Extrinsic and Intrinsic CPT Asymmetries in Neutrino Oscillations
Abstract
We reconsider the extrinsic and possible intrinsic CPT violation in neutrino oscillations, and point out an identity, i.e., , among the CP, T, and CPT asymmetries in oscillations. For threeflavor oscillations in matter of constant density, the extrinsic CPT asymmetries , , , and caused by Earth matter effects have been calculated in the plane of different neutrino energies and baseline lengths. It is found that two analytical conditions can be implemented to describe the main structure of the contours of vanishing extrinsic CPT asymmetries. Finally, without assuming intrinsic CPT symmetry in the neutrino sector, we investigate the possibility to constrain the difference of the neutrino CPviolating phase and the antineutrino one using a lowenergy neutrino factory and the superbeam experiment ESSSB. We find that in the former case and in the latter case can be achieved at the confidence level if is assumed.
I introduction
Recent years have seen great progress in experimental neutrino physics. In particular, neutrino oscillations have been well established and leptonic mixing parameters have been measured with an acceptable degree of accuracy. Under the assumption of conservation of the fundamental CPT symmetry, both threeflavor neutrino and antineutrino oscillations can be described by the same set of parameters, namely three leptonic mixing angles , one leptonic Dirac CPviolating phase , and two independent masssquared differences , where and with being the three neutrino masses. The primary goals of present and future neutrino oscillation experiments are to perform precision measurements of the neutrino parameters, determine the neutrino mass ordering (i.e., the sign of ), and probe . In the future, one could also try to establish if there is fundamental or intrinsic CPT violation in the neutrino sector.
Previously, various theoretical models based on violation of the fundamental CPT symmetry have been proposed in the literature. Such models, that naturally also break Lorentz invariance Greenberg (2002), include works by Coleman & Glashow Coleman and Glashow (1997, 1999) and Kostelecký et al. Kostelecký and Mewes (2004a, b); Diaz et al. (2009); Diaz and Kostelecký (2012). On the more phenomenological side, studies of CPT violation have recently been performed in Refs. Barger et al. (2000); Murayama and Yanagida (2001); Barenboim et al. (2002a); Bilenky et al. (2002); Barenboim et al. (2002b); Bahcall et al. (2002); Barenboim et al. (2002c); Datta et al. (2004); Minakata and Uchinami (2005); Dighe and Ray (2008); Antusch and FernándezMartínez (2008); Barenboim and Lykken (2009); Samanta (2010); Giunti and Laveder (2010); Chatterjee et al. (2014). Indirect limits on CPT violation for specific models in the neutrino sector have also been presented Mocioiu and Pospelov (2002). Finally, experimental collaborations have searched for signals of CPT violation in neutrino oscillation experiments, which include LSND Auerbach et al. (2005), MiniBooNE AguilarArevalo et al. (2013); Katori (2014), MINOS Adamson et al. (2008, 2010, 2012); Cao (2014), and SuperKamiokande Abe et al. (2011a); Kaji (2011); Takeuchi (2012).
In a phenomenological way, if the CPT symmetry is not assumed a priori, we need two separate sets of parameters to describe neutrino and antineutrino oscillations. Now, the neutrino flavor eigenstates are related to the neutrino mass eigenstates by a unitary leptonic mixing matrix (see e.g. Ref. Bilenky and Petcov (1987))
(1) 
and the three neutrino masses are (for ). Similarly, for antineutrinos, we have
(2) 
and the antineutrino masses are denoted by (for ). Therefore, the masssquared differences of antineutrinos are defined as and . Although microcausality may be violated if the masses of particles and the masses of their corresponding antiparticles are different from each other, the results in our phenomenological approach can actually be applied to the scenario of spontaneous CPT violation in Refs. Coleman and Glashow (1997, 1999); Kostelecký and Mewes (2004a, b). In principle, neutrino oscillation experiments can be used to place restrictive constraints on the CPTviolating parameters in the neutrino sector.
However, in longbaseline neutrino oscillation experiments, such as a future neutrino factory, neutrinos and antineutrinos will traverse Earth matter, and therefore, matter effects on neutrino and antineutrino oscillations will induce fake or extrinsic CPTviolating effects.
