Simple and compact expressions for neutrino oscillation probabilities in matter

Journal of High Energy Physics, Jan 2016

We reformulate perturbation theory for neutrino oscillations in matter with an expansion parameter related to the ratio of the solar to the atmospheric Δm 2 scales. Unlike previous works, we use a renormalized basis in which certain first-order effects are taken into account in the zeroth-order Hamiltonian. We show that the new framework has an exceptional feature that leads to the neutrino oscillation probability in matter with the same structure as in vacuum to first order in the expansion parameter. It facilitates immediate physical interpretation of the formulas, and makes the expressions for the neutrino oscillation probabilities extremely simple and compact. We find, for example, that the ν e disappearance probability at this order is of a simple two-flavor form with an appropriately identified mixing angle and Δm 2. More generally, all the oscillation probabilities can be written in the universal form with the channel-discrimination coefficient of 0, ± 1 or simple functions of θ 23. Despite their simple forms they include all order effects of θ 13 and all order effects of the matter potential, to first order in our expansion parameter.

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Simple and compact expressions for neutrino oscillation probabilities in matter

Received: May Simple and compact expressions for neutrino oscillation probabilities in matter Hisakazu Minakata 0 2 Stephen J. Parke 0 1 0 P. O. Box 500, Batavia, IL 60510 , U.S.A 1 Theoretical Physics Department, Fermi National Accelerator Laboratory 2 Instituto de F sica, Universidade de Sa~o Paulo We reformulate perturbation theory for neutrino oscillations in matter with an expansion parameter related to the ratio of the solar to the atmospheric Unlike previous works, we use a renormalized basis in which certain rst-order e ects are taken into account in the zeroth-order Hamiltonian. We show that the new framework has an exceptional feature that leads to the neutrino oscillation probability in matter with the same structure as in vacuum to rst order in the expansion parameter. It facilitates immediate physical interpretation of the formulas, and makes the expressions for the neutrino oscillation probabilities extremely simple and compact. We nd, for example, that the e disappearance probability at this order is of a simple two- avor form with an appropriately identi ed mixing angle and m2. More generally, all the oscillation probabilities can be written in the universal form with the channel-discrimination coe cient of 0; functions of 23. Despite their simple forms they include all order e ects of 13 and all order e ects of the matter potential, to rst order in our expansion parameter. Neutrino Physics; Beyond Standard Model - 3 Results of our perturbative expansion for all oscillation probabilities Mass squared eigenvalues in matter The mixing angle 13 and mixing matrix in matter Compact formulas for the oscillation probabilities in matter 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 e ! e disappearance channel e ! An example of the power of our oscillation probabilities Comparison with the existing perturbative frameworks oscillation probabilities 4 Formulating the renormalized helio-perturbation theory Choosing the basis for the renormalized helio-perturbation theory Hat basis S matrix and the oscillation probability Mass eigenstate in matter: V matrix method 5 More about the renormalized helio-perturbation theory Exact versus zeroth-order eigenvalues in matter Eigenvalues in vacuum and in the asymptotic regions a ! 1 Neutrino mixing matrix in matter and the 1 Introduction form matter 2 Structure of the neutrino oscillation probabilities in vacuum and in uni Simplicity and compactness Raison d'^etre and the requirements for perturbative treatment Neutrino oscillation based on the standard three- avor scheme provides the best possible theoretical framework available to date to describe most of the experimental results obtained in the atmospheric, solar, reactor, and the accelerator neutrino experiments. Although numerically calculated neutrino oscillation probabilities su ce to analyze experimental data, understanding of the framework, in particular the one in matter [1], has not yet reached a su cient level, in our opinion. Under the assumption of uniform matter density distribution, the exact expressions of the eigenvalues, mixing angles, and the oscillation probabilities in matter have been obtained [2{4]. Yet, the results for these quantities are generally too complicated to facilitate understanding of the structure of the three avor neutrino oscillations in matter primarily due to the complexities of solving the cubic eigenvalue characteristic equation. For a recent comprehensive treatment of neutrino oscillation in the matter, see ref. [5]. parameter in refs. [7{10]. Analytic approaches to the neutrino oscillation phenomenon, so far, are mostly based on variety of perturbative frameworks. If the matter e ect is small one can treat it as a small perturbation [6]. In the environments in which the matter e ect is comparable to the vacuum mixing e ect, the only available small expansion parameter known to us is the ratio of the solar-scale m2 to the atmospheric-scale m2 , m2 = m2 ' 0:03. sin 13 has been often used as an expansion parameter (there are enormous number of references, q see e.g., [7]), but it is now known that its value is not so small, sin 13 ' 0:15, which is of the order of m2 = m2 . Moreover, expansion around sin 13 = 0 misses the physics of the resonance which exists at an energy around E 10 GeV for earth densities. Therefore, it appears that the suitable perturbative framework is the one with the unique expansion m2 = m2 . This framework was indeed examined in the past, to our knowledge In this paper, we present a new framework of perturbative treatment of neutrino oscillation in matter. We follow the reasoning stated above which led to identi cation of the unique expansion parameter m2 = m2 . But, unlike the preceding works, we use a \renormalized basis" as the basis of perturbation. That is, we absorb certain terms of order to our \zeroth-order" Hamiltonian around which we perturb. Or, in other word, we take the zeroth-order eigenvalues in matter such that it matches the exact eigenvalues to order , see section 5. We will show that use of the renormalized basis makes the structure of the perturbation theory exceptionally transparent, as we will explain in the next section. It allows us to obtain simple, elegant and compact expressions for the oscillation probabilities, which have a universal form even to rst order in our expansion parameter. For example, e survival probability takes the form to order as denote the eigenvalues of the states which participate the 1-3 level crossing. Despite its extremely simple form, P ( e ! e) in (1.1) takes into account all order e ects of 13 and the matter potential. Since we will only consider terms up to order , in this paper, our results here are not applicable to the region near the solar MSW resonance [1, 11]. Our perturbative framework will be called as the \renormalized helio-perturbation theory" in the rest of this paper. The section plan of this paper is somewhat unusual: in the next section 2 we summarize the key features of the perturbative framework we develop in this paper. Then, in section 3 we describe the principle results of this work including the oscillation probabilities for all channels in matter. This section does not describe the derivations but provides a self contained summary of the results of this paper. Following this section, see section 4, we present a systematic exposition of our perturbative framework and how the results of the section 3 are derived. In the appendices A, B, and C we present, respectively, calculational details of the S matrix, the results of oscillation probabilities to order in the standardized form, and useful relationships to verify the equivalence of various expressions. 