Consistent description of angular correlations in β decay for Beyond Standard Model physics searches
CConsistent description of angular correlations in β decay for Beyond Standard Modelphysics searches L. Hayen
1, 2, ∗ and A. R. Young
1, 2 Department of Physics, North Carolina State University, Raleigh, 27607 North Carolina, USA Triangle Universities Nuclear Laboratory, Durham, 27710 North Carolina, USA (Dated: October 7, 2020)Measurements of angular correlations between initial and final particles in β decay remain one ofthe most promising ways of probing the Standard Model and looking for new physics. As experimentsreach unprecedented precision well into the per-mille regime, proper extraction of results requires oneto take into account a great number of nuclear structure and radiative corrections in a procedurewhich becomes dependent upon the experimental geometry. We provide here a compilation andupdate of theoretical results which describe all corrections in the same conceptual framework, pointout pitfalls and review the influence of the experimental geometry. Finally, we summarize thepotential for new physics reach. I. INTRODUCTION
Precision measurements of (nuclear) β decay observ-ables have to a significant extent defined the current sta-tus of the electroweak sector of the Standard Model [1–4].Additionally, they are one of several promising and com-plementary pathways of finding and studying possible ex-tensions to the latter in a theoretically relatively cleanenvironment [5]. In particular, due to the low energytransfers available in nuclear decays ( Q β (cid:46)
10 MeV),many of the intricacies contained in the Standard Modelare limited to higher-order effects and typically serve onlyto renormalize a number of coupling constants while leav-ing the bulk of the kinematic structure untouched. Thisis to the benefit of measurements of angular correlationsbetween initial and/or final states as these are by defini-tion relative effects, and generally do not require knowl-edge of all details of the decay distribution. Likewise,their measurement is experimentally promising as theirrelative nature allows for the cancellation of many oth-erwise dominant systematic uncertainties. Over the pastdecades, intense study in the neutron [6–9] and mirrorsystems [10, 11] have helped constrain and probe CKMuniversality and the presence of exotic scalar and tensorcurrents at a competitive level with those obtained fromthe LHC [12–14]. Already at the current experimentalprecision, however, several sources of theoretical higher-order input are required. We report here on a consistentdescription of the required corrections that experimen-tal analyses need to take into account as the precisionreaches and exceeds the per-mille level. While several ofthese results can be found in the literature, we argue thatit is beneficial to put these results together in a compre-hensive format as experimental analyses do not appearto be treated uniformly in the literature. This can leadto a incorrect comparison between different experimentalresults which in turn weakens their impact. ∗ Corresponding author: [email protected]
To leading order, exotic scalar or tensor currents inthe weak interaction typically manifest themselves in theappearance of the so-called Fierz interference term, b F .It modifies the total β decay rate through a multiplicativefactor d Γ dW e = d Γ SM dW e (cid:20) b F m e W e (cid:21) , (1)where d Γ SM is the Standard model differential decay ratewith W e the total β particle energy. In the more mod-ern language of β decay effective field theories (EFT), itdepends on new couplings (cid:15) according to [14–17] b F = ± γ
11 + ρ × Re (cid:26) g S (cid:15) S g V (1 + (cid:15) L + (cid:15) R ) + ρ g T (cid:15) T − g A (1 + (cid:15) L − (cid:15) R ) (cid:27) , (2)where the upper (lower) sign corresponds to β − ( β + )decay, γ = (cid:112) − ( αZ ) , g A is defined as positive and all (cid:15) i correspond to effective couplings arising due to newphysics, with (cid:15) i ∼ ( M W / Λ) , with M W the mass of the W boson and Λ the scale of new physics. Per definition,Λ (cid:29) M W and is typically at least of order TeV assumingnaturalness arguments. The form factors are defined as g i = (cid:104) p | ¯ uO i d | n (cid:105) , where g S = 0 . g T = 0 . β -asymmetry ( A β ), from which the discovery of parity vi-olation was made [19], while the second is the β - ν ( a βν )correlation, which helped solidify the V - A structure ofthe weak interaction [20, 21]. In an experimental settingone typically defines the differential decay rate accordingto their zeroth-order expressions d Γ SM dW e d Ω e d Ω ν = d Γ (cid:20) A β P β ˆ J · (cid:126)p e + a βν (cid:126)p e · (cid:126)p ν W e W ν (cid:21) (3) a r X i v : . [ nu c l - t h ] O c t where the isotropic decay rate is [22] d Γ = G F π F L C ( W e ) g ( W e , W ) × K ( W e , W ) p e W e ( W − W e ) . (4)Here ˆ J is a unit vector along the initial polarization, P = (cid:104) M (cid:105) /J is the effective polarization, W e and W arethe β particle total energy and endpoint energy in units ofthe electron rest mass, respectively, (cid:126)p e ( ν ) are the electron(antineutrino) three-momenta, and β = p e /W e = v/c isthe β -particle velocity. Additionally, G F ≈ − m − p isthe Fermi constant, F L is the Fermi function, C ( W )is the spin-independent shape factor, g ( W e , W ) is thewell-known O ( α ) energy-dependent radiative correctionby Sirlin, K ( W e , W ) correspond to higher-order correc-tions of varying nature and pW ( W − W ) is the phasespace factors [22].Naturally, correlations like those in Eq. (3) containdifferent higher-order corrections due to their differingkinematic signature. As a result, Eq. (3) describes effec-tive correlations with several sub-dominant effects foldedin. Some of these originate from nuclear structure, whileothers come about through electroweak radiative correc-tions or kinematic recoil. Additionally, like the appear-ance of the Fierz term, exotic scalar or tensor currentsmodify the effective values of A β and a βν [23]. Bothof these, however, depend only quadratically on exoticcouplings so that their measurement obtains new physicssensitivity mainly from the appearance of the Fierz term.Because of this, measurements are typically interpretedin terms of an effective correlation (cf. Eq. (1)) (cid:101) X = X b F (cid:104) m e /W e (cid:105) (5)where X is any correlation coefficient, and (cid:104) m e /W e (cid:105) isthe average of the m e /W e term weighted by the spectrum(Eq. (4)) over the experimental range. We already notethat the validity of Eq. (5) depends on the experimentalscheme, in particular for the β - ν correlation. This wasthe topic of Ref. [24] and will be reiterated below.The paper is organized as follows. In Sec. II we pro-vide the necessary theory input to calculate the StandardModel β correlations to high precision, with correctionsfrom nuclear structure, kinematics and radiative correc-tions. Sec. III discusses simplifications in mirror systemsand the benefit of near-cancellation for sensitivity to ρ ,the Fermi to Gamow-Teller mixing ratio. Precise mea-surements of the latter are additionally an ingredient inthe determination of | V ud | . Further, we discuss the wayreal experimental analyses are complicated due to a va-riety of effects in Sec. IV. Finally, we discuss the newphysics potential and sensitivity arising from precisionmeasurements of β decay correlations. We attach severalappendices treating kinematic recoil corrections typicallyneglected in multipole formalisms and provide compar-isons to other popular formalisms. II. GENERAL THREE-BODY DECAY RATE
We initiate our discussion through a definition of thegeneral decay rate based on angular momentum conser-vation and the symmetries of the electroweak interaction.While several first-order expressions are available in theliterature, in particular for the neutron [25–27], we aremainly interested here in arbitrary spin changes. We firstprovide the general expression to lay the foundation forour discussion, and provide its rationale in the followingsection.The general three-body β decay rate summed over thehelicities of the final states can then be written as d Γ = G F π F L K ( W e , W ) p e W e ( W − W e ) × f + (cid:88) k ≥ f βνk P k (cos θ βν ) + G k ( J i ) (cid:26) f σek P k (cos θ e )+ f σνk P k (cos θ ν ) + f σ × k P k (cos θ × ) (cid:27) + higher orders (cid:35) dW e d Ω e d Ω ν , (6)where G k is a polarization tensor of rank k of the initialstate, P k (cos θ ) a Legendre polynomial of degree k , f isthe isotropic shape factor and K ( W e , W ) common cor-rections [22]. The ‘higher orders’ in Eq. (6) stands forcorrelations involving more exotic combinations of mo-menta and higher powers (see, e.g., Ref. [27–29]), whichwe neglect here. The angles are defined as followscos θ βν = (cid:126)p e · (cid:126)p ν | (cid:126)p e || (cid:126)p ν | , cos θ e = ˆ J · (cid:126)p e | (cid:126)p e | , cos θ ν = ˆ J · (cid:126)p ν | (cid:126)p ν | , cos θ × = ˆ J · ( (cid:126)p e × (cid:126)p ν ) | (cid:126)p e || (cid:126)p ν | . (7)Comparing to Eq. (3), or more generally to the Jackson-Treiman-Wyld (JTW) categorization [23], one can rec-ognize the usual asymmetries when limiting ourselvesto k = 1. Specifically, the ratio of spectral functions f ik /f reduce to the well-known expressions of the β cor-relations, with f βν /f the β - ν correlation ( a βν ), G = (cid:104) M (cid:105) /J = P and f σe /f , f σν /f , and f σ × /f the β -asymmetry ( A β ), ν -asymmetry ( B ν ), and triple corre-lation ( D ), respectively. A. Nuclear structure and kinematics
