Decay amplitudes to three hadrons from finite-volume matrix elements
PPrepared for submission to JHEP
Decay amplitudes to three hadrons from finite-volumematrix elements
Maxwell T. Hansen , Fernando Romero-L´opez , and Stephen R. Sharpe Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edin-burgh, Edinburgh EH9 3FD, UK IFIC, CSIC-Universitat de Val`encia, 46980 Paterna, Spain Physics Department, University of Washington, Seattle, WA 98195-1560, USA
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We derive relations between finite-volume matrix elements and infinite-volumedecay amplitudes, for processes with three spinless, degenerate and either identical or non-identical particles in the final state. This generalizes the Lellouch-L¨uscher relation for two-particle decays and provides a strategy for extracting three-hadron decay amplitudes usinglattice QCD. Unlike for two particles, even in the simplest approximation, one must solveintegral equations to obtain the physical decay amplitude, a consequence of the nontrivialfinite-state interactions. We first derive the result in a simplified theory with three identicalparticles, and then present the generalizations needed to study phenomenologically relevantthree-pion decays. The specific processes we discuss are the CP-violating K → π weakdecay, the isospin-breaking η → π QCD transition, and the electromagnetic γ ∗ → π amplitudes that enter the calculation of the hadronic vacuum polarization contribution tomuonic g − a r X i v : . [ h e p - l a t ] F e b ontents A PV K π A PV K π to the physical decay amplitude 142.5 Isotropic approximation 17 γ ∗ → π η → π K → π A = A †
32B Alternative partial derivation following Lellouch-L¨uscher method 34C Relations between three-pion states 37D Formalism for K → π decays 39 – 1 – Introduction
The theoretical formalism for extracting three-hadron scattering amplitudes using latticeQCD has grown apace in recent years [1–20], and applications to simple systems have beensuccessfully undertaken [21–31]. In all such studies, the basic approach is to extract thespectrum of three-hadron states in a finite spatial volume, and to use this information,by means of general relations, to constrain the infinite-volume scattering amplitudes. Inparticular, the spectrum of three-pion and three-kaon states of maximal isospin has beenobtained in multiple calculations with different geometries, and with many values of to-tal momentum in the finite-volume frame. In the following we abbreviate the latter as“different frames”.A natural extension of this work is to consider electroweak transitions to three particles,e.g. the K → π decay. Although challenging, one can now conceive of undertaking alattice calculation of finite-volume matrix elements of the form h π, L |H W | K, L i , where H W is the weak Hamiltonian density, and h π, L | is a finite-volume state whose energyand momentum are tuned to match that of the initial kaon. Here we restrict attentionto a cubic, periodic spatial volume, and L denotes the periodicity (i.e. the box length) ineach of the three spatial dimensions. The question is then how to convert knowledge ofseveral such matrix elements (with different volumes and frames) into information on thecorresponding infinite-volume decay amplitude, including its dependence on the momentaof the three outgoing pions. In this work we answer this question, providing the formalismfor a first-principles calculation of the amplitudes for K → π and related decays.The corresponding problem for two-particle K → ππ decays was solved in a seminalpaper by Lellouch and L¨uscher (LL) [32], where it was shown, for the case of a kaonat rest in the finite-volume frame, that the relation between the squared finite-volumematrix element and the magnitude squared of the infinite-volume decay amplitude is anoverall multiplicative factor, the LL factor. This result was subsequently generalized inmany ways [33–49], with the most important extension for our purposes being the workof refs. [42, 44], in which an alternative and more general formalism was developed forcalculating the LL factors for arbitrary 1 → γ ∗ → π as well as the isospin breaking transition η → π . The former process is relevantfor quantifying finite-volume corrections to the hadronic-vacuum-polarization contributionto ( g − µ arising from the isoscalar part of the photon, along the same lines that γ ∗ → ππ is used for the isovector part as described in refs. [50, 51].The remainder of the paper is organized into two parts. In the first, contained insection 2, we derive the necessary formalism for decays to states containing three identicalparticles. To do so, we first summarize the three-particle scattering formalism in section 2.1.Then, in section 2.2, we derive the relation between the finite-volume matrix elementsand a scheme-dependent intermediate infinite-volume quantity, A PV K π . In section 2.3, weexplain how to systematically expand A PV K π about threshold based on symmetries, followingwhich we explain how A PV K π can be connected to the physical decay amplitude via integralequations (section 2.4). To conclude the discussion for identical particles, in section 2.5 weconsider the isotropic approximation in which a more explicit and much simpler expressioncan be given, results from which we illustrate with numerical examples.The second part of the paper, contained in section 3, concerns the case of decays tothree pions in isosymmetric QCD. We begin, in section 3.1, by presenting the appropriategeneralization of the formalism. We then consider the processes γ ∗ → π , η → π and K + → π in sections 3.2, 3.3 and 3.4, respectively. We present our conclusions and outlookin section 4.We included four appendices. Appendix A derives a technical result needed in themain text. Appendix B presents an alternative derivation of the relation obtained insection 2.2 using the method of Lellouch and L¨uscher. Appendix C collects relevant resultsconcerning the isospin decomposition of three-pion states. Finally, appendix D presentsthe generalization of the results of section 3.4 to the decays of neutral kaons.While this work was in preparation, a formalism for determining three-particle decayamplitudes to identical scalars in non-relativistic effective field theory (NREFT) was madepublic [52]. The authors considered only leading-order (non-derivative) couplings for thedecay and scattering vertices. The formalism presented here goes beyond that of ref. [52]in several ways: (i) it is valid for nonidentical particles, and thus for the three-pion system;(ii) no approximations concerning the couplings are made, and no truncation in angularmomenta is required; (iii) it is valid for generic moving frames; (iv) it is derived in a fullyrelativistic formalism. We include additional brief comments on the relationship betweenthe approaches in section 2.5. We consider first a simple theory consisting of two real scalar fields, the “kaon” K and“pion” φ , both having an associated Z symmetry that conserves particle number modulo2. Aside from this symmetry constraint, the interactions between these fields are arbitrary.The physical masses of the particles are m K and m π , respectively, and satisfy3 m π < m K < m π . (2.1)– 3 – igure 1 : Examples of the underlying diagrams describing the K → π decay and thecorresponding finite-volume matrix element. The left-most diagram represents a localone-to-three transition with only exponentially suppressed finite-volume effects. Bycontrast the middle two diagrams have power-like L dependence due to the on-shellintermediate states, indicated by the vertical dashed line. Finally, the rightmost diagramindicates a strong-interaction induced dressing to the weak vertex. All such interactions,as well as all dressing on the incoming and outgoing vertices are included in theformalism.Both the kaon and the pion are stable particles in this theory. To induce decays, we addan interaction Hamiltonian, suggestively denoted H W , that violates both Z symmetries,and is chosen to couple the kaon to the odd-pion-number sector. A simple example of therequired Hamiltonian density is H W ( x ) = c W K ( x ) φ ( x ) , (2.2)but we need not commit to a particular form; all that matters is that the interaction islocal and has the correct quantum numbers. We treat c W as small, such that we needonly work to first order in this parameter. Decays of the kaon to even numbers of pions,although kinematically allowed for two pions and possibly also for four pions, are forbiddenby symmetries. The potential decay K → π is kinematically disallowed for the mass rangein eq. (2.1).To understand the intuition behind the following analysis, consider a diagrammaticrepresentation of the K → π amplitude, to leading order in c W but to all orders in the Z preserving interactions. As we illustrate in figure 1, in such an expansion, the onlyon-shell intermediate states are those involving three pions. Arbitrary virtual interactionsbetween the incoming (dressed) kaon and the final-state pions are allowed, but do notlead to on-shell intermediate states. One can think of such virtual loops as resulting frompropagation that is localized near H W , and they lead to an effective renormalization of thebare coupling c W . This is the physics that one expects to be captured by a calculationof the matrix element in a finite volume. On the other hand, the final-state interactions,which involve long distance, near on-shell propagation, will be mangled in finite volume,and it is these distortions that are corrected by the formalism developed in this work.As stressed in the introduction, throughout this article we take finite volume to meana cubic box of side L with periodic boundary conditions on the fields K and φ . Thisrestricts momenta to lie in the finite-volume set p = n (2 π/L ), where n is a three-vectorof integers. In our derivation, we drop volume-dependent terms that fall as exp( − m π L )or faster. For typical volumes used in actual simulations, these exponentially-suppressedterms are much smaller than the power-law volume dependence that we keep. As is quite– 4 –tandard in these types of analyses, we take the temporal extent to be infinite. We alsowork in a continuum effective field theory with the assumption that the discretizationeffects entering a numerical lattice QCD calculation using these methods are small andincluded in the systematic uncertainties of the finite-volume matrix elements and energies. We make extensive use of the formalism developed to relate the finite-volume spectrum ofthree-particle states to the infinite-volume two- and three-particle scattering amplitudes.A general feature of the formalism is that it involves two steps. In the first, the finite-volume spectrum is related to an intermediate, unphysical infinite-volume three-particleK matrix ( K df , in the approach of this paper), while, in the second, the K matrix isrelated to the scattering amplitudes by solving integral equations. This two-step procedurecarries over naturally to the extension we develop here, with an intermediate, unphysicaldecay amplitude ( A PV K π below) determined from the finite-volume matrix elements, andthe physical decay amplitude then obtained from A PV K π via integral equations.As noted above, we use the approach developed in refs. [3, 4], and our aim in thissubsection is to recall its essential results. One important feature of this formalism forthe case of identical particles is that the intermediate three-particle K matrix, K df , , issymmetric under separate interchanges of initial and final momenta. This symmetry willcarry over to the intermediate one-to-three amplitude, A PV K π , that arises here. The central result of ref. [3] concerns the following three-particle finite-volume corre-lator: C M L ( E, P ) = Z ∞−∞ dx Z L d x e i ( Ex − P · x ) h | T σ ( x ) σ † (0) | i , (2.3)where the superscript indicates that the underlying correlation function is evaluated inMinkowski space, and T stands for time-ordering. Here σ ∼ φ couples to three pions,but is otherwise an arbitrary operator possibly containing derivatives. Assuming the Z symmetry described above, the kinematic range of interest is m π < E ∗ = p E − P < m π . (2.4)Within this range, it is shown in ref. [3] that the difference between the finite- and infinite-volume versions of this correlator takes the form C M L ( E, P ) − C M ∞ ( E, P ) = iA F − + K df , A . (2.5)Here all quantities have matrix indices { k‘m } , with A a row vector, A a column vector,while F and K df , are matrices. The index k is shorthand for the momentum k of one ofthe three particles, referred to as the spectator. The values of this index are drawn from It is also possible to derive a simpler (though equivalent) version of the three-particle formalism thatinvolves an asymmetric K matrix [16] or the asymmetric R matrix [17]. We do not use these results,however, as the resulting renormalized decay amplitude is less constrained by symmetry, leading to a morecomplicated parametrization. We are following the notation of ref. [18] since we use results from this work in the physical K → π case below. The notation differs slightly from that of refs. [3, 4]. – 5 –he finite-volume set. The indices ‘m give the decomposition into spherical harmonics ofthe angular dependence of the nonspectator pair, when boosted to the pair center-of-momentum frame (CMF). The sum over k is cut off by a smooth function contained in F and G , while the sum over ‘ is not cut off at this stage. All quantities are also implicitfunctions of E and P , with F also depending on L . F is given by F = 12 ωL " F − F
