Coupled-channel scattering on a torus

Before considering two-particle states on a torus, we will first review the scattering problem in infinite-volume, with further details given in Appendix A. After eliminating the three-particle states $|\phi^{+}\phi^{-}\theta\rangle$ from the coupled system (cf Eq.(2)) we are left with an equation for the center-of-mass frame momentum-space wavefunction, $\varphi_{JM}(\mathbf{q})$, which is a product of a radial wavefunction depending on the magnitude of the relative 3-momentum, $q=|{\bf q}|$, and the spherical harmonic of definite angular momentum, $(J,M)$,

Here, $E=\sqrt{s}$ is the energy of the two-particle system in the center-of-mass frame. The non-local potential, $V(\mathbf{q},\mathbf{k})$, induced by $\theta$-exchange is given explicitly in Eq. (26). Expressing this equation in coordinate space via a Fourier transform gives

where the free Green’s function is given by

The wavefunction satisfies a relativistic Schrödinger equation,

While Eq.(5) was derived in the context of a particular model, our subsequent derivation only requires the general form of the relativistic Lippmann-Schwinger equation. The asymptotic component of the two-body wavefunction relevant to scattering is given by the large distance behavior of the Green’s function. Evaluating the integral in Eq.(6) (cf. Appendix A), we find

with the first term on the right hand side dominating as $r\to\infty$. For a potential $\tilde{V}$ which falls at large separations, the solution to Eq (7) outside the range of the interaction is given by

where $f_{J}(k)$ is the partial wave scattering amplitude,

which up to the inelastic threshold can be parameterized in terms of a single real momentum-dependent parameter, the scattering phase-shift, $\delta_{J}(k)$, as

Now we consider the theory in a cubic box of volume $V=L^{3}$, with periodic boundary conditions. In Eq. (5) we split the integral over $\mathbf{r}^{\prime}$ into a sum of integrals over a set of boxes labelled by the integers $\mathbf{n}$ representing the location of one of its corners,

In general we can make the wavefunctions periodic up to a phase,

where the Bloch wave-vector, $\mathbf{Q}$, is related to the total momentum of the two-particle system Gottlieb:1995 by $\mathbf{P}=2\gamma\,\mathbf{Q}$. $\gamma=\sqrt{s+\mathbf{P}^{2}}/\sqrt{s}$ is the Lorentz contraction factor that reduces the effective size of the box in the direction parallel to $\mathbf{P}$. Using the periodicity of the potential, $\tilde{V}(\mathbf{r}^{\prime}+\mathbf{n}L,-\mathbf{z}^{\prime}-\mathbf{n}L)=\tilde{V}(\mathbf{r}^{\prime},-\mathbf{z}^{\prime})$, and the boundary condition on the wavefunction, we have

which is analogous to the infinite-volume equation, but with the Green’s function given by

Using the Poisson summation formula, $(2\pi)^{-3}\sum_{\mathbf{n}\in\mathbb{Z}^{3}}e^{i\mathbf{Q}\cdot\mathbf{n}L}=L^{-3}\sum_{\mathbf{n}\in\mathbb{Z}^{3}}\delta\left(\mathbf{Q}+\tfrac{2\pi}{L}\mathbf{n}\right)$, we obtain

where $P_{\mathbf{Q}}=\{\mathbf{q}\in\mathbb{R}^{3}|\,\mathbf{q}=\tfrac{2\pi}{L}\mathbf{n}+\mathbf{Q},\;\text{for}\;\mathbf{n}\in\mathbb{Z}^{3}\}$, and where we have retained only the leading term in the limit $L\gg|{\bf r}-{\bf r^{\prime}}|\gg m^{-1}$. Finally, expanding Eq.(11) for $r\gg r^{\prime}$ and using the definition of the scattering amplitude, Eq. (10) we can express the wavefunction as

