Radiative corrections to the three-body region of the Dalitz plot of baryon semileptonic decays with angular correlation between polarized emitted baryons and charged leptons
M. Neri, J. J. Torres, Ruben Flores-Mendieta, A. Martinez, A. Garcia
aa r X i v : . [ h e p - ph ] N ov Radiative corrections to the three-body region of the Dalitz plot of baryonsemileptonic decays with angular correlation between polarized emittedbaryons and charged leptons
M. Neri, J. J. Torres, Rub´en Flores-Mendieta, A. Mart´ınez, and A. Garc´ıa Escuela Superior de F´ısica y Matem´aticas del IPN,Apartado Postal 75-702, M´exico, D.F. 07738, Mexico Escuela Superior de C´omputo del IPN,Apartado Postal 75-702, M´exico, D.F. 07738, Mexico Instituto de F´ısica, Universidad Aut´onoma de San Luis Potos´ı,´Alvaro Obreg´on 64, Zona Centro, San Luis Potos´ı, S.L.P. 78000, Mexico Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados del IPN,Apartado Postal 14-740, M´exico, D.F. 07000, Mexico (Dated: November 23, 2018)
Abstract
We have calculated the radiative corrections to the Dalitz plot of baryon semileptonic decays with angularcorrelation between polarized emitted baryons and charged leptons. This work covers both charged andneutral decaying baryons and is restricted to the so-called three-body region of the Dalitz plot. Also itis specialized at the center-of-mass frame of the emitted baryon. We have considered terms up to order ( α/π )( q/M ) , where q is the momentum transfer and M is the mass of the decaying baryon, and neglectedterms of order ( α/π )( q/M ) n for n ≥ . The expressions displayed are ready to obtain numerical results,suitable for model-independent experimental analyses. PACS numbers: 14.20.Lq, 13.30.Ce, 13.40.Ks . INTRODUCTION Currently, experiments on spin 1/2-baryon semileptonic decays (BSD), A → Bℓν ℓ , where thepolarization ˆ s of the emitted baryon B is observed are underway [1]. The analysis of theseexperiments requires the inclusion of radiative corrections (RC) to the Dalitz plot when ˆ s isnonzero. Our previous work [2] does not cover this case. It is the purpose of this paper to producesuch RC.There are several requirements that must be met. In order to keep experimental analyses modelindependent it is necessary that RC are model independent themselves. There are many possiblecharge assignments to A and B and RC should be calculated so as to cover all the expectedassignments. The charged lepton ℓ should be allowed to be an e ± , µ ± , and even τ ± as the casemay be. Since RC depend on the form factors present in the uncorrected decay amplitude, itis also necessary that they be cast into a form that can produce numerical results which are notcompromised by fixing the form factors at prescribed values.The model independence of RC is achieved by following the generalization for hyperons [3] ofthe treatment of virtual RC in neutron beta decay [4] and of the application of the Low theorem[5] to the bremsstrahlung RC developed in Ref. [6]. There are six different charge assignmentspredicted by the light and heavy quark content of A and B . To cover all these cases it is necessaryto know only the RC to the neutral decaying baryon (NDB) A → B + ℓ − ¯ ν ℓ and to the chargeddecaying baryon (CDB) A − → B ℓ − ¯ ν ℓ cases. The other possibilities are obtained using the RCof the latter two [7]. The cases ℓ = e ± , µ ± , τ ± are included in the RC by keeping the mass m of ℓ uncompromised all along the calculation. In order to produce numerical values of RC thatare practical to use in the Monte Carlo simulation of an experimental analysis and that are notcommitted to fixed values of the form factors of the weak vertex, one can numerically calculatethe RC to the coefficients of the quadratic products of form factors that appear in the theoreticaldifferential decay rate of the decay being measured.Since current experiments are medium-statistics (of the order of thousands of events) experi-ments and in order to keep the effort of calculating RC within convenient bounds, we shall considercontributions of order ( α/π )( q/M ) n , with n = 0 only and neglect orders with n = 1 and higher.Here q is the four-momentum transfer and M is the mass of A . Also we shall exhibit our results ina form where the integration over the real photon variables are ready to be performed numerically,except for the finite terms that accompany the infrared divergence of the bremsstrahlung RC which2ill be given analytically. The virtual RC will be given fully analytically. Our final result will bespecialized to the center-of-mass frame of the emitted baryon B .In Sec. II we introduce our notation and conventions and discuss in detail the boundaries of theDalitz plot in the center-of-mass frame of B . We shall specialize our calculation to the three-bodyregion of this plot. Section III is devoted to the model-independent calculation of virtual RC. Wewill see that they can be put formally in the same form of our previous work, although now theywill be functions of the energies E of ℓ and E of A in the center-of-mass frame of B . The ratherlong expressions containing the form factors that appear in these corrections are exhibited in fullin Appendix A. The bremsstrahlung RC are obtained in Sec. IV also in a model-independent form.However, the detailed discussion of its infrared divergence and the finite terms that accompany itis presented in Appendix B. In Sec. V we collect our results in a final form and we discuss theirnumerical use. We will cover the NDB and the CDB cases but we will exhibit only the calculationof the CDB case and limit ourselves to present the final results for the NDB case. Section VI isdevoted to a brief discussion of our results. II. DALITZ PLOT IN THE CENTER-OF-MASS FRAME OF THE EMITTED BARYON
For definiteness, let us consider the BSD A − → B + ℓ − + ν ℓ . (1)The four-momenta and masses of the A − , B , ℓ − , and ν ℓ , will be denoted by p = ( E , p ) , p = ( E , p ) , l = ( E, l ) , and p ν = ( E ν , p ν ) , and by M , M , m , and m ν , respectively. Thereference system we shall use is the center-of-mass frame of B . Accordingly, E = M and p = . It must be kept in mind that all other variables are referenced to this frame now. Thereshould not arise any confusion with our previous work. A vanishing neutrino mass m ν will beassumed. Additionally, the direction of a vector p will be denoted by a unit vector ˆ p and wheneverthe expressions involved are not manifestly covariant, quantities like p , l , or p ν will also denotethe magnitudes of the corresponding three-momenta, unless stated otherwise.The uncorrected transition amplitude M for process (1) is given by the product of the matrixelements of the baryonic and leptonic currents, namely, M = G V √ u B ( p ) W µ ( p , p ) u A ( p )][ u ℓ ( l ) O µ v ν ( p ν )] , (2)3here u A , u B , u ℓ , and v ν are the Dirac spinors of the corresponding particles and W µ is the weakinteraction vertex given by W µ ( p , p ) = f ( q ) γ µ + f ( q ) σ µν q ν M + f ( q ) q µ M + (cid:20) g ( q ) γ µ + g ( q ) σ µν q ν M + g ( q ) q µ M (cid:21) γ . (3)Here O µ = γ µ (1 + γ ) , q ≡ p − p is the four-momentum transfer, and f i ( q ) and g i ( q ) are theconventional weak vector and axial-vector form factors, respectively, which are assumed to be realin this work. In Eq. (2) we have omitted the Cabibbo-Kobayashi-Maskawa factors. They shouldbe inserted once decay (1) is particularized.To cover the observation of the polarization of B , its spinor is modified through the replacement u B ( p ) → Σ( s ) u B ( p ) , (4)where Σ( s ) , the spin projection operator, is given by Σ( s ) = 1 − γ s , (5)and the polarization four-vector s satisfies the relations s · s = − and s · p = 0 . In thecenter-of-mass frame of B , s becomes the purely spatial unit vector ˆ s which points along thespin direction. In the present calculation the results will be organized to explicitly exhibit theangular correlation ˆ s · ˆ l .Energy and momentum conservation determines the allowed kinematical region in the variables E and E for process (1). This region, which is referred to as the Dalitz plot and is represented bythe shadowed area depicted in Fig. 