New method for calculating electromagnetic effects in semileptonic beta-decays of mesons
Chien-Yeah Seng, Xu Feng, Mikhail Gorchtein, Lu-Chang Jin, Ulf-G. Meißner
NNew method for calculating electromagnetic effects insemileptonic beta-decays of mesons
Chien-Yeah Seng , Xu Feng , , , , Mikhail Gorchtein , , ,Lu-Chang Jin , , and Ulf-G. Meißner , , Helmholtz-Institut f¨ur Strahlen- und Kernphysik and Bethe Center for Theoretical Physics,Universit¨at Bonn, 53115 Bonn, Germany School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Center for High Energy Physics, Peking University, Beijing 100871, China State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, China Helmholtz Institute Mainz, D-55099 Mainz, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung, 64291 Darmstadt, Germany Johannes Gutenberg University, D-55099 Mainz, Germany RIKEN-BNL Research Center, Brookhaven National Lab, Upton, NY, 11973, USA Physics Department, University of Connecticut,Storrs, Connecticut 06269-3046, USA Institute for Advanced Simulation,Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics,Forschungszentrum J¨ulich, 52425 J¨ulich, Germany and Tbilisi State University, 0186 Tbilisi, Georgia (Dated: October 27, 2020) a r X i v : . [ h e p - l a t ] O c t bstract We construct several classes of hadronic matrix elements and relate them to the low-energyconstants in Chiral Perturbation Theory that describe the electromagnetic effects in the semilep-tonic beta decay of the pion and the kaon. We propose to calculate them using lattice QCD, andargue that such a calculation will make an immediate impact to a number of interesting topicsat the precision frontier, including the outstanding anomalies in | V us | and the top-row Cabibbo-Kobayashi-Maskawa matrix unitarity. . INTRODUCTION The last few years have seen a rapid development in the theory of the electroweak radiativecorrections (RCs) in hadron and nuclear beta decay processes. In particular, a dispersionrelation analysis [1, 2] significantly reduced the hadronic uncertainty of the single-particleRCs in free neutron and superallowed nuclear beta decays, and led to a new status of the top-row Cabibbo-Kobayashi-Maskawa (CKM) matrix unitarity, as quoted in the 2020 ParticleData Group (PDG) [3]: | V ud | + | V us | + | V ub | = 0 . V ud (4) V us , (1)in contrast to the result in the 2018 PDG [4] with 0 . V ud (4) V us at the right hand side(RHS). The apparent violation of the top-row CKM unitarity at a 3 σ level and its impli-cations on the possible physics Beyond the Standard Model (BSM) [5–17] trigger renewedinterest from both the experimental and theoretical community in the precision frontier.The improvements in the recent years mainly concern the reduction of the Standard Model(SM) theory uncertainties in the extraction of V ud . And now, as indicated in Eq.(1), the nextbreakthrough must involve a similar reduction of the V us theory uncertainties. In particular,the outstanding disagreement between the V us extracted from the kaon semileptonic decay( K l ) and leptonic decay ( K l ) [3]: | V us | = . exp+RCs (6) lattice ( N f = 2 + 1 + 1 , K l )0 . N f = 2 + 1 + 1 , K µ ) (2)has to be understood. Apart from possible BSM explanations, such a disagreement couldoriginate either from unknown systematic errors in the SM input of the Kπ form factoror, although somewhat less likely, the RCs in K l . For the first case one simply needs abetter lattice Quantum Chromodynamics (QCD) calculation of the Kπ form factor at zeromomentum transfer, whereas the second case is much more complicated and will be thefocus in this paper. In particular, we will discuss the possible roles that lattice QCD canplay in this aspect.Recently lattice QCD has made a tremendous progress in first-principles studies of Quan-tum Electrodynamics (QED) corrections to hadronic processes, see e.g. [18–21]. In particu-lar, Ref. [20] presented, for the first time, the full lattice study of the QED RCs to the K µ and π µ decay rates, which involves a direct calculation of both the virtual and real photon3mission diagrams. The extension of the method above to semileptonic decay processes is,however, expected to be extremely challenging [22–24]. On the other hand, Ref. [25] adopteda completely different starting point, namely to calculate the so-called “axial γW box dia-gram” on the lattice, which resulted in a significant reduction of the theory uncertainty in π e [25], and also provided an independent cross-check of the dispersion relation analysis inthe neutron RCs [26]. This is the first time lattice QCD ever plays a decisive role in theunderstanding of RCs of semi-leptonic beta decays, so a natural question to ask is whetherthe same method is going to teach us anything useful about the RCs in K l , which is muchmore complicated than π e due to its larger Q-value.The answer is yes if we appropriately combine lattice QCD with the existing theoryframework. We first recall that the standard approach to deal with the electroweak RCsin K l is based on Chiral Perturbation Theory (ChPT) [27, 28], in which the theoreticaluncertainties are from two sources: (1) the neglected terms that scale as higher-order inthe chiral power counting, and (2) the unknown low-energy constants (LECs). The firstcan in principle be reduced by including higher-order loop corrections, whereas the secondrepresents a more fundamental issue: the LECs characterize the unknown dynamics of QCDat the chiral symmetry breaking scale Λ χ ∼ V us theory.There is also another motivation to get more reliable values of these LECs. In leptonicdecay processes, one extracts | V us /V ud | by considering the ratio R A = Γ K µ / Γ π µ [31], be-cause it turns out that the K µ and π µ decay rates share not only the same short-distanceelectroweak RCs, but also the same combination of LECs at O ( e p ) so they cancel out inthe ratio. This leads to a smaller theoretical uncertainty than the extractions of the indi-vidual | V us | and | V ud | themselves. Recently, a similar ratio R V = Γ K l / Γ π e was introducedfor the semileptonic decay processes [32], which provides another venue to extract | V us /V ud | and could shed new lights on the V us discrepancy mentioned above. However, we find thatΓ K l and Γ π e do not share the same LECs at O ( e p ) and so they do not fully cancel in theratio. Therefore, one could better make use of R V if its residual dependence on the LECscan be fixed through an extra lattice QCD calculation.4n this paper we demonstrate how all the LECs relevant for the RCs in K l and π e can bepinned down by calculating two types of rather simple hadronic matrix elements on lattice.The first type is just the axial γW box diagram, which has already been done for pion. Wederive a matching relation between this quantity and the relevant LECs, and show that thelattice QCD result differs significantly from the widely-adopted value based on resonancemodel estimation [30], which motivates us even further for a thorough re-analysis. A similarcalculation of the K e box diagram at the SU(3) symmetric point will eventually fix allthe needed LECs that describe the lepton-hadron electromagnetic interactions. Finally, forthe remaining LECs that do not involve a lepton, we propose a lattice calculation of thefour-point correlation functions based on the construction in Ref. [29].The contents in this paper are arranged as follows. In Sec. II we review the existingtheory frameworks to study the electroweak RCs in kaon and pion semileptonic decays,including the classical “Sirlin’s representation” and the modern ChPT representation. Weshow in Sec. III that comparing these two representations in the SU(3) limit gives an elegantmatching relation between a subset of LECs and the axial γW box diagram calculable onlattice. We discuss the implications of the lattice result in Ref. [25] and propose a similarcalculation in the Kπ system. In Sec. IV we construct a class of four-point correlationfunctions that enable a direct lattice determination of the lepton-free LECs. Our finalconclusions are given in Sec. V. II. RADIATIVE CORRECTIONS TO SEMILEPTONIC BETA DECAYS IN TWOREPRESENTATIONS
