Finite-volume effects in long-distance processes with massless leptonic propagators

In Ref. [1], a method was proposed to calculate QED corrections to hadronic self energies from lattice QCD without power-law finite-volume errors. In this paper, we extend the method to processes which occur at second-order in the weak interaction and in which there is a massless (or almost massless) leptonic propagator. We demonstrate that, in spite of the presence of the propagator of an almost massless electron, such an infinite-volume reconstruction procedure can be used to obtain the amplitude for the rare kaon decay $K^+\to\pi^+\nu\bar\nu$ from a lattice quantum chromodynamics computation with only exponentially small finite-volume corrections.

When analyzing matrix elements of bilocal operators, it is useful to insert a complete set of intermediate states between the two local operators. If the energy of the initial state is sufficiently large to create on-shell intermediate multi-particle states, power-law finitevolume effects can be generated 1 . Following Ref. [53], where the K L -K S mixing is analysed as an example, one can correct for such potentially large finite-volume effects. However, the situation changes when the intermediate multi-particle state involves a massless, or nearly massless, particle. Since the long-range massless propagator is distorted by the finite volume, power-law finite-volume effects appear even for states containing off-shell particles.
Such a situation happens, for example, in the rare kaon decay K + → Xe + ν e → π + ν eνe [7][8][9][10][11][12] where the intermediate states contain a positron, together possibly with additional hadronic particles specified here by the symbol X. The positron is effectively massless since its mass, m e , satisfies m e L ≪ 1, where L is the spacial extent of current lattices (with volume, V = L 3 ). An analogous procedure has been applied to the calculation of the amplitude for neutrinoless double-β decay π − π − → Xeν e → ee, in which there is the propagator of a massless neutrino [21]. 1 Throughout this paper we use the shorthand notation exponential finite-volume effects to denote ones which decrease exponentially with the spacial extent of the lattice L, and power-law finite-volume effects to denote those which decrease only as powers of L.
To completely remove the power-law finite-volume effects induced by the massless electron or neutrino, we adopt the infinite-volume reconstruction (IVR) method proposed in Ref. [1], which has been used to eliminate such effects in QED corrections to hadronic self energies. In that case the lightest intermediate hadron is the same as the stable hadron in the initial and final states. In this paper, we use the rare decay K + → π + νν ℓ to illustrate that the method is also applicable to processes in which the intermediate hadronic state is not degenerate with the initial state; indeed it can be either heavier or lighter than initial state.
The structure of the remainder of this paper is as follows. In the following section we discuss the structure and properties of the physical amplitude for the rare kaon decay K + → π + νν in Minkowski space and write the amplitude in a form which is convenient for continuation into Euclidean space. In Sec. III we present our proposed method for the evaluation of the amplitude from Euclidean correlation functions computed in a finite-volume, but with only exponentially small finite-volume corrections. Finally we present our conclusions in Sec. IV.
II. FINITE-VOLUME EFFECTS IN K + → π + ν ℓνℓ DECAYS As explained in Ref. [8], the K + → π + ν ℓνℓ decay amplitude, where ℓ represents the lepton quantum number, contains contributions from both Z-exchange diagrams and W -W diagrams. For the Z-exchange diagrams, for which the νν pair is emitted from the same vertex, there are no leptonic propagators in the amplitude and the dominant, power-law finite-volume effects are associated with the process K + → π + π 0 → π + ν ℓνℓ , which can be corrected using the formula provided in Ref. [53]. Here we focus on the contribution from the W -W diagrams illustrated in Fig. 1 in which the ν ℓ andν ℓ are emitted from separate vertices and which contain the propagator of the corresponding charged lepton ℓ + (e + , µ + or τ + ). The discussion of the properties and structure of the physical amplitude in this section is presented in Minkowski space.
The contribution to the amplitude from the W -W diagrams in Minkowski space, A M ℓ , is given by where the A M q,ℓ are defined by where q = u, c are the flavors of up-type quarks and ℓ = e, µ, τ are the flavors of leptons.
