Analytical Approximation of the Neutrino Oscillation Matter Effects at large
We argue that the neutrino oscillation probabilities in matter are best understood by allowing the mixing angles and mass-squared differences in the standard parametrization to ‘run’ with the matter effect parameter , where is the electron density in matter and is the neutrino energy. We present simple analytical approximations to these ‘running’ parameters. We show that for the moderately large value of , as discovered by the reactor experiments, the running of the mixing angle and the CP violating phase can be neglected. It simplifies the analysis of the resulting expressions for the oscillation probabilities considerably. Approaches which attempt to directly provide approximate analytical expressions for the oscillation probabilities in matter suffer in accuracy due to their reliance on expansion in , or in simplicity when higher order terms in are included. We demonstrate the accuracy of our method by comparing it to the exact numerical result, as well as the direct approximations of Cervera et al., Akhmedov et al., Asano and Minakata, and Freund. We also discuss the utility of our approach in figuring out the required baseline lengths and neutrino energies for the oscillation probabilities to exhibit certain desirable features.
Keywords:Neutrino, Oscillation Probability, Matter Effect, Jacobi Method
IPMU13-0030 Department of Chemistry and Physics, Western Carolina University, Cullowhee, NC 28723, USA \arxivnumber1302.6773
When performing long-baseline neutrino oscillation experiments on the Earth with accelerator based beams, or when detecting atmospheric neutrinos coming from below, the neutrinos necessarily traverse the Earth’s interior Minakata:1998bf (); Banuls:2001zn (); Huber:2002mx (); Gandhi:2004bj (); Huber:2005ep (); Akhmedov:2006hb (); Agarwalla:2012uj (); Blennow:2013rca (). This makes the understanding of matter effects Wolfenstein:1977ue (); Mikheev:1986gs (); Mikheev:1986wj (); Barger:1980tf () on the neutrino oscillation probabilities an indispensable part of analyzing such experiments. These matter effects can of course be calculated numerically for arbitrary matter profiles, but approximate analytical expressions are useful not only for making initial estimates on the requirements one must place on long-baseline experiments, but in obtaining a deeper understanding of the physics involved.
The exact three-flavor neutrino oscillation probabilities in constant-density matter
can be expressed analytically
Barger:1980tf (); Toshev:1986fs (); Petcov:1986qg (); Kim:1986vg (); Zaglauer:1988gz (); Krastev:1988yu (); Toshev:1991ku (); Dick:1999ed (); Ohlsson:1999xb (); Ohlsson:1999um (); Freund:2001pn (); Kimura:2002wd ().
This requires the diagonalization of the
effective Hamiltonian in matter whose -element in the
flavor basis is shifted by ,
where is the electron density and is the neutrino energy.
The eigenvalues of the effective Hamiltonian yield the effective neutrino mass-squared differences in matter
Consequently, various analytical approximations have been devised to better understand the physics potential of various neutrino experiments Barger:1980tf (); Kuo:1986sk (); Arafune:1997hd (); Cervera:2000kp (); Freund:2001pn (); Peres:2003wd (); Akhmedov:2004ny (); Akhmedov:2004rq (); Akhmedov:2008nq (); Asano:2011nj (). These approximations relied on expansions in the small parameters and/or in one form or another, a systematic treatment of which can be found in Ref. Akhmedov:2004ny (). In some cases the matter-effect parameter was also assumed to be small Barger:1980tf (); Arafune:1997hd (). For instance, the formula of Cervera et al. in Ref. Cervera:2000kp () and that of Ahkmedov et al. in Ref. Akhmedov:2004ny () include terms of order , , and . Unfortunately, the accuracies of these formulae suffer when the value of is as large as was measured by Daya Bay An:2012eh (); An:2012bu () and RENO Ahn:2012nd (), consistent with both earlier and later determinations by T2K Abe:2011sj (), MINOS Adamson:2011qu (); Adamson:2013ue (), and Double Chooz Abe:2011fz (); Abe:2012tg (). Given that the current world average of is about GonzalezGarcia:2012sz (), the terms included are not all of the same order. Asano and Minakata Asano:2011nj () have derived the order and corrections to the Cervera et al. formula, but the simplicity of the original expressions is lost. Further improvements in accuracy are possible at the expense of simplicity, as was shown by Freund in Ref. Freund:2001pn () where an approximate expression for the oscillation probability including all orders of was derived.
