B_{\mu}-i{g}\frac{\sigma^{a}}{2}W_{\mu}^{a}-ig_{s}T^{A}G_{\mu}^{A}\,\,.$
(12)

Here $B_{\mu}$, $W^{a}_{\mu}$, and $G^{A}_{\mu}$ are the gauge fields of the
$U(1)_{Y}$, $SU(2)_{L}$ and $SU(3)_{C}$ symmetries respectively with
$B_{\mu\nu}$, $W^{a}_{\mu\nu}$ and $G^{A}_{\mu\nu}$ and
$(g^{{}^{\prime}},g\,,g_{s})$ their corresponding gauge field strengths and
couplings. Also, $Y$ is the hypercharge, $\sigma^{a}$ are the Pauli matrices,
and $T^{A}$ the Gell-Mann matrices. The $SU(2)_{L}$ doublets
$\mathcal{Q}_{i}^{L}$ and $\mathcal{L}_{i}^{L}$ contain the left-handed quarks
and leptons, respectively, while $u_{i}^{R}$, $d_{i}^{R}$, $l_{i}^{R}$ are
$SU(2)_{L}$ singlets corresponding to the right-handed SM fermions. The
hypercharge values for the SM fields are gathered in the table below:

Fields $H$ $\mathcal{Q}^{L}$ $u^{R}$ $d^{R}$ $L^{L}$ $l^{R}$ $Y$ 1 1/3 4/3
-2/3 -1 -2

The Higgs-gauge sector of the SM is described by

$\displaystyle\mathcal{L}_{H}=(D_{\mu}H)^{\dagger}(D^{\mu}H)-m^{2}H^{\dagger}H-\frac{\lambda}{6}\left(H^{\dagger}H\right)^{2}\
,$ (13)

with $H$ the $SU(2)_{L}$ doublet

$H=\left(\begin{array}[]{c}\phi_{1}\\\ \phi_{2}\end{array}\right)\ .$ (14)

Last but not least the Lagrangian describing Yukawa interactions reads

$\mathcal{L}_{\text{Yukawa}}=-\left(Y_{u}^{ij}\left(\bar{\mathcal{Q}}_{i}^{L}H^{c}\right)u_{j}^{R}+Y_{d}^{ij}\left(\bar{\mathcal{Q}}_{i}^{L}H\right)d_{j}^{R}+Y_{l}^{ij}\left(\bar{L}_{i}^{L}H\right)l_{j}^{R}+\text{h.c.}\right)\,,$
(15)

where $H^{c}=i\sigma_{2}H^{\dagger}$ is the charge-conjugated scalar field.
The complex Yukawa matrices encode the flavor structure of the SM. All the
couplings $\kappa_{I}$ are renormalized as
$\kappa_{I0}=M^{\epsilon}Z_{\kappa_{I}}\kappa_{I}$ with $M$ the RG scale. We
will consider a minimal subtraction renormalization scheme and make use of the
one-loop value of the renormalization factors $Z_{\kappa_{I}}$ or
equivalently, the one-loop beta functions $\beta_{\kappa_{I}}$ which are
summarized in Appendix A. Finally, we redefine the couplings as
$\lambda\to(4\pi)^{2}\lambda$, $g\to 4\pi g$, $g^{\prime}\to 4\pi g^{\prime}$,
$g_{s}\to 4\pi g_{s}$, $Y_{u}\to 4\pi Y_{u}$, $Y_{d}\to 4\pi Y_{d}$, $Y_{l}\to
4\pi Y_{l}$.

##  4 Operator identification

The goal of this section is to identify the lowest-lying operator
$\mathcal{O}_{Q}$ with hypercharge $Q$ whose anomalous dimension $\Delta_{Q}$
will be determined in this work. Since the precise form of the operator
depends on which symmetries are gauged we start with the simple case of
vanishing gauge couplings. Since the ground state we consider is spinless we
are dealing with scalar operators. These are built out of tensor products of
the Higgs fields decomposed according to irreducible representations of the
$SU(2)_{L}$ symmetry as follows:

$\mathbf{2}^{\otimes n}=\sum_{k=0}^{[n/2]}a_{k}(\mathbf{n+1-2k})\,,$ (16)

where the coefficients $a_{k}$ are the multiplicity of the representation in
the tensor product decomposition. In the absence of the Yukawa couplings, the
Higgs sector of the Standard Model is invariant under the symmetry group
$O(4)$. As shown in Antipin:2020abu , the lowest-lying $O(4)$ operators with
total charge $Q$ (defined as the sum of the individual $O(4)$ charges) have
classical dimension $Q$ and are, therefore, built by multiplying the Higgs
field with itself $Q$ times and do not contain derivatives. Moreover, they
transform according to the traceless symmetric $O(4)$ representations with $Q$
indices. Then our operators appear in the branching of the latter in the
$SU(2)_{L}\times U(1)_{Y}$ subalgebra of $O(4)$. In terms of the Dynkin labels
for $O(4)$ ($D_{2}$) and $SU(2)_{L}$ ($A_{1}$), the branching rule reads

$[Q,0]_{D_{2}}\Rightarrow\sum_{k=0}^{Q}[Q-k]_{A_{1}}^{(Q-2k)}\,,$ (17)

where the superscript denotes the $U(1)_{Y}$ charge. Since the operators
appearing on the right hand side have all classical dimension $Q$ by
construction, the lowest-lying operators with $U(1)_{Y}$ charge $Q$ are
obtained when $k=0$. We, therefore, conclude that the lowest-lying operators
with $U(1)_{Y}$ charge $Q$ have no derivatives, classical scaling dimension Q,
and transform according to the $SU(2)_{L}$ representations
$[Q]_{A_{1}}=\mathbf{Q+1}$. Note that this corresponds to taking $n=Q$ and
$k=0$ in eq.(16). These operators can be compactly written as

$\mathbf{Q+1}\Rightarrow\mathcal{O}_{Q}=H^{I_{1}}\dots H^{I_{Q}}\,.$ (18)

For instance, the operator with $Q=1$ is the Higgs field itself, while the one
with $Q=2$ transforms in the triplet (the adjoint) of $SU(2)_{L}$.

Now consider turning on the hypercharge gauge coupling $g^{\prime}$. As
mentioned, the form of the relevant charged operator changes. In particular,
in a CFT the $\Delta_{Q}$ characterizes the scaling of the two-point function
of $\mathcal{O}_{Q}$ at criticality according to

$G(x_{i},x_{f})=\braket{\mathcal{O}_{Q}(x_{f})\mathcal{O}_{Q}(x_{i})}=\frac{1}{\rvert
x_{f}-x_{i}\rvert^{2\Delta_{Q}}}\,.$ (19)

However, for $\mathcal{O}_{Q}$ given by eq.(18), the above correlator is
gauge-dependent and evaluates to zero due to Elitzur’s theorem Elitzur:1975im
. It is therefore necessary to dress the two-point function via the
introduction of a gauge line connecting the two external points as

$G(x_{i},x_{f})=\braket{H^{I_{1}}\dots H^{I_{Q}}(x_{f})\exp\left(i\
g^{\prime}\,Q\int_{x_{i}}^{x_{f}}d^{D}x\,J^{\mu}(x)B_{\mu}(x)\right)H^{I_{1}}\dots
H^{I_{Q}}(x_{i})}\,,$ (20)

where $J^{\mu}$ satisfies the Ward identity

$\partial_{\mu}J^{\mu}=\delta(x-x_{f})-\delta(x-x_{i})\,,$ (21)

which ensures the gauge invariance of the two-point function. Therefore, the
problem of determining the lowest-lying operator with hypercharge $Q$ turned
into the problem of determining $J^{\mu}$ such that $G(x_{i},x_{f})$ scales
according to the smallest conformal dimension. It has been shown in
Antipin:2022hfe that the smallest scaling dimension is achieved by considering
a _Dirac line_ Dirac . The latter satisfies the $\partial^{2}J_{\mu}=0$
condition while the explicit form of the non-local current is
$J_{\mu}=J^{{}^{\prime}}_{\mu}(z-x_{f})-J^{{}^{\prime}}_{\mu}(z-x_{i})$ where

$J^{{}^{\prime}}_{\mu}(z)=-i\int\frac{d^{D}k}{(2\pi)^{D}}\frac{k_{\mu}}{k^{2}}e^{ik\cdot
z}=-\frac{\Gamma(D/2-1)}{4\pi^{D/2}}\partial_{\mu}\frac{1}{z^{D-2}}\,\,.$ (22)

This construction allows us to identify $\Delta_{Q}$ as the anomalous
dimension of the dressed operator

$\mathcal{O}_{Q}(x)=e^{-ig^{\prime}Q\int
d^{D}zJ^{{}^{\prime}}_{\mu}(z-x)B^{\mu}(z)}H^{I_{1}}\dots H^{I_{Q}}\,,$ (23)

whose two-point function at criticality is given by eq.(19). The above
operator can be interpreted as the insertion of $Q$ units of charge at the
point $x$ dressed with a coherent state of photons describing the induced
Coulomb field. Interestingly, since $J^{{}^{\prime}}_{\mu}$ is a total
derivative, the two-point function (20) reduces to the gauge-dependent
correlator $\braket{H^{I_{1}}\dots H^{I_{Q}}(x_{f})H^{I_{1}}\dots
H^{I_{Q}}(x_{i})}$ in the Landau gauge $\partial^{\mu}B_{\mu}=0$. As a
consequence, the perturbative results for $\Delta_{Q}$ match the scaling
dimension of the local operator (18) in the Landau gauge.

