Sterile Neutrino Production Through a Matter Effect Enhancement at Long Baselines
SSterile Neutrino Production Through a Matter Effect Enhancement at Long Baselines
Joseph Bramante
Department of Physics and Astronomy, University of Hawaii,2505 Correa Rd., Honolulu HI, USA ∗ (Dated: October 31, 2018)If sterile neutrinos have a neutral coupling to standard model fermions, matter effect resonant tran-sitions to sterile neutrinos and excess neutral-current events could manifest at long baseline experi-ments. Assuming a single sterile neutrino with a neutral coupling to fermionic matter, we re-examinebounds on sterile neutrino production at long baselines from the MINOS result P ν µ → ν s < .
22 (90%CL). We demonstrate that sterile neutrinos with a neutral vector coupling to fermionic matter couldevade the MINOS limit, allowing a higher fraction of active to sterile neutrino conversion at longbaselines. Scanning the parameter space of sterile neutrino matter effect fits of the LSND and Mini-BooNe data, we show that in the case of a vector singlet coupling of sterile neutrinos to matter, somefavored parametrizations of these fits would create neutral-current event excesses above standardmodel predictions at long baseline experiments (e.g. MINOS and OPERA).
PACS numbers: 14.60.Pq, 14.60.StKeywords: Sterile Neutrinos, Neutrino Matter Effect, Long Baseline
I. INTRODUCTION
Although many proposals of extra generations of neu-trinos apply to small neutrino mixing anomalies, the firstdetection of a sterile neutrino could come in the form ofa very large mixing anomaly at a long baseline.A recent result from the OPERA experiment [1] mea-sured a difference in the speed of light in vacuum andthe speed of muon neutrinos along a 730 km baseline:( v ν µ − c ) /c = (2 . ± . × − [OPERA]. Thisdata turned out to be an experimental error. Never-theless, the OPERA result prompted proposals of newphysics[2–6] and phenomenological constraints on muonneutrino Lorentz invariance violation [2, 7, 8]. One pop-ular superluminal mechanism for neutrinos involved ster-ile neutrino transport through a higher dimensional bulk[9–13], which supposed active neutrinos confined to a D3brane oscillate to sterile neutrinos, whose lack of gaugecharge leaves them free to travel through large extra di-mensions. However, a strong constraint on these ster-ile neutrino models comes from measurements of neu-trinos and photons arriving from SN1987a. The detec-tion of 24 neutrino events at three sites [14–16] arriving ∼ v ν e − c ) /c ∼ × − [IMB , KII , Baksan] . Although OPERAdetected muon neutrinos and SN1987a produced elec-tron neutrino data, leading to the possibility of a flavoranomaly, additional experimental constraints on neutrinomass eigenstate velocity differences [11, 17–20] ruled outactive flavour-dependent velocity anomalies, short of re-placing the standard PMNS matrix with a different for-malism [3–6].Although it is settled that the OPERA experiment ∗ Electronic address: [email protected] did not observe a superluminal anomaly [1], neverthelesssome of the phenomenological studies of sterile neutrinoproduction at long baselines are applicable to future neu-trino studies. In particular, among the many OPERAconstraints papers that arose in the wake of the anoma-lous OPERA data [7] pointed out that the fraction ofsterile neutrinos required to explain the anomaly was atodds with a prior study of sterile neutrinos at MINOS[21].In this paper we show that if sterile neutrinos havenon-standard interactions with fermionic matter, thisinduces matter effect resonance transitions and excessneutral-current events that would simultaneously allowfor the MINOS result and a sizeable production of ster-ile neutrinos at another long baseline. We develop thephenomenology of a matter-dependent increase of ster-ile neutrino production through a sterile neutrino neu-tral U(1) vector coupling to fermions. Similar modelsemploying new sterile neutrino interactions via a B-Lgauge boson have been developed in [22–25] to fit neu-trino disappearance anomalies at short and long baselineexperiments [26–29] . In this paper we consider a sterileneutrino matter effect model for which sterile neutrinoneutral interactions with matter would be detectable.The structure of this paper is as follows: In section IIwe find the evolution equation and transition probabilityfor a sterile neutrino with a new neutral U(1) coupling tofermions. In section III we examine constraints on sterileneutrinos with neutral current interactions and addition-ally comment on specific constraints on sterile neutrinosas an explanation for the OPERA anomaly. In section IVwe determine what parametrizations of sterile neutrinomass and coupling to standard model fermions wouldproduce detectable abundances at current long baselinestudies. Additionally we compare these parametrizationswith other studies fitting a similar model to short base-line anomalies. In section V we conclude. a r X i v : . [ h e p - ph ] J un FIG. 1: ν s elastic scattering through a vector singlet B . II. STERILE NEUTRINOS WITH A NEUTRALVECTOR SINGLET COUPLING
