Renormalization-group equations of neutrino masses and flavor mixing parameters in matter

Journal of High Energy Physics, May 2018

Abstract We borrow the general idea of renormalization-group equations (RGEs) to understand how neutrino masses and flavor mixing parameters evolve when neutrinos propagate in a medium, highlighting a meaningful possibility that the genuine flavor quantities in vacuum can be extrapolated from their matter-corrected counterparts to be measured in some realistic neutrino oscillation experiments. Taking the matter parameter \( a\equiv 2\sqrt{2}\kern0.5em {G}_{\mathrm{F}}{N}_eE \) to be an arbitrary scale-like variable with N e being the net electron number density and E being the neutrino beam energy, we derive a complete set of differential equations for the effective neutrino mixing matrix V and the effective neutrino masses \( {\tilde{m}}_i \) (for i = 1, 2, 3). Given the standard parametrization of V , the RGEs for \( \left\{{\tilde{\theta}}_{12},\kern0.5em {\tilde{\theta}}_{13},\kern0.5em {\tilde{\theta}}_{23},\kern0.5em \tilde{\delta}\right\} \) in matter are formulated for the first time. We demonstrate some useful differential invariants which retain the same form from vacuum to matter, including the well-known Naumov and Toshev relations. The RGEs of the partial μ-τ asymmetries, the off-diagonal asymmetries and the sides of unitarity triangles of V are also obtained as a by-product.

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Renormalization-group equations of neutrino masses and flavor mixing parameters in matter

HJE Renormalization-group equations of neutrino masses and avor mixing parameters in matter Zhi-zhong Xing 0 1 3 4 5 Shun Zhou 0 1 3 5 Ye-Ling Zhou 0 1 2 0 Durham DH1 3LE , U.K 1 Beijing 100049 , China 2 Institute for Particle Physics Phenomenology, Department of Physics, Durham University 3 School of Physical Sciences, University of Chinese Academy of Sciences 4 Center for High Energy Physics, Peking University 5 Institute of High Energy Physics, Chinese Academy of Sciences We borrow the general idea of renormalization-group equations (RGEs) to understand how neutrino masses and avor mixing parameters evolve when neutrinos propagate in a medium, highlighting a meaningful possibility that the genuine ties in vacuum can be extrapolated from their matter-corrected counterparts to be measured in some realistic neutrino oscillation experiments. Taking the matter parameter a 2 2 GFNeE to be an arbitrary scale-like variable with Ne being the net electron number density and E being the neutrino beam energy, we derive a complete set of di erential equations for the e ective neutrino mixing matrix V and the e ective neutrino masses mei (for i = 1; 2; 3). Given the standard parametrization of V , the RGEs for fe12; e13; e23; eg in matter are formulated for the rst time. We demonstrate some useful di erential invariants which retain the same form from vacuum to matter, including the well-known Naumov and Toshev relations. The RGEs of the partial - asymmetries, the o -diagonal asymmetries and the sides of unitarity triangles of V are also obtained as a by-product. Neutrino Physics; Renormalization Group 1 Introduction 2 3 4 5 Renormalization-group equations Mixing angles and CP-violating phase Some further discussions Concluding remarks which is conventionally parametrized in terms of three avor mixing angles f 12; 13; 23g 0Ue1 Ue2 Ue3 1 0 c12c13 s12c13 s13 1 s12c23e i U 1 U 2 U 3 c12s13c23 + s12s23e i s12s13c23 c12s23e i c13c23 s12s13s23 + c12c23e i c13s23AC ; (1.1) 31 i.e., m23 m21 where cij cos ij and sij sin ij (for ij = 12; 13; 23) have been introduced. The latest global- t analysis of neutrino oscillation data yield [ 2 ] the best- t values of four avor mixing parameters 12 10 3 eV2. Although there exists currently a slight preference for the normal neutrino mass ordering (NO, i.e., 31 > 0), the inverted mass ordering (IO, 31 < 0) is still allowed. Some preliminary hints on the maximal CP-violating phase 270 arise from the long-baseline accelerator neutrino experiments [ 3, 4 ], which needs to be con rmed when more data are available in the near future. The determination of neutrino mass ordering and leptonic CP-violating phase in the long-baseline accelerator neutrino oscillation experiments calls for an excellent understanding of the Mikheyev-Smirnov-Wolfenstein (MSW) matter e ects [5{7], which becomes 1Throughout this work we do not consider the possible Majorana phases, simply because they are irrelevant to neutrino oscillations in both vacuum and matter. { 1 { its counterpart in matter are de ned via where e ij m2 e i mej2 (for ij = 12; 23; 31), and the Jarlskog invariant in vacuum [14] and Je = J Ve1 Ue1 Ve2 Ue2 Ve3 = Ue3 0 0 me21 0 10 5 GeV 2 being the Fermi constant, Ne the net electron number density and E the neutrino beam energy. For antineutrino oscillations in matter, one may simply replace U by U and a by a in the e ective Hamiltonian. In avor mixing matrix V and neutrino masses mei (for i = 1; 2; 3) in matter have been de ned. For any realistic pro le of the matter density, it is possible to numerically calculate neutrino oscillation probabilities by solving the evolution