2 Structure of the neutrino oscillation probabilities in vacuum and in uniform matter probabilities for following form1 We start by giving a precise de nition of what we mean by \simplicity and compactness" of the expressions for the neutrino oscillation probabilities. Then, we explain the need for reformulating perturbative treatment, followed by outlining \what is new" in this paper. 2.1 Simplicity and compactness In vacuum and in matter with constant density, it is well known that the neutrino oscillation for three- avor mixing (i; j = 1; 2; 3) can be written in the P ( ! ) = X V iV i e i 2iEL 4 X Re[V iV iV j V j ] sin2 ( j i)L 2 = ( 3 4E j>i sin 2)L ( 2 4E 1)L sin ( 1 4E 3)L (2.1) 4E : ! 3 i=1 + 8 Im[V 1V 1V 2V 2] sin The avor mixing matrix elements, V i, relate the avor states, , to the eigenstates of the Hamiltonian, i, with eigenvalues i=2E by = V i i. Equation (2.1) applies both in vacuum and in uniform matter since the matrix V diagonalizes the full Hamiltonian, which includes the Wolfenstein matter potential, a [1]. In vacuum, i = mi2 where mi denotes the mass of the i-th neutrino state and V = UMNS. Notice that the identical form of the oscillation probabilities in vacuum and in matter, eq. (2.1), apart from replacement of of the formulas in matter is as transparent as in vacuum. mi2j by ( i j ), implies that physical interpretation 1We have used for the last term in (2.1) Im[V 1V 1V 2V 2] = Im[V 2V 2V 3V 3] = Im[V 1V 1V 3V 3] which follows from unitarity, and the identity (C.2). { 3 { What is less well appreciated is that the expressions of the oscillation probabilities in (2.1) are maximally simple and compact. That is, they contain 5 (including the constant term) functions of L=E which are linearly independent. This property can be easily veri ed by showing the Wronskian is nonvanishing, which implies that none of the functions can be written in terms of the other functions for all L=E. Hence, in a true mathematical sense, eq. (2.1) and it's equivalents give the simplest and most compact form for the complete set of three avor oscillation probabilities in vacuum and in uniform matter.2 A special feature of the oscillation probabilities which is also worth noting is that each term in eq. (2.1) factorizes into the characteristic sin [( j i)L=4E] factor and the products of the V matrix elements which control the amplitude of the oscillation. Both the eigenvalues, i and the matrix elements of V are independent of the baseline, L, but are functions of the mixing angles, 's, the mj2i, and the product of the energy of neutrino times the matter density via the matter potential.3 CP and T violation is described by the last term in (2.1), which has the universal, channel-independent form in the three neutrino mixing. 2.2 Raison d'^etre and the requirements for perturbative treatment With the simplest form of the oscillation probability eq. (2.1) and by knowing both the exact form of the eigenvalues, i=2E, and the elements of the V matrix, see [3], one might expect we have all that is needed for theoretical discussions. Unfortunately, the analytic expressions for these eigenvalues, i=2E, as well as the V matrix elements are notoriously complex and give no analytic insight into the oscillation physics in uniform matter. This is true even when one of m221, sin 13, sin 12 or a is set equal to zero. In any one of these limits, the characteristic equation for the eigenvalues factorizes. But, the form of the general solution does not simplify trivially to yield the correct eigenvalues, even though it must. The structure of the general solutions of the cubic characteristic equation is the root cause of this rather unusual behaviour. Hence, there is a need for a reformulated perturbative framework so that we can obtain approximate but much simpler expressions for the eigenvalues, i, and the mixing matrix elements, V i, which provide the necessary physics insight. In this paper, we formulate perturbation theory by which we can calculate the eigenvalues i and the elements of V matrix as a simple power series of the small expansion parameter, a renormalized m2 = m2 . At the same time the structure-revealing form of the oscillation probabilities (2.1) is kept intact. While the existing frameworks do not satisfy the latter requirement, the key to the success in our case is due to the correct decomposition of the Hamiltonian into the unperturbed and the perturbed terms. Use of the renormalized zeroth-order basis, allows us to correctly determine the eigenvalues to the appropriate order in our expansion parameter. The resultant renormalized eigenvalues and 3 a / 2There are equivalent ways to write this set of oscillation probabilities with 5 independent L=E functions, e.g one could use the trigonometric identity 2 sin2 x = 1 cos 2x to replace the sin2 x. Any other way of identity (C.2) increases the number of independent functions by 2. the mixing parameters e ectively absorb the additional terms that arise in the conventional perturbative frameworks, as extra functions to the minimally required 5 linearly independent functions. This occurs automatically and we believe that such a framework has never before been formulated.4 Because of the structural simplicity of the L=E dependence, the expressions of the oscillation probabilities calculated by our method are extremely simple and compact, as will be fully demonstrated in the next section. 3 Results of our perturbative expansion for all oscillation probabilities In this section, we describe the main results of this paper without derivations and with only minimal discussion. In later sections we provide the derivation and more in depth discussions. We start with the approximate eigenvalues of the Hamiltonian, the approximate neutrino mixing matrix and then give the oscillation probabilities for all channels to rst order in the expansion parameter, , see eq. (3.3) for the precise de nition. 3.1 Mass squared eigenvalues in matter In vacuum the three eigenvalues of the full Hamiltonian which governs the neutrino oscillation is given in the form mi2=2E, where mi is the mass of i-th mass eigenstate of neutrinos, i = 1; 2; 3. Similarly, in matter we write the three eigenvalues as where the state label runs over i = ; 0; + for the approximate Hamiltonian of three avor mixing system. To treat the normal and the inverted mass orderings (NO and IO respectively) in a uni ed way, we de ne the eigenvalues as follows5 mr2en + a sign( mr2en) ( mr2en mr2en + a + sign( mr2en) ( mr2en mmr222e1n ; In eq. (3.1), the renomalized stein matter potential [1], a, are de ned as follows:6 m2 mr2en, the expansion parameter , and the Wolfen4See, however, the clarifying remark at the end of subsection 3.3.6. 5We note that the eigenvalues in (3.1) above appear in ref. [5]. See section 4.1 for the derivation and a comment on the treatment in [5]. 