Solving forward and inverse problems of the logarithmic nonlinear Schrödinger
equation with $\mathcal{PT}$-symmetric harmonic potential via deep learning

Zijian Zhou and Zhenya Yan∗ ∗Email address<EMAIL_ADDRESS>(Corresponding author)

Key Laboratory of Mathematics Mechanization, Academy of Mathematics and
Systems Science,
Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China

Abstract: In this paper, we investigate the logarithmic nonlinear Schrödinger
(LNLS) equation with the parity-time ($\mathcal{PT}$)-symmetric harmonic
potential, which is an important physical model in many fields such as nuclear
physics, quantum optics, magma transport phenomena, and effective quantum
gravity. Three types of initial value conditions and periodic boundary
conditions are chosen to solve the LNLS equation with $\mathcal{PT}$-symmetric
harmonic potential via the physics-informed neural networks (PINNs) deep
learning method, and these obtained results are compared with ones deduced
from the Fourier spectral method. Moreover, we also investigate the
effectiveness of the PINNs deep learning for the LNLS equation with
$\mathcal{PT}$ symmetric potential by choosing the distinct space widths or
distinct optimized steps. Finally, we use the PINNs deep learning method to
effectively tackle the data-driven discovery of the LNLS equation with
$\mathcal{PT}$-symmetric harmonic potential such that the coefficients of
dispersion and nonlinear terms or the amplitudes of $\mathcal{PT}$-symmetric
harmonic potential can be approximately found.

Keywords: Logarithmic nonlinear Schrödinger equation; $\mathcal{PT}$-symmetric
potentials; physics-informed neural networks; solitons

Phys. Lett. A 387 (2021) 127010.

##  1 Introduction

With the rapid development of modern science and high technology, more and
more massive data sets are generated in many scientific fields and real lives.
It is still a huge and important issue that how well to mine and further
powerfully apply them in the scientific fields and even our lives. Though the
artificial intelligence (AI) has experienced several stages of development,
but now it seems that the new generation of AI is facing a new challenge or
revolution, and changing our lives. The deep (machine) learning based on the
neural networks (NNs), as one of most significant approaches realizing AI, has
been applied in various of fields, such as image recognition, computer vision,
natural language processing, solving equations, and other business fields [1,
2]. Recently, based on the neural networks and automatic differentiation (AD)
[3, 4, 5], some deep learning approaches had been presented to approximate the
solutions of some types of differential equations (see, e.g., Refs. [6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and references therein), in which
the physics-information neural networks (PINNs) deep learning method [11] is a
simple and effective method to solving the partial differential equations
(PDEs) via deep learning, and was extended to other types of differential
equations such as integro-differential equations, fractional PDEs, and
stochastic PDEs [12, 13, 14, 15, 16, 17, 18]. More recently, a Python library
DeepXDE was presented for some PINNs approaches [18]. The same characteristic
of these deep neural network methods is to use the NNs instead of the
traditional numerical discretization (ND) methods [18]. Moreover, the designs
of these NNs seem to be easily carried out than the traditional ND methods.

The PINNs deep learning method [11, 18] can be used to study not only data-
driven solutions of some types of differential equations, but data-driven
discovery of PDEs. Some other methods were also presented for the data-driven
discovery of PDEs [20, 21, 22]. In particular, this PINNs deep learning
approach [11, 18] can be used to study the nonlinear wave equations such as
the physically interesting soliton equations. In the following, we simply
introduce the PINNs method solving data-driven solutions of the PDEs with
initial-boundary value conditions [11]. For the given nonlinear wave equation
with the initial-boundary value (IBV) conditions

$\displaystyle\left\\{\begin{array}[]{ll}iu_{t}={\mathcal{L}}{\mathcal{N}}[x,u,u^{*};\Lambda_{0}]),&x\in
D\setminus\partial D\subset\mathbb{R}^{m},\,\,t\in(0,T],\vspace{0.1in}\\\
I[u(x,t)]_{t=0}=u_{I}(x),&x\in D\,\,\text{(initial
condition)},\vspace{0.1in}\\\ B[u(x,t)]|_{x\in\partial
D}=u_{B}(t),&t\in[0,T],\,\,\text{(boundary conditions)},\end{array}\right.$
(4)

where $u=u(x,t)$ stands for the latent (hidden) solution,
${\mathcal{L}}{\mathcal{N}}$ is the combination of linear and nonlinear
operators parametrized by the vector $\Lambda_{0}$, $I[\cdot],\,B[\cdot]$ are
the operators, $I[u(x,t)]_{t=0}=u_{I}(x)$ and $B[u(x,t)]|_{x\in\partial
D}=u_{B}(t)$ stand for the initial conditions, and the Dirichlet (or Neumann,
or Robin) boundary conditions, respectively. Let a deep hidden physics model
$F(x,t)$ be

$\displaystyle
F(x,t):=iu_{t}-{\mathcal{L}}{\mathcal{N}}[x,u,u^{*};\Lambda_{0}]).$ (5)

Then one can obtain the derivatives of the NN $u$ with respect to time $t$ and
space $x$ by applying the chain rule for differentiating compositions of
functions using automatic differentiation [3, 4, 5]. It is worth emphasizing
that automatic differentiation is different from, and in several aspects
superior to, numerical or symbolic differentiation; two commonly encountered
techniques of computing derivatives [3, 4, 5]. To perform the derivatives in
the above-mentioned definition one can depend on the Tensorflow [23], which is
a popular and relatively well documented open source software library for
automatic differentiation and deep learning computations. The PINNs approach
exhibits the structure of the NN, which has the $i$ hidden Layers, and every
hidden layer has $n$ neurons (see Fig. 1). The nonlinear activation function
in the NN is chosen as follows [11]

$A_{j}=\tanh(w_{j}*A_{j-1}+b_{j}),$ (6)

where the $w_{j}$ is a ${\rm dim}(A_{j})\times{\rm dim}(A_{j-1})$ matrix,
$A_{0},A_{i+1},b_{i+1}\in\textbf{R}^{2}$, $A_{j},b_{j}\in\textbf{R}^{n}$. The
training loss function is given by

$\displaystyle{\rm
TL}=\frac{1}{N_{I}}\sum_{j=1}^{N_{I}}|I[u(x_{I}^{j},t)]_{t=0}-u_{I}(x_{I}^{j})|^{2}+\frac{1}{N_{B}}\sum_{j=1}^{N_{B}}|B[u(x_{B},t_{B}^{j})]|_{x_{B}\in\partial
D}-u_{B}(t_{B}^{j})|^{2}+\frac{1}{N_{S}}\sum_{j=1}^{N_{S}}|F(x_{S}^{j},t_{S}^{j})|^{2},$
(7)

where $\\{x_{I}^{j},u_{I}^{j}\\}_{j=1}^{N_{I}}$ and
$\\{t_{B}^{j},u_{B}^{j}\\}_{j=1}^{N_{B}}$ denote the initial and boundary
training data on $u(x,t)$, respectively, and
$\\{x_{S}^{j},t_{S}^{j},\,F(x_{S}^{j},t_{S}^{j})\\}_{j=1}^{N_{S}}$ specify the
collocations points for $F(x,t)$. The optimization method for all loss
functions was chosen as the L-BFGS algorithm [24].

Figure 1: The deep learning frame based on the PINNs.

The main steps of the PINNs deep learning method solving the PDE (4) with IBV
conditions (see Fig. 1) are summarized as follows [11, 18]:

  * •

Step 1. Introduce a neural network $u_{\rm NN}(x,t;{\bf w},{\bf b})$ with
parameters ${\bf w}=\\{w_{j}\\}_{1}^{i}$ and ${\bf b}=\\{b_{j}\\}_{1}^{i}$
being the weight matrices and bias vectors, and the PINN $F(x,t)$ given by Eq.
(5) in terms of the back-propagating AD [25];

  * •

Step 2: Extract the three training sets ${\rm TS}_{I},\,{\rm TS}_{B},\,{\rm
TS}_{E}$ for the initial value condition, boundary value conditions, and
considered model;

  * •

Step 3: Extract a training loss function ${\rm TL}$ given by Eq. (7) by
summing the weighted $\mathbb{L}^{2}$-norm error of both the $F(x,t)$ and IBV
residuals;

  * •

Step 4: Train the NN to find the best parameters $\\{\widehat{\bf
w},\,\widehat{\bf b}\\}$ by minimizing the loss function ${\rm TL}$ in terms
of the L-BFGS optimization algorithm [24].

Since the concept of the $\mathcal{PT}$ symmetry was introduced in 1998 [26],
the $\mathcal{PT}$ symmetry plays an more and more important role in the
generalized Hamiltonian systems and nonlinear wave equations (see, e.g., Refs.
[27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 40, 41, 44, 47, 48, 49, 50, 38, 43,
39, 42, 46, 45, 34] and references therein). It is still a significant subject
to solve the nonlinear wave equations with $\mathcal{PT}$-symmetric potentials
[40]. To the best of our knowledge, the deep learning method was not used to
study the nonlinear wave equations with $\mathcal{PT}$-symmetric potentials
yet. In this paper we would like to study the solutions of the logarithmic
nonlinear Schrödinger (LNSL) equation with $\mathcal{PT}$-symmetric potentials
via the PINNs deep learning method.

The rest of this paper is arranged as follows. In Sec. 2, we firstly introduce
the deep learning scheme for the LNLS equation with $\mathcal{PT}$-symmetric
potentials. And then we use the effect deep learning scheme to investigate the
LNLS equation with $\mathcal{PT}$-symmetric potentials under the distinct
initial-boundary value conditions. In Sec. 3, we investigate the effectiveness
of the PINNs deep learning method in the LNLS equation (9) with IBV conditions
by comparing distinct space widths or optimized steps. In Sec. 4, we use the
PINNs deep learning method to tackle the data-driven discovery of the LNLS
equation with $\mathcal{PT}$-symmetric harmonic potentials. Finally, we give
some conclusions and discussions.

##  2 Data-deriven solutions of the LNLS equation with
$\mathcal{PT}$-symmetric potential

###  2.1 The scheme for the PINNs deep learning method

The well-known nonlinear Schrödinger (NLS) equation is a fundamental model,
and of important significance in some physical fields such as nonlinear
optics, Bose-Einstein condensates, deep ocean, DNA, even finance. In 1975,
Bialynicki-Birula and Mycielski [51] first presented an extended NLS equation,
that is, the NLS equation with a logarithmic nonlinear term
$g\ln(a|\varphi|^{2})\varphi$ (simply called the LNLS equation)

$i\hbar\varphi_{t}=-\frac{\hbar^{2}}{2m}\varphi_{xx}+V(x)\varphi+\sigma\ln(a|\varphi|^{2})\varphi,$
(8)

where $\varphi=\varphi(x,t)$ stands for the complex field, $\hbar$ is the
reduced Planck constant, $m$ is the mass, $\sigma$ is the nonlinear
coefficient corresponding to the focusing $(\sigma<0)$ and defocusing
$(\sigma>0)$ interactions, $V(x)$ usually denotes the real-valued potential,
$a$ is a positive, and real constant and can be chosen as $a=1$, without loss
of generality, by adding a constant to the potential [52]. Eq. (9) can be
applied to quantum mechanics [51, 52, 53], dissipative systems [54], nuclear
physics [55, 56], quantum optics [57], magma transport phenomena [58], open
quantum systems, effective quantum gravity, theory of superfluidity and Bose-
Einstein condensation [59, 61, 62, 60, 63].

