Shifts of neutrino oscillation parameters in reactor antineutrino experiments with non-standard interactions
SShifts of neutrino oscillation parameters in reactor antineutrinoexperiments with non-standard interactions
Yu-Feng Li ∗ Ye-Ling Zhou † Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918, Beijing 100049, China
Abstract
We discuss reactor antineutrino oscillations with non-standard interactions (NSIs) at the neu-trino production and detection processes. The neutrino oscillation probability is calculated with aparametrization of the NSI parameters by splitting them into the averages and differences of theproduction and detection processes respectively. The average parts induce constant shifts of theneutrino mixing angles from their true values, and the difference parts can generate the energy (andbaseline) dependent corrections to the initial mass-squared differences. We stress that only the shiftsof mass-squared differences are measurable in reactor antineutrino experiments. Taking JiangmenUnderground Neutrino Observatory (JUNO) as an example, we analyze how NSIs influence the stan-dard neutrino measurements and to what extent we can constrain the NSI parameters.
PACS number(s): 14.60.Pq, 13.10.+q, 25.30.PtKeywords: non-standard interaction, reactor antineutrino, JUNO ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ h e p - ph ] M a r Introduction
After the observation of non-zero θ from recent reactor [1, 2, 3, 4, 5] and accelerator [6, 7] neutrinoexperiments, we have established a standard picture of three active neutrino oscillations with threemixing angles and two independent mass-squared differences [8]. Therefore the remaining neutrinomass ordering and CP-violating phase, which manifest themselves as the generic properties of threeneutrino oscillations, constitute the main focus of future neutrino oscillation experiments. On the otherhand, the probe of new physics beyond the Standard Model (SM) is another motivation for futureprecision oscillation measurements.Experiments using reactor antineutrinos have played important roles in the history of neutrinophysics, which can be traced back to the discovery of neutrinos [9], to establishment [10] of the LargeMixing Angle (LMA) Mikheyev-Smirnov-Wolfenstein (MSW) solution of the long-standing solar neu-trino problem, and more recently to the discovery of non-zero θ [1, 2, 4, 5]. Moreover, future reactorexperiments would keep their competitive roles in the determination of the neutrino mass hierarchy,precision measurement of oscillation parameters, and search for additional neutrino types and interac-tions. The survival probability for the reactor antineutrino ν e → ν e oscillation in the three neutrinoframework can be written as P ee = 1 − c sin θ sin ∆ − c sin θ sin ∆ − s sin θ sin ∆ , (1)with c ij = cos θ ij , s ij = sin θ ij , and ∆ ji = ∆ m ji L/ (4 E ) where L is the baseline distance between thesource and detector, E is the antineutrino energy, and ∆ m ji = m j − m i is the mass-squared differencebetween the i th and j th mass eigenstates. Because there is a large hierarchy between different mass-squared differences, 30∆ m ∼ | ∆ m | ∼ | ∆ m | , (2)different reactor antineutrino experiments may measure different oscillation terms of ∆ or (∆ , ∆ ),which can be categorized into three different groups: • Long baseline reactor antineutrino experiments, such as KamLAND [10, 11]. The aim of theseexperiments is to observe the slow oscillation with ∆ and measure the corresponding oscillationparameters ∆ m and θ . • Short baseline reactor antineutrino experiments, such as Daya Bay [1, 2, 3], Double CHOOZ [4],RENO [5]. They are designed to observe the fast oscillation with ∆ and ∆ (or equivalently,∆ ee [3]) and measure the corresponding oscillation parameters ∆ m ee , θ . • Medium baseline reactor antineutrino experiments. They stand for the next generation experi-ments of reactor antineutrinos, with typical representatives of Jiangmen Underground NeutrinoObservatory (JUNO) [12] and RENO-50 [13]. They can determine the neutrino mass ordering( m < m < m or m < m < m ). In addition, they are expected to provide the precisemeasurement for both the fast and slow oscillations and become a bridge between short baselineand long baseline reactor antineutrino experiments.High-dimensional operators originating from new physics can contribute to the neutrino oscillationin the form of non-standard interactions (NSIs) [14, 15]. They induce effective four-fermion interactions2fter integrating out some heavy particles beyond the SM, where the heavy particles can be