6The following notation is used throughout: mi2j mi2 mj2, sij = sin ij and cij = cos ij where ij are the standard neutrino mixing angles and GF is the Fermi constant, Ne is the number density of electrons, E is the energy of the neutrino, Ye the electron fraction and is the density of matter. q q i 2E ; (3.3) of 3.2 p and a = 2 2GF NeE g:cm 3 GeV This choice of mr2en is crucial to the compact formulas for the oscillation probabilities that will be given in this paper. Note also that the sign of mr2en signals the mass ordering, both mr2en and are positive (negative) for NO (IO). However, for both orderings m221 > 0, as required by nature. Notice that 0 is the same for the both mass orderings, and when we switch from NO to IO we also switch the sign in front of the square root in eq. (3.1). The nicest feature of the sign choice is that the oscillation probability has a uni ed expression and the solar resonance is in - 0 level crossing for the both mass orderings. mr2en is equal to the e ective atmospheric disappearance experiment in vacuum, me2e m2 measured in a electron (anti-) neutrino me2e recently measured by the reactor 13 experiments [13{15] up to e ects m231)2. Whether the coincidence between me2e and mr2en re ects a deep aspect of neutrino oscillation or not will be judged depending upon what happens at second order in . This point as well as the relevance of the other e ective me2e, will be discussed in depth in a forthcoming communication. The mixing angle 13 and mixing matrix in matter We use the angle to represent the mixing angle 13 in matter. With the de nitions of the eigenvalues (3.1), the following mass-ordering independent expressions for cosine and sine 2 (see section 4.2) are given by (3.5) (3.6) (3.7) (3.8) and as a goes from + 1 to 1 for the IO. In vacuum (a = 0), the atmospheric resonance, when a = mr2en cos 2 13, for both mass orderings. 1 to + 1 for the NO = 13 and = =4 at The mixing matrix in matter, V , relates the avor eigenstates, e , , to the perturbatively de ned matter mass eigenstates, , 0 , + as follows (see section 4.4): where the matrix V is unitary. It is convenient to split V into a zeroth order term, V (0), and a rst order term, V ( 1 ) in our expansion, where the zeroth order matrix is given by V (0) = 6 2 4 0 B + V (0) + V ( 1 ); 0 c23 with being the CP violating phase, whereas the rst order correction is given by c by (3.7) with (3.8) and (3.9) must be unitary to order . In fact it is, since the following two conditions are satis ed Of course, none of what follows is self consistent without unitarity here. With the matter eigenvalues, 's , de nite by eq. (3.1) and the matter mixing matrix, V , given by eq. (3.7), simple and compact expressions can be easily derived for the oscillation probabilities in matter for all channels, to leading order in , as will be shown in the next section. 3.3 Compact formulas for the oscillation probabilities in matter In this section we start by presenting the shortest path to the oscillation probabilities of the e-related channel by using the eigenvalues, ;0, and mixing matrix, V , given in the previous section to order . Then, we derive the universal expression of the oscillation probabilities which is applicable to all channels. 3.3.1 e ! e disappearance channel The derivation of the e survival oscillation probability, P ( e ! e), in our renormalized helio-perturbation theory is extremely simple. We start from the general expression 4E 4E where, L, is the baseline. Because jVe0j2 = O( 2 ), we obtain to order eq. (1.1), or where j + { 7 { Notice that the formula in eq. (3.11) takes into account the matter e ect as well as the e ect of s13 to all orders. Nonetheless, it keeps an exceptional simplicity, an e ective twoavor form in matter which consists of single term with the unique eigenvalue di erence + , the feature we believe to be unique in the market. The feature stems from the fact that there is no e component at zeroth order in in the \0" state in matter. It is expressed in the zero in the Ve0 element of the zeroth-order V matrix as in (3.8), see also section 4.4. 3.3.2 e ! and e ! appearance channels Now, we discuss the appearance channels e ! simplest way to derive the formulas for the oscillation probabilities starting from the V matrix by using unitarity. The oscillation probability P ( e ! ) can be computed as appearance channel probability is quite compact, despite that it contains all-order contributions of s13 and a. In particular, it keeps the similar structure as the one derived by the Cervera et al. [7], which retains terms of order 2 but is expanded by s13 only up to second order. This method of computing P ( e ! expression of the oscillation probability which is manifestly free from the apparent singularity as ! 0 because 1=( 0) always appears adjacent to sin [( 0)L=4E]. We will refer this method as the \shortcut method" in the rest of this paper. ) in the above o ers the shortest path to the { 8 { where the common shorthand notation for the kinematic phase ij = ( i j )L=4E is used. Again, since jVe0j2 = O( 2 ), we have to order (3.13) (3.14) (3.15) here Jr, the reduced Jarlskog factor, is ( + 0) ( + sin ( + 0) 4E a) sin )L ( + 4E mr2en 0)L cos 2 ( + ), eq. (3.14), is not quite of the form of eq. (2.1) because of ( + 4E 0)L . This can be easily remedied, 2 sin + sin 0 cos +0 = sin2 +0 sin2 0 sin2 + (3.16) which leads to exactly the form of eq. (2.1). However, if this is used then some of the terms are singular when = 0, yet the total expression is equivalent to eq. (3.14) and is nite. This is the reason why we prefer the form of eq. (3.14). The oscillation probability for e ! channel can be obtained by all of the following three methods: ( 1 ) the similar calculation by the shortcut method, ( 2 ) using the unitarity relation P ( e ! s23; s23 ! c23) [10], whose derivation is sketched in section 4.1. We note that Jr changes sign by the transformation. In general the oscillation probability in arbitrary channel can be obtained by ( 1 ) the shortcut method, or by ( 2 ) rewriting the expressions in appendix B using the formulas in appendix C. 3.3.3 The general form of the oscillation probabilities ! The expressions of the oscillation probabilities in the sector can be derived by one of the methods mentioned in the previous subsection. Then, one observes a remarkable feature that all the oscillation probabilities P ( written in a universal form: ! ) (including the e sector) can be sin ( + ( + 4E { 9 { " " " P ( ! ) = +4 fA+ g s2 c2 + fB+ g (Jr cos ) +4 fA+0g c2 + fB+0 g Jr cos =c123 +4 fA 0g s2 + fB 0g Jr cos =c123 +8 Jr ( + )( + 0)( 0) ( mr2en)2 ( + ( + 1, or the simple functions of 23. and S are given in table 1. Notice that they In table 1, we observe that three relations hold between the coe cients A 0 = A+0; B 0 = B+0 and B+ This invariance must be respected by the oscillation probabilities because the two cases in eq. (3.19) are both equally valid ways of diagonalizing the zeroth-order Hamiltonian. Look at eq. (3.5) to con rm that it is invariant under (3.19). Then, the former two identities in (3.18) trivially follow, but the last one requires use of the kinematic relationship7 sin + sin +0 cos 0 = sin + sin 0 cos +0 + sin2 + : We note that 23 dependence in the oscillation probabilities, apart from that in Jr, is con ned into the ve independent coe cients. All the other parameters, m2's, 12, 13, , E and a, are contained in the remaining part of the probabilities, which takes the universal form for all the oscillation channels. The antineutrino oscillation probabilities P ( ) can be easily obtained from the neutrino oscillation probabilities. One can show that by taking complex conjugate of the evolution equation for anti-neutrinos of energy E that it is equivalent to the evolution equation for neutrino with energy equal to E. Therefore, ! : E) = P ( : E) : (3.20) Here, we give some critical observations regarding the formulas (3.17) and the associated table 1: The oscillation probability P ( e ! e), to rst order in , is of a simple two- avor form with the matter-modi ed mixing angle 13 and the mass squared di erence given by of P ( e ! frameworks so far studied. mr2en. As far as we know, having single two- avor form as the