Field Theory of the Fermi Function

The Fermi function $F(Z,E)$ accounts for QED corrections to beta decays that are enhanced at either small electron velocity $\beta$ or large nuclear charge $Z$. For precision applications, the Fermi function must be combined with other radiative corrections and with scale- and scheme-dependent hadronic matrix elements. We formulate the Fermi function as a field theory object and present a new factorization formula for QED radiative corrections to beta decays. We provide new results for the anomalous dimension of the corresponding effective operator complete through three loops, and resum perturbative logarithms and $\pi$-enhancements with renormalization group methods. Our results are important for tests of fundamental physics with precision beta decay and related processes.

QED corrections are dramatically enhanced relative to naive power counting in the fine structure constant α ≈ 1/137 for large-Z nuclei and for small-β leptons (Z denotes the nuclear charge, and β the lepton velocity).The Fermi function [45] in beta decay describes the enhancement (suppression) for negatively (positively) charged leptons propagating in a nuclear Coulomb field.For a nuclear charge Z and electron energy E it is traditionally defined by solving the Dirac equation in a pointlike Coulomb field.The result is then given as [45,46]: 2 e πξ (2pr) 2(η−1) , (1) where η ≡ 1 − (Zα) 2 , ξ = Zα/β, p = √ E 2 − m 2 and m is the electron mass.The quantity r denotes a short distance regulator identified approximately as the nuclear size [47].Several questions arise in the applica-tion of F (Z, E) to physical processes: 1) What is the scale r −1 and how does it relate to conventional renormalization in quantum field theory?2) How can other radiative corrections be included systematically?3) What is the relation between the Fermi function with Z = 1 and the radiative correction to neutron beta decay?Answering these questions is important for the interpretation of precision beta decay experiments.For example, corrections at order α(Zα) 2 must be included at the current precision (∼ 3 × 10 −4 ) of |V ud | extractions [21].These corrections require a theoretically self-consistent treatment of both the Fermi function and other radiative corrections, but have previously been treated only in a heuristic ansatz [38,48].To answer these questions, we re-formulate the Fermi function in effective field theory (EFT), and study its interplay with subleading radiative corrections.Factorization and all-orders matching.Factorization arises from the separation of different energy scales involved in a physical process [49][50][51].Nuclear beta decays involve physics at the weak scale ∼ 100 GeV, the hadronic scale ∼ 1 GeV, the scale of nuclear structure Λ nuc.∼ 100 MeV, and the kinematic scales relevant for beta decay E ∼ 1 MeV.The methods of EFT allow for each scale to be treated separately, and facilitate the calculation of higher order radiative corrections.In a sequence of EFTs, the components of a factorization formula are identified with a corresponding sequence of matching coefficients, and a final low-energy matrix element.In the context of nuclear beta decays, the longdistance (or outer) radiative corrections can be computed in the low-energy effective theory, while structure dependent and short-distance (or inner) radiative corrections are absorbed into the Wilson coefficient.Real radiation is straightforwardly included [52].
Consider the corrections to a tree-level contact interaction with a relativistic electron in the final state.Ladder diagrams from a Coulomb potential with source charge +Ze correct the tree level amplitude, M tree , with explicit loop integrals given by (see Ref. [43] for more details) Integrals are evaluated in dimensional regularization with D = 3 − 2ϵ dimensions, and we have included a photon mass, λ, to regulate infrared divergences [53].
In contrast to the non-relativistic problem [54], the relativistic expression (2) is UV divergent beginning at twoloop order, indicating sensitivity to short-distance structure.The factorization theorem reads [43] counting p ∼ m ∼ E and where Λ denotes the scale of hadronic and nuclear structure.We retain separate factorization scales µ S and µ H for clarity; conventional single scale matrix elements are obtained by setting µ S = µ H = µ.After MS renormalization, to all orders in Zα, the soft function is given by M S = exp iξ log µ S λ [55,56].Our result for the hard function is new [43], and is given (again to all orders in Zα) by [57] where γ 0 is a Dirac matrix, and γ E ≈ 0.577 is the Euler constant.
