Finite-volume effects in long-distance processes with massless leptonic propagators
Norman H. Christ, Xu Feng, Lu-Chang Jin, Christopher T. Sachrajda
aa r X i v : . [ h e p - l a t ] S e p Finite-volume effects in long-distance processes with masslessleptonic propagators
Norman H. Christ, ∗ Xu Feng,
2, 3, 4, 5, † Lu-Chang Jin,
6, 7, ‡ and Christopher T. Sachrajda § Physics Department, Columbia University, New York, NY 10027, USA School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China State Key Laboratory of Nuclear Physics andTechnology, Peking University, Beijing 100871, China Physics Department, University of Connecticut, Storrs, Connecticut 06269-3046, USA RIKEN BNL Research Center, BrookhavenNational Laboratory, Upton, New York 11973, USA Department of Physics, University of Southampton, Southampton SO17 1BJ, UK (Dated: September 18, 2020)
Abstract
In Ref. [1], a method was proposed to calculate QED corrections to hadronic self energies fromlattice QCD without power-law finite-volume errors. In this paper, we extend the method toprocesses 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 thepropagator of an almost massless electron, such an infinite-volume reconstruction procedure canbe used to obtain the amplitude for the rare kaon decay K + → π + ν ¯ ν from a lattice quantumchromodynamics computation with only exponentially small finite-volume corrections. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION Lattice Quantum Chromodynamics (QCD) has been successful in precision flavor physics,where observables such as the decay constants f K , f π and semileptonic form factors f + (0) canbe calculated with sub-percent precision [2]. The quantities mentioned above, which occurat leading order in the weak interaction, provide important constraints for CKM matrixelements. With the development of supercomputers, algorithms and new ideas, the rangeof lattice QCD calculations has been extended to include many second-order electroweakprocesses, where the calculations involve the construction of 4-point correlation functionsand the treatment of bilocal matrix elements with the insertion of two operators from theeffective Hamiltonian. Examples include K L - K S mixing [3–6] and ǫ K [5], rare kaon decays [7–17], neutrinoless double-beta decays [18–25], electroweak box contributions to semileptonicdecays [26], inclusive B -meson decays [27–29], nucleon Compton amplitudes [30–33], as wellas isospin-breaking effects in hadronic spectra [34–43], the hadronic vacuum polarizationfunction [44–47], leptonic decays [48–50] and K → ππ decays [51, 52].When analyzing matrix elements of bilocal operators, it is useful to insert a completeset of intermediate states between the two local operators. If the energy of the initial stateis sufficiently large to create on-shell intermediate multi-particle states, power-law finite-volume effects can be generated . Following Ref. [53], where the K L - K S mixing is analysedas 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, ornearly massless, particle. Since the long-range massless propagator is distorted by the finitevolume, 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 e ¯ ν e [7–12]where the intermediate states contain a positron, together possibly with additional hadronicparticles 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 ). An analogous procedure has been applied to the calculation of the amplitudefor neutrinoless double- β decay π − π − → Xe ¯ ν e → ee , in which there is the propagator of amassless neutrino [21]. Throughout this paper we use the shorthand notation exponential finite-volume effects to denote oneswhich 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 .