In previous papers Honda:2006hp (); Honda:2006gv (), we had argued the advantage of not expressing the neutrino oscillation probabilities in matter directly in terms of the vacuum parameters, but to maintain their expressions in terms of the effective parameters in matter which ‘ran’ with the parameter . Further, it was shown that the Jacobi method Jacobi:1846 () could be utilized to find approximate expressions for the ‘running’ parameters in a systematic fashion, leading to fairly simple and compact expressions. In particular, it was shown that the effective values of and the CP violating phase do not ‘run’ to the order considered, maintaining their vacuum values at all neutrino energies and baselines. (The non-running of and has also been discussed in Ref. Krastev:1988yu ().) The -dependence of the resulting expressions for the oscillation probabilities in terms of the approximate running parameters could be analyzed in a simple manner, facilitating the understanding of matter effects.
The approximation of Refs Honda:2006hp (); Honda:2006gv () worked extremely well except when was very small, a possibility that could not be ignored until the Daya Bay/RENO measurements. In this paper, we reintroduce the method with further refinements which improve the accuracy of the approximation for large , while maintaining its ease of use.
This paper is organized as follows. In section 2, we explain our approach to the matter effect problem, and list all the formulae necessary to calculate the approximate ‘running’ parameters in our approach. Approximate oscillation probabilities are obtained by replacing the mass-squared differences and mixing angles in the vacuum oscillation probabilities with their effective ‘running’ values. In section 3, we demonstrate the accuracy of our approximation at various baseline lengths, different mass hierarchies, and different values of the CP violating phase . Comparisons with the approximations of Cervera et al. Cervera:2000kp (), Akhmedov et al Akhmedov:2004ny (), Asano-Minakata Asano:2011nj (), and Freund Freund:2001pn () are also made. In section 4, we show how simple calculations using our approximation can be used to derive the baselines and energies at which the oscillation probabilities exhibit desirable features. We conclude in Section 5. Detailed derivation of our approximation is given in appendices B and C.
2 The Approximation
In the following, we use the conventions and notation reviewed in Appendix A.
2.1 Diagonalization of the Effective Hamiltonian
If the matter density along the baseline is constant
where is the neutrino mixing matrix in vacuum, and
Here, is the electron number density, the matter mass density along the baseline, and the neutrino energy. The above term appearing in the -component of is due to the interaction of with the electrons in matter via -exchange, and Eq. (2) assumes in Earth matter. It also assumes since the -exchange interaction is approximated by a point-like four-fermion interaction. -exchange effects are flavor universal and only contribute a term proportional to the unit matrix to , which can be dropped.
If we write the eigenvalues of as () and the diagonalization matrix as , that is
then the neutrino oscillation probabilities in matter are obtained by simply taking their expressions in vacuum and replacing the elements of the mixing matrix and the mass-square differences with their effective ‘running’ values in matter Wolfenstein:1977ue (); Mikheev:1986gs (); Mikheev:1986wj () :
Note that is -dependent, which means that both and are also -dependent. They also depend on the baseline length since the average matter density along a baseline varies with . The -dependence of the average and the corresponding value of are shown in Fig. 1.
For anti-neutrino beams, the flavor-evolution Hamiltonian in matter is
In comparison to Eq. (1), the CP violating phase in and the matter-effect term both acquire minus signs. Let us write the eigenvalues of as () and the diagonalization matrix as , that is
Note that the tilde above here is flipped to distinguish it from in Eq. (3). The anti-neutrino oscillation probabilities in matter are then obtained by making the replacements
in the vacuum expressions.
2.2 Effective Running Mixing Angles
While it is possible to write down exact analytical expressions for and , as well as their anti-neutrino counterparts Zaglauer:1988gz (), simpler and more transparent approximate expressions are often desirable. One popular approach is to expand the probability formulae in terms of small parameters such as and . Our approach, however, utilizes the Jacobi method Jacobi:1846 (). Instead of obtaining approximations for the probabilities directly, we derived the approximations for the effective mixing parameters. In the following two sections, we list the expressions necessary to calculate the effective running mixing angles and the effective running mass-squared differences for the neutrino and anti-neutrino cases separately. Detailed derivation of our approximation is given in Appendix B.
2.3 Neutrino Case
We first recognize that the mixing matrix in matter can be parameterized in the same fashion as in the vacuum case:
The effective mixing angles can be approximated by
where and are given by
while the angle and the CP-violating phase at kept at their vacuum values Krastev:1988yu ().
