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Analytical Theory of Neutrino Oscillations in Matter with CP violation Mikkel B. Johnson Los Alamos National Laboratory, Los Alamos, NM 87545 Ernest M. Henley 6 Department of Physics, University of Washington, Seattle, WA 98195 1 Leonard S. Kisslinger 0 2 Department of Physics, Carnegie-Mellon University, Pittsburgh, PA 15213 n Wedevelopanexactanalyticalformulationofneutrinooscillationsinmatterwithintheframework u oftheStandard NeutrinoModel assuming 3Dirac Neutrinos. OurHamiltonian formulation, which J includes CP violation, leads to expressions for the partial oscillation probabilities that are linear 0 combinations of spherical Bessel functions in the eigenvalue differences. The coefficients of these 1 BesselfunctionsarepolynomialsintheneutrinoCKMmatrixelements,theneutrinomassdifferences squared, the strength of the neutrino interaction with matter, and the neutrino mass eigenvalues ] in matter. We give exact closed-form expressions for all partial oscillation probabilities in terms h of these basic quantities. Adopting the Standard Neutrino Model, we then examine how the exact p - expressions for the partial oscillation probabilities might simplify by expanding in one of the small p parameters α and sinθ13 of this model. We show explicitly that for small α and sinθ13 there are e branchpointsintheanalyticstructureoftheeigenvaluesthatleadtosingularbehaviorofexpansions h nearthesolar andatmosphericresonances. Wepresentnumericalcalculations thatindicatehowto [ usethe small-parameter expansions in practice. 3 v 3 PACS Indices: 11.3.Er,14.60.Lm,13.15.+g and future neutrino facilities, including our earlier 9 work [7–10]. Keywords: 0 The present paper was undertaken, and used, for 4 the purpose of independently confirming the results 0 of Refs. [7–10]. We find that the accuracy of the . 1 I. INTRODUCTION expanded oscillation probability is restricted by the 0 presenceofbranchpointsintheanalyticstructureof 5 the eigenvalues of neutrinos propagating in matter. 1 In this paper, we develop an exact analytical We also show that the regions where the expanded : representation of neutrino oscillations [1] in mat- v resultsarereliableisdifferentforexpansionsinα[4] ter within the framework of the Standard Neutrino Xi Model (SNM) [2] with 3 Dirac Neutrinos. The ex- and sin2θ13 [5, 6]. We then map out regions where the expanded results are reliable by comparing nu- r act closed-form expressions we give for the time- a evolution operator S(t,t′) are obtained from H merical results to the exact results of our Hamilto- ν nian formulation. using the Lagrange interpolation formula given in Ref. [3]. The resulting expressions are easily evalu- Another recent study [11] takes a complementary ated without any approximations. approach and finds that the predictions of Refs. [7– The paper is divided into two main parts. In 10]canbeimprovedincertainregionsusinganexact the first, we summarize the main results of our the- evaluation of the integral Iα∗ rather than the ap- ory. Details underlying the derivation are given in proximateonefoundthere. Itconcludesthatwithin Appendices. We also retrieve the well-known two- these regions, predictions of (µ, e) oscillations im- neutrinoflavorresultsasaspecialcaseofourgeneral prove for certain values of the experimental param- results. eters. In the second part we address other analytical The dimensionalities of the neutrino Hamiltonian formulations found in the literature. The expan- H andtheparameterspacecharacterizingthemix- ν sion of the neutrino oscillation probability in one of ingofthreeneutrinopairsaresourcesofdifficultyfor the small parameters α and sin2θ of the standard finding a tractable representation of the oscillation 13 neutrino model (SNM) for H is of particular inter- probability. The Lagrange interpolation formula [3] ν est. The seminal work along these lines is found in is enormously helpful, providing an exact and for- Refs. [4–6]. This work underlies many of the analy- mally elegant expression for the exponentiation of sesandexploratorystudiesofexperimentsatpresent an n n matrix. × 2 The descriptionoftwo-flavorneutrino oscillations With the momentum dependence factored out, is elementary by comparison. In that case, H is a three basis states M(i)>, i=(1,2,3) are then re- ν | 2 2matrix,andthe mixingisdescribedbyasingle quired to describe three neutrinos. The basis states × real parameter. correspondtoaspecificrepresentation,asindescrip- tionsofaspin-1object. Thebasisshould,ofcourse, be orthogonal, II. NEUTRINO DYNAMICS <M(i)M(j)>=δ (6) ij | We will be interested in the dynamics of the and complete, threeknownneutrinosandtheircorrespondinganti- neutrinos in matter. This dynamics is determined M(k)><M(k) =1 . (7) by the time-dependent Schro¨dinger equation, | | k X d Oncethebasisischosen,wavefunctionsforaneu- i ν(t)>=H ν(t)> , (1) ν dt| | trino state are naturally introduced as the components of this state in the chosenbasis. For example, where the neutrino Hamiltonian, withtheeigenstatesofEq.(3)expandedinthebasis, H =H +H , (2) ν 0v 1 ν >= M(i)>mi , (8) consists of a piece H describing neutrinos in the | mj | j 0v i X vacuum and a piece H describing their interaction 1 with matter. the wave functions of νmj > would be the set mi, i = (1,2,3). With| the plane wave factored ThesolutionsofEq.(1)maybeexpressedinterms j of stationary-state solutions of the eigenvalue (EV) out, the wave function is just a set of three numbers. Additionally, introduction of a basis makes it equation possible to represent neutrino states and operators H ν >=E ν > , (3) such as H in matrix form, with each entry in the ν mi i mi ν | | matrix corresponding to a projection of the object where the label “m” indicates neutrino mass eigen- being described onto the basis. states,asdistinguishedfromtheirflavorstatessome- InthispaperwetaketheHamiltonianinEq.(2)to times denoted the label “f”. In operator form, this be expressed in the standard representation, where dynamics may be expressed in terms of the time- themassbasisstates M(i)>aretakenasthesetof evolution operator S(t′,t), which describes com- | states that diagonalize the neutrino vacuum Hamil- pletely the evolution of states from time t to t′ and tonian H , i.e. M(i)> ν0 >= ν¯0 >, also satisfies the time-dependent Schro¨dinger equa- 0v | ≡| mi | mi tion. H ν0 >=E0 ν0 > . (9) 0v| mi i| mi We will examine neutrinos propagating in a uni- form medium for interactions constant not only in In matrix form space but also time. Because the Hamiltonian is E0 0 0 then translationally invariant, attention may be re- 3 H = 0 E0 0 , (10) strictedtostates,bothinthevacuumandinmatter, 0v  2  0 0 E0 characterizedby momentum p~ and therefore having 1   the overall r-dependence eip~·~r. In this case expres- withtheEV’stakentobeorderedE0 E0 E0 as sions may be simplified by suppressing the overall 1 ≤ 2 ≤ 3 in the normal mass hierarchy. In the literature, the plane wave, a convention we adopt. Hamiltonian is often expressed in a different basis For time-independent interactions, S(t′,t), obtained by rotating to one in which the complete S(t′,t)=e−iHν(t′−t) , (4) neutrino Hamiltonian is diagonal as in Ref. [4]. We assume here that that neutrinos and anti- neutrinosrepresentedby ν0 > and ν¯0 >,respec- depends on time only through the time difference | mi | mi t′ t. Then, writtenintermsofthe stationarystate tively, are the structureless elementary Dirac fields − of the the Standard Neutrino Model [2]. For this solutions ν > of Eq. (1), mi | reason the theory is invariant under CPT, so the S(t′,t)= ν >e−iEi(t′−t) <ν . (5) mass of an anti-neutrino in the vacuum is the same mi mi | | as that for its corresponding neutrino. i X 3 A. Flavor and Mass States with V = √2G n and n the electron number F e e ± density in matter. Neutrinos are produced and detected in states of For electrically neutral matter consisting of pro- tons, neutrons, and electrons, the electron density good flavor, ν >. The three flavors, electron (e), fi muon (µ), an|d tau (τ) correspond, respectively, to ne is the same as the proton density np, the index values i = (1,2,3). In the vacuum, each n =n flavor state is a specific linear combination of the e p =RN , (15) three mass eigenstates M(i)> of the neutrino vac- | uum Hamiltonian H . This linear combination is 0v whereN =n +n istheaveragetotalnucleonnum- expressed in terms of the same set of numbers U n p ij ber density and R = n /N is the average proton- for both neutrinos and anti neutrinos p nucleon ratio. In the earth’s mantle, the domi- ν0 >= U∗ M(j)> nant constituents of matter are the light elements | fi ij| soR 1/2;inthesurfaceofaneutronstarR<<1. j ≈ X Matrix elements of H are thus 1 ν¯0 >= U M(j)> , (11) | fi Xj ij| <M(k)|H1|M(k′)>=U1∗kVU1k′ . (16) where Uij are the elements of a unitary operatorU, Matrix elements of H1 are thus the neutrino analog of the familiar CKM matrix. It is standard to express Uij in terms of three mixing <M(k)|H1|M(k′)>=U1∗kVU1k′ . (17) angles (θ ,θ ,θ ) and a phase δ characterizing 12 13 23 cp CP violation, C. Dimensionless variables c12c13 s12c13 s13e−iδcp U U s c , (12) The results are most naturally expressed in di- 21 22 23 13   U U c c mensionlessvariables. Wefirsttakeadvantageofthe 31 32 23 13 global phase invariance to express all energies rela-   where tive to the vacuum EV E0 of the same momentum. 1 We indicate that a quantity is expressed relative to U21 = s12c23 c12s23s13eiδcp E0 by placing a bar over it, e.g., − − 1 U =c c s s s eiδcp 22 12 23− 12 23 13 E¯0 E0 E0 . (18) U31 =s12s23 c12c23s13eiδcp i ≡ i − 1 − U = c s s c s eiδcp . (13) We follow the same conventionfor the Hamiltonian, 32 12 23 12 23 13 − − Weuseherethestandardabbreviations12 ≡sinθ12, H¯ν ≡Hν −1E10 , (19) c cosθ , etc. The parameters θ and δ are 12 ≡ 12 cp so the EV equation Eq. (3) becomes determined from experiment. thaBtetchaeusreelUatiijon→shiUpi∗ijnwEiqth. (1δc1p) b→etw−eδecnpflitavfoorllaownds (H¯0v+H1)|νmi >=E¯i|νmi >, (20) mass states for anti-neutrinos and neutrinos in the where vacuum is equivalent to δ δ . cp ↔− cp H¯ H 1E0 . (21) 0v ≡ 0v− 1 Then,toexpressthe theoryindimensionlessvari- B. Neutrino Interacting Hamiltonian ables we divide all energies, including the Hamil- tonian, by E¯0 = E0 E0. The stationary-states 3 3 − 1 TheinteractionH ,determinedbytakingtheelec- ν > are also be determined from the dimension- 1 mi | tronflavorstatesscatteringfromtheelectronsofthe less Hamiltonian Hˆ¯ , ν mediumtomediatetheinteraction,isthenexpressed as an operator in the standard representation, Hˆ¯ =Hˆ¯ +Hˆ , (22) ν 0v 1 V 0 0 i.e., from the solutions of H =U−1 0 0 0 U , (14) 1  0 0 0 Hˆ¯ ν >=Eˆ¯ ν > , (23) ν mi i mi | |   4 where the “hat” placed over a quantity indicates it The time-evolution operator, Eq. (4), expressed in is dimensionless. Thus dimensionless variables is, Eˆ¯ E¯i S(L)=e−iHν(t′−t) i ≡ E¯0 Hˆ¯ H¯30v , (24) =e2iEˆ¯01∆Le−2iHˆ¯ν∆L , (33) 0v ≡ E¯30 where Hˆ¯ is given in Eq. (26), and where ν and Hˆ1 is obtained from H1 by replacing L(m2 m2) ∆ 3− 