The two operators in Eq. (2) are given by where, for example, (sq In this paper we focus on the transition K + → Xℓ + ν ℓ → π + ν ℓνℓ . We denote the potentially large, i.e. the power-law, finite-volume effects in the spatial integral over a finite volume of size L 3 , by A Xℓ where A Xℓ + (L) and A Xℓ + (∞) are the amplitudes in finite and infinite volumes respectively. The label Xℓ + indicates that the correction comes from the Xℓ + intermediate states, where X can represent (i) the vacuum, (ii) the stable single-hadron states, π 0 or D 0 , or (iii) multi-hadron states of which the lightest ones are two-pion states. The neutrino in the Xℓ + ν ℓ intermediate state is the one which appears in the final state, and its energy and momentum determine those of Xℓ + .
For the transition K + → Y ℓ −ν ℓ → π + ν ℓνℓ , charge conservation requires Y to be a multihadron state. The corresponding finite-volume effects are largely similar to those from multi-hadron states X in K + → Xℓ + ν ℓ → π + ν ℓνℓ transitions apart from the presence of disconnected diagrams as discussed in Sec. III B. Figure 2. Illustration of the process K + → Xℓ + ν ℓ → π + ν ℓνℓ .
We now consider the three possibilities for X in turn. When X is the vacuum, the momentum of ℓ + is completely fixed by momentum conservation, There are no power-law finite-volume effects in this case.
When X is a single stable hadron with four momentum k, as shown in Fig. 2, the finitevolume effects in the amplitude, A Xℓ + FV , can be expressed as [8,12] A Xℓ where k is the momentum carried by the intermediate hadron X and P = p K − p ν ℓ is the total momentum flowing into the Xℓ + loop. The terms A K + →X α and A X→ π + β represent the transition matrix elements indicated by the superscripts and α, β are the Lorentz indices of the weak currents.
Although the present study is focussed on rare kaon decays, the main ideas are more general. Equation (4) is an example of the generic form of the expression for finite-volume effects: where, in the present calculation, P = p K − p ν ℓ ≡ (E, P ), m 1 = m X and m 2 = m ℓ . We can evaluate the k 0 integration using Cauchy's theorem, including the contributions from the poles in the two propagators shown in Eq. (5) and ignoring contributions from any other k 0 singularities in f (k 0 , k) since these will result from other more massive intermediate states than those we have chosen to study. For simplicity of notation, the dependence of f (k 0 , k) on the external momenta is not shown explicitly. Performing the integral over k 0 we obtain the integrand Using the Poisson summation formula, it can be shown that two singularities of the integrand contribute finite-volume effects which are not exponentially small in the volume. In the first term in Eq. (6) there is a singularity when the condition E = E 1 + E 2 is satisfied and two on-shell particles are created [54]. In the second term of Eq. (6), if m 2 is very small then there is an additional singularity from the factor 1/E 2 ≈ 1/| P − k|. For example, using a QED L -style regularization and omitting the zero-momentum mode from the allowed finite-volume lepton states, the region around | P − k| = 0 leads to a 1/L 2 difference between the finite-volume summation and infinitevolume integration [49,55]. This is the situation for rare kaon decays when the lepton is the electron where, since in practice m e L ≪ 1, the electron is effectively massless in lattice computations. Therefore, no matter how heavy is the hadron X, power-law finite-volume effects associated with the massless electron always exist. In the following section we will discuss how to remove these two sources of power-law finite-volume effects by extending the method developed in Ref. [1].
When X is a multi-hadron state, the situation is more complicated. For multi-hadron states with energies that are larger than the energy of the initial hadron, the contributions are exponentially suppressed at large time separations and the corresponding power-law finite-volume effects can be safely eliminated using our proposed method. For multi-hadron states with energies which are smaller than that of the initial hadron, it is unclear in general how to remove all the power-law finite-volume effects. Fortunately, for the K + → π + ν ℓνℓ decay, the contribution from low-lying multihadron states, e.g.
can safely be neglected due to the significant phase space suppression.

A. Structure of the amplitude
Before explaining how to obtain the physical decay amplitude from a computation on a finite lattice we formulate it in an expression suitable for continuation into Euclidean space.
