Pion-to-vacuum vector and axial vector amplitudes and weak decays of pions in a magnetic field

We propose a model-independent parametrization for the one-pion-to-vacuum matrix elements of the vector and axial vector hadronic currents in the presence of an external uniform magnetic field. It is shown that, in general, these hadronic matrix elements can be written in terms of several gauge covariant Lorentz structures and form factors. Within this framework we obtain a general expression for the weak decay $\pi^- \to l\,\bar\nu_l$ and discuss the corresponding limits of strong and weak external magnetic fields.


I. INTRODUCTION
The effect of intense magnetic fields on the properties of strongly interacting matter has gained significant interest in recent years [1][2][3]. This is mostly motivated by the realization that strong magnetic fields might play an important role in the study of the early Universe [4], in the analysis of high energy noncentral heavy ion collisions [5], and in the description of physical systems such as compact stellar objects called magnetars [6]. It is well known that magnetic fields also induce interesting phenomena like the enhancement of the QCD vacuum (the so-called "magnetic catalysis") [7], and the decrease of critical temperatures for chiral restoration and deconfinement QCD transitions [8].
In this work we concentrate on the effect of a magnetic field B on the weak pion-tolepton decay π − → lν l . In fact, the study of weak decays of hadrons in the presence of strong electromagnetic fields has a rather long history. Already in the mid 1960s, the effect of some particularly simple electromagnetic field configurations on leptonic decays of charged pions was considered [9,10]. Some years later, the decay of a neutron in the presence of a magnetic field was also studied [11,12]. In most of the existing calculations of the decay rates, however, the effect of the external field on the internal structure of the participating particles has not been taken into account. Only in recent years has the B-dependence of pion masses actively been investigated from the theoretical point of view. This has been done using approaches like e.g. chiral perturbation theory [13,14], quark effective models [15][16][17][18][19][20][21][22][23][24][25][26] and lattice QCD (LQCD) simulations [8,27,28]. In addition, some of these works [13][14][15][16][17][18][19][20][21] (see also Refs. [29,30]) considered, for the case of the neutral pion, the B-dependence of the decay constant f π 0 . For the charged pion, such an effect has been analyzed in the context of chiral perturbation theory [14], quark-antiquark effective chiral models [30] and very recently through LQCD calculations [31]. An interesting observation was made in Ref. [29], where it is claimed that, due to the explicit breaking of rotational invariance caused by the magnetic field, one can define two different decay constants, one of them associated with the direction parallel to B and another one associated with the perpendicular directions.
A further relevant statement has been pointed out in Ref. [31]. In that work it is noted that the existence of the background field opens the possibility of a nonzero pion-to-vacuum transition via the vector piece of the electroweak current, which implies the existence of a further decay constant. Taking into account this new constant, the authors of Ref. [31] obtained an expression for the decay width under the assumption that the decaying pion is at rest. However, one should note that in the case of a charged pion this condition cannot be exactly fulfilled, due to the presence of the magnetic field. Even in its lowest energy level (hence, in the lowest Landau level), the charged pion still keeps a zero point motion. In this context, the aim of the present work is twofold. Firstly, considering an external uniform magnetic field, we determine the form of the most general model-independent hadronic matrix elements, written in terms of gauge covariant Lorentz structures. In particular, we show that in the presence of a magnetic field several independent form factors can be in principle defined. Secondly, given the general form of the hadronic matrix elements, we obtain an expression for the π − → lν l decay width, taking into account the effect of the magnetic field on both pion and lepton wave functions. This paper is organized as follows. In Sec. II we discuss the structure of the pion-tovacuum matrix elements in the presence of a uniform static magnetic field from a general point of view. We start by identifying the relevant gauge covariant Lorentz structures, and then we proceed to obtain the hadronic matrix elements for neutral and charged pions.
In Sec. III we obtain an explicit expression for the π ± leptonic weak decay width. After considering the general case of an arbitrary magnetic field strength, we discuss the particular cases of both strong and weak magnetic fields. Finally, in Sec. IV we present our main conclusions. We also include Appendixes A, B, C and D to quote technical details of our calculations.

