Coupled-channel scattering in dimensional lattice model
Based on the Lippmann-Schwinger equation approach, a generalized Lüscher’s formula in dimensions for two particles scattering in both the elastic and coupled-channel cases in moving frames is derived. A 2D coupled-channel scattering lattice model is presented, the model represents a two-coupled-channel resonant scattering scalars system. The Monte Carlo simulation is performed on finite lattices and in various moving frames. The 2D generalized Lüscher’s formula is used to extract the scattering amplitudes for the coupled-channel system from the discrete finite-volume spectrum.
In recent years, remarkable progresses have been made on hadrons scattering in lattice QCD from both the theoretical algorithm of extracting scattering amplitudes from lattice data Lusher:1991 (); Gottlieb:1995 (); Lin:2001 (); Christ:2005 (); Bernard:2007 (); Bernard:2008 (); Liu:2005 (); Doring:2011 (); Aoki:2011 (); Briceno:2012yi (); Hansen:2012tf (); Guo:2013cp () and the practical lattice QCD computational algorithm aspect Michael:1985ne (); Luscher:1990ck (); Blossier:2009kd (); Jo:2010 (); Edwards:2011 (). Since Lüscher proposed the elastic scattering formalism in a finite volume Lusher:1991 (), the framework has been quickly extended to moving frames Gottlieb:1995 (); Lin:2001 (); Christ:2005 (); Bernard:2007 (); Bernard:2008 (), and to coupled-channel scattering Liu:2005 (); Doring:2011 (); Aoki:2011 (); Briceno:2012yi (); Hansen:2012tf (); Guo:2013cp (). The finite volume scattering formalism has been successfully used by the lattice community to extract elastic hadron-hadron scattering phase shifts Aoki:2007rd (); Sasaki:2008 (); Feng:2011 (); Jo_scatt:2011 (); Beane_scatt:2012 (); Lang_scatt:2011 (); Aoki:2011yj (); Jo_scatt:2012 (); Jo_scatt:2013 (). Realistic lattice QCD computations on coupled-channel hadron-hadron scattering are under way.
For the purpose of demonstrating the feasibility of extracting coupled-channel scattering amplitudes from lattice data and discussing some issues, such as, finite size effects, in this work, we present a coupled-channel scattering lattice model in 2D. Our model is a direct generalization of a 2D single channel scattering lattice model in Gatteringer:1993 (). The advantage of scattering in 2D is that only finite numbers of scattering amplitudes contribute in one spatial dimensional scattering theory, and the relation between phase shift and energy level in Lüscher’s formula in 2D Luscher:1990ck () appears more transparent. Our 2D lattice model represents a coupled-channel resonant scattering system with three species of scalar fields , where the scalar field acts as a resonance which couples to both and channels. The Monte Carlo simulation are carried out on various lattice sizes and in different moving frames. We also present the derivation of 2D Lüscher’s formulae in a general moving frame and for a coupled-channel system. The derivation is based on the Lippmann-Schwinger equation approach presented in Guo:2013cp (). These formulae are used in the end to extract the scattering amplitudes from Monte Carlo simulation data. The finite size effect on extracting scattering amplitudes (phase shifts and inelasticity) from lattice data is also addressed in this work. Although, our model is formulated and computed in dimensions, it still captures many of the features of hadrons scattering in a real dimensional QCD computation, and sheds some light on the future coupled-channel hadron-hadron scattering lattice QCD calculation.
The paper is organized as follows. A discussion of elastic scattering in a finite volume is given in Section II, with extension to the coupled channel system in Section III. The 2D lattice model, the Monte Carlo simulation and data analysis are described in Section IV. The summary and outlook are given in Section V.