The leading-in-Z radiative correction to unpolarized observables from the soft and hard functions is given by where we define r −1 H = µ H e γE .The angle brackets denote contraction with lepton spinors, M H → ēM H γ 0 ν L , sum over final state spins, and division by the same expression in the absence of Coulomb corrections.Note that there is a finite multiplicative correction relating the MS hard function to F (Z, E).

Effective operators and anomalous dimension.
The structure-dependent factor M UV appearing in Eq. (3) depends on the process of interest.Important examples are beta decay transitions [A, Z] → [A, Z +1]e − νe or [A, Z +1] → [A, Z]e + ν e .Superallowed beta decays are governed by an EFT consisting of QED for electrons, and heavy charged scalar fields [58][59][60][61], (6) where ϕ [A,Z] v denotes a heavy scalar with electric charge Z whose momentum fluctuations are expanded about p µ = M [A,Z] v µ , with v µ = (1, 0, 0, 0) in the nuclear rest frame.For neutron decay, the EFT involves spin-1/2 heavy fields [58][59][60][61], where h (p) v and h (n) v denote spin-1/2 heavy fields with electric charge 1 and 0, respectively.Matching to the EFT represented by Eqs. ( 6) or (7), we identify the components of (3) in terms of operator coefficients and matrix elements: M UV is proportional to (a linear combination of) C i , while M H and M S give the hard and soft contributions to the EFT matrix element.In M H , at each order in α, the leading power of Z is given by the explicit expression (4), obtained from the amplitudes (2).In particular, the leading-in-Z anomalous dimension is obtained from the µ H dependence of Eq. ( 4), cf.Eq. ( 9) below.
We may proceed to analyze the renormalization group properties of weak-current operators in the EFT.Radiative corrections enhanced by large logarithms, L ∼ log(Λ nuc./E), are determined by the anomalous dimensions of the operators in ( 6) and (7), which are spinstructure independent, i.e., γ we note several interesting all-orders properties: • Powers of Z greater than the power of α do not appear [62].
• The leading series involving (Zα) n sums to This result is obtained by differentiating Eq. ( 4) with respect to µ H .
When Z = 0, the problem reduces to a heavy-light current operator.Using our new result for γ (1) [42] and property (10), the complete result through three-loop order at arbitrary Z is n−1 , n = 1, 2, 3, are known from the heavy quark literature [64].Our result for γ (1) 2 disagrees with Ref. [39], which did not include the full set of relevant diagrams at O(Z 2 α 3 ) [42].Note that properties ( 9) and (10) also determine the anomalous dimension at order Z 4 α 4 and Z 3 α 4 .
Renormalization group analysis.An important advantage of identifying the Fermi function as a field theory object is the ability to resum large logarithms, ∼ log(Λ nuc./E), at high perturbative orders using renormalization group methods.Consider the solution to the renormalization group equation where α is the MS QED coupling (for one dynamical electron flavor) and  [21].Illustrative values E = 2 MeV, Em = 5 MeV, Λ = 100 MeV are used for the electron energy, maximum electron energy (which enters the one-loop matrix element [35]), and renormalization scale µH = Λ, respectively.The width of the curves is given by varying me/2 < µL < 2Em.Analytic expressions can be obtained using Eq. ( 16) [65].
