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LA-UR-04-5646, YITP-SB-04-43 Atmospheric neutrinos as probes of neutrino-matter interactions Alexander Friedland,1,∗ Cecilia Lunardini,2,† and Michele Maltoni3,‡ 1Theoretical Division, T-8, MS B285, Los Alamos National Laboratory, Los Alamos, NM 87545 2 School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540 3 C. N. Yang Institute for Theoretical Physics, SUNY at Stony Brook, Stony Brook, NY 11794-3840 (Dated: August 24, 2004) Neutrino oscillation experiments provide a unique tool for probing neutrino-matter interactions, especially those involving the tau neutrino. We describe the sensitivity of the present atmospheric neutrino data to these interactions in the framework of a three-flavor analysis. Compared to the two-flavorνµ−ντ analyses,wefindqualitativelynewfeatures,inparticular,thatlargenon-standard interactions, comparable in strength to those in the Standard Model, can be consistent with the data. The existence of such interactions could imply a smaller value of the neutrino mixing angle and larger value of the mass-squared splitting than in the case of standard interactions only. This 5 and other effects of non-standard interactions may be tested in the next several years by MINOS, 0 KamLANDand solar neutrinoexperiments. 0 2 PACSnumbers: 13.15.+g,14.60.Lm,14.60.Pq n a J I. INTRODUCTION uncertainty in the determination of the solar oscillation 2 parameters. Whether a combined analysis of the solar 1 and atmospheric data eliminates this uncertainty is an Historically, the studies of the neutrino properties important outstanding question. 2 have lead to severalimportant breakthroughsin particle Inthe following,we performthe firststudy ofthe sen- v physics. The discovery of the neutral current processes 4 played a key role in establishing the Standard Model sitivityoftheatmosphericdatatoNSIinthethree-flavor 6 framework. Our goals are: first, to describe the physics (SM), while the recent confirmation of neutrino oscilla- 2 behind the sensitivity of the data to NSI; second, to tionsprovidedthefirstrealsignofnon-standardphysics. 8 presentthe bounds on the interactionparameters,based To this day, however, neutrino interactions remain one 0 on a detailed fit to the data; and finally, to discuss how 4 of the least tested aspects of the SM. The importance of the possible presence of modified interactions can affect 0 theirdirectexperimentaldetermination,whetheritleads the measurement of the oscillation parameters. We in- / tofurtherconfirmationoftheSMortoadiscoveryofnew h tend to publish the results of a complete scan of the pa- physics, can hardly be overstated. p rameter space elsewhere [7]. - In this paper, we explore the possibility of measuring p neutrino-matter interactions using the atmospheric neu- e h trinodata. Akeyfeatureofthesedataistheirsensitivity : to the interactions of the tau neutrino. Indeed, atmo- II. SETUP v spheric muon neutrinos are believed to oscillate into tau i X neutrinos; modified interactions of the latter may make The atmospheric neutrino flux is comprised of both r the oscillation pattern incompatible with the data. In neutrinos and antineutrinos, with the energy spectrum a contrast, accelerator-based experiments have restricted spanning roughly five orders of magnitude, from 10−1 to mainly the interactions of the muon neutrino. 