Non-Standard Interaction Effects at Reactor Neutrino Experiments

We study non-standard interactions (NSIs) at reactor neutrino experiments, and in particular, the mimicking effects on \theta_13. We present generic formulas for oscillation probabilities including NSIs from sources and detectors. Instructive mappings between the fundamental leptonic mixing parameters and the effective leptonic mixing parameters are established. In addition, NSI corrections to the mixing angles \theta_13 and \theta_12 are discussed in detailed. Finally, we show that, even for a vanishing \theta_13, an oscillation phenomenon may still be observed in future short baseline reactor neutrino experiments, such as Double Chooz and Daya Bay, due to the existences of NSIs.


I. INTRODUCTION
Neutrino oscillations have successfully turned into the most plausible description of neutrino flavor transitions. At the moment, the most important task in neutrino physics is to accurately determine the neutrino parameters, especially the neutrino mass hierarchy and the leptonic mixing angle θ 13 . In this work, we will concentrate on the leptonic mixing parameters, and in particular, the parameter θ 13 . Recently, so-called non-standard interactions (NSIs) have been extensively studied in the literature. Such NSIs could affect neutrino oscillations in a similar way as normal matter affects them. Therefore, if present, NSIs will affect the determination of the fundamental neutrino parameters.
In this work, we will mainly investigate measurements of the fundamental leptonic mixing angles θ 13 and θ 12 at reactor neutrino experiments. Since reactor neutrino experiments such as the future Double Chooz [1] and Daya Bay [2] experiments as well as the existing KamLAND experiment [3] have relatively short baseline lengths, normal matter effects are negligible. This also holds for NSI effects during propagation of neutrinos. Thus, we will only assume that the NSI effects are present at sources and detectors.
In Ref. [4], the basic formalism and different neutrino states for source and detector including NSIs (or "new physics") were first presented. Later, NSIs in sources and detectors have been discussed using amplitudes that describe the neutrino sources and detectors. Such works have been carried out for long baseline neutrino oscillation experiments in general [5] as well as for neutrino factories in particular [6]. Recently, a study on the impact of NSIs at reactor and accelerator neutrino experiments has been performed [7]. Especially, the authors of this work derive first-order series expansions for oscillation probabilities including NSIs from sources and detectors. Explicit upper bounds on parameters describing NSIs from sources and detectors exist. However, these bounds are only generic and given by ε s αβ = O(0.1) for NSIs at sources from universality in lepton decays and ε d αβ = O(0.2) for NSIs at detectors from universality in pion decays [8,9]. This paper is organized as follows. In Sec. II, we will present general formulas for parameter mappings between the fundamental leptonic mixing parameters and the effective leptonic mixing parameters due to the effects of NSIs, and we will give expressions for oscillation probabilities. Then, in Sec. III, we will discuss reactor experiments and how these could be influenced by NSIs and what the outcome would be for the mixing angles θ 13 and θ 12 . Finally, in Sec. IV, we will summarize our results and present our conclusions.