equations of neutrino avor states. However, the analytical relations or identities between the e ective mixing parameters in matter and the fundamental ones in vacuum are very helpful. For instance, the well-known Naumov [8{11] and Toshev [12] relations can be summarized as [13] crucially important when the neutrino beam propagates in the Earth matter for a long distance. For the three- avor neutrino oscillations in matter, the e ective Hamiltonian reads and mei2 (for with and ijk being totally antisymmetric tensors, and ( ; ; ) and (i; j; k) being the unitarity triangles of V in matter [15{17]. cyclic permutations of (e; ; ) and (1; 2; 3), respectively. Moreover, some interesting sum rules for mei2 and the matrix elements of V have been derived in ref. [11] and used to study In this paper we emphasize that the dependence of the e ective mixing parameters V i and i = 1; 2; 3) on the matter term a can perfectly be described by a complete set of di erential equations, which are analogous to the renormalizationgroup equations (RGEs) associated with the dependence of fundamental parameters on the renormalization energy scale or distance in quantum eld theories [18, 19], solid-state physics [20, 21] and other elds of modern physics [22].2 Although this interesting analogy has already been pointed out in refs. [23, 24], it deserves some highlights and a further study. We argue that the introduction of e ective neutrino mass-squared di erences and e ective avor mixing parameters guarantees the form invariance of neutrino oscillation probabilities in vacuum and in a medium with arbitrary values of a. Such a form invariance 2Although E in a denotes the kinetic energy of a neutrino beam, it is also a re ection of the energy scale associated with weak charged-current interactions between the electron neutrino (or antineutrino) avor and the electrons in matter. In this sense it should be reasonable to treat a as a scale-like variable. { 2 { (or self-similarity) exactly re ects the spirit of the RGEs [18{20], and thus it implies the validity of the RGE-like approach for neutrino oscillations in matter. It is worth remarking that our present work di ers from refs. [23, 24] in several nontrivial aspects. First, we explain why the RGE language can be applied to the description of neutrino oscillation parameters in matter changing with the scale-like variable a. With this key point in mind, we derive the RGEs for neutrino masses mei, the squared-moduli of avor mixing matrix elements jV ij2 and even the matrix elements V i themselves. Second, we demonstrate that the standard parametrization of V is most convenient for the derivation of the RGEs of three avor mixing angles and one CP-violating phase, because it makes the rst row of V so simple that the coherent forward scattering between electrons and electron neutrinos (or antineutrinos) via weak charged-current interactions can be described in a very simple way. The RGEs of such mixing parameters will also be numerically solved, and the salient features of their evolution with respect to the matter parameter a will be discussed. Third, the RGEs of Je and some other interesting quantities, such as the partial - asymmetries, the o -diagonal asymmetries and the sides of unitarity triangles of V , are derived as a by-product. Fourth, we compare the newly obtained di erential results with some previously obtained integral results, and highlight the complementarity of both approaches in describing and understanding matter e ects on neutrino oscillations. In particular, we highlight that the RGEs for neutrinos running in matter may hopefully provide a meaningful possibility that the genuine (or fundamental) avor quantities in vacuum can be extrapolated from their matter-corrected (or e ective) counterparts to be measured in some realistic neutrino oscillation experiments. The remaining part of our paper is structured as follows. In section 2, we derive the RGEs of the e ective mixing parameters and neutrino masses explicitly and establish our conventions and notations. Adopting the standard parametrization of V , we further present the explicit expressions of the RGEs for the mixing parameters fe12; e13; e23; eg in section 3. Section 4 is devoted to further discussions on the RGEs of other phenomenologically interesting quantities. Finally, we summarize our main results in section 5. 2 Renormalization-group equations The essential idea of ours is to study the dependence of the avor mixing parameters on the scale-like matter term a by following the normal RGE approach. Di erentiating both sides of eq. (1.2) with respect to a, we immediately obtain D_ + hV yV_ ; D i 0 2 Ve2Ve23AC ; Ve3Ve1 Ve3Ve2 jVe3j 0 jVe1j diagf me12; me22; me23g and [A; B] da d mei2 = jVeij2 ; { 3 { (2.1) (2.2) V iV i + V_ iV i _ = X V iV j + V_ iV j _ = 0 ; (2.5) summarized below: lently, where 6 = and i 6= j should be noticed in the rst and second identities, respectively. With the help of the above equations, we are now ready to derive the RGEs for the matrix elements of V and the relevant rephasing invariants. The main results are Starting with the orthogonality condition V 1V 1 + V 2V 2 + V 3V 3 = 0, or equivaX (2.3) (2.4) HJEP05(218) (2.6) (2.7) for i = 1; 2; 3 by equating the diagonal elements on both sides of eq. (2.1); and X V iV j = VeiVej e ji1 ; _ for i 6= j by identifying the o -diagonal elements. In addition, we have a few useful identities from the normalization and orthogonality conditions for the unitary matrix V , namely, V iV i + V_ iV i _ = X V iV i + V_ iV i _ = 0 ; which can be recast into X Re V iV i _ = X Re V iV i _ = 0, where i = 1; 2; 3 and = e; ; are implied; and i X V j V j = j6=i V iV i ; we multiply both sides of eq. (2.6) by V_ i and sum over the avor index . Then, by using eq. (2.3), we arrive at _ V i = X V_ iV iV i + X VeiVej V j e ij1 : j6=i Note that the rst term on the right-hand side of eq. (2.7) is rephasing-dependent, and it can be arranged to vanish in a special phase convention without altering any physical results [24], as one has noticed in deriving the RGEs of quark avor mixing parameters [25]. We shall con rm that the terms associated with X V_ iV i can always be cancelled out in our subsequent calculations. Since there will be unphysical phases in the mixing matrix V , it is more interesting to present the RGEs for the rephasing invariants. The simplest ones are just the squared-moduli jV ij2, whose RGEs can be directly derived from eq. (2.7): d = 2 X Re VeiV j Vej V i e ij1 ; (2.8) j6=i d da V i { 4 { where the second identity in eq. (2.4) has been used. In principle, the RGEs in eqs. (2.2) and (2.8) are su cient to investigate the evolution of all physical quantities with respect to the matter term a, since the moduli jV ij of four independent matrix elements can unambiguously determine all three mixing angles and one CP-violating phase. Specifying = e and i = 1; 2; 3 in eq. (2.8), we explicitly have d d d da jVe1j2 = 2jVe1j2 jVe2j2 e 121 da jVe2j2 = 2jVe2j2 jVe3j2 e 231 da jVe3j2 = 2jVe3j2 jVe1j2 e 311 from eq. (2.2). Note that the RGEs in eqs. (2.9) and (2.10) are closed for fjVe1j2; jVe2j2; jVe3j2g and f e 12; e 23; e 31g, and completely symmetric under the cyclic permutations among the subscripts (1; 2; 3). Due to the normalization condition jVe1j2 + jVe2j2 + jVe3j2 = 1 and the identity e 12 + e 23 + e 31 = 0, there are only four independent di erential equations in eqs. (2.9) and (2.10). However, two redundant equations have been included in order to put them in a more symmetric form. For comparison, we quote the existing sum rules for jVeij2 and jUeij2 (for i = 1; 2; 3) from ref. [16]: jVe3j2 = b 23 b 33 jUe1j2 + b 13 b 33 jUe2j2 + b 13 b 23 jUe3j2 ; solutions to the RGEs of jVeij2 in eq. (2.9) with the mixing matrix elements jUeij2 and neutrino masses mi2 in vacuum as initial conditions. Substituting jVeij2 in eq. (2.11) mej2. Note that eq. (2.11) can be regarded as the formal (integral) into eq. (2.10), one can in principle obtain the solutions for e ij . Given eqs. (2.9) and (2.10), it is also straightforward to prove [24] d h da ln jVe1j2jVe2j2jVe3j e 12 e 23 e 31 2 2 2 2 i = 3 X i=1 d da ln jVeij 2 + X j>k d da 2 ln e jk = 0 ; (2.12) { 5 { which reproduces the second identity in eq. (1.3). In fact, eq. (2.12) indicates that the product jVe1jjVe2jjVe3j e 12 e 23 e 31 is a di erential invariant, so its value in matter and that in vacuum (i.e., corresponding to a = 0) should be equal to each other. This identity has previously been proved in ref. [26] by using a di erent approach. Then we come to the Jarlskog invariant Je, whose RGE can be found by starting with its original de nition in eq. (1.4) and implementing the derivatives of the mixing matrix elements in eq. (2.7). For instance, we have Je = Im Ve1V 2Ve2V 1 and thus its derivative d i h da Je = +Im hV_e1V 2Ve2V 1 + Im Ve1V 2Ve2V_ 1i h +Im Ve1V_ 2Ve2V 1 + Im Ve1V 2V_e2V 1 : i h i (2.13) According to eq. (2.7) and its complex conjugate, we can get _ After inserting eq. (2.14) into eq. (2.13), one can immediately observe that the rst and second lines on the right-hand side of eq. (2.13) become Im hV_e1V 2Ve2V 1i + Im hVe1V 2Ve2V_ 1i = Je h+jVe2j2 e 121 h Im Ve1V_ 2Ve2V 1 + Im Ve1V 2V_e2V 1 i h i = Je h leading to the following simple result d da Je = h Je jVe1j 2 jVe2j 2 e 121 + jVe2j 2 jVe3j 2 e 231 + jVe3j 2 jVe1j 2 Combining eq. (2.10) and eq. (2.16), one arrives at d da ln hJe e 12 e 23 e 31 = 0 ; i implying the well-known Naumov relation [8]. The corresponding identity Je e 12 e 23 e 31 = J 12 23 31 has previously been derived in the literature by implementing the commutators of e ective lepton mass matrices [9, 10]. In addition to the Naumov relation, it is easy to verify that X mei2V iV i = X mi2U iU i holds for arbitrary and except for = = e. i i { 6 { (2.14) e 311i ; e 231i ; (2.15) e 311i : (2.16) (2.17) For completeness, we explicitly write down the RGEs of jV ij2, which can also be expressed in terms of jV ij2 and e ij . Based on eq. (2.8) and the results from ref. [24], one can nd d d d da jV 1j2 = jV 1j da jV 2j2 = jV 2j da jV 3j2 = jV 3j 2 " jVe2j 2 2 " jVe3j 2 " jVe1j The RGEs of jV ij2 can be obtained from eq. (2.18) by simply exchanging jV ij with jV ij2 for i = 1; 2; 3. It is now evident that the evolution of jV ij2 (or jV ij2) 2 is governed not only by jV ij2 (or jV ij2) and e ij , but also by jV ij2 (or jV ij2) and jVeij2. Comparing