1 . (34) V L ≡ 4E V Aˆ . (25) → ≡ E¯0 3 [The similar quantity ∆L as defined in Ref. [7] is The quantity Aˆ is the same as that defined in exactly one-half of that appearing in Eq. (34).] Refs. [4, 7–11]. The connection of the Hamiltonian Hˆ¯ to the full Hamiltonian H =H +H is then ν ν 0v 1 E. Neutrino Mass Eigenvalues H =1E0+E¯0Hˆ¯ . (26) ν 1 3 ν The neutrino mass eigenstates in a medium are solutions to the EV equation for Hˆ¯ , Eq. (23). In ν D. Neutrino Vacuum Hamiltonian Hˆ¯0v many familiar formulations [4–6] the full solutions, including both the eigenstates ν > and EV’s E¯ˆ, i i | The case of main interest for many situations is are required to find the neutrino oscillation proba- the ultra-relativistic limit,~p >> m2 (we take the bilities. | | speedoflightc=1). Forultra-relativisticneutrinos in the laboratory frame, the energy of a neutrino in 1. Diagonalization of Neutrino Hamiltonian the vacuum becomes m2 E0 p~ + i , (27) The energies E¯ˆ are solutions of the cubic equa- i ≈| | 2E i tion [12] where m is its mass the vacuum. Similarly, E ap- i i pearing in Eq. (3) may be written E¯ˆ3+aE¯ˆ2+bE¯ˆ +c=0 , (35) i i i M2 E p~ + i , (28) where i ≈| | 2E a= (1+α+Aˆ) where M is its mass in the medium. Thus, in this i − limit, b=α+Aˆcos2θ +AˆαC(+) 13 2 Eˆ¯ Mi2−m21 (29) c=−Aˆαcos2θ12cos2θ13 . (36) i → m2 m2 3− 1 We have expressed b in terms of a frequently recur- and ring combination of mixing angles, 0 0 0 C(±) cos2θ sin2θ sin2θ . (37) Hˆ¯ 0 α 0 (30) 2 ≡ 12± 12 13 0v →  0 0 1 Note that the mass eigenstate energies are indepen-   dent of δcp and θ23 for both neutrinos and antineu- with trinos. m2 m2 The solutions of Eq. (35) are expressed conve- α 2− 1 . (31) ≡ m2 m2 niently in terms of the quantity d, 3− 1 Inthislimit,thedistanceLfromthesourcetothe d=ψ+ ψ2 4γ3 detector corresponding to S(t′,t) in Eq. (4) is γ a2 3b − p ≡ − L=t′ t . (32) ψ a3 27c 3aγ . (38) − ≡ − − 5 These solutions are real when An expression for Σ[ℓ], d1/3 2 =22/3γ >0 , (39) Σ[ℓ]= a E¯ˆ , (45) | | − − ℓ which requires follows from Eq. (42). An equivalent expression for ψ2 <4γ3 , (40) Π[ℓ] in terms of E¯ˆℓ is found by subtracting Eq. (35) for E¯ˆℓ and that for E¯ˆℓ′ and dividing through by eannedr,gtiehsusis, trheaqtuidrebdebcyomHperlemxi.ticBietycaoufsethheavnienugtrrienaol ∆Eˆ¯ℓℓ′. We find Halalmpailrtaomnieatne,rEseqtss.i(n39te,4r0m)saomfowuhnitchtoHconisdditeiofinnsedo.n 0=(E¯ˆℓ2+E¯ˆℓE¯ˆℓ′ +E¯ˆℓ′2) ν We find +a(E¯ˆℓ+E¯ˆℓ′)+b E¯ˆ = a 1 (√3Im[d1/3]+Re[d1/3]) =(E¯ˆℓ+E¯ˆℓ′)2 E¯ˆℓE¯ˆℓ′ E¯ˆ1 =−a3 −+ 3·211/3(√3Im[d1/3] Re[d1/3]) +a(E¯ˆℓ+E¯ˆℓ′)−+b , (46) 2 −3 3 21/3 − giving · E¯ˆ3 = a + 22/3Re[d1/3] . (41) Σ[ℓ]2 Π[ℓ]+aΣ[ℓ]+b=0 . (47) −3 3 − The masses are ordered so that m > m > m . Then, using Eq. (45), 3 2 1 Because EV do not cross, Eˆ¯ > Eˆ¯ > Eˆ¯ for all 3 2 1 Π[ℓ]=Σ (Σ +a)+b Aˆ. A simple constraint among Eˆ¯ is found from ℓ ℓ t|he| trace of Eq. (22), i =b+aE¯ˆ +E¯ˆ 2 . (48) ℓ ℓ TrHˆ¯ =E¯ˆ +E¯ˆ +E¯ˆ Finally, havingobservedthat powersof the quan- ν 1 2 3 titiesgiveninEq.(43)willappearinvariousexpres- =TrHˆ¯0v+TrHˆ1 sions, we note that Π[ℓ]p and Σ[ℓ]q with p 2 and =1+α+Aˆ≡−a . (42) q 3involvelinearcombinationsofeigenval≥uesE¯ˆℓn ≥ withpowersn 3. Suchtermsareequivalentlyrep- ≥ resentedby alinearcombinationofthree terms,one 2. Using neutrino mass eigenvalues in our Hamiltonian Formulation proportionaltoE¯ˆℓ2,oneproportionaltoE¯ˆℓ,andone independent of E¯ˆ , obtained by using the equation ℓ In our formulation, the entire dependence of the ofmotionrepeatedly. We later makeuse ofthis fact time evolution operator on the neutrino eigenvalues to simplify various expressions. occurs through three eigenvalue combinations, ∆Eˆ¯ℓℓ′ E¯ˆℓ E¯ˆℓ′ III. THE S-MATRIX IN OUR ≡ − Σℓℓ′ E¯ˆℓ+E¯ˆℓ′ HAMILTONIAN FORMULATION ≡ Πℓℓ′ ≡E¯ˆℓE¯ˆℓ′ , (43) TheprobabilityP(νa →νb)forneutrinostooscil- late from the initial state of flavor a to a final state with ℓ > ℓ′ (and powers thereof). We denote such offlavorb isfoundfromthe time-evolutionoperator quantities using a bracket notation, For example, S(t′,t) as ∆Eˆ¯[1]=Eˆ¯3−Eˆ¯2 (νa νb)= Sab(t′,t)2 ∆Eˆ¯[2]=Eˆ¯3 Eˆ¯1 P → |Pab(t′ t|) , (49) − ≡ − ∆Eˆ¯[3]=Eˆ¯ Eˆ¯ , (44) 2 1 where − inthecaseof∆Eˆ¯. Wewillgenerallyusethisbracket Sab(t′,t)2 = <ν0 S(t′,t)ν0 > 2 . (50) | | | fb| | fa | notationalsofor other quantities in ourformulation that depend on two indices (ℓ,ℓ′), such as Σℓℓ′ and We accordingly determine here Pab(t′ t) from Πℓℓ′. S(t′,t) defined in Eq. (4). − 6 In this section we review the formulation of neu- Using the convention that Oab, written without trino oscillations based on the Lagrange interpo- parentheses enclosing ab, denotes the matrix ele- lation formula as used in Ref. [3]. This formula- ments of the operator O, tion leads to exact, closed-form expressions for the Oab <M(b)OM(a)> , (56) time-evolution operator and the partial oscillation ≡ | | probabilities that are linear combinations of spher- thematrixelementsFab ofF giveninEq.(55)may ℓ ℓ ical Bessel functions in the eigenvalue differences be compactly written whose coefficients are polynomials in the neutrino CKMmatrixelements,theneutrinomassdifferences <M(b)Wˆ¯[ℓ]M(a)> Fab = | | , (57) squared, the strength of the neutrino interaction ℓ Dˆ¯[ℓ] with matter, and the neutrino mass eigenvalues in matter. We are led quite naturally to such expres- where [3], sions for all the partial oscillation probabilities in terms of these basic quantities. The numerical re- Wˆ¯[1] (UHˆ¯νU−1 1Eˆ¯3)(UHˆ¯νU−1 1Eˆ¯2) ≡ − − sults given later in this paper are based on this for- Wˆ¯[2] (UHˆ¯ U−1 1Eˆ¯ )(UHˆ¯ U† 1Eˆ¯ ) mulation. ≡ ν − 3 ν − 1 Wˆ¯[3] (UHˆ¯ U−1 1Eˆ¯ ) ν 2 ≡ − (UHˆ¯ U† 1Eˆ¯ ) (58) A. Time-Evolution Operator × ν − 1 and The overallphase in Eq. (33) does not contribute to Sab(t′,t)2, so for the purpose of calculating the Dˆ¯[1]=(Eˆ¯3−Eˆ¯1)(Eˆ¯2−Eˆ¯1) osc|illation p|robability, we may take Dˆ¯[2]=(Eˆ¯ Eˆ¯ )(Eˆ¯ Eˆ¯ ) 1 2 3 2 − − S(L) e−iHˆ¯ν∆L . (51) Dˆ¯[3]=(Eˆ¯1 Eˆ¯3)(Eˆ¯2 Eˆ¯3) . (59) → − − Then, with neutrinos created and detected in flavor Equations(58,59) use the same bracketnotationin- states,whicharecoherentlinearcombinationsofthe troduced in Eq. (34). The result in Eqs. (53,54,55) neutrino vacuum mass eigenstates given in Eq. (9), is immediately verified to be correct by inserting a complete set of intermediate eigenstates of H in ν |νf0a >= Ua∗j|M(j)> , (52) Eq. (58). Xj It follows from the unitarity of U that Wˆ¯[ℓ] is we see that the mass eigenstate components of Hermitian, the flavor states contribute coherently to the time- evolution operator. Thus, Wˆ¯[ℓ]† =Wˆ¯[ℓ] (60) <ν0 e−iHˆ¯ν∆L ν0 > and that the two factors in Eqs.(58) commute with fb| | fa each other. We find from Eq. (58) that =<M(b)Ue−iHˆ¯ν∆LU† M(a)> . (53) | | TrWˆ¯[ℓ]=Dˆ¯[ℓ] . (61) This coherenceleadsto the oscillationphenomenon. The elegant formulae for S(L) e−iHˆ¯ν∆L are Equation (60) establishes the reflection symmetry ≡ obtained from the Lagrange interpolation formula, Fab∗ =Fba . (62) Eqs. (9,11) of Ref. [3], ℓ ℓ Ue−iHˆ¯ν∆LU−1 = Fℓexp−iEˆ¯ℓ∆L , (54) terEmxsploifciHt e.xpTrehsesieonntsirfeordeWpˆ¯e[nℓd]eanrceeeoafsiWlˆ¯y[fℓo]uonndδin ν cp ℓ X occurs through three operators independent of δ , cp where T =L=t′ t and W [ℓ], W [ℓ] and W [ℓ], − 0 cos sin Fℓ ≡Πj6=ℓUHˆ¯νEˆ¯Uℓ−−1Eˆ−¯j1Eˆ¯j . (55) Wˆ¯[ℓ]=+Wiˆ¯si0n[ℓδ]+Wˆ¯cosδ[cℓp]W,ˆ¯cos[ℓ] (63) cp sin For three neutrinos, the sum in Eq. (54) runs over three values of ℓ and the product in Eq. (55) over