Multiple-particle interaction in $1+1$ dimensional lattice model

Finite volume multiple-particle interaction is studied in a two-dimensional complex $\phi^4$ lattice model. The existence of analytical solutions to the $\phi^4$ model in two-dimensional space and time makes it a perfect model for the numerical study of finite volume effects of multi-particle interaction. The spectra from multiple particles are extracted from the Monte Carlo simulation on various lattices in several moving frames. The $S$-matrix of multi-particle scattering in $\phi^4$ theory is completely determined by two fundamental parameters: single particle mass and the coupling strength of two-to-two particle interaction. These two parameters are fixed by studying single-particle and two-particle spectra. Due to the absence of the diffraction effect in the $\phi^{4}$ model, three-particle quantization conditions are given in a simple analytical form. The three-particle spectra from simulation show remarkable agreement with the prediction of exact solutions.

A three-particle lattice simulation was recently performed based on a complex φ 4 toy model [71], the data analysis was carried out by adopting effective theory framework. However, the simulation and analysis are limited solely to ground state energy levels where all three particles are nearly at rest, and the three-particle signals are quite noisy. In present work, we aim to perform a simulation on multiple-particle interaction also using φ 4 model, and study the finite volume effect on multipleparticle spectra in a better controlled environment and a more systematic way. For this purpose, multiple numbers of multi-particle operators are used in our simulation and variational analysis [72][73][74] is implemented to extract excited state energy levels. The exact scattering solutions of φ 4 theory in 1 + 1 dimensions are known in both free space [75][76][77] and finite volume [64]. Taking advantage of existing analytic multiple-particle scattering solutions, the simulation is therefore performed in 1 + 1 dimensional space and time for various lattice sizes and moving frames. The exact scattering solutions are used in data analysis of multi-particle simulation. In principle, the multiple particle scattering S-matrices are completely determined by only two free parameters: single particle mass and coupling strength of two-to-two particle interaction. The single particle mass is obtained from single particle correlation functions, and the coupling strength of pair-wise interaction is extracted by studying two-particle scattering spectra in a lattice. The comparison between three-particle scattering spectra and predicted three-particle energies by using analytic expression of three-particle quantization conditions are presented in the end.
The paper is organized as follows. The exact solutions of φ 4 theory for two-body and three-body interaction are summarized in Section II. The algorithm of the Hybrid Monte Carlo simulation of lattice model and strategy of data analysis are briefly discussed in Section III. The construction of multi-particle operators, multi-particle spectra in lattice simulation and data analysis are described in Section IV. The summary and outlook are given in Section V.
In this section, we summarize some results of the twodimensional φ 4 model. Classical action of the complex φ 4 model in two-dimensional Euclidean space is where x = (x 0 , x 1 ) are temporal and spatial coordinates in two-dimensional Euclidean space, respectively. It is known [75] that the complex φ 4 model in Eq.(1) is equivalent to a non-relativistic one-dimensional N -body interaction problem of particles interacting with pair-wise δ-function potentials, where x 1,i refers to the spatial position of i-th particle, and m stands for the mass of identical bosons. The coupling strength of δ-function potential, V 0 , differs from renormalized g in Eq.
(1) by a constant factor. The exact solutions of multi-particle interaction with δ-function potentials were studied and obtained in both free space [75][76][77] and finite volume [64]. In fact, the particles interacting with δ-function potential in 2D is only one of few exactly solvable multi-particle scattering problems. The multi-particle wave function is described completely by the linear superpositions of plane waves with all possible permutation on particle momenta. No new momenta are generated by collision, all the diffraction effects are canceled out as the consequence of Bethe's hypothesis [78,79]. The multi-particle S-matrix therefore is factorized into the product of a number of two-particle scattering amplitudes, as if the process of multi-particle scattering would be a succession of separated elastic two-particle collisions [64]. In finite volume for two-particle scattering, only one quantization condition is required [64,80] cot δ(k) + cot where P = p 1 + p 2 and k = p1−p2 2 denote center of mass and relative momenta of two particles. The phase shift δ(k) for δ-function potential is given by ). The L stands for the size of the square box in 2D, and center of mass momentum is discretized because of the periodic boundary condition of lattice: For three-body scattering in finite volume, three quantization conditions are obtained [64]. Only two of them are independent, where all the relative momenta are given in terms of two independent particle momenta: p 1 and p 2 , The momentum of particle-3 is constrained by momentum conservation, p 3 = P − p 1 − p 2 . Again, center of mass momentum of three-particle is quantized in the periodic box: The lattice φ 4 action is obtained from Eq.(1) by replacing the continuous derivative with discrete difference: , wheren denotes the unit vector in direction x i on a periodic square lattice. In addition, by introducing two new parameters: κ 2 , and also rescaling the φ field by φ → √ 2κφ, we thus obtain (5) where x = (x 0 , x 1 ) now refers to discrete coordinates of Euclidean T × L lattice site.
(iii) accept the proposed new fields, (φ(τ ), π(τ )), with probability: The simulations are performed with the choice of the parameters: κ = 0.1286, and λ = 0.01. The temporal extent of the lattice is fixed at T = 80, and the spatial extent of lattice, L, are from 10 up to 45. For each set of lattice size and moving frame, one million measurements are generated. The length of trajectory is fixed at τ = 8, the (φ, π) fields evolve from τ = 0 up to τ = 8 over 100 discrete steps.