Consider a medium Z nucleus, counting log 2 (Λ/m) ∼ Z 2 ∼ α −1 .This is relevant for superallowed beta decays contributing to |V ud | extraction, which range from Z = 6 ( 10 C) to Z = 37 ( 74 Rb).Through O(α 3/2 ), the scale dependence is log + γ (2) 1 γ (1) 0 where a H,L = α(µ H,L )/(4π) and the square brackets account for effects at order α 2 , etc. Achieving permille precision demands proper treatment of terms through resummed order α 3 2 .This result ( 16) replaces (and disagrees with) logarithmically enhanced contributions at order Z 2 α 3 in the "heuristic estimate" of Sirlin and Zucchini[69].Using our new result for γ (1) 2 [42] we compare to this heuristic estimate, and investigate the convergence of perturbation theory in Fig. 1.Here we fix µ H , and compute the product of |C(µ L )/C(µ H )| 2 and the squared operator matrix element at µ L , varying µ L as an estimate of perturbative uncertainty [70].Normalizing to the leading Fermi function (known analytically to all orders) this quantity corresponds to the outer radiative correction appearing in the beta decay literature, (cf.Eq. ( 11) of Supplemental Material).We note that Eq. ( 11) is in fact sufficient for a resummation of C(µ H )/C(µ L ) through O(α 2 ), although for practical applications one would also need currently unknown operator matrix elements at O(Zα 2 ) [71].
Neutron beta decay corresponds to the case Z = 0 (in our convention); we therefore define where the first term is of order α 1 2 , and the second term is of order α 3 2 .The complete result, correct through order α 3 2 , is obtained using (17) together with the one-loop low-energy matrix element.
Even after resumming logarithms in the ratio of hadronic and electron mass scales, log(Λ/m), large coefficients remain in the perturbative expansion of the hard matrix element.While the class of amplitudes summed in the Fermi function are enhanced at small β and large Z, neither limit holds for neutron beta decay [73].The large coefficients can instead be traced to an analytic continuation of the decay amplitude from spacelike to timelike values of momentum transfers.The enhancements are systematically resummed by renormalization of the hard factor M H in the factorization formula (3) from negative to positive values of µ 2 S (cf.Refs.[74,75]), with the result [44] where µ 2 S± = ±4p 2 − i0 and M H (µ 2 S− ) is free of πenhancements.This analysis systematically resums π-enhanced contributions, and does not rely on a non-relativistic approximation.
Discussion.At the outset of our discussion we posed three questions, which are now answered: 1) The scale r −1 appearing in the Fermi function ( 1) is unambiguously related to a conventional MS subtraction point, cf.Eq. ( 5). 2) The Fermi function is identified as the leading-in-Z contribution to the matrix element from the effective Lagrangian (6).Other radiative corrections are systematically computed using the same Lagrangian.3) Numerically enhanced contributions in neutron beta decay arise from perturbative logarithms | log[(−p 2 − i0)/(p 2 + i0)]| = π, and can be resummed to all orders.The result (18) differs from the nonrelativistic Fermi function ansatz [37,76] beginning at two loop order.
Our EFT analysis allows us to systematically resum large perturbative logarithms, and to incorporate corrections that are suppressed by 1/Z or E/Λ.New results include: 1. New coefficients in the expansion of the anomalous dimension for beta decay operators.We have computed the order Z 2 α 3 coefficient for the first time [77], and found a new symmetry linking leading-Z and subleading-Z terms in the perturbative expansion.Using our new result, and the existing HQET literature, we show that the first unknown coefficient occurs at four loops, at order 2. New results for the large-Z asymptotics of QED radiative corrections to beta decay.We supply the infinite series of terms of order α(Zα) 2n+1 log(Λ/E), replacing Wilkinson's ansatz [78], and present a new result for the term of order α(Zα) 2 log(Λ/E), replacing Sirlin's heuristic estimate [38].We provide the EFT matrix element to all orders in Zα and clarify its relation to the historically employed Fermi function [43].
3. An all-orders resummation of "π-enhanced" terms in neutron beta decay, replacing the Fermi function ansatz.This substantially improves the convergence of perturbation theory, and is important for modern applications to neutron beta decay [44].
The same formalism applies to any situation involving charged leptons and nuclei, provided the lepton energy is small compared to the inverse nuclear radius.
