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Finite Volume Corrections to the Two-Particle Decay of States with Non-Zero Momentum

We study the effects of finite volume on the two-particle decay rate of an unstable state with non-zero momentum. First Luscher's field-theoretic relation between the infinite volume scattering phase shifts and the quantized energy levels of a finite volume, two-particle system is generalized to the case of non-zero total momentum and compared with the earlier results of Rummukainen and Gottlieb. We then use this result and the method of Lellouch and Luscher to determine the corrections needed for a finite-volume calculation of a two-particle decay amplitude when the decaying particle has non-vanishing center-of-mass momentum.


I. INTRODUCTION
A central problem in making Standard Model predictions for K meson decays, including the important CP violating amplitudes, is calculation of the decay K → ππ into the I = 0 final state of two pions. In a standard calculation in which the K meson is at rest, the contribution of the two pion final state is expected to be very difficult to extract from a lattice QCD calculation. The physical state has the same quantum numbers as the state with the two pions at rest (the difficulty emphasized by Maiani and Testa [1]). Of even greater concern is the fact that the QCD vacuum also has these same quantum numbers. A very appealing approach to deal with these problems is to study the decay of a K meson with non-zero momentum. For a calculation in finite volume, it is possible to adjust the momentum of the K meson and the box size so that both the transition amplitude to the vacuum must vanish (since the vacuum has zero momentum) and the final two-pion state must have physical relative momentum and total energy equal to that of the decaying K meson.
For such a calculation to be useful we must be able to relate the decay amplitudes computed in finite volume using this technique to the infinite volume matrix elements determined by experiment. To a large extent the effects of finite volume on the particles involved are relatively mild. Typically the volumes used in such a calculation can be chosen sufficiently large that they do not significantly distort the K and π particles whose physical size will be much smaller than that of the box employed. The most obvious finite-volume effect will be the quantization of the energy levels of the two-pion final state. This is actually an advantage, permitting the box size to be adjusted to make one of the discrete, two-pion energies match precisely that of the K meson. For the case at hand, this can even be the lowest energy π − π state.
However, the violation of rotational symmetry by the finite-volume boundary conditions does induce an important distortion in the computed decay rate. Because of the resulting non-conservation of angular momentum, the two-particle state into which the decay occurs is actually a mixture of states with many angular momenta. For typical lattice volumes the actual decay of the K meson into these higher angular momentum states will be very small-angular momentum is effectively conserved at the short distances over which the decay occurs. However, the presence of these extra angular momentum states effects the normalization of the physical J = 0 amplitude which appears in the matrix element.
This finite volume normalization problem has been solved by Lellouch and Lüscher [2] for the case of the decay of a K meson at rest. This problem has also been solved by a different approach in Ref. [3]. In this paper we will generalize the approach of Lellouch and Lüscher to obtain a result for states with non-zero total momentum. Central to their argument is an earlier treatment of Lüscher [4,5] which determines the allowed, finite-volume, two-particle energy eigenvalues in terms of the infinite volume, two-particle scattering phase shifts for energies below all inelastic thresholds. This discussion must also be generalized to the case of non-zero center-of-mass momentum.
This topic has been studied earlier by Rummukainen and Gottlieb [6]. Their treatment involves an application of relativistic two-particle quantum mechanics. We believe that it is also important to study this problem starting from the equations of quantum field theory.
Our strategy follows closely that of Lüscher [4,5] and Lellouch and Lüscher [2]. We first discuss the energy quantization of finite-volume, interacting, two-particle states with non-zero center-of-mass momentum. Following Lüscher, this is first done in standard, twoparticle, non-relativistic quantum mechanics in Sec. II. In Sec. III we begin with the Bethe-Salpeter equation of relativistic field theory and, again following Lüscher, show how this equation when restricted to a particular 7-dimensional subspace of the 8-dimensional, 2particle momentum space reduces to the standard Lippmann-Schwinger equation describing the earlier non-relativistic system. Finally, in Sec IV we use this result to generalize the argument of Lellouch and Lüscher to the determine the finite volume corrections to the decay amplitude computed for states with non-zero total momentum.
The issues addressed in this paper have also been considered by Kim, Sachrajda and Sharpe. Using a related but different approach, they have also confirmed the validity of the results of Ref. [6] and derived the generalization of the result of Lellouch and Lüscher for the case of non-zero total momentum. Their paper [7] is being released simultaneously with the present article.