We here use the above-mentioned PINNs deep learning method [11] to study the
data-driven solutions of the dimensionless LNLS equation with the complex
$\mathcal{PT}$-symmetric potential

$\displaystyle
i\varphi_{t}=-\varphi_{xx}+[V(x)+iW(x)]\varphi+\sigma\ln(|\varphi|^{2})\varphi,\quad
x\in(-L,L),\quad t\in(0,T),$ (9)

with the following initial-boundary value conditions

$\displaystyle\begin{array}[]{l}\varphi(x,0)=\varphi_{0}(x),\quad
x\in[-L,L],\vspace{0.1in}\\\ \varphi(-L,t)=\varphi(L,t),\quad
t\in[0,T],\end{array}$ (12)

where $\varphi=\varphi(x,t)$ is a complex field, and $V(x)$ and $W(x)$ are
real-valued functions of space, and denote the external potential and the
gain-and-loss distribution, respectively, The required conditions $V(x)=V(-x)$
and $W(x)=-W(-x)$ imply that the complex potential $V(x)+iW(x)$ is
$\mathcal{PT}$-symmetric [26], and are chosen as the physically interesting
potentials (also called the $\mathcal{PT}$-symmetric harmonic potential) [46]

$\displaystyle V(x)=V_{0}x^{2},\qquad W(x)=W_{0}x,$ (13)

where $V_{0}>0,\,W_{0}$ are real-valued constants, and used to modulate the
effect of $\mathcal{PT}$-symmetric potential, $V(x)$ is the well-known
harmonic potential, and the gain-and-loss distribution $W(x)$ is not limited
(except for $W_{0}=0)$, and always has the effect on the LNLS equation (9).
The constant $\sigma$ stands for the nonlinear coefficients, and can be chosen
as $\sigma=\pm 1$ without loss of generality. The cases $\sigma=-1$ and
$\sigma=1$ correspond to the self-focusing and defocusing interactions,
respectively.

Since the function $\varphi(x,t)$ in Eq. (9) is complex, thus it can be
rewritten as $\varphi(x,t)=u(x,t)+iv(x,t)$ with $u(x,t),\,v(x,t)$ being its
real and imaginary parts, respectively. Introduce the associated complex-
valued PINNs $F(x,t)=iF_{u}(x,t)-F_{v}(x,t)$ with $-F_{v}(x,t),\,F_{u}(x,t)$
being its real and imaginary parts, respectively, as

$\displaystyle\begin{array}[]{l}F(x,t):=i\varphi_{t}+\varphi_{xx}-[V(x)+iW(x)]\varphi-\sigma\varphi\ln(|\varphi|^{2}),\vspace{0.1in}\\\
F_{u}(x,t):=u_{t}+v_{xx}-W(x)u-V(x)v-\sigma
v\ln(u^{2}+v^{2}),\vspace{0.1in}\\\
F_{v}(x,t):=v_{t}-u_{xx}+V(x)u-W(x)v+\sigma u\ln(u^{2}+v^{2}),\end{array}$
(17)

and proceed by approximating $\varphi(x,t)$ by a complex-valued deep neural
network. The complex neural network $\varphi(x,t)=(u(x,t),v(x,t))$ can be
defined as

⬇

def psi(x, t):

psi = neural_net(tf.concat([x,t],1), weights, biases)

u = psi[:,0:1]

v = psi[:,1:2]

return u, v

Moreover, the physics-informed neural network $F(x,t)$ is chosen the form

⬇

def F(x, t):

u, v = psi(x, t)

u_t = tf.gradients(u, t)[0]

u_x = tf.gradients(u, x)[0]

u_xx = tf.gradients(u_x, x)[0]

v_t = tf.gradients(v, t)[0]

v_x = tf.gradients(v, x)[0]

v_xx = tf.gradients(v_x, x)[0]

F_u = u_t + v_xx - W*u - V*v - sigma*tf.ln(u**2 + v**2)*v

F_v = v_t - u_xx + V*u - W*v + sigma*tf.ln(u**2 + v**2)*u

return F_u, F_v

The shared parameters between the neural network $\varphi(x,t)=u(x,t)+iv(x,t)$
and $F(x,t)=iF_{u}(x,t)-F_{v}(x,t)$ can be trained by minimizing the training
loss (TL), that is, the sum of the TLes on the initial data (${\rm TL}_{I}$),
boundary data (${\rm TL}_{B}$), and the whole equation (${\rm TL}_{S}$)

$\displaystyle{\rm TL}={\rm TL}_{I}+{\rm TL}_{B}+{\rm TL}_{S},$ (18)

where the mean squared errors are chosen as

$\displaystyle\begin{array}[]{l}{\rm
TL}_{I}=\displaystyle\frac{1}{N_{I}}\sum_{j=1}^{N_{I}}\left(\left|u(x_{I}^{j},0)-u_{0}^{j}\right|^{2}+\left|v(x_{I}^{j},0)-v_{0}^{j}\right|^{2}\right),\vspace{0.1in}\\\
{\rm
TL}_{B}=\displaystyle\frac{1}{N_{B}}\sum_{j=1}^{N_{B}}\left(\left|u(-L,t_{B}^{j})-u(L,t_{B}^{j})\right|^{2}+\left|v(-L,t_{B}^{j})-v(L,t_{B}^{j})\right|^{2}\right),\vspace{0.1in}\\\
{\rm
TL}_{S}=\displaystyle\frac{1}{N_{S}}\sum_{j=1}^{N_{S}}\left(\left|F_{u}(x_{S}^{j},t_{S}^{j})\right|^{2}+\left|F_{v}(x_{S}^{j},t_{S}^{j})\right|^{2}\right)\end{array}$
(22)

with $\\{x_{I}^{j},\,u_{0}^{j},\,v_{0}^{j}\\}_{j=1}^{N_{I}}$ standing for the
initial data ($\varphi_{0}(x)=u_{0}(x)+iv_{0}(x)$), $\\{t_{B}^{j},\,u(\pm
L,t_{B}^{j}),\,v(\pm L,t_{B}^{j})\\}_{j=1}^{N_{B}}$ standing for the boundary
data,
$\\{x_{S}^{j},\,t_{S}^{j},\,F_{u}(x_{S}^{j},t_{S}^{j}),\,F_{v}(x_{S}^{j},t_{S}^{j})\\}_{j=1}^{N_{S}}$
representing the collocation points within a spatio-temporal region on
$F(x,t)=iF_{u}-F_{v}$. In other words, ${\rm TL}_{I}$ represents the
$\mathbb{L}^{2}$-norm error in initial points, ${\rm TL}_{B}$ denotes the
$\mathbb{L}^{2}$-norm error in the periodic boundary condition, and ${\rm
TL}_{S}$ corresponds the $\mathbb{L}^{2}$-norm loss in the interior of the
spatio-temporal region. All of these sampling points can be generated using a
space filling Latin Hypercube Sampling strategy [64].

To solve Eq. (9) with the above-mentioned initial-boundary value conditions
via the PINNs deep learning method [11], we need to generate a LNLS.mat file
by utilizing the Fourier spectral method to Eq. (9) with the initial-boundary
conditions (12) in Matlab. In fact, we here choose the Fourier spectral method
[65] with a special Fourier discretization with 256 space modes and a fourth-
order explicit Runge-Kutta temporal integrator with a time step length $\Delta
t=10^{-4}$ to simulate it. To further study Eq. (9) via the deep learning, we
choose a $201$ temporal sampling points with the same time interval such that
$\varphi(x,t)=u(x,t)+iv(x,t)$ is a $256\times 201$ matrix. All of these
sampling points will be packaged as a LNLS.mat file.

###  2.2 Applications in the distinct initial value conditions and same
boundary value condition

In the follows, we would like to use the above-mentioned PINNs deep learning
approach to investigate the solitons of the LNLS equation with
$\mathcal{PT}$-symmetric harmonic potential (9) with distinct initial-boundary
value problems (12).

####  2.2.1 The initial value condition arising from the exact soliton

We can find that Eq. (9) with the $\mathcal{PT}$-symmetric harmonic potential
(13) admits the exact bright soliton

$\displaystyle\varphi(x,t)=A\exp\\!\left(-\omega
x^{2}\right)e^{-i[W_{0}x/(4\omega)+\mu t]},$ (23)

with $A\in\mathbb{R},\,A\not=0$ and

$\displaystyle\omega=\frac{1}{4}\left(\sqrt{\sigma^{2}+4V_{0}}-\sigma\right)>0,\quad\mu=2\omega\\!+\\!\sigma\ln
A^{2}\\!+\\!\frac{W_{0}^{2}}{16\omega}.$ (24)

When $|x|\to\infty$, $|\varphi(x,t)|\to 0$ and
$\int_{-\infty}^{\infty}|\varphi(x,t)|^{2}dx=A^{2}\sqrt{\frac{\pi}{2\omega}}$.

Case 1. The focusing case $\sigma=-1$ and other parameters
$W_{0}=0.2,\,V_{0}=1,\,A=1$.

We first use the exact solution (23) to choose the initial conditions:
$\varphi_{0}(x)=\varphi(x,0)=u_{0}(x)+iv_{0}(x)$ with

$\displaystyle u_{0}(x)=e^{-\omega x^{2}}\\!\cos(0.2x/(4\omega)),\quad
v_{0}(x)=-e^{-\omega x^{2}}\\!\sin(0.2x/(4\omega)),$ (25)

with $\omega=(\sqrt{5}+1)/4$ and the periodic boundary condition
$\varphi(-L,t)=\varphi(L,t)$ such that we can generate a high-resolution data
set by using the Fourier spectral method to simulate Eq. (9). Under the data-
derived setting, all measurements we observe are
$\\{x_{I}^{j},\,u_{0}^{j},\,v_{0}^{j}\\}_{j=1}^{N_{I}}$ of the latent function
$\varphi(x,t)=u(x,t)+iv(x,t)$ at initial time $t=0$ with the periodic boundary
conditions $\\{t_{B}^{j},\,u(\pm L,t_{B}^{j}),\,v(\pm
L,t_{B}^{j})\\}_{j=1}^{N_{B}}$. The training set consists of a total of
$N_{I}=100$ data points on the initial conditions $\\{u_{0}(x),\,v_{0}(x)\\}$
randomly deduced from the full high-resolution data set, and $N_{B}=200$
randomly sampled collocation points on the periodic boundary conditions
$\\{u(\pm L,t),\,v(\pm L,t)\\}$. We here choose $N_{s}=10,000$ randomly
sampled collocation points used to study the LNLS Eq. (9) inside the solution
region. Notice that all randomly sampled point locations are obtained via the
Latin Hypercube Sampling strategy [64].

