document: undoubtedly the axial anomaly represents a fundamental issue for understanding the basic aspects of quantum field theory . 
   this issue has been analysed deeply over the years . 
   + the anomaly problem has been treated by means of renormalization procedure , giving the interpretation of its origin in terms of ultraviolet divergences @xcite . 
   a more formal analysis of the axial anomaly can be made by using the path integral formalism @xcite . 
   + dolgov and zakharov @xcite have shown an alternative approach to the axial anomaly , by studying the @xmath2 triangular diagram through dispersion relations . from this approach 
   follows the interpretation of the axial anomaly as an infrared phenomenon . 
   it appears as due to a singularity present in the chiral limit in the absorbitive part of the triangular diagram . 
   + the infrared aspect of the axial anomaly , rised in this paper , is complementary to the more familiar ultraviolet one , which emerges from the renormalization procedure . 
   a particularly interesting feature of this approach is that it allows to shed light upon the physical meaning of the anomalous chiral symmetry breaking , which is connected to a non conservation of helicity . 
   + the connection between the anomaly and the breaking of a given symmetry has received a lot of attention in the literature and this subject has been discussed and developed in several papers . 
   gribov @xcite , in a seminal work , has described the source of the anomalies as a collective motion of particles with arbitrarily large momenta in the vacuum . 
   related to this work , in ref . 
   @xcite , mueller has discussed the manifestation of the axial anomaly as a flow of landau levels . in the papers of refs . 
   @xcite the origin of the axial anomaly has been studied in two dimensions and , again , as a level crossing phenomenon . 
   the infrared interpretation of the axial anomaly , according to the dolgov and zakharov approach , has been possibly advanced in @xcite . in a series of papers @xcite the leading terms in the chiral limit 
   have been correctly evaluated . 
   furthermore the dispersive analysis of the triangular @xmath2 diagram is fundamental in the formulation of the t hooft consistency condition @xcite . 
   the axial anomaly plays an essential role also in the interpretation of spin dependent parton distribution ( see @xcite ) . 
   + in this work we attempt to relate the dolgov and zakharov approach to the axial anomaly to some effects in the dynamics of physical reactions , as the radiative decays of @xmath0 and @xmath1 . 
   we will try to show that the axial anomaly can be related to polarized radiative decays , as in the usual ultraviolet interpretation it is connected to the @xmath3 decay . 
   we calculate the corresponding decay rates for the cases where the outgoing leptons are in a definite helicity state and we examine in some detail the structure of the mass singularities and their cancellation . 
   we study how the kinoshita - lee - nauenberg ( kln ) theorem @xcite applies and we consider the analogies and the differences with respect to the corresponding unpolarized decay rates . 
   + the paper is organized as follows . in the section [ anomaly ] 
   we briefly reconsider the dispersive approach to the axial anomaly . 
   in particular we concentrate on its physical origin . 
   + in section [ helicity ] we extend the dolgov and zakharov approach to the study of the radiative pion decay . 
   we calculate the differential decay rate for the process , where the outgoing lepton undergoes an helicity flip and we interpret its behaviour in the chiral limit , as a manifestation of the axial anomaly . + in section [ mass ] 
   we study the behaviour of the inner bremsstrahlung contribution to the pion decay rate in the collinear and infrared limits . 
   we consider separately the unpolarized decay rate and both the cases of right - handed and of left - handed outgoing lepton . 
   we find that the mass singularities cancellation mechanism occurs in different ways , according to the polarization of the outgoing lepton . 
   we discuss the various realizations of the kln theorem . 
   + we also consider a more general process , i.e. the radiative @xmath1 decay in a lepton - antilepton pair , with a right - handed polarized lepton . 
   we examine how the mass singularities cancel in this case and discuss the differences with respect to the pion decay . 
   + finally , in the conclusions , we summarize our arguments . + a short discussion of the main results obtained has been already given in @xcite . 
   the dolgov and zakharov approach @xcite to derive the axial anomaly is based on the dispersion relation method . in this framework , 
   the triangular diagram with two vector and one axial vertices is seen as the lowest order contribution to the process : @xmath4 as described by the diagrams of fig . 1 .    [ cols="^ " , ]     [ fig2 ]    in terms of dispersion relations , 
   the functions @xmath5 , @xmath6 can be expressed as follows : @xmath7 we can use unsubtracted dispersion relations , because the integrals contained in @xmath5 , @xmath6 are convergent , since these functions are multiplied by three and two powers of momentum , respectively . 
   + the imaginary part of the invariant scalar functions can be derived from the absorbitive part of the triangular diagram , calculated by cutting the diagram as shown in fig . 
   [ fig2 ] and by using the cutkosky rules or the perturbative unitarity relation . 
   we obtain @xcite : @xmath8 using eqs . 
   ( [ def1 ] ) , ( [ def2 ] ) , ( [ dr ] ) and the above expressions , we derive the complete triangular diagram  contribution : @xmath9 and the anomalous axial - vector ward - takahashi identity : @xmath10 thus , in the dispersive approach we find the same result obtained by using the renormalization procedure @xcite . 
   the method above allows a more direct treatment , since we avoid evaluating divergent integrals and introducing regularization schemes . 