We start by rewriting the bilocal matrix element in the integrand of A M q,ℓ in Eq. (2) as a product of two factors: where α and β are Lorentz indices. The hadronic factor H M,(q) αβ (x) and the leptonic factor Here S ℓ (x, 0) is a free lepton propagator. By inserting a complete set of energy eigenstates, A M q,ℓ can further be written as where the first and second terms on the right-hand sides of Eqs. (10) and (11) where (13). In both cases the energy of the charged lepton is given by E ℓ ± = p 2 ℓ ± + m 2 ℓ . In deriving Eq. (10) we have used the space-time translation property where (E i , p i ) and (E f , p f ) are the four-momenta of the initial and final states respectively, and have defined the quark V − A currents by Figure 3. Schematic drawing of the two time orderings in the correlation function. In (a) we have The principal objective of this paper is to demonstrate how to obtain the expression on the right-hand side of Eq. (10) from computations of correlation functions on a finite Euclidean lattice with only exponentially small finite-volume effects. We explain how to achieve this in the following section.

III. A M q,ℓ FROM EUCLIDEAN CORRELATION FUNCTIONS
Hadronic matrix elements are obtained in lattice QCD computations from calculations of finite-volume Euclidean correlation functions. In this section we present a detailed discussion of the evaluation of A M u,ℓ , since the evaluation of A M c,ℓ is considerably more straightforward and can readily be deduced from that of A M u,ℓ (as we briefly explain at the appropriate points in the discussion). We consider separately the two time-orderings t < 0 and t > 0 corresponding to each of the two terms on the right-hand side of Eq. (10). The correlation functions for the two time-orderings are sketched schematically in Fig. 3.

A. The time-ordering t < 0
Consider the finite-volume Euclidean correlation function where J † K and J π are interpolating operators for the creation of a kaon and annihilation of a pion respectively. The correlation function in Eq. d,β at times t and 0 respectively, with t < 0 and the annihilation of the pion at time t π ≫ 0. This is illustrated in Fig. 3(a). For compactness of notation we suppress the dependence of C (u) αβ (t) on t K , t π and the momenta. In this section we show how to obtain the A M,− u,ℓ component of the amplitude from the evaluation of C (u) αβ (t, x) up to exponentially small finite-volume corrections. Assuming, as is standard, that t − t k and t π are sufficiently large for the correlation function to be dominated by a kaon of momentum p K propagating in the time interval (t K , t) and for a single pion to be propagating in the interval (0, t π ) we have The energies of the kaon and pion, E K and E π respectively, and the matrix elements Z K = K( p K )|J † K (0)|0 and Z π = 0|J π (0)|π( p π ) can be obtained in the standard way from twopoint meson correlation functions using our normalization conventions, e.g. for the finitevolume state |π( p π ) , We then rewrite Eq. (17) as where and H E,(u) αβ (t, x) is the Euclidean equivalent of the bilocal hadronic matrix element in Eq. (8) at t < 0 where the sum is over a complete set of non-strange states |n . The basis of the infinitevolume reconstruction method is that we perform the integral in Eq. (7) using the hadronic matrix element H E,(u) αβ (t, x) calculated using lattice methods on a finite spatial volume and the leptonic tensor L E,αβ calculated in an infinite spatial volume which for t < 0 is given by In order to allow the external kaon and pion to propagate over sufficiently large time intervals to eliminate excited external states and obtain H E,(u) αβ (t, x) we imagine performing the time integration over the interval (T A , T B ), where t K ≪ T A ≪ 0 ≪ T B ≪ t π , as illustrated in Fig. 3. In this subsection we are considering the contribution from the region t < 0 and so the range of integration is (T A , 0) and the aim here is to compute in such a way as to reproduce the first term on the right-hand side of Eq. (10), A M,− u,ℓ , with only exponentially small finite-volume effects 2 . However, the presence of states |n in the sum in the second line of Eq. (21) with energies which are smaller than those of the external states leads to exponentially growing terms in |T A | and power-law finite-volume effects.