We choose an $11-$layer deep neural network with $50$ neurons per layer and a
hyperbolic tangent activation function given by Eq. (6) to jointly represent
the latent function $\varphi(x,t)=u(x,t)+iv(x,t)$. We choose the space
interval as $[-5,5]$ (i.e., $L=5$), and the time interval as $[0,5]$ (i.e.,
$T=5$). As a consequence, Figs. 2(a1-d1) illustrate the results of the
simulations. Fig. 2(a1) displays the magnitude of the predicted solution
$|\varphi(x,t)|=\sqrt{u^{2}(x,t)+v^{2}(x,t)}$ for the given locations of the
initial-boundary conditions. The whole training time of this result is $18475$
seconds on a matebook x pro with a 1.6GHz four-cores i5 processor (the same
hereinafter). The relative $\mathbb{L}^{2}-$norm error of $\varphi(x,t)$ in
this case is $1.1791\cdot 10^{-3}$. The relative $\mathbb{L}^{2}-$norm errors
of $u(x,t)$ and $v(x,t)$, respectively, are $1.4040\cdot 10^{-3}$ and
$1.3955\cdot 10^{-3}$. Figs. 2(b1-d1) exhibit the comparisons between exact
and predicted solutions for distinct time instants $t=1.25,\,2.5,\,3.75$.

Case 2. The defocusing case $\sigma=1$ and other parameters
$W_{0}=0.1,V_{0}=A=2$.

Similarly to Case 1, we use the exact solution (23) to choose the initial
conditions: $\varphi_{0}(x)=\varphi(x,0)=u_{0}(x)+iv_{0}(x)$ with

$\displaystyle u_{0}(x)=2e^{-0.5x^{2}}\\!\cos(0.05x),\quad
v_{0}(x)=-2e^{-0.5x^{2}}\\!\sin(0.05x),$ (26)

and the periodic boundary condition $\varphi(-L,t)=\varphi(L,t)$ such that we
can generate a high-resolution data set by using the Fourier spectral method
to simulate Eq. (9).

The training set is composed of a total of $N_{I}=100$ data points on the
initial conditions $\\{u_{0}(x),\,v_{0}(x)\\}$ randomly deduced from the full
high-resolution data set, and $N_{B}=200$ randomly sampled collocation points
on the periodic boundary conditions $\\{u(\pm L,t),\,v(\pm L,t)\\}$. We here
choose $N_{s}=10,000$ randomly sampled collocation points used to study the
LNLS Eq. (9) inside the solution region. Notice that all randomly sampled
point locations were obtained via the Latin Hypercube Sampling strategy [64].

Similarly to Case 1, we still choose an $11$-layer deep neural network with
$50$ neurons per layer and a hyperbolic tangent activation function given by
Eq. (6) to jointly represent the latent function
$\varphi(x,t)=u(x,t)+iv(x,t)$. We take the space interval as $[-5,5]$ (i.e.,
$L=5$), and the time interval as $[0,5]$ (i.e., $T=5$). As a consequence,
Figs. 2(a2-d2) illustrate the results of the simulations. Fig. 2(a2) displays
the magnitude of the predicted solution
$|\varphi(x,t)|=\sqrt{u^{2}(x,t)+v^{2}(x,t)}$ for the given locations of the
initial-boundary conditions. The whole training time of this result is $42417$
seconds. The relative $\mathbb{L}^{2}-$norm error of $\varphi(x,t)$ in this
case is $1.7889\cdot 10^{-4}$. The relative $\mathbb{L}^{2}-$norm errors of
$u(x,t)$ and $v(x,t)$, respectively, are $8.3298\cdot 10^{-3}$ and
$6.8020\cdot 10^{-3}$. Figs. 2(b2-d2) exhibit the comparisons between exact
and predicted solutions for distinct time instants $t=1.25,\,2.5,\,3.75$.

Figure 2: The LNLS equation with $\mathcal{PT}$-symmetric potential and exact
solution (23): (a1) $\sigma=-1$ The predicted solution $|\varphi(x,t)|$ with
$N_{I}=100$ data points on initial conditions and $N_{B}=200$ data points on
periodic boundary conditions (see the cross), and $N_{S}=10,000$ collocation
points deduced by Latin Hypercube Sampling strategy. The relative
$\mathbb{L}^{2}-$norm error of $\varphi(x,t)$ in this case is $1.1791\cdot
10^{-3}$. (b1)-(d1) $\sigma=-1$ and (b2)-(d2) $\sigma=1$: Comparisons between
the exact and predicted solutions by neural network at $t=1.25,t=2.50,t=3.75$.
(a2) $\sigma=1$ The predicted solution $|\varphi(x,t)|$ with $N_{I}=100$ data
points on initial conditions, $N_{B}=200$ data points on periodic boundary
conditions (see the cross), and $10,000$ collocation points deduced by Latin
Hypercube Sampling strategy. The relative $\mathbb{L}^{2}-$norm error of
$\varphi(x,t)$ in this case is $1.7889\cdot 10^{-3}$. (b2)-(d2) $\sigma=1$:
Comparisons between the exact and predicted solutions by neural network at
$t=1.25,t=2.50,t=3.75$.

####  2.2.2 The initial condition arising from the non-stationary solutions

In the second example, we consider Eq. (9) with another initial condition
(which does not satisfy the stationary equation of Eq. (9))

$\displaystyle\varphi_{0}(x)=A\exp\left(-\omega
x^{2}\right)e^{i\frac{W_{0}}{4\omega}x},$ (27)

with $\omega$ given by Eq. (24), the periodic boundary conditions
$\varphi(-3,t)=\varphi(3,t)$, and parameters $W_{0}=0.2$, $V_{0}=1$, $A=2$,
$\sigma=-1$ such that we have $\varphi_{0}(x)=\varphi(x,0)=u_{0}(x)+iv_{0}(x)$
with

$\displaystyle u_{0}(x)=2e^{-\omega x^{2}}\\!\cos(0.2x/(4\omega)),\quad
v_{0}(x)=2e^{-\omega x^{2}}\\!\sin(0.2x/(4\omega)),$ (28)

with $\omega=(\sqrt{5}+1)/4$.

To solve Eq. (9) with the above-mentioned initial-boundary value conditions
via the deep learning, we firstly obtain a high-resolution data set by using
the above-mentioned Fourier spectral method to simulate Eq. (9) with the
initial condition (27) and periodic boundary conditions. As a result, we find
that the amplitude of vibration of solitary wave grows as $|W_{0}|$ of gain-
and-loss distribution increases. Generally speaking, we can select a
200-dimensional temporal sampling points, which may be of the equal interval.
All of these sampling points will be packaged as a .mat file.

We now choose a $5$-layer deep neural network with $90$ neurons per layer and
a hyperbolic tangent activation function (6) to jointly represent the latent
function $\varphi(x,t)=u(x,t)+iv(x,t)$ where $u(x,t),\,v(x,t)$ represent the
real and imaginary parts of $\phi(x,t)$, respectively. Fig. 3a displays the
magnitude of the predicted solution
$|\varphi(x,t)|=\sqrt{u^{2}(x,t)+v^{2}(x,t)}$ for the given locations of the
initial-boundary conditions. The whole training time of this result is $46397$
seconds. The relative $\mathbb{L}^{2}-$norm error of $\varphi(x,t)$ in this
case is $4.6204\cdot 10^{-3}$. The relative $\mathbb{L}^{2}-$norm errors of
$u(x,t)$ and $v(x,t)$, respectively, are $8.4528\cdot 10^{-3}$ and
$8.6211\cdot 10^{-3}$. Figs. 3(b-d) exhibit the comparisons between exact and
predicted solutions for distinct time instants $t=2.5,\,5.0,\,7.5$.

Figure 3: The focusing LNLS equation with $\mathcal{PT}$-symmetric potential
and initial condition (27): (a) The predicted solution $|\varphi(x,t)|$ with
$N_{I}=100$ data points on initial conditions and $N_{B}=200$ data points on
periodic boundary conditions (see the cross). We choose $N_{S}=10,000$
collocation points deduced by Latin Hypercube Sampling strategy. (b)-(d)
Comparisons between the exact and predicted solutions by neural network at
$t=2.5,\,5.0,\,7.5$. The relative $\mathbb{L}^{2}-$norm error of
$\varphi(x,t)$ in this case is $4.6204\cdot 10^{-3}$.

Figure 4: The focusing LNLS equation with $\mathcal{PT}$-symmetric potential
and initial condition (29): (a) The predicted solution $|\varphi(x,t)|$ with
$N_{I}=100$ data points on initial conditions and $N_{B}=200$ data points on
periodic boundary conditions (see the cross). We choose $N_{S}=10,000$
collocation points deduced by Latin Hypercube Sampling strategy. (b)-(d)
Comparisons between the exact and predicted solutions by neural network at
$t=0.4,\,0.8,\,1.2$. The relative $\mathbb{L}^{2}-$norm error of
$\varphi(x,t)$ in this case is $1.1713\cdot 10^{-1}$.

####  2.2.3 The periodic initial condition

In this case, we consider Eq. (9) with the periodic initial condition

$\displaystyle\varphi_{0}(x)=|\cos(x)|,\quad x\in[-3\pi/2,\,3\pi/2],$ (29)

the periodic boundary conditions $\varphi(-3\pi/2,t)=\varphi(3\pi/2,t)$, and
parameters $W_{0}=0.02$, $V_{0}=1$, $\sigma=-1$ such that we have
$\varphi_{0}(x)=\varphi(x,0)=u_{0}(x)+iv_{0}(x)$ with $u_{0}(x)=|\cos(x)|$ and
$v_{0}(x)=0$.

We can obtain a training and test data-set by using the above-mentioned
Fourier spectral method to simulate Eq. (9) with
$W_{0}=0.02,\,V_{0}=1,\,\sigma=-1$, the initial condition (29), and periodic
boundary conditions $\varphi(-3\pi/2,t)=\varphi(3\pi/2,t)$.

We choose an $11$-layer deep neural network with $50$ neurons per layer and a
hyperbolic tangent activation function (6) to jointly represent the latent
function $\varphi(x,t)=u(x,t)+iv(x,t)$ where $u(x,t),\,v(x,t)$ represent the
real and imaginary parts of $\varphi(x,t)$, respectively. Fig. 4a displays the
magnitude of the predicted solution
$|\varphi(x,t)|=\sqrt{u^{2}(x,t)+v^{2}(x,t)}$ for the given locations of the
initial-boundary conditions. The whole training time of this result is $40083$
seconds. The relative $\mathbb{L}^{2}-$norm error of $\varphi(x,t)$ in this
case is $1.1713\cdot 10^{-1}$. The relative $\mathbb{L}^{2}-$norm errors of
$u(x,t)$ and $v(x,t)$, respectively, are $1.6250\cdot 10^{-1}$ and
$1.6204\cdot 10^{-1}$. Figs. 4(b-d) exhibit the comparisons between exact and
predicted solutions for distinct time instants $t=0.4,\,0.8,\,1.2$.

