IIsospin Breaking Effects in K (cid:96) Decays
Peter StofferAlbert Einstein Center for Fundamental Physics,Institute for Theoretical Physics, University of Bern,Sidlerstr. 5, CH-3012 Bern, Switzerland
Abstract
In the framework of chiral perturbation theory with photons and leptons, the one-loop isospin breaking effects in K (cid:96) decays due to both the photonic contribution and the quark and meson mass differences are computed.A comparison with the isospin breaking corrections applied by recent high statistics K e experiments is performed.The calculation can be used to correct the existing form factor measurements by isospin breaking effects that have notyet been taken into account in the experimental analysis. Based on the present work, possible forthcoming experimentson K e decays could correct the isospin breaking effects in a more consistent way. Contents K (cid:96) Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Definition of the Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Matrix Element, Form Factors and Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Radiative Decay K (cid:96) γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Definition of the Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Matrix Element, Form Factors and Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . 8 χ PT Calculation of the Amplitudes 8 × PHOTOS . . . . . . . . . . . . . . . . . . . . . . . . 30 a r X i v : . [ h e p - ph ] D ec cknowledgements 32A Loop Functions 33 A.1 Scalar Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33A.2 Tensor-Coefficient Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34A.3 Infrared Divergences in Loop Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
B Kinematics 35
B.1 Lorentz Frames and Transformations in K (cid:96) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35B.2 Lorentz Frames and Transformations in K (cid:96) γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 C Decay Rates 40
C.1 Decay Rate for K (cid:96) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40C.1.1 Isospin Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40C.1.2 Broken Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42C.2 Decay Rate for K (cid:96) γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 D χ PT with Photons and Leptons 46E Feynman Diagrams 48
E.1 Mass Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48E.1.1 Loop Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48E.1.2 Counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52E.1.3 External Leg Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52E.2 Photonic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54E.2.1 Loop Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54E.2.2 Counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59E.2.3 External Leg Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
References 61 Introduction
High-precision hadron physics at low energies pursues mainly two aims: a better understanding of the stronginteraction in its non-perturbative regime and the indirect search for new physics beyond the standard model.As perturbative QCD is not applicable, one has to use non-perturbative methods like effective field theories,lattice simulations or dispersion relations. The effective theory describing the strong interaction at low energy ischiral perturbation theory ( χ PT, [1, 2, 3]). In order to render it predictive, one has to determine the parametersof the theory, the low-energy constants (LECs), either by comparison with experiments or with the help of latticecalculations.The K (cid:96) decay is for several reasons a particularly interesting process. The physical region starts at the ππ threshold, i.e. at lower energies than Kπ scattering, which gives access to the same low-energy constants. χ PT,being an expansion in the meson masses and momenta, should therefore give a better description of K (cid:96) than Kπ scattering. Besides providing a very clean access to some of the LECs, K (cid:96) is, due to its final state, one ofthe best sources of information on ππ interaction [4, 5, 6].The recent high-statistics K (cid:96) experiments E865 at BNL [7] and NA48/2 at CERN [6, 8] have achieved animpressive accuracy. The statistical errors at the sub-percent level ask for a consistent treatment of isospinbreaking effects. Usually, theoretical calculations are performed in an ideal world with intact isospin, the SU (2) symmetry of up- and down-quarks. The quark mass difference and the electromagnetism break isospin symmetryat the percent level.Isospin breaking effects in K (cid:96) have been studied in the previous literature and played a major role concerningthe success of standard χ PT. In [9], the effect of quark and meson mass differences on the phase shifts wasstudied. The inclusion of this effect brought the NA48/2 measurement of the scattering lengths into perfectagreement with the prediction of the χ PT/Roy equation analysis [10]. For a review of these developments, see[11]. The mass effects on the phases at two-loop order have been recently studied in an elaborate dispersiveframework [12], which confirms the previous results. In both works, the photonic effects are assumed to betreated consistently in the experimental analysis. The earlier work [13, 14] treats both mass and photoniceffects. However, the calculation of virtual photon effects is incomplete and real photon corrections are takeninto account only in the soft approximation.The experimental analysis of the latest experiment [6, 8] treats photonic corrections with the semi-classicalGamow-Sommerfeld factor and PHOTOS Monte Carlo [15], which assumes phase space factorisation.The need for a theoretical treatment of the full radiative corrections was pointed out in [9] and consideredas a long-term project. With the present work, I intend to fill this gap. The obtained results enable a bettercorrection of isospin effects in the data: • as I will explain below, one can improve already now the handling of isospin effects in the data analysis; • in the future, an event generator which incorporates the matrix element calculated here should be writtenand used to perform the data analysis.The paper is organised as follows. In section 2, I define the kinematics, matrix elements and form factors of K (cid:96) and the radiative decay K (cid:96) γ . In section 3, I calculate the matrix elements within χ PT including leptonsand photons [16, 17]. In section 4, I present the strategy of extracting the isospin corrections and performthe phase space integration for the radiative decay. The cancellation of both infrared and mass divergencesis demonstrated. In section 5, the isospin corrections are evaluated numerically. I compare the full radiativeprocess with the soft photon approximation and with the strategy used in the experimental analysis. Theappendices give details on the calculation and explicit results for the matrix elements.It should be noted that large parts of this work assume a small lepton mass and are therefore not applicableto the muonic mode of the process. K (cid:96) Decay K (cid:96) is the semileptonic decay of a kaon into two pions, a lepton and a neutrino. Let us consider here thefollowing charged mode: K + ( p ) → π + ( p ) π − ( p ) (cid:96) + ( p (cid:96) ) ν (cid:96) ( p ν ) , (1)3here (cid:96) ∈ { e, µ } is either an electron or a muon.The kinematics of the decay (1) can be described by five variables. The same conventions as in [18] willbe used, first introduced by Cabibbo and Maksymowicz [5]. There appear three different reference frames: therest frame of the kaon Σ K , the π + π − centre-of-mass frame Σ π and the (cid:96) + ν centre-of-mass frame Σ (cid:96)ν . Thesituation is sketched in figure 1. K + π − π + ℓ + ν ℓ φθ π θ ℓ Σ π Σ ℓν ~c ~d~v Figure 1:
The reference frames and the kinematic variables for the K (cid:96) decay. The five kinematic variables are then: • s , the total centre-of-mass squared energy of the two pions, • s (cid:96) , the total centre-of-mass squared energy of the two leptons, • θ π , the angle between the π + in Σ π and the dipion line of flight in Σ K , • θ (cid:96) , the angle between the (cid:96) + in Σ (cid:96)ν and the dilepton line of flight in Σ K , • φ , the angle between the dipion plane and the dilepton plane in Σ K .The (physical) ranges of these variables are: M π + ≤ s ≤ ( M K + − m (cid:96) ) ,m (cid:96) ≤ s (cid:96) ≤ ( M K + − √ s ) , ≤ θ π ≤ π, ≤ θ (cid:96) ≤ π, ≤ φ ≤ π. (2)Following [18], I define the four-momenta: P := p + p , Q := p − p , L := p (cid:96) + p ν , N := p (cid:96) − p ν . (3)Total momentum conservation implies p = P + L .I will use the Mandelstam variables s := ( p + p ) , t := ( p − p ) , u := ( p − p ) (4)and the abbreviation z (cid:96) := m (cid:96) /s (cid:96) ,X := 12 λ / K(cid:96) ( s ) := 12 λ / ( M K + , s, s (cid:96) ) , λ ( a, b, c ) := a + b + c − ab + bc + ca ) ,σ π := (cid:115) − M π + s ,ν := t − u = − σ π X cos θ π , Σ := s + t + u = M K + + 2 M π + + s (cid:96) . (5)In the appendix B.1, the Lorentz transformations between the three reference frames are determined and theLorentz invariant products of the momenta are computed.4 .1.2 Matrix Element, Form Factors and Decay Rate2.1.2.1 K (cid:96) in the Isospin Limit After integrating out the W boson in the standard model, we end up with a Fermi type current-current inter-action. If we switch off the electromagnetic interaction, the matrix element of K (cid:96) (cid:10) π + ( p ) π − ( p ) (cid:96) + ( p (cid:96) ) ν (cid:96) ( p ν ) (cid:12)(cid:12) K + ( p ) (cid:11) in = (cid:10) π + ( p ) π − ( p ) (cid:96) + ( p (cid:96) ) ν (cid:96) ( p ν ) (cid:12)(cid:12) iT (cid:12)(cid:12) K + ( p ) (cid:11) = i (2 π ) δ (4) ( p − P − L ) T (6)splits up into a leptonic times a hadronic part: T = G F √ V ∗ us ¯ u ( p ν ) γ µ (1 − γ ) v ( p (cid:96) ) (cid:10) π + ( p ) π − ( p ) (cid:12)(cid:12) ¯ sγ µ (1 − γ ) u (cid:12)(cid:12) K + ( p ) (cid:11) . (7)The hadronic matrix element exhibits the usual V − A structure of weak interaction. Its Lorentz structureallows us to write the two contributions as (cid:10) π + ( p ) π − ( p ) (cid:12)(cid:12) V µ (0) (cid:12)(cid:12) K + ( p ) (cid:11) = − HM K + (cid:15) µνρσ L ν P ρ Q σ , (8) (cid:10) π + ( p ) π − ( p ) (cid:12)(cid:12) A µ (0) (cid:12)(cid:12) K + ( p ) (cid:11) = − i M K + ( P µ F + Q µ G + L µ R ) , (9)where V µ = ¯ sγ µ u and A µ = ¯ sγ µ γ u . The form factors F , G , R and H are functions of s , s (cid:96) and cos θ π (or s , t and u ).In order to write the decay rate in a compact form, it is convenient to introduce new form factors as linearcombinations of F , G , R and H (following [19, 18]) that correspond to definite helicity amplitudes: F := XF + σ π ( P L ) cos θ π G,F := σ π √ ss (cid:96) G,F := σ π X √ ss (cid:96) HM K + ,F := − ( P L ) F − s (cid:96) R − σ π X cos θ π G. (10)The partial decay rate for the K (cid:96) decay is given by d Γ = 12 M K + (2 π ) (cid:88) spins |T | δ (4) ( p − P − L ) d p p d p p d p (cid:96) p (cid:96) d p ν p ν . (11)Since the kinematics is described by five phase space variables, seven integrals can be performed. This leads to d Γ = G F | V us | (1 − z (cid:96) ) σ π ( s ) X π M K + J ( s, s (cid:96) , θ π , θ (cid:96) , φ ) ds ds (cid:96) d cos θ π d cos θ (cid:96) dφ. (12)The explicit expression for J is derived in the appendix C.1.1. F corresponds to the helicity amplitude of the spin 0 or temporal polarisation of the virtual W boson. Itscontribution to the decay rate is therefore helicity suppressed by a factor m (cid:96) and invisible in the electron mode.In the chiral expansion, F appears due to the chiral anomaly, which is at the level of the Lagrangian an O ( p ) effect. Thus, the important form factors for the experiment are F and F , or equivalently F and G . K (cid:96) in the Case of Broken Isospin In the presence of electromagnetism, the above factorisation of the K (cid:96) matrix element into a hadronic and aleptonic part is no longer valid. In addition to the V − A structure, a tensorial form factor has to be taken intoaccount [13, 14]: T = G F √ V ∗ us (cid:0) ¯ u ( p ν ) γ µ (1 − γ ) v ( p (cid:96) ) ( V µ − A µ ) + ¯ u ( p ν ) σ µν (1 + γ ) v ( p (cid:96) ) T µν (cid:1) , V µ := − HM K + (cid:15) µνρσ L ν P ρ Q σ , A µ := − i M K + ( P µ F + Q µ G + L µ R ) , T µν := 1 M K + p µ p ν T, (13)5here σ µν = i [ γ µ , γ ν ] . The form factors F , G , R , H and T depend now on all five kinematic variables s , s (cid:96) , θ π , θ (cid:96) and φ .I follow [14] and introduce in addition to (10) the form factor F (with a slightly different normalisation): F := σ π ( s ) ss (cid:96) M K + m (cid:96) T. (14)Still, the phase space is parametrised by the same five kinematic variables and the differential decay rate can bewritten as in (12). In the isospin broken case, the presence of the additional tensorial form factor changes thefunction J . We will see that F is finite in the limit m (cid:96) → . Its contribution to the decay rate is suppressedby m (cid:96) . Details are given in the appendix C.1.2.It is convenient to define the following additional Lorentz invariants [14]: t (cid:96) := ( p − p (cid:96) ) , u (cid:96) := ( p − p ν ) , s (cid:96) := ( p + p (cid:96) ) , s (cid:96) := ( p + p (cid:96) ) . (15) K (cid:96) γ If we consider electromagnetic corrections to K (cid:96) , we have to take into account contributions from both virtualphotons and real photon emission, because only an appropriate inclusive observable is free of infrared singular-ities. As long as we restrict ourselves to O ( e ) corrections, the radiative process with just one additional finalstate photon is the only one of interest (each additional photon comes along with a factor e in the decay rate).The radiative process K (cid:96) γ is defined as K + ( p ) → π + ( p ) π − ( p ) (cid:96) + ( p (cid:96) ) ν (cid:96) ( p ν ) γ ( q ) . (16)There are several possibilities to parametrise the phase space. I find it most convenient to replace the dileptonsub-phase space of K (cid:96) by a convenient three particle phase space.I describe the kinematics still in three reference frames: the rest frame of the kaon Σ K , the dipion centre-of-mass frame Σ π and the dilepton-photon centre-of-mass frame Σ (cid:96)νγ . In total, we need eight phase spacevariables: • s , the total centre-of-mass squared energy of the two pions, • s (cid:96) , the total centre-of-mass squared energy of the dilepton-photon system, • θ π , the angle between the π + in Σ π and the dipion line of flight in Σ K , • θ γ , the angle between the photon in Σ (cid:96)νγ and the (cid:96)νγ line of flight in Σ K , • φ , the angle between the dipion plane and the ( (cid:96)ν ) γ plane in Σ K . • q , the photon energy in Σ (cid:96)νγ , • p (cid:96) , the lepton energy in Σ (cid:96)νγ , • φ (cid:96) , the angle between the (cid:96)ν plane in Σ (cid:96)νγ and the ( (cid:96)ν ) γ plane in Σ K .The variables s , s (cid:96) , θ π are defined in analogy to the K (cid:96) decay. The reason for the chosen parametrisation ofthe (cid:96)νγ subsystem is that p (cid:96) and φ (cid:96) are of purely kinematic nature, i.e. the dynamics depends only on the sixother variables.Instead of the q and p (cid:96) , I will mostly use the dimensionless variables x := 2 Lqs (cid:96) , y := 2 Lp (cid:96) s (cid:96) , (17)where L := p (cid:96) + p ν + q and s (cid:96) = L . They are related to q and p (cid:96) by x = 2 q √ s (cid:96) , y = 2 p (cid:96) √ s (cid:96) . (18)6 give the photon an artificial small mass m γ in order to regulate the infrared divergences. The ranges ofthe phase space variables are: M π + ≤ s ≤ ( M K + − m (cid:96) − m γ ) , ( m (cid:96) + m γ ) ≤ s (cid:96) ≤ ( M K + − √ s ) , ≤ θ π ≤ π, ≤ θ γ ≤ π, ≤ φ ≤ π, ≤ φ (cid:96) ≤ π. (19)Let us determine in the following the ranges of the two variables x and y . Introducing the variable s (cid:96)ν :=( p (cid:96) + p ν ) , I find the relations q = s (cid:96) + m γ − s (cid:96)ν √ s (cid:96) , x = 1 + m γ s (cid:96) − s (cid:96)ν s (cid:96) . (20)The range of s (cid:96)ν is obviously m (cid:96) ≤ s (cid:96)ν ≤ ( √ s (cid:96) − m γ ) , (21)which leads to √ z γ ≤ x ≤ z γ − z (cid:96) , (22)where I have defined z (cid:96) := m (cid:96) s (cid:96) , z γ = m γ s (cid:96) . (23)The range of y for a given value of x can be found as follows. Determine the boost from Σ (cid:96)νγ to the (cid:96)ν centre-of-mass frame Σ (cid:96)ν by considering the vector p (cid:96) + p ν in both frames. Define z = cos ˆ θ (cid:96) with ˆ θ (cid:96) being theangle between the lepton momentum in Σ (cid:96)ν and the dilepton line of flight in Σ (cid:96)νγ . Then, with the help of theinverse boost, you will find y in terms of z and x : y = z (cid:112) x − z γ (1 + z γ − z (cid:96) − x ) + (2 − x )(1 + z γ + z (cid:96) − x )2(1 + z γ − x ) . (24)In the limit m γ → , I obtain the following range: − x + z (cid:96) − x ≤ y ≤ z (cid:96) . (25)Similar to K (cid:96) , I introduce for the radiative process the momenta P := p + p , Q := p − p , L := q + p (cid:96) + p ν , N := q + p (cid:96) − p ν . (26)It will be useful to define also the momenta ˆ L := p (cid:96) + p ν = L − q, ˆ N := p (cid:96) − p ν = N − q. (27)Total momentum conservation implies p = P + L . I will use the Lorentz invariants s := ( p + p ) , t := ( p − p ) , u := ( p − p ) , s γ := ( p (cid:96) + q ) = s (cid:96) ( x + y − . (28)In the appendix B.2, the Lorentz transformations between the three reference frames are determined and allthe Lorentz invariant products are computed. 7 .2.2 Matrix Element, Form Factors and Decay Rate The matrix element of the radiative decay (16) has the form (in analogy to K (cid:96) γ [20]) T γ = − G F √ eV ∗ us (cid:15) µ ( q ) ∗ (cid:20) H µν ¯ u ( p ν ) γ ν (1 − γ ) v ( p (cid:96) ) + H ν p (cid:96) q ¯ u ( p ν ) γ ν (1 − γ )( m (cid:96) − /p (cid:96) − /q ) γ µ v ( p (cid:96) ) (cid:21) =: (cid:15) µ ( q ) ∗ M µ , (29)where H ν = V ν − A ν is the hadronic part of the K (cid:96) matrix element. The second part of the matrix elementstems from diagrams where the photon is radiated off the lepton line, the first part contains all the rest. Thehadronic tensor H µν = V µν − A µν is defined by I µν = i (cid:90) d x e iqx (cid:104) π + ( p ) π − ( p ) | T { V µ em ( x ) I ν (0) }| K + ( p ) (cid:105) , I = V , A , I = V, A, (30)and satisfies the Ward identity q µ H µν = H ν , (31)such that the condition q µ M µ = 0 required by gauge invariance is fulfilled.If the contributions from the anomalous sector are neglected, the hadronic tensor can be decomposed intodimensionless form factors as (the photon is taken on-shell) H µν = iM K + g µν Π + iM K + ( P µ Π ν + Q µ Π ν + L µ Π ν ) , Π νi = 1 M K + ( P ν Π i + Q ν Π i + L ν Π i + q ν Π i ) . (32)Gauge invariance requires the following relations: M K + F − P q Π − Qq Π − Lq Π = 0 ,M K + G − P q Π − Qq Π − Lq Π = 0 ,M K + R − P q Π − Qq Π − Lq Π = 0 ,M K + Π +
P q Π + Qq Π + Lq Π = 0 , (33)where F , G and R are the K (cid:96) form factors.The partial decay rate for K (cid:96) γ is given by d Γ γ = 12 M K + (2 π ) (cid:88) spinspolar . |T γ | δ (4) ( p − P − L ) d p p d p p d p (cid:96) p (cid:96) d p ν p ν d q q . (34)Seven integrals can be performed, leaving the integrals over the eight phase space variables: d Γ γ = G F | V us | e s (cid:96) σ π ( s ) X π M K + J ( s, s (cid:96) , θ π , θ γ , φ, x, y, φ (cid:96) ) ds ds (cid:96) d cos θ π d cos θ γ dφ dx dy dφ (cid:96) . (35)The procedure how to find the explicit expression for J in terms of the form factors is explained in appendix C.2. χ PT Calculation of the Amplitudes
Isospin symmetry is the symmetry under SU (2) transformations of up- and down-quarks. In nature, thissymmetry is realised only approximately. The source of isospin symmetry breaking is twofold: on the onehand, u - and d -quarks do not have the same mass, on the other hand, their electric charge is different. Onthe fundamental level of the standard model, we can therefore distinguish between quark mass effects andelectromagnetic effects.Usually, calculations of processes can be simplified substantially if isospin symmetry is assumed to be exact.In order to link such calculations to real word measurements, the effects of isospin breaking have to be known.The aim of this work is to compute such isospin breaking corrections to the K (cid:96) decay.8s K (cid:96) is a process that happens at low energies, the hadronic part of the matrix element can not becomputed perturbatively in QCD. The low-energy effective theory of QCD, chiral perturbation theory ( χ PT)[1, 2, 3], does not treat quarks and gluons but the Goldstone bosons of the spontaneously broken chiral symmetryof QCD as the degrees of freedom. In this effective theory, the isospin breaking effects show up as differences inthe masses of the charged and neutral mesons and in form of photonic corrections. The meson mass differencesare due to both isospin breaking sources, the quark mass difference as well as electromagnetism. I compute theisospin breaking effects in K (cid:96) within χ PT including virtual photons and leptons [16, 17]. As this is a well-knownframework, I abstain from giving a review but only collect the most important formulae in appendix D in orderto settle the conventions.I take into account only first order corrections in the isospin breaking parameters and effects up to one loop.The leading-order form factors behave as O ( p ) , i.e. I consider effects of O ( p ) , O ( (cid:15) p ) , O ( e p ) , where e = + | e | is the electric unit charge and (cid:15) := √ m u − m d ˆ m − m s , ˆ m := m u + m d . (36)Since the chiral anomaly shows up first at next-to-leading chiral order, I do not compute isospin breakingcorrections to the form factor H . In contrast to the photonic effects that appear as O ( e ) corrections in my calculation, the ‘non-photonic’electromagnetic effects due to the different meson masses in the loops give corrections of the order O ( Ze ) ,where Z is the low-energy constant in L e . This allows for a separation of the mass effects from purely photoniccorrections (a subtlety concerning the counterterms will be discussed later). Let us thus first discuss the masseffects, i.e. the isospin corrections in the absence of virtual photons.These O ( (cid:15) p ) and O ( Ze p ) effects have been considered in [13, 14, 9]. The present calculation agrees withthe results given in [13, 14]. For completeness, I give the explicit expressions in my conventions. At leading order, we have to compute two tree-level diagrams, shown in figure 2. K + π + π − ℓ + ν ℓ (a) K + π + π − ℓ + ν ℓ K + (b)Figure 2: Tree-level diagrams for the K (cid:96) decay. Diagram 2a contributes to the form factors F , G and R , whereas diagram 2b only contributes to the formfactor R . This is true for all diagrams with an intermediate kaon pole, also at one-loop level.The leading-order results for the form factors are: F LOME = G LOME = M K + √ F ,R LOME = M K + √ F M K + − s − s (cid:96) − ν − π M K + − s (cid:96) ,T LOME = 0 . (37)Only the form factor R gets at leading order an isospin correction due to the mass effects.9 .1.2 Next-to-Leading Order Since the contributions of both R and T to the decay rate are suppressed by m (cid:96) and experimentally inaccessiblein the electron mode, I will calculate only corrections to the form factors F and G . Thus, I neglect at next-to-leading order all diagrams that contribute only to the form factor R , i.e. diagrams with a kaon pole in the s (cid:96) -channel. It is convenient to write the NLO expressions for the form factors as F NLOME = F LOME (cid:0) δF NLOME (cid:1) ,G NLOME = G LOME (cid:0) δG NLOME (cid:1) . (38)Since the LO contribution is of O ( p ) , the order of the NLO corrections considered here is δF NLOME , δG
NLOME = O ( p ) + O ( (cid:15) p ) + O ( Ze ) . (39)Of course, the loop integrals appearing at NLO are UV-divergent. I will regularise them dimensionally andrenormalise the theory as usual in the M S scheme. The divergent parts of the loop integrals are cancelled bythe divergent parts of the LECs.The explicit NLO results are rather lengthy and can be found in appendix E.1.