The eigenvalues () of are also given approximate running expressions:
where the upper(lower) sign is for the normal(inverted) hierarchy, with
and . For the inverted hierarchy case, , the above expressions simplify to
Thus, to take matter effects into account when calculating neutrino oscillation probabilities, all that is necessary is to take their expressions in terms of the mixing angles and CP-phase in vacuum as is, and replace the two angles as well as the mass-squared differences with their effective running values in matter: , , . This simplifies the calculation considerably, and allows for a transparent understanding of how matter-effects affect neutrino oscillation by looking at the -dependence of the effective parameters.
2.4 Anti-Neutrino Case
Similarly, in the anti-neutrino case, the mixing matrix can be parameterized by:
Note that the sign in front of the matter effect parameter is flipped relative to the neutrino case, so these effective mixing angles will be different. Our approximation is given by
Again, and are unaffected while and are replaced by their effective running values in matter.
The eigenvalues () of are given approximate running expressions as in the neutrino case. The three eigenvalues of the effective Hamiltonian are approximated by
where the upper(lower) sign is for the normal(inverted) hierarchy, with
and . For the normal hierarchy case, , the above expressions simplify to
Thus, the calculation of matter effects for anti-neutrino beams entails the replacements , , .
2.5 The -dependence of Mixing Parameters
We show plots depicting how our various
effective parameters run with the matter-effect parameter .
Due to the wide separation in scale
between and ,
we find it convenient to introduce the parameter via
and plot our effective running parameters as functions of instead of . Here corresponds to , to , and so on. The dependence of the effective mixing angles on are shown in Fig. 2 and that of the sines of twice these angles in Fig. 3. The -dependence of approximate eigenvalues of the effective Hamiltonian are shown in Fig. 4.
3 Demonstration of the Accuracy of the Approximation
In this section, we plot neutrino oscillation probabilities in several scenarios to demonstrate the accuracy of our approximation. As seen in the previous section, our formulae for both the neutrino and anti-neutrino cases are fairly compact and easy to code. In particular, the effective mixing angles for the neutrino and anti-neutrino cases can be calculated with the same code by simply flipping the sign of the matter-effect parameter , cf. Eqs. (14) and (27). The same can be said of and defined in Eqs. (19) and (32). In the case of and , one also needs to make the swap but otherwise the code will be essentially the same. For the vacuum values of the mixing angles and mass-squared differences, we use the global fit values from Ref. GonzalezGarcia:2012sz () listed in Table 1. All plots are generated assuming constant Earth matter density.
We begin by comparing our approximation to Eq. (16) of Cervera et al. Cervera:2000kp (), Eq. (3.5) of Akhmedov et al. Akhmedov:2004ny (), sum of Eqs. (4.2) to (4.4) of Asano and Minakata Asano:2011nj (), and Eq. (36) of Freund Freund:2001pn (). Note that both Cervera et al. and Akhmedov et al. expand the oscillation probabilities to the same order, so their expressions are quite similar except for a minor difference: Eq. (16) of Cervera et al is the same as Eq. (38) of Freund, while Eq. (3.5) of Akhmedov et al. is obtained from the same by setting while keeping non-zero.
In Fig. 5(a), we plot the approximate oscillation
probabilities calculated using these three approximations against the exact numerical result
for the baseline length .
This is the distance used by Asano and Minakata in Ref. Asano:2011nj () to
demonstrate the strength of their formula.
The line-averaged constant Earth matter density
The comparison at a shorter baseline length of , which is the distance from Fermilab to NOA, is shown in Fig. 6. There, all five approximations work well, with our approximation being the most accurate.
The situation changes at the longer baseline length of , which is the distance from CERN to Kamioka Agarwalla:2012zu (), as can be seen in Fig. 7. There, the Cervera et al. and the Akhmedov et al. formulae greatly overestimate , while the Asano-Minakata formula leads to negative probability for . The Freund formula is accurate up until where a kink occurs at . In comparison, our approximation remains accurate for all energies.
The accuracy of our approximation for both the neutrino and anti-neutrino cases, and both mass hierarchies, for different values of the CP violating phase , is demonstrated in Figs. 8 and 9 for the two baselines and , respectively. These distances correspond to those between Fermilab and Homestake (1300 km), and CERN and Pyhäsalmi (2300 km) Stahl:2012exa (). As is evident, our approximation maintains its accuracy for all energy ranges and mass densities.
4.1 Determination of the Mass Hierarchy from Oscillations
Consider the survival probability in matter which is given by
where we have assumed that so that is a good approximation. Similarly, we find:
From Fig. 3, it is clear that the factor
in these expressions
behaves quite differently depending on the mass hierarchy.