scalars,pseudo-scalars, vectors, axial-vectors, or tensors [16]. For reactor antineutrino experiments NSIs mayappear in the antineutrino production and detection processes, and can modify the neutrino oscillationprobability. Therefore, the neutrino mixing angles and mass-squared differences can be shifted andthe mass ordering (MO) measurement will be affected. There are some previous discussions on NSIsin reactor antineutrino experiments [17, 18, 19] and other types of oscillation experiments [20]. Inthis work, we study the NSI effect in reactor antineutrino oscillations in both specific models and alsothe most general case. Taking JUNO as an example, we apply our general framework to the mediumbaseline reactor antineutrino experiment. We discuss how NSIs influence the standard 3-generationneutrino oscillation measurements and to what extent we can constrain the NSI parameters.The remaining part of this work is organized as follows. Section 2 is to derive the analytical for-malism. We develop a general framework on the NSI effect in reaction antineutrino oscillations, andcalculate the neutrino survival probability in the presence of NSIs. In section 3, we give the numeri-cal analysis for the JUNO experiment. We analyze the NSI impacts on the precision measurement ofmass-squared differences and the determination of the neutrino mass ordering, and present the JUNOsensitivity of the relevant NSI parameters. Finally, we conclude in section 4. NSIs may occur in the neutrino production, detection and propagation processes in neutrino oscillationexperiments. The neutrino and antineutrino states produced in the source and observed in the detectorare superpositions of flavor states, | ν s α (cid:105) = 1 N s α (cid:16) | ν α (cid:105) + (cid:88) β (cid:15) s αβ | ν β (cid:105) (cid:17) , | ν s α (cid:105) = 1 N s α (cid:16) | ν α (cid:105) + (cid:88) β (cid:15) s ∗ αβ | ν β (cid:105) (cid:17) , (cid:104) ν d β | = 1 N d β (cid:16) (cid:104) ν β | + (cid:88) α (cid:15) d αβ (cid:104) ν α | (cid:17) , (cid:104) ν d β | = 1 N d β (cid:16) (cid:104) ν β | + (cid:88) α (cid:15) d ∗ αβ (cid:104) ν α | (cid:17) , (3)in which the superscripts ‘s’ and ‘d’ denote the source and detector, respectively, and N s α = (cid:113)(cid:80) β | δ αβ + (cid:15) s αβ | , N d β = (cid:113)(cid:80) α | δ αβ + (cid:15) d αβ | (4)are normalization factors.In general, NSIs in different physical processes may have distinct contributions. For a certain typeof neutrino experiments, the same set of effective NSI parameters can be introduced to describe the NSIeffect. But when one turns to anther type of neutrino experiments, neutrinos can have totally differentorigins and one should use another set of NSI effective parameters to parametrize the NSI effect. Theparameters (cid:15) s , d αβ used here are strongly experiment-dependent, and in principle also energy-dependent.However, they are usually considered as the averaged effects and treated as constant values.In order to measure the average and difference between neutrino production and detection processes,we introduce two sets of NSI parameters as˜ (cid:15) αβ = ( (cid:15) s αβ + (cid:15) d ∗ βα ) / , δ(cid:15) αβ = ( (cid:15) s αβ − (cid:15) d ∗ βα ) / , (5)to rewrite the NSI effect. One should note that the NSI parameters δ(cid:15) αβ are negligibly small comparedwith ˜ (cid:15) αβ when neutrinos are purely left-handed particles [16, 20]. However, if neutrinos are right-handed3articles and the four-fermion interactions are mediated by heavy scalars beyond SM, δ(cid:15) eα can reachthe percent level [16].The effective Hamiltonian that describes the vacuum neutrino oscillation is given by H = 12 E (cid:2) U ∗ diag( m , m , m ) U T (cid:3) , (6)where U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [21] and can be expressed in theform [22] U = P † l c c s c s − s c e − iδ − c s s c c e − iδ − s s s s c s s e − iδ − c c s − c s e − iδ − s c s c c P ν , (7)where s ij = sin θ ij , P l = diag { e iφ , e iφ , e iφ } , and P ν = diag { , e iρ , e iσ } . P l is unphysical since chargedleptons are Dirac particles. On the other hand, P ν can not be neglected for Majorana neutrinos butdoes not contribute to neutrino oscillations. We write the PMNS matrix in the form as in Eq. (7) tokeep the first row and the third column of P l U P † ν real. The mixing angles θ , θ , θ and CP phase δ take the same values as those in the PDG parametrization [8], respectively.In the presence of NSI effects at the source and detector, the amplitude of the ν α → ν α transition is˜ A αα = (cid:104) ν d α | e − iHL | ν s α (cid:105) = (cid:88) βγ (cid:104) ν d α | ν γ (cid:105)(cid:104) ν γ | e − iHL | ν β (cid:105)(cid:104) ν β | ν s α (cid:105) = ( δ γα + (cid:15) d ∗ γα ) A βγ ( δ αβ + (cid:15) s ∗ αβ ) , (8)where L is the baseline, and A βγ = (cid:104) ν γ | e − iHL | ν β (cid:105) = (cid:88) i U ∗ γi U βi exp (cid:18) − i m i L E (cid:19) (9)is the amplitude of ν β → ν γ without NSIs. It is useful to define˜ U αi = 1˜ N α (cid:88) β ( δ αβ + ˜ (cid:15) ∗ αβ ) U βi , δU αi = 1˜ N α (cid:88) β δ(cid:15) ∗ αβ U βi , (10)where ˜ N α = (cid:113)(cid:80) β | δ αβ + ˜ (cid:15) αβ | = (cid:112) N s α N d α + O ( δ(cid:15) ) , (11)and (cid:80) i | ˜ U αi | = 1 is required. Thus we can obtain ˜ A αα as˜ A αα = (cid:88) i ( ˜ U − δU ) ∗ αi ( ˜ U + δU ) αi exp (cid:18) − i m i L E (cid:19) + O ( δ(cid:15) ) , (12)and the survival probability for ν α → ν α is expressed as˜ P αα = | ˜ A αα | = 1 − (cid:88) i 0, we can restore the original PMNS matrix ofthree active neutrinos as˜ θ → θ , ˜ θ → θ , ˜ θ , ˜ θ (cid:48) → θ , ˜ δ, ˜ δ (cid:48) → δ, ˜ α µ , ˜ α µ , ˜ α τ , ˜ α τ → . (20)One should notice that although the effective PMNS matrix takes a simple form as in Eq. (19), thecorresponding parameters are in general dependent on the type of experiments. We should use different˜ U to characterize different realizations of the effective PMNS matrix for reactor and accelerator neutrinoexperiments. This is different from the non-unitary effect of the PMNS matrix, where ˜ (cid:15) is used toparametrize the universal mixing between the active and sterile neutrinos and δ(cid:15) = 0 by definition. Inthis case, ˜ U is an effective PMNS matrix for all neutrino oscillation experiments. In reactor antineutrino oscillations, only the electron antineutrino survival probability is relevant becauseof the high threshold of the µ / τ production. With the parametrization of ˜ U in Eq. (19), we can rewrite P ee with these effective mixing parameters as˜ P ee = 1 − ˜ c sin θ [sin ∆ + ( δε − δε ) sin 2∆ ] − ˜ c sin θ [sin ∆ + ( δε − δε ) sin 2∆ ] − ˜ s sin θ [sin ∆ + ( δε − δε ) sin 2∆ ] + O ( δ(cid:15) )= 1 − ˜ c sin θ sin ˜∆ − ˜ c sin θ sin ˜∆ − ˜ s sin θ sin ˜∆ + O ( δε ) (21)with ˜∆ ji = ∆ ji + δε i − δε j ,δε = Im( δU e )˜ c ˜ c , δε = Im( δU e )˜ s ˜ c , δε = Im( δU e )˜ s , (22)6here the superscript α = e in ˜∆ eji has been ignored.The average part ˜ (cid:15) can be treated as constant shifts to mixing angles θ and θ , and the differencepart δ(cid:15) leads to energy- and baseline-dependent shifts to the mass-squared differences ∆ m ji as∆ ˜ m ji ( E/L ) = ∆ m ji + ( δε i − δε j )4 E/L . (23)However, only two combinations of the three parameters δε i contribute to the oscillation probabilitythanks to the relation ( δε − δε ) = ( δε − δε ) − ( δε − δε ). It is notable that one cannot distinguishthe effect of mixing angle shifts from the scenario of three neutrino mixing using reactor antineutrinooscillations. This degeneracy can only be resolved by including different types of neutrino oscillationexperiments, where the NSI parameters and their roles in neutrino oscillations are totally distinct. Onthe other hand, the shifts of mass-squared differences are clearly observable due to the baseline- andenergy-dependent corrections in the reactor antineutrino spectrum.Different kinds of reactor antineutrino oscillation experiments have their own advantages in mea-suring the NSI parameters. The long baseline reactor antineutrino experiment (e.g., KamLAND) canmeasure the slow oscillation term ∆ and thus are sensitive to the measurement of δε − δε . Since thefast oscillation terms ∆ and ∆ are averaged out, the oscillation probability ˜ P ee is reduced to˜ P ee = ˜ s + ˜ c (cid:8) − sin θ (cid:2) sin ∆ + ( δε − δε ) sin 2∆ (cid:3)(cid:9) . (24)The short baseline reactor antineutrino experiments (e.g., Daya Bay) are designed to measure the fastoscillation terms ∆ and ∆ (or equivalently, ∆ ee ), and are effective to constrain δε − δε . Since theslow oscillation term ∆ is negligible, the oscillation probability ˜ P ee can be simplified as˜ P ee = 1 − sin θ sin ∆ ee − η (cid:104) ˜ c ( δε − δε ) + ˜ s ( δε − δε ) (cid:105) sin θ sin 2∆ ee . (25)For the reactor antineutrino oscillation at the medium baseline (e.g., 52 . P ee = 1 − 12 sin θ − ˜ c sin θ [sin ∆ + ( δε − δε ) sin 2∆ ]+ 12 sin θ ( C cos 2∆ ee − ηS sin 2∆ ee ) , (26)where ∆ ee = ∆ m ee L E , ∆ m ee = ˜ c | ∆ m | + ˜ s | ∆ m | , (27)and C = ˜ c cos(2˜ s ∆ ) + ˜ s cos(2˜ c ∆ ) − δε − δε )˜ c sin(2˜ s ∆ ) + 2( δε − δε )˜ s sin(2˜ c ∆ ) ,S = ˜ c sin(2˜ s ∆ ) − ˜ s sin(2˜ c ∆ )+ 2( δε − δε )˜ c cos(2˜ s ∆ ) + 2( δε − δε )˜ s cos(2˜ c ∆ ) , (28)with η = ± δε − δε and δε − δε , S varies with the NSI parameters, which can further alter the7ifference of oscillation probabilities between NMO and IMO. Since NSIs are constrained to the percentlevel [24], we expect that NSIs will not significantly affect the mass ordering measurement.Finally, one should keep in mind that there is an additional correction from terrestrial matter effectsduring the neutrino propagation inside the Earth. In addition to the Hamiltonian in Eq. (6), there is amatter potential term written as H mat = 12 E diag( − A CC , , , (29)where − A CC characterizes the contribution of charged-current interactions between antineutrinos andelectrons in matter and A CC = 2 √ G F N e E , with N e being the electron number density. For reactorantineutrino experiments, A CC is sufficiently small compared with the kinetic term ∆ m ji where thematter corrections to oscillation parameters of the solar sector are given bysin θ M12 (cid:39) sin θ (1 − A CC ∆ m cos 2 θ ) , ∆ m (cid:39) ∆ m (1 + A CC ∆ m cos 2 θ ) , (30)and the magnitude of these corrections is estimated as A CC ∆ m cos 2 θ (cid:39) . × E × ρ / cm , (31)with ρ being the matter density along the antineutrino trajectory in the Earth. In comparison, thecorrection to parameters of the atmospheric sector is only the order of 10 − . In our following stud-ies, matter effects during the neutrino propagation will be neglected in the analytical calculation buteffectively included in our numerical analysis.The N e E suppression leads to negligible NSI effects in the propagation process. The scenario is verydifferent from other types of neutrino oscillation experiments, where NSIs during propagation lead tomuch larger corrections to oscillation probabilities than those at the source and detector. In reactorantineutrino oscillations NSIs during propagation can be safely neglected. In this section, following the JUNO nominal setup in Ref. [12], we will numerically show the NSI effectin the medium baseline reactor antineutrino experiment. We will illustrate how the NSI effect shifts themass-squared differences, how it influences the mass ordering measurement and to what extent we canconstrain the NSI parameters. We employ the true power and baseline distribution in Table 1 of Ref. [12]. The weighted average ofthe baseline is around 52.5 km with baseline differences less than 500 m. We use the nominal runningtime of 1800 effective days for six years, and detector energy resolution 3% / (cid:112) E (MeV) as a benchmark.A NMO is assumed to be true otherwise mentioned explicitly. The relevant oscillation parameters are˜ θ , ˜ θ , ∆ m and ∆ m ee and the NSI parameters are δε − δε , δε − δε .8e directly employ the mixing angles measured in recent reactor antineutrino experiments as oureffective mixing angles, which can be shown assin θ = sin θ D13 = 0 . , tan ˜ θ = tan θ K12 = 0 . . (32)The measured mixing angles θ D13 and θ K12 are from Daya Bay [25] and KamLAND [26], respectively. Forsome recent discussions on how the true mixing angles are modified by the average of the NSI effectin a simplified version, see [18, 27]. However, the measured mass-squared differences ∆ m ee from DayaBay [25] and ∆ m from KamLAND [26] can be viewed as the true parameters rather than effectiveoscillation parameters, i.e.,∆ m ee = ∆ m ee = 2 . × − eV , ∆ m = ∆ m = 7 . × − eV . (33)The reason is that although the mass-squared differences have energy/baseline-dependent corrections asshown in Eq. (23), these shifts are sufficiently small compared with the current level of uncertainties. Indetails, δε i may be in the percent level and the ratio E/L for Daya Bay is around ( E/L ) D (cid:39) (cid:39) × − eV , and for KamLAND is around ( E/L ) K (cid:39) . 02 MeV/km (cid:39) × − eV . Therefore theshifts are still below the sensitivity of Daya Bay [25] and KamLAND [26], and the measured parameterscan be approximate to the true values for ∆ m ee and ∆ m , respectively.In our following numerical analysis, two different treatments for the NSI parameters will be explored. • The first treatment is a class of specific models with democratic entries for δ(cid:15) αβ , and for simplicitywe assume that these models have identical magnitude in Im( δU ei ) but different relative signs.The configurations of { Im( δU e ) , Im( δU e ) , Im( δU e ) } for these specific models are defined asS1 : ( δU, + δU, + δU ) , S2 : ( δU, + δU, − δU ) , S3 : ( δU, − δU, + δU ) , S4 : ( δU, − δU, − δU ) , (34)respectively. Therefore from Eq. (22), δε can be roughly larger than δε due to θ < ◦ andboth of them