II. ANALYTIC FORMALISM
For a realistic neutrino oscillation experiment, in the presence of non-standard neutrino interactions, the neutrino states produced in the source and observed at the detector can be treated as superpositions of pure orthonormal flavor states: where the superscripts 's' and 'd' denote the source and the detector, respectively, with the normalization factors being given by 1 Note that the states |ν s α and ν d β | are no longer orthonormal states because of NSIs. Since different physical processes take place at the source the and detector, the NSI parameter matrices ε s and ε d are arbitrary and non-unitary in general. In the minimal unitarity violation model (MUV) [10,11,12,13,14,15], where the unitarity of the leptonic mixing matrix [16,17] is slightly violated by possible new physics effects, the non-unitary effect can be regarded as one type of NSIs with the requirement ε s = ε d † . 2 Since in a terrestrial neutrino oscillation experiment, the Earth matter effects [18,19] are more or less involved, the propagation of neutrino flavor states in matter is governed by the effective Hamiltonian 1 Note that, in calculating the number of events, the normalization factors are canceled with the NSI factors in charged-current cross-sections. However, for running short baseline reactor neutrino experiments, the neutrino fluxes are directly measured by using a near detector, and not a Monte Carlo simulation. Hence, the normalization factors should be taken into account. (See also Ref. [10] for a detailed discussion.) 2 In the MUV model, a neutral-current contribution cannot, in principle, be rewritten as a global phase in the oscillation amplitude, and thus, it affects the oscillation process.
where V CC = √ 2G F N e arises from coherent forward scattering and N e denotes the electron number density along the neutrino trajectory in the Earth. Different from ε s and ε d , ε m is an exact Hermitian matrix describing NSIs in matter, and its current experimental bounds can be found in Refs. [20,21,22]. Here the superscript 'm' is used in order to distinguish NSI effects in the Earth matter from those in neutrino sources and detectors. The vacuum leptonic mixing matrix U is usually parametrized in the standard form by using three mixing angles and one CP violating phase [23] where U δ = diag(1, 1, e iδ ), and O ij is the orthogonal rotation matrix in the i, j plane with c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12, 13 and 23). In analogy to the vacuum Hamiltonian wherem 2 i (i = 1, 2, 3) denote the effective mass squared eigenvalues of neutrinos andÛ is the effective leptonic mixing matrix in matter. Now, we include all the NSI effects into the oscillation processes, and arrive at the amplitude for the process ν s where L is the propagation distance and the explicit form of A is a coherent sum over the contributions of all the mass eigenstates ν i Inserting Eq. (9) into Eq. (8), one can directly obtain In order to compare Eq. (10) with the standard oscillation amplitude given in Eq. (9), we It can be clearly seen that only the αth row of ε s and the βth column of ε d are relevant to the transition amplitude. In the limit ε → 0, Eq. (11) is reduced to the standard oscillation amplitude in matter.
With the definitions above, the oscillation probability is given by A salient feature of Eq. (13) is that, when α = β, the first term in Eq. (13) is, in general, not vanishing, and therefore, a flavor transition would already happen at the source even before the oscillation process and is known as the zero-distance effect [24]. Although the effective mixing matrix in matterÛ is still unitary, the presences of NSIs in the source and detector prevent us from defining a unique CP invariant quantity like the standard Jarlskog invariant [25]. New CP non-conservation terms, which are proportional to the NSI parameters and have different dependences on L/E, will appear in the oscillation probability. Another peculiar feature in the survival probability is that, in the case of α = β, CP violating terms in the last line of Eq. (13) should, in principle, not vanish. Note that Eq. (13) is also valid in the MUV model and could be very instructive for analyzing the CP violating effects in the MUV model in future long baseline experiments.

