Constraints on subleading interactions in beta decay Lagrangian

We discuss the effective field theory (EFT) for nuclear beta decay. The general quark-level EFT describing charged-current interactions between quarks and leptons is matched to the nucleon-level non-relativistic EFT at the O(MeV) momentum scale characteristic for beta transitions. The matching takes into account, for the first time, the effect of all possible beyond-the-Standard-Model interactions at the subleading order in the recoil momentum. We calculate the impact of all the Wilson coefficients of the leading and subleading EFT Lagrangian on the differential decay width in allowed beta transitions. As an example application, we show how the existing experimental data constrain the subleading Wilson coefficients corresponding to pseudoscalar, weak magnetism, and induced tensor interactions. The data display a 3.5 sigma evidence for nucleon weak magnetism, in agreement with the theory prediction based on isospin symmetry.


Conclusions 23
A

Introduction
Nuclear beta transitions have been at the center of the particle physics research program since over a hundred years. Historically they have been essential for understanding various ingredients of the Standard Model (SM), such as the existence of neutrinos, nonconservation of parity, or the Lorentz structure of weak interactions [1][2][3][4][5][6]. From the vantage point of a particle physicist today, their main role is twofold. On one hand they offer an opportunity for precision measurements of fundamental constants of the SM, notably of the V ud CKM matrix element [7][8][9][10][11]. They also provide insight into the complex non-perturbative dynamics emerging from the SM, for instance through phenomenological determinations of the axial nucleon charge g A [7,10,11]. On the other hand, they yield important constraints on physics beyond the SM (BSM), such as leptoquarks and other hypothetical particles contributing to scalar and tensor currents in weak interactions [8,11,12]. On a more theoretical side, because of a wide range of scales and physical processes involved, beta transitions are a perfect laboratory to research and develop the concepts of Effective Field Theory (EFT).
The main ingredients of the theory of beta transitions were worked out by the end of 1950s, see in particular Refs. [4,13,14]. The flip side is that the habitual language in the literature may sometimes be unfamiliar to contemporary QFT practitioners. One of the goals of this paper is to reformulate beta transitions in the modern EFT language. The advantage, apart from the conceptual side, is that the theory can be smoothly incorporated into the ladder of EFTs spanning various energy scales, from the TeV scale down to MeV. In particular, the EFT for beta transitions can be matched to the so-called WEFT (the general EFT of SM degrees of freedom below the electroweak scale), and via this intermediary to the SMEFT (the general EFT of SM degrees of freedom above the electroweak scale). This way, the general effects of heavy non-SM particles can be naturally incorporated, along with the more studied SM effects, into the low-energy effective theory of beta decay.
An appropriate EFT framework to describe beta decay is the pionless EFT [15]. The relevant degrees of freedom are the nucleons (protons and neutrons) and leptons (electrons and electron neutrinos). Most of the details of the pionless EFT Lagrangian, such as the nucleon self-interactions describing the nuclear forces, are not relevant for our discussion. For the sake of this paper we will focus instead on the interactions mediating beta transitions, which are quartic terms connecting the proton, neutron, electron, and neutrino fields and their derivatives. Amplitudes for the neutron decay can be directly calculated starting from this Lagrangian. As for the beta decay of nuclei with the mass number A > 1, the amplitudes involve matrix elements of the nucleon bilinears between the nuclear states. This is analogous to the usual treatment of hadrons in QCD, where the amplitudes involve matrix elements of quark operators between hadronic states.
The important difference between the present EFT approach and hadrons in QCD is in power counting. Since the 3-momentum transfer in beta transitions is much smaller than the nucleon mass m N , the EFT Lagrangian can be organized in a non-relativistic expansion in powers of ∇/m N . 1 Focusing on the part of the Lagrangian mediating beta transitions, the leading O(∇ 0 ) term in this expansion encodes the usual Fermi and Gamow-Teller contributions to the allowed beta decays. This includes the SM-like contributions from the vector and axial currents, as well as the non-SM ones from the scalar and tensor currents. The resulting beta decay observables, such as the lifetime or angular correlations, are described exactly by the formulas obtained by Jackson, Treiman, and Wyld in the seminal Ref. [13]. 2 The subleading corrections to these leading order expressions are the main focus of this paper. They vanish in the limit where 3-momenta of the parent and daughter nuclei are zero, hence in the literature they are referred to as the recoil corrections. We restrict to discussing the effects linear in 3-momenta of the nuclei. These originate from two sources. One is the O(∇ 1 ) terms in the EFT Lagrangian, which gives a complete description of recoil effects in neutron decay. For nuclei with A > 1 the other source is the O(∇ 0 ) Lagrangian with the nuclear matrix elements expanded to linear order in the 3-momenta. We give the full expressions for the amplitudes at the linear recoil level, as well as the corresponding differential decay width in a convenient parametrization. In the limit where non-standard currents are absent, our results can be matched to those in Ref. [16]. The novelty of this paper is that we also present a complete treatment on non-standard corrections at the linear recoil level. We describe how the quark-level scalar and pseudoscalar interactions enter the beta decay observables. Moreover, we give a complete description of the effects of quark-level tensor interactions, which lead a number of distinct terms in the leading and subleading Lagrangian of our low-energy EFT.
Parameters of the leading-order EFT interactions have been fit from data for more than 70 years. The existing experimental data on beta transitions [7][8][9][10]17] are nowadays precise enough to be sensitive to recoil corrections. Our formalism can be employed to place meaningful constraints on Wilson coefficients of leading and subleading EFT operators. We construct a global likelihood function for the Wilson coefficients based on the state-of-the art measurements of superallowed 0 + → 0 + transitions, neutron decay, mirror decay, and other allowed transitions. This likelihood encodes confidence intervals for all the Wilson coefficients. In addition to constraints on the leading Wilson coefficients, already obtained in Ref. [11], we derive constraints on certain Wilson coefficients of subleading EFT operators generated by BSM physics. In particular, we analyse the effects of pseudoscalar interactions on beta transitions, and obtain simultaneous constraints on non-standard pseudocalar, scalar, tensor, and right-handed currents. Next, we discuss the Wilson coefficient of the EFT operator describing the nucleon-level weak magnetism. Usually, its magnitude is determined by theory using isospin symmetry, which in this context is referred to as the conserved vector current (CVC) hypothesis. We show that this Wilson coefficient is now efficiently constrained by the global data, which provides the first evidence for the nucleonlevel weak magnetism. Finally, we also discuss a subleading EFT operator describing the so-called induced tensor interactions (one of the second class currents in the classification of [14]). While isospin symmetry predicts that this Wilson coefficient should be negligibly small, the data show a 1.8 sigma preference for its non-zero value.
This paper is organized as follows. In Section 2 we lay out the formalism connecting the general quark-level EFT below the electroweak scale to the low-energy EFT describing weak charged-current interactions of nucleons. Based on the latter EFT, in Section 3 we calculate the recoil corrections to the beta transition amplitudes and observables (the lifetime and correlation coefficients). Global fits to the Wilson coefficients are presented in Section 4. Our conclusions are contained in Section 5. Appendix A contains some useful mathematical details about spin representations, while the contributions of all onederivative EFT operators to the beta decay correlations coefficients are summarized in Appendix B.
2 Effective Lagrangian for beta decay