Since the operator (23) transforms nontrivially under $SU(2)_{L}$, when the
latter symmetry is gauged due to a non-vanishing value of $g$, $\Delta_{Q}$ no
longer corresponds to its conformal dimension. As a consequence, although the
computations are still valid, the identification of the lowest-lying operator
with hypercharge $Q$ becomes a subtle issue that goes beyond the scope of this
work 444For instance, we note that the construction of a non-Abelian
counterpart of the Abelian Dirac line, which has been suggested as the order
parameter for the confining phase of QCD, is an interesting open problem
Caudy:2007sf ; Greensite:2017ajx .. In what follows, we will, however, make
use of the $g\to 0$ limit, for which we can identify the relevant two-point
function as eq.(20), to partly check the validity of our calculations against
diagrammatic calculations. Concretely, for $g=0$ and in the Landau gauge we
will show that our results for $\Delta_{Q}$ match the three-loop anomalous
dimension of the operators defined in (18) sasha . Moreover, the $Q=1$ case
yields the anomalous dimension of the Higgs field which has been computed to
high precision in e.g. Bednyakov:2013eba ; Chetyrkin:2012rz ;
Bednyakov:2013cpa . Finally, we stress that no operator identification issue
arises when the weak symmetry is global providing relevant results for testing
higher-order computations stemming from $\lambda$, $g_{S}$, and $y_{t}$.

##  5 Scaling dimensions at leading order

Our goal is to compute the scaling dimension $\Delta_{Q}$ of the lowest-lying
operator with a certain value of the hypercharge $Q$. According to the
discussion in Sec. 2, in the double scaling limit (5) the scaling dimension
takes the form (6) with the leading term given by the classical ground state
energy of the conformal theory on $\mathbb{R}\times S^{D-1}$. In what follows,
we will measure all the dimensionful quantities in units of the radius of
$S^{D-1}$ which we set equal to unity. As previously discussed, conformal
symmetry on the cylinder is achieved by tuning the Higgs mass term to
$m^{2}=\left(\frac{d-2}{2}\right)^{2}$ and engineering a complex zero for all
the SM beta functions (which we summarize in App.A) in $D=4-\epsilon$
dimensions. We stress again that this procedure is just a computational trick
and the final results will apply to the real-world SM as well. The hypercharge
is fixed by introducing into the SM Lagrangian (10) the associated chemical
potential as the temporal component of a background $U(1)_{Y}$ gauge field,
i.e. by upgrading the covariant derivative as

$D_{\mu}\to D_{\mu}-i\mu Y\,.$ (24)

The relevant equations of motion are

$\displaystyle\partial_{\nu}B^{\mu\nu}$ $\displaystyle=0\ ,$ (25)
$\displaystyle-(D_{\mu}-i\mu\delta_{\mu 0})(D^{\mu}-i\mu\delta^{\mu
0})H-m^{2}H-\frac{(4\pi)^{2}}{3}\lambda(H^{\dagger}H)H$ $\displaystyle=0\ ,$
(26) $\displaystyle\partial^{\mu}W_{\mu\nu}^{(a)}+4\pi
g\epsilon^{abc}A^{\mu(b)}W_{\mu\nu}^{(c)}+4\pi
ig\left[H^{\dagger}\frac{\tau^{a}}{2}\partial_{\nu}H-\partial_{\nu}H^{\dagger}\frac{\tau^{a}}{2}H\right]$
$\displaystyle+\frac{g^{2}(4\pi)^{2}}{2}W_{\nu}^{(a)}H^{\dagger}H+8\pi
g\mu\delta_{\nu 0}H^{\dagger}\frac{\tau^{a}}{2}H$ $\displaystyle=0\ .$ (27)

As discussed in Sec. 2, we enforced charge neutrality by introducing a
background current which makes the RHS of the first EOM vanishing. Moreover,
we assumed a zero classical value for both the gluon field (since it does not
couple directly to the Higgs) and the fermions. This trivializes all the EOMs
except the above. As shown in Gusynin:2003yu , the global minimum of the
action is achieved by a vacuum solution that breaks rotational invariance and
is described by the following ansatz

$\displaystyle W_{3}^{(+)}=(W_{3}^{(-)})^{*}=C\neq 0\ ,\quad W_{0}^{(3)}=P\neq
0\ ,\quad H=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
v\end{array}\right)\ ,$ (30) $\displaystyle v\neq 0\ ,\quad
W_{1,2}^{(\pm)}=W_{0}^{(\pm)}=W_{1,2}^{(3)}=W_{3}^{(3)}=0\ ,\quad
B_{\mu}=0\,,$ (31)

where we introduced $W_{\mu}^{(\mp)}=\frac{1}{\sqrt{2}}(W_{\mu}^{(1)}\pm
iW_{\mu}^{(2)})$. The various vacuum expectation values are determined by the
EOMs as

$\displaystyle\left(P^{2}-\frac{v^{2}}{4}\right)C$ $\displaystyle=0\ ,$ (32)
$\displaystyle\left(2|C|^{2}+\frac{v^{2}}{4}\right)P-\frac{\mu v^{2}}{8\pi g}$
$\displaystyle=0\ ,$ (33) $\displaystyle\left[\left(\mu-2\pi
gP\right)^{2}-m^{2}-\frac{(4\pi)^{2}}{6}\lambda
v^{2}-\frac{(4\pi)^{2}}{2}g^{2}|C|^{2}\right]v$ $\displaystyle=0\,,$ (34)

with solution

$P=\frac{v}{2}\ ,\quad|C|^{2}=v\left(\frac{\mu}{8\pi g}-\frac{v}{8}\right)\
,\quad\tilde{v}\equiv 4\pi
v=\frac{9g\mu-\sqrt{96\lambda\left(\mu^{2}-m^{2}\right)+9g^{2}\left(\mu^{2}+8m^{2}\right)}}{3g^{2}-4\lambda}\,.$
(35)

The classical action evaluated on the above solution reads

$\mathcal{S}^{(0)}=\frac{m^{2}}{16}\tilde{v}^{2}-\frac{1}{16}\left(\frac{g\tilde{v}}{4}-\mu\right)^{2}\tilde{v}^{2}+\frac{\lambda}{192}\tilde{v}^{4}+\mu
Q\,.$ (36)

Finally, the relation between $Q$, $\mu$, and $v$ is obtained via the last EOM
$\partial\mathcal{S}^{(0)}/\partial\mu=0$, which yields

$\mu\tilde{v}^{2}-\frac{g\tilde{v}^{3}}{4}=8Q\,.$ (37)

One can easily check that in the limit $g\to 0$ vector condensation does not
occur, i.e. $C=P=0$. In this case the ground state is isotropic, homogeneous,
and characterized by scalar condensation with the vev of the scalar field $v$,
and the classical action $\mathcal{S}$ given by eq.(35) and eq.(36) with
$g=0$. The solution with $g=0$ and the associated symmetry-breaking pattern
define a phase of matter dubbed _conformal superfluid phase_ describing, along
with its non-Abelian generalization, the large-charge sector of most CFTs
considered in the literature Hellerman:2015nra ; Monin:2016jmo ; Badel:2019khk
; Jack:2021lja ; Antipin:2020rdw ; Badel:2019oxl ; Antipin:2022naw ;
Antipin:2022hfe ; Antipin:2020abu . On the other hand, the large-charge sector
of the Weinberg-Salam model realizes a different phase characterized by vector
condensation and the associated breaking of spatial rotations Sannino:2002wp ;
Gusynin:2003yu . Indeed, the solution (31) is spatially homogeneous but
anisotropic and breaks $SO(3)\times SU(2)_{L}\times U(1)_{Y}$ down to $SO(2)$
(spatial rotations in the $x_{1}-x_{2}$ plane). The spectrum of excitations
has been studied in Sannino:2002wp for massive vectors fields and in
Gusynin:2003yu for the SM. It contains two massless relativistic modes. In
other words, the Goldstone theorem works as usual when $SO(3)$ is viewed as a
global symmetry555For a general counting of the Goldstone bosons versus
different phases that relativistic vector condensation can feature see
Sannino:2002wp .

A few comments are in order.

  * •

The phase we consider is not the phase usually discussed in the context of SM
thermodynamics Loewe:2004zw ; Kapusta:1990qc ; Kapusta:1981aa . In fact, when
the Higgs mass term is negative, the SM is in the broken phase already at
$\mu=0$ and it is possible to realize a phase with unbroken rotational
invariance. The corresponding classical solution reads

$\tilde{v}^{2}=-\frac{6m^{2}}{\lambda}\,,\quad W_{0}^{(3)}=\frac{\mu}{2\pi
g}\,,\quad W_{0}^{(\pm)}=0\,,\quad B_{\mu}=0\,.$ (38)

This solution provides the ground state of the SM for low values of the
chemical potential $\mu^{2}<\frac{3g^{2}}{8\lambda}\rvert m^{2}\rvert$.
However, at the critical point $\mu^{2}=\frac{3g^{2}}{8\lambda}\rvert
m^{2}\rvert$ a second order phase transition to a phase with broken rotational
invariance occurs Gusynin:2003yu .

  * •

The vev for the Higgs field (35) diverges at $g^{2}=\frac{4}{3}\lambda$
signaling a phase transition. While for $g^{2}>\frac{4}{3}\lambda$, the
solution (35) is still a local minimum of the potential, the latter becomes
unbounded from below. In what follows we will therefore assume
$g^{2}<\frac{4}{3}\lambda$. However, this restriction will not affect our
results for the anomalous dimensions which hold for arbitrary values of the
coupling.

  * •

One may try to avoid spontaneous breaking of rotational invariance by
enforcing non-Abelian charge neutrality via the introduction of a non-Abelian
background current $J_{\mu}^{a}$. However, this leads to inconsistencies in
the spectrum of physical modes Hama:2011rt . In fact, as pointed out in
Watanabe:2014qla , in the non-Abelian case, it is not possible to include a
non-dynamical charge background without violating the symmetry of the theory.