At NLO, we have to compute the tadpole diagram 3a with all possible mesons ( π , π + , K , K + and η ) in theloop as well as the diagrams 3b-3d with two-meson intermediate states in the s -, t - and u -channel. K + π + π − ℓ + ν ℓ (a) K + π + π − ℓ + ν ℓ (b) K + π + π − ℓ + ν ℓ (c) K + π + π − ℓ + ν ℓ (d)Figure 3: One-loop diagrams contributing to the K (cid:96) form factors F and G . The contributions of the meson loop diagrams can be expressed in terms of the scalar loop functions A and B (which should not be confused with the low-energy constant B ). K + π + π − ℓ + ν ℓ Figure 4:
Counterterm diagram contributing to the K (cid:96) form factors F and G . I express the one-loop corrections in terms of the scalar loop functions A and B . These loop functionscontain UV divergences that have to cancel against the UV divergences in the counterterms and the field strengthrenormalisation. The only relevant counterterm diagram is shown in figure 4. It contains a vertex from theNLO Lagrangian. Now, a subtlety arises here. As we are interested in the mass effects, we have neglected pure O ( e ) loop corrections, but kept O ( Ze ) contributions. If we used the same prescription for the counterterms,the UV divergences would not cancel. The reason is that some of the electromagnetic LECs K i contain a factor Z in their beta function, hence their divergent part is multiplied by Z and contributes at O ( Ze ) . We thereforehave to assign also these LECs to the mass effects. 10 .1.2.3 External Leg Corrections (a) (b)Figure 5: Meson self-energy diagrams.
The last contribution at NLO are the external leg corrections. We have to compute only the field strengthrenormalisation of the mesons (the lepton propagators get no corrections). For the self-energy of the mesonsat NLO, the corrections to the propagator shown in figure 5 have to be taken into account. All the Goldstonebosons π + , π , K + , K and η have to be inserted in the tadpole diagram. The complete expressions for the form factors at NLO including the mass effects are X NLOME = X LOME (cid:0) δX NLOME (cid:1) , (38)with δX NLOME = δX NLOtadpole + δX NLO s -loop + δX NLO t -loop + δX NLO u -loop + δX NLOct + δX NLO Z , (40)where X ∈ { F, G } . The explicit expressions for the individual contributions can be found in the appendix E.1.The form factors have to be UV-finite, hence, we should check that in the above sum, all the UV divergencescancel. If I replace the LECs with help of (D.14) and the loop functions with (A.2), I find indeed that all theterms proportional to the UV divergence λ (D.15) cancel. In a next step, I calculate in the effective theory the effects due to the presence of photons. I include virtualphoton corrections up to NLO, i.e. I have to compute again one-loop diagrams, counterterms and external legcorrections. The sum of these contributions will be UV-finite but contain IR and collinear (in the limit m (cid:96) → )singularities. As it is well known, the IR divergences will cancel in the sum of the decay rates of K (cid:96) and the softreal photon emission process K + → π + π − (cid:96) + ν (cid:96) γ soft . The collinear divergence is in the physical case regulatedby the lepton mass, which plays the role of a natural cut-off. It cancels in the sum of the decay rates of K (cid:96) and the (soft and hard) collinear real photon emission process. (Note that at O ( e ) , the emission of only onephoton has to be taken into account.) The fully inclusive decay rate K + → π + π − (cid:96) + ν (cid:96) ( γ ) is free of IR and massdivergences and does not depend on a cut-off, in accordance with the KLN theorem [21, 22, 23].As in the case of the mass effects, also the photonic effects have already been computed in [13, 14]. However,in these works a whole gauge invariant class of diagrams appearing at NLO has been overlooked . The presentcalculation confirms the results for the diagrams calculated in [14] (in [13], eq. (72) gives a wrong result for oneof the diagrams) and completes it with the missing class of diagrams.For the calculation of the photonic effects, I take into account NLO corrections of O ( e ) , but I neglectcontributions of O ( Ze ) as well as O ( (cid:15) p ) , since they are treated by the mass effects. Photonic effects appear already at leading order in the effective theory, i.e. at O ( e p − ) , as diagrams with avirtual photon splitting into two pions. In addition to the O ( e p ) tree-level diagrams in figure 2, the diagramsshown in figure 6 have to be calculated.The diagram 6a contributes to the form factors G , R and the tensorial form factor T . However, thecontribution to G gets exactly cancelled by the diagram 6b. Diagram 6c only contributes to R .Therefore, the contribution of the diagrams in figure 6 does not alter the form factors F and G : F LOvirt .γ = M K + √ F , G LOvirt .γ = M K + √ F . (41) I thank V. Cuplov for confirming this. + π + π − ℓ + ν ℓ (a) K + π + π − ℓ + ν ℓ (b) K + π + π − ℓ + ν ℓ (c)Figure 6: Tree-level diagrams for the K (cid:96) decay with a virtual photon. The other form factors read (in agreement with [13]) R LOvirt .γ = M K + √ F (cid:18) M K + − s − s (cid:96) − νM K + − s (cid:96) + 4 e F s (cid:18) s (cid:96) − s (cid:96) u (cid:96) − m (cid:96) + νM K + − s (cid:96) (cid:19)(cid:19) ,T LOvirt .γ = 2 √ e F M K + m (cid:96) s ( u (cid:96) − m (cid:96) ) . (42)We see that the tensorial form factor F , which was defined above, F = σ π ( s ) ss (cid:96) M K + m (cid:96) T, (14)stays finite in the limit m (cid:96) → . This shows that its contribution to the decay rate (see (C.20) and (C.21) inthe appendix) is suppressed by m (cid:96) . In the following, I will therefore only consider the form factors F and G . In order to regularise the IR divergence of loop diagrams with virtual photons, I introduce an artificial photonmass m γ and use the propagator of a massive vector field: ˜ G µν ( k ) = − ik − m γ + i(cid:15) (cid:18) g µν − k µ k ν m γ (cid:19) . (43)The same regulator has to be used in the calculation of the radiative process. In the end, one has to take thelimit m γ → , which restores gauge invariance. Terms that do not contribute in this limit are neglected.For the NLO calculation of photonic effects, I consider all contributions to the form factors F and G of O ( e p ) . They consist of loop diagrams with virtual photons, counterterms and external leg corrections for K (cid:96) .On the other hand, tree diagrams for the radiative process K (cid:96) γ have to be included at the level of the decayrate.It is again convenient to write the NLO contribution in the form F NLOvirt .γ = F LOvirt .γ (cid:0) δF NLOvirt .γ (cid:1) ,G NLOvirt .γ = G LOvirt .γ (cid:0) δG NLOvirt .γ (cid:1) . (44)The explicit results are collected in the appendix E.2. A first class of loop diagrams is obtained by adding a virtual photon to the tree diagrams in figure 2. Alldiagrams contributing to F and G are shown in figure 7. Again, diagrams with a virtual kaon pole are omitted,as they contribute only to R .I choose to express most of the results in terms of the basic scalar loop functions A , B , C and D .12 a) (b) (c) (d)(e) (f ) (g) (h) (i) (j)(k) (l) (m) (n) (o)(p) (q) (r)Figure 7: First set of one-loop diagrams with virtual photons: they are obtained by a virtual photon insertion in thetree diagrams in figure 2 (I drop the labels for the external particles as they are always the same). Diagrams contributingonly to the form factor R are omitted. (a) (b) (c) (d)(e) (f ) (g)Figure 8: Second set of one-loop diagrams with virtual photons: they are obtained by a meson loop insertion in thetree diagrams in figure 6. Diagrams contributing only to the form factor R are omitted. F and G vanish, the NLO diagrams give a finite contribution to G . To my knowledge, they have not been considered inthe previous literature.In diagrams 8a - 8c, we have to insert charged mesons in the loop. In the tadpole loops, all octet mesonshave to be included. In order to renormalise the UV divergences in the loop functions, we have to compute the counterterm contri-bution, i.e. tree-level diagrams with one vertex from L p , L e p or L lept . These diagrams are shown in figure 9.The loop diagrams of the first class, figure 7, need only the counterterm 9a, the remaining four countertermdiagrams renormalise the meson loops of the second class, figure 8. (a) (b) (c) (d) (e)Figure 9: Counterterms needed to renormalise the loops with virtual photons.
In order to complete the NLO calculation, we need the external leg corrections at O ( e p ) . At this order, thecorrections for both charged mesons and lepton have to be taken into account. (a) (b) (c) (d)Figure 10: Meson and lepton self-energy diagrams.
The calculation of the field strength renormalisation and its contribution to the form factors can be foundin the appendix E.2.3.
The form factors at O ( e p ) are given by X NLOvirt .γ = X LOvirt .γ (cid:0) δX NLOvirt .γ (cid:1) , X ∈ { F, G } , (45)where the NLO corrections consists of δX NLOvirt .γ = δX NLO γ − loop + δX NLO γ − pole + δX NLO γ − ct + δX NLO γ − Z . (46)The individual contributions are all given explicitly in the appendix E.2. With these expressions, it can be easilyverified that the contributions stemming from the additional term k µ k ν /m γ in the propagator for a massivegauge boson (with respect to a massless propagator in Feynman gauge) cancel in the above sum (in the limit m γ → ). In appendix C.2, I show that the radiative decay rate only gets O ( m γ ) contributions from the14dditional term in the propagator. Hence, in the limit m γ → , the longitudinal modes decouple and gaugeinvariance is restored [24].Next, let us check that the UV-divergent parts cancel in the sum of all NLO contributions. Working in the M S scheme, I replace the LECs according to (D.14) with their renormalised counterparts. Introducing also therenormalised loop functions (A.2) and tensor coefficient functions (A.8), I find that all the terms proportionalto the UV divergence λ cancel. (a) (b) (c) (d) (e)(f ) (g) (h)(i) (j) (k) (l)Figure 11: Tree-level diagrams for the decay K (cid:96) γ . As explained before, an IR-finite result can only be obtained for a sufficiently inclusive observable. In thepresent case, we have to add the O ( e ) contribution of the radiative process at the decay rate level. Let ustherefore compute the O ( e ) tree-level amplitude for K (cid:96) γ .The relevant diagrams are shown in figure 11. If we use the decomposition of the matrix element defined insection 2.2.2, the diagrams 11e and 11l just reproduce the second term in (29), where the hadronic part is givenby the LO K (cid:96) form factors in the isospin limit.The diagrams 11d and 11k, where the photon is emitted off the vertex, correspond to the form factor Π : Π = M K + √ F (cid:18) − s + νM K + − s (cid:96) (cid:19) , (47)where ν = t − u .The form factors Π ij correspond to the remaining 8 diagrams, where the photon is emitted off a meson lineor a mesonic vertex. The form factors multiplying ¯ u ( p ν ) /P (1 − γ ) v ( p (cid:96) ) or ¯ u ( p ν ) /Q (1 − γ ) v ( p (cid:96) ) have a simple15orm: Π = Π = − M K + √ F (cid:18) m γ − pq + 1 m γ + 2 p q − m γ + 2 p q (cid:19) , Π = Π = − M K + √ F (cid:18) m γ + 2 p q + 1 m γ + 2 p q (cid:19) , Π = Π = − M K + √ F m γ − pq . (48)The remaining form factors are a bit more complicated. They satisfy the relations Π = − Π − M K + √ F m γ + 2 p q , Π = − Π − M K + √ F m γ + 2 p q , Π = − Π . (49)In the limit m γ → , I find Π = M K + √ F M K + − s (cid:96) + 2 Lq (cid:32) M K + − s − t + u − s (cid:96) (cid:18) pq − p q + 1 p q (cid:19) + Lq − Qqpq − Lqp q − Qqp q (cid:33) , Π = M K + √ F M K + − s (cid:96) + 2 Lq (cid:32) M K + − s − t + u − s (cid:96) (cid:18) − p q − p q (cid:19) − Lqp q + Qqp q + 3 (cid:33) , Π = M K + √ F M K + − s (cid:96) + 2 Lq (cid:32) M K + − s − t + u − s (cid:96) (cid:18) pq + 2 M K + − s (cid:96) (cid:19) + Lq − Qqpq + 1 (cid:33) . (50)These expressions fulfil the relations (33) as required by gauge invariance.The contribution of the diagrams 11f-11j to the decay rate is helicity suppressed by a factor of m (cid:96) . Thesuppression at leading chiral order also works for the diagrams 11k and 11l. One could therefore omit alldiagrams with a kaon pole in the limit m (cid:96) → . However, from a technical point of view, this barely reduces thecomplexity of the calculation. Hence, I have given here the results for the form factors using the complete set ofdiagrams. Moreover, at higher chiral order, one has to expect structure dependent contributions not suppressedby m (cid:96) . This section discusses the extraction of the isospin breaking corrections to the K (cid:96) form factors and decay rate.While the experiments are performed in our real world, where isospin is broken, it is useful to translate measuredquantities into the context of an ideal world with conserved isospin, i.e. a world with no electromagnetism andidentical up- and down-quark masses. The motivation for doing such a transformation is that in an isospinsymmetric world, calculations may become much easier. The isospin breaking corrections for K (cid:96) will be usedin a forthcoming dispersive treatment of this decay [25, 26, 27], which is performed in the isospin limit.Correcting the isospin breaking effects in existing experimental data on the K (cid:96) form factors is a delicatematter: the K (cid:96) form factors are in the real world themselves not observable quantities, because they are notinfrared-safe. As explained above, any experiment will measure a semi-inclusive decay rate of K (cid:96) and K (cid:96) nγ ,typically with some cuts on the real photons. These cuts are detector specific and naturally defined in the labframe. It is almost impossible to handle such cuts in an analytic way. Therefore, one must rely on a MonteCarlo simulation of the semi-inclusive decay that models the detector geometry and all the applied cuts in orderto extract isospin corrected quantities. I suggest for future experiments the inclusion of the here presentedamplitudes for K (cid:96) and K (cid:96) γ in a Monte Carlo simulation like PHOTOS [15].The isospin corrections due to the mass effects can be extracted directly for the form factors. For thephotonic effects, I calculate the radiative corrections to the (semi-)inclusive decay rate.16 .1 Mass Effects I define the isospin breaking corrections to the form factors as follows.The measured semi-inclusive differential decay rate d Γ exp( γ, cut) (neglecting experimental uncertainties) equalsthe result from the presented NLO calculation up to higher order in the chiral expansion or the isospin breakingparameters: d Γ exp( γ, cut) = d Γ NLO( γ, cut) + h.o. = d Γ NLOiso + ∆ d Γ NLOME + ∆ d Γ NLOvirt .γ + (cid:90) cut d Γ γ + O ( p , (cid:15) p , Ze p , e p ) + O ( (cid:15) , (cid:15) e , e ) , (51)where the real photon in the radiative decay rate is integrated using the same cuts as in the experiment. Iexpect the contribution of higher order in the breaking parameters to be negligible, while the O ( p ) contributionis certainly not. The different terms are of the following order: d Γ NLOiso = O ( p ) , ∆ d Γ NLOME = O ( (cid:15) p , Ze p ) , ∆ d Γ NLOvirt .γ = O ( e p ) , (cid:90) cut d Γ γ = O ( e p ) . (52)The NA48/2 analysis assumes the following isospin breaking effects: d Γ exp( γ, cut) = d Γ exp + ∆ d Γ Coulomb + ∆ d Γ cutPHOTOS . (53)If I assume ∆ d Γ Coulomb + ∆ d Γ cutPHOTOS ≈ ∆ d Γ NLOvirt .γ + (cid:90) cut d Γ γ + O ( e p ) , (54)(an approximation that I will test later), the form factors given by the experiment contain only the isospinbreaking mass effects (note that X LO = O ( p ) ): X exp = X NLOME + O ( p , (cid:15) p , Ze p ) . (55)The relative isospin corrections to the form factors due to the mass effects are δ ME X := 1 − X iso X ME = 1 − X NLOiso X NLOME + O ( (cid:15) p , Ze p ) . (56)The uncertainty can be naïvely estimated to be O ( (cid:15) p , Ze p ) ≈ . . The mass effects at NNLO in the chiralexpansion have been studied in a dispersive treatment [12] and found to be small given the present experimentalaccuracy.The definition of the isospin limit is a convention. I choose here X NLOiso := lim (cid:15) → ,e → lim M π → M exp π + ,M K → M exp K + X NLOME . (57) In this section, I calculate the (semi-)inclusive decay rate for K (cid:96) γ ) . This will allow on the one hand for a moreprecise treatment of photonic corrections in future experiments (compared to previous treatments that do notmake use of the matrix elements). On the other hand, I will be able to study the approximation ∆ d Γ Coulomb + ∆ d Γ cutPHOTOS ≈ ∆ d Γ NLOvirt .γ + (cid:90) cut d Γ γ + O ( e p ) , (58)although not for the experimental cuts, but for a simplified cut that can be handled analytically.17 .2.1 Strategy for the Phase Space Integration I have introduced a finite photon mass as a regulator and will eventually send this regulator to zero (in theinclusive decay rate). We are not interested in the full dependence of the decay rate on the photon mass, butonly in terms that do not vanish in the limit m γ → , i.e. in the IR-singular and finite pieces.I use this fact to simplify the calculation as follows. I split the phase space of the radiative decay into a softphoton and a hard photon region. In the soft region, I use the soft photon approximation (SPA) to simplifythe amplitude. This region will produce the IR singularity, which cancels against the divergence in the virtualcorrections. The hard region gives an IR-finite result. Here, the limit m γ → can be taken immediately. Thedependence on the photon energy cut ∆ ε that separates the soft from the hard region must cancel in the sumof the two contributions. The hard region either covers the whole hard photon phase space, or alternatively, anadditional cut on the maximum photon energy in the dilepton-photon system can be introduced rather easily. Let us calculate the soft photon amplitude. In the real emission amplitude T γ = − G F √ eV ∗ us (cid:15) µ ( q ) ∗ (cid:20) H µν L ν + H ν ˜ L µν (cid:21) (59)where L ν := ¯ u ( p ν ) γ ν (1 − γ ) v ( p (cid:96) ) , ˜ L µν := 12 p (cid:96) q ¯ u ( p ν ) γ ν (1 − γ )( m (cid:96) − /p (cid:96) − /q ) γ µ v ( p (cid:96) ) , H ν := iM K + ( P ν F + Q ν G + L ν R ) , H µν := iM K + g µν Π + iM K + ( P µ Π ν + Q µ Π ν + L µ Π ν ) , Π νi := 1 M K + ( P ν Π i + Q ν Π i + L ν Π i + q ν Π i ) , (60)I neglect according to the SPA terms with a q in the numerator, i.e. the /q in ˜ L µν and the q ν in Π νi . If I insertthe tree-level expressions for the form factors and consistently keep only terms that diverge as q − , I find thatthe soft photon amplitude factorises as T soft γ = e T LOiso (cid:18) − p(cid:15) ∗ ( q ) pq + p (cid:96) (cid:15) ∗ ( q ) p (cid:96) q + p (cid:15) ∗ ( q ) p q − p (cid:15) ∗ ( q ) p q (cid:19) , (61)where T LOiso is the tree-level K (cid:96) matrix element in the isospin limit.In the SPA, also the photon momentum in the delta function of the phase space measure is neglected. Thismeans that we can essentially use K (cid:96) kinematics to describe the other momenta: d Γ soft γ = 12 M K + (cid:103) dp (cid:103) dp (cid:102) dp (cid:96) (cid:103) dp ν (cid:102) dq δ (4) ( p − p − p − p (cid:96) − p ν ) (cid:88) spins , polar . (cid:12)(cid:12) T soft γ (cid:12)(cid:12) = e d Γ LOiso (cid:90) | (cid:126)q |≤ ∆ ε (cid:102) dq (cid:88) polar . (cid:12)(cid:12)(cid:12)(cid:12) − p(cid:15) ∗ ( q ) pq + p (cid:96) (cid:15) ∗ ( q ) p (cid:96) q + p (cid:15) ∗ ( q ) p q − p (cid:15) ∗ ( q ) p q (cid:12)(cid:12)(cid:12)(cid:12) = − e d Γ LOiso (cid:90) | (cid:126)q |≤ ∆ ε (cid:102) dq (cid:20) M K + ( pq ) + m (cid:96) ( p (cid:96) q ) + M π + ( p q ) + M π + ( p q ) − pp (cid:96) ( pq )( p (cid:96) q ) − pp ( pq )( p q ) + 2 pp ( pq )( p q )+ 2 p p (cid:96) ( p q )( p (cid:96) q ) − p p (cid:96) ( p q )( p (cid:96) q ) − p p ( p q )( p q ) (cid:21) , (62)where I use the abbreviation (cid:102) dk := d k (2 π ) k . (63)18hese are standard bremsstrahlung integrals, which have been computed in [28] (see also [29]). They amountto I ( k ) := (cid:90) | (cid:126)q |≤ ∆ ε (cid:102) dq kq ) = 18 π k (cid:34) (cid:18) εm γ (cid:19) − k | (cid:126)k | ln (cid:32) k + | (cid:126)k | k − | (cid:126)k | (cid:33) (cid:35) + O ( m γ ) . (64)The integrals with two different momenta are more complicated: I ( k , k ) := (cid:90) | (cid:126)q |≤ ∆ ε (cid:102) dq k q )( k q ) = α π (cid:20) k − k ln (cid:18) k k (cid:19) ln (cid:18) εm γ (cid:19) + ˜ I ( k , k ) (cid:21) + O ( m γ ) , ˜ I ( k , k ) = 1 k − k v (cid:34)
