Radiative corrections to the lepton flavor mixing in dense matter

One-loop radiative corrections will lead to a small difference between the matter potentials developed by $\nu_\mu^{}$ and $\nu_{\tau}$ when they travel in a medium. By including such radiative corrections, we derive the exact expressions of the corresponding effective mass-squared differences and the moduli square of the lepton flavor mixing matrix elements $|\widetilde U_{\alpha i}^{}|^2$ (for $\alpha=e,\mu,\tau $ and $i=1,2,3$) in matter in the standard three-flavor mixing scheme and focus on their asymptotic behaviors when the matter density is very big (i.e., the matter effect parameter $A\equiv 2\sqrt{2} G_{\rm F}^{} N_e^{}E$ is very big). Different from the non-trivial fixed value of $|\widetilde U_{\alpha i}^{}|^2$ in the $A\to \infty$ limit in the case without radiative corrections, we get $|\widetilde U_{\alpha i}^{}|^2=0~{\rm or}~1$ under this extreme condition. The radiative corrections can significantly affect the lepton flavor mixing in dense matter, which are numerically and analytically discussed in detail. Furthermore, we also extend the discussion to the $(3+1)$ active-sterile neutrino mixing scheme.


Introduction
In 1978, Wolfenstein firstly pointed out that when neutrinos travel in matter, the coherent forward scattering of them with electrons and nucleons must be considered and the induced matter potentials will change the neutrino oscillation behaviors [1]. In 1985, Mikheev and Smirnov put forward that the effective mixing angle can be significantly amplified in matter (such as inside the sun) even if the corresponding mixing angle in vacuum is small. This is the wellknown Mikheev-Smirnov-Wolfenstein (MSW) effects, which successfully explain the flavor conversion behaviors of solar neutrinos in the sun [2]. Such matter effects have been proved very important in a number of reactor, solar, atmospheric, accelerator neutrino oscillation experiments aiming to accurately extract the intrinsic neutrino oscillation parameters in vacuum [3]. A lot of efforts have been made to make the neutrino oscillation probabilities in matter more intuitive [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The language of renormalization-group equation was also introduced to describe the effective neutrino masses and flavor mixing parameters in matter [18][19][20][21]. In this paper, we mainly focus on the neutrino flavor mixing in very dense matter, which has been discussed in Refs. [22][23][24][25]. We further include the radiative corrections in this connection.
In the standard three-flavor mixing scheme, the Hamiltonian responsible for the propagation of neutrinos in matter can be expressed as where E is the neutrino beam energy, m i (for i = 1, 2, 3) and U stand respectively for neutrino masses and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton flavor mixing matrix, m i (for i = 1, 2, 3) and U denote effective neutrino masses and PMNS matrix in matter, respectively, and V α (α = e, µ, τ ) represent the matter potentials arsing from charged-and neutralcurrent coherent forward scattering of ν α with electrons, protons and neutrons in matter. Considering the one-loop radiative corrections to V α , we have [26] where G F is the Fermi coupling constant, α is the fine-structure constant; N e , N p and N n denote the number density of electrons, protons and neutrons in matter, respectively; m τ and m W are the masses of τ lepton and W boson, respectively; and sin 2 θ W ≡ 1 − m 2 W /m 2 Z with m Z being the Z boson mass. To be more intuitive, Eq. (1) can be rewritten as where ∆ ij ≡ m 2 i −m 2 j , ∆ ij ≡ m 2 i − m 2 j , A = 2 √ 2G F N e E, B = m 2 1 −m 2 1 −2EV µ , and I denotes a 3 × 3 identity matrix. According to Eq. (2) and assuming N e = N p = N n , ǫ ≃ 5 × 10 −5 is a small quantity but matters a lot in dense matter (i.e., A is very big).