expression e) in matter at this order is the unique case among all the perturbative 7Notice that the relation B+ = C needs to be satis ed only to order 0 because these terms are already suppressed by . Formulating the perturbation theory of neutrino oscillation with automatic implementation of the oscillation probability formulas (2.1), which is to be carried out in section 4, is the principal result of this paper. However, the formula (3.17) goes one step further beyond eq. (2.1) by displaying the channel-independent universal function of L=E, up to the coe cients A , B , etc. which depend only on 23. in the limit perfectly nite. ! With the universal form of the probabilities, unitarity can be trivially checked just by adding up the appropriate columns of table 1 with proper care of using the correct signs in the last row for the intrinsic CP-violating terms.8 One can easily con rm it by adding, for example, the e ! e , e ! , and e ! columns to give zero in If one uses the identity (3.16) to transform the sin ( + term to the right-hand side of (3.16), then the transformational invariance given by eq. (3.19) will be manifest and eq. (3.17) will become the form of eq. (2.1). However, 0 some terms will diverge even though the total expression is In the universal form, the amplitude of a given oscillation frequency, to leading (zeroth) order in , is controlled by . The amplitudes of the frequencies ( + 0), ( 0) and ( + ) are given proportional to A+0 cos2 , A 0 sin2 A+ sin2 cos2 , respectively. For the channels involving e, only the ( + frequency appears whereas in the sector all frequencies appear and are of equal magnitude near the 13 resonance, when =4, but every where else either and ) the frequency ( + 0) mr2en or ( 0) mr2en dominates. Finally, we give more clarifying remarks on generic features of the oscillation probability formulas presented in this section 3. The term proportional to sin in P ( ), in eq. (3.17), is equal to 0) ! sin 8 Jr sin ( + sin ( 4E 0)L sin ( + 0)L 4E up to an overall sign. This is because the quantity in f g is just the Jarlskog factor in matter due to the Naumov-Harrison-Scott identity, [17, 18]. Due to the above theorem sin terms are always accompanied by the Jarlskog factor Jr = c12s12c23s23c123s13. Though less well known, it can be proven that cos must be proportional to the reduced Jarlskog factor Jr=c213 [19]. It can also be shown terms that both of the -dependent terms must come with the suppression factor of [17{ 19]. It is nice to observe that these general properties are realized in our formulas. Association of the factor and the all angle factors to the -dependent terms indicates that they are genuine three avor e ects. In our general formula (3.17), all the terms 8Here, we assume that the oscillation probability formulas are derived without using unitarity, excluding one of the options we mentioned earlier to derive them. with explicit factor are of this -dependent terms. Notice that other corrections of order , though they do not show up explicitly, are implicitly contained in mr2en, ;0 and which do not contain any or 23 dependence. In vacuum, a = 0, the above oscillation probabilities reproduce the standard results, to rst order in . The form is somewhat unusual but we have checked that the matter e ects are negligible, the coe cients of the (L=E)n, for n expressions are identical. Also, we have checked that in the limit L=E ! 0, where 3, terms in eq. (3.17), are identical to those in vacuum for all oscillation channels. Range of applicability of the formulas To discuss the range of applicability of our expressions, it is useful to rst consider the vacuum expressions to rst order in the expansion parameter . For all channels, the expansion of the vacuum oscillation probabilities to rst order in does not include terms proportional to sin2 21 which starts at second order in , sin2 =2, that is at the rst atmospheric oscillation 0:002 which is small for the channels e ! x where x = e; , or since 1 at the second atmospheric oscillation maximum, ) are all of order sin2 2 13 31 = 3 =2 and 2 2 2 31 0:1. However, 0:02, which is signi cant compared to the sin2 2 13 term. So in vacuum our rst order expansion is only a good approximation for the other channels, ! 2 , ! or L=E . 1000 km/GeV for these e channels. For and ! , our rst order expansion is a good approximation to somewhat beyond L/E = 1000 km/GeV because the leading terms are not suppressed by the smallness of sin2 2 13. Then, what about the validity in matter? In section 5 we will argue that our perturbative description is valid outside the solar resonance. Notice that the region without validity (no guarantee for approximation being good) is rather wide and includes the vacuum because the solar resonance width j aj = p3(sin 2 12= cos2 13) m221 is larger than the solar resonance position a = (cos 2 12= cos2 13) m221. We expect then that our helioperturbation theory works for the matter potential a larger than a few tenth of j mr2enj. To give the reader a sense of the precision of our approximation we have plotted in gure 1, the contours of equal probability for the exact and the approximate solutions for the channels e ! , e ! e and ! . The right (left) half plane of each panel of gure 1 corresponds to the neutrino (anti-neutrino) channel. As expected, for large values of the matter potential, jaj > 13 j mr2enj we nd we have no restrictions on L/E, to have a good approximation to the exact numerical solutions. Whereas for small values of the matter potential, jaj < 13 j mr2enj we still need the restriction L=E . 1000 km/GeV.9 9Of course, the boundary between these two regions should be interpreted as an approximate one. In fact, an exact boundary would sensitively depend on the de nition of the di erence allowed between the exact and the approximate probabilities. oscillation probabilities for upper left, e ! e, upper right, e ! and lower, ! . The upper (lower) half plane is for normal ordering (inverted ordering), whereas positive (negative) L/E is for neutrinos (antineutrinos). For treatment of antineutrinos, see section 3.3.3. The order of the contours given in the title is determined from the line L/E=0. The discontinued as one crosses Ye jEj = 0 is because we are switching mass orderings at this point. In most of parameter space the approximate and exact contours sit on top of one another so the lines appear to alternate blue-red dashed. Note that, for L/E >1000 km/GeV and jYe Ej < 5 g.cm 3.GeV, the di erence between the exact and approximate contours becomes noticeable at least for e ! e and e ! . We note that most of the settings for the ongoing and the proposed experiments, except possibly for the one which utilizes the second oscillation maximum, fall into the region L=E . 1000 km/GeV. To improve the accuracy to larger values of L/E, especially for values of jaj < 13 j mr2enj, second order perturbation theory in be the subject of a future publication. is needed, which will An example of the power of our oscillation probabilities In this section we present an example of the power of our compact expression of the oscillation probabilities in understanding the physics of oscillations to rst order in our expansion parameter . For simplicity we consider the e disappearance channel. In gure 2 the e disappearance probability is shown, as a function of neutrino energy E, for baselines, L, of 3000 km (upper panel) and 5000 km (lower panel). As in gure 1, the solid blue curves are drawn by using the exact expression of P ( e ! e), and the dashed red curves by our compact formula in eq. (3.11). The black dotted curves are for the vacuum case. We rst notice that our approximate formula agrees well with the exact expression in particular at higher energy. Secondly, because of the long baselines, the matter e ect is sizeable, producing not only a large shift in the position of the rst oscillation minimum but also an signi cant increase in the depth of the minimum. These two e ects are correlated by the energy dependent quantity, ( + depth of the oscillation minimum changes from ) which can be seen graphically in gure 3: the whereas the energy at which the the rst oscillation minimum occurs changes from ! + mr2en 2 Because of the simplicity of our expression for P ( e ! e), eq. (3.11), these shifts are accurate to rst order in the expansion parameter . This simple understanding of the features of P ( e ! e) is new to this paper. 