104 GeV [8]. Initially created as the electron and muon The sensitivity of the atmospheric neutrino data to flavorstates, these neutrinos (depending on the distance non-standard interactions (NSI) was studied in [1, 2, 3]. traveled)undergoflavoroscillations. Theoscillationsare Itwas foundthat, inthe contextofatwo-flavorνµ ντ governedbytheHamiltonianH,whichhasakinetic(vac- ↔ analysis, these data are quite sensitive to NSI. The sen- uum) part and an interaction part. The former is given, sitivity to possible νe ντ conversion, however, was at the leading order, by ↔ not systematically explored beyond some specific exam- ples [2]. Solar neutrino data constrain this channel [4], 1 0 0 but at the same time leave room for non-trivial effects,  −  H ∆ 0 cos2θ sin2θ , (1) vac bothonthesolarandatmosphericneutrinos. Inthesolar ≃ −  0 sin2θ cos2θ  case, the ν ν interaction gives rise to a new discon- e τ ↔ nected solution LMA-0 [4], [5, 6], thus introducing an where we omit the smaller, “solar”mass splitting, ∆m2 ⊙ [9],andtheremainingtwomixingangles,θ andθ . In 12 13 Eq.(1), ∆ ∆m2/(4E ), E is the neutrino energyand ν ν ∆m2 isthe≡largestmasssquaredifferenceintheneutrino ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] spectrum; the flavor basis, νe,νµ,ντ, is assumed. The ‡Electronicaddress: [email protected] interaction part (up to an irrelevant constant) will be 2 parameterized as are available. These oscillations are driven by the off- diagonal ν ν mixing in Eq. (1) and the introduction µ τ 1+ǫ ǫ∗ ǫ∗ −  ee eµ eτ of sufficiently largeNSI for the tau neutrino will, in gen- Hmat =√2GFne ǫeµ ǫµµ ǫ∗µτ . (2) eral, suppress that mixing. Since the vacuum Hamilto-  ǫeτ ǫµτ ǫττ nian scales as E−1, this suppressionshould be especially ν strongathighenergy,inthethrough-goingmuonsample. Here G is the Fermiconstant,n is the number density F e As a simple illustration, consider the case when only of electrons in the medium. ǫ is nonzero. Clearly, ǫ introduces a diagonal split- ThephysicaloriginoftheepsiloncontributionsinH ττ ττ mat ting between the ν and ν states, thereby decreasing can be the exchange of a new heavy vector or scalar [22] µ τ the effective mixing angle in matter. The correspond- particle. We parameterize the resulting NSI with the ing bound can be estimated by comparing the mat- effective low-energy four-fermion Lagrangian ter term √2 ǫ G n to the vacuum oscillation term ττ F e LNSI = 2√2G (ν¯ γ ν )(ǫff˜Lf¯ γρf˜ +ǫff˜Rf¯ γρf˜ ) ∆m2/(2Eν). For neutrinos going through the center − F α ρ β αβ L L αβ R R of the Earth, the highest energy at which an oscilla- + h.c. (3) tion minimum occurs in the standard case is around E 20 30 GeV. If the matter term is sufficiently 0 Here ǫff˜L (ǫff˜R) denotes the strength of the NSI be- large∼, √2−ǫ G n & ∆m2/(2E ), the mixing in mat- αβ αβ ττ F e 0 tween the neutrinos ν of flavors α and β and the left- ter and hence the oscillation amplitude are expected to handed(right-handed)componentsofthefermionsf and be suppressed. Substituting numerical values, we find a f˜. The epsilons in Eq. (2) are the sum of the contri- bound ǫττ.0.2. butions from electrons (ǫe), up quarks (ǫu), and down Next, we generalize this argument to the case of non- quarks (ǫd) in matter: ǫαβ ≡ Pf=u,d,eǫfαβnf/ne. In Hvanishcianngbǫeee,diǫaegτo.nTalhizeedmbaytterortpaatirntgoifntthheeHνamiνltonsuiabn- turn, ǫf ǫfL + ǫfR and ǫfP ǫffP. Notice that mat e− τ the maαttβer≡effeαcβts