###### Abstract

Given an accelerator-based neutrino experiment with the beam energy GeV, we expand the probabilities of and oscillations in matter in terms of two small quantities and , where and are the neutrino mass-squared differences, and measures the strength of terrestrial matter effects. Our analytical approximations are numerically more accurate than those made by Freund in this energy region, and thus they are particularly applicable for the study of leptonic CP violation in the low-energy MOMENT, ESSSM and T2K oscillation experiments. As a by-product, the new analytical approximations help us to easily understand why the matter-corrected Jarlskog parameter peaks at the resonance energy GeV (or GeV) for the normal (or inverted) neutrino mass hierarchy, and how the three Dirac unitarity triangles are deformed due to the terrestrial matter contamination. We also affirm that a medium-baseline neutrino oscillation experiment with the beam energy lying in the range is capable of exploring leptonic CP violation with little matter-induced suppression.

Analytical approximations for matter effects on CP violation in

the accelerator-based neutrino oscillations with GeV

Zhi-zhong Xing ^{*}^{*}*E-mail:
and Jing-yu Zhu
^{†}^{†}†E-mail:

Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Center for High Energy Physics, Peking University, Beijing 100080, China

PACS number(s): 14.60.Pq, 13.10.+q, 25.30.Pt

Keywords: CP violation, matter effects, unitarity triangles, neutrino oscillations

## 1 Introduction

In the past two decades we have witnessed a booming period in neutrino physics thanks to a number of indisputable observations of atmospheric, solar, reactor and accelerator neutrino oscillations [1], and thus achieved a smoking gun for the incompleteness of the standard model (SM) in particle physics — the neutrinos actually have finite rest masses and the lepton flavors are significantly mixed, motivating us to explore the other unknowns of massive neutrinos beyond the SM and search for their possible consequences in nuclear physics, particle astrophysics and cosmology.

In the standard three-flavor scheme there are six neutrino oscillation parameters: two independent neutrino mass-squared differences (e.g., and ), three lepton flavor mixing angles (i.e., , and ) and one CP-violating phase (i.e., ). Among them, the sign of and the size of remain unknown [2, 3, 4]. But some preliminary hints for and have recently been seen by combining the T2K [5, 6] and NOA [7] data on oscillations with the Daya Bay (reactor oscillation [8, 9]) and Super-Kamiokande (atmospheric oscillation [10]) data [11]. Provided is really around or takes a nontrivial value far away from and , then remarkable CP- and T-violating effects will emerge in some upcoming long-baseline neutrino oscillation experiments.

Among a number of ongoing and proposed accelerator-based experiments
which aim to probe or constrain CP violation in neutrino
oscillations [12], those with the beam energy
GeV (e.g., T2K [5], MOMENT [13] and ESSSM
[14]) are expected to involve much smaller terrestrial matter
effects. To understand the salient features of the matter-corrected
and oscillations in this energy region, it is
important to expand their probabilities in terms of two small
expansion parameters
and , where with being the Fermi constant
and being the background density of electrons. But the
previous analytical approximations in this connection, such as the
popular one developed by Freund [15], are usually subject
to GeV and will become invalid when approaches
vanishing
^{1}^{1}1Xu has noticed that the approximate formulas obtained by
Freund [15] are still valid even near the solar neutrino
resonance in matter (i.e., ) [16], but we are going to show that they will
become problematic in the GeV region and definitely
turn to be invalid in the GeV region..
The reason is simply that mainly the long-baseline neutrino
oscillation experiments with GeV were considered
in those works.

Hence our present work is well motivated to offer the hitherto most systematic and useful analytical approximations for terrestrial matter effects on CP violation in the medium-baseline neutrino oscillation experiments with the beam energy GeV.

