Limits on Tensor Coupling from Neutron β -Decay
LLimits on Tensor Coupling from Neutron β -Decay R. W. Pattie Jr.,
1, 2
K. P. Hickerson, and A. R. Young
1, 2 Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA ∗ Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, Calfornia, 90095, USA (Dated: September 21, 2018)Limits on the tensor couplings generating a Fierz interference term, b, in mixed Gamow-TellerFermi decays can be derived by combining data from measurements of angular correlation parametersin neutron decay, the neutron lifetime, and G V = G F V ud as extracted from measurements of the F t values from the 0 + → + superallowed decays dataset. These limits are derived by comparingthe neutron β -decay rate as predicted in the standard model with the measured decay rate whileallowing for the existence of beyond the standard model couplings. We analyze limits derived fromthe electron-neutrino asymmetry, a , or the beta-asymmetry, A , finding that the most stringent limitsfor C T /C A under the assumption of no right-handed currents is − . < C T /C A < . A . PACS numbers: 23.40.-s,23.40.Bw,12.60.Cn
I. INTRODUCTION
In the following we present an analysis of a “lifetime-consistency test” for neutron beta-decay, from which wederive relevant limits for beyond the standard modelphysics, in particular for new scalar and tensor couplings.Our analysis utilizes high precision data from 0 + → + decays and neutron decay and does not supplant moregeneral fitting procedure to obtain limits from all beta-decay data[1, 2]. We note, however, that our limitsare comparable to those obtained from fits to the en-tire beta decay set when similar assumptions are made(no right-handed neutrinos). This brief report was in-spired by comments in Bhattacharya et al. [3] and be-gun as a part of thesis research [4]; however, we notethat additional details have subsequently been publishedby Ivanov, Pitschmann, and Troitskaya [5].The general method compares the measured value ofthe neutron lifetime, whose current Particle Data Group(PDG) value is τ n = 880 . ± . + → + and the value of the axial-vector coupling constant, λ ≡ g A /g V , extracted fromangular correlations measurements. Because this com-parison requires the interpretation of specific angularcorrelations measurements to consistently extract lim-its, we analyze some specific cases of interest associatedwith the electron-neutrino correlation, a , and the beta-asymmetry, A . We understand that this treatment isnot exhaustive, nor should it supplant direct limits onthe Fierz term in the neutron system, but it is intendedto indicate the utility of these limits. ∗ Electronic address : [email protected]
II. DERIVATION OF IMPACT OF THE FIERZTERM ON THE NEUTRON DECAY RATE β -decay can be represented, using all possible Lorentz-invariant couplings, by the Hamiltonian density H = (¯ pn )(¯ e ( C S + C (cid:48) S γ ) ν )+(¯ pγ µ n )(¯ eγ µ ( C V + C (cid:48) V γ ) ν )+ 12 (¯ pσ λµ n )(¯ eσ λµ ( C T + C (cid:48) T γ ) ν ) − (¯ pγ µ γ n )(¯ eγ µ γ ( C A + C (cid:48) A γ ) ν )+(¯ pγ n )(¯ eγ ( C P + C (cid:48) P γ ) ν ) + H.c., (1)where σ λµ = − i/ γ λ γ µ − γ µ γ λ ) and p, n, e, and ν repre-sent the hadronic and leptonic fields [6]. The strength ofeach type of interaction in the lepton current is given bya coupling constant C i and C (cid:48) i where i ∈ { V, A, S, T, P } are the vector, axial-vector, scalar, tensor, and pseudo-scalar interactions, respectively. In the scenario where | C i | = | C (cid:48) i | , parity is maximally violated, and in the stan-dard model | C V | = | C (cid:48) V | and | C A | = | C (cid:48) A | and C S = C (cid:48) S = C T = C (cid:48) T = C P = C (cid:48) P = 0. These restrictions are ex-perimentally determined, leaving the possibility for devi-ations below the current experimental precision.Limits on tensor couplings can be derived by notingthat the decay rate for neutron β -decay can be writtenas (ignoring, at present, the possibility of a Fierz term )1 τ n = G π (cid:126) (1 + 3 λ ) f n (1 + ∆ RC ) , (2)where, under the conserved vector current hypothesis, G V = G F | V ud | , f n is the statistical rate function for theneutron defined as f n = I ( x )(1 + ∆ f ) = 1 . , (3) I k ( x ) = (cid:90) x x − k ( x − x ) (cid:112) x − x, (4) a r X i v : . [ nu c l - t h ] S e p and where x and x are the electron total energy andend-point energy in terms of the electron rest mass,and ∆ f is the Coulomb and recoil correction for thephase-space integral I ( x ) = 1 . RC = 3 . × − [7]. G F is the Fermi coupling con-stant as extracted