areαβsensitiveαβon≡ly tαoβthe interactions space. In the new basis (νe′,νµ,ντ′), Hmat has the form thatpreservethe flavorofthe backgroundfermionf (re- diag(λe′,0,λτ′), with λe′,τ′ = √2GFne(1+ ǫee + ǫττ (1+ǫ ǫ )2+4 ǫ 2)/2. It straightforwardly qveucirteodr pbayrtcoohfetrheantcein[t1e0r]a)ctainodn,. furthermore, only to the f±opllows theaet−if τλτe′,τ′ | ∆eτm|2/(2E0), the oscillations of | | ≪ the muon neutrinos proceed unimpeded, while in oppo- Neutrinoscatteringtests,likethoseofNuTeV[11]and CHARM [12], mainly constrainthe NSI couplings of the site case, λe′,τ′ &∆m2/(2E0), they are suppressed. | | muonneutrino,e.g., ǫ .10−3, ǫ .10−3 10−2. It is very important to consider the intermediate eµ µµ Thelimitstheyplace|onǫ| ,ǫ ,and| ǫ |arerathe−rloose, regime, when the spectrum has the hierarchy (a) λτ′ < e. g., ǫuuR < 3, 0.4ee<eǫτuuR <ττ0.7, ǫuu < 0.5, ∆m2/(2E0) λe′ or (b) λe′ < ∆m2/(2E0) | λ|τ′ . ǫdd <|0τ.5τ [1|3]. Str−onger conseteraints exist|oτne|the cor- In both cases≪, th|e o|scillation|s b|etween νµ and th≪e c|orre|- | τe| sponding light eigenstate are allowed to proceed while responding interactions involving the charged leptons. those between ν and the heavy eigenstates are sup- Those, however, cannot, in general, be extended to the µ pressed. Remarkably, the resulting oscillation pattern is neutrinos, for example when the underlying operators indistinguishable from the standard case at high energy, contain the Higgs fields [14], and hence will not be con- where only muon neutrinos are detected. sidered here. From now on we specialize to hierarchy (a), which is Giventheaboveboundswewillsetǫ andǫ tozero eµ µµ smoothly connected to the origin ǫ =ǫ =ǫ = 0 ((b) in our analysis. Furthermore, for simplicity we will also ee eτ ττ is realized only if ǫ + ǫ is a large negative number). set ǫ to zero. The earlier analyses of the atmospheric ee ττ µτ When it is satisfied, muon neutrinos oscillate into the neutrino data [1] have indicated that this parameter is quite constrained (ǫ < 10−2 10−1). Corrections due state µτ − to non-vanishing ǫ will be described in [7]. Here, we have a three-dimenµsτional NSI parameter space, spanned ντ′ =−sβe2iψ νe +cβ ντ , (4) by ǫ , ǫ , and ǫ . ee eτ ττ where c = cosβ, s = sinβ, 2ψ = Arg(ǫ ), tan2β = β β eτ 2 ǫ /(1+ǫ ǫ ). eτ ee ττ III. CONVERSION EFFECTS AND |The|condition−|λτ′|.∆m2/(2E0) implies SENSITIVITY TO THE NSI 1+ǫ +ǫ (1+ǫ ǫ )2+4 ǫ 2 .0.4. (5) | ee ττ −p ee − ττ | eτ | | The physics of the sensitivity of the atmospheric neu- This equation gives our analytical prediction for the trino data to ǫee, ǫeτ, and ǫττ can be understood as fol- bound on ǫee,ǫeτ,ǫττ. When ǫee=ǫeτ= 0, it reduces to lows. The data are known to be very well fit by large- the bound ǫ .0.2 given above. ττ amplitudeoscillationsbetweentheνµandντ states. This The regionEq.(5)describes extends to largevalues of holds both at high energy (Eν & 10 GeV), where only ǫeτ, ǫττ. To see this, note that in the limit λτ′ =0, or the muon neutrino flux is measured, and at lower energies, where both the muon and electron neutrino data ǫ = ǫ 2/(1+ǫ ) , (6) ττ eτ ee | | 3 theνµ ντ′ oscillations,thoughdependentonthematter fluxes given in Ref. [16]. The statistical analysis of the ↔ angle β, are independent of the absolute size of the NSI. data follows the appendix of Ref. [3]. As already mentioned, these oscillations have the same TheresultsoftheK2Kexperimenthavebeenincluded. dependence