All of the spectral functions of Eq. (6) depend on acombination of nuclear structure and QED correctionsfolded in together. Taking for now only the O ( αZ ), low-energy part of the virtual photon exchange (i.e. theCoulomb interaction), the matrix element for β decaycan then be written down in a simple quantum mechanicspicture with initial and final state interaction as [30, 31] M = − πiδ ( E f − E i ) (cid:104) f | T (cid:20) exp (cid:18) − i (cid:90) ∞ dt H Z f ( t ) (cid:19)(cid:21) × H β (0) T (cid:20) exp (cid:18) − i (cid:90) −∞ dt H Z i ( t ) (cid:19)(cid:21) | i (cid:105) (8)with T the time-ordered product, H Z the Hamiltoniandensity describing the Coulomb interaction and H β (0) = G F √ V ud H µ (0) L µ (0) (9)is the Fermi current-current description of β decay, with L µ = ¯ u ( p e ) γ µ (1 − γ ) v ( p ν ) the lepton current. Regardlessof the description of the hadronic current, H µ , it is intu-itively clear from Eq. (8) that the final result depends ona convolution of the initial and final nuclear wave func-tions with the lepton current,all of which are modified be-cause of the Coulomb interaction. Electroweak radiativecorrections beyond the Coulomb interaction depend onlyat higher orders ( O ( α n Z n − ) for n >
1) on details of thenuclear wave functions [22, 32–35], and will be discussedin further detail in Sec. II B for angular correlations.In the simplest case of the J i = 1 / J f = 1 / H µ can be written down explicitly H µ = i ¯ u ( p f ) (cid:26) g V γ µ + i (cid:101) g M M σ µν q ν + (cid:101) g S M q µ − g A γ µ γ + i (cid:101) g T M σ µν q ν γ + (cid:101) g P M q µ γ (cid:27) u ( p i ) (10)from the requirement of Lorentz-invariance and initialand final spinors being on-shell. Here all g i ( q ) are di-mensionless form factors and a function of q = ( p f − p i ) , σ µν = i [ γ µ , γ ν ] and M is the nucleon mass. The absenceof second-class currents and the conserved vector cur-rent (CVC) hypothesis requires (cid:101) g S = (cid:101) g T = 0. Addition-ally, the application of CVC together with the Ademollo-Gatto theorem [36] sets g V = 1 up to corrections of( q/M ) . CVC further allows for the interchange of weakand electromagnetic form factors. For example, for theneutron one can set (cid:101) g M = µ p − µ n = 3 . µ p,n arethe anomalous magnetic moment of proton and neutron,respectively. Finally, the partially conserved axial cur-rent relates g A and (cid:101) g P through the Goldberger-Treimanrelation (cid:101) g P ( q ) = − g A (0) (2 M ) m π − q (11)assuming pion-pole dominance. In the limit of zero mo-mentum transfer (as is appropriate in β decay) one ob-tains (cid:101) g P ≈ − . In this simple system then, one is left While the large magnitude of (cid:101) g P offsets somewhat the strongattenuation of the pseudoscalar matrix element, (cid:104) p | γ | n (cid:105) ∼ ( v n /c ) , its influence is felt only at the 10 − level. In the caseof strong cancellations, however, this can become relevant (seeApp. B). with only a single independent form factor, g A ( q ), to bedetermined either from experiment or from lattice QCD[18, 37].In order to generalize Eq. (10), several options havebeen explored in the literature. Almost all of these weredeveloped half a century ago using the so-called elemen-tary particle approach [31, 38, 39], using form factors cou-pled to specific angular momentum operators . Becauseof the small number of dominant operators in allowed β decay, Holstein expanded the scalar product of Eq. (9)into a manifestly covariant form similar in spirit to Eq.(10). While this has some clear advantages, it does notgeneralize well to arbitrary spin-parity changes and theCoulomb interaction has to be put in post-hoc. The sec-ond option takes inspiration from multipole decomposi-tions in classical electrodynamics. Already in the 1960’sit was shown that a Lorentz-invariant decomposition of afour-current exists [40], which becomes particularly sim-ple in the Breit frame, where (cid:126)p i = − (cid:126)p f . The ramificationsof choosing the Breit frame are discussed in Appendix C,and are related to the appearance of kinematical recoilcorrections. The time component of Eq. (10) can, e.g.,be written as [41] H = (cid:88) LM C J i J f ; Lm i m f ; M Y ML (ˆ q ) ( qR ) L (2 l + 1)!! F L ( q ) , (12)where C contains a Wigner-3 j symbol, Y LM is a sphericalharmonic and R is the nuclear radius so that qR (cid:28) L µ , the matrix element of Eq. (8)can be calculated systematically for any spin-parity tran-sition using spherical tensor algebra. As a consequence,in the analysis one never finds only a single term propor-tional to some cos θ , but rather a Legendre polynomial P l (cos θ ), where each l couples to a spherical tensor op-erator (and form factor) of rank l . The result is Eq. (6),where each spectral function f i is a combination of formfactors.In what follows, we will provide an outline of the calcu-lation and report on the final results. All nuclear struc-ture corrections were calculated in the Behrens-B¨uhringformalism [31] and reported here using a shorthand no-tation explained in Appendix A. Kinematic recoil correc-tions were obtained following the discussion in AppendixC. Sec. III discusses some qualitative results for experi-mentally interesting cases. Final expressions obtained this way are somewhat unsurprisinglyvery similar to those obtained by more modern EFT techniques.In that sense, using form factors near zero momentum with, e.g.,a dipole formulation can be considered the phenomenologicalanalog of separation of scales and the appearance of low-energyconstants in current EFTs.
1. Isotropic spectral function
Since it normalizes the spectrum and thus appears inevery β -correlation, we start with the isotropic spectralfunction, f . Together with the prefactors defined in Eq.(6), this is simply the β spectrum when no other vari-ables are measured. The isotropic spectral function wasstudied in great detail in Ref. [22] in the context of β -spectrum measurements aimed at directly measuring theFierz term energy-dependence of Eq. (1), and we simplywrite the general result f = V V C ( W e ) V R N ( W e , M )+ A A C ( W e ) A R N ( W e , M ) , (13)where C ( W e ) is the so-called shape factor and R N cap-tures the kinematic recoil corrections from a decay withnuclear mass M (see also Appendix C). The dominantform factors in Eq. (13) reduce to the well-known ex-pressions at zeroth order, with V ≡ V F ( q = 0) (cid:39) g V M F (14a) A ≡ A F ( q = 0) (cid:39) − g A M GT (14b)where M F ( M GT ) is the Fermi (Gamow-Teller) matrixelement. Since the shape factor is 1 + O (10 − ), and thekinematic recoil corrections are at most O (10 − ), to lead-ing order (LO) we have f LO = | g V M F | + | g A M GT | + O (10 − ) , (15)with the percent-order corrections arising from finite sizecorrections and induced currents [22]. β -Asymmetry Following the discussion on Eqs. (3) and (6), we definean effective β -asymmetry according to A β P β cos θ e ∼ = (cid:88) k ≥ G k ( J i ) f σek f P k (cos θ e ) , (16)where we note an approximate equivalence to stress thefact that the angular structure is different in both sides.In the simplest case where k = 1, the polarization tensoris simply G = (cid:104) M i (cid:105) /J i and P (cos θ e ) = cos θ e . Aftersome tedious algebra we find f σe = (cid:114) J i J i + 1 Λ ( W e ) β (cid:104) S A ± (cid:112) / V A + α + α W e + α W e (cid:3) (17)where the upper (lower) sign refers to β − ( β + ), S is aspin-coupling coefficient, all the α i are of order O (10 − )and are listed in full in Appendix B. The factor Λ ( W e )in Eq. (17) is a type of Coulomb function originally de-fined by Behrens and J¨anecke [42] which is O (1) and not present in most other formalisms [39]. It was cal-culated numerically long ago [42], and shows deviationsfrom unity only at the few 10 − level for energies belowa few MeV. At the current and future level of precision,however, its influence can already be felt.Analogous to Eq. (15), we can write f σe to leadingorder to find f σe ∼ ∓ (cid:114) J i J i + 1 β (cid:16) S A ± (cid:112) / V A (cid:17) + O (10 − ) . (18)Extracting now a factor V from both f and f σe anddefining the Fermi to Gamow-Teller mixing ratio (see Sec.III and Appendix B) ρ ≡ A V (cid:39) − g A M GT g V M F , (19)(taking g A positive as before) we recover the usualleading-order result for the β -asymmetry [43], A LOβ = ∓ (cid:114) J i J i + 1 S ρ ± (cid:112) / ρ ρ . (20)For J → J transitions we have S = { J ( J + 1) } − / (Appendix B), so that for J = 1 / A LOβ J =1 / = ∓ √ ρ / √ ± ρ ρ . (21)The next spectral function, f σe , typically denoted bythe ‘anisotropy’, can similarly be calculated. Followingthe spherical tensor algebra, it couples with the polariza-tion tensor of rank 2 G ( J i ) = 1 J i (cid:20) (cid:104) M i (cid:105) − J i ( J i + 1) (cid:21) . (22)The anisotropy is then f σe = ( p e R ) βν S (cid:2) α + α W e (cid:3) (23)where once again all α i are O (10 − ), and ν = 1 + O{ ( αZ ) } another Coulomb function [31, 42]. The pref-actor ( p e R ) (cid:46) . − effect depending on G . Since the Legendre polynomial is, however, P (cos θ e ) = 12 (3 cos θ e −
1) (24)the influence of f σe depends on potential cancellations in f σe and on the experimental geometry and solid angle.We will get back to this in Sec. III and IV. β - ν Correlation
Analogously to Eq. (16) we define the effective β - ν correlation coefficient as a βν β cos θ βν ∼ = (cid:88) k ≥ f βνk f P k (cos θ βν ) (25)where like the β -asymmetry the r.h.s. has a richer struc-ture than the traditional l.h.s. The first order result for k = 1 can similarly be found f βν = β Λ (cid:20) V − A + (cid:101) α + (cid:101) α W + (cid:101) α W (cid:21) (26)where it is well-known that no vector-axial vector crossterms appear like in Eq. (17). Likewise, all (cid:101) α i are O (10 − ) and are listed in Appendix B. In the same spirit,it is well-known [39] that (cid:101) α i contains a smaller set ofinduced currents than, for example, α , such as the so-called induced tensor form factor.Similarly to Eqs. (15) and (18), the LO behavior of f βν is f βν ∼ β (cid:18) | g V M F | − | g A M GT | (cid:19) + O (10 − ) , (27)so that the LO β - ν asymmetry is the well-known expres-sion a LOβν = 1 − ρ ρ . (28)The anisotropy in the β - ν correlation can likewise becalculated f βν = β ( p e R )[ (cid:101) α + (cid:101) α W e ] (29)and where once again all (cid:101) α i are of order O (10 − ), mak-ing this at most a few 10 − effect with the same angularstructure as Eq. (24) and sensitivity as discussed be-fore for the β -asymmetry. Note that from symmetry re-quirements, f βν contains no nuclear form factors, and isinstead only a kinematic feature arising from the three-body decay (see also Ref. [27]). B. Radiative corrections