II. FINITE VOLUME, NON-RELATIVISTIC, TWO-PARTICLE STATES
We begin by considering a simple, non-relativistic system of two distinguishable particles confined in a cubic box of side L and obeying periodic boundary conditions. The system is described by a wave function ψ( r 1 , r 2 ) which is periodic in r 1 and r 2 separately. An eigenstate of energy ψ E obeys the Schrödinger equation: where m is the identical mass of the two particles and V (| r 1 − r 2 |) their rotationally invariant interaction potential.
Such an equation is conventionally simplified by changing to center-of-mass and relative coordinates: with conjugate momenta where p i is the momentum conjugate to the coordinate r i .
Using ψ (rel) E ( r, R) to represent the original wave function expressed in terms of r and R, we can write down the standard equation which it obeys: The periodicity of ψ( r 1 , r 2 ) (rel) E under the simultaneous translation r i → r i +ê k L (wherê e k is a unit vector parallel to one of the edges of the box) implies the periodicity of the wave function ψ r under a translation of R by L. Thus, the conserved total momentum P must obey the quantization condition: P = 3 k=1 2πn kêk /L for integer n k . It is easy to see that in a direction k for which the integer n k is even, the corresponding component of the relative momentum p k = 2πn ′ k /L while if n k is odd then p k = 2π(n ′ k + 1 2 )/L, where n ′ k is an integer. With this change of coordinates and a specific choice of P , our two-particle problem reduces to the quantum mechanics of a single particle in an L 3 box obeying either periodic or antiperiodic boundary conditions on each of its three opposing faces.
In infinite volume, this can be viewed as a scattering problem often phrased as a Lippmann-Schwinger integral equation. One defines an energy eigenstate ψ in p whose incoming part (that term with radial dependence e −ipr ) is that of a plane wave with momentum p. Necessarily, the outgoing part of ψ in p is more general, being created by scattering from the potential V . Such a state must have energy E cm = p 2 /m. Manipulation of Eq. 6 easily produces the desired integral equation: where φ p is the plane wave solution φ p ( r) = e i p· r of the free Schrödinger equation and H 0 = −∇ 2 /m, the free Hamiltonian of a particle with the "reduced mass" m/2. Of course the full solution to Eq. 6, ψ (rel) , is a product of a plane-wave depending on the centerof-mass coordinate R and the wave function above: and the total energy E is related to the energy in the center-of-mass system by E = E cm + P 2 /4m.
Examining the asymptotic behavior of Eq. 7, one derives the standard relation between the conventional scattering amplitude f (θ) and the matrix element of V between the plane wave state φ p ′ and ψ in p : where p ′ · p = p 2 cos(θ). We obtain an equation closer to the relativistic Bethe-Salpeter equation by defining the T matrix as: Here we are using the conventional Dirac bra-ket notation to represent momentum eigenstates: φ p ≡ | p . Note, Eq. 10 defines matrix elements of T even when | p ′ | = | p| and energy is not conserved.
Using the matrix T we can rewrite Eq. 7 in a form that will be useful later, if we multiply by V and transform to momentum space: This equation explicitly involves the matrix elements of T between states with different energies. Equation 11 is a component of standard scattering theory and applies only to the case of infinite volume. Since we are interested also in the eigenfunctions and energies for the finite volume problem it will be helpful to cast the standard Schrödinger equation, obeyed by an eigenstate, ψ n ( r) with the discrete energy E cm into a similar form: This equation demonstrates that the state V ψ En solves the homogenous Lippmann-Schwinger equation if the energy argument in the denominator, E cm = p 2 /m is continued to that state's actual energy E cm = E cm n . Equation 13 can be used in infinite volume to determine the energy of a possible bound state. It also can be applied to the case of finite volume if one makes a simple replacement of the integral over the relative momentum k by an appropriate discrete sum.
