Neutron beta decay in effective field theory

Radiative corrections to the lifetime and angular correlation coefficients of neutron beta-decay are evaluated in effecitive field theory. We also evaluate the lowest order nucleon recoil corrections, including weak-magnetism. Our results agree with those of the long-range and model-independent part of previous calculations. In an effective theory the model-dependent radiative corrections are replaced by well-defined low-energy constants. The effective field theory allows a systematic evaluation of higher order corrections to our results to the extent that the relevant low-energy constants are known.


Introduction
The radiative corrections for beta decay have been intensively investigated by a number of authors, and the prime issue for such studies has been to deduce the value of the Cabbibo-Kobayashi-Maskawa (CKM) matrix element V ud from nuclear beta-decay data. An accurate value for V ud is important for testing the unitarity of the CKM matrix. The most precise values of V ud have been obtained from the accurate data of super-allowed 0 + → 0 + nuclear beta-decays [1]. Neutron beta-decay measurements provide an alternative method of determining V ud , a method which does not depend on the accuracy of nuclear models. Neutron beta-decay experiments also provide the most precise determination of the axial-vector coupling constant, g A , which plays an important role in hadronic weak-interaction reactions including many astrophysical processes. Theoretically, pion beta-decay can also be used for determining V ud . Unfortunately, however, the currently available experimental data on pion beta-decay are not accurate enough to allow us to take full advantage of this merit, see e.g. Cirigliano et al. [2].
To extract an accurate value of V ud from neutron decay data, the theoretical expression for the neutron decay rate including radiative corrections must be known with sufficient accuracy. The usual convention is to decompose radiative corrections of order α into two parts, the "outer" and the "inner" corrections [3,4,5]. The "outer" correction is a universal function of the electron energy, independent of the details of the strong interactions. The "inner" correction stems from short-range terms and hadronic structure effects. This hadronic structure dependence (and additional nuclear structure dependence in the case of nuclear beta-decay) causes uncertainties in extracting fundamental quantities like V ud from experimental data 1 .
In this communication we present the first calculation of radiative corrections to neutron beta-decay based on a low-energy effective field theory (EFT). EFT provides symmetry constraints required by the underlying theory and a systematic expansion scheme for the evaluation of the hadron current. As suggested by Weinberg [7], low-energy hadronic physics can be described by an effective field theory of QCD known as "chiral perturbation theory" (χPT). The effective chiral Lagrangian, L χ , reflects the symmetries and the pattern of symmetry breaking of the underlying QCD. For massless quarks the QCD Lagrangian is chirally symmetric, but chiral symmetry is spontaneously broken generating the pions as massless Goldstone bosons. Since the u and d quark masses are very small compared with the QCD scale Λ QCD , and since the finite pion mass generated by the quark masses is small compared to a typical strong interaction scale, it is reasonable to treat the explicit chiral symmetry breaking terms as small perturbations. L χ is expanded in powers of Q/Λ χ ≪ 1 where Q denotes the typical four-momentum of the process in question or the pion mass, m π , which represents the small explicit chiral symmetry breaking scale. The chiral scale, Λ χ ≃ 4πf π ≃ 1 GeV (f π = 92.4 MeV is the pion decay constant), is associated with the "high-energy" processes that have been integrated out in arriving at L χ and with pion loops. The parameters appearing in L χ , called the low-energy constants (LEC's), effectively subsume the high-energy physics that has been integrated out. In principle, these LEC's could be determined from the underlying theory, but in practice the LEC's are determined phenomenologically from experimental data. Once the LEC's are determined from appropriate empirical data, then L χ represents a complete Lagrangian up to a specified chiral order. Furthermore, starting from L χ , one can develop, for the amplitude of a given process, a well-defined perturbation scheme by organizing the relevant Feynman diagrams according to powers in Q/Λ χ . If all the Feynman diagrams up to a given power, ν, in Q/Λ χ are taken into account, then the results depend only on the LEC's up to this order, with the contributions of higher order terms suppressed by an extra power of Q/Λ χ .