   + the fact that the axial anomaly can be derived without using the renormalization procedure , suggests that this should not be considered as the only origin of the anomalous breaking of the chiral symmetry . 
   moreover , by studying the axial anomaly with the renormalization procedure , that is by considering its ultraviolet interpretation , an important aspect of this phenomenon remains obscure and we are bound by a formal derivation only . 
   as stressed in @xcite and @xcite , the dispersive approach shows that the anomaly is related to the chiral limit and therefore , it can be interpreted as an infrared effect . in this work 
   we are close to this infrared interpretation , which , as we will see , can help to shed light on some aspects of the physics connected to the axial anomaly . 
   + let us briefly discuss the physics involved in the amplitudes contributing to the absorbitive part of the triangular diagram . 
   the two born diagrams , obtained from the cut of the triangular diagram ( see fig . [ fig1 ] ) , describe the following two processes :    1 .   the production of a fermion - antifermion pair ( for example @xmath11 ) by an axial - vector source ; 2 . 
   the subsequent annihilation of the pair into two real photons . 
   in both these processes there occur helicity flips , thus the chirality is not conserved in the zero mass limit . 
   + let us go to the center of mass frame of the two final photons . in the first process the axial - vector source produces an @xmath11 pair of total spin zero , since a spin 1 state can not annihilate into a two real photons state . 
   thus @xmath12 and @xmath13 must have the same helicity and hence opposite chirality in the massless limit . in the process 
   b ) the @xmath11 pair annihilates into two real photons by going through an intermediate virtual state . 
   there are four possible virtual states @xcite : one is drawn in fig . 
   3    c    [ fig3 ]    ( @xmath14 and @xmath15 are the @xmath13 and @xmath12 linear momenta ) , a second one is obtained by reversing the virtual state helicity ; the remaining two are obtained replacing the outgoing virtual fermion with an incoming virtual antifermion . 
   let us study the case shown in fig . 3 and assume that @xmath12 and @xmath13 are both right - handed . in the vertex b the chirality is conserved in the massless limit , while in the vertex a there is an helicity flip , thus the chiral symmetry is broken . 
   as can be easily checked , for all the remaining virtual states we always have an allowed vertex and a forbidden one . 
   + at the born level , these reactions are described by the classical qed lagrangian , which , in the @xmath16 limit , is invariant under chiral transformations . 
   thus it seems at first sight that the absorbitive part of the triangular diagram , being proportional to the product of amplitudes relative to processes forbidden by chiral invariance , vanishes . on the contrary 
   , one sees that taking the limit @xmath16 in ( [ g2 ] ) gives a finite result @xcite : @xmath17 therefore , by studying the absorbitive part of the triangular diagram , one establishes that a non - conservation of helicity occurs , becoming , in the massless limit , a non - conservation of chirality . 
   we interpret this as related to the presence of the anomalous term in the divergence of the axial - vector current . 
   thus the axial anomaly can be derived by studying the properties of the amplitude in the infrared region . 
   + even if in this work we will analyse the cancellation of mass singularities in polarized processes , we will not discuss the physical implication of the zero fermion mass limit . 
   + as stated in refs . 
   @xcite , the result in eq . 
   ( [ limit ] ) indicates that there occurs a cancellation of the suppression factor @xmath18 , due to the terms coming from the vertices with helicity flip . in ( [ g1 ] ) 
   the logarithmic factor conspires to give a finite result . 
   this logarithm is a collinear one ; we shall discuss about this kind of logarithms in section [ mass ] . 
   its presence is a manifestation of the singularity occurring in the fermion propagator as @xmath16 , which exactly cancels the suppression factor @xmath18 in the numerator . + we observe that the behaviour of the absorbitive part of the triangular diagram  given in eq . 
   ( [ limit ] ) shows that the @xmath16 limit is not smooth ; if we evaluate the amplitudes with the massless theory , they identically vanish . if , on the contrary , we take the chiral limit after summing their product over the intermediate states , we obtain a result different from zero . 
   in the process b ) contributing to the absorbitive part of the triangular diagram  a charged polarized fermion , changes helicity by emitting a photon . 
   thanks to the collinear singularity within the propagator relative to this fermion , the chiral suppression factor @xmath18 is cancelled and the absorbitive part of the triangular diagram   does nt vanish in the chiral limit . 
   + let us now extend these remarks to physical processes having the same features as the ones characterizing the absorbitive part of the triangular diagram . 
   this means that we want to investigate about a manifestation of the axial anomaly in reactions where a fermion changes helicity by emitting a photon . 
   we call these processes helicity changing processes . 
   they have been considered by lee and nauenberg @xcite , who observed that states with opposite helicity do nt decouple in the massless limit , provided that this limit is taken after having summed the transition probability over the final phase space . in the process 
   b ) the incoming @xmath12 and @xmath13 are in a definite helicity state , as discussed above . in the reactions we want to study , 
   we calculate the probability that an outgoing fermion assumes an helicity opposite to the one required by the interaction before the emission of the photon . in other words , 
   we evaluate the probability that in the fermion - photon vertex occurs an helicity flip . according to the dolgov and zakharov analysis 
   , we interpret the presence of a term independent of the fermion mass in the corresponding cross sections as a manifestation of the axial anomaly . 
   due to this term , the probability for a process with helicity flip does nt vanish in the chiral limit . +      we first examine the non radiative pion decay @xmath19 where @xmath20 is an antilepton ( @xmath21 or @xmath22 ) and @xmath23 is the associated neutrino . at the born level 
   the total decay rate is given by : @xmath24 where @xmath25 is the fermi coupling constant , @xmath26 is the pion decay constant , @xmath27 is the ckm matrix element , @xmath28 and @xmath29 are the lepton and pion masses , respectively . 
   the decay rate ( [ born ] ) is proportional to @xmath30 , since , due to angular momentum conservation , the pion produces a left - handed lepton , while the structure of the weak coupling requires the @xmath31 to be right - handed for @xmath32 . 