Figure 5: The space width influence on the accuracy of deep learning in the
LNLS equation with $\mathcal{PT}$-symmetric potential and IBV in Case 2.2.2:
(a)-(d) exhibit the deep learning results in four interval widths
$L=3,\,4,\,5,\,6$, respectively. We choose randomly $N_{I}=50$ sampling points
on the initial boundary, $N_{B}=100$ sampling points on the periodic boundary
conditions, and $N_{S}=8,000$ collocation points deduced by Latin Hypercube
Sampling strategy. The comparisons of the exact and predicted solutions for
three times $t=1.25,\,2.5\,,\ 3.75$ in (e1)-(e3): $L=3$; (f1)-(f3): $L=4$;
(g1)-(g3): $L=5$; (h1)-(h4): $L=6$.

##  3 Comparisons of applications of the PINNs in the LNLS equation

###  3.1 The effects of spatial widths on the learning ability of PINNs

In what follows, we will discuss the effect of spatial widths on the learning
ability of PINNs. In order to simplify this problem, we use the same example
in Case 1 of Sec. 3.1. We consider the wave evolution on the distinct
intervals including $L=3,\,4,\,5,\,6$, and require the 5000 Adam optimizations
and 5000 L-BFGS optimizations.

We use an $11$-layers deep neural network with $50$ neurons per layer and a
hyperbolic tangent activation function (6) to jointly represent the latent
function $\varphi(x,t)$. In the first two rows of Fig. 5, we exhibit the wave
propagations for distinct spatial widths and same parameters via the deep
learning of PINNs. Obviously, the accuracy of PINNs learning decreases as the
space width increases. Their training times are 5001.1s, 4758.5s, 5050.3s, and
4717.5s, respectively.

In the other four rows of Fig. 5, we respectively exhibit the states at three
moments for four space widths, which also display that the predicted solutions
are better agree with the exact solutions as the space width become larger.
Table 1 shows all accuracy of the experimental results.

Figure 6: The optimized step influence on the accuracy of deep learning in the
LNLS equation with $\mathcal{PT}$-symmetric potential and IBV in Case 2.2.2:
(a)-(d) The deep learning results in four optimized steps
$N=15,000,30,000,45,000,60,000$. We select randomly $N_{I}=100$ sampling
points on initial boundary, $N_{B}=200$ sampling points on spatial boundary
and $N_{S}=10000$ sampling points in the interior of the region. The
comparisons of the exact and predicted solutions for three times
$t=2.50,\,5.00,\,7.50$ and (e1)-(e3): $N=15,000$; (f1)-(f3): $N=30,000$;
(g1)-(g3): $N=45,000$; (h1)-(h4): $N=60,000$.

###  3.2 The effects of optimized steps on the learning ability of PINNs

In this subsection, we will discuss the effect of optimized steps on PINNs
learning ability in the LNLS equation with $\mathcal{PT}$-symmetric potential.
To simplify this problem, we use the same example in (27). We consider the
evolution in different optimized steps: $N=60000,45000,30000,15000$. We using
a 5 hidden layers deep neural network with 90 neurons per layer and a tangent
activation function to jointly represent $\varphi(x,t)$. The training steps of
these four cases are $60000,45000,30000,15000$ Adam optimizations, and
$60000,45000,30000,15000$ L-BFGS optimizations. There training time are
36767s, 32852s, 28295s, and 15512s, respectively (see the first two rows of
Fig. 6). In the other four rows of Fig. 6, we respectively exhibit the states
at three moments for four optimized steps, which also display that the
predicted solutions are better agree with the exact solutions as the optimized
step become larger. Table 2 shows all accuracy of the experimental results.

Table 1: The comparisons between the errors and widths. $\mathbb{L}^{2}$ error Width | L=3 | L=4 | L=5 | L=6
---|---|---|---|---
loss function | 4.34$\times 10^{-5}$ | 2.09$\times 10^{-4}$ | 8.32$\times 10^{-3}$ | 8.69$\times 10^{-3}$
$u$ | 1.84$\times 10^{-2}$ | 2.02$\times 10^{-2}$ | 8.79$\times 10^{-1}$ | 8.95$\times 10^{-1}$
$v$ | 1.28$\times 10^{-2}$ | 1.19$\times 10^{-2}$ | 8.92$\times 10^{-1}$ | 9.08$\times 10^{-1}$
$\varphi$ | 5.64$\times 10^{-3}$ | 8.62$\times 10^{-3}$ | 7.13$\times 10^{-1}$ | 8.03$\times 10^{-1}$
Table 2: The comparisons between the errors and steps. $\mathbb{L}^{2}$ error Step | N=15000 | N=30000 | N=45000 | N=60000
---|---|---|---|---
loss function | 2.76$\times 10^{-3}$ | 7.27$\times 10^{-5}$ | 5.70$\times 10^{-6}$ | 5.16$\times 10^{-6}$
$u$ | 9.65$\times 10^{-1}$ | 1.25$\times 10^{-1}$ | 1.47$\times 10^{-2}$ | 1.21$\times 10^{-2}$
$v$ | 9.69$\times 10^{-1}$ | 1.29$\times 10^{-1}$ | 1.52$\times 10^{-2}$ | 1.23$\times 10^{-2}$
$\varphi$ | 6.77$\times 10^{-1}$ | 9.80$\times 10^{-2}$ | 1.18$\times 10^{-2}$ | 1.04$\times 10^{-2}$

##  4 Data-derived discovery of the LNLS equation with
$\mathcal{PT}$-symmetric harmonic potential

In what follows, we apply the PINNs deep learning method of data-driven
discovery of PDEs [11] to identify the parameters of the focusing
($\sigma=-1$) LNLS equation with $\mathcal{PT}$-symmetric harmonic potential
(9). The method of data-driven discovery of PDEs is similar to one solving the
data-driven solutions of PDEs.

###  4.1 Discovering the coefficients of $\mathcal{PT}$-symmetric harmonic
potential

Here we identify the coefficient of potential $V_{0},W_{0}$ in the focusing
LNLS equation with $\mathcal{PT}$-symmetric harmonic potential

$\displaystyle
i\psi_{t}=-\psi_{xx}+(V_{0}x^{2}+iW_{0}x)\psi-\psi\ln(|\psi|^{2}),$ (30)

where $V_{0},\,W_{0}$ are the unknown real-valued parameters.

Let $\varphi(x,t)=u(x,t)+iv(x,t)$ with $u(x,t),\,v(x,t)$ being its real and
imaginary parts, respectively, and the PINNs $F(x,t)=iF_{u}(x,t)-F_{v}(x,t)$
with $-F_{v}(x,t),\,F_{u}(x,t)$ being its real and imaginary parts,
respectively, be

$\displaystyle\begin{array}[]{l}F(x,t):=i\varphi_{t}+\varphi_{xx}-(V_{0}x^{2}+iW_{0}x)\varphi+\varphi\ln(|\varphi|^{2}),\vspace{0.1in}\\\
F_{u}(x,t):=u_{t}+v_{xx}-W_{0}xu-
V_{0}x^{2}v+v\ln(u^{2}+v^{2}),\vspace{0.1in}\\\
F_{v}(x,t):=v_{t}-u_{xx}+V_{0}x^{2}u-W_{0}xv-u\ln(u^{2}+v^{2}),\end{array}$
(34)

Then the deep neural network is used to train $\\{u(x,t),\,v(x,t)\\}$ and
parameters $(V_{0},\,W_{0})$ by minimizing the mean squared error loss

$\displaystyle{\rm TL}={\rm TL}_{uv}+{\rm TL}_{D}$ (35)

with

$\displaystyle{\rm TL}_{uv}$
$\displaystyle=\frac{1}{N_{D}}\sum_{j=1}^{N_{D}}\left(|u(x^{j},t^{j})-u^{j}|^{2}+|v(x^{j},t^{j})-v^{j}|^{2}\right),$
(36) $\displaystyle{\rm TL}_{D}$
$\displaystyle=\frac{1}{N_{D}}\sum_{j=1}^{N_{D}}\left(|F_{u}(x^{j},t^{j})|^{2}+|F_{v}(x^{j},t^{j})|^{2}\right).$

where $\\{x^{j},\,t^{j},\,u^{j},\,v^{j}\\}_{i=1}^{N_{D}}$ represents the
training data on $u(x,t),\,v(x,t)$, and $u(x^{j},t^{j}),\,v(x^{j},t^{j})$ are
real and imaginary parts of the exact solution $\varphi(x,t)=u(x,t)+iv(x,t)$
given by Eq. (23) with $A=V_{0}=1,\,W_{0}=0.2,\,\sigma=-1$ in
$(x,t)\in[-3,3]\times[0,10]$.

To study the data-driven discovery of the LNLS equation (30) for
$V_{0},\,W_{0}$, we generate a training data-set by randomly choosing
$N_{D}=10,000$ points in the $(x,t)$ domain, $[-3,3]\times[0,10]$ from the
exact solution $\psi(x,t)$ for $V_{0}=1,\,W_{0}=0.2$. Then the obtained data-
set is applied to train an $11$-layer deep NN with $50$ neurons per hidden
layer to find the approximate parameters $V_{0},\,W_{0}$ in terms of
minimizing the mean squared loss of (35). We here use the 20000-step Adam
optimization and 20000-step L-BFGS optimization.

For the unknown $V_{0},\,W_{0}$, Table 3 illustrates the correct LNLS equation
and identified LNLS equation under the distinct sense of the clean data or the
data with $1\%$ noise, and the errors of $V_{0},\,W_{0}$.