B. Strategy of data analysis
As already mentioned in Section II, the twodimensional φ 4 model is exactly solvable, the solutions of the model are given in terms of only two free parameters: particle mass, m, and the coupling strength of δ-function potential, V 0 . The mass of identical particles, m, can be extracted from one-particle spectra of the lattice simulation. The second parameter, V 0 , can be fixed by twoparticle spectra from the simulation. Taking advantage of the existence of exact solutions of the two-dimensional φ 4 model provides an excellent playground and controlled environment for a systematic study of finite volume effects of multi-particle scattering in lattice simulation. In present work, we are not aiming at obtaining any new fundamental information from three-body spectra, such as three-body force effect, etc. Instead, after fixing m and V 0 from one-and two-particle spectra, we tend to study how well the three-body spectra from simulation match the prediction of exact solutions. In real QCD simulation, the significant difference between simulation results of three-body spectra and prediction based on pair-wise interaction may signal the effect of three-body forces or something more fundamental. The present work serves only as a testbed for more realistic future lattice studies of multi-particle interaction.
To accomplish the goal of this work mentioned above, the following steps are taken in data analysis of the simulation results: 1. measure one-particle spectra for various sizes of lattice and moving frames, and extract contin-  C(x 0 +1) , for one particle (black), two particles (blue) and three particles (green) at L = 40 and P = 0, and corresponding fitting curves (red band). uum limit particle mass, m, by using relation [83]: m(L) = m + c √ L e −mL . 2. measure two-particle spectra for various sizes of lattice and moving frames, and extract the coupling strength, V 0 , from lattice data. 3. three-particle spectra are measured for various sizes of lattice and moving frames as well, three-particle spectra are thus compared with predicted three-particle spectra. The predicted three-particle spectra, E (d) 3b (L), are given in terms of two independent particle momenta, p 1 and p 2 , by where p 1 and p 2 are the solutions of Eq.(4), and The particles spectra are extracted by fitting exponential multi-particle correlation functions as a function of time x 0 : C(x 0 ) ∝ e −Ex0 . See the example of one-, two-and three-particle correlation functions and effective mass, ln C(x0) C(x0+1) , in Fig.1 and Fig.2, respectively. The construction of multi-particle operators and correlation functions will be explained later in Section IV.

IV. PARTICLES SPECTRA AND DATA ANALYSIS
In this section, we present significant results for multiparticle scattering. Some details on multi-particle operator construction and data analysis are also given.
A. One particle spectra The one-particle spectra are extracted from the exponential decay of the correlation functions where the one particle propagator, φ n (x 0 ), is defined by Single particle energy E 1b,n (L) is obtained for multiple lattice sizes, L = 10 up to 45. By fitting single particle energies in multiple lattice sizes with relation where c and m are used as fitting parameters, we thus find the mass of single particle: m = 0.2708 ± 0.0002, see Fig.3. The excited single particle energy levels are used to check the energy-momentum dispersion relations in a finite lattice, The comparison between lattice results and the lattice dispersion relation is presented in Fig.4.