On the other hand, given the implications of extra light sterile neutrinos in short-baseline neutrino oscillation experiments [27], we extend our discussion to the scheme of (3 + 1) flavor mixing with one more sterile neutrino ν s . The corresponding Hamiltonian describing the propagation of neutrinos in a medium turns out to be where V and V denote the 4 × 4 lepton flavor mixing matrix in vacuum and matter, respectively, A ′ = −2EV µ = √ 2G F N n E, ∆ 41 = m 2 4 − m 2 1 , and ∆ 41 = m 2 4 − m 2 1 with m 4 and m 4 being the sterile neutrino mass in vacuum and matter, respectively. The existence of an extra sterile neutrino can make the neutrino flavor mixing in dense matter very different from the standard three-flavor mixing scheme.
The remaining parts of this paper are organized as follows. In section 2, we include the radiative corrections and derive the corresponding expressions of ∆ ij , | U αi | 2 and U αi U * βi in matter in the standard three-flavor mixing scheme. The asymptotic behaviors of ∆ ij and | U αi | 2 and neutrino oscillations in dense matter are analytically and numerically investigated in detail. In section 3, we extend our discussion to the (3 + 1) flavor mixing scheme. Section 4 is devoted to a brief summary.

The standard three-flavor mixing scheme
In the standard three-flavor mixing scheme, the exact formulas of ∆ ij without radiative corrections have been given in Refs. [28][29][30]. Considering radiative correction effects, we derive the eigenvalues of H m in Eq. (3) and express the two independent effective masssquared differences ∆ ij (for ij = 21, 31) as for the case of normal mass ordering (NMO) with m 1 < m 2 < m 3 ; or for the case of inverted mass ordering (IMO) case with m 3 < m 1 < m 2 , where The unitarity conditions of U and the sum rules derived from H m and H 2 m , constitute a set of linear equations of U αi U * βi (for α, β = e, µ, τ and i = 1, 2, 3): where A αβ stand for the (α, β) element of the matter potential matrix A ≡ Diag{A, 0, Aǫ}. Taking α = β and solving Eq. (9), we obtain with where α = e, µ, τ and i, j, k = 1, 2, 3. Similarly, in the case of α = β, U αi U * βi can be derived from Eq. (9): with where (α, β, γ) run over (e, µ, τ ) and n = m = 1, 2. Note that U α1 U * β1 , U α2 U * β2 and U α3 U * β3 for α = β constitute the effective Dirac leptonic unitarity triangle in the complex plane. From Eq. (12), it is straightforward to check that the Naumov relation J ∆ 21 ∆ 31 ∆ 32 = J ∆ 21 ∆ 31 ∆ 32 [31] still holds, where J and J are the Jarlskog invariants [32] in vacuum and in matter, respectively, with ε αβγ and ε ijk being three-dimension Levi-Civita symbols. The only difference due to the radiative corrections in the exact formulas of ∆ ij , | U αi | 2 and U αi U * βi above is the appearance of the term A τ τ = Aǫ. By setting ǫ = 0, one can turn off the radiative corrections and get the corresponding expressions of ∆ ij , | U αi | 2 and U αi U * βi in the previous literature [30,[33][34][35]. With the help of Eqs. (5), (6) and (12), we can directly write out the probabilities of the ν α → ν β (for α, β = e, µ, τ ) oscillations in matter where α, β = e, µ, τ ; i, j = 1, 2, 3; and L is the neutrino oscillation length. Note that the results in Eqs. (5)- (15) are only valid for a neutrino beam. When it comes to an antineutrino beam, we need to do the replacements U → U * and A → −A. According to the exact expressions of ∆ ij , | U αi | 2 and U αi U * βi in Eqs. (5), (6) (10) and (12), we study the neutrino flavor mixing in dense matter in the standard three-flavor mixing scheme. Both neutrinos and antineutrinos with the normal or inverted mass ordering (i.e., cases (NMO, ν), (IMO, ν), (NMO, ν) and (IMO, ν)) will be considered separately. Numerically, we take the standard parametrization of U, and input the best-fit values of (θ 12 , θ 13 , θ 23 , δ, ∆ 21 , ∆ 31 ) in Ref. [36]: Analytically, we treat ∆ 21 /A, ∆ 31 /A and ǫ as small quantities and make perturbative expansions of ∆ ij and | U αi | 2 . Thus the analytical approximations in this section only apply to the range A ≫ ∆ 31 .