Over the past decade χPT has been successfully applied to many processes; for reviews, see, e.g., Refs. [8,9]. Chiral Lagrangians including the photon field have been developed and applied to, e.g., pion-nucleon scattering, see Refs. [10,11]. Our present calculation of the radiative corrections to neutron beta-decay is an EFT based on the spirit of the chiral Lagrangian approach. Thus we write down an effective Lagrangian, appropriate to neutron beta-decay, obeying chiral symmetry and involving a minimum set of LEC's and use the Lagrangian to estimate the relevant amplitudes to leading, next-to-leading, and next-to-next-to leading orders (LO, NLO, N 2 LO) in the Q/Λ χ expansion. In fact, since the typical energy transfer of the reaction is much smaller than the pion mass, the "Q/Λ χ expansion" here has a special feature to be explained in the next section.
The results of our EFT calculation confirm the expression for the model-independent universal function derived by Sirlin [3]. Furthermore, our calculation provides expressions for corrections of order α to the angular correlation coefficients in neutron beta-decay. We will show that the short-distance phenomena including the model-dependent hadronic radiative corrections can be condensed into two LEC's, one relevant to the Fermi constant G F and the other to the axial coupling constant g A . The values of these LEC's need to be determined by experiments. In order to have crude order-of-magnitude estimates of our LEC's, we also compare our results with the "inner" radiative corrections obtained in the standard calculations. Furthermore, we shall argue that, provided the LEC's involved in our calculation are of a "natural" size, the neutron-decay rate and angular correlation coefficients calculated here are expected to have a precision better than 10 −3 .

Effective theory for neutron beta-decay
Since neutron beta-decay is a low energy process, it is natural to use here heavy-baryon chiral perturbation theory (HBχPT), see, e.g., Refs. [8,9]. In fact the appropriate amplitude, however without radiative corrections, can be obtained from HBχPT calculations of muon capture on a proton, µ + p → n + ν, which have been carried out including N 2 LO correction terms [12,13,14,15]. Neutron beta-decay, however, has a feature not shared by muon capture, namely several different expansion scales. In particular, the maximum energy release, ∆M = m n − m p − m e = 0.782 MeV, is very small compared to the pion mass m π and the nucleon mass m N = (m p + m n )/2. Correspondingly, if we denote bȳ Q the typical four-momentum transfer of the process,Q ∼ ∆M is also very small. We therefore introduce here a particular "Q/Λ χ " expansion in which Q, unlike most HBχPT calculations, only representsQ. The chiral symmetry breaking scale, m π /Λ χ ≃ 0.14, will be accounted for separately. The nucleon recoil terms are governed by the scalē Q/m N ≃ 0.8 × 10 −3 , and they are NLO corrections to the LO expression. The scalē Q/m N ≃Q/Λ χ is numerically of the same magnitude as α/(2π) ∼ 10 −3 , governing the radiative corrections, which are our primary interest (α is the fine structure constant). Therefore, for our present purposes, we consider the α/(2π) andQ/m N corrections to be of the same order.
The relevant effective Lagrangian, L β , for the neutron decay process reads where L eνγ is the lepton-photon Lagrangian, L N N γ describes the heavy nucleon interacting with a photon, and L eνN N gives the effective V −A interaction between the lepton and the heavy nucleon current. Since the pion mass is much heavier than the typical momentum scale of the reaction,Q ≪ m π , we suppress the pion fields of the chiral Lagrangian, L χ , and in L β we have retained only the interactions between the heavy nucleon field, lepton current, and photons. Later in the text, we will discuss the role of the pions in the present calculation. Thus one has, through LO and NLO, where F µν = ∂ µ A ν − ∂ ν A µ and D µ is the covariant derivative of QED. The ξ A is the gauge parameter and we choose the Feynman gauge ξ A = 1. The v µ is the velocity vector of the heavy-baryon formalism, which we take as v µ = (1, 0), and S µ is the nucleon spin operator 2S µ = (0, σ). The isovector magnetic moment in the NLO Lagrangian is The quantities e 1 , e 2 , e V and e A are defined as the LEC's of the theory. The LEC's e 1 and e 2 are the α-order corrections related to the wave-function normalization factors of the electron and proton, respectively. The LEC's e V and e A are the α-order corrections to the Fermi and Gamow-Teller amplitudes, where we have factored out the common coefficient ( Those LEC's are used to absorb infinities coming from the virtual photonloops and take into account short-range radiative effects. We remark that some of those LEC's contain contributions from, e.g., g i 's for the one nucleon sector without leptons [10] and X i 's for the meson sector with leptons [16] in χPT 2 . As is conventional, the parameters of the initial Lagrangian, e.g. the Fermi constant • GF and the axial coupling constant • g A , 2 Unfortunately the connection between the LEC's e V and e A and the g i and X i of Refs. [10,16] is not are taken as the coupling constants in the absence of radiative corrections and in the chiral limit, m π = 0. Thus in particular, we assume that the Fermi constant, • GF → G F = 1.166 ×10 −5 GeV −2 , as determined from muon-decay. As we discuss in the next paragraph, higher order hadronic corrections, i.e., pion-loops, renormalize these "bare" couplings to their physical values in the absence of electromagnetic effects, e.g.