with Wˆ¯ [ℓ], Wˆ¯ [ℓ] and Wˆ¯ [ℓ] real and indepen- 0 cos sin two values of j. dent of δ . Details are given in Appendix A. cp 7 B. Total Oscillation Probability with Pab linear in sinδ , Pab linear in cosδ , sinδ cp cosδ cp Pab quadratic in cosδ , and Pab independent of cos2δ cp 0 δ . Although only the overall oscillation probabil- Expressions for (ν ν ) may be obtained di- cp a b rectly from S(L), P → ity is a true probability, guaranteed to be strictly positive everywhere, we find it convenient to refer (ν ν )= Sab(t′,t)2 to these four terms as “partial oscillation probabili- a b P → | | =Re[Sab(L)]2+Im[Sab(L)]2 .(64) ties.” Approximate expressions for the partial oscil- lationprobabilitiesexpandedinthesmallparameter Convenient expressions for Re[Sab(L)] and α of the SNM in Ref. [4]. Im[Sab(L)] defined by Eq. (54) are presented Obtaining expressions for the partial oscillation inAppendix A.InourHamiltonianformulation,the probabilities from the time-evolution operator re- dependence of S(T) on the CP violating phase δ quires additional analysis, given in Appendix B. In cp is very simple and follows from Eqs. (54,63) noting terms of spherical Bessel functions, we find there, that Fab =Wˆ¯ab[ℓ]/Dˆ¯[ℓ], Eq. (57). We thus find, 4∆ ℓ Pab (∆ ,Aˆ)=sinδ L ( 1)ℓwab [ℓ] sinδ L cp Dˆ − sin Re[Sab(t′,t)]=δ 2 Wˆ¯a0b[ℓ]sin2Eˆ¯ ∆ j (2∆ˆ[ℓ]) , Xℓ (68) ab− Dˆ¯[ℓ] ℓ L × 0 ℓ X wherewab [ℓ],andthereforePab ,isanti-symmetric −2cosδcpXℓ Wˆ¯Dˆ¯acob[sℓ][ℓ]sin2Eˆ¯ℓ∆L bup.nrodWbeaerbfiailnsitid↔niesba.reTinhdeivoitdhuearllythssriyenmeδmpaerttriiacluonsdceilrlaatio↔n +sinδcp ℓ Wˆ¯Dˆ¯asib[nℓ][ℓ]sin2Eˆ¯ℓ∆L(,65) Pcaobsδ(∆L,Aˆ)=−cosδcp4∆Dˆ¯2L (−1)ℓwcaobs[ℓ] X ℓ X and ∆Eˆ¯[ℓ]j2(∆ˆ[ℓ]) × 0 Im[Sab(t′,t)]=−2sinδcp Wˆ¯Dˆ¯asib[nℓ][ℓ]sin2Eˆ¯ℓ∆L Pcaobs2δ(∆L,Aˆ)=−cos2δcp4∆Dˆ¯2L Xℓ (−1)ℓwcaobs2[ℓ] Xℓ ∆Eˆ¯[ℓ]j2(∆ˆ[ℓ]) −Xℓ Wˆ¯Dˆ¯a0[bℓ[]ℓ]sin2Eˆ¯ℓ∆L P0ab(∆L,Aˆ)=×−4∆Dˆ¯2L0ℓ (−1)ℓw0ab[ℓ] −cosδcp Wˆ¯Dˆ¯acob[sℓ][ℓ]sin2Eˆ¯ℓ∆L (,66) ×∆Eˆ¯[ℓ]j02X(∆ˆ[ℓ]) , (69) Xℓ where all sums run over ℓ= 1 ,2 ,3, ∆ˆ[ℓ] is defined where ∆ is defined in Eq. (34). Approximate ex- as L opfreHssνiownesrfeorobPt(aνinae→d fνrob)minSt(eLr)misnoRfetfhse. [p5a,r6a]mbeytearns ∆ˆ[ℓ]≡∆Eˆ¯[ℓ]∆L , (70) expansion in sinθ13. with ∆Eˆ¯[ℓ] defined in Eq. (44), Dˆ¯ is defined as Dˆ¯ ∆Eˆ¯[1]∆Eˆ¯[2]∆Eˆ¯[3] , (71) C. Partial Oscillation Probabilities ≡ andthe matrixelementswab[ℓ]aregivenintermsof i Using somewhat different techniques, the oscilla- the mixing angles and Aˆ in Appendix B. Since we tion probability may be expressed through a set of order the energies so that Eˆ¯ >Eˆ¯ > Eˆ¯ , ∆Eˆ¯[ℓ] as 3 2 1 functions that express how P(νa → νb) ≡ Pab de- well as Dˆ¯ are all positive. pends on the CP violating phase δ [4]. In our cp We begin our derivation with the expression for Hamiltonian formulation there are four such terms, the oscillation probability written in terms of S(T), Eq. (54), Pab =δ(a,b)+Pab+Pab +Pab 0 sinδ cosδ +Pab , (67) (ν ν ) Pab cos2δ P a → b ≡ 8 = <M(b)Ue−iHν(t′−t)U† M(a)> 2 It follows from Eq. (77) that the coefficients of | | | | = Fℓaℓb′exp−i(E¯ℓ−E¯ℓ′)L . (72) i∆ngEˆ¯[tℓo]ja0(s2i∆mˆ[pℓl]e)ienxpErqe.s(s6io8n)aforrePalalbpr,oportional,lead- ℓℓ′ sinδ X Here Fℓaℓb′ is defined as Psaibnδ(∆L,Aˆ)=∆3Lα(1−α)sinδcpǫasibn wab cosθ13sin2θ12sin2θ13sin2θ23 Fℓaℓb′ ≡FℓabFℓa′b∗ = Dˆ¯[ℓ]ℓDˆ¯ℓ′[ℓ′] (73) ××j0(∆ˆ[3])j0(∆ˆ[2])j0(∆ˆ[1]) . (80) Note that Pab is antisymmetric under a b. with Dˆ¯[ℓ] given in Eq. (59), and sinδ ↔ Analytic expressions for all other partial oscilla- wℓaℓb′ ≡<<MM((bb))|WWˆ¯ˆ¯[[ℓℓ′]]|MM((aa))>>∗ . (74) itwoiohnni,ctphhreoesxbepaprbeailsristteiiaesslFofoℓsaℓlcb′liolilwnatftiroeonrmmpEsroqob.fa(wb7i2ℓalℓ)ib′t.uiessIinnagrteEhaiqsl.sf(oa7es3hx)--, × | | pressed in terms of the parameters