In Eq. 11 we have followed the standard procedure, exploiting the separation of relative and center-of-mass variables permitted by Eq. 6, and written that integral equation as an equation obeyed by functions of a single, three-dimensional, relative momentum k or p. The three-momentum of the center-of-mass, P , disappears from the problem once we use the energy in center-of-mass system, E cm . However, we could create a more explicit analogy with the relativistic discussion to follow if we viewed the states | p and | k as functions of the four-momentum of the center-of-mass as well: P ( p) = ( P 2 4m + p 2 /m, P ) and P ( k) = ( P 2 4m + k 2 /m, P ) respectively. Had we done this, Eq. 11 would be an equation obeyed by functions of seven variables. Each factor in this equation would be explicitly diagonal in P and the difference of these total energy variables P 0 ( p) and P 0 ( k) would replace the present denominator in that equation. (A similar remark applies to Eq. 13 as well.) The final step to be reviewed in this section is the connection between the infinite volume scattering problem, defined by Eq. 11, and the finite volume energy eigenvalues of the original Schrödinger equation, Eq. 6 or Eq. 13. This is the problem solved by Lüscher in Refs. [4,5].
Recall that the scattering amplitude f (θ) can be written as a sum over partial waves as: where the δ l are the standard scattering phase shifts and the P l (cos(θ)) the usual Legendre polynomials. In Refs. [4,5], Lüscher examines the case of a potential of finite range, V (r) = 0 for r > R max , and for the case L > 2R max derives a relation between the allowed energies E cm = p 2 /m in the finite box and the phase shifts δ l . For the simplest case where all δ l ≈ 0 for l > 0, this finite-volume quantization condition becomes: where n is an integer, p = √ mE, q = pL/2π and the function φ(q) is a known kinematic function given by: with the zeta function Z 00 (s; q 2 ) defined by The zeta function defined above applies to the case that the integers appearing in the center of mass momentum P are even. For the case that one or more is odd, the finite volume problem will obey anti-periodic boundary conditions in those directions and an appropriate offset of 1/2 must be added in the summation in Eq. 17, as discussed in Ref. [6]. We conclude that the total energy of a finite volume system with total momenta P , is given by Thus, Lüscher's relation for non-relativistic quantum mechanics between the scattering phase shifts and the energy eigenvalues for that same system in a finite volume is straight forward to generalize to the case that the two-particle system carries non-zero total momentum. We now consider the generalization of the next step to non-zero total momentum: the connection between the Bethe-Salpeter equation of relativistic quantum field theory and Eq. 11 above, obeyed by the non-relativistic T matrix. This represents the new result of this paper.

III. FINITE VOLUME RELATIVISTIC TWO-PARTICLE STATES
In this section we will generalize to the case of non-zero total momentum the procedure introduced by Lüscher to reduce the Bethe-Salpeter equation of relativistic field theory to an equation whose form is identical to the non-relativistic Eqs. 11 and 13. This will permit the quantization condition described in Eqs. 15-17 to be applied to quantum field theory, and QCD in particular. As discussed in Refs. [4,5], we must do this by carefully distinguishing effects of finite volume which fall as powers or exponentially in the system size. Note: our objective is to derive an equation which is both similar in form to Eqs. 11 and 13 and also accurate for both finite and infinite volume so that it can be used to relate the finite volume spectrum and the infinite volume scattering amplitude.
We begin with the standard Bethe-Salpeter equation, which connects the amputated four-point function, T (p ′ 1 , p ′ 2 ; p 1 , p 2 ), the two-particle irreducible kernel K(p ′ 1 , p ′ 2 ; p 1 , p 2 ) and the single particle propagator, ∆(k 2 ): where P = p 1 +p 2 = p ′ 1 +p ′ 2 is the total four-momentum. For simplicity, we have also removed an overall δ-function for the conservation of total four-momentum from where φ(x) is the quantized scalar field used in this example (normalized so that the single particle pole in the 2-point function has unit residue), . . . conn indicates that only connected diagrams are to be included and we are following the conventions of Peskin and Schoeder [8].
For simplicity we write the sum over the internal four-momentumk in Eq. 18 as an integral. However, this equation applies equally well to a finite volume system if the spatial part of this continuous integral is replaced by an appropriate discrete sum. Specifically, in finite volume the total momentum operator P op is conserved and, given periodic spatial boundary conditions, has components which are quantized in units of 2π/L. The singleparticle three momenta, p, p 1 ′ , P /2 +k, etc. correspond to indices in a finite-volume Fourier series and are also quantized in units of 2π/L. Equation 18 is more general than needed, constraining the off-shell, two-particle scattering amplitude, a matrix acting on a space of functions of eight momentum components. We must specialize this equation, reducing the number of momentum variables from eight to seven. The choice of this seven-dimensional restriction of Eq. 18 is the key step in the desired generalization to non-zero total 3-momentum, P = 0. Once that has been done, the resulting equation will still be quite different from our non-relativistic target, Eq. 11.