• g A → g A . Furthermore, radiative effects give rise to additional corrections to the coupling constants, G F and g A which depend on the process being considered. These radiative corrections will be displayed explicitly in the present work.
We calculate the Feynman diagrams shown in Fig. 1, where the vertices are determined by the Lagrangian, L β , given above. Several remarks are in order on the diagrams in Fig.  1. Consider first diagram (a), which does not involve radiative corrections. Diagram (a) is a tree-diagram for the LO and NLO amplitudes. As regards the LO contribution, one may wonder why we do not consider here the pion-pole diagram (not shown). The pion-pole diagram, which is responsible for the induced pseudoscalar coupling, formally belongs to LO and hence would be included in normal circumstances. However, the extremely small momentum transfer involved in neutron beta-decay (Q ≪ m π ) drastically suppresses straightforward. The g i and X i originate in Lagrangians involving only subsets of the degrees of freedom considered here and thus generate radiative corrections to only particular vertices in the diagrams for neutron beta decay. Their contribution can be absorbed in e V and e A , but e V , e A would also contain contributions from the LEC's of a yet-to-be-calculated Lagrangian involving the nucleon, lepton current, and photons simultaneously. the pion-pole diagram contribution. Due to the presence of the pion propagator and a momentum of orderQ at each vertex, the pion-pole diagram scales like (Q/m π ) 2 ≃ 3×10 −5 relative to the dominant LO terms. The accuracy of our present treatment does not warrant the inclusion of this tiny pion-pole contribution, and we will not consider it in the main body of our calculation. In the concluding section, however, we will briefly discuss the LO pion-pole term and its radiative corrections. Diagram (a) in Fig. 1 also includes the NLO vertex coming from the nucleon recoil terms ∝Q/m N featuring in Eq. (4). Since we are treating theQ/m N and α/(2π) corrections as contributions of the same order, we will discuss these recoil terms later in the text; however, in evaluating radiative corrections, we need not consider the recoil terms since these corrections would be of higher order ∼ α/(2π) × µ VQ /(2m N ) ∼ 10 −6 . At order N 2 LO there occur two kinds of contributions. Higher order recoil corrections scale as (Q/m N ) 2 ≃ 10 −6 and therefore can be neglected. The remaining N 2 LO terms (diagrams not shown) come from pion-loops and the corresponding hadronic LEC's which would appear in HBχPT Lagrangian at this order, see e.g. Refs. [8,9]. The pion-loop diagrams which generate terms proportional toQ 2 i.e., terms representing the hadronic vertex form factor effects, can be neglected, since their contributions are suppressed by a factor of (Q/Λ χ ) 2 ≃ 10 −6 relative to the dominant LO terms. The remaining contributions of the pion-loops, which contain terms proportional to (m π /Λ χ ) 2 , renormalize the bare quantities such as the "bare" axial vector coupling constant • g A . These (m π /Λ χ ) 2 terms and the corresponding hadronic LEC's are absorbed into the renormalized g A so that to N 2 LO order, , see e.g. Eq. (4.50) in Ref. [8] or Eq. (50) in Ref. [12]. Radiative corrections to the pion loop diagrams are suppressed by a scale (m π /Λ χ ) 2 ≃ 2 × 10 −2 relative to the leading radiative corrections, and therefore their contributions can be ignored in the present calculation.
