The Effective $\Delta m^2_{ee}$ in Matter

In this paper we generalize the concept of an effective $\Delta m^2_{ee}$ for $\nu_e/\bar{\nu}_e$ disappearance experiments, which has been extensively used by the short baseline reactor experiments, to include the effects of propagation through matter for longer baseline $\nu_e/\bar{\nu}_e$ disappearance experiments. This generalization is a trivial, linear combination of the neutrino mass squared eigenvalues in matter and thus is not a simple extension of the usually vacuum expression, although, as it must, it reduces to the correct expression in the vacuum limit. We also demonstrated that the effective $\Delta m^2_{ee}$ in matter is very useful conceptually and numerically for understanding the form of the neutrino mass squared eigenstates in matter and hence for calculating the matter oscillation probabilities. Finally we analytically estimate the precision of this two-flavor approach and numerically verify that it is precise at the sub-percent level.


I. INTRODUCTION
Since the discovery that neutrinos oscillate [1,2] tremendous progress has been made in understanding their properties. The oscillation parameters are all either well-measured or will be with the advent of next generation experiments. As the final parameters are measured, precision in the neutrino sector becomes more important than ever.
In vacuum, an effective two-flavor oscillation picture was presented in [3] for calculating the ν e → ν e disappearance probability which introduced an effective ∆m 2 , ∆m 2 ee ≡ cos 2 θ 12 ∆m 2 31 + sin 2 θ 12 ∆m 2 32 , which precisely and optimally determines the shape of the disappearance probability around the first oscillation minimum. That is, even in the three favor framework, for ν e disappearance in vacuum (P 0 ), the two-flavor approximation P 0 (ν e → ν e ) : ≈ 1 − sin 2 2θ 13 sin 2 ∆ ee , where ∆ ee ≡ ∆m 2 ee L/(4E) , is an excellent approximation at least over the first oscillation. ∆m 2 ee has been widely used by the short baseline reactor experiments, Daya Bay [4] and RENO [5] in their shape analyses around the first oscillation minimum and will be precisely measured to better than 1% in the medium baseline JUNO [6] experiment.
The matter generalization of the three-flavor ν e disappearance probability in matter (P a ) can also be ade- * peterbd1@gmail.com; 0000-0002-5209-872X † parke@fnal.gov; 0000-0003-2028-6782 quately approximated by a two-flavor disappearance oscillation probability in matter where ∆ ee ≡ ∆ m 2 ee L/(4E) , and x denotes the exact matter version of a variable and is a function of the Wolfenstein matter potential [7]. This new ∆ m 2 ee would be the dominant frequency, over the first few oscillations, for ν e disappearance at a potential future neutrino factory [8] in the same way that ∆m 2 ee is for short baseline reactor experiments. As we will find in section II, satisfies all of the necessary criteria to describe ν e disappearance in matter in the approximate two-flavor picture of eq. 3 above and trivially reproduces eq. 1 in vacuum. We will also discuss an alternate expression ∆ m 2 EE which numerically behaves quite similarly, but is somewhat less useful analytically.
The layout of this paper is as follows. In section II we define the matter version of ∆m 2 ee denoted ∆ m 2 ee . We review the connection between the three-flavor and twoflavor expressions in section III which naturally leads to a slightly different expression dubbed ∆ m 2 EE . In section IV we show how the natural definition of ∆ m 2 ee matches the expression given from a perturbative description of oscillation probabilities. We analytically and numerically show that both expressions are very close in section V. We perform the numerical and analytical calculations to show the precision of this definition of ∆ m 2 ee compared with other definitions of ∆m 2 ee in matter in section VI. Finally, we end with our conclusions in section VII, and some details are included in the appendices.