WEFT
The starting point is the so-called weak EFT (WEFT) Lagrangian, which is defined at the scale µ 2 GeV and organized in an expansion in ∂/m W , where the W boson mass m W plays the role of the cutoff scale. The leading order term describing charged-current interactions between quarks and leptons is [18] where u, d, e, and ν L ≡ (1 − γ 5 )ν/2 are the up quark, down quark, electron, and lefthanded electron neutrino fields, σ µν ≡ i 2 [γ µ , γ ν ], V ud is the CKM matrix element, and v ≡ ( √ 2G F ) −1/2 ≈ 246.22 GeV. The central assumption is that, below 2 GeV, there are no other light degrees of freedom except for those of the SM. We treat the neutrino as massless, its tiny mass having no discernible effects on the observables studied in this paper. The Wilson coefficients X X = L, R, S, P, T , parametrize possible effects of non-SM particles heavier than 2 GeV. In the SM limit, X = 0 for all X.
The Lagrangian in Eq. (2.1) is convenient to connect to new physics at high scales. For example, integrating out the so-called S 1 leptoquark [19,20] with mass M and Yukawa couplings y can be approximated by the Wilson coefficients S ≈ − P ≈ − T = |y| 2 4V ud M 2 , up to loop and RG corrections. More generally, X can be matched at the scale µ m W to the Wilson coefficients of the SMEFT (see e.g. [21]), which captures a broad range of new physics scenarios with heavy particles [22]. In this paper, however, we are interested in low-energy physics of beta transitions. In these processes, the relevant degrees of freedom are not quarks but nucleons (protons and neutrons) or composite states thereof. In the following we connect the EFT in Eq. (2.1) to another EFT describing charged-current interactions of nucleons and leptons.

Nucleon matrix elements
As a first step toward this end, we define the matrix elements of quark currents between nucleon states [16,18] Above p and k are the momenta of an incoming neutron and an outgoing proton, P = p + k is to the total momentum of the nucleon system, and q = p − k is the momentum transfer. Next, m N ≡ (m n + m p )/2 is the nucleon mass, and u N is the Dirac spinor wave function of the neutron or proton, which implicitly depends on the respective momentum and polarization. The matrix elements take the most general form allowed by the Lorentz symmetry and the discrete C and P symmetries of QCD. The dependence on the momentum transfer is encoded in the form factors g X (q 2 ). We also define nucleon charges g X ≡ g X (0), which are the relevant parameters for beta transitions, where q 2 m 2 N . Symmetries impose important restrictions on the possible values of the charges coming from the different quark currents: • Vector current. The Ademollo-Gatto theorem implies that, up to second order in isospin breaking, g V = 1 [23,24]. The induced scalar coupling, g IS , vanishes in the isospin limit and then an O(10 −3 ) value is expected for it. Finally, g M can be related, through an isospin rotation, to the response of the nucleons to an external magnetic field and can be fixed, up to relative O(10 −3 ) isospin breaking corrections, from the experimentally known difference of the magnetic moment of the nucleons [16,25], where µ N is the nuclear magneton e 2mp .
• Axial current. The axial charge g A is not known from symmetry considerations alone and in practice is fixed from experimental data or by lattice calculations. As for the latter, the FLAG'21 average quotes g A = 1.246(28) [26][27][28][29]. The induced pseudoscalar 3 For the vector and axial matrix elements our notation is close to that of Ref. [16], except that we trade gII → gIT , and gS,P → gIS,IP (to remove the conflict with the gS,P form factors in the scalar and pseudoscalar currents). The notation for the scalar, pseudoscalar, and tensor matrix elements follows that of Ref. [18], up to i and mN factors to make the g (i) T form factors real and dimensionless.
charge, g IP , only enters into the observables at second order in the recoil expansion and then, in principle, it is beyond the scope of this analysis. Notice however that g IP is enhanced by the pion pole. Indeed, using partial conservation of axial current one has [30] − where m q is the average of the light quark masses. Using that g A (q 2 ) is a very smooth function of q 2 for q 2 m 2 π and that g P (q 2 ) is dominated by the pion pole contribution (e.g. see [31,32]), one obtains, up to a few per-cent level correction, where the displayed uncertainty is due to the error on the lattice determination of g A . Finally the induced tensor, g IT , which by itself enters suppressed by one power in the recoil expansion, vanishes in the isospin limit, and then its expected size is O(10 −3 ).