The above discussion indicates that, to the accuracy in question, we need only consider radiative corrections of the following type. Of the contributions topologically represented by diagram (a), consider those involving the LO vertex and evaluate all possible radiative corrections applied to these LO diagrams. Diagrams (b), (d), (f) in Fig. 1 are onephoton loop corrections for the electron propagator, the nucleon propagator, and the four-point vertex function, respectively. Meanwhile, diagrams (c), (e) and (g) represent the contributions of the counter terms, the e 1 , e 2 , e V and e A terms, in the Lagrangian. These LEC's remove the ultraviolet divergence arising from the loop diagrams (b), (d) and (f). As is well known, the infrared divergences contained in diagrams (b), (d), (f) should be canceled by the infrared divergences in the bremsstrahlung diagrams (h) and (i) 3 , and we have confirmed this cancellation explicitly.

The correlation coefficients and the decay rate from EFT.
A general expression for the differential neutron decay rate dΓ is well known [18] for a case wherein only the neutron is polarized, and in which the nucleon recoil and radiative corrections are ignored: Here E e and p e (E ν and p ν ) are the electron (neutrino) energy and momentum,n is the neutron spin polarization vector, β = p e /E e , and a, b, A, B, D are the correlation coefficients. If we calculate diagram (a) in Fig.1 in the LO approximation, and if we neglect the nucleon recoil terms in the phase space factor, then our calculation reproduces Eq. (5), and furthermore we recover the standard lowest order expressions for the correlation coefficients as given in [18]: where g A is the physical axial coupling constant. The coefficient b in Eq. (5), which reflects the presence of scalar and tensor weak couplings, vanishes in our LO calculation, since our Lagrangian only contains the standard vector and axial vector weak interaction. The parameter D in Eq. (5) is related to time-odd correlations and hence it also should vanish in the LO calculation since our Lagrangian is T invariant. However, "induced" D terms can appear at higher orders. For instance, interference between the weak magnetism and the radiative corrections would generate a D term of order 10 −5 [19].
As we proceed to include the higher order radiative diagrams generated by the Lagrangian of Eq. (1), we encounter infinities coming from the photon-loop diagrams in Fig. 1. In order to eliminate these infinities, we need to introduce counter terms with the corresponding LEC's in our Lagrangian. We renormalize these LEC's in the usual effective field theoretical method based on the dimensional regularization of loop integrals [8,9]. The finite LEC's renormalized at the scale µ are given by: This renormalization is adequate to remove all the infinities associated with virtual photons which we encounter in this calculation. The differential neutron decay rate including the radiative corrections and 1/m N corrections is found to be: where we have retained the relativistic expression for the phase factor, and The explanation of the quantities appearing in this expression will be given below. We remark that, in order to arrive at this factored form, we have freely exploited the fact that terms of order (α/2π) 2 , (α/2π)(Q/m N ) and (Q/m N ) 2 can be ignored to the order of accuracy of our concern. In Eq. (8) the Coulomb part of the radiative correction has been extracted as an overall factor and incorporated into the usual Fermi function F (Z, E e ) ≃ 1 + (α/2π)δ (Coul) α = 1 + απ/β, for Z = 1. In Eq. (9) the finite LEC, e R V , featuring in the factor (1 + α 2π e R V ) subsumes those short-range radiative corrections to the Fermi constant G 2 F which have been integrated out in arriving at our effective Lagrangian. This point will be further discussed in the final section. The axial coupling constant, g A , which has been renormalized by pion loops, is multiplied by short-range radiative corrections involving the finite LEC e R A as well as e R V . For convenience, and to simplify the results, we incorporate this radiative correction to g A intog A defined bỹ and thisg A has been used in Eq. (9). Recall that g A corresponds to the physical value, with all short-range radiative corrections removed. In Eq. (9), δ (1) α represents the model-independent radiative correction to G F , which depends only on the kinematics of the electron, while δ (2) α gives the model-independent radiative corrections to the coefficients of the angular correlation terms, β ·p ν andn · β. The explicit expressions for δ (1) α and δ (2) α are: Here E max e = (m 2 n − m 2 p + m 2 e )/2m n is the maximum electron energy, and L(z) is the Spence function defined