Physics of parameter correlations around the solar-scale enhancement in neutrino theory with unitarity violation
NNUHEP-TH/19-09, FERMILAB-PUB-19-397-T
Prepared for submission to JHEP
Physics of parameter correlations around thesolar-scale enhancement in neutrino theory withunitarity violation
Ivan Martinez-Soler a,b,c
Hisakazu Minakata d a Theoretical Physics Department, Fermi National Accelerator Laboratory, P.O. Box 500, BataviaIL 60510, USA b Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA c Colegio de Física Fundamental e Interdisciplinaria de las Américas (COFI), 254 Norzagaraystreet, San Juan, Puerto Rico 00901 d Center for Neutrino Physics, Department of Physics, Virginia Tech, Blacksburg, Virginia 24061,USA
E-mail: [email protected] , [email protected] Abstract:
We discuss physics of the three neutrino flavor transformation with non-unitary mixing matrix, with particular attention to the correlation between the ν SM- andthe α parameters which represent effect of unitarity violating (UV) new physics. Towardthe goal, a new perturbative framework is created to illuminate the effect of non-unitarity inregion of the solar-scale enhanced oscillations. We refute the skepticism about the physicalreality of the ν SM CP δ - α parameter phase correlation by analysis with the SOL conventionof U MNS in which e ± iδ is attached to s . Then, a comparative study between the solar-and atmospheric-scale oscillation regions allowed by the framework reveals a dynamical δ − (blobs of the α parameters) correlation in the solar oscillation region, in sharp contrastto the “chiral” type phase correlation [ e − iδ ¯ α µe , e − iδ ¯ α τe , ¯ α τµ ] in the PDG convention seenin the atmospheric oscillation region. An explicit perturbative calculation to first order inthe ν µ → ν e channel allows us to decompose the UV related part of the probability intothe unitary evolution part and the genuine non-unitary part. We observe that the effect ofnon-unitarity tends to cancel between these two parts, as well as between the different α βγ parameters. a r X i v : . [ h e p - ph ] M a y ontents ν SM extendedsettings 4 ν SM and NSI parameters 53.2 Dynamical nature of the parameter correlation 63.3 Phase correlation through NSI-UV parameter correspondence? 6 F and K matrices 114.5 Formulating perturbation theory with the hat basis 114.6 Calculation of ˆ S and ˜ S matrices 134.7 The relations between various bases and computation of the flavor basis S matrix 154.8 Effective expansion parameter with and without the UV effect 15 P ( ν µ → ν e ) (1) EV P ( ν µ → ν e ) (1) UV ν SM and the UV α parameters 20 α parameter correlation 206.2 Correlations between ν SM phase δ and the α parameters 216.3 Nature of the δ − α parameter correlation: Are they real? 216.4 Clustering of the α parameters 22 U MNS ν SM - α parameter phase correlation: The atmospheric vs. solar regions 26 – i – Explicit expressions of F ij , K ij and Φ ij
33B The first order tilde basis unitary evolution ˜ S matrix elements 34C The zeroth-order ν SM S matrix elements 36D The neutrino oscillation probability in the ν µ → ν e and the other channels 36 D.1 The neutrino oscillation probability in the ν µ → ν e channel: Rest of theunitary evolution part 36D.2 The neutrino oscillation probability in the ν µ − ν τ sector 38 The discovery of neutrino oscillation and hence neutrino mass [1, 2] under the framework ofthree-generation lepton flavor mixing [3] created a new field of research in particle physics.It led to construction of the next-generation accelerator and underground experiments withthe massive detectors, Hyper-Kamiokande [4] and DUNE [5]. They are going to establishCP violation due to the lepton Kobayashi-Maskawa phase [6], possible lepton counterpartof the quark CP violation [7]. They will also determine the neutrino mass ordering athigh confidence level by utilizing the earth matter effect [8, 9]. Of course, the flagshipprojects will be challenged by the ongoing [10–12] and the other upcoming experiments, forexample, ESS ν SB [13], JUNO [14], T2KK [16], INO [17], IceCube-Gen2/PINGU [18], andKM3NeT/ORCA [19], which compete for the same goals.Toward establishing the three-flavor mixing scheme, in particular in the absence ofconfirmed anomaly beyond the neutrino-mass embedded Standard Model ( ν SM), one ofthe most important topics in the future would be the high-precision paradigm test. Inthis context, leptonic unitarity test, either by closing the unitarity triangle [21], or by analternative method of constraining the models of unitarity violation (UV) at high-energy[22, 23], or at low-energy scales [24–26] are extensively discussed. It includes the subsequentdevelopments, for example in [27–35]. A summary of the current constraints on UV is givene.g., in refs. [26, 36].It was observed that in the × active neutrino subspace the evolution of the systemcan be formulated in the same footing in low-scale as well as high-scale UV scenarios [25, 26].Nonetheless, dynamics of the three neutrino system with non-unitary mixing in matter hasnot been investigated in a sufficient depth. Apart from numerically implemented calculationdone in some of the aforementioned references, only a very limited effort was devoted foranalytical understanding of the system so for. It has a sharp contrast to the fact that A possible acronym for the setting, “Tokai-to-Kamioka observatory-Korea neutrino observatory”, up-dated from the one used in ref. [15]. For possible candidates of the anomalies which suggest physics beyond the ν SM see e.g., ref. [20]. It is appropriate to mention that in the physics literature UV usually means “ultraviolet”. But, in thispaper UV is used as an abbreviation for “unitarity violation” or “unitarity violating”. – 1 –reat amount of efforts were devoted to understand the three-flavor neutrino oscillation. A general result known to us so far is the exact S matrix with non-unitarity in matter withconstant density [25] calculated by using the KTY-type construction [37]. It allows us toobtain the exact expression of the oscillation probability with non-unitarity.In a previous paper [38], we have started a systematic investigation of analytic struc-ture of the three neutrino evolution in matter with non-unitarity. We have used so calledthe α parametrization [23] to implement non-unitarity in the three neutrino system. Usinga perturbative framework dubbed as the “helio-UV perturbation theory” (a UV extendedversion of [39]) with the two kind of expansion parameters, the helio-to-terrestial ratio (cid:15) ≈ ∆ m / ∆ m and the α parameters, we computed the oscillation probability valid tofirst-order in the expansion parameters. The region of validity of the perturbative frame-work spans the one around the atmospheric-scale enhanced oscillations which covers therelevant region for the ongoing and the next generation long-baseline (LBL) acceleratorneutrino oscillation experiments. Possibility of application to the data from the near futurefacilities and the currently almost un-understood properties of the system may justify theexamination even though it is to first order in expansion.To our view, the most significant observation in ref. [38] is that the ν SM CP phase δ and the complex α parameters have an the intriguing phase correlation of the form [ e − iδ ¯ α µe , e − iδ ¯ α τe , ¯ α τµ ] in the Particle Data Group (PDG) convention of U MNS [40]. What isunique in the phase correlation is that it universally holds in all the oscillation channels aswell as unitary and non-unitary parts of the oscillation probability. One should note thatthe definition of the α parameters, and consequently the precise form of the correlationbetween the CP phases, depends on the phase convention of the lepton flavor mixing MNSmatrix. A puzzling feature of the δ - α parameter phase correlation in ref. [38] is that it dis-appears in the SOL convention of U MNS in which e ± iδ is attached to s . It triggered askepticism of the nature of phase correlation, which may allow the following two alternativeinterpretations:1. Existence of the SOL phase convention of U MNS in which δ and α phase correlationis absent implies that the CP phase correlation is not physical, but an artifact ofinadequate choice of U MNS phase convention.2. Physics must be U MNS convention independent. In all the other convention of U MNS except for the SOL, there exists δ − α parameter phase correlation. Therefore, theexistence of phase correlation is generic and it must be physical.If the interpretation 1 and the reasoning behind it are correct, δ and α phase correlationmust be absent under the SOL convention of U MNS everywhere in the allowed kinematical Here, we give a cautious remark that when the term “neutrino oscillation” is used in this paper, oroften in many other literatures, it may imply not only the original meaning, but also something beyondsuch as “neutrino flavor transformation”, or “neutrino flavor conversion”, depending upon the contexts. In the ATM phase convention of U MNS in which e ± iδ is attached to s , the phase correlation takesthe form [ e − iδ α µe , α τe , e iδ α τµ ] . – 2 –egions. Conversely, if we see a non-vanishing phase correlation in the oscillation probabilitycalculated with the SOL convention somewhere, it implies that the interpretation 1 cannotbe true. We will show throughout this paper that the interpretation 2 holds by investigationof the system in