2o completely remove the power-law finite-volume effects induced by the massless electronor 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. Inthat case the lightest intermediate hadron is the same as the stable hadron in the initial andfinal states. In this paper, we use the rare decay K + → π + ν ¯ ν ℓ to illustrate that the methodis also applicable to processes in which the intermediate hadronic state is not degeneratewith 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 sectionwe discuss the structure and properties of the physical amplitude for the rare kaon decay K + → π + ν ¯ ν in Minkowski space and write the amplitude in a form which is convenient forcontinuation into Euclidean space. In Sec. III we present our proposed method for the evalu-ation of the amplitude from Euclidean correlation functions computed in a finite-volume, butwith only exponentially small finite-volume corrections. Finally we present our conclusionsin Sec. IV. II. FINITE-VOLUME EFFECTS IN K + → π + ν ℓ ¯ ν ℓ DECAYS
As explained in Ref. [8], the K + → π + ν ℓ ¯ ν ℓ decay amplitude, where ℓ represents thelepton quantum number, contains contributions from both Z -exchange diagrams and W - W diagrams. For the Z -exchange diagrams, for which the ν ¯ ν pair is emitted from the samevertex, there are no leptonic propagators in the amplitude and the dominant, power-lawfinite-volume effects are associated with the process K + → π + π → π + ν ℓ ¯ ν ℓ , which can becorrected using the formula provided in Ref. [53]. Here we focus on the contribution fromthe W - W diagrams illustrated in Fig. 1 in which the ν ℓ and ¯ ν ℓ are emitted from separatevertices 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 sectionis presented in Minkowski space.The contribution to the amplitude from the W - W diagrams in Minkowski space, A Mℓ , isgiven by A Mℓ = A Mu,ℓ − A Mc,ℓ (1)3 du π + ¯ su K + O ∆ S =1 O ∆ S =0 ν ℓ ¯ ν ℓ ℓ + Type 1 ¯ du π + ¯ s K + O ∆ S =1 O ∆ S =0 ν ℓ ¯ ν ℓ ℓ + ¯ u, ¯ c Type 2
Figure 1. Quark and lepton contractions for the W - W diagrams. The quark flavors are as indicatedand the lepton ℓ = e, µ or τ . The quark (leptons) are represented by solid (dashed) lines.Theoperators O ∆ S =1 and O ∆ S =0 are shorthand representations of O M, ∆ S =1 qℓ and O M, ∆ S =0 qℓ defined inEq. (3). where the A Mq,ℓ are defined by A Mq,ℓ = i Z d x h π + ν ℓ ¯ ν ℓ | T (cid:8) O M, ∆ S =1 qℓ ( x ) O M, ∆ S =0 qℓ (0) (cid:9) | K + i , (2)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 O M, ∆ S =1 qℓ = (¯ sq ) V − A (¯ ν ℓ ℓ ) V − A , O M, ∆ S =0 qℓ = (¯ qd ) V − A (¯ ℓν ℓ ) V − A , (3)where, for example, (¯ sq ) V − A (¯ ν ℓ ℓ ) V − A ≡ (cid:0) ¯ sγ µ (1 − γ ) q (cid:1)(cid:0) ¯ ν ℓ γ µ (1 − γ ) ℓ (cid:1) .In this paper we focus on the transition K + → Xℓ + ν ℓ → π + ν ℓ ¯ ν ℓ . We denote the po-tentially large, i.e. the power-law, finite-volume effects in the spatial integral over a finitevolume of size L , by A Xℓ + FV = A Xℓ + ( L ) − A Xℓ + ( ∞ ), where A Xℓ + ( L ) and A Xℓ + ( ∞ ) are theamplitudes in finite and infinite volumes respectively. The label Xℓ + indicates that the cor-rection comes from the Xℓ + intermediate states, where X can represent (i) the vacuum, (ii)the stable single-hadron states, π or D , or (iii) multi-hadron states of which the lightestones are two-pion states. The neutrino in the Xℓ + ν ℓ intermediate state is the one whichappears in the final state, and its energy and momentum determine those of Xℓ + .For the transition K + → Y ℓ − ¯ ν ℓ → π + ν ℓ ¯ ν ℓ , charge conservation requires Y to be a multi-hadron state. The corresponding finite-volume effects are largely similar to those frommulti-hadron states X in K + → Xℓ + ν ℓ → π + ν ℓ ¯ ν ℓ transitions apart from the presence of disconnected diagrams as discussed in Sec. III B.4 + X π + ℓ + ν ℓ ¯ ν ℓ Figure 2. Illustration of the process K + → Xℓ + ν ℓ → π + ν ℓ ¯ ν ℓ . We now consider the three possibilities for X in turn. When X is the vacuum, themomentum of ℓ + is completely fixed by momentum conservation, p ℓ + = p K − p ν ℓ ≡ P .