Sterile neutrino models usually suppose that the activeneutrinos have a small mixing angle with extra genera-tions of non-interacting, or sterile neutrinos. Here weexamine how these models change with the introductionof a straightforward U(1) vector coupling of sterile neu-trinos to standard model fermions. In short, we find thatthe coupling causes an enhancement of an otherwise smallmuon-sterile mixing term in matter. The contribution ofthe process in figure 1 to the effective potential of sterileneutrino propagation is given by H ( B ) eff = − g s g f m B [ ¯ ν s γ µ ν s ][ ¯ f R,L γ µ f R,L ] , (1)where f is a fermion abundant in matter, e.g. (e − , u , d),and B is a neutral vector boson singlet. In general, B willcouple to (u , d) L , ( ν e , e) L , u R , d R , and e R with strength g f and couple to ν s with strength g s . Depending onthe scale of (1), the couplings to sterile neutrinos andfermions will need to be unequal to avoid precision elec-troweak constraints on standard model fermion couplingsto a new vector singlet boson ( g f ).The proposed active-sterile matter mixing enhance-ment will affect all active-sterile neutrino oscillations inmatter, but to simplify our analysis and focus on pa-rameters which maximize muon neutrino oscillation tosterile neutrinos, we here set the vacuum mixing anglesfor ν e,τ ↔ ν s to zero, so that the only appreciable sterileneutrino production in matter will come from ν µ ↔ ν s .For this simple system with only ν µ mixing with ν s , theflavor evolution equation in matter is [30–33] i ddt (cid:18) A ν µ → ν µ A ν µ → ν s (cid:19) = (cid:32) − ∆ m E cos 2 θ − √ N G s ∆ m E sin 2 θ ∆ m E sin 2 θ ∆ m E cos 2 θ + √ N G s (cid:33) × (cid:18) A ν µ → ν µ A ν µ → ν s (cid:19) , (2) where G s ≡ √ g s g f m B , N = n e + n u + n d is the num-ber density of matter fermions, θ is the vacuum mixingangle and ∆ m is the squared mass difference betweenthe mass eigenstates of ν µ and ν s in vacuum. Standardmodel MSW terms in (2) have a small effect over a ∼ km baseline and have been omitted. Diagonalizing theevolution Hamiltonian yields∆ m M E (cid:18) − cos 2 θ M sin 2 θ M sin 2 θ M cos 2 θ M (cid:19) (3)where the mixing angle and squared mass difference inmatter aresin 2 θ M = ∆ m ∆ m M sin 2 θ (4)∆ m M = ∆ m (cid:32) cos 2 θ − √ N EG s ∆ m (cid:33) + sin θ / (5)and the corresponding matter ν µ → ν s transition proba-bility over a distance D is P ν µ → ν s = sin θ M sin (cid:18) ∆ m M D E (cid:19) = sin θ (cid:16) cos 2 θ − √ NEG s ∆ m (cid:17) + sin θ × sin ∆ m D E (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) cos 2 θ − √ N EG s ∆ m (cid:33) + sin θ . (6) III. CONSTRAINTS ON STERILE NEUTRINOSWITH A VECTOR SINGLET COUPLING
A strong bound on muon neutrino oscillation to sterileneutrinos in matter comes from the MINOS measurementof neutral-current (NC) interactions of the NuMI muonneutrino beam at the end of a 730 km baseline [21]. TheMINOS result of 802 NC events against an expected 754 ± stat ± sys event background excludes P ν µ → ν s > . ν s to standard model fermions (1). If the scale ofthe interaction considered is on the order of the Fermiconstant, G s ≈ G F , sterile neutrino interactions withstandard model fermions would contribute to the neutralcurrent event counts at long baseline experiments.Assuming that the mass of the new neutral vector bo-son is much greater than the momentum of the sterileneutrino, at current long baseline energies the four-fermiapproximation is valid for active, σ NCν a ∝ G F E ν N e ,as well as sterile neutrino neutral current interactions, σ NCν s ∝ G s E ν N e , where N e is the density of electronsin matter. The contribution of a sterile neutrino to theneutral current event rate will be equal to the contribu-tion of an active neutrino multiplied by a proportionalityconstant α ≡ G s /G F , where the factor of 7 arises be-cause the singlet vector coupling of ν s to matter does nothave V-A diagram cancellations [30–33]. Thus one wayto construct a sterile neutrino matter effect model thatallows for P ν µ → ν s (cid:38) .