1 and labeled as I , is bounded in E by E min1 ≤ E ≤ E max1 , (6)where E max , min1 = ( M + E ∓ l ) + M M + E ∓ l ) , (7)while the charged lepton energy falls within the interval m ≤ E ≤ E m , (8)where E m = M − M − m M . (9)4 I I EE FIG. 1: Kinematical region as a function of E and E for baryon semileptonic decays. The areas I and I + II correspond to the Dalitz plots of the processes A → B + ℓ + ν ℓ and A → B + ℓ + ν ℓ + γ , respectively. Similarly, area II in Fig. 1 is bounded by M ≤ E ≤ E min1 , m ≤ E ≤ E c , (10)where E c = ( M − M ) + m M − M ) . (11)The distinction between these two areas has important physical implications that should beclarified. Finding an event with energies E and E in area II demands the existence of a fourthparticle which in our case will be a photon and will carry away finite energy and momentum. Incontrast, in area I this photon may or may not do so. In consequence, area II is exclusively a four-body region whereas area I is both a three- and a four-body region. We will refer loosely to areas I and II as the three- and four-body regions (TBR and FBR) of the Dalitz plot, respectively. III. VIRTUAL RADIATIVE CORRECTIONS
The method to calculate the virtual RC to the Dalitz plot of unpolarized and polarized decayingbaryons has been discussed in detail in Refs. [3] and [4]. It can be readily adapted to our case hereof nonzero ˆ s , so only a few salient facts will be repeated now. The virtual RC can be separatedinto a model-independent part M v which is finite and calculable and into a model-dependent onewhich contains the effects of the strong interactions and the intermediate vector boson. To order ( α/π )( q/M ) , the latter amounts to two constants ( α/π ) c and ( α/π ) d which can be absorbed into f and g of M , respectively, through the definition of effective form factors, hereafter referred to5s f ′ and g ′ . Thus, the decay amplitude M V with virtual RC is given by M V = M ′ + M v , (12)where M v = α π h M ˆ φ + M p ˆ φ ′ i , (13)and M p = (cid:18) EmM (cid:19) G V √ u B W λ u A ][ u ℓ p O λ v ν ] . (14)The prime on M in Eq. (12) will be used as a reminder that the effective form factors appearexplicitly in this amplitude. Also, to order ( q/M ) the amplitudes M and M p in Eq. (13) [butnot in Eq. (12)] are limited to contain only the leading form factors f and g . The calculation ofthe model-independent functions ˆ φ ( E ) and ˆ φ ′ ( E ) shows that they formally retain the same formgiven in previous work [2]. The hats over them denote they are now given in the center-of-massframe of the emitted baryon B . These functions read ˆ φ ( E ) = 2 (cid:20) β tanh − β − (cid:21) ln (cid:20) λm (cid:21) − β (tanh − β ) + 1 β L (cid:20) β β (cid:21) + 1 β tanh − β − π β + 32 ln M m ( NDB )32 ln M m ( CDB ) (15)and ˆ φ ′ ( E ) = (cid:20) β − β (cid:21) tanh − β, (16)where β ≡ l/E , L is the Spence function, λ is the infrared-divergent cutoff and with CDB andNDB we distinguish the results for the charged and neutral decaying baryon cases. The divergentterm in Eq. (15) will be canceled by its counterpart in the bremsstrahlung contribution.At this point we can construct the Dalitz plot with virtual RC by leaving the energies E and E as the relevant variables in the differential decay rate for process (1). After making the replacement(4) in (12), squaring it, averaging over initial spins, summing over final spin states, and rearrangingterms we can express the differential decay rate as d Γ V = d ˆΩ n ˆ A ′ + απ ( ˆ A ′ ˆ φ + ˆ A ′′ ˆ φ ′ ) − ˆ s · ˆ l h ˆ A ′′ + απ ( ˆ A ′ ˆ φ + ˆ A ′′ ˆ φ ′ ) io , (17) We shall also use hats over other expressions to emphasize that the center-of-mass frame of B is being used. d ˆΩ = 12 (cid:20) M M (cid:21) G V dEdE d Ω ℓ dϕ (2 π ) M . (18)Let us notice that this expression of d ˆΩ has some differences with respect to the one of previouswork [2]. These are the factor / , which results from averaging over the spin of the initial baryon,and the factor ( M /M ) , which arises out of the Lorentz transformation to the new referenceframe. To recover the unpolarized