III. REACTOR NEUTRINO EXPERIMENTS
Reactor neutrino experiments with short or medium baselines are only sensitive to the survival probability P (ν s e →ν d e ). The typical energy of antineutrinos produced in nuclear reactors is around a few MeV, which indicates that the Earth matter effects are extremely small and can safely be neglected. Hence, we take (Û ≃ U,m i ≃ m i ) or effectively set V CC = 0 in Eq. (5). As mentioned above, among all the NSI parameters, only ε s eα and ε d αe are relevant to our discussion. It has been pointed out that for realistic reactor neutrino experiments, the leading-order NSIs are of the V ± A type, and the relation ε s eα = ε d * αe holds well [7]. Therefore, we assume ε s eα = ε d * αe = |ε eα |e iφeα in the current consideration and neglect the superscript 's' throughout the following parts of this work. It can be seen from Eq. (12) that the imaginary parts of the parameters J i ee disappear, and hence, the correspondinḡ ν s e →ν d e oscillation is a CP conserved process. Similar to the case without NSIs, one may define the effective mixing anglesθ 13 and θ 12 , in which all the NSI effects are included. For the smallest mixing angleθ 13 , we take α = β = e and i = 3 in Eq. (12) together with the standard parametrization defined by Eq. (6), and obtain the mapping betweenθ 13 and θ 13 where the third-order terms in ε are neglected. As for the effective mixing angleθ 12 , we take α = β = e and i = 2, and obtaiñ Since the NSI parameters should not be comparable to the sizable mixing angle θ 12 , only the first-order terms in ε are taken into account in Eq. (15).
With the help of the effective mixing anglesθ 13  .
Since NSIs are only sub-leading order effects, higher-order terms proportional to εs 2 13 can be ignored, and then the effective mixing angleθ 13 Note thats 2 13 is invariant with respect to the exchange ε eµ ↔ ε eτ , and obviously, in the limit ε → 0,θ 13 equals θ 13 . Equation (18) clearly shows how the mixing angle θ 13 is modified by NSIs. Some comments are in order: • The contributions coming from the NSI parameter ε ee are always correlated with higher-order corrections, and hence cannot be well constrained in a reactor experiment.
However, it induces an enhancement of the total neutrino flux at a near detector, which appears as an overall factor in the oscillation probability if we do not normalize neutrino states as in Eq. (1). Due to the flux uncertainty in reactor experiments, it is very hard for this enhancement to be observed [7].
• For a given set of NSI parameters, sinθ 13 is a quadratic function of sin θ 13 . Thus, there exists a minimum of sin 2θ 13 at the position and the minimum value of sin 2θ 13 is given bỹ Since the fundamental θ 13 cannot be well distinguished from the effectiveθ 13 measured in an experiment, the mimicking effects of NSIs play a very important role in the small θ 13 region. Even if the true value of θ 13 is too tiny to be detected, we may still hope to obtain an oscillation phenomenon in reactor experiments. On the other hand, compared to θ 13 ,θ 13 may also be remarkably suppressed by NSIs, which makes the current experiments quite pessimistic. Note that mimicking (or "fake") values of θ 13 due to so-called damping effects have been investigated in Ref. [26]. Such damping effects could arise from decoherence-like damping signatures (e.g. wave-packet decoherence related to production and detection processes). Thus, damping could fake values of θ 13 , and therefore the value of θ 13 would turn out to be smaller than one expects.
• We illustrate the mappings betweenθ 13 , ε, and θ 13 in Fig. 1. In our numerical calculations, we use the exact analytical formulas and do not make any approximations.
We also adopt the central values of other relevant parameters from the global fit given in Ref. [27]. Without loss of generality, we take |ε eµ | = |ε eτ | = |ε| in our analysis 3 , and allow all the CP violating phases to vary from 0 to 2π. For a given value ofθ 13 , which is in fact the parameter measured in experiments, the true values of θ 13 may be remarkably different, i.e., there exists a degeneracy in θ 13 . Therefore, one has to there is still a widely allowed range 1 • < θ 13 < 8 • with respect to a large ε. Even ifθ 13 is too small to be measured in a reactor experiment, i.e.,θ 13 < 3 • , a discovery search of a non-vanishing θ 13 may still be carried out at future neutrino factories, where the source of neutrinos is a muon storage ring with very clean muon decay and quite limited room for NSIs [29].
• In Fig. 2, we show the oscillation probabilities with respect to the NSI parameters.
The upper plot in Fig. 2 indicates that mimicking oscillation effects, which are induced by sizable NSIs, can be observed in despite of a negligible θ 13 . Once other type of neutrino oscillation experiments can help us to fix the true value of θ 13 , the mimicking effects will provide us with the opportunity to search for NSIs in neutrino sources and detectors.
The oscillation process expressed in Eq. (17) is actually CP conserved. However, the CP violating phase δ enters the oscillation probability explicitly, and so does the leptonic mixing The shadings correspond ε < 0.05, ε < 0.01, and ε < 0.001, respectively. The values of θ 13 are labeled on the plots. For other mixing parameters, we use the central values given in Ref. [27].
Here we take the average energy of reactor neutrinos E = 3 MeV.
angle θ 23 . It is then very helpful to extract information on leptonic CP violation and θ 23 by analyzing the corresponding disappearance channel together with future long-baseline appearance experiments.