3.3.6 Comparison with the existing perturbative frameworks As we emphasized in section 2, our machinery has an advantage over the existing perturbative frameworks by having the minimum number of terms composed of sin [( j i)L=4E] (i; j = 1; 2; 3) in the oscillation probabilities. This contrasts with the features of the existing perturbative frameworks in which much larger number of terms than those minimally necessary as in (2.1) are produced. They include, typically, the terms with either extra L=E dependences or di erent frequencies in the sine functions, or often both, which easily obscures the physical interpretation. To give a feeling to the readers on how simple and compact our formulas are, we compare our expressions to the ones in the existing literatures to the same order in expansion. For de niteness, we pick the ones in ref. [10] to make the comparison, the most recent one among the reference list given in section 1. (upper panel) and 5000 km (lower panel). We have used the earth matter density 2.8 g/cm3. Our expression of P ( e ! e) in (3.11) which has only a single term (ignoring unity) may be compared with eqs. (4.6) and (4.7) which consist of total 3 terms. With regard to ), if we count numbers of terms with di erent L=E dependence we have one independent and 2 -dependent terms in (3.14), total 3. Whereas eqs. (4.8) and (4.9) in [10] have total 7 terms, 3 -independent and 4 -dependent ones. Our expression of P ( in (3.17) has 3 -independent and 2 -dependent terms, adding up to total 5. On the other hand, eqs. (4.10) and (4.11) in [10] contain total 8 -independent and 10 -dependent terms (2 sin and 8 cos terms), which adds up to 18. So not only do our expressions for the oscillation probabilities have less than half the number of L=E structures, but the form of ! ) the L=E dependence is made manifest and identical to the vacuum form. We note that when our formulas are expanded by m221= m231, they agree, of course, with the existing ones calculated previously. Therefore, our formalism may be regarded as a systematic way of organizing the equivalent expressions to order neater, structure-revealing ones. The bene t for having such simple expressions is that the physical interpretation of the formulas is transparent, as emphasized in section 2. m221= m231 into 4 Formulating the renormalized helio-perturbation theory In this section, we formulate the helio-to-terrestrial ratio perturbation theory, for short the helio-perturbation theory, which has the unique expansion parameter (4.1) We will show that use of its renormalized version is the key to the very simple formulas of the oscillation probabilities exhibited in section 3.3 and appendix B. In fact, there are mmr222e1n : two ways of deriving the oscillation probabilities, the S matrix method and the wave function method. Here we sketch both of them, leaving technical or computational parts into appendices A and B. The meaning of the agreement between results obtained by both the S matrix and the wave function methods will be discussed at the end of this section. The S matrix describes neutrino avor changes after traversing a distance L, and the oscillation probability is given by (L) = S (0); P ( ! When the neutrino evolution is governed by the Schrodinger equation, i ddx = H , S matrix is given as S = T exp dxH(x) where T symbol indicates the \time ordering" (in fact \space ordering" here). In the standard three- avor neutrinos, Hamiltonian is given by H = 8 1 <> 2E > : where the symbols are de ned in an earlier footnote. For the case of constant matter density, the right-hand side of (4.4) may be written as e iHx. We recapitulate here the earlier footnote: in (4.5) mj2i mj2 mi2 where mi denotes the mass of i-th mass eigenstate neutrinos. Position dependent function a(x) 2 2GF Ne(x)E is a coe cient for measuring the matter e ect on neutrinos propagating in medium of electron number density Ne(x) [1] where GF is the Fermi constant and E is the neutrino energy. The neutrino avor mixing matrix U is usually taken to be the standard form UPDG given by Particle Data Group. We, however, prefer to work in a slightly di erent basis, for this paper, in which the avor mixing matrix has a form (with the obvious notations sij sin ij etc. and being the CP violating phase) (4.2) (4.3) (4.4) (4.5) under the understanding that the left phase matrix in the rst line in eq. (4.6) is to be absorbed into the neutrino wave functions. Being connected by the phase matrix rotation, it is obvious that our U and UPDG give rise to the same neutrino oscillation probabilities. i Z L 0 3 0 0 ! j2: It is convenient to work with the tilde basis de ned as ~ Hamiltonian is related to the avor basis one as = (U2y3) , in which the H~ = U2y3HU23; where U23 is de ned in eq. (4.6). The S matrix in the avor basis is related to the S matrix in the tilde basis S~ as S(L) = U23S~(L)U2y3; with (4.10) and (4.11) is identical with the tilde-Hamiltonian in (4.9). Note the simplicity of the perturbing Hamiltonian, H~1, especially the zeros on the diagonal elements. The diagonalization of H~0 leads to the eigenvalues given in section 3.1, eq. (3.1), and because of the zeros along the diagonal of H~1, there are no corrections to these eigenvalues at rst order in a perturbative expansion. The authors of [5] treat the order e ect in the Hamiltonian as a renormalization of the matter potential, whereas we regard it as a renormalization of m231. mm222311 as H~ (x) = Notice that the matter term in the Hamiltonian (4.5) is invariant under the U23 rotation. Hence, the dynamics of neutrino propagation in matter is governed by only the two mixing angles 12 and 13 [22], which is also independent of thanks to our convention of U in (4.6). From the rst equation in (4.8), the relationships P ( e ! c23 ! s23; s23 ! c23) etc. simply follow as used in section 3.3. The simplest formulation of helio-perturbative treatment in the tilde-basis includes decomposition of H~ into the zeroth and the rst order terms in the expansion parameter ) = P ( e ! : m231 >< 2E 6 4 > : To derive the compact formulas of oscillation probabilities we use slightly di erent basis, a renormalized basis, to formulate the perturbation theory. That is, we absorb a certain order terms into the zeroth order Hamiltonian, H~ (x) = H~0(x) + H~1(x): 2 0 mr2en 64 c13 2E 0 c13 0 s13 m221= 3 0 0 (4.7) (4.8) Our treatment can be easily generalized to cases with matter density variation as far as the adiabatic approximation holds. See e.g., [10] for such a treatment. However, we prefer to remain, for simplicity and compactness of the expressions, into the formulas with constant matter density approximation in the rest of this paper. We transform the Hamiltonian H^ = H^0 + H^1, from the\tilde" basis to the \hat" basis, H^0 = U yH~0U ; H^1 = U yH~1U where the unperturbed Hamiltonian H^0 is diagonal, We take the following form of unitary matrix U to diagonalize H~0: H^0 = 2 1 i Z L 0 (L) = eiH^0LS^(L): (4.12) (4.13) (4.14) (4.15) (4.16) (4.18) HJEP01(26)8 The