from muon decay [8], and V ud is thefirst element of the Cabibbo-Kobyashi-Maskawa (CKM)quark mixing matrix. One can also predict the neutrondecay rate from 0 + → + decays, by using the extractedvalue of ˜ G V from the average F t + → + and λ from neu-tron angular correlation measurements, where if the Fierzterm is zero ˜ G = G τ + = ˜ G π (cid:126) (1 + 3 λ ) f n (1 + ∆ RC ) . (5)A non-zero Fierz term will alter the neutron decayrate, τ n , via a (cid:104) m e /E (cid:105) term in the phase-space integraland modify the value of G V extracted from superallowedFermi decays to˜ G = G (cid:28) b F γ I (˜ x ) I (˜ x ) (cid:29) , (6)where γ = (cid:112) − ( Zα ) , Z is the atomic number, α is thefine structure constant, and ˜ x is the end point energy forthe superallowed Fermi decay isotopes, and I (˜ x ) /I (˜ x )corresponds to the ratio of phase-space integrals over thesuperallowed decay used in the determination of V ud and b F = 2 Re ( C S /C V ) [9]. For the moment we will ignorethe changes in λ induced by b , this will be addressed inthe following sections. In Table I, the 13 isotopes in-cluded in the determination of the average F t are listedwith the absolute uncertainty on the measurement andthe statistical rate function and the ratio I (˜ x ) /I (˜ x )[9]. The reported values include both recoil and Coulombcorrections. Writing Eq. 2 and Eq. 5 in terms of G V , b F and b , we have1 τ n = G π (cid:126) (1 + 3 λ ) f n (1 + ∆ RC ) (1 + κb ) , (7)and 1 τ + = G π (cid:126) (1 + 3 λ ) f n (1 + ∆ RC ) (1 + ζb F ) , (8)where κ = I ( x ) /I ( x ), ˜ κ = I (˜ x ) /I (˜ x ), and ζ = (cid:104) γ ˜ κ (cid:105) ∼ . κb ) arises fromthe neutron phase-space integral when b (cid:54) = 0, and the (1+ ζb F ) term in Eq. 8 is from substitution of measured G V using Eq. 6. Taking the difference between the measuredneutron decay rate and the decay rate predicted from0 + → + decays in terms of measured quantities gives τ n K (1 + 3 λ ) = 1 + ζb F κb (9)where all the constants have been combined into K = ˜ G f n (1 + ∆ RC )2 π (cid:126) = 1 . × − s − , we express Eq. 9 in terms of the measured vlue of theweak coupling constant ˜ G V using Eq. 6, and we are ne-glecting any affect on λ due to b . Critically, leading orderdifferences in the predicted versus measured decay ratesmust come from scalar and tensor-induced couplings inthe Fierz term, and any new physics which adjusts thevalue of G F and V ud affects both rates uniformly. Ad-ditionally, the impact of the scalar coupling determinedin the superallowed decays is suppressed by ζ due to themuch higher endpoint energy of these decays relative toneutron β -decay.Under of the assumptions of this analysis the Fierz in-ference term in neutron β -decay can be approximated interms of the scalar C S /C V and tensor C T /C A couplings[1] b = 2 √ − α λ (cid:20) Re (cid:18) C S C V (cid:19) + 3 λ Re (cid:18) C T C A (cid:19)(cid:21) . (10)At this point, we already have a reasonably strong con-straint on new physics by using Eq. 9, Eq. 10, and thedefinition of b F C T C A (6 λ γ ) = δbτ n Kκ − γ C S C V − (1 + 3 λ ) (cid:18) δbκ − ζ C S C V (cid:19) , (11)where δb = (cid:104) C S /C V )˜ κ (cid:105) . Using the PDG valuesfor λ = − . τ n = 880 . .
1) s, and V ud =0 . C S /C V =0 . σ (95% C.L.) limits of − . < C T /C A < . λ = − . − . < C T /C A < . III. b DEPENDENCE OF λ The limits obtained from Eq. 11 have ignored the factthat λ is determined experimentally by measuring cor-relation coefficients, which typically are modified by theexistence of a Fierz term as in X m ( E e ) = X ( E e )1 + b m e /E e , (12)where X m ( E e ) ∈ { a, A, B, ... } is the measured value ofcoefficient as a function of electron energy. For this anal-ysis we will focus on a and A since closed form expressionscan be obtained for limits on tensor couplings and theyhave the highest sensitivity to λ . The method used toextract the correlation coefficient and the energy rangeof the analysis will impact the sensitivity to b , as willbe shown explicitly in the case of a . Note: we are alsoexplicitly ignoring imaginary couplings and the affect oftensor and scalar couplings on the angular correlations a and A , which are second order in C S and C T . TABLE I. The statistical weighting and ratio of the phase-space factors is presented for each of the 13 isotopes used in the0 + → + superallowed dataset to calculate the average F t value. I k (˜ x ) are the statistical rate functions defined in Eq. 4 andcalculated by Towner and Hardy [9, 10]. Isotope F t I (˜ x ) I (˜ x ) I (˜ x ) /I (˜ x ) C 3067 . .