We therefore cannot simply evaluate the integral in Eq. (23) using H E,(u) αβ (t, x) computed directly on a finite Euclidean lattice for all t ∈ (T A , 0) and a modified procedure must be introduced. We now explain in some detail the presence of power-law finite-volume effects and the exponentially growing behaviour with |T A | in Eq. (23) together with our proposed method for eliminating them.
Using Eqs. (21) and (22) we see that the integration over time (under the sum over |n and integration over p ℓ + ) is given by The difficulty arises because there are states |n for which E n + E ℓ + + E ν ℓ − E K < 0 leading to an unphysical contribution that is exponentially growing in T A and power-law finitevolume effects due to the singularity in the denominator of the right-hand side of Eq. where The vacuum contribution H vac αβ (t, x) can be determined with only exponential finite-volume errors using the matrix elements π|O The subtraction of the vacuum contribution analogous to that needed to obtain H had, (u) from Eq. (25) has been performed successfully in a study of neutrinoless double β-decay [23] and here we also envisage exploiting the statistical correlations between H where the ≃ symbol indicates the equality of the two sides of the equation up to excitedstate contributions which are assumed to be negligible. Although |t s | is large enough for the ground state to dominate for |t| ≥ |t s |, it is nevertheless finite so that the finite-volume corrections in H had,(u) αβ (t s , x) are exponentially suppressed and we have therefore replaced the sum over finite-volume |π 0 by the infinite-volume phase-space integral.
In the original presentation of the IVR method [1], the integration region over t, i.e.
t ∈ (−∞, 0), is divided into two intervals (t s , 0) and (−∞, t s ), labeled as regions s (for short) and l (long) respectively. Here we start by evaluating I (s) , the integral over the region t ∈ (t s , 0) (see Eq. (29) below). We then show that the corresponding contribution to the physical amplitude A M,− u,ℓ (see Eqs. (10) and (11)) is obtained from I (s) +Ĩ (l) , wherẽ I (l) is an appropriately modified contribution from the integration region (−∞, t s ). The hadronic components of I (s) andĨ (l) can both be determined from lattice computations.
The integration over t in the interval (t s , 0) is defined by since for finite t s the finite-volume effects are exponentially small 3 . Nevertheless I (s) does not reproduce the corresponding contribution to the Minkowski amplitude, A (M,−) u,ℓ defined in Eq. (11). Instead I (s) is given by where a sum over intermediate states |n is implied. For excited states with E n + E ℓ + + E ν ℓ − E K > 0 the contributions from Euclidean matrix elements reproduce the corresponding Minkowski results up to the exponentially suppressed term e −(En+E ℓ + +Eν ℓ −E K )|ts| . This is not the case however, for the ground-state, |π 0 , contribution which has to be treated differently. (Note that even in this case, the integrand on the right-hand side of Eq. (30) has no singularity at E π 0 + E ℓ + + E ν ℓ − E K = 0 since the numerator also vanishes at this point.) In order to reproduce A M,− u,ℓ we must remove the |π 0 contribution from I (s) in (30), and replace it by the corresponding term (i.e. the term with |n = |π 0 ) in Eq. (10). To this end we define the quantityĨ (l) by 4 where E π 0 = m 2 π + p 2 π 0 . In the second line of Eq. (31) the first term is the Minkowski contribution from the |π 0 intermediate state (see Eq. (10)) and the second term is the |π 0 contribution to I (s) (see Eq. (30)). Since in this second term there is no singularity, 3 The suffix ∞ in Eq. (29) indicates that the integral is performed in infinite volume. 4 The tilde onĨ (l) is introduced to denote the fact thatĨ (l) is not simply the integral over the region t < t s .
it is possible and also convenient to add −iε to the denominator. In this way we remove the unphysical contribution in I (s) and replace it with the missing term in the physical amplitude. Combining Eqs. (28) and (32) we havẽ whereL αβ 1 (t s , x) is defined as In the integrand of Eq. (34), p π 0 = p K − p ℓ + − p ν ℓ and E π 0 = p 2 π 0 + m 2 π 0 . Thus we see that the quantities I (s) andĨ (l) can be approximated using the quantities H had αβ (t, x) calculated for |t| ≤ |t s | in a lattice computation as inputs. The finite-volume effects induced by this approximation are exponentially suppressed.
