On the effective theory of neutrino-electron and neutrino-quark interactions

We determine the four Fermi effective theory of neutrino interactions within the Standard Model including one-loop electroweak radiative corrections, in combination with the measured muon lifetime and precision electroweak data. Including two-loop matching and three-loop running corrections, we determine lepton coefficients accounting for all large logarithms through relative order $\cal{O}(\alpha \alpha_s)$ and quark coefficients accounting for all large logarithms through ${\cal{O}}(\alpha)$. We present four-fermion coefficients valid in $n_f=3$ and $n_f=4$ flavor quark theories, as well as in the extreme low-energy limit. We relate the coefficients in this limit to neutrino charge radii governing matter effects via forward neutrino scattering on charged particles.

scales. Electroweak scale physics is computed perturbatively and is represented by four-fermion operator coefficients. The effective operator matrix elements provide a rigorous starting point for nonperturbative (e.g. lattice QCD) evaluation and/or experimental analysis of the relevant hadronic amplitudes.
In this work, we construct the low-energy effective field theory of neutrino-lepton and neutrino-quark interactions suitable for predictions with sub-percent accuracy. Predictions for low-energy processes including bremsstrahlung and virtual QED corrections can be evaluated using this theory as starting point (see e.g. Ref. [45] for an application to neutrino-electron scattering). We match the effective theory to the Standard Model (SM) at the electroweak (EW) energy scale, evaluating all effective couplings and determining scale-independent combinations. We then solve renormalization group equations (RGEs) and heavy quark threshold matching conditions, and present four-fermion coefficients at the GeV energy scale in n f = 3 and n f = 4 flavor QCD. We also present coefficients in the extreme low-energy QED limit and determine neutrino charge radii.

Effective theory and tree level matching
Neglecting corrections of order αm 2 f /m 2 W and |q 2 |/m 2 W , where m f denotes a fermion mass, and q 2 denotes an invariant momentum transfer, the structure of neutrino interactions with quarks and charged leptons in the Standard Model is severely constrained. 2 Besides the usual kinetic, mass, and QED+QCD gauge coupling terms for neutrinos, charged leptons, and quarks (including the mass mixing matrix of neutrinos), the effective Lagrangian consists of dimension 6 four-fermion operators and neutrino-photon couplings, where e denotes the positron charge. The sums in Eq. (1) run over 3 lepton flavors (ℓ = e, µ, τ ) and n f active quark flavors (q = u, c and q ′ = d, s, b for n f = 5). P L = (1 − γ 5 )/2 and P R = (1 + γ 5 )/2 are projection operators onto left-and right-handed states, respectively. For neutrino scattering applications, it is convenient to replace the neutrino-photon operator by an equivalent combination of four-fermion operators in the effective theory, obtained by the field redefinition Under Eq. (2), charged current coefficients c and c qq ′ remain unchanged while in the neutral current sector The four-fermion coefficients can be determined order-by-order in perturbation theory by matching amplitudes in the full (SM) theory and the effective theory. In the MS renormalization scheme [46], the lepton coefficients may be written 2 Power corrections to four-fermion theory can be readily estimated for specific processes. For neutrino-electron scattering, these corrections are negligible, at the 10 −6 level for Eν 10 GeV [45]. For neutrino-nucleon scattering, leading corrections to four-fermion theory scale as |q 2 |/M 2 W 2mN Eν/M 2 W where mN ≈ 1 GeV is the nucleon mass and Eν is the neutrino beam energy. For Eν ∼ few × GeV, these corrections can amount to few-permille contributions. However, form factors in hadronic amplitudes typically suppress the region of large |q 2 | making these corrections further subdominant.
For the quark coefficients, we have where for up-type quarks q, and down type quarks q ′ , V qq ′ is the corresponding element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [47,48]. We write M W and M Z for the mass of the W ± and Z 0 bosons, related as M W = M Z cos θ W , using the notation s W = sin θ W and c W = cos θ W . The neutrino-photon coupling is expressed as At tree level, we have the well-known expressions [40,41] for lepton couplings, and for quark couplings, The neutrino-photon coupling vanishes at tree level, g ν ℓ γ = 0 and c ν ℓ γ = O(αG F ) . Here Q q is the quark electric charge in units of the positron charge (Q u = 2/3, , and α(µ) is the electromagnetic coupling constant.