In this work, we investigate the extrinsic and intrinsic CPT asymmetries in oscillations. First, some general remarks are given on the relationship among the CP, T, and CPT asymmetries. An identity is derived. Second, we explore the conditions under which the extrinsic CPT asymmetries induced by matter effects vanish. In this case, if intrinsic CPT violation exists, it will be made more apparent. Finally, we illustrate the experimental sensitivity to the CPTviolating parameters by taking a lowenergy neutrino factory and a superbeam experiment as examples.
Ii CP, T, and CPT Asymmetries
First of all, we present some general discussion on the CP, T, and CPT asymmetries in neutrino and antineutrino oscillations in vacuum and matter (see e.g. Ref. Jacobson and Ohlsson (2004) and references therein). We denote the oscillation probabilities for neutrinos in the channels by , while those for antineutrinos by . Here the neutrino flavor indices and run over , , and . Note that the oscillation probabilities are dependent due to the unitarity conditions:
(3) 
and likewise for . It is straightforward to verify that four out of nine oscillation probabilities in the threeflavor case are independent Jacobson and Ohlsson (2004). However, in the twoflavor case, there is only one independent oscillation probability. Based on neutrino and antineutrino oscillation probabilities, the CP, T, and CPT asymmetries can be defined as
(4)  
(5)  
(6) 
Hence, any CP, T, and CPT violation will be characterized by a nonzero value of , , and , respectively. In a similar way, one can also define the corresponding asymmetries for antineutrinos, i.e., , , and . Obviously, and are dependent quantities, since they are related to the CP and CPT asymmetries for neutrinos, i.e., and . However, the T asymmetries are in general independent.
Subtracting Eq. (4) from Eq. (6), one obtains an interesting relation among the CP, T, and CPT asymmetries, viz.
(7) 
and similarly, for the antineutrino counterpart of Eqs. (4) and (6), we find that
(8) 
Now, it is straightforward to derive
(9) 
and a similar relation among the corresponding asymmetries for antineutrinos. It is worthwhile to emphasize that the relation in Eq. (9) is valid even if the fundamental CPT symmetry is not preserved. Some comments are in order:

From the definition in Eq. (5), we can observe that , and thus, the T asymmetry vanishes in the disappearance channels , i.e., Krastev and Petcov (1988); Akhmedov et al. (2001). Furthermore, Eq. (3) implies , so we have , which is the unique T asymmetry in the threeflavor case Krastev and Petcov (1988). This conclusion applies to oscillations both in vacuum and matter.

For oscillations in vacuum, there is no extrinsic CPT violation Jacobson and Ohlsson (2004). Then, if the intrinsic CPT symmetry holds, we can see that from Eq. (9), implying a unique CP asymmetry . Explicitly, we can calculate the unique CP or T asymmetry Krastev and Petcov (1988)
(10) where is the Jarlskog invariant Jarlskog (1985a, b) and . Here is the neutrino beam energy and is the baseline length. However, in the presence of matter effects or intrinsic CPT violation, we have four independent CP asymmetries , as indicated by and . This applies also to the CPT asymmetries .
In the following section, we will calculate the CPT asymmetries for oscillations in matter, assuming constant matter density. Furthermore, the conditions, under which the extrinsic CPT asymmetries vanish, will be derived and discussed.
Iii Extrinsic CPT Asymmetries
We proceed to consider CPT asymmetries in two and threeflavor neutrino and antineutrino oscillations in matter. For oscillations in vacuum, the CPT asymmetries vanish exactly if the fundamental CPT symmetry is preserved. It has been pointed out that one can test the intrinsic CPT symmetry in a longbaseline neutrino oscillation experiment, such as a future neutrino factory Bilenky et al. (2002). As we will show in the next section, future neutrino superbeam experiments and lowenergy neutrino factories are very powerful in constraining intrinsic CPT violation, particularly for the case of CPviolating phases. However, the extrinsic CPT asymmetry induced by Earth matter effects will mimick the intrinsic one, reducing experimental sensitivity to the CPTviolating parameters. Therefore, it is interesting to explore the conditions for the extrinsic CPT asymmetries to vanish.