expressions of the zeroth order eigenvalues Similarly, cosine and sine are given in eq. (3.5). Also the perturbing Hamiltonian, H^1, retains it's simple form thanks to that the U rotation keeps \zeros" in H~1 unchanged both on the diagonal and in the top-right and i Z L 0 To calculate S^(L) we de ne (L) as In fact, H^1 is identical to H~1 with 13 replaced by ( 13 4.3 S matrix and the oscillation probability The S matrix in the avor basis is related to the S matrix in the tilde and the hat bases as S(L) = U23S~(L)U2y3 = U23U S^(L)U yU2y3 where we have used explicitly the fact that the matter density is constant: Then, (L) obeys the evolution equation where Then, (x) can be computed perturbatively as i d dx H1 (x) = H1 (x) eiH^0xH^1e iH^0x: Collecting the formulas the S matrix can be written as Z L 0 (L) = 1 + ( i) dxH1(x) + O( 2 ): S(L) = U23U e iH^0L (L)U yU2y3 Thus, we are left with perturbative computation of (L) with use of H1 in (4.20) to calculate the S matrix. With the S matrix in hand it is straightforward to compute the oscillation probabilities by using (4.3). We leave these tasks to appendices A and B. Mass eigenstate in matter: V matrix method H^0 is diagonal, we have In this section we calculate the V -matrix directly using our perturbation theory. If we switch o the perturbation H^1, the mass eigenstates in matter, to lowest order, are given by the hat-basis wave function ^i(0), which are the eigenstates of H^0 in (4.12), and since ^(0) = (U23U )iy i : Thus, the V matrix is given to zeroth order by V (0) = U23U whose explicit form is given in section 5.3, and also in eq. (3.8). In order to obtain the mass eigenstates in matter to rst order in , i = ^i(0) + ^i( 1 ), let us compute the rst order correction to the hat basis wave functions. Using the familiar perturbative formula for the perturbed wave functions i ^( 1 ) = 2E j6=i i X (H^1)ji ^(0) j j with H^1 in (4.12), and the i's are given by the eigenvalues of H^0, see (3.1). Then the mass eigenstate in matter i can be written to rst order in as: 0 1 0 13) mr2en c12s12c( mr2en c12s12s( + 1 0 0 13) 13) 0 1 mr2en c12s12s( + 0 1 0 ^(0) 1 Using (4.23), this equation is of the form i = V y which can be inverted to easily obtained the V -matrix given in eq. (3.7), 2 4 V = U23U 61 + c12s12 mr2en 8 >< c( > : 0 0 13) B 1 0 0 C + A > = > ; A 5 7 : (4.25) (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) HJEP01(26)8 This can be used to directly compute the oscillation probabilities as was performed in section (3.3), or by inserting the V matrix elements into the expressions of the oscillation probabilities in (4.6). This V matrix method was used to calculate the matter e ect correction in the oscillation probabilities [20]. For a di erent approach toward simpler expressions of the V matrix elements see [21]. In sections 4.3 and 4.4, we have sketched the two di erent methods for calculating the oscillation probabilities, the S-matrix method (section 4.3) and the V -matrix method (section 4.4). The results obtained by the both methods agree with each other, and the expressions of the oscillation probabilities are of the form given in eq. (2.1). Let us make a comment on what happens if we use a di erent decomposition of the Hamiltonian into unperturbed and perturbed parts as in (4.9). Then, the perturbed Hamiltonian has nonzero diagonal terms, and this produces rst order corrections to the eigenvalues. If we use these expressions for the eigenvalues and expand by the small expansion parameter we obtain terms that do not exist in (2.1), such as the ones proportional to sin [( j i)L=2E]. Of course, the S-matrix method will give the same expressions. Therefore, absence of the explicit rst-order correction to the eigenvalues is essential to keep perturbative expressions of the oscillation probabilities of the form (2.1) to order . In our case it is guaranteed by vanishing diagonal elements of the perturbed Hamiltonian (4.11). 5 More about the renormalized helio-perturbation theory In this section, we critically examine the framework of the renormalized helio-perturbation theory. Despite a drawback of the current framework (which is to be described below) we argue that our perturbation theory works apart from the vicinity of the solar resonance crossing. 5.1 Exact versus zeroth-order eigenvalues in matter The three eigenvalues of the Hamiltonian are written as 2Ei , where i runs over 1; 2; 3 for the exact eigenvalues, and i = ; 0; + for the zeroth-order eigenvalues in our perturbative framework. In gure 3 the i are plotted as a function of a, the Wolfenstein matter potential, for both the exact and the zeroth-order ones given by eq. (3.1). It is clear in gure 3 that our zeroth-order eigenvalues fail to treat the solarm2 scale level crossing correctly. As one goes through the solar resonance in the exact solution the two eigenvalues involved, the red and green, repel one another, whereas in our perturbative solution the two corresponding eigenvalues cross with each other. We point out here that the drawback is the common feature among the similar perturbative treatments of neutrino oscillation available in the market. In this section we argue, for the rst time, that despite the drawback, our perturbative framework successfully treat the avor content of these two states in region reasonably far from the solar resonance. We rst note that, despite the feature, the atmospheric and solar resonances occur at the correct values of the matter potential with our zeroth-order eigenvalues. The atmospheric resonance occurs when + is a minimum, which is at a = panel) and in the inverted mass ordering, abbreviated as IO, (right panel). The exact eigenvalues are depicted with colored lines, green for 1, red for 2, and ight blue for 3 . The eigenvalues calculated by using our renormalized helio-perturbation theory are drawn by the black dashed lines whose state labels are marked on the gures. The approximate eigenvalues for 0 and cross at the solar resonance, whereas the exact eigenvalues for 1 and 2 repelled at solar resonance. To make these features visible we used a solar m221 three times as large as the measured value. and the minimum di erence between + and resonance occurs when 0 is a minimum; this occurs when is mr2en sin 2 13 as expected. The solar a = mr2en cos 2 12= cos2 13 and the minimum di erence is zero! This is not the value determined by the full Hamiltonian which is mr2en sin 2 12. Therefore, while our perturbative scheme treats the atmospheric resonance correctly to order , it misses the e ects of the solar resonance. 5.2 Eigenvalues in vacuum and in the asymptotic regions a ! 1 In this and the next subsections, we will give the arguments to indicate that our renormalized helio-perturbation theory works apart from the vicinity of the solar resonance despite the issue mentioned above. We rst show that the zeroth-order eigenvalues given in (3.1) agrees with the exact ones to order in the asymptotic regions a ! 1. We use the characteristic equation of the full Hamiltonian (4.5) to derive (i = 1; 2; 3) X i X i;j i = a + m321 + m221 ; i j = m221 m321 + a (c122 + s122s213) m221 + c123 m321 ; 1 2 3 = c122c213a m221 m321: (5.1) Then, by using the asymptotic expansion of 's one can obtain to leading order in the 1=a expansion (at a ! +1 in the NO case) 1 = with our zeroth-order eigenvalues given in (3.1): must be interchanged. They are identical to the ones computed For the IO, the analytic expressions of i (i = i at a ! 1 for the case of NO.10 the both mass orderings by When the matter potential vanishes, a = 0, the mass squared eigenvalues are given for ; 0; +) at a ! 