30 and is consistent with the MI-NOS measurement is to set G s (cid:38) G F / m B >> E ν .While this study uses a single active-sterile mixing an-gle and squared mass difference to identify possible activeto sterile mixing resonances at long baselines, any modi-fication of muon neutrino mixing in matter is subject toconstraints from measurements of the atmospheric mix-ing angle [34]. Most parametrizations of this model areruled out by these measurements. However, very smallvacuum mixing angles would create active to sterile mix-ing resonances over a small range of neutrino energies,as shown in figure 2. With a small enough vacuum mix-ing angle, it would be possible to identify a resonance ata baseline experiment, while the signal of this neutrinodisappearance (and extra flux of neutral current events)would not be evident in more broadly binned energy dataat atmospheric experiments. A. Constraints on Superluminal Sterile Neutrinosat OPERA
Although the OPERA anomaly was an experimentalerror, nevertheless there is continued interest in Lorentz-violating neutrinos [3–6] and the constraints developedfor OPERA may be applicable to future neutrino anoma-lies. Here we briefly evaluate constraints on the OPERAsuperluminal anomaly for sterile neutrinos with a neutralvector coupling.In [7] it was demonstrated that there is a minimumfraction of neutrinos which must travel superluminallyin order to reproduce the OPERA anomaly. The spec-tral flatness of time-binned neutrino events requires thesuperluminal fraction χ = Σ ν c + | U µ → ν c + | / Σ i | U µi | to beat least ∼ σ and 0.28 at 2 σ confidence. Fur-thermore, [8] showed that superluminal active neutri-nos would undergo ν f → ν f e + e − Cherenkov-like ra-diation forcing an effective energy cutoff above ∼ P ν µ → ν s > .
18, though a more promising model wouldallow for P ν µ → ν s (cid:38) .
30. Although we previously spec-ified that the mixing ν e → ν s would be very small, inprinciple there must be some mixing of all active neutri-nos with sterile through shared mass eigenstates. Thuseven a weak scale coupling ( G s ≈ G F ) reintroduces the E (GeV)P υ μ→ υ s FIG. 2: Sterile neutrino transition probability plotted againstneutrino energy for parameters indicated in the text.
Cerenkov radiation cutoff problem, because the superlu-minal sterile neutrino will oscillate to electron flavor.The OPERA experiment completed an additionalstudy in which the proton bunches and resulting neu-trino packets were more tightly spaced in time [1]. Theanalysis in [7] was updated to show that P ν µ → ν s (cid:38) .
40 6 σ exclusion P ν µ → ν s (cid:38) .
80 3 σ exclusion . (7)Although a sterile neutrino fraction of this magnitudemay seem unrealistic, one interesting feature of ster-ile neutrinos with a weak scale vector singlet interac-tion is that at terrestrial baseline lengths and ener-gies, there are parametrizations which cause energy-dependent resonant transitions P ν µ → ν s ∼
1. In figure2 we plot the transition probability (6) for the param-eters ∆ m = 0 .
45 eV , A s = 10 − eV , sin θ =0 .
05 (solid) , sin θ = 0 .
005 (dashed) , and D = 730 km.If sterile neutrinos have a neutral-current interactioncross-section seven times larger than active neutrino NCinteractions, an oscillation of transition probability peak-ing at 5% in figure 2 would be observable as an oscillationof NC event flux of ∼ ±
20% around SM expectations.