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 finite-volume effects in the amplitude, A Xℓ + FV , can be expressed as [8, 12] A Xℓ + FV = L X ~k Z dk π − Z d k (2 π ) (cid:26) A K + → Xα ( p K , k ) ik − m X + iε A X → π + β ( k, p π ) (cid:27) × (cid:26) ¯ u ( p ν ℓ ) γ α (1 − γ ) i ( /P − /k ) − m ℓ + iε γ β (1 − γ ) v ( p ¯ ν ℓ ) (cid:27) , (4)where k is the momentum carried by the intermediate hadron X and P = p K − p ν ℓ is thetotal momentum flowing into the Xℓ + loop. The terms A K + → Xα and A X → π + β represent thetransition matrix elements indicated by the superscripts and α, β are the Lorentz indices ofthe weak currents.Although the present study is focussed on rare kaon decays, the main ideas are moregeneral. Equation (4) is an example of the generic form of the expression for finite-volumeeffects: I FV = I ( L ) − I ( ∞ ) = L X ~k Z dk π − Z d k (2 π ) f ( k , k )( k − m + iε )(( P − k ) − m + iε ) , (5)where, in the present calculation, P = p K − p ν ℓ ≡ ( E, ~P ), m = m X and m = m ℓ . We canevaluate the k integration using Cauchy’s theorem, including the contributions from thepoles in the two propagators shown in Eq. (5) and ignoring contributions from any other k singularities in f ( k , k ) since these will result from other more massive intermediate statesthan those we have chosen to study. For simplicity of notation, the dependence of f ( k , k )5n the external momenta is not shown explicitly. Performing the integral over k we obtainthe integrand − i f ( E , ~k )2 E (cid:0) ( E − E ) − E + iε (cid:1) − i f ( E + E , ~k )2 E (cid:0) ( E + E ) − E + iε (cid:1) (6)with E = q m + ~k and E = q m + ( ~P − ~k ) . Using the Poisson summation formula, itcan be shown that two singularities of the integrand contribute finite-volume effects whichare not exponentially small in the volume. In the first term in Eq. (6) there is a singularitywhen the condition E = E + E is satisfied and two on-shell particles are created [54]. Inthe second term of Eq. (6), if m is very small then there is an additional singularity fromthe factor 1 /E ≈ / | ~P − ~k | . For example, using a QED L -style regularization and omittingthe zero-momentum mode from the allowed finite-volume lepton states, the region around | ~P − ~k | = 0 leads to a 1 /L difference between the finite-volume summation and infinite-volume integration [49, 55]. This is the situation for rare kaon decays when the lepton isthe electron where, since in practice m e L ≪
1, the electron is effectively massless in latticecomputations. Therefore, no matter how heavy is the hadron X , power-law finite-volumeeffects associated with the massless electron always exist. In the following section we willdiscuss how to remove these two sources of power-law finite-volume effects by extending themethod developed in Ref. [1].When X is a multi-hadron state, the situation is more complicated. For multi-hadronstates with energies that are larger than the energy of the initial hadron, the contributionsare exponentially suppressed at large time separations and the corresponding power-lawfinite-volume effects can be safely eliminated using our proposed method. For multi-hadronstates with energies which are smaller than that of the initial hadron, it is unclear in generalhow to remove all the power-law finite-volume effects. Fortunately, for the K + → π + ν ℓ ¯ ν ℓ decay, the contribution from low-lying multihadron states, e.g. K + → πℓ + ν ℓ → π + ν ℓ ¯ ν ℓ ,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 afinite 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 Mq,ℓ in Eq. (2) as a6roduct of two factors: A Mq,ℓ = i Z d x H M, ( q ) αβ ( x ) L M,αβ ( x ) , (7)where α and β are Lorentz indices. The hadronic factor H M, ( q ) αβ ( x ) and the leptonic factor L M,αβ ( x ) are defined by H M, ( q ) αβ ( x ) = h π + ( p π ) | T (cid:8)(cid:2) ¯ s ( x ) γ α (1 − γ ) q ( x ) (cid:3) (cid:2) ¯ q (0) γ β (1 − γ ) d (0) (cid:3) (cid:9) | K + ( p K ) i (8) L M,αβ ( x ) = ¯ u ( p ν ℓ ) γ α (1 − γ ) S ℓ ( x, γ β (1 − γ ) v ( p ¯ ν ℓ ) e ip νℓ · x . (9)Here S ℓ ( x,
0) is a free lepton propagator. By inserting a complete set of energy eigenstates, A Mq,ℓ can further be written as A Mq,ℓ = Z dφ n h π + | O ( q ) d,β (0) | n ih n | O ( q ) s,α (0) | K + i E n + E ℓ + + E ν ℓ − E K − iε L αβ ( ~p n )+ Z dφ n s h π + | O ( q ) s,α (0) | n s ih n s | O ( q ) d,β (0) | K + i E n s + E ℓ − + E ¯ ν ℓ − E K L αβ ( ~p n s ) (10) ≡ A M, − q,ℓ + A M, + q,ℓ , (11)where the first and second terms on the right-hand sides of Eqs. (10) and (11) are the con-tributions from the regions x < x > | n i and | n s i representnon-strange and strangeness S = 1 intermediate states respectively and in each case φ n and φ n s is the corresponding phase space and a sum over all such states is implied. The denom-inator in the second term on the right-hand side of Eq. (10) is always positive and hencewe omit the − iε . The three-momenta of the charged leptons are fixed by the momentum-conserving δ -functions obtained after the integration over ~x , so that L αβ ( ~p n ) = 12 E + ℓ ¯ u ( p ν ℓ ) γ α (1 − γ )( /p ℓ + − m ℓ ) γ β (1 − γ ) v ( p ¯ ν ℓ ) , (12) L αβ ( ~p n s ) = 12 E − ℓ ¯ u ( p ν ℓ ) γ α (1 − γ )( /p ℓ − + m ℓ ) γ β (1 − γ ) v ( p ¯ ν ℓ ) , (13)where ~p ℓ + = ~p K − ~p ν ℓ − ~p n in Eq.(12) and ~p ℓ − = ~p K − ~p ¯ ν ℓ − ~p n s in Eq.(13). In both cases theenergy of the charged lepton is given by E ℓ ± = p ~p ℓ ± + m ℓ .In deriving Eq. (10) we have used the space-time translation property h f | O ( t, ~x ) | i i = e − i ( E i − E f ) t + i ( ~p i − ~p f ) · ~x h f | O (0 , ~ | i i , (14)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 O ( q ) s,α = ¯ sγ α (1 − γ ) q, O ( q ) d,β = ¯ qγ β (1 − γ ) d. (15)7 † K J π O ( q ) d,β O ( q ) s,α nK + π + t K t π T A T B t time (a) J † K J π O ( q ) d,β O ( q ) s,α n s K + π + t K t π T A T B t time (b) Figure 3. Schematic drawing of the two time orderings in the correlation function. In (a) we have t < t > The principal objective of this paper is to demonstrate how to obtain the expressionon the right-hand side of Eq. (10) from computations of correlation functions on a finiteEuclidean lattice with only exponentially small finite-volume effects. We explain how toachieve this in the following section.
III. A Mq ,ℓ FROM EUCLIDEAN CORRELATION FUNCTIONS
Hadronic matrix elements are obtained in lattice QCD computations from calculations offinite-volume Euclidean correlation functions. In this section we present a detailed discussionof the evaluation of A Mu,ℓ , since the evaluation of A Mc,ℓ is considerably more straightforwardand can readily be deduced from that of A Mu,ℓ (as we briefly explain at the appropriatepoints in the discussion). We consider separately the two time-orderings t < t >
A. The time-ordering t < Consider the finite-volume Euclidean correlation function C ( u ) αβ ( t, ~x ) = X ~x π ,~x K h | J π ( t π , ~x π ) O ( u ) d,β (0 , ~ O ( u ) s,α ( t, ~x ) J † K ( t K , ~x K ) | i e i~p K · ~x K e − i~p π · ~x π , (16)where J † K and J π are interpolating operators for the creation of a kaon and annihilation ofa pion respectively. The correlation function in Eq. (16) describes the creation of a kaon ata large negative time t K , the insertion of the weak operators O ( u ) s,α and O ( u ) d,β at times t and 0respectively, with t < t π ≫
0. This is illustrated in8ig. 3(a). For compactness of notation we suppress the dependence of C ( u ) αβ ( t ) on t K , t π andthe momenta. In this section we show how to obtain the A M, − u,ℓ component of the amplitudefrom 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 correlationfunction 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 C ( u ) αβ ( t, ~x ) = Z K Z π e E K t K E K e − E π t π E π h π ( ~p π ) | O ( u ) d,β (0 , ~ O ( u ) s,α ( t, ~x ) | K ( ~p K ) i . (17)The energies of the kaon and pion, E K and E π respectively, and the matrix elements Z K = h K ( ~p K ) | J † K (0) | i and Z π = h | J π (0) | π ( ~p π ) i can be obtained in the standard way from two-point meson correlation functions using our normalization conventions, e.g. for the finite-volume state | π ( ~p π ) i , h π ( ~p ′ π ) | π ( ~p π ) i = 2 E π (cid:18) L π (cid:19) δ ~p ′ π ,~p π . (18)We then rewrite Eq. (17) as C ( u ) αβ ( t, ~x ) ≡ Z Kπ H E, ( u ) αβ ( t, ~x ) , (19)where Z Kπ = Z K Z π e E K t K E K e − E π t π E π (20)and H E, ( u ) αβ ( t, ~x ) is the Euclidean equivalent of the bilocal hadronic matrix element in Eq. (8)at t < H E, ( u ) αβ ( t, ~x ) = h π + ( p π ) | (cid:2) ¯ u (0) γ β (1 − γ ) d (0) ¯ s ( x ) γ α (1 − γ ) u ( x ) (cid:3) | K + ( p K ) i = X n h π ( p π ) | O ( u ) d,β (0) | n ( p n ) i h n ( p n ) | O ( u ) s,α (0) | K ( p K ) i e i ( ~p K − ~p n ) · ~x e − ( E K − E n ) t , (21)where the sum is over a complete set of non-strange states | n i . The basis of the infinite-volume reconstruction method is that we perform the integral in Eq. (7) using the hadronicmatrix element H E, ( u ) αβ ( t, ~x ) calculated using lattice methods on a finite spatial volume andthe leptonic tensor L E,αβ calculated in an infinite spatial volume which for t < L E,αβ ( x ) = Z d p ℓ + (2 π ) e ( E νℓ + E ℓ + ) t e − i ( ~p ℓ + + ~p νℓ ) · ~x E ℓ + ¯ u ( p ν ℓ ) γ α (1 − γ )( /p ℓ + − m ℓ ) γ β (1 − γ ) v ( p ¯ ν ℓ ) . (22)In order to allow the external kaon and pion to propagate over sufficiently large time intervalsto eliminate excited external states and obtain H E, ( u ) αβ ( t, ~x ) we imagine performing the time9ntegration over the interval ( T A , T B ), where t K ≪ T A ≪ ≪ T B ≪ t π , as illustrated inFig. 3. In this subsection we are considering the contribution from the region t < T A ,
0) and the aim here is to compute A E, − u,ℓ = Z T A dt Z L d x H E, ( u ) αβ ( t, ~x ) L E,αβ ( t, ~x ) , (23)in such a way as to reproduce the first term on the right-hand side of Eq. (10), A M, − u,ℓ , withonly exponentially small finite-volume effects . However, the presence of states | n i in thesum in the second line of Eq. (21) with energies which are smaller than those of the externalstates 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 ) computeddirectly on a finite Euclidean lattice for all t ∈ ( T A ,
0) and a modified procedure must beintroduced. We now explain in some detail the presence of power-law finite-volume effectsand the exponentially growing behaviour with | T A | in Eq. (23) together with our proposedmethod for eliminating them.Using Eqs. (21) and (22) we see that the integration over time (under the sum over | n i and integration over ~p ℓ + ) is given by Z T A dt e ( E n + E ℓ + + E νℓ − E K ) t = 1 E n + E ℓ + + E ν ℓ − E K (cid:2) − e − ( E n + E ℓ + + E νℓ − E K ) | T A | (cid:3) . (24)The difficulty arises because there are states | n i for which E n + E ℓ + + E ν ℓ − E K < T A and power-law finite-volume effects due to the singularity in the denominator of the right-hand side of Eq. (24).This singularity is present in the range of the summation over intermediate states. Thisdemonstrates that it is not possible to achieve the stated aim by performing the integral inEq. (23) using H E, ( u ) αβ ( x ), computed on the lattice over the full time interval ( T A , L E,αβ ( t, ~x ) in Eq. (22). We now discuss the modifications wepropose in order to complete the determination of the first term of the physical amplitude A M, − u,ℓ , i.e. the first term on the right-hand side of Eq. (10), with only exponential finite-volume effects.We separate H E, ( u ) αβ ( t, ~x ) into the contributions from the vacuum and the hadronic inter-mediate states writing H E, ( u ) αβ ( t, ~x ) = H vac αβ ( t, ~x ) + H had , ( u ) αβ ( t, ~x ) , (25) The L suffix in Eq. (23) indicates that the integral is performed over the finite spatial volume. H vac αβ ( t, ~x ) ≡ h π | O ( u ) d,β (0) | ih | O ( u ) s,α (0) | K i e i~p K · ~x e − E K t . (26)The vacuum contribution H vac αβ ( t, ~x ) can be determined with only exponential finite-volumeerrors using the matrix elements h π | O ( u ) d,β (0) | i latt and h | O ( u ) s,α (0) | K i latt determined in latticecomputations of two-point correlation functions in the standard way. We can thereforeobtain the contribution from the purely leptonic intermediate state ( | n i = | i ) to the physicalamplitude in Eq. (10) A vac = h π | O ( u ) d,β (0) | i latt h | O ( u ) s,α (0) | K i latt E ℓ + + E ν ℓ − E K − iε L αβ ( ~ . (27)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 E, ( u ) αβ and H vac αβ inorder to obtain sufficiently precise values of H had , ( u ) αβ .We now consider H had , ( u ) αβ ( t, ~x ) for t <