between eq. (2.9) and eq. (2.18), one can easily notice the special role played by the electron avor in studying the matter e ects on the neutrino avor mixing parameters. The central results for the RGEs of the leptonic avor mixing matrix V and neutrino of jV ij2 against the dimensionless parameter a= 21 in masses mei in matter are given in eqs. (2.2) and (2.8). For illustration, we show the evolution gures 1 and 2, where the best- t values of all the neutrino mixing parameters from ref. [ 2 ] have been used in our numerical calculations. Throughout this paper, the blue solid (dashed) curves are referred to the results for neutrino (antineutrino) oscillations in the NO case, whereas the red solid (dashed) curves to those for neutrino (antineutrino) oscillations in the IO case. The main features of the evolution of jV ij2 can be understood by using the RGEs in eqs. (2.9) and (2.10) together with the general properties of matter e ects themselves: 1. First of all, it should be stressed that the evolution of jV ij2 is qualitatively identical to that of jV ij2 for i = 1; 2; 3, comparing the plots in the second row and those in the third row of gure 1. This behavior can be well understood by noticing that the muon and tau avors are indistinguishable, since muon and tau neutrinos (antineutrinos) experience only the universal neutral-current interactions in ordinary matter. In addition, the initial values of jV ij2 and jV ij2 at a = 0, namely, the mixing matrix elements in vacuum, approximately respect the - symmetry jU ij2 = jU ij2 for i = 1; 2; 3. The slight breaking of this symmetry will be responsible for the quantitative di erence between the evolution of jV ij2 and that of jV ij2. This conclusion is also applicable to antineutrinos. For this reason, we shall only concentrate on the electron and muon avors. 2. As the matrix elements have to ful ll the unitarity condition jVeij2 +jV ij2 +jV ij2 = 1 for i = 1; 2; 3, it is then necessary to consider only jVeij2 in the rst row of gure 1. First, the evolution of jVe1j2 is governed by the rst equation in eq. (2.9). At the { 7 { The evolution of the e ective mixing matrix elements in matter jV ij2 (for and i = 1; 2; 3) with respect to the parameter a= 21, where the best- t values of neutrino mixing parameters from ref. [ 2 ] in the NO case are input and the blue solid (dashed) curves correspond to the results of neutrino (antineutrino) oscillations. jVe1j2 + jVe2j2 + jVe3j2 = 1. beginning, we have e 12 = is approximately given by 21 < 0 and e 31 = 31 21, so the derivative of jVe1j 2 2jUe1j 2 211 < 0, indicating that jVe1j2 decreases with the increasing a. Similarly, one can observe from the second equation in eq. (2.9) that jVe2j2 is increasing. On the other hand, at the early stage, the evolution of jVe3j2 is highly suppressed by both jVe3j2 = jUe3j2 itself and the large neutrino mass-squared di erence 31 32, as one can see from the third equation of eq. (2.9). Then, the resonance corresponding to 21 is reached around a= 21 = 1, where jVe1j2 = jVe2j2 is satis ed and the changing rates of jVe1j2 and jVe2j2 maximize. Looking at again the RGE of jVe2j2, we nd that as jVe1j 2 decreases and e 21 increases, the right-hand side of the second equation of eq. (2.9) rst approaches zero and then changes its sign. This means that jVe2j2 reaches its maximum and decreases to zero afterwards. The decreasing rate is maximal around the MSW resonance corresponding to 31, namely, a= 21 30. Finally, since both jVe1j2 and jVe2j2 become vanishing for an extremely large a, we have jVe3j2 close to one due to the unitarity condition 3. Now we consider the results for antineutrinos in the NO case, as represented by the dashed curves in and a ! gure 1. It is worth emphasizing that the replacements U ! U a have been made and thus the matter term a itself keeps positive for { 8 { and i = 1; 2; 3) with respect to the matter term a= 21, where the best- t values of neutrino mixing parameters from ref. [ 2 ] in the IO case are input and the red solid (dashed) curves correspond to the results of neutrino (antineutrino) oscillations. both neutrinos and antineutrinos. As a consequence, the right-hand sides of the RGEs in eqs. (2.9) and (2.10) should be multiplied by a negative sign when applied to antineutrinos. Furthermore, in the NO case, there are no MSW resonances for antineutrinos, so the evolution of jVe1j2 and jVe2j2 seems to be milder and in the opposite directions, compared with the results for neutrinos. In particular, jVe3j2 is monotonically decreasing from the initial value to zero in the end. The numerical results for neutrinos and antineutrinos in the IO case have been given in gure 2. One can analyze the evolution of jVeij2 in a very similar way to the NO case. The di erence between these two cases is the location of the MSW resonances. As is well known, the 21-driven resonance remains for neutrinos in the IO case, while the 31-driven resonance is absent. But the opposite is true for antineutrinos. Bearing these general features in mind, one can easily understand the behaviors of jV ij2 evolving with an increasing a= 21. The running behavior of the Jarlskog invariant Je, normalized by its vacuum value J , is given in gure 3 in both NO and IO cases. For neutrinos, as we have mentioned, both 21and 31-driven resonances take place in the NO case, corresponding to two local maxima of Je=J . The existence of these two maxima can be partly understood by examining the right-hand side of eq. (2.16). In the early stage of evolution, e.g., a . 