by: The factor C 0 (E e ) contains the recoil corrections to the overall rate. It is given by (14) where we have used The other coefficients C i (E e ) (i = 1, 2, . . . , 5) are given by whereã,Ã,B are given by Eq. (6) with the substitution g A →g A . It is to be noted that Eq. (9) exhibits angular dependences that are missing in Eq. (5). These extra angular dependences arise from the NLO contributions that have been included in the 1/m N corrections (which leads to Eq. (9)) but ignored in the LO evaluation (which leads to Eq. (5)). It has been a common practice to approximate the overall kinematic factor in Eq. (8) by applying an expansion in 1/m N . If convenient, one could use the following approximation: where we have used The angular dependence appearing in Eq. (20) needs to be considered simultaneously with the angular dependences contained in Eq. (9). The model independent radiative correction δ (1) α in Eq. (11) agrees with that obtained by Sirlin [3], while δ (2) α in Eq. (12) also agrees with the result reported by Garcia and Maya [20]. We note that recoil corrections have also been calculated in the literature using the conventional methods. For instance, Wilkinson [21] evaluated corrections to the decay rate and the correlation coefficient A, and Bilen'kii et al. [22] computed corrections to the decay rate and the correlation coefficient a. Furthermore, Holstein [23] considered recoil corrections to all the observables for general nuclear beta-decays. Our results for the recoil corrections agree with those found in these previous studies.

Discussion and conclusions
As mentioned in the introduction, a prime issue in the studies of neutron beta-decay is to deduce the precise value of V ud from the experimental data. Another issue is the extraction of the value of g A from the data. We shall discuss here the significance of our present calculation in connection with these two issues.
To obtain the actual numerical values of V ud and g A we need to know the values of the LEC's, e R V and e R A , pertaining to the lepton-current nucleon-current vertex. These LEC's parameterize short-distance physics not explicitly included in the effective Lagrangian, L β , and they need to be determined empirically using appropriate observables. This is an important line of studies for the future. Here, instead, we discuss simple orderof-magnitude estimates of the LEC e R V , which is the most important LEC in neutron beta-decay. Based on the general estimation of a photon loop diagram, one may expect the natural scale for this parameter to be of the order of (α/2π)e R V ∼ 2 × 10 −2 , with e R V ∼ ln(m e /Λ χ ). To obtain another rough estimate of e R V we may compare our result for the neutron decay rate obtained from Eq. (8) with Eq. (6) of Marciano and Sirlin [5]. Thus we introduce the premise where m W , m Z are the masses of the W, Z bosons and m A is the axial mass scale. As is customary, we define the Fermi constant G F of muon decay by absorbing the factor 1 + (3α/4π)ln(m W /m Z ) into G F [24]. The contribution ln(m W /m Z ) in Eq. (22) is actually the difference between the contribution of the Z-box diagrams 5 in neutron beta-decay and the contribution of the Z-box diagrams in muon decay. In Eq. (22), the major contributions to the right-hand side originate from the short-range virtual photon corrections to the Fermi transition from the weak vector and axial-vector vertices. The former gives the contribution, 3 ln(m W /m N ), and the latter ln(m W /m A ). The C in the expression is the long-range model-dependent correction coming from the axial-current and anomalous magnetic moments of the nucleon, and is proportional to (µ S g A ) where µ S is the isoscalar magnetic moment of the nucleon. A value of 2C = 1.77 was found in Ref. [5]. In an HBχPT calculation, however, we have verified that a correction estimated from the diagrams of C is of higher order ∝ α/(2π) (Q/m N ) 2 and can be neglected (see section 2). Finally, the A g term, which includes a short-range strong-interaction correction, is very small: A g ≃ −0.34 [5]. In this connection, it might be of interest to decompose, following Cirigliano et al. [2], our e R V into two parts: e R V = e SD V +ẽ R V . The e SD V term describes the universal short-distance physics of electroweak theory discussed by Sirlin [24], while thẽ e R V term describes short-distance hadronic physics. It is possible that the A g term is associated with theẽ R V term. The above considerations lead to a rough estimate, e R V ≃ 20, i.e., [α/(2π)]e R V ∼ 4 × 10 −2 , which is of a natural size as discussed above. The above comparison also leads us to expect that the dominant contribution to e R V comes from the short-range electroweak corrections.