However, the final steps connecting Eq. 18 with Eqs. 11 and 13 proceed in a fashion very similar to those in Lüscher's original derivation: We extract from the second term in Eq. 18 particular pieces which are regular functions of the total energy P 0 . By a re-arrangement procedure, these terms are incorporated in a modified kernel K(p ′ 1 , p ′ 2 ; p 1 , p 2 ). After this step, our re-arranged equation will have the same form as the non-relativistic Eqs. 11 and 13.
In contrast to the non-relativistic case, our new equation will have a volume dependent potential. However, if we remain below the four-particle threshold and have introduced into our re-arranged kernel K only regular factors, the difference between the finite and infinite volume kernels will vanish exponentially in the box size. Similar exponentially small errors will come from the failure of the resulting potential to have a truly finite range. Thus, we will be assured that the quantization condition in Eq. 15 will apply to the relativistic case with P = 0 up to exponentially small corrections.
We begin by restricting the general Bethe-Salpeter equation of Eq. 18 to a carefully defined seven-dimensional momentum surface, making it similar to the target Eq. 11. (As discussed earlier, Eq. 11 can also be viewed as involving functions of seven dimensions provided the trivial dependence on the total four-momentum is included.) First express the pairs of two particle momenta p ′ 1 , p ′ 2 and p 1 , p 2 using relative and total momenta: In Eq. 18 we have already imposed four-momentum conservation, removing an over-all momentum conserving delta function from both T and K. Next we impose a further condition on the four-momenta k and k ′ requiring that k 0 = βk , where β = | P |/P 0 and k is the spatial component of the four-vector k in the direction of P . A similar condition defines a restricted value for k ′ and the integration variablek in Eq. 18. This condition is not immediately useful since, while we can consistently impose it on the external k and k ′ vari-ables in Eq. 18, the integral (or sum) overk does not obey any such restriction. Note, the restriction k 0 = βk can be applied equally well in finite or infinite volume since, while k will be discrete for the finite volume case, the time component, k 0 , is always continuous and can be chosen to obey such a relation.
To make progress we must remove from the integral overk some terms which are not singular as P 0 approaches an allowed two-particle energy, P 0 → ω + + ω − , where the single particle energies are given by ω ± = ( P /2 ± k) 2 + m 2 . This can be done by generalizing a discussion in Ref. [4] and arguing that the portion of the product ∆((P/2+k which is a singular function of P 0 comes from the product of the single-particle singularities in each of the ∆ factors. This singular term, arising when these poles pinch thek 0 contour, can be viewed as a distribution ink 0 , allowing us to write this product as: where the function R(P,k) is a regular function of P 0 in the interval below the four-pion threshold: and the singular part, S(P, k), is defined by Eq. 21. Equation 21 is derived in the Appendix and is a generalization of Lüscher's Eq. 3.16 from Ref. [4] to the case of P = 0.
The next step absorbs the contribution from the regular function R(P, k) as follows. First rewrite Eq. 18 in a more symbolic form exploiting the decomposition in Eq. 21: The second term on the right-hand-side can then be moved to the left hand side: (1 − KR)T = K + KST.
Finally we divide by the factor (1 − KR) and define The resulting equation can then be written: Here we have replaced the variables p i and p ′ i with the total and relative four-momenta P , k ′ and k and suppressed the variable P .
It is now easy to see that Eq. 26 has a form identical to the original non-relativistic Lippmann-Schwinger equation, Eq. 11. First we observe that the delta function δ(k · P ) forces the integration four-momentumk to obey our restriction: This permits us to impose this relation between the time and parallel components of the relative four-momenta everywhere in this equation, effectively reducing it to a three-dimensional integral equation as is the case for the non-relativistic problem. (As in that case, this equation is diagonal in the total four momentum, P , the remaining four of our seven-dimensional momentum variables.) Second, we observe that the denominator has a non-relativistic form if we rescale the axis parallel to P by a factor of γ = 1/ 1 − β 2 : Thus, if we change variables fromk to the denominator has the normal Laplacian form. We obtain a complete match between the denominators in Eqs. 11 and 13 and that in Eq. 26 if we remove a factor of m from the denominator of Eq. 26 and identify the non-relativistic energy E cm with (P 2 0 − P 2 )/4m − m. Here we should recall the standard connection between Bethe-Salpeter equation, Eq. 18 and the version of the Schrödinger equation given by the homogenous Eq. 13, which determines the discrete, finite-volume energies. Since the Bethe-Salpeter equation, e.g. Eq. 26, holds in finite volume, the energy eigenvalues, E n for the finite-volume, relativistic system will correspond to poles in the 2-particle scattering amplitude T obeying that equation.