14 ln (cid:18) u − | (cid:126)u | u + | (cid:126)u | (cid:19) + Li (cid:18) v − u + | (cid:126)u | v (cid:19) + Li (cid:18) v − u − | (cid:126)u | v (cid:19) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = k u = k , (65)where k = αk and α is the solution of ( k − αk ) = 0 such that αk − k has the same sign as k . Further, v is defined as v := k − k k − k ) . (66)I find it most convenient to evaluate the soft photon contribution in the rest frame of the two leptons andthe photon, Σ (cid:96)νγ . The particle momenta in this frame are given by p = M K + − s + s (cid:96) √ s (cid:96) , | (cid:126)p | = λ / K(cid:96) ( s )2 √ s (cid:96) ,p (cid:96) = √ s (cid:96) z (cid:96) ) , | (cid:126)p (cid:96) | = √ s (cid:96) − z (cid:96) ) ,p = P L + σ π X cos θ π √ s (cid:96) , | (cid:126)p | = (cid:113) ( p ) − M π + ,p = P L − σ π X cos θ π √ s (cid:96) , | (cid:126)p | = (cid:113) ( p ) − M π + . (67)The bremsstrahlung integrals become I ( p ) = 18 π M K + (cid:34) (cid:18) εm γ (cid:19) − M K + − s + s (cid:96) λ / K(cid:96) ( s ) ln (cid:32) M K + − s + s (cid:96) + λ / K(cid:96) ( s ) M K + − s + s (cid:96) − λ / K(cid:96) ( s ) (cid:33) (cid:35) ,I ( p (cid:96) ) = 18 π m (cid:96) (cid:34) (cid:18) εm γ (cid:19) + 1 + z (cid:96) − z (cid:96) ln( z (cid:96) ) (cid:35) ,I ( p ) = 18 π M π + (cid:34) (cid:18) εm γ (cid:19) − p | (cid:126)p | ln (cid:18) p + | (cid:126)p | p − | (cid:126)p | (cid:19) (cid:35) ,I ( p ) = 18 π M π + (cid:34) (cid:18) εm γ (cid:19) − p | (cid:126)p | ln (cid:18) p + | (cid:126)p | p − | (cid:126)p | (cid:19) (cid:35) . (68)The evaluation of the integrals with two momenta is straightforward but a bit tedious. I give here therespective values of α ( k , k ) : α ( p, p (cid:96) ) = λ / ( t (cid:96) , M K + , m (cid:96) ) + M K + + m (cid:96) − t (cid:96) m (cid:96) , α ( p, p ) = λ / Kπ ( t ) + M K + + M π + − t M π + ,α ( p , p (cid:96) ) = λ / ( s (cid:96) , M π + , m (cid:96) ) − m (cid:96) − M π + + s (cid:96) m (cid:96) , α ( p, p ) = λ / Kπ ( u ) + M K + + M π + − u M π + ,α ( p , p (cid:96) ) = λ / ( s (cid:96) , M π + , m (cid:96) ) − m (cid:96) − M π + + s (cid:96) m (cid:96) , α ( p , p ) = sσ π + s − M π + M π + . (69)19 .2.3 Hard Region The hard region is defined as the phase space region where | (cid:126)q | > ∆ ε , i.e. x > x min = 2∆ ε √ s (cid:96) =: ˜ x min (1 − z (cid:96) ) , (70)where the variable ˜ x min is independent of s (cid:96) .Here, the full K (cid:96) γ kinematics has to be applied. However, as the hard region does not produce any IRsingularity, the limit m γ → can be taken at the very beginning.In the appendix C.2, I have derived the expression for the decay rate d Γ hard γ = G F | V us | e s (cid:96) σ π ( s ) X π M K + J ds ds (cid:96) d cos θ π d cos θ γ dφ dx dy dφ (cid:96) , (71)where J = M K + (cid:88) polar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) (cid:34) H ν H ∗ σ (cid:88) spins ˜ L µν ˜ L ∗ ρσ + H µν H ∗ ρσ (cid:88) spins L ν L ∗ σ + 2Re (cid:18) H µν H ∗ σ (cid:88) spins L ν ˜ L ∗ ρσ (cid:19)(cid:35) . (72)Since the form factors only depend on the first six phase space variables, the integrals over y and φ (cid:96) can beperformed without knowledge of the dynamics. The K (cid:96) form factors and the form factor Π depend on s , s (cid:96) and cos θ π only (at the order we consider). I therefore split the hadronic tensor into two pieces H µν = iM K + g µν Π + iM K + ˜ H µν , ˜ H µν = P µ Π ν + Q µ Π ν + L µ Π ν (73)and write J as follows: J = J (cid:96)(cid:96) + J hh + J int8 ,J (cid:96)(cid:96) = M K + (cid:88) polar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) (cid:34) H ν H ∗ σ (cid:88) spins ˜ L µν ˜ L ∗ ρσ + 1 M K + g µν g ρσ | Π | (cid:88) spins L ν L ∗ σ + iM K + (cid:18) g µν Π H ∗ σ (cid:88) spins L ν ˜ L ∗ ρσ − g µν Π ∗ H σ (cid:88) spins L ∗ ν ˜ L ρσ (cid:19)(cid:35) ,J hh = (cid:88) polar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) (cid:34) ˜ H µν ˜ H ∗ ρσ (cid:88) spins L ν L ∗ σ (cid:35) ,J int8 = M K + (cid:88) polar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) (cid:34) M K + (cid:16) g µν ˜ H ∗ ρσ Π + ˜ H µν g ρσ Π ∗ (cid:17) (cid:88) spins L ν L ∗ σ + i (cid:18) ˜ H µν H ∗ σ (cid:88) spins L ν ˜ L ∗ ρσ − ˜ H ∗ µν H σ (cid:88) spins L ∗ ν ˜ L ρσ (cid:19)(cid:35) . (74)The first term, J (cid:96)(cid:96) , denotes the absolute square of the contributions where the photon is attached to the leptonline (either the external line or the vertex). Here, the hadronic part is described by the K (cid:96) form factors and Π . I can therefore integrate directly over the five phase space variables cos θ γ , φ , x , y and φ (cid:96) .The second term, J hh , is the absolute square of the contributions with the photon emitted off the hadrons.The form factors Π ij describe here the hadronic part. As they depend on six phase space variables, I performfirst the integral over φ (cid:96) and y , then insert the explicit tree-level expressions for the form factors Π ij , given insection 3.2.3. I further integrate the decay rate and keep it differential only with respect to s , s (cid:96) and cos θ π .The same strategy applies to the third term, J int8 , the interference of off-lepton and off-hadron emission.It is important to note that for a vanishing lepton mass m (cid:96) , the phase space integrals containing H µ produce asingularity for collinear photons. The lepton mass plays the role of a natural cut-off for this collinear divergence,which emerges as a ln m (cid:96) mass singularity. In those integrals, the limit m (cid:96) → must not be taken before theintegration.Let us now consider the three parts separately. 20 perform the five phase space integrals in the (cid:96)(cid:96) -part and apply an expansion for small values of ˜ x min ,keeping only the logarithmic term. Only after the integration, it is safe to expand the result for small values of m (cid:96) : d Γ hard ,(cid:96)(cid:96)γ dsds (cid:96) d cos θ π = e G F | V us | σ π ( s ) X · π M K + (cid:18) (cid:0) | F | + sin θ π | F | (cid:1) (12 ln ˜ x min − z (cid:96) + 5)+ 3 | F + s (cid:96) Π | (cid:19) + O ( z (cid:96) ln z (cid:96) ) . (75)The soft photon contribution corresponding to the square of the off-lepton emission amplitude is given by I ( p (cid:96) ) .In the sum of the soft and the hard photon emission, the dependence on ∆ ε drops out: d Γ (cid:96)(cid:96)γ dsds (cid:96) d cos θ π = e G F | V us | σ π ( s ) X · π M K + (cid:18) (cid:0) | F | + sin θ π | F | (cid:1) (5 + 6 ln z γ − z (cid:96) )+ 3 | F + s (cid:96) Π | (cid:19) + O ( z (cid:96) ln z (cid:96) ) . (76)I can introduce an additional cut on the photon energy in Σ (cid:96)νγ by integrating x only over a part of the hardregion: ˜ x min (1 − z (cid:96) ) < x < ˜ x max (1 − z (cid:96) ) . (77)Instead of (76), I find then d Γ (cid:96)(cid:96)γ, cut dsds (cid:96) d cos θ π = e G F | V us | σ π ( s ) X · π M K + (cid:18) | F | + sin θ π | F | ) · (cid:16) ˜ x max (9 − ˜ x max (3 + ˜ x max )) + 6 ln z γ − x ) ln z (cid:96) − − ˜ x ) ln(1 − ˜ x max ) −
12 ln(˜ x max ) (cid:17) + 3˜ x (3 − x max ) | F + s (cid:96) Π | (cid:19) + O ( z (cid:96) ln z (cid:96) ) . (78)The integration of the hh -part is more involved. I perform the integrals over φ (cid:96) and y analytically, insertthe explicit form factors Π ij and integrate over x analytically, too (either with or without the energy cut ˜ x max ).Although, with some effort, the integrals over φ and cos θ γ could be performed analytically, I choose to integratethese two angles numerically: since they only describe the orientation of the dilepton-photon three-body systemwith respect to the pions, these two integrals contain nothing delicate. The dependence on the cuts ˜ x min and ˜ x max is manifest after the integration over x and collinear singularities cannot show up in the remaining integrals.I therefore write the hh -part as d Γ hard ,hhγ, cut dsds (cid:96) d cos θ π = e G F | V us | s (cid:96) σ π ( s ) X π M K + (cid:32) ln (cid:18) ˜ x min ˜ x max (cid:19) (cid:90) − d cos θ γ (cid:90) π dφ j hh ( s, s (cid:96) , cos θ π , cos θ γ , φ )+ (cid:90) − d cos θ γ (cid:90) π dφ j hh , cut ( s, s (cid:96) , cos θ π , cos θ γ , φ ) (cid:33) . (79)The function j hh is given by j hh ( s, s (cid:96) , cos θ π , cos θ γ , φ ) = 32 πM K + F (cid:0) ( P L + Xσ π cos θ π ) − s (cid:96) M π + (cid:1) · (cid:32) sA + sA + 2 P L + s + s (cid:96) ( P L + s (cid:96) + cos θ γ X ) + 2( P L + s ) A ( P L + s (cid:96) + cos θ γ X ) − P L + s ) A ( P L + s (cid:96) + cos θ γ X ) − s + 4 cos θ π Xσ π A A + 4 cos θ π Xs (cid:96) σ π A A ( P L + s (cid:96) + cos θ γ X ) − sσ π ( P L + cos θ γ X ) A A (cid:33) , (80)21here the φ -dependence is hidden in A = P L + cos θ γ X − cos θ γ cos θ π P Lσ π − cos θ π Xσ π + cos φ σ π (cid:113) (1 − cos θ γ )(1 − cos θ π ) ss (cid:96) ,A = P L + cos θ γ X + cos θ γ cos θ π P Lσ π + cos θ π Xσ π − cos φ σ π (cid:113) (1 − cos θ γ )(1 − cos θ π ) ss (cid:96) . (81)The integrand j hh , cut of the second numerical integral is a lengthy expression that I do not state here explicitly.The soft photon contribution to this second part contains the six bremsstrahlung integrals I ( p ) , I ( p ) , I ( p ) , I ( p, p ) , I ( p, p ) and I ( p , p ) . It is easy to verify numerically that in the sum of the contributionsfrom soft and hard region, the dependence on ∆ ε (i.e. on ˜ x min ) again drops out. The analytic result of theintegral over j hh can therefore be inferred from the soft photon hh -part (note that these bremsstrahlung integralsdo not depend on φ or cos θ (cid:96) ).The interference term of off-lepton and off-hadron photon emission is the last and most intricate part of thephase space integral calculation. On the one hand, the explicit form factors Π ij have to be inserted after the φ (cid:96) -and y -integration. On the other hand, while the part of the interference term containing Π is free of collinearsingularities and independent of ˜ x min , the contrary is true for the part involving the K (cid:96) form factors. I againintegrate over φ (cid:96) , y and x analytically, expand the result for small m (cid:96) and obtain the structure d Γ hard , int γ, cut dsds (cid:96) d cos θ π = e G F | V us | s (cid:96) σ π ( s ) X π M K + · (cid:32) ln z (cid:96) (cid:18) ˜ x max + ln (cid:18) ˜ x min ˜ x max (cid:19)(cid:19) (cid:90) − d cos θ γ (cid:90) π dφ j int1 ( s, s (cid:96) , cos θ π , cos θ γ , φ )+ ln (cid:18) ˜ x min ˜ x max (cid:19) (cid:90) − d cos θ γ (cid:90) π dφ j int2 ( s, s (cid:96) , cos θ π , cos θ γ , φ )+ (cid:90) − d cos θ γ (cid:90) π dφ j int3 , cut ( s, s (cid:96) , cos θ π , cos θ γ , φ ) (cid:33) . (82)I perform the integrals over φ and cos θ γ numerically. The expressions for the integrands j int i are too lengthy tobe given explicitly. j int3 , cut depends on the cut ˜ x max .Again, the sum of the soft and hard photon contribution must not depend on ∆ ε . I expand the softcontribution, given by the remaining bremsstrahlung integrals I ( p, p (cid:96) ) , I ( p , p (cid:96) ) and I ( p , p (cid:96) ) , in m (cid:96) andneglect terms that vanish for m (cid:96) → : d Γ soft , int γ dsds (cid:96) d cos θ π = − e (cid:90) − d cos θ (cid:96) (cid:90) π dφ d Γ LOiso π (cid:32) ln (cid:18) εm γ (cid:19) (cid:20) z (cid:96) + b int1 ( s, s (cid:96) , cos θ π , cos θ (cid:96) , φ ) (cid:21) + ln z (cid:96) + b int2 ( s, s (cid:96) , cos θ π , cos θ (cid:96) , φ ) (cid:33) , (83)where the b int i are again rather lengthy expressions.I perform the integrals over cos θ (cid:96) and φ numerically and find that the dependence on ∆ ε drops out indeedin the sum of soft and hard photon contribution. Both the virtual corrections and the real emission contain infrared divergences. These divergences, which areregulated by the artificial photon mass m γ , must vanish in the inclusive decay rate. In the radiative process,the IR divergence is generated in the soft region, which I have treated in the soft photon approximation.Furthermore, collinear (or mass) divergences arise in the virtual corrections and in the soft and hard regionof the radiative process. They are regulated by the lepton mass m (cid:96) that acts as a natural cutoff. According tothe KLN theorem [21, 22, 23], there must not be any divergences in the fully inclusive decay rate. Since thelimit m (cid:96) → is usually taken in experimental analyses, I apply the same approximation to the inclusive decayrate. Here, however, it is crucial that the collinear divergences indeed cancel.22ote that I use everywhere the physical lepton mass, which can be identified (up to higher order effects)with the renormalised mass. A necessary condition for the KLN theorem to hold in this representation is thatthe mass renormalisation does not diverge in the limit m (cid:96) → . This condition is fulfilled by (E.56). In the virtual corrections, the six triangle diagrams 7e-7j and the external leg corrections are IR-divergent. Therelevant loop functions are given in appendix A.3.A priori, one would expect that the box diagrams 7p-7r also give rise to an IR singularity, because the scalarfour point loop function D is IR-divergent as well. However, as can be shown with Passarino-Veltman reductiontechniques [28, 30] and the explicit expressions for the IR-divergent scalar box integral [31], the contribution ofthe box diagrams to the form factors F and G are IR-finite. This can be understood rather easily: consider thefour-loop kaon self-energy diagram in figure 12. This diagram is an IR-finite quantity and so must be the sum ofits four- and five-particle cuts. Each of the four cuts corresponds to a phase space integral of the product of twodiagrams, shown in figure 13. Now, as the IR divergence in the radiative process is generated in the soft region,where the matrix element factorises into the LO non-radiative process times the soft photon factor (61), the IRdivergence has to drop out already in the differential inclusive decay rate, where the photon is integrated. Thephase space products 13b-13d can only contribute to the term RF ∗ , RG ∗ and | R | . Therefore, the phase spaceproduct 13a cannot give an IR-divergent contribution to | F | or | G | . Hence, the box diagram on the left-handside of the product can only give IR-divergent contributions to R . An analogous argument works for the twoother box diagrams. (a)(b)(c) (d) Figure 12:
Four-loop kaon self-energy diagram with four- or five-particle cuts. · † (a) · † (b) · † (c) · † (d)Figure 13:
Phase space products corresponding to the four cuts of the kaon self-energy diagram.
Let us now turn our attention to the IR divergences of the virtual corrections. Summing all the IR-divergent23ontributions (after UV renormalisation), I find δF NLO , IRvirt .γ = δF NLO , IR γ − loop ,e − j + δF NLO , IR γ − Z = δG NLO , IRvirt .γ = δG NLO , IR γ − loop ,e − j + δG NLO , IR γ − Z = 2 e (cid:32) ( M K + + M π + − t ) C IR0 ( M π + , t, M K + , m γ , M π + , M K + ) − ( M K + + M π + − u ) C IR0 ( M π + , u, M K + , m γ , M π + , M K + )+ ( M K + + m (cid:96) − t (cid:96) ) C IR0 ( m (cid:96) , t (cid:96) , M K + , m γ , m (cid:96) , M K + )+ (2 M π + − s ) C IR0 ( M π + , s, M π + , m γ , M π + , M π + ) − ( M π + + m (cid:96) − s (cid:96) ) C IR0 ( m (cid:96) , s (cid:96) , M π + , m γ , m (cid:96) , M π + )+ ( M π + + m (cid:96) − s (cid:96) ) C IR0 ( m (cid:96) , s (cid:96) , M π + , m γ , m (cid:96) , M π + ) − π ln z γ (cid:33) =: δX NLO , IRvirt .γ , (84)where C IR0 ( m , s, M , m γ , m , M ) = − π x s mM (1 − x s ) ln x s ln z γ ,x s = − − (cid:113) − mMs − ( m − M ) (cid:113) − mMs − ( m − M ) . (85)The infrared-divergent part of the NLO decay rate is given by d Γ NLO , IR = d Γ LOiso δX NLO , IRvirt .γ ) + O ( z (cid:96) ln z (cid:96) ) . (86)By extracting the IR divergence (terms proportional to ln z γ ) out of the soft photon contribution to the radiativedecay rate (62), it is now easy to verify that the sum of virtual corrections and soft bremsstrahlung (where thephoton is integrated) and hence the inclusive decay rate is free of infrared divergences: d Γ NLO , IR + d Γ soft , IR γ = 0 . (87) Both the soft and the hard region of the radiative process give rise to collinear singularities, terms proportionalto ln z (cid:96) . Let us now check that these mass divergences cancel in the fully inclusive decay rate (the cut on thephoton energy must be removed for this purpose, i.e. I take the limit ˜ x max → ). Virtual photon correctionscan produce a collinear divergence if one end of the photon line is attached to the lepton line. Since themass divergence in the radiative process is produced in the collinear region of the phase space (soft and hard),where the matrix element could be factorised similarly to the soft region [32], one can argue in an analogousway as for the IR divergences that the contribution of the box diagrams to the form factors F and G has nomass divergence. This is confirmed by the explicit expressions for the diagrams. The only collinear divergentcontributions stem from the external leg correction for the lepton and the three diagrams 7g, 7i and 7j.The external leg correction for the lepton contains the following collinear divergence: δF NLO , coll γ − Z = δG NLO , coll γ − Z = 3 e π ln z (cid:96) , (88)contributing to the decay rate as d Γ NLO , coll Z = d Γ LOiso e π ln z (cid:96) . (89)This cancels exactly the mass divergence in the (cid:96)(cid:96) -part of the real photon corrections (76).Next, I collect the mass divergent terms contained in the three relevant loop diagrams: δF NLO , coll γ − loop = δG NLO , coll γ − loop = e π ln z (cid:96) (cid:18)
12 ln z (cid:96) − ln z γ − (cid:19) , (90)24esulting in a collinear divergence in the decay rate of d Γ NLO , collloop = d Γ LOiso e π ln z (cid:96) (ln z (cid:96) − z γ − . (91)This singularity must cancel with the mass divergence in the interference term of the radiative decay rate. Thedivergent contribution from the soft photon region is given by d Γ soft , int γ, coll = − d Γ LOiso e π ln z (cid:96) (cid:18) ln z (cid:96) + 4 ln (cid:18) εm γ (cid:19)(cid:19) = − d Γ LOiso e π ln z (cid:96) (cid:18) ln z (cid:96) − z γ + 4 ln (cid:18) ε √ s (cid:96) (cid:19)(cid:19) . (92)In the sum of virtual and soft real corrections, the double divergences (double collinear and soft-collinear) cancel: d Γ NLO , collloop + d Γ soft , int γ, coll = − d Γ LOiso e π ln z (cid:96) (1 + ln ˜ x min ) . (93)This single divergence must cancel against the one in the hard real corrections (82). By evaluating numericallythe integral over j int1 , I have checked that this cancellation takes place.I have now verified that the fully inclusive decay rate d Γ ( γ ) dsds (cid:96) d cos θ π = d Γ NLOvirt .γ dsds (cid:96) d cos θ π + d Γ soft γ dsds (cid:96) d cos θ π + d Γ hard γ dsds (cid:96) d cos θ π (94)does not depend on the energy cut separating the soft from the hard region and contains neither infrared norcollinear (mass) singularities. The calculation is therefore in accordance with the KLN theorem. Note that thisis a necessary but highly non-trivial consistency check, since the two regions of the radiative phase space areparametrised differently. The existing high statistics experiments on K (cid:96) , E865 [7, 33] and NA48/2 [6, 8], have applied isospin correctionsto a certain extent and with different approximations. In the NA48/2 experiment, the data was corrected by thesemi-classical Gamow-Sommerfeld (or Coulomb) factor and with help of PHOTOS [15]. The E865 experimentused the same analytic prescription by Diamant-Berger [34] as the older Geneva-Saclay experiment [35]. Bothtreatments did not make use of the full matrix element and relied on factorisation of the tree-level amplitudeas it happens in a soft and collinear photon approximation. The isospin breaking due to the mass effects wasnot taken into account.Unfortunately, in the case of NA48/2, an analysis without the effect of PHOTOS is not available. Hence,it seems almost impossible to make use of the here calculated photonic effects for a full a posteriori correctionof the form factors. Nevertheless, I have a program at hand that calculates the effect of PHOTOS on the(partially) inclusive decay rate . This enables me to perform a comparison of the here presented calculationwith the effect of PHOTOS, using the simple photon energy cut in Σ (cid:96)νγ described in the previous section.I pursue therefore two aims in the following sections. First, the isospin corrections due to the mass effects canbe extracted directly for the form factors. Second, for the photonic effects, I calculate the radiative correctionsto the (semi-)inclusive decay rate. These isospin breaking effects are then compared with the correction appliedby NA48/2. As explained in the previous chapter, the isospin breaking effects due to the quark and meson mass differencescan be extracted on the level of the amplitude or form factors. I now evaluate these corrections numerically.The form factors have the partial wave expansions [18] F + σ π P LX cos θ π G = ∞ (cid:88) l =0 P l (cos θ π ) f l ( s, s (cid:96) ) ,G = ∞ (cid:88) l =1 P (cid:48) l (cos θ π ) g l ( s, s (cid:96) ) , (95) I am very grateful to B. Bloch-Devaux for providing me with this program. P l are the Legendre polynomials. The NA48/2 experiment [8] uses the expansion F = F s e iδ s + F p e iδ p cos θ π + . . . ,G = G p e iδ p + . . . (96)and defines ˜ G p = G p + Xσ π P L F p . (97)Hence, I identify F s = | f | , ˜ G p = Xσ π P L | f | , G p = | g | (98)and calculate the partial wave projections f l = 2 l + 12 (cid:90) − d cos θ π P l (cos θ π ) (cid:18) F + σ π P LX cos θ π G (cid:19) ,g l = (cid:90) − d cos θ π P l − (cos θ π ) − P l +1 (cos θ π )2 G. (99)At the order that I consider, the isospin correction due to the mass effects to the norms and phases of the partialwaves is then given by δ ME F s := 1 − | f | lim isospin | f | = 1 − | Re( f ) | lim isospin | Re( f ) | + O ( p ) ,δ ME ˜ G p := 1 − | f | lim isospin | f | = 1 − | Re( f ) | lim isospin | Re( f ) | + O ( p ) ,δ ME G p := 1 − | g | lim isospin | g | = 1 − | Re( g ) | lim isospin | Re( g ) | + O ( p ) , ∆ ME δ := arg( f ) − lim isospin arg( f ) = Im( f ) f LO0 − lim isospin Im( f ) f LO0 + O ( p ) , ∆ ME δ := arg( f ) − lim isospin arg( f ) = Im( f ) f LO1 − lim isospin Im( f ) f LO1 + O ( p )= arg( g ) − lim isospin arg( g ) = Im( g ) g LO1 − lim isospin Im( g ) g LO1 + O ( p ) . (100)The isospin correction to the P -wave phase shift vanishes at this order. Using the inputs described in [9, 12], Ireproduce their NLO results for the S -wave phase shift.The correction to the phase depends on the pion decay constant and the breaking parameters. In thecorrection to the norm of the partial waves, also the low-energy constants L r , K r , K r and K r appear ( K r onlyappears in the correction to the P -wave).I have presented the analytic results of the loop calculation in terms of the decay constant in the chiral limit F . Unfortunately, different lattice determinations do not yet agree on its value [36]. For the numerics, I convertthe results to an expansion in /F π using the relation between F and F π in pure QCD at O ( p , (cid:15)p ) [37], F π = F (cid:34) F (cid:16) L r ( µ )( M π + 2 M K ) + L r ( µ ) M π (cid:17) − π ) F (cid:18) M π ln (cid:18) M π µ (cid:19) + M K ln (cid:18) M K µ (cid:19)(cid:19) (cid:35) , (101)where M π,K denote the masses in the isospin limit, defined as M π = M π , M K = 12 (cid:0) M K + + M K − M π + + M π (cid:1) . (102)For F π and the meson masses, I use the current PDG values [38].26nother strategy would be to work directly with F and assign a large error that covers the differentdeterminations, as done in [12]. I use the solution based on the expansion in /F with a central value of F = 75 MeV for a very rough estimate of higher order corrections.The correction to the norms of the partial waves depends rather strongly on the value of L r . The O ( p ) fits in [39, 40] give the large value L r = 1 . · − . I decide however, to rely on the lattice estimate of [41],recommended in [36], but to use a more conservative uncertainty of ± . · − (see table 1).For the NLO constants of the electromagnetic sector, I use the estimates of [42] and assign a 100% error.For the isospin breaking parameter (cid:15) , I take the latest recommendation in the FLAG report [36], (cid:15) = √ R , R = 35 . ± . , (103)where I added the lattice and electromagnetic errors in quadrature.I fix the electromagnetic low-energy constant Z with the LO relation to the pion mass difference (D.8). · L r ( µ ) 0 . ± . [36] · L r ( µ ) 0 . ± . [36] · K r ( µ ) 0 . ± . [42] · K r ( µ ) 1 . ± . [42] · K r ( µ ) 2 . ± . [42] F π (92 . ± . MeV [38] R . ± . [36] Table 1:
Input parameters for the evaluation of the mass effects ( µ = 770 MeV).