are the same order of δU . δε can be several times larger due to the smallness of θ . • The second one is the general treatment for the NSI parameters. As shown in Eq. (21), onlytwo combinations of the NSI parameters δε i are independent, thus we can treat ( δε − δε ) and( δε − δε ) as free parameters to cover the full parameter space. Note that the non-unitary effectis identical with NSIs for the case of δ(cid:15) αβ = 0. If we can observe the splittings of ˜∆ αji comparedwith ∆ ji , we may distinguish NSIs from the non-unitary effect. To calculate the expected reactor ν e spectrum in the presence of NSIs, we need first deal with thestandard case without the NSI effect. The energy spectrum of detected events S ( E vis ), as a function ofthe visible energy E vis of the inverse β -decay ν e + p → e + + n (IBD), is parametrized as S ( E vis ) = (cid:90) ∞ m e dE e (cid:34)(cid:90) ∞ E T dE (cid:16) (cid:88) i N i Φ i ( E ) P ee ( E/L i ) (cid:17) dσ ( E, E e ) dE e (cid:35) r ( E e + m e , E vis ) , (35)where Φ i ( E ) is the antineutrino flux with i standing for different reactor cores and E the antineutrinoenergy, N i is the corresponding normalization and conversion factor, P ee ( E/L i ) is the oscillation prob-ability of ν e → ν e with different baseline L i from the ν e source i to the detector, dσ ( E, E e ) /dE e is the9BD differential cross section with E e being the true positron energy, r ( E e + m e , E vis ) is the Gaussianenergy resolution function with a standard deviation σ E defined as σ E E e + m e = 3% (cid:112) ( E e + m e ) / MeV . (36)In the presence of NSIs, we can use the following replacements to show the NSI effect during theantineutrino oscillation, production and detection processes, respectively P ee → ˜ P ee , Φ i → ˜Φ i = ( N s e ) Φ i , σ → ˜ σ = ( N d e ) σ. (37)Therefore, we can obtain the NSI-modified reactor ν e spectrum as˜ S ( E vis ) = ( ˜ N e ) (cid:90) ∞ m e dE e (cid:34)(cid:90) ∞ E T dE (cid:16) (cid:88) i N i Φ i ( E ) ˜ P ee ( E/L i ) (cid:17) dσ ( E, E e ) dE e (cid:35) r ( E e + m e , E vis ) + O ( δ(cid:15) ) , (38)where N s e , N d e and ˜ N e are defined in Eqs. (4) and (11), and can be absorbed by redefining the couplingsof nuclear matrix elements in the reactor ν e production and detection processes. From Eq. (11), ˜ N e is related to the average parts of NSIs, and contributes to the normalization factor of the reactorantineutrino flux. In this work we take ˜ N e = 1 . ν e spectra at a baseline of 52.5 km in Fig. 1, where theinfluences of δε − δε and δε − δε are presented in the upper panel and lower panel, respectively.The oscillation parameters are taken as in Eqs. (32) and (33). The scenario of 3-generation neutrinooscillations with δU = 0 is also shown for comparison. In the upper panel, we fix δε − δε = 0 andfind that non-zero δε − δε introduces the spectral distortion to the slow oscillation term ∆ . For δε − δε = 0 . 02, the spectrum is suppressed in the high energy region with E > < E < δε − δε gives the oppositeeffect on the spectrum distortion. In the lower panel, we set δε − δε = 0 and observe that δε − δε canaffect the spectral distribution for the fast oscillation term ∆ . Non-trivial NSI effect will contribute asmall phase advancement or retardance to the fast oscillation depending upon the sign of δε − δε . In this part, we shall implement the standard χ statistical method to do the numerical analysis withthe above setup. A general χ function using the spectrum calculated in Eq. (38) can be defined as χ = N bin (cid:88) i =1 (cid:2) ˜ M i ( p M , δε M ) − ˜ T i ( p T , δε T )(1 + (cid:80) k α ik (cid:15) k ) (cid:3) ˜ M i ( p M , δε M ) + (cid:88) k (cid:15) k σ k , (39)where ˜ M i and ˜ T i are the measured and predicted (with oscillation) reactor ν e fluxes in the i -th energybin respectively. The definition of bin sizes is identical to that assumed in Ref. [12]. The systematicuncertainties σ k together with the corresponding pull parameters (cid:15) k for the nominal setups are alsothe same as those in Ref. [12], which include the correlated (absolute) reactor uncertainty (2%), theuncorrelated (relative) reactor uncertainty (0 . p and δε are for the standard oscillation parameters and NSIparameters respectively with p = { ˜ θ , ˜ θ , ∆ m , ∆ m ee } and, δε = { δU/ ˜ c ˜ c , ± δU/ ˜ s ˜ c , ± δU/ ˜ s } for specified models, or δε = { δε − δε , δε − δε } for the general model defined in section 3.1.10igure 1: The effect of NSIs in reactor ν e spectra at a baseline of 52.5 km. For visualization, we set δε − δε = 0 , ± . 