21 31, one can safely assume e 32 = e 31 e 21 and thus ignore the last two terms. This leads to { 9 { 0.8 0.6 J 1.0 0.8 respect to a= 21. The numerical results in the NO and IO cases are given in the left and right panel, respectively, where the same input values as before are adopted. jVe2j2. On the other hand, when a= 21 jVe2j2)= e 21, which is vanishingly small at the resonance around 10, one can read from the rst row of gure 1 that jVe1j 2 0 and jVe2j 2 1 jVe3j2. The term (jVe2j 2 jVe3j2)= e 32 expressions of jVeij2 and e ij are needed to becomes dominant at the late stage as e 32 ! 21 and e 21 of Je is obtained at the 31-driven resonance with jVe2j 2 maxima. See, e.g., ref. [13], for the discussions about the rst local maximum. The numerical results in the IO case or those for antineutrinos in both cases can be understood by studying the appearance of the MSW resonances. e 31, so the second maximum jVe3j2. However, the explicit gure out the exact values of a for the local 3 Mixing angles and CP-violating phase Although the RGEs for the mixing matrix elements V i are su cient to explore their dependence on the matter term a, it will be instructive to derive the RGEs for the e ective mixing angles fe12; e13; e23g and the CP-violating phase e in the standard parametrization [ 1 ]. The motivation for such an investigation is two-fold. First of all, the RGEs of neutrino mixing parameters due to radiative corrections have been extensively studied [27]. A detailed comparison between the RGEs arising from quantum corrections and those from matter e ects in neutrino oscillations will be very helpful. Second, as neutrino oscillation behaviors are usually understood in terms of neutrino mixing parameters and neutrino mass-squared di erences, the impact of matter e ects on neutrino oscillations can be conveniently represented by the e ective avor mixing angles and CP-violating phase in matter. The e ective mixing matrix V , which is a 3 3 unitary matrix, can in general be parametrized in terms of three mixing angles and six phases, namely, V = Q U 0 P with Q diagfei'1 ; ei'2 ; 1g and P diagfei 1 ; ei 2 ; ei 3 g. The unitary matrix U 0 takes on the same form as in eq. (1.1) with the mixing angles and CP-violating phase replaced by fe12; e13; e23g and e. As V DV y = Q U 0DU 0y Qy, it is obvious that the diagonal phase matrix P disappears from eq. (2.1). Therefore, we can just ignore P , but have to retain Q, in which two unphysical phases '1 and '2 are involved. Taking V = QU 0 and noticing X U 0 iU_ 0 j + i '_ 1Ue0i Ue0j + '_ 2U 0 iU 0 j = Ue0i Ue0j e ji1 ; (3.1) for ij = 12; 13; 23. The diagonal elements give rise to me_i2 = jUe0ij2 as before. It is worthwhile to stress that eq. (3.1) resembles the salient features of the ordinary RGEs for quantum corrections to the lepton avor mixing parameters in the case of massive Dirac neutrinos, particularly in the limit of so-called tau-lepton dominance [28]. In comparison with the taulepton dominance due to ye2 y 2 y2, where y (for = e; ; ) stand for the chargedlepton Yukawa couplings, the case of matter e ects under consideration corresponds to the electron dominance, since the coherent forward scattering of neutrinos in the normal matter singles out the electron avor. As a consequence, the standard parametrization and the original Kobayashi-Maskawa parametrization [29, 30] with the simplest matrix elements in the rst row will be most convenient for us to derive the RGEs of relevant avor mixing parameters. Adopting the standard parametrization of U 0, we get the equation array for the deriva_ _ _ _ tives of the avor mixing parameters fe12; e13; e23; e; '_ 1; '_ 2g from both imaginary and real parts of eq. (3.1) for ij = 12; 13; 23. After a lengthy but straightforward calculation, we nd the RGEs for four physical mixing parameters (3.2) (3.3) (3.4) and those for two unphysical phases _ _ _ Using the last two equations in eq. (3.2), one can easily verify d da sin 2e23 sin e = 2 cos 2e23 sin e e23 + sin 2e23 cos e e = 0 ; _ _ which is just the Toshev relation sin 2e23 sin e = sin 2 23 sin in the standard parametrization. Some comments in the RGEs in eqs. (3.2) and (3.3) are in order. If there exists a - symmetry in the lepton avor mixing matrix U in vacuum, namely, jU ij2 = jU ij2 for i = 1; 2; 3, the mixing parameters should satisfy 23 = =4 and = =2. As has been proved in ref. [31], the matter e ects preserve the symmetry jV ij2 = jV ij2 (for i = 1; 2; 3), i.e., e23 = =4 and e = =2. This conclusion can be understood via the RGEs of e23 and e in eq. (3.2). For instance, the initial conditions e23ja=0 = 23 = =4 and eja=0 = = beta functions of e_23 / cos e and e / cos 2e23 are vanishing, indicating that the _ =2 guarantee that the symmetry with e23 = =4 and e = =2 is fully stable against matter e ects. _ _ Furthermore, let us look for possible xed points of other avor mixing