Let us recap the calculation of $\Delta_{-1}$ in the $g\to 0$ limit where the
solution to the EOMs is

$\mu=\frac{y^{2/3}+3^{1/3}}{3^{2/3}y^{1/3}}\ ,\quad y=6\lambda
Q+\sqrt{(6\lambda Q)^{2}-3}\,,$ (39)

whereas the classical action reads

$\mathcal{S}^{(0)}=\frac{Q\left(3\mu^{2}+m^{2}\right)}{4\mu}\,.$ (40)

Then the leading order in the semiclassical expansion is obtained by
substituting the solution of eq.(39) into the above yielding

$Q\,\Delta_{-1}=\frac{6\ 3^{1/3}y^{2/3}-3y^{4/3}+3^{1/3}y^{8/3}+2\
3^{2/3}y^{2}+3\ 3^{2/3}}{48y^{4/3}\lambda}\ .$ (41)

This result holds in many models with $O(N)$ symmetry in $D=4-\epsilon$
dimensions Badel:2019oxl ; Antipin:2020abu ; Antipin:2022hfe ; Antipin:2022naw
; Jack:2021lja where it resums the terms with the highest power of $Q$ to all
orders in perturbation theory. For the $g\neq 0$ case, we obtain

$\displaystyle Q\Delta_{-1}$
$\displaystyle=\frac{3}{32\left(3g^{2}-4\lambda\right)^{3}}\Bigg{(}9g^{4}\left(3\mu_{0}^{4}-20\mu_{0}^{2}+8\right)-48g^{2}\lambda\left(15\mu_{0}^{4}-\mu_{0}^{2}+4\right)$
$\displaystyle+128\lambda^{2}\left(-3\mu_{0}^{4}+2\mu_{0}^{2}+1\right)+3g\mu_{0}^{3}\left(3g^{2}+32\lambda\right)\sqrt{9g^{2}\left(\mu_{0}^{2}+8\right)+96\lambda\left(\mu_{0}^{2}-1\right)}\Bigg{)}\,,$
(42)

where $\mu_{0}$ is the real solution of

$\frac{\mu_{0}\left(9g\mu_{0}-\sqrt{9g^{2}\left(\mu_{0}^{2}+8\right)+96\lambda\left(\mu_{0}^{2}-1\right)}\right)^{2}}{8\left(3g^{2}-4\lambda\right)^{2}}-\frac{g\left(9g\mu_{0}-\sqrt{9g^{2}\left(\mu_{0}^{2}+8\right)+96\lambda\left(\mu_{0}^{2}-1\right)}\right)^{3}}{32\left(3g^{2}-4\lambda\right)^{3}}=Q\,,$
(43)

which minimizes $\Delta_{-1}$. Unlike the $g=0$ case, it is not possible to
obtain a compact expression for $\Delta_{-1}$ as a function of the ’t Hooft-
like couplings $\lambda Q$ and $g\sqrt{Q}$. However, it is possible to solve
eq.(43) perturbatively for small values of the ’t Hooft-like couplings and
determine $\Delta_{-1}$ to arbitrarily high orders in such expansion. As
discussed in Sec. 2, this is the regime where we recover ordinary diagrammatic
results. In Sec. 7 we will provide an explicit expression for the small ’t
Hooft-like coupling expansion of $\Delta_{-1}$ up to $3$ loops and check it
against perturbation theory. Novel additional results up to the six-loop order
are collected in App.C.

##  6 Scaling dimensions at next-to-leading order

The next-to-leading order $\Delta_{0}$ consists of the Gaussian path integral
over the fluctuations on top of the classical solution. Concretely, it can be
written as the sum of the zero-point energies. For this involved contribution
the $SU(2)_{L}$ gauge coupling $g$ is set to zero. This allows us to provide a
more transparent presentation of the key issues related to this computation
while setting the stage for the complete result in a forthcoming work.

Since bosons and fermions do not mix at the quadratic level, we can separate
their contributions as

$\displaystyle\Delta_{0}=\Delta_{0}^{(bos)}-\Delta_{0}^{(fm)}\,,$ (44)

where Badel:2019oxl ; Antipin:2020abu

$\displaystyle\Delta_{0}^{(fm)}=\frac{1}{2}\sum_{i=\text{fermions}}\sum_{\ell=\ell_{0}}^{\infty}n_{i}(\ell)\omega_{i}(\ell)\,,\qquad\Delta_{0}^{(bos)}=\frac{1}{2}\sum_{i=\text{bosons}}\sum_{\ell=\ell_{0}}^{\infty}n_{i}(\ell)\omega_{i}(\ell)\,,$
(45)

where the index $i$ runs over all the nontrivial dispersion relations
($\omega\neq p$ with $p$ the spatial momentum) $\omega_{i}$ of the spectrum.
Here $\ell$ labels the eigenvalues of the relevant differential operator
(Laplacian and Dirac operators for bosons and fermions, respectively) on
$S^{D-1}$ which have multiplicity $n_{i}(\ell)$. The relevant expressions for
fields of different spin are collected in App.B. In the next section, we
illustrate the calculation of $\Delta_{0}^{(fm)}$ in the case of a single
generation of fermions.

###  6.1 Warm-up: single generation Yukawa contribution

The Yukawa interactions for the third generation of fermions read

$\displaystyle\mathcal{L}_{\text{Yukawa}}=$ $\displaystyle-4\pi
y_{t}\bar{\mathcal{Q}}^{L}H^{c}t_{R}-4\pi
y_{b}\bar{\mathcal{Q}}^{L}Hb_{R}-4\pi
y_{\tau}\bar{L}^{L}H\tau_{R}+\text{h.c.}$ $\displaystyle=$ $\displaystyle-4\pi
y_{t}\left[(\bar{t}P_{R}t)\phi_{2}^{*}+(\bar{t}P_{L}t)\phi_{2}-(\bar{b}P_{R}t)\phi_{1}^{*}-(\bar{t}P_{L}b)\phi_{1}\right]$
$\displaystyle-4\pi
y_{b}\left[(\bar{b}P_{R}b)\phi_{2}+(\bar{b}P_{L}b)\phi_{2}^{*}-(\bar{t}P_{R}b)\phi_{1}-(\bar{b}P_{L}t)\phi_{1}^{*}\right]$
$\displaystyle-4\pi
y_{\tau}\left[(\bar{\tau}P_{R}\tau)\phi_{2}+(\bar{\tau}P_{L}\tau)\phi_{2}^{*}-(\bar{\nu}_{\tau}P_{R}\tau)\phi_{1}-(\bar{\tau}P_{L}\nu_{\tau})\phi_{1}^{*}\right]\
.$ (46)

Here $\mathcal{Q}^{L}=(t,b)_{L}$ and $L^{L}=(\nu_{\tau},\tau)_{L}$ are the
$SU(2)_{L}$ left-handed doublets and
$P_{L/R}=\frac{1}{2}\left(1\pm\gamma_{5}\right)$. For the quadratic Lagrangian
of the fermions we have

$\displaystyle\mathcal{L}_{fm}^{(2)}$ $\displaystyle=i\bar{t}\not{\nabla}t-\mu
Y_{\mathcal{Q}_{L}}\bar{t}\gamma_{0}P_{L}t-\mu
Y_{u_{r}}\bar{t}\gamma_{0}P_{R}t+\frac{y_{t}}{\sqrt{2}}\tilde{v}\bar{t}t+i\bar{b}\not{\nabla}b-\mu
Y_{\mathcal{Q}_{L}}\bar{b}\gamma_{0}P_{L}b-\mu
Y_{d_{r}}\bar{b}\gamma_{0}P_{R}b$
$\displaystyle+\frac{y_{b}}{\sqrt{2}}\tilde{v}\bar{b}b+i\bar{\tau}\not{\nabla}\tau-\mu
Y_{L_{L}}\bar{\tau}\gamma_{0}P_{L}\tau-\mu
Y_{l_{r}}\bar{\tau}\gamma_{0}P_{R}\tau+\frac{y_{\tau}}{\sqrt{2}}\tilde{v}\bar{\tau}\tau+i\bar{\nu_{\tau}}\not{\nabla}\nu_{\tau}-\mu
Y_{L_{L}}\bar{\nu_{\tau}}\gamma_{0}P_{L}\nu_{\tau}\,,$ (47)

where the scalar vev is given by eq.(35) for $g=0$, that is

$\tilde{v}^{2}=\frac{6}{\lambda}\left(\mu^{2}-m^{2}\right)\,.$ (48)

It is possible to redefine the left-handed neutrino field as $\nu_{\tau}\to
e^{-Y_{L_{L}}\mu t}\nu_{\tau}$, $\bar{\nu}_{\tau}\to e^{Y_{L_{L}}\mu
t}\bar{\nu}_{\tau}$ to eliminate the last term from the Lagrangian. Then the
neutrino has a trivial dispersion relation $\omega=p$ and does not contribute
to $\Delta_{0}$. The remaining fermions do not mix among themselves and have
the same quadratic Lagrangian except for the hypercharge assignation and the
relevant Yukawa coupling. The quadratic Lagrangian for the top quark field
entails the following inverse propagator

$D^{-1}=\not{p}-\mu Y_{\mathcal{Q}_{L}}\gamma_{0}P_{L}-\mu
Y_{u_{R}}\gamma_{0}P_{R}+\frac{\tilde{v}}{\sqrt{2}}y_{t}\mathbbm{1}_{4\times
4}\,.$ (49)

The dispersion relations are found by solving $\det D_{i}^{-1}=0$ for every
fermion. Noticeably, despite the different values of the hypercharge, the
dispersion relations have all the same form up to an irrelevant term that does
not contribute to $\Delta_{Q}$. After dropping the latter, we arrive at the
following dispersion relations

$\omega_{\pm}^{(i)}(\ell)=\sqrt{\left(\frac{\mu}{2}+\lambda_{f\pm}\right)^{2}+\frac{y_{i}^{2}\tilde{v}^{2}}{2}}\,\,.$
(50)

where $i=t,b,\tau$ and $\lambda_{f\pm}$ denotes the eigenvalues of the Dirac
operator on $S^{D-1}$ whose value and degeneracy are given in eq.(115). Since
all the dispersion relations have the same form the Yukawa contributions to
$\Delta_{0}$ can be divided as

$\displaystyle\Delta_{0}^{(fm)}=N\Delta_{0}^{(t)}+N\Delta_{0}^{(b)}+\Delta_{0}^{(\tau)}\,,$
(51)

where $N$ stands for the number of colors and

$\Delta_{0}^{(t)}=\frac{1}{2}\sum_{\ell=0}n_{f}(\ell)\left(\omega_{+}^{(t)}(\ell)+\omega_{-}^{(t)}(\ell)\right)\,,\qquad\Delta_{0}^{(b)}=\Delta_{0}^{(t)}(y_{t}\to
y_{b})\,,\qquad\Delta_{0}^{(\tau)}=\Delta_{0}^{(t)}(y_{t}\to y_{\tau})\,.$
(52)