Table 3: Comparison of the correct LNLS equation and the identified ones deduced by deep learning $V_{0}$ and $W_{0}$, and their errors. PDE Item | Equation | Error of $V_{0}$ | Error of $W_{0}$
---|---|---|---
Correct | $i\psi_{t}=-\psi_{xx}+(x^{2}+0.2ix)\psi-\psi\ln(|\psi|^{2})$ | 0 | 0
identified (clean data) | $i\psi_{t}\\!=\\!-\psi_{xx}\\!+\\!(1.00601x^{2}\\!+\\!0.18655ix)\psi\\!-\\!\psi\ln(|\psi|^{2})$ | 6.01$\times 10^{-3}$ | 1.34$\times 10^{-2}$
identified ($1\%$ noise) | $i\psi_{t}\\!=-\\!\psi_{xx}\\!+\\!(0.99249x^{2}\\!+\\!0.19985ix)\psi\\!-\\!\psi\ln(|\psi|^{2})$ | 7.51$\times 10^{-3}$ | 1.49$\times 10^{-4}$
Table 4: Comparison of the correct LNLS equation and the identified ones deduced by deep learning $\alpha$ and $\sigma$, and their errors. PDE Item | Equation | Error of $\alpha$ | Error of $\sigma$
---|---|---|---
Correct | $i\psi_{t}=-\psi_{xx}+(x^{2}+0.2ix)\psi-\psi\ln(|\psi|^{2})$ | 0 | 0
identified (clean data) | $i\psi_{t}\\!=\\!-1.00054\psi_{xx}\\!+\\!(x^{2}\\!+\\!0.2ix)\psi\\!-\\!0.99641\psi\ln(|\psi|^{2})$ | 5.43$\times 10^{-4}$ | 3.59$\times 10^{-3}$
identified (1% noise) | $i\psi_{t}\\!=\\!-1.00306\psi_{xx}\\!+\\!(x^{2}\\!+\\!0.2ix)\psi\\!-\\!0.99003\psi\ln(|\psi|^{2})$ | 3.06$\times 10^{-3}$ | 9.97$\times 10^{-3}$

###  4.2 Discovering the dispersive and nonlinear coefficients of the LNLS
equation

In what follows we will identify the dispersive and nonlinear coefficients of
the focusing LNLS equation with $\mathcal{PT}$-symmetric harmonic potential in
the form

$\displaystyle
i\psi_{t}=-\alpha\psi_{xx}+(x^{2}+0.2ix)\psi+\sigma\psi\ln(|\psi|^{2}),$ (37)

where $\alpha,\,\sigma$ are the unknown real-valued parameters.

Let $\varphi(x,t)=u(x,t)+iv(x,t)$ with $u(x,t),\,v(x,t)$ being its real and
imaginary parts, respectively, and the PINNs $F(x,t)=iF_{u}(x,t)-F_{v}(x,t)$
with $-F_{v}(x,t),\,F_{u}(x,t)$ being its real and imaginary parts,
respectively, be

$\displaystyle\begin{array}[]{l}F(x,t):=i\varphi_{t}+\alpha\varphi_{xx}-[x^{2}+0.2ix]\varphi-\sigma\varphi\ln(|\varphi|^{2}),\vspace{0.1in}\\\
F_{u}(x,t):=u_{t}+\alpha v_{xx}-0.2xu-x^{2}v-\sigma
v\ln(u^{2}+v^{2}),\vspace{0.1in}\\\ F_{v}(x,t):=v_{t}-\alpha
u_{xx}+x^{2}u-0.2xv+\sigma u\ln(u^{2}+v^{2}),\end{array}$ (41)

Then the deep neural network is used to train $\\{u(x,t),\,v(x,t)\\}$ and
parameters $(\alpha,\,\sigma)$ by minimizing the mean squared error loss given
by Eq. (35).

To study the data-driven discovery of the LNLS equation (30) for
$\alpha,\,\sigma$, we generate a training data-set by randomly choosing
$N_{D}=10,000$ points in the $(x,t)$ domain, $[-3,3]\times[0,10]$ from the
exact solution $\psi(x,t)$ for $\alpha=-\sigma=1$. Then the obtained data-set
is applied to train an $11$-layer deep NN with $50$ neurons per hidden layer
to find the approximate parameters $\alpha,\,\sigma$ in terms of minimizing
the mean squared loss of (35). We here also choose the 20000-step Adam
optimization and 20000-step L-BFGS optimization.

For the unknown $\alpha,\,\sigma$, Table 4 illustrates the correct LNLS
equation and identified LNLS equation under the distinct sense of the clean
data or the data with $1\%$ noise, and the errors of $\alpha,\,\sigma$.

##  5 Conclusions and discussions

In conclusion, we have used the PINNs deep learning method to solve the LNLS
equation with $\mathcal{PT}$-symmetric harmonic potential and distinct
initial-boundary value conditions. We find that the deep learning method is
powerful in the LNLS equation by comparing these results and ones deduced from
the Fourier spectral method. Moreover, we also verify the PINNs learning
ability by solving the LNLS equation with $\mathcal{PT}$-symmetric harmonic
potential under the distinct conditions: 1) we fix the optimized step and
change the space width; 2) we fix the space width and change the optimized
step. Moreover, we also apply the PINNs deep learning method to effectively
investigate the data-driven discovery of the LNLS equation with
$\mathcal{PT}$-symmetric harmonic potential. Moreover, the PINNs deep learning
method can also be extended to other nonlinear wave equations with
$\mathcal{PT}$-symmetric potentials.

Acknowledgements

This work is partially supported by the NSFC under Grant Nos. 11731014 and
11925108.

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# Class numbers and self-conjugate 7-cores

Ken Ono and Wissam Raji  Department of Mathematics, University of Virginia,
Charlottesville, VA 22904<EMAIL_ADDRESS>Department of Mathematics,
American University of Beirut, Beirut, Lebanon<EMAIL_ADDRESS>
###### Abstract.

We investigate $\operatorname{sc}_{7}(n)$, the number of self-conjugate
$7$-core partitions of size $n$. It turns out that
$\operatorname{sc}_{7}(n)=0$ for $n\equiv 7\pmod{8}$. For $n\equiv
1,3,5\pmod{8}$, with $n\not\equiv 5\pmod{7},$ we find that
$\operatorname{sc}_{7}(n)$ is essentially a Hurwitz class number. Using recent
work of Gao and Qin, we show that

$\operatorname{sc}_{7}(n)=2^{-\varepsilon(n)-1}\cdot H(-D_{n}),$

where $-D_{n}:=-4^{\varepsilon(n)}(7n+14)$ and
$\varepsilon(n):=\frac{1}{2}\cdot(1+(-1)^{\frac{n-1}{2}})$. This fact implies
several corollaries which are of interest. For example, if $-D_{n}$ is a
fundamental discriminant and $p\not\in\\{2,7\\}$ is a prime with
$\mathrm{ord}_{p}(-D_{n})\leq 1$, then for every positive integer $k$ we have

(1)
$\operatorname{sc}_{7}\left((n+2)p^{2k}-2\right)=\operatorname{sc}_{7}(n)\cdot\left(1+\frac{p^{k+1}-p}{p-1}-\frac{p^{k}-1}{p-1}.\left(\frac{-D_{n}}{p}\right)\right),$

where $\left(\frac{-D_{n}}{p}\right)$ is the Legendre symbol.

###### Key words and phrases:

Class numbers, partitions

The first author thanks the support of the Thomas Jefferson Fund and the
National Science Foundation (DMS-1601306 ).The second author thanks the
support of the Center for Advanced Mathematical Sciences (CAMS) at the
American University of Beirut.

##  1\. Introduction and statement of results

A partition of a non-negative integer $n$ is any nonincreasing sequence of
positive integers which sum to $n$. The partition function $p(n)$, which
counts the number of partitions of $n$, has been studied extensively in number
theory. In particular, Ramanujan proved that

$\begin{split}p(5n+4)&\equiv 0\pmod{5},\\\ p(7n+5)&\equiv 0\pmod{7},\\\
p(11n+6)&\equiv 0\pmod{11}.\end{split}$

Partitions also play a significant role in representation theory (for example,
see [10]). Indeed, partitions of size $n$ are used to define Young tableaux,
and their combinatorial properties encode the representation theory of the
symmetric group $S_{n}$. Moreover, the $t$-core partitions of size $n$ play an
important role in number theory (for example, see [6, 7, 12]) and the modular
representation theory of $S_{n}$ and $A_{n}$ (for example, see Chapter 2 of
[10], and [5, 7]). Recall that a partition is a $t$-core if none of the hook
numbers of its Ferrers-Young diagram are multiples of $t$. If $p$ is prime,
then the existence of a $p$-core of size $n$ is equivalent to the existence of
a defect 0 $p$-block for both $S_{n}$ and $A_{n}$.

For positive integers $t$, we let $c_{t}(n)$ denote the number of $t$-core
partitions of size $n$. If $t\in\\{2,3\\}$, then it is well-known that
$c_{t}(n)=0$ for almost all $n\in\mathbb{N}.$ However, if $t\geq 4$, then
$c_{t}(n)>0$ for every positive integer $n$ (see [7]).

The case where $t=4$ is particularly interesting, as these partitions arise
naturally in algebraic number theory. As usual, let $H(-D)$ denote the
discriminant $-D<0$ Hurwitz class number. For the discriminants considered
here, $H(-D)$ is the number of inequivalent (not necessarily primitive)
positive definite binary quadratic forms with discriminant $-D<0$. In
particular, if $-D\not\in\\{-3,-4\\}$ is a fundamental discriminant, then
$H(-D)$ is the class number of the imaginary quadratic field
$\mathbb{Q}(\sqrt{-D})$. Sze and the first author proved (see Theorem 2 of
[12]) that

$c_{4}(n)=\frac{1}{2}H(-32n-20).$

Furthermore, for primes $p$ and $N\in\mathbb{N}$ with $\mathrm{ord}_{p}(N)\leq
1$, they proved (see Corollary 2 of [12]) for positive integers $k$ that

(2)
$c_{4}\left(\frac{Np^{2k}-5}{8}\right)=c_{4}\left(\frac{N-5}{8}\right)\cdot\left(1+\frac{p^{k+1}-p}{p-1}-\frac{p^{k}-1}{p-1}\cdot\left(\frac{-N}{p}\right)\right),$

where $\left(\frac{-N}{p}\right)$ is the Legendre symbol. These formulas
implied earlier conjectures of Hirschhorn and Sellers [9].

Further relationships between integer partitions and class numbers are
expected to be extremely rare. In this note we find one more instance where
$t$-cores and class numbers are intimately related. To this end, we let
${\text{\rm sc}}_{t}(n)$ denote number of the self-conjugate $t$-core
partitions of size $n$. These are $t$-core partitions which are symmetric with
respect to the operation which switches the rows and columns of a Ferrers-
Young diagram.