iii.1 TwoFlavor Case
For twoflavor oscillations, say and , only one probability for neutrinos is independent, and we choose . Similarly, we take for antineutrinos. In this case, we have , so the T asymmetries are and . Note that by definition. However, there is only one CPT asymmetry
(11) 
One can further verify that . According to Eq. (9), we find that . It is well known that there is no intrinsic CP violation in the twoflavor case, i.e., there are no physical CPviolating phases. For oscillations in vacuum, both and vanish. However, for oscillations in matter, if the intrinsic CPT symmetry is preserved, we conclude that the matterinduced extrinsic CP and CPT asymmetries are equal to each other in the twoflavor case.
It is straightforward to calculate the extrinsic CPT and CP asymmetries in twoflavor oscillations in matter of constant density. The survival probability is given by Wolfenstein (1978); Mikheyev and Smirnov (1985)
(12) 
where with characterizes the matter effects, and the plus (minus) sign refers to the antineutrino (neutrino) oscillation channel. In the limit of a small matter potential, namely , the extrinsic CPT asymmetry turns out to be
(13) 
with . Obviously, the CPT and CP asymmetries are proportional to the matter potential and will vanish for oscillations in vacuum. For a nonzero , the leadingorder term in Eq. (13) becomes zero if is satisfied.
In Fig. 1, the contour curves for a vanishing CPT asymmetry are shown, where constant matter density of , electron fraction , and typical neutrino oscillation parameters and are used. The dashed (green) curves are the exact calculation using neutrino and antineutrino oscillation probabilities given in Eq. (12), while the solid (red) curves correspond to the one using Eq. (13) and assuming the leadingorder term to vanish, namely, . It can be observed that the condition is no longer satisfied for a higher neutrino energy and a longer baseline. However, for an extremelylong baseline, the dashed curves obtained from the exact oscillation probabilities approach the solid curves corresponding to two neighboring solutions to .
If the intrinsic CPT symmetry is not preserved, the leptonic mixing angle and masssquared difference for neutrinos are generally different from those for antineutrinos. Thus, in order to quantify deviations from the intrinsic CPT symmetry, we express the mixing parameters for antineutrinos as follows
(14)  
(15) 
and expand the CPT asymmetry in terms of perturbation parameters , , and . Then, we obtain
(16) 
where the first term refers to the extrinsic CPT asymmetry given in Eq. (13). Therefore, an experimental setup for a vanishing or extremelysmall extrinsic CPT asymmetry will be sensitive to the intrinsic CPT asymmetry in neutrino oscillations, except for the case where the oscillation terms proportional to the CPTviolating parameters and become extremely small as well.
iii.2 ThreeFlavor Case
Now, we turn to the case of threeflavor oscillations in matter. In general, there are four independent CPT asymmetries, which will be taken as , , , and in the following discussion Jacobson and Ohlsson (2004). For constant matter density, the relevant neutrino oscillation probabilities are given by Cervera et al. (2000); Freund (2001); Freund et al. (2001); Barger et al. (2002); Akhmedov et al. (2004)
(17)  
(18)  
(19)  
to second order in and first order in . Here and for have been defined. In addition, we have defined the oscillation phase driven by the large neutrino masssquared difference as , and that measures the importance of matter effects. Given current neutrino oscillation data, we have , so it is safe to neglect terms. Note that the series expansion of the oscillation probabilities in Eqs. (17)–(19) is valid as long as , or equivalently, . Under this condition, the oscillation terms mainly driven by the small masssquared difference are negligible. One can verify that this condition is satisfied by the ongoing and forthcoming longbaseline neutrino oscillation experiments, which make use of intensive neutrino beams of energies around a few GeV and baselines shorter than the diameter of the Earth.