The strength of CP and T violation in neutrino oscillations is measured by a universal and rephasing-invariant quantity of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix [17, 18, 19], the so-called Jarlskog parameter [20] defined via

(1) |

where the Greek and Latin subscripts run over and
, respectively. When a neutrino beam travels through a
medium, it can see two kinds of refractive indices because of its
interactions with the constituents of the medium (i.e., electrons,
protons and neutrons) via the weak neutral current (NC) and charged
current (CC) [21, 22]. All the three neutrino flavors share a
common “matter” phase due to the refractive index arising from the
NC forward scattering, but the electron neutrinos develop an extra
“matter” phase owing to the CC forward scattering. The latter is
nontrivial, and hence it is likely to change the neutrino
oscillation behavior. In this case one may define the
matter-corrected neutrino masses and the
corresponding PMNS matrix , so as to express the
probabilities of neutrino oscillations in matter in the same way as
those in vacuum. For example, the T-violating asymmetry between the
probabilities of and oscillations in matter is given by [23, 24]
^{2}^{2}2Since an ordinary medium (e.g., the Earth) only consists
of electrons, protons and neutrons instead of both these particles
and their antiparticles, the matter background is not symmetric
under the CP transformation. Hence the expression of the
CP-violating asymmetry between and
is not so simple
as that of in Eq. (2), as one can
clearly see in section 4.

(2) |

in which denotes the neutrino beam energy, is the distance between a neutrino source and the detector, and are the matter-corrected counterparts of and (for ), respectively. It is known that exactly holds for a constant matter profile [25, 26, 27]. But a more transparent relationship between and , which can directly tell us why or how CP violation in matter is enhanced or suppressed as compared with that in vacuum, has been lacking. It should be noted that (or ) is in principle a measurable quantity, but in practice it is not directly observable since it is always correlated with the oscillation terms as shown in Eq. (2).

However, a careful study of the ratio changing with the neutrino (or antineutrino) beam energy is not only conceptually interesting but also practically indispensable for expanding the matter-corrected oscillation probabilities and in terms of the afore-defined small parameters and in the GeV region. So we plan to organize the remaining parts of this paper in an easy-to-follow and step-by-step way: starting from the analytical approximation of , passing through those of , and (for ), and ending with those of and .

In section 2 we aim to reveal a unique range of the neutrino beam energy in which the size of the effective Jarlskog invariant can be enhanced as compared with its fundamental counterpart . We find that will hold if is below the upper limit GeV in a realistic oscillation experiment. In particular, we find that peaks at the resonance energy

(3) |

which is about GeV (or GeV) for (or ), corresponding to the normal (or inverted) neutrino mass ordering. Accordingly, we arrive at the maximum value

(4) |

which is roughly (or ) for (or
). As for an antineutrino beam,
decreases monotonically in the GeV region and thus does not undergo any resonances. In
this sense one may draw the conclusion that a medium-baseline
neutrino oscillation experiment with being in the range should be able to explore leptonic CP
violation with little matter-induced suppression
^{3}^{3}3Note that Minakata and Nunokawa have discussed a similar
possibility and obtained the leading-order analytical result of
in Ref. [28]. In comparison, our analytical
result in Eq. (3) has a much higher degree of accuracy and thus the
new result in Eq. (4) can explain the sensitivity of
to the neutrino mass ordering..

In section 3 we concentrate on a geometrical description of leptonic
CP violation in matter and make some analytical approximations for
this intuitive and useful language. Namely, we show how the three
Dirac unitarity triangles (UTs) in the complex plane [29]
^{4}^{4}4The other three unitarity triangles (defined as
, and ), the
so-called Majorana UTs [30, 31], will not be discussed here
because they have nothing to do with leptonic CP and T violation in
normal neutrino-neutrino and antineutrino-antineutrino
oscillations.,
defined through the orthogonality relations

(5) |

are modified (either enlarged or suppressed) by terrestrial matter effects in a low-energy medium-baseline neutrino oscillation experiment. We find that the third side of each UT (i.e., , or ) is essentially insensitive to the matter-induced corrections when the neutrino beam energy is low, but the other two sides — both their sizes and orientations — can get appreciable corrections. Besides some new and useful analytical results to be obtained in a reasonably good approximation, a numerical illustration of the real shapes of the effective Dirac UTs in matter (denoted as , and ) changing with will also be presented.

In section 4 we aim to combine our new results about and (for ) with the probabilities of neutrino oscillations in matter. In particular, the effective probabilities and are expanded in the whole GeV region with the help of the small quantities and . We show that our analytical approximations are numerically more accurate than those made by Freund in this energy region, and thus they are particularly applicable for the study of leptonic CP violation in the low-energy MOMENT, ESSSM and T2K oscillation experiments. We also affirm that a medium-baseline neutrino oscillation experiment with the beam energy lying in the range is capable of exploring leptonic CP violation with little matter-induced suppression.