The sum over $\ell$ needs to be regularized and renormalized. The former step
can be easily performed by subtracting the divergent terms in the expansion of
the summand around $\ell=\infty$. The sum over subtracted terms is then zeta-
regularized and added back to the result. However, the "logarithmic
divergence" $\sim\sum_{\ell=1}\frac{1}{\ell}$ needs to be further regularized
using dimensional regularization as
$\sum_{\ell=1}\frac{1}{\ell}=\zeta(1+\epsilon)=\frac{1}{\epsilon}+\gamma_{E}$
with $\gamma_{E}$ the Euler-Mascheroni constant. This allows isolating a
divergent term scaling as $\frac{1}{\epsilon}$ which is canceled when
renormalizing the classical contribution $\Delta_{-1}$. We obtain

$\displaystyle\Delta_{0}^{(t)}$
$\displaystyle=\sqrt{\frac{12\left(\mu^{2}-1\right)y_{t}^{2}}{\lambda}+(\mu-3)^{2}}+\sqrt{\frac{12\left(\mu^{2}-1\right)y_{t}^{2}}{\lambda}+(\mu+3)^{2}}$
$\displaystyle-\frac{3\left(\mu^{2}-1\right)y_{t}^{2}\left(\lambda\left(13-3\mu^{2}\right)+3\left(3\mu^{2}+1\right)y_{t}^{2}\right)}{8\lambda^{2}}-6+\frac{1}{2}\sum_{\ell=1}^{\infty}\sigma^{(t)}\,,$
(53)

where

$\begin{split}\sigma^{(t)}(\ell)=&\frac{1}{2\lambda^{2}\ell}\Big{(}-4\lambda^{2}\ell(\ell+1)(\ell+2)(2\ell+3)-3\lambda\left(\mu^{2}-1\right)y_{t}^{2}\left(4\ell^{2}+\mu^{2}+6\ell-1\right)+9\left(\mu^{2}-1\right)^{2}y_{t}^{4}\Big{)}\\\
&+2(\ell+1)(\ell+2)\left(\sqrt{\frac{3\left(\mu^{2}-1\right)y_{t}^{2}}{\lambda}+\left(\frac{2\ell+3-\mu}{2}\right)^{2}}+\sqrt{\frac{3\left(\mu^{2}-1\right)y_{t}^{2}}{\lambda}+\left(\frac{2\ell+3+\mu}{2}\right)^{2}}\right)\,.\end{split}$
(54)

Having solved the one-generation case we can now tackle the full flavour
sector of the SM.

###  6.2 Complex Yukawa matrices

The flavour structure of the SM is encoded in the Yukawa couplings summarised
as three complex $n_{G}\times n_{G}$ matrices where $n_{G}$ is the number of
generations yielding the following Lagrangian

$\mathcal{L}_{\text{Yukawa}}=-4\pi\left(Y_{u}^{ij}\left(\mathcal{Q}_{i}^{L}H^{c}\right)u_{j}^{R}+Y_{d}^{ij}\left(\mathcal{Q}_{i}^{L}H\right)d_{j}^{R}+Y_{l}^{ij}\left(L_{i}^{L}H\right)l_{j}^{R}\right)\,.$
(55)

As we shall see, our results will depend on the following traces of Yukawa
matrices

$\mathcal{Y}_{f}=(4\pi)^{2}{\rm
Tr}Y_{f}Y_{f}^{\dagger}\,,\quad\mathcal{Y}_{ff}=(4\pi)^{4}{\rm
Tr}(Y_{f}Y_{f}^{\dagger})^{2}\,,\quad\mathcal{Y}_{fff}=(4\pi)^{6}{\rm
Tr}(Y_{f}Y_{f}^{\dagger})^{3}\,,\quad f=u,d,l\,.$ (56)

Analogously to the $n_{G}=1$ case, the fermionic contribution can be divided
as

$\displaystyle\Delta_{0}^{(fm)}$
$\displaystyle=N\left(\Delta_{0}^{(f=u)}+\Delta_{0}^{(f=d)}\right)+\Delta_{0}^{(f=l)}\,,$
(57) $\displaystyle\Delta_{0}^{(f)}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{n_{G}}\sum_{\ell=0}n_{f}(\ell)\left(\omega^{(i)}_{f+}(\ell)+\omega^{(i)}_{f-}(\ell)\right)\,.$
(58)

For each value of $f$ the inverse propagator is a $4n_{G}\times 4n_{G}$ matrix
where the off-diagonal $4\times 4$ blocks contain the mixing among different
generations owed to the Yukawa structure. For instance, the inverse propagator
for the up quarks in the $n_{G}=3$ case reads

$\displaystyle D^{-1}=\left(\begin{array}[]{ccc}U_{1}&V_{12}&V_{13}\\\
V_{21}&U_{2}&V_{23}\\\ V_{31}&V_{32}&U_{3}\end{array}\right)\,,$ (62)

where

$\displaystyle U_{i}$ $\displaystyle=\not{p}-\mu
Y_{\mathcal{Q}_{L}}\gamma_{0}P_{L}-\mu
Y_{u_{R}}\gamma_{0}P_{R}+\frac{\tilde{v}}{\sqrt{2}}Y_{u}^{ii}P_{L}+\frac{\tilde{v}}{\sqrt{2}}\left(Y_{u}^{ii}\right)^{*}P_{R}\,,\qquad
i=1,2,3$ (63) $\displaystyle V_{ij}$
$\displaystyle=\frac{\tilde{v}}{\sqrt{2}}\text{diag}\left(Y_{u}^{ij},Y_{u}^{ij},\left(Y_{u}^{ji}\right)^{*},\left(Y_{u}^{ji}\right)^{*}\right)\,.$
(64)

For the two-generation case, the dispersion relations read

$\displaystyle\omega_{f\pm}^{(1)}(\ell)$
$\displaystyle=\sqrt{\left(\frac{\mu}{2}+\lambda_{f\pm}\right)^{2}+\frac{\tilde{v}^{2}}{4}\left(\mathcal{Y}_{f}+\sqrt{2\mathcal{Y}_{ff}-\mathcal{Y}_{f}^{2}}\right)}\,,$
(65) $\displaystyle\omega_{f\pm}^{(1)}(\ell)$
$\displaystyle=\sqrt{\left(\frac{\mu}{2}+\lambda_{f\pm}\right)^{2}+\frac{\tilde{v}^{2}}{4}\left(\mathcal{Y}_{f}-\sqrt{2\mathcal{Y}_{ff}-\mathcal{Y}_{f}^{2}}\right)}\,,$
(66)

whereas in the complete $n_{G}=3$ case we obtain

$\displaystyle\omega_{f\pm}^{(1)}(\ell)$
$\displaystyle=\sqrt{\left(\frac{\mu}{2}+\lambda_{f\pm}\right)^{2}-\frac{\tilde{v}^{2}}{3A}\left(A^{2}-2A\mathcal{Y}_{f}-2\mathcal{Y}_{f}^{2}+6\mathcal{Y}_{ff}\right)}\,,$
(67) $\displaystyle\omega_{f\pm}^{(2)}(\ell)$
$\displaystyle=\sqrt{\left(\frac{\mu}{2}+\lambda_{f\pm}\right)^{2}+\frac{\tilde{v}^{2}}{24A}\left(4A\mathcal{Y}_{f}+\left(1-i\sqrt{3}\right)A^{2}-2\left(1+i\sqrt{3}\right)\left(\mathcal{Y}_{f}^{2}-3\mathcal{Y}_{ff}\right)\right)}\,,$
(68) $\displaystyle\omega_{f\pm}^{(3)}(\ell)$
$\displaystyle=\sqrt{\left(\frac{\mu}{2}+\lambda_{f\pm}\right)^{2}+\frac{\tilde{v}^{2}}{24A}\left(4A\mathcal{Y}_{f}+\left(1+i\sqrt{3}\right)A^{2}-2\left(1-i\sqrt{3}\right)\left(\mathcal{Y}_{f}^{2}-3\mathcal{Y}_{ff}\right)\right)}\,,$
(69)

where

$\displaystyle A=$
$\displaystyle\bigg{(}6\sqrt{2}\sqrt{\mathcal{Y}_{f}^{6}-9\mathcal{Y}_{f}^{4}\mathcal{Y}_{ff}+8\mathcal{Y}_{f}^{3}\mathcal{Y}_{fff}+21\mathcal{Y}_{f}^{2}\mathcal{Y}_{ff}^{2}-36\mathcal{Y}_{f}\mathcal{Y}_{ff}\mathcal{Y}_{fff}-3\mathcal{Y}_{ff}^{3}+18\mathcal{Y}_{fff}^{2}}$
$\displaystyle+36\mathcal{Y}_{f}\mathcal{Y}_{ff}-36\mathcal{Y}_{fff}-8\mathcal{Y}_{f}^{3}\bigg{)}^{1/3}\,.$
(70)

By regularizing the sum over $\ell$ in eq.(57) and renormalizing the resulting
expression as explained in the previous section we can write
$\Delta_{0}^{(f)}$ in terms of a convergent sum as

$\displaystyle\Delta_{0}^{(f)}$
$\displaystyle=\frac{9\left(-3\mu^{4}+2\mu^{2}+1\right)\mathcal{Y}_{ff}}{8\lambda^{2}}+\frac{3\left(3\mu^{4}-16\mu^{2}+13\right)\mathcal{Y}_{f}}{8\lambda}-18$
$\displaystyle+\frac{1}{2}\sum_{i=1}^{3}n_{f}(0)\left(\omega_{f+}^{(i)}(0)+\omega_{f-}^{(i)}(0)\right)+\frac{1}{2}\sum_{\ell=1}\sigma(\ell)\,,$
(71)

where

$\displaystyle\sigma(\ell)$
$\displaystyle=-6(\ell+1)(\ell+2)(2\ell+3)-\frac{\mu^{2}-1}{2\lambda^{2}\ell}\left(3\lambda\mathcal{Y}_{f}\left(4\ell^{2}+6\ell+\mu^{2}-1\right)+9\left(\mu^{2}-1\right)\mathcal{Y}_{ff}\right)$
$\displaystyle+2(\ell+1)(\ell+2)\sum_{i=1}^{3}\left[\omega_{f+}^{(i)}(\ell)+\omega_{f-}^{(i)}(\ell)\right]_{d=4}\,.$
(72)