The residual sum over all angular momenta reflects the broken rotational invariance induced by the finite cubic volume, with the volume-dependent matrix elements ${\cal M}$ given in Appendix B. In the infinite-volume case, the most general solution of the relativistic Schrödinger equation, Eq.(7), outside the range of the potential is $\sum_{JM}c_{JM}\,\psi_{JM}(\mathbf{r})$ for $\psi_{JM}(\mathbf{r})$ given by Eq. (9). Correspondingly in finite-volume, the most general solution is given by $\sum_{JM}c_{JM}\,\psi^{(L,\mathbf{Q})}_{JM}(\mathbf{r})$ for $\psi^{(L,\mathbf{Q})}_{JM}(\mathbf{r})$ given by Eq.(12). Matching the two wavefunctions at a fixed $r$, larger than the range of the interaction, we obtain

which has a non-trivial, $c_{JM}\neq 0$, solution provided

This condition expresses the relationship between the asymptotic behavior of the two-particle wavefunction on a torus, expressed through the matrix elements $\mathcal{M}^{(\mathbf{Q})}_{JM,J^{\prime}M^{\prime}}$, and the effect of the interaction on the wavefunction determined by the phase shifts, $\delta_{J}$. In practice for a given set of elastic phase-shifts, $\delta_{J}(k)$, it determines a discrete spectrum of states in a finite volume.

The analysis presented here can be generalized to an arbitrary shaped box. In general the three edges of box are spanned by three arbitrary vectors $\mathbf{L}_{1,2,3}$. The volume of the cube $L^{3}$ is replaced by $(\mathbf{L}_{1}\times\mathbf{L}_{2})\cdot\mathbf{L}_{3}$ and the vector $\mathbf{n}L$ by $\sum_{i=1,2,3}n_{i}\mathbf{L}_{i}$, $n_{i}\in\mathbb{Z}$. Finally the momentum $\mathbf{q}=2\pi\mathbf{n}/L,\mathbf{n}\in\mathbb{Z}^{3}$ is replaced by generalized momentum $2\pi\sum_{i=1,2,3}n_{i}(\mathbf{L}_{j}\times\mathbf{L}_{k})/|(\mathbf{L}_{1}\times\mathbf{L}_{2})\cdot\mathbf{L}_{3}|,\,n_{i}\in\mathbb{Z}$ , where indices $(i,j,k)$ follow the cyclic permutation. Such a generalization has to be considered when using the moving center-of-mass frame since the symmetric shape of a cubic box in the rest frame is deformed due to Lorentz contraction Gottlieb:1995 . In this case if $\mathbf{P}=2\pi\mathbf{d}/L,\mathbf{d}\in\mathbb{Z}^{3}$, is the center-of-mass momentum, the volume of the box becomes $\gamma L^{3}$, and the vectors $\mathbf{n}L$ and $2\pi\mathbf{n}/L$ are replaced by $\gamma\mathbf{n}L$ and $2\pi\gamma^{-1}\mathbf{n}/L$, respectively (using the notation defined in Gottlieb:1995 ). With these substitutions and the relation $\mathbf{P}=2\gamma\mathbf{Q}$, our definition of the matrix elements $\mathcal{M}^{(\mathbf{Q})}_{JM,J^{\prime}M^{\prime}}$ becomes identical to the matrix elements $M^{\mathbf{d}}_{lm,l^{\prime}m^{\prime}}$ in Eq.(89) of Gottlieb:1995 .

Typically, as discussed in Lusher:1991 , for the low-energy region that we are interested in, higher partial waves become progressively smaller and can be ignored, so that the partial wave basis can be truncated at a certain maximal angular momentum $J_{\mathrm{max}}$. For a finite-volume with cubic boundaries, the continuous rotation symmetry is reduced to the little-group of allowed cubic rotations that leave the centre-of-mass momentum invariant - the matrices in Eq.(13) become block-diagonal if subduced according to the irreducible representations of these little groups. Details of subduction in general moving frames can be found in Christopher_subduction:2012 .

We consider a model in which a single $S$-wave resonance coupled to both scattering channels interferes with a non-resonant background. The two-channel scattering amplitude is parametrized in terms of a $K$-matrix,

which is related to the $t$-matrix by

and to the scattering amplitude defined in Eq. (15) by $t^{\alpha\beta}_{J}(s)=\sqrt{s}f^{\alpha\beta}_{J}(s)/8\pi$ with ($\alpha=\phi,\sigma$). Here $I_{\alpha}(s)$ is the Chew-Mandelstam form Basdevant:1977 ,

whose imaginary part above threshold, i.e. for $s>4m_{\alpha}^{2}$, is the negative of the phase-space, $\mathrm{Im}\big{[}I_{\alpha}(s)\big{]}=-\rho_{\alpha}(s)$. This form ensures the unitarity of the amplitude and provides a smooth transition across the kinematic threshold. We have opted to subtract the integral once, and it is convenient to choose $I_{\alpha}(0)$ such that $\text{Re}\,I_{\alpha}(M^{2})=0$ so that we have an amplitude which for real $s$ near $M^{2}$ is close to the Breit-Wigner form with mass $M$. The $t$-matrix thus constructed is an analytical function in the complex $s$-plane with the discontinuity across the right-hand cut determined by unitarity.