We start by reviewing the existing theoretical frameworks in the treatment of the semilep-tonic decay of a generic spinless particle φ , and its corresponding electroweak RCs. First,the electromagnetic and charged weak currents in the quark sector are defined as: J µ em = 23 ¯ uγ µ u −
13 ¯ dγ µ d −
13 ¯ sγ µ s, J µW = V ud ¯ uγ µ (1 − γ ) d + V us ¯ uγ µ (1 − γ ) s, (3)and the matrix element of the charged weak current can be expressed in terms of two formfactors: F µfi ( p (cid:48) , p ) = (cid:104) φ f ( p (cid:48) ) | J µ † W (0) | φ i ( p ) (cid:105) = F fi + ( t )( p + p (cid:48) ) µ + F fi − ( t )( p − p (cid:48) ) µ , (4)5here t = ( p − p (cid:48) ) . Notice that in the definition above the form factors contain the CKMmatrix elements. It is useful to remember that the contribution from F fi − to the decay rateis suppressed at tree level by the factor m l /M φ i , where l is the emitted charged lepton.Now let us consider the decay process φ i ( p ) → φ f ( p (cid:48) ) e + ( p e ) ν e ( p ν ), where φ i,f are spinlessparticles. At tree level the decay amplitude is given by: M = − G F √ u ν γ λ (1 − γ ) v e F λfi ( p (cid:48) , p ) . (5)Here, G F = 1 . × − GeV − is the Fermi constant measured in muon decay.This definition has a natural advantage as it absorbs a large portion of the electroweak RCsthat is common to both the muon and hadron semileptonic beta decays into the definitionof G F .Next we discuss the two different representations of the electroweak RCs in this decayprocess, namely Sirlin’s representation and the effective field theory (EFT) representation.We will show later that the comparison between the results in these two representations leadsto useful relations between the LECs in ChPT and hadronic matrix elements calculable onlattice. To avoid discussing issues such as the gauge-dependence of the LECs, throughoutthis paper we simply adopt the Feynman gauge which is the standard choice in all papersof similar topics. A. Sirlin’s representation
Earliest theory analysis of electromagnetic RCs in Fermi interactions can be traced backto the seminal work by Kinoshita and Sirlin in 1958 [33], and later by Sirlin. He derivedthe universal function g ( E, E m , m ) that summarizes the infrared (IR) physics of the RCs ingeneric beta decay processes [34]. The analysis was then extended to the full electroweakRCs, where the muon decay rate was taken as a normalization [35]. All these were laterintegrated into a complete theory framework based on current algebra [36] and the on-shell renormalization of the SM electroweak sector [37], which we shall name as Sirlin’srepresentation. Despite being gradually superseded by the EFT representation, recently itwas re-introduced in the study of K l RCs in a hybridized form with EFT, which aims tofurther reduce the existing theory uncertainty [38].In Sirlin’s representation, the O ( G F α ) electroweak RCs to the amplitude of a semi-6eptonic decay process of a spinless particle φ i ( p ) → φ f ( p (cid:48) ) e + ( p e ) ν e ( p ν ) can be summarizedas [38]: δM = (cid:20) − α π (cid:18) ln M W M Z + 14 ln M W m e −
12 ln m e M γ + 98 + 34 a pQCD (cid:19) + 12 δ QEDHO (cid:21) M − G F √ u ν γ λ (1 − γ ) v e δF λfi ( p (cid:48) , p ) + δM γW . (6)The first line in the equation above represents the contributions from the “weak” RCs(see Ref. [38] for rigorous definition) including its perturbative QCD (pQCD) corrections a pQCD ≈ . M γ as an IR regulator), as well as the contribution from the resummationof the large QED logs, which is formally of higher order but numerically sizable: δ QEDHO =0 . γW box diagram. Employing the on-mass-shell formula [40] and Ward identities, the form factor correction splits into two pieces: δF λfi = δF λfi, + δF λfi, , among which the “two-point function” contribution reads: δF λfi, ( p (cid:48) , p ) = − e (cid:90) d q (cid:48) (2 π ) T µfi µ ( q (cid:48) ; p (cid:48) , p ) ∂∂q (cid:48) λ (cid:18) q (cid:48) − M γ M W M W − q (cid:48) (cid:19) , (7)where we have defined the “generalized Compton tensor” that consists of the interferencebetween the electromagnetic and charged weak current as: T µνfi ( q (cid:48) ; p (cid:48) , p ) = (cid:90) d xe iq (cid:48) · x (cid:104) φ f ( p (cid:48) ) | T { J µ em ( x ) J ν † W (0) } | φ i ( p ) (cid:105) . (8)On the other hand, the explicit form of the “three-point function” contribution δF λfi, is notof our concern. One needs only to know that it vanishes when the vector charged weakcurrent is conserved and p − p (cid:48) = 0. Finally, the γW box diagram contribution is given by: δM γW = − G F e √ (cid:90) d q (cid:48) (2 π ) ¯ u ν γ ν (1 − γ )( /q (cid:48) − /p e + m e ) γ µ v e ( p e − q (cid:48) ) − m e q (cid:48) − M γ M W M W − q (cid:48) T fiµν ( q (cid:48) ; p (cid:48) , p ) . (9)An important point to notice is that all the integrals above are ultraviolet (UV)-finite, sothere is no need to introduce any extra UV-regulators and unknown counterterms.Further simplifications can be made to the expressions above. First, using the on-shellformula ( /p e + m e ) v e = 0 and the Dirac matrix identity: γ µ γ ν γ α = g µν γ α − g µα γ ν + g να γ µ − i(cid:15) µναβ γ β γ , (10)7ith (cid:15) = − u ν γ ν (1 − γ )( /q (cid:48) − /p e + m e ) γ µ v e = ¯ u ν γ λ (1 − γ ) v e (cid:2) g λν q (cid:48) µ + g λµ q (cid:48) ν − g µν q (cid:48) λ − g λν p µe + i(cid:15) µναλ q (cid:48) α (cid:3) . (11)With this, the box diagram contribution in Eq. (9) splits into two parts: δM γW = δM VγW + δM AγW , (12)where δM VγW and δM AγW include the contribution from the first four terms and the last termat the RHS of Eq.(11), respectively.Next, we recall that the generalized Compton tensor satisfies the following Ward identi-ties: q (cid:48) µ T µνfi ( q (cid:48) ; p (cid:48) , p ) = − iF νfi ( p (cid:48) , p ) q ν T µνfi ( q (cid:48) ; p (cid:48) , p ) = − iF µfi ( p (cid:48) , p ) − i Γ µfi ( q (cid:48) ; p (cid:48) , p ) , (13)where q = p (cid:48) + q (cid:48) − p , andΓ µfi ( q (cid:48) ; p (cid:48) , p ) = (cid:90) d xe iq (cid:48) · x (cid:104) φ f ( p (cid:48) ) | T { J µ em ( x ) ∂ · J † W (0) } | φ i ( p ) (cid:105) . (14)These Ward identities are derived from the equal-time commutation relation between the J † W and J µ em , i.e. the current algebra relation, which is protected from perturbative QuantumChromodynamics (pQCD) corrections to all orders.With the identities above, the two-point function contribution (i.e. Eq.(7)) and δM VγW sums up to give: δM + δM VγW = α π (cid:20) ln M W m e + 34 + 12 ˜ a res g (cid:21) M + G F e √ u ν γ λ (1 − γ ) v e (cid:90) d q (cid:48) (2 π ) M W M W − q (cid:48) × p e − q (cid:48) ) − m e (cid:26) p e · q (cid:48) q (cid:48) λ ( q (cid:48) − M γ ) T µfi µ ( q (cid:48) ; p (cid:48) , p ) + 2 p eµ q (cid:48) − M γ T µλfi ( q (cid:48) ; p (cid:48) , p ) − ( p − p (cid:48) ) µ q (cid:48) − M γ T λµfi ( q (cid:48) ; p (cid:48) , p ) + iq (cid:48) − M γ Γ λfi ( q (cid:48) ; p (cid:48) , p ) (cid:27) . (15)Here, ˜ a res g ≈ .