11 + M ,L G M ,L F , M − ,L = K − + F , (2.6)where ω , F , G , and K are matrices defined in ref. [3], and (with the exception of ω ) arealso implicit functions of E , P and, in the case of F and G , also L . The only detail weneed to know now is that F , G and K pick out one of the three particles as the spectator,so that these are intrinsically asymmetric quantities, an asymmetry that is inherited by F .By contrast, the endcaps A and A , as well as K df , , are intrinsically symmetric quantitiesthat are being expressed in terms of asymmetric variables.The endcaps play an important role in the determination of the decay amplitude, as wewill see below. The derivation of ref. [3] defines these quantities by an all-orders constructiveprocedure, the key feature of which is that it involves loop integrals regulated by a principalvalue (PV) scheme. Thus one can think of the endcaps as, roughly speaking, the sum of allvacuum to three-pion diagrams in which only the short distance contributions from loopsare kept. The long distance part, which leads to final state interactions, and the associatedcomplex phases, is removed by the use of the PV prescription. We stress, however, thatthis qualitative interpretation of the endcaps is not needed to carry through the derivationdescribed below. A technical result that is important below is that, if the creation andannihilation operators in C M L are related by hermitian conjugation, then A = A † . Weprove this fact in appendix A.From the result (2.5) for the correlator, the quantization condition is seen to bedet( F − + K df , ) = 0 . (2.7)As written here, this equation ignores the residual symmetries of the finite-volume systemthat can be used to block diagonalize the matrix F − + K df , . The relevant symmetrygroup depends on the value of P . For the purposes of this work it suffices to note that foreach group one can identify a set of irreducible representations (irreps), denoted by Λ, andfor each irrep a row index, denoted µ . Each set of Λ µ then corresponds to a block so thateq. (2.7) breaks into a set of independent quantization conditions of the formdet Λ µ (cid:2) P Λ µ · ( F − + K df , ) · P Λ µ (cid:3) = 0 , (2.8)where P Λ µ projects out a given irrep and row.To give the definition of P Λ µ , we introduce R as a unitary matrix with the propertythat R † · ( F − + K df , ) · R , (2.9)is block diagonal with one block corresponding to each possible value of Λ µ . The con-struction of this matrix is a standard group-theoretic exercise, described, for example, in– 6 –ef. [12]. We then define e P Λ µ as a diagonal matrix of ones and zeroes that annihilates allblocks besides that corresponding to the target irrep and row. Finally we define P Λ µ = R · e P Λ µ · R † , (2.10)which projects to the target irrep while preserving the { k‘m } matrix space. The matrix P Λ µ · ( F − + K df , ) · P Λ µ will always have vanishing determinant, since the projection amountsto setting all eigenvalues with eigenvectors outside the Λ µ subspace to zero. For this reason,we include the Λ µ subscript on the determinant, indicating that this is evaluated only overthe nontrivial subspace.We stress that eqs. (2.7)-(2.10) are formal relations involving infinite-dimensional ma-trices and must be truncated in practice. This is done by assuming that the two- andthree-particle interactions vanish above some value of ‘ . For a given P , Λ µ and L , thisequation will be satisfied for a discrete set of values of E , which we label E Λ n ( P , L ) andoften abbreviate as E n .The final result we need concerns the finite-volume three-particle scattering amplitude, M ,L , defined in ref. [4]. This is the finite-volume version of the amputated, connectedinfinite-volume amplitude M . What will be important here is how M ,L can be obtainedfrom C L by an amputation procedure discussed in refs. [4, 9]. The idea is that, as we movein from the endcaps we may encounter a factor of F , and this sets the three particles onshell. An unsymmetrized form of the scattering amplitude, M ( u,u )3 ,L , is then obtained bykeeping terms in C L that have at least two factors of F —one for incoming and the otherfor outgoing particles—and dropping all but the contributions between the two outermost F s. In fact, this includes some disconnected three-particle diagrams that must also bedropped. In a final step, the resulting connected amplitude is symmetrized.We now explain the resulting procedure in detail. We first remove the factors of i , A and A , and rewrite the result as F − + K df , = F − F
11 + K df , F K df , F , (2.11)= F ωL − F ωL
11 + M ,L G M ,L ωL F ωL − F
11 + K df , F K df , F . (2.12)We drop the first term on the right-hand side as it contains a single F , and complete theamputation by multiplying by the inverse of iF/ (2 ωL ) on both ends. This leads to11 + M ,L G M ,L ωL + (cid:18) F ωL (cid:19) − F
11 + K df , F K df , F (cid:18) F ωL (cid:19) − . (2.13)Expanding out the first term in a geometric series, the leading contribution, M ,L ωL , isdisconnected and thus dropped, leading to the final result for M ( u,u )3 ,L , M ( u,u )3 ,L = D ( u,u ) + L ( u ) L
11 + K df , F K df , R ( u ) L , (2.14) We remove the i since the result of removing A and A alone is i M ,L . – 7 – ( u,u ) = −
11 + M ,L G M ,L G M ,L ωL , (2.15) L ( u ) L = (cid:18) F ωL (cid:19) − F = 13 −
11 + M ,L G M ,L F , (2.16) R ( u ) L = F (cid:18) F ωL (cid:19) − = 13 − F M ,L
11 + G M ,L . (2.17)The full amplitude is then given by M ,L = S n M ( u,u )3 ,L o , (2.18)where the symmetrization operator is defined in ref. [4], and discussed in more detail inref. [18]. We also note that, following ref. [4], M can be obtained from M ,L by takingthe L → ∞ limit in which poles in F and G are shifted from the real axis by the usual i(cid:15) prescription. The approach we follow is adapted from that of ref. [44], and also draws from ref. [42]. Thematrix elements that can be determined in finite volume are h E n , P , Λ µ, L |H W (0) | K, P , L i . (2.19)Here | K, P , L i is a single kaon state, with momentum P drawn from the finite-volume set,while | E n , P , Λ µ, L i is a three-particle finite-volume state with the same momentum P ,and with energy E n . It transforms in the irrep Λ and in the row µ of that irrep. Bothstates are normalized to unity. The energy of the kaon state is E K ( P ) = ( P + m K ) / ,with no volume dependence aside from exponentially suppressed effects. The energy of thethree-particle state, by contrast, has a power-law dependence on L . In order to obtain amatrix element related to the infinite volume decay amplitude, L should be tuned so that E Λ n ( P , L ) = E K ( P ), implying that four-momentum is conserved. There can be manysuch matrix elements, each corresponding to a different finite-volume level, with a differentchoice of L needed in each case.It is useful to sketch how the matrix elements (2.19) would be determined from asimulation of the theory, carried out necessarily in Euclidean space. We idealize the setup byassuming an infinite Euclidean time direction, and work with correlators fully transformedto momentum space. The three correlators that are needed are C K,L ( P ) = Z K Z ∞−∞ dx Z L d x e − iP x h | T E K ( x , x ) K (0) | i , (2.20) C π,L ( P ) = Z ∞−∞ dx Z L d x e − iP x h | T E A π ( x , x ) A † π (0) | i , (2.21) C K π,L ( P ) = Z ∞−∞ dx Z L d x e − iP x h | T E A π ( x , x ) B K π (0) | i , (2.22) If one were interested in the matrix element (2.19) in which H W (0) inserted energy, then the subsequentderivation would still hold in an appropriate kinematic regime. The analysis can also be straightforwardlygeneralized to the case where H W (0) inserts momentum. – 8 –here P = ( P , P ) and x = ( x , x ) are Euclidean four-vectors, whose inner product isdenoted P x , and T E denotes Euclidean time ordering.The correlator C K,L determines the normalization constant Z K . It should be chosenso that lim P → iE K ( P ) ( P + m K ) C K,L ( P ) = 1 , (2.23)which implies that the renormalized kaon field satisfies |h K, P , L | p Z K K (0) | i| = 1 p E K ( P ) L . (2.24)The correlator C π,L determines the coupling of the operator A π to the finite-volumestates | E n , P , Λ µ, L i . Here, A π is an operator chosen to couple to three-pion states in aparticular row of the desired finite-volume irrep. In practice, A π will involve pion fieldswith phase factors such that they have appropriate relative momenta, and thus will becomplex. Other details of the operator are not relevant in the following. The correlatorwill consist of a sum of poles, and we pick out the contribution of the desired state fromthe residue R π ( E n , P , Λ µ, L ) ≡ lim P → iE n ( E n + iP ) C π,L ( P ) = L |h |A π (0) | E n , P , Λ µ, L i| . (2.25)The final correlator, C K π,L , can then be used to determine the desired matrix element.Here, following ref. [44], we use a composite operator B K π that both creates the initialkaon (implicitly having momentum P ) and includes the action of the weak Hamiltonian, B K π ( x ) = p Z K lim P → iE K ( P ) h P + m K i Z d y e iP y H W ( x ) K ( x + y ) , (2.26)where P = ( P , P ). The limit picks out the incoming kaon pole, while the factor of P + m K amputates the kaon propagator. Including all factors we obtain R K π ( E n , P , Λ µ, L ) ≡ lim P → iE n ( E n + iP ) C K π,L ( P ) , (2.27)= L h |A π (0) | E n , P , Λ µ, L ih E n , P , Λ µ, L |H W (0) | K, P , L i q E K ( P ) L . (2.28)Without loss of generality, we can choose the phase of the operator and state such that h |A π (0) | E n , P , Λ µ, L i is real and positive. Then, combining eqs. (2.25) and (2.28), weobtain h E n , P , Λ µ, L |H W (0) | K, P , L i q E K ( P ) L = R K π ( E n , P , Λ µ, L ) p L R π ( E n , P , Λ µ, L ) . (2.29)This matrix element will only be nonvanishing if Λ and µ are chosen to match the trans-formation properties of H W (0) | K, P , L i . If not, then the correlator C K π,L ( P ) and the Note that a subtlety arises here due to the fact that the operator B K π is not local in time. This is notan issue because the P → iE K ( P ) limit is dominated by early y so that the K ( x + y ) operator is orderedfar to the right. Thus only one time-ordering arises, that with the intermediate finite-volume states thatwe analyze explicitly. – 9 –esidue R K π will vanish. For a rotationally invariant H W , only the trivial irrep of the lit-tle group for momentum P will appear (or else the corresponding parity conjugate irrep),but we develop the formalism allowing for more general cases.We now evaluate this ratio using the results from the previous subsection. To do so wefirst generalize the correlator C L of eq. (2.3) by replacing σ and σ † with general operators A and B that couple the vacuum to three-pion states, but are, in general, unrelated to eachother: C M AB,L ( E, P ) = Z ∞−∞ dx Z L d x e i ( Ex − P · x ) h | T A ( x ) B (0) | i . (2.30)The analysis of ref. [3] remains valid for C M AB,L , since it requires only that the allowed on-shell intermediate states involve three pions. Thus the expression (2.5) still holds, exceptthat the endcaps A and A are replaced by new quantities that we call, respectively, A PV and B PV . The superscript is a reminder that loops in these quantities are defined using aPV prescription.We next do a Wick rotation ( x → − ix ) on the underlying correlation function, sothat it is evaluated in Euclidean space-time. This results in C M AB,L ( E, P ) = − iC AB,L ( P ) (cid:12)(cid:12) P = iE , (2.31) C AB,L ( P ) = Z ∞−∞ dx Z L d x e − iP x h T E A ( x ) B (0) i , (2.32)where again P = ( P , P ). It follows that C AB,L can be written C AB,L ( P ) = C AB, ∞ ( P ) − A PV F − + K df , B PV , (2.33)where now A PV , F , K df , and B PV are written as functions of P by setting E = − iP .The poles now lie on the