II. PION-TO-VACUUM AMPLITUDES IN THE PRESENCE OF AN EXTERNAL UNIFORM MAGNETIC FIELD: GENERAL STRUCTURE
In this section we analyze the general form of the pion-to-vacuum matrix elements of vector and axial vector quark currents. Throughout this work we use the Minkowski metric g µν = diag(1, −1, −1, −1), as well as the convention ǫ 0123 = +1 for the totally antisymmetric tensor ǫ µναβ .
Let us start by considering the hadronic matrix elements for the case of a neutral pion in the absence of the external magnetic field. The amplitudes for the vector and axial vector quark currents are where ψ(x) includes the u and d quark fields, and τ 3 is a Pauli matrix that acts on flavor space. To deal with the matrix elements in Eqs. (1) it is possible to hadronize the quark currents, i.e., to consider matrix elements of hadronic field operators carrying appropriate Lorentz indices and quantum numbers. In the low energy limit (typically, below the ρ meson threshold), the relevant hadronic field is the pion field φ π 0 (x), and (in absence of external fields) the only available vectorlike differential operator is the momentum operatorp µ = i∂ µ . Since the pion field is pseudoscalar, only the matrix element of the axial-vector hadronic current can be nonzero. In this way, one has Here, the function f (p 2 ) contains all the information of nonperturbative QCD contributions.
Using the explicit form of φ π 0 (x) and the commutation rules for the corresponding creation and annihilation operators [see Eqs. (A1) and (A2)], one immediately finds As usual, the four-momentum p µ is defined by p µ = (E π , p ), with E π = m 2 π + | p | 2 . Similar expressions can be obtained for charged pions. It can be seen that the invariance of strong interactions under P, C and T transformations implies that f (p 2 ) is a real function. In the absence of external fields, the pion decay constant is given by f π = f (m 2 π ) ≃ 92.3 MeV [32]. We turn now to the situation in which a static external electromagnetic field is present.
In this case, other tensor structures are possible. For a particle of charge Q the relevant basic tensors are the gauge covariant derivative D µ and the gauge invariant electromagnetic field strength F µν , defined as Taking them as building blocks, one can in principle obtain an infinite number of differential operators with different Lorentz tensor structures. However, for the particular case of a uniform static magnetic field B, it is well known (see e.g. Ref. [33]) that only a few independent tensors exist. Noting that F 0i = 0 and For definiteness, and without losing generality, in what follows we take B k = B δ k3 . Using the above relations it is easy to prove that one has only two independent scalars, apart from the particle electric charge Q and F µν F µν = 2B 2 . These can be taken to be In addition, it is possible to find four independent four-vectors. One possible choice is the (notice that the last of these tensors transforms in fact as an axial vector).
From the above expressions for Lorentz scalars and four-vectors, we can write now a general form for the hadronic currents we are interested in. We consider first the case of the neutral pion, for which Q = 0 and the operator D µ reduces to the usual derivative ∂ µ .
Taking into account once again the intrinsic parity of the pion field, one has wheref are complex functions of the magnetic field and the scalar differential operators ∂ 2 || and ∂ 2 ⊥ . The hadronic matrix elements can be readily obtained using Eqs. (A1) and (A2). We find it convenient to define some linear combinations of the respective Lorentz components, namely and with ǫ = ±. Here, f Notice that, although it is not indicated explicitly, the pion mass m π 0 is a function of the magnetic field B. As in the B = 0 case, it is important to consider the constraints on the form factors arising from the discrete symmetries of the interaction Lagrangian in the presence of the magnetic field. This is discussed in some detail in Appendix B, where it is shown that these symmetries lead to f (A2) π 0 = 0 while the remaining form factors turn out to be real. In this way, we conclude that the most general forms of the vector and axial vector pion-to-vacuum matrix elements, in the presence of an external constant and homogenous magnetic field along the 3-axis, are and where all form factors are real. The results in Eqs. (17) and (18) are in agreement with the observation made in Ref. [29] that, due to the explicit breaking of rotational invariance caused by the magnetic field, one can define for the neutral meson two different form factors related to the axial current. One of them can be associated with the direction parallel to B, and the other one with the perpendicular directions. In addition, according to Eq. (15) we find that a further form factor related to the vector current can be defined as well.

