Probing Non-Standard Interactions at Daya Bay
Sanjib Kumar Agarwalla, Partha Bagchi, David V. Forero, Mariam Tortola
PPrepared for submission to JHEP
IFIC/14-41, IP/BBSR/2014-10
Probing Non-Standard Interactions at Daya Bay
Sanjib Kumar Agarwalla, a Partha Bagchi, a David V. Forero, b,c
Mariam T´ortola b a Institute of Physics, Sachivalaya Marg, Sainik School Post, Bhubaneswar 751005, India b AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val`encia, Parc Cientificde Paterna. C/ Catedratico Jos´e Beltr´an, 2 E-46980 Paterna (Val`encia) - Spain c Center for Neutrino Physics, Virginia Tech, Blacksburg, VA 24061, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
In this article we consider the presence of neutrino non-standard interactions(NSI) in the production and detection processes of reactor antineutrinos at the Daya Bayexperiment. We report for the first time, the new constraints on the flavor non-universaland flavor universal charged-current NSI parameters, estimated using the currently released621 days of Daya Bay data. New limits are placed assuming that the new physics effectsare just inverse of each other in the production and detection processes. With this specialchoice of the NSI parameters, we observe a shift in the oscillation amplitude withoutdistorting the
L/E pattern of the oscillation probability. This shift in the depth of theoscillation dip can be caused by the NSI parameters as well as by θ , making it quitedifficult to disentangle the NSI effects from the standard oscillations. We explore thecorrelations between the NSI parameters and θ that may lead to significant deviationsin the reported value of the reactor mixing angle with the help of iso-probability surfaceplots. Finally, we present the limits on electron, muon/tau, and flavor universal (FU) NSIcouplings with and without considering the uncertainty in the normalization of the totalevent rates. Assuming a perfect knowledge of the event rates normalization, we find strongupper bounds ∼ Keywords:
Neutrino, Reactor Experiments, Daya Bay, Non-Standard Interactions
ArXiv ePrint: a r X i v : . [ h e p - ph ] J u l ontents ε seγ = ε d ∗ γe θ : iso-probability plots 12 ε seγ (cid:54) = ε d ∗ γe A.1 Presence of the NSI parameters only at the production stage 27A.2 NSI at the source and detector with the same magnitude and different phases 28
The recent discovery of the smallest neutrino mixing angle θ by the modern reactor an-tineutrino experiments Daya Bay [1–3] and RENO [4] has firmly established the three-flavorneutrino paradigm [5–7] and signifies an important development towards our understandingof the structure of the neutrino mass matrix, whose precise reconstruction would shed lighton the underlying new physics that gives rise to neutrino mass and mixing [8–10]. Anotherreactor electron antineutrino disappearance experiment: Double Chooz [11, 12], and thetwo accelerator electron (anti-)neutrino appearance experiments: MINOS [13] (completed)and T2K [14, 15] (presently running) have also confirmed the non-zero and moderatelylarge value of θ in the standard three-flavor oscillation scenario. It is quite remarkableto see that with 621 days of data taking and using the merit of identical multi-detectorsetup, the Daya Bay experiment reveals a non-zero value of θ at more than 16 σ andsuggests a best-fit value of sin θ = 0 . ± .
005 [3]. This data provides a relative 1 σ – 1 –recision of 6% on sin θ which is already better than the precision achieved on sin θ .Undoubtedly, this high precision measurement of the 1-3 mixing angle has speeded up thesearch for the neutrino mass ordering and the possible presence of a CP-violating phase incurrent and future neutrino oscillation experiments [16–20].To explain the presence of small neutrino masses and relatively large neutrino mixingsas indicated by neutrino oscillation data, various neutrino mass models have been proposed.These neutrino mass models come in various categories such as the cases where neutrinosacquire mass via the popular seesaw mechanism [21–31]. We find also models where neu-trinos get mass radiatively due to the presence of extra Higgs bosons [32–34] or low energysupersymmetric hybrid models with spontaneous or bilinear breaking of R-parity [35, 36].The structure of the standard electroweak neutral and charged currents gets affected by thepresence of these mechanisms responsible for the neutrino mass generation [25]. In most ofthe cases, in the low energy regime, these effects are known as non-standard interactions(NSI). Various extensions of the Standard Model (SM), such as left-right symmetric modelsand supersymmetric models with R-parity violation, predict NSI of neutrinos with otherfermions [37–45]. The NSI in these models are usually generated via the exchange of newmassive particles at low energies.Neutrino NSI may be of charged-current (CC) or neutral-current (NC) type, and theycan be classified in two main categories: flavor-changing NSI, when the flavor of the leptoniccurrent involved in the process is changed, or flavor-conserving non-universal NSI, whenthe lepton flavor is not changed in the process but the strength of the interaction dependson it, violating the weak universality. In the low energy regime, these new interactionsmay be parameterized in the form of effective four-fermion Lagrangians: L CC − NSI = G F √ (cid:88) f,f (cid:48) ε s d,ff (cid:48) αβ (cid:2) ¯ ν β γ ρ (1 − γ ) (cid:96) α (cid:3) (cid:2) ¯ f (cid:48) γ ρ (1 ± γ ) f (cid:3) , (1.1) L NC − NSI = G F √ (cid:88) f ε m,ffαβ (cid:2) ¯ ν β γ ρ (1 − γ ) ν α (cid:3) (cid:2) ¯ f γ ρ (1 ± γ ) f (cid:3) . (1.2)where G F is the Fermi constant, α and β are neutrino or lepton flavor indices, f and f (cid:48) label light SM fermions, and the dimensionless coefficients ε parametrize the strength ofthe interaction. Here we denote the CC-NSI couplings as ε s d,ff (cid:48) αβ since they affect in generalthe source (s) and detector (d) interactions at neutrino experiments, while ε m,ffαβ refers tothe NC-NSI couplings generally affecting the neutrino propagation in matter (m). Notethat in Eqs. (1.1) and (1.2) we have assumed for the new interactions the same Lorentzstructure as for the SM weak interactions, (V ± A). Even though more general expressionsare possible within a generic structure of operators, as shown in Ref. [46], these are thedominant contributions for reactor experiments, where we will focus our attention in thiswork.NSI effects may appear at three different stages in a given neutrino experiment, namelyneutrino production, neutrino propagation from the source to the detector, and neutrinodetection. In a short-baseline reactor experiment, the effects on the neutrino propagationare negligible since it happens mainly in vacuum. Therefore, the new generation of short-– 2 –aseline reactor experiments, such as Daya Bay, offers an excellent scenario to probe thepresence of NSI at the neutrino production and detection, free of any degeneracy withNSI propagation effects. Actually, some work has already been done in the context of NSIat short-baseline reactor experiments. In particular, forecasts for the sensitivity of DayaBay to NSI have been published, for instance, in Ref. [47]. More recently, another articlepresented constraints on neutrino NSI using the previous Daya Bay data set [48]. Theresults derived there are in general agreement with some of the cases we discuss in Sec. 4.Nevertheless, our paper also studies the phenomenology of some other interesting casesnot considered in Ref. [48], and provides a detailed description of the effect of NSI in theneutrino survival probability. Finally, we also discuss the fragility of the bounds on theNSI couplings derived using Daya Bay data against the presence of an uncertainty on thetotal event rate normalization in the statistical analysis of reactor data.Given the production and detection neutrino processes involved in short-baseline re-actor neutrino experiments ( β -decay and inverse β -decay), the NSI parameters relevantfor these experiments are ε udeα , i.e., the CC-NSI couplings between up and down quarks,positrons and antineutrinos of flavor α . In the literature we can find the following 90%C.L. bounds on these parameters [49]: | ε ud,Veα | < . , | ε ud,Leµ | < . , | ε ud,Reµ | < . , (1.3)coming from unitarity constraints on the CKM matrix as well as from the non-observationof neutrino oscillations in the NOMAD experiment. Here the couplings with differentchirality are related by : ε ff (cid:48) ,Vαβ = ε ff (cid:48) ,Lαβ + ε ff (cid:48) ,Rαβ . Other constraints on neutrino NSIcouplings using solar and reactor neutrinos have been given in [50–55]. On the other hand,NSI have also been studied in the context of laboratory experiments with accelerators inRefs. [53, 56–58], while bounds from atmospheric neutrino data have been presented inRefs. [59, 60]. Recently a forecast for the sensitivity to NSI of the future PINGU detectorhas been presented in [61]. Future medium-baseline reactor experiments like JUNO canalso serve as test bed to look for NSI [62].This paper is organized as follows. In Sec. 2, we describe the procedure for imple-menting the NSI effect in the modern reactor experiments. There, we derive the effectiveantineutrino survival probability expressions which we use later to analyze the Daya Baydata. We also give plots to discuss in detail the impact of the NSI parameters on theeffective probability considering a special case where ε seγ = ε d ∗ γe . We also show the possiblecorrelations between the NSI parameters and θ with the help of iso-probability surfaceplots. Sec. 3 describes the numerical methods adopted to analyze the reactor data. Apartfrom this, a brief description of the Daya Bay experiment and the important features of itspresent data set which are relevant for the fit are also given in this section. Sec. 4 presentsthe constraints on the NSI parameters imposed by the current Daya Bay data assuming aperfect knowledge of the event rates normalization. Next we derive the bounds on the NSIparameters taking into account the uncertainty in the normalization of event rates with a The NSI parameters probed in our analysis get contributions from the (V ± A) operators in Eq. (1.1)and therefore they can be generally expressed as ε ud, V ± A αβ or equivalently ε ud, L ± R αβ . However, for simplicity,we have dropped the chirality indices all over the paper. – 3 –rior of 5% in Sec. 5. In Sec. 6, we compare the constraints on the NSI parameters obtainedwith the current 621 days of Daya Bay data with the limits derived using the previouslyreleased 217 days of Daya Bay data. Finally, we summarize and draw our conclusions inSec. 7. In Appendix A, we give the effective antineutrino survival probability expressionsfor the physical situations where ε seγ (cid:54) = ε d ∗ γe . According to the usual procedure followed in the non-standard analyses of reactor data [47,63], we start by re-defining the neutrino flavour states in the presence of NSI in the sourceand detection processes. For the initial (at source) and final (at detector) neutrino flavorstates, we have [64–67]: | ν sα (cid:105) = 1 N sα (cid:32) | ν α (cid:105) + (cid:88) γ ε sαγ | ν γ (cid:105) (cid:33) , (cid:104) ν dβ | = 1 N dβ (cid:32) (cid:104) ν β | + (cid:88) η ε dηβ (cid:104) ν η | (cid:33) , (2.1)while the redefinition of the antineutrino flavor states is given by: | ¯ ν sα (cid:105) = 1 N sα (cid:32) | ¯ ν α (cid:105) + (cid:88) γ ε s ∗ αγ | ¯ ν γ (cid:105) (cid:33) , (cid:104) ¯ ν dβ | = 1 N dβ (cid:32) (cid:104) ¯ ν β | + (cid:88) η ε d ∗ ηβ (cid:104) ¯ ν η | (cid:33) . (2.2)The normalization factors required to obtain an orthonormal basis can be expressed as: N sα = (cid:113) [(1 + ε s )(1 + ε s † )] αα , N dβ = (cid:113) [(1 + ε d † )(1 + ε d )] ββ , (2.3)and the neutrino mixing between flavor and mass eigenstates is given by the usual expres-sions: | ν α (cid:105) = (cid:88) k U ∗ αk | ν k (cid:105) , | ¯ ν α (cid:105) = (cid:88) k U αk | ¯ ν k (cid:105) . (2.4)The correct normalization of the neutrino states in presence of NSI is a very importantpoint, required to obtain a total neutrino transition probability normalized to 1. How-ever, one has to consider that when dealing with a non-orthonormal neutrino basis, thenormalization of neutrino states will affect not only the neutrino survival probability butalso the calculation of the produced neutrino fluxes and detection cross sections. In thiscase, as shown in Ref. [68], all the normalization terms coming from N sα and N dβ will cancelwhile convoluting the neutrino oscillation probabilities, cross sections, and neutrino fluxesto estimate the number of events in a given experiment such as Daya Bay. This is due tothe fact that the SM cross sections and neutrino fluxes used in our simulation have beentheoretically derived assuming an orthonormal neutrino basis and therefore they need to becorrected. Then, from here we can consider the following effective redefinition of neutrinoand antineutrino states: | ν sα (cid:105) eff = | ν α (cid:105) + (cid:88) γ ε sαγ | ν γ (cid:105) , (cid:104) ν dβ | eff = (cid:104) ν β | + (cid:88) η ε dηβ (cid:104) ν η | , (2.5)– 4 – ¯ ν sα (cid:105) eff = | ¯ ν α (cid:105) + (cid:88) γ ε s ∗ αγ | ¯ ν γ (cid:105) , (cid:104) ¯ ν dβ | eff = (cid:104) ¯ ν β | + (cid:88) η ε d ∗ ηβ (cid:104) ¯ ν η | , (2.6)where we have dropped the normalization factors that will cancel in the Monte Carlosimulation of Daya Bay data. From these effective neutrino states we will calculate an effective neutrino oscillation probability, that will be used all along our analysis. Note thatwhen we will discuss the features of the probability prior to the simulation of a particularexperiment, we will always refer to the effective probability , that might be greater than one. The effective antineutrino transition probability from flavor α to β after traversing a dis-tance L from source to detector is defined as: P ¯ ν sα → ¯ ν dβ = |(cid:104) ¯ ν dβ | exp ( − i H L ) | ¯ ν sα (cid:105)| . (2.7)In terms of the neutrino mass differences and mixing angles, this transition probability invacuum may be written as: P ¯ ν sα → ¯ ν dβ = (cid:88) j,k Y jαβ Y k ∗ αβ − (cid:88) j>k R{ Y jαβ Y k ∗ αβ } sin (cid:32) ∆ m jk L E (cid:33) + 2 (cid:88) j>k I{ Y jαβ Y k ∗ αβ } sin (cid:32) ∆ m jk L E (cid:33) , (2.8)where ∆ m jk = m j − m k . In the case of standard oscillations, Y jαβ is defined as: Y jαβ ≡ U ∗ βj U αj . (2.9)In presence of NSI, however, according to the definition of neutrino states in Eq. (2.2), thisexpression is modified as follows [63]: Y jαβ ≡ U ∗ βj U αj + (cid:88) γ ε s ∗ αγ U ∗ βj U γj + (cid:88) η ε d ∗ ηβ U ∗ ηj U αj + (cid:88) γ,η ε s ∗ αγ ε d ∗ ηβ U ∗ ηj U γj . (2.10)To obtain the ¯ ν e survival probability in a reactor experiment, where an electron antineutrinois produced at the source and a positron is detected inside the detector, one has to replace α and β by e in Eq. (2.8). For the NSI parameters, we adopt the following parametrizationby splitting the new couplings into its absolute value and its phase: ε seγ ≡ | ε seγ | e i φ seγ and ε dηe ≡ | ε dηe | e i φ dηe . (2.11)Now expanding the various terms of the general transition probability as given in Eq. (2.8)and using the parametrization above, we obtain the effective ¯ ν e survival probability: P ¯ ν se → ¯ ν de = P SM¯ ν e → ¯ ν e + P NSInon-osc + P NSIosc-atm + P NSIosc-solar (2.12)+ O (cid:34) ε , s , ε s , εs , εs (cid:18) ∆ m L E (cid:19) , ε (cid:18) ∆ m L E (cid:19) , s (cid:18) ∆ m L E (cid:19)(cid:35) , – 5 –here the Standard Model (SM) contribution is given by P SM¯ ν e → ¯ ν e = 1 − sin θ (cid:0) c sin ∆ + s sin ∆ (cid:1) − c sin θ sin ∆ , (2.13)with s ij = sin θ ij , c ij = cos θ ij , and ∆ ij = ∆ m ij L/ E . The various NSI terms of theeffective survival probability in Eq. (2.12) can be written as: P NSInon-osc = 2 (cid:16) | ε dee | cos φ dee + | ε see | cos φ see (cid:17) + | ε dee | + | ε see | + 2 | ε dee || ε see | cos( φ dee − φ see ) (2.14)+ 2 | ε dee || ε see | cos( φ dee + φ see ) + 2 | ε seµ || ε dµe | cos( φ seµ + φ dµe ) + 2 | ε seτ || ε dτe | cos( φ seτ + φ dτe ) ,P NSIosc-atm = 2 (cid:110) s s (cid:104) | ε seµ | sin( δ − φ seµ ) − | ε dµe | sin( δ + φ dµe ) (cid:105) + s c (cid:104) | ε seτ | sin( δ − φ seτ ) − | ε dτe | sin( δ + φ dτe ) (cid:105) − s c (cid:104) | ε seµ || ε dτe | sin( φ seµ + φ dτe ) + | ε seτ || ε dµe | sin( φ seτ + φ dµe ) (cid:105) − c | ε seτ || ε dτe | sin( φ seτ + φ dτe ) − s | ε seµ || ε dµe | sin( φ seµ + φ dµe ) (cid:111) sin (2∆ ) − (cid:110) s s (cid:104) | ε seµ | cos( δ − φ seµ ) + | ε dµe | cos( δ + φ dµe ) (cid:105) + s c (cid:104) | ε seτ | cos( δ − φ seτ ) + | ε dτe | cos( δ + φ dτe ) (cid:105) + s c (cid:104) | ε seµ || ε dτe | cos( φ seµ + φ dτe ) + | ε seτ || ε dµe | cos( φ seτ + φ dµe ) (cid:105) + c | ε seτ || ε dτe | cos( φ seτ + φ dτe ) + s | ε seµ || ε dµe | cos( φ seµ + φ dµe ) (cid:111) sin (∆ ) , (2.15) P NSIosc-solar = 2 sin 2 θ ∆ (cid:110) − c ( | ε seµ | sin φ seµ + | ε dµe | sin φ dµe )+ s ( | ε seτ | sin φ seτ + | ε dτe | sin φ dτe ) (cid:111) . (2.16)The linear coefficients of the terms of order | ε | in Eqs. (2.14, 2.15, 2.16) are the same asgiven in Ref. [46], with a good agreement between the calculated probabilities here andthere. However, as it can be seen in the expressions above, here we also include newterms up to second order in | ε | in the effective neutrino probability. The relevance of thesecorrections will be discussed later in the paper. In Eq. (2.15), note the presence of a termlinear in the sine of ∆ m L/ E which therefore depends on the choice of neutrino massordering. This term does not appear in the standard ¯ ν e → ¯ ν e oscillation expression and itcan affect the L/E dependence of the probability in the presence of neutrino NSI. ε seγ = ε d ∗ γe In this work we assume that, likewise the mechanisms responsible for production (via β -decay) and detection (via inverse β -decay) of reactor antineutrinos are just inverse ofeach other, this is also true for the associated NSI [46, 47]. This assumption allows us to In the appendix, we have given the effective probability expressions for the physical situations where ε seγ (cid:54) = ε d ∗ γe . In such cases, the spectral analysis of the reactor data becomes inevitable since the NSIparameters not only cause a shift in θ i.e. the change of the depth of the first oscillation maximum butalso modify the L/E pattern of the oscillation probability due to the shift in its energy. A detailed analysisof the Daya Bay data under such scenarios will be performed in [69]. – 6 –rite ε sγ = ε d ∗ γ ≡ | ε γ | e i φ γ where we drop the universal e index for simplicity. With theseassumptions, Eq. (2.12) takes the form (keeping the terms up-to the second order in smallquantities): P ¯ ν se → ¯ ν de (cid:39) − sin θ (cid:0) c sin ∆ + s sin ∆ (cid:1) − c sin θ sin ∆ (cid:124) (cid:123)(cid:122) (cid:125) Standard Model terms + 4 | ε e | cos φ e + 4 | ε e | + 2 | ε e | cos 2 φ e + 2 | ε µ | + 2 | ε τ | (cid:124) (cid:123)(cid:122) (cid:125) non − oscillatory NSI terms − { s | ε µ | + c | ε τ | + 2 s c | ε µ || ε τ | cos( φ µ − φ τ ) } sin ∆ (cid:124) (cid:123)(cid:122) (cid:125) oscillatory NSI terms − { s [ s | ε µ | cos ( δ − φ µ ) + c | ε τ | cos( δ − φ τ )] } sin ∆ (cid:124) (cid:123)(cid:122) (cid:125) oscillatory NSI terms . (2.17)For this special case of NSI parameters, there is no linear sine-dependent term in Eq. (2.17)and two striking features are emerging from the effective probability expression which areresponsible for a change in the oscillation amplitude. First we can see the presence of somenon-oscillatory NSI terms which are independent of L and E and are given by1 + 4 | ε e | cos φ e + 4 | ε e | + 2 | ε e | cos 2 φ e + 2 | ε µ | + 2 | ε τ | , (2.18)and, second, there is a shift in the effective s → s + s | ε µ | + c | ε τ | + 2 s c | ε µ || ε τ | cos( φ µ − φ τ )+ 2 s [ s | ε µ | cos( δ − φ µ ) + c | ε τ | cos( δ − φ τ )] . (2.19)These two features, which are brought about by the NSI parameters, are responsible fora shift in the oscillation amplitude without distorting the L/E pattern of the oscillationprobability as can be clearly seen from Fig. 1 and Fig. 2 that we will discuss in the nextsection. Eq. (2.19) suggests that it will be quite challenging to discriminate the effect of true θ and NSI parameters in the modern reactor experiments. It is also interesting tonote that there are some CP conserving terms in Eq. (2.19) which come into the picturedue to the presence of NSI parameters. One of the most important consequences of thenew non-oscillatory NSI terms (see Eq. (2.18)) is that they can cause a flavor transition atthe source ( L = 0) even before neutrinos start to oscillate. In the literature, this featureis known as “zero-distance” effect [46, 70]. In modern reactor experiments, this effect canbe probed using the near detectors which are placed quite close to the source.For definiteness, in this work we have restricted our analysis to the following choicesof the NSI parameters: • Lepton number conserving non-universal
NSI parameters which depend on the flavorcharacterizing the violation of weak universality. Under this category, we study thefollowing two cases: – 7 –. Considering only the NSI parameters | ε e | and φ e which are associated with ¯ ν e .In the presence of these flavor conserving NSI parameters, Eq. (2.17) takes theform: P NSI-e¯ ν se → ¯ ν de (cid:39) P SM¯ ν e → ¯ ν e + 4 | ε e | cos φ e + 4 | ε e | + 2 | ε e | cos 2 φ e . (2.20)At first order in | ε e | and neglecting the effect of the solar mass splitting, thenew non-oscillatory NSI terms appearing at the survival probability produce atotal shift in the effective θ mixing angle given by:˜ s ≈ s − | ε e | cos φ e sin ∆ . (2.21)This expression will be very useful to discuss the behavior of the effective prob-ability as well as the correlations between θ and the NSI parameters in thenext subsections.2. Considering only the NSI parameters | ε µ | and φ µ which are associated with ¯ ν µ .In the presence of these flavor violating NSI parameters, Eq. (2.17) takes theform: P NSI- µ ¯ ν se → ¯ ν de (cid:39) P SM¯ ν e → ¯ ν e + 2 | ε µ | − { s | ε µ | + 2 s s | ε µ | cos( δ − φ µ ) } sin ∆ . (2.22)Note that, for the NSI parameters | ε τ | and φ τ which are associated with ¯ ν τ ,the effective survival probability will be exactly the same as Eq. (2.22) with thereplacements | ε µ | → | ε τ | and φ µ → φ τ provided that the 2-3 mixing angle ismaximal , i.e. , sin θ = 0 .
5. As before, the new non-standard oscillatory termsin the neutrino transition probability may be interpreted as a global redefinitionof the effective θ mixing angle:˜ s ≈ s + 2 s s | ε µ | cos ( δ − φ µ ) . (2.23) • Lepton number conserving universal
NSI parameters which do not depend on flavor.In this case, we have | ε e | = | ε µ | = | ε τ | = | ε | and φ e = φ µ = φ τ = φ and the probabilityin Eq. (2.17) takes the form: P NSI- α ¯ ν se → ¯ ν de (cid:39) P SM¯ ν e → ¯ ν e + 4 | ε | cos φ + 2 | ε | (4 + cos 2 φ ) − {| ε | + 2 s c | ε | + 2 s | ε | cos( δ − φ )( s + c ) } sin ∆ . (2.24)In this case, the effective mixing angle in the presence of oscillatory and non-oscillatoryNSI terms will be given by:˜ s ≈ s − | ε | (cid:20) cos φ sin ∆ − s ( s + c ) cos ( δ − φ ) (cid:21) . (2.25)As stated above, the expressions given in this subsection to illustrate the shift in theeffective reactor angle in the presence of NSI, Eqs. (2.21), (2.23) and (2.25), contain onlyfirst order corrections in the NSI couplings. Note, however, that terms of second order– 8 –arameter sin θ sin θ sin θ ∆ m (eV ) ∆ m (eV ) δ Value 0.32 0.5 0.023 7.6 × − × − π Table 1 : Benchmark values of the neutrino oscillation parameters used in this work, takenfrom Refs. [5, 71].in | ε | have been included in all the numerical results shown along the paper. Here wewill briefly discuss the relevance of second order corrections in our analysis. Clearly, thesecorrections to the effective neutrino probability are only significant in the cases where firstorder corrections are very small or totally cancelled. This happens for φ e = ± ◦ in thecase of electron-NSI couplings, for ( δ − φ µ,τ ) = ± ◦ in the case of muon/tau-NSI couplingsand for ( δ = 0, φ = ± ◦ ) in the flavour-universal case. From Eqs. (2.20), (2.22) and (2.24),it is straightforward to evaluate the size of the second order terms for the three cases understudy. Taking into account the baselines and neutrino energies probed at the Daya Bayexperiment, we find that second order corrections (for vahishing first order corrections) areapproximately given by: 0 . | ε e | (electron-NSI case), 0 . | ε µ,τ | (muon/tau-NSI case) and0 . | ε | (flavour-universal-NSI case). The small size of the corrections in the muon/tau-NSI,one order of magnitude smaller than for the two other cases, comes from the smallness ofthe coefficient responsible for second order corrections in the expression of the effectivereactor angle, (cid:16)
12 sin ∆ − s (cid:17) , very close to zero for the energies and baselines studiedin Daya Bay. This result can be observed in the correlation plots in Figs. 3 and 4 (seedashed blue line) in Sec. 2.4. There, one sees that second order corrections are small butvisible for electron-NSI and flavour-universal case, while they are almost negligible for themuon/tau-NSI case. The impact of the second order corrections over the results presentedin this work is discussed in Sec. 7. Now we will study the possible impact of the NSI parameters at the effective probabilitylevel. Table 1 depicts the benchmark values of the various oscillation parameters that areconsidered to generate the oscillation probability plots. These choices of the oscillationparameters are in close agreement with the best-fit values that have been obtained in therecent global fits of the world neutrino oscillation data [5–7]. Here we would like to mentionthat for the 2-3 mixing angle, we have taken the maximal value i.e. sin θ = 0 .
5, thoughin the global fit studies, there is a slight hint for a non-maximal value of θ . We have alsoassumed normal mass ordering, i.e. , ∆ m positive. In Fig. 1, we present the standard andthe NSI-modified three-flavor oscillation effective probability as a function of the electronantineutrino energy with a source-detector distance of 1.58 km. In the left panel of Fig. 1,the band shows how the probability changes if we vary | ε e | in the range [0, 0.04] and φ e over the range [-180 ◦ , 180 ◦ ] simultaneously. For the NSI parameters which are onlyassociated with ¯ ν e , the probability is independent of the CP phase δ (see Eq. (2.13) andEq. (2.20)). The solid black line shows the standard oscillation probability without theNSI terms (see Eq. (2.13)). The other four lines have been drawn considering particular– 9 – E ff ec ti v e P r ob a b ilit y Energy [MeV] P sm ❘ ε e ❘ =0.02, φ e =0 ❘ ε e ❘ =0.04, φ e =0 ❘ ε e ❘ =0.02, φ e =180 ❘ ε e ❘ =0.04, φ e =180 E ff ec ti v e P r ob a b ilit y Energy [MeV] P sm ❘ ε µ , τ ❘ =0.02,( δ - φ µ , τ )=0 ❘ ε µ , τ ❘ =0.04,( δ - φ µ , τ )=0 ❘ ε µ , τ ❘ =0.02,( δ - φ µ , τ )=180 ❘ ε µ , τ ❘ =0.04,( δ - φ µ , τ )=180 Figure 1 : Effective ¯ ν se → ¯ ν de survival probability as a function of neutrino energy in thepresence of NSI with L = 1.58 km. The band in the left panel has been generated byvarying | ε e | in the range [0, 0.04] and φ e over the range [-180 ◦ , 180 ◦ ] simultaneously. Thesimultaneous variation of | ε µ,τ | in the range [0, 0.04] and ( δ − φ µ,τ ) over the range [-180 ◦ ,180 ◦ ] is responsible for the band in the right panel. In both the panels, the solid black linesdepict the probability without new physics involved (SM case).choices of | ε e | and φ e which are mentioned in the figure legends. If φ e = 180 ◦ , then theoscillation probability is less compared to the standard value because the contributionfrom the non-oscillatory NSI terms takes the form − | ε e | + 6 | ε e | which always gives anoverall negative contribution to the full probability if | ε e | ≤ .
66. On the other hand, ifwe consider φ e = 0 ◦ , then the oscillation probability is above the standard value becausethe contribution from the non-oscillatory terms takes the form 4 | ε e | + 6 | ε e | which alwaysgives an overall positive contribution to the full probability for any choice of | ε e | . Wecan also see that even for a small value of | ε e | of 0 .
02, the effective oscillation probabilitycan be more than unity for most of the energies of interest. This is the sign of the non-unitarity effects [68, 72–75], caused by the presence of neutrino NSI at the source anddetector of reactor experiments. In the right panel of Fig. 1, the band shows the changesin the effective probability after varying | ε µ,τ | in the range [0, 0.04] and ( δ − φ µ,τ ) overthe range [-180 ◦ , 180 ◦ ] simultaneously. Note that in Eq. (2.22), the phases appear in theform of cosine of ( δ − φ µ,τ ) and also the NSI terms have a non-trivial L/E dependency.The standard oscillation probability without the NSI terms is shown by the solid blackline and the other four lines have been drawn considering particular choices of | ε µ,τ | and( δ − φ µ,τ ) which are mentioned in the figure legends. If ( δ − φ µ,τ ) = 0 ◦ (180 ◦ ), then theeffective oscillation probability is less (more) compared to the standard value for almostall the choices of neutrino energy as opposed to the case of the NSI parameters associatedwith ¯ ν e .Fig. 2 shows the impact of the NSI parameters at the probability level with L =1.58 km for the flavor-universal NSI case where we consider | ε e | = | ε µ | = | ε τ | = | ε | and– 10 – E ff ec ti v e P r ob a b ilit y Energy [MeV] P sm x ε x =0.02, φ =0 x ε x =0.05, φ =0 x ε x =0.02, φ =180 x ε x =0.05, φ =180 Figure 2 : Effective ¯ ν se → ¯ ν de survival probability as a function of neutrino energy with L = 1.58 km for the flavor-universal NSI case (see Sec. 2.2 for details). The dark salmonregion shows the combined effect of the variation of the new physics parameters | ε | and φ with δ = 0 ◦ . The extended probability band in light grey has been obtained by varyingthe CP phase δ in the range [-180 ◦ , 180 ◦ ] along with the other two parameters | ε | and φ .The solid black line displays the probability without new physics involved (SM case). φ e = φ µ = φ τ = φ . In this plot, the dark salmon region has been generated by varying theNSI parameter | ε | in the range [0, 0.05] and φ in the range [-180 ◦ , 180 ◦ ] simultaneously,keeping the CP phase δ fixed to 0 ◦ . Next, we vary δ in its entire range from -180 ◦ to180 ◦ along with the NSI parameters | ε | and φ and obtain the extended probability bandin the form of the light grey region. In Fig. 2, the solid black line depicts the standardprobability without considering the NSI parameters. The other four lines in this plotdisplay the effective oscillation probability for particular combinations of | ε | and φ with δ = 0 ◦ which are mentioned in the figure legends. It is quite clear from Eq. (2.24) thatthe non-oscillatory terms dominate over the oscillatory terms in the flavor-universal NSIcase. Therefore, the dark salmon region of Fig. 2 closely resembles the left panel of Fig. 1where we consider the NSI parameters which are only associated with ¯ ν e , namely | ε e | and φ e . Next, we discuss the possible correlations between the NSI parameters and θ withthe help of iso-probability plots. – 11 – | ε e | sin θ φ e = 0 φ e = 90,-90 φ e = 180 | ε µ , τ | sin θ ( δ - φ µ , τ ) = 0 ( δ - φ µ , τ ) = 90,-90 ( δ - φ µ , τ ) = 180 Figure 3 : Left panel shows the iso-probability surface contours in the (sin θ – | ε e | ) planefor different choices of φ e as mentioned in the figure legends. Here we consider sin θ =0.023 and | ε e | = 0 as benchmark choices. Right panel displays the same in the (sin θ – | ε µ,τ | ) plane for different choices of ( δ − φ µ,τ ) considering sin θ = 0.023 and | ε µ,τ | = 0as true choices. For both the panels, we consider a fixed neutrino energy E = 4 MeV andthe baseline L = 1.58 km. θ : iso-probability plots We consider the neutrino energy E = 4 MeV and the source-detector distance L = 1.58 kmto draw the iso-probability surface plots. Left panel of Fig. 3 shows the iso-probability sur-face contours in the (sin θ – | ε e | ) plane for four different choices of φ e considering sin θ = 0.023 and | ε e | = 0 as best-fit choices. Our best-fit choices correspond to the standardoscillation probability without considering the NSI parameters as given by Eq. (2.13). Nowas we consider the finite value of | ε e | with φ e = 0 ◦ (see the solid red line), the NSI termsincrease the overall probability (see Eq. (2.20)). Then, we need to increase the value ofsin θ to reduce the SM probability in order to compensate the enhancement due to theNSI contribution. For φ e = 180 ◦ case (see the dot-dashed magenta line), a finite value of | ε e | decreases the overall probability demanding a lower value of sin θ so as to enhancethe contribution from the standard probability. In the cases with φ e = 90 ◦ or -90 ◦ (see thedashed blue line), the non-oscillatory NSI term which is linear in | ε e | drops out from theprobability expression and the remaining NSI contribution is 2 | ε e | (see Eq. (2.20)). Due tothis weak quadratic dependence on | ε e | , a very large value of | ε e | is needed to compensateeven a very small increment in sin θ . Right panel displays the same in the (sin θ – | ε µ,τ | ) plane for different choices of ( δ − φ µ,τ ). Here sin θ = 0.023 and | ε µ,τ | = 0 are con-sidered as true choices. In this panel, the combination of phases ( δ − φ µ,τ ) shows oppositefeatures for 0 ◦ and 180 ◦ as compared to φ e in the left panel. Note that the impact of | ε µ,τ | on sin θ is weaker compared to | ε e | . If we examine the terms which are linear in | ε e | (seeEq. (2.20)) and | ε µ,τ | (see Eq. (2.22)) then we can see that the contribution coming from– 12 – | ε | sin θ δ = 0 δ = 90,-90 δ = 180 | ε | sin θ φ = 0 φ = 90,-90 φ = 108 φ = 180 Figure 4 : Left panel shows the iso-probability surface contours for the flavor-universalNSI case in the (sin θ – | ε | ) plane for different choices of the CP phase δ with φ = 0 ◦ .Here we consider sin θ = 0.023 and | ε | = 0 as benchmark choices. Right panel displaysthe same for various choices of φ assuming δ = 0 ◦ . For both the panels, we take a fixedneutrino energy E = 4 MeV and the baseline L = 1.58 km. | ε µ,τ | is sin θ suppressed even if we work at the first oscillation maximum. For ( δ − φ µ,τ )= 90 ◦ or -90 ◦ , the | ε µ,τ | -dependent terms completely disappear from Eq. (2.22) if sin θ =0.5 and sin ∆ = 1. Therefore, we do not see any correlation between sin θ and | ε µ,τ | for ( δ − φ µ,τ ) = 90 ◦ or -90 ◦ in the right panel of Fig. 3.In Fig. 4, we show the correlation between sin θ and | ε | for the flavor-universal NSIcase with the help of iso-probability surface contours. In the left panel, we consider fourdifferent choices of the CP phase δ keeping φ fixed to 0 ◦ . In the right panel, we take fivedifferent choices of φ assuming δ = 0 ◦ . In both panels, the iso-probability surface contoursare the same for 90 ◦ and -90 ◦ choices of phases (see the dashed blue lines) because the phasesappear in the form of cosines in Eq. (2.24). For the φ = 0 ◦ case (left panel), the contributionfrom the non-oscillatory NSI terms is maximum and takes the form 4 | ε | + 10 | ε | . Thus,it enhances the overall oscillation probability to a great extent even for a small value of | ε | . Now to compensate this enhancement, we need to increase the value of sin θ toreduce the contribution coming from the standard survival oscillation probability. Rightpanel of Fig. 4 ( δ = 0 ◦ case) closely resembles the left panel of Fig. 3 because for theflavor-universal NSI case, the contribution from the non-oscillatory NSI terms dominateswhich is also true for the NSI parameters associated with ¯ ν e and also these NSI parametersare δ independent. In the right panel, we present a special case of φ = 108 ◦ to explainthe features emerging from the extreme right panel of Fig. 7 (see later in Sec. 4) wherewe have displayed the allowed region in (sin θ – | ε | ) plane using the current data fromthe Daya Bay experiment allowing the flavor-universal NSI phase φ to vary in the entirerange of [-180 ◦ , 180 ◦ ] with the CP phase δ to be fixed to 0 ◦ . Eq. (2.24) suggests that wecan always choose some values of φ such that the non-oscillatory terms are canceled. This– 13 –appens, for instance, for φ = 108 ◦ and | ε | = 0 . θ by the same quantity. This is exactly the feature that we can see for φ = 108 ◦ case. Reactor antineutrinos are produced by the fission of the isotopes U, Pu,
Pu and
U contributing to the neutrino flux with a certain fission fraction f k . Reactor antineu-trinos are detected via inverse β -decay process (IBD), ¯ ν e + p → e + + n . The techniqueused is a delayed coincidence between two gamma rays: one coming from the positron( prompt signal) and the other coming from the neutron capture in the innermost part ofthe antineutrino detector (AD), containing gadolinium-doped liquid scintillator. The lightcreated is collected by the photo-multipliers (PMTs) located in the outermost mineraloil-region. The antineutrino energy E ¯ ν is reconstructed from the positron prompt energy E prompt following the relation: E ¯ ν = E prompt + ¯ E n + 0 .
78 MeV, where ¯ E n is the averageneutron recoil energy.The expected number of IBD events at the d -th detector, T d , can be estimated summingup the contributions of all reactors to the detector: T d = (cid:88) r T rd == (cid:88) r (cid:15) d N p πL rd P rth (cid:80) k f k (cid:104) E k (cid:105) (cid:88) k f k (cid:90) ∞ dE Φ k ( E ) σ IBD ( E ) P ee ( E, L rd ) , (3.1)where N p is the number of protons in the target volume, P rth is the reactor thermal power, (cid:15) d denotes the efficiency of the detector and (cid:104) E k (cid:105) is the energy release per fission for a givenisotope k taken from Ref. [76]. The neutrino survival probability P ee depends also on thedistance from r -th reactor to d -th detector, L rd . For the antineutrino flux prediction Φ k ( E )we use the parameterization given in Ref. [77] as well as the new normalization for reactorantineutrino fluxes updated in Ref. [78]. The inverse beta decay cross section σ IBD ( E ν ) istaken from Ref. [79]. Daya Bay Experiment
Daya Bay is a reactor neutrino experiment with several antineutrino detectors (ADs),arranged in three experimental halls (EHs). Electron antineutrinos are generated in sixreactor cores, distributed in pairs, with equal thermal power (P rth =2.9 GW th ) and detectedin the EHs. The effective baselines are 512 m and 561 m for the near halls EH1 and EH2and 1579 m for the far hall EH3 [2]. With this near-far technology Daya Bay has minimizedthe systematic errors coming from the ADs and thus provided until now the most precisedetermination of the reactor mixing angle. In the last Neutrino conference, Daya Bay hasreported its preliminary results considering 621 days of data taking combining their results– 14 –or two different experimental setups [3]: one with six ADs as it was published in Ref. [2]and the other after the installation of two more detectors, eight ADs in total. This newcombined data set has four times more statistics in comparison with the previous DayaBay results. Thus, the precision in the determination of the reactor mixing angle has beenimproved, and it is now of the order of 6%.In this work we will consider the most recent data release by the Daya Bay Collabora-tion described above and we will concentrate on the total observed rates at each detector,that will be analyzed using the following χ expression: χ = (cid:88) d =1 (cid:2) M d − T d (cid:0) a norm + (cid:80) r ω dr α r + ξ d (cid:1) + β d (cid:3) M d + B d + (cid:88) r =1 α r σ r + (cid:88) d =1 (cid:18) ξ d σ d + β d σ B (cid:19) + a σ a . (3.2)Here T d corresponds to the theoretical prediction in Eq. (3.1), M d is the measured numberof events at the d -th AD with its backgrounds ( B d ) subtracted and ω dr is the fractionalcontribution of the r -th reactor to the d -th AD number of events, determined by thebaselines L rd and the total thermal power of each reactor. The pull parameters, used toinclude the systematical errors in the analysis, are given by the set ( α r , ξ d , β d ) representingthe reactor, detector and background uncertainties with the corresponding set of errors( σ r , σ d , σ B ). Uncertainties in the reactor related quantities are included in σ r (0 . σ d (0 . σ B isthe quadratic sum of the background uncertainties taken from Ref. [3]. Finally, we alsoconsider an absolute normalization factor a norm to account for the uncertainty in the totalnormalization of events at the ADs, given by σ a , and coming for instance from uncertaintiesin the normalization of reactor antineutrino fluxes. In our analysis we will follow twodifferent approaches concerning this parameter. In Sec. 4 we will take it equal to zero,assuming perfect knowledge of the events normalization. This hypothesis will be relaxedin Sec. 5, where we will allow for a non-zero normalization factor in the statistical analysis,being determined from the fit to the Daya Bay data. As we will see, the results obtainedin our analysis are strongly correlated with the treatment of the total normalization ofreactor neutrino events in the statistical analysis of Daya Bay data and therefore it is ofcrucial importance to do a proper treatment of this factor. In this section we will present the bounds on the NSI couplings we have obtained usingcurrent Daya Bay reactor data. In all the results, we have assumed maximal 2-3 mixingand we have marginalized over atmospheric splitting with a prior of 3%. For definitenesswe will start considering only the couplings relative to electron neutrino: ( | ε e | , φ e ), for whatwe will switch all the other NSI parameters to zero. Next we will do the same for ( | ε µ | , φ µ )and ( | ε τ | , φ τ ), that are equivalent for maximal value of θ . Finally, we will consider thepossibility of having all NSI couplings with the same value: ε e = ε µ = ε τ = ε . In all cases– 15 –
0 0.01 0.02 0.03 0.04 sin θ | ε e | φ e =0
0 0.01 0.02 0.03 0.04 sin θ | ε e | φ e =free Figure 5 : Allowed region in the sin θ - | ε e | plane. Left panel is obtained by setting thephase φ e to zero, while in the right panel φ e is marginalized, varying freely between -180 ◦ and 180 ◦ . The regions correspond to 68% (black dashed line), 90% (green line) and 99%C.L. (red line) for 2 d.o.f.we will discuss the bounds arising from Daya Bay data in comparison with existing bounds.We will also consider the robustness of the θ measurement by Daya Bay in the presenceof NSI. According to the expression in Eq. (2.20), the effective survival probability in the casewhen only NSI with electron antineutrinos are considered is independent of the standardCP phase δ . Therefore, in our analysis we consider only two cases, one with the onlyrelevant phase φ e set to zero, and a second case where we allow this phase to vary freely.Our results are presented in Fig. 5. From the left panel in this figure, we can confirmthe behaviors shown by the iso-probability curves in the Sec. 2.4, namely, the presence ofa non-zero ε e coupling has to be compensated with a slightly larger value of the reactormixing angle θ . In this panel one also sees how current Daya Bay data constrain verystrongly the magnitude of the NSI coupling | ε e | , improving the current bound in Eq. (1.3)by one order of magnitude: | ε e | ≤ . × − (90% C.L.) . (4.1)However, the situation changes dramatically when the phase φ e is allowed to vary freely,as shown in the right panel of Fig. 5. In this case, the strong bound on | ε e | disappearsdue to the presence of a correlation bewteen | ε e | and cos φ e in the | ε e | -linear term in theneutrino survival probability (see Eq.(2.20)). The presence of second order terms in | ε e | isnot enough to break this degeneracy and, therefore, the sensitivity to | ε e | disappears andno bound can be obtained from reactor data.– 16 –
0 0.012 0.024 0.036 0.048 sin θ | ε µ , τ | δ - φ µ , τ = 0
0 0.012 0.024 0.036 0.048 sin θ | ε µ , τ | δ - φ µ , τ = free Figure 6 : Allowed region in the sin θ - | ε µ,τ | plane. Left panel is obtained setting therelevant phase ( δ - φ µ,τ ) to zero, while in the right panel ( δ - φ µ,τ ) is marginalized, varyingfreely between -180 ◦ and 180 ◦ . The regions correspond to 68% (black dashed line), 90%(green line), and 99% C.L. (red line) for 2 d.o.f.Concerning the determination of the reactor mixing angle, the presence of the NSI-coupling with electron antineutrinos results in the following allowed range for θ :0 . ≤ sin θ ≤ .
024 (90% C.L.) . (4.2)The same interval is obtained for the two panels at Fig. 5 and it also coincides exactly withthe allowed range in absence of NSI. In consequence, we can say that the reactor angledetermination by Daya Bay is robust in this specific case. In this subsection we present the results obtained considering only the NSI parametersassociated with muon and tau neutrinos. As we have discussed in Sec. 2.2, in this case, thephases δ and φ µ,τ do not appear separately in the expression of the survival probability,see Eq. (2.22). Therefore, it is enough to consider in our calculations the effective phase( δ − φ µ,τ ).The results corresponding to this particular case are shown in Fig. 6. Here again we cansee how the regions presented in the left panel of the figure agree with the behavior shownin the iso-probability plots (right panel of Fig. 3) where there is an anticorrelation betweenthe reactor angle and the NSI coupling | ε µ,τ | . Thus, an increase in | ε µ,τ | is compensated bya shift of the preferred value of the reactor mixing angle toward smaller values, differentlyto what happens with the NSI coupling | ε e | in Fig. 5. The allowed interval for θ in thiscase is given by: 0 . ≤ sin θ ≤ .
024 (90% C.L.) , (4.3)while the obtained bound for the NSI coupling is the following | ε µ,τ | ≤ . × − (90% C.L.) . (4.4)– 17 –n this case reactor data can not improve the present constraints on the NSI couplings atEq. (1.3), and we get a limit of the same order of magnitude of the ones derived at Ref. [49].However, in both cases the limits have been derived using different data and assumptions,and therefore, they can be regarded as complementary bounds coming from different datasets.In the right panel of Fig. 6 we show the results obtained when the phase ( δ − φ µ,τ ) isallowed to vary. In this case, a wider range in the reactor mixing angle is allowed:0 . ≤ sin θ ≤ .
036 (90% C.L.) . (4.5)The reason is that, in addition to the anticorrelation shown in the left panel, a correlationbetween the reactor angle and | ε µ,τ | is also possible when the cosine function in Eq. (2.23)is negative. Note, however, that both correlations are not symmetric, what results inthe asymmetric behaviour of the allowed sin θ region with a bigger enlargement in thedirection of increasing θ . This can be explained by the presence of a linear term in sin θ in the redefinition of the effective reactor angle in the presence of NSI given in Eq. (2.23).Nevertheless, even though there is a wider allowed region in the reactor angle, the boundon the | ε µ,τ | NSI coupling is nearly the same as the one obtained when the phase ( δ − φ µ,τ )is set to zero, namely: | ε µ,τ | ≤ . × − (90% C.L.) . (4.6) Here we present the results obtained under the hypothesis of flavor-universal NSI, thatis, we assume all NSI couplings are present and they take the same value. Therefore, weconsider only two NSI parameters: | ε | and φ , in addition to the standard model parametersentering in the calculations. In this case, the effective survival probability is given by theexpression in Eq. (2.24), with separate dependence on the phases δ and φ . Therefore wehave considered four different cases in our analysis: one with all the phases set to zero, twocases varying only one of the phases with the other set to zero and a last case varying thetwo phases simultaneously. Our results are presented at Fig. 7.The left panel in Fig. 7 shows the tight constraint obtained for the magnitude of theflavor-universal NSI coupling | ε | when the phases are set to zero: | ε | ≤ . × − (90% C.L.) . (4.7)This result follows directly from the tendency already observed for | ε e | at Fig. 5. This hap-pens because, even though the NSI couplings | ε µ,τ | are also present in the flavor-universalcase, the dominant contribution comes from the non-oscillatory term depending on | ε e | .The same behavior is also present in the middle panel of Fig. 7, where the Dirac phase δ isallowed to freely vary. In this case, the effect of the Dirac phase is just a scaling in the lastterm in Eq. (2.25), driven by cos δ , but again the total effect on θ is dominated by thenon-oscillatory term in Eq. (2.25). Under this assumption, we obtain the following boundon the magnitude of the flavor-universal coupling: | ε | ≤ . × − (90% C.L.) . (4.8)– 18 –
0 0.01 0.02 0.03 0.04 sin θ | ε | δ =0, φ =0
0 0.01 0.02 0.03 0.04 sin θ | ε | δ =free, φ =0
0 0.01 0.02 0.03 0.04 sin θ | ε | δ =0, φ =free Figure 7 : Allowed region in the sin θ - | ε | plane for different assumptions concerningthe phases δ and φ . The left panel is obtained switching all phases to zero, whereas in themiddle panel φ = 0 and δ is left free. In the right panel δ is taken equal to zero while φ has been marginalized. The conventions for the lines is the same as in Fig. 5Note that in the two former cases the allowed range for the mixing angle θ is very close tothe one obtained in the standard case, and therefore the Daya Bay determination is barelyaffected by the presence of NSI. However, when δ = 0 and the NSI phase φ is allowed to takedifferent values, a completely different behavior results, as it is shown in the right panel ofFig. 7. In this case, in the fit of Daya Bay data, the phase φ takes a value such that thenon-oscillatory term in Eq. (2.24) is cancelled up to order | ε | . Most importantly, with apreferred value of cos φ (cid:39) − . | ε | , new terms of second order in | ε | appear at Eq. (2.24) andtherefore the first order expression in Eq. (2.25) can not satisfactory explain the degeneracybetween θ and | ε | . Actually, the shift in the effective reactor angle is given now by thisother expression (up to order | ε | ):˜ s ≈ s + | ε | (cid:16) − √ s (cid:17) . (4.9)Then, it is possible to see that, even with s = 0, one can reproduce the measured valueof θ in Daya Bay with | ε | ∼ .
11, as shown in the corresponding plot. As a consequenceof the degeneracy in the plane s - | ε | , this specific case is not very restrictive and we geta loose bound on | ε | : | ε | ≤ . × − (90% C.L.) . (4.10)and only an upper bound on θ :sin θ ≤ .
024 (90% C.L.) . (4.11)As commented above, the presence of flavor universal NSI implies that the reactor mixingangle may be compatible with zero. Nevertheless, the degeneracy between the mixing angleand the new physics parameter | ε | observed here may be lifted by a combined analysiswith accelerator long-baseline neutrino experiments. A global analysis of neutrino dataassuming the simultaneous presence of NSI in reactor and accelerator neutrino data wouldbe very useful for this purpose. Besides solving the degeneracy, the combined analysismight provide further constraints on the NSI couplings as well as improve the agreement– 19 –hases sin θ | ε | electron-type NSI coupling φ e = 0 0 . ≤ sin θ ≤ . | ε e | ≤ . φ e free 0 . ≤ sin θ ≤ . | ε e | unboundmuon or tau-type NSI couplings( δ − φ µ,τ ) = 0 0 . ≤ sin θ ≤ . | ε µ,τ | ≤ . δ − φ µ,τ ) free 0 . ≤ sin θ ≤ . | ε µ,τ | ≤ . δ = φ = 0 0 . ≤ sin θ ≤ . | ε | ≤ . δ free, φ = 0 0 . ≤ sin θ ≤ . | ε | ≤ . δ = 0, φ free sin θ ≤ . | ε | ≤ . Table 2 : 90% C.L. bounds (1 d.o.f) on sin θ and the NSI couplings from current DayaBay data without considering any uncertainty in the normalization of reactor event ratesin the statistical analysis ( a norm = 0).between the preferred θ value from reactors and long-baseline experiments, as discussedin Ref. [80–82]. However, since the production, detection and propagation of neutrinos isquite different in both kind of experiments, a global analysis would require a very detailedstudy with many new physics parameters involved, besides the consideration of a specificmodel for NSI. In any case, this point is out of the scope of the present analysis and it willbe considered elsewhere.Finally, let us comment that we have also considered the case of flavor-universal NSIwith all the phases different from zero. However, we have not presented the results obtainedfor this setup because, in this case, the confusion between θ and | ε | is complete, andtherefore no information on any of the parameters can be extracted from the analysis ofDaya Bay data. All the results obtained in this section are summarized in Table 2. Needlessto mention, to obtain all the limits on sin θ presented along this section as well as in thetable, we have marginalized over the NSI couplings over a wide range. Similarly, to placebounds on the NSI parameters, sin θ has also been allowed to float over a wide range. In the previous section, we have not considered any normalization error in the statisticalanalysis of Daya Bay reactor data. This means that we have assumed a perfect knowledge ofthe event normalization at the experiment, disregarding the presence of uncertainties in theflux reactor normalization or in the detection cross section, among others. This procedurehas been followed in most of the previous phenomenological analyses of Daya Bay data inpresence of NSI, see for instance Ref. [47]. In the more recent work at Ref. [48], the authorshave considered small uncertainties in the reactor flux and in the detector properties,although they did not take into account an uncertainty in the overall normalization of theevent rates. Needless to say that a more detailed analysis of reactor data can not ignore thepresence of such normalization errors. Therefore, in this section we present a χ analysis– 20 –f Daya Bay data using the expression defined at Eq. (3.2), where a free normalizationfactor is considered in order to account for the uncertainties in the total event numbernormalization. This point is very relevant in the study of NSI with reactor experiments,since the uncertainty in the event normalization presents a degeneracy with the zero-distance effect due to NSI. In consequence, the far over near technique exploited by DayaBay in order to reduce the dependence upon total normalization does not work equallyfine in the presence of NSI, where the non-oscillatory zero-distance effect, simultaneouslypresent at near and far detectors, does not totally cancel. Actually, in the standard modelcase without NSI, the number of events expected at the near detector is given by: N SMND (cid:39) N (1 + a norm ) P SM ee ( L = 0) = N (1 + a norm ) , (5.1)while, in the presence of NSI, the event number at the near detector is calculated as follows: N NSIND (cid:39) N (1 + a norm ) P NSI ee ( L = 0) = N (1 + a norm )(1 + f ( ε )) (cid:39) N (1 + f ( ε ) + a norm ) . (5.2)Here a norm controls the normalization of far and near detector events in the fit and, togetherwith the NSI couplings ε , it fixes the total zero-distance effect. As a result, if we set a norm to zero, we artificially increase the power of Daya Bay data to constrain the zero-distanceeffect due to NSI, getting non-realistic strong bounds on the NSI couplings. On the otherhand, we can not leave the factor a norm totally free in our statistical analysis, as it is usuallydone in the standard Daya Bay analysis, where the factor is kept small thanks to the farover near technique. Actually, we have found that leaving the normalization factor totallyfree, and due to the degeneracy with the NSI couplings, it could achieve very large values,of the order of 10-20%. For this reason it is necessary the use of a prior on this magnitude.Recent reevaluations of the reactor antineutrino flux indicate an uncertainty on the totalflux of about 3% [77, 83]. However, an independent analysis in Ref. [84] claims that thisuncertainty may have been underestimated due to the treatment of forbidden transitionsin the antineutrino flux evaluation, and proposes a total uncertainty of 4%. Since the totalnormalization errors may also include uncertainties coming from other sources, we followthe conservative approach of taking a total uncertainty on the reactor event normalizationof 5%. This is the value we have assumed for σ a in Eq. (3.2).To illustrate the differences with respect to the results obtained in the previous section,assuming no uncertainties in the event rate normalization, here we have considered onlythe cases where all phases are set to zero . The results obtained with these assumptionsare presented in Fig. 8 and Table 3. In the left panel of Fig. 8 we present the allowed regionin the plane sin θ - | ε e | when only NSI with electron antineutrinos are present. In thiscase, the range for the reactor mixing angle is rather similar to the one shown in the leftpanel of Fig. 5: 0 . ≤ sin θ ≤ .
025 (90% C.L.) , (5.3) Note that, in this case, the correction terms to the effective reactor angle at first order in | ε | dominateand, therefore, second order corrections are not relevant. – 21 –
0 0.01 0.02 0.03 0.04 sin θ | ε e |
0 0.01 0.02 0.03 0.04 sin θ | ε µ , τ |
0 0.01 0.02 0.03 0.04 sin θ | ε | Figure 8 : Allowed regions in the sin θ - NSI coupling plane for the different casesconsidered: | ε e | (left panel), | ε µ,τ | (middle panel) and | ε | (right panel) setting all thephases equal to zero and assuming 5% uncertainty on the total event rate normalization ofDaya Bay events. The conventions for the lines is the same as in Fig. 5.while the bound on | ε e | , however, is much weaker than the one given at Eq. (4.1) (althoughstill slightly better than the one at Eq. (1.3)): | ε e | ≤ . × − (90% C.L.) . (5.4)The same bound is also obtained in the flavor-universal case for the NSI parameter | ε | , seethe right panel of Fig. 8. In this case, the allowed range for the reactor mixing angle is abit enlarged with respect to the previous one:0 . ≤ sin θ ≤ .
024 (90% C.L.) , (5.5)due to the presence of NSI oscillation terms driven by the new physics couplings with muonand tau antineutrinos. As commented above, the loss of sensitivity to the NSI couplings isdue to the degeneracy between the normalization uncertainty and the zero-distance termsinduced by the presence of NSI. In this way, a larger value of the NSI parameters canbe compensated with a non-zero normalization factor a norm , without spoiling the goodagreement with experimental reactor data.Finally, the middle panel of Fig. 8 shows the results obtained when only NSI with muonor tau antineutrinos are considered. In this case, the cancellation between the normalizationterm and the zero-distance effect due to terms of second order in | ε µ,τ | results in an extendedregion in the sin θ - | ε µ,τ | plane with an upper bound of:sin θ ≤ .
024 (90% C.L.) . (5.6)The bound on the NSI coupling is given by: | ε µ,τ | ≤ .
176 (90% C.L.) . (5.7)– 22 –ase sin θ | ε | φ e = 0 0 . ≤ sin θ ≤ . | ε e | ≤ . δ − φ µ,τ ) = 0 sin θ ≤ . | ε µ,τ | ≤ . δ = φ = 0 0 . ≤ sin θ ≤ . | ε | ≤ . Table 3 : 90% C.L. bounds (1 d.o.f) on sin θ and the NSI couplings using current DayaBay data with a 5% uncertainty on the total event normalization. | ε e | sin θ
621 days Daya Bay217 days Daya Bay | ε µ , τ | sin θ
621 days Daya Bay217 days Daya Bay
Figure 9 : Left panel shows the allowed parameter space in sin θ – | ε e | plane at 90% C.L.(2 d.o.f). Right panel depicts the same in sin θ – | ε µ,τ | plane. Here the solid (dashed)lines correspond to the new (old) 621 (217) days of Daya Bay data. Here all the phasesare considered to be zero and the normalization of events is fixed in the statistical analysiswith a norm = 0. The new high-precision data from Daya Bay with 621 days of running time [3] has improvedthe measurement of sin θ dramatically with a relative 1 σ precision of ∼
6% as comparedto ∼
11% obtained using the previously published 217 days of data [2]. This clearly showsthe impact of the four time more statistics that the Daya Bay experiment has accumulatedwith the help of eight ADs in comparison with the previously released Daya Bay data setwith six ADs. Now it would be quite interesting to see how much we can further constrainthe allowed ranges for these NSI parameters under consideration using the new 621 days ofDaya Bay data in comparison with the old 217 days of Daya Bay run. In Fig. 9, we comparethe performance of the current and the previous data sets of Daya Bay in constraining theallowed regions in sin θ – | ε e | plane (left panel) and in sin θ – | ε µ,τ | plane (right panel).In both the panels, the solid (dashed) lines portray the results with the new (old) 621 (217)days of Daya Bay data. For the sake of illustration, we have only chosen the cases of NSIparameters which are associated with ¯ ν e (left panel) and ¯ ν µ (right panel) assuming all thephases to be zero. In particular, we have focused on the situations which are presented in– 23 –aya Bay data sin θ | ε | electron-type NSI parametersCurrent (621 days) 0 . ≤ sin θ ≤ . | ε e | ≤ . . ≤ sin θ ≤ . | ε e | ≤ . . ≤ sin θ ≤ . | ε µ,τ | ≤ . . ≤ sin θ ≤ . | ε µ,τ | ≤ . . ≤ sin θ ≤ . | ε | ≤ . . ≤ sin θ ≤ . | ε | ≤ . Table 4 : 90% C.L. (1 d.o.f) bounds on sin θ and the NSI parameters using the old(new) 217 (621) days of Daya Bay data. Here all the phases are considered to be zero. Wealso do not consider the uncertainty on the normalization of reactor events and set a norm = 0 in the statistical analysis.sections 4.1 and 4.2, where we do not consider the normalization uncertainty on the reactorevents and set a norm = 0 in the statistical analysis. As compared to the old data set, withthe current data, the improvement in constraining the allowed parameter space betweensin θ and the NSI parameters is quite significant as can be readily seen from Fig. 9.In Table 4, we present the 90% C.L. (1 d.o.f) constraints on sin θ and the NSIparameters using the previous (current) 217 (621) days of Daya Bay data. Here all thephases are considered to be zero. We do not consider the normalization uncertainty on thereactor events and take a norm = 0 in the statistical analysis. We can see from Table 4 thatin case of electron-type and universal NSI parameters, the bounds on | ε e | and | ε | are thesame while analyzing 621 days of Daya Bay data. This feature is also there in the caseof 217 days of Daya Bay run. The constraints on | ε e | and | ε | get improved by factor oftwo when we consider the current 621 days of Daya Bay data compared to its previous217 days of data. We also get better bounds on | ε µ,τ | using the current Daya Bay data ascompared to the old data set. These results suggest that the future data from the Daya Bayexperiment with more statistics is going to play an important role to further constrain theallowed parameter space for the NSI parameters. There are also marginal improvementson the bounds to sin θ with the current 621 days of Daya Bay data.Finally in Table 5, we compare the the 90% C.L. (1 d.o.f) limits on sin θ and the NSIparameters obtained using the old 217 days and new 621 days of Daya Bay data allowingthe phases or their certain combinations to vary freely as we consider in Table 2 in Sec. 4.Like in Table 4, here also we do not take into account the normalization uncertainty onthe reactor events and consider a norm = 0 in the statistical analysis. Table 5 depicts thateven if we allow the phases to vary freely, we obtain better limits on the NSI parameterswith the new data set as compared to the previous 217 days of Daya Bay data, exceptfor the case of | ε e | which remains unbounded. Table 5 indicates that the new data set ofDaya Bay also reduces the allowed ranges for sin θ in the presence of NSI parameters as– 24 –aya Bay data sin θ | ε | electron-type NSI parameters [ φ e free]Current (621 days) 0 . ≤ sin θ ≤ . | ε e | unboundPrevious (217 days) 0 . ≤ sin θ ≤ . | ε e | unboundmuon or tau-type NSI parameters [( δ − φ µ,τ ) free]Current (621 days) 0 . ≤ sin θ ≤ . | ε µ,τ | ≤ . . ≤ sin θ ≤ . | ε µ,τ | ≤ . δ free, φ = 0]Current (621 days) 0 . ≤ sin θ ≤ . | ε | ≤ . . ≤ sin θ ≤ . | ε | ≤ . δ = 0, φ free]Current (621 days) sin θ ≤ . | ε | ≤ . θ ≤ . | ε | ≤ . Table 5 : 90% C.L. (1 d.o.f) bounds on sin θ and the NSI parameters using the old (new)217 (621) days of Daya Bay data. Here we allow the phases or their certain combinationsto vary freely. We do not consider the uncertainty on the normalization of reactor eventsand set a norm = 0 in the statistical analysis.compared to the old data. This improvement is quite significant in the case of | ε µ,τ | evenif the effective phase ( δ − φ µ,τ ) is allowed to vary freely in the fit. The success of the currently running Daya Bay, RENO, and Double Chooz reactor an-tineutrino experiments in measuring the smallest lepton mixing angle θ with impressiveaccuracy signifies an important advancement in the field of modern neutrino physics withnonzero mass and three-flavor mixing. With this remarkable discovery, the neutrino oscil-lation physics has entered into a high-precision era opening up the possibility of observingsub-dominant effects due to possible new physics beyond the Standard Model of particlephysics. At present, undoubtedly the Daya Bay experiment in China is playing a leadingand an important role in this direction. The recent high-precision and unprecedentedlycopious data from the Daya Bay experiment has provided us an opportunity to probe theexistence of the non-standard interaction effects which might crop up at the productionpoint or at the detection stage of the reactor antineutrinos.In this paper for the first time, we have reported the new constraints on the flavor non-universal and also flavor universal NSI parameters obtained using the currently released621 days of Daya Bay data. While placing the bounds on these NSI parameters, we haveassumed that the new physics effects are just inverse of each other in the production anddetection processes of the reactor antineutrino experiment , i.e. , ε seγ = ε d ∗ γe . Considering thisspecial case, we have discussed in detail the impact of the NSI parameters on the effectiveantineutrino survival probability expressions which we ultimately use to analyze the DayaBay data. With this special choice of the NSI parameters, we have observed a shift in– 25 –he oscillation amplitude without altering the L/E pattern of the oscillation probability.This shift in the depth of the oscillation dip can be caused due to the NSI parameters andas well as θ , making it quite difficult to disentangle the NSI effects from the standardoscillations. Before presenting the final results, we have studied the correlations betweenthe NSI parameters and θ with the help of iso-probability surface plots in Sec. 2. Thisstudy has been quite useful to understand the final bounds on the NSI parameters that wehave obtained from the fit. Since the shape of the oscillation probability is not distortedwith the special choice of the NSI parameters considered in this paper, an analysis basedon the total event rate at the Daya Bay experiment is sufficient to obtain the limits on theNSI parameters.As far as the flavor non-universal NSI parameters are concerned, first we have consid-ered the NSI parameters | ε e | and φ e which are associated with ¯ ν e . Assuming φ e = 0 ◦ anda perfect knowledge of the normalization of the event rates, the current Daya Bay dataplaces a strong constrain on | ε e | ≤ . × − (90% C.L.) improving the present bound on | ε e | by one order of magnitude. Now if we consider the uncertainty in the normalization ofevent rates with a prior of 5%, then this limit changes to 15 × − , diluting the constraintby almost one order of magnitude. No limits can be placed on | ε e | if we allow φ e to varyfreely in the fit. We have also observed that the determination of the 1-3 mixing angle isquite robust in this specific case and it is almost independent of the issue of uncertainty inthe normalization of event rates and the choice of φ e . In fact, the allowed range of sin θ coincides exactly with the allowed range in absence of NSI. Next we turn our attention tothe NSI parameters | ε µ | and φ µ which are associated with ¯ ν µ . With ( δ − φ µ ) = 0 ◦ and aperfect knowledge of the event rates normalization, the current Daya Bay data sets a limitof | ε µ | ≤ . × − (90% C.L.) which is comparable and complementary to the existingbound obtained using a different data set under different assumptions. This limit on | ε µ | becomes 17 . × − once we consider the uncertainty in the normalization of event rateswith a prior of 5%. These limits on | ε µ | remain almost unchanged even if we allow ( δ − φ µ )to vary freely in the fit. This is not true while placing the constraint on sin θ . For anexample, in the case when we do not consider any uncertainty in the normalization of eventrates, the upper bound on sin θ increases from 2 . × − to 3 . × − due to freelyvarying ( δ − φ µ ) in the fit instead of setting it to zero. We cannot place a lower bound onsin θ when we consider the uncertainty in the normalization of event rates with a priorof 5% even if ( δ − φ µ ) is taken to be zero in the fit. The above mentioned limits on | ε µ | andsin θ are also valid for the NSI parameters | ε τ | and φ τ as long as the 2-3 mixing angle ismaximal , i.e. , sin θ = 0 . universal NSI parameters, we have placed limits on | ε | and sin θ under certain assumptions on δ and φ . With δ = φ = 0 ◦ and perfect knowledge ofthe normalization of the event rates, the upper bound on | ε | at 90% C.L. (1 d.o.f) is1 . × − . Though this limit does not change much when we marginalize over δ in the fit,it deteriorates by almost ninety times when we allow φ to vary freely in the fit providing alimit of | ε | ≤ × − (90% C.L.). With a 5% uncertainty in the normalization of totalevent rates and δ = φ = 0 ◦ , the constraint on | ε | becomes ≤ × − at 90% C.L. Withthe same assumptions, we can restrict θ within a range of 0 . ≤ sin θ ≤ .
024 (90%– 26 –.L.).One of the novelties of this work is the inclusion of correction terms of second order inthe NSI couplings | ε | in the effective neutrino probability in Eq. (2.17). The role of thesesecond order corrections at the effective probability level has been analyzed in Sec. 2.2 aswell as in the discussion of the probability and correlation plots in Sec. 2.3 and Sec. 2.4.The impact of second order terms on the determination of sin θ is specially relevant forthe flavour-universal-NSI case, as commented in Sec. 4.3.One of the interesting studies that we have performed in this paper is the comparisonof the constraints on the NSI parameters placed with the current 621 days of Daya Baydata with the limits obtained using the previously released 217 days of Daya Bay run. Inthis analysis, all the phases are considered to be zero and the normalization of events is alsokept fixed in the statistical analysis with a norm = 0. We have observed that the constraintson | ε e | and | ε | get improved by factor of two when we analyze the current 621 days ofDaya Bay data compared to its previous 217 days data. This comparative study revealsthe merit of the huge statistics that Daya Bay has already accumulated. It also suggeststhat the future high-precision data from the Daya Bay experiment with enhanced statisticsis inevitable to further probe the sub-leading effects in neutrino flavor conversion due tothe presence of the possible NSI parameters beyond the standard three-flavor oscillationparadigm. Acknowledgments
S.K.A. acknowledges the support from DST/INSPIRE Research Grant [IFA-PH-12], De-partment of Science and Technology, India. The work of D.V.F. and M.T. was supportedby the Spanish grants FPA2011-22975 and MULTIDARK CSD2009-00064 (MINECO) andPROMETEOII/2014/084 (Generalitat Valenciana). This work has also been supported bythe U.S. Department of Energy under award number DE-SC0003915.
A Effective Survival Probability with the NSI parameters: ε seγ (cid:54) = ε d ∗ γe In this appendix, we present the effective probability expressions for the physical scenarioswhere ε seγ (cid:54) = ε d ∗ γe . In such cases, the spectral study of the reactor data plays an importantrole since the NSI parameters are not only responsible for a shift in θ i.e. the changeof the depth of the first oscillation maximum but they also modify the L/E pattern ofthe probability due to the shift in its energy. We have already mentioned that a detailedanalysis of the Daya Bay data considering such interesting physical cases will be performedin [69].
A.1 Presence of the NSI parameters only at the production stage
Here we assume that the NSI parameters only affect the production mechanism of theantineutrinos in the reactor experiment. It allows us to write (dropping the universale-index): ε sγ = | ε γ | e i φ γ , and ε dγ = 0 . (A.1)– 27 –ith this assumption, we get the effective neutrino survival probability as follows: P ¯ ν se → ¯ ν de (cid:39) P SM¯ ν se → ¯ ν de + | ε e | + 2 | ε e | cos φ e + 2 s [ s | ε µ | sin( δ − φ µ ) + c | ε τ | sin( δ − φ τ )] sin (2∆ ) − s [ s | ε µ | cos( δ − φ µ ) + c | ε τ | cos( δ − φ τ )] sin (∆ )+ sin 2 θ [ − c | ε µ | sin φ µ + s | ε τ | sin φ τ ] sin (2∆ ) . (A.2) A.2 NSI at the source and detector with the same magnitude and differentphases
In this case, we assume that the magnitude of the NSI parameters is the same at theproduction and detection level, but the phases associated with the NSI parameters aredifferent at the source and detector. Under this situation, we can write: ε sγ = | ε γ | e i φ sγ , and ε dγ = | ε γ | e i φ dγ . (A.3)Under this assumption, the effective neutrino survival probability takes the form: P ¯ ν se → ¯ ν de (cid:39) P SM¯ ν e → ¯ ν e + P NSI-IIbnon-osc + P NSI-IIbosc-atm + P NSI-IIbosc-solar , (A.4)where the non-standard terms are given by: P NSI-IIbnon-osc = 2 (cid:110) | ε e | (cid:16) cos φ de + cos φ se (cid:17) + | ε e | (cid:104) φ de − φ se ) + cos( φ de + φ se ) (cid:105) + | ε µ | cos( φ sµ + φ dµ ) + | ε τ | cos( φ sτ + φ dτ ) (cid:111) , (A.5) P NSI-IIbosc-atm = 2 (cid:110) s s | ε µ | (cid:104) sin( δ − φ sµ ) − sin( δ + φ dµ ) (cid:105) + s c | ε τ | (cid:104) sin( δ − φ sτ ) − sin( δ + φ dτ ) (cid:105) − s | ε µ | sin( φ sµ + φ dµ ) − c | ε τ | sin( φ sτ + φ dτ ) − c s | ε µ || ε τ | (cid:104) sin( φ sτ + φ dµ ) + sin( φ sµ + φ dτ ) (cid:105)(cid:111) sin (2∆ ) − (cid:110) s s | ε µ | (cid:104) cos( δ − φ sµ ) + cos( δ + φ dµ ) (cid:105) + c s | ε τ | (cid:104) cos( δ − φ sτ ) + cos( δ + φ dτ ) (cid:105) + s | ε µ | cos( φ sµ + φ dµ ) + c | ε τ | cos( φ sτ + φ dτ )+ c s | ε µ || ε τ | (cid:104) cos( φ sτ + φ dµ ) + cos( φ sµ + φ dτ ) (cid:105)(cid:111) sin (∆ ) , (A.6) P NSI-IIbosc-solar = sin 2 θ (cid:104) − c | ε µ | (sin φ sµ + sin φ dµ ) + s | ε τ | (sin φ sτ + sin φ dτ ) (cid:105) sin (2∆ ) . (A.7) References [1]
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