The contribution A M,− c,ℓ is much more straightforward to evaluate as the intermediate states now have charm quantum number C = 1, and so have larger energies than m K . In this case one simply performs the integral We now consider the case t > 0 and the evaluation of A M,+ u,ℓ , for which the elimination of the power-law finite-volume effects is a little more straightforward but which nevertheless contains a new subtlety. We start by following the same steps as for t < 0, relating the Euclidean correlation function illustrated in Fig. 3(b) to the bilocal hadronic matrix element: where Z Kπ is given in Eq. (20) and for t > 0 , αβ (t, x) and they need to be treated separately in an analogous way to the vacuum contribution in Eq. (26). We call these contributions disconnected 5 .
to demonstrate that it is the disconnected contribution to A M,+ u,ℓ in Eq. (10). Note that for the disconnected contribution p ns = p K + p π .
For the connected contribution, there are two important points to note, both resulting from the observation that the intermediate states |n s all have energies which are larger than those of the external states. The first point is that the finite-volume effects in H conn,(u) αβ (t, x) computed on a finite lattice are exponentially suppressed. The second related point is that even in infinite volume this hadronic matrix element is exponentially suppressed at large t and | x|.
In evaluating the contribution to the integral of Eq. (7) we need to combine H conn,(u) αβ with the corresponding leptonic tensor L E,αβ (t, x), where for t > 0 In this subsection we are considering the contribution from the region t > 0, so the range of integration is (0, T B ) and we arrive at the following contribution to the decay amplitude: The contribution A conn u,ℓ is equal to the connected contribution to A M,+ u,ℓ up to exponentially suppressed terms in the volume and in T B .
Combining Eqs. (41) and (43) we obtain where the equality holds up to exponentially small finite-volume corrections.
The evaluation of the corresponding contribution from the charmed intermediate states, i.e. to A M + c,ℓ , follows in the same way except that there are no Type 1 diagrams and hence there is no disconnected contribution.
The discussion of the contribution from the region t > 0 is considerably simplified because the necessary presence of multi-hadron S = 1 intermediate states implies that there are no power-law finite-volume effects arising from on-shell intermediate states. In addition, the use of the infinite-volume leptonic tensor (42) in the integration in Eq. (43) avoids power-law finite-volume effects which would arise due to the factor of 1/2E l − in the difference between a finite-volume sum over p ℓ − and the corresponding infinite-volume integration. The above discussion is a particular illustration of how to avoid power-law finite-volume corrections in the second term on the right-hand side of Eq. (6), which applies to general processes with a massless (or almost massless) leptonic propagator.

C. Summary
In summary therefore, we propose to calculate the decay amplitude A M u,ℓ , using the following form where all the hadronic quantities can be obtained from lattice simulations with exponential finite-volume effects: The first three terms on the right-hand side of Eq. (45) come from the region t < 0 and are respectively (i) the vacuum contribution obtained using H vac αβ in Eq. (26) and the contributions from (ii) I s in Eq. (25) and (iii)Ĩ l in Eqs. (33) and (34). The final two terms in αβ (t, x) L E,αβ (t, x) to obtain A M c,ℓ . An important point to note is that it still requires further investigations to extend the method developed in Ref. [1] and in this work to multi-hadron intermediate states with energies smaller than the external ones, as such states induce branch cuts which cannot be simply described by discrete QCD eigenstates in infinite volume.

IV. CONCLUSION
In this work, we extend the infinite-volume reconstruction method proposed in Ref. [1] to long-distance processes with massless (or almost massless) leptonic propagators. Using the rare K + → π + ν ℓνℓ decay as an example, we show that the power-law finite-volume effects induced by the massless electron can safely be removed using the form in Eq. (45) in lattice computations. A similar approach has been applied to the amplitude for neutrinoless double-β decay in which there is the propagator of a massless neutrino [23]. We are also performing exploratory studies with the aim of extending the method to the evaluation of electromagnetic corrections to leptonic and semileptonic decays [56] and applying it in numerical lattice QCD calculations.