0, obtained after subtracting the vacuum contri-bution from H E, ( u ) αβ ( t, ~x ). We assume that for some sufficiently large | t s | the hadronic factor H had , ( u ) αβ ( t, ~x ) with | t | ≥ | t s | is dominated by the | π i intermediate state, so that in particular H had , ( u ) αβ ( t s , ~x ) ≃ Z d p π (2 π ) E π h π ( p π ) | O ( u ) d,β (0) | π ( p π ) i h π ( p π ) | O ( u ) s,α (0) | K ( p K ) i × e i ( ~p K − ~p π ) · ~x e − ( E K − E π ) t s , (28)where the ≃ symbol indicates the equality of the two sides of the equation up to excited-state contributions which are assumed to be negligible. Although | t s | is large enough forthe ground state to dominate for | t | ≥ | t s | , it is nevertheless finite so that the finite-volumecorrections in H had , ( u ) αβ ( t s , ~x ) are exponentially suppressed and we have therefore replaced thesum over finite-volume | π i by the infinite-volume phase-space integral.In the original presentation of the IVR method [1], the integration region over t , i.e. t ∈ ( −∞ , 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 theregion t ∈ ( t s ,
0) (see Eq. (29) below). We then show that the corresponding contributionto the physical amplitude A M, − u,ℓ (see Eqs. (10) and (11)) is obtained from I ( s ) + ˜ I ( l ) , where˜ I ( l ) is an appropriately modified contribution from the integration region ( −∞ , t s ). Thehadronic components of I ( s ) and ˜ I ( l ) can both be determined from lattice computations.11he integration over t in the interval ( t s ,
0) is defined by I ( s ) ≡ Z t s dt Z L d x H had , ( u ) αβ ( t, ~x ) L E,αβ ( t, ~x ) ≃ Z t s dt Z ∞ d x H had , ( u ) αβ ( t, ~x ) L E,αβ ( t, ~x ) , (29)since for finite t s the finite-volume effects are exponentially small . Nevertheless I ( s ) doesnot reproduce the corresponding contribution to the Minkowski amplitude, A ( M, − ) u,ℓ definedin Eq. (11). Instead I ( s ) is given by I ( s ) = Z dφ n h π | O ( u ) d,β (0) | n ( ~p n ) ih n ( ~p n ) | O ( u ) s,α (0) | K i E n + E ℓ + + E ν ℓ − E K L αβ ( ~p n ) (cid:2) − e − ( E n + E ℓ + + E νℓ − E K ) | t s | (cid:3) , (30)where a sum over intermediate states | n i is implied. For excited states with E n + E ℓ + + E ν ℓ − E K > e − ( E n + E ℓ + + E νℓ − E K ) | t s | . Thisis not the case however, for the ground-state, | π i , contribution which has to be treateddifferently. (Note that even in this case, the integrand on the right-hand side of Eq. (30) hasno singularity at E π + 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 i contribution from I ( s ) in (30), andreplace it by the corresponding term (i.e. the term with | n i = | π i ) in Eq. (10). To this endwe define the quantity ˜ I ( l ) by ˜ I ( l ) = Z d p π (2 π ) E π h π | O ( u ) d,β (0) | π ( ~p π ) ih π ( ~p π ) | O ( u ) s,α (0) | K i L αβ ( ~p π ) × (cid:20) E π + E ℓ + + E ν ℓ − E K − iε − − e − ( E π + E ℓ + + E νℓ − E K ) | t s | E π + E ℓ + + E ν ℓ − E K − iε (cid:21) , (31)= Z d p π (2 π ) E π h π | O ( u ) d,β (0) | π ( ~p π ) ih π ( ~p π ) | O ( u ) s,α (0) | K i L αβ ( ~p π ) × e − ( E π + E ℓ + + E νℓ − E K ) | t s | E π + E ℓ + + E ν ℓ − E K − iε , (32)where E π = p m π + ~p π . In the second line of Eq. (31) the first term is the Minkowskicontribution from the | π i intermediate state (see Eq. (10)) and the second term is the | π i contribution to I ( s ) (see Eq. (30)). Since in this second term there is no singularity, The suffix ∞ in Eq. (29) indicates that the integral is performed in infinite volume. The tilde on ˜ I ( l ) is introduced to denote the fact that ˜ I ( l ) is not simply the integral over the region t < t s .