One-loop electroweak matching
We perform the matching onto the effective theory (1) by first integrating out heavy vector and scalar bosons, W ± , Z 0 and H, and the top quark, t, in the Standard Model. Before the field redefinition (2), the neutrino-photon coupling is determined by computing the neutrino scattering process ν ℓ (p) → ν ℓ (p ′ ) in a background electromagnetic field. 3 After MS renormalization, the one-loop contribution in Feynman-t' Hooft gauge is After the field redefinition (2), this coefficient vanishes g ν ℓ γ → 0 and all other couplings change according to Eq. (3). Reproducing one-loop EW radiative corrections in the Standard Model [15,16,[18][19][20] in Feynman-'t Hooft gauge with massless leptons and quarks, besides the top, and subtracting the corresponding diagrams in the effective theory, we obtain the following reduced couplings for neutrino-lepton interactions after the field redefinition (3): where r t = m 2 t /M 2 Z and r H = M 2 H /M 2 Z with masses of the top quark m t and the Higgs boson M H . Tree level couplings g 0 L,R have been specified above and depend on renormalization scale µ through s 2 W . To perform the matching, we have enforced the vanishing of the Higgs tadpole (the amputated and renormalized Higgs field one-point function) as a renormalization condition, but use MS renormalization for non-tadpole counterterms. This hybrid renormalization scheme leads to compact, but gaugedependent, expressions for the reduced couplings (10) and for MS masses. The prefactor, αM −2 W s −2 W , in Eq. (4), and the tree level reduced couplings in (7) must be evaluated in this scheme in order to employ the expressions (10). Here α(µ), as well as M W , M Z and s W , refer to the full SM particle content. We have verified that the complete effective couplings (5) are gauge independent and have obtained the same one-loop matching onto the effective Lagrangian of Eq. (1) in an arbitrary R ξ gauge. 4 We have also obtained the one-loop coefficients using on-shell renormalization, first expressing the effective couplings (5) in terms of on-shell (pole) M W , M Z , and low-energy (Thomson limit) α, and then expressing these on-shell quantities in terms of M W (µ), M Z (µ) and α(µ). In performing the matching, we have regulated infrared divergences in both full and effective theories with small photon and fermion masses. The matching could be performed in the exact massless limit but would require consideration of new operator structures in the effective theory to account for Fierz rearrangements in d = 4.
Similar to Eq. (10), we can determine neutrino-quark couplings as where ω t = m 2 t /M 2 W . Here a stands for the scheme parameter defining the behavior of γ 5 in d = 4. It appears in the reduction of Dirac matrices resulting from two-boson exchange diagrams, e.g.
The starting point for our renormalization analysis below is n f = 5 flavor QCD, with four-fermion coefficients determined at µ = M Z 5 through O(αG F ) in Eqs. (10) and (11). For lepton coefficients (10), we also include complete corrections through O(αα s G F ) [55][56][57][58][59][60][61], arising from gluons attached to closed quark loops. Details about this correction, and the determination of appropriate MS masses, are discussed in the Appendix A. Uncomputed corrections of O(αα s G F ) to quark coefficients are left to future work; these corrections are small compared to hadronic uncertainties in current neutrino scattering analyses. Note that in Eqs. (10) and (11) we have performed the field redefinition (2) and we maintain the condition c ν ℓ γ = 0 for renormalized coefficients.