of the SNM and Allresultsneededfordeterminingexact,closed-form the neutrino eigenvalues, Eˆ¯ . expressions for the partial oscillation probabilities ℓ The usefulness of the partial oscillation probabil- are found in Appendix B. ities can be seen as follows. It is a general result As we explain in Appendix B, wab may be ex- ℓℓ′ that the exchange of initial and final states in the pressed through four operators, oscillation probability or neutrinos (antineutrinos) wℓaℓb′ =w0aℓbℓ′ +cosδcpwcaobs ℓℓ′ +cos2δcp is equivalent to letting δcp → −δcp. Thus, the re- ×wcaobs2ℓℓ′ +isinδcpwsaibnℓℓ′ . (75) esuxlcthafonrgtinhge(ian,vbe)rsienrEeqa.ct(i6o7n).PS(iνnbce→Pνaab) iissfaonutnisdymby- sinδ found from the decomposition of Wˆ¯[ℓ] given in metric under the exchange of (a,b), and P0ab, Pcaobsδ Eq. (63). The quantities wiab[ℓ] appearing in and Pcaobs2δ symmetric, it follows that Pba is given Eqs. (68,69) are the same as wab written in the by iℓℓ′ streamlined notation, (ν ν )=δ(a,b)+Pab Pab +Pab P b → a 0 − sinδ cosδ wiab[1]=wiaℓbℓ′ for (ℓ,ℓ′)=(3,2) +Pcaobs2δ . (81) wiab[2]=wiaℓbℓ′ for (ℓ,ℓ′)=(3,1) In analogy to Eq. (67), we may express the oscil- wiab[3]=wiaℓbℓ′ for (ℓ,ℓ′)=(2,1) , (76) lation probability for antineutrinos as which takes advantage of ℓ>ℓ′. (ν¯ ν¯ ) P¯ab An analytic expression for wab [ℓ], P a → b ≡ sin =δ(a,b)+P¯ab+P¯ab +P¯ab +P¯ab , (82) 0 sinδ cosδ cos2δ wab [ℓ]=Ksin2θ ∆Eˆ¯[ℓ]ǫab , (77) sin 23 sin where the bared probabilities for anti neutrinos are follows directly fromEq.(63). Here, ǫ is the anti- obtained fromthe unbarredfor neutrinos by replac- sin symmetric matrix ing δcp δcp and Aˆ Aˆ. Because the energies →− → − of antineutrinos are different from those of the neu- 0 1 1 trinos in matter, we can expect Pab = P¯ab in this − ǫ 1 0 1 (78) 6 sin situation. ≡ −  1 1 0 Again applying the rule that exchange of initial −   and final states is accomplished by letting δ and cp → δ , the oscillation probability (ν¯ ν¯ ) is ex- cp b a α(1 α) − P → pressed in terms of the same four quantities, K = − − 8 ×cosθ13sin2θ12sin2θ13 . (79) P(ν¯b →ν¯a) =δ(a,b)+P¯ab P¯ab +P¯ab +P¯ab . (83) Equation (77) is one of the more striking results. 0 − sinδ cosδ cos2δ Analytic formulae for the other wiab[ℓ] follow from It is worth noting that the entire dependence of Eq. (57). These are all given in Appendix B, where the oscillation probabilities given in Eqs. (68,69) on they are expressed in terms of the parameters spec- theneutrinobeamenergyE,thebaselineL,andthe ifying Hˆ¯ and Hˆ = U−1VÛ. These are the same medium properties occurs throughthe variables∆ 0v 1 L parameters defining SNM. and Aˆ defined in Eqs. (34,25), respectively. Since 9 we will be most interested in how the neutrino os- with∆ giveninEq.(34). Thetwo-flavoroscillation L cillation probability depends on the beam energy, probability P12(∆ ,Aˆ) is then 2f L baseline,andmediumproperties,the partialoscilla- tion probabilities have been expressed as functions P12(∆ ,Aˆ)= S12 2 of ∆ and Aˆ. 2f L | 2f| L sin22θ Because the vacuum result is also easily obtained = sin2∆ φ , (90) by less sophisticated arguments, the vacuum limit φ2 L provides an opportunity to verify our Hamiltonian whichwe will next compare to our Hamiltonian for- formulation in a well-known special case. mulation. To find S12 =< M(2)S[L]M(1) >, it is 2f | | easytooverlookthatthe2 2matrixH inEq.(84) ν × D. Special Cases is not diagonal in the flavor basis . The oscillation probability in the three neutrino mixing of our Hamiltonian formulation is matched Wenextexaminespecialcaseschosentohighlight to the two-flavor case just discussed by setting two specific aspects of our formulation. Because these ofthe mixinganglesinEq.