In using Eq. (8) we have only taken into account theCoulomb interaction, i.e., the large-wavelength behaviourof the of virtual photon exchanges between initial and fi-nal states. That is not the only O ( α ) correction thatshows up, however, which are more generally known aselectroweak radiative corrections. The topic of radia-tive corrections has a rich history which lies at the heartof our current understanding of electroweak interactionsand the Standard Model, and has been reviewed in sev-eral excellent works [3, 44]. Instead, we shall again be brief, and only summarize results available in the litera-ture.The order α photonic radiative corrections are the re-sult of three processes: ( i ) virtual photon exchange be-tween initial and final states, ( ii ) real photon emissionfrom external lines, and ( iii ) wave function renormal-ization of the external legs. The results of these pro-cesses have typically been calculated in the way pro-posed by Sirlin [45], with a separation of the processesaccording to photon momentum. It was shown that arelatively clean, gauge-invariant separation could be ob-tained between contributions for high photon momentum( k (cid:29) p e ), resulting in a renormalization of the couplingconstants [45] g V → g (cid:48) V ≡ g V (cid:16) α π c (cid:17) (30a) g A → g (cid:48) A ≡ g A (cid:16) α π d (cid:17) (30b)(known as the inner radiative correction, ∆ V ( A ) R = α/πc ( d ) [35, 46, 47]) and those at low photon momen-tum ( k ≤ p e ) with a dependence on final state kinemat-ics (known as the outer radiative corrections, δ R ( W e ))[22, 45, 48]. This is possible because results are domi-nated by either infrared divergences ( δ R ) or high-energy( (cid:29) m e ) electroweak and strong physics (∆ V,AR ). Practi-cally all other calculations have been constructed in thesame way [50–53].In terms of these renormalized coupling constants, thekinematic structure of the radiative corrections can bewritten down for the lowest order results d Γ ≈ d Γ (cid:34) f (cid:48) (cid:16) α π g (cid:17) + f βν (cid:48) (cid:16) α π h (cid:17) cos θ βν + P (cid:26) f σe (cid:48) (cid:16) α π h (cid:17) cos θ e + f σν (cid:48) (cid:16) α π g (cid:17) cos θ ν + f σ ×(cid:48) cos θ × (cid:27) + (cid:88) k ≥ f i (cid:48) k (cid:35) dW e d Ω e d Ω ν (31)where all primed f (cid:48) k correspond to the usual expressions,but using the renormalized coupling constants of Eqs.(30a) and (30b), and the outer radiative corrections arewell known [50, 51] g ( W e , W ) = 3 log M p −
34 + 4 β L (cid:18) β β (cid:19) + 4 (cid:18) tanh − ββ − (cid:19) (cid:20) W − W e W e −
32 + ln[2( W − W e )] (cid:21) + tanh − ββ (cid:20) β ) + ( W − W e ) W e − − β (cid:21) (32) The notable exception is, of course, the γW box which is alsosensitive to physics at the nuclear scale. For the purpose of thisdiscussion, however, we consider it fully part of ∆ R (see Ref.[49]). and h ( W e , W ) = 3 log M p −
34 + 4 β L (cid:18) β β (cid:19) + 4 (cid:18) tanh − ββ − (cid:19) (cid:20) ln[2( W − W e )] −
32 + W − W e W e β + ( W − W e ) W e β (cid:21) + 4 β tanh − β (1 − tanh − β ) (33)where L ( x ) = (cid:82) x (log(1 − t ) /t ) dt is the Spence functionand M p is the mass of the proton. It is well-known thatthe triple correlation, f σ × , contains no outer radiativecorrections [54]. We have neglected additional outer ra-diative corrections to higher-order terms, as they wouldconstitute only a O ( α/ π ) ∼ − shift on top of analready small effect (see Eqs. (23) and (29)), but forconsistency treat them using the renormalized couplingconstants.The measurement of a β correlation has the particu-lar advantage of being sensitive only to the relative dif-ferences between the isotropic and correlation spectralfunctions. As a consequence, at first sight there is no ad-ditional kinematic structure arising from radiative correc-tions to the ν -asymmetry ( B ν ), while the β -asymmetry( A β ) and β - ν correlation ( a βν ) are modified by R ≡ α π h α π g ≈ α π (cid:26) (1 − β ) 4( W − W e )3 W e β (cid:0) β − tanh − β − (cid:1) + ( W − W e ) W e β (cid:2) (1 − β ) β − tanh − − (cid:3) + 2(1 − β ) β − tanh − β (cid:41) . (34)It is interesting to note that as β → g and h reduces to a single term in the secondline. In the case of the neutron, this has been numericallyestimated by Fukugita and Kubota [55] R ≈ (cid:0) − .
63 + 4 . W − e + 0 . W e (cid:1) · − . (35)Figure 1 shows the different radiative corrections andtheir ratio for two β transitions with a 1 and 3 MeVendpoint. Interesting to note that is that R > β → a βν and B ν from experiment it is typically not appro-priate to use the above expressions. We will discuss thisfurther in Sec. IV. R C g h R a t i o - E = 3 MeV E = 1 MeV Figure 1. Radiative corrections due to g (Eq. (32)), h (Eq.(33)) and their ratio, R (Eq. (34)) for two different endpointenergies. III. MIRROR DECAYS AS A TESTINGGROUND
The previous section summarized the required theo-retical input arising from nuclear structure and radiativecorrections. In all measurements of aforementioned cor-relations the largest unconstrained parameter for a mixed( J → J , J >
0) is the mixing ratio, ρ , traditionally de-fined as [10, 11] ρ = A V (cid:20) AR VR (cid:21) / ≈ g A M GT g V M F (cid:20) (1 + δ ANS − δ AC )(1 + ∆ AR )(1 + δ VNS − δ VC )(1 + ∆ VR ) (cid:21) / (36)where δ NS ( C ) are nuclear structure and isospin breakingcorrections to the Fermi ( F ) and Gamow-Teller ( GT ) ma-trix elements in the limit of isospin symmetry, denotedby the “0” superscript. [56]. The latter are typically as-sumed to be equal for axial and vector parts, meaningexperiments measure the ratio of many-body matrix el-ements and renormalized coupling constants. Note thatwe have also neglected the presence of second-class cur-rents here, which are briefly discussed in Appendix B.An accurate determination of ρ is extremely interest-ing from a physics point of view (see Sec. V), howeverthe precision that can be obtained depends both on thesensitivity of the correlation coefficient to ρ and the un-certainty on the remaining theory input. In both cases,so-called isospin T = 1 / β decays are a primecandidate [10, 11].It is important to note, however, that the notation inEq. (36) can be somewhat deceiving. The reason for theseparation of M F into M F and isospin breaking correc-tions is because CVC and isospin symmetry allow for acertain determination of M F , with the former addition-ally guaranteeing that no additional corrections appearbeyond the impulse approximation result. The leadingGamow-Teller form factor, however, is very different. Notonly is M GT not determined by any symmetry, the ab-sence of the conservation of the axial current means ad-ditional contributions beyond the impulse approximationresult necessarily enter, traditionally denoted by core-polarization and meson-eschange effects. Phenomenolog-ically, this is often obfuscated by a so-called quenchingfactor to the axial vector coupling constant [44, 57, 58].As ab initio calculations ramp up their capabilities inthis regard [59], however, using more sophisticated waysof solving the N -body Schr¨odinger equation, a more cor-rect way of presenting ρ would be ρ ≈ g QCDA F GT (0) g V M F (cid:20) AR − ∆ VR δ VNS − δ VC (cid:21) / (37)where g QCDA is the renormalized value solely due to stronginteraction effects, and F GT (0) is the normalized nuclearresponse to a nucleonic Gamow-Teller operator near zeromomentum transfer. If g A is instead taken from an ex-perimental measurement in the neutron g nA = g QCDA (cid:2) n ∆ AR − n ∆ VR (cid:3) / , (38)a partial cancellation occurs with the inner radiative cor-rections to the mirror Gamow-Teller transition, whichare recently found to contain transition-dependent terms[35, 49]. Note that all of these effects require and inpart originate from an internally consistent set of defini-tions used both in experimental extraction and theoret-ical analysis [35]. With the isospin-breaking correctionsbeing an 0 .
2% to 1% effect [11], and differences between∆ AR and ∆ VR on the 10 − level [35, 49], such differencesbecome relevant in the neutron and low-mass systems. A. Cancellation for precision
Looking at Eqs. (17) and (26) there is a potentialfor cancellation between the two main terms for a mixeddecay. In particular, when ρ ≈ (cid:26) ∓ ρ (cid:112) / S − ( A β )3 ( a βν ) (39)significant cancellation occurs. This is interesting sincetypically the values of correlation coefficients are verysensitive to the value of ρ near such a turnover point.For J → J transitions like nuclear mirrors, ( S ) − = (cid:112) J ( J + 1) (see Appendix B). The neutron, for exam-ple, has ρ = g A √ A β and a βν are close tocancellation, since ( S ) − = 3 / √
2. In this case, one finds δA/A ≈ . δρ/ρ and δa/a ≈ . δρ/ρ . This results in anenhancement factor on ρ of a factor 4, which is of partic-ular interest for V ud and CKM unitarity tests discussedin Sec. V. Table I shows the enhancement factor for all a / a n3H 11C13N15O17F 19Ne2 1 0 1 22010010 A / A Figure 2. (Top) Calculated sensitivities to δρ/ρ from δa/a .The sensitivity is symmetric w.r.t. ρ and spin-independent.(Bottom) Calculated sensitivities to δρ/ρ from δA/A for β − , J = 1 / β + , J = 1 / β + , J = 3 / β + , J = 5 / nuclear mirrors up to mass 19, where advances in nu-clear ab initio theory are also likely to make significantprogress in the near future [60]. Nucleus n H C N O F Ne ρ − . − .
10 0 .
75 0 . − . − .