Standard Model inputs
Having computed expressions for the four-fermion operator coefficients, let us define the numerical inputs to these expressions. We begin by isolating coefficient combinations that are independent of renormalization scale, and combinations that vanish up to neglected O(αm 2 f /m 2 W ) corrections. As a consequence of Ward-Takahashi identities [62,63] and the partial conservation of vector and axial-vector currents in QED+QCD gauge theory, the evolution equation for couplings in the neutral current sector is governed by vacuum polarization diagrams as in Fig. 1, where f denotes an active lepton or quark flavor in the theory, N c,f is the number of color degrees of freedom (N c,f = 1 for leptons, N c,f = 3 for quarks), and S = L, R. In particular, the running of c f S /Q f does not depend on f or S. This property of the anomalous dimension matrix holds also at higher orders so that 6 All couplings in the neutral current sector may thus be written as linear combinations of scale-invariant quantities and a single scale-dependent coupling. Neglecting fermion mass corrections of order αm 2 f /m 2 Z , 5 For definiteness, we employ the mass of Z boson, M PDG Z from Eq. (19), as the scale µ for all MS quantities at µ = MZ. 6 Diagrams as in Fig. 1 with the exchanged photon replaced by a gluon vanish by color symmetry. Diagrams with two photons exchanged between the loop in Fig. 1 and exterior fermion legs vanish in the sum of direct and crossed contributions according to Furry's theorem. The sum of three-photon exchange contributions is UV finite [64,65] and does not affect the running. Similar arguments apply to two-and three-gluon exchange diagrams. While not phenomenologically important, it may be formally interesting to investigate this structure to higher orders in QED+QCD couplings.
In the charged current leptonic sector described by coefficient c(µ), the operator anomalous dimension vanishes, and we may define the scale-independent Fermi constant [66,67] In the charged current semileptonic sector, the presence of three external charged fermion lines results in scale dependence of c qq ′ (µ), cf. Eq. (23) below. The remaining independent coefficients are the scale- The quantities G q (q = u, d) are differences of left-handed couplings c q L and c νµe L , while tilded quantities Having reduced the scale dependence to c R (µ) in the neutral current sector and c qq ′ (µ) in the charged current sector, we proceed to specify numerical inputs to the intial conditions of RGE. For numerical evaluation, we employ high order running and threshold matching corrections for MS QCD and QED couplings 7 with input values [1] α (5) s (µ = M Z ) = 0.1187 (16) , α (5) (µ = M Z ) −1 = 127.955 (10) .
The MS quantities α (5) and α (5) s in Eq. (17) have been defined in the theory containing W ± , Z 0 and H, i.e., the complete SM particle content except the top quark. For values at lower scales and in the theory with top quark, see Appendix B. We use the Fermi constant determined from muon decay, The quantity M PDG Z represents a fit parameter in Z 0 lineshape analysis; its relation to the corresponding pole mass and MS mass is discussed in the Appendix A. The quantity m PDG t represents the top quark pole mass. The translation to the corresponding MS mass, relevant for O(αα s ) corrections to fourfermion coefficients, is discussed in the Appendix A. The quantities (17), (18), and (19), together with CKM elements for semileptonic charged current processes, fully determine the electroweak scale matching coefficients.
Leptonic and semileptonic four-fermion coefficients at µ = M Z are displayed in Tables 1 and 2, respectively. For reference, we also display the determination of s 2 W (µ = M Z ) in the tables. 8 9 To 7 Evolution of αs(µ) is computed with five-loop QCD running and four-loop threshold matching corrections, ignoring QED. Evolution of α(µ) is computed with two-loop QED and O(α 3 s ) corrections to running, and one-loop QED and O(α 2 s ) corrections to threshold matching. 8 Note that we employ the MS scheme for s 2 W , in contrast to the quantityŝ 2 Z of Ref. [1] which includes also a finite subtraction [56]. Our convention corresponds to the quantityŝ 2 ND of Ref. [1]. 9 Choosing the right-handed coefficient cR (µ) as the relevant scale-dependent coupling avoids difficulties in defining s 2 W at low energies [68][69][70]. Table 1: Neutrino-lepton four-fermion coefficients. G F andG e are in units 10 −5 GeV −2 . The second and third lines of the table provide our evaluation using inputs (M W , M Z , α (5) ) and (G F , M Z , α (5) ) respectively. Uncertainty due to Standard Model input parameters is denoted with index i. Perturbative uncertainty is denoted with index p.  (83) Tables 1 and 2 correspond to Standard Model input parameters (denoted by index "i"), and to uncalculated higher-order perturbative corrections (denoted by index "p"). The leading unaccounted corrections appear at O αα 2 s , α 2 for leptonic coefficients and at O (αα s ) for quark coefficients. 10 The perturbative uncertainty, at the level 10 −4 , is estimated by varying the matching scale in the range M 2 Z /2 ≤ µ 2 ≤ 2M 2 Z for scale-invariant quantities. Note that the scale dependence is larger forG b due to the large CKM matrix element V tb . This uncertainty is subdominant to SM parameter input uncertainty when using EW inputs, represented by the second line of Tables 1 and 2. With the G F constraint, represented by the third line of Tables 1 and 2, perturbative uncertainty dominates but is well below current hadronic uncertainties for neutrino scattering applications.

Running
The RGE in the neutral current sector mixes all quark and lepton operators involving a given neutrino flavor: Here β f denotes the contribution of fermion f to the QED beta function: where for leptons and quarks we have with N c = 3. The O(α 3 s ) contribution does not change results of this work within significant digits. In the charged current sector, the coefficient c is scale independent while the anomalous dimension of c qq ′ is not zero [52,[85][86][87][88][89][90] Accounting for the scale-dependence of α and MS vector-boson masses M W and M Z , it is readily verified that the µ-dependence of the one-loop matching coefficients, (4) and (5), obeys the evolution equations derived from the UV structure of effective theory operators, Eqs. (20) and (23). We solve the RGE between particle thresholds numerically for the coefficients c R (µ), c qq ′ (µ), using high order running of α (µ) and α s (µ). 7 Integrating out heavy b and c quarks, we account for small threshold matching effects through O(αα s ). We use the following values for bottom and charm quark masses [1], The running from EW scale to GeV energies yields the effective couplings in Table 3. We have employed the weak scale couplings from the last row of each of Tables 1 and 2. The uncertainty from higher-order perturbative corrections is estimated by varying matching scales µ 2