The LEC e R A enters only as a radiative correction to g A in Eq. (10), and therefore it may seem that there is no significant motivation to remove the radiative correction α 4π (e R A −e R V ) fromg A defined in Eq. (10) and deduce the values of g A . Indeed, if we limit ourselves to neutron beta-decay, all the observables can be expressed usingg A without referring to g A . However, since radiative corrections are specific to individual processes, there should be cases wherein the removal of α 4π (e R A − e R V ) fromg A has physical consequences and hence e R A does play a significant role. A possible example is the Goldberger-Treiman relation, g A m N = f π g πN , where g πN is the pion-nucleon coupling constant. To elaborate on this point, it is useful to illustrate processes which necessitate the introduction of the LEC, e R A . To this end, we consider diagrams containing the exchange of a pion (pion-pole) plus a virtual photon. These diagrams involve three distinct one-particle-irreducible vertex functions. The first type is a nucleon-nucleon-lepton-lepton four-point vertex in which a virtual photon couples to both the nucleon and the leptonic currents. This class of diagrams requires a counter term involving e A associated with g A . The second type is a lepton-lepton-pion three-point vertex wherein a virtual photon only couples to the pion, the pion-lepton vertex or the lepton, and this vertex is related to the pion decay constant f π . Some of the LEC's arising from this type of diagrams can be found in the chiral Lagrangian considered by Knecht et al. [16]. These LEC's are also related to the "inner" radiative corrections calculated for pion beta-decay, see e.g. [25]. The third type is a nucleon-nucleon-pion vertex in which a virtual photon only couples to the pion, the pionnucleon vertex or the nucleon, and this vertex is related to g πN . The corresponding LEC's are the g i 's appearing in Müller and Meißner's work [10]. To our knowledge, however, no systematic HBχPT study of the Goldberger-Treiman relation including the radiative corrections associated with each of the vertices has been done so far. In fact the radiative correction e R A really has not been fully studied yet in the standard approach. Instead it has usually been assumed that e R A ≃ e R V , which makes the radiative correction to g A small. Such radiative corrections could contribute to the evaluation of the Goldberger-Treiman discrepancy, but there is clearly not yet enough information to determine whether they turn out to be significant in comparison with the chiral symmetry breaking term.
As discussed in section 2, we have not included in our work radiative corrections involving the NLO vertex or pion loop diagrams. The former should be suppressed at least by a factor of µ VQ /(2m N ) ≃ 2 × 10 −3 , and the latter by a factor of (m π /Λ χ ) 2 ≃ 2 × 10 −2 relative to the leading radiative corrections. Also omitted from our work are the isospin breaking effects, which are naturally incorporated in the N 2 LO heavy-baryon chiral Lagrangian [8] not explicitly written in this paper. Recently, Kaiser [26] studied isospin violation corrections to G F V ud using HBχPT and found that the isospin breaking corrections are of the order of 10 −5 . To the accuracy of our present concern, we can safely neglect the isospin violation corrections.
We now summarize. Using the effective field theory for neutron beta-decay, we have calculated the decay rate of the neutron and the angular correlation coefficients including recoil corrections and radiative corrections of order α. We have included all non-radiative terms through N 2 LO except those which are negligible because of the extremely small value ofQ for neutron beta-decay. Our results reproduce the model-independent radiative corrections and recoil corrections in the literature. The short-range radiative corrections of the earlier calculations are replaced in our theory by the two finite radiative LEC's, e R V and e R A , where e R V affects G F and the difference, e R V − e R A , affects g A . Via comparison with the results of the existing model calculations, we have argued that the value of e R V is of a natural scale. An advantage of our EFT approach is the possibility of evaluating higher order corrections in a systematic way, and the possibility to parameterize the strong interaction dependent contributions in terms of well-defined LEC's, which can in principle be obtained from independent experiments. The next order corrections in the EFT for neutron beta-decay are estimated to be of the order 10 −5 or smaller. Therefore, to the extent that the LEC's involved in the present calculation are of a "natural" size (as discussed above), we expect our expressions for the rate and the angular correlation coefficients to be accurate to better than 10 −3 .