28 1 . J / / / / / / / δA β /A β − . − . δa βν /a βν − . − . − . − . − . δρ/ρ for the lowest massmirrors, with approximate ρ values taken from [10] and theleading order expressions. As expected from Eq. (39), mirrors with | ρ | ∼ √ J = 1 /
2, thelargest sensitivity to ρ is likely to come from a measure-ment of the β - ν correlation. A turning point sensitivityis reached for ρ = 2 /
3, where δa/a = δρ/ρ .A clear disadvantage of such a cancellation, however, isthat as leading order effects become small, initially sub-dominant corrections gain in relative importance. Taking Ne as an example, since its leading order β -asymmetryis about -4% [61], the relative importance of all subdom-inant corrections is now enlarged by a factor 25, whichputs more stringent constraints on additional theory in-put. B. Remaining theory uncertainty
Regardless of a potential cancellation in any of the co-efficients, the experimental precision is such that in anycase subdominant effects must be taken into account tovarying degree. These have been summarized in the pre-vious section, with additional complications due to theexperimental geometry and detection scheme treated inthe following section.From a theory point of view, the precision bottlenecklies in the accurate calculation of nuclear matrix ele-ments, in particular those stemming from induced cur-rents. Because the β decay occurs within an isospin mul-tiplet, however, mirrors have a distinct advantage. Dueto the conserved vector current, all vector form factorscan be determined exactly in the limit of isospin symme-try. The Fermi matrix element, M F , is equal to unity for T = 1 /
2, with isospin breaking corrections calculated ina many-body code (see Eq. (36)) [56]. As mentioned be-fore, all induced scalar form factors are identically equalto zero. Further, the invocation of CVC trivializes mostof the additional theory input, as most recoil form factorsare either zero or known to very high precision. This isthe case for the so-called ’weak magnetism’ form factor, V ( ∼ b ( q ) in Holstein’s notation, see Appendix D),which can be related to the isovector magnetic momentof initial and final states for T = 1 / M γ transition for T = 1.Finally, the first-class part of the induced tensor form fac-tor, A ( ∼ d ( q ) in Holstein’s notation, see Appendix D),is identically equal to zero within an isospin multiplet.Besides the mixing ratio, ρ , this leaves at least twomore subdominant sources of nuclear structure input,since axial form factors are not protected by any sym-metry. In particular, the so-called induced pseudoscalarcoupling, A ( ∼ h ( q ) in Holstein’s notation, see Ap-pendix D), must be calculated by a many-body methodunless it is trivially equal to zero [22, 62]. Higher-orderform factors such as A can additionally contribute fortransitions with J ≥
1, and must be calculated usingmany-body methods. From the expressions in the ap-pendix, one can estimate their influence to be at the few10 − level. Finally, there is an induced pseudoscalar con-tribution proportional to (cid:101) g P (see Eq. (11)), which is dis-cussed to some depth in Ref. [22] and the appendix, andcan also contribute up to the 10 − level. IV. EXPERIMENTAL CONDITIONS
Section II contained some foreshadowing and caveatsconcerning the validity of the equations presented or theconclusions taken from it. The formulae written abovecorrespond to an ideal situation, i.e., a perfect cancella-tion of all terms but the one of interest, 4 π solid angle,measurement of the (anti)neutrino rather than the recoil-ing nucleus, perfect energy measurements, and so on.An analysis attempting an extraction of the correct quantity runs into at least three conceptual difficultiesdue to experimental conditions: ( i ) relative rate mea-surements in an open geometry folds in other observablesand higher-order polarization effects ( ii ) real photons inradiative β decay change the kinematics and must beaccounted for ( iii ) a measurement of a correlation maynot allow for the effective parametrization (cid:101) X (Eq. (5))[24]. Besides this, several systematic effects emerge re-lated to detector performance, e.g., through linearity andefficiency. Finally, measurements not relying on initialpolarization can contain contamination from experimen-tal residual polarization which may not be known to greatprecision.Additional complication arises because of the experi-mental scheme and which final states are detected. Eventhough any three body decay allows for only two indepen-dent degrees of freedom, several combinations are typi-cally used in the literature. Due to the additional rich-ness it brings, several modern experiments measure, e.g.,both the β particle and the recoiling nucleus. This opensup the Dalitz distribution for analysis d Γ dW e dW f = G F π W f W e q × (cid:40) b F W e + 1 F (cid:88) k F βνk P k (cos θ βν ) (cid:41) , (40)where all F k correspond to the modified spectral func-tions due to radiative corrections, e.g., F = f (cid:48) [1 +( α/ π ) g ], for brevity. The β - ν angle is then simplycos θ βν = p f − p e − q W e q (41)where q = W − W e − W f is the antineutrino energyand W f is the recoil energy (neglecting the real photonmomentum).The following sections summarize results arising fromhigher-order effects and discuss the complications dueto real photons and the detection scheme on StandardModel comparisons. Following the discussion in Sec. III,these argument become particularly relevant in the caseof strong cancellations such as several mirror systems. A. Solid angle
In a typical experiment one measures the difference inintegrated count rates X = N ↑ − N ↓ N ↑ + N ↓ , (42)or with some more complicated super-ratio, where N ↑ ( ↓ ) are integrated count rates either in separate (usually op-posite w.r.t. the maximum of X ) or a single detector andinstead changing, e.g., the polarization direction. As aconsequence, everything in Eq. (6) besides X also foldsinto N , where now the residual effect depends on theexperimental conditions and geometry.The full decay rate results from an integration over allremaining variables of Eq. (6)Γ = 1(4 π ) (cid:90) W dW e d Γ F (cid:90) − d cos θ e (cid:90) π dφ e × (cid:90) − d cos θ ν (cid:90) π dφ ν D (43)where the z -axis is along the initial polarization if presentand random otherwise and D = 1 + b F W e + 1 F (cid:88) k ≥ F βνk P k (cos θ βν )+ G k ( J i ) (cid:26) F σek P k (cos θ e ) + F σνk P k (cos θ ν )+ F σ × k P k (cos θ × ) (cid:27) . (44)In practice, the angular integration limits depend on theexperimental geometry and the energy integration re-quires a convolution with a calibrated detector responsefunction, e.g., R ( W e , E ) as the probability of measuring E for a β particle with real energy W e , to find Γ exp = 1(4 π ) (cid:90) E max E min dE (cid:90) W dW e R ( W e , E ) d Γ F × (cid:90) Ω exp e d Ω e E (Ω e ) (cid:90) Ω exp ν d Ω ν E (Ω ν ) D . (45)where E is the detected energy, E ∈ [ E min , E max ] cor-responds to the experimental analysis window, Ω exp e,ν isthe effective solid angle for detection of electrons and(anti)neutrinos and E the detection efficiencies. This in-tegration is in principle non-trivial and should ideally beperformed numerically unless a high degree of symmetryexists in the experimental set-up. Additionally, since itis typically not the (anti)neutrino which is measured butinstead the nuclear recoil, an additional detector responsefor its detection function must be introduced analogousto that of the β particle.For simplicity, we consider a perfect detector, i.e., R ( W e , E ) = δ ( W e − E ) and E (Ω) = 1. If the experi-mental geometry is symmetric around, e.g., the axis ofinitial polarization, ˆ J , we can simply perform the az-imuthal integration for the β particle and (anti)neutrino, (cid:82) d Ω → π (cid:82) d cos θ , leaving only the polar angle inte-gration. Since P (cos θ × ) is odd under φ → φ + π , theazimuthal integration resolves to zero. In general the detector response function depends not only onthe particle energy but also, e.g., on its angle of incidence intothe detector face.
Since at this point all further analysis depends on in-tegration of Legendre polynomials, we introduce the fol-lowing property (cid:90) x dx (cid:48) P k ( x (cid:48) ) = 1 − x k ( k + 1) dP k ( x ) dx (46) ≡ I k ( x )for k (cid:54) = 0, so that I k ( −
1) = 0 for all k , and I k (0) is 1 for k odd, and 0 for k even.
1. Fierz cancellation in A β The simplest effects can be shown in a measurement ofthe β -asymmetry with a single detector. Let us assumeonce more a perfect detector, with the ability to changethe polarization direction externally. Assuming only the β particle is detected, the integral simplifies significantly,and only F σek terms remain, A ↑ ( ↓ ) = 1+ b F W e + 1 F (cid:88) k ≥ G k ( J i ) F σek P k ( ± cos θ e ) . (47)The integrated count rates N ↑ ( ↓ ) are then N ↑ ( ↓ ) = 12 (cid:90) E max E min dEd Γ F (cid:90) x d cos θ e A ↑ ( ↓ ) , (48)where x denotes the polar extent of the detector. Us-ing Eq. (46), the experimental asymmetry definition, X ,(Eq. (42)) becomes X = (cid:82) dEd Γ F (cid:80) k odd I k ( x ) F σek F (cid:82) dEd Γ F (cid:104) Q + (cid:80) k even I k ( x ) F σek F (cid:105) , (49)where Q = (1 − x )(1 + b F /W e ). It is now interesting tonote that since F σe F ≈ p e W e Rν S (cid:26) α + α W e V + A (cid:27) (50)using p e = W e −
1, additional 1 /W e , W e and W e ap-pear. The Fierz term in the denominator of Eq. (49)consequently gets modified to1 W e (cid:18) b F − G (1 + x ) xRν S α V + A (cid:19) . (51)If an integrated measurement is performed, also the ef-fects of additional G W e and G W e interfere. Even inthe case of a differential measurement, since 1 /W e ≈ − W e contributions from an additional G W e may notbe experimentally distinguishable. Assuming perfect po-larization, i.e. (cid:104) M i (cid:105) = J i , then G = (2 − J − i ) /
3. Re-membering that α ∼ O (10 − ), cancellations on the levelof 10 − to 10 − can occur for systems with J i ≥
1. Thislies in the expected sensitivity range of modern experi-ments.0
2. Coincidence coupling
For most other correlations one typically measures thenuclear recoil in coincidence either with the emitted β particle or a subsequent nuclear γ decay. Even in thecase where no energy measurement is made of the β par-ticle or γ , the acceptance solid angle of the secondaryparticle couples all other angular correlations besides theintended one, either through, e.g., the β - ν correlation or β - γ correlation, respectively. We follow the approach byGluck [63].When detecting the recoiling nucleus rather than the(anti)neutrino, we use the following identity in the centerof mass frame (cid:90) dφ ν (cid:126)p f · ˆ J = − (cid:90) dφ ν ( (cid:126)p e + (cid:126)p ν ) · ˆ J = − π ( W ν cos θ ν + βW e cos θ e ) (52)to perform the integration of Eq. (43). The latter thendepends on the signs of Eq. (52) and cos θ e , leading tofour different electron spectra and integrated rates. An-alytical formulae for k = 1 can be found, e.g., in Ref.[63, 64]. We can extend the results to higher orders of k using the same techniques. For example, for k = 2 theadditional terms are Q ++ [ r <
1] = Q [ r < − f βν f (cid:0) r + r (cid:1) + G (cid:26)(cid:18) r − r (cid:19) f σν f − r f σe f (cid:27) , (53) Q ++ [ r >