To formulate these results, for a positive odd integer $n$ we define the
negative discriminant

(3) $-D_{n}:=\begin{cases}-28n-56\ \ \ \ \ &{\text{\rm if}}\ n\equiv
1\pmod{4},\\\ -7n-14\ \ \ \ \ &{\text{\rm if}}\ n\equiv 3\pmod{4}.\end{cases}$

######  Theorem 1.

If $n\not\equiv 5\pmod{7}$ is a positive odd integer, then we have

$\operatorname{sc}_{7}(n)=\begin{cases}\frac{1}{4}H(-D_{n})\ \ \ \ \ \ \
&{\text{\rm if}}\ n\equiv 1\pmod{4}\\\ \frac{1}{2}H(-D_{n})\ \ \ \ \
&{\text{\rm if}}\ n\equiv 3\pmod{8}\\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
&{\text{\rm if}}\ n\equiv 7\pmod{8}.\end{cases}$

######  Example.

Here we consider the case where $n=9$. According to Theorem 1, we have that
$\operatorname{sc}_{7}(9)=H(-308)/4.$ One readily finds that there are eight
equivalence classes of discriminant $-308$ binary quadratic forms. The reduced
forms representing these classes are:

$\begin{matrix}2X^{2}+2XY+39Y^{2},&3X^{2}-2XY+26Y^{2},&3X^{2}+2XY+26Y^{2},&6X^{2}-2XY+13Y^{2},\\\
6X^{2}+2XY+13Y^{2},&7X^{2}+11Y^{2},&9X^{2}-4XY+9Y^{2},&9X^{2}+4XY+9Y^{2}.\end{matrix}$

There are fourteen $7$-core partitions of $n=9$. However, only two of them are
self-conjugate. They are (subscripts are the hook numbers):

$\begin{matrix}\bullet_{5}&\bullet_{4}&\bullet_{3}\\\
\bullet_{4}&\bullet_{3}&\bullet_{2}\\\
\bullet_{3}&\bullet_{2}&\bullet_{1}\end{matrix}\ \ \ \ \ \ \ \ {\text{\rm
and}}\ \ \ \ \ \ \ \
\begin{matrix}\bullet_{9}&\bullet_{4}&\bullet_{3}&\bullet_{2}&\bullet_{1}\\\
\bullet_{4}\\\ \bullet_{3}\\\ \bullet_{2}\\\ \bullet_{1}\end{matrix}.$

This example illustrates the conclusion that
$\operatorname{sc}_{7}(9)=2=H(-308)/4$.

######  Example.

For $n=25$, we have that $-D_{n}=-756=-3^{2}\cdot 84$ and $H(-756)=16$.
Therefore, Theorem 1 implies that $\operatorname{sc}_{7}(25)=H(-756)/4=4.$

Theorem 1 implies simple short finite formulas for $\operatorname{sc}_{7}(n)$.
To this end, we recall the standard Kronecker character for a discriminant
$D$. Define the Kronecker character $\chi_{D}(n)$ for positive integers $n$ by

$\chi_{D}(n)=\left(\frac{D}{n}\right):=\prod\left(\frac{D}{p_{i}}\right)^{a_{i}},$

where $n=\prod p_{i}^{a_{i}}$ and $\left(\frac{D}{p}\right)$ is the Legendre
symbol when $p$ is an odd prime and

$\left(\frac{D}{2}\right):=\begin{cases}0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \mbox{if $D$ is even}\\\ (-1)^{(D^{2}-1)/8}\ \ \ \ \ \ \ \ \mbox{if $D$
is odd}.\end{cases}$

######  Corollary 2.

If $n\not\equiv 5\pmod{7}$ is a non-negative odd integer for which $-D_{n}$ is
a fundamental discriminant, then

$\operatorname{sc}_{7}(n)=\begin{cases}-\frac{1}{4D_{n}}\sum_{m=1}^{D_{n}}\left(\frac{-D_{n}}{m}\right)m\
\ \ \ \ \ &{\text{\rm if}}\ n\equiv 1\pmod{4},\\\ \ \ \\\
-\frac{1}{2D_{n}}\sum_{m=1}^{D_{n}}\left(\frac{-D_{n}}{m}\right)m,\ \ \ \ \ \
&{\text{\rm if}}\ n\equiv 3\pmod{8},\\\ \ \ \\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &{\text{\rm if}}\ n\equiv
7\pmod{8}.\\\ \end{cases}$

######  Example.

If $n=11$, then $-D_{11}=-91$. We then find that

$\operatorname{sc}_{7}(11)=-\frac{1}{182}\sum_{m=1}^{91}\left(\frac{-91}{m}\right)m=1.$

Hurwitz class numbers enjoy a host of multiplicative properties which can be
formulated in terms of the Möbius function $\mu(d)$ and the divisor function
$\sigma_{1}(n):=\sum_{1\leq d\mid n}d$. These imply the following simple
corollary which is analogous to (2).

######  Corollary 3.

If $n\not\equiv 5\pmod{7}$ is a positive odd integer, and $-D_{n}$ is a
fundamental discriminant, then for all odd integers $f$ coprime to $7$, we
have

$\operatorname{sc}_{7}((n+2)f^{2}-2)=\operatorname{sc}_{7}(n)\sum_{1\leq d\mid
f}\mu(d)\left(\frac{-D_{n}}{d}\right)\sigma_{1}(f/d).$

######  Remark.

The cases where $f=p^{k}$ is a prime power coincide with (1).

######  Example.

If $n=11$, then we have that $-D_{11}=-91$. Now suppose that $f=15$. Corollary
3 implies that

$\operatorname{sc}_{7}(2923)=\operatorname{sc}_{7}(11)\sum_{1\leq d\mid
15}\mu(d)\left(\frac{-91}{d}\right)\sigma_{1}(15/d).$

By direct calculation, since $\operatorname{sc}_{7}(11)=1$, the right hand
side of this expression equals

$\sigma_{1}(15)+\sigma_{1}(5)-\sigma_{1}(3)-\sigma_{1}(1)=25.$

Using the $q$-series identities in the next section (for example (4)), one can
indeed check directly that $\operatorname{sc}_{7}(2923)=25$.

##  2\. Proofs

Here we prove Theorem 1 and Corollaries 2 and 3.

###### Proof of Theorem 1.

If $t$ is a positive odd integer, then the generating function for
$\mathrm{sc}_{t}(n)$ (see (2) of [8]) is

(4)
$\sum_{n=0}^{\infty}\mathrm{sc}_{t}(n)q^{n}=\prod_{n=1}^{\infty}\frac{(1-q^{2tn})^{\frac{t-1}{2}}(1+q^{2n-1})}{(1+q^{t(2n-1)})}.$

Therefore, in terms of Dedekind’s eta-function
$\eta(\tau):=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})$, where $q:=e^{2\pi
i\tau}$, we have that

$S(\tau):=\sum_{n=0}^{\infty}\operatorname{sc}_{7}(n)q^{n+2}=\frac{\eta(2\tau)^{2}\eta(14\tau)\eta(7\tau)\eta(28\tau)}{\eta(4\tau)\eta(\tau)}.$

In particular, by the standard theory of modular forms (for example, see
Chapter 1.4 of [11]), it follows that $S(\tau)$ is a holomorphic modular form
of weight 3/2 on $\Gamma_{0}(28)$ with Nebentypus
$\chi_{7}(n):=\left(\frac{7}{\bullet}\right)$. In terms of theta functions, we
have that111This corrects the statement of Theorem 9 of [1].

$S(\tau)=\frac{1}{14}\Theta_{1}(\tau)-\frac{1}{7}\Theta_{2}(\tau)+\frac{1}{14}\Theta_{3}(\tau),$

where

$\Theta_{1}(\tau)=\sum_{(x,y,x)\in\mathbb{Z}^{3}}q^{Q_{1}(x,y,z)}=\sum_{(x,y,x)\in\mathbb{Z}^{3}}q^{x^{2}+y^{2}+2z^{2}-yz},$
$\Theta_{2}(\tau)=\sum_{(x,y,x)\in\mathbb{Z}^{3}}q^{Q_{2}(x,y,z)}=\sum_{(x,y,x)\in\mathbb{Z}^{3}}q^{x^{2}+4y^{2}+8z^{2}-4yz}$

and

$\Theta_{3}(\tau)=\sum_{(x,y,x)\in\mathbb{Z}^{3}}q^{Q_{3}(x,y,z)}=\sum_{(x,y,x)\in\mathbb{Z}^{3}}q^{2x^{2}+2y^{2}+3z^{2}+2yz+2xz+2xy}.$

As a result, we get that

(5)
$\operatorname{sc}_{7}(n)=\frac{1}{14}R(Q_{1};n+2)-\frac{1}{7}R(Q_{2};n+2)+\frac{1}{14}R(Q_{3};n+2),$

where $R(Q_{i};n+2)$ is the number of integral representations of $n+2$ by
$Q_{i}(x,y,z)$.

To prove the theorem, we must relate these three theta functions to the
Eisenstein series in the space $M_{\frac{3}{2}}(28,\chi_{7})$. Thankfully, Gao
and Qin [4] have already carried out these calculations. They produce three
Eisenstein series which form a basis of this space (see Theorem 3.2 of [4]).
These series are given by their Fourier expansions:

$\displaystyle g_{1}(\tau)$ $\displaystyle=$ $\displaystyle
1+2\pi\sqrt{7}\sum_{n=1}^{\infty}\lambda(7n,28)\alpha(7n)\left(A(7,7n)-\frac{1}{7}\right)\sqrt{n}q^{n},$
$\displaystyle g_{2}(\tau)$ $\displaystyle=$
$\displaystyle\frac{2}{49}\pi\sqrt{7}\sum_{n=1}^{\infty}\lambda(7n,28)\alpha(7n)\sqrt{n}q^{n},$
$\displaystyle g_{3}(\tau)$ $\displaystyle=$ $\displaystyle
2\pi\sqrt{7}\sum_{n=1}^{\infty}\lambda(7n,28)\left(A(7,7n)-\frac{1}{7}\right)\sqrt{n}q^{n}.$

The quantities $\alpha(m),A(p,m),\lambda(7m,28)$, $h_{p}(m)$ and
$h^{\prime}_{p}(m)$ are defined by

$\alpha(m):=\begin{cases}3\cdot 2^{-\frac{1+h_{2}(m)}{2}}\ \ \ \ \ &{\text{\rm
if}}\ h_{2}(m)\ {\text{\rm is odd}},\\\ 3\cdot 2^{-1-\frac{h_{2}(m)}{2}}\ \ \
\ &{\text{\rm if}}\ h_{2}(m)\ {\text{\rm is even and}}\
h_{2}^{\prime}(m)\equiv 1\pmod{4},\\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
&{\text{\rm if}}\ h_{2}(m)\ {\text{\rm is even and}}\ \
h_{2}^{\prime}(m)\equiv 3\pmod{8},\\\ 2^{-\frac{h_{2}(m)}{2}}\ \ \ \ \ \ \ \ \
\ &{\text{\rm if}}\ h_{2}(m)\ {\text{\rm is even and}}\
h_{2}^{\prime}(m)\equiv 7\pmod{8},\\\ \end{cases}$

$A(p,m):=\begin{cases}p^{-1}-(1+p)p^{-\frac{3+h_{p}(m)}{2}}\ \ \ &{\text{\rm
if}}\ h_{p}(m)\ {\text{\rm is odd,}}\\\ p^{-1}-2p^{-1-\frac{h_{p}(m)}{2}}\ \ \
&{\text{\rm if}}\ h_{p}(m)\ {\text{\rm is even
and}}\left(\frac{-h^{\prime}_{p}(m)}{p}\right)=-1,\\\ p^{-1},\ \ \ &{\text{\rm
if}}\ h_{p}(m)\ {\text{\rm is even and}}\
\left(\frac{-h^{\prime}_{p}(m)}{p}\right)=1,\end{cases}$

where $h_{p}(m)$ is the non-negative integer for which $p^{h_{p}(m)}||m$ and
$h^{\prime}_{p}(m):=\frac{m}{p^{h_{p}(m)}}$, and

$\lambda(7m,28)=\begin{cases}\frac{49}{4\pi\sqrt{7m}}\cdot H(-7m),\ \ \
&{\text{\rm if}}\ m\equiv 5\pmod{8},\\\ \frac{49}{12\pi\sqrt{7m}}\cdot
H(-7m),\ \ \ &{\text{\rm otherwise.}}\end{cases}$

The formula for $\alpha(m)$, when $h_{2}^{\prime}(m)\equiv 3,7\pmod{8},$
corrects a typographical error in [4].