imaginary axis, at the positions P = iE n , where E n is a solutionof the quantization condition eq. (2.7).The reason for these manipulations is that the two correlators that enter into theexpression (2.29) for the desired matrix element, C π and C K π , are in the class for whicheq. (2.33) holds. In particular, we can use the results of ref. [3] to write these correlatorsas C π,L ( P ) = C π, ∞ ( P ) − A PV3 π F − + K df , A PV † π , (2.34) C K π,L ( P ) = C K π, ∞ ( P ) − A PV3 π F − + K df , A PV K π . (2.35)In eq. (2.34) we are using the result, demonstrated in appendix A, that if the source andsink operators are related by hermitian conjugation, then the same holds for the endcapfactors. Note that this only holds because the latter are defined with the PV prescription.We next evaluate the residues that enter eq. (2.29). Since the infinite-volume correla-tors and the endcaps are smooth, infinite-volume functions, L -dependent poles only arisefrom the zero eigenvalues in F − + K df , . The required residues are thus R Λ µ (cid:0) E Λ n , P , L (cid:1) = lim P → iE Λ n − ( E Λ n + iP ) P Λ µ · F − + K df , · P Λ µ , (2.36)– 10 –here the minus sign is for later convenience, and E Λ n is one of the finite-volume three-pionenergies for the given choice of P , Λ and L . R Λ µ is a matrix in the { k‘m } space, whichcan be evaluated explicitly given expressions for K (contained in F ) and K df , . The ideahere is that these quantities have been previously determined (or, more realistically, con-strained within some truncation scheme) by using the two- and three-particle quantizationconditions applied to the spectrum of two- and three-particle states.An important property of R Λ µ is that it has rank one. This is because only one of theformally infinite tower of eigenvalues of P Λ µ · ( F − + K df , ) · P Λ µ will vanish for a givenfinite-volume energy E Λ n ( P , L ). Denoting the relevant eigenvalue by λ ( E, P , Λ µ, L ) andthe corresponding normalized eigenvector by e ( E, P , Λ µ, L ), one finds R Λ µ (cid:0) E Λ n , P , L (cid:1) = (cid:18) ∂λ ( E, P , Λ µ, L ) ∂E (cid:12)(cid:12)(cid:12)(cid:12) E = E Λ n ( P ,L ) (cid:19) − e ( E, P , Λ µ, L ) e † ( E, P , Λ µ, L ) . (2.37)This rank one property of R Λ µ was first described in the two-particle case in refs. [42, 44].As is discussed, e.g. in refs. [10, 12, 38], the eigenvalue must satisfy the inequality (cid:18) ∂λ ( E, P , Λ µ, L ) ∂E (cid:12)(cid:12)(cid:12)(cid:12) E = E Λ n ( P ,L ) (cid:19) − > , (2.38)Thus, defining v ( E Λ n , P , Λ µ, L ) ≡ (cid:18) ∂λ ( E, P , Λ µ, L ) ∂E (cid:12)(cid:12)(cid:12)(cid:12) E = E Λ n ( P ,L ) (cid:19) − / e ( E, P , Λ µ, L ) , (2.39) R Λ µ can be written as a simple outer product R Λ µ ( E Λ n , P , L ) = v ( E Λ n , P , Λ µ, L ) v † ( E Λ n , P , Λ µ, L ) . (2.40)Since F − + K df , is a real, symmetric matrix (assuming that we use real spherical harmon-ics), the elements of each v are relatively real, with only the overall phase undetermined.Using these results, we can immediately evaluate the required residues, obtaining R π ( E Λ n , P , Λ µ, L ) = | A PV3 π v | , (2.41) R K π ( E Λ n , P , Λ µ, L ) = ( A PV3 π v )( v † A PV K π ) , (2.42)where v is an abbreviation for v ( E Λ n , P , Λ µ, L ). All quantities on the right-hand side are(implicitly) evaluated at P = ( P , iE Λ n ), with E Λ n = E K ( P ). The overall sign in eq. (2.36)can now be justified. From eq. (2.25), we know that R π is positive, and thus the overallsign in eq. (2.41) must be positive, as shown. Choosing the phase of v such that A PV3 π v is real and positive, and inserting these resultsinto eq. (2.29), we obtain q E K ( P ) L h E n , P , Λ µ, L |H W (0) | K, P , L i = v † A PV K π . (2.43) This is in fact the criterion introduced in ref. [10], and studied in refs. [12, 15], to determine whethersolutions to the three-particle quantization condition are physical. – 11 –his achieves the aim of relating the finite-volume decay matrix element (which could bedetermined by a numerical simulation) to a quantity in the generic relativistic field theory,namely a projection of the quantity A PV K π . By using multiple matrix elements, one coulddetermine the parameters in a truncated approximation to A PV K π . The result (2.43) canalso be derived by a generalization of the method of Lellouch and L¨uscher [32], as we showin appendix B.Before turning to parametrizations of A PV K π , we close this section with a few morecomments on the phase conventions entering the various relations on matrix elements. Wefirst review the requirements we have imposed above. First, we have fixed the phase of thestate A π (0) | E n , P , Λ µ, L i by requiring that h |A π (0) | E n , P , Λ µ, L i is real and positive.Second, we have required that, while A PV3 π and v ( E Λ n , P , Λ µ, L ) may individually carryphases, these must cancel such that A PV3 π v is real and positive. We have then demonstratedthat, with these two convention choices, the finite-volume matrix element appearing ineq. (2.43) must have the same phase as the combination v † A PV K π . Finally, to extract thevalue of A K π , we must establish the phase of v itself, which has been left open so far.The most natural convention is to simply require A PV3 π and v to be individually real. Inthis convention v † is also real, so any phase in the finite-volume matrix element on theleft-hand side of eq. (2.43) (resulting, for example, from a CP-violating phase in H W ) willbe inherited by A PV K π .As was already discussed in refs. [42, 44], the utility in carefully tracking this phaseinformation is that it allows one to extract relative phases between various matrix elements.For example, if the weak Hamiltonian density is decomposed into operators O ( x ) and O ( x ), it follows from eq. (2.43) that h E n , P , Λ µ, L |O (0) | K, P , L ih E n , P , Λ µ, L |O (0) | K, P , L i = v † A PV K π [ O ] v † A PV K π [ O ] . (2.44)The overall phase in v † cancels, so the phase in the ratio of PV amplitudes on the right-hand side is given by that of the ratio of the matrix elements on the left-hand side. Thisphase information will be passed on to the decay matrix elements by solving the integralequations described below in section 2.4. A PV K π Since A PV K π is an unfamiliar quantity, we discuss its properties in this brief subsection. Werecall that it is an infinite-volume on-shell quantity, given, crudely speaking, by calculatingall K → π diagrams with PV regulation for the poles. Thus it is an analytic function ofthe kinematic variables, symmetric under interchange of any pair of final-state momenta.A useful parametrization of A PV K π is given by the threshold expansion, which is anexpansion in powers of relativistic invariants that vanish at threshold, for instance∆ = m K − m π m π . (2.45)For the decays K + → π + π + π − and K + → π + π π , for example, ∆ ≈ .
39 and 0 . p , p , and p , so that P = p K = p + p + p ,– 12 –he three Mandelstam variables are s i = ( p j + p k ) = ( P − p i ) , X i =1 s i = m K + 3 m π , (2.46)where { i, j, k } are ordered cyclically. We will expand in dimensionless quantities that vanishat threshold, namely ∆ and ∆ i = s i − m π m π , (2.47)which satisfy P i ∆ i = ∆. Using this sum rule, and enforcing particle-interchange symmetryand smoothness, we find A PV K π = A iso + A (2) X i ∆ i + A (3) X i ∆ i + A (4) X i ∆ i + O (∆ ) . (2.48)Here “iso” refers to the isotropic limit, in which the amplitude is independent of the mo-menta of the decay products. To obtain a strict expansion in powers of ∆, one would needto expand the coefficients, e.g. A iso = ∞ X n =0 ∆ n A iso , n , (2.49)keeping only the appropriate number of terms (e.g. the first five terms if working to fourthorder in ∆).To use the threshold expansion (2.48) in the result from the previous subsection,eq. (2.43), one must convert A PV K π to the { k‘m } basis. We recall here how this is done [3].We first note that the on-shell three-particle phase space with fixed total four-momentum(and ignoring Lorentz invariance) is five-dimensional. We can parametrize this space invarious ways, one choice being to use a set of five momentum coordinates: p ,x , p ,y , p ,z , p ,x , p ,y . The remaining five coordinates are then set by the fixed total energy and mo-mentum. To connect to the { k‘m } basis we make a different choice, labelled { k , b a ∗ } . Here k is one of the three momenta, e.g. k = p , while b a ∗ is the result of boosting the remainingtwo particles to their CMF and picking the direction of one of them, say particle 2. Herewe are using the notation that a quantity with a superscript ∗ is evaluated in a boostedframe. We then decompose the amplitude into spherical harmonics in the pair CMF, A PV K π ( k , b a ∗ ) = X ‘m √ π Y ‘m ( b a ∗ ) A PV K π ( k ) ‘m . (2.50)To use the result of the previous subsection we must restrict k to lie in the finite-volumeset, A PV K π ; k‘m ≡ A PV K π ( k ) ‘m (cid:12)(cid:12)(cid:12) k =2 π n /L . (2.51)The decomposition of the terms in the threshold expansion into the { k‘m } basis isstraightforward but tedious, and we do not present it here. It follows closely the corre-sponding decomposition of K df , worked out in ref. [12]. The presence of only a single term in each of the second, third and fourth orders is a pattern that doesnot continue to higher orders. – 13 – .4 Relating A PV K π to the physical decay amplitude In this subsection we show how the physical K → π decay amplitude can be obtained bysolving appropriate integral equations, once the endcap A PV K π has been determined usingthe results of the previous two subsections. This is the second step of the general proceduredescribed in section 2.1, and involves relations between infinite-volume quantities. Themethod we use follows the strategy introduced in ref. [4]: we consider a finite-volumecorrelator whose infinite-volume limit produces the physical decay amplitude, and writethis correlator in terms of K , K df , , and in particular A PV K π .We begin by recalling that the infinite-volume decay matrix element can be defined by T K π = h π, out | H W (0) | K, P i , (2.52)where states are defined using the standard relativistic normalization. The decay rate isthen given by Γ = 13! 12 m K Z dLIPS | T K π | , (2.53)where 1 /
3! is the identical-particle symmetry factor, and dLIPS is the Lorentz-invariantphase-space measure. We will use the { k , b a ∗ } variables introduced above, in terms of whichthe measure becomes dLIPS = d k ω k (2 π ) a ∗ πω a ∗ d Ω ˆ a ∗ π . (2.54)Here a ∗ = q ∗ ,k is the squared momentum of one of the nonspectator pair in their CMF,with q ∗ ,k = ( E K ( P ) − ω k ) − ( P − k ) , (2.55)and ω a ∗ = p a ∗ + m π is the corresponding energy.In order to obtain an expression for T K π in terms of A PV K π , we consider the finite-volume decay matrix element, T K π,L . This is defined as the sum of all Feynman diagramscontributing to T K π , including appropriate amputations, but evaluated with finite-volumeFeynman rules. A subtlety arises because the energies of three external on-shell pions, eachwith a momentum from the finite-volume set will, not, in general, sum to E K ( P ). To havean energy-conserving process, the external momenta in T K π,L must be adjusted. This on-shell projection is done using the method introduced in ref. [3]. The spectator momentum, k , is held fixed at a finite-volume value, while the magnitude of a ∗ (the momentum of one ofthe nonspectator pair boosted to the pair CMF) is adjusted until energy is conserved. Thisrequires setting a ∗ = q ∗ ,k b a ∗ , and leads to the third particle having momentum − a ∗ in thepair CMF. This is the on-shell projection that appears in all quantities adjacent to factorsof F and G . The projection only affects the external momenta for T K π,L —when writtenas a skeleton expansion in terms of Bethe-Salpeter kernels, the internal loop momenta areall drawn from the finite-volume set. This point is discussed at length in ref. [4]. The resultis the quantity T K π,L ( k , b a ∗ ).We will need a variant of this quantity in the following, namely T ( u ) K π,L ( k , b a ∗ ), whichwe refer to as the asymmetric decay amplitude. This is defined as the sum of the sameset of amputated diagrams with two restrictions: First, if the final interaction involves a– 14 –wo-particle Bethe-Salpeter kernel, then k is chosen as the momentum for the spectatorparticle. Second, if the final interaction involves a three-particle kernel, then the diagramis multiplied by 1 /