With the following choice of parameters,

we obtain the phase-shifts and inelasticity shown in Fig. 1. The parameters have been chosen in such a way that there is a narrow resonance near $\sqrt{s}=1.1\,\mathrm{GeV}$. It is usual to analyse scattering in terms of the most relevant singularities of the $t$-matrix on the nearby unphysical sheets. Poles on unphysical sheets are often identified with hadron resonances. In this case the four sheets ($\mathsf{sheet\,I}$ is the physical sheet) can be defined by

The model amplitude has a single pole on the lower half-plane¹¹1and a conjugate pole on the upper half-plane of each of sheets $\mathsf{II}$ and $\mathsf{III}$, with the $t$-matrix in the neighbourhood of the pole at $s_{0}$ behaving like,

with

on $\mathsf{sheet\,II}$ and

on $\mathsf{sheet\,III}$. Our aim is to use the finite-volume spectrum determined on a set of volumes and total momenta, $\mathbf{P}$, to reproduce the pole positions of this scattering amplitude.

The finite-volume spectrum corresponding to the model defined in the previous section can be obtained by solving Eq. (23) (or equivalently Eq. (19)), where for $E=\sqrt{s}<2m_{\sigma}$ we require the analytic continuation of $t$-matrix elements featuring channel $\sigma$. Assuming that partial waves higher than $S$-wave are negligible reduces Eq. (23), for $E>2m_{\sigma}$, to,

where $\mathcal{M}_{\phi}\equiv\mathcal{M}^{(\mathbf{Q})}_{J=0\,M=0,J^{\prime}=0\,M^{\prime}=0}(k_{\phi})$ with a similar expression for $\mathcal{M}_{\sigma}$. $\mathbf{Q}$ is a function of $\mathbf{d}$, $L$, $E$ as discussed in Sec. II.2. In Fig.2 we show the finite-volume spectrum²²2this would be the spectrum in irrep $A_{1}$ Jo_scatt:2012 obtained by solving the determinant condition as a function of the volume in a region $L=16-24\mbox{ GeV}^{-1}$ (or $L=3.2-4.7\mbox{ fm}$).

In the region below $2m_{\sigma}$ the $\phi\phi$ scattering is elastic and it is tempting to use Eq (13), completely ignoring the existence of the kinematically closed $\sigma\sigma$ channel. Doing so leads to the points shown in pink in Fig 3 which successfully reproduce the input model except in a region just below the threshold. As previously discussed, in a finite-volume the effect of a closed channel can be felt within a limited energy region immediately below the threshold. With this in mind, we must be careful not to use the simple elastic relation, Eq (13), close to a threshold.

One approach to determining the phase-shifts and inelasticities at discrete values of energy above $2m_{\sigma}$ is to locate multiple energy levels (in different volumes and/or different $\mathbf{d}$) that appear at approximately the same energy. As an example consider the three levels labeled $\mathsf{A}$ in Fig. 2 which all lie within 3 MeV of $\sqrt{s_{\mathsf{A}}}=830\,\mathrm{MeV}$. For the three levels we can build three independent copies of Eq. (23) which each feature approximately the same values of $\delta^{\phi}(s_{\mathsf{A}}),\delta^{\sigma}(s_{\mathsf{A}}),\eta(s_{\mathsf{A}})$, which can be determined by solving the set of simultaneous equations. Since the energies are not exactly degenerate, there need not be an exact solution to the equations and hence we seek to find the solution which minimizes

with $\Omega$ defined in Eq. (25) and where the sum is over the three energy levels. For the levels $\mathsf{A}$, the obtained solution, as shown in Fig. 3, is within $3\%$ of the exact value of $\delta^{\phi}(s_{\mathsf{A}}),\delta^{\sigma}(s_{\mathsf{A}}),\eta(s_{\mathsf{A}})$. We emphasize that this procedure is not guaranteed to successfully reproduce the true scattering amplitudes - the $\mathcal{M}$ matrices can in some circumstances vary rather rapidly over a narrow energy region.