B. Two particles spectra
In moving frames, the matrix element of the two particle correlation function read where d ∈ Z is related to center of mass momentum by P = 2π L d, and two-particle operators are constructed by Four two-particle operators are used in our simulation: n = (0, 1, 2, 3), so the size of matrix of two-particle correlation functions are 4 × 4, 3 × 3 and 2 × 2 for d = 0, 1, 2.
The spectral decomposition of the correlation function matrices are usually given by where v 2b,i (0)|0 , and n labels the n-th energy eigenstate E (d) 2b,n . In order to extract excited energy states, a generalized eigenvalue method [73] is proposed wherex 0 is a small reference time. Mixing of multiparticle states is protected by the conservation of charge quantum number in the complex φ 4 model. Also, diagonalized correlation functions barely show the contamination of higher energy states in λ vs. free two-particle energy levels (red curve): Dotted curves represent four-particle threshold.
x 0 ∈ [0, 10]. The two particle spectra for various lattice sizes and d are presented in Fig.5.
The phase shift of two-body scattering is extracted from two-particle energy levels, E (d) 2b,n , by using relation: where the relative momentum of two particles, k, is given by the solutions of two-particle energy momentum dispersion relation where p 1 = π L d + k , p 2 = π L d − k, see extracted phase shift in Fig.6. The exact expression of phase shift, is used to fit lattice results, δ (d) lat (k), and to fix the coupling strength, V 0 . We thus find mV 0 = 0.170 ± 0.015.

C. Three particles spectra
For three-particle operators with d = 0, 1, 2, four operators are used in present work: and Similar to the two-particle correlation function matrix, the matrix element of the three particle correlation function is given by In the two-particle sector, the generalized eigenvalue method is also applied to extract three-body energy levels, where λ (d) 3b,n (x 0 , 0) = e −E (d) 3b,n x0 . An example of the threeparticle correlation function, λ (d) 3b,n (x 0 , 0), and effective mass, ln λ 3b,n (x 0 + 1, 0) is given in Fig.1 and Fig.2.
Given the values of particle mass, m, and coupling strength, V 0 , that we learned from discussion in previous sections, three-particle spectra does not provide any new insight into the fundamental parameters of φ 4 theory due to the absence of three-body force. However, in general, three-particle spectra are still considered a useful tool to explore and understand the dynamics of threeparticle interaction. In reality, it also provides opportunities to investigate the possibility of more fundamental parameters of lattice QCD theory. Nevertheles, since the exact solutions are known, we only tend to demonstrate the consistence of predicted three-particle spectra compared to simulation results. The predicted three-particle predicted three-particle energy levels (red band): spectra are determined by three-body energy-momentum dispersion relations in terms of two independent particle momenta, say p 1 and p 2 , where p 3 = 2π L d − p 1 − p 2 . Two independent particle mo-menta, p 1 and p 2 , are the solutions of three-body quantization conditions given in Eq. (4). The main results of three-particle spectra are presented in Fig.7. As we can see, the agreement of predicted spectra against simulation results is quite remarkable.

V. SUMMARY
In summary, the lattice simulation of multi-particle interaction is studied by using a complex φ 4 lattice model. The simulation is performed in 1 + 1 dimensional space and time for various sizes of lattices and multiple moving frames. The two dimensional φ 4 model is exactly solvable and analytical expressions of multi-particle quantization conditions are known in finite volume [64]. This feature makes it a perfect testbed for studying multiparticle interaction in a lattice. The typical 3 − 4 numbers of multi-particle operators are used in our simulation, and a variational approach is implemented to extract excited state energy levels. Two parameters of φ 4 theory, single particle mass and coupling strength, are extracted from single particle and two particles spectra, respectively. Then, extracted φ 4 theory parameters are applied to predict three-particle spectra by using analytical three-body quantization conditions compared with three-particle spectra from simulations. Predicted threeparticle spectra and lattice results show quite remarkable agreement.

VI. ACKNOWLEDGMENTS
We acknowledge support from the Department of Physics and Engineering, California State University, Bakersfield, CA. We also thank Vladimir Gasparian and David Gross for numerous fruitful discussions. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958.