Using constant matter density, it is possible to derive the oscillation probabilities for antineutrinos from those for neutrinos by flipping the signs of the matter potential (i.e., ) and the CPviolating phase (i.e., ). Furthermore, the probabilities for the Tconjugate channels can be obtained by changing the sign of , if the matter density profile is symmetric Akhmedov et al. (2001), which is obviously the case for constant matter density. Therefore, one can calculate the oscillation probabilities , , , , and from Eqs. (17)–(19) by applying the aforementioned rules. Then, with all the relevant oscillation probabilities, we readily compute the four independent CPT asymmetries
(20)  
(21)  
(22)  
(23)  
where the higherorder terms of and have been neglected. Although it is impossible to obtain a universal condition for all four CPT asymmetries to vanish, one can easily figure out if the following identity
(24) 
is fulfilled, holds at leading order. This equality is trivially satisfied for , i.e., for oscillations in vacuum. However, there exist nontrivial solutions to Eq. (24), as we will show later. The asymmetry is generally nonzero under this condition, but it can be further reduced to
(25) 
which is proportional to and becomes extremely small for low neutrino energies. Moreover, one can observe that holds if another condition
(26) 
is satisfied. In this case, we expect , , and to be suppressed as well, since the terms of in the first lines of Eqs. (21)–(23) vanish and the much smaller terms of survive.
In order to illustrate the above observations, we plot the contour lines for vanishing CPT asymmetries for a variety of neutrino energies and baseline lengths, as shown in Fig. 2. In our calculations, constant matter density , electron fraction , and the neutrino parameters , , , , , and are assumed. The dashed (green) curves are determined using the exact probabilities for threeflavor oscillations in matter. The solid (gray) curves correspond to the identity in Eq. (24), while the dotted (gray) ones to that in Eq. (26).
As one can observe from Fig. 2 (a), the main structure of the contours for can be perfectly described by the two analytical conditions in Eqs. (24) and (26), i.e., the solid and dotted curves. For each pair of two curves, there are two intersecting points, one of which is along the line of and the other . For neutrino energies around a few GeV, we find that Eqs. (24) and (26) are equivalent to (i) and (ii) . On the other hand, we have
(27) 
Therefore, for the given matter density and electron fraction, , implying for and for . Since the baseline length cannot exceed the diameter of the Earth, only these two possibilities are allowed. In the first case with and , we can further fix neutrino energies at the intersecting points by requiring , or equivalently, , where is a nonnegative integer. With the help of Eq. (27), we obtain , leading to , , , and for , respectively. In the second case with and , the neutrino energies at the intersecting points are further determined by , or equivalently, with being a nonnegative integer. In a similar way, one can figure out the energies by setting . However, it is worthwhile to point out that the oscillation probabilities themselves are also highly suppressed at these points, rendering them not useful in searching for intrinsic CPT violation.
In Fig. 2 (b) and (c), the solid curves from the condition in Eq. (24) coincide with the dashed curves from the exact numerical calculations. However, the dotted curves from the condition in Eq. (26) significantly deviate from the dashed ones. In addition, the analytical conditions in Eqs. (24) and (26) cannot provide a satisfactory description of , as shown in Fig. 2 (d). For a baseline length below , we have verified that the numerical results with constant matter density in Fig. 2 are essentially unchanged when a realistic density profile (e.g., the Preliminary Reference Earth Model Dziewonski and Anderson (1981)) is used.
In analogy to the case of twoflavor oscillations, one can introduce different mixing parameters for antineutrinos and investigate the CPT asymmetries in the presence of intrinsic CPT violation. However, with six additional mixing parameters for antineutrinos, the approximate and analytical expressions of will be rather lengthy and less instructive. In the next section, we will summarize the current experimental constraints on the antineutrino parameters , , and , whose deviations from the neutrino parameters are clear signatures of intrinsic CPT violation. Moreover, we focus on a future lowenergy neutrino factory and a superbean experiment, and study their sensitivities to the difference between the CPviolating phase in the neutrino sector and in the antineutrino sector.