Pionless EFT
Beta transitions are characterized by a 3-momentum transfer much smaller than the nucleon mass, typically |q| ∼1-10 MeV. Therefore, the nucleon degrees of freedom are non-relativistic (in an appropriate reference frame), and can be described by non-relativistic quantum fields ψ N , N = p, n, which are 2-components spinors. On the other hand, we continue describing leptons by the relativistic fields ν L and e. The effective Lagrangian is constructed as the most general function of ψ N , e, and ν L and their derivatives, respecting the rotational symmetry and Galilean boosts. It is organized in an expansion in ∇/m N , where ∇ denotes spatial derivatives. This framework is known as the pionless EFT [15]. In this paper we are interested in the subset of the pionless EFT Lagrangian relevant for beta transitions, that is in the quartic interactions between a proton, a neutron, an electron, and a neutrino. 4 We organize these interactions as where L (n) refers to O(∇ n /m N ) terms. 5 At the zero-derivative level, the most general Lagrangian respecting the rules mentioned above is where σ k are the Pauli matrices, γ µ = 0 σ μ σ µ 0 , σ µ = (σ 0 , σ k ),σ µ = (σ 0 , −σ k ). At this level we have only two distinct nucleon bilinears, ψ † p ψ n and ψ † p σ k ψ n , which, up to two-body current effects, mediate the so-called allowed Fermi and Gamow-Teller (GT) transitions, respectively. The labels and normalization of the Wilson coefficients C + X are chosen such that they simply relate to the familiar parameters of the Lee-Yang Lagrangian [4]: C + X ≡ C X + C X for X = V, A, S, T . In order to match C + X to the parameters of the quark-level Lagrangian we calculate the n → pe −ν amplitude in two ways: using Eq. (2.8), and using Eq. (2.1) together with Eq. (2.2). We then demand that both calculations give the same result in the limit q → 0. At tree level, this procedure leads to the matching equations T + 2g (2) T T , up to O(q 2 /m 2 N ) effects which are consistently neglected in our analysis. On the other hand, we keep track of O(q/m N ) effects, which appear in this matching as the terms suppressed by m e /m N . One important thing to notice is that the quark-level pseudoscalar interactions, parametrized by P in Eq. (2.1), do not affect the leading order Lagrangian of pionless EFT. In the matching equations for the vector and axial Wilson coefficients we included the short-distance (inner) radiative corrections, where we use the numerical values ∆ V R = 0.02467(22) [41] and ∆ A R − ∆ V R = 0.00026(26) [42]. Other radiative corrections in this matching are not relevant from the phenomenological point of view and are omitted.
At the next-to-leading (one-derivative) order we consider the following interactions: For brevity, we do not display the overall e ±i(mp−mn)t factors multiplying the Lagrangian terms.
Again, we calculate the n → pe −ν amplitude in two ways: using Eq. (2.10) and using Eq. (2.1) together with Eq. (2.2), but now we concentrate on linear terms in q in the limit |q|/m N 1. Matching these linear terms requires the following identification of the nucleon-level and quark-level Wilson coefficients: where all the equalities are true up to O(q/m N ) corrections, i.e up to terms suppressed by m e /m N or (m n − m p )/m N . Note that we should not keep such terms in the matching of the Wilson coefficients of the subleading Lagrangian, as they correspond to O(q 2 /m 2 N ) effects in our EFT counting. We can see that quark-level pseudoscalar interactions, which arise only beyond the SM, induce the interaction term proportional to C + P in Eq. (2.10). 6 The interaction term proportional to C + M is known as the weak magnetism, while that proportional to C + E is referred to as the induced tensor term. 7 The Wilson coefficients C + T 1 -C + T 3 are all proportional to T parametrizing tensor interactions at the quark-level. They are less important phenomenologically, as we expect to first observe effects of T through its contributions to the leading order Lagrangian in Eq. (2.8). The interactions in the first two lines of Eq. (2.10) depend on the same nucleon bilinears ψ † p ψ n and ψ † p σ k ψ n as the leading Lagrangian in Eq. (2.8). On the other hand, the last line contains bilinears where the derivative acts on the nucleon field, thus they appear in the subleading Lagrangian only. 6 Within the SM one has the contribution to C + P through the induced pseudoscalar coupling, ∆C + P = V ud v 2 me 2m N gIP . This is however suppressed by an additional factor of q/mN , and thus is neglected here as an O(q 2 /m 2 N ) effect. 7 This is a misnomer because it has nothing to do with the bona fide tensor interactions at the quark level, which contribute to completely different structures in the non-relativistic EFT Lagrangian. We will however use this name, to conform with the bulk of the beta decay literature.

Recoil corrections to beta decay observables
Starting from the EFT Lagrangian in Eq. (2.7) one can calculate the associated amplitude and differential distributions for the nuclear beta transitions N → N eν. Here, N (N ) denotes the parent (daughter) nucleus, e denotes the beta particle (e ∓ for β ∓ transitions), and ν denotes the antineutrino (for β − ) or the neutrino (for β + ). In this paper we focus on the allowed beta transitions where N and N have the same spin, J = J; the discussion of the allowed decays with J = J ± 1 is analogous. We first introduce the general structure of the amplitude including the recoil effects, and then move to discussing differential distributions.