An immediate conclusion of our study is that the existing estimate for O(Z 2 α 3 ) corrections is incorrect.Focusing on the dominant logarithmically enhanced terms, the coefficient "a" in Sirlin's heuristic estimate [38,39], changes.For the 9 transitions with smallest Ft uncertainty (at or below permille level), this leads to shifts ranging from 1.1×10 −4 for 14 O to 1.4×10 −3 for 54 Co [96] i.e., an order of magnitude larger than the estimated uncertainty on the outer radiative correction [21].We observe that these shifts are comparable in magnitude to the CKM discrepancy, [21], and with a sign that goes in the direction of resolving the discrepancy.Accounting for these strongly Z-dependent corrections should also impact New Physics constraints such as on scalar currents beyond the Standard Model [21].A complete determination of the long-distance radiative corrections at the 10 −4 level will require revisiting the O(Zα 2 ) matrix element in the point-like EFT considered here; this work is ongoing.Future work will address factorization at subleading power, and investigate the impact on phenomenology including hadronic [12,40] and nuclear [15,97] matching uncertainties.
Feynman rules for the hard-scale Lagrangian (19), with gauge parameter ξ and photon mass λ Consider the Lagrangian (Eq.( 6) and Eq. ( 7) of the main text) describing physics below hadronic and nuclear scales: where v µ = (1, 0, 0, 0) in the nuclear rest frame.For the superallowed beta decay case we have scalar heavy particle fields , and Dirac structures Γ UV = C and Γ e = / v(1 − γ 5 ).For the neutron beta decay case we have fermionic heavy particle fields v , and Dirac structures Γ UV = C V γ µ + C A γ µ γ 5 and Γ e = γ µ (1 − γ 5 ).Coefficients C, C V and C A are determined by matching the effective theory (19) to the quark-level Standard Model Lagrangian.Feynman rules corresponding to this Lagrangian are shown in Fig. 2.
For physics below the electron mass scale, electrons of a given four velocity v ′ µ = p µ /m (plus soft radiation) are described by a Lagrangian where the electron is also represented by a heavy-particle field [98], Feynman rules corresponding to this Lagrangian are shown in Fig. 3.It is readily seen that the heavy particle effective theory Feynman rules ensure that the most general Dirac structure is given by the square bracket in Eq. ( 20) (note that we have From the soft Lagrangian (20) we may read off the complete amplitude in the factorized form, Eq. (3) of the main text.Writing the complete amplitude as we identify and the soft matrix element is given by 3. Feynman rules for the soft-scale Lagrangian (20) involving the heavy electron field.

REAL RADIATION
Real photon radiation does not contribute to the leading-in-Z radiative correction described by the Fermi function (cf.Eq. ( 1) of the main text), but does contribute at subleading orders.The Lagrangian (19) describes physics below hadronic and nuclear scales, including arbitrary photon radiation.In particular, the solution to the renormalization group equation, Eq. ( 12) of the main text (cf.Eqs. ( 13), ( 16) of the main text), applies to both processes with and without real radiation, and systematically resums large logarithms ∼ log(Λ/m).
To demonstrate the application of ( 19) to real photon radiation, let us consider the leading, order α, correction, computed from the diagrams shown in Fig. 4. Let us write The virtual and real soft contributions are given by Feynman diagrams in the soft theory, cf.(20) and Fig. 3: where real photons of energy smaller than ∆E are included (we may take ∆E ≪ m).The remaining hard contributions are where E m is the nuclear energy difference, equal to the maximum possible electron energy.It is readily seen that the dependence on the photon mass regulator cancels in the sum of ∆ S,V and ∆ S,R , and the soft photon threshold ∆E cancels in the sum of ∆ S = ∆ S,V + ∆ S,R and ∆ H = ∆ H,V + ∆ H,R .Further, the dependence on the arbitrary factorization scale µ S cancels between ∆ S and ∆ H , and the dependence on µ H cancels between ∆ H and σ tree (µ H ).