###  6.3 Scalars and vector bosons contributions

In this section, we present the contributions to the anomalous dimensions from
the bosonic sector of the theory by expanding around the classical solutions
as

$\phi_{1}=\frac{1}{\sqrt{2}}e^{\mu\tau}(\varphi_{1}(x)+i\varphi_{2}(x))\,,\qquad\phi_{1}^{*}=\frac{1}{\sqrt{2}}e^{-\mu\tau}(\varphi_{1}(x)-i\varphi_{2}(x))\,,$
(73)

and

$\phi_{2}=\frac{1}{\sqrt{2}}(v+r(x))e^{\mu\tau+i\pi(x)/v}\,,\qquad\phi_{2}^{*}=\frac{1}{\sqrt{2}}(v+r(x))e^{-\mu\tau-i\pi(x)/v}\,,$
(74)

where $r,\pi$ and $\varphi_{(1,2)}$ are real valued fields. Using the
expressions (73) and (74), the quadratic part of the Lagrangian is

$\mathcal{L}^{(2)}\left(\varphi,r,\pi,B_{\mu}\right)=\mathcal{L}^{(2)}(\varphi_{1},\varphi_{2})+\mathcal{L}^{(2)}(r,\pi,B_{\mu})\,,$
(75)

where

$\mathcal{L}^{(2)}(\varphi_{1},\varphi_{2})=\frac{1}{2}\varphi_{1}(-\nabla^{2})\varphi_{1}+\frac{1}{2}\varphi_{2}(-\nabla^{2})\varphi_{2}+\frac{1}{2}2i\mu\varphi_{2}\dot{\varphi}_{1}-\frac{1}{2}2i\mu\varphi^{1}\dot{\varphi}^{2}\,,$
(76)

and

$\begin{split}\mathcal{L}^{(2)}(r,\pi,B_{\mu})&=\frac{1}{4}B_{\mu\nu}^{2}+\frac{1}{2}r(-\nabla^{2})r+\frac{1}{2}\pi(-\nabla^{2})\pi+i\mu\,g{{}^{\prime}}\tilde{v}YB_{0}r\\\
&-\frac{1}{2}g{{}^{\prime}}\tilde{v}YB_{\mu}\nabla^{\mu}\pi-\frac{i}{2}4\,\mu\,r\dot{\pi}-(m^{2}-\mu^{2})r^{2}+\frac{1}{8}(g{{}^{\prime}}\tilde{v}Y)^{2}B_{\mu}^{2}\,\,.\end{split}$
(77)

Note that at the quadratic level the $\varphi_{(1,2)}$ decouple from the rest
of the action. Therefore, our results simplify as follows

$\mathcal{Z}_{\text{1-loop }}=\mathcal{Z}^{\varphi}_{\text{1-loop
}}\mathcal{Z}^{B}_{\text{1-loop }}\,,$ (78)

where

$\mathcal{Z}^{\varphi}_{\text{1-loop
}}=\frac{1}{\mathcal{Z}^{\varphi}}\int\mathcal{D}\varphi_{1}\mathcal{D}\varphi_{2}\,e^{-\int\mathcal{L}^{(2)}(\varphi_{1},\varphi_{2})}\,,$
(79)

and

$\mathcal{Z}^{B}_{\text{1-loop
}}=\frac{1}{\mathcal{Z}^{B}}\int\mathcal{D}r\mathcal{D}\pi\mathcal{D}B_{\mu}\,e^{-\int\mathcal{L}^{(2)}(r,\pi,B_{\mu})}\,\,.$
(80)

The partition functions $\mathcal{Z}^{B}$ and $\mathcal{Z}^{\varphi}$ serve as
normalizations and are evaluated around the trivial vacuum. Computing (79)
yields

$\omega^{\varphi}_{\pm}=\sqrt{J_{\ell(s)}^{2}+\mu^{2}}\pm\mu\,\,.$ (81)

The evaluation of the partition function (80) has been described in
Antipin:2022hfe and therefore, here we only highlight the main steps of the
computation process. First note that the action (75) enjoys gauge invariance

$\delta r=0,\quad\delta\pi=v\,\alpha(x),\quad\delta
B_{\mu}=-\frac{1}{g^{{}^{\prime}}}D_{\mu}\beta\,,$ (82)

which is expected from Elitzur’s theorem Elitzur:1975im which states that we
can spontaneously break only the global part of a compact local symmetry.
Hence, for a finite result we should fix the gauge. Consequently, we employ
$R_{\xi}$-gauge fixing

$\mathcal{Z}=\frac{1}{\mathcal{Z}^{B}}\int\mathcal{D}r\mathcal{D}\pi\mathcal{D}B_{\mu}\,e^{-\int
d^{D}x\left(\mathcal{L}^{(2)}(r,\pi,B)+\frac{1}{2}G^{2}\right)}\det\left(\frac{\delta
G}{\delta\beta}\right)\,,$ (83)

where

$G=\frac{1}{\sqrt{\xi}}\left(\nabla_{\mu}B^{\mu}+\frac{1}{2}g{{}^{\prime}}\tilde{v}\,\pi\right)\,,\qquad\frac{\delta
G}{\delta\beta}=\frac{1}{g{{}^{\prime}}\sqrt{\xi}}\left((-\nabla^{2})+\frac{1}{4}({g{{}^{\prime}}}\tilde{v})^{2}\right)\,\,.$
(84)

As usual, the $\det(\delta G/\delta\beta)$ is presented using a set of Fadeev-
Popov ghosts $\bar{c},c$

$\det\left(\frac{\delta
G}{\delta\beta}\right)=\int\mathcal{D}\bar{c}\,\mathcal{D}c\,\,e^{-\int
d^{D}x\,\,\bar{c}\,(-\nabla^{2}+\frac{1}{4}(g{{}^{\prime}}v)^{2})\,c}\,\,.$
(85)

In addition, we split the gauge field $B_{i}$ in a transverse
$\mathcal{B}_{i}$ and a longitudinal $C_{i}$ components

$B_{i}=\mathcal{B}_{i}+C_{i}\,,$ (86)

where $\nabla_{i}\mathcal{B}^{i}=0$ and $C_{i}=\nabla_{i}f$ for $f$ a scalar
function (for more details see Appendix B). Expanding the action in eq. (83)
in terms of eq. (86), the part containing the $\mathcal{B}_{i}$ field
decouples and contributes to independent dispersion relations. Performing the
integration with respect to the fluctuating fields for the gauge-fixed action
we obtain

$\int\mathcal{D}r\mathcal{D}\pi\mathcal{D}B_{0}\mathcal{D}C_{i}\,\,e^{-\mathcal{L}_{fix}(r,\pi,B_{0},C_{i})}=\left(\det\mathcal{U}\right)^{1/2}\,,$
(87)

where $\mathcal{U}$ stands for the matrix

$\mathcal{U}\footnotesize=\left(\begin{array}[]{cccc}-\omega^{2}+J_{\ell(s)}^{2}+2(\mu^{2}-m^{2})&-2i\mu\omega&-ig{{}^{\prime}}v\mu&0\\\
2i\mu\omega&-\omega^{2}+J_{\ell(s)}^{2}+\frac{1}{4\xi}(g{{}^{\prime}}\tilde{v})^{2}&-\frac{1}{2}g{{}^{\prime}}v\left(1-\frac{1}{\xi}\right)\omega&-\frac{i}{2}g{{}^{\prime}}v\left(1-\frac{1}{\xi}\right)|J_{\ell(s)}|\\\
-ig{{}^{\prime}}v\mu&\frac{1}{2}g{{}^{\prime}}v\left(1-\frac{1}{\xi}\right)\omega&-\frac{1}{\xi}\omega^{2}+J_{\ell(s)}^{2}+\frac{1}{4}(g{{}^{\prime}}\tilde{v})^{2}/4&i\left(1-\frac{1}{\xi}\right)\omega|J_{\ell(s)}|\\\
0&\frac{i}{2}g{{}^{\prime}}v\left(1-\frac{1}{\xi}\right)|J_{\ell(s)}|&i\left(1-\frac{1}{\xi}\right)\omega|J_{\ell(s)}|&-\omega^{2}+\frac{1}{\xi}J_{\ell(s)}^{2}+\frac{1}{4}(g{{}^{\prime}}\tilde{v})^{2}\\\
\end{array}\right)\,\,.$ (88)

We can now obtain the determinant in the following factorized form

$\xi\,\det\mathcal{U}=(\omega^{2}+\omega_{+}^{2})(\omega^{2}+\omega_{-}^{2})(\omega^{2}+\omega_{1}^{2})^{2}\,\,.$
(89)

The detailed expressions of the dispersion relations for all fields are given
in Table 1.

Field | $d_{\ell}$ | $\omega_{\ell}$ | $\ell_{0}$
---|---|---|---
$\mathcal{B}_{i}$ | $n_{A}(\ell)$ | $\sqrt{J_{\ell(v)}^{2}+(D-2)+\frac{1}{4}(g^{{}^{\prime}}\tilde{v})^{2}}$ | 1
$C_{i}$ | $n_{b}(\ell)$ | $\sqrt{J_{\ell(s)}^{2}+\frac{1}{4}(g^{{}^{\prime}}\tilde{v})^{2}}$ | 1
$(c,\bar{c})$ | $-2n_{b}(\ell)$ | $\sqrt{J_{\ell(s)}^{2}+\frac{1}{4}(g^{{}^{\prime}}\tilde{v})^{2}}$ | 0
$B_{0}$ | $n_{b}(\ell)$ | $\sqrt{J_{\ell(s)}^{2}+\frac{1}{4}(g^{{}^{\prime}}\tilde{v})^{2}}$ | 0
$\phi$ | $n_{b}(\ell)$ | $\sqrt{J_{\ell(s)}^{2}+3\mu^{2}-m^{2}+\frac{1}{8}(g^{{}^{\prime}}\tilde{v})^{2}\pm\sqrt{\left(3\mu^{2}-m^{2}-\frac{1}{8}(g^{{}^{\prime}}\tilde{v})^{2}\right)^{2}+4J_{\ell(s)}^{2}\mu^{2}}}$ | 0
Table 1: The fields, their degeneracy and dispersion relations in the bosonic
sector of the theory at fixed hypercharge

Note that $\xi$ will be canceled with the evaluation of $\mathcal{Z}_{B}$ in
eq. (80) in the denominator and we arrive at a gauge invariant result. Of
course, this is expected for the computation of a physical object as is the
scaling dimension of an operator.