decay rate one makes the factor 1/2 disappear by inserting inEq. (2) the operator Σ( − s ) = (1 + γ s ) / instead of (5) and adding the result to (17).The functions ˆ A ′ and ˆ A ′′ , which emerge in the uncorrected amplitude M , read ˆ A ′ = EE ν ˆ Q − Ep ( p − ly ) ˆ Q − l ( l − p y ) ˆ Q − E ν p ly ˆ Q + p ly ( p − ly ) ˆ Q , (19)and ˆ A ′′ = Ep y ˆ Q + El ˆ Q , (20)where y is defined as the scalar product ˆ p · ˆ l and can be expressed as y = p + l − E ν p l , (21)and also, by energy conservation, the neutrino energy E ν is given by E ν = E − E − M . (22)The ˆ Q i are new functions of the form factors and are listed in Appendix A. The hat is used toavoid confusing them with the ones of Ref. [2]. The functions ˆ A ′ , ˆ A ′′ , ˆ A ′ , and ˆ A ′′ that emerge inthese virtual RC read ˆ A ′ = D EE ν − D l ( l − p y ) , (23) ˆ A ′′ = M M EE ν D , (24) ˆ A ′ = E ( l − p y ) D − lE ν D , (25)and ˆ A ′′ = EM M ( l − p y ) D , (26)where the coefficients D i are quadratic functions of the effective form factors. Explicitly, they are D = f ′ + 3 g ′ , D = f ′ − g ′ , (27a) D = 2( − g ′ + f ′ g ′ ) , D = 2( g ′ + f ′ g ′ ) . (27b)7ere we use also the effective form factors, so that our result is uniformly expressed. This is arearrangement of second order in α/π , which we are free to make within our approximations.To stress the parallelism with our previous work we have used the same notation, but thereshould arise no confusion. The expressions given here apply to the present case only. IV. BREMSSTRAHLUNG RADIATIVE CORRECTIONS
To obtain the bremsstrahlung RC we have to consider the four-body decay A − → B + ℓ − + ν ℓ + γ, (28)where γ represents a massive photon with four-momentum k = ( ω, k ) and k = λ . Toobtain the bremsstrahlung RC in a model-independent way we shall use the Low theorem [5, 6],which asserts that the radiative amplitudes of order /k and ( k ) can be determined in termsof the nonradiative amplitude without further structure dependence. Then, we can express thebremsstrahlung amplitude M B as M B = M B + M B + M B , (29)with M B = e M (cid:20) l · ǫ l · k + λ + 2 p · ǫλ − p · k (cid:21) , (30)and M B = eG V √ ǫ µ [ u B W λ u A ][ u ℓ γ µ k l · k + λ O λ v ν ] . (31) M B contains terms of order /k and M B contains terms of order ( k ) . Although M B containsalso some terms of order ( k ) , it can be ignored because its contribution to the decay rate is oforder q/M [2] so we do not need its explicit form. The infrared-divergent terms are all containedin M B .Next, we have to replace Eq. (4) in Eq. (29), square the resulting M B , average over the initialspins and sum over the final spins and over the photon polarization. To perform the latter sum,we proceed in two ways. First, we can use the rule of Coester [8] to account for the longitudinaldegree of polarization of the photon, namely, X ǫ ( ǫ · a )( ǫ · b ) = a · b − ( a · k )( b · k ) ω (32)8here ω = k + λ (here k is the magnitude of k ) and a = ( a , a ) and b = ( b , b ) arearbitrary 4-vectors. Second, the infrared-convergent contributions can be calculated using theusual summation over the photon polarization, namely, P ǫ ( ǫ · a )( ǫ · b ) = − a · b , with ω = k .After a standard calculation, the bremsstrahlung differential decay rate d Γ B corresponding tothe TBR becomes d Γ B = d Γ ′ B − d Γ ( s ) B , (33)where d Γ ′ B denotes half of the unpolarized decay rate whereas d Γ ( s ) B contains the spin of theemitted baryon. Explicitly, using the D i of Eqs. (27a) and (27b) , they are, d Γ ′ B = 12 απ G V M (2 π ) (cid:20) M M (cid:21) d p E d lE d kω d p ν E ν δ ( p − p − l − p ν − k ) × (cid:26) β (1 − k x /ω )( ω − βkx ) ( D EE ν + D l · p ν )+ 1 E ( ω − βkx ) (cid:20) D E ν (cid:18) ω + 2 E − m ωE ( ω − βkx ) − E ( ω − βkx ) ω (cid:19) + D (cid:20) k · p ν (cid:18) − m E ( ω − βkx ) + Eω (cid:19) + l · p ν (cid:21)(cid:21)(cid:27) , (34)and d Γ ( s ) B = − απ G V M (2 π ) (cid:20) M M (cid:21) d p E d lE d kω d p ν E ν δ ( p − p − l − p ν − k ) (ˆ s · ˆ l ) × (cid:26) β (1 − k x /ω )( ω − βkx ) ( D E ˆ l · p ν + D E ν l )+ 1 E ( ω − βkx ) (cid:20) D ˆ l · p ν (cid:18) ω + 2 E − m ωE ( ω − βkx ) − E ( ω − βkx ) ω (cid:19) + D E ν (cid:20) k · ˆ l (cid:18) − m E ( ω − βkx ) + Eω (cid:19) + l (cid:21)(cid:21)(cid:27) . (35)The above expression of d Γ ( s ) B is specialized to the angular correlation ˆ s · ˆ l . To achieve this wehave used the replacement ˆ s · p → (ˆ s · ˆ l )(ˆ l · p ) , with p = p , k , p ν , which is valid over theDalitz plot after all other variables are integrated. Both expressions (34) and (35) contain theinfrared divergence in their first summand within the curly brackets, which we analyze in detailin Appendix B. The λ that appears in Eq. (30) contributes in Eqs. (34) and (35) with linear andhigher powers, which will become zero in the λ → limit. For the sake of the uniformity of our results, the effective form factors f ′ and g ′ may be used here, too. Again, thisamounts to a rearrangement of order ( α/π ) , valid within our approximations. δ ( p − l − p ν − k ) in Eqs. (34) and (35) istrivial, but leaves a nontrivial argument inside the last δ ( E − M − E − E ν − ω ) , which allowsone to integrate over the photon momentum. Without further ado, the resulting expressions can becast into d Γ ′ B = απ d ˆΩ h ˆ A ′ ˆ I + ( ρ + ρ ′ ) D + ( ρ + ρ ′ ) D i , (36)and d Γ ( s ) B = απ d ˆΩ ˆ s · ˆ l h ˆ A ′ ˆ I + ( ρ + ρ ′ ) D + ( ρ + ρ ′ ) D i , (37)where d ˆΩ , ˆ A ′ , ˆ A ′ , D , D , D , and D were already defined. The integrals over the angularvariables of the photon, x = ˆ k · ˆ l and ϕ k , and over y = ˆ p · ˆ l are left to be performed numerically,except in ˆ I which contains the infrared divergence and is obtained analytically. Its result is foundin Appendix B. The functions ρ , ρ ′ , ρ , ρ ′ , ρ , ρ ′ , ρ , and ρ ′ thus read ρ = β p lE π Z y dy Z − dx Z π dϕ k D − x (1 − βx ) , (38) ρ ′ = − p l π Z y dy Z − dx Z π dϕ k D E ν − βx (cid:20) β (1 − x )1 − βx + ωE (cid:21) , (39) ρ = − β p l π Z y dy Z − dx Z π dϕ k − x (1 − βx ) (cid:20) − lxD (cid:21) , (40) ρ ′ = − p l π Z y dy Z − dx Z π dϕ k D − βx (cid:20) ˆ k · p ν (cid:18) − − β − βx + ωE (cid:19) + β ˆ l · p ν (cid:21) , (41) ρ = βp l π Z y dy Z − dx Z π dϕ k − x (1 − βx ) (cid:20) − lxD (cid:21) (42) = − ρ β , (43) ρ ′ = − p l π Z y dy Z − dx Z π dϕ k D l − p y + ωx − βx (cid:20) β (1 − x )1 − βx + ωE (cid:21) , (44) ρ = − β p l π Z y dy Z − dx Z π dϕ k D − x (1 − βx ) (45) = − βρ , (46) ρ ′ = p l π Z y dy Z − dx Z π dϕ k E ν D − βx (cid:20) β + x (cid:18) − − β − βx + ωE (cid:19)(cid:21) , (47)where ω = p l ( y − y ) D , (48)with D = E ν − ( p − l ) · ˆ k . (49)10e have obtained the bremsstrahlung RC to the differential decay rate to order ( α/π )( q/M ) .In the next section we present the total differential decay rate by gathering both virtual andbremsstrahlung contributions together. V. FINAL RESULTS AND NUMERICAL FORM OF THE RADIATIVE CORRECTIONS
We have reached our first goal: To obtain the complete RC to the Dalitz plot with the ˆ s · ˆ l correlation to order ( α/π )( q/M ) restricted to the TBR. The final result is obtained by summingthe virtual RC, Eq. (17), and the bremsstrahlung RC, Eq. (33). The latter is obtained whenEqs. (36) and (37) are put together. Thus d Γ = d Γ V + d Γ B . (50)We can rearrange this expression into a simple form as d Γ = d ˆΩ n ˆ A ′ + απ ˆΘ I − ˆ s · ˆ l h ˆ A ′′ + απ ˆΘ II io , (51)with ˆΘ I = ˆ A ′ ( ˆ φ + ˆ I ) + ˆ A ′′ ˆ φ ′ + ( ρ + ρ ′ ) D + ( ρ + ρ ′ ) D , (52)and ˆΘ II = ˆ A ′ ( ˆ φ + ˆ I ) + ˆ A ′′ ˆ φ ′ + ( ρ + ρ ′ ) D + ( ρ + ρ ′ ) D , (53)where all the ingredients are given in the previous sections.Notice that the only difference between the NDB and the CDB cases is found in the function ˆ φ ( E ) , Eq. (15). The bremsstrahlung RC does not make any difference between both cases becauseof the order of approximation in this work.We now come to our second goal in this paper. This Eq. (51) has triple integrals over someangular variables ready to be performed numerically. It requires that the numerical integrals inthe RC be calculated within a Monte Carlo simulation every time E , E , and the form factors arevaried, a task that represents a