(NMO, ν)
Let us first consider the case of a neutrino beam with normal mass ordering. The corresponding evolution of ∆ ij and | U αi | 2 with the matter effect parameter A are illustrated in the upper left panel of Fig. 1 and Fig. 2, respectively. We find that the radiative corrections Figure 1: In the standard three-flavor mixing scheme, the illustration of how the effective neutrino mass-squared differences ∆ 21 and | ∆ 31 | evolve with the matter effect parameter A in the cases with or without radiative corrections, where the best-fit values of (θ 12 , θ 13 , θ 23 , δ, ∆ 21 , ∆ 31 ) in Ref. [36] have been input.
may significantly affect the values of ∆ ij and | U αi | 2 only if A is big enough (for example, A > 1 eV 2 ). This can be revealed more clearly by expanding the exact expressions of ∆ ij and | U αi | 2 in terms of ∆ 21 /A, ∆ 31 /A and ǫ. Only keeping the first order of these quantities, with or without radiative corrections, where the best-fit values of (θ 12 , θ 13 , θ 23 , δ, ∆ 21 , ∆ 31 ) in Ref. [36] have been input.
we simplify ∆ ij in Eq. (5) as According to Eq. (17), it becomes clear that if A is big enough, ∆ 21 will increase with A instead of taking a fixed value in the case without radiative corrections. The reason why the radiation corrections to ∆ 31 are not significant is just that the much smaller Aǫ term appears in the next-to-leading order with the leading order being A. By performing perturbative expansions of Eq. (10) in terms of ∆ 21 /A, ∆ 31 /A and ǫ and only keeping the leading order, | U αi | 2 are approximately expressed as This means that the neutrino flavor mixing in dense matter can be approximately described by only one degree of freedom, as having been pointed out in Ref. [24]: where θ ∈ [0, π/2] and Similarly, the neutrino oscillation probability P αβ in Eq. (15) can be approximately written as where ∆ 21 is taken from Eq. (17) and sin 2 2θ = 4| U µ1 | 2 (1 − | U µ1 | 2 ) with | U µ1 | 2 being taken from Eq. (19). Note that Eq. (22) is similar to Eq. (8) in Ref. [25] except that we include radiative corrections in θ and ∆ 21 . In order to numerically test the accuracies, we define the absolute error of P αβ as ∆ P αβ = |( P αβ ) Exact − ( P αβ ) Approximate |, where ( P αβ ) Exact stand for the exact results of P αβ and ( P αβ ) Approximate represent the approximate results of P αβ . The absolute errors of P αβ in Eq. (22) with different L/E and A/∆ 31 are demonstrated in Fig. 3. Similar to the case without radiative corrections discussed in Ref. [25], the analytical expressions of P αβ in Eq. (22) are accurate enough in most of the parameter space. For the upper left part in each subgraph of Fig. 3, we need to keep higher orders of ∆ 21 /A, ∆ 31 /A and ǫ, or just make perturbative expansions in terms of ∆ 21 /A and ǫ to improve the accuracies of P αβ .
To be more explicit, if the Aǫ term is not bigger than the ∆ 21 term in Eq. (19), we can further simplify | U αi | 2 (for αi = µ1, µ2, τ 1, . This is equivalent to the asymptotic values of | U αi | 2 (for αi = µ1, µ2, τ 1, τ 2) in the A → ∞ limit when radiative corrections are not taken into account (the blue dashed line in Fig. 2). As the increase of A, the Aǫ term in Eq. (19) becomes non-negligible. If the Aǫ term and ∆ 31 term are of the same order, the relation can be derived. This means θ = arctan(| U µ2 |/| U µ1 |) will decrease with the increase of A due to the existence of the radiative correction parameter ǫ. In the A → ∞ limit, it is easy to infer from Eqs. (19) and (21) that | U αi | 2 trivially take 0 or 1 and θ is approaching zero, implying that all the three flavors do not oscillate into one another. Thus it makes no sense to discuss lepton flavor mixing in this extreme case. Considering the four cases (NMO, ν), (IMO, ν), (IMO, ν) and (IMO, ν) separately, we summarize the corresponding analytical expressions of ∆ ij (for ij = 21, 31) and U in the A → ∞ limit in Table 1 while the other three cases will be discussed later.