The plots in figures 14 and 15 show the relative isospin correction due to the mass effects for the norm ofthe partial waves. I separately show the error band due to the variation of the input parameters and the errorband that also includes the estimate of higher order corrections, given by the difference between the F π - andthe F -solution, added in quadrature. The error due to the input parameters is dominated by the uncertaintyof the low-energy constant L r . The LECs of the electromagnetic sector and the isospin breaking parameter R play a minor role. δ M E F s i n % √ s/ MeVcentral solutionerror due to input parameterserror including higher order estimate-0.500.511.522.5 300 350 400 450 500
Figure 14:
Relative value of the mass effect correction to the S -wave F s for s (cid:96) = 0 . The exact meaning of the errorbands is explained in the text. In contrast to the S -wave, where the isospin corrections are at the percent level, the effect in the two P -wavesis within the uncertainty compatible with zero. The dependence on s (cid:96) is rather weak and covered by the errorbands.To conclude this section, I suggest to apply the additional isospin breaking corrections to the NA48/2measurement [8] shown in table 2. In order to obtain the partial waves of the form factors in the isospin limit,one has to subtract the given corrections. The corrections to the P -waves are certainly negligible. However, forthe S -wave, the isospin correction (and also its uncertainty, unfortunately) is much larger than the experimentalerrors. 27 M E ˜ G p i n % √ s/ MeVcentral solutionerror due to input parameterserror including higher order estimate-0.4-0.200.20.40.60.81 300 350 400 450 500 δ M E G p i n % √ s/ MeVcentral solutionerror due to input parameterserror including higher order estimate-0.4-0.200.20.40.6 300 350 400 450 500
Figure 15:
Relative value of the mass effect corrections to the P -waves ˜ G p and G p for s (cid:96) = 0 . √ s/ MeV √ s (cid:96) / MeV F s [6, 8] δ ME F s · F s ˜ G p [6, 8] δ ME ˜ G p · ˜ G p G p [6, 8] δ ME G p · G p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2:
Isospin breaking corrections due to the mass effects, calculated for the bins of the NA48/2 measurement [6, 8].For comparison, I quote the values of the partial waves with their uncertainties (statistical and systematic errors addedin quadrature) without including the dominant error of the normalisation. Note that the uncertainties of F s are takenfrom [6], as the values displayed in [8] are not correct . The first error to the isospin correction is due to the inputparameters, the second is a rough estimate of higher order corrections. For the numerical evaluation of the photonic effects, I compute the (semi-)inclusive decay rate, differentialwith respect to s , s (cid:96) and cos θ π . After some general considerations and tests, I compare the resulting O ( e ) correction to the one applied in the NA48/2 experiment [8], i.e. the Gamow-Sommerfeld factor combined withPHOTOS [15].For the numerical evaluation of the inclusive decay rate d Γ ( γ ) , I need several input parameters. As I aminterested in O ( e ) effects but work only at leading chiral order, I directly replace F by the physical pion decayconstant F π . When calculating the fully inclusive decay rate, I take advantage of the cancellation of collinearsingularities and send the lepton mass m (cid:96) to zero, while I use the physical masses of the charged mesons [38].In the calculation of the semi-inclusive decay rate with the photon energy cut ∆ x , I neglect terms that vanishin the limit m (cid:96) → and evaluate the large logarithm ln z (cid:96) with the physical electron mass [38].In the NLO counterterm corrections, the low-energy constants L r and L r of the strong sector enter. Thelattice determinations of these LECs have not yet reached ‘green status’ in the FLAG report [36]. For L r ,I use the value of [39], for L r , I take the O ( p ) fit of [43], which is compatible with the available latticedeterminations.As for the case of the mass effects, I again use the estimates of [42, 44] for the electromagnetic LECs witha 100% error assigned to them.The ‘leptonic’ LECs X r and X r are unknown. X r contains the universal short-distance contribution [45], I thank B. Bloch-Devaux for the confirmation thereof. X r ( µ ) = ˜ X r ( µ ) + X SD6 , e X SD6 = 1 − S EW ( M ρ , M Z ) = − e π ln (cid:18) M Z M ρ (cid:19) , (104)such that ˜ X r is of the typical size of a LEC in χ PT. I use the naïve dimensional estimate that those LECs areof the order / (4 π ) . For the short-distance contribution, I take the value that includes leading logarithmic andQCD corrections [45]. · L r ( µ ) 5 . ± . [39] · L r ( µ ) − . ± . [43] · K r ( µ ) − . ± . [42] · K r ( µ ) 2 . ± . [42] · K r ( µ ) 11 . ± . [42] · K r ( µ ) − . ± . [44] · X r ( µ ) 0 ± . · ˜ X r ( µ ) 0 ± . S EW . [45] F π (92 . ± . MeV [38]
Table 3:
Input parameters for the evaluation of the photonic effects ( µ = 770 MeV).
In a first step, I want to quantify the importance of considering the full (hard) matrix element for the radiativeprocess instead of relying on the soft photon approximation. To this end, I compare the semi-inclusive totaland differential decay rates (using the photon energy cut ˜ x max ) with the decay rate, where the radiative processis just given by the SPA with a finite ∆ ε . The same energy cut in the two descriptions is obtained by setting ˜ x min = ˜ x max ⇒ ∆ ε = √ s (cid:96) x max (1 − z (cid:96) ) . (105)In this prescription, the photon energy cut is not constant but respects the bounds given by the phase space.The maximum photon energy is ∆ ε max = ˜ x max ( M K + − M π + ) − m (cid:96) M K + − M π + ) . (106)I compare in the following the corrections to the total decay rate, defined by Γ cut( γ ) = Γ LO (cid:16) δ Γ cut( γ ) (cid:17) . (107)In figure 16, the correction to the decay rate δ Γ cut( γ ) is shown as a function of the photon energy cut. The virtualcorrections are evaluated using the central values of the input parameters. The soft photon approximationdepends logarithmically on the energy cut (reflecting the IR divergence at low energies), whereas the correctionusing the full matrix element is somewhat smaller. Since I use a cut in the dilepton-photon rest frame, the resultcannot be applied directly to the experiment, where an energy cut is present in the lab frame. However, I expectthat the picture of the difference between full matrix element and soft photon approximation will look similarin the kaon centre-of-mass frame. In the relative form factor measurement of NA48/2, a 3 GeV photon energycut was applied in the lab frame [6]. This translates into a minimal detectable photon energy of 11.7 MeV inthe centre-of-mass frame. For such a low photon energy, the soft approximation can be expected to still workwell (the deviation in Σ (cid:96)νγ is ≈ . of the total rate). However, the experimental cut is not sharp: at theouter edge of the calorimeter, the minimal detectable centre-of-mass photon energy is about 36.8 MeV and ofcourse, only photons flying in the direction of the calorimeter can be detected. At larger photon energies, theerror introduced by using a SPA is quite substantial (up to 1.6% of the total rate for hard photons). This canbe understood in terms of the collinear singularity: the SPA alone does not produce the correct dependence onthe lepton mass, hence, the large logarithm does not cancel.29 Γ c u t( γ ) i n % ∆ ε max / MeVsoft photon approximationfull real emission amplitudedifference-20246 0 20 40 60 80 100
Figure 16:
Comparison of the O ( e ) photonic correction (virtual and real photons) to the semi-inclusive total decayrate as a function of the photon energy cut in Σ (cid:96)νγ , using the soft photon approximation vs. the full radiative matrixelement. As explained before, the gauge invariant class of loop diagrams in figure 8 together with the correspondingcounterterms has been neglected in the previous literature [13, 14]. To judge the influence of these diagrams, Icompute the total inclusive decay rate, remove the cut ( ˜ x max = 1 ) and sum the uncertainties due to the inputparameters in quadrature. Using all the diagrams for the virtual corrections, I find δ Γ ( γ ) = (4 . ± . , (108)whereas neglecting the mentioned class of diagrams results in δ Γ negl . ( γ ) = (4 . ± . . (109)The uncertainty is completely dominated by X r ( µ ) . Note that approximately half of the correction (2.32%) isdue to the short-distance enhancement. × PHOTOS
The Gamow-Sommerfeld (or Coulomb) factor is defined by d Γ Coulomb = d Γ · (cid:89) i Figure 17: Comparison of the photonic corrections to the fully inclusive differential decay rate. The right plot excludesthe short-distance enhancement factor. δ d Γ c u t( γ ) / d s i n % √ s/ MeVsoft photon approximationfull real emission amplitudeCoulomb factor × PHOTOS-20246810 300 350 400 450 500 δ d Γ c u t( γ ) / d s i n % √ s/ MeVsoft photon approximation, S EW = 1 full real emission amplitude, S EW = 1 Coulomb factor × PHOTOS-20246810 300 350 400 450 500 Figure 18: Comparison of the photonic corrections to the semi-inclusive total decay rate with a photon energy cut of ∆ ε max = 40 MeV in Σ (cid:96)νγ . The rise of the PHOTOS factor at large s could be a numerical artefact, as the decay rateapproaches zero in this phase space region. The plots in figures 17 and 18 show the corrections to the differential decay rate. The divergence at the ππ threshold is the Coulomb singularity, reproduced in all descriptions. The rise of the PHOTOS factor at largevalues of s , however, could be a numerical artefact, because the differential decay rate drops to zero at theupper border of the phase space.The comparison without the short-distance enhancement shows that the Coulomb factor × PHOTOS ap-proach is relatively close to the soft photon approximation, which overestimates the radiative corrections.However, the short-distance factor has not been included in the experimental analysis, such that in total, theradiative corrections are underestimated. B. Bloch-Devaux, private communication. × PHOTOS approach is an acceptable description of the radiative corrections(a free normalisation factor corresponds to a free additive constant in the correction, hence the slopes of thecorrections have to be compared).I suggest to replace in a matching procedure the O ( e ) part of the Coulomb factor and the 0.69% PHOTOSeffect (or rather artefact) with the result of the here presented fixed order calculation, i.e. to apply the followingcorrection to the norm of the form factors X ∈ { F, G } : | X | = | X exp | (cid:18) (cid:16) δ Γ e Coulomb + δ Γ PHOTOS − δ Γ ( γ ) (cid:17)(cid:19) = | X exp | (0 . ± . , ⇒ δ | X | = ( − . ± . . (114)Note that replacing the systematic PHOTOS uncertainty with the above error increases the . uncertaintyof the NA48/2 norm measurement [8] to 0.70%.The fact that the a posteriori correction is so small is at least partly accidental: as argued above, I havethe strong suspicion that the estimate δ Γ PHOTOS = 0 . is simply the outcome of statistical fluctuations. Bychance, this number leads to a result close to the estimate obtained by Diamant-Berger in his analytic treatmentof radiative corrections. For this reason, it has been considered so far as a reliable estimate . In the present work, I have computed the one-loop isospin breaking corrections to the K (cid:96) decay within χ PTincluding leptons and photons. The corrections can be separated into mass effects and photonic effects. Themass effects for the S -wave are quite substantial but the result for the norm of the form factors suffers fromlarge uncertainties, on the one hand due to the uncertainty in the LEC L r , on the other hand due to higherorder corrections. The mass effects for the P -waves are negligible.For the photonic corrections, I have compared the fixed order calculation with the Coulomb × PHOTOSapproach used in the experimental analysis of NA48/2. An a posteriori correction of the data is possible for thenormalisation but not for the relative values of the form factors. The present calculation includes for the firsttime a treatment of the full radiative process and compares it with the soft photon approximation.For possible forthcoming experiments on K (cid:96) , I suggest that photonic corrections are applied in a MonteCarlo simulation that includes the exact matrix element. This can be done e.g. with PHOTOS. The mass effectscan be easily corrected a posteriori.This work goes either beyond the isospin breaking treatments in previous literature or is complementary:I confirm the largest part of the amplitude calculation of [13, 14], but correct their results by a neglectedgauge invariant class of diagrams. I have included the full radiative process and shown that the soft photonapproximation is not necessarily trustworthy and certainly not applicable for the fully inclusive decay.I reproduce the NLO mass effect calculation for phases of the form factors done in [9, 12], but concentrate hereon the absolute values of the form factors. As the NLO mass effect calculation suffers from large uncertainties,an extension of the dispersive framework of [12] to the norm of the form factors would be desirable.To judge the reliability of the photonic corrections, one should ideally calculate them to higher chiral orders,which is however prohibitive (and would bring in many unknown low-energy constants). Here, I have assumedimplicitly that the photonic corrections factorise and therefore modify the higher chiral orders with the samemultiplicative correction as the lowest order. It is hard to judge if this assumption is justified: for this reason,I have attached a rather conservative estimate of the uncertainties to the photonic corrections presented here. Acknowledgements I cordially thank my thesis advisor, Gilberto Colangelo, for his continuous support during the last years (in manyrespects and also at unearthly hours) and for having drawn my attention to the isospin breaking correctionsto K (cid:96) , which became a side-project to my thesis. I am very grateful to Brigitte Bloch-Devaux for her time B. Bloch-Devaux, private communication. A Loop Functions A.1 Scalar Functions I use the following conventions for the scalar loop functions: A ( m ):= 1 i (cid:90) d n q (2 π ) n q − m ] ,B ( p , m , m ):= 1 i (cid:90) d n q (2 π ) n q − m ][( q + p ) − m ] ,C ( p , ( p − p ) , p , m , m , m ):= 1 i (cid:90) d n q (2 π ) n q − m ][( q + p ) − m ][( q + p ) − m ] ,D ( p , ( p − p ) , ( p − p ) , p , p , ( p − p ) , m , m , m , m ):= 1 i (cid:90) d n q (2 π ) n q − m ][( q + p ) − m ][( q + p ) − m ][( q + p ) − m ] . (A.1)The loop functions A and B are UV-divergent. The renormalised loop functions are defined in the M S scheme by A ( m ) = − m λ + ¯ A ( m ) + O (4 − n ) ,B ( p , m , m ) = − λ + ¯ B ( p , m , m ) + O (4 − n ) , (A.2)where λ = µ n − π (cid:18) n − − 12 (ln(4 π ) + 1 − γ E ) (cid:19) . (A.3) µ denotes the renormalisation scale.The renormalised loop functions are given by [48] ¯ A ( m ) = − m π ln (cid:18) m µ (cid:19) , ¯ B ( p , m , m ) = − π m ln (cid:16) m µ (cid:17) − m ln (cid:16) m µ (cid:17) m − m + 132 π (cid:18) (cid:18) − ∆ p + Σ∆ (cid:19) ln (cid:18) m m (cid:19) − νp ln (cid:18) ( p + ν ) − ∆ ( p − ν ) − ∆ (cid:19)(cid:19) , (A.4)where ∆ := m − m , Σ := m + m ,ν := (cid:112) ( s − ( m + m ) )( s − ( m − m ) ) = λ / ( s, m , m ) . (A.5)33 .2 Tensor-Coefficient Functions Although all the loop integrals can be expressed in terms of the basic scalar loop functions by means of aPassarino-Veltman reduction [28, 30], this produces sometimes very long polynomial coefficients. I thereforealso use the tensor coefficient functions. The