||
and D 2 ⊥ , whilep = (ℓ, p 2 , p 3 ), ℓ being a non-negative integer (Landau index), denotes a set of quantum numbers that characterize the charged pion state in the presence of the magnetic field (see Appendix A.2 for details). The sign s in the term carryingf (A2) π σ has been conventionally introduced for later convenience. Notice that in Eqs. (19) and (20) and where D ǫ = D 0 + ǫ D 3 and D ǫ ⊥ = D 1 + iǫD 2 . We use here the notationp = (E π σ ,p), with E π σ = m 2 π σ + (2ℓ + 1)B e + (p 3 ) 2 (in fact, one has E π + = E π − , the charge index σ being kept in order to distinguish E π ± from E π 0 ). The functions F ± p (x) are eigenfunctions of the charged pion Klein-Gordon equation in the presence of the magnetic field, for our choices of gauge and magnetic field direction. Their explicit expressions are given in Eq. (A12).
Notice that on the rhs of Eqs. (21) and (22) we have differential operators acting on these functions. Using the relations and wherep ± 1 ≡ (E π σ ,p ± 1), withp ± 1 = (ℓ ± 1, p 2 , p 3 ), we finally obtain and In the above expressions, the form factors f arise from the action of the opera- on the functions F s p (x). These operators are, in general, gauge-dependent functions of the operators D 2 and D 2 ⊥ . However, for an on-energy-shell pion, taking into account Eqs. (25) and (26) it can be seen that the resulting form factors f π σ and f (Ai) π σ turn out to be gauge-independent functions of the pion mass, m π σ ; the third component of the momentum, p 3 ; the Landau index, ℓ; and the magnetic field, B (both explicitly and through the charged pion mass). In the so-called symmetric gauge, Eqs. (25) and (26) should be modified by changing the functions F σ p (x) by the corresponding ones in that gauge, which involve associated Laguerre polynomials [34]. On the other hand, notice that the eigenvalues in Eqs. (25) and (26)  π σ and f (Ai) π σ . Indeed, as shown in Appendix B, they have to be real and independent of the sign of the pion charge. Finally, let us point out that the form factors f (A1) π σ appearing in Eqs. (29) and (30) are the same. In fact, this is a consequence of our choice of ordering in Eq. (20).
Had we put the operator functionsf In any case, it should be stressed that this would also imply a redefinition of f (A2) π σ and f (A3) π σ , in such a way that the contribution of H σ, ǫ ⊥,A (x,p) to any physical quantity remains unchanged.