Given a neutrino beam with inverted mass ordering, the evolution of ∆ ij and | U αi | 2 with A are illustrated in the lower left panel of Fig. 1 and Fig. 4, respectively. Note that there is no intersections between ∆ 21 and ∆ 31 in cases (IMO, ν) and (IMO, ν) in Fig. 1 with | ∆ 31 | = − ∆ 31 being shown in fact. In the case (IMO, ν), we also note that ∆ 21 > | ∆ 31 | holds when the matter effect parameter A is big enough. Analytically, expanding Eq. (6) in ∆ 21 /A, ∆ 31 /A and ǫ directly leads to with ξ being defined in Eq. (18). Consistent with Fig. 1, ∆ 31 approaches −Aǫ instead of a constant value in the A → ∞ limit. To understand the asymptotic behaviors of | U αi | 2 in the A → ∞ limit shown in Fig. 4, we expand Eq. (10) and get So one can use only one parameter to approximately describe lepton flavor mixing,  22), where the best-fit values of (θ 12 , θ 13 , θ 23 , δ, ∆ 21 , ∆ 31 ) in Ref. [36] and ǫ ≃ 5 × 10 −5 [26] have been input.
where θ ∈ [0, π/2] and The analytical approximations of P αβ in Eq. (15) turn out to be where ∆ 31 comes from Eq. (24) and sin 2 2θ = 4| U µ1 | 2 (1 − | U µ1 | 2 ) with | U µ1 | 2 coming from Eq. (25). For simplicity, we do not show the accuracies of Eq. (28), which are very similar to the case (NMO, ν) in Fig. 3. Comparing Eq. (28) with Eq. (22), we find that it is impossible to discriminate the normal mass ordering from the inverted mass ordering from neutrino oscillations if the matter density is very big. We also notice that if the ∆ 21 term and Aǫ term in Eq. (25) are of the same order, one can omit them and get . This corresponds to the fixed values of | U αi | 2 (for αi = µ1, µ3, τ 1, τ 3) in the A → ∞ limit if radiative corrections are not taken into account (the blue dashed line in Fig. 4). As the increase of A, the Aǫ term will gradually dominate and the neutrino oscillation behaviors can be very sensitive to A. In the limit of A → ∞, θ approaches π/2 and there will be no neutrino oscillation phenomenon.

(NMO, ν)
Considering an antineutrino beam with normal mass ordering, we make the replacements A → −A and U → −U * in Eqs. (5) and (10), and draw the corresponding evolution of ∆ ij and | U αi | 2 with A in the upper right panel of Fig. 1 and Fig. 5, respectively. Note that we always have ∆ 31 > ∆ 21 in this case although the difference between them is too small to be shown clearly in Fig. 1 if A is big enough. The radiative corrections to both ∆ 21 and ∆ 31 are very small, which can be analytically understood. By performing perturbative expansions, ∆ ij (for ij = 21, 31) in Eq. (5) are reduced to From Eq. (29), it is clear that the leading order of ∆ 21 and ∆ 31 is A and the radiative corrections in the next-to-leading order do not matter a lot. The only difference between Eq. (30) (the expression of ξ in the case (NMO, ν)) and Eq. (18) (the expression of ξ in the case (NMO, ν)) is the sign of ǫ. Similarly, | U αi | 2 can be expanded as namely, with or without radiative corrections, where the best-fit values of (θ 12 , θ 13 , θ 23 , δ, ∆ 21 , ∆ 31 ) in Ref. [36] have been input.