The bosonic contribution to $\Delta_{0}$ is then obtained by plugging the
dispersion relations in Table 1 plus the ones given in eq.(81) into the
general expression eq.(45) and computing the sum over $\ell$. By proceeding as
in the previous sections, we arrive at

$\displaystyle\Delta_{0}^{(bos)}$
$\displaystyle=\frac{1}{128\lambda^{2}}\left[9g^{\prime
4}\left(5+2\mu^{2}-7\mu^{4}\right)+12g^{\prime
2}\lambda\left(5-14\mu^{2}+9\mu^{4}\right)-16\lambda^{2}\left(3\mu^{2}+1\right)^{2}\right]$
$\displaystyle+\frac{1}{2}\sqrt{6\mu^{2}-2}+\mu+\frac{1}{2}\sum_{\ell=1}\sigma^{(bos)}(\ell)\,,$
(90)

with

$\displaystyle\sigma^{(bos)}(\ell)$
$\displaystyle=\frac{1}{32\lambda^{2}\ell}\Big{[}27g^{\prime
4}\left(\mu^{2}-1\right)^{2}-36g^{\prime
2}\lambda\left(\mu^{2}-1\right)\left(\mu^{2}+2\ell(\ell+1)-1\right)$
$\displaystyle+16\lambda^{2}\left(3\mu^{4}-6\mu^{2}-6\ell(\ell+1)\mu^{2}-2\ell(\ell+1)(6\ell(\ell+2)+1)+3\right)\Big{]}+(\ell+1)^{2}$
$\displaystyle\times\Bigg{[}\sqrt{\frac{3g^{\prime
2}\left(\mu^{2}-1\right)}{4\lambda}-\sqrt{\left(\frac{3g^{\prime
2}\left(\mu^{2}-1\right)}{4\lambda}-3\mu^{2}+1\right)^{2}+4\ell(\ell+2)\mu^{2}}+3\mu^{2}+\ell(\ell+2)-1}$
$\displaystyle+\sqrt{\frac{3g^{\prime
2}\left(\mu^{2}-1\right)}{4\lambda}+\sqrt{\left(\frac{3g^{\prime
2}\left(\mu^{2}-1\right)}{4\lambda}-3\mu^{2}+1\right)^{2}+4\ell(\ell+2)\mu^{2}}+3\mu^{2}+\ell(\ell+2)-1}$
$\displaystyle+2\sqrt{\mu^{2}+\ell(\ell+2)}\Bigg{]}+2\ell(\ell+2)\sqrt{\frac{3g^{\prime
2}\left(\mu^{2}-1\right)}{2\lambda}+\ell(\ell+2)+1}\,.$ (91)

##  7 Explicit perturbative results and discussion

We are now ready to combine our findings to check them against known
perturbative results and provide novel predictions to all orders in the
couplings. We start by highlighting the relevant expressions which, in the
double scaling limit defined in eq.(5), at the leading order is given in
eq.(5) and at the next-to-leading in (6.2), (6.2), (6.3), and (6.3). These
overall contribute to the scaling dimension $\Delta_{Q}$ of the lowest-lying
scalar operators with fixed hypercharge $Q$. We now move to compare our
infinite orders computations with known perturbative results.

###  7.1 Three-loop results for $g=0$

In the $g\to 0$ limit we can compare our findings to the anomalous dimensions
of the operators defined in eq.(23) which, in the Landau gauge, become
eq.(18). At the three-loop order, we obtain

$\displaystyle\Delta_{Q}$
$\displaystyle=Q+\Bigg{\\{}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{3}\lambda
Q^{2}}+\Bigg{[}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}N\mathcal{Y}_{u}+N\mathcal{Y}_{d}+\mathcal{Y}_{l}}{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}{-\frac{3}{4}g^{\prime
2}-\frac{\lambda}{3}}}\Bigg{]}Q\Bigg{\\}}-\Bigg{\\{}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{2}{9}\lambda^{2}Q^{3}}-\Bigg{[}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2N\mathcal{Y}_{uu}+2N\mathcal{Y}_{dd}+2\mathcal{Y}_{ll}}$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\frac{2}{3}\lambda(N\mathcal{Y}_{u}+N\mathcal{Y}_{d}+\mathcal{Y}_{l})}{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}{-\frac{1}{3}\lambda
g^{\prime 2}+\frac{g^{\prime
4}}{16}+\frac{\lambda^{2}}{9}}}\Bigg{]}Q^{2}+C_{22}Q\Bigg{\\}}+\Bigg{\\{}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{8}{27}\lambda^{3}Q^{4}}+\Bigg{[}{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}{\frac{1}{16}g^{\prime
6}(9\zeta(3)-1)}}$
$\displaystyle{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}{-\frac{1}{6}g^{\prime
4}\lambda(1+3\zeta(3))+\frac{1}{3}g^{\prime
2}\lambda^{2}(3-2\zeta(3))+\frac{4}{27}\lambda^{3}(9\zeta(3)-8)}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+\frac{4}{27}\left(3N\left(\lambda^{2}\mathcal{Y}_{u}-3\lambda\mathcal{Y}_{uu}\right.\right.}$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left.\left.+9\zeta(3)(\lambda\mathcal{Y}_{uu}-2\mathcal{Y}_{uuu})\right)+3N\left(\lambda^{2}\mathcal{Y}_{d}-3\lambda\mathcal{Y}_{dd}+9\zeta(3)(\lambda\mathcal{Y}_{dd}-2\mathcal{Y}_{ddd})\right)+3\left(\lambda^{2}\mathcal{Y}_{l}-3\lambda\mathcal{Y}_{ll}\right.\right.}$
$\displaystyle{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\left.\left.+9\zeta(3)(\lambda\mathcal{Y}_{ll}-2\mathcal{Y}_{lll})\right)\right)}\Bigg{]}Q^{3}+C_{23}Q^{2}+C_{33}Q\Bigg{\\}}+{\cal
O}\left(\kappa_{I}^{4}Q^{5}\right)\,.$ (92)

The curly brackets separate different loop orders while red, blue, and orange
colors highlight the terms stemming from the small $\kappa_{I}Q$ expansion of
$\Delta_{-1}$, $\Delta_{0}^{(fm)}$, and $\Delta_{0}^{(bos)}$, respectively
666While at two loops and higher the small $\kappa_{I}Q$ expansion of
$\Delta_{0}$ yields directly the perturbative results, the one-loop
contribution deserves more attention. In fact, the leading order of the small
$\kappa_{I}Q$ expansion of $\Delta_{0}$ is a rational function of the
couplings that need to be converted into the polynomial one-loop term. This is
achieved by first determining the result at the fixed point as a function of
$\epsilon$ and then inferring the polynomial in the coupling constants
yielding the same result.. The above has been checked via an independent
diagrammatic calculation sasha . We used the coefficients $C_{kl}$ introduced
in eq.(7) to write the terms that carry subleading powers of $Q$ and,
therefore, have not been computed in the present work, since they appear at
higher orders of the semiclassical expansion (6). However, as explained in
Sec. 2, these terms may be fixed by requiring consistency between the general
structure (7) and known perturbative results for fixed values of $Q$. In our
case, the only anomalous dimension known in the literature for the family of
operators eq.(18) is the one of the Higgs field corresponding to $Q=1$
Bednyakov:2013eba ; Chetyrkin:2012rz ; Bednyakov:2013cpa . By combining our
results with this information we can fix the following coefficient

$C_{22}=20g_{s}^{2}(\mathcal{Y}_{d}+\mathcal{Y}_{u})+\frac{5\lambda^{2}}{18}+\frac{2}{3}\lambda(3\mathcal{Y}_{d}+3\mathcal{Y}_{u}+\mathcal{Y}_{l})-\frac{1}{4}(51\mathcal{Y}_{uu}+51\mathcal{Y}_{dd}-6\mathcal{Y}_{ud}+17\mathcal{Y}_{ll})\,,$
(93)

where

$\mathcal{Y}_{ud}=(4\pi)^{4}{\rm Tr}Y_{u}Y_{u}^{\dagger}Y_{d}Y_{d}^{\dagger}\
,$ (94)

allowing us to arrive, for the first time, at the full two-loop anomalous
dimension of the Higgs family of operators for arbitrary $Q$. To ease the
comparison with future diagrammatic computations we provide additional
explicit results up to six loops in App.C.

We conclude this section by observing that all the terms scaling as
$g^{\alpha}{g^{\prime}}^{\beta}\lambda^{\delta}Q^{\alpha+\beta+\delta}$ with
$\beta>0$ in the anomalous dimension of the family of operators (18) can be
obtained from $\Delta_{0}^{(bos)}$ in eq.(6.3) by replacing
${g^{\prime}}^{2}\rightarrow{g^{\prime}}^{2}+g^{2}$. This can be traced back
to the absence of mixing between electrically charged and neutral modes when
calculating the quadratic Lagrangian in the bosonic sector.