   + this situation is confirmed by the expression of the total decay rate for the process in which the antilepton is polarized : @xmath33 where @xmath34 and @xmath35 are the linear momentum and the spin vector of the lepton , respectively , giving @xmath36 , for right - handed and left - handed lepton , respectively . 
   ( [ bornpol ] ) indicates that the lepton is mandatory left - handed , as requested by angular momentum conservation . 
   + we now consider the radiative correction , due to the emission of a real photon , to the pion decay , @xmath37 when the outgoing antilepton is polarized , i.e. when it is in a definite helicity state . as we have seen in the non - radiative case , due to angular momentum conservation , the @xmath0 
   is coupled to a left - handed lepton . 
   we calculate the probability that the lepton flips its helicity and becomes right - handed , by emitting a real photon . 
   + the amplitude describing the radiative pion decay can be divided into two parts , the inner bremsstrahlung and the structure dependent amplitudes @xcite : @xmath38 the inner bremsstrahlung amplitude , where the photon is radiated from the external charged particles , can be calculated using the rules of qed , with a point like pion coupling ; the structure - dependent amplitude is governed by the strong interactions . 
   + clearly , the relevant part for the problem we are considering is the inner bremsstrahlung contribution described by the diagrams of fig . 
   c    [ innerb ]    the term associated to the @xmath39 diagram is the so called contact term and it is introduced to ensure gauge invariance ( see , for example , @xcite ) . 
   + we consider only tree level diagrams , since , to reveal the effects of the axial anomaly , it is sufficient to take into account the contribution of the vertex with the helicity flip . for the moment , we may neglect the corrections due to the emission of virtual photons ; these will be discussed in section [ mass ] . 
   finally , at this order in perturbation theory , we retain terms of all powers in the lepton mass . 
   we will argue that the manifestation of the axial anomaly is strictly connected to these terms . 
   + the inner bremsstrahlung amplitude is given by @xcite : @xmath40 v(p_l , s_{lr } ) , \label{e:4.5}\end{aligned}\ ] ] where @xmath41 and @xmath42 are the dirac spinors for the neutrino and the lepton , respectively , @xmath43 is the pion momentum , @xmath44 and @xmath45 are the momentum and the polarization vector of the lepton , @xmath46 and @xmath47 are the momentum and the polarization vector of the photon . + we see that @xmath48 is proportional to the lepton mass @xmath28 , thus the decay rate is proportional to @xmath30 . as we have said above , this factor is a consequence of the structure of the weak coupling . 
   thus , if we remove it by normalizing the radiative decay rate with respect to the non radiative one , we can emphasize the mass dependence of the radiation emission process . 
   + the differential inner bremsstrahlung contribution for right - handed lepton is : @xmath49 \nonumber \\ & & + \,\big[(a+2r)(1+r)^2 + ay(y-4r ) + 2ry(1+y)-y(1+y^2 + 5r^2)\big]\,\ln{\frac{y+a}{y - a } } \nonumber \\ & & + \,(1-y+r)^2(y-2r - a ) \ln{\frac{y+a-2}{y - a-2}}\bigg\}. \label{ibr}\end{aligned}\ ] ] the dimensionless variable @xmath50 is defined as @xmath51 and @xmath52 the physical region for @xmath50 is : @xmath53 we see that , in eq . 
   ( [ ibr ] ) , there is a term independent of the lepton mass , the one in the first square brackets . owing to this term , the differential decay rate does nt vanish in the limit @xmath54 . 
   we have : @xmath55 this indicates that there occurs an helicity flip and thus a chirality non conservation in the limit of zero lepton mass . 
   according to the interpretation given above , this term can be interpreted as connected to the axial anomaly . 
   it corresponds to the anomalous term present in the divergence of the axial current . 
   + since the polarized radiative process with the right - handed @xmath31 is forbidden , in the limit @xmath54 , by the chiral invariance of the massless qed lagrangian , the appearance of a term different from zero , in this limit , indicates the action of a cancellation mechanism , analogous to the one acting in the absorbitive part of the triangular diagram . 
   + to see how this mechanism acts , let us examine the decay rate differential with respect to the lepton energy and to the emission angle : @xmath56 \ , \frac{1}{(y - a\cos{\theta } ) } \nonumber \\ & & + ( 1-y+r)^2(y - a-2r)\ , \frac{1}{(y - a\cos{\theta } -2 ) } \bigg\}. \label{integrale}\end{aligned}\ ] ] the first term is proportional to the lepton propagator squared . 
   thanks to this term , when we carry out the final integration over the emission angle , we obtain , besides logarithmic collinear divergences , also power collinear divergences . 
   these power terms are essential for the convergent behaviour of the distribution . 
   the cancellation of the chiral suppression would not take place were these terms absent . the term related to the axial anomaly 
   ( [ massa0 ] ) ) originates from this cancellation . to obtain the differential decay rate 
   , we have to evaluate integrals of the form : @xmath57 where the propagator has a power mass singularity that exactly cancels the factor @xmath30 , coming from the vertex lepton - photon . 