If A is small enough, we can ignore the smaller terms of ∆ 21 and Aǫ in Eq. (31), and This is consistent with the fixed values of | U αi | 2 (for αi = µ2, µ3, τ 2, τ 3) in the A → ∞ limit if radiative corrections are not included (the blue dashed line in Fig. 5). If the term Aǫ too big to be abandoned, the neutrino flavor mixing can be significantly affected by A. In the A → ∞ limit, | U αi | 2 trivially take 0 or 1 and θ approaches π/2, leading to no neutrino oscillations.

(IMO, ν)
Similarly, in the case (IMO, ν), i.e. an antineutrino beam with inverted mass ordering, we illustrate the evolution of ∆ ij and | U αi | 2 in the lower right panel of Fig. 1 and Fig. 6, respectively. Through making perturbative expansions, ∆ ij and | U αi | 2 can be approximately expressed as Figure 6: In the standard three-flavor mixing scheme, the illustration of how | U αi | 2 (for α = e, µ, τ and i = 1, 2, 3) evolve with the matter effect parameter A in the case (IMO, ν) with or without radiative corrections, where the best-fit values of (θ 12 , θ 13 , θ 23 , δ, ∆ 21 , ∆ 31 ) in Ref. [36] have been input. and where ξ has been defined in Eq. (30). From either Fig. 1 where θ ∈ [0, π/2] and The corresponding analytical approximations of P αβ are the same as Eq. (22) . This coincides with the fixed values of | U αi | 2 (for αi = µ1, µ2, τ 1, τ 2) in the A → ∞ limit if radiative corrections are not included (the blue dashed line in Fig. 6). If the term Aǫ is too big to be omitted, it will affect the neutrino flavor mixing a lot. In the A → ∞, θ approaches π/2 and no neutrino oscillations between ν e , ν µ and ν τ will happen.
Specifically, if Aǫ is negligible in Eqs. (54) and (57), one may abandon the smaller terms of ∆ 21 and ∆ 31 , and get where |V µ4 | 2 = cos 2 θ 14 sin 2 θ 24 , |V τ 4 | 2 = cos 2 θ 14 cos 2 θ 24 sin 2 θ 34 from the parametrization of V in Eq. (53). This is equivalent to the asymptotic behaviors of ∆ 21 and | V αi | 2 (for αi = µ1, µ2, τ 1, τ 2) in very dense matter in the case without radiative corrections (i.e. the blue dashed lines in Fig. 8). Due to the typical value θ 34 = 0 inputted in Fig. 8, we get By choosing a non-zero value of θ 34 , the neutrino flavor mixing can be very different. With the increase of A, the Aǫ term in Eqs. (54) and (57) will become dominate. In the A → ∞ limit, With | V αi | 2 taking 0 or 1 (or θ s → 0 or π/2), there will be no neutrino oscillation between the four flavors and it makes no sense to discuss lepton flavor mixing.

Summary
With the coming of the precision measurement era of neutrino physics, we are committed to digging the underlying physics behind the lepton flavor mixing [40] and on the other hand to conducting cosmological and astronomical researches with neutrinos being a good probe. As preliminarily discussed in Ref. [25], it is possible to explore the density and size of a hidden compact object in the universe by observing its effects on the neutrino flavor mixing. In this paper, we point out that radiative corrections to the matter potentials can significantly affect the neutrino flavor mixing in dense matter. Considering the standard three-flavor mixing scheme with radiative corrections, we derive the exact expressions of the effective neutrino mass-squared differences ∆ ij , the moduli square of the nine lepton flavor mixing matrix elements | U αi | 2 , the vector sides of the Dirac leptonic unitarity triangles U αi U * βi in a medium. From these exact formulas, the neutrino flavor mixing in dense matter are numerically and analytically discussed. Different from the fixed value of | U αi | 2 in dense matter in the case without radiative corrections, | U αi | 2 can be very sensitive to the value of A and trivially approach 0 or 1 in the A → ∞ limit if radiative corrections are taken into account. When it comes to the (3 + 1) flavor mixing scheme, the neutrino flavor mixing will be very different from the standard three-flavor scheme if A is big enough but not infinite. However it is meaningless to discuss the lepton flavor mixing in both schemes in the A → ∞ limit.