tensor integrals that I use are defined by B µν ( p ; m , m ) := 1 i (cid:90) d n q (2 π ) n q µ q ν [ q − m ][( q + p ) − m ] ,C µ ( p , p ; m , m , m ) := 1 i (cid:90) d n q (2 π ) n q µ [ q − m ][( q + p ) − m ][( q + p ) − m ] ,C µν ( p , p ; m , m , m ) := 1 i (cid:90) d n q (2 π ) n q µ q ν [ q − m ][( q + p ) − m ][( q + p ) − m ] ,D µ ( p , p , p ; m , m , m , m ) := 1 i (cid:90) d n q (2 π ) n q µ [ q − m ][( q + p ) − m ][( q + p ) − m ][( q + p ) − m ] ,D µν ( p , p , p ; m , m , m , m ) := 1 i (cid:90) d n q (2 π ) n q µ q ν [ q − m ][( q + p ) − m ][( q + p ) − m ][( q + p ) − m ] . (A.6)The tensor coefficients are then given by a Lorentz decomposition: B µν ( p ; m , m ) = g µν B ( p , m , m ) + p µ p ν B ( p , m , m ) ,C µ ( p , p ; m , m , m ) = p µ C ( p , ( p − p ) , p , m , m , m )+ p µ C ( p , ( p − p ) , p , m , m , m ) ,C µν ( p , p ; m , m , m ) = g µν C ( p , ( p − p ) , p , m , m , m )+ (cid:88) i,j =1 p µi p νj C ij ( p , ( p − p ) , p , m , m , m ) ,D µ ( p , p , p ; m , m , m , m ) = (cid:88) i =1 p µi D i ( p , ( p − p ) , ( p − p ) , p , p , ( p − p ) , m , m , m , m ) ,D µν ( p , p , p ; m , m , m , m ) = g µν D ( p , ( p − p ) , ( p − p ) , p , p , ( p − p ) , m , m , m , m )+ (cid:88) i,j =1 p µi p νj D ij ( p , ( p − p ) , ( p − p ) , p , p , ( p − p ) , m , m , m , m ) . (A.7)Only some of those tensor coefficient functions are UV-divergent: B ( p , m , m ) = − λ (cid:18) m + m − p (cid:19) + ¯ B ( p , m , m ) + O (4 − n ) ,B ( p , m , m ) = − λ + ¯ B ( p , m , m ) + O (4 − n ) ,C ( p , ( p − p ) , p , m , m , m ) = − λ C ( p , ( p − p ) , p , m , m , m ) + O (4 − n ) . (A.8) A.3 Infrared Divergences in Loop Functions The following explicit formulae are used to extract the IR divergence in the loop functions.The derivative of the two-point function is IR-divergent: ¯ B (cid:48) ( M , M , m γ ) = − π M (cid:32) (cid:32) m γ M (cid:33)(cid:33) + O ( m γ ) , ¯ B ( M , M , m γ ) = 116 π (cid:18) − ln (cid:18) M µ (cid:19)(cid:19) + O ( m γ ) , ¯ B (0 , M , m γ ) = − π ln (cid:18) M µ (cid:19) + O ( m γ ) . (A.9)34he IR-divergent three-point function is given by [31] C ( m , s, M , m γ , m , M ) = 116 π x s mM (1 − x s ) (cid:32) ln x s (cid:18) − 12 ln x s + 2 ln(1 − x s ) + ln (cid:18) mMm γ (cid:19)(cid:19) − π ( x s ) + 12 ln (cid:16) mM (cid:17) + Li (cid:16) − x s mM (cid:17) + Li (cid:18) − x s Mm (cid:19) (cid:33) + O ( m γ ) , (A.10)where x s = − − (cid:113) − mMs − ( m − M ) (cid:113) − mMs − ( m − M ) . (A.11) B Kinematics B.1 Lorentz Frames and Transformations in K (cid:96) Let us first look at the kaon rest frame Σ K . From the relations P = p + p = (cid:18)(cid:113) s + (cid:126)P , (cid:126)P (cid:19) ,L = p (cid:96) + p ν = (cid:18)(cid:113) s (cid:96) + (cid:126)P , − (cid:126)P (cid:19) ,p = P + L = (cid:16) M K + ,(cid:126) (cid:17) , (B.1)one finds (cid:126)P = λ K(cid:96) ( s )4 M K + , (B.2)where λ K(cid:96) ( s ) := λ ( M K + , s, s (cid:96) ) and λ ( a, b, c ) := a + b + c − ab + bc + ca ) .I choose the x -axis along the dipion line of flight: P = (cid:32) M K + − s (cid:96) + s M K + , λ / K(cid:96) ( s )2 M K + , , (cid:33) ,L = (cid:32) M K + + s (cid:96) − s M K + , − λ / K(cid:96) ( s )2 M K + , , (cid:33) . (B.3)In the dipion centre-of-mass frame Σ π , the boosted dipion four-momentum is P (cid:48) = Λ − P = (cid:16) √ s,(cid:126) (cid:17) . (B.4) Λ is just a boost in the x -direction. Thus, I find Λ = M K + + s − s (cid:96) M K + √ s λ / K(cid:96) ( s )2 M K + √ s λ / K(cid:96) ( s )2 M K + √ s M K + + s − s (cid:96) M K + √ s . (B.5)Analogously, in the dilepton centre-of-mass frame Σ (cid:96)ν , the boosted dilepton four-momentum is L (cid:48)(cid:48) = Λ − L = (cid:16) √ s (cid:96) ,(cid:126) (cid:17) . (B.6)35 is given by a rotation around the x -axis and a subsequent boost in the x -direction. I find Λ = M K + − s + s (cid:96) M K + √ s (cid:96) − λ / K(cid:96) ( s )2 M K + √ s (cid:96) − λ / K(cid:96) ( s )2 M K + √ s (cid:96) M K + − s + s (cid:96) M K + √ s (cid:96) φ sin φ − sin φ cos φ . (B.7)Let us determine the momenta of the four final-state particles in the kaon rest frame. In Σ π , the pionmomenta p (cid:48) = (cid:18)(cid:113) M π + + (cid:126)p , (cid:126)p (cid:19) ,p (cid:48) = (cid:18)(cid:113) M π + + (cid:126)p , − (cid:126)p (cid:19) (B.8)satisfy P (cid:48) = p (cid:48) + p (cid:48) = (cid:16) √ s,(cid:126) (cid:17) . (B.9)Therefore, we find (cid:126)p = s − M π + , (B.10)leading to p (cid:48) = (cid:18) √ s , (cid:114) s − M π + cos θ π , (cid:114) s − M π + sin θ π , (cid:19) ,p (cid:48) = (cid:18) √ s , − (cid:114) s − M π + cos θ π , − (cid:114) s − M π + sin θ π , (cid:19) . (B.11)The pion momenta in Σ K are then given by p = Λ p (cid:48) = (cid:32) M K + + s − s (cid:96) M K + + λ / K(cid:96) ( s )4 M K + σ π ( s ) cos θ π ,λ / K(cid:96) ( s )4 M K + + M K + + s − s (cid:96) M K + σ π ( s ) cos θ π , (cid:114) s − M π + sin θ π , (cid:33) ,p = Λ p (cid:48) = (cid:32) M K + + s − s (cid:96) M K + − λ / K(cid:96) ( s )4 M K + σ π ( s ) cos θ π ,λ / K(cid:96) ( s )4 M K + − M K + + s − s (cid:96) M K + σ π ( s ) cos θ π , − (cid:114) s − M π + sin θ π , (cid:33) , (B.12)where σ π ( s ) = (cid:113) − M π + s .Again, the analogous procedure for the dilepton system leads to the lepton momenta in the kaon system. In Σ (cid:96)ν , the lepton momenta are p (cid:48)(cid:48) (cid:96) = (cid:18)(cid:113) m (cid:96) + (cid:126)p (cid:96) , (cid:126)p (cid:96) (cid:19) , p (cid:48)(cid:48) ν = ( | (cid:126)p (cid:96) | , − (cid:126)p (cid:96) ) , (B.13)satisfying L (cid:48)(cid:48) = p (cid:48)(cid:48) (cid:96) + p (cid:48)(cid:48) ν = (cid:16) √ s (cid:96) ,(cid:126) (cid:17) , (B.14)with the solution (cid:126)p (cid:96) = (cid:0) s (cid:96) − m (cid:96) (cid:1) s (cid:96) , (B.15)36ence p (cid:48)(cid:48) (cid:96) = (cid:18) s (cid:96) + m (cid:96) √ s (cid:96) , − s (cid:96) − m (cid:96) √ s (cid:96) cos θ (cid:96) , s (cid:96) − m (cid:96) √ s (cid:96) sin θ (cid:96) , (cid:19) ,p (cid:48)(cid:48) ν = (cid:18) s (cid:96) − m (cid:96) √ s (cid:96) , s (cid:96) − m (cid:96) √ s (cid:96) cos θ (cid:96) , − s (cid:96) − m (cid:96) √ s (cid:96) sin θ (cid:96) , (cid:19) . (B.16)I obtain the lepton momenta in Σ K by applying the Lorentz transformation Λ : p (cid:96) = Λ p (cid:48)(cid:48) (cid:96) = (cid:32) (1 + z (cid:96) ) M K + − s + s (cid:96) M K + + (1 − z (cid:96) ) λ / K(cid:96) ( s )4 M K + cos θ (cid:96) , − (1 + z (cid:96) ) λ / K(cid:96) ( s )4 M K + − (1 − z (cid:96) ) M K + − s + s (cid:96) M K + cos θ (cid:96) ,s (cid:96) − m (cid:96) √ s (cid:96) sin θ (cid:96) cos φ, − s (cid:96) − m (cid:96) √ s (cid:96) sin θ (cid:96) sin φ (cid:33) ,p ν = Λ p (cid:48)(cid:48) ν = (cid:32) (1 − z (cid:96) ) (cid:32) M K + − s + s (cid:96) M K + − λ / K(cid:96) ( s )4 M K + cos θ (cid:96) (cid:33) , − (1 − z (cid:96) ) (cid:32) λ / K(cid:96) ( s )4 M K + − M K + − s + s (cid:96) M K + cos θ (cid:96) (cid:33) , − s (cid:96) − m (cid:96) √ s (cid:96) sin θ (cid:96) cos φ, s (cid:96) − m (cid:96) √ s (cid:96) sin θ (cid:96) sin φ (cid:33) , (B.17)where z (cid:96) = m (cid:96) /s (cid:96) .With these explicit expressions for the particle momenta, I calculate in the following all the Lorentz invariantproducts in terms of the five phase space variables.The Lorentz invariant squares of the vectors (3) are given by P = p + 2 p p + p = 2 M π + + 2 p p = s,Q = p − p p + p = 4 M π + − s,L = p (cid:96) + 2 p (cid:96) p ν + p ν = m (cid:96) + 2 p (cid:96) p ν = s (cid:96) ,N = p (cid:96) − p (cid:96) p ν + p ν = 2 m (cid:96) − s (cid:96) . (B.18)The remaining Lorentz invariant products are: P Q = p − p = 0 ,P L = 12 (cid:0) p − P − L (cid:1) = 12 (cid:0) M K + − s − s (cid:96) (cid:1) ,P N = 12 (cid:0) ( p − p ν ) − P − N (cid:1) = 12 z (cid:96) (cid:0) M K + − s − s (cid:96) (cid:1) + (1 − z (cid:96) ) X cos θ (cid:96) ,QL = Qp = σ π X cos θ π ,QN = z (cid:96) σ π X cos θ π + σ π (1 − z (cid:96) ) (cid:26) (cid:0) M K + − s − s (cid:96) (cid:1) cos θ π cos θ (cid:96) − √ ss (cid:96) sin θ π sin θ (cid:96) cos φ (cid:27) ,LN = ( p (cid:96) + p ν )( p (cid:96) − p ν ) = m (cid:96) , (cid:104) LN P Q (cid:105) := (cid:15) µνρσ L µ N ν P ρ Q σ = − (1 − z (cid:96) ) σ π X √ s (cid:96) s sin θ π sin θ (cid:96) sin φ. (B.19) B.2 Lorentz Frames and Transformations in K (cid:96) γ For the radiative process, I copy the results for the dipion subsystem from the K (cid:96) kinematics and thereforefind the following expressions for the momenta in the kaon rest frame Σ K : P = (cid:32) M K + − s (cid:96) + s M K + , λ / K(cid:96) ( s )2 M K + , , (cid:33) ,L = (cid:32) M K + + s (cid:96) − s M K + , − λ / K(cid:96) ( s )2 M K + , , (cid:33) . (B.20)37 = (cid:32) M K + + s − s (cid:96) M K + + λ / K(cid:96) ( s )4 M K + σ π ( s ) cos θ π ,λ / K(cid:96) ( s )4 M K + + M K + + s − s (cid:96) M K + σ π ( s ) cos θ π , (cid:114) s − M π + sin θ π , (cid:33) ,p = (cid:32) M K + + s − s (cid:96) M K + − λ / K(cid:96) ( s )4 M K + σ π ( s ) cos θ π ,λ / K(cid:96) ( s )4 M K + − M K + + s − s (cid:96) M K + σ π ( s ) cos θ π , − (cid:114) s − M π + sin θ π , (cid:33) . (B.21)We still need to determine the momenta of the photon and the two leptons. The photon and charged leptonmomenta in Σ (cid:96)νγ are given by q (cid:48)(cid:48) = (cid:32) √ s (cid:96) x, − √ s (cid:96) (cid:113) x − z γ cos θ γ , √ s (cid:96) (cid:113) x − z γ sin θ γ , (cid:33) ,p (cid:48)(cid:48) (cid:96) = (cid:32) √ s (cid:96) y, √ s (cid:96) (cid:112) y − z (cid:96) (sin θ γ sin θ (cid:96)γ cos φ (cid:96) − cos θ γ cos θ (cid:96)γ ) , √ s (cid:96) (cid:112) y − z (cid:96) (cos θ γ sin θ (cid:96)γ cos φ (cid:96) + sin θ γ cos θ (cid:96)γ ) , √ s (cid:96) (cid:112) y − z (cid:96) sin θ (cid:96)γ sin φ (cid:96) (cid:33) , (B.22)where θ (cid:96)γ denotes the angle between photon and lepton in Σ (cid:96)νγ : cos θ (cid:96)γ = x ( y − 2) + 2(1 − y + z (cid:96) + z γ ) (cid:112) x − z γ (cid:112) y − z (cid:96) . (B.23)The neutrino momentum is then easily found by p (cid:48)(cid:48) ν = L (cid:48)(cid:48) − q (cid:48)(cid:48) − p (cid:48)(cid:48) (cid:96) .The momenta in the kaon rest frame Σ K are given by q = Λ q (cid:48)(cid:48) , p (cid:96) = Λ p (cid:48)(cid:48) (cid:96) , p ν = Λ p (cid:48)(cid:48) ν , (B.24)where Λ is defined in (B.7). I do not state here the expressions explicitly, as they are rather long. I use themto calculate in the following all the Lorentz invariant products in terms of the eight phase space variables.The Lorentz invariant squares of the vectors (26) are P = p + 2 p p + p = 2 M π + + 2 p p = s,Q = p − p p + p = 4 M π + − s,L = ( p (cid:96) + q ) + 2( p (cid:96) + q ) p ν + p ν = s γ + 2( p (cid:96) + q ) p ν = s (cid:96) ,N = ( p (cid:96) + q ) − p (cid:96) + q ) p ν + p ν = 2 s γ − s (cid:96) = s (cid:96) (2 x + 2 y − . (B.25)38he remaining Lorentz invariant products involving the vectors (26) are given by: P Q = 0 , P L = 12 (cid:0) M K + − s − s (cid:96) (cid:1) , QL = σ π X cos θ π , LN = s (cid:96) ( x + y − ,P N = ( x + y − 1) 12 (cid:0) M K + − s − s (cid:96) (cid:1) + X (cid:16)(cid:113) x − z γ cos θ γ + (cid:112) y − z (cid:96) (cos θ (cid:96)γ cos θ γ − sin θ (cid:96)γ sin θ γ cos φ (cid:96) ) (cid:17) ,QN = ( x + y − σ π X cos θ π + σ π (cid:26) (cid:0) M K + − s − s (cid:96) (cid:1) cos θ π (cid:16)(cid:113) x − z γ cos θ γ + (cid:112) y − z (cid:96) (cos θ (cid:96)γ cos θ γ − sin θ (cid:96)γ sin θ γ cos φ (cid:96) ) (cid:17) − √ ss (cid:96) sin θ π (cid:20) cos φ (cid:16)(cid:113) x − z γ sin θ γ + (cid:112) y − z (cid:96) (cos θ (cid:96)γ sin θ γ + sin θ (cid:96)γ cos θ γ cos φ (cid:96) ) (cid:17) + sin φ (cid:112) y − z (cid:96) sin θ (cid:96)γ sin φ (cid:96) (cid:21)(cid:27) , (cid:104) LN P Q (cid:105) := (cid:15) µνρσ L µ N ν P ρ Q σ = − σ π X √ ss (cid:96) sin θ π (cid:16)(cid:113) x − z γ sin φ sin θ γ + (cid:112) y − z (cid:96) (cid:0) sin φ (cos θ (cid:96)γ sin θ γ + sin θ (cid:96)γ cos θ γ cos φ (cid:96) ) − cos φ sin θ (cid:96)γ sin φ (cid:96) (cid:1)(cid:17) . (B.26)In addition, we need the Lorentz invariant products involving q : P q = x (cid:0) M K + − s − s (cid:96) (cid:1) + X (cid:113) x − z γ cos θ γ ,Qq = σ π (cid:20) xX cos θ π + (cid:113) x − z γ (cid:18) cos θ π 12 ( M K + − s − s (cid:96) ) cos θ γ − sin θ π √ ss (cid:96) sin θ γ cos φ (cid:19)(cid:21) ,Lq = s (cid:96) x,N q = s (cid:96) x + 2( y − z γ − z (cid:96) )) , (cid:104) LN P q (cid:105) = 12 Xs (cid:96) (cid:113) x − z γ (cid:112) y − z (cid:96) sin θ (cid:96)γ sin θ γ sin φ (cid:96) , (cid:104) LN Qq (cid:105) = 12 σ π s (cid:96) (cid:113) x − z γ (cid:112) y − z (cid:96) sin θ (cid:96)γ · (cid:18) 12 ( M K + − s − s (cid:96) ) cos θ π sin θ γ sin φ (cid:96) − √ ss (cid:96) sin θ π (sin φ cos φ (cid:96) − cos φ sin φ (cid:96) cos θ γ ) (cid:19) , (cid:104) LP Qq (cid:105) = − σ π X √ ss (cid:96) sin θ π (cid:113) x − z γ sin θ γ sin φ, (cid:104) N P Qq (cid:105) = 12 σ π √ ss (cid:96) · (cid:26)(cid:113) x − z γ (cid:112) y − z (cid:96) sin θ (cid:96)γ (cid:18) − √ ss (cid:96) cos θ π sin θ γ sin φ (cid:96) + 12 ( M K + − s − s (cid:96) ) sin θ π (sin φ cos φ (cid:96) − cos φ sin φ (cid:96) cos θ γ ) (cid:19) + X sin θ π (cid:18) x (cid:112) y − z (cid:96) (cid:0) − sin θ (cid:96)γ (cos φ sin φ (cid:96) − sin φ cos φ (cid:96) cos θ γ ) + cos θ (cid:96)γ sin φ sin θ γ (cid:1) − ( y − (cid:113) x − z γ sin φ sin θ γ (cid:19)(cid:27) . (B.27)39 Decay Rates C.1 Decay Rate for K (cid:96) C.1.1 Isospin Limit The partial decay rate for the K (cid:96) decay is given by d Γ = 12 M K + (2 π ) (cid:88) spins |T | δ (4) ( p − P − L ) d p p d p p d p (cid:96) p (cid:96) d p ν p ν . (C.1)The kinematics of the decay is described by the 5 variables s , s (cid:96) , θ π , θ (cid:96) and φ . The remaining 7 integrals canbe performed explicitly [5]. Let us review the reduction of the partial decay rate to the five-dimensional phasespace integral.The spin summed square of the matrix element T = G F √ V ∗ us ¯ u ( p ν ) γ µ (1 − γ ) v ( p (cid:96) ) (cid:10) π + ( p ) π − ( p ) (cid:12)(cid:12) ¯ sγ µ (1 − γ ) u (cid:12)(cid:12) K + ( p ) (cid:11) = G F √ V ∗ us L µ H µ , (C.2)where L µ := ¯ u ( p ν ) γ µ (1 − γ ) v ( p (cid:96) ) and H µ := (cid:10) π + ( p ) π − ( p ) (cid:12)(cid:12) ¯ sγ µ (1 − γ ) u (cid:12)(cid:12) K + ( p ) (cid:11) , can be written as (cid:88) spins |T | = G F | V us | H µ H ∗ ν (cid:88) spins L µ L ∗ ν . (C.3)The spin sum can be performed with standard trace techniques: L µν := (cid:88) spins L µ L ∗ ν = (cid:88) spins ¯ u ( p ν ) γ µ (1 − γ ) v ( p (cid:96) )¯ v ( p (cid:96) ) γ ν (1 − γ ) u ( p ν )= Tr (cid:104) /p ν γ µ (1 − γ )( /p (cid:96) − m (cid:96) ) γ ν (1 − γ ) (cid:105) = − g µν ( L − N ) + 4( L µ L ν − N µ N ν ) + 4 i(cid:15) µνρσ L ρ N σ = 4 (cid:0) g µν ( m (cid:96) − s (cid:96) ) + L µ L ν − N µ N ν + i(cid:15) µνρσ L ρ N σ (cid:1) . (C.4)After the contraction with the hadronic matrix element, expressed in terms of the form factors, H µ = − HM K + (cid:15) µνρσ L ν P ρ Q σ + i M K + ( P µ F + Q µ G + L µ R ) , (C.5)all the scalar products can be expressed in terms of the five phase space variables s , s (cid:96) , θ π , θ (cid:96) and φ .Let us now consider the phase space measure: dI := δ (4) ( p − p − p − p (cid:96) − p ν ) d p p d p p d p (cid:96) p (cid:96) d p ν p ν = δ (4) ( p − p − p − p (cid:96) − p ν ) d p p d p p d p (cid:96) p (cid:96) d p ν p ν · δ (4) ( p + p − P ) δ (4) ( p (cid:96) + p ν − L ) d P d L θ ( P ) θ ( L )= ds ds (cid:96) δ (4) ( p − P − L ) d P δ ( s − P ) θ ( P ) d Lδ ( s (cid:96) − L ) θ ( L ) · δ (4) ( p + p − P ) d p p d p p δ (4) ( p (cid:96) + p ν − L ) d p (cid:96) p (cid:96) d p ν p ν . (C.6)The phase space integral can be split into three separately Lorentz invariant pieces: dI = dI dI dI ,dI := ds ds (cid:96) δ (4) ( p − P − L ) d P δ ( s − P ) θ ( P ) d Lδ ( s (cid:96) − L ) θ ( L ) ,dI := δ (4) ( p + p − P ) d p p d p p ,dI := δ (4) ( p (cid:96) + p ν − L ) d p (cid:96) p (cid:96) d p ν p ν . (C.7)40ach of these three pieces can be evaluated in a convenient frame. For dI , I choose the kaon rest frame: dI = ds ds (cid:96) δ (3) ( (cid:126)p − (cid:126)P − (cid:126)L ) δ (cid:18) p − (cid:113) (cid:126)P + s − (cid:113) (cid:126)L + s (cid:96) (cid:19) d P (cid:112) (cid:126)P + s d L (cid:113) (cid:126)L + s (cid:96) = ds ds (cid:96) δ (3) ( (cid:126)P + (cid:126)L ) δ (cid:18) M K + − (cid:113) (cid:126)P + s − (cid:113) (cid:126)L + s (cid:96) (cid:19) d P (cid:112) (cid:126)P + s d L (cid:113) (cid:126)L + s (cid:96) = ds ds (cid:96) δ (cid:18) M K + − (cid:113) (cid:126)P + s − (cid:113) (cid:126)P + s (cid:96) (cid:19) d P (cid:112) (cid:126)P + s (cid:113) (cid:126)P + s (cid:96) = πds ds (cid:96) δ (cid:18) M K + − (cid:113) (cid:126)P + s − (cid:113) (cid:126)P + s (cid:96) (cid:19) (cid:126)P (cid:112) (cid:126)P + s (cid:113) (cid:126)P + s (cid:96) d | (cid:126)P | = πds ds (cid:96) δ (cid:32) | (cid:126)P | − λ / ( M K + , s, s (cid:96) )2 M K + (cid:33) | (cid:126)P | (cid:112) (cid:126)P + s + (cid:113) (cid:126)P + s (cid:96) d | (cid:126)P | = πds ds (cid:96) λ / K(cid:96) ( s )2 M K + = πds ds (cid:96) XM K + . (C.8)I have used that the integrand depends on (cid:126)P only through (cid:126)P .The second piece is evaluated in the dipion frame: dI = δ (3) ( (cid:126)p + (cid:126)p − (cid:126)P ) δ (cid:18)(cid:113) (cid:126)p + M π + + (cid:113) (cid:126)p + M π + − P (cid:19) d p (cid:113) (cid:126)p + M π + d p (cid:113) (cid:126)p + M π + = δ (3) ( (cid:126)p + (cid:126)p ) δ (cid:18)(cid:113) (cid:126)p + M π + + (cid:113) (cid:126)p + M π + − √ s (cid:19) d p (cid:113) (cid:126)p + M π + d p (cid:113) (cid:126)p + M π + = δ (cid:18) (cid:113) (cid:126)p + M π + − √ s (cid:19) d p (cid:126)p + M π + ) = δ (cid:18) | (cid:126)p | − (cid:114) s − M π + (cid:19) π σ π ( s ) d cos θ π d | (cid:126)p | = π σ π ( s ) d cos θ π , (C.9)and the third piece analogously in the dilepton frame: dI = δ (3) ( (cid:126)p (cid:96) + (cid:126)p ν − (cid:126)L ) δ (cid:18)(cid:113) (cid:126)p (cid:96) + m (cid:96) + | (cid:126)p ν | − L (cid:19) d p (cid:96) (cid:112) (cid:126)p (cid:96) + m (cid:96) d p ν | (cid:126)p ν | = δ (3) ( (cid:126)p (cid:96) + (cid:126)p ν ) δ (cid:18)(cid:113) (cid:126)p (cid:96) + m (cid:96) + | (cid:126)p ν | − √ s (cid:96) (cid:19) d p (cid:96) (cid:112) (cid:126)p (cid:96) + m (cid:96) d p ν | (cid:126)p ν | = δ (cid:18)(cid:113) (cid:126)p (cid:96) + m (cid:96) + | (cid:126)p (cid:96) | − √ s (cid:96) (cid:19) d p (cid:96) | (cid:126)p (cid:96) | (cid:112) (cid:126)p (cid:96) + m (cid:96) = δ (cid:18) | (cid:126)p (cid:96) | − s (cid:96) − m (cid:96) √ s (cid:96) (cid:19) 18 (1 − z (cid:96) ) d cos θ (cid:96) dφd | (cid:126)p (cid:96) | = 18 (1 − z (cid:96) ) d cos θ (cid:96) dφ. (C.10)Putting the three pieces together, I find dI = λ / K(cid:96) ( s ) M K + π 64 (1 − z (cid:96) ) σ π ( s ) ds ds (cid:96) d cos θ π d cos θ (cid:96) dφ, (C.11)and for the differential decay rate d Γ = 12 π λ / K(cid:96) ( s ) M K + (1 − z (cid:96) ) σ π ( s ) (cid:88) spins |T | ds ds (cid:96) d cos θ π d cos θ (cid:96) dφ = G F | V us | (1 − z (cid:96) ) σ π ( s ) X π M K + H µ H ∗ ν L µν ds ds (cid:96) d cos θ π d cos θ (cid:96) dφ =: G F | V us | (1 − z (cid:96) ) σ π ( s ) X π M K + J ( s, s (cid:96) , θ π , θ (cid:96) , φ ) ds ds (cid:96) d cos θ π d cos θ (cid:96) dφ. (C.12)41 rather tedious calculation yields (in accordance with [18]) J ( s, s (cid:96) , θ π , θ (cid:96) , φ ) = M K + H µ H ∗ ν L µν = 2(1 − z (cid:96) ) (cid:20) I + I cos(2 θ (cid:96) ) + I sin ( θ (cid:96) ) cos(2 φ ) + I sin(2 θ (cid:96) ) cos( φ )+ I sin( θ (cid:96) ) cos( φ ) + I cos( θ (cid:96) ) + I sin( θ (cid:96) ) sin( φ ) + I sin(2 θ (cid:96) ) sin( φ )+ I sin ( θ (cid:96) ) sin(2 φ ) (cid:21) , (C.13)where I := 14 (cid:18) (1 + z (cid:96) ) | F | + 12 (3 + z (cid:96) ) sin ( θ π ) (cid:0) | F | + | F | (cid:1) + 2 z (cid:96) | F | (cid:19) ,I := − 14 (1 − z (cid:96) ) (cid:18) | F | − 12 sin ( θ π ) (cid:0) | F | + | F | (cid:1)(cid:19) ,I := − 14 (1 − z (cid:96) ) sin ( θ π ) (cid:0) | F | − | F | (cid:1) ,I := 12 (1 − z (cid:96) ) sin( θ π )Re ( F ∗ F ) ,I := − sin( θ π ) (Re ( F ∗ F ) + z (cid:96) Re ( F ∗ F )) ,I := z (cid:96) Re ( F ∗ F ) − sin ( θ π )Re ( F ∗ F ) ,I := sin( θ π ) ( z (cid:96) Im ( F ∗ F ) − Im ( F ∗ F )) ,I := 12 (1 − z (cid:96) ) sin( θ π )Im ( F ∗ F ) ,I := − 12 (1 − z (cid:96) ) sin ( θ π )Im ( F ∗ F ) . (C.14) C.1.2 Broken Isospin In the case of broken isospin, the Lorentz structure of the K (cid:96) matrix element is modified by the presence ofthe additional tensorial form factor. The expression for the spin sum has to be adapted. This is, however, theonly necessary modification. The phase space is still parametrised by the same five kinematic variables.The T -matrix element is given by (see also (13)) T = G F √ V ∗ us (cid:0) ¯ u ( p ν ) γ µ (1 − γ ) v ( p (cid:96) ) H µ + ¯ u ( p ν ) σ µν (1 + γ ) v ( p (cid:96) ) T µν (cid:1) , H µ = V µ − A µ , T µν = 1 M K + p µ p ν T. (C.15)Let us calculate the spin sum of the squared T -matrix: (cid:88) spins |T | = G F | V us | (cid:32) H µ H ∗ ν (cid:88) spins L µ L ∗ ν + T µν T ∗ ρσ (cid:88) spins ˆ L µν ˆ L ∗ ρσ + 2Re (cid:20) H µ T ∗ ρσ (cid:88) spins L µ ˆ L ∗ ρσ (cid:21)(cid:33) , (C.16)where again L µ = ¯ u ( p ν ) γ µ (1 − γ ) v ( p (cid:96) ) and ˆ L µν := ¯ u ( p ν ) σ µν (1 + γ ) v ( p (cid:96) ) .The differential decay rate is given by d Γ = 12 π λ / K(cid:96) ( s ) M K + (1 − z (cid:96) ) σ π ( s ) (cid:88) spins |T | ds ds (cid:96) d cos θ π d cos θ (cid:96) dφ =: G F | V us | (1 − z (cid:96) ) σ π ( s ) X π M K + J ( s, s (cid:96) , θ π , θ (cid:96) , φ ) ds ds (cid:96) d cos θ π d cos θ (cid:96) dφ, (C.17)42here now J := J V − A + J T + J int5 ,J V − A := M K + H µ H ∗ ν (cid:88) spins L µ L ∗ ν ,J T := M K + T µν T ∗ ρσ (cid:88) spins ˆ L µν ˆ L ∗ ρσ ,J int5 := M K + (cid:20) H µ T ∗ ρσ (cid:88) spins L µ ˆ L ∗ ρσ (cid:21) . (C.18) J V − A agrees with J in the isospin limit, but with the form factors F , . . . , F replaced by the isospin correctedones. J T is due to the tensorial form factor only, J int5 is the interference of the tensorial and the V − A part. J can still be written in the form J ( s, s (cid:96) , θ π , θ (cid:96) , φ ) = 2(1 − z (cid:96) ) (cid:20) I + I cos(2 θ (cid:96) ) + I sin ( θ (cid:96) ) cos(2 φ ) + I sin(2 θ (cid:96) ) cos( φ )+ I sin( θ (cid:96) ) cos( φ ) + I cos( θ (cid:96) ) + I sin( θ (cid:96) ) sin( φ ) + I sin(2 θ (cid:96) ) sin( φ )+ I sin ( θ (cid:96) ) sin(2 φ ) (cid:21) , (C.19)where I i = I V − Ai + I Ti + I int i . I V − Ai correspond to the functions I i in the isospin limit (C.14). The additionalpieces are given by I T = 14 z (cid:96) (cid:18) (1 + z (cid:96) ) + sin ( θ π ) (cid:18) (1 + 3 z (cid:96) ) X ss (cid:96) − 12 (1 − z (cid:96) ) (cid:19)(cid:19) | F | ,I T = 14 z (cid:96) (1 − z (cid:96) ) (cid:18) − sin ( θ π ) (cid:18) X ss (cid:96) + 32 (cid:19)(cid:19) | F | ,I T = 14 z (cid:96) (1 − z (cid:96) ) sin ( θ π ) | F | ,I T = − z (cid:96) (1 − z (cid:96) ) sin(2 θ π ) P L √ ss (cid:96) | F | ,I T = − z (cid:96) sin(2 θ π ) X √ ss (cid:96) | F | ,I T = − z (cid:96) sin ( θ π ) P L Xss (cid:96) | F | ,I T = I T = I T = 0 (C.20)and I int1 = z (cid:96) (cid:18) − cos( θ π )Re( F ∗ F ) − P L √ ss (cid:96) sin ( θ π )Re( F ∗ F ) − X √ ss (cid:96) sin ( θ π )Re( F ∗ F ) (cid:19) ,I int2 = I int3 = I int4 = 0 ,I int5 = z (cid:96) (cid:18) X √ ss (cid:96) sin( θ π )Re( F ∗ F ) + sin( θ π ) cos( θ π )Re( F ∗ F ) − P L √ ss (cid:96) sin( θ π )Re( F ∗ F ) (cid:19) ,I int6 = z (cid:96) (cid:18) X √ ss (cid:96) sin ( θ π )Re( F ∗ F ) + P L √ ss (cid:96) sin ( θ π )Re( F ∗ F ) − cos( θ π )Re( F ∗ F ) (cid:19) ,I int7 = z (cid:96) (cid:18) P L √ ss (cid:96) sin( θ π )Im( F ∗ F ) − sin( θ π ) cos( θ π )Im( F ∗ F ) + X √ ss (cid:96) sin( θ π )Im( F ∗ F ) (cid:19) ,I int8 = I int9 = 0 . (C.21)These results agree with [14] apart from the different normalisation of F .43 .2 Decay Rate for K (cid:96) γ The partial decay rate for the K (cid:96) γ decay is given by d Γ γ = 12 M K + (2 π ) (cid:88) spinspolar . |T γ | δ (4) ( p − P − L ) d p p d p p d p (cid:96) p (cid:96) d p ν p ν d q q . (C.22)The kinematics of the decay is described by the 8 variables s , s (cid:96) , θ π , θ γ , φ , x , y and φ (cid:96) . The remaining 7integrals can be performed explicitly. The reduction of the partial decay rate to the eight-dimensional phasespace integral is performed in the following.The spin summed square of the matrix element T γ = − G F √ eV ∗ us (cid:15) µ ( q ) ∗ (cid:20) H µν L ν + H ν ˜ L µν (cid:21) , (C.23)where L ν := ¯ u ( p ν ) γ ν (1 − γ ) v ( p (cid:96) ) , ˜ L µν := 12 p (cid:96) q ¯ u ( p ν ) γ ν (1 − γ )( m (cid:96) − /p (cid:96) − /q ) γ µ v ( p (cid:96) ) , (C.24)can be written as (cid:88) spinspolar . |T γ | = e G F | V us | (cid:88) polar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) (cid:34) H ν H ∗ σ (cid:88) spins ˜ L µν ˜ L ∗ ρσ + H µν H ∗ ρσ (cid:88) spins L ν L ∗ σ + 2Re (cid:18) H µν H ∗ σ (cid:88) spins L ν ˜ L ∗ ρσ (cid:19)(cid:35) . (C.25)All the spin sums can be performed with standard trace techniques. As I give the photon an artificial smallmass m γ , I have to use the polarisation sum formula for a massive vector boson: (cid:88) polar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) = − g µρ + q µ q ρ m γ . (C.26)Using the Ward identity, I find that the second term in the polarisation sum formula does only contributeat O ( m γ ) : q µ q ρ m γ (cid:34) H ν H ∗ σ (cid:88) spins ˜ L µν ˜ L ∗ ρσ + H µν H ∗ ρσ (cid:88) spins L ν L ∗ σ + 2Re (cid:18) H µν H ∗ σ (cid:88) spins L ν ˜ L ∗ ρσ (cid:19)(cid:35) = 1 m γ Re (cid:34) H ν H ∗ σ (cid:88) spins (cid:16) q µ q ρ ˜ L µν ˜ L ∗ ρσ + L ν L ∗ σ + 2 q ρ L ν ˜ L ∗ ρσ (cid:17) (cid:35) = 4 m γ ( ˆ Lq + ˆ N q ) Re (cid:34) H ν H ∗ σ (cid:32) g νσ ˆ N − ˆ L L ν ˆ L σ − ˆ N ν ˆ N σ + i(cid:15) νσαβ ˆ L α ˆ N β (cid:33) (cid:35) . (C.27)I therefore find the following results for the spin and polarisation sums: (cid:88) spinspolar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) ˜ L µν ˜ L ∗ ρσ = 8 Lq + N q (cid:18) g νσ ( N q − Lq ) + q ν L σ + q σ L ν − q ν N σ − q σ N ν + i(cid:15) νσαβ L α q β − i(cid:15) νσαβ N α q β (cid:19) − m (cid:96) ( Lq + N q ) · (cid:16) g νσ N − L L ν L σ − N ν N σ + i(cid:15) νσαβ L α N β (cid:17) + O ( m γ ) , (C.28) (cid:88) spinspolar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) L ν L ∗ σ = − g µρ (cid:18) g νσ ˆ N − ˆ L L ν ˆ L σ − ˆ N ν ˆ N σ + i(cid:15) νσαβ ˆ L α ˆ N β (cid:19) + O ( m γ ) , (C.29)44 spinspolar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) L ν ˜ L ∗ ρσ = 4 Lq + N q (cid:34) L µ L ν L σ − N µ N ν N σ + N µ L ν L σ − L µ N ν N σ − q µ L ν L σ + q µ N ν N σ − q ν L µ L σ + q ν N µ N σ + q σ L µ N ν − q σ L ν N µ + g µν (cid:18) N − L q σ − N q L σ + Lq N σ (cid:19) + g µσ (cid:18) L − N q ν − Lq L ν + N q N ν (cid:19) + g νσ (cid:18) N − L L µ + N µ − q µ ) + Lq L µ − N q N µ (cid:19) − ig νσ (cid:15) µαβγ L α N β q γ + ( L σ − N σ ) i (cid:15) µναβ ( L α + N α ) q β + ( L ν − N ν ) i (cid:15) µσαβ ( L α + N α ) q β + ( L µ + N µ ) i (cid:15) νσαβ ( − L α + N α ) q β + ( L µ + N µ − q µ ) i(cid:15) νσαβ L α N β + i (cid:15) µνσα ( L α − N α )( Lq + N q ) (cid:35) + O ( m γ ) . (C.30)I perform the contraction with the hadronic part and express all the scalar products in terms of the eightphase space variables. Neglecting the contribution form the anomalous sector, one can express the hadronicmatrix elements in terms of the following form factors: H µ = iM K + ( P µ F + Q µ G + L µ R ) , H µν = iM K + g µν Π + iM K + ( P µ Π ν + Q µ Π ν + L µ Π ν ) , Π νi = 1 M K + ( P ν Π i + Q ν Π i + L ν Π i + q ν Π i ) . (C.31)The K (cid:96) form factors F , G , R depend on scalar products of P , Q and L , hence, they can be expressed asfunctions of s , s (cid:96) and θ π . The K (cid:96) γ form factors Π and Π ij depend on the scalar products of P , Q , L and q .They are therefore functions of the six phase space variables s , s (cid:96) , θ π , θ γ , φ and x .I consider now the phase space measure: dI γ := δ (4) ( p − p − p − p (cid:96) − p ν − q ) d p p d p p d p (cid:96) p (cid:96) d p ν p ν d q q = δ (4) ( p − p − p − p (cid:96) − p ν − q ) d p p d p p d p (cid:96) p (cid:96) d p ν p ν d q q · δ (4) ( p + p − P ) δ (4) ( p (cid:96) + p ν + q − L ) d P d L θ ( P ) θ ( L )= ds ds (cid:96) δ (4) ( p − P − L ) d P δ ( s − P ) θ ( P ) d Lδ ( s (cid:96) − L ) θ ( L ) · δ (4) ( p + p − P ) d p p d p p δ (4) ( p (cid:96) + p ν + q − L ) d p (cid:96) p (cid:96) d p ν p ν d q q . (C.32)The phase space integral can again be split into three separately Lorentz invariant pieces: dI γ = dI γ dI γ dI γ ,dI γ := ds ds (cid:96) δ (4) ( p − P − L ) d P δ ( s − P ) θ ( P ) d Lδ ( s (cid:96) − L ) θ ( L ) ,dI γ := δ (4) ( p + p − P ) d p p d p p ,dI γ := δ (4) ( p (cid:96) + p ν + q − L ) d p (cid:96) p (cid:96) d p ν p ν d q q . (C.33)Each of these three pieces can be evaluated in a convenient frame. dI γ and dI γ can be evaluated in completeanalogy to K (cid:96) , i.e. in the kaon and dipion rest frames: dI γ = πds ds (cid:96) λ / K(cid:96) ( s )2 M K + = πds ds (cid:96) XM K + , dI γ = π σ π ( s ) d cos θ π . (C.34)45he third piece represents now a three body decay. I first perform the neutrino momentum integrals in thethree body rest frame: dI γ = δ (3) ( (cid:126)p (cid:96) + (cid:126)p ν + (cid:126)q − (cid:126)L ) δ (cid:18)(cid:113) (cid:126)p (cid:96) + m (cid:96) + | (cid:126)p ν | + (cid:113) (cid:126)q + m γ − L (cid:19) d p (cid:96) (cid:112) (cid:126)p (cid:96) + m (cid:96) d p ν | (cid:126)p ν | d q (cid:113) (cid:126)q + m γ = δ (cid:18)(cid:113) (cid:126)p (cid:96) + m (cid:96) + | (cid:126)p (cid:96) + (cid:126)q | + (cid:113) (cid:126)q + m γ − √ s (cid:96) (cid:19) d p (cid:96) (cid:112) (cid:126)p (cid:96) + m (cid:96) | (cid:126)p (cid:96) + (cid:126)q | d q (cid:113) (cid:126)q + m γ = δ (cid:18)(cid:113) | (cid:126)p (cid:96) | + m (cid:96) + (cid:113) | (cid:126)p (cid:96) | + | (cid:126)q | + 2 | (cid:126)p (cid:96) || (cid:126)q | cos θ (cid:96)γ + (cid:113) | (cid:126)q | + m γ − √ s (cid:96) (cid:19) · | (cid:126)p (cid:96) | d | (cid:126)p (cid:96) | d cos θ (cid:96)γ dφ (cid:96) | (cid:126)q | d | (cid:126)q | d cos θ γ dφ (cid:112) | (cid:126)p (cid:96) | + m (cid:96) (cid:112) | (cid:126)p (cid:96) | + | (cid:126)q | + 2 | (cid:126)p (cid:96) || (cid:126)q | cos θ (cid:96)γ (cid:113) | (cid:126)q | + m γ = | (cid:126)p (cid:96) || (cid:126)q | (cid:113) | (cid:126)p (cid:96) | + m γ (cid:113) | (cid:126)q | + m γ d | (cid:126)p (cid:96) | d | (cid:126)q | dφ (cid:96) d cos θ γ dφ = 18 dp (cid:96) dq dφ (cid:96) d cos θ γ dφ = s (cid:96) dxdydφ (cid:96) d cos θ γ dφ, (C.35)where I have used the angle θ (cid:96)γ between the photon and the lepton.Putting the three pieces together, I find dI γ = λ / K(cid:96) ( s ) M K + π σ π ( s ) s (cid:96) ds ds (cid:96) d cos θ π d cos θ γ dφ dx dy dφ (cid:96) , (C.36)and for the differential decay rate d Γ γ = 12 M K + (2 π ) (cid:88) spinspolar . |T γ | dI γ = 12 π λ / K(cid:96) ( s ) M K + σ π ( s ) s (cid:96) (cid:88) spinspolar . |T γ | ds ds (cid:96) d cos θ π d cos θ γ dφ dx dy dφ (cid:96) = G F | V us | e s (cid:96) σ π ( s ) X π M K + J ds ds (cid:96) d cos θ π d cos θ γ dφ dx dy dφ (cid:96) , (C.37)where J = M K + (cid:88) polar . (cid:15) µ ( q ) ∗ (cid:15) ρ ( q ) (cid:34) H ν H ∗ σ (cid:88) spins ˜ L µν ˜ L ∗ ρσ + H µν H ∗ ρσ (cid:88) spins L ν L ∗ σ + 2Re (cid:18) H µν H ∗ σ (cid:88) spins L ν ˜ L ∗ ρσ (cid:19)(cid:35) . (C.38) D χ PT with Photons and Leptons In order to settle the conventions, I collect here the most important