   + the integration of the terms in equation ( [ integrale ] ) containing the lepton propagator gives rise to the collinear logarithms : @xmath58 in the differential decay rate ( [ ibr ] ) there are also terms proportional to the logarithm @xmath59 this one is another collinear logarithm ; it diverges in the limit @xmath60 and @xmath54 and corresponds to the possibility that the photon is emitted parallel to the pion . 
   + finally , we point out that the chiral limit ( @xmath54 ) is not smooth . 
   in fact we get different results depending upon whether we describe an helicity changing process using the massless theory or we take the @xmath54 limit , after carrying out the integration over the final phase space . 
   the radiative pion decay is not a good process to see this , since , owing to the angular momentum conservation in the pion vertex , this process can not take place within a massless theory , given the structure of the @xmath61 coupling of the electroweak theory . to see that the chiral limit is not a smooth one , we consider the scattering of a polarized electron with a proton ( treated as a point like particle ) of initial and final momenta @xmath62 and @xmath63 respectively , accompanied by the emission of a real photon . 
   we consider a left - handed incoming electron with momentum @xmath43 and spin @xmath64 : we calculate the probability that the electron makes an helicity flip , emitting a real photon with momentum @xmath46 and polarization @xmath47 and thus becoming right - handed electron with momentum @xmath65 and spin @xmath66 : @xmath67 we study this process with the massless qed . 
   the left - handed and right - handed spinors are given respectively by : @xmath68 the corresponding transition amplitude identically vanishes : @xmath69 ( l^2 + i \epsilon ) } \nonumber \\ & & \bar{u}(p')\frac{1-\gamma_5}{2 } { \not\!}\epsilon({\not\!}p ' + { \not\!}k ) \gamma_{\rho } \frac{1-\gamma_5}{2}u(p ) \bar{u}(q')\gamma^{\rho}u(q ) = 0,\end{aligned}\ ] ] [ qed0 ] since it contains the product of different chirality projectors . 
   + we now calculate the cross sections for processes with helicity flip using the massive qed and then we take the massless limit after having summed the transition probability over the final phase space . 
   an example can be found in @xcite , where the cross section for the process @xmath70 is calculated . 
   here @xmath71 is the target ( for example another fermion ) , @xmath72 is a bremsstrahlung photon , assumed almost collinear with respect to the direction of the incident electron and @xmath73 is a set of particles produced in the reaction . 
   + the helicity flip cross section is given by : @xmath74 where @xmath75 , @xmath76 is the energy of the incoming electron and @xmath77 is the cross section for the born process @xmath78 we see that the expression ( [ hf ] ) does not vanish in the massless limit . 
   + the result in eq . 
   ( [ hf ] ) coincides with the one in eq . 
   ( [ massa0 ] ) at leading order . indeed , in @xcite the subdominant terms are not accounted for . 
   as it will become apparent in the following section , these terms are essential for the cancellation of mass singularities . 
   @xcite , the authors give a different interpretation of the helicity changing processes and hence come to a different conclusion about the smoothness of the zero mass limit . + 
   as it is well know , there are two types of divergences occurring in a theory when the mass of a particle goes to zero , which will be comprehensively call in the following mass singularities . 
   + the first type of divergences appears when we reach the phase space region , where the momentum of the massless particles vanishes : these are called infrared divergences . 
   they occur for example in qed when the energy of the photon goes to zero . 
   the block - nordsieck theorem @xcite assures that the infrared divergences cancel out in any inclusive cross section . 
   + the other type of mass singularities occurs in theories with massless coupled particles , like in qed when the photon couples to a fermion , in the limit of zero fermion mass . 
   the origin is purely kinematical : when two massless particles , say with momenta @xmath46 and @xmath79 , move parallel to each other , they have combined invariant mass equal to zero : @xmath80 even though neither @xmath46 nor @xmath79 are soft . 
   these divergences are called collinear singularities . 
   + if we keep the fermion mass finite and integrate over the photon emission angle , the collinear divergence does nt occur , but the possibility of a divergence in the limit @xmath81 results in the presence of the collinear logarithms , that is logarithms of the form @xmath82 , diverging for @xmath16 . + in the case of collinear singularities , the theorem by kinoshita , lee and nauenberg @xcite guarantees that these divergences cancel out if we sum the transition probability over the set of degenerate states , order by order in perturbation theory . 
   this cancellation mechanism is analogous to that of the infrared divergences , as stated by the block - nordsieck theorem . 
   both types of mass singularities arise because the states of a theory with massless particles are highly degenerate . 
   the infrared divergences can be interpreted as a consequences of the fact that a state with a single charged particle is degenerate with a state made of the same particle plus a number of soft photons ; this correspond to the impossibility of distinguishing experimentally a charged particle from one accompanied by soft photons , owing to the finite resolution of the measurement apparatus . 
   the situation of the collinear singularities is analogous : the state with a massless charged particle is degenerate with the states containing the same particle and a number of collinear photons . 
   this corresponds to the fact that , as a consequence of the finite angular resolution , we can not establish if a massless charged particle is accompanied by collinear photons . 