12t is possible and also convenient to add − iε to the denominator. In this way we removethe unphysical contribution in I ( s ) and replace it with the missing term in the physicalamplitude. Combining Eqs. (28) and (32) we have˜ I ( l ) = Z d x H had , ( u ) αβ ( t s , ~x ) ˜ L αβ ( t s , ~x ) , (33)where ˜ L αβ ( t s , ~x ) is defined as˜ L αβ ( t s , ~x ) = Z d p ℓ + (2 π ) e − i ( ~p ℓ + + ~p νℓ ) · ~x L αβ ( ~p π ) e − ( E ℓ + + E νℓ ) | t s | E π + E ℓ + + E ν ℓ − E K − iǫ . (34)In the integrand of Eq. (34), ~p π = ~p K − ~p ℓ + − ~p ν ℓ and E π = p ~p π + m π . Thus we see thatthe quantities I ( s ) and ˜ I ( l ) can be approximated using the quantities H had αβ ( t, ~x ) calculatedfor | t | ≤ | t s | in a lattice computation as inputs. The finite-volume effects induced by thisapproximation are exponentially suppressed.Combining the contribution from the vacuum intermediate state in Eq.(27), with thosefrom the hadronic intermediate states in Eqs. (29) and (33), we obtain the contribution tothe physical amplitude in Eq.(10) from the region x < A M, − u,ℓ = A vac + I ( s ) + ˜ I ( l ) . (35)The contribution A M, − c,ℓ is much more straightforward to evaluate as the intermediatestates now have charm quantum number C = 1, and so have larger energies than m K . Inthis case one simply performs the integral A E, − c,ℓ = Z T A dt Z d x H E, ( c ) αβ ( t, ~x ) L E,αβ ( t, ~x ) (36)with the hadronic matrix elements computed directly in lattice QCD. In this case A E, − c,ℓ = A M, − c,ℓ up to exponential finite-volume effects. B. The time-ordering t > We now consider the case t > A M, + u,ℓ , for which the elimination ofthe power-law finite-volume effects is a little more straightforward but which neverthelesscontains a new subtlety. We start by following the same steps as for t <
0, relating the13uclidean correlation function illustrated in Fig. 3(b) to the bilocal hadronic matrix element: C ( u ) αβ ( t, ~x ) = X ~x π ,~x K h | J π ( t π , ~x π ) O ( u ) s,α ( t, ~x ) O ( u ) d,β (0 , ~ J † K ( t K , ~x K ) | i e i~p K · ~x K e − i~p π · ~x π = Z Kπ H E, ( u ) αβ ( t, ~x ) , (37)where Z Kπ is given in Eq. (20) and for t > H E, ( u ) αβ ( t, ~x ) = h π + ( p π ) | (cid:2) ¯ s ( x ) γ α (1 − γ ) u ( x ) (cid:3) (cid:2) ¯ u (0) γ β (1 − γ ) d (0) (cid:3) | K + ( p K ) i = X n s h π ( p π ) | O ( u ) s,α (0) | n s ( p n s ) ih n s ( p n s ) | O ( u ) d,β (0) | K ( p K ) i e i ( ~p ns − ~p π ) · ~x e − ( E ns − E π ) t , (38)and the sum is over the finite-volume multi-hadron strangeness S = 1 states | n s i . Thereare therefore no on-shell intermediate states | n s i and consequently there is no unphysicalexponentially growing behaviour in T B in the integral over t . Nevertheless there is a subtletywhich must be taken into account. Consider the Type 1 diagram in Fig. 1. Although wehave drawn the two loops as only being connected by a lepton propagator, it is to beunderstood that gluonic and vacuum polarization effects, although not drawn explicitly,are also implicitly included. However, the functional integral over the gluon and sea-quarkfields contains contributions in which the strong-interaction effects are restricted to each loopseparately and do not connect the two loops. For these contributions there is no suppressionin | ~x | when evaluating H E, ( u ) αβ ( t, ~x ) and they need to be treated separately in an analogousway to the vacuum contribution in Eq. (26). We call these contributions disconnected .Analogously to Eq. (25) we write for t > H E, ( u ) αβ ( t, ~x ) = H disc αβ ( t, ~x ) + H conn , ( u ) αβ ( t, ~x ) (39)where the labels disc and conn represent the disconnected and connected contributions re-spectively. The disconnected contribution has the same form as H vac ( t, ~x ), but now t > H disc αβ ( t, ~x ) ≡ h | O ( u ) s,α (0) | K i h π | O ( u ) d,β (0) | i e i~p K · ~x e − E K t . (40)Each of the two local matrix elements in Eq. (40) can be computed independently as anaverage over the gauge configurations with only exponential finite-volume corrections. Wewrite the corresponding contribution to the amplitude in the form A disc u,ℓ = h π | O ( u ) d,β (0) | ih | O ( u ) s,α (0) | K i ( E K + E π ) + E ¯ ν ℓ + E ℓ − − E K L ( ~p K + ~p π ) (41) In lattice QCD literature “disconnected” frequently refers to diagrams in which quark loops are onlyconnected by gluons. We stress that our use of “disconnected” here is different and denotes diagrams withno strong interactions between the quark loops.