non-negligible computer effort. We shall now discuss a secondform of the RC that should be more practical to use.For fixed values of E and E , Eqs. (52) and (53) take the form ˆΘ r = a r f + b r f g + c r g , (54)11ecause they are quadratic in the form factors. The subindex r takes the values r = I , II . Thesecond form of RC we propose consists of calculating arrays of the a r , b r , and c r coefficientsdetermined at fixed values of ( E, E ) and that these pairs of ( E, E ) cover a lattice of points onthe Dalitz plot.To calculate the coefficients a r , b r and c r it is not necessary to rearrange our final results totake the form (54). One can calculate them following a systematic procedure. One chooses fixed ( E, E ) points. Then one fixes f = 1 and g = 0 , and obtains a r , one repeats this calculation for g = 1 , f = 0 , to obtain c r . Next, one repeats the calculation with f = 1 , g = 1 , and fromthese results one subtracts a r and c r , this way one obtains the coefficient b r . The arrays of thesecoefficients should be fed into the Monte Carlo simulation. Within this simulation the repetitivetriple integrations are reduced into a form of matrix multiplication.We may close this section by stressing that none of the forms of our RC results is compromisedto fixing from the outset values for the form factors when such RC are applied in a Monte Carlosimulation. VI. DISCUSSIONS
Our final result for the RC to the Dalitz plot of BSD with the angular correlation ˆ s · ˆ l betweenthe polarization of the emitted baryon and the direction of the charged lepton is given in Eq. (51).It is valid to order ( α/π )( q/M ) and it covers the TBR of this plot. It meets the requirementsdiscussed in the introductory section, namely, it is model independent, it can be used in all chargedassignments in different BSD, the charged lepton may be e ± , µ ± , or τ ± , and it is not compromisedto fixed values of the form factors of the uncorrected decay amplitude. The finite terms thataccompany the infrared divergence in the bremsstrahlung RC are given in analytical form. Theother terms in this correction are presented with triple integrations ready to be performed.This result may be used in a Monte Carlo simulation of an experimental analysis. However,performing the triple integrations every time E , E , and the form factors are varied may representa very heavy computer effort. A more practical use of our result is through Eq. (54). Numericalarrays of the RC to the coefficients of the quadratic products of form factors may be first obtainedat a lattice of points ( E, E ) covering the Dalitz plot and afterwards be fed in the Monte Carlosimulation. The computer effort within it would then be reduced to a sort of matrix multiplication.The procedure of this paper may be followed in the future to extend the calculation of RC to12SD with the observation of polarization of the emitted baryon to cover the angular correlation ˆ s · ˆ p and the FBR. The precision of RC may be improved, while still preserving their modelindependence, by including terms of order ( α/π )( q/M ) n with n = 1 . It may be the case thatexperimental analyses should be limited to the center-of-mass frame of the decaying baryon A .Our results should be then adapted to this frame. Each one of these possibilities requires furtherserious efforts. They should be attempted as the need for them arises. Acknowledgments
The authors are grateful to Consejo Nacional de Ciencia y Tecnolog´ıa (Mexico) for partialsupport. J. J. T. and A. M. were partially supported by Comisi´on de Operaci´on y Fomento deActividades Acad´emicas (Instituto Polit´ecnico Nacional). R. F.-M. was also partially supportedby Fondo de Apoyo a la Investigaci´on (Universidad Aut´onoma de San Luis Potos´ı).
APPENDIX A: THE ˆ Q i COEFFICIENTS
The ˆ Q i factors contained in Eqs. (19) and (20) are quadratic functions of the form factors. Forthe spin-independent contribution they read ˆ Q = E M ( F + G − F G + F F − G G ) + (1 − β ) EM ( F F − G G ) + E ˆ Q − ˆ Q , (A1) M ˆ Q = 4 M M F G + (1 − β ) EM (cid:20) F F + G G + E + M M F F + E − M M G G (cid:21) + M ˆ Q , (A2) ˆ Q = M M ( F − G ) −