II. DEFINING ∆ m 2 ee IN MATTER
In this section we create a qualitative picture to derive the ∆ m 2 ee presented in the previous section. We then verify that it passes the necessary consistency checks. Figure 1 gives the neutrino mass squared eigenvalues in matter, m 2 i , as a function of the neutrino energy as well as the value of their electron neutrino content, | U ei | 2 . Neutrinos (anti-neutrinos) are positive (negative) energy in this figure and vacuum corresponds to E = 0. From the ν e content, it is clear that for energies greater than a few GeV that ∆ m 2 32 will dominate the L/E dependence of ν e disappearance and similarly ∆ m 2 31 will dominate for energies less than negative, a few GeV, that is, where a = 2 √ 2EG F N e is the matter potential, G F is Fermi's constant, N e is the electron density, and the m 2 i /2E are the exact eigenvalues which are calculated in [9], see also appendix A. This is independent of mass ordering.
It is also useful to note that m 2 0 can be written as Then, as suggested by eq. 4, ∆ m 2 ee can also be written in the following simple and easy to remember form, where recovery of the vacuum limit is manifest. In the following sections we will address in more detail why the definition of eq. 4 works for all matter potentials including |a/∆m 2 21 | 1. Here we will use eq. 4 to re-write the m 2 i 's in matter as a function of the two relevant ∆ m 2 's: ∆ m 2 ee and ∆ m 2 21 . By properties of the trace of the Hamiltonian 2 , we have m 2 3 + m 2 2 + m 2 1 = ∆m 2 31 + ∆m 2 21 + a .
Then together with eq. 6 above We make the typical definition ∆ m 2 21 ≡ m 2 2 − m 2 1 , then 1 Note that m 2 0 is identical to λ b = λ 0 from [10].    [11] or Table 4 of [10]. Adding the same constant to all entries in this table, does not effect oscillation physics. Our convention is that in vacuum m 2 We can also use ∆ m 2 ee to estimate ∆ m 2 21 except near where we have made the natural definition, as the effective matter potential for the 12 sector as was used in [12]. For this derivation eq. 11 is needed. The asymptotic eigenvalues in Table I, can also be used to obtain a simple approximate expression for ∆ m 2 ee , when |a| ∆m 2 ee : These two asymptotic expressions for ∆ m 2 ee and ∆ m 2 21 , eqs. 16 and 14 respectively, which were obtained with only general information of the neutrino mass squareds in matter here, will be compared to the expressions obtained using the approximations of [11] & [10] in section IV.
Note 5 that m 2 3 (a → ∞) → m 2 a and m 2 1 (a → −∞) → m 2 a . In the low energy limit, when | m 2 3 | | m 2 j | for j = (1, 2, a), a first order perturbative expansion in m 2 j / m 2 3 gives consistent with our previous definition, eq. 6. In fact, ∆ m 2 ee and ∆ m 2 EE differ by less than < 0.3% for all values of matter potential.
In vacuum (E = 0), it is known that eq. 2 is an excellent approximation over the first couple of oscillations see e.g. [15], further verifying the use of this two-flavor approximation. The analysis of this paper can be trivially extend away from vacuum region using the matter oscillation parameters.

