Limits from neutron beta decay on parameters describing physics beyond the
Standard Model are presented. New Physics is described by the most general
Lorentz invariant effective Hamiltonian involving vector, scalar and tensor
operators and Standard Model fields only. Two-parameter fits to the decay
parameters measured in free neutron beta decay have been done, in some cases
indicating rather big dependence of the results on g_A/g_V ratio of nucleon
form factors at zero four-momentum transfer.Comment: 6 pages, 6 figures, presented at the XXXV International Conference of
  Theoretical Physics "Matter to the Deepest 2011", Ustron, Poland, September
  12-18, 201

Holeczek, Jacek

Ochman, Michal

Stephan, Elzbieta

Zralek, Marek

Acta Physica Polonica B

English

arXiv

Limits from neutron beta decay on parameters describing physics beyond the Standard Model are presented. New Physics is described by the most general Lorentz invariant effective Hamiltonian involving vector,
scalar and tensor operators and Standard Model fields only. Two-parameter fits to the decay parameters measured in free neutron beta decay have been done, in some cases indicating rather big dependence of the results on gA=gV ratio of nucleon form factors at zero four-momentum transfer

Ochman, Michał

Stephan, Elżbieta

Zrałek, Marek

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Physics Beyond Standard Model in Neutron Beta Decay  Author: Jacek Holeczek, Michał Ochman, Elżbieta Stephan, Marek Zrałek  Citation style: Holeczek Jacek, Ochman Michał, Stephan Elżbieta, Zrałek Marek. (2011). Physics Beyond Standard Model in Neutron Beta Decay. "Acta Physica Polonica B" (Vol. 42, no. 11 (2011), s. 2493-2499), doi 10.5506/APhysPolB.42.2493   Vol. 42 (2011) ACTA PHYSICA POLONICA B No 11PHYSICS BEYOND STANDARD MODELIN NEUTRON BETA DECAY∗Jacek Holeczek, Michał Ochman, Elżbieta StephanMarek ZrałekInstitute of Physics, University of SilesiaUniwersytecka 4, 40-007 Katowice, Poland(Received October 18, 2011)Limits from neutron beta decay on parameters describing physics be-yond the Standard Model are presented. New Physics is described bythe most general Lorentz invariant effective Hamiltonian involving vector,scalar and tensor operators and Standard Model fields only. Two-parameterfits to the decay parameters measured in free neutron beta decay have beendone, in some cases indicating rather big dependence of the results on gA/gVratio of nucleon form factors at zero four-momentum transfer.DOI:10.5506/APhysPolB.42.2493PACS numbers: 12.60.Cn, 13.30.Ce, 14.20.Dh, 14.60.St1. IntroductionFor many years nuclear β-decays have been exploited as laboratories fortesting the Standard Model (SM) in the domain of low energies. Alongwith developments of intense sources of cold and ultracold neutrons andimprovements of experimental techniques, the precision of measurements inthe simplest of such systems: the β-decay of a free neutron, is constantlyincreasing. It opens the way to study the limits on physics beyond SM setsolely by the parameters of the neutron β-decay.We assume that at the quark–lepton level β-decay is described by thegeneral 4-point Hamiltonian [1]Hβ = 4∑k,l=L,R{akle¯γµPkν(k)u¯γµPld+Akle¯Pkν(k)u¯Pld+αkke¯σµν√2Pkν(k)u¯σµν√2Pkd}+ h.c. , (1)∗ Presented by M. Ochman at the XXXV International Conference of TheoreticalPhysics “Matter to the Deepest”, Ustroń, Poland, September 12–18, 2011.(2493)2494 J. Holeczek et al.where u, d are quark fields, e stands for electron field and PL = 12 (1− γ5),PR = 12 (1 + γ5), σµν =i2 [γµ, γν ], where our metric and gamma matricesare the same as e.g. in [2]. We work in the basis in which mass matrix ofcharged leptons is diagonal and the left (L) and right (R) neutrino fields aregiven byν(L) =∑iUeiPLνi , (2a)ν(R) =∑iVeiPRνi , (2b)where νi is the i-th neutrino field with a certain mass, U and V are respec-tively the Pontecorvo–Maki–Nakagawa–Sakata matrix and similar mixingmatrix for right-handed neutrinos. SM is restored when akl = Akl = αkk = 0for k, l = L, R except aLL = VudGF/√2, where GF is the usual Fermi con-stant and Vud is the element of quark Cabibbo–Kobayashi–Maskawa mixingmatrix.When calculating the amplitudes for neutron beta decay at small four-momentum transfer q2 ≈ 0 we have used the relations [1]gVu¯pγµun = 〈p|u¯γµd|n〉 , (3a)gAu¯pγµγ5un = 〈p|u¯γµγ5d|n〉 , (3b)gSu¯pun = 〈p|u¯d|n〉 , (3c)gTu¯pσµνun = 〈p|u¯σµνd|n〉 (3d)with 〈p|, up and |n〉, un being proton and neutron states and bispinors,respectively. From conserved vector current hypothesis one gets gV = 1.In the quark model with spherically symmetric wave functions of quarksthe following relations have been derived [3]: gS = −12 + 910gA and gT =53(12 +310gA). Substituting the SM value for gA ' 1.27 into the aboverelations leads to: gS ' 0.64 and gT ' 1.47. However, in our derivationsand fits we treat gS and gT as free parameters (independent of gA).2. Decay parametersFrom Eq. (1) the five-fold differential decay width for polarized neutronwithout measurement of final electron and proton polarization is given by(in analogy to [4])dΓdEedΩedΩν∼ peEeE2ν{1 + a~pe · ~pνEeEν+ bmeEe+~λn ·[A~peEe+B~pνEν+D~pe × ~pνEeEν]}, (4)Physics Beyond Standard Model in Neutron Beta Decay 2495where ~λn is the neutron polarization vector, me, pe = |~pe|, Ee are, respec-tively, the mass, momentum and total energy of electron, E0 is the maxi-mum value of Ee, |~pν | = Eν = E0 − Ee is the antineutrino energy1. TheΩe, Ων denotes the solid angles of electron and antineutrino emission. Wehave worked at tree-level (except: calculation of 〈E−1e 〉 — see below) andwith approximations such that terms proportional to u¯pγ5un are not presentin (4). Furthermore, we will consider only cases when: gV, gA, gS, gT, aswell as akl, Akl, αkk for k, l = L, R are real — then D ≡ 0 and time re-versal symmetry is preserved, that is well motivated experimentally (PDGaverage [5]: D = (−4± 6)× 10−4).We express the decay parameters a, b, A, B, where B has the form ofB = B0+bνme/Ee, in terms of the ratio gA/gV and the following parameters(see also [1]) for k, l = L, RVkl =aklaLLκk , Skl =AklaLLgSgVκk , Tkl =αklaLLgTgVκk , (5)whereκL = 1 , κR =(∑′i |Vei|2∑′i |Uei|2)1/2(6)with summation∑′i running only over kinematically allowed antineutrinostates. In SM and for some cases of physics beyond SM b = 0 and bν = 0.As a result of the applied approximations formulas for a, b, A, B0, bν dependin general case only on two combinationssL = SLL + SLR , (7a)sR = SRR + SRL . (7b)Next, following the approach applied in [6, 7, 8], we definea¯(〈W−1〉)=a1 + b 〈W−1〉 , (8a)A¯(〈W−1〉)=A1 + b 〈W−1〉 , (8b)B¯(〈W−1〉)=B0 + bν〈W−1〉1 + b 〈W−1〉 , (8c)where 〈W−1〉 = me〈E−1e 〉, and apply these quantities in the fits to theexperimental data. The χ2, which we will minimize with the fit procedure,1 The effect of nonzero neutrino masses enters only trough presence of mixing matricesU and V .2496 J. Holeczek et al.is of the formχ2 =∑iai − a¯( 〈W−1〉i)δai2+∑jAj − A¯(〈W−1〉j)δAj2+∑kBk − B¯( 〈W−1〉k)δBk2 , (9)where the selected data are presented in Table I: ai, Aj , Bk and δai, δAj , δBkdenote the central value and the error of the respective decay parameter in acertain experiment. We calculate the particular value of 〈W−1〉i = me〈E−1e 〉iTABLE IWe have followed the PDG [5] data selection but took only the most precise data(the error in measurements of a is less than 6% of central value, for A and B — it isless than 2%). When experiment reports statistic and systematic error separatelywe add these two errors in quadrature. In the case of asymmetric errors, we havetaken the larger of the reported errors. Most of presented values of the 〈W−1〉 havebeen taken from [7]. We have used all 11 “data points” in the table below in everyfit presented in this paper. Because of unsolved experimental ambiguity of neutronlifetime measurements (see [5]) we have not included this quantity in our analyzes.Par. Value Error 〈W−1〉 Paper ID (PDG)a −0.1054 0.0055 0.655 Byrne 02 [12]−0.1017 0.0051 0.655 Stratowa 78 [13]A −0.11966 0.00166 0.557 Liu 10 [14]−0.1189 0.0007 0.534 Abele 02 [15]−0.1160 0.0015 0.582 Liaud 97 [16]−0.1135 0.0014 0.558 Yerozolimsky 97 [17]−0.1146 0.0019 0.581 Bopp 86 [18]B 0.980 0.005 0.599 Schumann 07 [19]0.967 0.012 0.600 Kreuz 05 [20]0.9801 0.0046 0.594 Serebrov 98 [21]0.9894 0.0083 0.554 Kuznetsov 95 [22]Physics Beyond Standard Model in Neutron Beta Decay 2497from 〈E−1e〉i=Emaxi∫EminidEedΓdEeE−1e/ Emaxi∫EminidEedΓdEe, (10)where Emini and Emaxi in general are different for different experiments. Atthis stage of calculation, Fermi function F (Ee) (that is a leading order QEDcorrection) [9, 10,11] has been incorporated and SM was assumeddΓdEe=(g2V + 3g2A) G2F|Vud|22pi3peEe(E0 − Ee)2F (Ee) , (11)F (Ee) =2piαEe/pe1− e−2piαEe/pe. (12)3. ResultsIn SM the formulas derived for decay parameters depend on λ = gA/gValone and simplify toa =1− λ23λ2 + 1, A =2λ(1− λ)3λ2 + 1, B =2λ(λ+ 1)3λ2 + 1. (13)In this case, the one-parameter fit is performed, which results in χ2min = 25.42with± 0.0014 ( 68.27% C.L. ) (14a)λ = 1.2703 ± 0.0023 ( 90% C.L. ) (14b)± 0.0028 ( 95.45% C.L. ) (14c)that is in a good agreement with the PDG average [5]: λ = 1.2701 ± 0.0025(error scaled by PDG by 1.9).The above results (14a)–(14c) apply also when: Vkl = Skl = Tkk = 0 fork, l = L, R except VLR (and VLL = 1 by definition — see Eqs. (5) and (6)).In this case the formulas (13) hold for modified λλ =gAgV1− VLR1 + VLR. (15)In the next step one of the parameters: VRk, sk, Tkk for k = L, R isnonzero and fitted together with the ratio gA/gV. Among these cases onlywhen the nonzero parameter is sL or TLL we have b 6= 0 and bν 6= 0. Theresults of such two-parameter fits are presented in Fig. 1.2498 J. Holeczek et al.V / gAg1.26 1.27 1.28 1.29 1.3 |RL| V00.020.040.060.080.1 = 22.012minχ 68.27 % C.L.90 % C.L.95.45 % C.L.V / gAg1.26 1.27 1.28 1.29 1.3 |RR| V00.020.040.060.080.1 = 20.222minχ 68.27 % C.L.90 % C.L.95.45 % C.L.V / gAg1.26 1.27 1.28 1.29 1.3LR +  SLLS-0.2-0.15-0.1-0.0500.050.1 = 21.482minχ 68.27 % C.L.90 % C.L.95.45 % C.L.V / gAg1.26 1.27 1.28 1.29 1.3 |RL +  SRR| S00.050.10.150.20.250.3 = 23.012minχ 68.27 % C.L.90 % C.L.95.45 % C.L.V / gAg1.26 1.27 1.28 1.29 1.3 1.31 1.32 1.33 1.34LLT-0.1-0.08-0.06-0.04-0.020 = 17.362minχ 68.27 % C.L.90 % C.L.95.45 % C.L.V / gAg1.26 1.27 1.28 1.29 1.3 |RR| T00.020.040.060.080.1 = 20.182minχ 68.27 % C.L.90 % C.L.95.45 % C.L.Fig. 1. Results of the two-parameter fits (the cross denote the position of χ2min ineach case). The line — marked areas correspond to the λ = gA/gV intervals: (14a)— the narrow one and (14c) — the wider one. Note that | · | is the absolute value— not the module of a complex number, as all parameters are real.Physics Beyond Standard Model in Neutron Beta Decay 2499In conclusion, Standard Model describes the neutron beta decay verywell. The fits are minimally better if New Physics is included, especially iftensor terms are present. In some cases there is rather big dependence ofthe results on gA/gV ratio.This work has been supported by the Polish Ministry of Science andHigher Education under grant No. N N202 064936.REFERENCES[1] P. Herczeg, Prog. Part. Nucl. Phys. 46, 413 (2001).[2] C. Giunti, C.W. Kim, Fundamentals of Neutrino Physics and Astrophysics,Oxford University Press, 2007.[3] S.L. Adler et al., Phys. Rev. D11, 3309 (1975).[4] J.D. Jackson, S.B. Treiman, H.W. Wyld, Jr., Phys. Rev. 106, 517 (1957).[5] K. Nakamura et al. [Particle Data Group], J. Phys. G 37, 075021 (2010) and2011 partial update for the 2012 edition, http://pdglive.lbl.gov[6] F. Gluck, I. Joo, J. Last, Nucl. Phys. 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Physics Beyond Standard Model in Neutron Beta Decay

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