Ii Lus̈cher’s formula in dimensions
For completeness, we first present the basic scattering theory in dimensions. Based on the Lippmann-Schwinger equation approach, a generalized Lüscher’s formula in dimensions for two particles elastic scattering in moving frames is presented in the end of this Section.
ii.1 Two-particle scattering in infinite volume
We consider spinless particles scattering in a symmetric potential , the mass of scalar particles is . The wave function of scattering particles in center of mass frame satisfies the relativistic Lippmann-Schwinger equation
where the center of mass frame energy is and the free-particle Green’s function is given by
The Green’s function can be further written as a oscillating term and an exponentially decaying term over the separation of two particles. The singularities of integrand in Eq.(2) on the complex plane are two poles on real axis and two branch cuts on imaginary axis , see Fig.1. Therefore, for , we choose the contour to include pole and cross the cut , and for , we choose the contour to include pole and cross the cut , as shown in Fig.1. Thus, contour integral leads to
where is momentum of particle in CM frame. At large separations, the free Green’s function can be approximated by the oscillating term only
where the functions and are defined by
Such that and resemble the spherical harmonic and Bessel functions in three spatial dimensions, and is the parity eigenstate with eigenvalue . The continuous rotation symmetry in three dimensions reduces to discrete spatial reflection in one spatial dimension, thus, the partial wave expansion of wave function in three dimensions reduce to the expansion of the wave function in terms of parity eigenstates , where .
For a potential which falls at large separations, Eq.(1) is solved outside the range of the potential by
where denotes to the effective range of potential and the free solutions has been also included in Eq.(7). The scattering amplitudes are defined by
which up to the inelastic threshold can be parametrized by scattering phase shift
ii.2 Two-particle scattering on a torus
Now we consider the theory in a one spatial dimensional box with periodic boundary conditions. In lattice QCD calculations, the computations are usually done in the moving frame of the two-particle system Gottlieb:1995 (). After the system is boosted back to the CM frame, the shape of cubic box in moving frame is deformed in CM frame due to Lorentz contraction. Similarly, in the one spatial dimension, the volume of a one dimensional box, , in a moving frame with total momentum becomes in CM frame, where is the Lorentz contraction factor.
Taking into account the Lorentz contraction effect as well, we divide the integral over into a sum of integrals over each translated box in Eq.(1), giving,
The wave function in CM frame satisfies the boundary condition Gottlieb:1995 () of
Using the periodicity of the potential , we have
where the periodic Green’s function is given by
By using the Poisson summation formula, , Eq.(13) can be reexpressed as
As in the infinite volume case, the periodic Green’s function Eq.(14) can be shown to consist of an oscillatory part and an exponentially decaying part which can be neglected for large volume . The remaining oscillatory part takes the form
Using the definition of scattering amplitudes in Eq.(8), we can express the wave function as
Iii Coupled-channel scattering in dimensions
The previous discussion in section II can be generalized to a coupled channel system by including another species of scalar fields, let’s name two species of particles and , the masses are . The coupled channel wave function has the form of , and satisfies equation
A matrix of coupled-channel scattering amplitudes can be defined by
where is the CM frame scattering momentum in channel . Neglecting exponentially decaying terms and also include the free solution, we have for the wave function in channel
Extending the single channel derivation in finite-volume to the two-channel system, one obtains,
The scattering amplitudes can be parametrized by three real parameters: two phase shifts and an inelasticity ,
Thus, we can also write the Eq.(24) as
Iv The Ising model for coupled channel scattering
To simulate a coupled channel scattering system, we build a model with two light mass fields coupled to a heavier mass field with two 3-point couplings, and . The physical masses of the fields are calibrated to be at the region . For elastic scattering, the Ising model has been used and tested in both Gatteringer:1993 () and Gottlieb:1995 () dimensions by coupling two Ising fields, and , together through a 3-point nonlocal interaction. For our purpose, we could introduce one more species of Ising field and another 3-point term to couple and together, where field gives rise to the resonant behavior in both and channels.