IV. RELATION TO DMP APPROXIMATION
While eq. 6 is a compact expression that behaves as we expect ∆ m 2 ee ought to, it is not simple due to the complicated expressions for the eigenvalues, in particular the cos( 1 3 cos −1 . . . ) part of each eigenvalue, see appendix A. In order to both verify the behavior of ∆ m 2 ee for |a/∆m 2 ee | 1 and provide an expression that is simple we look to approximate expressions of the eigenvalues.
In refs. [11], [10] & [12] (DMP) simple, approximate, and precise analytic expressions were given for neutrino oscillations in matter. In the DMP approximation 6 through zeroth order, the definition of ∆ m 2 ee given in eq. 6 can be shown to be = ∆m 2 ee (cos 2θ 13 − a/∆m 2 ee ) 2 + sin 2 2θ 13 , where θ 12 and θ 13 are excellent approximations for the matter mixing angles θ 12 and θ 13 and ∆ m 2 31 and ∆ m 2 32 are the corresponding approximate expressions for ∆ m 2 31 and ∆ m 2 31 from [10] and reproduced in appendix B below 7 . The approximation has corrections to the eigenvalues of O( 2 ) where = sin( θ 13 − 5 Also note that m 2 a is identical to λa from [10]. 6 In the notation of DMP, ∆ m 2 ee ≡ ∆λ +− = cos 2 ψ ∆λ 31 + sin 2 ψ ∆λ 32 , see eq. A.1.7 of [10]. Also, θ 12 = ψ and m 2 i = λ i in DMP; see [12]. 7 The notation is such that while both x and x are quantities in matter, x denotes the exact quantity and x denotes the zeroth order approximation from DMP, and x is an excellent approximation for x. θ 13 )s 12 c 12 ∆m 2 21 /∆m 2 ee . | | < 0.015 and is equal to zero in vacuum. Equation 23 provides a very simple means to modify the vacuum ∆m 2 ee to get the corresponding expression in matter.
In the DMP approximation, all three expressions, eq. 23, for ∆ m 2 ee can be shown to be analytically identical. This is however not true for the exact eigenvalues and mixing angles in matter, there are small differences between these expressions (quote fractional differences.). We use the first line of eq. 23 for our definition ∆m 2 ee in matter, because this definition allows us a general understanding of the three neutrino eigenvalues in matter (see eqs. 12 and 13). We now verify that this definition of ∆m 2 ee in matter meets all the other criteria we need it to.
In the next section we will analytically and then numerically show that the fractional difference between the two expressions, ∆ m 2 ee and ∆ m 2 EE , are small.
The two expressions can be seen as two choices for the how to relate these to the matter version: one is to elevate each eigenvalue to its matter equivalent (everything except m 2 0 ) and the other is to elevate each term including the mixing angles. We refer to the former as ∆ m 2 ee and the latter as ∆ m 2 EE .
To understand how these expressions differ, we carefully examine their difference, We now quantify the difference between these expressions using DMP. If both expressions provide good approximations for the two flavor frequency in matter then the difference between them should be small. At zeroth order the difference is so these expressions are exactly equivalent at zeroth order. At first order the eigenvalues receive no correction, but θ 12 does. From [14] we have that the first order correction is where t ij = tan θ ij . This leads to a correction of, (31) As expected ∆ Ee ∝ a for small a. Also, we can verify that ∆ Ee /∆ m 2 ee is always small by seeing that a/∆ m 2 ee remains finite and the only case where t 13 ∝ a for a → ∞, but ∆ m 2 32 ∆ m 2 31 ∝ a 2 , thus the difference between the two expressions is always small. ∆ Ee provides an adequate approximation of the difference between ∆ m 2 ee and ∆ m 2 EE as shown in fig. 2. A precise estimate of the difference requires the second order correction to θ 12 given explicitly in [14] along with the second order corrections to the eigenvalues from DMP. This is because this difference ∆ Ee depends strongly on the asymptotic behavior of θ 12 which only becomes precise beyond the atmospheric resonance at second order. The result of this is also shown in fig. 2 which shows that first order is not sufficient to accurately describe the difference, but second order is. We see that for neutrinos the expressions agree to 0.3%, and the agreement is ∼ 3 orders of magnitude better for anti-neutrinos.
In the next section we will investigate how well the twoflavor approximation, eq. 3, works numerically for both the depth and position over the first oscillation minimum for ν e disappearance for all values of the neutrino energy.

VI. PRECISION VERIFICATION
The goal of ∆ m 2 ee is to provide the correct frequency such that the two-flavor disappearance expression, eq. 3, is an excellent approximation for ν e disappearance over the first oscillation in matter. In particular, we want this expression to reproduce the position and depth of the first oscillation minimum at high E (small L) correctly compared to the complete three-flavor picture.