###  7.2 $\Delta_{Q}$ for $g\neq 0$

We now study the scaling dimension $\Delta_{Q}$ for non-vanishing $SU(2)_{L}$
gauge coupling $g$. We start by solving eq.(43) in the limit of small ’t
Hooft-like couplings. To this end, we note that only two out of the six
solutions of eq.(43), feature the correct $g\to 0$ limit. These are

$\displaystyle\mu_{0}^{(\pm)}$
$\displaystyle=1\pm\frac{3g}{2\sqrt{2}}\sqrt{Q}+Q\left(\frac{2\lambda}{3}-\frac{g^{2}}{8}\right)\pm\frac{5g\left(3g^{2}-64\lambda\right)}{384\sqrt{2}}Q^{3/2}+\left(-\frac{3g^{4}}{512}+\frac{3g^{2}\lambda}{8}-\frac{2\lambda^{2}}{3}\right)Q^{2}$
$\displaystyle\pm\frac{7g}{16384\sqrt{2}}\left(3g^{4}-640g^{2}\lambda+4096\lambda^{2}\right)Q^{5/2}+\frac{1}{108}\lambda\left(9g^{4}-144g^{2}\lambda+128\lambda^{2}\right)Q^{3}+\mathcal{O}\left(Q^{7/2}\right)\,,$
(95)

and lead to the following expressions for $\Delta_{-1}$

$\displaystyle\Delta_{-1}^{(\pm)}$
$\displaystyle=Q\pm\frac{gQ^{3/2}}{\sqrt{2}}+\left(\frac{\lambda}{3}-\frac{g^{2}}{16}\right)Q^{2}\pm\frac{g}{192\sqrt{2}}\left(3g^{2}-64\lambda\right)Q^{5/2}$
$\displaystyle+\left(-\frac{g^{4}}{512}+\frac{g^{2}\lambda}{8}-\frac{2\lambda^{2}}{9}\right)Q^{3}+\mathcal{O}\left(Q^{7/2}\right)\,.$
(96)

The free energy in eq.(5) is a unique function of the chemical potential $\mu$
while the apparent multivaluedness of $\Delta_{-1}$ arises via a naive
definition of the Legendre transform. To resolve this issue we recall that for
a real-valued function $f(x)$ of a real variable $x$, the standard definition
of the Legendre transform is

$L[f](y)=\underset{x}{\text{sup}}(xy-f(x))\,,$ (97)

that naturally selects the lowest energy state. While at the classical level,
the theory is conformal for arbitrary values of $g$ and $\lambda$, the Wilson-
Fisher fixed point occurs at complex values of the couplings defining a pair
of complex CFTs with complex conjugate CFT data Gorbenko:2018ncu . In this
case, the problem of defining the Legendre transform is not immediate due to
the presence of complex saddles. This issue is addressed via the Picard-
Lefschetz theory Witten:2010cx ; Cherman:2014ofa ; Tanizaki:2014xba and we
plan to revisit it in the future. However, we observe that the average over
the two solutions, i.e.

$\Delta_{-1}^{(\text{Avg})}=\frac{1}{2}\left(\Delta_{-1}^{(+)}+\Delta_{-1}^{(-)}\right)=Q+\left(\frac{\lambda}{3}-\frac{g^{2}}{16}\right)Q^{2}+\left(-\frac{g^{4}}{512}+\frac{g^{2}\lambda}{8}-\frac{2\lambda^{2}}{9}\right)Q^{3}+\mathcal{O}\left(Q^{4}\right)\,,$
(98)

uniquely yields the correct $g\to 0$ limit without featuring unexpected non-
integer powers of $Q$.

We provided the first determination of the anomalous dimensions of the family
of composite operators made out of Higgs fields with fixed hypercharge at all
orders in the SM couplings and to leading and subleading orders in the charge.
Future directions include a more detailed identification of the composite
operators, the computation of $\Delta_{0}$ at nonvanishing weak gauge
coupling, and the integration of our results into the computation of the
multi-Higgs production Goldberg:1990qk ; Rubakov:1995hq ; Degrande:2016oan for
colliders applications.

## Acknowledgments

The work of J.B. was supported by the World Premier International Research
Center Initiative (WPI Initiative), MEXT, Japan; and also supported by the
JSPS KAKENHI Grant Number JP23K19047. The work of P. P is supported by the
National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (No. 2023R1A2C1006975) and from the JRG Program at the APCTP
through the Science and Technology Promotion Fund and Lottery Fund of the
Korean Government. The work of F.S. is partially supported by the Carlsberg
Foundation, grant CF22-0922. We thank Alexander Bednyakov for sharing with us
his three-loops perturbative results for the Standard Model anomalous
dimensions of the family of Higgs operators with hypercharge $Q$.

##  Appendix A One-loop SM $\beta$ functions

Below we collect the one-loop SM beta functions needed for this work:

$\displaystyle\beta_{Y_{u}}$
$\displaystyle=-\frac{9g^{2}}{4}-\frac{17g^{\prime
2}}{20}-8g_{S}^{2}+3(\mathcal{Y}_{d}+\mathcal{Y}_{u})+\mathcal{Y}_{l}+\frac{3(Y_{u}-Y_{d})}{2}\,,$
(99) $\displaystyle\beta_{Y_{d}}$
$\displaystyle=-\frac{9g^{2}}{4}-\frac{g^{\prime
2}}{4}-8g_{S}^{2}+3(\mathcal{Y}_{d}+\mathcal{Y}_{u})+\mathcal{Y}_{l}+\frac{3(Y_{d}-Y_{u})}{2}\,,$
(100) $\displaystyle\beta_{Y_{l}}$
$\displaystyle=-\frac{9g^{2}}{4}-\frac{9g^{\prime
2}}{4}+3(\mathcal{Y}_{d}+\mathcal{Y}_{u})+\mathcal{Y}_{l}+\frac{3Y_{l}}{2}\,,$
(101) $\displaystyle\beta_{\lambda}$
$\displaystyle=-12\left(N\mathcal{Y}_{uu}+N\mathcal{Y}_{dd}+\mathcal{Y}_{ll}\right)+4\lambda\left(N\mathcal{Y}_{u}+N\mathcal{Y}_{d}+\mathcal{Y}_{l}\right)$
$\displaystyle+\frac{9g^{\prime 4}}{4}+\frac{9g^{\prime 2}g^{2}}{2}-3g^{\prime
2}\lambda+\frac{27g^{4}}{4}-9g^{2}\lambda+4\lambda^{2}\,,$ (102)
$\displaystyle\beta_{g^{\prime}}$ $\displaystyle=\frac{41g^{\prime
3}}{6}\,,\qquad\beta_{g}=-\frac{19}{6}g^{3}\,,$ (103)

where the beta functions for the Yukawa matrix couplings are defined as

$M\frac{d}{dM}Y_{f}=\beta_{Y_{f}}Y_{f}\,,\qquad
M\frac{d}{dM}Y_{f}^{\dagger}=Y_{f}^{\dagger}\beta_{Y_{f}^{\dagger}}\,,\qquad
f=u,d,l\,,$ (104)

with $M$ the RG scale. At the one-loop level we have
$\beta_{Y_{f}}=\beta_{Y_{f}^{\dagger}}$.

##  Appendix B Scalar, Gauge and Spinor fields on $S^{D-1}$

Here, we provide details about the Laplacian for scalar, vector, and spinor
fields on $S^{D-1}$. Although in the main text we set the radius of the
$S^{D-1}$ sphere to one here we re-introduce and define it as $R$. Therefore,
the conformally invariant action for a real scalar field $H$ on a curved
manifold $\mathcal{M}$ with Ricci scalar $\mathcal{R}$ is

$S=\frac{1}{2}\int
d^{D-1}x\sqrt{-g}\,H\left(-\nabla^{2}_{S^{D-1}}+\frac{D-2}{4(D-1)}\mathcal{R}\right)H\,,$
(105)

where $\nabla^{2}\equiv\nabla_{\mu}\nabla^{\mu}$. The Laplacian $\Delta$ of a
scalar field $H$ on curved background $\mathcal{M}$ is defined as

$\Delta
H=-\frac{1}{\sqrt{g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}H\right)=-\nabla^{2}_{\mathcal{M}}H\,.$
(106)

The eigenvectors of the scalar Laplacian on $S^{D-1}$ are given by the
spherical harmonics $Y_{\ell}$. These are labeled by the angular momentum
quantum number, $\ell\in\mathbb{Z}\geq 0$. The eigenvalue equation reads

$-\nabla^{2}_{S^{D}}\,Y_{\ell}=J^{2}_{\ell(s)}Y_{\ell}\,,$ (107)

where the eigenvalues are given by

$J^{2}_{\ell(s)}=\frac{1}{R^{2}}\ell(\ell+D-2)\,,$ (108)

and have degeneracy

$n_{b}(\ell)=\frac{(2\ell+D-2)\Gamma(\ell+D-2)}{\Gamma(D-1)\Gamma(\ell+1)}\,\,.$
(109)

Proceeding with gauge bosons, the pure Maxwell theory on the sphere reads

$\begin{split}S(A)=&\frac{1}{4}\int d^{D-1}x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}\\\
=&\frac{1}{2}\int
d^{D-1}x\sqrt{-g}\,A^{\nu}\left(-\delta^{\mu}_{\nu}\,\nabla^{2}+\mathcal{R}^{\mu}_{\nu}+\nabla_{\nu}\nabla^{\mu}\right)A_{\mu}\,,\end{split}$
(110)

where in the last step we used the fact that
$[\nabla^{\mu},\nabla_{\nu}]A_{\mu}=\mathcal{R}^{\mu}_{\nu}A_{\mu}$ with
$\mathcal{R}^{\mu}_{\nu}=\frac{D-2}{R^{2}}\,\delta^{\mu}_{\nu}$ is the Ricci
tensor on $S^{D-1}$. For the component $A_{0}$ which is a scalar on $S^{D-1}$
the Ricci tensor does not contribute and $\nabla^{2}$ is equivalent to the
scalar Laplacian. The vector field, as usual, can be decomposed as the kernel
plus the image of $\nabla^{i}$, namely $A^{i}=B^{i}+C^{i}$ with

$\nabla_{i}B^{i}=0,\qquad C^{i}=\nabla^{i}f\,,$ (111)

where $f$ is an arbitrary function. These two modes have different eigenvalues
when Laplacian acts on them. In particular, $f$ is a scalar while the $B^{i}$
modes are spanned by $V_{\ell}^{i}$ labelled by $\ell\in Z>0$ and satisfying

$-\nabla^{2}_{S^{D}}V_{\ell}^{i}=J_{\ell(v)}^{2}V_{\ell}^{i}\,,$ (112)

where the eigenvalues read

$J_{\ell(v)}^{2}=\frac{1}{R^{2}}\left(\ell(\ell+D-2)-1\right)\,\,,$ (113)

and have the following degeneracy

$n_{A}(\ell)=\frac{\ell(\ell+D-2)(2\ell+D-2)\Gamma(\ell+D-3)}{\Gamma(\ell+2)\Gamma(D-2)}\,\,.$
(114)

We conclude with the action of the Laplacian on $S^{D-1}$ on spinor fields
yields the following eigenvalues and degeneracy

$\lambda_{f\pm}(\ell)=\pm\frac{1}{R}\left(\ell+\frac{D-1}{2}\right),\qquad
n_{f}(\ell)=\frac{4\,\Gamma(\ell+D-1)}{\Gamma(D-1)\Gamma(\ell+1)}\,\,.$ (115)

##  Appendix C Explicit $6$-loops results for the anomalous dimensions

Here we provide explicit results for $\Delta_{Q}$ in the $g\to 0$ limit. We
use the notation introduced in eq.(7) and list the values of the coefficients
$C_{0l}$ and $C_{1l}$ for $l=4,5,6$ while the coefficients with $l<4$ can be
read off from eq.(7.1).