The action is given by
where is coordinates of Euclidean lattice site and denotes the unit vector in direction . The values of the fields are restricted to , and the periodic boundary condition has been applied in Monte Carlo simulation. In the scaling limit, the Ising model represent a lattice theory, thus, the action in Eq.(IV) effectively describes an interacting theory of Gottlieb:1995 ()
in Euclidean space. By adjusting the masses and coupling constants, we could have an resonance sit above both and thresholds, and couple to both channels by interaction terms and respectively. Therefore, the lattice Monte Carlo simulation by using the action in Eq.(IV) is expected to imitate a coupled-channel scattering model: . Due to the Bose-symmetry, only scattering amplitudes with positive parity contribute in this model.
iv.1 Cluster algorithm for coupled-channel Ising model
An generalized cluster algorithm is used in our simulation, similar to the cluster algorithm in Gatteringer:1993 (), we update , and fields alternately.
Updating the field: Bonds between neighbored spins of equal sign are kept with the probability . After identification of the connected clusters, the spin of cluster is flipped with probability
Updating the fields: Bonds between like-sign neighbors are kept with the probability , the spin of cluster is flipped with probability .
In our simulation, the parameters are chosen as , and , the masses of and fields are measured through single particle propagators, the values are given by and respectively in lattice unit. The mass of resonance is established from phase shifts, and the approximate value is given by .
In this work, we use and various spatial extensions between 15 and 50. For each set of lattice size and moving frame, we generated typically one million measurements.
iv.2 Particles spectrum
As shown in the elastic scattering case in dimensions Gatteringer:1993 (), one particle propagator can be constructed by operators
where and . The spectrum of single particle fields is extracted from exponential decay of the correlation functions
The two particles operators in the moving frame with total momentum of are constructed from single particle operators
The two particles correlation function matrices read
where short hand notation denotes the different sets of or . The disconnected contribution has to be subtracted in CM frame (). The spectral decomposition of the correlation function matrices has the form,
where and labels the energy eigenstate . The energy levels are determined by solving generalized eigenvalue problem Luscher:1990ck ()
where and is a small reference time, in our analysis, is set to be zero. In our simulation, the size of the matrices varies according to the volume, the number of operators we are using is always two or three more than the number of energy eigenstates in the region . The values of the energy levels are determined by fitting for with the form
where and are fitting parameters. This form allows a second exponential, however, we found that the value of is typically times of the value of , so that it decreases rapidly and the first exponential becomes dominant at around .
We show the measured two-particle energy spectra from our simulations in Fig.3 for various volumes and total momenta of two particles system .
iv.3 A coupled-channel -matrix model
In order to extract the scattering amplitudes (phase shifts and inelasticity) from the discrete finite volume spectra of the Monte Carlo simulation, we consider a -matrix model for a coupled-channel S-wave scattering system.
where represent the normal phase shift in which a resonance may appear at value of . Thus, the unitarized -matrix may be defined by
where and -matrix are parametrized by phase shifts and inelasticity respectively. The -matrix is related to the-matrix defined in Eq.(25) by equation . The unitarity of the -matrix is guaranteed by the unitarity of the -matrix. The Ising model suggestion is to parameterize a resonance coupling to both channels using a pole interfering with a polynomial in an -wave -matrix,
where the inverse of the -matrix is given by
Here is the Chew-Mandelstam form Basdevant:1977 () whose imaginary part above threshold () is the phase-space,
We have opted to subtract the integral once, and it is convenient to choose such that so that we have an amplitude which for real near is close to the Breit-Wigner form with mass .
Given an explicit model for the scattering amplitudes, we can solve Eq.(IV.3) for the finite volume spectra in various volumes and total momenta .
Eq.(IV.3) is derived from Eq.(24) by replacing with (), where is the relative momentum of two particles in a moving frame: and . To compensate for the ultraviolet cut-off effect from the finite lattice spacing, the dispersion relation Gatteringer:1993 () is used in this work, accordingly, in Eq.(IV.3), the relative momenta of two particles in a moving frame are solved by the equations
The lattice dispersion relation and finite size effects are further discussed in Appendix B.
iv.4 Data analysis
within the space of -matrix parameters, , where denotes the energy levels from Monte Carlo simulation, and are the solutions of Eq.(IV.3).