3. In fact, what appears in the expressions below is T ( u ) K π,L,k‘m , whichresults when we decompose the b a ∗ dependence into spherical harmonics as in eq. (2.50).To obtain the desired expression for T K π,L,k‘m , we begin from the correlator C K π,L ( P ),introduced in eq. (2.22), which describes a finite-volume K → π process. We consider theMinkowski version of this correlator, given by C M K π,L ( E, P ) = C M K π, ∞ ( E, P ) + A PV3 π iF − + K df , A PV K π . (2.56)We obtain T K π,L by keeping contributions that have at least one factor of F (since thisputs the intermediate three-particle state on shell) and amputating all that lies to the leftof the left-most F . Only the second term on the right-hand side contains F s, and weamputate it as described in section 2.1 by removing A PV3 π and multiplying by the inverse of iF/ (2 ωL ), leading to T ( u ) K π,L = (cid:18) iF ωL (cid:19) − F i K df , F A PV K π , (2.57)= L ( u ) L
11 + K df , F A PV K π , (2.58)where L ( u ) L is given in eq. (2.16). Note that, unlike in the construction of M ( u,u )3 ,L describedin section 2.1, here there are no disconnected terms to drop.With the expression for T ( u ) K π ; k‘m in hand, we next note, following ref. [4], that theresult can be extended to an arbitrary choice of k , not just one in the finite-volume set. Theform of eq. (2.58) remains unchanged, and the various quantities extend simply to arbitrary k , as explained in ref. [4]. The result, T ( u ) K π,L ( k ) ‘m , is still a finite-volume quantity, sinceinternal loops remain summed. We now insert i(cid:15) factors to regulate the poles in F and G ,and take the infinite-volume limit holding k fixed T ( u ) K π ( k ) ‘m = lim (cid:15) → + lim L →∞ T ( u ) K π,L ( k ) ‘m (cid:12)(cid:12)(cid:12)(cid:12) E → E + i(cid:15) . (2.59)This gives the correct asymmetric infinite-volume decay amplitude because, in the limit, allsums in Feynman diagrams that run over a pole (which are those in which three particlescan go on shell) are replaced by integrals in which the pole is regulated by the standard i(cid:15) prescription.The final step is to obtain the complete decay amplitude by symmetrizing, whichcorresponds to adding all possible attachments of the momentum labels to the Feynmandiagrams. This is effected by T K π ( k , b a ∗ ) ≡ S { T K π ( k ) ‘m } , (2.60)= T ( u ) K π ( k , b a ∗ ) + T ( u ) K π ( a , b b ∗ ) + T ( u ) K π ( b , b k ∗ ) , (2.61)where T ( u ) K π,L ( k , b a ∗ ) is obtained by combining T ( u ) K π,L ( k ) ‘m with spherical harmonics as ineq. (2.50). The notation in eq. (2.61) is the natural generalization of that given above:– 15 –ust as ( ω a ∗ , a ∗ ) is the result of boosting ( ω a , a ) to the CMF of the { a , b } pair (with b = P − k − a ), so ( ω b ∗ , b ∗ ) is the result of boosting ( ω b , b ) to the CMF of the { b , k } pair,while ( ω k ∗ , k ∗ ) is the result of boosting ( ω k , k ) to the CMF of the { k , a } pair.Applying this procedure to the result eq. (2.58) for T ( u ) K π,L leads to a set of integralequations. Since the steps are very similar to those in ref. [4], we simply quote the finalresults. As for T ( u ) K π , the { k‘m } indices used in finite volume go over in infinite-volumeto a dependence on the continuous spectator momentum, k , as well as an unchangeddependence on ‘ and m . Thus the matrix indices ‘m remain, and will be implicit in thefollowing equations, while the dependence on k will be explicit.The combination (1 + M ,L G ) − M ,L , which appears in L ( u ) L and in F , goes over ininfinite volume to D ( u,u )23 ( p , k ) ‘ m ; ‘m (using the notation of ref. [16]), which satisfies D ( u,u )23 ( p , k ) = δ ( p − k ) M ( k ) − M ( p ) Z r G ∞ ( p , r ) D ( u,u )23 ( r , k ) , (2.62)where G ∞ is defined in eq. (81) of ref. [4], and includes an i(cid:15) -regulated pole, while δ ( p − k ) = 2 ω p (2 π ) δ ( p − k ) , (2.63) M ( k ) ‘ m ; ‘m = δ ‘ ‘ δ m m M ( ‘ )2 ( q ∗ ,k ) , (2.64) Z r = Z d r ω r (2 π ) . (2.65)Here M ( ‘ )2 is the ‘ th partial wave of M , evaluated for the CMF momentum of one of thescattering pair. Given a solution to the integral equation (2.62), and the relation of F to L ( u ) L , eq. (2.16), the equation satisfied by the infinite-volume limit of X = (1 + K df , F ) − is X ( p , k ) = δ ( p − k ) − Z r , s K df , ( p , r ) e ρ PV ( r ) L ( u ) ( r , s ) X ( s , k ) . (2.66)In the first term there is an implicit identity matrix in ‘m space. The quantity e ρ PV resultsfrom the infinite-volume limit of F , and is e ρ PV ( r ) ‘ m ; ‘m = δ ‘ ‘ δ m m e ρ ( ‘ )PV ( q ∗ ,r ) , (2.67)where ρ ( ‘ )PV is a modified phase space factor given in eq. (B6) of ref. [16]. Finally, L ( u ) ( r , s ) = 13 δ ( r − s ) − D ( u,u )23 ( r , s ) e ρ PV ( s ) , (2.68)which is the infinite-volume limit of L ( u ) L .With these ingredients we can write down the relationship of the asymmetric decayamplitude to A PV K π , T ( u ) K π ( k ) = Z r , s L ( u ) ( k , r ) X ( r , s ) A PV K π ( s ) . (2.69)The full amplitude is then given by symmetrization T K π ( k , b a ∗ ) = S n T ( u ) K π ( k ) ‘m o , (2.70)– 16 –sing the definition in eq. (2.60) above. This completes the procedure for determining thedecay amplitude from the finite-volume decay matrix elements. The physical interpretationof the factors in eq. (2.69) is as follows. L ( u ) incorporates pairwise final state interactions,through multiple factors of M alternating with switch factors G ∞ . T ( u ) K π becomes complexboth because M itself is complex, and due to the i(cid:15) in G ∞ . The quantity X incorporatesfinal state interactions involving all three particles, with intermediate pairwise scattering.Since this result derives from an all-orders diagrammatic derivation, the amplitude T K π will automatically satisfy the required unitarity constraints, and in particular those thatlead to Khuri-Treiman relations describing final-state interactions [53]. We close this section by giving an explicit example of how the formalism works when makingthe simplest approximations to the decay and scattering amplitudes. We assume that onlythe leading, isotropic term in the threshold expansion of the decay amplitude, A iso , isnonvanishing—see eq. (2.48). This implies that A PV K π ; k‘m is only nonzero for ‘ = m = 0,and is independent of k . In addition, it couples only to three-pion states in the trivial irrepof the appropriate little group, e.g., the A − irrep for P = 0 (for pions with negative intrinsicparity). For the amplitudes M and K df , , we assume that only the s -wave contributes(so again ‘ = m = 0) and that K df , is independent of the spectator momentum. This isequivalent to keeping only the isotropic term in the threshold expansion of K df , [10, 12].Given these approximations, all quantities entering the definition of F depend onlyon the spectator momenta. The isotropic nature of A PV K π and K df , is represented byintroducing the vector | i in spectator-momentum space, which equals unity for all choicesof k in the finite-volume set that lie below the cutoff. Specifically, A PV K π −→ | i A iso and K df , −→ | i K iso df , h | , (2.71)where A iso and K iso df , are constants. Using eq. (2.11), one then finds that1 F − + K df , −→ F − F | i F iso + ( K iso df , ) − h | F , (2.72)where F iso is the isotropic component of F , F iso ≡ h | F | i . (2.73)It follows that the only poles in three-particle correlators [e.g. C M L of eq. (2.5)] that dependon K iso df , occur when the isotropic quantization condition is satisfied, i.e. F iso = − ( K iso df , ) − . (2.74)There are also solutions at free energies resulting from the F terms in eq. (2.72), but theseare an artifact of the isotropic approximation, as discussed in Appendix F of ref. [12]. Fromeq. (2.72), we can determine the residue using eq. (2.36), finding R iso n = F | i r iso n h | F , (2.75)– 17 –here we have abbreviated the arguments of R Λ µ ( E n , P , L ), and defined r iso n = − (cid:18) ∂F iso ( E, P , L ) ∂E + ∂ [1 / K iso df , ( E ∗ )] ∂E (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) E = E A n ( P ,L ) . (2.76)Here all derivatives are evaluated at the energy E A n ( P , L ), a solution to the isotropicquantization condition. The quantity r iso n is real in general, and positive for a physicalsolution. Thus we can read off the vector v ( E n , P , Λ µ = A , L ) defined in eq. (2.40),( v iso n ) † = ( r iso n ) / h | F . (2.77)Here we have chosen the overall phase according to the convention discussed above, so that v iso n is real. Using eq. (2.43) we now obtain q E K ( P ) L h E n , P , A , L | H W (0) | K, P , L i = ( r iso n ) / F iso A iso . (2.78)This can be massaged into a simple form for determining A iso A iso ( E ∗ n ) = 2 E K ( P ) L h E n , P , A , L | H W (0) | K, P , L i × ∂F iso ( E, P , L ) − ∂E + ∂ K iso df , ( E ∗ ) ∂E ! E = E A n ( P ,L ) . (2.79)Thus, in the isotropic approximation, we need to measure the matrix element to only asingle three-pion state in order to determine A iso at that energy. In figure 2 we plot theconversion factor appearing on the second line of this equation for the case of constant K iso df , , implying ∂ K iso df , ( E ∗ ) /∂E = 0.The relationship of A iso to T K π is also substantially simplified in the isotropic approx-imation. We first note that eq. (2.58) simplifies to T ( u ) , iso K π,L = L ( u ) L | i
11 + K iso df , F iso A iso . (2.80)Taking the infinite volume limit as before, we obtain T ( u ) , iso K π ( k , b a ∗ ) = S n T ( u ) , iso K π ( k ) o , (2.81)where T ( u ) , iso K π ( k ) = L ( u ) , iso ( k ) A iso K iso df , F ∞ , iso . (2.82)Here the momentum dependence arises solely from the final-state interactions in L ( u ) , iso ( k ) = 13 − Z s D ( u,u )23 ( k , s ) e ρ PV ( s ) , (2.83)where D ( u,u )23 ( k , s ) still satisfies eq. (2.62), but now with all quantities restricted to ‘ = m = 0, and F ∞ , iso = Z r e ρ PV ( r ) L ( u ) , iso ( r ) . (2.84)– 18 – igure 2 : Plot of the conversion factor appearing in eq. (2.79) (rescaled as indicated bythe plot label) in the vicinity of the three-particle threshold for the case of constant K iso df , .The factor is plotted versus energy E for P = and mL = 6. The two-particle K matrix,entering F iso3 , determined by keeping only the scattering length, a , in the the effectiverange expansion. The three curves correspond to three values of the scattering length, asindicated by the legend, and each unfilled marker corresponds to the ground-state energyfor the corresponding ma value when K iso df , = 0. In particular, the blue squarecorresponds to the non-interacting limit. The fact that the conversion factor is unity inthe latter case indicates that the non-interacting matrix elements are equal in finite andinfinite volume, up to a trivial normalization. More generally, once the scattering lengthis determined, these types of curves allow one to directly relate—within the isotropicapproximation—any value of measured three-particle energy (horizontal axis) to a matrixelement conversion factor (vertical axis).In this case, the only integral equation that has to be solved is that for D ( u,u )23 , as has beendone recently in refs. [27, 54]. We note that F ∞ , iso and L ( u ) , iso are, in general, complex.The expressions in the isotropic approximation are sufficiently simple that one canreadily combine eqs. (2.79) and (2.82) to display the direct relation between the finite-volume matrix element and the physical amplitude. Unpacking the compact notation usedabove slightly, we reach | T iso K π ( E ∗ , m , m ) | = 2 E K ( P ) L (cid:12)(cid:12)(cid:12) h E n , P , A , L | H W (0) | K, P , L i (cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12) L iso ( E ∗ , m , m ) 11 + K iso df , ( E ∗ ) F ∞ , iso ( E ∗ ) (cid:12)(cid:12)(cid:12)(cid:12) ∂F iso ( E, P , L ) − ∂E + ∂ K iso df , ( E ∗ ) ∂E ! , (2.85)where E (and thus E ∗ ) is fixed by the value of finite-volume energy, tuned to E ∗ = M K for a physical decay amplitude. We have emphasized that the right-hand side depends on– 19 –he two squared invariant masses m and m , defined by m = ( E − ω k ) − ( P − k ) , (2.86) m = ( E − ω a ) − ( P − a ) , (2.87)and have also introduced the symmetrized final-state interaction factor. L iso ( E ∗ , m , m ) ≡ L ( u ) , iso ( k ) + L ( u ) , iso ( a ) + L ( u ) , iso ( b ) . (2.88)At this stage we can comment on the relationship of our result to that of ref. [52]. Weexpect that the isotropic limit, given in eq. (2.85), is equivalent to the result of ref. [52],aside from differences in the schemes used to define the short-distance quantities. In-deed, the equations have the same basic structure, with a contribution resulting from finalstate interactions (the term involving L iso ) and a Lellouch-L¨uscher-like correction factor.Demonstrating the precise equivalence, however, is nontrivial, since our approach basedin short-distance quantities, K df , and A PV K π , that are symmetric under particle exchange,whereas the