1] = Q [ r > − f βν f (cid:18) − r (cid:19) + G (cid:26) r f σν f + (cid:18) − r r (cid:19) f σe f (cid:27) , (54)where r = p e /W ν and Q are, e.g., Eqs. (3.14) and(3.15) in Ref. [64] with the appropriate substitutions,and (++) denotes both the β particle and recoil goingalong the positive symmetry axis. We have calculatedthe additional terms in the infinite nuclear mass approx-imation, with corrections due to recoil and radiative cor-rections reported in Ref. [63]. The results assume perfectdetection efficiency in the positive hemi-sphere, but cus-tom results can trivially be obtained. Note that since f βν contains purely kinematical terms (see Eq. (29)), italways contributes regardless of the spin change of thetransition. For the neutron, however, it shows up only atthe few 10 − level [27]. B. Real photons
The regular β decay process is technically al-ways accompanied by emitted photons, so-called innerbremsstrahlung or radiative β decay. While the branch-ing ratio drops off steeply with increasing photon energy, the presence of the latter changes the kinematics, therebyturning β decay into a four-body process. While this isin principle contained in the kinematic radiative correc-tions discussed in Sec. II B, the analysis leading to theseexpressions assumes the photon is either perfectly iden-tifiable, or not detected at all. Besides measurementsin calorimetric systems, a complication arises, however,when an experiment aims to measurement a correlationinvolving an (anti)neutrino. Taking the β - ν correlationas an example, the decay rate of Eq. (6) specifies a cor-relation for P k (cos θ βν ), wherecos θ βν = (cid:126)p e · (cid:126)p ν | (cid:126)p e || (cid:126)p ν | . (55)Experimentally, however, (cid:126)p ν can typically not be mea-sured and often one measures insteadcos θ exp βν = − (cid:126)p e · ( (cid:126)p e + (cid:126)p f ) | (cid:126)p e || (cid:126)p e + (cid:126)p f | , (56)with (cid:126)p f the 3-momentum of the recoiling nucleus. Whilethese expressions are equivalent for a three-body decay,it is clear that this is not the case in the presence of anadditional photon. This discrepancy was noted alreadya long time ago [52, 54]. As a consequence, however, theformulae presented in Sec. II B involving an antineutrinoare not appropriate for use in an experimental settingunless one can additionally measure the photon momen-tum with great accuracy. Since this is typically not afeasible option, other expressions must be derived for theradiative corrections when the emerging recoil is mea-sured rather than the (anti)neutrino.A bremsstrahlung photon can arise in regular β de-cay through three processes: emission from either thecharged lepton, the initial or final hadronic states andfrom the weak vertex itself. The latter corresponds tothe emission of a photon by the W boson, which repre-sents an O ( G F ) process and can therefore be neglected.Due to the enormous difference in mass between the emit-ted charged lepton and initial and final hadronic states,practically all γ emission arises from the charged lepton.This process is well-known to contain an infrared (IR) di-vergence [65], which is cancelled by the corresponding IRdivergence in the virtual photon exchange diagram [45],so that the two processes cannot be calculated separately.Due to its usefulness in experimental analyses, Gl¨uck[66] has split up the photon energy integral into a softand hard part , with an interface defined at ω (cid:28) m e .The integration over soft photons contains the IR diver-gence, but the very low-energy photons do not apprecia-bly change the kinematics, i.e. Eqs. (55) and (56) arequasi-identical. The radiative correction to the angular Not to be confused with the separation introduced by Sirlin [45],which occurs at a scale between M A and M W to split low-energyQED processes from electroweak and strong physics at M W . δa V Sβν ≈ a βν tanh − ββ (1 − β ) p f W e ( W − W e ) . (57)The occurrence of hard photons is responsible for adiscrepancy between Eqs. (55) and (56), and the totaldecay rate can be written as [53] ρ H = G F π α π (cid:90) W dW e (cid:90) E γ ω dK × (cid:90) (cid:90) (cid:90) d Ω e d Ω ν d Ω γ KβW ν W e | M γ | , (58)where the matrix element can be written as | M γ | = f LO (cid:20) H ( K µ ) + a βν H ( K µ ) (cid:21) (59)with K µ the photon four-momentum, f LO the leading-order isotropic spectral function (Eq. (15)) and the ex-pressions for H i can be found, e.g., in Refs. [53, 66].More specifically, H ( K µ ) is a simple scalar and H ( K µ ) = A , (60a) H ( K µ ) = (cid:126)p e · (cid:126)p ν B + (cid:126)p ν · (cid:126)K C , (60b)where A , B and C are scalars depending on kinematics,and A ∼ B (cid:29) C .From an experimental point of view, it is most inter-esting to note that both H and H contain collinearpeaks due to the charged particle propagators. Since the β particle is mainly responsible for the emission of realphotons, it implies that the photon distribution is peakedalong the β particle direction. For 1 (cid:28) W e and θ βγ (cid:28) K | M γ | ∼ θ βγ + W − e , (61)where θ βγ is the angle between the photon and outgoinglepton in the center of mass frame .Figure 3 shows the behavior of H as a function of β -energy and cos θ βγ . As expected, a maximum is reachedfor nearly-collinear β particle and photon, with an ap-proximate parabolic behaviour near cos θ βγ = 1 as in Eq.(61). Due to the addition of two extra degrees of free-dom (cos θ βν and cos θ νγ ), H is not as straightforward toshow graphically. The kinematic structure is very similarto H , however, as was shown in Sec. II B.Unfortunately, however, analytical solutions to Eq.(58) are typically not available when one detects the re-coiling nucleus. That is because the analytical results ob-tained by several authors [51, 67–69] calculate the inner While for regular β decay 1 (cid:28) W e is often not valid due to thelow energy transfer, the peaking of the photon distribution alongthe β momentum arises naturally from Lorentz invariance. e n e r g y [ M e V ] c o s H [ a r b . u . ] E = 100 keV Figure 3. Behaviour of H from Eq. (59) for a fixed photonenergy of 100 keV in a 1 MeV β -transition, as a function of β energy and cos θ βγ . bremsstrahlung corrections for a constant (anti)neutrinoenergy and treat the final nucleus as infinitely massive,thereby simplifying Eq. (58) substantially. This corre-sponds to integrating over the photon energy and direc-tion keeping (cid:126)p e and (cid:126)p ν constant, making Eqs. (60a) and(60b) straightforward. Those results give rise to Eq. (31),with the small difference in A and B resulting in the near-equality of g ( W, W ) and h ( W, W ) in Eqs. (32) and (33)(see also Fig. 1). While semianalytical results have beenreported [66], those assume perfect reconstruction of β particle and recoil energies in a closed 4 π geometry withperfect detectors. The only way then to take into accountexperimental conditions is through a numerical proce-dure, such as that outlined in Ref. [53, 70], or throughexplicit event generation of the additional photon accord-ing to Eq. (59) [71].While the above discussion focused on a measurementof the β - ν correlation, the same argument applies to the ν -asymmetry, B ν and the β - f correlation. Some pub-lished results are available for the neutron [54, 72] andrecoil spectra of He and Ar [73]. Further analysis isplanned for future work.
C. Effective coefficient
Equation (5) introduced an effective correlation coeffi-cient, (cid:101) X , as the way many experiments analyse the sensi-tivity to exotic scalar or tensor currents through the Fierzterm. When the correlation involves the (anti)neutrino,however, some caveats once again emerge as one typicallymeasures the nuclear recoil rather than the antineutrino.The case that was extensively described in Ref. [24] dis-cusses the measurement of the β - ν asymmetry throughthe measurement of the recoil energy distribution (seeEq. (40)). Due to the dependence of the β energy on the2 β - ν angle (Eq. (41)), a measurement of only the latterdoes not allow for a parametrization of Fierz in terms of (cid:101) X . More generally, Eq. (5) is only valid when a corre-lation described as XR ( W e , θ ) with R any function, theobservables W e and θ are separable, i.e., when measuringonly the angular variable d Γ dθ = (cid:90) dW e G ( θ ) H ( W e ) (cid:20) b F W e + XR ( W e , θ ) (cid:21) = CG ( θ ) (cid:18) b F (cid:104) W e (cid:105) (cid:19) (cid:104) (cid:101) X (cid:104) R ( W e , θ ) (cid:105) W e (cid:105) . (62)When a βν is determined from the recoil momentum only,this is not the case as θ and W e are coupled throughmomentum conservation (Eq. (41)). In that case, aparametrization like (cid:101) X is not valid, and one has to prop-erly integrate over the Dalitz distribution of Eq. (40)according to the experimental geometry to obtain thecorrect result. Alternatively, the β energy is determinedand a fit is performed either on the Dalitz distribution,or on slices of constant β energy [74, 75]. D. Practical difficulties
In any real experiment there’s a potentially large num-ber of additional practical difficulties, such as detectornon-linearities, detection efficiencies, energy losses out-side of the active detector area, unresolved polarizationand/or alignment, etc. Regardless of the conceptualproblems posed above, all of these must be overcome inorder to extract meaningful results. Clearly, the preciseoccurrence of each of these effects is unique to each ex-periment. In this section we describe two critical contri-butions to asymmetry measurements, and demonstratetheir particular advantage compared to, e.g., spectrummeasurements.As an example, consider an experiment in which thecalibrated energy defines the boundaries of energy bins,with a β -correlation calculated in each bin according toEq. (42). In the case of any physical detector, therewill generally be some remaining systematic differencebetween the deposited energy E and the reconstructedenergy (cid:15) , such that E = (cid:15) + (cid:88) i =0 c i (cid:15) i , (63)where c i are parameters that can be constrained basedon the calibration procedure in place. Clearly, a largernumber of calibration points and good representationthroughout the region of interest will force the different c i to be smaller. In the following we assume this uncer-tainty to be sufficiently small, i.e. | ( (cid:15) − E ) /E | (cid:28)