A. Numerical Comparison
Using the definition of ∆ m 2 ee given in eq. 6, we plot in fig. 3 ∆ m 2 ee ∆m 2 ee 2 (1 − P a (ν e → ν e )) verses ∆ ee , (32) for various values of the neutrino energy. Here P a (ν e → ν e ) is evaluated using the exact oscillation probability given in [9]. We see that this behaves like sin 2 ∆ ee as expected, with increasing precision for increasing energy. Note the approximate neutrino energy independence of this figure, demonstrating the universal nature of the approximation given in eq. 3 using our definition of ∆ m 2 ee .
Next, we want to check that this two-flavor expression reproduces the first oscillation minimum at high E (small L) correctly compared to the complete three-flavor picture. The minimum occurs when the derivate of P is zero. We now have a choice: we can define the minimum when dP a /dL = 0 or dP a /dE = 0. Since both θ and ∆ m 2 ee are nontrivial functions of E, the correct option is to use dP a /dL = 0.
In order to numerically test the various expressions, we find the location L of the first minimum by solving dP a /dL = 0 for a given E using the full three-flavor expressions. We then convert the (L, E) pair at the first minimum into the corresponding ∆ m 2 ee using Next, we compare the difference between this numeric solution and the expressions presented in this paper, eqs. 4, 19, and 23. We also compare to the approximate analytic solution from [16] (HM), see appendix C. This comparison is shown in fig. 4. When determining the minimum from the exact expression, a two-flavor expression using only ∆ m 2 ee will get the ∆m 2 31 and ∆m 2 32 terms correct including matter effect, but will always be off by ∆m 2 21 terms. Thus in fig. 4 we don't include the effect of the 21 term which will affect any two-flavor approximation comparably.
We see that for either eq. 6 or eq. 23 the agreement is excellent with relative error < 0.2%. In addition, the two expressions clearly agree with each other to a higher level of precision than is necessary. For the HM expression the agreement is good for anti-neutrinos and in the high energy limit, but is poor in a broad range near the atmospheric resonance for neutrinos. In addition, we have modified the HM expression by taking the absolute value so that the HM expression asymptotically returns to the correct expression past the atmospheric resonance for neutrinos.  4. We show the fractional error (δx/x) of various different ∆ m 2 ee expressions with the precise numerical one determined at the point where dPa/dL = 0, see eq. 33. For the exact numerical expression we ignore the ∆ m 2 21 term as no definition will get it correct. The ee curve uses the formula from eq. 4 and the EE curve uses the formula from eq. 19. The DMP curve uses the zeroth order expressions [10] in the same formula which leads to the simple expression shown in eq. 23. The HM curve uses the expression from [16] and takes the absolute value to get the sign correct for large E, see appendix C. We have fixed ρ = 3 g/cc and assumed the NO. E > 0 corresponds to neutrinos, E < 0 corresponds to antineutrinos, and E = 0 corresponds to the vacuum.
We have also compared ∆ m 2 ee with the exact solution including the ∆m 2 21 term and found agreement to better than 1%.

B. Analytic Comparison
We now analytically estimate the precision of the twoflavor expression, for both the small E (large L) limit and the large E (small L) limit.
First, if ∆ m 2 21 |∆ m 2 ee | then at the n th oscillation minimum the ratio of the 21 term to the ee term is well approximated by as derived in appendix D. For the first (second) oscillation peak this yields an error estimate of < 2% (16%); this two-flavor approach breaks down for n > 5 when the ratio is > 1. The second case is when ∆ m 2 21 |∆ m 2 ee |, which occurs away from vacuum (high E, low L), and the ratio of the 21 coefficient to the ee coefficient is which is small away from vacuum as desired. In particular, it is < 1 for |E| > 1 GeV. See appendix D for details and numerical confirmation of each region.