   + let us now discuss the structure of mass singularities and their cancellation in the decay rates for the radiative pion and @xmath1 decays and how the kln theorem applies to this cases . +      it is useful to separate the cases of unpolarized , right - handed and left - handed outgoing lepton . 
   + let us consider first the mass singularity cancellation mechanism for the familiar case of unpolarized radiative @xmath0 decay to the first order in @xmath83 . 
   the differential inner bremsstrahlung contribution is given by : @xmath84\,\ln{\frac{y+a}{y - a } } \nonumber \\ & - & ( 1-y+r)^2\,\ln{\frac{y+a-2}{y - a-2 } } \bigg\}. \label{irnp}\end{aligned}\ ] ] one can easily see that the eq . 
   ( [ irnp ] ) is divergent both in the collinear and in the infrared limits . 
   the coefficients of the collinear logarithms do nt go to zero in the limit @xmath85 . 
   there are also infrared divergences , because if we let @xmath50 reach its kinematical limit @xmath86 , corresponding to the photon energy going to zero , the expression ( [ irnp ] ) diverges . 
   + the decay rate is made free from mass singularities in the ordinary way : the divergences cancellation occurs in the total inclusive decay rate , when we add all the first order contributions to the perturbative expansion , i.e. those relative to real and virtual photon emission . 
   + the diagrams describing the real photon emission contribution were already given in fig . 
   4 ; the diagram for the virtual correction is drawn in fig . 
   c    [ fig5 ]    the expression for the lepton energy spectrum , including the inner bremsstrahlung contribution and the virtual photon one , calculated to the leading order in @xmath87 , is given by @xcite : @xmath88 . 
   \label{spettroy}\ ] ] @xmath89 is the lepton distribution function , given , to the first order in @xmath83 , by : @xmath90p^{(1)}(y ) , \label{defd}\ ] ] where @xmath91 is the logarithm @xmath92 diverging in the collinear limit ; if the lepton is an electron , @xmath93 . 
   @xmath94 is the gribov - lipatov - altarelli - parisi kernel @xcite , which can be expressed in the form : @xmath95 @xmath96 is a finite term , free from infrared and collinear singularities , which has the expression : @xmath97 the differential decay rate to order @xmath83 therefore becomes : @xmath98 to calculate the inclusive decay rate , we have to integrate the expression in eq . 
   ( [ ordalfa ] ) over @xmath50 ; to the leading order in the lepton mass , the physical region for @xmath50 is @xmath99 . 
   the kernel @xmath100 has the property that : @xmath101 thus , when we calculate the inclusive decay rate , the coefficient of the collinear logarithm vanishes and the resulting expression is finite in the zero mass limit . + carrying out the integration over @xmath50 , we obtain the well known inclusive decay rate to order @xmath83 @xmath102 . \label{tot}\ ] ] as expected , the expression ( [ tot ] ) is finite in the collinear limit and is also free from infrared divergences , because , as usual , the infrared divergences present in the soft photon contribution and in the virtual photon contribution have cancelled each other . + 
   let us now discuss the mass singularities in the case of the right - handed inner bremsstrahlung contribution , given in eq . 
   ( [ ibr ] ) . 
   it is easy to see that @xmath103 is finite in the limit @xmath104 , i.e. it is free from collinear singularities . in this limit 
   the coefficients of both the collinear logarithms vanish . indeed , as we have observed in section [ pionpol ] , only the term related to the axial anomaly survives in the zero mass limit , that is the part of the decay rate independent of the lepton mass . 
   + we observe that the right - handed inner bremsstrahlung contribution is free from infrared divergences as well . if we make the lepton energy @xmath50 reach its kinematical limit @xmath105 , we obtain a finite result : @xmath106 the result ( [ limir ] ) shows that the soft photon emission does not contribute to the radiative @xmath0 decay with a right - handed lepton . 
   this is a consequence of the fact that the soft photon contribution factorizes with respect to the born decay rate , but this vanishes in the case of right - handed @xmath31 ( see eq . 
   ( [ bornpol ] ) ) . physically , eq . 
   ( [ limir ] ) is due to the fact that soft photons do nt carry spin , thus they can not contribute to the angular momentum balance ; therefore the process with the right - handed lepton emitting a soft photon is forbidden by angular momentum conservation . 
   + for the same reason of angular momentum conservation , in the right - handed case also the virtual contribution identically vanishes . 
   the virtual photon diagram ( see fig . 
   5 ) interferes with the born one ; the corresponding correction to the total decay rate for @xmath107 was calculated long ago by kinoshita @xcite and is given by : @xmath108   + \frac{r}{1-r } \ln{r } + 1\bigg\}\gamma_0 , \label{kin}\end{aligned}\ ] ] where @xmath109 @xmath110 is the ultraviolet cutoff and @xmath111 is the infrared one . + as usual , the virtual correction is factorized with respect to the born decay rate , but , as we have already seen , if the lepton is right - handed , this is identically zero . 
   + in the right - handed case , the mass singularities cancellation occurs trough a mechanism different from the one working in the unpolarized decay rate . 
   the infrared and the collinear limits give separately a finite result . 
   in particular , the coefficient of the collinear logarithms is the lepton mass , instead of the usual correction factor coming from the soft and the virtual photon contributions , as in eq . 