B. Medium baseline reactor experiments and θ 12
The current medium baseline reactor neutrino experiment KamLAND receivesν e from nuclear reactors located at an average distance L ≃ 180 km. In order to improve the accuracy of current measurements, the next generation experiments should take the baseline length of about 50 km, which is close to the first minimum of the survival probability related with the small mass squared difference ∆m 2 21 . In neglecting contributions from θ 13 , the corresponding oscillation probability reads where the effective mixing angleθ 12 approximates tõ We now discuss how the NSIs affect the leptonic mixing angle θ 12 : • Similarly, there is no contribution coming from ε ee at leading order. Thus, we can observe that reactor experiments are not sensitive to ε ee .
• Compared to the tiny θ 13 , the magnitude of θ 12 is more sizable. Hence the NSI effects cannot mimic an effectiveθ 12 with a vanishing θ 12 . However, NSI effects may dramatically modify the observed mixing angleθ 12 . We plot θ 12 as a function ofθ 12 and ε in Fig. 3. In the large ε regions, the true value of θ 12 may be close to the bi-maximal mixing value 45 • [30,31,32]. On the other hand, the lower bound θ 12 > 26 • deviates much from its tri-bimaximal mixing pattern [33,34]. Figure 3 indicates that, in the presence of NSIs, even ifθ 12 can be well measured, there is still a large room of θ 12 for various flavor symmetric models.
• The oscillation probabilities of medium baseline reactor experiments are illustrated in Fig. 4. We take the best-fit values of θ 13 and θ 23 in our numerical calculations [27].
Hence, the oscillation behavior around L ≃ 0 is mainly induced by ∆m 2 31 . It can be clearly seen that NSI corrections are more significant for a smaller θ 12 .  12 . For the right-hand plot, |ε| is allowed to vary from 0 to 0.05, and the gray shadings represent |ε| < 0.05, |ε| < 0.01, and |ε| < 0.001, respectively, with darker regions for smaller |ε|. As in Fig. 1, we allow all the CP violating phases to vary from 0 to 2π.
Unlike the mapping betweenθ 13 and θ 13 , only θ 23 is entangled in Eq. (22). Hence, we may also acquire useful constraints on θ 23 through precision measurements of θ 12 and NSI parameters in future experiments.

C. Correlations between θ 13 and θ 12
A crucial question for future experiments is how to distinguish real mixing parameters from NSI effects. As discussed above, there is no hope to extract the fundamental mixing angles from a single reactor neutrino experiment. Since the NSI effects bring in intrinsic correlations between the effective mixing parameters, a combined analysis of both appearance and disappearance channels should be able to help us to determine NSI effects.
In Fig. 5, we show the confidence regions of θ 13 and θ 12 , constrained by the current global fit of neutrino oscillation data [27]. Because of the experimental uncertainties associated withθ 13 andθ 12 , the allowed parameter spaces for θ 13 and θ 12 are quite wide. The true value of θ 13 can achieve the range of the Cabibbo angle within 1σ confidence level, which shades some light on the quark-lepton complementary models [32,35,36]. We want to stress that Finally, we briefly discuss the NSI corrections to θ 23 and δ. One may define an effective mixing angleθ 23 by using the analogous way that we performed above for θ 13 and θ 12 . However, since reactor neutrino experiments are only sensitive to the first row of the leptonic mixing matrix, the effective θ 23 loses its meaning. In future long baseline β-beam experiments or neutrino factories, where different types of NSIs are involved in the production, propagation, and detection processes, one cannot simply employ the language of effective mixing parameters as in reactor neutrino experiments. However, the generic formulas given in Eq. (13) are still valid and very helpful for us in order to figure out NSI effects. A detailed and joint numerical analysis based on Eq. (13) should be very meaningful and will be elaborated elsewhere.

IV. SUMMARY
In this work, we have studied NSI effects in reactor neutrino experiments, and in particular, the mimicking effects on θ 13 . We first presented the most general formulas of oscillation probabilities with all NSI effects at production, propagation, and detection processes being considered. Instead of directly discussing oscillation probabilities, we took use of a more straightforward method, which started from the effective amplitude and derived instructive mappings between fundamental mixing angles (θ 13 , θ 12 ) and effective NSI corrected mixing angles (θ 13 ,θ 12 ) in reactor neutrino experiments. The analytical relations clearly show how these mixing angles are affected by NSIs. We have also illustrated the NSI effects at short and medium baseline reactor experiments. We found that the mixing angles measured in