A. Basic relations and decay kinematics
Let us analyze the decay width for the process π − → lν l , with l = µ, e, in the presence of the external magnetic field. Following the notation introduced in the previous section, the initial charged pion state is determined by the quantum numbersp = (ℓ, p 2 , p 3 ), the associated energy being E π − = m 2 π − + (2ℓ + 1)B e + (p 3 ) 2 . The quantum numbers corresponding to the outgoing lepton state are taken to beq = (n, q 2 , q 3 ), together with a polarization index r l . In this case the energy is given by E l = m 2 l + 2nB e + (q 3 ) 2 (see Appendix A.3). Finally, being electrically neutral, the outgoing antineutrino is taken to be in a state of momentum k = (k 1 , k 2 , k 3 ) and polarization rν l , with energy Eν l = ( On general grounds, the decay width for the process is given by where (S − 1) f i is the relevant S-matrix element between the initial and final states, and the particle number n π − associated with the initial π − state is given in Eq. (A17). Thus, one has where L W is the usual V − A four-fermion effective weak interaction Langrangian integrated over space-time, while T and S are the time interval and the surface on the x 2 x 3 plane in which the interaction is active. At the end of the calculation, the limit S, T → ∞ will be taken. Let us recall that, according to our gauge choice, the motion in the x 1 axis is bounded. Using the notation introduced in Appendix A, the matrix element in Eq. (32) is given by where H −,µ L (x,p) stands for the matrix element of the hadronic current [see Eqs. (19) and (20)], Thus, from Eqs. (A6), (A20) and (A25), the decay amplitude is given by where k µ = (Eν l , k) andq = (E l ,q). Given our gauge choice for the external magnetic field, the electromagnetic potential depends only on the x 1 coordinate. Consequently, the momentum is conserved in the 0, 2 and 3 directions, leading to the corresponding delta functions when the space-time integration in Eq. (35) is performed. We obtain where the M −,µ λ functions will be explicitly given below. In this way, the amplitude can be written as where Thus, replacing in Eq. (32) and taking the limit of infinite space-time volume, the decay width is given by where As customary, when putting the expression on the rhs of Eq. (37) into Eq. (32) we have Now, as it is usually done, we concentrate on the situation in which the decaying pion is in the state of lowest energy. This corresponds to the case ℓ = 0 and p 3 = 0. Moreover, as will be shown below, neither the pion energy nor the decay width depend on the value of p 2 . The expression for the decay width can be worked out, leading to where we have used the definitions In addition, in the amplitude M π − → lν l one should take q 3 = −k 3 .

B. Evaluation of the weak decay amplitude
As a second step we concentrate on the evaluation of the sum of the squared amplitudes for all possible lepton polarizations, M π − → lν l 2 . Using the properties of electron and neutrino spinors in Eqs. (A7) and (A23), a somewhat tedious but straightforward calculation leads where we have used the definitions To proceed we need to calculate M −, ǫ ,λ and M −, ǫ ⊥,λ , which are related to the matrix elements of the hadronic current through Eq. (36). Defining from the expressions in Eqs. (27)(28)(29)(30) we get Now the integral over space-time variables in Eq. (36) can be carried out as described in Appendix C. In this way one gets where the functions G −,λ ℓ,n (k ⊥ ) are explicitly given in Eq. (C9). Notice that M −, ǫ ,λ and M −, ǫ ⊥,λ depend on p 2 and q 2 only through a global common phase, therefore the decay width turns out to be p 2 and q 2 independent.
Let us consider once again the case in which the decaying pion is in the state of lowest energy. Taking ℓ = 0, from Eq. (C9) one gets The result obtained for the sum of the squared amplitudes, taking ℓ = 0 and p 3 = 0, can be replaced in the expression for the partial decay width Γ − l (B), Eq. (41). Notice that for a pion in the lowest energy state the "perpendicular" amplitudes M −, ǫ ⊥,λ vanish for ǫ = −1. Thus, from Eq. (49), it is seen that the partial decay width will depend on three form factor combinations, namely We recall that the form factors are in general functions of ℓ, p 3 and B (to be evaluated in this case at ℓ = 0, p 3 = 0). In this way, after some algebra one arrives at where we have introduced the dimensionless variable x = k 2 ⊥ /(2B e ) and used the definitions and In addition, from Eq. (44) one has while n max is given by Eq. (42). Notice that for ℓ = 0, p 3 = 0, the π − energy is given by It is worth remarking that the above expression for Γ − l (B) corresponds to the case B > 0, s = σ = −, in Eqs. (27)(28)(29)(30). For B < 0, s = −σ = +, we obtain an expression for Γ − l (B) similar to that in Eq. (52), where in the function A π − has to be an odd function of B, it is seen that the partial decay width is invariant under the exchange B → −B, as expected.