Structure of the decay amplitude
For concreteness, we present the formulas for the β − decay amplitude, . The amplitude can be expanded in powers of recoil momenta: In this expansion, all 3-momenta involved (p N , k N , k e , k ν ), the lepton energies and electron mass (E e , m e , E ν ≡ E max e − E e ) as well as the nuclear mass difference (∆ ≡ m N −m N ) count as one order in recoil, and we denote these collectively as O(q/m N ).
Only the leading order Lagrangian in Eq. (2.8) contributes to the leading part of the amplitude M Above, C + X are the Wilson coefficients in the Lagrangian of Eq. (2.8). The leptonic currents are defined as where k = 1 . . . 3, and u and v are the spinor wave functions of the outgoing electron and antineutrino (the L/R subindex denote the chirality projection). Finally, · ≡ N (k N )| · |N (p N ) denotes matrix elements of the non-relativistic nucleon fields sandwiched between the daughter and parent nuclear states in relativistic normalization N (p ) |N (p) = 2E N (2π) 3 δ 3 (p − p ). The matrix elements entering into Eq. (3.1) are labeled as Fermi and GT, respectively. Rotational symmetry together with parity imply that they take the most general form [16]: are the spin-J generators of the rotation group (see Appendix A). For J = 0 one should take the limit r → 0 while ignoring all apparent 1/J singularities. Terms linear in p or p cannot appear in Eq. (3.2) due to parity conservation. The common normalization factor M F is defined in the isospin limit and can be calculated when parent and daughter nuclei are members of the same isospin multiplet. For β ∓ transitions one has M F = δ j j δ j 3 ,j 3 ±1 j(j + 1) − j 3 (j 3 ± 1), where (j, j 3 ) and (j , j 3 ) are the isospin quantum numbers of the parent and daughter nuclei. The parameter r, which is real by time-reversal invariance, is referred to as the ratio of GT and Fermi matrix elements in the literature. For the neutron decay r = √ 3. For decays with A > 1, it cannot currently be calculated from first principles with high accuracy, and instead has to be extracted from experiment or estimated in nuclear models. In this notation the so-called mixing ratio is given by ρ = r C + A /C + V , up to radiative corrections. Finally, O(q 2 ) refers to corrections of the second order in recoil, which are consistently neglected in this paper.
The subleading Lagrangian in Eq. (2.10) contributes at the next-to-leading order in the recoil expansion. The amplitude at this order takes the form The matrix elements in the first two lines are the Fermi and GT ones, already discussed around Eq. (3.2). In the last line, three new nuclear matrix elements enter at the next-toleading order in recoil. They can be parametrized as is approximately the mass number of the parent nucleus, and the matrix T jk (J) is defined in Eq. (A.2). The coefficients of the terms proportional to P are related by Lorentz invariance to those in Eq. (3.2) [16]. 8 On the other hand, the form factors β F V , γ F V , β F T and γ F T are transition-dependent and are determined by strong dynamics. 9 Time-reversal invariance implies they are real, while isospin symmetry (CVC) implies γ F V = 0 and relates β F V to the magnetic moments of the parent and daughter nuclei [16]: where µ + and µ − denote the magnetic moments of the nuclei with larger and smaller j 3 quantum number respectively. 8 For example, in the relativistic theory Lorentz invariance implies pγ µ n = cP µ + . . . . Taking the nonrelativistic limit, one has pγ 0 n → ψ † p ψn = cP 0 + · · · ≈ 2mN c + . . . , and pγ k n ← → ∇ k ψn matrix element can be related to the ψ † p ψn matrix element. 9 In this paper we do not calculate the effects proportional to γF V and γF T because they are not relevant for our subsequent phenomenological analysis.
Given the leading and subleading amplitudes in Eq. (3.1) and Eq. (3.3) it is straightforward if tedious to calculate the differential decay width. In the next subsection we discuss the general parametrization of the differential width appropriate to incorporate the subleading corrections in the recoil expansion.