approach of ref. [52] does not symmetrize until the very end. Presumably,the mapping can be determined using the relation between symmetric and asymmetricapproaches explained in refs. [16, 17], but this is beyond the scope of the present work.In closing, we note that eq. (2.85) is analogous to the original Lellouch-L¨uscher relationpresented in ref. [32]. In particular, the two-particle result is reached by making thereplacements T iso K π ( E ∗ , m , m ) −→ T K π ( E ) , (2.89) L iso ( E ∗ , m , m ) −→ , (2.90) K iso df , ( E ∗ ) −→ K ( E ) , (2.91) F ∞ , iso ( E ∗ ) −→ − iρ ( E ) ≡ − i q πE , (2.92) F iso ( E, P , L ) −→ F ( E, L ) , (2.93)where we have also restricted attention to the P = frame. On the right-hand side wehave introduced the physical K → ππ amplitude T K π ( E ), extended to allow for final-stateenergies different from the kaon mass. We have also used the two-particle K-matrix, K ,and the two-particle finite-volume function, F , both restricted to the s -wave. These areessentially the same quantities as appearing in eq. (2.6), in the definition of F , but withoutthe implicit sub-threshold regulator used there and without the spectator-momentum index.We have also introduced the two-particle phase-space, ρ ( E ), with q = p E / − m .Making the indicated substitutions into eq. (2.85) yields | T K π ( E ) | = 2 M K L h E n , A , L | H W (0) | K, L i × (cid:12)(cid:12)(cid:12)(cid:12) − i K ( E ) ρ ( E ) (cid:12)(cid:12)(cid:12)(cid:12) ∂F ( E, L ) − ∂E + ∂ K ( E ) ∂E ! . (2.94)Substituting the definitions of the scattering phase δ ( E ) and the L -dependent, so-calledpseudophase φ ( E, L ) K ( E ) = 16 πE tan δ ( E ) q , F ( E, L ) − = 16 πE tan φ ( E, L ) q , (2.95)– 20 –ne can easily reach eq. (4.5) of ref. [32], after some algebraic manipulations.This completes our discussion of the formalism in the context of the simplified theory.We now turn to realistic applications of these results. In this section, we describe the generalization of the previous analysis to processes involvingthree-pion final states in isosymmetric QCD. This allows our results to be applied to severalprocesses of phenomenological interest: (i) the electromagnetic transition γ ∗ → π , whichcontributes to the hadronic vacuum polarization piece of the muon’s magnetic momentum,( g − µ ; (ii) the isospin-violation strong decay η → π ; and (iii) the weak decay K → π ,which has both CP-conserving and violating amplitudes.The generalization presented here requires the generic three-pion quantization condi-tion derived in ref. [18]. We start this section by recalling some results from that work,and presenting the generalization of the formulae derived above to the three-pion system.We then describe the specific applications to the three processes listed above. In the derivation in section 2, the “kaon” and “pion” fields were taken to be real scalars withseparate Z symmetries. Here we consider the physical kaon and pion fields. The former,which can be either charged or neutral, are complex fields with strangeness conservationplaying the role of the Z symmetry. The pions are represented by a triplet of fields,with two complex fields in the definite charge basis ( π + and π − ) and one real filed ( π ),with the Z symmetry being G parity. Both kaons and pions are stable particles in QCD,with masses satisfying the required inequality, eq. (2.1). The form of the weak Hamiltoniandepends on the decay being considered, but its essential property, unchanged from above, isthat it annihilates one of the kaons and creates three pions. The new feature is the presenceof multiple three-pion intermediate states, e.g. π + π π − and π π π in the neutral sector,and it is this feature that the derivation of ref. [18] takes into account.We stress again that, since the weak interactions are added by hand as external op-erators, we can choose to separately consider operators that create three and two pions,with G parity ensuring that these two sectors do not mix. We can also consider one at atime operators that create three pions in states of definite isospin. Indeed, the quantiza-tion condition of ref. [18] decomposes into separate results for each choice of total isospin.Finally, we note that, although we couch the discussion in this subsection in terms of the K → π decay, the essential aspects of the discussion apply equally well if the kaon isreplaced by a γ ∗ or η , and the weak operator is replaced by the electromagnetic current orthe isospin-breaking Hamiltonian, respectively.A generic three-pion state can have total isospin I = 0 , , I = 0 : (cid:8) | ρπ i (cid:9) ,I = 1 : (cid:8) | σπ i , | ρπ i , | ( ππ ) π i (cid:9) ,I = 2 : (cid:8) | ρπ i , | ( ππ ) π i (cid:9) ,I = 3 : (cid:8) | ( ππ ) π i (cid:9) , (3.1)where “ σ ”,“ ρ ”,“( ππ ) ” label a two-pion combination with isospin 0,1, and 2, respectively,and the subscripts on the kets denotes the total isospin. Explicit expressions for thesestates for the charge zero ( I = 0) sector are given in appendix C of ref. [18].The order of pion fields in each state of eq. (3.1) is a shorthand for the interplay ofmomentum and isospin assignment. In particular, if we consider asymptotic states withfixed total energy and momentum ( E, P ) then the remaining degrees of freedom, ‘m and k ,are assigned to the leading pion pair and the third pion field, respectively. As emphasizedin section 2.1, the asymmetric description is natural from the perspective of the finite-volume formalism, since many of the quantities appearing there, in particular F , G , K and F , single out a pion pair in their definition. The result is that there are additionalflavor spaces with dimensions one, three, two and one, for I = 0 , , , C AB,L ( P ), defined in eq. (2.32). The operators A and B now respectively destroy andcreate a three-pion state of definite isospin. The expression for this correlator, previouslygiven by eq. (2.33), now becomes C [ I ] AB,L = C [ I ] AB, ∞ − i A PV , [ I ] F [ I ]3 ] − − K [ I ]df , B PV , [ I ] . (3.2)The notation for bold-faced quantities is taken over from ref. [18]: they contain a factorof i compared to those used for identical particles [which explains differences in signs andfactors of i compared to eq. (2.33)] and also have an additional index corresponding tothe flavor space described above. For example, for I = 1, the endcap A PV , [ I ] is a three-dimensional row vector in these indices (in addition to being a row vector in the k‘m indices), while F [ I ]3 and K [ I ]df , are 3 × k‘m indices). The explicit expressions for the flavor structure of F [ I ]3 are given in Table 1of ref. [18] and we do not repeat them here.With eq. (3.2) in hand, the derivation in section 2.2 goes over almost verbatim. Oneuses the same three correlators, eqs. (2.20)-(2.22), except for the above-described changes One difference compared to ref. [18] is that the endcaps in that work are matrices in flavor space, whilethose here are row or column vectors. This reflects the fact that creation and annihilation operators inref. [18] were chosen to create three-pion states of all isospins, whereas here we consider single operatorswith definite three-pion isospin. – 22 –o the kaon field and the three-pion operators. The final result is a generalization ofeq. (2.43): q E K ( P ) L h E Λ , [ I ] n , P , I, I , Λ µ, L |H W (0) | K, P , L i = v † A PV , [ I ] K π . (3.3)The matrix element on the left-hand side is obtained from the lattice simulation with thekaon state having the desired quantum numbers, and E Λ , [ I ] n being the energy of a three-pionstate of chosen isospin and hypercubic-group irrep. We assume that the weak Hamiltoniancouples the kaon to this state, for otherwise the equation is trivially satisfied as both sidesvanish. On the right-hand side the column vector v is an abbreviation for v ( E Λ , [ I ] n , P , I, I , Λ µ, L ) , (3.4)which is a row vector having both { k‘m } and flavor indices, and includes a factor of i relative to the v of section 2.2 in order to cancel the factor of i in A PV , [ I ] K π . It is aneigenvector of [ F [ I ]3 ] − − K [ I ]df , with vanishing eigenvalue, and is defined by the generalizationof eq. (2.40): R [ I,I ]Λ µ ( E Λ , [ I ] n , P , L ) = lim P → iE Λ , [ I ] n − ( E Λ , [ I ] n + iP ) P [ I,I ]Λ µ ( − i )1 / F [ I ]3 − K [ I ]df , P [ I,I ]Λ µ ≡ v v † . (3.5)We stress that we do not include a relative factor of i between the definitions of R [ I,I ]Λ µ and R Λ µ of section 2.2. The bold quantity defined here thus differs from the R Λ µ only by theaddition the flavor index.The workflow for using eq. (3.3) is as follows: First, one chooses the initial kaonquantum numbers and the form of H W based on the physical process under consideration.This determines the allowed values of I and I for the three-pion final states. Second,one calculates the three-pion energy spectrum for one of the allowed values of { I, I } ,using a range of choices of P , and picking irreps/rows Λ µ such that the desired K → π matrix element is nonvanishing. Third, one compares this spectrum to the result from thequantization condition of ref. [18],det (cid:16) [ F [ I ]3 ] − − K [ I ]df , (cid:17) = 0 , (3.6)and uses this to determine (a parameterized form of) K [ I ]df , . Fourth, with this form in handone uses eq. (3.5) to determine the vectors v for levels that have their energies matched to E K ( P ). Finally, one uses eq. (3.3) to provide a constraint on the row vector A PV , [ I ] K π . Bycombining several such constraints can determine a (parametrized form of) A PV , [ I ] K π .The second step—connecting to the physical decay amplitude—also mirrors that foridentical particles, which was described in section 2.4. One first introduces an asymmetricfinite-volume amplitude that generalizes eq. (2.58), T [ I ]( u ) K π,L = (cid:16) F [ I ] (cid:17) − F [ I ]3 − K [ I ]df , F [ I ]3 A PV , [ I ] K π , (3.7)– 23 –here F [ I ] is iF/ (2 ωL ) tensored with the identity in the corresponding flavor space [18].Here again the boldfaced quantity T [ I ]( u ) K π,L differs from the T ( u ) K π,L used in section 2.4 bothby the addition of flavor indices and by a factor of i . The physical amplitude is thenobtained by taking the appropriate ordered limit and symmetrizing, T [ I ] K π = S (cid:26) lim (cid:15) → + lim L →∞ T [ I ]( u ) K π,L (cid:27) . (3.8)This limit leads to integral equations that are simple generalizations of those presented insection 2.4, and which we do not display explicitly. The only subtlety that is introduced bythe flavor indices is the need to generalize the definition of symmetrization, as is explainedin section 2.3 of ref. [18]. We stress that the symmetrization here acts on a column vectorwith a single index, rather than on a matrix as in ref. [18].The results of these steps are the infinite-volume decay amplitudes in the isospin basis.To convert to a measurable amplitude, e.g. that for K + → π + π π , one must combine theisospin amplitudes appropriately. The results needed to do this are collected in appendix C.In this regard there is a further subtlety concerning the amplitudes that have a multi-dimensional flavor space, i.e. those with I = 1 and 2. To explain this point (which is notdiscussed in ref. [18]) we focus on the example of I = 1. The result from eq. (3.8) is then three K → [3 π ] I =1 amplitudes, each expressed as a function of the three pion momenta.The issue is that, when one has the full momentum dependence, these three amplitudesare not independent. In fact, as we explain below, one needs to know only two of the threein order to completely reconstruct the I = 1 amplitude. Similarly, for the I = 2 case,only one of the two amplitudes is needed. This redundancy does not, however, lead to anysimplification in the solution of the integral equations implicit in eq. (3.8). γ ∗ → π The electromagnetic process γ ∗ → π is of phenomenological interest as it contributes, viathe hadronic vacuum polarization (HVP) and the hadronic light-by-light scattering (HLbL),to the anomalous magnetic moment of the muon [55–59]. Our formalism allows one todetermine the infinite volume amplitude using a finite volume lattice QCD calculation. Inparticular, although this is not a decay, the results above are readily adapted—one simplytakes advantage of the fact that one can allow the final three-particle state to take on anyenergy and momentum in the relations given above. This then corresponds to a timelikephoton with virtuality q = E Λ n ( L, P ) − P . The analogous two-particle process, γ ∗ → ππ ,and its relation to finite-volume matrix elements is discussed in ref. [37].The replacement of the kaon with a virtual photon simplifies the required lattice cal-culation. The composite operator B K π ( x ) in eq. (2.26) is replaced by the electromagneticcurrent J ν ( x ), and the kaon correlator is not required. We consider here only the part ofthis current that involves up and down quarks, J ν = 23 ¯ uγ ν u −