12 (1 − β ) E M (cid:20) E + M M F + E − M M G (cid:21) + E M ( F F − G G ) + M Q , (A3) ˆ Q = 1 M ( F + G − F G + F F − G G ) + E ˆ Q , (A4) M ˆ Q = 2 M M ( F F + G G ) + E + M M F + E − M M G , (A5)13hereas for the spin-dependent contribution they read ˆ Q = (cid:20) E − M − βp y M (cid:21) F + (cid:20) E + M − βp y M (cid:21) G − βp y M ( F F + G G ) − (cid:20) E − βp y M (cid:21) F G + (cid:20) − β ) EM (cid:18) − EM (cid:19) − M + 2 E − βp y M (cid:21) F G − (cid:20) − β ) EM (cid:18) EM (cid:19) + M + 2 E − βp y M (cid:21) F G + (1 − β ) EM (cid:20)(cid:18) − M M − EM (cid:19) F G − (cid:18) M M + EM (cid:19) F G (cid:21) + (cid:20) − β ) EM + E − βp y M (cid:18) − E E − p ly ) + 2 M E M (cid:19)(cid:21) F G + (1 − β ) EM (cid:20) M M − E M − EM E − βp y M (cid:21) ( F G + F G )+ (1 − β ) E M (cid:20) E ν − β ( p y − l ) M (cid:21) F G , (A6)and ˆ Q = (cid:20) M M (cid:21) (cid:20) E − M E (cid:21) F + (cid:20) − M M (cid:21) (cid:20) E + M E (cid:21) G + p M E ( F F + G G )+ (cid:20) (1 − β ) EM − E ν − EE (cid:21) (cid:20) F G + E − M M F G + E + M M F G (cid:21) + (1 − β ) EM (cid:20) E − M M F G + E + M M F G (cid:21) . (A7)In these expressions we have used the form factors F i and G i , which read F = f ′ + (cid:18) M M (cid:19) f , F = − f , F = f + f ,G = g ′ − (cid:18) − M M (cid:19) g , G = − g , G = g + g . APPENDIX B: EXTRACTION OF THE INFRARED DIVERGENCE
Here we discuss the procedure we followed to identify and isolate the infrared divergencecontained in the ˆ I function introduced in Eqs. (36) and (37). We only show how the infrared-divergent term is calculated in the spin-independent contribution, because in the spin-dependentone the procedure is analogous. The term where the infrared divergence is contained is X spins ,ǫ | M ′ B | = e G V X ǫ (cid:20) l · ǫ l · k + λ + 2 p · ǫλ − p · k (cid:21) M M mm ν ( N + N ′ ) , (B1)14here the factors N and N ′ are N = D EE ν − D l ( l − p y ) , N ′ = − ω ( ED + lxD ) , (B2)and y = ˆ p · ˆ l and x = ˆ k · ˆ l . N ′ is proportional to ω so that it is infrared-convergent, and it isabsorbed into Eqs. (38) and (40). Therefore, we here only consider the contribution of N . With allgenerality, we can orient the coordinate axes so that the momentum of the charged lepton is alongthe z -axis and so that A is in the first or fourth quadrant of the plane ( x, z ) . Thus, performing thesum over the photon polarization in the Coester representation and rearranging terms yields d Γ ir B = απ d ˆΩ p l π Z − dx Z π dϕ k Z k m dk k ω g ( θ ) N β (1 − k x /ω )( ω − βkx ) , (B3)where ω = k + λ , k m is the maximum value of the photon momentum, k m = E ν − ( p − l ) E ν − ( p − l ) cos θ k ] , (B4)and g ( θ ) emerges from the last δ function. It is given by g ( θ ) = sin θ a sin θ − b cos θ , (B5) θ is the polar angle of the decaying baryon, a = 2 p ( l + k cos θ k ) , b = 2 p k sin θ k cos ϕ k , θ k and ϕ k are the polar and the azimuthal angles of the photon, respectively.We find it convenient to consider the partition (0 , ∆ k ) , (∆ k, k m ) of the integration interval (0 , k m ) for dk , with ∆ k arbitrary. Thus d Γ ir B = απ d ˆΩ β p l π Z − dx Z π dϕ k Z ∆ k dk k ω g ( θ ) N − k x /ω ( ω − βkx ) + απ d ˆΩ β p l π Z − dx Z π dϕ k Z k m ∆ k dk k ω g ( θ ) N − k x /ω ( ω − βkx ) = d Γ ir(1) B + d Γ ir(2) B . (B6)We must stress that our result does not depend on ∆ k . The divergence is now in the first integraland because ∆ k is arbitrary, we can make it slightly larger than λ , i.e., ∆ k > ∼ λ . Then we canapproximate g ( θ ) in the first integral, by allowing k ≃ . Thus, g ( θ ) ≃ p l . (B7)Also, in N we expand y in powers of k up to first order, y ≃ y + f ′ k + O ( k ) , (B8)15here f ′ = 1 p l (cid:2) E ν + ( l − p y ) x − p (1 − y ) / (1 − x ) / cos θ k (cid:3) , (B9)so that N takes on the form N ≃ ˆ A ′ + D p lf ′ k. (B10)The first integral in Eq. (B6) becomes d Γ ir(1) B = απ d ˆΩ β p l π Z − dx Z π dϕ k Z ∆ k dk p l k ω ( ˆ A ′ + D p lf ′ k ) 1 − k x /ω ( ω − βkx ) , (B11)and then the infrared divergence is finally contained in the first summand of the above equation.The second summand picks up a factor of k so it is infrared-convergent and we can use ω = k in it.It will cancel away with one term of d Γ ir(2) B [see comment after Eq. (B21)]. A further simplificationis obtained by identifying in Eq. (B11) the integral of Kinoshita and Sirlin [9] π β Z − dx Z π dϕ k Z ∆ k dk k ω − k x /ω ( ω − βkx ) = 2 ln (cid:20) ∆ kλ (cid:21) (cid:18) tanh − ββ − (cid:19) + ˆ C, (B12)where ˆ C = 2 ln( I −