## 2 The matter-enhanced Jarlskog parameter

Given the effective neutrino masses and the effective lepton flavor mixing matrix which have accommodated the matter-induced corrections to and , the effective Hamiltonian responsible for the propagation of a neutrino beam in matter can be written as [21, 22]

(6) |

in which denotes the charged-current contribution to the coherent forward scattering in matter. When a constant terrestrial matter profile is concerned, as in the present work, Eq. (6) allows one to derive the following relation between the fundamental Jarlskog invariant and its matter-corrected counterpart :

(7) |

which is a reflection of both the Naumov relation
[25, 26, 27] and the Toshev relation [32].
The latter means and . Namely
^{5}^{5}5For the sake of simplicity, we have omitted the Majorana
CP-violating phases of massive neutrinos in this parametrization
simply because they have nothing to do with neutrino oscillations
under discussion.,
in the
standard parametrization of

(8) |

with and (for ). The parametrization of is exactly the same as that of in Eq. (8), and hence one may obtain and the same expression of as a function of , , and . Note, however, that Eq. (7) is actually a parametrization-independent result. We shall use it to establish an approximate but more transparent relationship between and later on.

In fact, the exact relations between and (for ) have been derived by several authors with the help of Eq. (6) [33, 34, 35], but only the normal neutrino mass ordering (i.e., ) was assumed in those works. Here we consider both normal and inverted (i.e., ) neutrino mass hierarchies. To be explicit, we have

(9) |

in the case; or

(10) |

in the case, where

(11) |

with the definitions and . When an antineutrino beam is taken into account, the corresponding oscillation behaviors depend on and . In this case the above formulas remain valid but the replacements and (i.e., and ) are required. Eq. (7) tells us that both and flip their signs in the above replacements, and thus their ratio remains positive.

Although Eqs. (9)—(11) are exact, they are unable to reveal the
dependence of on in a
transparent way. It is therefore important to make reasonable
analytical approximations in this connection, so as to simplify the
relations between and .
The remarkable analytical approximations made by Freund
[15] have been popularly applied to the studies of various
long- or medium-baseline neutrino oscillation experiments with GeV
^{6}^{6}6See, also, the analytical expansions made in Refs.
[16, 24, 36]. When the unitarity of the
PMNS matrix is directly or indirectly violated in
the presence of light or heavy sterile neutrinos, the similar
analytical expansions of neutrino oscillation probabilities have
been done by Li and Luo [37]..
Given the fact that holds at the
confidence level [4] and the dependence of
terrestrial matter effects on the neutrino beam energy can be
effectively expressed as for a realistic ongoing or upcoming
neutrino oscillation experiment [38]
^{7}^{7}7To be more specific, the “matter” parameter is given
as , where is
the electron fraction and is the
typical matter density for a neutrino trajectory through the Earth’s
crust.,
the limit GeV is essentially equivalent to the
requirement .

But we stress that the case of is also interesting in neutrino phenomenology, especially in the aspect of probing leptonic CP and T violation in a low-energy medium-baseline oscillation experiment [28]. In fact, there will be no way to obtain if the neutrino beam energy is higher than about GeV. To see this point, we calculate the ratio of to by using Eqs. (7)—(11) and inputting the best-fit values of , , and listed in Table 1 [4]. Allowing to vary from to GeV, we plot the numerical change of with in Fig. 1, where both the neutrino (with ) and antineutrino (with ) beams are considered, together with both the normal () and inverted () neutrino mass hierarchies. Some observations and discussions are in order.