019 is a small pQCD correction to the two-point function. Using the free-field operator product expansion (OPE) of the hadronic tensors, it is easy to see that theremaining integrals in the equation above do not depend on physics at the scale q (cid:48) ∼ M W .8 . The EFT representation The second and more commonly adopted representation in studies of the RCs in betadecays is based on the EFT of the SM at low energy. In such a formalism, one constructs themost general Lagrangian consistent with the symmetry properties of the underlying theoryin terms of the relevant low-energy degrees of freedom (DOFs). UV-divergences due to loopintegrals are first regularized using dimensional regularization (DR) and then canceled by thecorresponding LECs. A power counting scheme is defined to ensure the finiteness of termsin the Lagrangian for any given precision that one wants to achieve. Finally, a matchingwith the perturbative calculation in the SM at the UV-end is carried out to determine thedependence of the LECs on the UV-physics, e.g. large electroweak logarithms.For the decay processes we are discussing in this paper, i.e. K l and π e , the correspondingEFT is simply the three-flavor ChPT with dynamical photons and leptons. Here we shallsimply quote the involved chiral Lagrangian for future reference. First, the pseudo-Nambu-Goldstone boson (pNGB) octet is contained in the usual matrix U . To describe its couplingwith the dynamical photon field A µ , we introduce the following covariant derivative: D µ U = ∂ µ U − i ( r µ + q R A µ ) U + iU ( l µ + q L A µ ) , (16)where we have introduced the left/right-handed external sources { l µ , r µ } and spurion fields { q L , q R } that are traceless, Hermitian matrices in the quark flavor space. We also define u = √ U , and u µ = i [ u † ( ∂ µ − ir µ − iq R A µ ) u − u ( ∂ µ − il µ − iq L A µ ) u † ] , (17)as well as the covariant derivatives on the spurion fields: ∇ µ q R = ∂ µ q R − i [ r µ , q R ] , ∇ µ q L = ∂ µ q L − i [ l µ , q L ] . (18)Finally, for the SM charged weak interaction Lagrangian, the external sources should beidentified as: q R = q L = − eQ em , l µ = (cid:88) l (¯ lγ µ ν lL Q wL + h.c. ) , r µ = 0 , (19)where Q em = / − / − / , Q wL = − √ G F V ud V us . (20)9ne sees that the dynamical leptons enter through the left-handed source field l µ .Now we can write down the chiral Lagrangian. In a consistent chiral power countingscheme, p (a typical small momentum of the pNGBs) and e should carry the same chiralorder. Therefore at leading order (LO) we have: L (2) = F (cid:10) D µ U ( D µ U ) † + U χ † + χU † (cid:11) + ZF (cid:10) q L U † q R U (cid:11) − F µν F µν − ξ ( ∂ µ A µ ) + 12 M γ A µ A µ + (cid:88) l [¯ l ( i /∂ + e / A − m l ) l + ¯ ν lL i /∂ν iL ] , (21)where F is the pion decay constant in the chiral limit, F µν is the photon field strengthtensor, χ = 2 B M q with M q the quark mass matrix, and Z ≈ . π ± − π mass splitting. The notation (cid:104) ... (cid:105) represents the trace over the flavor space. Asstated above, throughout this work we choose ξ = 1, the Feynman gauge.To absorb the UV-divergences generated from L (2) at one loop, one needs to introducethe next-to-leading order (NLO) chiral Lagrangian, which could either scale as O ( p ) or O ( e p ). The former is just the standard Gasser-Leutwyler Lagrangian [41] so we shallconcentrate on the latter. There are two types of chiral Lagrangian at O ( e p ). The firsttype characterizes the short-distance electromagnetic effects of hadrons [42, 43]: L e p { K } = F (cid:26) K (cid:10) D µ U ( D µ U ) † (cid:11) (cid:104) q R q R + q L q L (cid:105) + K (cid:10) D µ U ( D µ U ) † (cid:11) (cid:10) q R U q L U † (cid:11) + K (cid:0)(cid:10) ( D µ U ) † q R U (cid:11) (cid:10) ( D µ U ) † q R U (cid:11) + (cid:10) D µ U q L U † (cid:11) (cid:10) D µ U q L U † (cid:11)(cid:1) K (cid:10) ( D µ U ) † q R U (cid:11) (cid:10) D µ U q L U † (cid:11) + K (cid:10) q L q L ( D µ U ) † D µ U + q R q R D µ U ( D µ U ) † (cid:11) + K (cid:10) ( D µ U ) † D µ U q L U † q R U + D µ U ( D µ U ) † q R U q L U † (cid:11) + 12 K (cid:10) χ † U + U † χ (cid:11) (cid:104) q R q R + q L q L (cid:105) + K (cid:10) χ † U + U † χ (cid:11) (cid:10) q R U q L U † (cid:11) + K (cid:10) ( χ † U + U † χ ) q L q L + ( χU † + U χ † ) q R q R (cid:11) + K (cid:10) ( χ † U + U † χ ) q L U † q R U + ( χU † + U χ † ) q R U q L U † (cid:11) + K (cid:10) ( χ † U − U † χ ) q L U † q R U + ( χU † − U χ † ) q R U q L U † (cid:11) + K (cid:10) ( D µ U ) † [ ∇ µ q R , q R ] U + D µ U [ ∇ µ q L , q L ] U † (cid:11) + K (cid:10) ∇ µ q R U ∇ µ q L U † (cid:11) + K (cid:104)∇ µ q R ∇ µ q R + ∇ µ q L ∇ µ q L (cid:105) (cid:27) , (22)although the lepton fields may still enter through the covariant derivatives. The second typeinvolves explicit leptonic degrees of freedom. The part relevant to K l and π e RCs is given10 Kl em (%) K e . ± . e p ± . LEC K ± e . ± . e p ± . LEC K µ . ± . e p ± . LEC K ± µ . ± . e p ± . LEC
Table I: δ Kl em calculated in ChPT [28]. by [44]: L e p { X } = e F (cid:88) l (cid:8) X ¯ lγ µ ν lL (cid:104) u µ {Q emR , Q wL }(cid:105) + X ¯ lγ µ ν lL (cid:104) u µ [ Q emR , Q wL ] (cid:105) + X m l ¯ lv lL (cid:104)Q wL Q emR (cid:105) + h.c. (cid:9) + e (cid:88) l X ¯ l ( i /∂ + e / A ) l , (23)where Q emR = u † Q em u and Q wL = uQ wL u † .The LECs { K i , X i } are generically UV-divergent, and their corresponding renormalizedLECs are defined as: K ri ( µ ) = K i − Σ i λ, X ri ( µ ) = X i − Ξ i λ, (24)where λ = µ d − π (cid:18) d − −
12 [ln 4 π − γ E + 1] (cid:19) , (25)with µ the scale introduced in DR, d the number of the space-time dimensions, and γ E theEuler-Mascheroni constant. The values of { Σ i , Ξ i } are given in Refs. [42, 44], respectively.In connection with the SM electroweak sector, we find that X r and K r are sensitive tophysics at the scale q ∼ M W (in another word, they carry the large electroweak logarithms).It is customary to define the combination X phys6 ( µ ) ≡ X r ( µ ) − K r ( µ ) and take µ = M ρ inthe numerical analysis.With the effective Lagrangian above, the RCs to K l and π e were computed to O ( e p ) [27, 28, 45], and we shall briefly discuss the main results. First, the master for-mula of the K l decay rate is given by:Γ K l = C K G F M K π S EW | F π − K + (0) | I (0) Kl ( λ i ) (cid:0) δ Kl em + δ Kπ SU(2) (cid:1) , (26)11mong which the short-distance electroweak factor S EW is defined as : S EW ≡ − e (cid:20) − π ln M Z M ρ + ( X phys6 ) α s (cid:21) + δ QEDHO = 1 . , (27)where we take M ρ = 0 .