As one approaches the singularity at P 0 → E n the inhomogeneous term, proportional to the kernel K, is not singular and can be dropped from the equation leaving a homogenous equation identical in form to Eq. 13.
Finally we must investigate the rotational symmetry of the kernel, K(k ′ ; k). Since this function also depends on the four-vector P , there is possible rotationally asymmetric dependence on arguments of the form P · k ′ and P · k in addition to the acceptable dependence on ( k ′ ) 2 , k 2 and k ′ · k. Fortunately, if the components of k ′ and k parallel to P are rescaled as described in Eq. 29, it is easy to see that the resulting function K(k ′ ; k) becomes rotationally symmetric. This can be demonstrated by exploiting the Lorentz invariance of the function K(k ′ ; k) which is assured by the covariant separation of regular and singular parts used in Eq. 21. We use this Lorentz symmetry to equate K(k ′ ; k) to its value in the rest system: where for clarity we now also display the dependence on the total four-momentum P .
Since in the center of mass system P cm = 0, the right hand side of Eq. 30 is a manifestly rotationally invariant function of the center of mass variables k ′ cm and k cm . However we can easily express these variable in terms of the original k ′ and k. For example: where the second and third lines follow from the constraint obeyed by k 0 , imposed when we specialized to this three-dimensional equation. The third equation for (k cm ) 0 is necessary to insure that rotational asymmetry is not introduced through a non-symmetric dependence of this variable on the original laboratory variables.
Note that the rescaled variables on which K(P ; k ′ ; k) depends symmetrically, (k cm ) = (1/γ)k and (k ′ cm ) = (1/γ)k ′ , are precisely the variables in terms of which the denominator in Eq. 26 contains the standard Laplacian.
For clarity, we now rewrite the resulting three-dimensional integral equation, Eq. 26, in terms of these translated momentum variables, labeled suggestively k cm : While the variables k ′ cm , k cm and k cm may appear to be Lorentz transformed, center-of-mass variables, in fact, they are simply the original variables defined in the laboratory system except for a transformation of scale for the single component parallel to P . This rescaling is equally well defined if the corresponding variable is continuous or discrete. The only explicit use of Lorentz symmetry is to constrain the possible dependence of the function K(P, k ′ , k) on its arguments.
Thus, we have demonstrated that our original, field-theoretic Bethe-Salpeter equation can be rewritten as a non-relativistic Lippmann-Schwinger equation if we change to rescaled variables (k cm ) = (1/γ)k . We must now ask if such a rescaled set of momenta corresponds to an actual finite volume problem. Recall that we began by examining a two-particle problem in a finite, cubic box of side L. The transformation to the coordinates R and r of Eqs. 2 and 3 and the choice of total momenta P then requires that the relative momenta k or k ′ that appear in the Bethe-Salpeter Eq. 26 have the form 2π(n 1 , n 2 , n 3 )/L where n i is an integer or half-integer depending on whether LP i /2π is an even or odd integer. Now, we have recognized that this problem is equivalent to a non-relativistic problem with transformed momenta given by Eqs. 31 and 32. If these k cm momenta correspond to those for a finite volume problem, then we have succeeded in casting the original relativistic problem into a non-relativistic problem which can be solved using the techniques developed by Lüscher in Refs. [4,5].