6) 2 . . . O 3071 . .
3) 42 . . . Mg 3078 . .
4) 418 . . . Ar 3069 . .
5) 3414 . .
5) 724 . . Al m . .
4) 478 . . . Cl 3070 . .
1) 1995 . . . K m . .
4) 3297 . . . Sc 3072 . .
7) 4472 . .
15) 895 . . V 3073 . .
7) 7209 . . . Mn 3070 . .
8) 10745 . . . Co 3069 . .
3) 15766 . .
9) 2470 . . Ga 3071 . .
2) 26400 . .
3) 3719 . .
2) 0 . Rb 3078 . .
0) 47300 . . .
4) 0 . . . A. λ derived from the measured β -asymmetryparameter, A m Single parameter fits to the energy dependence ofthe β -asymmetry will modify the measured quantity by A / (1+ b (cid:104) I ( x , x ) /I ( x , x ) (cid:105) ), where x and x are thelimits of the energy range used in analysis. Table II showsthe ratio of the phase-space integrals over the reportedanalysis energy range for several of the experiments thatmeasure the asymmetry and would be subject to the typeof dilution analyzed here. The leading order expression of A o (where one has already corrected the measured asym-metry for small radiative and recoil order corrections [6])in terms of λ is A = 2 | λ | − | λ | λ , (13)which, combined with Eq. 10, Eq. 12, and Eq. 11, givesan relation for C T /C A in terms of purely measured quan-tities3 λ = (cid:20) hC T /C A γ + y (cid:21) = 3 − − (cid:114) − A m (cid:16) C T + 2 /A m (cid:17) ˜ C S A m ˜ C T + 2 . (14)In Eq. 14, we have made the following substitutions˜ C x = C x γ + 1, y = ( δb/κ ) − ζ ( C S /C V ), and h =( τ n K ) − − γκ − y . We assume here that the BSM scalarand tensor couplings make negligible contributions to theradiative and recoil-order corrections. For BSM cou-plings at the ≈ C T /C A and are ruled outby current experimental limits (see Appendix A). The re-maining solution gives a 2- σ limit on the tensor coupling of − . < C T /C A < . λ includes the result of Mostovoiet al [16], which is determined by simultaneous measure-ment of the β -asymmetry, A , and the neutrino asym-metry, B , and is therefore not consistent with the ap-proach presented here. Careful analysis of a simultane-ous experiment would be required to determine the im-pact a non-zero Fierz term would have on the extracted λ . Using the results from Perkeo II [12] and UCNA[17], A m = − . − . 1% [21–23]. Determining the a coefficientcan be performed by directly measuring the angular dis-tribution of emitted electron and proton in coincidence,in which case the a / (1 + bm e /E e ) scaling can apply(directional method), or via a measurement of the pro-ton energy spectrum. Directly fitting the proton spec-trum or using discrete points from the spectrum (spec-tral fit method) as in Stratowa et al [24] will result in a m = a + x f b , where x f ∼ . 09 as determined by MonteCarlo. An alternate method of analyzing proton spectraldata is an integral analysis, which has the advantage of TABLE II. Experimental results for λ from measurements of A using a single parameter fit to energy dependence of theasymmetry are summarized. For each measurement the reported analysis window and the phase-space integral ratios over thatrange are listed. The last column estimates the change in the asymmetry where b = 0 . Experiment A λ Energy Range I ( x , x ) /I ( x ) I ( x , x ) /I ( x ) ∆ A [%]Perkeo [18] -0.1146(19) -1.262(5) > 200 keV 0.801 0.581 0.06Perkeo II [12] -0.11951(50) -1.2755(13) 325-675 keV 0.843 0.534 0.05Ill TPC [19] -0.1160(9)(12) -1.266(4) 200-700keV 0.807 0.583 0.06UCNA [17] -0.11952(110) -1.2756(30) 275-625 keV 0.828 0.557 0.06Yerozolimsky [20] -0.1135(14) -1.2594(38) 250-780 keV 0.824 0.561 0.06 being much less sensitive to the presence of a Fierz term,as presented in [25].In such analysis one compares theintegral rate over a fixed energy range to the total decayrate. This results in a linear scaling of the form a m = a + x I b = 1 − λ λ + x I b, (15)where a m = − . x I ∼ . 008 from [25], which wasconfirmed by this analysis using Monte Carlo. UsingEq. 