Since the three Eisenstein series form a basis of this space, it is trivial to
deduce that

$\Theta_{1}(\tau)=g_{1}(\tau)-3g_{2}(\tau),\
\Theta_{2}(\tau)=g_{1}(\tau)-\frac{3}{2}g_{3}(\tau),\
\Theta_{3}(\tau)=g_{1}(\tau)+14g_{2}(\tau).$

Using $h_{7}(7n)=1$, $h_{7}^{\prime}(7n)=n$, $h_{2}(7n)=0$ and
$h_{2}^{\prime}(7n)=7n$ for $(n,14)=1$, we find that

$A(7,7n)=\frac{1}{7}-\frac{8}{49},$

and

$\alpha(7n):=\begin{cases}\frac{3}{2}\ \ \ \ &{\text{\rm if}}\ n\equiv
3\pmod{4},\\\ 1\ \ \ \ &{\text{\rm if}}\ n\equiv 1\pmod{8},\\\ 0\ \ \
&{\text{\rm if}}\ n\equiv 5\pmod{8}.\end{cases}$

Combining these facts, for positive integers $n$ we deduce that

$R(Q_{1};n)=2\pi\sqrt{7n}\lambda(7n,28)\left(A(7,7n)-\frac{1}{7}\right)(\alpha(7n)-3),$
$R(Q_{2};n)=2\pi\sqrt{7n}\lambda(7n,28)\left(A(7,7n)-\frac{1}{7}\right)(\alpha(7n)-\frac{3}{2}),$

and

$R(Q_{3};n)=2\pi\sqrt{7n}\lambda(7n,28)\alpha(7n)\left(A(7,7n)-\frac{1}{7}+\frac{14}{49}\right).$

Finally, these expressions, for positive odd $n\not\equiv 5\pmod{7}$, simplify
to

$R(Q_{1};n+2)=\begin{cases}2H(-D_{n})\ \ \ &{\text{\rm if}}\ n\equiv
1\pmod{4},\\\ 8H(-D_{n})\ \ \ &{\text{\rm if}}\ n\equiv 3\pmod{8},\\\
4H(-D_{n})\ \ \ &{\text{\rm if}}\ n\equiv 7\pmod{8},\end{cases}$
$R(Q_{2};n+2)=\begin{cases}0\ \ \ &{\text{\rm if}}\ n\equiv 1\pmod{4},\\\
2H(-D_{n})\ \ \ &{\text{\rm if}}\ n\equiv 3\pmod{8},\\\ 2H(-D_{n})\ \ \
&{\text{\rm if}}\ n\equiv 7\pmod{8},\end{cases}$

$R(Q_{3};n+2)=\begin{cases}\frac{3}{2}H(-D_{n})\ \ \ &{\text{\rm if}}\ n\equiv
1\pmod{4},\\\ 3H(-D_{n})\ \ \ &{\text{\rm if}}\ n\equiv 3\pmod{8},\\\ 0\ \ \
&{\text{\rm if}}\ n\equiv 7\pmod{8}.\end{cases}$

The claimed formulas now follow from (5). ∎

###### Proof of Corollary 2.

If $-D<0$ is a fundamental discriminant, then it is well-known (for example,
see (7.29) of [3]) that

$H(-D)=-\frac{|O_{K}^{\times}|}{2D}\sum_{m=1}^{D}\left(\frac{-D}{m}\right)m,$

where $K=\mathbb{Q}(\sqrt{-D})$ and $O_{K}^{\times}$ denotes the units in its
corresponding ring of integers. The claim now follows from Theorem 1. ∎

###### Proof of Corollary 3.

If $-D<0$ is a fundamental discriminant of an imaginary quadratic field, then
for every integer $f$ it is known (p. 273, [2]) that

$H(-Df^{2})=\frac{H(-D)}{w(-D)}\cdot\sum_{1\leq d\mid
f}\mu(d)\left(\frac{-D}{d}\right)\sigma_{1}(f/d),$

where $w(-D)$ denotes half of the number of roots of unity in
$\mathbb{Q}(\sqrt{-D})$. The claim now follows from Theorem 1.

∎

## References

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[a]Maria Laura Piscopo

# Status of the lifetimes of heavy hadrons within the HQE

###### Abstract

We present the theoretical status of the lifetimes of weakly decaying heavy
hadrons containing a bottom or a charm quark, and discuss the current
predictions, based on the framework of the Heavy Quark Expansion (HQE), for
both mesons and baryons. Potential improvements to reduce the theoretical
uncertainties are also highlighted.

##  1 Introduction

The total lifetime $\tau$, or the total decay width $\Gamma=\tau^{-1}$, is one
of the fundamental properties of particles. In the study of lifetimes, a
special role is occupied by heavy hadrons: QCD bound states containing a heavy
quark $Q$ with $m_{Q}\gg\Lambda_{\rm QCD}$, where $\Lambda_{\rm QCD}$ denotes
a typical hadronic non-perturbative scale of the order of few hundreds MeV. As
their weak decays involve the interplay between the weak and the strong
interactions over the wide range of scales $m_{W}\gg m_{Q}\gg\Lambda_{\rm
QCD}$, heavy hadrons are interesting systems to test the Standard Model (SM)
and also, given enough precision in both theory and experiments, perform
indirect searches of new physics (NP).

Lifetimes measurements are by now very precise [1, 2]. On the theoretical
side, the Heavy Quark Expansion (HQE) [3] 111See e.g. the review [4] for a
more comprehensive list of references. provides a well established framework
to compute inclusive decay widths of heavy hadrons in terms of a systematic
expansion in inverse powers of $m_{Q}$. As the reliability of the HQE strongly
depends on the assumption that $Q$ is heavy, bottom hadrons are clearly the
most suited systems to be described within this framework. The charm quark
mass, on the other hand, lies at the boundary between the heavy and the light
quark regime, and if charmed systems pose notoriously more challenges to
precise theoretical studies, they also give an opportunity to test even
further the applicability of the HQE.

##  2 The HQE in the bottom sector

The total decay width of a bottom hadron $H_{b}$ is given by

$\Gamma(H_{b})=\frac{1}{2m_{H_{b}}}\sum_{n}\int_{\rm
PS}(2\pi)^{4}\delta^{(4)}(p_{n}-p_{H_{b}})|\langle n|{\cal H}_{\rm
eff}|H_{b}\rangle|^{2}\,,$ (1)

where the sum runs over all possible final states into which $H_{b}$ can
decay, ${\rm PS}$ is the corresponding phase space, and ${\cal H}_{\rm eff}$
the effective Hamiltonian [5]. Using the optical theorem, eq. (1) becomes

$\Gamma(H_{b})=\frac{1}{2m_{H_{b}}}{\rm Im}\langle H_{b}|{\cal
T}|H_{b}\rangle\,,\quad{\rm with}\quad{\cal T}=i\\!\\!\int\\!\\!d^{4}x\,\,{\rm
T}\\{{\cal H}_{\rm eff}(x),{\cal H}_{\rm eff}(0)\\}\,.$ (2)

The transition operator ${\cal T}$ contains the time-ordered product of ${\cal
H}_{\rm eff}$ and is thus a non-local operator; its expression however,
simplifies in the limit of a heavy decaying quark. In fact, by exploiting the
hierarchy $m_{b}\gg\Lambda_{\rm QCD}$, the $b$-quark momentum and field can be
respectively parametrised as

$p_{b}^{\mu}=m_{b}v^{\mu}+k^{\mu}\,,\qquad b(x)=e^{-im_{b}v\cdot
x}b_{v}(x)\,,$ (3)

where $v=p_{H_{b}}/m_{H_{b}}$ is the hadron velocity, $k\ll m_{b}$ a residual
momentum of the order of $\Lambda_{\rm QCD}$, and we have introduced the
rescaled field $b_{v}(x)$ containing only low oscillation frequencies of the
order of $k$.

Semileptonic modes (SL)
---
$\Gamma_{3}^{(3)}$ | Fael, Schönwald, Steinhauser [6]
Czakon, Czarnecki, Dowling [7]
$\Gamma_{5}^{(1)}$ | Alberti, Gambino, Nandi [8]
Mannel, Pivovarov, Rosenthal [9]
$\Gamma_{6}^{(1)}$ | Mannel, Moreno, Pivovarov [10]
$\Gamma_{7}^{(0)}$ | Dassinger, Mannel, Turczyk [11]
$\Gamma_{8}^{(0)}$ | Mannel, Turczyk, Uraltsev [12]
$\tilde{\Gamma}_{6}^{(1)}$ | Lenz, Rauh [13]

Non-leptonic modes (NL)
---
$\Gamma_{3}^{(2)}$ | Czarnecki, Slusarczyk, Tkachov [14]∗
$\Gamma_{3}^{(1)}$ | Ho-Kim, Pham [15]; Altarelli, Petrarca [16]; Bagan et al. [17]
Lenz, Nierste, Ostermaier [18]; Krinner, Lenz, Rauh [19]
$\Gamma_{5}^{(0)}$ | Bigi, Uraltsev, Vainsthtein [20]; Blok, Shifman [21]
$\Gamma_{6}^{(0)}$ | Lenz, Piscopo, Rusov [22]; Mannel, Moreno, Pivovarov [23]
$\tilde{\Gamma}_{6}^{(1)}$ | Beneke, Buchalla, Greub, Lenz, Nierste [24]
Franco, Lubicz, Mescia, Tarantino [25]
$\tilde{\Gamma}_{7}^{(0)}$ | Gabbiani, Onishchenko, Petrov [26]

Table 1: Status of the perturbative corrections in the HQE. The work with ∗
contains only partial results.