VII. CONCLUSIONS
In this paper, we have demonstrated that is the matter generalization of vacuum ∆m 2 ee that has been widely used by the short baseline reactor experiments Daya Bay and RENO and will be precisely measured (< 1%) in the medium baseline JUNO experiment. The exact and approximate expressions in the above equation differ by no more than 0.06%. Another natural choice called ∆ m 2 EE is numerically very close to ∆ m 2 ee but does not provide the ability to simply rewrite the eigenvalues as ∆ m 2 ee does.
For ν e disappearance in matter the position of the first oscillation minimum, for fixed neutrino energy E, is given by and the depth of the minimum is controlled by ≈ sin 2 2θ 13 (cos 2 2θ 13 − a/∆m 2 ee ) 2 + sin 2 2θ 13 .
This two-flavor approximate expression is not only simple and compact, but it is precise to within < 1% precision at the first oscillation minimum 8 . The combination of ∆ m 2 ee and ∆ m 2 21 is very powerful for understanding the effects of matter on the eigenvalues and the mixing angles of the neutrinos. In this article we have illuminated the exact nature of ∆ m 2 ee and ∆ m 2 21 which were extensively used in DMP [10,12].
The remaining two oscillation parameters, θ 23 = θ 23 and δ = δ, remain unchanged in this approximation. We note that for each parameter above x provides an excellent approximation for x.
We also note two additional useful expressions, An alternate approximate expression was previously provided in [16], the expression from that paper is where r A ≡ a/∆m 2 31 . This expression clearly has a pole at a = ∆m 2 31 which is the atmospheric resonance for neutrinos. In addition, past the resonance, for a > ∆m 2 31 , the sign is incorrect as ∆ m 2 ee,HM < 0 for the NO. Thus we take the absolute value in our numerical studies.
In fig. 2 of [16], the author compared eq. C1 with the minimum obtained via solving dP a /dE = 0 whereas we have argued in section VI that a better comparison is obtained by solving dP a /dL = 0 for fixed E.

Appendix D: Precision in Different Ranges
In this appendix we further expand upon the discussion in subsection VI B. 3) compared to the full three-flavor expression (P3 from eq. 17) in matter is shown in the solid curves for the first several oscillation minima. The dashed lines are the simple approximation from eq. 34. As expected eq. 34 performs well near vacuum at |E| few GeV.
The exact three-flavor expression in matter from eq. 17 can be written as, 1 − P a = sin 2 2θ 13 ∆m 2 ee ∆ m 2 ee 2 sin 2 ∆ ee + C(E)c 4 13 sin 2 2 θ 12 sin 2 ∆ 21 , where C(E) 1 contains the correction between the first and second term. For the two-flavor approximation to be valid, the 21 term, C(E)c 4 13 sin 2 2 θ 12 sin 2 ∆ 21 must be small compared to the two-flavor ee term, sin 2 2 θ 13 sin 2 ∆ ee . As in section VI B, we consider two cases.
First, if ∆ m 2 21 |∆ m 2 ee | then at the n th oscillation minimum the ratio R 1 of the 21 term to the ee term is where the approximation uses the DMP zeorth order expression, the θ 13 ≈ θ 13 approximation of eq. B6, and s 2 13 ≈ ∆m 2 21 /∆m 2 ee . The C(E) term contains the effect of combining the ∆ 31 and ∆ 32 terms and is just under one within a few GeV of the vacuum. Since all of the terms in the right square bracket are < 1, We numerically confirmed that eq. 34 is correct to within ∼ 10% near vacuum as shown in fig. 5. 1. See eq. 35 in the text. We also show the ratio of the mass squared differences in matter in red.
The second case is when ∆ m 2 21 |∆ m 2 ee |, which occurs away from vacuum. In this case we compare the ratio R 2 of the coefficients which is Away from vacuum, θ 12 π/2 (0) for neutrinos (antineutrinos) (see e.g. fig. 1 of [10]) which makes the numerator of R 2 very small. The remaining part is 1/(4 tan 2 θ 13 ). This part is large only when θ 13 → 0. Since θ 12 → 0 faster than θ 13 , we always have R 2 1 as desired. See fig. 6 for a numerical verification that R 2 is small away from the vacuum.