1 is the same as those of 0 B 0 B 2 c12 mr2en a + (s213 + s122 ) mr2en 0 a + (s213 + s212 ) mr2en mr2en mr2en 1 C A 1 C A 0 B 0 s212 mr2en AC m231 (5.2) (5.3) (5.4) (5.5) (5.6) terms are di erent from the exact values, 1 = 0, 2 = m231. To understand the meaning of the failure at order we need to discuss what happens to the avor content of the matter mass eigenstates as the matter potential changes from m221, and 3 = below to above the solar resonance. This will be done in the next subsection. Similarly, the asymptotic behaviour of the angle , i.e., 13 in matter can be easily worked out. With the de nition of in (3.5), it is easy to show that takes on the following values as a is varied from 1 to + 1. In the case of normal mass ordering, NO = < 8 > > > > > > > > > > : 0; 13; 4 2 ; ; 2 13; 1 a = a = 0 a = a = 2 10In fact, the rst line in (5.1) holds exactly. while in the case of inverted mass ordering, IO = <>> 2 8 > > > > 13; 2 4 ; ; 13 0; a = a = 2 a = a = 0 (5.7) (5.8) (5.9) It re ects a natural view that physics at a ! +1 for the normal mass ordering corresponds to the one at a ! 1 for the inverted mass ordering at least to leading order in . Neutrino mixing matrix in matter and the Since the two levels cross at the solar resonance in our perturbative treatment, one may expect that our treatment fails completely beyond the solar resonance, i.e., in the region with matter density higher than the resonance. However, we will show in this subsection that the avor contents of the three eigenstates are correctly reproduced at least in the asymptotic region. That is, the zeroth-order V matrix describes correctly the asymptotic behaviour of the exact eigenstates in matter. Suppose we denote the exact avor mixing matrix in matter as the (almost standard) U matrix de ned in (4.6). Let us discuss the case of NO but V (0) in (3.8). Notice that s23 in matter is frozen to its vacuum value for < 0:1, and s12 ' 0 at a ! 1 [3]. At a ! +1, s12 ' 1 and c12 ' 0 the matter U matrix is identical to (3.8) if we interchange 1 and 2 apart from re-phasing factor 1 for the new second rst. At a ! 1 U is nothing mass eigenstate. To make the meaning of this feature clearer we write down here the avor content of the states i (i = 1; 2; 3) at a ! 1. Noticing that i = (V y)i at a ! 1 is given at zeroth order by , the avor composition Whereas the composition at a ! +1 is given by 1 = c e 2 = fc23 s (s23e i s23e i g = 0 + c23 ) = 3 = s e + c (s23e i + c23 ) = + 1 = 2 = c e fc23 s23e i s (s23e i 3 = s e + c (s23e i g = 0 + c23 ) = + c23 ) = + The avor compositions given in (5.8) and (5.9) imply that the avor content of the lower two mass eigenstate in matter is correctly described in our perturbative framework despite the failure of describing the solar level crossing. Note, the combinations of and in the ( ) and f g in (5.8) and (5.9) are orthogonal with each other. In the case of IO, essentially the same discussion goes through. The asymptotic behavior of 12 at a ! 1 is the same as that in NO. The asymptotic behavior of 13 at a ! 1 for NO is mapped into the one at a ! 1 for IO. But, this is already taken care of by the de nition of given in (3.5). Therefore, the same U matrix in matter as the one in NO are obtained at a ! 1. Then, the same avor compositions as in (5.8) and (5.9) follow for IO. Again it is consistent with those we expect from the level crossing diagram. We add that our + state always corresponds to 3 state. At a ! +1 in NO and at a ! 1 in IO the electron neutrino component is all in this + = 3 state. At a ! 1 in NO and at a ! +1 in IO the electron neutrino component is all in for IO) state. In vacuum for both NO and IO, Ve(+0) = s = s13, which is the correct value ( 1 for NO, 2 To summarize: despite that the eigenvalues calculated by our helio-perturbation theory do not show the correct behavior at around the solar level crossing, the avor composition of the states are correctly represented by our zeroth-order states. It is the very reason why HJEP01(26)8 our perturbative formulas work at much higher and lower values of Ye E compared to the one at the solar resonance. This point escaped detection in the previous literature to our knowledge. We have seen that they work practically whole regions except for the vicinity of the solar resonance. 6 Concluding remarks We have developed a new perturbative framework of neutrino oscillations, which allows us to derive compact expressions of the oscillation probabilities in matter, which we examined in this paper to order m221= mr2en ' m221= m231. Although vacuum like in their forms, our compact results keep all-order e ects of s13 and the matter potential a, while using as the unique expansion parameter. The characteristic features of our perturbative framework (dubbed as the renormalized helio-perturbation theory), in contrast to the ones previously studied by various authors, are as follows: We use the renormalized basis (4.10) de ned with the atmospheric mass-squared di erence corrected by an order quantity, mr2en m231 2 s12 m221. The decomposition of the Hamiltonian into the unperturbed (4.10) and perturbed terms (4.11), is done in such a way that there is no diagonal entries in the perturbed one. It makes the eigenvalues of the zeroth order Hamiltonian the correct ones to order . Usage of this decomposition is crucial to obtain the form of the oscillation probabilities (2.1) akin to the form in vacuum. This facilitates an immediate physical interpretation that the frequencies of the oscillations are determined by ( + ), ( + 0), and ( 0), where the 's are the eigenvalues of the zeroth order Hamiltonian, see eq. (3.1). They determine the oscillation pattern, the functional form of L=E dependence, for all the oscillation probabilities as in eq. (2.1) to this order. The amplitudes of the sin [( j i)L=4E] terms in the oscillation probabilities, (2.1), can be determined either by the S-matrix method, or the V -matrix method. The latter may be simpler because of no rst order correction to the eigenvalues. In contrast to the complexity of the exact solution, this perturbative approach ensures calculability of the V matrix elements in the form of power series, see eq. (4.25). It leads to the simple analytic expressions of the oscillation probabilities which have a universal form to rst order in , as shown in eq. (3.17). The coe cients in this universal form are 0; 1 or simple functions of 23, see table 1. From the universal form of the oscillation probabilities, eq. (3.17), we readily observe the followings: ( 1 ) Each one of the zeroth-order terms in the oscillation probabilities, in all channels, takes the same form as the corresponding two- avor oscillation probability in vacuum but with use of the eigenvalues and 13 in matter. In the special case of e ! e, only the term with the particular frequency / + is non-zero, so the L=E dependence is of the two avor form. The dominance of this frequency also occurs in the oscillation probability for e ! . ( 2 ) All the rst-order correction terms with the explicit factor ( m2 = m2 ) in the oscillation probabilities are proportional to either cos or sin . They are also suppressed by the angle factor / c12s12c23s23s13. Hence, they are the genuine three- avor e ects. The explicit expressions of the -dependent terms in our formulas are in agreement with the general theorems. (3) Unitarity can be trivially checked. Further comments are made in section 3.3.3. Despite the success of our current framework in almost all regions including the one around the rst oscillation maximum, it has a clear drawback. It fails to accommodate the physics at around the solar-scale resonance. This is related to the unphysical feature that the two eigenvalues ( and 0) cross with each other at the solar resonance, the universal fault in all the perturbative treatment which include the expansion