77 GeV. Here ( X ) phys α s ≈ . × − [30] summarizes the O ( α s ) pQCDcontribution to X phys6 (but not from higher-order contributions such as O ( α (2) s ), which weshall discuss later). This value is consistent with that quoted in Ref. [46] as well as the morecommonly cited value of 1.0232 by Marciano and Sirlin [47] . Meanwhile, the long-distanceEM correction is represented by the quantity δ Kl em . The ChPT estimations of their numericalvalues in different channels are summarized in Tab. I. We see that there are two sourcesof uncertainties in δ Kl em , namely (1) the neglected higher-order terms in the chiral powercounting, and (2) the LECs { K ri , X ri } . Here we are only interested in its dependence on thenon-unsuppressed LECs (i.e. those contributing to δF πK + ) : δ K ± l em = 2 e (cid:20) − X −
12 ˜ X phys6 ( M ρ ) − K r ( M ρ ) + K r ( M ρ ) + 23 K r ( M ρ ) + 23 K r ( M ρ ) (cid:21) + ...,δ K l em = 2 e (cid:20) X −
12 ˜ X phys6 ( M ρ ) (cid:21) + ..., (28)where ˜ X phys6 ( M ρ ) ≡ X phys6 ( M ρ ) + (2 π ) − ln( M Z /M ρ ) − ( X phys6 ) α s removes the large elec-troweak logarithm and the O ( α s ) pQCD correction from X phys6 . As a comparison, we candefine a similar quantity for π e , and its LEC-dependence reads: δ π ± e em = 2 e (cid:20) − X −
12 ˜ X phys6 ( M ρ ) (cid:21) + ... . (29)It is useful to contrast the results above with the case of the kaon and pion leptonic betadecay. We notice that both the K l and π l decay rate depend on the same combination ofLECs [44]: E r ≡ K r + 83 K r + 209 K r + 209 K r − X − X r + 4 X r − X phys6 , (30)so it will be canceled out in the ratio R A = Γ K µ / Γ π µ , which results in a reduced theoryuncertainty in the extraction of the ratio | V us /V ud | . This is, however, not the case in the There is a typo in Eq. (94) of Ref. [30], the factor 1/2 in front of e should not be there. On the other hand, the quoted value of S EW = 1 . V us phenomenology in the same paper, and therefore should not be used. Notice that X is scale-independent, so X r = X . The same goes for K , K and K in the Feynmangauge. R V = Γ K l / Γ π e recently introduced in Ref. [32], as we see that Eqs. (28) and (29)are not identical (except the ˜ X phys6 term which is common to all channels). Therefore, toreduce the theoretical uncertainty in R V we propose a first-principles calculation of X and − K r + K r + (2 / K r + K r ) and outline an appropriate method below. III. LATTICE QCD CALCULATION OF X AND X phys6 VIA THE γW BOX
We start by discussing the LECs X and X phys6 . They describe the electromagneticinteraction between leptons and pNGBs, so it is natural to expect that they could be relatedto the hadronic matrix element that occurs in the γW box diagram, Eq. (9). This sectionserves to derive such a relation.We first consider the electroweak RCs in the decay process φ i → φ f e + ν e in Sirlin’srepresentation, and restrict ourselves to the case where M φ i ≈ M φ f (cid:29) m e . In this limit, wecan define a power counting where p − p (cid:48) , p e and p ν all scale as a small expansion parameter∆. An enormous amount of simplification is observed if we retain the terms in δM only upto O (∆ ):1. The three-point function contribution to δF µfi vanishes;2. The weak axial charged-current contribution to the integrals in Eq. (15) vanishes. Thevector contribution does not vanish, but it survives only in the region where q (cid:48) ∼ ∆, soit is sufficient to replace T µνfi and Γ µfi by their respective “convection terms” [49] thatdescribe the IR behavior of these quantities. By doing so, the integrals in Eq. (15) areanalytically calculable.3. The remainder of the γW box contribution simplifies to δM AγW = (cid:3) V AγW ( φ f , φ i ) M ,where (cid:3) V AγW ( φ f , φ i ) ≡ ie M φ i (cid:90) d q (2 π ) q ) M W M W − q (cid:15) µναβ q α p β T fiµν ( q ; p, p ) F fi + (0) (31)shall be denoted as the “forward axial γW box”, as it probes the axial charged weakcurrent in T fiµν .From the above, we see that in the ∆ → δM is (cid:3) V AγW ( φ f , φ i ) which depends on the details of the non-perturbative QCD at the hadron scale.13t is, however, a well-defined hadronic matrix element which is calculable on lattice. Infact, Ref.[25] presented a first-principles calculation of (cid:3) V AγW ( π , π + ) by combining the directcomputation of the relevant four-point contraction diagrams at small Q = − q and a pQCDcalculation to O ( α s ) at large Q , achieving an impressive 1% overall accuracy. Other possiblemethods include the application of the Feynman-Hellmann theorem on lattice [50–52].Now it is clear how one could obtained the LECs X and X phys6 on the lattice: One repeatsthe calculation of δM in the ChPT and take the ∆ → a priori are the LECs. Therefore, comparing the expression of δM in the ∆ → { X , X phys6 } and (cid:3) V AγW . Of course, one needs to calculate the latter atleast in two different channels to fix X and X phys6 individually. In what follows we choose π e and K e to fulfill this task. A. Axial γW box diagram in π e decay In the π e channel, since the strong isospin breaking effects are small, the ∆ → T µν and Γ µ by their convectionterms: T µνπ π + ( q (cid:48) ; p (cid:48) , p ) → i (2 p − q (cid:48) ) µ F νπ π + ( p (cid:48) , p )( p − q (cid:48) ) − M π Γ µπ π + ( q (cid:48) ; p (cid:48) , p ) → − (2 p − q (cid:48) ) µ ( p (cid:48) − p ) · F π π + ( p (cid:48) , p )( p − q (cid:48) ) − M π . (32)With these, the total one-loop electroweak RCs to the decay amplitude in Sirlin’s represen-tation read ( u = ( p − p e ) , β = | (cid:126)p e | /E e ): δM = M (cid:26) α π (cid:20)
32 ln M W m e − M W M Z + 2 ln m e M γ −
114 + ˜ a g + 4 p e · pC ( u, M π , m e ) + 1 β ln 1 + β − β (cid:21) + (cid:3) V AγW ( π , π + ) + 12 δ QEDHO (cid:27) + α π G F √ u ν /p e (1 − γ ) v e p · F π π + p · p e β ln 1 + β − β + O (∆) . (33)Here, ˜ a g = − (3 / a pQCD + ˜ a res g ≈ − .