on the neutrino energy and on the distance Theiradditionhasaminimalimpactonourresults,pro- Lasvacuumoscillationsandthereforemimictheir effect viding some constraint at high ∆m2. The details of the in the distortion of the neutrino energy spectrum and of K2K analysis can be found in Sec. 2.2 of Ref. [17]. the zenith angle distribution. More specifically, we get the oscillation probability: 1 ε =−0.15 P(νµ→ντ′)=sin22θmsin2[∆m2mL/(4Eν)] , (7) ee where the effective mixing and mass square splitting are derived to be ε 0.5 ττ ∆m2 =∆m2 (c (1+c2) s2)2/4+(s c )2 1/2 , m (cid:2) 2θ β − β 2θ β (cid:3) tan2θ =2s c /(c (1+c2) s2) . (8) m 2θ β 2θ β − β 0 If NSI are present, but not included in the data analysis, a fit of the highest energy atmospheric data, i.e. the through-going muon ones, would give ∆m2m and θm in- −1 −0.5 0 0.5 1 stead of the corresponding vacuum quantities. If we fix ε a set of NSI and – to reproduce the no-NSI case – re- eτ quire that θ π/4 and ∆m2 2.5 10−3 eV2, from Eqs. (8) wemge≃t that the vacumum≃mixi·ng would not be FIG. 1: A 2-D section (ǫee= −0.15) of the allowed region of the NSI parameters (shaded). We assumed ∆m2 = 0 and maximal; in particular we have cos2θ≃s2β/(1+c2β) and θ13 =0, and marginalized over θ and ∆m2. Thed⊙ashed con- ∆m2 ≃∆m2m(1+cos−2β)/2. toursindicateouranalytical predictions. Seetextfordetails. In the intermediate energy range, E 1 10 GeV, ∼ − when matter and vacuum terms are comparable, the re- Uponscanningtheparameterspaceandmarginalizing ductiontoatwo-neutrinosystemisnotpossible,andthe over∆m2 andθ weobtainthethree-dimensionalallowed problem does not allow a simple analytical treatment. region in the space (ǫ ,ǫ ,ǫ ). As an illustration, in ee eτ ττ The neutrino conversionprobability in this energy range Fig.1 we show a sectionof this regionby the plane ǫ = ee depends onthe signofthe neutrinomasshierarchy(nor- 0.15(the choicemotivated by the solaranalysisin [4]). mal, ∆m2 > 0, or inverted, ∆m2 < 0). At the sub-GeV −The χ2 minimum occurs at ǫ = 0.07,ǫ = 0.01; the eτ ττ energies, we expect vacuum-domination, and therefore value at the minimum, χ2 = 48.50, is virtually the min smalldeviationswithrespecttovacuumoscillations[23]. sameasattheorigin(noNSI),χ2 =48.57. Theshaded orig Finally, we observe that for θ13 =0, ∆m⊙ =0, as has regionscorrespond,from the innermostcontour, to χ2 beenassumedhere,thereisnosensitivitytoψ,thephase χ2 7.81, 11.35, and 18.80. They represent the 95%−, of ǫ [24]. This is unlike the case of the solar neutrinos, min ≤ eτ 99%, and 3.6σ confidence levels (C.L.) for three degrees where ψ plays a crucial role [4]. Corrections due to θ 13 of freedom (d.o.f.). The last contour also corresponds to and ∆m⊙ = 0 break the phase degeneracy and will be the 95%C.L.for50d.o.f.. Forthe purposeofhypothesis 6 presented elsewhere [7]. testing this means that atheory whichgivesNSI outside of this region should be rejected. The dashed-dotted parabola illustrates the condition IV. NUMERICAL RESULTS of zero eigenvalue, Eq. (6); the two outer curves give the predicted bound according to Eq. (5). For both, We performed a quantitative analysis of the atmo- the agreement between the theory and numerical results spheric neutrino data with five parameters: two “vac- is quite convincing. Moreover, we have verified that uum” ones, (∆m2,θ), and three NSI quantities (ǫ ,ǫ the agreement remains very good for ǫ in