Neutrino mixing in matter S. H. Chiu, $^{1,}$∗ T. K. Kuo, $^{2,}$† and Lu-Xin Liu $^{2, 3,}$‡ $^{1}$Physics Group, CGE, Chang Gung University, Kwei-Shan 333, Taiwan $^{2}$Department of Physics, Purdue University, West Lafayette, IN 47907, USA $^{3}$National Institute for Theoretical Physics, Department of Physics and Centre for Theoretical Physics, University of the Witwatersrand, Wits, 2050, South Africa Three-neutrino mixing in matter is studied through a set of evolution equations which are based on a rephasing invariant parametrization. Making use of the known properties of measured neutrino parameters, analytic, approximate solutions are obtained. Their accuracy is confirmed by compar- ison with numerical integration of these equations. The results, when expressed in the elements squared of the mixing matrix, exhibit striking patterns as the matter density varies. PACS numbers: 14.60.Pq, 14.60.Lm, 13.15.+g It is well-established that neutrino mixing is modified by the presence of matter [1]. Their effect has been used in the analyses of solar neutrinos, and is expected to im- pact those of the supernova neutrinos, when and if they become available. Closer to home, there is a plethora of long baseline experiments either in operation or in the planning stage. For these studies, it is essential to in- clude the matter effects in order to understand neutrino mixing at the fundamental level. In the literature, effort has been devoted to solving problems along this line [2]. However, the process in- volves the complication of the cubic eigenvalue problems, and the results are usually far from transparent for a clear extraction of the physical implications. In this work we derive a set of evolution equations for the neutrino parameters, using a rephasing invari- ant parametrization which was developed for three-flavor quark mixing. The same formalism can be used in the neutrino sector, as long as it is used for lepton num- ber conserving processes, such as in neutrino oscillation, which will be studied here. It will be shown that the coupled equations have simple, analytic, solutions which, when compared to the complete numerical solutions, are quite accurate. For the neutrino mixing (PMNS) matrix ( V ), we adopt the parametrization introduced earlier [3]. Briefly, with- out loss of generality, one can demand det V = +1. There are then a set of rephasing invariants where their common imaginary part can be identified with the Jarlskog invariant J [4]. Their real parts were defined as where D$_{i}$ = neutrino mass squared, L is the length of baseline, E is the neutrino energy, and Physical measurables can always be expressed in terms of ( x, y ). For instance, the ν$_{µ}$ → ν$_{e}$ transition probability is given by One can readily obtain the parameters ( x, y ) from W by computing its cofactors, which form the matrix w with w $^{T}$W = (det W ) I , and is given by The ( x, y ) parameters are related to | V$_{ij}$ | 2 by In addition, it is found that These variables are bounded by ± 1: − 1 ≤ ( x$_{i}$, y$_{j}$ ) ≤ +1, with y$_{j}$ ≤ x$_{i}$ for any ( i, j ). They satisfy two constraints $^{∗}$Electronic address: schiu@mail.cgu.edu.tw $^{†}$Electronic address: tkkuo@purdue.edu $^{‡}$Electronic address: Luxin.Liu@wits.ac.za ( R$_{123}$, R$_{231}$, R$_{312}$ ; R$_{132}$, R$_{213}$, R$_{321}$ ) = ( x$_{1}$, x$_{2}$, x$_{3}$ ; y$_{1}$, y$_{2}$, y$_{3}$ ) . (2) Γ$_{ijk}$ = V$_{1}$$_{i}$V$_{2}$$_{j}$V$_{3}$$_{k}$ = R$_{ijk}$ − iJ, (1) arXiv:1001.1469v3 [hep-ph] 7 Apr 2010 det V = ( x$_{1}$ + x$_{2}$ + x$_{3}$ ) − ( y$_{1}$ + y$_{2}$ + y$_{3}$ ) = 1 , (3) ( x$_{1}$x$_{2}$ + x$_{2}$x$_{3}$ + x$_{3}$x$_{1}$ ) − ( y$_{1}$y$_{2}$ + y$_{2}$y$_{3}$ + y$_{3}$y$_{1}$ ) = 0 . (4) J 2 = x$_{1}$x$_{2}$x$_{3}$ − y$_{1}$y$_{2}$y$_{3}$. (5) W = [ | V$_{ij}$ | $^{2}$] =   x$_{1}$ − y$_{1}$ x$_{2}$ − y$_{2}$ x$_{3}$ − y$_{3}$ x$_{3}$ y$_{2}$ x$_{1}$ y$_{3}$ x$_{2}$ y$_{1}$ x$_{2}$ − y$_{3}$ x$_{3}$ − y$_{1}$ x$_{1}$ − y$_{2}$   . (6) w =   x$_{1}$ + y$_{1}$ x$_{2}$ + y$_{2}$ x$_{3}$ + y$_{3}$ x$_{3}$ + y$_{2}$ x$_{1}$ + y$_{3}$ x$_{2}$ + y$_{1}$ x$_{2}$ + y$_{3}$ x$_{3}$ + y$_{1}$ x$_{1}$ + y$_{2}$   . (7) P ( ν$_{µ}$ → ν$_{e}$ ) = − 4[ F µe 21 sin $^{2}$( D$_{2}$ − D$_{1}$ 4 E/L ) + F µe 31 sin $^{2}$( D$_{3}$ − D$_{1}$ 4 E/L ) + F µe 32 sin $^{2}$( D$_{3}$ − D$_{2}$ 4 E/L )] + 8 J sin( D$_{2}$ − D$_{1}$ 4 E/L ) sin( D$_{3}$ − D$_{1}$ 4 E/L ) sin( D$_{3}$ − D$_{2}$ 4 E/L ) , (8) x$_{2}$x$_{3}$ + x$_{1}$y$_{2}$ − y$_{1}$y$_{2}$ − y$_{2}$y$_{3}$ ≡ F µe $_{21}$, − x$_{1}$x$_{3}$ − x$_{2}$x$_{3}$ + x$_{3}$y$_{1}$ + y$_{2}$y$_{3}$ F µe $_{31}$, x$_{1}$x$_{3}$ + x$_{2}$y$_{3}$ − y$_{1}$y$_{3}$ − y$_{2}$y$_{3}$ ≡ F µe $_{32}$. (9)