Parametrization of the differential width
We focus on observables summed over the daughter and β particle polarizations. We are interested in the differential decay width at the leading and subleading order in recoil momenta. In this observable, the effects of the Wilson coefficients in the EFT Lagrangian fit into the following template: where F is the Fermi function: (3.7) R is the nuclear radius, α is the fine structure constant, δ R stands for radiative corrections 10 , Z is the daughter nucleus charge, p e ≡ E 2 e − m 2 e , m e is the electron mass, E e is the electron energy in the range E e ∈ [m e , E max e ], the endpoint electron energy is E max and hereinafter, the upper (lower) sign applies to β − (β + ) transitions. Finally, k e and k ν denote the 3-momenta of the beta particle and neutrino, J is the polarization vector of the parent nucleus, and j is the unit vector in the polarization direction (if J = 0 then all terms proportional to 1/J should be set to zero).
The bullets below offer some comments and rationale regarding our parametrization.
• Only the first line in Eq. (3.6) contributes to the total beta decay width (or lifetime/halflife) of the nucleus. The overall normalizationξ and the Fierz term b are related to 10 Customarily, one splits 1 where δ R is the long-distance (outer) radiative correction, δNS is the nucler-structure dependent correction, and δC is the isospin breaking correction. the parameters of the leading order Lagrangian in Eq. (2.8) as [13] and by definition do not receive any recoil-order corrections. The latter are encoded Ee me , with b −1,0,1 independent of E e . Above and hereafter we use a bar to denote complex conjugation. The relation with the traditional notation of Ref. [13] is simply ξ = M 2 Fξ /2.
• For unpolarized decays, the differential distribution should be averaged over the possible quantized values of J in the range J ∈ [−J, J]. Only the first two lines of Eq. (3.6) survive the averaging. In addition to the parameters affecting the total width, the relevant parameters for unpolarized decays include the β-ν correlation a(E e ), and a parametrizing recoil effects quadratic in cos(k β , k ν ). The former can be represented as a(E e ) = a 0 + ∆a(E e ), where a 0 is generated by the leading order Lagrangian in Eq. (2.8) and is independent of E e [13]: while ∆a(E e ) arises at the recoil level and in general does depend on the energy of the beta particle. As for a , at the linear order in recoil we have a = b a Ee me with b a independent of E e .
while ∆A(E e ), ∆B(E e ), ∆D(E e ), and also A , B , D arise only at the recoil level. Note that the latter group of coefficients also depends on the beta particle energy in general, but we do not stress this fact in our notation.
• Out of the terms linear in J , only A(E e ) survives when the distribution is integrated over the neutrino direction dΩ ν . Thus, the remaining coefficients will not affect experiments measuring the β-asymmetry, where the neutrino momentum is not reconstructed. Similarly, only B(E e ) survives when these terms are integrated over the electron direction dΩ e , and the remaining coefficients will not affect experiments measuring the β-asymmetry once the information about e ± kinematics is integrated out. These features make our parametrization more convenient in practice than e.g. the one used in Refs. [16,18,43].
• The last three lines in Eq. (3.6) describe effects relevant only for polarized decays with J ≥ 1 (thus in particular they are absent in neutron decay). We haveĉ(E e ) = c 0 + ∆c(E e ) whereĉ 0 generated by the leading order Lagrangian [13]: 11 while ∆c(E e ), c , and c 1,2,3,4 arise at the recoil level. To our knowledge these correlations have never been experimentally measured, even the leading orderĉ 0 , and thus their phenomenological role is currently null. We nevertheless quote them here for completeness.
• The coefficients a , c and D are actually not generated by the interactions in Eq. (2.10). They are however generated via kinematic recoil corrections to the 3-body decay phase space, which we treat on the same footing.
• The leading electromagnetic corrections are included in Eq. (3.6) via the Fermi function F (Z, E e ) and via δ R . The correlation coefficients receive other O(α) corrections ("outer radiative corrections"), see e.g. [43]. In this paper we do not discuss these explicitly, but they are taken into account in the experimental analyses that we use as input of our phenomenological analysis in the next section, whenever they are required for the theory predictions to match the experimental accuracy.
The complete dependence of all the correlation coefficients in Eq. (3.6) on the Wilson coefficients in the subleading EFT Lagrangian in Eq. (2.10) is summarized in Appendix B. Compared to earlier works [16,18,43,44] our results include complete BSM effects arising at the subleading order in the EFT, that is at the linear order in recoil. Let us note that our results include terms that are quadratic in BSM couplings. These results will be employed in the next section to perform global fits constraining the EFT Wilson coefficients. We will focus on the effects of the pseudoscalar interactions (C + P ), nucleon-level weak magnetism (C + M ), and induced tensor interactions (C + E ), but our result allow one to constrain any pattern of the Wilson coefficient entering the leading and subleading EFT Lagrangian in Eq. (2.10).

Observables
Before embarking on that analysis, let us first discuss how common experimental observables are related to the correlation coefficients in Eq. (3.6). The total width is given by the expression 12 where and (3.14) Concerning the β-asymmetry, experiments typically measure the number of events N ↑ and N ↓ with the beta particle emitted, respectively, into the northern and southern hemisphere in reference to the parent polarization direction j. The corresponding up-down asymmetry is given by where P the parent polarization fraction. We substitute A(E e ) = A 0 + ∆A(E e ) and expand the above to linear order in recoil: (3.16) Experimental collaborations often translate the observable asymmetry into a measurement of A 0 , assuming vanishing Fierz term and the recoil corrections calculated in the absence of new physics beyond the SM. BSM physics can contribute as an E e -independent shift of A 0 and b, or as an E e -dependent shift of ∆A(E e ) and ξ b (E e ). We split the SM and BSM contributions at the recoil level as ∆A(E e ) = ∆A SM + ∆A BSM , ξ i = ξ SM i + ξ BSM i . With this notation, we reinterpret the experimental extraction of A 0 as a measurement ofÃ, defined as 1+b me/Ee [11,46]. Eq. (3.17) generalizes this prescription to the subleading order in recoil.
By the same logic one can define the tilde prescription for the neutrino asymmetry and for the β-ν asymmetry: If the correlations are measured as a function of energy of the beta particle, we can likewise defineX(E e ) with the energy averages · and · P removed (or, more realistically, with the dE e integration restricted to a given energy bin). Note also that experimental conditions often imply that only a part of the phase space is effectively detected, and that should be taken into account when the · averages are calculated [46].

Global fit to recoil effects
In our phenomenological analysis we use the experimental data summarized in Appendix C. The differences of this dataset with the one used in the recent analysis of leading-order effects [11] are the following: 1. An older measurement of the β-ν correlation of the neutron [47] by the aCORN collaboration is superseded by the new resultã n = −0.1078 (18) Table 1. Sensitivity of the correlation coefficients to pseudoscalar interactions parametrized by the Wilson coefficient C + P . The entries are in units of 10 −4 and correspond to the averaged function multiplying C + P in Eq. (4.2) for different parent nuclei. For the sake of this table, C + V , C + A , and r are set to their minimum values in the SM fit (although they are kept as free parameters in the fits of this section). For ξ b (E e ) and ∆B(E e ) the average is defined in Eq. (3.14), while in the case of ∆a(E e ) and ∆A(E e ) the pseudoscalar contribution is independent of the β particle energy and thus no average is needed.
For the sake of these fits we assume all Wilson coefficients are real, since the sensitivity of the imaginary parts to the considered observables is limited. 13