(12)tozero,sayθ 0 cases are well known in other contexts, they pro- 13 → andθ 0andidentifyingθwithθ . Accordingly, videusefulcrosschecks. Wefirstconsidertwo-flavor 23 → 12 we find, mixing and then the vacuum limit. cosθ sinθ 0 U sinθ cosθ 0 . (91) 1. Two Flavor Mixing ≡ −  0 0 1   Expressionsfortwo-neutrinooscillationsareeasily This is easily recognized as a two-flavormixing ma- obtained from the Hamiltonian, trix by noting that one of the three neutrinos does not mix with the other two. We see that it not only Hˆ¯ =Hˆ¯ +U−1Hˆ¯ U , (84) ν 0ν 1ν dependsonjustonemixingangleθ,butalsothatthe dependenceontheCPviolatingphaseδhasdropped where out. Hˆ¯0ν ≡ 00 01 , (85) uuNmatHuarmaliclthooniicaensfaonrdai3n×te3ratcwtioo-nflaavroer,neutrinovac- (cid:18) (cid:19) 0 0 0 Hˆ¯1ν ≡ A0ˆ 00 . (86) Hˆ¯0ν ≡0 1 0 , (92) (cid:18) (cid:19) 0 0 0 The standard mixing matrix U is   and cosθ sinθ U . (87) Aˆ 0 0 ≡(cid:18)−sinθ cosθ (cid:19) Hˆ¯1 0 0 0 . (93) ≡  The neutrino Hamiltonian is easily diagonalized to 0 0 0 findthemedium-modifiedtwo-flavorneutrinoeigen-   values, Whenthe3 3neutrinoHamiltonianHˆ¯ =Hˆ¯ + ν 0ν E¯ˆ2 = 1 1+Aˆ+φ mU−od1Hiˆ¯fie1νdUneisutdr×iiangooneiagleiznevda,lutweso,ofthethreemedium- 2 E¯ˆ1 = 21(cid:16)1+Aˆ−φ(cid:17) , (88) E¯ˆ3 = 1 1+Aˆ+φ 2 (cid:16) (cid:17) where φ= 1+Aˆ2 2Aˆcos2θ. E¯ˆ = 1(cid:16)1+Aˆ φ(cid:17) − 2 2 − Expressed in dimensionless variables, the time- evolution opperator Sab for the transition a b be- E¯ˆ =0 .(cid:16) (cid:17) (94) 1 → comes are identical to those found above in the two-flavor Sab =<M(b)Uexp−2iHˆ¯ν∆LU−1 M(a)> ,(89) case. Note that the eigenvalues have no branch | | 10 pointsforAôntherealaxis. Fromtheseeigenvalues, Thus, for all two-flavortransitions, we find the differences, ∆Eˆ¯[3]=E¯ˆ2 E¯ˆ1 Pab(∆L,Aˆ)=δ(a,b)+ 4∆Dˆ2Lw0ab[3]∆E¯ˆ[3] − = 12 1+Aˆ−φ ×j02(∆E¯ˆ[3]∆L) , (102) ∆Eˆ¯[2]=E¯ˆ(cid:16) E¯ˆ (cid:17) where Dˆ is from Eq. (103), 3 1 − 1 = 2 1+Aˆ+φ Dˆ ∆Eˆ¯[1]∆Eˆ¯[2]∆Eˆ¯[3] ≡ ∆Eˆ¯[1]=E¯ˆ(cid:16) E¯ˆ =φ(cid:17). (95) ∆Eˆ¯[1] 3− 2 = ((1+Aˆ)2 1 Aˆ2+2Aˆcos2θ) 4 − − Finally, consider the oscillationprobability in our Hamiltonianformulation. BecausePab , Pab , and =Aˆcos2θ∆Eˆ¯[1] . (103) sinδ cosδ Pab areproportionaltosin2θ (andθ hasbeen To compare our result to to the well-known two- cos2δ 23 23 setto0),thesetermsdonotcontributefortwo-flavor flavor oscillation probability given in Eq. (90), we mixing. Thus, evaluate Eq. (102) for 1 2 transitions obtaining, → Pab(∆ ,Aˆ)=δ(a,b)+Pab(∆ ,Aˆ) (96) sin22θ L 0 L P12(∆ ,Aˆ)= sin2∆ φ . (104) L φ2 L with Pab(∆ ,Aˆ) takenfromEq.(69). We find after 0 L a straightforwardcalculation, As expected, this is in complete agreement with Eq. (90). w11[ℓ]= w12[ℓ] 0 − 0 w21[ℓ]=w12[ℓ] 0 0 w022[ℓ]=−w012[ℓ] , (97) 2. Vacuum Oscillation Probability where, In our Hamiltonian formulation, an expression w12[ℓ]=w(12)(Aˆcos2θ (1+Aˆ)Eˆ¯ for the time-evolution operator in the vacuum limit 0 02 − ℓ Aˆ 0 is found from Eqs. (53), (54), (57), and (59), +Eˆ¯2) . (98) → ℓ <M(b)S0(t′,t)M(a)> S0ab(t′ t) Taking Eˆ¯ from Eq. (94), we evaluate | | ≡ − ℓAˆcos2θ (1+Aˆ)Eˆ¯ +Eˆ¯2 , (99) = <M(b)|WD¯¯ˆˆ00[[ℓℓ]]|M(a)>exp−iE¯ℓ0∆L (.105) − ℓ ℓ Xℓ to find Here,E¯0 andD¯ˆ0[ℓ]arethevacuumvaluesofE¯ and ℓ ℓ w12[1]=w12[2]=0 , (100) Dˆ¯[ℓ], respectively. In the vacuum, of course, there 0 0 is no distinction between the energy differences for and neutrinos and anti-neutrinos. w12[3]=Aˆcos2θw(12) Evaluating Eq. (105) by inserting a complete set 0 = Aˆcos2θs0in,222θ . (101) owfesatarrtievseinattermediate states |n ><n| inside Wˆ¯0[ℓ], 4 <M(b)|S0(t′,t)|M(a)>= Ubn(Eˆ¯0n−Eˆ¯D¯0aˆ)0([Eℓˆ¯]0n−Eˆ¯0b)Un∗ae−iE¯ℓ0(t′−t) . (106) ℓ n XX Theindices (a,b,n)runoverallpermutationsofthe three integers (1,2,3), from which it follows that