Low-energy leptonic theory
For certain processes, the momentum transfer entering the loop of Fig. 1 is below the scale where perturbative QCD is applicable. An important example is the neutrino-electron scattering process where momentum transfer is bounded by Q 2 ≤ 2m e E ν 0.01 GeV 2 for incoming neutrino energy E ν 10 GeV. This process is described by an effective theory in which hadrons and heavy leptons are integrated out and only neutrinos and electrons remain as dynamical fields. Here we determine the four-fermion couplings in this low-energy leptonic theory. We perform the matching in a series of steps, first integrating Table 3: Effective couplings in the Fermi theory of neutrino-fermion scattering with n f quark flavors, at different renormalization scales, in units 10 −5 GeV −2 . τ is not present in the theory described by last three rows. The final row gives couplings to ν τ .  Table 4: Effective couplings (in units 10 −5 GeV −2 ) in the Fermi theory of neutrino-lepton scattering, at renormalization scale µ = m µ (in the theory with neutrinos, e and µ) and in the low-energy limit at µ = m e (in the theory with neutrinos and e). The error is dominated by the light-quark contribution. out hadrons, then muons. For the hadronic contribution to effective couplings, we employ results from Ref. [45]. In a final step, we also describe the extreme low-energy limit applicable to forward scattering on nonrelativistic electrons. Note that integrating out a heavy charged lepton violates the equality of couplings c ν ℓ ℓ ′ R for any ℓ and ℓ ′ , and of c ν ℓ ℓ ′ L for any ℓ = ℓ ′ . Consequently, the conventional definition of Weinberg angle in terms of effective couplings is flavor-independent only above the τ -lepton mass scale. Here we describe details for decoupling the µ lepton; a similar procedure involving τ resulted in the last rows of Table 3.
In the theory valid below the hadron mass scale, i.e., µ m π , only neutrino couplings to muons and electrons contribute to running. The renormalization group equations in the theory above the muon mass scale, i.e., m µ ≤ µ ≤ m π , are given by where c νee In the theory below the muon mass scale and above the electron mass scale, i.e., m e ≤ µ ≤ m µ , RGEs for different neutrino flavors decouple: resulting in distinct right-handed couplings in the scattering of electron-and muon-type neutrinos: 12 For momentum transfers below the electron mass scale, the effective theory in the one-electron sector describes the interaction of neutrinos with a static electron source. 13 In the theory valid for µ ≤ m e , the electron is no longer involved in dynamics and the four-fermion couplings are scale invariant: These effective couplings may be interpreted as effective radii, analogous to the description of low momentum neutron scattering on charged particles [92,93]. In detail, we may define the effective radius of interaction as where e 2 0 = 4πα 0 denotes the QED coupling in the low-energy limit. The matching from the n f = 3 or n f = 4 quark level theory to the leptonic theory involves nonperturbative QCD. Taking the relevant photon vacuum polarization tensorΠ γγ (0) from experimental data, the result for electron neutrino effective radius is The uncertainty 0.08 fm 2 results fromΠ γγ (0) = 1 ± 0.2 as in Ref. [45]. The difference between radii for different neutrino flavors, ν ℓ and ν ℓ ′ , is independent of the target particle T, when T is distinct from the charged lepton partner of the incident neutrinos. Examples include ν ℓ = ν µ and ν ℓ ′ = ν τ scattering on electrons, T = e; or any flavors ν ℓ and ν ℓ ′ scattering on protons or neutrons. In these cases, the difference r 2 represents a radius that is intrinsic to the neutrino species. 14 Starting from the theory with three active charged leptons (e.g. the n f = 4 flavor theory renormalized at µ = 2 GeV in Table 3), the differences in radii are induced by loops of charged leptons. Summing over leptons in the loop and using the relations (4), we obtain 15 In particular, 12 It is convenient to perform the matching between effective theories with and without the muon degree of freedom at exactly µ = mµ. In this case, threshold matching corrections to effective couplings and running α vanish up to neglected corrections of relative order α 2 . 13 Corrections to the static limit may be described by nonrelativistic EFT. 14 Our definition of radii, in terms of four Fermi operators of left-handed neutrinos, is independent of the neutrino mass sector, in particular whether right-handed neutrinos are introduced to form a Dirac mass, or a Majorana mass term is included. For a general discussion of electromagnetic interactions of Majorana neutrinos, see Ref. [94,95]. 15 Our normalization for the lepton vacuum polarization function is as in Ref. [45]: Π(0, m, µ) = Nc 1 3 ln µ 2 m 2 + αs 4π CF ln µ 2 m 2 + 15 4 , in terms of pole mass m. For QED, we take Nc → 1, CF → 1, αs → α. We also have replaced α(µ) by α0 including vacuum polarization, electron vertex and field renormalization corrections.
For the special case of ℓ = τ and ℓ ′ = µ, the same result is obtained from Eq. (31) upon substitution of low-energy coefficients from Table 4 Table 4, hadronic corrections cancel in the difference. The evaluations differ by matching corrections of O(α 2 ), present in Eq. (33) but omitted in the RG analysis; this difference impacts digits not displayed in Table 4.
The effective potential in matter can be expressed in terms of the effective radii as a sum over all target particles with particle charge Q T and number density n T . In a charge-neutral medium consisting of protons, neutrons and electrons, differences r 2 ντ T − r 2 νµT enter with an opposite sign for positively and negatively charged particles resulting in V ντ = V νµ up to corrections suppressed by powers m 2 τ /M 2 W [96]. Such differences would appear as corrections to the weak scale matching coefficients in (1), and as O(G 2 F ) corrections to forward scattering computed using (1).