083 summarizes the O ( α s ) pQCD correction to allone-loop diagrams except the axial γW box . Meanwhile, C is the well-known IR-divergent See Ref.[53] for an early attempt to compare these two representations. This pQCD correction is small because it is not attached to a large electroweak logarithm, so it is notnecessary to include terms with higher powers in α s . In fact this term is usually discarded in most papers. C ( z, m , m ) = (cid:90) d qiπ q − M γ + i(cid:15) )(( q + p ) − m + i(cid:15) )(( q − p ) − m + i(cid:15) ) , (34)with p = m , p = m and z = ( p + p ) . On the other hand, taking the ∆ → O ( e p ) ChPT expression [45] gives: δM = M (cid:26) α π (cid:20) − −
32 ln m e µ + 2 ln m e M γ + 4 p e · pC ( u, M π , m e ) + 1 β ln 1 + β − β (cid:21) + 12 δ QEDHO + e (cid:18) − X − X phys6 (cid:19)(cid:27) + α π G F √ u ν /p e (1 − γ ) v e p · F π π + p · p e β ln 1 + β − β + O (∆) . (35)We see that Eq. (33) and (35) agree completely in their IR behavior, which is of courseexpected.We now want to equate these two expressions to obtain the relation between X i and (cid:3) V AγW . In doing so, we find the definition of ˜ X phys6 to be not particularly convenient, because(1) in Ref.[25] the pQCD correction is evaluated up to O ( α s ) instead of just O ( α s ), and (2)in the first-principles evaluation of Eq. (31), one requires a smooth connection between thepQCD-corrected integrand in the asymptotic region and the non-perturbative integrand atsmall Q . Thus, the procedure to “remove the pQCD correction” becomes rather unnatural.Therefore, we choose instead to express our result in terms of¯ X phys6 ( M ρ ) ≡ X phys6 ( M ρ ) + 12 π ln M Z M ρ , (36)that removes only the large electroweak logarithm but retains the full pQCD corrections toall orders. With this we obtain43 X + ¯ X phys6 ( M ρ ) = − πα (cid:18) (cid:3) V AγW ( π , π + ) − α π ln M W M ρ (cid:19) + 18 π (cid:18) − ˜ a g (cid:19) , (37)which is the first central result in this paper: It matches a specific linear combination of X and ¯ X phys6 to the axial γW box in π e decay. We observe that in the first bracket at theright of Eq. (37), the large electroweak logarithm contribution to (cid:3) V AγW has been subtractedout due to the use of ¯ X phys6 at the left.Substituting the lattice QCD result (cid:3) V AγW ( π , π + ) = 2 . × − [25] gives:43 X + ¯ X phys6 ( M ρ ) = 0 . box (8) ChPT , (38) Here we retain it for completeness. π − η mixing terms which scaleas M π / ( M η − M π ) ∼ X = − . × − and ¯ X phys6 = ˜ X phys6 +( X phys6 ) α s = (10 . . × − , with no robust estimation of the theory uncertainty. Thatimplies 43 X + ¯ X phys6 ( M ρ ) = 0 . , (resonance model) (39)which is significantly below the lattice result. This suggests that a careful first-principlesstudy of the LECs could lead to a visible change in the central values of δ em . B. Axial γW box diagram in K e deacy The same matching can in principle also be done on K e deacy in order to determineanother linear combination of X and ¯ X phys6 . The only extra complication is that M K issignificantly larger than M π so the ∆ → M K ≈ M π ≡ M φ ,which is always achievable on the lattice, the well-known SU(3) limit. In this limit allthe simplifications in Sirlin’s representation work again, provided that the axial γW boxdiagram for K e decay is now evaluated at the SU(3) symmetric point (i.e. m u = m d = m s )rather than on the physical point. Despite such an unphysical setting, the LECs extractedfrom this procedure can still be applied to physical processes because they are by definitionindependent of the quark masses.To evaluate the integrals in Eq. (15), one again replaces T µν and Γ µ by their convectionterms. In this case they read: T µνπ − K ( q (cid:48) ; p (cid:48) , p ) → − i (2 p (cid:48) + q (cid:48) ) µ F νπ − K ( p (cid:48) , p )( p (cid:48) + q (cid:48) ) − M φ , Γ µπ − K ( q (cid:48) ; p (cid:48) , p ) → (2 p (cid:48) + q (cid:48) ) µ ( p (cid:48) − p ) · F π − K ( p (cid:48) , p )( p (cid:48) + q (cid:48) ) − M φ . (40)With these, the total one-loop electroweak RCs to the K e decay amplitude in Sirlin’s16epresentation with the unphysical setting reads ( s = ( p (cid:48) + p e ) , β = | (cid:126)p e | /E e ) : δM = M (cid:26) α π (cid:20)
32 ln M W m e − M W M Z + 2 ln m e M γ −
114 + ˜ a g − p e · p (cid:48) C ( s, M φ , m e ) − β ln (cid:32) − (cid:115) − β β + i(cid:15) (cid:33)(cid:35) + (cid:0) (cid:3) V AγW ( π − , K ) (cid:1) SU(3) + 12 δ QEDHO (cid:41) − α π G F √ u ν /p e (1 − γ ) v e p (cid:48) · F π − K p (cid:48) · p e β ln (cid:32) − (cid:115) − β β + i(cid:15) (cid:33) + O (∆) . (41)Here, the subscript in (cid:0) (cid:3) V AγW ( π − , K ) (cid:1) SU(3) reminds us that it should be evaluated at theSU(3) symmetric point. On the other hand, in the ∆ → δM = M (cid:26) α π (cid:20) − −
32 ln m e µ + 2 ln m e M γ − p e · p (cid:48) C ( s, M φ , m e ) − β ln (cid:32) − (cid:115) − β β + i(cid:15) (cid:33)(cid:35) + 12 δ QEDHO + e (cid:18) X − X phys6 (cid:19)(cid:41) − α π G F √ u ν /p e (1 − γ ) v e p (cid:48) · F π − K p (cid:48) · p e β ln (cid:32) − (cid:115) − β β + i(cid:15) (cid:33) + O (∆) . (42)Matching the two expressions gives: − X + ¯ X phys6 ( M ρ ) = − πα (cid:18)(cid:0) (cid:3) V AγW ( π − , K ) (cid:1) SU(3) − α π ln M W M ρ (cid:19) + 18 π (cid:18) − ˜ a g (cid:19) , (43)which is the second central result in this paper. Therefore, a future lattice calculation of (cid:0) (cid:3) V AγW ( π − , K ) (cid:1) SU(3) allows a simultaneous determination of X and ¯ X phys6 ( M ρ ) from firstprinciples. A point to remember is that the matching above is valid only up to O ( e p ),therefore taking M φ (cid:28) Λ χ in the lattice calculation will help suppressing the theory un-certainties from the neglected O ( e p ) terms. In the flavor SU(3) limit, the K e γ W-boxdiagrams share the same types of quark contractions as π e in the lattice calculation. There-fore, it is straightforward to extend the calculation of γ W-box diagrams from the pion tothe kaon sector.One may wonder if calculating the axial γW box diagrams in more channels, such as K + e ,will also give us information about other LECs, for example the { K ri } that appear in δ K ± l em We take this opportunity to point out that the definition of the quantity X in Eq.(B.1) of Ref.[27] isincorrect. The correct definition follows Eq.(2.7) in Ref.