Normal mass ordering (NMO) | Inverted mass ordering (IMO) | |||

best-fit | range | best-fit | range | |

— | — | |||

— | — | |||

— | — | |||

— | — | |||

— | — | |||

— | — |

(1) Except the extreme case of (i.e., or
) which makes the ratio of to
nonsense, the profile of changing with
is stable and independent of the value of and
the large uncertainties of itself. In all the four
possibilities shown in Fig. 1, the size of goes down quickly when becomes larger than about GeV.
As for the case of an antineutrino beam plus the normal mass
hierarchy, decreases in a
monotonic way and does not develop any maxima or minima. In
comparison, can have one maximum in
the case of a neutrino beam plus the inverted mass hierarchy, or one
maximum and one minimum in the case of an antineutrino beam plus the
inverted mass hierarchy, or two maxima and one minimum in the case
of a neutrino beam plus the normal mass hierarchy. But we are mainly
interested in the peaks of in the
latter three cases, where the corresponding values of are
summarized as follows
^{8}^{8}8In the case of a neutrino (or antineutrino) beam with the
normal (or inverted) mass ordering, the minimum of is about (or ) appearing at GeV (or GeV). The magnitude of such an extreme is
actually similar to the suppressed peak at GeV
(or GeV).:

(12) |

Of course, the suppressed peaks with are not within the scope of our interest in this work, because the corresponding beam energies are far above GeV.

(2) But a suppressed peak and its resonance energy can be well understood by following the analytical approximations made in Ref. [15] for GeV. Namely,

(13) |

in which correspond to the neutrino and antineutrino beams, respectively. We find that this ratio peaks at

(14) |

with , where the smallness of has been taken into account. Now that itself is positive, the plus (or minus) sign in front of in Eq. (14) must correspond to the neutrino (or antineutrino) beam with the normal (or inverted) mass ordering. Given the best-fit value of in Table 1, it is straightforward to obtain GeV in the case or GeV in the case. Such approximate results are in agreement with the exact numerical results shown in Eq. (12) to a reasonably good degree of accuracy.

From now on let us concentrate on the first (matter-enhanced) peak around GeV in Fig. 1 and understand why it appears in an approximate but more transparent way. Fig. 2 is a clearer illustration of this peak, where the ranges of , , and are also taken into account. One can see that the numerical uncertainties associated with the four input parameters do not change the lineshape of , implying that our analytical approximations to be made below will keep valid when the relevant neutrino oscillation parameters are measured to a much higher degree of accuracy in the near future. In the low-energy region under consideration the magnitude of is comparable with or smaller than that of , and thus both of them can serve for the small expansion parameters in our analytical approximations for , and . We first consider the neutrino beam. A tedious but straightforward calculation leads us to the results

(15) |

for the case; and

(16) |

for the case, where

(17) |

is a small parameter, and the smallness of is already implied. Then we obtain the effective neutrino mass-squared differences from Eq. (9) or Eq. (10):

(18) |

for the case; or

(19) |

for the case. Given the standard parametrization of the PMNS mixing matrix in Eq. (8), the small parameter in Eq. (17) can be reexpressed as

(20) |

so its magnitude is apparently of or . With the help of Eqs. (7), (18) and (19), we arrive at the approximate analytical results for the ratio of to as follows:

(21) |

for the case; or

(22) |

for the case, in which the terms proportional to in have been omitted thanks to the smallness of . Since has a minimum value at , we expect that the ratio takes its maximum value in the leading-order approximation, no matter whether the neutrino mass ordering is normal or inverted. As for an antineutrino beam, the matter parameter is actually , and thus the replacement must be made for the analytical results obtained above. In other words, does not develop a minimum value in the antineutrino case — that is why does not undergo any resonances in this case, a conclusion independent of the neutrino mass ordering. So we only concentrate on the neutrino beam in the subsequent discussions.

Let us go beyond the leading-order approximation to calculate the extreme value of , which is a function of (or equivalently, the matter parameter or the neutrino beam energy ). To do so, we take the first derivative of with respect to the variable in Eq. (21) or (22) and set it to equal zero, and find that such a treatment leads to the same equation in these two cases:

(23) |

The solution to Eq. (23) turns out to be

(24) |

with , from which one can easily obtain the resonance energy that has been given in Eq. (3). Substituting Eq. (24) into Eq. (21) or (22), we immediately arrive at the maximum value of on the resonance:

(25) |

an interesting and instructive result whose leading and next-to-leading-order parts have been shown in Eq. (4). Taking the best-fit values of , , and for example, we obtain GeV (or GeV) and (or ) for the normal (or inverted) neutrino mass ordering from the analytical formulas in Eqs. (24) and (25), in good agreement with the more exact numerical results that have been listed in Eq. (12).