02 in the upper panel and δε − δε = 0 , ± . δε − δε is fixedat zero in the upper panel, and δε − δε is fixed at zero in the lower panel. The oscillation parametersare taken as in Eqs. (32) and (33). The NMO is assumed for illustration. Neglecting the existing non-zero NSIs, we may get biased best-fit oscillation parameters. In this partwe shall evaluate the sizes and properties on the shifts of mass-squared differences due to the NSI effect.In the numerical simulation, we use the spectrum with non-zero NSIs as the true spectrum, and thespectrum of the standard neutrino oscillation without NSIs as the predicted spectrum. In other word,11igure 2: The NSI-induced shifts for ∆ m and ∆ m in the four specific models defined in Eq. (34)with δU = 0 . 01. The red stars stand for the true values of ∆ m and ∆ m , and contours are the 68.3%(1 σ ), 95.5% (2 σ ), 99.7% (3 σ ) allowed regions for ∆ m and ∆ m when the NSI effect is neglected. TheNMO is assumed for illustration.the true spectrum is defined in Eq. (38) with the oscillation probability given in Eq. (21), and thepredicted spectrum is given in Eq. (35) with the oscillation probability in Eq. (1). Then we minimizethe χ function and find out the best-fit mixing angles and mass-squared differences.The effects of mass shifts are shown in Figs. 2 and 3, corresponding to the first and second treatments,respectively, where NMO has been assumed. In the first treatment, the mass-squared differences havedifferent shift sizes and directions in each specific models, dependent upon the sign of δU . The best-fitvalue of ∆ m decreases from its true value in the models of S1, S2 or increases for S3, S4, and thebest-fit value of ∆ m decreases in S1, S3 or increases in S2, S4. A simple explanation can be found inthe following estimation. The NSI parameters in these models are numerically given byS1 : δε − δε = − . δU, δε − δε = − . δU, S2 : δε − δε = − . δU, δε − δε = +8 . δU, S3 : δε − δε = +3 . δU, δε − δε = − . δU, S4 : δε − δε = +3 . δU, δε − δε = +8 . δU, (40)12igure 3: The best-fit (b.f.) mass-squared differences for ∆ m (left panel) and ∆ m (right panel) asthe functions of the true values of δε − δε or δε − δε respectively in the generic treatment of the NSIparameters. The best-fit values of ∆ m and ∆ m are obtained by the minimization of the χ functionwithout the NSI effect. The horizontal dashed lines are for true values of the mass-squared differences,and the NMO is assumed for illustration.where δU is fixed at 0.01 in Fig. 2. The sign of δε − δε is “ − ” in S1, S2 and “+” in S3, S4, whichreduces or increases the measured value of ∆ ˜ m in the LHS of Eq. (23), respectively. Similar analysisis also valid for explaining the shift of ∆ m as shown in Fig. 2. Moreover, due to the smallness of thecoefficients of δε − δε in models S1, S2, the shift of ∆ m in S1, S2 is much smaller than that in S3, S4.The relative mass shift for ∆ m is around 0.4% in S1, S2 and 2% in S3, S4. Although the magnitudeof δε − δε is in general larger than δε − δε , the absolute value of ∆ m is much larger than ∆ m ,and thus the relative shift of ∆ m is not significant, just roughly around 0.2% for the best-fit data infour models.The effects of NSI-induced mass shifts in the general case are presented in Fig. 3, without anyassumptions on the relation of NSI parameters. Our simulation results can be understood using therelation in Eq. (23) that the fitted mass-squared differences ∆ m and ∆ m in JUNO are linearlydependent upon δε − δε and δε − δε , respectively. With the JUNO nominal setup, we can simplifyEq. (23) into the following formulae∆ m = ∆ ˜ m (( E/L ) J ) = ∆ m + ( δε − δε )4( E/L ) J ∆ m = ∆ ˜ m (( E/L ) J ) = ∆ m + ( δε − δε )4( E/L ) J (41)where 4( E/L ) J (cid:39) × − eV roughly holds. The relative mass shift of ∆ m is about ( δε − δε ),in the same level of δε − δε , i.e., in the same order as Im( δU ei ) and δ(cid:15) αβ . The relative mass shift of∆ m is about 0 . δε − δε ). Keeping in mind δε = Im( δU ei ) / ˜ s , we obtain that the relative massshift of ∆ m is one order smaller than Im( δU ei ) and δ(cid:15) αβ . In the case of the IMO, as ∆ m = −| ∆ m | holds, the shift of | ∆ m | will go in the opposite direction to the NMO.13igure 4: The MO sensitivity for different true values of the NSI parameter δU in the four differentspecific models defined in Eq. (34). The NMO is assumed for illustration.Figure 5: The iso-∆ χ contours for the MO sensitivity in the generic NSI model as a function of twoeffective NSI parameters δε − δε and δε − δε . The NMO is assumed for illustration.14 .3.2 Impacts on the MO measurement When fitting