[54]. γW box cannot carry any information of these LECs. To study them, we mustconstruct another type of correlation functions calculable on lattice, which we shall discussin the following section. IV. THE SETUP OF A LATTICE QCD CALCULATION OF THE { K ri } As far as the unsuppressed contribution to the K l decay rate is concerned, the only extraLEC we need to calculate is the combination − K r + K r + (2 / K r + K r ) (see Eq.(28)).However, if we wish to be more precise by also studying the RCs to the form factor F πK − ,then we need to know K r , ..., K r individually [27]. At the same time, K r and K r are alsointeresting because in the large- N c limit they satisfy the relations K r = − K r and K r = 2 K r ,[42, 55], so by calculating them one could test the precision of the large- N c predictions fromfirst principles. Therefore in this section we shall outline a strategy to calculate K r , ..., K r on the lattice. While the remaining { K ri } are also interesting by themselves (e.g., K r , ..., K r contribute to the K ± − K mass splitting at O ( e p ) [42, 56]), we will not discuss them here.Ref. [29] expressed the { K ri } in terms of a series of four-point functions, which they latercalculated using resonance models to obtain an estimate of the LECs. We find that sucha formalism is indeed a good starting point to motivate a realistic lattice QCD calculationupon appropriate modifications (for instance, the chiral limit, which is not attainable onlattice). In what follows, we shall derive the modified four-point function representation ofthe LECs. Of course we could work on the physical point, but since the variation of non-zeroquark masses do not give rise to extra singularities in these correlation functions (which canbe seen from the Feynman diagrams in Fig.4, 5 and 6), here we shall present our result inthe SU(3) limit, M π = M K = M η ≡ M φ , which brings a great simplification to the involvedloop functions. 18 . Lepton-free Lagrangian with external sources and spurions We start again by discussing the SM Lagrangian responsible for the semi-leptonic betadecay processes, which was explained in some detail in Ref. [38]. First, the UV-divergencesin the electroweak sector are reabsorbed into the respective coupling constants and massparameters following the on-shell renormalization scheme [37]. Next, since here we areonly interested in the LECs that do not involve the lepton-hadron interaction, we can take G F → L nl = L QCD − e ¯ ψQ em / A < ψ − F <µν F µν< − ξ ( ∂ µ A µ< ) + 12 M γ A <µ A µ< , (44)with ψ = ( u, d, s ) T , and ξ = 1 for the Feynman gauge. Here A µ< represents the photon fieldwith its propagator being multiplied by a Pauli-Villars regulator with Λ = M W : D µν< ( z ) = (cid:90) d q (2 π ) e − iq · z − ig µν q − M γ M W M W − q . (45)This extra regulator comes from the splitting of the full photon propagator in the on-shellrenormalization scheme: 1 q = 1 q − M W + 1 q M W M W − q . (46)To make a connection with the chiral Lagrangian in Sec. II B, we generalize L nl by intro-ducing external sources { l µ , r µ } and spurion fields { q L , q R } :˜ L nl = L QCD + ¯ ψ L γ µ (cid:0) l µ + q L A <µ (cid:1) ψ L + ¯ ψ R γ µ (cid:0) r µ + q R A <µ (cid:1) ψ R − F <µν F µν< − ξ ( ∂ µ A µ< ) + 12 M γ A <µ A µ< . (47)However, unlike Sec. II B, here we do not identify the external sources and spurions withthe charge matrices and the fermion bilinears, but rather define l µ = v µ − a µ , r µ = v µ + a µ , q L = q V − q A , q R = q V + q A , and decompose them into flavor octet components: v µ = v aµ λ a , a µ = a aµ λ a , q V = q aV λ a , q A = q aA λ a , (48)where { λ a } are the Gell-Mann matrices. We may also define flavor-octet vector and axialcurrents as: V aµ = ¯ ψγ µ λ a ψ, A aµ = ¯ ψγ µ γ λ a ψ . (49)19hus we can write: ˜ L nl = L QCD + V aµ (cid:0) v aµ + q aV A <µ (cid:1) + A aµ (cid:0) a aµ + q aA A <µ (cid:1) − F <µν F µν< − ξ ( ∂ µ A µ< ) + 12 M γ A <µ A µ< . (50) B. Defining the four-point correlation functions
Using the generating functional of the action ˜ S nl in the presence of external sources andspurions: W ( v, a, q V , q A ) = 1 Z (cid:90) D ( ¯ ψ, ψ, A < ) exp { i ˜ S nl ( v, a, q V , q A ) } , (51)we can define three types of four-point correlation functions [29]: (cid:10) A aα A bβ Q cV Q dV (cid:11) ≡ (cid:90) d xd yd ze ik · y δ W ( v, a, q V , q A ) δa aα ( x ) δa bβ ( y ) δq cV ( z ) δq dV (0) (cid:12)(cid:12)(cid:12)(cid:12) , (cid:10) A aα A bβ Q cA Q dA (cid:11) ≡ (cid:90) d xd yd ze ik · y δ W ( v, a, q V , q A ) δa aα ( x ) δa bβ ( y ) δq cA ( z ) δq dA (0) (cid:12)(cid:12)(cid:12)(cid:12) , (cid:10) V aα V bβ Q cV Q dV (cid:11) ≡ (cid:90) d xd yd ze ik · y δ W ( v, a, q V , q A ) δv aα ( x ) δv bβ ( y ) δq cV ( z ) δq dV (0) (cid:12)(cid:12)(cid:12)(cid:12) , (52)where k is a freely-chosen external momentum. The “ | ” means that we take v µ = a µ = q V = q A = 0 after the functional derivative, which decouples the quarks from the photon.Obviously, the only possible Lorentz structures of these correlation functions are g αβ and k α k β .Using Eq. (50), it is straightforward to show that the correlation functions above can bewritten as: (cid:10) A aα A bβ Q cV Q dV (cid:11) = (cid:90) d xd yd ze ik · y (cid:104) | T { A aα ( x ) A bβ ( y ) V cρ ( z ) V dσ (0) } | (cid:105) D ρσ< ( z ) , (cid:10) A aα A bβ Q cA Q dA (cid:11) = (cid:90) d xd yd ze ik · y (cid:104) | T { A aα ( x ) A bβ ( y ) A cρ ( z ) A dσ (0) } | (cid:105) D ρσ< ( z ) , (cid:10) V aα V bβ Q cV Q dV (cid:11) = (cid:90) d xd yd ze ik · y (cid:104) | T { V aα ( x ) V bβ ( y ) V cρ ( z ) V dσ (0) } | (cid:105) D ρσ< ( z ) . (53)Note that (cid:104) | T { ... } | (cid:105) are pure QCD matrix elements, so the RHS of the equations aboveare in principle calculable on the lattice. For instance, the hadronic part in the correlationfunctions defined in Eq. (53) can be calculated using the sequential-source propagators.Combining the hadronic part with the photonic weight function of D ρσ< ( z ), the whole 4-point correlation functions can be constructed in lattice simulations.20 Figure 1: Diagrammatic representation of (cid:68) A aα A bβ Q cV Q dV (cid:69) . The other correlation functions can berepresented in a similar way. There is a simple diagrammatic interpretation of the correlation functions. Take (cid:10) A aα A bβ Q cV Q dV (cid:11) as an example: It is nothing but the amplitude iM ( q cV (0) q dV ( k ) → a aα (0) a bβ ( k ))calculated using the action ˜ S nl ( v, a, q V , q A ), see Fig. 1 (notice that v, a, q V , q A are not dy-namical fields and do not