parameters in their running with a. First, starting from the mixing angles and CP-violating phase in vacuum, we can see that e12 other term tan2 e13 e 221=( e 31 e 32) sin 2e12 cos2 e13 e 211=2 is positive, where the 1 at a = 0 has been neglected. Therefore, e12 increases as the matter density or neutrino energy becomes larger. Second, if 13 = 0 is assumed, then one can observe e13 = 0 and e23 = 0. In this case, only the mixing angle e12 will be a ected by matter e ects, and the CP-violating phase e is not wellde ned and thus irrelevant. Third, we assume = 0 or in vacuum, corresponding to the case of CP conservation, and then obtain e / sin e = 0, implying that the CP-violating phase e is xed. However, the CP asymmetries between neutrino and antineutrino oscillation probabilities could exisit since the mass-squared di erences for neutrinos and antineutrinos will be di erent due to the opposite signs in front of a. This is just the fake CP violation induced by matter e ects. We can numerically solve the RGEs of the mixing parameters fe12; e13; e23; eg and the neutrino mass-squared di erences f e 21; e 31g. However, as the evolution of jV ij (for and i = 1; 2; 3) have been obtained, we extract the results of mixing 2 parameters from the calculations of jV ij2 for gures 1 and 2 and summarize them in gures 4 and 5, where the input values are the same as before. The running behaviors of e12 and e13 are directly extracted from those of jVe2j2 = cos2 e13 cos2 e12 and jVe3j2 = sin2 e13 in the standard parametrization. For instance, we have tan2 e12 = jVe2j2=jVe1j2. Note that e12 ! 0 or 90 after crossing the rst MSW resonance, while e13 ! 0 or 90 after crossing the second resonance no matter how small the value of 13 in vacuum is. As for e23 and e, since current neutrino oscillation data prefer nearly-maximal mixing angle and CP-violating phase, the matter e ects have very little in uence on their values in matter, which is well consistent with the Toshev relation. As shown in gures 4 and 5, all the e ective mixing angles become constant in the limit _ a= 21 ! +1. In other words, the in nity serves as a special xed point of the RGEs of mixing angles. In both NO and IO cases, one can observe that e13 always approaches either 0 or 90 for both neutrinos and antineutrinos in this limit. These values are asymptotically stable because e13 vanishes at either e13 = 0 or 90 . However, the limits of e12 and e23 depend upon the asymptotic value of e13. Following the perturbation calculations in refs. [32, 33] in the limit of 21 31 a, we obtain cot e12 ! 31 c123s13 21 c12s12 ; tan e23 ! tan 23 + ei for neutrinos in the NO case. For antineutrinos in the IO case, the results can be obtained by replacing cot e12 by tan e12 but keeping tan e23 unchanged. In the NO case for neutrinos, ill 0 O 50 10 5 1 0.5 236 235 e[ ] 60 40 20 0 234 233 50 10 5 1 0.5 31 21 10 angles fe12; e13; e23g, one CP-violating phase e and two mass-squared di erences f e 21; e 31g, with respect to a= 21 in the NO case. The same convention and input values as in gure 1 are taken. with the best- t values of the mixing angles and the CP-violating phase in vacuum, one can gure out the asymptotic values e12 84:6 and e23 44:3 in the limit a= 21 ! +1, which are in excellent agreement with the numerical results in the rst row of gure 4. For the mass-squared di erences in the NO case, their evolution can be understood by using the RGEs in eq. (2.10): For e 21, the beta function is given by jVe2j but turns to be positive after crossing the 2 jVe1j2, which is initially negative rst MSW resonance. This is why the 1 ratio e 21= 21 gets its minimum at about a= 21 = 1. As jVe2j2 increases rapidly to jUe3j2 afterwards and becomes stable until the second resonance is reached, e 21 is linearly proportional to a during this stable region. The ultimate value of e 21 is xed to 31 for a= 21 ! +1. For e 31, the beta function is jVe3j 2 jVe1j2, which is negative as jUe3j 2 the beginning, so e 31 decreases for the increasing a. But jVe1j2 is reduced to zero quickly, while jVe3j2 keeps almost unchanged, so the evolution of e 31 is negligible. The situation changes when the second resonance is encountered and jVe3j2 approaches one rapidly. Hence, e 31 turns out to be linearly proportional to a ultimately. jUe1j2 at 0.01 0.10 1 10 ill 0 O 50 10 5 1 0.5 279 e[ ] 60 40 20 0 278 277 50 10 5 1 0.5 31 angles fe12; e13; e23g, one CP-violating phase e and two mass-squared di erences f e 21; e 31g, with respect to a= 21 in the IO case. The same convention and input values as in gure 2 are taken. The results for antineutrinos and the IO case can be discussed in a similar way. As indicated in gure 3, the Jarlskog invariant Je will be vanishing as a= 21 ! +1. This can be explained via the Naumov relation Je = J 21 31 32=( e 21 e 31 e 32), in which the denominator is approaching in nity. On the other hand, as e13 ! =2 for a= 21 ! +1, the Jarlskog invariant is Je / sin 2e13 cos e13 ! 