14o demonstrate that it is the disconnected contribution to A M, + u,ℓ in Eq. (10). Note that forthe disconnected contribution p n s = p K + p π .For the connected contribution, there are two important points to note, both resultingfrom the observation that the intermediate states | n s i all have energies which are larger thanthose 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 thateven 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 ( t, ~x )with the corresponding leptonic tensor L E,αβ ( t, ~x ), where for t > L E,αβ ( x ) = Z d p ℓ − (2 π ) e − ( E νℓ − E ℓ − ) t e i ( ~p ℓ − − ~p νℓ ) · ~x E ℓ − ¯ u ( p ν ℓ ) γ α (1 − γ )( /p ℓ − + m ℓ ) γ β (1 − γ ) v ( p ¯ ν ℓ ) . (42)In this subsection we are considering the contribution from the region t >
0, so the range ofintegration is (0 , T B ) and we arrive at the following contribution to the decay amplitude: A conn u,ℓ = Z T B dt Z L d x H conn , ( u ) αβ ( t, ~x ) L E,αβ ( t, ~x ) . (43)The contribution A conn u,ℓ is equal to the connected contribution to A M, + u,ℓ up to exponentiallysuppressed terms in the volume and in T B .Combining Eqs. (41) and (43) we obtain A M, + u,ℓ = A disc u,ℓ + A conn u,ℓ , (44)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 hencethere is no disconnected contribution.The discussion of the contribution from the region t > S = 1 intermediate states implies that there are nopower-law finite-volume effects arising from on-shell intermediate states. In addition, theuse of the infinite-volume leptonic tensor (42) in the integration in Eq. (43) avoids power-lawfinite-volume effects which would arise due to the factor of 1 / E l − in the difference betweena finite-volume sum over ~p ℓ − and the corresponding infinite-volume integration. The above15iscussion is a particular illustration of how to avoid power-law finite-volume corrections inthe second term on the right-hand side of Eq. (6), which applies to general processes with amassless (or almost massless) leptonic propagator. C. Summary
In summary therefore, we propose to calculate the decay amplitude A Mu,ℓ , using the fol-lowing form where all the hadronic quantities can be obtained from lattice simulations withexponential finite-volume effects: A Mu,ℓ = h π | O ( u ) d,β (0) | i latt h | O ( u ) s,α (0) | K i latt E ℓ + + E ν ℓ − E K − iε L αβ ( ~ Z t s dt Z L d x H had , (u) , latt αβ ( t, ~x ) L E,αβ ( t, ~x )+ Z L d x H had , (u) , latt αβ ( t s , ~x ) ˜ L αβ ( t s , ~x ) + h π | O ( u ) d,β (0) | i latt h | O ( u ) s,α (0) | K i latt ( E K + E π ) + E ¯ ν ℓ + E ℓ − − E K L αβ ( ~p K + ~p π )+ Z T B dt Z L d x H conn , (u) , latt αβ ( t, ~x ) L E,αβ ( t, ~x ) . (45)The first three terms on the right-hand side of Eq. (45) come from the region t < H vac αβ in Eq. (26) and the con-tributions from (ii) I s in Eq. (25) and (iii) ˜ I l in Eqs. (33) and (34). The final two terms inEq. (45) come from the region region t > latt underlines theobservation that the local and bilocal matrix elements are calculable in a lattice computationwith only exponential finite-volume effects.The focus of this paper has been on the demonstration of the presence of power-law finite-volume effects in the amplitudes A Mu,ℓ for rare-kaon decays K → πν ℓ ¯ ν ℓ and the presentationof a proposed method to eliminate them. Such effects are absent from A Mc,ℓ , where theintermediate states carry C = 1 charm quantum number and Type 1 diagrams do notcontribute. There are therefore no vacuum or disconnected contributions, nor any fromother intermediate states lighter then the kaon and so H E, ( c ) αβ ( t, ~x ) as computed on a finiteEuclidean lattice can be used directly in the integral R T B T A dt R L d x H E, ( c ) αβ ( t, ~x ) L E,αβ ( t, ~x ) toobtain A Mc,ℓ .An important point to note is that it still requires further investigations to extend themethod developed in Ref. [1] and in this work to multi-hadron intermediate states with16nergies smaller than the external ones, as such states induce branch cuts which cannot besimply 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. Usingthe rare K + → π + ν ℓ ¯ ν ℓ decay as an example, we show that the power-law finite-volumeeffects induced by the massless electron can safely be removed using the form in Eq. (45) inlattice computations. A similar approach has been applied to the amplitude for neutrinolessdouble- β decay in which there is the propagator of a massless neutrino [23]. We are alsoperforming exploratory studies with the aim of extending the method to the evaluationof electromagnetic corrections to leptonic and semileptonic decays [56] and applying it innumerical lattice QCD calculations. ACKNOWLEDGMENTS
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