   ( [ ordalfa ] ) . in this sense , since the soft and collinear radiation factorizes with respect to the born helicity changing decay rate , the double logarithm sudakov term can be equally factorized . 
   it could be useful to investigate the impact of the higher order terms on the radiative correction to the born amplitude . 
   it is an open question if say hard collinear photons can be factorized and resummed . 
   + the particular mass cancellation mechanism occurring in the right - handed radiative decay is the consequence of the combination of two constraints : the angular momentum conservation in the pion vertex and the helicity flip in the photon - lepton vertex . 
   + the situation is completely different if we consider the radiative process with the outgoing left - handed lepton , i.e. the process without helicity flip . 
   the differential inner bremsstrahlung contribution for left - handed outgoing lepton is given by : @xmath112 \nonumber \\ & & + \big[(a-2r)(1+r)^2 + ay(y-4r ) -2ry(1+y ) + y(5r^2+y^2 + 1)\big ] \ 
   , \ln{\frac{y+a}{y - a } } \nonumber \\ & & + ( 1-y+r)^2(2r - a - y ) \ , \ln{\frac{y+a-2}{y - a-2 } } \bigg\ } \label{ibl}\end{aligned}\ ] ] the expression ( [ ibl ] ) contains collinear singularities , since the coefficients of the collinear logarithms do nt vanish in the limit @xmath104 , as one can see from eq . 
   ( [ ibl ] ) . 
   the decay rate ( [ ibl ] ) is also infrared divergent , as one can verify by taking the limit @xmath113 . in this case 
   we do nt have the constraint constituted by the helicity flip in the photon - lepton vertex and in the pion vertex the angular momentum is conserved for soft and virtual photon emission . 
   thus , in the left - handed case , the mass singularity cancellation occurs in the ordinary way , as in the unpolarized case , i.e. in the total inclusive decay rate , obtained by adding all the order @xmath83 contributions . + let us show how the cancellation takes place . 
   as we have already seen ( eq . 
   ( [ bornpol ] ) ) , in the born @xmath0 decay the outgoing lepton is left - handed , due to angular momentum conservation . 
   thus the unpolarized and the left - handed born decay rate coincide : @xmath114 because of the factorization with respect to the born decay rate , also the unpolarized and the left - handed virtual contributions are equal : @xmath115 expressing the left - handed inner bremsstrahlung contribution in terms of the unpolarized and the right - handed ones , the total contribution to order @xmath83 to the left - handed process is given by : @xmath116 the expression ( [ totl ] ) is finite both in the infrared and in the collinear limit , because the mass singularities present in the terms between brackets cancel each other , as we have seen ( see eq . 
   ( [ tot ] ) and @xmath117 is free from mass singularities . 
   + let us now discuss the origin of these different cancellation mechanisms . 
   the presence of mass singularities is a consequence of the fact that the states of a theory containing massless particles are highly degenerate . 
   the kln theorem states that the mass singularities disappear from the transition probability when we average it over the ensemble of degenerate states . 
   this theorem contains the block - nordsieck theorem as a special case , when we consider only the cancellation of the infrared divergences . 
   + we can define two degeneration ensembles , one relative to the infrared divergences and one relative to the collinear singularities . 
   we call them the infrared and the collinear ensembles , respectively . 
   if we sum the transition probability over the states contained in the former , the infrared divergences cancel out and if we average it over the latter , we obtain a quantity free from both infrared and collinear singularities . 
   the infrared ensemble is the one prescribed by the block - nordsieck theorem , while the second is the one prescribed by the kln theorem and contains the first as a subset . 
   + let us now examine how the infrared and collinear ensemble are composed for the radiative pion decay in the cases of unpolarized , left - handed and right - handed outgoing lepton . 
   this discussion concerns the issue of the degeneration of states already addressed for the unpolarized case @xcite . 
   this issue in the case of helicity changing processes presents peculiar features . 
   + we have seen in section [ pion ] that in the unpolarized and left - handed inner bremsstrahlung contributions there are mass singularities , indicating that we have not summed the transition probability over the entire ensemble of degenerate states . in these cases 
   the infrared ensemble contains , to order @xmath83 , the state with a pion , a charged lepton and a neutrino and all the other states differing from this for the presence of a soft virtual or real photon , i.e. a photon with an energy @xmath118 , where @xmath119 is an infrared cut off , tipically the measurement apparatus resolution . 
   the collinear ensemble is constituted by all the states of the infrared ensemble plus the states with a hard photon moving parallel to the pion or the lepton . clearly the degeneration arises in the limit @xmath120 . + according to the kln theorem the fact that @xmath103 is finite both in the infrared and in the collinear limits means that in the right - handed case , calculating the differential decay rate ( i.e. summing over the photon polarization and integrating over the photon energy and emission angle ) , we have already averaged over the set of degenerate states relative to this process . 
   let us now consider how the degeneration ensemble for the right - handed radiative decay is composed . to obtain @xmath121 
   we have not averaged over the infrared subspace of the collinear ensemble , but this is enough to render the transition probability free from mass singularities . 
   indeed in this case the infrared ensemble is empty , owing to the constraints imposed both by the angular momentum conservation in the pion vertex and by the helicity flip in the photon - lepton vertex . 
   thus in this case the degeneration ensemble contains only the states with the outgoing lepton accompanied by hard collinear photons . 