the range ee eτ ee ,ǫ ). The goodness-of-fit for a given point is deter- 0.7 <ǫ < 0.3 [7]. For the case when only ǫ is non- ττ ee eτ − mined by performing a fit to the data. We use the zero we find the bounds ǫ < 0.38 at 99% C.L. and eτ | | complete 1489-day charged current Super-Kamiokande ǫ <0.5 at 3.6σ. eτ | | phase I data set [15], including the e-likeand µ-like data The extent of the allowed region along the parabola is samples of sub- and multi-GeV contained events (each beyond the scope of our analytical treatment. Indeed, groupedinto 10bins in zenith angle)as wellas the stop- sinceathighenergythe leadingNSI effectiscanceledby ping(5angularbins)andthrough-going(10angularbins) construction, the fit quality is determined by subdomi- upgoing muon data events. This amounts to a total of nantNSIeffectsinallenergysamples. Remarkably,these 55 data points. For the calculation of the expected rates effects are rather small, especially for the inverted mass we use the new three-dimensional atmospheric neutrino hierarchy,where the regionχ2 χ2 7.81 extends up − min ≤ 4 4 Theeffectis evenmoredramaticwhenonecomparesthe ] bounds on the flavor-changing interactions: the bound 2 V ǫ < a few 10−2 [1] does not extend to ǫ , for which µτ eτ e × 3 order one values are allowed. The bound ǫ < 0.38 at 3 eτ − 99% C.L. – found for the case when the other epsilons 0 H 1 vanish – is likewise new. We also stress that the effects [ 1 2 of these large NSI are absolutely non-trivial. In particu- 2m3 lar, the best-fit values of the oscillation parameters shift ∆ to smaller angles and higher ∆m2. The physical mecha- nism behind this is well understood analytically. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Thesepropertieshaveimportantimplicationsforother neutrino experiments. For example, the MINOS exper- sin2θ iment will measure the mixing angle θ with good pre- 23 cision; assuming the true value θ = π/4 one expects sin2θ > 0.4 at 99% C.L. [18]. Given its baseline of FIG.2: TheeffectoftheNSIontheallowedregion andbest- only 735 km, MINOS will be almost free from matter fit values of the oscillation parameters; see text for details. effects and will therefore determine the vacuum value of θ. Thus, it can look for the large NSI effects described here: a measurementof sin2θ >0.4 and low ∆m2 would to ǫ =1, as the Figure shows. The extent is somewhat eτ contribute to disfavor large NSI, while in the opposite smaller (to ǫ = 0.6) for the normal hierarchy [7]. eτ ± case (sin2θ 0.3) one would have indications of non- As mentioned, the symmetry of Fig. 1 with respect to ∼ ǫ ǫ follows from setting θ and ∆m2 to zero. standard physics. Weτe↔ha−ve cehτeckedthat for θ =0 t1h3e region do⊙es indeed Further tests can be performed at future experiments 13 develop a small asymmetry [76]. withlongerbaseline(afew 103km)andsufficientlyhigh × energy,E 20 30GeV.Thesewillbedirectlysensitive Finally, we notice that as the NSI point is moved ν ∼ − totheNSImattereffectsandadiscrepancybetweenthem along the parabola (6) away from the origin, the best- and MINOS could be a sign of NSI. fit oscillation parameters change. This behavior is illus- trated in Fig. 2. The four contours plotted are the sec- It is worth mentioning that, in principle, another sig- tions of the five-dimensional allowed region in the space nature of large NSI could be an unexpected number of (∆m2,θ,ǫ ,ǫ ,ǫ )forthe values: (i) ǫ = 0.15,ǫ = neutral current events in a detector relative to the num- ee eτ ττ ee eτ 0,ǫ = 0; (ii) ǫ = 0.15,ǫ = 0.30,ǫ =−0.106; (iii) ber of charged current events for a given oscillation sce- ττ ee eτ ττ ǫ = 0.15,ǫ = 0.60−,ǫ = 0.424; (iv) ǫ = 0.15,ǫ = nario. Thiseffectwouldbeduetothemodificationofthe ee eτ ττ ee eτ 0.90,ǫ−ττ= 0.953. The contours are drawn−for ∆χ2 = detection neutral current cross section by large ǫeτ and 11.07 with respect to the global minimum, marked by ǫττ. Unfortunately, no firm prediction on the size of the a star (this corresponds to 95% CL for 5 d.o.f.). The effect can be made, since the effect additionally depends dots representthe best fit points obtained in each of the onthe valuesofthe axialcouplingsthatdonotenterthe fourcasesabove,whentheNSIareassumedknown. The propagationHamiltonian [25]. figure confirms the trend predicted in Eq. (8): as the Finally,thereareimportantimplicationsforsolarneu- NSI increase the best-fit mixing angle becomes smaller, trinos. Indeed, ǫee, ǫeτ, and ǫττ also enter the solar neu- θ <π/4, while the best-fit ∆m2 increases. trino survival probability, and may alter the interpreta- The shaded region in the background represents the tionofthecurrentdata[4,5,6]. Ourresultsheredemon- result of marginalizing with respect to ǫ and ǫ (for strate that the LMA-0 solution, obtained with non-zero eτ ττ ǫee= 0.15). It was drawnfor ∆χ2 =11.83,correspond- ǫeτ as in [4], is compatible with the atmospheric data. ing to−99.73% (3σ) C.L. for 2 d.o.f. While the best-fit On the other hand, solar neutrinos, being sensitive to point is nominally unchanged by the NSI, the shape of the phase of ǫeτ, may rule out a sizable portion of the the χ2 function is significantly altered,so that smallval- parameter space with ǫeτ< 0 [4, 7]. This exclusion is ues of the mixing angle are now much less disfavored. impacted by the latest results from KamLAND [19] and it would be very interesting to see the combined bound from the atmospheric, solar, and KamLAND data. V. CONCLUSION AND IMPLICATIONS In conclusion, the three-flavor oscillation analysis ex- Acknowledgments hibitsqualitativelynewfeaturescomparedwiththeν µ ↔ ν studies. While giving no positive evidence for non- We are very grateful to Thomas Schwetz for sharing τ standard physics, the data do not exclude large NSI. with us his K2K analysis. We thank the Aspen Center The old bound ǫ < 0.15 at 99% C.L. is signifi- for Physics where part of this research was completed ττ | | cantly relaxed by the addition of nonzero ǫ : if ǫ and for hospitality. A. F. was supported by the Department eτ eτ ǫ are varied simultaneously, even ǫ 1 is allowed. of Energy, under contract W-7405-ENG-36, C. L. by a ττ ττ ∼ 5 grant-in-aid from the W. M. Keck Foundation and the PHY-0354776. NSF grant PHY-0070928, and M. M. by the NSF grant [1] N.Fornengo,M.Maltoni,R.T.BayoandJ.W.F.Valle, transparencies/07/index.php. Phys.Rev.D65,013010(2002)[arXiv:hep-ph/0108043]. [16] M. Honda, T. Kajita, K. Kasahara and S. Midorikawa, [2] M.Guzzo, P.C. deHolanda,M. Maltoni, H.Nunokawa, astro-ph/0404457. M. A. Tortola and J. W. F. Valle, Nucl. Phys. B 629, [17] M. Maltoni, T. Schwetz, M. A. Tortola and 479 (2002) [arXiv:hep-ph/0112310]. J. W. F. Valle, arXiv:hep-ph/0405172. [3] M. C. Gonzalez-Garcia and M. Maltoni, [18] V. D.Barger, A. M. Gago, D. Marfatia, W.J. C. 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