Iv Experimental Constraints
If the fundamental CPT symmetry is not assumed, one has to fit neutrino and antineutrino oscillation experiments separately using different mixing parameters and masssquared differences. In this section, we present a brief summary of current experimental constraints, and emphasize that the future neutrino facilities offer a new possibility to constrain the difference between neutrino and antineutrino CPviolating phases.
iv.1 Current Constraints
First, we consider the most precise measurements of and in solar neutrino experiments, and and in the longbaseline reactor neutrino experiment, i.e., KamLAND. In Ref. Aharmim et al. (2013), a combined analysis of three phases of solar neutrino data from the SNO experiment has been performed. If the solar neutrino rates in Gallium Abdurashitov et al. (2009) and Chlorine Cleveland et al. (1998) experiments, Borexino Bellini et al. (2010, 2011) and SuperKamiokande Hosaka et al. (2006); Cravens et al. (2008); Abe et al. (2011b) solar data are further included, a global analysis in the framework of threeflavor oscillations yields Aharmim et al. (2013)
(28) 
where the errors are attached to the bestfit values, and at confidence level (C.L.). Given in the currentlyfavored region, the neutrinos with relatively high energies experience adiabatic flavor conversion in solar matter and the survival probability is just determined by the mixing angle . Thus, the ratio of chargedcurrent neutrino events and the neutralcurrent ones from the SNO experiment will be very sensitive to the mixing angle , but not the masssquared difference . For solar neutrinos of lower energies, matter effects are negligible and the vacuum oscillation probability averaged over the long distance between the Sun and the Earth is applicable. A nonzero leads to an energyindependent suppression of the survival probability in the threeflavor case, so solar neutrino experiments also place a bound on . The KamLAND experiment is designed to observe the disappearance of from nuclear reactors at an averaged distance of , so it is sensitive to and , and also constrains . The latest threeflavor analysis of oscillation data in KamLAND indicates Gando et al. (2011)
(29) 
where the bestfit values with errors are given, and at C.L. The energy spectrum of neutrino events measured in KamLAND allows us to probe with a high precision, while the uncertainty in the flux normalization limits the sensitivity to .
Note that the bound on from KamLAND should be superseded by the precise measurements from the shortbaseline reactor experiments. The determination of is dominated by the Daya Bay experiment, which has recently published the rate An et al. (2012, 2013) and spectral An et al. (2014a) measurements of reactor antineutrinos, and an independent measurement via neutron capture on Hydrogen An et al. (2014b). The combined analysis of both rate and spectral data from Daya Bay gives
(30) 
where is assumed and the tiny difference is neglected. In addition, we assume normal mass hierarchy in both neutrino and antineutrino sectors throughout this work. The information on can be extracted from a threeflavor analysis of solar and atmospheric neutrino data, and from the appearance data in the accelerator neutrino experiments. The T2K collaboration has carried out a combined analysis of the disappearance and appearance data in the threeflavor oscillation case Abe et al. (2014a, b); de Perio (2014), and obtained , , and , where the CPviolating phase is set to be free in the fit.
Then, we come to the measurements of and in atmospheric and accelerator neutrino experiments, where both and disappearance channels are dominant. In Ref. Abe et al. (2011a), a search for differences between the neutrino and the antineutrino oscillation parameters has been performed for all three phases of atmospheric neutrino data in SuperKamiokande, indicating and at C.L. On the other hand, the MINOS experiment has operated in both neutrino and antineutrino channels, and accumulated about 38 kilotonyears of atmospheric neutrinos Adamson et al. (2013). The simultaneous fit to neutrino and antineutrino data at MINOS yields Cao (2014)
(31) 
where the slightly worse sensitivity to antineutrino parameters can be ascribed to a factor of three lower exposure in the accelerator data, and a smaller antineutrino cross section in the atmospheric data.
Finally, by combining Eqs. (28)–(31), we summarize the most conservative constraints at C.L. from current oscillation data:
(32) 
and there is essentially no constraint on at C.L. In deriving the above limits, we have assumed Gaussian errors and chosen the larger absolute value of errors on the relevant oscillation parameters.