C. Large magnetic field limit
It is interesting to study the case of a large external magnetic field. As stated, since the pion is built with charged quarks, the pion mass will depend in general on the magnetic field. Now, if the mass growth is relatively mild, for large magnetic fields one should get B e > m 2 π − m 2 l . In fact, this is what is obtained from lattice QCD calculations [31] as well as from effective approaches like the Nambu-Jona-Lasinio model [26], for values of B e say 0.05 GeV 2 . According to Eq. (42), this implies n max = 0; hence the outgoing muon or electron (assuming that the energy is below the τ production threshold) has to lie in its lowest Landau level n = 0. In this case the expression for the partial width simplifies to where Eν l = E π − − E l = k3 2 + k 2 ⊥ and E l = k3 2 + m 2 l , with A further simplification can be obtained in the case where the squared lepton mass can be neglected in comparison with B e (or, equivalently, in comparison with E 2 π − , which is expected to grow approximately as B e ). Setting m l = 0, one has E l =k 3 , and the integral over k ⊥ extends up to E π − . Thus, the decay width is given by where, according to the definitions in Eq. (51), It is worth stressing that the decay width does not vanish in the limit m l = 0; i.e. it does not show the helicity suppression found in the B = 0 case. In fact, it turns out to grow with the magnetic field as B 2 e /E π − , with some suppression due to the factor in square brackets. Moreover, it is seen that the contributions of the usual pion decay form factor, f (A1) π − , get canceled in Eq. (59); hence the decay width is proportional to a combination of form factors that do not contribute to the hadronic amplitude in the absence of the external magnetic field. Clearly, the relevance of Eq. (58) depends on whether these form factors are nonnegligible for magnetic fields that are much larger than the lepton mass squared. While this is likely to happen for the π − decay to eν e , in the case of the muon (and of course, the tau) the situation is less clear, and the corrections arising from a finite lepton mass should be taken into account. Interestingly, it is possible to obtain relatively simple expressions for the π − → lν l decay width at leading order in the ratio m l /E π − . From Eq. (56) one gets where with α = E 2 π − /2B e and I(α) = It is also interesting at this point to compare our result in Eq. (56) with the expression quoted in Eq. (5) of Ref. [31], which also corresponds to the limit of a large external magnetic field. The authors of that work make some approximations for the motion of a charged pion in the presence of the magnetic field, concluding that only one of the two possible antineutrino polarizations can contribute to the decay amplitude. Moreover, based on considerations of angular momentum conservation, they assume that the antineutrino momentum in the perpendicular plane k ⊥ vanishes. In fact, it can be seen that if one imposes such a condition in Eq. (56), the result quoted in Ref. [31] can be recovered. However, we find that if one takes full account of the effect of the magnetic field on charged pion wave functions, conservation laws do not imply k ⊥ = 0; therefore one should integrate over all possible values of the antineutrino momentum, as in Eq. (56). Another main difference between our work and the analysis in Ref. [31] is that our calculations include a perpendicular piece of the hadronic amplitude (related to c π − ), which arises due to the presence of a π − zero point motion in the perpendicular plane, even in the ℓ = 0 state.
The study of angular momentum in the presence of magnetic fields has been addressed in the recent literature (see e.g. Ref. [34]) and deserves some discussion. As suggested in Ref. [34], the consequences of the axial symmetry of the problem, as well as the physical meaning of angular momenta, can be better understood if one works in the symmetric gauge.
Having this in mind, we have rederived the π − → lν l decay width in this gauge, considering for simplicity the case of a charged pion in its lowest energy state and a charged lepton in the lowest Landau level. For the spatial wave functions of the pion and charged lepton states we have used the functions given e.g. in Eq. (17) of Ref. [34] (see also Ref. [35]). These functions are eigenstates of the z component of the orbital angular momentum operator, L 3 , with eigenvalues say M p and M q for the pion and the charged lepton, respectively. If the particles are in the lowest Landau level, it is found that M p and M q can take any integer value ≤ 0. In order to deal with right-handed antineutrino states we have used spinors that are eigenstates of the z component of the total angular momentum, J 3 , with eigenvalue j 3 .