Instead of establishing the resonance pole position in our toy model, the purpose of this work is to demonstrate (1) the methodology of extracting scattering amplitudes from data of coupled-channel Monte Carlo simulations, (2) predictability of scattering amplitudes extracted from a set of lattice data, and (3) the validity of our formalism while taking into account of the finite size effect presented in Appendix B. Therefore, in this work, we choose to fit the spectra below threshold for only, then for a consistency check, we compare our predicted spectra for to the spectra from the Monte Carlo simulation. We show the spectra of -matrix model (red bands) in Fig.4 with the comparison of spectra from the Monte Carlo simulation (filled black circles, filled blue squares and filled green triangles). The -matrix we used in the fitting has nine free parameters, the polynomial of is taken up to . The value of parameters we find from fitting read
The extracted phase shifts , and inelasticity are shown in Fig.5.
As demonstrated in the middle and the right plots in Fig.4, our predicted energy spectra from -matrix model (red bands) for agree with the spectra from the Monte Carlo simulation (filled blue squares and filled green triangles) within a reasonable precision. Therefore, we accomplished our goals, (1) we proved that our formalism gives the consistent result in different moving frames while taking into account of finite size effect, and (2) we have shown that the global fitting method is a valid and fairly reliable means for extracting scattering amplitudes from Monte Carlo simulation data.
Based on the Lippmann-Schwinger equation approach, in Section II and III, we first derived a generalized Lüscher’s formula in 2D for two particles scattering in both the elastic and coupled-channel cases in moving frames. In Section IV, we presented a 2D coupled-channel scattering lattice model. The model simulates a two-coupled-channel resonant scattering system, in which a resonance couples to both channels. Next, we performed Monte Carlo simulations on various finite lattice sizes and in different moving frames. The discrete finite-volume spectra were extracted by fitting two-particle correlation functions. Finally, we used the 2D generalized Lüscher’s formula to extract the scattering amplitudes for the coupled-channel system from the discrete finite-volume spectra. We have shown that the global fitting method can be used to reliably extract scattering amplitudes from Monte Carlo simulation data. The finite size effects on the solution of the generalized Lüscher’s formula were discussed in details in Section IV and Appendix B. We demonstrated that while taking into account of finite size effects, our formulae produce consistent results in different moving frames.
We thank Dru B. Renner, Robert G. Edwards and Han-Qing Zheng for useful discussions, and the special thanks go to Dru B. Renner for inspiration of this work and for his encouragement. We also thank David J. Wilson for carefully reading through this manuscript. PG acknowledges support from U.S. Department of Energy contract DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and operates Jefferson Laboratory.
Appendix A One dimensional infinite sum
Let’s consider the one dimensional infinite sum in Eq.(15),
where . In the region which we are interested in: and , Eq.(44) can be rewritten to
With the help of equations
where the infinite sums are performed by using the property of polylogarithmic function , thus, we find
for and .
Appendix B Lattice dispersion relation
For the determination of the lattice spectrum at a precise level, the finite size effect has to be taken into consideration by using the lattice dispersion relation Gatteringer:1993 ()
where and are the total energy of system in moving frames and the CM frame respectively, and the total momentum of system is given by . In the limit of vanishing lattice spacing, Eq.(47) reduces to the relativistic dispersion relation: .
The relative momentum of two particles in a moving frame () is related to the relative momentum of two particles in the CM frame () by Lorentz transformation relation . In the limit of vanishing lattice spacing, the Lorentz contraction factor is given by . However, due to the ultraviolet cut-off effect from finite lattice spacing, the definition of the Lorentz contraction factor is inconsistent with lattice dispersion relation Eq.(47). This inconsistency leads to the large discrepancies of phase shifts and inelasiticy computed in different frames. To resolve this problem, we may use the relation to rewrite Eq.(19), Eq.(24) and Eq.(26) to
for single channel scattering, and
So that, rather than solving Eq.(19), Eq.(24) and Eq.(26) with the Lorentz contraction factor given by , we use Eq.(48) and Eq.(IV.3) with the solution of relative momentum of two particles given by Eq.(B) for single and coupled-channel scattering respectively in this work.
As a simple demonstration how the above proposal works, let’s consider a non-interacting two-particle system in dimensions. Two particles in an arbitrary moving frame, , have individual momenta