IV. PARAMETRIZATION FOR RESONANCEAT LONG BASELINES
Sterile neutrino matter effects have recently been con-sidered as an explanation of short and long baselineanomalies [22–25, 35, 36]. Most fits indicate a 4th neu-trino with a mass of about 0.5 eV. Particularly, [35, 36]uses a “model agnostic” ν s matter effect to fit a 3+1model to the LSND and MiniBooNE datasets. Theactive-sterile mixing angles, matter effect potentials, andsquared mass differences of [35, 36] are consistent with aparametrization which would cause substantial ν µ → ν s mixing at long baseline experiments for the model weoutline here.To demonstrate what parameters are required to createa large active to sterile matter effect resonance for thesimplified mixing equations here, we will use the OPERAbaseline and neutrino energies and require P ν µ → ν s (cid:38) . θ = 0 . ν µ → ν s transition probability of 0.025. Inspection of the firstterm in P ν µ → ν s (6) sin θ (cid:16) cos 2 θ − √ NEG s ∆ m (cid:17) + sin θ (8)produces a squared-mass difference-coupling resonancecondition 2 √ N EG s ∆ m = 2 EA s ∆ m ∼ O (1); (9)If EA s ∆ m >> P ν µ → ν s will diminish rapidly, and if EA s ∆ m << P ν µ → ν s cannot exceed a value of 0.05. Insert-ing the OPERA values into the second term of P ν µ → ν s ,D = 730 km and E ∼
17 GeV,sin ∆ m eV − (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) cos 2 θ − √ N EG s ∆ m (cid:33) + sin θ (10)a minimum value of ∆ m (0.04 eV ) becomes apparent.This bound follows from the maximum value of the pref-actor of (6), which is unity. If (cid:32) cos 2 θ − √ N EG s ∆ m (cid:33) ∼ θ (cid:16) cos 2 θ − √ NEG s ∆ m (cid:17) + sin θ ∼ P ν µ → ν s = sin (cid:16) . √ . (cid:17) ∼ . . (11)As ∆ m increases substantially from this value, the sec-ond term in P ν µ → ν s will average to and the first termwill have to resonate at ∼ to produce P ν µ → ν s = 0.30.In figure 3 have plotted a band of black dashes whichincludes the region P ν µ → ν s > .
18 to illustrate the reso-nance ∆ m ∼ EA s . The band of black dashes centerson the maximum resonance (0 . m = 2 E ν A s for 17GeV muon neutrinos, for which the transition probability P ν µ → ν s can approach unity. The same curve is plottedfor 3 GeV neutrinos, which is the central neutrino energyat MINOS. The dotted vertical line indicates the value A s (GeV) − − − − − − −
90% CL99% CL
FIG. 3: Lines of maximum resonant active to sterile mixingfor sterile neutrino matter effect models as detailed in thetext. The underlaid scatter plot fit is taken from [35, 36]. of A s which corresponds to sterile neutrinos having weakscale neutral interactions with standard model fermions.The dotted horizontal line indicates the smallest possible∆ m value which yields P ν µ → ν s (cid:38) .
30 for 17 GeV neu-trinos when sin θ = 0 .
05. Underlaid is a scatter plottaken from [35, 36], which fits ∆ m and A s = √ G s N to LSND and MiniBooNE neutrino oscillation data. Itshould also be noted when examining figure 3 that thechoice of sin θ affects the location and shape of the res-onance lines. Increasing the vacuum sterile mixing an-gle from the value sin θ = 0 .
05 will both elevate andbroaden the P ν µ → ν s > .
18 inclusion band.If sterile neutrinos interact via neutral-currents with allmatter fermions, in the case of a straightforward singletvector coupling the parameter space around the 3 GeV(MINOS) resonance line is certainly ruled out. (MINOSwas not bombarded with NC events in [21]). However,there are intermediate values of A s along and beside the3 GeV line, where the cross-section for sterile neutrinoNC events exceeds that of active neutrinos, and sterileneutrinos are produced in significant quantities. A possi-ble signal of this in a long baseline neutral current eventstudy would be an oscillation above the SM backgroundof NC event counts with respect to energy (see figure 2). V. CONCLUSION
In conclusion we have developed the phenomenologyof a matter effect enhanced model of interacting ster-ile neutrinos at long baselines. We have shown that forparameters commonly used in sterile neutrino matter ef-fect models of short and long baseline anomalies, sterilemodels with a neutral vector singlet coupling could beobserved at a long baseline experiment as a severe over-production of neutral current events or as an oscillation ofneutral current events over the expected SM background.We demonstrated that this model avoids prior constraints[7] on sterile neutrinos as an explanation for the now de-funct OPERA superluminal anomaly.While we have shown what signals might arise in thecase of a sterile neutrino coupling to standard modelfermions via a neutral vector singlet boson, a more com-plete treatment of this model would need to demonstratethat the new coupling arises from a fully renormaliz-able theory [37–39]. In addition, it is not clear whetherthe model outlined here would be consistent with atmo-spheric neutrino data under any parametrization. Weleave the fitting of neutral-coupled sterile neutrinos to atmospheric, baseline, and reactor neutrino data to fu-ture work.
Acknowledgments
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