propagate internally). Therefore, the strategy is to make use ofthe ChPT representation of ˜ S nl to calculate the correlation functions. The results obvi-ously depend on the unknown LECs { K ri } . Comparing the ChPT expression and the latticecalculation of the correlation functions then allows us to determine the unknown LECs.Before proceeding with the ChPT calculation, we make a final comment on the correlationfunctions in Eq. (53). Due to the existence of the Pauli-Villars regulator in D ρσ< ( z ), all thespace-time integrals with respect to x, y, z are convergent. Still, if the LECs probe the physicsat the scale q ∼ M W , then the corresponding correlation functions are not fully computedby lattice QCD alone because this will require a lattice spacing of the size a ∼ /M W whichis not achievable in practice. Fortunately, unlike K r (see the discussion in Sec. II B), noneof the LECs K r , ..., K r is sensitive to physics at the UV-scale, so the use of a typical latticespacing is sufficient. C. ChPT representation of the four-point functions
The four-point functions defined in Eq. (52) were already calculated in ChPT to O ( e p )in Ref. [29], but there they worked in the chiral limit and retained only the g αβ structure,making the results not directly applicable for the lattice. Here, we redo the calculation atthe SU(3) symmetric point with non-zero M φ and include both the g αβ and k α k β structures.21 Figure 2: O ( p ) contributions to (cid:68) A aα A bβ Q cA Q dA (cid:69) . Following that reference, we cast our results in terms of the four flavor basis defined below:ˆ e = f acg f bdg + f adg f bcg , ˆ e = δ ac δ bd + δ ad δ bc , ˆ e = d acg d bdg + d adg d bcg , ˆ e = f abg f cdg . (54)Up to O ( e p ), the four-point functions read: (cid:10) A aα A bβ Q cV Q dV (cid:11) = iF g αβ (cid:88) i =1 α ( i ) AV ˆ e i + iF k α k β k − M φ (cid:88) i =1 β ( i ) AV ˆ e i + (cid:10) A aα A bβ Q cV Q dV (cid:11) φ + (cid:10) A aα A bβ Q cV Q dV (cid:11) γ , (cid:10) A aα A bβ Q cA Q dA (cid:11) = iF g αβ (cid:20) δ ad δ bc F k − M γ − δ ac δ bd F M γ (cid:21) + iF g αβ (cid:88) i =1 α ( i ) AA ˆ e i + iF k α k β k − M φ (cid:88) i =1 β ( i ) AA ˆ e i + (cid:10) A aα A bβ Q cA Q dA (cid:11) φ + (cid:10) A aα A bβ Q cA Q dA (cid:11) γ , (cid:10) V aα V bβ Q cV Q dV (cid:11) = iF g αβ α (1) V V ˆ e + (cid:10) V aα V bβ Q cV Q dV (cid:11) φ . (55)Let us explain the results above. First, the square bracket in (cid:10) A aα A bβ Q cA Q dA (cid:11) represents theonly O ( p ) contribution that comes from the diagrams shown in Fig. 2 which is, for somereason, missing in Ref. [29]. All the others are O ( e p ). The coefficients α ( i ) and β ( i ) containthe contributions from the LECs (as depicted in Fig. 3) as well as the UV-divergent part ofthe loop contributions . The remaining parts that carry the subscript φ and γ denote theUV-finite contributions of the meson and photon loop diagrams, further detail can be foundin Appendix A. We find that Eq. (2.15) in Ref. [29] is wrong by a sign. Figure 3: LEC contributions to the correlation functions. The gray dot represents the countertermvertex.
Let us concentrate on the coefficients α ( i ) and β ( i ) . They read: α (1) AV = 2 K r + 2 K r + 2 K r + 2 K r + 4 K r − K + 2 K + 5 Z − π ln µ M φ ,α (2) AV = − K r + 2 K r + 43 K r + 43 K r + 3 Z π ln µ M φ ,α (3) AV = 6 K r + 6 K r + 2 K r + 2 K r + 9 Z π ln µ M φ , (56) β (1) AV = − K r − K r − K r − K r − K r + K + 5 − Z π ln µ M φ ,β (2) AV = 4 K r − K r − K r − K r − Z π ln µ M φ ,β (3) AV = − K r − K r − K r − K r − Z π ln µ M φ ,β (4) AV = − K r + K − π ln µ M φ , (57) α (1) AA = 2 K r − K r + 2 K r − K r + 4 K r + K + 2 K − Z + 232 π ln µ M φ ,α (2) AA = − K r − K r + 43 K r − K r − Z π ln µ M φ ,α (3) AA = 6 K r − K r + 2 K r − K r − Z π ln µ M φ , (58)23 (1) AA = − K r + 2 K r − K r + 2 K r − K r − K + 5 + 10 Z π ln µ M φ ,β (2) AA = 4 K r + 2 K r − K r + 43 K r + 3 Z π ln µ M φ ,β (3) AA = − K r + 6 K r − K r + 2 K r + 9 Z π ln µ M φ ,β (4) AA = − K r − K − π ln µ M φ , (59)and finally, α (1) V V = K + 2 K . They provide an over-complete set of equations to solve forthe needed LECs, an example of solutions is given in Appendix B. So in principle one couldcalculate each correlation function with several flavor combinations to extract the neededcoefficients α ( i ) and β ( i ) , and with them one could determine all the { K ri } individually.However, if we are only interested in the unsuppressed combination of { K ri } that enters δ K ± l em (see Eq. (28)), things are much simpler: It can be obtained from a single four-pointfunction at zero external momentum: (cid:10) A α A β Q V Q V (cid:11) k =0 = iF g αβ (cid:34) − K r + 2 K r + 43 K r + 43 K r + 3 Z π (cid:32) − µ M φ (cid:33)(cid:35) , (60)which is the last central result of this paper.This completes the setup of the problem for the future lattice calculation. The chiralLEC’s are unambiguously related to a 4-point correlation function and the axial γW box.Using lattice QCD simulations, one can expect to determine the LECs with controlled un-certainties and provide useful information for the electromagnetic corrections to K l decays. V. CONCLUSIONS
We have entered a new era where lattice QCD becomes increasingly important in thestudies of high-precision electromagnetic effects in low-energy phenomena. In particular, itis now timely to extend its impact to the field of semileptonic beta decays which plays adecisive role in the precision test of the top-row CKM matrix unitarity and the implicationsfor BSM physics therein.It is expected to be extremely challenging to perform a full lattice QCD calculation to thevirtual + real QED corrections to the kaon semileptonic decay rate, of which the estimatedtime span is of the order of 10 years. Given the current status of the CKM unitarity, it is24ighly desirable to look for an alternative starting point such that lattice QCD can makeimmediate impact to the field. In this paper we propose a strategy of such kind. We firstpoint out that, at O ( e p ) in chiral power counting, there are only three combinations ofLECs that are relevant for K l and π e decays: X , ¯ X phys6 and − K r + K r + (2 / K r + K r ). Based on a careful comparison between the Sirlin’s representation and the ChPTrepresentation of the QED effects, we show that these LECs can all be pinned down bycalculating a few simple quantities on the lattice.To obtain the LECs X and ¯ X phys6 , we need to calculate the axial γW -box diagrams forthe π π + and π − K systems in the degenerate limit. The former was already performedin Ref. [25], which translates into a determination of (4 / X + ¯ X phys6 with 10% accuracy.We observe that the outcome is significantly different from the resonant model calculationwidely adopted in the existing K l RC analysis, which adds to the urgency of our proposedcalculations. The π − K axial box can be computed in exactly the same way, and in fact itsresult will be available in the near future.On the other hand, the extraction of the LECs { K ri } will be based on the lattice calcu-lation of the four-point correlation functions defined in Eq.