1, andneglect losses causing E to be smaller than the initialenergy. We assume a general decay rate d Γ dE = K ( E, θ ) (cid:20) b F E + χ ( E, θ ) (cid:21) , (64) where θ is some angle and χ switches sign for “up” and“down” detectors. The measured bin counts are then N ↑ = ∆ t (cid:90) d Ω ↑ (cid:90) f ( (cid:15) h ) f ( (cid:15) l ) dEK ( E, θ ) (cid:20) b F E + χ ( E, θ ) (cid:21) (65)where ∆ t is the measurement time and f is Eq. (63)for the low and high bin edges (cid:15) l , (cid:15) h , respectively. The β correlation after performing the angular integration isthen X (¯ (cid:15) ) = N ↑ − N ↓ N ↑ + N ↓ = (cid:82) E h E l (cid:104) K ( E ) χ ( E ) (cid:105) Ω dE (cid:82) E h E l (cid:104) K ( E ) (cid:105) Ω (1 + b F /E ) dE , (66)where ¯ (cid:15) denotes the bin center, and E h,l ≡ f ( (cid:15) h,l ). Weintroduce some additional notational simplicity A ( E ) = (cid:104) K ( E ) χ ( E ) (cid:105) Ω (67a) S ( E ) = (cid:104) K ( E ) (cid:105) Ω (1 + b F /E ) (67b)to denote the (a)symmetric (numerator) denominator in-tegrand. We then perform a Taylor expansion of A , S around E l so that X (¯ (cid:15) ) = (cid:80) n A ( n ) ( E l )(∆ E ) n +1 / ( n + 1)! (cid:80) n S ( n ) ( E l )(∆ E ) n +1 / ( n + 1)! (68)with∆ E ≡ E h − E l = (cid:15) h − (cid:15) l + (cid:88) i =1 c i (cid:2) ( E h ) i − ( E l ) i (cid:3) (69)and superscript ( n ) denoting the n -th derivative. Thesystematic difference induced by the possible calibrationerrors can be then written as∆ X = (cid:80) n,m A ( n ) ( E l ) S ( m ) ( (cid:15) l )(∆ E ) n +1 (∆ (cid:15) ) m +1 − E ↔ (cid:15) ( n +1)!( m +1)! (cid:80) n,m S ( n ) ( E l ) S ( m ) ( (cid:15) l )(∆ E ) n +1 (∆ (cid:15) ) m +1 ( n +1)!( m +1)! (70)with ∆ (cid:15) = (cid:15) h − (cid:15) l . Equation (70) is not terribly enlight-ening, so it is worthwhile to consider some examples.In the case where χ is energy-independent, is it obvi-ous from Eqs. (67a) and (67b) that the numerator onlysurvives for non-zero b F . As a consequence, the observ-able effect is O ( b F { ∆ E − ∆ (cid:15) } ) and can be neglected.This is in stark contrast to when one measures only thespectrum for a Fierz extraction, where a handle on Eq.(63) is crucial [76]. Taking the β -asymmetry as anotherexample, we have up to leading order χ ( E, θ ) ≈ βA LOβ (cid:104) P (cid:105) cos θ e , (71)with A LOβ as in Eq. (20) and (cid:104) P (cid:105) the average polariza-tion. The energy-dependence at this order comes onlyfrom β , so that the effects of calibration uncertaintyare mainly relevant at lower energies, since β (1) → β →
1. Going beyond leading order, additional en-ergy dependence shows up coming from induced currents3(Eq. (17)) and radiative corrections (Eq. (34)). As theseare themselves small corrections of O (10 − ), assumingthe calibration to be sufficiently under control, these canagain be neglected.The situation changes when one takes into account de-tection efficiencies. This efficiency, typically determinedby scattering effects and the detector threshold, can dif-fer significantly from unity, particularly near-threshold.This fact is exacerbated by the typically strong angu-lar dependence of the efficiency due to, e.g., dead layerlosses and backscattering. As a consequence the angularintegration in Eq. (65) should be replaced (cid:90) d Ω (cid:90) dE → (cid:90) (cid:90) d Ω dE E ( E, Ω) (72)leading to corresponding changes in Eqs. (67a) and(67b). Due to the different angular weighting of the(a)symmetric distributions, efficiency differences show upto first order when using X as in Eq. (42). If one for-mulates the measurement in terms of a ratio of rates fortwo different spin states for a single detector or a ”super-ratio” of two or more detectors arranged with the appro-priate spin dependence, however, one expects first ordercancellation for constant efficiency factors shared by bothisotropic and cos θ -weighted distributions. Relative cor-rections for the isotropic to angular distributions (dueto, e.g., backscattering effects) do not cancel, however.As with calibration errors, these appear in the measuredasymmetry at the level of the difference in these efficien-cies and energy dependent effects for isotropic vs. cos θ -weighted decays, and the differences in the integrals overenergy (which vanish as β → β decay . V. NEW PHYSICS SENSITIVITY
As already mentioned in the introduction, precisionmeasurements of correlation coefficients in (nuclear) β decay are an attractive option for new physics searchesboth from a theory and experimental point of view dueto their relative nature. The latter allows for a cancella-tion of many systematic uncertainties in an experimental This advantage provided significant motivation for early workon the PERKEO experiment [77, 78] and the 19Ne β asymmetrymeasurement [61]. setting, and theoretically it is often easier to reliably esti-mate ratios of matrix elements than their absolute mag-nitude. As a consequence, the reach for these measure-ments can be fairly broad. We focus here on two possiblecases, namely the search for exotic currents through theappearance of a Fierz term, and V ud determinations forcertain mirror systems and CKM unitarity. A. V ud and CKM unitarity Common to all semileptonic β decays, the decay rate isdetermined at the coupling level by the following productΓ semi − l ∝ G F V ud g V (1 + ∆ R ) F (73)where G F is the Fermi coupling constant, V ud is the up - down quark mixing matrix element, g V (1 + ∆ R ) is therenormalized vector coupling constant and where F takesinto account additional transition-specific information.If all other information can be either experimentally ortheoretically sufficiently determined, a measurement ofthe lifetime in semileptonic systems gives access to | V ud | .The β -decay of the muon is theoretically extremely clean,which allows one to calculate both F µ and ∆ µR very accu-rately. The latter is lumped together with G F to definethe traditional Fermi coupling constant which is experi-mentally found to be G F = 1 . × − GeV − [79]. In neutron or nuclear systems then, the conservedvector current hypothesis sets g V = 1 up to higher-ordercorrections [36], and one needs to calculate only that partof ∆ R which is unique to nuclei [3, 80]. Finally, in the nu-clear sector F can be calculated to high precision in twodifferent cases: ( i ) superallowed 0 + → + Fermi decays[81], and ( ii ) T = 1 / M F equal to √ F in both casesconsists of the half-life of the β transition, the branchingratio, and the endpoint energy. Because mirror decayshave both non-zero Fermi and Gamow-Teller componentsand the latter is not constrained by symmetry, this needsto additionally be experimentally determined from, e.g.,the measurement of a β -correlation (see Sec. III).One can construct a so-called F t value, analogous tothe F t for superallowed decays [81], which according toCVC must be equal for all nuclear mirrors [10] F t ≡ f V t (1 + δ R )(1 + δ NS − δ C ) (cid:18) ρ f A f V (cid:19) = Kg V G F V ud |M F | (1 + ∆ VR ) (74)where K = 8120 . × − GeV − s is a combina-tion of constants, δ i correspond to radiative ( R ), isospin-breaking ( C ) and nuclear structure ( N S ) corrections, and f V,A are so-called phase space integrals [22, 35, 82]. Thefirst line shows all transition-specific factors, while thesecond line consists only of common constants.4 A of initial state0.96750.97000.97250.97500.97750.9800 | V u d | n Ne Na P Ar K | V ud | mirror old | V ud | mirror | V ud |
0+ 0+
Figure 4. Summary of current status and the influence of the-ory changes on the mirror | V ud | extraction compared to thesuperallowed 0 + → + data set and the neutron. The blue ar-row signifies the shift in | V ud | mirror due to the change in ∆ VR [83], while the red arrow signifies the shift due to updated f A /f V values [35], which is also shown in grey for individualresults. The uncertainty in Ne is significantly smaller be-cause of the reduced theory uncertainty from f A /f V results[35]. Because of strong cancellations in some mirror transi-tions such as the neutron and Ne, great sensitivity canbe obtained for a determination of ρ from a β correlationcoefficient. Following a reduction in uncertainty and re-moval of double counting in f A /f V calculations [35], pre-cise measurements of mirror transitions can shed lightboth on the shift in inner radiative corrections through-out the lower mass region, and be competitive in settingconstraints on CKM unitarity,∆ CKM = | V ud | + | V us | + | V ub | − . (75)Figure 4 shows the current status and summary ofchanges in the past years for the | V ud | extraction fromnuclear mirrors and its comparison to the neutron andsuperallowed decays [81, 83]. Recently, renewed atten-tion has been devoted to the calculation of the inner ra-diative correction [46, 49], with a significant shift in itscentral value since its last evaluation in 2006 [84]. Whilethis shift is consequential for the superallowed decays andto an extent the neutron, the results of the mirror decaysare most significantly impacted by change in f A /f V val-ues [35, 47]. Because of the reduced uncertainty on thelatter, the new experimental measurement of ρ for Ne[85] is significantly lower and dominated by experiment[47].In addition to unitarity tests, the required internal con-sistency of the F t values in mirror and super-allowed de-cays provides a number of paths to very clean constraintsfor new physics, some of which can evade the precisionlimits imposed by the vertex corrections. Examples in-clude the Marciano’s axial coupling relationship for theneutron [86], a neutron lifetime consistency test [87], theratio of | V ud | values extracted from the neutron and thesuperallowed decays articulated by [88]. B. Exotic currents
Traditionally the search for exotic currents in low en-ergy nuclear β experiments have been interpreted interms of the Lee-Yang Hamiltonian [89]. The past decadehas seen tremendous progress in the development of ef-fective field theories at the quark level, which allows oneto directly compare obtained limits to LHC constraintsif the new physics lies above the LHC energy scale. Ne-glecting right-handed neutrinos and writing only linearBSM couplings, one can write [14, 90] L eff = − G F ˜ V ud √ (cid:26) ¯ eγ µ (1 − γ ) ν e · ¯ uγ µ [1 − (1 − (cid:15) R ) γ ] d + (cid:15) S ¯ e (1 − γ ) ν e · ¯ ud − (cid:15) P ¯ e (1 − γ ) ν e · ¯ uγ d + (cid:15) T ¯ eσ µν (1 − γ ) ν e · ¯ uσ µν (1 − γ ) d (cid:27) + h.c. , (76)with ˜ V ud ≈ V ud (cid:18) (cid:15) R + (cid:15) L − δG F G F (cid:19) (77)and (cid:15) i are linear BSM effects of order O ( M W / Λ BSM )and δG F contains new physics contributions specific tomuon decay. Equation (77) is what causes a deviationfrom CKM unitarity at the quark level, i.e. Eq. (75),which can, e.g., be investigated using the F t values formirror and superallowed decays.From Eq. (76) one can see that the axial coupling con-stant is renormalized at the quark level, so that a mea-surement of ρ in different systems cannot reveal someBSM physics, as one always measures ˜ ρ = ρ (1 − (cid:15) R ).An exception to this is when the coupling constant canbe calculated to high precision from theory, which isonly feasible for the neutron using lattice QCD (LQCD).The determination of ˜ g A from a measurement of ˜ ρ = √ g A /g V in the neutron is theoretically a clean channelfor looking for right-handed currents through the com-parison with LQCD [14, 37]˜ g A = g LQCDA [1 − (cid:15) R )] , (78)where care must be taken to take into account the dif-ference in inner radiative corrections between vector andaxial vector parts [35]. The constraints from Eq. (78)are currently limited by the uncertainty on LQCD re-sults, which vary between 1% and 4% [18, 37].We note in passing that while both the older [61] andnewer [85] measurements of A β in Ne allow for a non-zero induced tensor component - a so-called Second-ClassCurrent (SCC), see App. B - through a two-parameterfit of the slope in A β , their findings are of opposite signand the modern result is not statistically significant (1 σ ).An SCC would additionally show up as a difference in λ extracted from a βν and A β in the neutron [91]. Whilethere is currently some tension between the results of5PERKEO III [7] and aSPECT [8], a simplified analysisshows that SCC effects would need to be about threetimes larger than that expected from weak magnetismand result in a significantly higher value of λ ≈ .