13 ¯ dγ ν d , (3.9)as this leads to the dominant contribution to γ ∗ → π . No tuning of the volume is neededto match a given energy; instead, each finite-volume three pion state with appropriate– 24 –uantum numbers leads to a result for the desired amplitude with photon virtuality givenby the energy of the state.The electromagnetic current contains both isoscalar and isovector parts. The latterhas positive G parity and thus, in isosymmetric QCD, couples only to even numbers ofpions, and in particular to the ρ resonance. What is of interest here is the isoscalar part, J ν = 16 (cid:16) ¯ uγ ν u + ¯ dγ ν d (cid:17) , (3.10)which has negative G parity and thus couples to three pions. The dominant contributionin the energy range of interest for muonic g − ω (782) resonance.The desired amplitude is obtained using the two-step process explained above. Eachmatrix element obtained from a lattice calculation is related to the intermediate PV am-plitude by L / h E Λ , [0] n , P , I = 0 , Λ µ, L |J ν (0) | i = v † A PV , [0] γ π,ν , (3.11)where v = v [0] ( E Λ , [0] n , P , I = 0 , Λ µ, L ) is obtained from the spectrum of I = 0 three pionstates using eq. (3.5). The irreps Λ and rows µ that lead to nonzero matrix elements dependon the total momentum P and the Lorentz index ν . Note that for I = 0 the flavor spaceis one dimensional, so A PV , [0] γ π,ν and v can be viewed as vectors in { k‘m } space alone. Wealso comment that the left-hand side of eq. (3.11) differs from the corresponding resultsfor kaon decays, eqs. (2.43) and and (3.3), by the absence of a factor of (2 E K ( P ) L ) / .This is because, in contrast to the unit normalized finite-volume kaon state, there is noneed to correct the normalization of the vacuum, which matches between the finite- andinfinite-volume theories.To implement eq. (3.11), the infinite-volume PV amplitude A PV , [0] γ π,ν must be parametrized.This is most easily done by using eq. (2.50) to convert from { k‘m } space to a function ofthree on-shell momenta, p , p and p . Up to the overall factor of i , the amplitude is a real,smooth function of momenta, antisymmetric under the interchange of any pair of momenta,and transforming as an axial vector. Expanding about threshold as in section 2.3 (with m K → q ), the general form satisfying these properties is A PV , [0] γ π,µ = i(cid:15) µνρσ p ν p ρ p σ A (0) γ π + A (2) γ π X i ∆ i + . . . ! . (3.12)Here the ∆ i are the threshold expansion parameters defined in eq. (2.47), and the coef-ficients A ( n ) γ π are functions of ∆ = q / (9 m π ) −
1. For a consistent threshold expansion, A (0) γ π should be a quadratic function of ∆, while A (2) γ π should be a constant. The ellipsisrepresents higher order terms. We observe that the threshold expansion begins at higherorder than for the symmetric amplitude discussed in section 2.3.The second step is to solve the integral equations encoded in the I = 0 versions ofeqs. (3.7) and (3.8), which convert A PV , [0] γ π,ν into the γ ∗ → [3 π ] I =0 amplitude, T [0] γ π,ν ( p , p , p ). If the intrinsic negative parity of the pions is included the amplitude transforms as a vector, as requiredto couple to the electromagnetic current. – 25 –ecalling from ref. [18] that the I = 0 state is given by1 √ (cid:16) | π + π π − i − | π π + π − i + | π π − π + i − | π − π π + i + | π − π + π i − | π + π − π i (cid:17) , (3.13)with the three pions in each ket having the momenta p , p and p , respectively, and notingthat only the I = 0 amplitude is nonzero, we obtain the physical amplitude as i T h γ ∗ → π + ( p ) π ( p ) π − ( p ) i = r T [0] γ π,ν ( p , p , p ) , (3.14)where the index ν refers to the polarization of the virtual photon. η → π The decay η → π provides an example where our formalism can be used within thecontext of the strong interactions. The key point is that the η is stable in isosymmetricQCD, but can decay to three pions in the presence of isospin violation. The decay has avery small partial width, Γ( η → π ) ≈ . α due to the neutrality of the η . Thus this process is uniquely sensitive to theup-down quark mass difference. We refer the reader to ref. [61] for a recent review of thestatus of phenomenological predictions for these decays.A natural approach for a first-principles lattice QCD calculation of these decay ampli-tudes is to simulate isosymmetric QCD with mass term H ∆ I =0 = m u + m d (cid:16) ¯ uu + ¯ dd (cid:17) , (3.15)but introduce isospin violation through the insertion of the mass difference operator H ∆ I =1 = m u − m d (cid:16) ¯ uu − ¯ dd (cid:17) . (3.16)This brings the calculation into the same class as that for K → π decays, with the initialkaon replaced by the η and H W replaced by H ∆ I =1 . We observe that, although isospin-breaking is being included only at leading order, our formalism includes all rescatteringeffects due to final state interactions. Thus it provides an alternative to the dispersivemethods used in present analyses [64, 65].Since the initial η has I = 0, the final three pion state has I = 1. Thus to obtain the η → π amplitude we can use the results of section 3.1, by simply making the replacement This can also be obtained from the bottom row of the matrix R given in eq. (C.3). Since only the I = 0amplitude is nonzero, the rightmost entry in this row gives the relevant factor. Potential decays to 2 π and 4 π that are allowed by G parity and kinematics are forbidden by parityconservation, irrespective of isospin breaking. We note that this method of calculating isospin-violating effects is similar to the perturbative methodintroduced in refs. [62, 63], but differs in that here we imagine inserting the operator at a single positionrather than over the entire volume. – 26 – → η , and taking I = 1. In this way, we can use the formalism to determine theintermediate PV amplitude A PV , [1] η π and the final, physical amplitude T [1] η π . We note thatthese amplitudes have a three-dimensional flavor space. For a practical implementationone needs a parametrization of A PV , [1] η π , and the relation of T [1] η π to the amplitudes intocharged and neutral pions. We provide these results in the remainder of this subsection.To present the parametrization of A PV , [1] η π , it is convenient to use a different basisfor the flavor space of three-pion states than that of eq. (3.1). The new basis, which wedenote the χ basis, uses states that lie in irreps of the symmetric group S correspondingto permutations of the three particles. It is given by [18] (cid:8) | χ s i , | χ i , | χ i (cid:9) = ( | ( ππ ) π i + √ | σπ i , − √ | ( ππ ) π i + 23 | σπ i , | ρπ i ) , (3.17)where | χ s i transforms in the trivial irrep of S , while {| χ i , | χ i} transform in the two-dimensional standard irrep. We refer to appendix C in ref. [18] for explicit expressions forthe isospin states, as well as further discussion of the group properties.We now adapt the results obtained in ref. [18] for the parametrizations of scatteringamplitudes to that of the intermediate PV amplitude. Working to quadratic order in thethreshold expansion, we find A PV , [1] η π = i A s , η π + A s , η π ∆ + A s , η π ∆ + A s , aη π X i ∆ i ! + i (cid:16) A d , η π + A d , η π ∆ (cid:17) P · ξ P · ξ + iA d , aη π P · ξ ) − ( P · ξ ) P · ξ P · ξ + . . . , (3.18)where A s , η π , etc. are real coefficients. The notation is as in section 2.3, except for thereplacement m K → m η , and the use of the new quantities ξ = 1 √ p − p − p ) , and ξ = 1 √ p − p ) . (3.19)The superscripts s and d refer to the “singlet” symmetric and “doublet” standard irrep of S , respectively. We observe that the symmetric part of the amplitude begins at leadingorder in the threshold expansion, while that transforming in the doublet enters only atlinear order.Finally we describe the reconstruction of the decay amplitudes into final states com-posed of pions with definite charges, which are T η ( p , p , p ) ≡ T [ η → π ( p ) π ( p ) π ( p )] , (3.20) T +0 − η ( p , p , p ) ≡ T [ η → π + ( p ) π ( p ) π − ( p )] . (3.21)– 27 –ur formalism yields the I = 1 amplitude, which, expressed in the χ basis, is T [1] η π ( p , p , p ) = i T [1] s ( p , p , p ) T [1] d , ( p , p , p ) T [1] d , ( p , p , p ) . (3.22)The relation between the χ basis and that involving particles of definite charge is given ineq. (C.3). Using this result, and the fact that the amplitudes for I = 0, 2, and 3 vanish,we obtain T η ( p , p , p ) = − r T [1] s ( p , p , p ) , (3.23) T +0 − η ( p , p , p ) = 1 √ T [1] s ( p , p , p ) − √ T [1] d , ( p , p , p ) + 12 T [1] d , ( p , p , p ) . (3.24)We note that all three I = 1 amplitudes are invariant under the interchange p ↔ p ,so that T +0 − η ( p , p , p ) = T +0 − η ( p , p , p ), which is consistent with the positive chargeconjugation parity of the pseudoscalar mesons.As noted earlier, the two doublet amplitudes are not independent when one uses thefreedom to permute the momenta. A convenient form of this relationship is T [1] d , ( p , p , p ) = 1 √ T [1] d , ( p , p , p ) + 2 √ T [1] d , ( p , p , p ) , (3.25)where we stress that the order of the momentum arguments differs in the last term. Usingthis result, eq. (3.24) can be rewritten as T +0 − η ( p , p , p ) = 1 √ T [1] s ( p , p , p ) + 1 √ T [1] d , ( p , p , p ) . (3.26) K → π Finally, we turn to the K → π decays that are the primary motivation for this work. Wehave left these processes to the end as they are the most complicated to analyze. The mainreason for developing the formalism for a lattice calculation of the K → π amplitudes isto provide a method for determining the CP-violating contribution, so as to allow furthertests of the Standard Model. This is analogous to the situation with K → π decays,where the well-measured CP-violating quantity (cid:15) /(cid:15) can now be predicted reliably in theStandard Model using lattice QCD [66–68].In the three-particle case, the decay amplitudes are T +00 K ( p , p , p ) ≡ T [ K + → π + ( p ) π ( p ) π ( p )] , T − ++ K ( p , p , p ) ≡ T [ K + → π − ( p ) π + ( p ) π + ( p )] , (3.27)together with their charge conjugates, and the neutral kaon amplitudes T + − K S ( p , p , p ) ≡ T [ K S → π + ( p ) π ( p ) π − ( p )] , T K S ( p , p , p ) ≡ T [ K S → π ( p ) π ( p ) π ( p )] , T + − K L ( p , p , p ) ≡ T [ K L → π + ( p ) π ( p ) π − ( p )] , T K L ( p , p , p ) ≡ T [ K L → π ( p ) π ( p ) π ( p )] . (3.28)– 28 –n the absence of CP violation, all are nonzero except for T K S . All have been measuredexcept for those for neutral kaon decays to 3 π [60]. The effects of CP violation that aremeasurable at present involve the charged kaon decays. Specifically, CP violation showsup as a difference between Dalitz plot slope parameters in K + and K − decays (see ref. [69]for a review). Experimentally, these differences are on the edge of observability [70, 71].Phenomenological predictions for CP violating observables achieve a comparatively higheraccuracy [72, 73]. In light of this situation, we focus here on the formalism for the decaysof charged kaons, and specifically on the K + decay. The generalization to the K − decayis straightforward, and that for the neutral kaon decays is summarized in appendix D.The operators needed for a lattice study of this process are those of the effectiveelectroweak Hamiltonian, H W . The set of operators that are relevant after running toscales below the charm mass is given for instance in refs. [74, 75]. Since H W containsonly operators that change isospin by 1 / /