Pseudoscalar
This subsection is focused on the pseudo-scalar interactions: It is instructive to first discuss a simpler setting where, with the exception of C + P , only the leading order Wilson coefficients C + V and C + A are present. 14 In particular, we assume C + S = C + T = 0 in the leading order Lagrangian in Eq. (2.8). In this limit the pseudoscalar contributions to the correlations simplify: There are also contributions to ∆c(E e ) and c 1 , which are not relevant for our analysis. One can observe that, in each case, the pseudoscalar contribution is multiplied by m e /m N ∼ 13 Constraints on the imaginary parts can be improved by including in the analysis experimental measurements of CP-violating beta decay observables, such as the D [50] or R [51] parameter of the neutron [8]. This is left for future work. 14 We note that SM recoil corrections are included, since they have been taken into account in the experimental analyses that we use as inputs. . Constraints on C + P from a subset of most sensitive observables. We show the fit to C + P marginalized over C + V and C + A , assuming other Wilson coefficients vanish. The error bars show the 68% CL intervals obtained using Ft(0 + → 0 + ), neutron's lifetime, and one of the following: neutron's beta asymmetry, neutron's β-ν correlation, Ft and beta asymmetry of 19 Ne, or Ft, beta asymmetry, and neutrino asymmetry of 37 K. The pink band corresponds to the global fit in Eq. (4.3).
10 −3 , which is a typical suppression factor for recoil corrections in beta transitions. For this reason one expects much weaker constraints on C + P compared to those on C + V and C + A . Furthermore, the pseudoscalar corrections vanish in the limit r → 0, that is they are non-zero only for mixed Fermi-GT or pure GT transitions. In particular, they do not affect the superallowed 0 + → 0 + transitions, currently the most accurate nuclear measurement, which further diminishes the experimental sensitivity to C + P . Let us note that the ξ b (E e ) expression given above agrees with the result of Ref. [30] for the neutron decay case.
Comparing Eq. (4.2) with the leading contributions to correlations, cf. Section 3, one can see clearly that the pattern of the pseudoscalar contributions to the correlation is distinct than that of C + V or C + A . Therefore, a global fit with enough different observables can discriminate C + P from other Wilson coefficient. Indeed we obtain the following simultaneous constraints on all three Wilson coefficients: As expected, the uncertainty of C + P is O(10 4 ) larger than that of C + V of C + A . The relative sensitivity of various observables contributing to this result is visualized in Fig. 1.
We can relax our assumptions by allowing new physics to enter also via scalar and tensor interactions at the quark level, which leads to C + S and C + T being non-zero in the leading order Lagrangian in Eq. (2.8). The existing experimental data allows for a simultaneous fit of all five Wilson coefficients: Introduction of C + S and C + T increases the degeneracy of the parameter space, leading to the constraints on C + V , C + A and C + P being relaxed by approximately a factor of two compared to the simplified scenario in Eq. (4.3). This is enough to disentangle the five independent WEFT couplings present in Eq. (2.1). At linear order in new physics, they can be identified with the new physics parameters R,S,P,T and the "polluted" CKM element corresponding to the combinationV ud ≡ V ud (1 + L + R ) [11]. 15 Indeed, using the matching in Eqs. (2.9) and (2.11), we can translate the fit above into constraints on the parameters of the quarklevel Lagrangian in Eq. (2.1), in such a way that all the new physics parameters except for L can be disentangled. We obtain This is the first time such general constraints, including those on the pseudoscalar parameter P , are extracted from nuclear data. Inclusion of P in the fit does not increase substantially the uncertainty on the remaining new physics parameters (see the fit without P in Ref. [11]), as also indicated by moderate off-diagonal entries in the last row/column of the correlation matrix. Curiously, the uncertainty on P is only percent-level, which is even slightly smaller than that on R . The lower sensitivity to the latter is in part a consequence of the relatively large uncertainty of the lattice input for g A when we match to Eq. (2.9), but it also shows that beta transitions have decent sensitivity to pseudocalar interactions, even though their effects are suppressed by O(m e /m N ) ∼ 10 −3 , thanks to the large value of the pseudoscalar charge g P [30]. All in all, the magnitudes of X effectively probed by beta transitions is well within the validity range of the quark-level EFT. However, the sensitivity of beta transitions to P is well inferior to that of pion decays. Very recently, Ref. [52] obtained the constraint P = 3.9(4.3) × 10 −6 in the combined fit to beta and pion decay data. This is more than three orders of magnitude better than the result in Eq. (4.5) based nuclear beta transitions alone. 16 15 Let us stress that one cannot simultaneously extract V ud and L from any set of observables associated to the Lagrangian of Eq. (2.1). Beyond the linear order, one may take {V ud ,ˆ R,S,P,T ≡ R,S,P,T 1+ L + R } as a basis of WEFT couplings. 16 If one allows for cancellations between the linear and quadratic pseudoscalar contributions to pion decays, then the bound is relaxed to | P | ∼ 4 × 10 −4 [8].