formulae needed to define χ PT with photonsand leptons [1, 2, 3, 16, 17].We consider SU (3) χ PT, where the Goldstone bosons are collected in the SU (3) matrix U = exp (cid:32) i √ F φ (cid:33) , (D.1)with φ = (cid:88) a =1 λ a φ a = π (cid:16) √ + (cid:15) √ (cid:17) + η (cid:16) √ − (cid:15) √ (cid:17) π + K + π − π (cid:16) (cid:15) √ − √ (cid:17) + η (cid:16) √ + (cid:15) √ (cid:17) K K − ¯ K − η (cid:113) − π (cid:113) (cid:15) . (D.2)46t leading order, the Lagrangian is given by L LOeff = L p + L e + L QED , L p = F (cid:104) D µ U D µ U † + χU † + U χ † (cid:105) , L e = e F Z (cid:104) U QU † Q (cid:105) , L QED = − F µν F µν + (cid:88) (cid:96) (cid:2) ¯ (cid:96) ( i/∂ + e /A − m (cid:96) ) (cid:96) + ¯ ν (cid:96)L i/∂ν (cid:96)L (cid:3) , (D.3)where D µ U = ∂ µ U − ir µ U + iU l µ ,χ = 2 B ( s + ip ) , r µ = v µ + a µ , l µ = v µ − a µ ,F µν = ∂ µ A ν − ∂ ν A µ ,ν (cid:96)L = 1 − γ ν (cid:96) . (D.4)The external fields are fixed by s + ip = M = diag( m u , m d , m s ) ,r µ = − eA µ Q,l µ = − eA µ Q + (cid:88) (cid:96) (cid:16) ¯ (cid:96)γ µ ν (cid:96)L Q wL + ¯ ν (cid:96)L γ µ (cid:96)Q w † L (cid:17) ,Q = 13 diag(2 , − , − ,Q wL = − √ G F T, T = V ud V us . (D.5)By expanding L LOeff in the meson fields, we can extract the mass terms. At leading order, I find: M π = 2 B ˆ m,M π + = 2 B ˆ m + 2 e ZF ,M K = B (cid:18) m s + ˆ m + 2 (cid:15) √ m s − ˆ m ) (cid:19) ,M K + = B (cid:18) m s + ˆ m − (cid:15) √ m s − ˆ m ) (cid:19) + 2 e ZF ,M η = 43 B (cid:18) m s + ˆ m (cid:19) . (D.6)At this order, the masses obey the Gell-Mann – Okubo relation: M K + + 2 M K − M π + + M π = 3 M η . (D.7)Let us define ∆ π := M π + − M π = 2 e ZF , ∆ K := M K + − M K = 2 e ZF + B ( m u − m d ) . (D.8)The next-to-leading-order Lagrangian is given by L NLOeff = L LOeff + L p + L e p + L lept + L γ , (D.9) I denote by (cid:104)·(cid:105) the flavour trace. L p = L (cid:104) D µ U D µ U † (cid:105)(cid:104) D ν U D ν U † (cid:105) + L (cid:104) D µ U D ν U † (cid:105)(cid:104) D µ U D ν U † (cid:105) + L (cid:104) D µ U D µ U † D ν U D ν U † (cid:105) + L (cid:104) D µ U D µ U † (cid:105)(cid:104) χU † + U χ † (cid:105) + L (cid:104) D µ U D µ U † ( χU † + U χ † ) (cid:105) + L (cid:104) χU † + U χ † (cid:105) + L (cid:104) χU † − U χ † (cid:105) + L (cid:104) U χ † U χ † + χU † χU † (cid:105) − iL (cid:104) F µνR D µ U D ν U † + F µνL D µ U † D ν U (cid:105) + L (cid:104) U F µνL U † F Rµν (cid:105) + H (cid:104) F µνR F Rµν + F µνL F Lµν (cid:105) + H (cid:104) χχ † (cid:105) , (D.10) L e p = e F (cid:26) K (cid:104) QQ (cid:105)(cid:104) D µ U D µ U † (cid:105) + K (cid:104) QU † QU (cid:105)(cid:104) D µ U D µ U † (cid:105) + K (cid:0) (cid:104) QU † D µ U (cid:105)(cid:104) QU † D µ U (cid:105) + (cid:104) QU D µ U † (cid:105)(cid:104) QU D µ U † (cid:105) (cid:1) + K (cid:104) QU † D µ U (cid:105)(cid:104) QU D µ U † (cid:105) + K (cid:104) QQ ( D µ U † D µ U + D µ U D µ U † ) (cid:105) + K (cid:104) U QU † QD µ U D µ U † + U † QU QD µ U † D µ U (cid:105) + K (cid:104) QQ (cid:105)(cid:104) χU † + U χ † (cid:105) + K (cid:104) QU † QU (cid:105)(cid:104) χU † + U χ † (cid:105) + K (cid:104) QQ ( U † χ + χ † U + χU † + U χ † ) (cid:105) + K (cid:104) QU † Qχ + QU Qχ † + QU † QU χ † U + QU QU † χU † (cid:105)− K (cid:104) QU † Qχ + QU Qχ † − QU † QU χ † U − QU QU † χU † (cid:105) + iK (cid:104) (cid:2) [ l µ , Q ] , Q (cid:3) D µ U † U + (cid:2) [ r µ , Q ] , Q (cid:3) D µ U U † (cid:105)− K (cid:104) [ l µ , Q ] U † [ r µ , Q ] U (cid:105) + 2 K (cid:104) l µ [ l µ , Q ] Q + r µ [ r µ , Q ] Q (cid:105) (cid:27) , (D.11) L lept = e (cid:88) (cid:96) (cid:26) F (cid:104) X ¯ (cid:96)γ µ ν (cid:96)L i (cid:104) D µ U Q wL U † Q − D µ U † QU Q wL (cid:105)− X ¯ (cid:96)γ µ ν (cid:96)L i (cid:104) D µ U Q wL U † Q + D µ U † QU Q wL (cid:105) + X m (cid:96) ¯ (cid:96)ν (cid:96)L (cid:104) Q wL U † QU (cid:105) + X ¯ (cid:96)γ µ ν (cid:96)L (cid:104) Q wL l µ Q − Q wL Ql µ (cid:105) + X ¯ (cid:96)γ µ ν (cid:96)L (cid:104) Q wL U † r µ QU − Q wL U † Qr µ U (cid:105) + h.c. (cid:105) + X ¯ (cid:96) ( i/∂ + e /A ) (cid:96) + X m (cid:96) ¯ (cid:96)(cid:96) (cid:27) , (D.12) L γ = e X F µν F µν . (D.13)The low-energy constants (LECs) are UV-divergent. Their finite part is defined by L i = Γ i λ + L ri ( µ ) ,H i = ∆ i λ + H ri ( µ ) ,K i = Σ i λ + K ri ( µ ) ,X i = Ξ i λ + X ri ( µ ) , (D.14)where λ = µ n − π (cid:18) n − − 12 (ln(4 π ) + 1 − γ E ) (cid:19) . (D.15)The coefficients Γ i , ∆ i , Σ i and Ξ i can be found in [3, 16, 17]. E Feynman Diagrams E.1 Mass Effects E.1.1 Loop Diagrams The meson loop diagrams contribute as follows to the form factors F and G : δF NLOtadpole = 112 F (cid:2) A ( M π ) + 4 A ( M π + ) + 8 A ( M K ) + 8 A ( M K + ) + 9 A ( M η ) (cid:3) ,δG NLOtadpole = 14 F (cid:2) A ( M π ) + 4 A ( M π + ) + 4 A ( M K + ) + A ( M η ) (cid:3) , (E.1)48 F NLO s -loop = 1 F (cid:20) s − M π ) B ( s, M π , M π ) + 3( s + 4∆ π ) B ( s, M π + , M π + )+ (cid:18) s + ∆ K − ∆ π (cid:19) B ( s, M K , M K ) + 3(4∆ π + s ) B ( s, M K + , M K + )+ 3 M π B ( s, M η , M η ) − A ( M π ) − A ( M π + ) − A ( M K ) − A ( M K + )+ 2 √ (cid:15) (cid:16) s − M π ) B ( s, M π , M π ) + 23 ( M K − M π ) B ( s, M K , M K )+ (4 M π − s ) B ( s, M η , M π ) − M π B ( s, M η , M η ) − A ( M π ) + A ( M η ) (cid:17)(cid:21) ,δG NLO s -loop = 16 F (cid:20) ( s − M K + ) B ( s, M K + , M K + ) − 12 ( s − M K ) B ( s, M K , M K )+ ( s − M π + ) B ( s, M π + , M π + ) − A ( M K + ) + A ( M K ) − A ( M π + )+ 2 M K − M K + − M π + + s π (cid:21) , (E.2)49 F NLO t -loop = 16 F (cid:34) t (cid:18) M K + (cid:0) t − M η (cid:1) + 6 M η M π + + 3 M η t + 6 M K (cid:0) M K + − M π + − t (cid:1) − M π t − M π + t (cid:19) (cid:0) M K − M η (cid:1) B (cid:0) , M η , M K (cid:1) + 14 t (cid:18) M K (cid:0) M K + (cid:0) M η − t (cid:1) − M η (cid:0) M π + + 3 t (cid:1) + t (cid:0) M π + 2 M π + − t (cid:1)(cid:1) + (cid:0) M η − t (cid:1) (cid:0) M K + (cid:0) t − M η (cid:1) + 3 M η (cid:0) M π + + t (cid:1) − t (cid:0) M π + 2 M π + (cid:1)(cid:1) + 6 M K (cid:0) − M K + + M π + + t (cid:1) (cid:19) B (cid:0) t, M η , M K (cid:1) + 12 t (cid:18) M K (cid:0) M K + − M π + (cid:1) − M K + (cid:0) M π + 2 t (cid:1) + M π M π + + 3 M π t + 2 M π + t (cid:19) (cid:0) M K − M π (cid:1) B (cid:0) , M K , M π (cid:1) + 12 t (cid:18) M K (cid:0) M K + (cid:0) M π + t (cid:1) − M π (cid:0) M π + + 3 t (cid:1) + t (cid:0) t − M π + (cid:1)(cid:1) + M K (cid:0) M π + − M K + (cid:1) − (cid:0) M π + M π t − t (cid:1) (cid:0) M K + − M π + − t (cid:1) (cid:19) B (cid:0) t, M K , M π (cid:1) + 1 t (cid:18) M K + − M K + (cid:0) M π + + t (cid:1) + 3 M π t + M π + − M π + t (cid:19) (cid:0) M K + − M π + (cid:1) B (cid:0) , M K + , M π + (cid:1) + 1 t (cid:18) − M K + + M K + (cid:0) M π + + 2 t (cid:1) − M K + (cid:0) t (cid:0) M π + t (cid:1) + 3 M π + (cid:1) + M π + t (cid:0) M π − t (cid:1) + 3 M π t + M π + − M π + t (cid:19) B (cid:0) t, M K + , M π + (cid:1) − (cid:0) M K + − M π + + t (cid:1) t A (cid:0) M η (cid:1) + (cid:0) − M K + + M π + + 3 t (cid:1) t A (cid:0) M π (cid:1) + (cid:0) M π + − M K + (cid:1) t A (cid:0) M π + (cid:1) − A (cid:0) M K + (cid:1) + (cid:0) M K + − M π + − t (cid:1) (cid:0) M η + 4 M K + 2 M K + + M π + 2 M π + − t (cid:1) π t (cid:35) + 16 F √ (cid:15) (cid:34) t (cid:18) M K − M K (cid:0) M π + t (cid:1) + M π − M π t (cid:19) ( M K − M π ) B (0 , M η , M K )+ 19 t (cid:18) − M K + M K (cid:0) M π + 13 t (cid:1) − M K (cid:0) M π + 14 M π t + 57 t (cid:1) + M π + M π t + 3 M π t + 27 t (cid:19) B ( t, M η , M K ) − t (cid:18) M K − M K (cid:0) M π + t (cid:1) + M π + 2 M π t (cid:19) ( M K − M π ) B (0 , M K , M π )+ 1 t (cid:18) M K − M K (cid:0) M π + t (cid:1) + M K (cid:0) M π + 2 M π t + t (cid:1) − (cid:0) M π − t (cid:1) (cid:0) M π + 3 t (cid:1) (cid:19) B ( t, M K , M π ) − (cid:0) M K − M π + t (cid:1) t A ( M η ) + (cid:0) M K − M π + t (cid:1) t A ( M π )+ (cid:0) M K − M π (cid:1) (cid:0) M K − M π − t (cid:1) π t (cid:35) , (E.3)50 G NLO t -loop = 16 F (cid:34) t (cid:18) M K + (cid:0) M η − t (cid:1) − M η M π + + 3 M η t − M K (cid:0) M K + − M π + (cid:1) + 3 M π t + 2 M π + t − t (cid:19) (cid:0) M K − M η (cid:1) B (cid:0) , M η , M K (cid:1) + 14 t (cid:18) − M K (cid:0) M K + (cid:0) M η − t (cid:1) + 3 M η (cid:0) t − M π + (cid:1) + t (cid:0) M π + 2 M π + (cid:1)(cid:1) + (cid:0) M η − t (cid:1) (cid:0) M K + (cid:0) M η − t (cid:1) + M η (cid:0) t − M π + (cid:1) + t (cid:0) M π + 2 M π + − t (cid:1)(cid:1) + 6 M K (cid:0) M K + − M π + (cid:1) (cid:19) B (cid:0) t, M η , M K (cid:1) − t (cid:18) M K (cid:0) M K + − M π + + t (cid:1) − M K + (cid:0) M π + 2 t (cid:1) + M π M π + + 2 M π t + 2 M π + t + t (cid:19) (cid:0) M K − M π (cid:1) B (cid:0) , M K , M π (cid:1) + 12 t (cid:18) M K (cid:0) − M K + (cid:0) M π + t (cid:1) + M π (cid:0) M π + + t (cid:1) + t (cid:0) M π + − t (cid:1)(cid:1) − (cid:0) M π − t (cid:1) (cid:0) M K + (cid:0) − (cid:0) M π + 2 t (cid:1)(cid:1) + M π (cid:0) M π + + 2 t (cid:1) + t (cid:0) M π + + 7 t (cid:1)(cid:1) + M K (cid:0) M K + − M π + + t (cid:1) (cid:19) B (cid:0) t, M K , M π (cid:1) − t (cid:18) M K + − M K + (cid:0) M π + + t (cid:1) + t (cid:0) M π + t (cid:1) + M π + − M π + t (cid:19) (cid:0) M K + − M π + (cid:1) B (cid:0) , M K + , M π + (cid:1) + 1 t (cid:18) M K + − M K + (cid:0) M π + + t (cid:1) + M K + (cid:0) t (cid:0) M π − t (cid:1) + 3 M π + − M π + t (cid:1) + 3 M π + t (cid:0) t − M π (cid:1) + t (cid:0) t − M π (cid:1) − M π + + 3 M π + t (cid:19) B (cid:0) t, M K + , M π + (cid:1) − (cid:0) − M K + + M π + + t (cid:1) t A (cid:0) M η (cid:1) + (cid:0) M K + − M π + − t (cid:1) t A (cid:0) M π (cid:1) + (cid:0) M K + − M π + − t (cid:1) t A (cid:0) M π + (cid:1) + A (cid:0) M K + (cid:1) − (cid:0) M K + − M π + + t (cid:1) (cid:0) M η + 4 M K + 2 M K + + M π + 2 M π + − t (cid:1) π t (cid:35) + 16 F √ (cid:15) (cid:34) t (cid:18) M K − M K M π + M π − M π t − t (cid:19) (cid:0) M π − M K (cid:1) B (cid:0) , M η , M K (cid:1) + 19 t (cid:18) M K − M K (cid:0) M π + 4 t (cid:1) + 3 M K (cid:0) M π + 4 M π t + 5 t (cid:1) − M π + 3 M π t − t (cid:19) B (cid:0) t, M η , M K (cid:1) + 1 t (cid:18) M K − M K M π + M π + M π t + t (cid:19) (cid:0) M K − M π (cid:1) B (cid:0) , M K , M π (cid:1) + 1 t (cid:18) − M K + 3 M K M π + M K (cid:0) t − M π (cid:1) + (cid:0) M π − t (cid:1) (cid:0) M π + 2 t (cid:1) (cid:19) B (cid:0) t, M K , M π (cid:1) − (cid:0) − M K + M π + t (cid:1) t A (cid:0) M η (cid:1) + (cid:0) − M K + M π + t (cid:1) t A (cid:0) M π (cid:1) − (cid:0) M K − M π (cid:1) (cid:0) M K − M π + t (cid:1) π t (cid:35) , (E.4) δF NLO u -loop = δG NLO u -loop = 12 F (cid:20) B ( u, M K + , M π + )( M K + + 3 M π + − M π − u ) + 13 A ( M K + ) + 13 A ( M π + ) (cid:21) . (E.5)51 .1.2 Counterterms The counterterm contribution to the form factors is given by: δF NLOct = 1 F (cid:34) s − M π + ) L + 8( M K + + s − s (cid:96) ) L + 4( M K + − M π + + 2 s − t ) L + 8 (cid:18) M K + 5 M π − √ (cid:15) M K − M π ) (cid:19) L + 4( M K + + 2 M π + − π ) L + 2 s (cid:96) L (cid:35) + 29 e (84 K + 37 K ) ,δG NLOct = 1 F (cid:34) t − u ) L − M K + + M π + − t ) L + 8 (cid:18) M K + M π − √ (cid:15) M K − M π ) (cid:19) L + 4( M K + + 2 M π + − π ) L + 2 s (cid:96) L (cid:35) + 29 e (12 K + 18 K + 25 K ) . (E.6) E.1.3 External Leg Corrections Let us first consider the pion self-energy: it is given by Σ π + ( p ) = i ( D loop π + + D ct π + ) , (E.7)where p denotes the external pion momentum.The value of the tadpole diagram is D loop π + = i F (cid:20) p (cid:0) A ( M K + ) + A ( M K ) + 2 A ( M π + ) + 2 A ( M π ) (cid:1) − M π + (cid:0) A ( M K + ) + A ( M K ) + A ( M η ) + 2 A ( M π + ) − A ( M π ) (cid:1) (cid:21) − i e Z (cid:0) A ( M K + ) − A ( M η ) + 12 A ( M π + ) + 3 A ( M π ) (cid:1) , (E.8)and the counterterm is given by D ct π + = p (cid:34) iF (cid:18) (2 M K + − M π + + 3 M π ) L + M π L + 4 √ (cid:15) M K + − M π + ) L (cid:19) + 4 i e (6 K + 5 K ) (cid:35) + 16 iF (cid:18) ( − M π M K + + 3 M π + − M π M π + ) L + M π + ( M π + − M π ) L − √ (cid:15) M π + ( M K + − M π + ) L (cid:19) − i e (cid:0) M K + + 5 M π + ) K + 23 M π + K (cid:1) . (E.9)Since the full propagator is ip − M π + − Σ π + ( p ) = iZ π + p − M π + , ph + regular , (E.10)the field strength renormalisation Z π + can be computed as Z π + = 1 + Σ (cid:48) π + ( M π + , ph ) + h.o. = 1 + Σ (cid:48) π + ( M π + ) + h.o., (E.11)where h.o. denotes higher order terms.The physical mass, i.e. the position of the pole is given by M π + , ph = M π + + δM π + , δM π + = Σ π + ( M π + , ph ) = Σ π + ( M π + ) + h.o. (E.12)52 find the following expression for the field strength renormalisation of the pion: Z π + = 1 − F (cid:18) (cid:0) A ( M K ) + A ( M K + ) + 2 A ( M π + ) + 2 A ( M π ) (cid:1) + 8(2 M K + − M π + + 3 M π ) L + 8 M π L + 32 √ (cid:15) M K + − M π + ) L (cid:19) − e (6 K + 5 K ) . (E.13)We still have to compute the external leg correction for the kaon. The values of the two self-energy diagramsfor a charged kaon are given by D loop K + = p (cid:34) i F (cid:0) A ( M K ) + 4 A ( M K + ) + 3 A ( M η ) + 2 A ( M π + ) + A ( M π ) (cid:1) − i √ (cid:15) F (cid:0) A ( M η ) − A ( M π ) (cid:1) (cid:35) − iM K + F (cid:18) A ( M K ) + 4 A ( M K + ) − A ( M η ) + 2 A ( M π + ) + A ( M π )+ 2 √ (cid:15) (cid:0) A ( M η ) − A ( M π ) (cid:1) (cid:19) − i e Z (cid:0) A ( M K + ) + A ( M η ) + 3 A ( M π + ) (cid:1) , (E.14) D ct K + = p (cid:34) iF (cid:18) (2 M K + − M π + + 3 M π ) L + ( M K + − M π + + M π ) L + 4 √ (cid:15) M K + − M π + ) L (cid:19) + 4 i e (6 K + 5 K ) (cid:35) + 16 iF (cid:18) ( M K + (4 M π + − M K + − M π ) + M π + ∆ π ) L − M K + ( M K + − π ) L − √ (cid:15) M K + ( M K + − M π + ) L (cid:19) − i e (cid:0) M K + + 3 M π + ) K + (20 M K + + 3 M π + ) K (cid:1) . (E.15)The field strength renormalisation of the kaon is given by Z K + = 1 − F (cid:18) (cid:0) A ( M K ) + 4 A ( M K + ) + 3 A ( M η ) + 2 A ( M π + ) + A ( M π ) (cid:1) + 8(2 M K + − M π + + 3 M π ) L + 8( M K + − ∆ π ) L + 32 √ (cid:15) M K + − M π + ) L − √ (cid:15) (cid:0) A ( M η ) − A ( M π ) (cid:1) (cid:19) − e (6 K + 5 K ) . (E.16)The contribution of the field strength renormalisation to the amplitude consists of the LO tree diagramsmultiplied by a factor of √ Z i for every external particle i . Therefore, the NLO external leg corrections to theform factors are given by δF NLO Z = δG NLO Z = Z π + (cid:112) Z K + − − F (cid:18) (cid:0) A ( M K ) + 8 A ( M K + ) + 3 A ( M η ) + 10 A ( M π + ) + 9 A ( M π ) (cid:1) + 12(2 M K + − M π + + 3 M π ) L + 4( M K + − M π + + 3 M π ) L − √ (cid:15) (cid:0) A ( M η ) − A ( M π ) (cid:1) + 16 √ (cid:15) ( M K + − M π + ) L (cid:19) − e (6 K + 5 K ) . (E.17)53 .2 Photonic Effects E.2.1 Loop Diagrams Here, I give the explicit expressions for the contributions of the loop diagrams shown in figures 7 and 8 to theform factors F and G .The first four diagrams only contain bulb topologies. Their contribution, expressed in terms of the scalarloop function B , is given by δF NLO γ − loop ,a = 43 e (cid:0) B (0 , M K + , m γ ) − B ( M K + , M K + , m γ ) (cid:1) ,δG NLO γ − loop ,a = 0 ,δF NLO γ − loop ,b = δG NLO γ − loop ,b = − δF NLO γ − loop ,c = δG NLO γ − loop ,c = 23 e (cid:0) B (0 , M π + , m γ ) − B ( M π + , M π + , m γ ) (cid:1) ,δF NLO γ − loop ,d = δG NLO γ − loop ,d = 0 , (E.18)hence, in total δF NLO γ − loop ,a − d = 43 e (cid:0) B (0 , M K + , m γ ) − B ( M K + , M K + , m γ ) (cid:1) ,δG NLO γ − loop ,a − d = 43 e (cid:0) B (0 , M π + , m γ ) − B ( M π + , M π + , m γ ) (cid:1) . (E.19)The next six diagrams consist of triangle topologies. My results agree with [14] up to the contribution of theadditional term in the massive gauge boson propagator (which cancels in the end), though I choose