In Fig. 3 we compare the result of obtained from our analytical approximation made in Eq. (21) or (22) with its exact numerical result by allowing the neutrino beam energy to vary from zero to GeV. We see that the two sets of results agree with each other in a perfect way. In contrast, the numerical result of obtained from Freund’s analytical approximation in Eq. (13) is not so good in the range, and it becomes out of control for GeV. Hence our analytical approximations stand out as a much better tool of understanding the salient features of the matter-corrected Jarlskog parameter in the GeV region. In fact, the typical neutrino beam energy of the realistic T2K long-baseline oscillation experiment [5] is about GeV, just within this region. So one may use the analytical formulas given in the present work to do a reliable phenomenological analysis of CP violation and the associated matter effects in the T2K experiment.

Given the resonance energy in Eq. (3) and the maximum value in Eq. (4), the profiles of in the left and right panels of Fig. 2 can easily be understood. Simply because the next-to-leading-order terms of and are both proportional to the expansion parameter , they exhibit a small but appreciable difference in Fig. 2 with respect to the normal and inverted neutrino mass hierarchies. This observation indicates that even a low-energy neutrino oscillation experiment could have the potential to probe not only the CP- and T-violating effects but also the neutrino mass ordering.

At this point it is worth stressing that the matter-induced amplification or enhancement of under discussion is actually associated with the sensitivity of to the matter-induced correction. It is well known that , and are almost insensitive to terrestrial matter effects (i.e., , and ) in the GeV region [24, 28, 39], and hence the first equality in Eq. (7) leads us to the relation

(26) |

where Eq. (21) or (22) has been taken into account. So the behavior of the ratio of to changing with is expected to be the same as that of shown in Fig. 2.

Last but not least, let us figure out the upper limit of which allows to hold. For this purpose, we take in Eq. (21) or (22) and then solve this equation. Besides the trivial solution , there is a nontrivial solution

(27) |

which is valid for both normal and inverted neutrino mass hierarchies. Namely, holds for — the region of which might be especially interesting for the study of leptonic CP violation in a low-energy medium-baseline neutrino oscillation experiment. If only the leading term in Eq. (27) is taken into account (i.e., omitting the term and taking ), we are then left with GeV by considering and inputting the best-fit values of and [4]. Given the best-fit values and ranges of , , and listed in Table 1, the more accurate results of can be obtained from solving in Eq. (7) in a numerical way: GeV (best-fit) and ( range) for the normal neutrino mass ordering, or GeV (best-fit) and ( range) for the inverted neutrino mass ordering. These results are consistent with those shown in Fig. 2.

Since holds as a good approximation, one could consider to set the neutrino beam energy in the range when designing a realistic medium-baseline oscillation experiment to probe the region of CP violation. In fact, the typical beam energies of the proposed MOMENT [13] and ESSSM [14] experiments just lie in such an interesting region.

## 3 The matter-deformed unitarity triangles

The three Dirac UTs defined in vacuum in Eq. (5) have their counterparts in matter, namely,

(28) |

Thanks to the unitarity of and , the areas of
and (for
) are equal to and
, respectively. Hence a change of the ratio
with the neutrino beam energy
implies that the three UTs must be deformed by terrestrial matter
effects. The exact analytical expressions of the three sides of
for a constant matter profile have
been derived in Ref. [40]. Here we find a more convenient
way to reexpress the previous results
^{9}^{9}9In the low-energy region under consideration we find that
the side of ,
where the subscripts , and run over ,
and cyclically, is least sensitive to terrestrial
matter effects. Hence it is appropriate to express the other two
sides in matter as (for or ), in which
the coefficients and deviate respectively from
and due to the matter-induced corrections.,
and take into account both normal and inverted neutrino mass
hierarchies
^{10}^{10}10In this connection only the possibility of a normal
neutrino mass hierarchy is discussed analytically and numerically in
the literature. The present work improves the previous ones by
taking account of both normal and inverted mass hierarchies and shows
the phenomenological differences between these two cases..
To be specific, we obtain the formulas for a neutrino beam as
follows: