hep-ph/9812531 KIAS-P98051 SNUTP 98-149 Bounds and Implications of Neutrino Magnetic 9 Moments from Atmospheric Neutrino Data 9 9 1 n a Sin Kyu Kang,a,∗ Jihn E. Kim,a,b,† and Jae Sik Leea,‡ J 8 3 (a) School of Physics, Korea Institute for Advanced Study, Seoul 130-012, Korea, and v 1 (b) Department of Physics and Center for Theoretical Physics, Seoul National University, 3 5 Seoul 151-742, Korea 2 1 (February 1, 2008) 8 9 / h p - p e Abstract h : v i X r The neutral current effects of the future high statistics atmospheric neutrino a data can be used to distinguish the mechanisms between a ν oscillation to a µ tau neutrino or to a sterile neutrino. However, if neutrinos possess large di- agonal and/or transition magnetic moments, the neutrino magnetic moments cancontributetotheneutralcurrenteffects whichcanbestudiedbythesingle π0 production events in the Super-K data. This effect should be included in the future analyses of atmospheric data in the determination of ν to tau or µ sterile neutrino oscillation. PACS number(s): 13.15.+g, 14.60.Pq, 14.60.St ∗[email protected] †[email protected] ‡[email protected] 1 I. INTRODUCTION Neutrinos might possess two properties which are feeble, but will be important barom- eters of physics beyond the standard model scale. These are neutrino mixings (masses and oscillations) [1] and neutrino magnetic moments [2]. Before 1998, experiments gave bounds on these properties, in general. But the recent results from the Super-Kamiokande Collaboration [3] have provided a strong evidence for a deficit in the flux of atmospheric neutrinos, which are presented in the form of the double ratio (N /N ) µ e obs R = , (1) (N /N ) µ e MC which implies the existence of ν oscillation. The measured value ofR for Super-Kamiokande µ is 0.61 0.06 0.05 for the sub-GeV data and 0.67 0.06 0.08 for the multi-GeV data, ± ± ± ± whileweexpect R = 1inaworldwithoutoscillations. Muonneutrinooscillationintoanother species of neutrino provides a natural explanation for the deficit and even the zenith angle dependence. The ν ν oscillation is the most favorable solution for the atmospheric µ τ → neutrino problem, whereas the ν ν oscillation is strongly disfavored by CHOOZ results µ e → [4]. The oscillations into sterile neutrinos (ν ) give a plausible solution as well [5]. This s evidence for the neutrino oscillations is also supported by the SOUNDAN2 [6] and by the Super-Kamiokande [7] and MACRO [8] data on upward-going muons. It is usually assumed that the neutral current effect in neutrino oscillation experiment is unchanged, since any standard model neutrino produced by oscillation has the same neutral current (NC) interaction. Therefore, the ratio of the neutral (NC) and charged currents (CC) events is important to investigate the neutrino neutral current. Thus the observation of single π0 events, induced by the neutral current, by Super-Kamiokande Collaboration [3] can lead to important physical implications. Since the π0 NC event is detected as two diffuse rings whereas the CC events due to ν e are detected as one diffuse ring due to e± and one sharp ring due to π± and the CC events due to ν are detected as two sharp rings from µ± and π± [9]. Thus, a NC event can be µ discriminated from a ν CC event and a ν CC event [10,11]. It has been considered difficult e µ to separate NC and CC events clearly. But the single π0 events described above can be used to discriminate the NC events from CC events. Indeed, it is believed that the cleanest way to identify NC events in Super-Kamiokande is to detect a single π0 from the process ν +N ν +N +π0, with N being either a neutron or a proton below Cerenkov threshold. → The π0 is detected via its decay into two photons which lead to two diffuse e-like rings whose 2 invariant mass is consistent with the π0 mass [9]. The ratio of π0-like events to e-like events compared to the same ratio of the Monte Carlo in the absence of the oscillation has been measured by the Super-Kamiokande Collaboration [7], (π0/e) data Rπ0/e = (π0/e) = 0.93±0.07stat ±0.19sys, (2) MC where the systematic error is dominated by the poorly known single π0 cross section, and the statistical error is based on 535 days of running. The ratio Rπ0/e is expected to be 1 for ν ν oscillations while 0.75 for ν ν oscillations or ν ν oscillations if one takes the µ τ µ s µ e − − − measured ν /ν ratio to be 0.65 [10]. The admixture of ν ν and ν ν oscillations also µ e µ τ µ e − − leads to a deviation of Rπ0/e from 1. Therefore, a precise measurement of the ratio will be used to distinguish ν ν from ν ν oscillation. µ τ µ s → → At first sight, it is likely that any deviation of Rπ0/e from 1 implies muon neutrino oscillation into a sterile neutrino. However, if there exists a large muon neutrino magnetic moment (diagonal or transition), it will produce an additional neutral current effect which has to be separated out to draw a definite conclusion. Indeed, right after the discovery of the neutral current, the upper bound on the muon neutrino magnetic moment was given [12]. Also, the experimental bounds on transition magnetic moments and other properties in view of NC data were presented [13]. The theoretical problem of obtaining a large neutrino magnetic moment has begun with interactions beyond the standard model [2]. In general, the loop diagram will have a (mass)2 suppression, presumably by M2 where M can be the W boson mass or a scalar mass. It X X is possible to have a large Dirac neutrino magnetic moment if the loop contains a heavy fermion [2], m µ (3) ν ∼ M2 X where m is the mass of the heavy fermion. This mechanism can be generalized in models with scalars [14]. But, the same loop without the external photon line would give a contribution to the neutrino mass matrix. Therefore, one expects, taking the coupling as 10−3, m m 1/3 µ 10−3 e µ µ 10−13µ (4) ν B B ∼ M M ∼ X (cid:18) X(cid:19) where µ = eh¯/2m c is the electron Bohr magneton and we used m O(1) eV for the B e ν ∼ numerical illustration. To suppress the contribution to the mass and still allow a large magnetic moment, continuous [15] and discrete [16] symmetries have been considered. In 3 this case, the neutrino magnetic moment can be as large as µ 10−10µ , which is not νµ ∼ B affected by the SN1987A constraint 10−13µ [17] since this bound applies to the electron B ∼ neutrino only. However, a large transition magnetic moment to a sterile neutrino is not forbidden that severely. For example, one can introduce a transition moment with an accompanying mass as large as several hundred MeV. Of course, the masses of the light neutrinos are bounded by eV. In this case, the transition neutrino magnetic moments can be as large as 10−7µ and B may contribute to NC events. In particular, we are interested in the single π0 production through a large transition magnetic moment, which would contribute to Rπ0/e. In this spirit, we will obtain the lower bound on the transition neutrino magnetic moment (to a sterile neutrino) from Rπ0/e. This paper is organized as follows: In Sec. 2, we describe the amplitude for the single π0 production. In Sec. 3, the kinematics and the differential cross section for the single π0 production are given. In section 4, we present the contribution to the cross section of the π0 production generated by a possible neutrino transition magnetic moment. In Sec. 5, we discuss the physical implications based on the numerical result. II. PRODUCTION OF THE SINGLE NEUTRAL PION The single π0 production has two contributions: one from the production and decay of the (3/2,3/2) baryon resonances and the other from the continuum contribution. At low energies (E < 2 GeV), the contribution from baryon resonance production is a dominant one for a single π0 production [18]: ν +N ν +N∗, → N∗ π0 +N, (5) → where N∗ represents baryon resonances. The cross section for the single π0 production in the region W < 1.6 GeV (W is the hadronic invariant mass in the final state) can be described following Fogli and Nardulli [18]. The effective Lagrangian for the neutrino neutral current is defined by 1 = G ν¯γλ(1+γ )νJNC, (6) LNC √2 F 5 λ by assuming the following general V,A structure of the hadronic part of the NC: JNC = g3V3 +g3A3 +g8V8 +g8A8 +g0V0 +g0A0, (7) λ V λ A λ V λ A λ V λ A λ 4 where Vi,Ai(i = 3,8,0) are the SU(3) nonet partners of the CC [19]. Neglecting the strange λ λ and charm NC’s, one can write JNC = g V3 +g A3 +g′ V′0 +g′ A′0, (8) λ V λ A λ V λ A λ where 1 V′0 = (V8 +√2V0), (9) λ s3 λ λ 1 A′0 = (A8 +√2A0), (10) λ s3 λ λ the electromagnetic current being given by Jem = V3 + 1V′0. From the Weinberg-Salam λ λ 3 λ model [19], 1 1 g = sin2θ , g = , (11) V W A 2 − 2 1 g′ = sin2θ , g′ = 0. (12) V −3 W A The isospin decomposition of one π0 channels is given by 1 1 A(νp νpπ0) = (2A +A )+ S, (13) 3 1 → 3 s3 1 1 A(νn νnπ0) = (2A +A ) S. (14) 3 1 → 3 −s3 The reduced matrix elements A ,A are given by 1 3 1 A = (A0 +A0 +A0 ), (15) 3 √2 ∆ π N 3 1 A = A0 √2A0 A0 +A0 +A0 +A0 , (16) 1 2√2 NNπ − π − 2√2 N S P D where A0,i = N,NNπ,P,S,D are given in the appendix and the indices S,P,D denote i S ,P ,D [20], respectively. The contributions totheamplitude S comefromthefollowing 11 11 11 amplitudes 1 1 1 2 2 2 S = √3 A0 + A0 + A0 + A0 + A0 . (17) 2 s2 N s2 NNπ 3 P 3 S 3 D   As is well known, the dominant contribution to the amplitude A comes from the ∆ reso- nance in this region (W < 1.6 GeV) [21]. Then, the amplitude for the single π0 production can be described by G A Fl Jα (18) NC α ≃ √2 5 where l = u¯(k′)γ (1+γ )u(k), (19) α α 5 Jα = g u¯(p′)qρD (gµαQ/ Qµγα)γ g C3V mπ π µρ − − 5 V MN (cid:20) (gµαQ p′ Qµp′α)γ g C4V gµαg CA u(p), (20) − · − 5 V MN2 − A 5 (cid:21) where M is the nucleon mass, D is the propagator of Rarita-Schwinger field which is N µρ given by /p+M′ 2p p 1p γ p γ 1 µ ρ µ ρ ν ρ D (p) = g + − γ γ , (21) µρ p2 M′2 +iM′Γ µρ − 3 M′2 3 M′ − 3 µ ρ − (cid:18) (cid:19) and q = p′ + q p = k k′ is the momentum transfer, Γ is the decay width, CV and π − − i CA(i = 3,4,5)arethevector andaxialvector transitionformfactorsasdefined byLlewellyn- i Smith [22], and M′ is mass of the ∆ resonance. As shown in Ref. [18], the form factors C (q2) can be obtained by comparison with the values of the helicity amplitudes given by i the relativistic quark model [23]. The explicit forms are given by 1.7 1 q2/4M2 C (q2) = − R , (22) 3 [1 q2/(M +Mq)2]3/2[1 q2/0.71 GeV2]2 R N − − M C (q2) = N C (q2), (23) 4 3 −√W2 C (q2) = 0. (24) 5 Then, the vector form factors CV(q2) used in Eq.(20) are given by i CV(q2) = √3C (q2). (25) i i The axial form factors CA(q2) is also taken to be the general formula i CA(0) CA(q2) = , (26) (1 q2/M2)2 − A where M = 0.65 GeV. A III. KINEMATICS Consider the process given in (5). It is convenient to choose the center of momentum frame. Without loss of generality, we can choose the initial four momenta of the neutrino and the nucleon in the CM frame as (p,p,0,0) and (E , p,0,0), respectively, where N − 6 E ν p = , (27) 1+2E /m ν N q m +E N ν E = , (28) N 1+2E /m ν N q where E is the incident neutrino energy in the LAB frame. The final four momenta of ν neutrino, nucleon and π0 are k′,p′ and q , respectively, where π k′ = Eν′ [1,(cosθ,sinθ,0)], (29) m2 p′ = EN′ 1,v1− E2N (−cosβ ·cosθ +sinβ ·sinθ·cosφ, u N′ u  t cosβ sinθ sinβ cosθ cosφ, sinβ sinφ) , (30) − · − · · − · (cid:21) m2 q = E 1, 1 π( cosα cosθ sinα sinθ cosφ, π π v − E2 − · − · · u π u  t cosα sinθ +sinα cosθ cosφ,sinα sinφ) , (31) − · · · · (cid:21) where ~k′2 ~p′2 +~q2 cosα = − π, (32) 2~k′ ~q π | |·| | ~k′2 +p~′2 ~q π cosβ = − . (33) 2~k′ p~′ | |·| | The angles θ,φ correspond to the rotations around z and x axes, respectively. Then, the − − differential cross section dσ can be expressed as (2π)4 A 2 NC dσ = | | dΦ 3 4(p k) · A 2 NC = 32(2|π)4M| E dEν′dEπdφd(cosθ), (34) N ν where G2 A 2 = FL Jµν, (35) NC µν | | 2 with 1 L = l†l , (36) µν 2 µ ν spins X and 1 J = J†J (37) µν 2 µ ν spins X where the summation is performed over the hadronic spins. 7 IV. CONTRIBUTION FROM THE NEUTRINO MAGNETIC MOMENT In this section, we consider the ∆ production arising from the Feynman diagram shown in Fig. 1. The decay of ∆ to a nucleon plus π0 follows in the detector, and one observes the ν′ + N + π0 final state. The ν ν′ γ vertex is parametrized by a transition magnetic µ − − moment ifν′νµµBu¯(l′)ν′σµνqνu(l)νµ, (38) where q = l − l′ = p′ + qπ − p is the momentum transfer. The coupling fν′νµ at q2 = 0 is the transition neutrino magnetic moment in units of the electron Bohr magneton and will be denoted as f′. The squared matrix element that describes the single π0 production, induced by a tran- sition neutrino magnetic moment f′, can be written as f′2µ2 A 2 = BMµνJem, (39) | M| q4 µν where 1 Mµν = [u¯(l′)σµαq u(l)][u¯(l)σνβq u(l′)] (40) α β 2 spins X and 1 Jem = JemJem (41) µν 2 µ ν spins X where Jem is the hadronic electromagnetic current given before by Jem = V3 +(1/3)V′0. µν µ µ µ V. RESULTS AND DISCUSSIONS In this section we present the numerical results of the cross section of the single π0 production for the NC interactions. The calculated cross section generated by the neutrino transition magnetic moment is shown in Fig. 2 as a function of the incident neutrino energy for thehadronicinvariant mass less than1.6 GeV.Wefind that thevalues of thecross section is of the order 10−40 cm2 for E 2 GeV. ν ≤ In order to see how the contribution to the cross section generated by the neutrino magnetic moment can be constrained by the experimental results of the ratio Rπ0/e, it is sufficient to calculate the ratio σσ0NfC′ where σf′,σ0NC are the cross sections of the single π0 production from the neutrino magnetic moment and the standard model NC interactions, 8 respectively. The reason is that e-like events (CC) is not affected by the presence of the neutrino magnetic moment. In Fig. 3, we plot this ratio σf′ rf′/NC = σNC (42) 0 as a function of the incident neutrino energy E for f′ = f′ 0.6 10−8. f′ is defined ν 0 ≡ × 0 as the value giving a similar contribution as the NC interaction. If f′ is ǫ times f′, Fig. 3 0 should be multiplied by a factorǫ2. Note that the contributions from the transition magnetic moment and from the standard model NC do not mix in the process ν+N ν′+N′ due the → unmixable γ matrix structure among these two. However, forν+N ν′+∆ there are terms → which mix these two contributions. This is because the form factors Γ , defined in ∆¯Γ N, µ µ can match the γ matrices through taking out one index by q . The dashed line corresponds µ to the case of no cut whereas the solid line corresponds to the hadronic invariant mass cut at 1.6 GeV. The current experimental result of the ratio Rπ0/e implies that the possible excess from 1 amounts to 0.13 from which we can obtain the constraint on the neutrino magnetic moment. For example, at Eν = 5 GeV a constraint rf′/NC 0.13 leads to ≤ f′ 2.2 10−9. (43) ≤ × The transition magnetic moment of this magnitude implies the muon neutrino and sterile neutrino mass matrix of the form m , m Mνµν′ = m11, m12 (44) (cid:18) 21 22(cid:19) where m m is roughly 104 times m O(10−2) eV. Thus m is of order 100 eV. 12 21 11 12 ∼ ∼ ≤ The diagonalization process should not change the mass of the muon neutrino drastically, i.e. m m2 /m 1 MeV. Therefore, a singlet neutrino at the intermediate scale with 22 ≥ 12 11 ∼ possible beyond the standard model interactions (scalar or gauge) can lead to a sizable transition magnetic moment. The effects of the muon neutrino transition magnetic moment should be separated out toward a final determination of the muon neutrino oscillation to the tau neutrino or to a sterile neutrino. The most promising method is to study the energy distribution of the final π0, since the kinematics for the magnetic moment is different fromthe NC interactions where the former has a 1/q2 dependence in the differential cross section while the latter has no q2 dependence at low energy. Indeed, ifthetransitionmagneticmoment isdiscovered bythemeasurement oftheenergy distribution, it will hint an intermediate scale physics. On the other hand, one can compare 9 this anticipation to the earlier expectation that neutrinos must oscillate due to the belief that singlet fermions, the remnants of grand unification or the standard model superstring, would be present at the intermediate scale [24,25]. Similarly, if singlet neutrinos are present much above the eV scale, there may be a large transition magnetic moment which can be detected by the future high statistics atmospheric neutrino experiments. Acknowledgments We thank S. Y. Choi for helpful discussions. One of us (JEK) is supported in part by KOSEF, MOE through BSRI 98-2468, and Korea Research Foundation. 10