2, the allowed total isospin of the 3 π state is I = 0, 1 and 2. For charged kaons only decays to I = 1 and 2 amplitudesare allowed. Using the formalism described above, a lattice calculation can determine(constraints on) the intermediate amplitudes A PV , [1] K π and A PV , [2] K π . We stress that this canbe done separately for each choice of total isospin, and for the CP-conserving and CP-violating parts of each operator contained in H W . To carry this out in practice one needs,as usual, parametrizations of the PV amplitudes. That for A PV , [1] K π is identical in formto the result given for the η → π amplitude in eq. (3.18), with only the labels on thecoefficients changing: A PV , [1] K π = i A [1] , s , K π + A [1] , s , K π ∆ + A [1] , s , K π ∆ + A [1] , s , aK π X i ∆ i ! + i (cid:16) A [1] , d , K π + A [1] , d , K π ∆ (cid:17) P · ξ P · ξ + iA [1] , d , aK π P · ξ ) − ( P · ξ ) P · ξ P · ξ + . . . , (3.29)The corresponding result for the I = 2 case is A PV , [2] K π = i (cid:16) A [2] , d , K π + A [2] , d , K π ∆ (cid:17) P · ξ P · ξ ! + iA [2] , d , aK π ( P · ξ ) − ( P · ξ ) P · ξ P · ξ ! + . . . (3.30)Here we are using the basis [18] {| χ i , | χ i } = {| ( ππ ) π i , | ρπ i } , (3.31)which is further discussed in appendix C. We have worked to quadratic order in the ex-pansions of A [ I ] K π , since fits to experimentally measured Dalitz plots usually work only tothis order.Given a determination of A PV , [1] K π and A PV , [2] K π , the second step of solving the integral– 29 –quations leads to the decay amplitudes in the isospin basis. There are five amplitudes T [1] K π ( p , p , p ) = i T [1] s ( p , p , p ) T [1] d , ( p , p , p ) T [1] d , ( p , p , p ) , T [2] K π ( p , p , p ) = i T [2] d , ( p , p , p ) T [2] d , ( p , p , p ) , (3.32)although, as above, only one from each doublet is independent. The form of this redundancyis exactly as in eq. (3.25) for both I = 1 and 2. The relationship of the isospin-basis statesto those with pions of definite charges is given in appendix C. Using these results, andsimplifying using the redundancy equation (3.25), we find T +00 ( p , p , p ) = − √ T [1] s ( p , p , p ) + 1 √ h T [1] d , ( p , p , p ) + T [2] d , ( p , p , p ) i + 1 √ h T [1] d , ( p , p , p ) + T [2] d , ( p , p , p ) i , (3.33) T − ++ ( p , p , p ) = 2 √ T [1] s ( p , p , p ) + 1 √ h T [1] d , ( p , p , p ) − T [2] d , ( p , p , p ) i + 1 √ h T [1] d , ( p , p , p ) − T [2] d , ( p , p , p ) i , (3.34)where we have used the vanishing of the I = 3 amplitude. In this article we have derived the formalism that allows the study of three-particle decayprocesses using input from lattice QCD calculations. This generalizes the well-establishedformalism for two-particle decays developed by Lellouch and L¨uscher [32] and its subsequentextensions. Specifically, our formalism applies for decays in which the three particles aredegenerate and spinless, although they do not need to be identical. Thus, in particular, thephenomenologically important K → π decays are now accessible to lattice methods in theisospin-symmetric limit. Our formalism applies not only to 1 → → γ ∗ → π , which is relevantfor lattice calculations of the hadronic vacuum polarization contribution to muonic g − A PV K π . This quantity plays a rolethat is analogous to that of K df , in the scattering formalism of refs. [3, 4]. The second step There is a potential confusion with the amplitudes for η decay that have the same names—see eq. (3.22).It should, however, be clear from the context to which process the amplitudes apply. – 30 –n the formalism is to relate A PV K π to the physical decay amplitude, and is analogous to therelation between K df , and the physical scattering amplitude [4]. This relation is achievedby solving integral equations in infinite-volume that incorporate the effects of two- andthree-particle final state interactions (entering through the two-particle K matrix K and K df , , respectively) and leads to a decay amplitude satisfying the constraints of unitarity.Our derivation is independent of the details of the effective theory, aside from theassumption of a Z symmetry analogous to G parity. It holds for decays of “kaons” withmasses up to the first inelastic threshold, m K < m π . The approach is relativistic, imply-ing, for one thing, that the intermediate amplitude A PV K π is Lorentz invariant. We use thisconstraint to develop an expansion of A PV K π about threshold.It is instructive to compare the two and three-particle formalisms in more detail. Thefirst step of our formalism is the analog of the multiplication by the LL factor that isrequired for two-particle decays involving only a single channel. In particular, the vector v that enters the key relation, eq. (2.43), is determined by a combination of scatteringamplitudes and kinematic factors, just as the LL factor is in the two-particle case. Themain new feature compared to the two-particle analysis is the need for the second step. Inthe original LL derivation, this step is essentially replaced by the the multiplication by thefinal-state phase required by Watson’s theorem. It is the more complicated nature of three-particle final-state interactions that necessitates the solution of integral equations. Anotherdifference from the original LL result is that, in general, each finite-volume three-particlematrix element serves only to constrain A PV K π , rather than provide a direct determination.This difference is, however, only due to the simplicity of the set-up considered in theoriginal LL work. If one considers a multiple-channel two-particle system, then each latticematrix element again only provides a constraint on physical decay amplitudes [38, 42, 44].Conversely, if we consider the simplest approximation for three-particle scattering anddecay amplitude, then, as shown in section 2.5, only a single finite-volume matrix elementis required to determine A PV K π .In the second part of our presentation, given in section 3, we generalize the formalism sothat it applies for decays to a general three-pion state in isosymmetric QCD. This buildsupon our recent generalization of the formalism for three-particle scattering to includeall three-pion isospin channels [18]. It allows us to address phenomenologically relevantprocesses, and we have discussed in detail three applications: the electromagnetic transition γ ∗ → π , the isospin-violating decay η → π , and the weak decay K → π . While most ofthe features of the formalism for identical particles also hold for three-pion decays, the keydifference is that all quantities have an additional isospin index. One impact of this changeis that the symmetry properties of the generalization of A PV differ from those for identicalparticles, and we have presented explicit expressions in a threshold expansion that shouldsuffice for realistic calculations.An important difference between the process γ ∗ → π on the one hand and the decays η → π and K → π on the other, is that the latter two have a clear physical interpretationonly when the initial and final state energies match, whereas the virtual photon transitionis meaningful for all final state energies. However, the formalism presented here also holdsfor matrix elements in which the kinematics are not perfectly matched. In practice, this– 31 –reedom can be used to extract A PV K π as a function of the final state energy, e.g. by fittingto multiple closely spaced states. This could be useful both for giving stronger constraintson the target amplitude and for interpreting the value, including the role of resonanceenhancement in the amplitude, by considering the result for energies away from physicalkinematics.Although a controlled computation of the K → π decay amplitude using lattice QCDhas only been achieved very recently [68], we are hopeful that the extension to K → π decays can be undertaken in the next few years. This will require a program of calculationsof the finite-volume three-pion spectrum with all allowed total isospins, in addition to thecalculation of the finite volume K → π matrix elements. We note that work on the secondstep of our formalism—which requires solving integral equations—can begin independentlyof lattice simulations, since the methods required do not depend on the functional formof the necessary input quantities ( K , K df , and A PV K π ). Indeed, methods for solving theclosely-related integral equations required for three-particle scattering are under activedevelopment [27, 54].Finally, we note that further generalizations of the formalism derived here will beneeded to allow lattice calculations of all three-particle decay amplitudes of interest. Forexample, to address isospin breaking in K → π decays requires formalism for three non-degenerate particles, as well as for multiple, nondegenerate channels. The recent extensionof the three-particle quantization condition to the case of nondegenerate particles is a firststep in this direction [19]. Acknowledgments
We thank Ra´ul Brice˜no and Toni Pich for useful discussions. The work of MTH issupported by UK Research and Innovation Future Leader Fellowship MR/T019956/1.FRL acknowledges the support provided by the European projects H2020-MSCA-ITN-2019//860881-HIDDeN, the Spanish project FPA2017-85985-P, and the Generalitat Va-lenciana grant PROMETEO/2019/083. The work of FRL also received funding fromthe European Union Horizon 2020 research and innovation program under the MarieSk lodowska-Curie grant agreement No. 713673 and “La Caixa” Foundation (ID 100010434,LCF/BQ/IN17/11620044). FRL also acknowledges financial support from Generalitat Va-lenciana through the plan GenT program (CIDEGENT/2019/040). The work of SRS issupported in part by the United States Department of Energy (USDOE) grant No. DE-SC0011637.
A Proof that A = A † In this appendix, we prove that the quantities A and A , introduced in eq. (2.5), arerelated by hermitian conjugation, provided that the same is true of the two operatorsentering the corresponding correlation function, eq. (2.3). This result is required to reacheq. (2.34), which is used, in turn, to derive the main result of section 2.– 32 – constructive definition of the quantities A and A is provided in ref. [3], but it iscumbersome and difficult to use in proving basic relations. Therefore, here we find it easierto pursue an indirect method. Our approach is in the spirit of ref. [4] in which K df , isrelated to the physical scattering amplitude via a finite-volume quantity, without makingdirect use of the complicated constructive definition of ref. [3].The key idea is to use the relation between A , A and their corresponding finite-volume decay amplitudes. To define the latter we first introduce matrix elements definedin terms of physical, asymptotic three-particle states: T ( E, k , b a ? ) = h | σ (0) | π, in i , (A.1) T ( E, k , b a ? ) = h π, out | σ † (0) | i , (A.2)where the arguments on the left-hand side provide a description of the three incoming oroutgoing pions, as described in the text following (2.61). Starting from these, one can givediagrammatic definitions of T ( u ) L and T ( u ) L , the asymmetric finite-volume decay amplitudescorresponding to A and A respectively. For concreteness, we focus on T ( u ) L ; the argumentfor T ( u ) L is analogous. The definition of T ( u ) L is essentially the same as that for T ( u ) K π,L given in section 2.4, except that the initial amputated kaon propagator is absent, so thatthe initial kaon state in eq. (2.52) is replaced in eq. (A.2) with the vacuum. In words, T ( u ) L ( E, k , b a ∗ ) is the asymmetric finite-volume vacuum to three pion amplitude in which,if the final interaction involves a 2 → k is the momentumassigned to the spectator, and if the final interaction involves the 3 → /
3. The amplitude T ( E, k , b a ? ) in eq. (A.2) is then obtained by takingthe appropriate L → ∞ limit and symmetrizing, just as for T K π in eqs. (2.59)-(2.61) ofthe main text.From the analysis given in section 2.4, it then follows that T ( u ) L = X L A , X L = L ( u ) L
11 + K df , F , (A.3)with T ( u ) L a column vector in { k‘m } space, and L ( u ) L is given in eq. (2.16). This has exactlythe same structure as eq. (2.58), with A PV K π replaced here with A . A similar analysis leadsto T ( u ) L = A X R , X R = 11 + F K df , R ( u ) L , (A.4)with T ( u ) L a row vector in { k‘m } space, and R ( u ) L given in eq. (2.17). The first key obser-vation is now that X R = X † L , (A.5)which follows because L ( u ) † L = R ( u ) L , F = F † and K df , = K † df , . These results themselvesfollow from the hermiticity of the building blocks F , K and (2 ωL ) − G .The second key relation that we need is T ( u ) L = ( T ( u ) L ) † , (A.6)– 33 –hich follows directly from the diagrammatic definitions of T ( u ) L and T ( u ) L [without referenceto eqs. (A.3) and (A.4)], assuming T and P invariance of the effective field theory, and Pinvariance of the operator σ (ignoring the intrinsic parity of the pion). To make theargument, we first we note that, aside from phases arising from the operators σ † and σ ,each diagram contributing to T ( u ) L and T ( u ) L is real. This is because we are working infinite volume. One way to show this result is to evaluate diagrams using time-orderedperturbation theory, in which case the only source of imaginary contributions is the i(cid:15) inthe energy denominators. But in finite volume, the sums over spatial momenta do notrequire that the poles from these denominators be regulated, so that (cid:15) can be set to zero.Next we note that T invariance implies the relation T ( u ) L ( E, k , b a ? ) = T ( u ) L ( E, − k , − b a ? ) ∗ ,where complex conjugation is only needed because of possible phases arising from σ † and σ . Now, using parity invariance, we have that T ( u ) L ( E, − k , − b a ? ) = T ( u ) L ( E, k , b a ? ). Finally,decomposing into the { k‘m } basis, and taking into account that T ( u ) L is a row vector and T ( u ) L a column vector, we obtain eq. (A.6).Combining eqs. (A.3), (A.4) and (A.6) yields A X R = A † X R . (A.7)The final step is to note that, for any total energy E , X R is well-defined and invertibleaway from a discrete set of values of L for which one of its eigenvalues vanishes or diverges.Away from these “singular” values of L , we can apply the inverse of X R to both sides ofeq. (A.7), and conclude that A = A † . This demonstrates the desired equality for all valuesof the spectator momentum k that lie in the finite-volume sets of the nonsingular valuesof L . Assuming that the nonsingular values of L form a dense set, then, given that A and A are continuous functions of the spectator momentum, we find that A = A † in general. B Alternative partial derivation following Lellouch-L¨uscher method