Replacing in the corresponding weak decay matrix element and performing the integral over space-time coordinates we obtain where M is a function that depends on the pion decay form factors f (V,A i ) π − and on particle quantum numbers. Hence, in this gauge it is clearly seen that angular momentum conservation leads to the selection rule M p = M q − 1/2 + j 3 , without requiring k ⊥ = | k ⊥ | = 0.
Using the explicit form of M and taking p 3 = 0, we have then calculated the sum/integral over all allowed outgoing states. The outcome (which, as expected, does not depend on M p ), leads to an expression for the decay width that coincides exactly with the one quoted in Eq. (56), confirming the gauge independence of our result. It is also worth mentioning that, in addition to the usual "canonical" angular momentum, one can define "mechanical" angular momenta replacing particle momenta P µ by P µ mech ≡ P µ − QA µ (a detailed comparison of both quantities for various gauge choices is given in Ref. [34]). Interestingly, for the π − → lν l decay in the limit of large B (when only the n = 0 state contributes) it is found that the third components of mechanical angular momenta of incoming and outgoing states only coincide when the transverse antineutrino momentum vanishes, i.e., for |k ⊥ | = 0.

D. B → 0 limit
In the B → 0 limit our expressions should reduce to the well-known result for the π − → lν l decay width obtained in the absence of external fields. For simplicity we consider the decay of a π − in its fundamental state, taking ℓ = 0 and p 2 = p 3 = 0. Then, in the B → 0 limit the decay width should reduce to that of a π − at rest, namely where p µ = (m π , 0); q µ = (E l , q); and k µ = (Eν l , k) stand for the pion, lepton and antineutrino four-momenta, respectively. It can be checked that for a given value of the lepton mass the numerical results for Eqs. (66) and (52), in the limit B e → 0, n max → ∞, , which is usually called the "magnetic length". In the B → 0 limit these have to be much larger than the time T and the distance 2L along which the interaction is active, i.e.
In addition, we assume that T and L are large enough so that the product of any momentum of the system times L or any energy of the system times T turns out to be much larger than 1.
In particular, We recall that the lepton energy is given by E l = m 2 l + 2nB e + (q 3 ) 2 , which implies that the magnetic field will contribute significantly to the energy only for very large values of n. In fact, from this expression we can infer that the term 2nB e will lead to the (q ⊥ ) 2 contribution to the energy in the B → 0 limit. As B decreases, the contributing leptonic states will have increasingly larger values of n, in such a way that nB remains finite.
From Eqs. (8), (19) and (20), we observe that in the B → 0 limit the only nonvanishing pion decay form factor is f (A1) π − . Thus, from Eqs. (45-48) we have where the functions I −,± ℓ,n are given by In fact, the latter correspond to the functions I −,λ ℓ,n (k ⊥ , p 2 , q 2 ) defined in Eq. (C3), taking p 2 = 0 and restricting the integral over x 1 to the interval (−L, L) in order to take into account the conditions in Eqs. (67) and (68).
Owing to the restriction of the integration interval in Eq. (70), in the B → 0 limit one has √ 2B e x 1 → 0. Thus, the factors N ℓ D ℓ ( √ 2B e x 1 ), where ℓ = 0 or 1, satisfy and the terms with I −,− 1,n in Eq. (69) can be neglected. We obtain The detailed calculation of the functions I −,λ 0,n and their contributions to the expression in Eq. (73) is given in Appendix D. We finally arrive at whereq n = 2nB e − (q 2 ) 2 . We recall that n has to be a large number, in such a way that nB e is kept finite for small B.
We introduce now a new variable q 1 , in the following way: Next, let us consider the decay width in Eq. (39). We need to treat with some care the pion density n π − , which appears in the definition of the width in Eq. (31). In fact, for a finite space length 2L, taking into account the approximation in Eq. (71), the pion density will be given by This result can be understood by writing the pion density in the form n π − = V 2E π − (4πB e ) 1/2 , where V is the volume in which the interaction occurs. It is seen that Finally, we can perform the integral over κ . The delta functions fix q 1 = ± κ − (q 2 ) 2 , leading to the expected result κ = 2nB e = q 2 ⊥ . Identifying f (A1) π − with f π in the B → 0 limit, we arrive at the expression in Eq. (65).