10, Eq. 11, and the two expressions for the measuredcoefficient including a Fierz term, we can find a solutionfor C T /C A in terms of measurement quantities C T C A = h ( a + ) − y (cid:16) − a m + x I γ C S C V (cid:17) γ (cid:16) − a m + x I γ C S C V (cid:17) − xγh (Linear) , h ( am ) − y (cid:16) am − − γκ C S C V (cid:17) γ (cid:16) am − − κγ C S C V − hκ (cid:17) (Inverse) , (16)where we have made use of the substitutions defined inthe previous section. This approach is more straightfor-ward due to the fact that, unlike A , there are no termswhich are linear in λ in Eq. 15. Using the current PDGvalue for a = − . x f = 0 . 09) we find limits of − . < C T C A < . C.L. ) . (17)While this limit is not currently competitive with thoseobtained through measurements of the β -asymmetry,Figure 1 shows that determining a from angular distri-butions is more sensitive than the spectral fit method totensor couplings, and the next generation experimentsshould improve upon current limits set by A . (Forexample, a measurement such as Nab [21], aiming fora precision of ∆ a/a (cid:39) − , could set constraints of | C T /C A | < . IV. CONCLUSION In this analysis we have presented a self-consistentderivation of limits to tensor couplings in the weak inter-action, using experimentally measured quantities fromneutron β -decay and 0 + → + superallowed Fermi de-cays. By calculating the difference of the measured andpredicted neutron β -decay rate, we are able to derive alimit to the tensor coupling − . < C T /C A < . C X = C (cid:48) X . Noting thata non-zero Fierz term would modify the experimentallyreported value of λ , we have shown that the measuredcorrelation coefficients a m and A m can be used to setlimits on the tensor couplings that are competitive withthose obtained from global fits to the available data [2].If the precision of A or a reaches the 0.1% level, thenthe accuracy of the neutron lifetime becomes the leadingcontribution to the derived limits.These results can be used to set constraints on theeffective couplings from Bhattacharya et al [3], where the ∆ X (cid:144) X ∆ (cid:72) C T (cid:144) C A (cid:76) Current a Limits (cid:64) (cid:68) a (cid:72) Nab (cid:64) (cid:68)(cid:76) a (cid:72) aSPECT (cid:64) (cid:68)(cid:76) a (cid:72) aCORN (cid:64) (cid:68)(cid:76) A (cid:72) UCNA (cid:64) (cid:68) (cid:43) Perkeo II (cid:64) (cid:68)(cid:76) A (cid:72) PERC (cid:64) (cid:68)(cid:76) A m (cid:61) A (cid:144)(cid:72) (cid:43) (cid:88) b (cid:92)(cid:76) a m (cid:61) a (cid:144)(cid:72) (cid:43) (cid:88) b (cid:92)(cid:76) a m (cid:61) a (cid:43) xb FIG. 1. The 2- σ limits on C T /C A are shown for the meth-ods of extracting a discussed in the text, along with thosefrom fitting the energy dependence of the β -asymmetry A .Current limits on a are taken from the PDG2012 [11], andfuture limits denote the proposed sensitivity of experimentssuch as Nab[21](directional method), aCORN [23](directionalmethod), and aSPECT [22](fit method). For this analysis weconsider the β -asymmetry from UCNA [17] and Perkeo II [12],and the proposed limits are from the PERC experiment [26]. tensor coupling is given as C T C A = − g T (cid:15) T g A (1 + (cid:15) L − (cid:15) R ) , (18)where (cid:15) R ( L ) represent the effective right and left handedcouplings and both are zero in the SM. In general, thisdirectly leads to a limit of − . × − < g T (cid:15) T / ( g A (1 + (cid:15) L − (cid:15) R ) < . × − . However, under the assumptionthat BSM physics arises from tensor couplings then (cid:15) R = (cid:15) L = 0, and this simplifies to a limit on g T (cid:15) T /g A . ACKNOWLEDGMENTS We would like to thank V. Cirigliano and J.C. Hardyfor many helpful discussions and I.S. Towner for provid- ing calculations of I /I for the superallowed dataset inTable I. This work was supported by NSF grant 1005233and DOE grant number DE-FG02-97ER41042. Appendix: Solutions for C T /C A ( A m ) The roots obtained from solving Eq. 14 are C T /C A = (cid:40) − A m A m γ , − A m (2+3 A m ) h +3 yA m (1+ C s γ ) + h [ A m { A m (1+ y ) } (1+ C s γ ) − ± hγ √ A m { A m ( y − − } (1+ h + C s γ )3 A m γ (1+ h + C s γ ) , (A.1)where h , y , and γ are defined in the text and C S ≡ C S /C V . 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