Within the energy region considered, $E=0.8-1.2\,\mathrm{GeV}$, considering only three volumes, $L=16,\,20,\,24\,\mathrm{GeV}^{-1}$, and three sets of center-of-mass momentum, $\mathbf{d}=(000),\,(100),\,(110)$, we can isolate a number of such sets. These sets of three near-degenerate energy levels are labelled $\mathsf{A}-\mathsf{J}$ in Fig. 2. In Fig. 3 the labels are shown on the plot of $\delta^{\phi}$ with the corresponding solutions for $\delta^{\sigma}$ and $\eta$ marked by solid dots.

With these points alone, in Fig. 3 we see strong hints of a signal for resonant behavior in the $\delta^{\phi}$ phase shift. While obviously reasonably successful, this approach does not make optimal use of the finite-volume spectral information, by failing to use any energy level which does not have two near-degenerate partners. To use somewhat more of the discrete levels we might consider building a system of two instances of Eq. (23) with one parameter from the set $\delta^{\phi},\delta^{\sigma},\eta$, estimated using interpolation between already determined values. For example consider the two levels near $1.009\mbox{ GeV}$ labeled $\mathsf{e}$ in Fig. 2. Linear interpolation between the energies of the $\mathsf{C}$ and $\mathsf{D}$ points in Fig. 3 gives $\delta^{\phi}(1.009\,\mathrm{GeV})=30.7^{\circ}$. Using this value and minimizing with respect to $\delta^{\sigma},\eta$ at the energy corresponding to point $\mathsf{e}$ we obtain $\delta^{\sigma}=50.6^{\circ},\eta=0.903$. A spline interpolation using all the points $\mathsf{A-J}$ yields $\delta^{\phi}(1.009\,\mathrm{GeV})=29.6^{\circ}$ which results in $\delta^{\sigma}=48.1^{\circ},\eta=0.934$ at the point $\mathsf{e}$. In Fig. 3 we show the results for sets of two degenerate levels labeled $\mathsf{a}-\mathsf{h}$ from Fig. 2. In each case the range shown indicates limiting values obtained using two methods of interpolation. Even though in some cases there is a considerable sensitivity to the interpolation method, overall the points are in reasonable agreement with the model input (solid light-colored curves).

The previously discussed “pointwise” strategy, while having the advantage of being largely model-independent, is reliant upon there being multiple energy levels which, through accident or design, are close to degenerate. Since it would be unusual to engineer lattice volumes purely for this purpose, and unusual in contemporary calculations to have such a high density of determined energy levels, it is appropriate to consider alternative methods of analysis. One such approach that makes full use of all determined levels, and which may require far fewer levels to be determined, involves parameterising the scattering amplitude and performing a minimisation to get the best description of the determined finite-volume spectrum by varying the parameters. In the current toy-model, even limited “pointwise” analysis would suggest that the phase $\delta^{\phi}$ is rapidly rising and would indicate that a resonance could be present. In practical calculations (e.g.Jo:2010 , Dudek:2012xn ), the presence of a sharp meson resonance can also be indicated by large overlap onto fermion bilinear operators. By including a pole (as well as polynomial behaviour) in a $K$-matrix parameterisation we are likely to get rapid convergence to a solution with a pole in the $t$-matrix corresponding to the resonance.

We will take this opportunity to make the toy model a slightly more realistic simulation of an actual lattice QCD calculation by introducing statistical uncertainty on the energy level values. In recent work Jo_scatt:2011 , Jo_scatt:2012 , Dudek:2012xn , the Hadron Spectrum Collaboration has obtained statistical errors on excited levels as small as 0.3% and we will assume that this remains practical. For each energy level below $1.2$ GeV on a single volume $L=16\,\mathrm{GeV}^{-1}\sim 3.2\,\mathrm{fm}$ with $\mathbf{d}=(000),\,(100),\,(110)$, we randomly generated an ensemble by drawing from a distribution whose mean is the exact value given in Fig. 2 and whose variance is chosen such that the ensemble has variance on the mean of 0.3% of the mean value. The resulting spectrum is shown in Fig. 4.

Parametrizing according to a form like that given in Eq. (24), we can minimise a function,

(where $E^{\text{det}}$ are the solutions of Eq. (19)), by varying the parameters of the $K$-matrix parameterisation, $\{a_{i}\}=\{M,g_{\phi},g_{\sigma},\gamma^{(n)}\ldots\}$. In practical lattice QCD calculations, the $\chi^{2}$ can be trivially redefined to deal with correlated data by replacing the inverse diagonal variance ($1/\sigma^{2}$) by the inverse of the data covariance matrix.