288 forboth to agree. The latter would be in strong violation ofCKM unitarity and the additional data sets as in, e.g.,Fig. 4 and point to the presence of right-handed currentsfrom Eq. (78) when using [37] at face value. We concludethat at this time there is no strong evidence for SCCs.Finally, we show a simple analysis demonstrating thephysics reach of mirror decays in the search for scalarand tensor currents. Several experimental programs arecurrently underway to measure or constrain b F to a fewparts in 10 using either the β -asymmetry or β - ν corre-lation, predominantly in the neutron [14]. Here we makeuse of the fact that the Fierz term changes sign for β ± decay as in Eq. (2) to compare the F t values of theneutron and Ne and obtain competitive limits.By turning on BSM physics, Eq. (74) is modified ac-cording to F t = f V t (1 + δ (cid:48) r ) (cid:0) δ VNS − δ VC (cid:1) (cid:18) f A f V ˜ ρ (cid:19) = Kg V G F | V ud M F | (cid:0) VR (cid:1) × (cid:104) (cid:15) L + 2 (cid:15) R − δG F G F + b F (cid:104) W − (cid:105) (cid:105) , (79)where all BSM physics is contained in the last line andthe Fierz contribution is transition-dependent due to theendpoint-dependence on (cid:104) W − (cid:105) . As a consequence, aratio of F t values for the neutron and Ne maintainssensitivity only to the Fierz term, but all other commontheoretical inputs cancel. The change in sign in b F en-hances its sensitivity.While Eq. (79) is correct, there is an additional sub-tlety involved when using experimental input for ˜ ρ . As anexample, we discuss its extraction from the β -asymmetry.As mentioned in Eq. (5), the presence of a non-zeroFierz term serves to dilute experimentally observed β -asymmetry A exp β = A SMβ b F (cid:104) W − (cid:105) exp (80)where the “exp” superscript serves as a reminder that theaverage is calculated over the experimentally analysedrange rather than the full spectrum. Our measured ratioof F t values must therefore be modified (where we use the subscript “m” now for measured values): F t ,m ≡ F t (cid:34) f A f V ˜ ρ m f A f V ˜ ρ (cid:35) ≈ F t (cid:20) ρ ρ d ˜ ρdb F b F (cid:21) = Kg V G F | V ud M F | (cid:0) VR (cid:1) × (cid:20) − ρ ρ A β b F (cid:104) W − (cid:105) exp d ˜ ρdA (cid:21)(cid:104) (cid:15) L + 2 (cid:15) R − δG F G F + b F (cid:104) W − (cid:105) (cid:105) , (81)where we take f A /f V ≈ ρb F = ˜ ρ m b F . The ratio of F t values for Neand the neutron can then be written as R m ≡ F t , Ne F t ,n = 1 + b nF (cid:20) (cid:104) W − (cid:105) + 2 ˜ ρ ρ d ˜ ρdA A SM β (cid:104) W − (cid:105) exp (cid:21) n b F (cid:20) (cid:104) W − (cid:105) + 2 ˜ ρ ρ d ˜ ρdA A SM β (cid:104) W − (cid:105) exp (cid:21) Ne . (82)Using the recent measurements of A β for the neutron[6, 7, 92] and Ne [85] as numerical input and evaluating b F using Eq. (2) with the latest lattice charges, we find R m = 1 + ( − . (cid:15) T + 0 . (cid:15) S )(0 .
65 + 0 . − ( − . (cid:15) T + 0 . (cid:15) S )(0 . − . ≈ − . (cid:15) T + 0 . (cid:15) S . (83)If we use values for Ne of F t , = 6142(17) and theneutron of F t ,n = 6155 . R =0 . Ne F t value, which it itself dominated by thaton ˜ ρ . Using a value of (cid:15) S = 1 . . × − from Hardyand Towner [81], we determine (cid:15) T = 4 . . × − (90%C.L.). The resulting limit for the BSM energy scale for anexotic tensor coupling at the 90% (C.L.) is Λ T > . F t = 6151(10), which yields a ratio R = 0 . ρ m,n . If the uncertainty due to Ne is broughtto the same level as that of the neutron (an improvementof a factor 2 .
5) through an improved measurement of ˜ ρ ,the tensor scale becomes Λ T > . We evaluate b F at ˜ ρ m and neglect the details of the energy depen-dence of the Fierz term in the one parameter fit to the asymme-try. These introduce a systematic error in our extracted value of (cid:15) T less than about 4% for the neutron and Ne for | b F | < . VI. CONCLUSION
Measurements of correlations in (nuclear) β decay havecontinuously been a central pillar in the exploration ofthe low energy electroweak sector of the Standard Model,and modern experiments are entering a regime where ad-ditional theory corrections become relevant. We havecompiled here a comprehensive summary of theory inputwith a special focus on the β -asymmetry ( A β ) and β - ν correlation ( a βν ). In particular, we have reviewed thekinematic and nuclear structure effects, including thoseof higher order in the relevant angle and electroweak ra-diative corrections. We have taken another look at themirror T = 1 / β -correlation measurements feeds into tests for new physicsin the electroweak sector. This was done with a par-ticular focus on mirror decays, which have a number ofpleasant features which make them prime candidates forhigh-impact measurements. Together with the neutron,these have undergone steady progress over the last decadeand have the potential to become as precise as the super-allowed data set, with different systematic uncertainties.Additionally, we have shown in a simplified analysis thatby using only the neutron and Ne a sensitivity on newtensor couplings lies above 5 . Ne. Given the subtleties atthis level of scrutiny, it is to the benefit of experimentsto use a comprehensive formalism.
ACKNOWLEDGMENTS
The authors would like to acknowledge useful discus-sion with and and inspiration from V. Cirigliano, A. Gar-cia, O. Naviliat-Cuncic, B. M¨arkisch, W. Marciano, D.Melconian, B. Plaster, G. Ron, and N. Severijns. Thisarticle was supported through the Department of Energy,Low Energy Physics grant DE-FG02-ER41042 and NSFgrant PHY-1914133.
Appendix A: Notation and conventions
In the entirety of the manuscript we use units suitableto β decay, i.e. (cid:126) = c = m e = 1 . (A1)As a consequence, typical β energies are of order unity,the nuclear radius R ∼ . A / and nuclear mass M ∼ A × V F KLs ( q ) → (cid:26) V K s = 0 (time) V KL s = 1 (space) (A2)and analogously for the axial vector form factors. In theBehrens-B¨uhring formalism one usually performs an ex-pansion of the form factors in terms of ( qR ), where thedifferent coefficients are denoted by F ( n ) KLs , with n the as-sociated power of ( qR ). In β decay one has ( qR ) (cid:28)
1, sothat one is only concerned with n = 0 for all form factorsand n = 1 for the dominant form factors. As such, wewill leave out this additional index and denote the n = 1component with a prime. Finally, the form factors inthe Behrens-B¨uhring formalism are typically encounteredwith a convolution with (parts of) the leptonic sphericalwave expansion . This results in a further complicationof notation, e.g., F ( n ) KLs ( ρ, k, m, n ). In the case of alloweddecays, ratios of such form factors can be calculated as-suming CVC which are then evaluated directly. Whilesuch results are included in the full calculation, we donot need to introduce additional notation.Our sign conventions, however, are slightly differentfrom those of the Behrens-B¨uhring results. Our met-ric and γ matrices follow the convention by Bjorken andDrell [93] when specified. We take the first-class ax-ial form factors to switch sign for β + / EC rather thantheir impulse approximation expressions ( g A → − g A ) asis done in Refs. [31, 62]. Appendix B: Coefficients and form factordecomposition
Here we report on the energy-independent factors oc-curring in the spectral functions as described in Sec. II,and comment on the impulse approximation and conse-quences for, e.g., second-class current searches. This is discussed as the convolutional finite size correction inRef. [22].
1. Spectral functions
This is a reproduction of the coefficients of the shapefactor in Ref. [22] entering the formulae above, with asmall caveat related to the inner radiative correction to g A .The vector coefficients, V C i , are as follows V C = − αZ ) −
15 ( W R ) ∓ αZW R, (B1a) V C = ∓ αZR + 415 W R , (B1b) V C − = 215 W R ± αZR, (B1c) V C = − R (B1d)where the upper (lower) sign corresponds to β − ( β + )decay, while the modified axial vector coefficients are A C = −
15 ( W R ) + 49 R (cid:18) − Λ20 (cid:19) + 13 √ W A R ( ∓ √ V + 2 A ) ± αZW R (1 − Λ) − αZ ) + Φ (cid:20) ± αZW R + 51250 ( αZ ) (cid:21) , (B2a) A C = 4 √ √ R V A + 49 W R (cid:18) − Λ10 (cid:19) ∓ αZR (cid:18) − Λ10 (cid:19) ± Φ (cid:20) αZR (cid:21) , (B2b) A C − = − √ RA ( ± √ V + 2 A ) − W R (1 − Λ) ∓ αZR
70 + Φ (cid:20) − W R ± αZR (cid:21) , (B2c) A C = − R (cid:18) − Λ10 (cid:19) , (B2d)where Φ = (cid:101) g P g A M N R ) ∼ O ( − .
1) (B3)denotes explicitly the induced pseudoscalar contributionas in Ref. [22]. Equations B2a-(B2d) use the notation wehave defined in Appendix A, while the results of Ref. [22]are written using Holstein’s form factors. The translationis discussed in Appendix D. We have left out the effects ofthe induced Coulomb recoil corrections (i.e. O ( αZ/M R )terms) to A C , which serve to renormalize the axial vec-tor form factor as discussed in Ref. [35, 47].The subleading terms to the β correlations discussed inthe main text use a combination of the preceding ones toprovide the full description. For the β -asymmetry (Eqs. (17) and (23)), these are α = Γ A A C ± − / V A ( A C + V C ) ± W M (cid:32) Γ A ± (cid:114) V A (cid:33) (B4a) α = Γ A A C ± − / V A ( A C + V C )+ √ R η Λ (cid:104) Γ A ∆ ∓ − / V A ∆ −√ A (cid:18) V + 13 W R ( V + A ) (cid:19)(cid:21) ∓ M (cid:32) Γ A ± (cid:114) V A (cid:33) (B4b) α = Γ A A C ± − / V A ( A C + V C ) . (B4c)Note that for each α i the first line(s) contain finite sizeand dynamical recoil order corrections, while the last lineoriginates from kinematical recoil corrections, discussedin Appendix C. Additionally, we define∆ = − (cid:114) A A ± (cid:114) V A + W R A A + 2 √ W R. (B5)The spin-coupling coefficients are written in terms ofthe Γ ij factors, with S = Γ (1). These follow the defi-nitions by Weidenm¨uller and later Behrens and B¨uhringand are given byΓ (1) = { J ( J + 1) } − / J → J −{ J/ J + 1) } / J → J + 1 { ( J + 1) / J } / J → J − (1) = − (cid:26) (2 J − J + 3)30 J ( J + 1) (cid:27) / J → J −{ ( J + 2) / J + 1) } / J → J + 1 −{ ( J − / J } / J → J − . (B7)Similarly, we can write down the coefficients for the β − ν correlation functions (Eqs. (26) and (29)). The8 k = 1 terms are as follows˜ α = V V C − A A C − √ W RA ∆ (cid:48) + 23 W M A (B8a)˜ α = V V C − A A C − √ η Λ W R V − √ RA (cid:18) η Λ ∆ (cid:48)(cid:48) − ∆ (cid:48) (cid:19) + 4 M A (B8b)˜ α = V V C − A A C + 2 √ η Λ W R V (B8c)where again the last line is a consequence of the kinematicrecoil corrections, and∆ (cid:48) = (cid:114) A A ± (cid:114) V A − W R A A ∓ αZ (cid:40) √
235 + 13 A A (cid:41) (B9a)∆ (cid:48)(cid:48) = (cid:114) A ∓ (cid:114) V A − W R (cid:40) √
25 + A A (cid:41) . (B9b)Finally, the k = 2 terms are then˜ α = 445 ν RW − M A R (cid:18) V − A (cid:19) (B10a)˜ α = 445 ν RW e , (B10b)where ν is a Coulomb function of O (1+( αZ ) ) as before[42].