Weak magnetism
The name weak magnetism refers to a sum of two distinct effects entering at the subleading order in recoil. One is the contribution to the decay amplitude due to the operator in the subleading EFT Lagrangian in Eq. (2.10). We will refer to this effect as the universal weak magnetism, because it is common for all nuclear transitions. It turns out that a part of the contribution of another operator in Eq. (2.10), has the same tensor structure as that originating from Eq. (4.6). Namely, using the parametrization of the ψ † p ← → ∇ k ψ n matrix element in Eq. (3.4) and retaining only the part proportional to β F V , the subleading decay amplitude is affected by the operators in Eq. (4.6) and Eq. (4.7) as We will refer to the contribution entering via the operator in Eq. (4.7) as the nucleusdependent contribution to weak-magnetism (because the form factor β F V depends on the nuclei participating in the transition) or, in short, nuclear weak magnetism. Once again, it is instructive to first display the correlations in a simplified setting where we only take into account the interference between weak magnetism and the leading SM effects proportional to C + V and C + A . Defining C + W M ≡ C + M + β F V C + F V , at the linear order in recoil the total (universal+nuclear) weak magnetism enters the relevant correlations as: See Appendix B.2 for the complete expressions including the interference with scalar and tensor currents and for the remaining correlations. Much as the pseudoscalar interactions in the previous subsections, weak magnetism is relevant only for mixed Fermi-GT or pure GT transitions, in particular it does not affect the superallowed 0 + → 0 + transitions. Unlike pseudoscalar, weak magnetism leads to all correlation coefficients picking up a dependence on the beta particle energy E e .  Table 2.
Sensitivity of the correlation coefficients to the nucleon-level weak magnetism parametrized by the Wilson coefficient C + M . The entries are in units of 10 −4 and correspond to the averaged function multiplying C + M in Eq. (4.9) for different parent nuclei. For the sake of this table, C + V , C + A , and r are set to their minimum values in the SM fit (although they are kept as free parameters in the fits of this section). The averages · (p) over the β particle energy E e are defined in Eqs. (3.14) and (3.18).
As discussed in Section 2, the numerical value of C + W M for each transition is fixed by isospin symmetry (CVC). More precisely, C +,CVC

.7, and β CVC
F V is transition-dependent and can be related to nuclear magnetic moments via Eq. (3.5). The goal of this subsection is to show that the existing experimental data is powerful enough to pinpoint universal weak magnetism parametrized by C + M , without a theoretical input from isospin symmetry. Notice first that β F V = 0 for the neutron decay. Therefore, restricting to the subset of data that includes only the 0 + → 0 + transitions and the neutron decay, we can directly determine C + M (treated as a free parameter) along with other Wilson coefficients in the EFT Lagrangian. In the restricted framework where only C + V , C + A , and C + M are non-zero, we obtain the simultaneous constraint on the remaining three Wilson coefficients: The fit is dominated by the measurements of Ft(0 + → 0 + ) (which fixes C V +), neutron's lifetime (which then fixes C + A ), and the neutron's beta asymmetry (which then fixes C + M ). We observe that the result is consistent with the CVC prediction C +,CVC M ≈ 4.6/v 2 , and that C + M = 0 is excluded at more than 3 sigma. To our knowledge this is the first experimental evidence for the universal (nucleon-level) weak magnetism.
We can sharpen this evidence somewhat by including data from mirror and GT transitions studied in Ref. [11]. In those cases β F V is non-zero, therefore those transitions do not give us an unobstructed access to the C + M Wilson coefficient. Ideally for this argument, β F V would be determined by first principle calculations, for example by the lattice. In absence of such, for the sake of the fits below we fix β F V from the CVC relation in Eq. (3.5). This admittedly makes the following argument a bit circular. Note however that the constraining power of the mirror and GT data is largely dominated by the 19 Ne decay [53,54], in which case the contribution proportional β F V is expected to be subleading with respect to that proportional to C + M , unless β F V for 19 Ne is larger by an O(1) factor than the value  where the uncertainty on C + M is improved by ∼ 20% and the experimental evidence for universal weak magnetism is strengthened to ∼ 4σ. The relative sensitivity of various observables contributing to this result is visualized in Fig. 2.
Finally, we can also make a more general fit by allowing for new physics entering as scalar and tensor current at the leading order: Even in this more general framework the preference for non-zero C + M is still at the ∼ 2σ level.
We remark that that all our fits in this section use only observables integrated over the energy spectrum of the beta particle, because only that information is provided by experimental collaborations in the readily usable form. On the other hand, as is clear from Eq. (4.9), weak magnetism predicts specific dependence of the correlations on E e . Exploring this energy dependence will allow one to further tighten the theory-independent bounds on C + M , possibly pushing the evidence for weak magnetism beyond the 5σ threshold.

Induced tensor
As the final example of application of our formalism, we present the fit to the Wilson coefficient parametrizing the so-called induced tensor interactions in the subleading EFT Lagrangian in Eq. (2.10): As discussed in Section 2, the UV matching relations for these Wilson coefficients imply C + E = C + E = −C + A g IT /g A , thus in the following we set C + E = C + E and assume C + E is real. We expect the nucleon-level parameter g IT to be suppressed by small isospin breaking effects, hence v 2 |C + E | ∼ 10 −3 . But, as in the preceding subsection, we can ask the question what does experiment tell about C + E without any theoretical input from isospin symmetry. At the level of interference with C + V and C + A , the operators in Eq. (4.13) affects the correlations as (4.14) See Appendices B.3 and B.4 for the complete expressions including the interference with scalar and tensor currents and for the remaining correlations. Assuming that C + E is the only free parameter in the fit in addition to the SM Wilson coefficients Clearly, the existing data are sensitive only to |C + E | ∼ 1/v 2 (similarly as for the other recoillevel Wilson coefficients, e.g. C + P or C + M ). This is 3 orders of magnitude larger than the theoretically expected magnitude and hence experimental detection of C + E is unlikely in the envisagable future. Furthermore, the current data show a mild 1.8σ preference for a nonzero C + E , driven largely by the measurement of neutron's β-ν correlation in the aSPECT  Table 3. Sensitivity of the correlation coefficients to the induced tensor interactions parametrized by the Wilson coefficients C + E = C + E . The entries are in units of 10 −4 and correspond to the averaged function multiplying C + E in Eq. (4.14) for different parent nuclei. For the sake of this table, C + V , C + A , and r are set to their minimum values in the SM fit (although they are kept as free parameters in the fits of this section). The averages · (p) over the β particle energy E e are defined in Eqs. (3.14) and (3.18); for ∆a(E e ) the induced tensor contribution is independent of E e . experiment [55]. It will be interesting to see if this preference goes away, as experiments acquire more precise data.

