Rediscovery of Numerical L uschers Formula from the Neural Network Yu Lu1Yi-Jia Wang1Ying Chen1 2and Jia-Jun Wu1 1School of Physical Sciences University of Chinese Academy of Sciences UCAS Beijing 100049 China

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Rediscovery of Numerical L¨uscher’s Formula from the Neural Network

Yu Lu,1, ∗Yi-Jia Wang,1, †Ying Chen,1, 2, ‡and Jia-Jun Wu1, §

1School of Physical Sciences, University of Chinese Academy of Sciences (UCAS), Beijing 100049, China

2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

We present that by predicting the spectrum in discrete space from the phase shift in continuous

space, the neural network can remarkably reproduce the numerical L¨uscher’s formula to a high

precision. The model-independent property of the L¨uscher’s formula is naturally realized by the

generalizability of the neural network. This exhibits the great potential of the neural network

to extract model-independent relation between model-dependent quantities, and this data-driven

approach could greatly facilitate the discovery of the physical principles underneath the intricate

data.

I. INTRODUCTION

Physicists are always going after a concise description

of data. Generally, this concise description boils down to

analytic expressions or conserved quantities, which are

usually dodging and hiding and cannot be trapped easily.

Nowadays the rapid progress of machine learning (ML)

techniques are helping physicists to meet their goals, as

manifested by the applications, such as AI Feynman[1,2]

and AI Poincar´e [3,4]. For a review of ML techniques in

physics, see [5] and several applications in hadron physics

in Refs. [6–12] and references therein.

In most cases of modern physics, a concise description

is generally realized at more abstract levels, such as ana-

lytic diﬀerential or integral equations whose solutions are

supposed to explain the data. If these equations are ex-

plicitly known but cannot be solved easily even through

numerical methods, ML may help to work out the solu-

tions through Physics-informed-neural-network (PINN)

approach[13]. In a more challenging case that there are

conceptually links between physical principles and real-

istic phenomena but we cannot write down the exact ex-

pressions, maybe we can also resort to the data driven

ML for uncovering the underneath connections.

A typical example is the study of the strong interaction

in the low energy regime. It is known that the properties

of hadrons are necessarily dictated by quantum chromo-

dynmics (QCD), the fundamental theory of the strong

interactions. However, due to the unique self-interacting

properties of gluons, the strong coupling constant is large

at the low energy regime and makes the standard per-

turbation theory inapplicable. Up to now, lattice QCD

(LQCD) is the most important ab initio non-perturbative

method for investigating the low energy properties of

the strong interactions. LQCD is deﬁned on the dis-

cretized Euclidean spacetime lattice and adopts the nu-

merical simulation as its major approach. The major

observables of LQCD are energies and matrix elements

∗ylu@ucas.ac.cn

†wangyijia18@mails.ucas.ac.cn

‡cheny@ihep.ac.cn, corresponding author

§wujiajun@ucas.ac.cn, corresponding author

of hadron systems. However, except for the properties of

ground state hadrons without strong decays, it is usually

non-trivial for lattice results (on the Euclidean spacetime

lattice) to be connected with experimental observables in

the continuum Minkowski spacetime. For example, most

hadrons are resonances observed in the invariant mass

spectrum of multi-hadron system in decay or scattering

processes, while what the lattice QCD can calculate are

the discretized energy levels of related hadron systems on

ﬁnite lattices. Therefore, the connection must be estab-

lished.

FIG. 1. The workﬂow of this work.

One successful approach to address this issue is called

L¨uscher’s formula [14–16], developed by L¨uscher and col-

laborators more than 30 years ago. By making use of

the ﬁnite volume eﬀects, L¨uscher’s formula describes re-

lation of the spectrum E(L) of a two-body system on the

ﬁnite lattice of size Lwith the scattering phase shift δ(E)

of this system in the continuum Minkowski space. The

extension of L¨uscher’s formula to three-body systems is

still undergoing [17–27]. L¨uscher’s formula and its ex-

tension are not only practically useful, but also is invalu-

ably model-independent on the theoretical side. Deriving

these model-independent theoretical approaches are very

challenging and require a lot of wisdom and insight.

For the multi-channel case, more than one free param-

eters in the scattering amplitude will show up in the inﬁ-

nite volume, while L¨uscher’s formula oﬀers only one con-

arXiv:2210.02184v2 [hep-lat] 8 Apr 2024

strain to connect the volume size and scattering ampli-

tude at the discrete energy levels. To extract the infor-

mation of scattering amplitude in the inﬁnite volume, we

will need several diﬀerent ﬁnite volume sizes which share

the same energy level. However, one cannot know a pri-

ori which volume size will produce the desired spectra

without doing the expensive lattice calculations. There-

fore, one practical way in the multi-channel process is

to build a model to relate the scattering amplitude at

diﬀerent energy levels in order to use the L¨uscher’s for-

mula to translate the lattice spectrum into phase shifts or

other information. As a consequence, model dependence

will inevitably enter into such calculation. In contrast,

since the neural network is trained via the data-driven

way, it is naturally model independent, or at least its

model dependence can be safely ignored. To this end, as

a ﬁrs step, we should answer whether the neural network

can rediscover the numerical L¨uscher’s formula in single

channel case.