(53) which can be done using,e.g., sequential-source propagators. In particular, we show an example in Appendix B whereall individual { K ri } are obtained from the coefficients { β ( i ) } in the four-point functions. Inpractice it is of course not so trivial, because these coefficients are associated to the k α k β structure that is sensitive to the direction of the external momentum k , which may lead toextra systematic uncertainties due to the breaking of the exact rotational symmetry on thelattice (it is not possible to solve for all individual { K ri } using only the simpler coefficients { α ( i ) } without imposing further assumptions, such as large- N c approximation, which onenormally avoids in first-principles calculations). Fortunately, as far as the relevant linearcombination − K r + K r + (2 / K r + K r ) is concerned, one needs only to calculate a singlefour-point correlation function with zero external momentum, as indicated in Eq.(60). Wewill defer the discussions of the actual lattice QCD setup needed for such a calculation to afuture work.Our proposed calculation will not only improve the precision of the | V us | extraction from K l alone, but will also reduce the theoretical uncertainty in the ratio R V = Γ K l / Γ π e thathelps us to better understand the disagreement between the K l and K l extractions of | V us | .25 igure 4: Contributions from meson loops. The black circle denotes the O ( e ) vertex. The thirddiagram contains a meson pole. Acknowledgements
We thank Vincenzo Cirigliano and Bachir Moussallam for many inspiring discussions.This work is supported in part by the DFG (Grant No. TRR110) and the NSFC (Grant No.11621131001) through the funds provided to the Sino-German CRC 110 “Symmetries andthe Emergence of Structure in QCD” (U-G.M and C.Y.S), by the Alexander von HumboldtFoundation through the Humboldt Research Fellowship (C.Y.S), by the Chinese Academy ofSciences (CAS) through a President’s International Fellowship Initiative (PIFI) (Grant No.2018DM0034) and by the VolkswagenStiftung (Grant No. 93562) (U-G.M), by EU Horizon2020 research and innovation programme, STRONG-2020 project under grant agreement No824093 and by the German-Mexican research collaboration Grant No. 278017 (CONACyT)and No. SP 778/4-1 (DFG) (M.G), by NSFC of China under Grant No. 11775002 (X.F)and by DOE grant DE-SC0010339 (L.C.J).
Appendix A: loop contributions to the four-point functions
In this Appendix we present the UV-finite parts of the one-loop contributions to thefour-point correlation functions in Eq. (55) . We acknowledge the power of Package-X that provides the fully analytic expressions of all loop integralsin terms of elementary functions [57, 58]. Figure 5: Contributions from photon loops without a meson pole. O ( e p ) contributions from meson loops The meson loop contributions are depicted in Fig. 4. The results are: (cid:10) A aα A bβ Q cV Q dV (cid:11) φ = − (cid:10) A aα A bβ Q cA Q dA (cid:11) φ = iF Z π (cid:18)
52 ˆ e + 3ˆ e + 92 ˆ e (cid:19) (cid:0) k , M φ ) (cid:1) (cid:32) g αβ − k α k β k − M φ (cid:33)(cid:10) V aα V bβ Q cV Q dV (cid:11) φ = 0 , (A1)where Λ( k , M φ ) = (cid:113) k ( k − M φ ) k ln (cid:113) k ( k − M φ ) − k + 2 M φ M φ . (A2) O ( e p ) contributions from photon loops The photon loop contributions involve more Feynman diagrams, so for the benefits offuture cross-check, we split them into two pieces: (cid:104) ... (cid:105) γ = (cid:104) ... (cid:105) γ + (cid:104) ... (cid:105) γ , where the twoterms on the RHS denote contribution without and with a meson pole, respectively. a. without meson pole The photon loop contributions without a meson pole are depicted in Fig. 5. The resultsread: (cid:10) A aα A bβ Q cV Q dV (cid:11) γ = − iF π ˆ e − ˆ e (cid:32) k − M φ k ln M φ M φ − k + 1 (cid:33) g αβ (cid:10) A aα A bβ Q cA Q dA (cid:11) γ = − iF π ˆ e (cid:32) M φ − k M φ Λ( k , M φ ) + ( k − M φ ) k M φ ln M φ M φ − k + 12 (cid:33) g αβ . (A3)27 Figure 6: Contributions from photon loops with a meson pole. b. with meson pole
The contributions from photon loops with a meson pole are depicted in Fig. 6. Theresults read: (cid:10) A aα A bβ Q cV Q dV (cid:11) γ = iF π ˆ e − ˆ e k + M φ k (cid:32) k − M φ k ln M φ M φ − k + 1 (cid:33) k α k β k − M φ (cid:10) A aα A bβ Q cA Q dA (cid:11) γ = iF π (cid:40) ˆ e (cid:34) M φ − k M φ Λ( k , M φ ) + 2( k ) − k ) M φ + 2 k M φ − M φ k ) M φ × ln M φ M φ − k + 3 k + M φ k (cid:35) + ˆ e (cid:34) − ( k ) + 4 k M φ − M φ k ) ln M φ M φ − k − k + 5 M φ k (cid:21)(cid:27) k α k β k − M b . (A4)28 ppendix B: Obtaining every K ri individually In this Appendix we present one (out of the many possible) set of solutions for K r , ..., K r in terms of the coefficients { α ( i ) , β ( i ) } defined in Eq. (55). Here we make use of only { β ( i ) } : K r = 18 (cid:32) β (1) AA − β (3) AA − β (4) AA + β (1) AV − β (3) AV − β (4) AV − π ln µ M φ (cid:33) K r = 18 (cid:32) − β (1) AA + β (3) AA + β (4) AA + β (1) AV − β (3) AV − β (4) AV − Z π ln µ M φ (cid:33) K r = 124 (cid:32) − β (1) AA + 3 β (2) AA + β (3) AA + 3 β (4) AA − β (1) AV + 3 β (2) AV + β (3) AV + 3 β (4) AV + 916 π ln µ M φ (cid:33) K r = 112 (cid:32) − β (1) AA + 3 β (2) AA + β (3) AA + 3 β (4) AA + 3 β (1) AV − β (2) AV − β (3) AV − β (4) AV − Z π ln µ M φ (cid:33) K r = 18 (cid:32) − β (1) AA + β (3) AA + 3 β (4) AA − β (1) AV + β (3) AV + 3 β (4) AV + 916 π ln µ M φ (cid:33) K r = 18 (cid:32) β (1) AA − β (3) AA − β (4) AA − β (1) AV + β (3) AV + 3 β (4) AV − Z π ln µ M φ (cid:33) . (B1) [1] C.-Y. Seng, M. Gorchtein, H. H. Patel and M. J. Ramsey-Musolf, Reduced HadronicUncertainty in the Determination of V ud , Phys. Rev. Lett. (2018), no. 24 241804[ ].[2] C. Y. Seng, M. Gorchtein and M. J. Ramsey-Musolf,
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