Taking into account eq. (3), the time ordered product in eq. (2) can be
systematically expanded into a series of local operators ${\cal O}_{d}$ of
increasing dimension $d$ and suppressed by $d-3$ powers of the heavy quark
mass. In this way, the total decay width takes the following form

$\Gamma(H_{b})=\Gamma_{3}+\Gamma_{5}\frac{\langle{\cal
O}_{5}\rangle}{m_{b}^{2}}+\Gamma_{6}\frac{\langle{\cal
O}_{6}\rangle}{m_{b}^{3}}+\ldots+16\pi^{2}\left[\tilde{\Gamma}_{6}\frac{\langle\tilde{\mathcal{O}}_{6}\rangle}{m_{b}^{3}}+\tilde{\Gamma}_{7}\frac{\langle\tilde{\mathcal{O}}_{7}\rangle}{m_{b}^{4}}+\dots\right]\,.$
(4)

The short-distance functions $\Gamma_{d}$ can be computed in terms of a
perturbative expansion in $\alpha_{s}(m_{b})$

$\Gamma_{d}=\Gamma_{d}^{(0)}+\left(\frac{\alpha_{s}}{4\pi}\right)\Gamma_{d}^{(1)}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\Gamma_{d}^{(2)}+\ldots\,,$
(5)

whereas $\langle{\cal O}_{d}\rangle\equiv\langle H_{b}|{\cal
O}_{d}|H_{b}\rangle/(2m_{H_{b}})$ are matrix elements parametrising the non-
perturbative effects. Note that in eq. (4) both two- and four-quark operators
contribute. While at LO-QCD the former are obtained from the discontinuity of
two-loop diagrams, the latter already arise at one-loop at the same order in
$\alpha_{s}$, hence the explicit phase space factor $16\pi^{2}$ in front of
the four-quark operators contributions, here labeled by a tilde. Lifetime
ratios of bottom hadrons can then be computed as

$\frac{\tau(H_{b})}{\tau(H_{b}^{\prime})}=1+\left[\Gamma(H_{b}^{\prime})^{\,\rm
HQE}-\Gamma(H_{b})^{\,\rm HQE}\right]\tau(H_{b})^{\rm exp.}\,,$ (6)

where the experimental value of $\tau(H_{b})$ is used to cancel out the
dependence on $\Gamma_{3}$, albeit this is not necessary and eq. (6) can be
also computed entirely within the HQE with only slightly larger uncertainties,
see e.g. the discussion in [27]. It is worth emphasising that while two-quark
operators present short-distance coefficients which are universal for a given
heavy quark, so that any differences in lifetimes of bottom hadrons would only
be induced by the value of their matrix elements, the Wilson coefficients of
the four-quark operators also depend on the spectator quark in the specific
hadron considered and this can in general lead to larger effects in lifetime
ratios.

The HQE is by now a very advanced framework. The theoretical status for both
the perturbative and non-perturbative side is summarised respectively in
tables 1 and 2.

| $B_{d},B^{+}$ | $B_{s}$ | $\Lambda^{0}_{b}$ |  $\Xi_{b}^{-}$, $\Xi_{b}^{0}$, $\Omega_{b}^{-}$
---|---|---|---|---
$\langle{\cal O}_{5}\rangle$ | Fits to SL data [28, 29, 30, 31] |  |  |
HQET sum rules [32, 33] | Spectroscopy [37] | Spectroscopy [20] | Spectroscopy [43]
Lattice QCD [34, 35] |  |  |
$\langle{\cal O}_{6}\rangle$ | Fits to SL data [28, 29, 30, 31] | Sum rules estimates [37] | EOM relation to $\langle\tilde{\cal O}_{6}\rangle$ | EOM relation to $\langle\tilde{\cal O}_{6}\rangle$
EOM relation to $\langle\tilde{\cal O}_{6}\rangle$ | EOM relation to $\langle\tilde{\cal O}_{6}\rangle$ |  |
$\langle\tilde{\cal O}_{6}\rangle$ | HQET sum rules [36] | HQET sum rules [38] | HQET SR [39]; NRCQM + | NRCQM +
|  | spectroscopy [41, 42] | spectroscopy [41, 42]
$\langle\tilde{\cal O}_{7}\rangle$ | Vacuum insertion approximation | —– | —-
Table 2: Status of the available determinations of the non-perturbative matrix
elements in the HQE for the bottom sector. The following abbreviations are
used: heavy quark effective theory (HQET), sum rules (SR), equation of motion
(EOM), non-relativistic constituent quark model (NRCQM).

###  2.1 Lifetimes of bottom mesons

The current HQE predictions for the total widths of the bottom mesons
$B_{d},B^{+},B_{s},$ and their lifetime ratios, are shown in fig. 1 and table
3 [27]. The two scenarios correspond to different choices of inputs, including
the value of the $b$-quark mass and of the non-perturbative parameters
associated to the matrix elements of two-quark operators, see for details
[27]. While the total decay widths and the ratio $\tau(B^{+})/\tau(B_{d})$ are
only mildly sensitive to this choice, the results for
$\tau(B_{s})/\tau(B_{d})$ are strongly affected, and in one case, a small
tension with the experimental value emerges. In this respect, we point out
that the two fits to experimental data on inclusive semileptonic $B$-meson
decays [30, 31] 222We stress that corresponding measurements for inclusive
semileptonic $B_{s}$-decays are still missing. find different values of the
Darwin parameter $\rho_{D}^{3}$, which plays a crucial role in the prediction
of $\tau(B_{s})/\tau(B_{d})$. Further insights on the origin of this
discrepancy are clearly of utmost importance given the potential that this
observable could have as an indirect probe of NP. Moreover, an investigation
by lattice QCD of the small size of the bag-parameters of the octet-operators
and of the so-called ‘eye-contractions’ as found in [36, 38], would also be
very desirable.

Scale variation in the leading term $\Gamma_{3}$, and the values of the non-
perturbative parameters, including SU(3)F breaking effects, are the main
sources of uncertainties for the total widths and the lifetime ratios,
respectively. Overall, there is very good agreement between HQE and data, and
the clean observable $\tau(B^{+})/\tau(B_{d})$ could already be used to
constrain certain NP operators [44].

| HQE Scenario A | HQE Scenario B | Exp. value
---|---|---|---
$\tau(B^{+})/\tau(B_{d})$ | $1.0855^{+0.0232}_{-0.0219}$ | $1.0851^{+0.0230}_{-0.0217}$ | $1.076\pm 0.004$
$\tau(B_{s})/\tau(B_{d})$ | $1.0279^{+0.0113}_{-0.0113}$ | $1.0032^{+0.0063}_{-0.0063}$ | $0.998\pm 0.005$
Table 3: HQE predictions of the lifetime ratios for two different choices of
theory inputs, see [27].

Figure 1: Comparison of data and HQE predictions for two different choices of
theory inputs, based on [27].

###  2.2 Lifetimes of bottom baryons

The most recent theoretical predictions for the total widths of the
$b$-baryons $\Lambda^{0}_{b},\Xi_{b}^{0},\Xi_{b}^{-},\Omega_{b}^{-},$ as well
as their lifetime ratios, are displayed in fig. 2 and in table 5, see [45].
Along all the observables considered there is very good agreement between
theory and experiments, while the dominant sources of uncertainties are again
scale variation and the values of the non-perturbative parameters. In fact,
despite the efforts in e.g. [39, 40], the four-quark dimension-six matrix-
elements have not yet been computed for all baryons either within lattice QCD
or QCD sum rules, and currently only spectroscopy relations based on
simplified models of QCD, like the NRCQM, can be consistently used. First
principle calculations would be of great importance in order to reduce the
overall uncertainties and obtain an independent determination of these non-
perturbative parameters.

| $\tau(\Lambda^{0}_{b})/\tau(B_{d})$ | $\tau(\Xi_{b}^{0})/\tau(\Xi_{b}^{-})$ | $\tau(\Omega^{-}_{b})/\tau(B_{d})$
---|---|---|---
HQE | $0.955\pm 0.014$ | $0.929\pm 0.028$ | $1.081\pm 0.042$
Exp. | $0.969\pm 0.006$ | $0.929\pm 0.028$ | $1.080^{+0.118}_{-0.112}$
Table 4:  Comparison of data and HQE predictions for selected lifetime ratios,
based on [45].

Figure 2: Comparison of HQE predictions and experimental data for $b$-baryons
[45].

##  3 The HQE at its limit: lifetimes of charmed hadrons

While the applicability of the HQE appears to be well established in the
bottom sector, it is questionable whether it also extends to the charm system,
given that $m_{c}\sim 1$ GeV. In this case the two expansion parameters
$\alpha_{s}(m_{c})$ and $\Lambda_{\rm QCD}/m_{c}$ become larger, making both
the perturbative and the power-correction series a priori less reliable.
Furthermore, many of the non-perturbative parameters of the HQE for charmed
hadrons are still poorly known and in most of the cases only rough estimates
based on symmetry arguments with respect to the bottom sector are available.

It is then interesting to confront the HQE framework with the experimental
data for the charm system. The current theoretical predictions for the total
widths of the charmed mesons $D^{0},D^{+},D_{s}^{+}$, their lifetime ratios,
as well as the semileptonic branching fractions, are shown in fig. 3 and table
5 [46]. Although with very large uncertainties, the HQE succeeds in correctly
describing the observed patterns, and no indication of a possible breakdown of
the framework seems to emerge. These results have been confirmed in the recent
study [43], which included as well predictions for the lifetimes of singly
charmed baryons. Also in this case, it is found that the HQE can consistently
accommodate the experimental data, albeit again within large uncertainties.

| $\tau(D^{+})/\tau(D^{0})$ | $\bar{\tau}(D_{s}^{+})/\tau(D^{0})$
---|---|---
HQE | $2.80^{+0.86}_{-0.90}$ | $1.00\pm 0.16$
Exp. | $2.54\pm 0.02$ | $1.30\pm 0.01$
Table 5: Comparison of data and HQE predictions of charmed mesons lifetime
ratios, based on [46]. Note that $\bar{\tau}(D_{s}^{+})$ does not include the
semileptonic mode $D_{s}^{+}\to\tau^{+}\nu_{\tau}$, see for details [46].

Figure 3: Comparison of HQE predictions and experimental data for charmed
mesons, based on [46].

##  4 Conclusions

We have presented the status of the lifetimes of heavy hadrons containing a
bottom or charm quark within the HQE 333We have not discussed hadrons
containing two heavy quarks. For studies of the $B_{c}$ lifetime see [47,
48].. Along all the observables considered the agreement with the experimental
data is good, confirming the HQE as a powerful framework for inclusive heavy
hadron decays, contrary to some conclusions in [49]. Moreover, the precision
already achieved for several clean quantities in the bottom system is high and
comparable with the experimental one. Further improvements in the computation
of higher order corrections, namely $\Gamma_{3,{\rm NL}}^{(2)}$,
$\Gamma_{7}^{(0)}$, $\tilde{\Gamma}_{6}^{(2)}$, as well as in the
determination of the non-perturbative parameters, will be crucial in order to
reduce the theoretical uncertainties, so that in future these observables
might also serve as interesting probes of NP.

##  5 Acknowledgments

I would like to thank the organisers of DISCRETE 2022 for the invitation, and
Alexander Lenz and Aleksey Rusov for helpful comments on the manuscript. This
work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - project number 500314741.

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