Another challenge comes from the L¨uscher’s formula it-

self. Unlike extracting the analytic expression of the con-

served quantities from the trajectory by ML approach [4],

the L¨uscher’s formula is beyond the elementary function,

therefore it can only be evaluated numerically. This prop-

erty also leaves a great challenge for neural network to

discovery it.

FIG. 2. The structure of our neural network. Green round

rectangles with integer nrepresent the linear layer with size n,

which consists of all the learning parameters. Orange circles

denote the input and output nodes and blue circles are layers

with operations marked in the middle. The yellow thick arrow

marks the “SoftPlus” activation function and the right brace

is a conjunction of the corresponding layers.

Encouraged by the achievement of machine learning in

various areas, it is intriguing to ask if the neural network

is able to discover the L¨uscher’s formula and its vari-

ance after fed by plenty data of spectra on lattice and

the corresponding phase shifts. If a model-independence

link does exist, in principle, a highly generalizable neu-

ral network will be a decent approximation of this link,

because of the universal approximation theorem [28–30].

In this paper, we will show that neural network is able

to rediscover the numerical L¨uscher’s formula to a high

precision.

This paper is organized as follows. Sec. II is devoted

to build the theoretical formalism to generate the data of

energy levels in the ﬁnite volume and phase shifts in the

inﬁnite space. Then we will elaborate on the construction

of neural network and its training setup in Sec. III. In

Sec. IV, we will analyse the result in detail, and provide

the evidence to show the numerical form of L¨uscher’s

formula is generated from neural network. Finally, we

give a brief conclusion in Sec. V.

II. THEORETICAL FORMALISM

It is known that L¨uscher’s formula connects the ﬁnite

volume energy level Eand the S-wave phase shift δ(E)

as [16]

δ(E) = arctan qπ3/2

Z00(1; q2)+nπ, (1)

where q=k0L

2πis deﬁned with k0being the on-shell mo-

mentum of energy E, and the generalized zeta function

Z00(1; q2) is deﬁned as

Z00 1; q2:= 1

√4πX

⃗n∈Z3

(⃗n2−q2)−1.(2)

The system which we check against L¨uscher’s formula

is the elastic ππ S-wave scattering process. In order to

generate the training and test set, which consists of the

phase shift δ(E) and the ﬁnite volume spectrum E(L) for

a given lattice size L, we model this scattering process

by Hamiltonian Eﬀective Field Theory (HEFT) [31].

Following Refs. [32,33], we assume that ππ scattering

can be described by vertex interactions and two-body po-

tentials. In the rest frame, the Hamiltonian of a meson-

meson system takes the energy-independent form as fol-

lows,

H=H0+HI.(3)

The non-interacting part is

H0=|σ⟩mσ⟨σ|+ 2 Zd⃗

k|⃗

k⟩ω(|⃗

k|)⟨⃗

k|,(4)

where |σ⟩is the bare state with mass mσ, and |⃗

k⟩is for

the ππ channel state with relative momentum 2⃗

kin the

rest frame of σ, and ω(k) = pm2

π+k2.

The interaction Hamiltonian is

HI= ˜g+ ˜v, (5)

where ˜gis a vertex interaction describing the decays of

the bare state into two-pion channel,

˜g=Zd⃗

k{|⃗

k⟩g∗(k)⟨σ|+h.c.},(6)

and the direct ππ →ππ interaction (only S-wave) is

deﬁned by

˜v=Zd⃗

kd⃗

k′|⃗

k⟩v(k, k′)⟨⃗

k′|.(7)

For the S-wave, the ππ scattering amplitude is then

deﬁned by the following coupled-channel equation,

t(k, k′;E) = V(k, k′) + Z∞

k2d˜

kV(k, ˜

k)t(˜

k, k′;E)

E−2ω(˜

k) + iϵ ,(8)

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RediscoveryofNumericalL¨uscher’sFormulafromtheNeuralNetworkYuLu,1,∗Yi-JiaWang,1,†YingChen,1,2,‡andJia-JunWu1,§1SchoolofPhysicalSciences,UniversityofChineseAcademyofSciences(UCAS),Beijing100049,China2InstituteofHighEnergyPhysics,ChineseAcademyofSciences,Beijing100049,ChinaWepresentthatbypredictingthe...

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Rediscovery of Numerical L uschers Formula from the Neural Network Yu Lu1Yi-Jia Wang1Ying Chen1 2and Jia-Jun Wu1 1School of Physical Sciences University of Chinese Academy of Sciences UCAS Beijing 100049 China

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