   + we conclude that imposing to the outgoing lepton a polarization opposed to the one prescribed by the vertex preceding the photon emission , implies a reduction of the degeneration subspace . 
   this fact has two consequences : the first is that both the infrared and the collinear limits are finite and the second is that these limits are disconnected , since the collinear degeneration subspace is constituted only by the states with the pion and the outgoing lepton accompanied by hard collinear photons . 
   thus in this case we have a particular application of the kln theorem . 
   +      in the radiative pion decay , due to the angular momentum conservation in the pion vertex , there is no room for a right - handed lepton . 
   for such a channel , soft and virtual photon contributions are zero . 
   this result is valid independently of the lepton mass . 
   + let us now consider a more general case , by loosing the value of the angular momentum of the decaying state . 
   as an example , we study the radiative @xmath1 decay in a lepton - antilepton @xmath122 pair , in which the lepton is in a definite helicity state . 
   + the @xmath1-leptons vertex is : @xmath123 with @xmath124 where @xmath125 is the weinberg angle and @xmath126 is the @xmath1 mass . 
   + we have chosen this process , since , by varying the constants @xmath127 and @xmath128 , we can control the structure of the @xmath1-leptons coupling ; thus it is possible to point out the role played by the conservation law occurring in this vertex in the chiral limit in the collinear singularity cancellation . 
   if we set @xmath129 , we require that in the limit of zero lepton mass , the @xmath1 couples to a left - handed lepton . 
   + we calculate the decay rate for the process in which the lepton is right - handed . at the born level this is given by : @xmath130 - 2g_vg_a(1 - 4r ) \bigg\}. \label{bornz}\end{aligned}\ ] ] if in eq . 
   ( [ bornz ] ) we set @xmath129 , @xmath131 vanishes in the chiral limit , since there is nt the term related to the axial anomaly . 
   + let us now study the decay process with the lepton emitting a real photon ( see fig . 
   6 )    c    [ fig6 ]    and evaluate the probability that the outgoing lepton is right - handed . 
   the electromagnetic interaction does nt couples states with different chirality , hence the decay rate is expected to vanish for @xmath104 . 
   + the decay rate for the process described by the diagram of fig . 6 , differential with respect to the lepton energy , is given by : @xmath132 \nonumber \\ & + & \frac{(1-y)}{2(1-y+r)}\ , \big[(g^2_v+g^2_a)a(2r - y ) + 2g_vg_a(y^2 - 2r)\big ] \nonumber \\ & + & \frac{2}{(y-1)}\ , \big[2(g^2_v-2g^2_a)ar + ( g^2_v+g^2_a)a + 2g_vg_a(4r - y^2+y-1)\big ] \nonumber \\ & + & \big[g^2_v+g^2_a + 2g_vg_a\,\frac{1}{a}\,\big(4r^2+r(y - y^2 + 1)-y\big)\big ] \,\big[\ln{\frac{y+a}{y - a } } - \ln{\frac{y+a-2}{y - a-2}}\big]\bigg\}. \nonumber \\ \label{rz}\end{aligned}\ ] ] here @xmath133 @xmath50 is the usual dimensionless variable : @xmath134 where @xmath135 is the lepton energy and @xmath136 the physical region for @xmath50 is @xmath137 the result obtained , as given by the emission of the photon by a single leg , is gauge dependent . to have a gauge independent amplitude , the contribution of the diagram b ) of fig . 
   7 must be added . for the purpose of the polarized amplitude , however , the helicity flip contribution of the diagram b ) of fig . 
   7 gives zero in the massless limit and is therefore negligible in our discussion . 
   + from now on we consider the case @xmath129 , to have the condition of chirality conservation in the @xmath1 vertex for @xmath120 . taking this limit in eq . 
   ( [ rz ] ) , we see that @xmath138 does not vanish : @xmath139 the result of this limit is the contribution related to the axial anomaly , which has the same form of the one found in the pion case . 
   + let us now discuss the mass singularities cancellation mechanism for the @xmath1 decay case . 
   if we keep the lepton mass different from zero , the helicity is not fixed by the interaction occurring before the photon emission , even if we set @xmath129 . 
   thus , for @xmath140 , the soft and virtual photons contribution are different from zero and diverge in the infrared limit . 
   indeed , if we let the lepton energy reach its kinematical limit , @xmath141 , we see that @xmath138 diverges . 
   we expect the infrared divergences to cancel , if we add all the first order contributions , given by the diagrams of fig . 
   6 and 7    c    [ fig7 ]    and calculate the totally inclusive decay rate . 
   ( [ collz ] ) shows that the collinear limit gives a finite result . 
   thus , we conclude that , as in the case of the radiative pion decay , evaluating @xmath138 we have already summed over all the collinear degeneration subspace . as a consequence , 
   the collinear and infrared limit are disconnected . 
   + if we take the limit @xmath142 in eq . 
   ( [ collz ] ) , we obtain : @xmath143 in general , if a quantity is finite in the collinear limit , it is finite also in the infrared limit , since the collinear subspace contains the infrared one . 
   ( [ irz ] ) indicates that in the massless limit the soft photon contribution is zero . 
   indeed , it is factorized with respect to the born decay rate , which , for @xmath104 and @xmath129 , vanishes . as we have discussed in section [ anomaly ] , the presence of the anomalous term is directly connected to the emission of the photon , hence it vanishes in the infrared limit . + the virtual photon contribution vanishes in the zero mass limit , as well . 
   indeed , it is factorized respect to the born decay rate , which goes to zero as @xmath104 . 
   the virtual correction factor can produce only a logarithmic collinear singularity , not a power - like one , needed for the cancellation of the chiral suppression . 
   + we observe that taking the infrared limit @xmath144 in eq . 
   ( [ hf ] ) , gives a finite result ( indeed the cross section vanishes ) . 
   this is a consequence of the fact that the cross section ( [ hf ] ) has been calculated to the leading order in the lepton mass . from the eq . 
   ( [ rz ] ) , we see that , for @xmath145 , the infrared divergent term is given by : @xmath146 and it is proportional to the lepton mass . performing the calculation , neglecting the mass terms , as done in ref . 
   @xcite , means imposing the chirality conservation law in the @xmath1 vertex ; thus the soft photon contribution is zero and the infrared divergences disappear . to the leading order in the lepton mass 
   , we have only the anomalous term , which vanishes in the infrared limit . + 
   as done for the pion case , we now examine the composition of the degeneration ensembles for the radiative @xmath1 decay with the right - handed lepton . 
   ( [ irz ] ) indicates that , in the chiral limit , the soft photons do nt contribute to the process with the right - handed outgoing lepton , since , not carrying spin , they can not contribute to the helicity flip . 
   + as we have already said , the virtual photon contribution vanishes in the zero mass limit . in the limit @xmath120 , 
   also the diagram with the photon emitted by @xmath147 does nt contribute , since , clearly , it violates the chirality conservation in the @xmath1 vertex . 
   + the collinear degeneration arises in the massless limit . 
   the result ( [ collz ] ) shows that , in this limit , the collinear ensemble is constituted only by the states with the lepton accompanied by hard collinear photons , just as in the case of the pion decay . 
   thus for @xmath120 the infrared ensemble is empty and also the states with the antilepton accompanied by a hard collinear photon do nt contribute . 
   we have shown that the dolgov and zakharov treatment of the axial anomaly can be extended to processes characterized by a lepton which changes helicity by emitting a photon , as it was already noticed in @xcite . 
   the corresponding decay rates do nt vanish in the chiral limit , due to a term independent of the lepton mass ; we interpret the presence of this term as related to the axial anomaly . 
   this can be seen as a signal of the anomalous symmetry breaking in processes different from the usual ones , like the @xmath3 decay . 
   + we have computed the rates corresponding to the @xmath0 and @xmath1 radiative decays . 
   we have analysed their infrared and collinear limits . 
   it results essential to keep the terms of all orders in the lepton mass , since the cancellation of the infrared and collinear divergences takes place among these terms . 
   we have examined the connection between the polarization of the outgoing leptons and the application of the kln theorem . 
   + we have shown that for the helicity changing processes the cancellation of the collinear singularities occurs through a mechanism different from the usual one of real and virtual compensation . 
   the coefficients in front of the collinear terms go to zero in the chiral limit , producing the finiteness of the distribution . 
   we have found , however , a difference between the pion case and the more general case of the @xmath1 decay . 
   the former represents a particular case due to the angular momentum conservation in the pion vertex . as a consequence , 
   the virtual and real soft photon contribution are zero , even if the lepton mass is kept different from zero . 
   the inner bremsstrahlung contribution is finite both in the infrared and in the collinear limits . 
   + in the @xmath1 case , the decay rate diverges in the infrared limit , since , for @xmath140 , the soft photon contributions are not zero . 
   however in the collinear limit , the result is finite , despite the fact that the collinear degenerate states arise only in the zero lepton mass limit . in this limit 
   the virtual and real soft photon contributions do vanish . in order to make the collinear limit finite , it is therefore sufficient to sum over degenerate states made of the changing helicity lepton accompanied by a hard collinear photon . 
   the transition probability becomes finite after summing over the photon final phase space . 
   + we noticed that in the helicity changing processes the collinear limit results disconnected from the infrared one . 
   the contributions coming from the virtual photon emission and from the emission of photons by particles different from the one changing helicity , are zero . 
   + this situation is due to the fact that the born part of the process fixes the fermion chirality in the zero mass limit , while , after the photon emission , it is in a state of opposite chirality ; this reduces the collinear ensemble . 
   + we conclude that the collinear singularity cancellation mechanism for helicity changing processes is controlled by the anomalous breaking of the chiral symmetry . 
   the axial anomaly implies that the collinear limit gives a finite result , independent of the fermion mass . 
   + the extension of the above remarks to other gauge theories like qcd , is possible . 
   it could allow a more systematic and complete treatment of the infrared and collinear singularities . 
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summary: we discuss a connection between axial anomaly and polarized radiative processes . by comparison with the corresponding unpolarized cases , we consider some physical outputs for the @xmath0 and @xmath1 polarized radiative decays . 
   we analyse in detail the pattern of mass singularity cancellation . 
   uprf-00 - 10 + june 2000    axial anomaly effects in pion and @xmath1 radiative decays 1.0 cm    luca  trentadue and michela  verbeni +    .2 cm _ dipartimento di fisica , universit di parma , + infn gruppo collegato di parma , 43100 parma , italy _ +    pacs 11.30.rd , 13.20.cz , 11.15.bt