IV. CONCLUSIONS
In this work we present a general method to parametrize the one-pion-to-vacuum matrix elements of the vector and axial vector hadronic currents in the presence of an external uniform static magnetic field B. Choosing this field to be orientated along the 3-axis, we show that for the case of the neutral pion the matrix elements of the parallel (0-and 3-) components of the vector current can be expressed in terms of one single real form factor, can be defined.
Alternatively, the latter can be written in terms of a parallel and a perpendicular form factor, in consistency with the result obtained in Ref. [29]. In the case of the charged pion, the situation is similar in what concerns the vector current. Namely, the matrix elements of the parallel components can be expressed in terms of a real (in general, nonvanishing) form factor f (V ) π σ , common to both π + and π − , while the perpendicular (i.e. 1-and 2-) components vanish. This is in agreement with the statement made in Ref. [31]. On the other hand, we find that three form factors, f π ± , are in general required to parametrize the matrix elements of the axial vector hadronic current. Once again, the three of them are real, and they are equal for both pion charges. The matrix elements of the charged pions in Eqs. (27)(28)(29)(30) can be viewed as a proper generalization of the corresponding expressions given in Ref. [31]. We have included here all possible gauge covariant structures, taking fully into account the effect of the magnetic field on the charged pion wave functions. It should be noticed that in the particular case of a charged pion lying in the lowest Landau level, only two combinations of these three form factors contribute to the decay width.
Using the above results we introduce a general, model-independent framework to study the weak decay π − → lν l in the presence of an arbitrary large external magnetic field. For the case in which the decaying pion lies in its state of minimum energy (i.e. in the lowest Landau level, with zero momentum along the 3-direction), we obtain an explicit expression for the π − → lν l decay width. The limits of this expression for the cases of strong and weak magnetic fields are also studied, checking that in the limit of B = 0 it reduces to the usual result. It is interesting to note that the expression obtained in the limit of large magnetic field, Eq. (56), is valid in most cases of physical interest. Namely, we estimate its range of validity to be 0.05 GeV 2 < eB < m 2 τ ≈ 3 GeV 2 . It is seen that our result shows some differences with the one given in Ref. [31], also obtained in the limit of large B. We understand that the discrepancies arise from some approximations made in Ref. [31] concerning the motion of a charged pion in the presence of the magnetic field. It is also worth noticing that the decay width does not vanish in the limit m l = 0; i.e. it does not show the helicity suppression found in the absence of the external magnetic field.