The $C_{0l}$ coefficients appear in the small ’t Hooft-like coupling expansion
of $\Delta_{-1}$ and are given by

$\displaystyle C_{04}=-\frac{14}{27}\lambda^{4}\,,\quad
C_{05}=\frac{256}{243}\lambda^{5}\,,\quad
C_{06}=-\frac{572}{243}\lambda^{6}\,.$ (116)

Analogously, the $C_{1l}$ coefficients stem from the small ’t Hooft-like
coupling expansion of $\Delta_{0}$ and read

$\displaystyle C_{14}$
$\displaystyle=\frac{1}{81}(6N\mathcal{F}_{4}^{(u)}+6N\mathcal{F}_{4}^{(d)}+6\mathcal{F}_{4}^{(l)}-2\lambda^{4}(89\zeta(3)+85\zeta(5)-138))$
$\displaystyle+\frac{1}{128}g^{\prime
8}(41\zeta(3)-125\zeta(5)+17)-\frac{1}{48}g^{\prime
6}\lambda(134\zeta(3)-140\zeta(5)+25)$ $\displaystyle+\frac{1}{24}g^{\prime
4}\lambda^{2}(80\zeta(3)-80\zeta(5)+33)-\frac{2}{27}g^{\prime
2}\lambda^{3}(7\zeta(3)-40\zeta(5)+37)\,,$ (117) $\displaystyle C_{15}$
$\displaystyle=\frac{1}{62208}\Bigg{[}512\left(3N\mathcal{F}_{5}^{(u)}+3N\mathcal{F}_{5}^{(d)}+3\mathcal{F}_{5}^{(l)}+2\lambda^{5}(280\zeta(3)+260\zeta(5)+231\zeta(7)-579)\right)$
$\displaystyle-243g^{\prime
10}(224\zeta(3)-50\zeta(5)-441\zeta(7)+95)+648g^{\prime
8}\lambda(413\zeta(3)-50\zeta(5)-525\zeta(7)+167)$ $\displaystyle-864g^{\prime
6}\lambda^{2}(164\zeta(3)-315\zeta(5)-140\zeta(7)+152)-576g^{\prime
4}\lambda^{3}(368\zeta(3)+7(60\zeta(5)-80\zeta(7)+31))$
$\displaystyle+768g^{\prime
2}\lambda^{4}(156\zeta(3)-130\zeta(5)-560\zeta(7)+559)\Bigg{]}\,,$ (118)
$\displaystyle C_{16}$
$\displaystyle=\frac{1}{729}\left(6N\mathcal{F}_{6}^{(u)}+6N\mathcal{F}_{6}^{(d)}+6\mathcal{F}_{6}^{(l)}-2\lambda^{6}(3934\zeta(3)+3522\zeta(5)+3500\zeta(7)+2730\zeta(9)-9935)\right)$
$\displaystyle+\frac{g^{\prime
12}}{2048}(5662\zeta(3)+462\zeta(5)-3514\zeta(7)-7434\zeta(9)+2487)$
$\displaystyle+\frac{1}{192}g^{\prime
10}\lambda(-2956\zeta(3)+319\zeta(5)+1680\zeta(7)+2583\zeta(9)-1270)$
$\displaystyle+\frac{1}{576}g^{\prime
8}\lambda^{2}(14932\zeta(3)-8350\zeta(5)-10185\zeta(7)-6300\zeta(9)+6286)$
$\displaystyle+\frac{1}{108}g^{\prime
6}\lambda^{3}(-2413\zeta(3)+1335\zeta(5)+910\zeta(7)+840\zeta(9)-650)$
$\displaystyle+\frac{1}{162}g^{\prime
4}\lambda^{4}(2645\zeta(3)+515\zeta(5)-490\zeta(7)-2520\zeta(9)+1421)$
$\displaystyle+\frac{1}{243}g^{\prime
2}\lambda^{5}(-1886\zeta(3)+46\zeta(5)+2464\zeta(7)+4032\zeta(9)-4633)\,,$
(119)

where

$\displaystyle\mathcal{F}_{4}^{(f)}$
$\displaystyle=45\zeta(5)\left(\mathcal{Y}_{f}^{4}-6\mathcal{Y}_{f}^{2}\mathcal{Y}_{ff}+8\mathcal{Y}_{f}\mathcal{Y}_{fff}+3\mathcal{Y}_{ff}^{2}-4\lambda\mathcal{Y}_{fff}\right)$
$\displaystyle-2\lambda\zeta(3)\left(\lambda^{2}\mathcal{Y}_{f}+45\lambda\mathcal{Y}_{ff}-108\mathcal{Y}_{fff}\right)+24\lambda^{2}\mathcal{Y}_{ff}-8\lambda^{3}\mathcal{Y}_{f}\,,$
(120) $\displaystyle\mathcal{F}_{5}^{(f)}$
$\displaystyle=63\zeta(7)\left(5\lambda\left(\mathcal{Y}_{f}^{4}-6\mathcal{Y}_{f}^{2}\mathcal{Y}_{ff}+8\mathcal{Y}_{f}\mathcal{Y}_{fff}+3\mathcal{Y}_{ff}^{2}\right)-6\left(\mathcal{Y}_{f}^{5}-5\mathcal{Y}_{f}^{3}\mathcal{Y}_{ff}+5\mathcal{Y}_{f}^{2}\mathcal{Y}_{fff}+5\mathcal{Y}_{ff}\mathcal{Y}_{fff}\right)\right)$
$\displaystyle+60\lambda\zeta(5)\left(-6\left(\mathcal{Y}_{f}^{4}-6\mathcal{Y}_{f}^{2}\mathcal{Y}_{ff}+8\mathcal{Y}_{f}\mathcal{Y}_{fff}+3\mathcal{Y}_{ff}^{2}\right)+\lambda^{2}\mathcal{Y}_{ff}+18\lambda\mathcal{Y}_{fff}\right)$
$\displaystyle+4\lambda^{2}\zeta(3)\left(5\lambda^{2}\mathcal{Y}_{f}+144\lambda\mathcal{Y}_{ff}-378\mathcal{Y}_{fff}\right)-126\lambda^{3}\mathcal{Y}_{ff}+42\lambda^{4}\mathcal{Y}_{f}\,,$
(121) $\displaystyle\mathcal{F}_{6}^{(f)}$
$\displaystyle=1701\zeta(9)\left(\mathcal{Y}_{f}^{6}-3\mathcal{Y}_{f}^{4}\mathcal{Y}_{ff}+4\mathcal{Y}_{f}^{3}\mathcal{Y}_{fff}-9\mathcal{Y}_{f}^{2}\mathcal{Y}_{ff}^{2}-2\lambda\left(\mathcal{Y}_{f}^{5}-5\mathcal{Y}_{f}^{3}\mathcal{Y}_{ff}+5\mathcal{Y}_{f}^{2}\mathcal{Y}_{fff}+5\mathcal{Y}_{ff}\mathcal{Y}_{fff}\right)\right.$
$\displaystyle\left.+12\mathcal{Y}_{f}\mathcal{Y}_{ff}\mathcal{Y}_{fff}+3\mathcal{Y}_{ff}^{3}+4\mathcal{Y}_{fff}^{2}\right)-315\lambda\zeta(7)\left(7\lambda\left(\mathcal{Y}_{f}^{4}-6\mathcal{Y}_{f}^{2}\mathcal{Y}_{ff}+8\mathcal{Y}_{f}\mathcal{Y}_{fff}+3\mathcal{Y}_{ff}^{2}\right)\right.$
$\displaystyle\left.-12\left(\mathcal{Y}_{f}^{5}-5\mathcal{Y}_{f}^{3}\mathcal{Y}_{ff}+5\mathcal{Y}_{f}^{2}\mathcal{Y}_{fff}+5\mathcal{Y}_{ff}\mathcal{Y}_{fff}\right)+4\lambda^{2}\mathcal{Y}_{fff}\right)-2\lambda^{2}\zeta(5)\left(-1440\left(\mathcal{Y}_{f}^{4}-6\mathcal{Y}_{f}^{2}\mathcal{Y}_{ff}\right.\right.$
$\displaystyle\left.\left.+8\mathcal{Y}_{f}\mathcal{Y}_{fff}+3\mathcal{Y}_{ff}^{2}\right)+\lambda^{3}\mathcal{Y}_{f}+345\lambda^{2}\mathcal{Y}_{ff}+3600\lambda\mathcal{Y}_{fff}\right)-2\lambda^{3}\zeta(3)\left(89\lambda^{2}\mathcal{Y}_{f}+2079\lambda\mathcal{Y}_{ff}-5760\mathcal{Y}_{fff}\right)$
$\displaystyle+768\lambda^{4}\mathcal{Y}_{ff}-256\lambda^{5}\mathcal{Y}_{f}\,.$
(122)

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