2) + 1 + 14 I (cid:20) − β (cid:21) + 1 β L (cid:20) β β (cid:21) − I ln 1 + β , (B13)and I = 2 β tanh − β. (B14)Let us now analyze the second summand in Eq. (B6), d Γ ir(2) B . We shall change the integral over k into an integral over y . For this purpose, we rewrite the photon momentum as k = F D , (B15)and N = ˆ A ′ + 12 D F, (B16)where F = 2 p l ( y − y ) , (B17)and, rearranging Eq. (49), D = E ν + lx − p ( xy + p (1 − y )(1 − x ) cos ϕ k ) . (B18)16hen we can replace dk = dy Dg ( θ ) . (B19)The integration limits change to y (∆ k ) ≃ y + f ′ ∆ k and y ( k m ) = 1 . Notice that the upper limit k m of (B6) is replaced in this variable by y = 1 .Putting all these changes together yields d Γ ir(2) B = απ d ˆΩ β p l π Z − dx Z π dϕ k Z y + f ′ ∆ k dy " ˆ A ′ F + 12 D − x (1 − βx ) , (B20)and the integration over y gives d Γ ir(2) B = απ d ˆΩ β π A ′ Z − dx − x (1 − βx ) Z π dϕ k ln (cid:20) M (1 − y ) M f ′ ∆ k (cid:21) + απ d ˆΩ β p l π Z − dx Z π dϕ k D (cid:20) (1 − y − f ′ ∆ k ) 1 − x (1 − βx ) (cid:21) . (B21)We notice that the term proportional to f ′ ∆ k in the second summand of this latter equation cancelsprecisely the second summand proportional to D k of Eq. (B11) once the integration over k isperformed in it.Finally, the resulting expression for d Γ ir B becomes d Γ ir B = d Γ ir(1) B + d Γ ir(2) B = απ d ˆΩ ˆ A ′ (cid:26) (cid:20) tanh − ββ − (cid:21) ln (cid:20) ∆ kλ (cid:21) + ˆ C (cid:27) + απ d ˆΩ ˆ A ′ (cid:26)(cid:20) (cid:20) tanh − ββ − (cid:21) ln (cid:20) M (1 − y )∆ k (cid:21) + ˆ C (cid:21) + D ˆ C (cid:27) = απ d ˆΩ ˆ A ′ (cid:20) (cid:20) tanh − ββ − (cid:21) ln (cid:20) M (1 − y ) λ (cid:21) + ˆ C + ˆ C (cid:21) + απ d ˆΩ D ˆ C = απ d ˆΩ h ˆ A ′ ˆ I + D ˆ C i . (B22)The function ˆ I is defined in Eq. (B22) as ˆ I = 2 (cid:20) tanh − ββ − (cid:21) ln (cid:20) M (1 − y ) λ (cid:21) + ˆ C + ˆ C , (B23)17here ˆ C = − β π Z π dϕ k Z − dx − x (1 − βx ) ln [ M f ′ ]= − (cid:26) ln (cid:12)(cid:12)(cid:12)(cid:12) ( a + − − a − )4 p /M (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) β − β (1 − βx ) − β − β − β ln 1 − βx β − (1 + x ) (cid:21) − ln (cid:12)(cid:12)(cid:12)(cid:12) ( a + + 1)( a − + 1)4 p /M (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) β − β (1 − βx ) + 1 + ββ + 2 β ln 1 − β − βx + (1 − x ) (cid:21) − ββ ln(1 + β ) − − ββ ln(1 − β ) + 2 + 2 + β (1 − x )1 − βx (1 − x ) ln(1 − x ) − − βx ) + 2 − β (1 + x )1 − βx (1 + x ) ln(1 + x ) + 2 β (cid:20) L (cid:18) − β − βx (cid:19) − L (cid:18) − β β (cid:19) + L (cid:18) − βx β (cid:19) + ln β β ln 1 − β β −
12 ln (cid:18) − βx β (cid:19)(cid:21)(cid:27) , (B24)where x = − l − p y E ν , and a ± = E ν ∓ p l , (B25)and ˆ C is given in Eq. (B13). It possesses the right coefficient to exactly cancel the infrared-divergent term in its counterpart in the virtual RC, Eq. (15).On the other hand, ˆ C given by ˆ C = p l β − y ) Z − dx − x (1 − βx ) , (B26)is absorbed in ρ and ρ . [1] M. Piccini, NA48 Collaboration (private communication)[2] M. Neri, A. Martinez, A. Garcia, J. J. Torres and R. Flores-Mendieta, Phys. Rev. D , 097301 (2007);Phys. Rev. D , 077501 (2006), and references therein.[3] A. Garcia and S. R. Juarez W., Phys. Rev. D , 1132 (1980); , 2923(E) (1980).[4] A. Sirlin, Phys. Rev. , 1767 (1967).[5] F. E. Low, Phys. Rev. , 974 (1958).[6] H. Chew, Phys. Rev. , 377 (1961)[7] A. Martinez, J. J. Torres, A. Garcia and R. Flores-Mendieta, Phys. Rev. D , 074014 (2002).[8] J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, Reading MA,1955). See Secs. 6-5 and 15-2.[9] T. Kinoshita and A. Sirlin, Phys. Rev. , 1652 (1959)., 1652 (1959).