That this is in fact the case can be seen by examining two simple examples. In the first example P = 2π(0, 0, 1)/L. In this case the transformed variables k cm i take on a very simple form: where n 1 , n 2 and n 3 + 1 2 are integers. These quantized momenta correspond to a simple finite volume of length L in the 1-and 2-directions and expanded length γL in the 3-direction. If periodic boundary conditions are imposed in the 1-and 2-directions and anti-periodic conditions imposed in the 3-direction, the resulting quantized momenta will correspond precisely with those in Eqs. 35-37. Thus, after generalizing Lüscher's non-relativistic technique to this sort of asymmetric box, we will obtain the desired relation between the infinite-volume scattering phase shifts and the discrete finite volume energies in this asymmetric box. Of course, our analysis above has determined that these same energies will be found in the original relativistic problem with total momentum P = 2π(0, 0, 1)/L and cubic box of side L. This agrees with the result obtained Rummukainen and Gottlieb [6] and corresponds to a case of immediate practical interest. Now let us examine a second case where P = (1, 1, 0)2π/L. Using Eqs. 31 and 32 the rescaled momentum k cm is given by where the first two terms correspond to the untransformed perpendicular component while the third term is the transformed parallel piece. Written in terms of the individual components this equation becomes: These quantization conditions can be easily realized if we impose the following (anti-)periodicity conditions on the wave function ψ cm ( r) on which the operators in the integral equation, Eq. 34, act: ψ cm ( r) = +ψ cm ( r +ê 3 L) where the displacement vector D i is chosen to pick out the integer n i from the dot product D i · k cm = 2πn i L: The quantization condition given in Eqs. 42 and 43 is equivalent to requiring that the wavefunction ψ cm ( r) obey anti-periodic boundary conditions on the faces of a rhombus whose sides are parallel to the vectors D i and whose diagonals have length |D 1 + D 2 | = √ 2γL and , precisely the earlier result of Rummukainen and Gottlieb [6]. From these two examples, it is clear that the case of general total momentum P = (n 1 , n 2 , n 3 )2π/L can also be realized by imposing (anti-)periodic on an appropriately distorted volume.
The results of this section can be summarized by returning to our first example, the case of a symmetrical L 3 box and a two-pion state with total momentum oriented in the 3-direction: Under these circumstances, the Bethe-Salpeter equation obeyed by the twoparticle, off-shell scattering amplitude has been shown to be equivalent to a Schrödinger-like wave equation obeyed in an asymmetrical box with sides L × L × γL with the longer γL side parallel to the 3-direction. Thus, up to exponentially small corrections, the energy eigenvalues of the original 2-particle, L 3 system can be predicted using this L × L × γL, Schrödinger-like system. As a result, the allowed energies E of this original system must obey a quantization condition similar to that given in Eq. 15 where the function φ(k) in that equation must be modified to describe the anti-periodic boundary conditions and expanded length in the 3-direction: where n is an integer, k cm = 1 2 P 2 − (2m) 2 and q = k cm L/2π. This is the original result of Rummukainen and Gottlieb [6].

IV. TWO-PARTICLE DECAY OF STATES WITH NON-ZERO MOMENTUM
The last part of this discussion is a generalization of the arguments of Lellouch and Lüscher in Ref. [2] to the case of non-zero total momenta. Fortunately, this is very straight forward because the methods employed in that paper work equally well for P = 0 once the formula relating energy levels and scattering phase shifts has been generalized to this non-zero momentum case.
We begin by reviewing the Lellouch-Lüscher approach, which is somewhat indirect. For our present purposes we will describe this method for the case of non-zero center-of-mass momentum, P = 0. One considers a finite volume system with both a K 0 meson and a degenerate π + π − state, each with total four-momentum P . Here the finite volume has been adjusted to insure that E ππ = m 2 K + P 2 , including the effects of the π − π interaction. Next, the effects of the weak interaction Hamiltonian, H W , mixing these two states, are then examined in perturbation theory. To zeroth order, the K 0 and π + π − states are degenerate and non-interacting. To first order these states mix and their energies can be determined in first order, degenerate perturbation theory: = P 0 ± π + π − |H W |K 0 (51) where the states appearing in this formula are finite-volume states with non-zero total momentum, normalized to unit probability. Thus, we must compute the variation in the π − π scattering phase shift caused by the resonant scattering into the K meson state. This infinite volume calculation is most easily done in the π − π center of mass system and follows easily from the single π − π scattering diagram with a K meson intermediate state shown in Fig. 1 giving: Here we have evaluated the addition to the π − π scattering phase shift coming from the resonant scattering into the K meson state at a center-of-mass energy which corresponds to the laboratory energy P 0 + ∆P 0 determined by Eqs. 52: . 1: The contribution of H W to π − π scattering involving resonant production of a K meson.
Because of the singular K-meson propagator, the amplitude corresponding to this graph will be first order in H W when evaluated at the center-of-mass energies m K ± γ|M | as required for our application.