the χ function in Eq. (39) with both NMO and IMO, we can take the difference of theminima to measure the sensitivity of neutrino mass ordering, where the discriminator is defined as∆ χ (MO) = (cid:12)(cid:12) χ (NMO) − χ (IMO) (cid:12)(cid:12) . (42)For these specific models defined in Eq. (34), we illustrate in Fig. 4 the NSI effect on the MO measurementby showing the dependence of the MO sensitivity on the true value of the NSI parameter δU , where theNMO is assumed for illustration. The NSI effect with a negative δU in S1, S3 or positive δU in S2, S4will decrease the ∆ χ (MO) value and thus degrade the sensitivity of the MO determination. Howeverin the other half possibilities, the NSI effect can increase the ∆ χ (MO) value and enhance the MOsensitivity. Moreover, the NSI effect shows stronger influence on the MO measurement in models S2, S3than S1, S4. On the other hand, we illustrate in Fig. 5 the iso-∆ χ contours for the MO sensitivity inthe generic NSI model as a function of two effective NSI parameters δε − δε and δε − δε . The NMOis assumed for illustration. We can learn from the figure that the smaller δε − δε and larger δε − δε will reduce the possibility of the MO measurement. If δε − δε decreases by 0.03 or δε − δε increasesby 0.05, ∆ χ will be suppressed by 2 units. In this part we shall discuss the constraints on the NSI parameters with the JUNO nominal setup. Inour numerical calculation, the true oscillation parameters are taken as in Eqs. (32) and (33), and thetrue NSI parameters are taken as δε − δε = δε − δε = 0. In the fitting process, we fix the oscillationparameters but take the NSI parameters as free. With the above simplification, we can obtain theconstraints on the considered NSI parameters. In Fig. 6, we show the limit on these two parametersat the 1, 2, 3 σ confidence levels. For δε − δε , the precision is much better than 1%. However, theprecision for δε − δε is around the 10% level. JUNO is designed for a precision spectral measurementat the oscillation maximum of ∆ m . From Eq. (21), the precision for δε − δε can be compatible withthat of sin θ , where a sub-percent level can be achieved [28]. On the other hand, the precision forsin θ is also at the 10% level, also consistent with that of δε − δε in our numerical simulation.Because δε − δε is suppressed by sin ˜ θ , the above two constraints are actually compatible if weconsider the physical NSI parameters δ(cid:15) αβ defined in Eq. (5). Notice that different assumptions (e.g.,the uncertainties of oscillation parameters) on the experimental systematics may alter the quantitativeprecision of the NSI parameters, but our qualitative conclusion is reasonable in any realistic systematicalassumptions. In this work we have presented a complete and new derivation on the generic NSI effects in reactorantineutrino oscillations, where the NSI parameters are divided into the average and difference partsof the antineutrino production and detection processes. The average part can induce an effective non-unitary PMNS matrix and shift the true values of the mixing angles. On the other hand, the differencepart of the NSI effect can be parametrized with only two independent parameters (i.e., δε − δε , δε − δε ), and give the energy- and baseline-dependent corrections to the mass-squared differences. Eq.(21) is our key formula for the reactor ν e → ν e survival probability, where15igure 6: The experimental constraints on the generic NSI parameters δε − δε and δε − δε , wherethe true values are fixed at δε − δε = δε − δε = 0, and the contours are the 68.3% (1 σ ), 95.5% (2 σ ),99.7% (3 σ ) allowed regions. The NMO is assumed for illustration. • we define the mixing angle shifts as the deviations of measured mixing angles ˜ θ , ˜ θ from theirtrue values θ , θ . However, we stress that these constant shifts are undetectable in reactorantineutrino experiments. • the two NSI parameters δε − δε and δε − δε can be absorbed into the mass-squared differencesand the corresponding E/L -dependent effective parameters ∆ ˜ m and ∆ ˜ m can be defined asthe shifts of mass-squared differences. These shifts are detectable in the spectral measurement ofreactor antineutrino oscillations.Our analytical formalism is applied to the future medium baseline reactor antineutrino experimentJUNO. Two different treatments (a class of specific models and the most general case with the fullparameter space) of the NSI parameters are employed in our numerical analysis. We analyze the NSIimpact on the precision measurement of mass-squared differences and the determination of the neutrinomass