0 in the standard parametrization. Finally, one may wonder whether the RGEs in eq. (3.2) can be analytically solved, so as to express e12, e13, e23 and e in terms of 12, 13, 23, and the relevant neutrino mass-squared di erences. This will be a challenge if the matter density is arbitrarily varying. Given a constant matter pro le, however, the exact analytical relations between fe12; e13; e23; eg and f 12; 13; 23; g have been established in refs. [34{36] in a di erent approach. But those relations are so complicated that they are not very helpful for understanding the behaviors of neutrino oscillations in matter. That is why some useful and more transparent analytical approximations have been made in the literature for longand medium-baseline neutrino oscillation experiments (e.g., ref. [13] for E . 1 GeV and refs. [32, 33] for E & 0:5 GeV). In this section we demonstrate that the RGEs derived in the previous sections can also be utilized to analyze the matter e ects on several phenomenologically interesting observables. Let us begin with the partial - asymmetries of V [37], Ai jV ij 2 jV ij 2 (for i = 1; 2; 3) and the o -diagonal asymmetries of V , AL AR jVe2j jVe2j 2 2 jV 1j2 = jV 3j jV 3j2 = jV 1j 2 2 jV 2j2 = jV 1j jV 2j2 = jV 3j 2 2 The phenomenological implications of the partial symmetry jU 1j2 = jU 1j2 or jU 2j2 = jU 2j2 for the leptonic CP-violating phase and mixing angles in vacuum have been investigated in ref. [37], in which it has been shown that the leptonic CP-violating phase is correlated with three mixing angles if such a symmetry is imposed. In the standard parametrization, jU 3j2 = j U 3j2 leads to the maximal mixing angle 23 = contrary, jU 1j2 = jU 1j2 or jU 2j2 = jU 2j2 allows for an appreciable deviation of 23 from =4 and that of from =2, which are compatible with current neutrino oscillation data. =4. On the Unlike the full - symmetry jV ij 2 = jV ij2, the partial symmetry is not preserved by matter e ects, which can be seen from the following RGEs: (4.1) (4.2) where eq. (2.8) has been used. It is evident that if Ai = 0 (for i = 1; 2; 3) hold exactly in vacuum (namely, jU ij2 = jU ij2), they remain to be vanishing in matter. This point has also been emphasized in the previous section with the standard parametrization of V . However, if only the partial - symmetry (say A1 = 0 or jU 1j2 = jU 1j2) is valid in vacuum, then we have d da A1 = 2jV 1j 2 h jV 2j 2 jV 2j 2 e 211 + jV 3j 2 jV 3j 2 (4.4) which is in general nonzero for jV 2j2 6= jV 2j2 and jV 3j2 6= jV 3j2. Therefore, the predictions from jU 1j2 = jU 1j2 in vacuum are invalidated in matter. In a similar way, one can calculate the RGEs for the o -diagonal asymmetries, dda AL = 2 hRe (V 1Ve2V 2Ve1) e 121 h d da AR = 2 jVe1j2jVe2j2 e 121 jVe2j2jVe3j2 e 231 + Re V 3Ve1V 1Ve3 e 311i ; Re (V 2Ve3V 3Ve2) e 23 1 Re V 3Ve1V 1Ve3 e 311i : (4.5) The latest neutrino oscillation data indicate that both AL and AR are nonzero for the avor mixing matrix in vacuum. We present the running behaviors of the partial - asymmetries and the o -diagonal asymmetries in gures 6 and 7, respectively. Assuming the initial values of neutrino mixing 0.15 0.10 respect to a= 21, where the same convention and input values as in gure 3 are taken. j V ij 2 jV ij2 for i = 1; 2; 3 with parameters in vacuum to be the best- t numbers, one can nd that the asymmetries jAij . 0:1, which are indeed modi ed by the matter e ects, but only slightly. On the other hand, however, the o -diagonal asymmetries can be signi cantly enhanced or suppressed during the evolution with respect to a= 21. It is straightforward to explain the primary features of the evolution of these asymmetries by using the numerical results in gures 1 and 2. Next, we focus on the sides of six leptonic unitarity triangles of V , which are de ned by the orthogonality conditions in the complex plane [ 38 ]: and 4e : V 1 V 1 + V 2 V 2 + V 3 V 3 = 0 ; 4 4 : V 1Ve1 + V 2Ve2 + V 3Ve3 = 0 ; : Ve1V 1 + Ve2V 2 + Ve3V 3 = 0 ; 41 : Ve2Ve3 + V 2 42 : Ve3Ve1 + V 3 43 : Ve1Ve2 + V 1 V 3 + V 2 V 3 = 0 ; V 1 + V 3 V 1 = 0 ; V 2 + V 1 V 2 = 0 : (4.6) (4.7) HJEP05(218) NO IO The evolution of the o -diagonal asymmetries AL and AR with respect to a= 21, where the same convention and input values as in gure 3 are taken. Taking the unitarity triangle 4 for example, one may gure out d da d da d da p Ve1V 1 = Ve1V 1 jVe2j2 e 12 1 Ve2V 2 = Ve2V 2 jVe3j2 e 23 1 Ve3V 3 = Ve3V 3 jVe1j2 e 31 1 where eq. (2.7) has been utilized to compute the derivatives of the matrix element and its complex conjugate. From eq. (4.8), we can observe how the three sides of 4 are changing with the matter term. Since the evolution of all the six leptonic unitarity triangles has been systematically studied in refs. [15, 16], we do not elaborate on this issue here. Last but not least, we give some remarks on the parameter a= 21, which has been chosen as an arbitrary dimensionless scale-like variable. Based on the de nition a 2 2 GFNeE, it is convenient to rewrite a= 21 as follows: a 21 = 0:02 Ne NA cm 3 E 10 MeV ; (4.9) where NA = 6:022 Ne is related to the matter density gure 8 the contours of a= 21 have been shown in the plane of (E; Ne), and three typical neutrino oscillation experiments have been indicated on the plot for the purpose of illustration: JUNO reactor antineutrinos at (4 MeV; 1:5NA=cm3) [ 39 ], solar neutrinos at (10 MeV; 102NA=cm3) [ 1 ] and DUNE with accelerator neutrinos at ( GeV; 1:5NA=cm3) [40]. For the reactor- and