The infinite volume decay amplitude A appearing in Eq. 53 is normalized following the conventions of Lellouch and Lüscher so that the corresponding decay width is given by: Finally, these results are used to evaluate the terms in the quantization condition, Eq. 46 for the original finite volume P = 0, π − π system. The terms in this equation which are first order in the matrix elements of H W are: where relation between ∆k cm and ∆P 0 is determined by k cm = P 2 /4 − m 2 which implies: Combining Eqs. 52 and 57 then gives the desired connection between A and M = π + π − |H W |K 0 : If this equation is solved for |A| 2 we obtain the desired generalization of the original Lellouch-Lüscher condition to the case of P = 0: This formula differs from that of Ref. [2] by the presence of the factor of γ 2 and the more complex, γ-dependent definition of the function φ(q) given in Eqs. 47 and 48.

V. CONCLUSION
In the preceding sections we have examined the case of two interacting particles confined in a finite spatial volume and carrying non-zero total momentum. We have determined a relation between the quantized energies of these finite-volume states and the two-particle scattering phase shifts. This result, first obtained by Rummukainen and Gottlieb in Ref. [6], is here obtained from the Bethe-Salpeter equation of relativistic quantum field theory using an extension of the methods that Lüscher applied to the case of zero total momentum in Refs. [4,5]. We then exploit these finite volume results to analyze two-particle decays. The result, an extension of earlier work of Lellouch and Lüscher, Ref. [2], to the case of nonzero total momentum, provides an explicit formula that relates finite volume decay matrix elements computed using lattice gauge theory techniques and the infinite-volume quantities that enter physical decay rates.
The ability to work with states with non-zero total momentum when computing such decay matrix elements offers two potentially important benefits to the study of K → ππ decay. First, by a proper choice of the total momentum, the corresponding two-pion state with lowest energy can be arranged to have an energy equal to that of the K meson, permitting the direct calculation of a physical, on-shell, decay matrix element. Second, by working with a K meson with non-zero momentum, we insure that the unphysical vacuum decay amplitude, normally dominant in such a Euclidean matrix element calculation, will vanish because of momentum conservation. We are now exploring this approach numerically.

APPENDIX A: SINGULAR PART OF TWO-PROPAGATOR PRODUCT
The singular contribution to the integral in Eq. 18 will come as the energy P 0 is adjusted to cause singularities present in both of the single-particle propagator factors, ∆((P/2±k) 2 ), to pinch thek 0 contour. Thus, we can obtain the singular part of the integral overk 0 in Eq. 18 by replacing both propagators by their singular part: ∆((P/2 ± k) 2 ) → i (P/2 ± k) 2 − m 2 . (A1) Equation 21 is based on the following expression for the singular part of the product of two free scalar propagators: where these two quantities are equivalent in the sense that they have the same pole in the total energy P 0 at the point P 0 = ω + + ω − Following Lüscher, this formula is interpreted as relating two distributions in the variable k 0 over the space of test functions f (k 0 ) analytic in k 0 within a band around the imaginary k 0 axis. Their equivalence can be demonstrated by multiplying by such a test function and then integrating k 0 along the imaginary axis. The left-hand-side of Eq. A2 is evaluated by moving the k 0 contour past one of the two pinching poles, k 0 = P 0 /2 − ω − or k 0 = −P 0 /2 + ω + and keeping the contribution of that pole given by Cauchy's theorem. Following this procedure the left-hand-side of Eq. A2 becomes: where we have simplified this expression by replacing P 0 in the residue of the P 0 = ω + + ω − pole by its value at that pole. Next we evaluate the right-hand-side of Eq. A2 by using the delta-function to evaluate the integral over k 0 . We obtain: −iπ f ( P · k P 0 ) P 0 P 2 0 /4 + ( P · k P 0 ) 2 − P 2 /4 − k 2 − m 2 + iǫ (A4) where for clarity the P 0 arguments have not been simplified. The final step requires computing the location and residue of the pole in P 0 found in the expression in Eq. A4. The result agrees precisely with that shown in Eq. A3 provided one is careful to include the effects of the ( P · k/P 0 ) 2 term when computing the residue and then replaces P 0 in the residue with ω + + ω − , it value at the pole. We forego a discussion of the specific region of analyticity for the test functions above and its consistency with the analytic properties of the amplitudes in the Bethe-Salpeter equation, Eq. 18, since these kinematics should be a direct Lorentz transform of those discussed by Lüscher in Ref. [4].