Summary
We have determined the low-energy neutrino-fermion effective field theory in MS renormalization scheme as the basis for neutrino scattering on electrons [45], nucleons and nuclei at sub-percent level. Electroweak scale coupling constants were determined by matching to the Standard Model including complete oneloop electroweak corrections, two-loop mixed QCD-electroweak corrections in the lepton sector, and the γ 5 scheme dependence of effective operators. Among the eleven independent effective couplings, only two depend on the scale. Solving the renormalization group equation for these couplings, we have determined all parameters in the quark level effective Lagrangian with n f = 3 or n f = 4 quark flavors. A complete error budget due to parametric inputs and higher-order perturbative corrections is presented. Using experimental data and SU(3) flavor symmetry constraints, we have evaluated hadronic contributions to determine the matching onto the low-energy theory involving leptons and the extreme low-energy theory describing neutrino interactions with static electric charge distributions. The hadronic correction provides the dominant source of uncertainty in low-energy neutral-current interactions. As a byproduct of our analysis, we revisited the comparison of the Fermi coupling constant G F evaluated at the electroweak scale to extractions from muon lifetime measurements; the comparison shows only 0.9σ tension with 6 × 10 −4 relative difference.
A MS masses of vector bosons and O(αα s G F ) corrections To relate pole and MS masses of vector bosons at some scale µ, we consider renormalization of the vector boson propagator D µν , using the example of the Z boson as illustration: with vector boson self-energy The pole mass is determined by the inverse propagator: We have cross checked expressions in terms of scalar integrals from Refs. [20,102] in Feynman-'t Hooft gauge and have verified the gauge independence of mass renormalization at one loop when including Higgs tadpoles. For numerical evaluations, we exploit analytical expressions for scalar integrals obtained following Refs. [103,104], work in Feynman-'t Hooft gauge and do not include Higgs tadpoles. We account for the leading QCD corrections of Refs. [56][57][58][59][60][61] representing the fermionic contribution to the following MS differences (cf. Ref. [56]) at renormalization scale µ, in terms of MS mass of top quark: The normalization at q 2 = 0 is given by [56,58,60,61] Σ W T µ, q 2 = 0 = The O(α) correction is given by with Λ (r t ) = √ 4r t − 1 sin −1 (4r t ) −1/2 . The O(αα s ) contribution is expressed in terms of B

B QED and QCD couplings
In order to evaluate the expressions (4) and (5) at µ = M Z , we require values for M W (µ), M Z (µ), s W (µ) and α(µ), evaluated in the MS scheme without tadpoles, with the full SM particle content. We begin by deriving the necessary QED and QCD couplings from Ref. [1], which were defined in the theory without top quark (cf. Refs. [55,68]). We use the same precision of running and matching as described in Section 3. 7 The translation from the inputs (17) For comparison, our inputs together with solution of RGEs and threshold matching conditions yield α (4) (µ = m τ ) −1 = 133.476 (7), and α (4) s (µ = m τ ) = 0.325 (15).