to employPassarino-Veltman reduction techniques to write everything in terms of the basic scalar loop functions A , B and C , even if this results in longer expressions. δF NLO γ − loop ,e = 12 e (cid:32) M K + + M π + − t ) C ( M π + , t, M K + , m γ , M π + , M K + )+ (cid:18) M K + − M π + − M K + M π + − t (3 M K + − M π + ) + 3 t (cid:19) B ( M π + , M π + , m γ ) λ ( t, M π + , M K + )+ (cid:18) M K + + 2 M K + ( M π + − t ) + 5( M π + − t ) (cid:19) B ( M K + , M K + , m γ ) λ ( t, M π + , M K + ) − (cid:18) ( M K + − M π + ) + t ( M K + − M π + + 2 M K + M π + ) − t (13 M K + + 7 M π + ) + 11 t (cid:19) B ( t, M π + , M K + ) tλ ( t, M π + , M K + ) − B (0 , M π + , m γ ) − B (0 , M K + , m γ ) + M K + − M π + t B (0 , M π + , M K + ) (cid:33) − e A ( m γ ) m γ , (E.20) δG NLO γ − loop ,e = 12 e (cid:32) M K + + M π + − t ) C ( M π + , t, M K + , m γ , M π + , M K + )+ (cid:18) M K + + 2 M K + ( M π + − t ) + 3( M π + − t ) (cid:19) B ( M π + , M π + , m γ ) λ ( t, M π + , M K + ) − (cid:18) M K + + 2 M K + (3 M π + − t ) − ( M π + − t ) (cid:19) B ( M K + , M K + , m γ ) λ ( t, M π + , M K + )+ (cid:18) ( M K + − M π + ) + t ( M K + − M π + + 2 M K + M π + )+ 3 t ( M K + + 3 M π + ) − t (cid:19) B ( t, M π + , M K + ) tλ ( t, M π + , M K + ) − B (0 , M π + , m γ ) − M K + − M π + t B (0 , M π + , M K + ) (cid:33) − e A ( m γ ) m γ , (E.21)54 F NLO γ − loop ,f = δG NLO γ − loop ,f = − e (cid:18) M K + + M π + − u ) C ( M π + , u, M K + , m γ , M π + , M K + )+ B ( M π + , M π + , m γ ) + B ( M K + , M K + , m γ ) − B ( u, M π + , M K + ) (cid:19) + e A ( m γ ) m γ , (E.22) δF NLO γ − loop ,g = e (cid:32) M K + + m (cid:96) − t (cid:96) ) C ( m (cid:96) , t (cid:96) , M K + , m γ , m (cid:96) , M K + )+ (cid:18) M K + − M K + (5 m (cid:96) + 6 t (cid:96) ) − m (cid:96) t (cid:96) − m (cid:96) + 3 t (cid:96) (cid:19) B ( m (cid:96) , m (cid:96) , m γ )3 λ ( t (cid:96) , m (cid:96) , M K + )+ (cid:18) M K + − M K + t (cid:96) + 7( m (cid:96) − t (cid:96) ) (cid:19) B ( M K + , M K + , m γ )3 λ ( t (cid:96) , m (cid:96) , M K + )+ (cid:18) M K + (3 t (cid:96) − m (cid:96) ) + m (cid:96) t (cid:96) + 2 m (cid:96) − t (cid:96) (cid:19) B ( t (cid:96) , m (cid:96) , M K + )3 λ ( t (cid:96) , m (cid:96) , M K + ) − B (0 , M K + , m γ ) (cid:33) − e A ( m γ ) m γ , (E.23) δG NLO γ − loop ,g = e (cid:32) M K + + m (cid:96) − t (cid:96) ) C ( m (cid:96) , t (cid:96) , M K + , m γ , m (cid:96) , M K + )+ (cid:18) M K + − M K + ( m (cid:96) + 2 t (cid:96) ) + t (cid:96) ( t (cid:96) − m (cid:96) ) (cid:19) B ( m (cid:96) , m (cid:96) , m γ ) λ ( t (cid:96) , m (cid:96) , M K + ) − (cid:18) M K + − ( m (cid:96) − t (cid:96) ) (cid:19) B ( M K + , M K + , m γ ) λ ( t (cid:96) , m (cid:96) , M K + )+ (cid:18) M K + + m (cid:96) − t (cid:96) (cid:19) t (cid:96) B ( t (cid:96) , m (cid:96) , M K + ) λ ( t (cid:96) , m (cid:96) , M K + ) (cid:33) − e A ( m γ ) m γ , (E.24) δF NLO γ − loop ,h = e (cid:32) M π + − s ) C ( M π + , s, M π + , m γ , M π + , M π + )+ 2 B ( M π + , M π + , m γ ) − B ( s, M π + , M π + ) (cid:33) − e A ( m γ ) m γ , (E.25) δG NLO γ − loop ,h = e (cid:32) M π + − s ) C ( M π + , s, M π + , m γ , M π + , M π + )+ 2(8 M π + − s )4 M π + − s B ( M π + , M π + , m γ ) − M π + − s )4 M π + − s B ( s, M π + , M π + ) − B (0 , M π + , m γ ) (cid:33) − e A ( m γ ) m γ , (E.26) δF NLO γ − loop ,i = δG NLO γ − loop ,i = e (cid:32) − M π + + m (cid:96) − s (cid:96) ) C ( m (cid:96) , s (cid:96) , M π + , m γ , m (cid:96) , M π + )+ (cid:18) m (cid:96) + m (cid:96) ( s (cid:96) + 5 M π + ) − M π + − s (cid:96) ) (cid:19) B ( m (cid:96) , m (cid:96) , m γ )3 λ ( s (cid:96) , m (cid:96) , M π + )+ (cid:18) M π + + 7 m (cid:96) − s (cid:96) (4 M π + + 7 m (cid:96) ) + 7 s (cid:96) (cid:19) B ( M π + , M π + , m γ )3 λ ( s (cid:96) , m (cid:96) , M π + ) − (cid:18) m (cid:96) − m (cid:96) (2 M π + − s (cid:96) ) − s (cid:96) ( s (cid:96) − M π + ) (cid:19) B ( s (cid:96) , m (cid:96) , M π + )3 λ ( s (cid:96) , m (cid:96) , M π + )+ 13 B (0 , M π + , m γ ) (cid:33) + e A ( m γ ) m γ , (E.27)55 F NLO γ − loop ,j = e (cid:32) M π + + m (cid:96) − s (cid:96) ) C ( m (cid:96) , s (cid:96) , M π + , m γ , m (cid:96) , M π + )+ (cid:18) m (cid:96) + m (cid:96) ( M π + − s (cid:96) ) + 3( M π + − s (cid:96) ) (cid:19) B ( m (cid:96) , m (cid:96) , m γ )3 λ ( s (cid:96) , m (cid:96) , M π + )+ (cid:18) m (cid:96) − M π + − m (cid:96) M π + − s (cid:96) ( m (cid:96) − M π + ) + s (cid:96) (cid:19) B ( M π + , M π + , m γ )3 λ ( s (cid:96) , m (cid:96) , M π + ) − (cid:18) m (cid:96) − m (cid:96) (7 s (cid:96) + 4 M π + ) − s (cid:96) ( M π + − s (cid:96) ) (cid:19) B ( s (cid:96) , m (cid:96) , M π + )3 λ ( s (cid:96) , m (cid:96) , M π + )+ 16 B (0 , M π + , m γ ) (cid:33) − e A ( m γ ) m γ , (E.28) δG NLO γ − loop ,j = e (cid:32) M π + + m (cid:96) − s (cid:96) ) C ( m (cid:96) , s (cid:96) , M π + , m γ , m (cid:96) , M π + )+ (cid:18) m (cid:96) − m (cid:96) ( M π + + 5 s (cid:96) ) + 3( M π + − s (cid:96) ) (cid:19) B ( m (cid:96) , m (cid:96) , m γ )3 λ ( s (cid:96) , m (cid:96) , M π + )+ (cid:18) m (cid:96) − m (cid:96) (5 s (cid:96) + 6 M π + ) + ( s (cid:96) − M π + )(5 s (cid:96) + M π + ) (cid:19) B ( M π + , M π + , m γ )3 λ ( s (cid:96) , m (cid:96) , M π + ) − (cid:18) m (cid:96) − m (cid:96) (5 s (cid:96) + 2 M π + ) − s (cid:96) ( M π + − s (cid:96) ) (cid:19) B ( s (cid:96) , m (cid:96) , M π + )3 λ ( s (cid:96) , m (cid:96) , M π + ) − B (0 , M π + , m γ ) (cid:33) − e A ( m γ ) m γ . (E.29)The remaining eight diagrams in this first set are loop corrections to the diagram 2b. Here, the Passarino-Veltman reduction [28, 30] produces too lengthy expressions, hence, I use the tensor-coefficient functions (seeappendix A.2): δF NLO γ − loop ,k = δG NLO γ − loop ,k = − e B ( s (cid:96) , M K + , m γ ) + 43 e B ( s (cid:96) , M K + , m γ ) m γ , (E.30) δF NLO γ − loop ,l = e (cid:32) B ( s (cid:96) , M K + , m γ ) + 13 B ( M K + , M K + , m γ ) − B (0 , M K + , m γ ) − ( s + ν ) C ( s (cid:96) , s, M K + , m γ , M K + , M K + ) − νC ( s (cid:96) , s, M K + , m γ , M K + , M K + ) − s + 3 ν C ( s (cid:96) , s, M K + , m γ , M K + , M K + ) − ν C ( s (cid:96) , s, M K + , m γ , M K + , M K + ) − ν C ( s (cid:96) , s, M K + , m γ , M K + , M K + ) (cid:33) − e B ( s (cid:96) , M K + , m γ ) m γ , (E.31) δG NLO γ − loop ,l = e C ( s (cid:96) , s, M K + , m γ , M K + , M K + ) − e B ( s (cid:96) , M K + , m γ ) m γ , (E.32) δF NLO γ − loop ,m = δG NLO γ − loop ,m = − e (cid:32) B ( s (cid:96) , M K + , m γ ) + 13 B ( M π + , M π + , m γ ) − B (0 , M π + , m γ )+ ( M K + + M π + − u ) C ( M π + , u, s (cid:96) , m γ , M π + , M K + )+ M K + + M π + − u C ( M π + , u, s (cid:96) , m γ , M π + , M K + ) (cid:33) + 13 e B ( s (cid:96) , M K + , m γ ) m γ , (E.33)56 F NLO γ − loop ,n = e (cid:32) − B ( M π + , M π + , m γ ) + 13 B ( s (cid:96) , M K + , m γ ) + 16 B (0 , M π + , m γ )+ ( s (cid:96) + M π + − u ) C ( M π + , t, s (cid:96) , m γ , M π + , M K + ) − M K + + 5 M π + − s − t C ( M π + , t, s (cid:96) , m γ , M π + , M K + )+ ( s (cid:96) + M π + − u ) C ( M π + , t, s (cid:96) , m γ , M π + , M K + )+ C ( M π + , t, s (cid:96) , m γ , M π + , M K + ) + s − M π + C ( M π + , t, s (cid:96) , m γ , M π + , M K + )+ s (cid:96) + M π + − u C ( M π + , t, s (cid:96) , m γ , M π + , M K + ) (cid:33) − e B ( s (cid:96) , M K + , m γ ) m γ , (E.34) δG NLO γ − loop ,n = − δF NLO γ − loop ,n + 2 e C ( M π + , t, s (cid:96) , m γ , M π + , M K + ) − e B ( s (cid:96) , M K + , m γ ) m γ , (E.35) δF NLO γ − loop ,o = δG NLO γ − loop ,o = 43 e m (cid:96) − s (cid:96) (cid:32) m (cid:96) B ( m (cid:96) , m (cid:96) , m γ ) − s (cid:96) B ( s (cid:96) , M K + , m γ )+ m (cid:96) ( M K + − s (cid:96) ) C ( m (cid:96) , , s (cid:96) , m γ , m (cid:96) , M K + ) (cid:33) − e B ( s (cid:96) , M K + , m γ ) m γ , (E.36) δF NLO γ − loop ,p = e (cid:32) B (0 , M K + , m γ ) − B ( M K + , M K + , m γ ) − B ( s (cid:96) , M K + , m γ )+ ( ν + s ) C ( M K + , s, s (cid:96) , m γ , M K + , M K + ) + 12 ( ν + s ) C ( M K + , s, s (cid:96) , m γ , M K + , M K + )+ νC ( M K + , s, s (cid:96) , m γ , M K + , M K + ) + 12 νC ( M K + , s, s (cid:96) , m γ , M K + , M K + )+ 13 m (cid:96) C ( m (cid:96) , , s (cid:96) , m γ , m (cid:96) , M K + ) + 13 m (cid:96) C ( M K + , t (cid:96) , m (cid:96) , m γ , M K + , m (cid:96) ) − m (cid:96) ( ν + s ) D ( M K + , t (cid:96) , , s (cid:96) , m (cid:96) , s, m γ , M K + , m (cid:96) , M K + ) − m (cid:96) νD ( M K + , t (cid:96) , , s (cid:96) , m (cid:96) , s, m γ , M K + , m (cid:96) , M K + )+ m (cid:96) ( s (cid:96) − s (cid:96) ) D ( M K + , t (cid:96) , , s (cid:96) , m (cid:96) , s, m γ , M K + , m (cid:96) , M K + ) − m (cid:96) νD ( M K + , t (cid:96) , , s (cid:96) , m (cid:96) , s, m γ , M K + , m (cid:96) , M K + ) (cid:33) + 13 e B ( s (cid:96) , M K + , m γ ) m γ , (E.37)where I use the abbreviation C i + j ( X ) := C i ( X ) + C j ( X ) ,D i + j ( X ) := D i ( X ) + D j ( X ) . (E.38) δG NLO γ − loop ,p = − e (cid:32) C ( M K + , s, s (cid:96) , m γ , M K + , M K + )+ 2 m (cid:96) D ( m (cid:96) , t (cid:96) , s, s (cid:96) , M K + , , m γ , m (cid:96) , M K + , M K + ) (cid:33) + e B ( s (cid:96) , M K + , m γ ) m γ , (E.39)57 F NLO γ − loop ,q = δG NLO γ − loop ,q = e (cid:32) − B (0 , M π + , m γ ) + 13 B ( M π + , M π + , m γ ) + 13 B ( s (cid:96) , M K + , m γ )+ ( M K + + M π + − u ) C ( M π + , u, s (cid:96) , m γ , M π + , M K + )+ 12 ( M K + + M π + − u ) C ( M π + , u, s (cid:96) , m γ , M π + , M K + ) − m (cid:96) C ( m (cid:96) , , s (cid:96) , m γ , m (cid:96) , M K + ) − m (cid:96) C ( m (cid:96) , s (cid:96) , M π + , m γ , m (cid:96) , M π + ) − m (cid:96) ( M K + + M π + − u ) D ( M π + , s (cid:96) , , s (cid:96) , m (cid:96) , u, m γ , M π + , m (cid:96) , M K + ) (cid:33) − e B ( s (cid:96) , M K + , m γ ) m γ , (E.40) δF NLO γ − loop ,r = e (cid:32) − B (0 , M π + , m γ ) + 23 B ( M π + , M π + , m γ ) − B ( s (cid:96) , M K + , m γ ) − ( M π + + s (cid:96) − u ) C ( M π + , t, s (cid:96) , m γ , M π + , M K + )+ 12 (3 M π + − s − s (cid:96) + u ) C ( M π + , t, s (cid:96) , m γ , M π + , M K + ) − ( M π + + s (cid:96) − u ) C ( M π + , t, s (cid:96) , m γ , M π + , M K + ) − C ( M π + , t, s (cid:96) , m γ , M π + , M K + )+ 12 (2 M π + − s ) C ( M π + , t, s (cid:96) , m γ , M π + , M K + ) − 12 ( M π + + s (cid:96) − u ) C ( M π + , t, s (cid:96) , m γ , M π + , M K + )+ 13 m (cid:96) C ( m (cid:96) , , s (cid:96) , m γ , m (cid:96) , M K + ) − m (cid:96) C ( m (cid:96) , s (cid:96) , M π + , m γ , m (cid:96) , M π + )+ m (cid:96) ( M π + + s (cid:96) − u ) D ( s (cid:96) , t, s (cid:96) , m (cid:96) , M π + , , m γ , M K + , M π + , m (cid:96) )+ m (cid:96) ( M π + + s (cid:96) − u ) D ( s (cid:96) , t, s (cid:96) , m (cid:96) , M π + , , m γ , M K + , M π + , m (cid:96) )+ m (cid:96) ( M π + + s (cid:96) − u ) D ( s (cid:96) , t, s (cid:96) , m (cid:96) , M π + , , m γ , M K + , M π + , m (cid:96) )+ m (cid:96) ( s − M π + ) D ( s (cid:96) , t, s (cid:96) , m (cid:96) , M π + , , m γ , M K + , M π + , m (cid:96) )+ m (cid:96) ( m (cid:96) + 2 M π + − s (cid:96) + s (cid:96) − u ) D ( s (cid:96) , t, s (cid:96) , m (cid:96) , M π + , , m γ , M K + , M π + , m (cid:96) )+ m (cid:96) ( s − M π + ) D ( s (cid:96) , t, s (cid:96) , m (cid:96) , M π + , , m γ , M K + , M π + , m (cid:96) )+ m (cid:96) ( m (cid:96) + M π + − s (cid:96) ) D ( s (cid:96) , t, s (cid:96) , m (cid:96) , M π + , , m γ , M K + , M π + , m (cid:96) ) (cid:33) + 43 e B ( s (cid:96) , M K + , m γ ) m γ , (E.41) δG NLO γ − loop ,r = − δF NLO γ − loop ,r − e C ( M π + , t, s (cid:96) , m γ , M π + , M K + ) + 2 e B ( s (cid:96) , M K + , m γ ) m γ . (E.42)I use the notation ν = t − u , λ K(cid:96) ( s ) = λ ( M K + , s, s (cid:96) ) , λ π(cid:96) ( s ) = λ ( M π + , s, s (cid:96) ) .Next, I give the explicit expressions for the diagrams of the second set, shown in figure 8. These diagramscontain a photon pole in the s -channel and mesonic loops. δF NLO γ − pole ,a = 0 ,δG NLO γ − pole ,a = − e s (cid:32) s − M K + ) B ( s, M K + , M K + ) + ( s − M π + ) B ( s, M π + , M π + ) − A ( M K + ) − A ( M π + ) − M K + + 2 M π + − s π (cid:33) , (E.43)58 F NLO γ − pole ,b = − δF NLO γ − pole ,c = − e s − M π + π m γ ,δG NLO γ − pole ,b = − δG NLO γ − pole ,c = − e s (cid:32) ( s − M K + ) B ( s, M K + , M K + ) + 2( s − M π + ) B ( s, M π + , M π + ) − A ( M K + ) − A ( M π + ) − M K + + 4 M π + − s π (cid:33) , (E.44) δF NLO γ − pole ,d = 0 ,δG NLO γ − pole ,d = − e s (cid:18) A ( M π ) + 8 A ( M π + ) + 2 A ( M K ) + 16 A ( M K + ) + 3 A ( M η ) (cid:19) , (E.45) δF NLO γ − pole ,e = 0 ,δG NLO γ − pole ,e = e s (cid:18) A ( M π ) + 2 A ( M π + ) + 2 A ( M K ) + 4 A ( M K + ) + 3 A ( M η ) (cid:19) , (E.46) δF NLO γ − pole ,f = δF NLO γ − pole ,g = 0 ,δG NLO γ − pole ,f = − δG NLO γ − pole ,g = − e s (cid:18) A ( M π ) + 8 A ( M π + ) + A ( M K ) + 4 A ( M K + ) (cid:19) . (E.47)In the sum of these diagrams, the contribution to F vanishes: δF NLO γ − pole = 0 ,δG NLO γ − pole = − e s (cid:32) s − M K + ) B ( s, M K + , M K + ) + ( s − M π + ) B ( s, M π + , M π + )+ 8 A ( M K + ) + 4 A ( M π + ) − M K + + 2 M π + − s π (cid:33) . (E.48) E.2.2 Counterterms The individual contributions of the counterterm diagrams, shown in figure 9, are given by δF NLO γ − ct ,a = 29 e (12 K + 19 K + 9 K − X ) ,δG NLO γ − ct ,a = 29 e (12 K + 36 K + 7 K + 9 K + 6 X ) ,δF NLO γ − ct ,b = − e t − u ) s ( L + L ) ,δG NLO γ − ct ,b = − e s (cid:18) ( M K + + s − s (cid:96) ) L + ( M K + − s − s (cid:96) ) L + 4(2 M K + + M π + ) L + 4 M K + L (cid:19) ,δF NLO γ − ct ,c = δF NLO γ − ct ,d = 0 ,δG NLO γ − ct ,c = − δG NLO γ − ct ,d = − e s (cid:0) M K + + M π + ) L + 4 M π + L + sL (cid:1) ,δF NLO γ − ct ,e = 0 ,δG NLO γ − ct ,e = e s (cid:0) (2 M K + + M π + ) L + M K + L (cid:1) . (E.49)Their sum is δF NLO γ − ct = 29 e (12 K + 19 K + 9 K − X ) − e t − u ) s ( L + L ) ,δG NLO γ − ct = 29 e (12 K + 36 K + 7 K + 9 K + 6 X ) − e s (cid:0) ( M K + + s − s (cid:96) ) L + ( M K + − s − s (cid:96) ) L (cid:1) . (E.50)59 .2.3 External Leg Corrections I first compute the external leg corrections for the mesons (figures 10a and 10b). The field strength renormali-sation of a charged meson is related to the self-energy by Z γm + = 1 + Σ γ (cid:48) m + ( M m + , ph ) + h.o. = 1 + Σ γ (cid:48) m + ( M m + ) + h.o., Σ γm + ( p ) = i ( D γ − loop m + + D γ − ct m + ) , Σ γ (cid:48) m + ( p ) = ddp Σ γm + ( p ) , (E.51)where p denotes the meson momentum and h.o. stands for higher order terms.I find the following field strength renormalisations: Z γπ + = 1 + e (cid:32) A ( m γ ) m γ + 2 B ( M π + , M π + , m γ ) + 4 M π + B (cid:48) ( M π + , M π + , m γ ) (cid:33) − e (6 K + 5 K ) ,Z γK + = 1 + e (cid:32) A ( m γ ) m γ + 2 B ( M K + , M K + , m γ ) + 4 M K + B (cid:48) ( M K + , M K + , m γ ) (cid:33) − e (6 K + 5 K ) . (E.52)Finally, we need the field strength renormalisation of the lepton. The two diagrams 10c and 10d contributeto the self-energy: Z γ(cid:96) = 1 + Σ γ (cid:48) (cid:96) ( m (cid:96) ) + h.o., Σ γ(cid:96) ( /p ) = i ( D γ − loop (cid:96) + D γ − ct (cid:96) ) , Σ γ (cid:48) (cid:96) ( /p ) = dd/p Σ γ(cid:96) ( /p ) . (E.53)Up to terms that vanish for m γ → , the lepton self-energy is given by Z γ(cid:96) = 1 + e (cid:32) A ( m γ ) m γ − A ( m (cid:96) ) m (cid:96) − X − π (cid:33) . (E.54)The contribution of the field strength renormalisation to the form factors is therefore δF NLO γ − Z = δG NLO γ − Z = Z γπ + (cid:113) Z γK + Z γ(cid:96) − e (cid:18) B ( M K + , M K + , m γ ) + 2 B ( M π + , M π + , m γ ) − A ( M K + ) M K + − A ( M π + ) M π + − A ( m (cid:96) )2 m (cid:96) + 6 A ( m γ ) m γ − K − K − X − π (cid:19) . 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