accelerator-based experiments, the matter density is usually taken to 1023 is the Avogadro constant, and the electron number density through Ne = NA cm 3 Ye [ =(1 g cm 3)]. In 102 3 101 100 10-1 JUNO DUNE 10-4 10-3 10-2 10-2 10-1 100 101 102 103 E [10 MeV] and the black triangle for DUNE at (2 GeV; 1:5NA=cm3). neutrino energy E and the net electron number density Ne, where the yellow disk stands for JUNO at (E; Ne) = (4 MeV; 1:5NA=cm3), the blue square for solar neutrinos at (10 MeV; 102NA=cm3), be = 3 g cm 3, i.e., the average density of the Earth crust or mantle. Thus, the evolution with respect to a can be realized by changing the neutrino beam energy or the matter density. As a potentially interesting application of the RGE approach developed above, one may rst express the neutrino oscillation probabilities relevant for those realistic experiments in terms of the e ective mixing parameters and then extract their values directly from the corresponding experimental data. The exact RGEs of those e ective mixing parameters can subsequently be implemented to run the measured values to a common scale of a= 21. In particular, the fundamental oscillation parameters (i.e., two neutrino mass-squared di erences and four avor mixing parameters) can be extrapolated from their matter-corrected counterparts in the vacuum limit of a= 21 ! 0. It is still unclear whether this procedure will work better than the usual treatment of matter e ects in the present neutrino oscillation experiments with reasonable analytical approximations, but its principle is de nitely on solid ground because the language of RGEs itself is completely model-independent. 5 Concluding remarks It is well known that the RGE approach has been serving as a powerful tool in a number of aspects of theoretical physics to systematically describe the changes of a physical system as viewed at di erent distances or energy scales, and its success in quantum eld HJEP05(218) theory is especially marvelous. In the present work we have applied this language to the description of neutrino masses and avor mixing parameters in a medium, which evolve with the arbitrary scale-like matter parameter a possibility that the genuine neutrino avor quantities in vacuum can be extrapolated from their matter-corrected counterparts to be measured in some realistic neutrino oscillation p 2 2 GFNeE, and highlighted a striking experiments. To be explicit, we have clearly demonstrated that the dependence of the e ective avor mixing parameters V i and mei2 on the matter parameter a can perfectly be described by a complete set of di erential equations, which are just referred to as the RGEs of those quantities. The point is that the introduction of e ective neutrino mass-squared di erences and avor mixing parameters guarantees the form invariance or self-similarity of neutrino oscillation probabilities in vacuum and in matter, and hence the RGE-like approach for jV ij2 [23, 24], we have also derived the RGEs of three describing neutrino oscillations in matter works well. In addition to the RGEs for mei and avor mixing angles and one CPviolating phase in the standard parametrization of V , and numerically illustrated some salient features of their evolution with respect to the matter parameter a. The RGEs of Je and some other interesting quantities, such as the partial - asymmetries, the o -diagonal asymmetries and the sides of unitarity triangles of V , have been derived and discussed as a by-product of this work. The Naumov and Toshev relations are reformulated too. In the long run, the RGE-like approach that we have developed may hopefully provide a generic framework for the systematic study of neutrino masses and avor mixing parameters in any possible matter environments. Although such a tool might be \scienti cally indistinguishable" from the conventional methods of dealing with matter e ects on neutrino oscillations, \they are not psychologically identical" in making the underlying physics more transparent [41]. In particular, tracing an analogy between the evolution of neutrino masses and avor mixing parameters in matter and their evolution with the energy scale is theoretically interesting. We therefore expect that our work can nd some useful applications in neutrino phenomenology. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under grant No. 11775231 (XZZ) and grant No. 11775232 (SZ), by the National Youth Thousand Talents Program (SZ), by the CAS Center for Excellence in Particle Physics (SZ), and by the European Research Council under ERC Grant NuMass (FP7-IDEASERC ERC-CG 617143). Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. { 19 { 087 [arXiv:1611.01514] [INSPIRE]. to three neutrino mixing: exploring the accelerator-reactor complementarity, JHEP 01 (2017) the T2K experiment including a new additional sample of e interactions at the far detector, Phys. Rev. D 96 (2017) 092006 [arXiv:1707.01048] [INSPIRE]. [6] S.P. Mikheev and A. Yu. 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Zhi-zhong Xing, Shun Zhou, Ye-Ling Zhou. Renormalization-group equations of neutrino masses and flavor mixing parameters in matter, Journal of High Energy Physics, 2018, 15, DOI: 10.1007/JHEP05(2018)015