Conclusions
In this paper we discussed the EFT for beta transitions. Working in the framework of the pionless EFT, with nucleons as degrees of freedom, the effective interactions between nucleons and leptons were organized in a non-relativistic expansion in powers of ∇/m N . The novelty of this paper is that the low-energy EFT was matched to the general quark-level EFT at higher energy. The latter, which we refer to as the WEFT, describes the effects of the SM weak interactions, as well as possible effects of new heavy particles from beyond the SM. We worked out the matching between the WEFT and the low-energy EFT up to the subleading order in ∇/m N , that is including the linear recoil effects. The results in Eq. (2.9) and Eq. (2.11) describe the matching conditions for the Wilson coefficients of the leading and subleading Lagrangian in Eq. (2.8) and Eq. (2.10). In particular, the matching of the non-standard tensor interactions in the WEFT to the recoil-level non-relativistic interactions at low energy was worked out for the first time.
The EFT framework allows us to systematically describe how the standard and nonstandard weak interactions affect beta decay observables, such as the lifetime, beta energy spectrum, and various angular correlations. We calculated the impact of all the terms in the leading and subleading Lagrangian on the differential decay width in allowed beta transitions summed over the beta particle and daughter nucleus polarizations. Partial results, most relevant for our numerical analysis, were displayed in Section 4, while the complete results are collected in Appendix B.
With the expressions for the observables at hand, we can use the existing experimental data on beta transitions to determine confidence intervals for the Wilson coefficients in the EFT Lagrangian. For the leading Lagrangian this exercise was already completed in Ref. [11], where O(10 −4 ) relative precision was found for the standard Wilson coefficients corresponding to the vector and axial currents, and stringent constraints were established on the non-standard scalar and tensor interactions. In this paper we extended this analysis to recoil-level Wilson coefficients. In particular, we performed the first ever comprehensive analysis of the pseudoscalar interactions in allowed beta decay. We find that nuclear decays set a robust constraint on the Wilson coefficient descending from the pseudoscalar interactions in the WEFT, even though it enters the observables only at the linear order in recoil. This translates into a percent-level constraint on the pseudoscalar WEFT parameter ( P in Eq. (2.1)), which is comparable to the sensitivity to the right-handed WEFT current ( R ), and one order of magnitude weaker than the sensitivity to the scalar and tensor currents ( S and T ). One should note, however, that within the WEFT framework, constraints on P from pion decays are 4 orders of magnitude stronger.
Weak magnetism is another recoil-level effect to which experiment is sensitive. Our global analysis of the allowed beta decay data showed a 3 sigma evidence for a non-zero value of the EFT Wilson coefficient corresponding to the universal (nucleon-level) weak magnetism. The evidence is dominated by the neutron decay measurements (lifetime and beta asymmetry), and is further strengthened by mirror decay data. We also discussed the recoil-level EFT operator describing the so-called induced tensor interactions. The isospin symmetry of QCD predicts that this Wilson coefficient should be suppressed so as to give negligible contributions to observables. Instead, our global analysis showed a small 1.8 σ preference for non-zero induced tensor interactions. Future measurements and better theoretical calculations will improve the understanding of the effects of these Wilson coefficients.

A Spin-J representation matrices
In this appendix we present the definition of the spin-J representation matrices T k (J) and T kl The tensor representation matrices are defined as with the rows corresponding to m running from +2 to −2.

B Subleading corrections to correlations
In this appendix we present the dependence of the beta decay correlations on the Wilson coefficients of the subleading effective Lagrangian in Eq. (2.10). To organize the presentation, each subsection below deals with the contribution of a single Wilson coefficient. The left-hand-sides refer to the correlation coefficients defined by Eq. (3.6), and we recall the . We are interested in O(q/m N ) effects, that is linear order in recoil, which are suppressed by one power of the nucleon mass m N or the nuclear mass m N . We neglect O(q 2 /m 2 N ) and higher order effects, in particular we neglect the contributions quadratic in the Wilson coefficients of Eq. (2.10). In expression containing ± or ∓, the upper (lower) sign refers to β − (β + ) transitions. We give the results for mixed Fermi-GT transitions with J = J; for pure GT transitions one should take the coefficient of the r 2 term and, for some correlations, multiply it by the appropriate group theory factor (see e.g. Eqs. A1,A2 of [13]).
These results are new because they include the effects of subleading non-SM Wilson coefficients (C + P , C + T 1 , C + T 2 , C + T 3 , C + F T ) at the linear order in recoil, as well as interference of the subleading Wilson coefficients with the non-SM leading order Wilson coefficients (C + S , C + T ). If the SM is the UV completion of our EFT, all these Wilson coefficients are zero (ignoring the tiny induced C + S ), and moreover C + Moreover, in Appendix B.11 we also quote the results for another class of recoil corrections to the correlations. They appear because the recoil corrections to the phase space and the relativistic normalization of the nuclear states result in the overall factor Eν m N + O(m −2 N ) multiplying the differential width. The contributions in Appendix B.11 result from multiplying the O(m −1 N ) terms in this expression with the zeroth order correlations discussed in Section 3.2. These effects are in fact relevant to establish the matching with the results in Ref. [16].
(B.4) The contribution of another Wilson coefficient C + F V multiplied by the form factor β F V in Eq. (3.4) is exactly the same as that of C + M . The joint effect can be described by replacing above (B.14) B.9 C + F A Wilson coefficient

C Data used in the analysis
In our numerical analysis we use the input from superallowed 0 + → 0 + beta transitions (Table 4), neutron decay (Table 5), mirror decays (Table 6), and correlation measurements in pure Fermi and pure GT decays (Table 7). Table 4. Input from superallowed 0 + → 0 + beta transitions [9] used in our analysis.  Table 6. Input from mirror beta decays used in our analysis.