region of the solar-scale enhanced oscillation. In this paper, we discuss physics of neutrino flavor transformation in region of solar-scaleenhanced oscillation. We will try to achieve the two goals: • To examine the system of the three-flavor neutrinos in the SOL convention ( e ± iδ attached to s ) of U MNS in region of the enhanced solar oscillation, which will testifyfor physical reality of the correlation between ν SM phase - UV α parameter phases. • To understand the ν SM - UV parameter correlation in more generic context and inwider kinematical region by combining the results of this and the previous works [38].A few words for the examination of the “solar region” are ready. We feel that animmense need exists for the real understanding of parameter correlation in theories withnon-unitarity, in particular outside the region investigated in ref. [38]. The natural “fieldof research” for this purpose is the region of solar-scale enhanced oscillation, the uniqueplace for enhancement other than the atmospheric one in our world of the three generationleptons. The feature can be seen clearly in the “terrestrial-friendly” region of E vs. L plote.g., in refs. [41, 42] whose latter also serves for a brief summary of recent activities onatmospheric neutrinos at low energies. We note that it has been the target of investigationfor a long time, see e.g., [43–46], and possibly others that we may miss, mainly in thecontext of atmospheric neutrino observation at low energies. It should also be mentionedthat this topic is now receiving the renewed interest [41, 42] given the new possibilities ofgigantic detectors such as JUNO [47], DUNE [48], and Hyper-K [49]. Thus, the second goalof this paper is to achieve a deeper understanding of parameter correlation by combiningknowledges in regions of the atmospheric-scale and the solar-scale enhanced oscillations.Very recently, we have formulated a perturbative framework in the ν SM, dubbed as the“solar resonance perturbation theory” [41], whose validity is around the very region of ourinterest. We extend this perturbative framework to include the effect of UV, by treating the α parameters as the additional expansion parameters. Using the framework, we investigatedynamics of the three neutrino evolution with non-unitary mixing matrix under the constantmatter density approximation, with particular attention to the parameter correlation. Wewill show that the system displays a rich, new phenomenon of clustering of the ν SM andthe UV variables. The feature of merely replacing the atmospheric oscillation to the solar one may trigger the questionto us: “Are you attempting another experiment replacing copper with iron?”. At this stage, we would liketo say that a mere change in the field of exercise brings new insights to us because the system is so rich indynamics with the extra nine UV parameters introduced into the ν SM system. In sections 6 and 7, thereaders will see our clear-cut full answer to this question. – 3 –onetheless, we find it not sufficient to rely on analytic treatment based on perturbationtheory to extract the characteristic feature of the system due to a new and intricate featureof the parameter correlation. For this reason we rely also on exact numerical analysesas well as the perturbative formula we derive in this paper to elucidate physics of theparameter correlation in region of enhanced solar-scale oscillation. It will be particularlyilluminating when our analysis is done in a style of comparative study between the solar-and atmospheric-scale oscillation regions, as will be done in section 7. We hope that suchunderstanding will eventually help analyzing data for leptonic unitarity test.In section 3, we introduce the concept of parameter correlations by describing a ped-agogical example of the three-neutrino system with the non-standard interactions (NSI).In section 4, we give a step-by-step formulation of the perturbative framework which tobe utilized in analyzing features of the three-neutrino evolution with non-unitary mixingmatrix. The prescription for computing S matrix elements is given with the help of thetilde basis ˜ S matrix elements summarized in appendix B. In section 5, a general formulafor the oscillation probability is derived, and applied to computation of the appearanceprobability in the ν µ → ν e channel. This section together with appendix D.1 contains theexplicit expression of the oscillation probability in the ν µ → ν e channel to first order inexpansion parameters. In section 6, we discuss the characteristic features of the correlationbetween the ν SM CP phase and UV α parameters in the region of validity of our perturba-tive framework. In section 7, physics of neutrino flavor transformation with UV is discussedpaying a particular attention to parameter correlation, contrasting between the regions ofthe solar- and atmospheric-scale enhanced oscillations. In section 8, we give the concludingremarks. ν SM ex-tended settings