In principle, a threeflavor globalfit analysis to all the above neutrino and antineutrino data is needed to derive statistically reliable constraints on the differences between the neutrino and the antineutrino oscillation parameters, which is beyond the scope of our work. Such a analysis was actually performed in Refs. GonzalezGarcia and Maltoni (2008); Maltoni and Schwetz (2007) for the oscillation data at that time, and the expected sensitivities of future betabeam experiment, mediumbaseline reactor experiments, and neutrino factories have been discussed in Ref. Antusch and FernándezMartínez (2008). The recent discovery of a nonzero has triggered tremendous discussion in the literature about future experimental sensitivities to the leptonic Dirac CPviolating phase at a lowenergy neutrino factory (LENF), which provides the unique possibility to probe the differences in neutrino and antineutrino CPviolating phases as we will show in the next subsection. See Ref. Choubey et al. (2011) for a detailed description of different neutrino factory setups.
iv.2 CPviolating Phases
In order to concentrate on the determination of CPviolating phases and , we assume normal mass hierarchy in both neutrino and antineutrino sectors (i.e., and ). For relatively large and , it has been proposed that a neutrino factory with neutrino energies of several GeV and baseline lengths around will be a powerful facility to pin down the CPviolating phases Geer et al. (2007); FernándezMartínez et al. (2007). Therefore, we examine the expected sensitivities of a LENF to both and .
First, it may be instructive to investigate the extrinsic CPT asymmetries at the probability level for a LENF. In Fig. 3, the four CPT asymmetries , , , and for different baseline lengths and neutrino energies are shown. In the numerical calculations, the averaged matter density along the trajectory is used and the full threeflavor oscillation probabilities are implemented. In addition, the neutrino oscillation parameters , , , , , and have been assumed for both neutrinos and antineutrinos. Two comments on the numerical results in Fig. 3 are in order:

As expected, the extrinsic CPT asymmetries are absent in the limit of a very short baseline, when matter effects are negligible. The conditions for vanishing CPT asymmetries, which have been discussed in the previous section, cannot be satisfied for a single baseline length and a wide range of neutrino energies. However, for a LENF with the stored muon energy , the CPT asymmetries are small around . One can observe from Fig. 3 that the zero point of CPT asymmetries for the neutrino energy of (solid black curves) is reached around , while the asymmetries for higher neutrino energies have not yet developed much at this baseline length.

In Fig. 3 (d), it is evident that is extremely small for the whole relevant energy range and its absolute value is less than up to the baseline length . This has already been observed in Refs Barger et al. (2000); Xing (2002); Bilenky et al. (2002), and it has been proposed that the disappearance channel is suitable to probe intrinsic CPT violation Barger et al. (2000); Bilenky et al. (2002), namely the differences between and . Due to , we have , as a consequence of the fact that sizable differences between neutrino and antineutrino oscillation probabilities appear only at long baselines.
However, it is difficult to conclude from the CPT asymmetries at the probability level that is the optimal baseline length to probe intrinsic CPT violation, since a shorter baseline means a larger number of neutrino events.
Then, we use the GLoBES software Huber et al. (2005, 2007) to perform numerical simulations to study the experimental sensitivity to intrinsic CPT violation at a LENF, in particular to the difference between and . In the simulation, the baseline length is , namely the distance between Fermilab to the Sanford Underground Laboratory at Homestake, South Dakota, USA. This choice is also motivated by our previous observations on the extrinsic CPT asymmetries. Following Ref. FernándezMartínez et al. (2010), for the neutrino beam, we assume a muon energy of with useful muon decays per year, running for ten years at each polarity. For the detector, we consider a totally active scintillating detector with a fiducial mass of 20 kiloton, a energy threshold of , and a energy resolution. At a neutrino factory, the combination of () channels and () channels can solve the problem of parameter degeneracies. Moreover, although the disappearance channels () and () are insensitive to the CPviolating phase, they are helpful in determining the other mixing parameters. Hence, we include all these signal channels in our simulations. Note that we have explicitly indicated the signals in the case of decays, whereas those in the case of decays are given in the parentheses.
In the and (dis)appearance channels, the detection efficiency of is set to be below and above. The main background arises from the charge misidentification and neutralcurrent events, for which we assume a constant fraction of the wrongsign rates and the neutralcurrent rates. The charge identification of in the lowenergy region is very challenging, and the pion background is difficult to subtract from the electron signals. Therefore, in the and (dis)appearance channels, the detection efficiency of is set to be below and above. Furthermore, we assume the same type of background as in the case, and choose a constant fraction of for the wrongsign rates and the neutralcurrent rates. For both cases, an uncorrelated systematic error of on signal and background is adopted.