NUMERICAL IMPACT OF RADIATIVE CORRECTIONS
Our analysis determines new contributions to structure-independent radiative corrections to superallowed nuclear beta decay rates.These corrections directly impact the associated extraction of |V ud |.In the notation of Ref. [21] we have where M is the effective operator matrix element and dΠ denotes integration over (electron/positron, neutrino and real photon) phase space and sum over lepton spin states [99].The product of denominators in Eq. ( 29) extracts the conventional integrated Fermi function prefactor appearing in traditional rate formulas [21].Here, as in the main text, the short distance regulator in F (Z, E) is identified as r H,L ≡ µ H,L e γ E .Taking µ H ∼ Λ and µ L ∼ m, the round bracket in Eq. ( 29) is free of large logarithms L = log(Λ/m).Further, because we have divided by the leading-in-Z expression, the perturbative expansion contains only terms Z m α n with m < n.In the "intermediate Z" power counting discussed in the main text, the relevant terms are at order α [35] (cf.Eq. ( 24)) and at order Zα 2 ∼ α 3 2 .The order Zα 2 matrix element has been considered using a different regulator scheme in Ref. [38].Further discussion of this matrix element will be presented elsewhere [44].
Logarithmic enhancements are contained in the square bracket in Eq. (29).Our results for the resummed coefficient, Eq. ( 16) of the main text, yield expressed in terms of the onshell coupling α.The terms at order αL and at order α 2 ZL have been included as part of the order α [35] and order Zα 2 [38] corrections.The term at order α 3 Z 2 L 2 is a correction to the two-loop Fermi function, and has been included in Ref. [38].The term at order α 3 Z 2 L corresponds to γ (1) 2 and corrects a previous result in the literature, as noted in the main text.The remaining terms have not to our knowledge been included in analyses of nuclear beta decay: the term at order α 2 L involves the two-loop anomalous dimension γ (2) 1 ; the terms at order α 2 L 2 and α 3 L 3 are higher-order corrections that involve the one-loop anomalous dimension γ (1) 0 ; the term at order α 3 ZL 2 is a higher order correction induced by the Zα 2 contribution (involving γ Let us focus on the α 3 Z 2 L correction and determine the numerical impact of our new calculation compared to the Jaus-Rasche estimate of the anomalous dimension.We use the updated value from their 1987 paper [39], which determines the coefficient a in Sirlin's heuristic estimate of δ 3 (i.e., a = (π 2 /3 − 3/2)/π ≈ 0.5697 as written in Footnote 10 of Ref. [48]).We find instead a = −γ (1) 2 /(32π 3 ) ≈ −0.4313.We define ∆a = −0.4313− 0.5697 ≈ −1.001.To estimate the size of the logarithm we set L = log(Λ/m) with Λ = √ 6/ ⟨r 2 ⟩ and ⟨r 2 ⟩ taken from Ref. [100].In the convention we follow in the main text, the two nuclei involved in the transition have charge Z and Z + 1.
The superallowed transitions that are important for |V ud | extractions are of the form [A, Z + 1] → e + ν e [A, Z] and for simplicity we will always choose transitions with a positron in the final state.This means that Z will always correspond to the daughter nucleus.For the transitions with relatively small errors [21] we find the results in Table I [101].In each case, the shift is larger than the error currently ascribed to δ ′ R .A definitive statement at an accuracy on the order of 100 ppm will require a renewed scrutiny of the matrix element at O(Zα 2 ) in the point-like effective theory.

FIG. 4 .
FIG.4.Feynman diagrams for first order corrections to neutron beta decay rate computed from Eq. (19).Wavefunction renormalization is not shown.

TABLE I .
[21]t in the outer radiative correction at order Z 2 α 3 log(Λ/m), for the 9 transitions with smallest Ft uncertainty in Ref.[21].