For normal hierarchy will peak around
but for the inverted hierarchy case it will not.
This will become manifest if the factor
also peaked at or near the same energy.
For the normal hierarchy case, when we have
From Fig. 1, it is clear that as long as the neutrino beam does not enter the core of the Earth, at which point the constant average matter density approximation breaks down. Therefore, in order to take as close as possible to while preventing the beam from entering the Earth’s core, we need .
For instance, if we take for which , we have . The value of at resonance is then
leading to an oscillation peak/dip factor of . Using Eq. (2), the neutrino beam energy at which is found to be
Indeed, in Fig. 10(a) we show the exact survival probabilities at for both hierarchies, and we can see that the normal hierarchy case dips by over 95% around . Thus, our rough estimates give a highly reliable result.
If we take a somewhat shorter baseline of , which is the distance between CERN and Kamioka Agarwalla:2012zu (), we have , and . The value of at resonance is then
leading to an oscillation peak/dip factor of , which is still fairly prominent. Using Eq. (2), the neutrino beam energy at which is found to be
The actual oscillation peak occurs slightly off resonance around as can already be seen in Fig. 7. Comparison of at with for the normal and inverted hierarchies are shown in Fig. 10(b). is compared in Fig. 11(b).
The differences between the normal and inverted hierarchies for both baselines is manifest, indicating that measuring these oscillation probabilities at this baseline would allow us to determine the mass hierarchy easily. (We consider the dependence on the CP violating phase in the next section.) Eqs. (41), (50), and (58) also suggest that the measurement may provide a better determination of .
4.2 The “Magic” Baseline
The “magic” baseline is the baseline at which the dependence
of on the CP violating phase vanishes Huber:2003ak ().
Therefore, the magic baseline condition is
If we are in the energy and mass-density range such that , we can see from Fig. 4 that , and the above condition reduces to
which is the usual magic baseline condition. Using Eq. (2), this condition for the case becomes
This is satisfied for as can be read off of Fig. 1. Indeed, in Fig. 11(a) we plot the bands that at sweeps for both mass hierarchies when is varied throughout its range of . We can see that the dependence on is very weak.
However, if we look at Eq. (48) carefully, it is clear that all terms that include the CP violating phase are multiplied by which goes to zero when . Indeed, this was why did not appear in Eq. (50). The condition demands
which is clearly satisfied around the oscillation peak for the case just discussed in the previous section. Thus, for this baseline is also only very weakly dependent on as shown in Fig. 11(b). We can conclude that, in general, as long as Eq. (71) is satisfied, one does not need to be at a specific “magic” baseline to suppress the -dependence of .
We have presented a new and simple approximation for calculating the neutrino oscillation matter effects. Our approximation was derived utilizing the Jacobi method Jacobi:1846 (), and we show in the appendix that at most two rotations are sufficient for approximate diagonalization of the effective Hamiltonian. The two rotation angles are absorbed into effective values of and .
As explained in detail in the appendix, the approximation works when , where , a condition which has been shown to be satisfied by Daya Bay An:2012eh () and RENO Ahn:2012nd (). Our formulae are compact and can easily be coded as well as be manipulated by hand. The application of our method to finding the resonance conversion condition, and that to the determination of the ‘magic’ baseline Huber:2003ak (); Smirnov:2006sm () have been demonstrated.
In this paper, only the matter effect due to Standard Model exchange between the neutrinos and matter was considered. New Physics effects which distinguish between neutrino flavor would add extra terms to the effective Hamiltonian, which would require further rotations for diagonalization. This has been discussed previously in Ref. Honda:2006gv (), and the potential constraints on New Physics from long baseline neutrino oscillations experiments in Refs. Honda:2006di (); Honda:2007yd (); Honda:2007wv (). Updates to these works will be presented in future publications.
We would like to thank Minako Honda and Naotoshi Okamura for their contributions to the earlier version of this work Honda:2006hp (); Honda:2006gv (). Helpful communications with W. Liao, T. Ohlsson, S. Petcov, P. Roy, and K. Takeuchi are gratefully acknowledged. SKA was supported by the DST/INSPIRE Research Grant [IFA-PH-12], Department of Science & Technology, India. SKA is also grateful for the support of IFIC-CSIC, University of Valencia, Spain, where some initial portions of this work was carried out. TT was supported by the U.S. Department of Energy, grant number DE-FG05-92ER40677 task A, and by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
Appendix A Conventions, Notation, and Basic Formulae
Here, we collect the basic formulae associated with neutrino oscillation in order to fix our notation and conventions.