ordering, and present the JUNO sensitivity of the relevant NSI parameters. Numerically, • we find that the relative mass shift of ∆ m is around ( δε − δε ), in the same order of theoriginal NSI parameters δ(cid:15) αβ ; and the relative shift of ∆ m is around 0 . δε − δε ), one ordersmaller than the magnitude of δ(cid:15) αβ . However, cancelations may appear in δε − δε and suppressthe mass shift of ∆ m (see the models S1 and S2). • a positive δε − δε or negative δε − δε may enhance the sensitivity of the neutrino MO mea-surement at JUNO. 16 due to the specific configuration of JUNO, the constraint on δε − δε can be better than 1%, but δε − δε can only be measured at the 10% precision level.Compared with long baseline and short baseline reactor antineutrino experiments (e.g., KamLANDand Daya Bay), the medium baseline reactor antineutrino experiment JUNO is more suitable for con-straining the NSI effect because both the slow and fast oscillation terms are measurable in the reactorantineutrino spectral measurement. Taking into account the complementary properties of reactor an-tineutrino experiments at different baselines, it is desirable to present a sophisticated global analysis ofall the reactor antineutrino experiments and therefore, we may obtain the most complete and precisiontesting of the NSIs or other new physics beyond the SM. Acknowledgements We are indebted to Z.Z. Xing for his continuous encouragement and reading the manuscript. We arealso grateful to S. Zhou for useful discussions. This work was supported in part by the National NaturalScience Foundation of China under Grant Nos. 11135009, 11305193, and in part by the Strategic PriorityResearch Program of the Chinese Academy of Sciences under Grant No. XDA10010100. References [1] Daya Bay Collaboration, (F. P. An et al. ), Phys. Rev. Lett. , 171803 (2012).[2] Daya Bay Collaboration, (F. P. An et al. ), Chin. Phys. C , 011001 (2013).[3] Daya Bay Collaboration, (F. P. An et al. ), Phys. Rev. Lett. , 061801 (2014).[4] Double CHOOZ Collaboration, (Y. Abe et al. ), Phys. Rev. Lett. , 131801 (2012).[5] RENO Collaboration, (J.K. Ahn et al. ), Phys. Rev. Lett. , 191802 (2012).[6] T2K Collaboration, (K. Abe et al. ), Phys. Rev. Lett. , 041801 (2011).[7] MINOS Collaboration, (P. Adamson et al. ), Phys. Rev. Lett. , 181802 (2011).[8] Particle Data Group, (K. A. Olive et al. ), Chin. Phys. C (9), 090001 (2014).[9] C. L. Cowan, F. Reines et al. , Science, , 103 (1956).[10] KamLAND Collaboration, (K. Eguchi et al. ), Phys. Rev. Lett. , 021802 (2003).[11] KamLAND Collaboration, (A. Gando et al. ), Phys. Rev. D , 052002 (2011).[12] Y. F. Li, J. Cao, Y. Wang and L. Zhan, Phys. Rev. D , 013008 (2013).[13] S. B. Kim, Proposal for RENO-50: detector design and goals, International Workshop on RENO-50toward Neutrino Mass Hierarchy, Seoul, June 13-14, (2013).[14] S. Antusch, J. P. Baumann and E. Fernandez-Martinez, Nucl. Phys. B , 369 (2009); M. B. Wiseand Y. Zhang, arXiv:1404.4663 [hep-ph]. 1715] M. Malinsky, T. Ohlsson and H. Zhang, Phys. Rev. D , 011301 (2009); T. Ohlsson, Rept. Prog.Phys. , 044201 (2013).[16] N. Severijns, M. Beck and O. Naviliat-Cuncic, Rev. Mod. Phys. , 991 (2006).[17] T. Ohlsson and H. Zhang, Phys. Lett. B , 99 (2009); R. Leitner, M. Malinsky, B. Roskovec andH. Zhang, JHEP , 001 (2011).[18] T. Ohlsson, H. Zhang and S. Zhou, Phys. Lett. B , 148 (2014).[19] A. N. Khan, D. W. McKay and F. Tahir, Phys. Rev. D , 113006 (2013); A. N. Khan, D. W. McKayand F. Tahir, arXiv:1407.4263 [hep-ph].[20] J. Kopp, M. Lindner, T. Ota and J. Sato, Phys. Rev. D , 013007 (2008); A. Bolanos, O. G. Mi-randa, A. Palazzo, M. A. Tortola and J. W. F. Valle, Phys. Rev. D , 113012 (2009); R. Adhikari,S. Chakraborty, A. Dasgupta and S. Roy, Phys. Rev. D , 073010 (2012); P. Bakhti and Y. Farzan,JHEP , 064 (2014); I. Girardi, D. Meloni and S. T. Petcov, Nucl. Phys. B , 31 (2014).[21] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. , 870 (1962); B. Pontecorvo, Sov.Phys. JETP , 984 (1968).[22] J. W. Mei and Z. Z. Xing, Phys. Rev. D , 073003 (2004).[23] H. Minakata, H. Nunokawa, S. J. Parke and R. Zukanovich Funchal, Phys. Rev. D , 053004(2007) [Erratum-ibid. D , 079901 (2007)].[24] C. Biggio, M. Blennow and E. Fernandez-Martinez, JHEP , 090 (2009).[25] C. Zhang, “Results from Daya Bay”, talk given at the conference “The XXVI International Con-ference on Neutrino Physics and Astrophysics”, Boston, 2014.[26] KamLAND Collaboration, (A. Gando et al. ), Phys. Rev. D , 033001 (2013).[27] I. Girardi and D. Meloni, arXiv:1403.5507 [hep-ph].[28] Y. F. Li, Int. J. Mod. Phys. Conf. Ser.31