parameter in the market. However, we have o ered, for the rst time to our knowledge, an explanation why our framework works beyond the resonance despite the failure in treating the solar level crossing. We hope that we can return to the problem as a whole in the future. As we noticed at the end of section 3.1 our renormalized mr2en is identical, to order , to the e ective m2 measurable in an (anti-) e disappearance experiment in vacuum. It is a tantalizing question whether it is just a coincidence, or is an indication of something deep. It is quite possible that the features of the current framework mentioned above naturally generalizes to higher orders. That is, one can demand that the oscillation probabilities of the general form in (2.1) calculated by the V -matrix method, with the carefully chosen zeroth-order eigenvalues, agree with the ones from the S-matrix method and are correct to certain order in . Our result in this paper may be regarded as an existence proof of this concept to order . Since we know that this is true in the exact form of the oscillation probabilities (assuming adiabaticity) [3], it is likely to be correct in each order in perturbation theory. It may or may not require higher order renormalization in mr2en. Acknowledgments We thank Nordic Institute for Theoretical Physics (NORDITA) and the Kavli Institute for Theoretical Physics in UC Santa Barbara for their hospitalities, where part of this work A A.1 pute H1(x) calculated as Calculation of S matrix elements Computation of S^ matrix elements Z x 0 2 4 1(x) = ( i) dx0H1(x0) ; By using H^1 in (4.15) and eiH^0x = diag ei x=2E; ei 0x=2E; ei +x=2E , one can easily comeiH^0xH^1e iH^0x. Then, using eq. (4.21), the rst order term of (x) can be where we have introduced the simpli ed notations, c( sin ( 13), etc. The simplicity in the structure of (A.1) with many zeros is the mathematical reason why the expressions of neutrino oscillation probabilities are so simple in 13), s( 13) our renormalized helio-perturbation theory. The S^ matrix is given by S^ = e iH^0x = e iH^0x [1 + matrix. Then, the elements of S^ matrix are given by: 1(x)] where 1 denotes the unit 0 0 s 13) 13) i( + ei( + 0)x=2E 1 0) 0) 13) ; ; s ( cos ( 0 0 13) e i( + 0)x=2E 1 7 i( + 0) was done. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. H.M. thanks Universidade de S~ao Paulo for the great opportunity of stay under \Programa de Bolsas para Professors Visitantes Internacionais na USP". He is grateful to Theory Group of Fermilab for supports and warm hospitalities during the several visits in 2012-2015 where this work was started and made corner turning progresses. S.P. acknowledges partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN- GA-2011- 289442). Fermilab is operated by the Fermi Research Alliance under contract no. DE-AC02-07CH11359 with the U.S. Department of HJEP01(26)8 ^ See = e i x=2E; Se = S^ e = 0; ^ Se = S^ e = ^ ^ S = S^ i i S = e i 0x=2E; S = e i +x=2E; mr2enc12s12c( mr2enc12s12s( 13) 13) e i 0x=2E e i x=2E e i 0x=2E e i +x=2E i( i( + 0) 0) ; : A.2 The relationships between S^, S~, and S matrices The relationships between S^, S~, and S matrices are summarized as S~(x) = U S^(x)U y S(x) = U23S~(x)U2y3: (A.1) 3 5 (A.2) (A.3) The left equation can be written explicitly as S 2 S~ee S~e Se ~ 6 S~ e S~ 4 ~ S e S~ S S 3 5 2 c 4 0 s 3 2 S^ee S^e Se ^ 3 2 c 0 s 3 7 = 6 0 1 0 7 6 S^ e S^ s 0 c 5 4 ^ S e S^ ^ S ^ S Hence, the elements of S~ can by written in terms of the elements of S^ as See = c2 S^ee + s2 S^ ~ + c s Se + S^ e ; ^ S s S^ e + c S^ ; s2 S^e + c2 S^ e s S^e + c S^ = s2 S^ee + c2 S^ c s = S~ c s ^ See ^ S ; ^ See ^ S = S~e ; Se + S^ e : ^ Similarly, the elements of S can be written in terms of the elements of S~ matrix as See = S~ee; S e = c23S~ e + s23ei S~ e = Se ( ); S e = c23S~ e s23e i S~ e = Se ( ); S S S S = c223S~ = c223S~ = c223S~ = s223S~ + s223S~ + c23s23(e i S~ + ei S~ ); s223e2i S~ + c23s23ei (S~ S~ ); s223e 2i S~ + c23s23e i (S~ S~ ) = S ( c23s23(e i S~ + ei S~ ): (A.4) (A.5) (A.6) ; L) = jS j2, eq. (4.3). B Expressions of neutrino oscillation probabilities With the expressions of S matrix elements obtained in appendix A, and using eq. (4.3) it is straightforward to calculate the neutrino oscillation probabilities. Similarly, one can insert the V matrix elements given in (3.7) into (2.1) to obtain the equivalent results. In this appendix we only give the resulting expressions, but of all the oscillation channels under the viewpoint that unitarity is not to be imposed but must be proven to show the consistency of the calculation. To compare these with the results of section (3.3), use of the identities given in appendix C may be of some help. The only comment worth to give here is about the simple method for transformation c23 ! P ( s23 and s23 ! c23 to obtain P ( e ! ) from P ( e ! ), or P ( ! ), which is utilized in section 3. Though we work with the rephased mixing matrix de ned in (4.6), the transformations produces Se from Se , and S S , up to an overall phase, see eq. (A.6). ) from avor from B.1 Oscillation probabilities in e row P ( e ! e) is extremely simple as where L is the baseline. The reasons for the simplicity is discussed in depth in section 3.3. 4E (B.1) It is almost trivial to verify unitarity in the e row: P ( e ! e) + P ( e ! ) + P ( e ! 1 1 c c( 8 4 4E ) = 4c2 s2 c223 sin2 ( + 4E 0)L mr2enc12s12c23s23c s cos c c( sin 4E ( 0 )L c c( 13) ( 1 13) ( 0) + s s( 0) + s s( +8 mr2enc12s12c23s23c s sin sin ( + 4E sin 4E 13) ( + 13) ( + c c( sin 0)L ( 0 4E 0)L 4E 0)L (B.2) (B.3) 1 13) ( 4E 1 13) ( 4E 1 0) 1 0) 0) 1 : 0) 1 : P ( P ( ) Oscillation probabilities in row e : ) = P ( e ! e) is related to the T-conjugate channel probability P ( e ! ) as P ( ! : ), whose latter can be obtained by replacing by in (B.2). Therefore, we only give the expressions of P ( ! ) and P ( ! ): s223c s c c( 13)( c s223s c c( 13)( 4E 1 1 +4 mr2enc12s12c23s23 c223 s223 cos 0)L + s2 sin2 ( 4E 4E 0)L With the above results the unitarity in row can also be veri ed: P( ! e) + P( ! ) + P( ! ) = 1. B.3 Oscillation probabilities in row P( ! e) and P( ) can be given by their T-conjugate channels: P( ! : ) = P( ! : ). Therefore, we only give the expressions of P( ! ) below. P( ! ) = 1 4c243c2s2 sin2 ( + 8 mr2enc12s12c23s23 cos c223c s c c( 13)( c c223s c c( 13)( 4E 1 1 + s s223 c223s2 c( 13)( )L 4c223s223 c2 sin2 ( + 0)L + s2 sin2 ( 4E 4E 0)L 0) s s( 13)( + 1 0) sin2 ( + )L 4E 0) + s223 c223c2 s( 13)( + 1 0) sin2 ( + 0)L 4E 1 0)+c223c s s( 13)( + 0) sin2 ( 1 4E 0)L : (B.6) The unitarity in row can also be veri ed: P( ! e) + P( ! ) + P( ! ) = 1. 2 X Im[V iV iV j V j ] sin ( j 2E i)L where i = 0; ; + (or 1; 2; 3). To cast this term to the one in (2.1), one needs the following identity HJEP01(26)8 sin ( + 2E sin 4E 0)L 2E sin + sin 4E 0)L 2E sin 0)L ( 0 4E +)L : Here, we list some more formulas which may be useful to understand the relationship between di erent expressions of the oscillation probabilities: Some useful identities A straightforward derivation of the general expressions of the oscillation probabilities contains the following form of CP- or T-violation terms (C.1) (C.2) (C.3) (C.4) (C.5) cos 2( c c( s s( c s( s c( 13) = 13) = 13) = 13) = 13) = a cos 2 13 ; sin 2( ) + ( mr2en ( mr2en ( mr2en + a) ; 13) = a) ; a) : ) + ( mr2en + a) ; 2 s12 2 s12 mr2en ; mr2en : ( + a sin 2 13 : 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. oscillations, Phys. Rev. D 22 (1980) 2718 [INSPIRE]. [3] H.W. Zaglauer and K.H. 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Hisakazu Minakata, Stephen J. Parke. Simple and compact expressions for neutrino oscillation probabilities in matter, Journal of High Energy Physics, 2016, 180, DOI: 10.1007/JHEP01(2016)180