We finally mention that an obvious application of our work would be to study how weak decay rates of charged pions get modified due to the presence of the magnetic field. To reach this goal, however, the behavior of the decay form factors as functions of the magnetic field should be determined. This would require either the use of LQCD simulations, as proposed in Ref. [31], or to rely on some hadronic effective model. We expect to report soon on such a calculation in the framework of Nambu-Jona-Lasinio-like models. where The neutrino field can be written as where Here k = (E ν l , k), with E ν l = | k|, while u ν l ( k, r) and v ν l ( k, r) are the usual Dirac spinors with polarization states r = 1 or 2. They satisfy Note that we are assuming that neutrinos are massless. The corresponding creation and annihilation operators satisfy the relations

Charged pion field
The charged pion fields can be written as where Q π σ = σ|e| is the pion charge with σ = ±, s = sign(Q π σ B) and B e = |Q π σ B|,. Note that if B > 0 then s = σ. As defined in the main text,p = (p 0 ,p), wherep = (ℓ, p 2 , p 3 ) and the pion energy is given by p 0 = E π σ = m 2 π σ + (2ℓ + 1)B e + (p 3 ) 2 . The functions F s p (x) are solutions of the eigenvalue equation where D µ = ∂ µ + isB e x 1 δ µ2 . Their explicit form is given by where N ℓ = (4πB e ) 1/4 / √ ℓ! , ρ s = √ 2B e x 1 − s 2/B e p 2 and D ℓ (x) are cylindrical parabolic functions. The latter are defined as H ℓ (x) being the Hermite polynomials, with the standard convention H −1 (x) = 0. It can be seen that the functions F s p (x) satisfy the orthogonality relations ∞ ℓ=0 dp 0 dp 2 dp 3 In addition, the creation and annihilation operators in Eq. (A10) satisfy the commutation Note that with these conventions the operators a σ (p) and a −σ (p) turn out to have different dimensions from the creation and annihilation operators that are usually defined in the absence of the external magnetic field [and also from those corresponding to the π 0 field; see It is also useful to calculate the particle number associated with the state |π σ (p) = a † (p)|0 in a volume V . Given our choice of the Landau gauge, in this case it is convenient to consider an infinite cylinder of section S lying along the x 1 axis. We obtain where the current is defined in a similar way to that corresponding to the neutral pion; see Eq. (A4). Note that we are normalizing to 4πE particles per unit surface, which differs from the usual normalization ρ = n/V = 2E.

Charged lepton field
Assuming the same conventions for the magnetic field and considering the Landau gauge, for the charged lepton fields we have Here, s = sign(Q l B) with Q l = −|e|. As indicated in the main text,q = (n, q 2 , q 3 ) and E l = m 2 l + 2nB e + (q 3 ) 2 , with B e = |Q l B|. The creation and annihilation operators satisfy In Eq. (A18) we have also used the definitions whereq = (q 0 ,q), with q 0 = E l . The spinors u s l and v −s l are given in the Weyl basis by where φ (1) † = −φ (2) † = (1, 0) and φ (2) † =φ (1) † = (0, 1). They satisfy the relations r=1,2 u s l (q, r)ū s l (q, r) = / q s + m l , whereq µ s = (E l , 0, −s √ 2nB e , q 3 ). In Eq. (A20), E s q (x) andẼ −s q (x) are Ritus functions that satisfy the eigenvalue equation where / D = / ∂ − isB e x 1 γ 2 . They can be written as where ∆ ± = (1 ± iγ 1 γ 2 )/2, and F s q (x) being the function defined in Eq. (A12). Here the integer index n sλ is related to the quantum number n by It can be seen that the spinors in Eq. (A20) satisfy where Following the same steps that led to Eq. (A17), it is found that the number of particles in an infinite cylinder of section S lying along the x 1 axis is given by n l = 2E l 2πS.

Appendix B: Discrete symmetries
In the Landau gauge, the electromagnetic interaction term between the light quarks and the external field (chosen to be orientated along the z axis) is given by where the sum extends over f = u, d, and Q f are the corresponding electric charges. It is easy to see that the action is separately invariant under P, CT and PCT , where P, C and T stand for parity, charge conjugation and time reversal transformations acting on the quark fields. Moreover, it can be seen that the Lagrangian density in Eq. (B1) is invariant under the transformation CR 1 , where R 1 is a spatial rotation by angle π about the x axis (i.e., a rotation that inverts the orientation of the magnetic field B).
The existence of these symmetries imposes constraints on the form factors in the pionto-vacuum hadronic matrix elements discussed in our work. As in the case of no external field, parity is responsible for selecting which Lorentz structures in Eq. (8) contribute to the matrix elements of the vector and axial-vector currents, as quoted in Eqs. (9,10,19,20).
Moreover, it is possible to use CT and CR 1 symmetries to show that the form factors are real and equal for both charged pions.
It is worth noticing that N ℓ D ℓ ( √ 2B e x 1 ) comes form the pion field, N n −λ D n −λ ( √ 2B e x 1 + (2/B e ) q 2 ) comes from the lepton field, and the exponential comes from the antineutrino field.