2. Impulse approximation and second-classcurrents
For an experimental analysis to discern new physicsphenomena from Standard Model input, one needs a wayof translating form factors into nuclear matrix elements,in particular for those which are not related by CVC. Theusual approach follows the so-called impulse approxima-tion, whereby the nuclear current is approximated as acoherent sum of individual, non-interacting, nucleon cur-rents. This couples nicely with usual methods of com-putation consisting of some form of Slater determinantsof single-particle wave functions. In practice, this trans-lation is often done by performing a Foldy-Wouthuysentransformation [94]. Some care is required here, how-ever, due to presence of second-class currents [95]. Thelatter has an opposite transformation under G -parity (i.e. G = Ce iπI [96]) compared to the main currents, so that V F → V F I ± V F IIA F → ± A F I + A F II (B11)for β − ( β + ), and where I ( II ) stands for first (second)class currents. While Eq. (B11) is general, the exact de-composition depends on the methods used and the framein which the decomposition is performed. The followingis a subset of the relevant form factors in impulse approx-imation in the Behrens-B¨uhring formalism, performed inthe Breit frame (see Appendix C), translated from Ref.[62] V = g V M F (cid:18) ± (cid:101) g S M N ∆ C (cid:19) (B12a) A = − g A M GT (cid:18) ∓ (cid:101) g T M N ∆ C (cid:19) (B12b) A = g A M ∓ √ (cid:101) g T M N R M GT (B12c) V = g V M + √ (cid:101) g M M N R M GT (B12d)where the matrix elements are those defined in Ref. [62]with the same notation as in App. A , and∆ C = W ± αZR (B13)is the difference between the endpoint and Coulomb dis-placement energy. For β + mirror transitions, ∆ C is fairlyclose to zero, resulting in a decreased sensitivity.The search for second-class currents in β decay has astoried history [97, 98], with initial experiments showingstrong effects. The A = 12 isospin triplet system in par-ticular has been an intense avenue of study through, e.g.,a comparison of F t values. Additional complications dueto nuclear structure make this comparison more complexthan it appears at first sight, and subsequent experimentshave found no strong evidence in favour of second-classcurrents. This remains the case in the study of the β -asymmetry in Ne [61, 85], which was identified as amore robust case through a measurement of the energydependence of the asymmetry.When performing the ratio of F t values of the neutronand Ne in Sec. V, there is additionally a contributiondue to second-class currents as evidenced by Eqs. (B12a)-(B12d). Due to the current constraints on second-classcurrents we do not take this into account and insteadfocus on scalar and tensor currents. Note that since we defined axial vector form factors to switchsigns for β + / EC, it is the second-class contributions whichchange sign in Eqs. (B12a)-(B12d). Appendix C: Kinematic recoil in form factordecomposition methods
In the treatment of any multi-body decay with en-ergy releases much smaller than at least one of the con-stituents, small recoil corrections appear, i.e. contribu-tions of O ( q/M ) (cid:28)
1, where q is the momentum transferduring the decay and M in the mass of the decaying par-ticle. In the case of β decay, the energy released almostnever exceeds 10 MeV, so that q/M ∼ − at most.At the current level of experimental precision, however,these terms are relevant. This fact is exacerbated whensignificant cancellations occur in the main matrix ele-ments, so that these recoil-order effects are significantlyboosted in relative precision (see Sec. III). Following Hol-stein [98], it is useful to categorize recoil-order terms fol-lowing their origin • Kinematical, O (1) × q/M • Dynamic, O ( A ) × q/M • Coulombic, O ( αZM R ) × q/M where A is the mass number of the decaying nucleus and R is the charge radius. Points two and three are con-tained in the proper description of the transition matrixelement. Here we are mainly concerned with the first,and show how the effects are treated differently in differ-ent descriptions of nuclear β decay at this time.The kinematical recoil order corrections arise in twodifferent parts of the calculation. The first occurs inthe evaluation of the nuclear current through a choiceof frame. We can most easily show this in the method ofHolstein, by explicitly expanding the product of leptonand nuclear currents as a set of Lorentz-scalars. In thecase of a pure J → J vector transition, we can write i M = l µ (cid:104) f | V µ | i (cid:105) = a ( q ) P · l M (C1)where P = p f + p i is the sum of initial and final four-momenta, and a ( q ) is a general form factor. In the restframe of the initial state, we can write P µ = (2 M + E R , − (cid:126)q ) (C2)where E R is the recoil energy of the final state and q = p i − p f = p + k. (C3)Taking the Hermitian square of Eq. (C1) one arrives at |M| = | a (0) | (cid:32) l + l (cid:126)q · (cid:126)lM (cid:33) (C4)up to first order in q/M and neglecting E R /M . Usingnow the conservation of the lepton current, ∂ µ l µ = 0, wefind |M| = l | a (0) | (cid:18) W M (cid:19) (C5) where W is the energy difference between initial andfinal states, and l | a (0) | represents the main transitionamplitude squared.Moving to the Breit frame now, where (cid:126)p i = − (cid:126)p f wefind P µ = (2 M + X,(cid:126)
0) (C6)by construction, and the second term in Eq. (C5) doesnot appear. This is of course no problem, since oneshould also evaluate the lepton current in this frame. Themultipole decomposition of the leptonic and hadroniccurrents is performed following standard methods in theBreit frame [40, 99]. In the usual multipole decomposi-tions [31, 41, 100], however, one neglects the differencebetween lab frame and Breit frame and considers the ex-pansion of the nuclear current correct only up to zerothorder in O ( q/M ) whether explicitly or implicitly stated.As a consequence, the results in the usual formalismsmust be corrected through a Lorentz transformation fromthe Breit frame to the lab frame. The corrections intro-duced are different for different spectral functions.A second contribution to kinematical recoil order cor-rections comes from the treatment of the energy integral.Rather than perform the three-body momentum integral,one sets the recoiling particle momentum to zero and in-stead introduces an effective correction to the transitionrate d Γ ∝ |M| (cid:32) W e − W − (cid:126)p e · ˆ kM (cid:33) (C7)where ˆ k is a unit vector in the direction of the(anti)neutrino three-momentum. Combining this resultwith, e.g., Eq. (C5) one then easily recovers the mainkinematical recoil order corrections for vector transitions.The term proportional to W /M cancels with Eq. (C5),and (cid:126)p · ˆ k integrates to zero unless combined with similarterms in |M| . To lowest order the latter is the β - ν cor-relation, f βν /f ( a βν ). Performing the angular integralsone obtains finally V R N ≈ W e M (3 − a LOβν ) . (C8)In the case of a pure vector transition one has to lowestorder a βν = 1, and one recovers the usual term, V R N ≈ W e M . (C9)Higher-order corrections and similar results for Gamow-Teller transitions can be found, e.g., in Refs. [22, 39, 101,102].
Appendix D: Comparison of popular formalisms
When comparing to other formalisms it is important toonce again take note of the fact that the form factor de-composition is non-unique (e.g. Eq. (12)), meaning that0the definition of, e.g., ρ (Eq. (36)) is too. Thankfully,for more complex nuclei typically only two systems arein widespread use, i.e. the multipole decomposition inthe Breit frame introduced by Stech and Sch¨ulke whichis followed here [41, 103], Donelly and Walecka (assum-ing infinitely heavy nuclei) [100, 104], and others, andthe manifest Lorentz-invariance expansion suitable to al-lowed decays by Holstein [39]. While in the neutronseveral different works exist by a multitude of authors[25–27, 29], the situation there is simple enough to allowexplicit spinorial calculations.Since both approaches have been in use for severaldecades, compilations of comparisons have already beenreported and we can be brief. The comparison betweenthe results here using the Behrens-B¨uhring formalism andothers employing the Breit frame multipole decomposi-tion is trivial and consists only of simple prefactors. Inparticular, that by Donnelly and Walecka [100, 104, 105]which is now being used by the Jerusalem group [106] canbe found in Ref. [31]. In the notation of the Jerusalemgroup, we can write (cid:104) J f || ˆ C J ( q ) || J i (cid:105) = C ( qR ) J (2 L + 1)!! F JJ ( q ) , (D1a) (cid:104) J f || ˆ L J ( q ) || J i (cid:105) = −C (cid:40) ( qR ) J − (2 J − (cid:114) J J + 1 F JJ − ( q ) − ( qR ) J +1 (2 J + 3)!! (cid:114) J + 12 J + 1 F JJ +11 ( q ) (cid:41) , (D1b) (cid:104) J f || ˆ M J || J i (cid:105) = C ( qR J )(2 J + 1)!! F JJ ( q ) , (D1c) (cid:104) J f || ˆ E J || J i (cid:105) = −C (cid:40) ( qR ) J − (2 J − (cid:114) J + 12 J + 1 F JJ − ( q )+ ( qR ) J +1 ( J (cid:114) J J + 1 F JJ +11 ( q ) (cid:41) , (D1d)with C = (cid:113) J i +14 π . Since qR (cid:28) β -decay, the secondand last expression can be reduced to their first term,so that there is a one-to-one translation between formfactors. For the leading order J = 0 terms this is par-ticularly trivial. Note that because the Walecka decom-position formally occurs with infinitely heavy initial andfinal nuclear states - meaning the lab frame coincides withthe Breit frame where such a decomposition is justified- additional kinematic recoil corrections must included a posteriori as discussed in Appendix C. These currentlydo not appear to be accounted for in Ref. [106].The translation of Behrens-B¨uhring form factors tothose of Holstein can also be found in a variety of placesin the literature [31, 39, 62]. Final expressions agree per-fectly if one takes into account the phase space recoilcorrection factor (Eq. (C7)). 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