Here we follow the approach of ref. [32], which provides an alternative to the first step of thederivation, which is presented in the main text in section 2.2. We consider the same theoryas in section 2 but now imagine determining the finite-volume spectrum in the two-pion andthree-pion sectors in the presence of the weak interaction, with Hamiltonian density H W ( x ).These sectors are still decoupled in the presence of H W , differing by whether the totalnumber of particles is even or odd. The logic of the approach is that the weak interactionsshift the spectrum, beginning at linear order, and these shifts can be calculated in twoways: (i) from the finite-volume matrix element; (ii) using the quantization condition, dueto a shift in the infinite-volume interactions that depends on the infinite-volume decayamplitude. Comparing the two results for the shift leads to the desired relation. We stressthat throughout this section we drop contributions of quadratic or higher order in H W from all equations.We begin with the two-pion sector. A key distinction here, as compared to the K → ππ case of ref. [32], is that H W only couples the single kaon to states with G parity minus.Thus, the lightest new intermediate state coupling to ππ via the weak interactions is the– 34 – π state, which, given the constraint eq. (2.1), has a CMF energy E ∗ that exceeds 4 m π .It follows that the spectrum in the energy range E ∗ < m π will only be shifted by second-order weak processes involving off-shell intermediate Kπ states. Since we work at linearorder, these can be ignored. Thus the energy levels are unchanged, which, using the two-particle quantization condition, implies that the two-particle scattering amplitude M isalso unchanged. The latter result can also be seen by studying the modifications to thisamplitude directly in infinite volume.The situation is different in the three-pion sector. Here the lightest new intermediatestate consists of a single kaon, and this is kinematically allowed; see again (2.1). Levelsaway from the kaon energy will be shifted only at second order in perturbation theory.However, if the volume is tuned so that there is a three-pion level in the theory withoutweak interactions whose CMF energy matches that of a finite-volume kaon, then we mustuse degenerate perturbation theory at leading order. We consider here only a rotationallyinvariant, local form of H W ( x ) [such as that of eq. (2.2)]. In this case, only the trivial irrepof the appropriate little group will be coupled to the kaon and thus only the tuned QCDlevel in this irrep is relevant. The degenerate sector is thus ( | K, P , L i , | E n , P , A , L i ), andthe Hamiltonian restricted to this sector is E K ( P ) M ( P ) M ∗ ( P ) E K ( P ) ! , M ( P ) = L h E n , P , A , L | H W (0) | K, P , L i , (B.1)where the factor of L arises due to the difference between Hamiltonian and Hamiltoniandensity. Diagonalizing, we obtain the energies to first order in H W , E K ( P ) → E ± K ( P ) ≡ E K ( P ) ± | M ( P ) | . (B.2)This is the first result for the energy shifts.To obtain the second result for the shifts we begin by noting that, when the total CMFenergy E ∗ lies within O ( H W ) of m K , the three-particle scattering amplitude is changed atlinear order in H W . This is because of the nearly on-shell process 3 π → K → π , whichleads to iδ M ( E ∗ ) ≡ i M [ H W =0]3 ( E ∗ ) − i M [ H W =0]3 ( E ∗ ) , (B.3)= h π, out | [ − i H W (0)] | K, P i iE ∗ − m K + i(cid:15) h K, P | [ − i H W (0)] | π, in i . (B.4)where we have used the superscripts [ H W = 0] and [ H W = 0] to indicate whether the3 π → K → π transition is present or absent. Here the dependence on the initial andfinal pion momenta is implicit. Although this appears to be of second order in H W , thedenominator of the propagator is E ∗ − m K = E ( P ) − E K ( P ) , (B.5) The difference between finite- and infinite-volume kaon energies is exponentially suppressed in L andthus neglected in this derivation. Therefore, strictly speaking, the approach described in this appendixis equally valid whether one tunes the three-pion level to the finite- or the infinite-volume kaon energy.However, in practice, the tuning should be to the finite-volume kaon as this is the quantity available in thelattice calculation. – 35 – 2 E K ( P ) (cid:2) E ( P ) − E K ( P ) (cid:3) + O (cid:2) ( E ( P ) − E K ( P )) (cid:3) , (B.6)and thus of O ( H W ) for E ( P ) = E ± K ( P ). It follows that the difference between theperturbed and unperturbed amplitudes at the shifted finite-volume energy is O ( H W ): δ ± M ≡ δ M (cid:0) [ E ± K ( P ) − P ] / (cid:1) , (B.7)= ∓ h π, out | H W (0) | K, P i h K, P | H W (0) | π, in i E K ( P ) | M ( P ) | . (B.8)Our next task is to determine the shift in K df , that corresponds to that in M , forthe former is the quantity that enters the quantization condition. For the sake of brevity,we write the following expressions in terms of finite-volume quantities, with the L → ∞ limit implied. We use the expression for M ( u,u )3 ,L , eq. (2.14), but need keep only the second,divergence-free term, since D ( u,u ) does not depend on K df , : δ M = S n δ M ( u,u )df , ,L o , (B.9) M ( u,u )df , ,L = L ( u ) L
11 + K df , F K df , R ( u ) L , (B.10) δ M ( u,u )df , ,L = L ( u ) L
11 + K df , F δ K df ,
11 + F K df , R ( u ) L . (B.11)Next we use eq. (2.58) for the decay amplitude, and the conjugate result for the 3 π → K amplitude, to rewrite eq. (B.8) as δ ± M = ∓S ( L ( u ) L
11 + K df , F A PV K π A PV † K π E K | M |
11 + F K df , R ( u ) L ) , (B.12)where we have suppressed the P dependence in E K and M . Matching eqs. (B.9) and(B.11) with eq. (B.12), we find δ ± K df , = ∓ A PV K π A PV † K π E K | M | . (B.13)The outer product structure reflects the factorization of the residue at the pole in M .The final step is to enforce the quantization condition with the shifted amplitude atthe shifted energies. To this end we define A ( E ) ≡ F ( E, P , L ) − + K df , ( E ∗ ) . (B.14)Then the unshifted quantization condition can be written as det[ A ( E K )] = 0, and theshifted version as det[ A ( E ± K ) + δ ± K df , ] = det[ A ( E K ) + δ ± A ] = 0 , (B.15)where we have introduced δ ± A = ±| M | dAdE (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E K + δ ± K df , . (B.16)– 36 –ecalling that v is the eigenvector of A ( E K ) with vanishing eigenvalue, and defining v + δ ± v as the corresponding eigenvector for A ( E K ) + δ ± A , we have( v † + δ ± v † ) · [ A ( E K ) + δ ± A ] · ( v + δ ± v ) = 0 . (B.17)Multiplying out this result, using A ( E K ) · v = 0 = v † · A ( E K ), and using the fact thatthe left-hand side of eq. (B.17) must vanish order by order in H W (in particular at linearorder) yields v † · δ ± A · v = 0 . (B.18)Substituting eqs. (B.13) and (B.16) then gives | M ( P ) | (cid:20) v † · dAdE (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E K · v (cid:21) = v † · A PV K π A PV † K π E K ( P ) | M ( P ) | · v . (B.19)To evaluate the quantity in square brackets we use eqs. (2.36) and (2.40) of the main text,which imply A ( E ) = ( E − E K ) vv † | v | + X ( E ) , (B.20)where the first term results from A ( E ) − = vv † E − E K + O (cid:2) ( E − E K ) (cid:3) , (B.21)and X ( E ) arises from the non-singular part of A − . Here we only require that X ( E )satisfies v · X ( E ) · v † = 0. This relies on the fact that the eigenvectors of A ( E K ) form acomplete set that can be used for any A ( E ). Then X ( E ) is built from the sum over alleigenvector pairs e ( i ) e ( j ) † , weighted by E -dependent coefficients, with at least one of thetwo vectors e ( i ) and e ( j ) orthogonal to v . From this it immediately follows that v † · dAdE (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E K · v = 1 . (B.22)Finally, inserting eqs. (B.1) and (B.22) into eq. (B.19), we obtain | v † A PV K π | = 2 E K ( P ) L |h E n , P , A , L | H W (0) | K, P , L i| . (B.23)This agrees with eq. (2.43) in the main text. C Relations between three-pion states
In ref. [18], we provided the isospin decomposition for all neutral ( I = 0) three-pion states,and described the decomposition into irreducible representations of the group S . Here weprovide a result for the neutral sector not given explicitly in ref. [18], since this is needed inthe discussion of the γ ∗ → π and η → π processes. In addition, we generalize the resultsto the charge 1 ( I = 1) sector, as these are needed in the discussion of K + decays.– 37 –he first result is for the matrix R defined by |− i| − + i|− + 0 i| i| + − i| −i| + 0 −i = R · | ( ππ ) π i | ( ππ ) π i = | χ i | ρπ i = | χ i | χ s i | χ i | χ i | ρπ i , (C.1)where we are using the shorthands |− i ≡ | π − ( p ) π ( p ) π + ( p ) i , | + 0 −i ≡ | π + ( p ) π ( p ) π − ( p ) i , etc . (C.2)We find R = √ − − √
12 1 √ − √
12 12 − √ √ −
12 1 √
12 1 √ − √ −
12 1 √ √ − √
12 1 √
15 2 √ √ √ − √ √ √
12 1 √
15 2 √ − √ √
10 12 − √
12 1 √ − √ − − √ √
10 12 1 √
12 1 √ − √
12 12 1 √ . (C.3)We use the last row of R in sections 3.2 and 3.3.We now turn to the charge 1 sector of three pions, giving our conventions for the statesand the relation between the isospin and definite-charge bases. In this sector, the totalisospin can only be I = 1, 2 or 3, with degeneracies 3, 2, 1, respectively [18]. The S irrepsthat appear are the symmetric irrep, labeled | χ s i I , and the two-dimensional standard irrep,labeled {| χ i I , | χ i I } .The relation to the states in the basis with definite isospin for the first pair is | χ s i +3 = | ( ππ ) π i +3 (C.4) | χ i +2 = | ( ππ ) π i +2 (C.5) | χ i +2 = | ρπ i +2 , (C.6) | χ s i +1 = 23 | ( ππ ) π i +1 + √ | σπ i +1 , (C.7) | χ i +1 = − √ | ( ππ ) π i +1 + 23 | σπ i +1 , (C.8) | χ i +1 = | ρπ i +1 . (C.9)– 38 –rom this, the relation to the states composed of pions of definite charges is simple toobtain. What we need in section 3.4 is this inverse of this relation, | + 0 0 i| i| i|− + + i| + − + i| + + −i = R · | χ s i +3 | χ i +2 | χ i +2 | χ s i +1 | χ i +1 | χ i +2 , (C.10)where R = √
15 1 √
12 12 − √
15 1 √
12 122 √
15 1 √ − − √
15 1 √ − √ − √ − √ − √ √ − √ −
12 2 √
15 1 √
12 121 √ − √
12 12 2 √
15 1 √ − √
15 2 √ √ − √ . (C.11) D Formalism for K → π decays For completeness, we collect here the results needed to apply the formalism to the decaysof neutral kaons. We do so for the K decay. That for K decay is identical in form, andby forming appropriate combinations one can determine the amplitudes for K S and K L decays.The major change compared to K + decays is the presence of the I = 0 final state inaddition to those with I = 1 and 2. The parametrization of the intermediate PV I = 0amplitude requires an antisymmetric combination of the pion momenta that is a Lorentzinvariant. In terms of the parameters defined in section 2.3, we find that the leading termis of cubic order in the threshold expansion, A PV , [0] K π = iA aK π h ∆ (∆ − ∆ ) + ∆ (∆ − ∆ ) + ∆ (∆ − ∆ ) i + . . . . (D.1)The parametrizations of the I = 1 and 2 amplitudes are as for the K + decay discussed insection 3.4.We use the same notation for the isospin-basis amplitudes as in eq. (3.32), but nowadd T [0] K π ( p , p , p ) ≡ i T [0] a ( p , p , p ) , (D.2)where the subscript “ a ” denotes the antisymmetric irrep of S . Using R in eq. (C.3) andthe redundancy result eq. (3.25) we obtain the relation between isospin amplitudes andthose for pions of definite charge, T ( p , p , p ) = − √ T [1] s ( p , p , p ) (D.3)– 39 – + − ( p , p , p ) = 1 √ T [1] s ( p , p , p ) + 23 T [2] d, ( p , p , p ) + 13 T [2] d, ( p , p , p )+ 1 √ T [1] d, ( p , p , p ) + 1 √ T [0] a ( p , p , p ) . (D.4) References [1] R. A. Brice˜no and Z. Davoudi,
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