It may be useful to start the description of this paper by briefly recollecting some knownfeatures of parameter correlation in neutrino oscillation, in particular, in an extended settingthat includes physics beyond the ν SM. In this context, a general framework that is mostfrequently discussed is the one which includes the neutrinos’ non-standard interactions(NSI) [8] H NSI = a E ε ee ε eµ ε eτ ε ∗ eµ ε µµ ε µτ ε ∗ eτ ε ∗ µτ ε ττ , (3.1)in the flavor basis Hamiltonian, where the ε parameters describe flavor dependent strengthsof NSI and a denotes the matter potential, see (4.4). We discuss only so called the “prop-agation NSI”. For a review of physics of NSI in wider contexts, see e.g., refs. [50–52]. Wenote that inclusion of the NSI Hamiltonian (3.1) brings the extra nine parameters into the ν SM Hamiltonian with six degrees of freedom, the two ∆ m , the three mixing angles, andthe unique CP phase, under the influence of the matter potential background.– 4 – .1 Emergence of collective variables involving ν SM and NSI parameters
With more than doubled, a large number of the parameters, it is conceivable that dynamicsof neutrino oscillation naturally involves rich correlations among these variables. Here, wediscuss only a particular type of correlation uncovered in ref. [55] because, we believe, itilluminates the point. In this reference, the authors formulated a perturbative frameworkof the system with NSI by using the three (the latter two assumed to be) small expansionparameters, (cid:15) ≡ ∆ m / ∆ m , s ≡ sin θ , and the ε parameters. They derived theformulas of the oscillation probability to second order (third order in ν µ → ν e channel) inthe expansion parameters, which is nothing but an extension of the Cervera et. al. formulas[56] to include NSI. In this calculation the PDG convention of U MNS [40] is used.An interesting and unexpected feature of the NSI-extended formulas is the emergenceof the two sets of “collective variables” Θ ≡ s ∆ m a + e iδ ( s ε eµ + c ε eτ ) , Θ ≡ (cid:18) c s ∆ m a + c ε eµ − s ε eτ (cid:19) e iδ , (3.2)where an overall e − iδ is factored out from the matrix element S eµ to make the s term δ freethrough which e iδ dependences in Θ in eq. (3.2) results. That is, if we replace s
13 ∆ m a and c s
12 ∆ m a in the original formulas by Θ and Θ , respectively, the extended second-orderformulas with full inclusion of NSI effects automatically appear [55]. In fact, the procedureworks for the third order formula for P ( ν µ → ν e ) as well. We note that the second ordercomputation of ref. [55] includes the ν µ − ν τ sector, and the additional corrective variablesare identified. But, for simplicity, we do not discuss them here and refer the interestedreaders ref. [55].Appearance of the cluster variables composed of the ν SM and NSI parameters in (3.2)implies that there exists strong correlations between the ν SM variables s - δ and the NSI ε eµ - ε eτ parameters in such a way that they form the collective variable Θ to convert theCervera et. al. formula to the NSI-extended version. The similar statement can be madefor the other cluster variable Θ as well. The NSI-extended second order formula derivedin this way serves for understanding the s - ε eµ confusion uncovered in ref. [60] in a morecomplete manner in such a way that the effects of ε eτ and CP phase δ are also included.It also predicts occurrence of the similar correlation among the variables to produce thecollective variable Θ , whose feature could be confirmed by experiments at low energies, ∆ m a ∼ O (1) , the possibility revisited recently [41, 42].Therefore, there is nothing strange in the parameter correlations among the ν SM andnew physics parameters. It appears that the phenomenon arises generically, at least underthe environment that the matter effect is comparable to the vacuum effect. They include the correlations between the NSI variables themselves. The examples include the ε ee - ε eτ - ε ττ correlation discussed in refs. [53, 54]. The Cervera et. al. formula is the most commonly used probability formula in the standard threeflavor mixing in matter for many purposes, e.g., in the discussion of parameter degeneracy [57–59]. – 5 – .2 Dynamical nature of the parameter correlation
We must point out, however, that the features of the parameter correlation depend on thevalues of the parameters involved, and also on the kinematical region of neutrino energyand baseline with background matter density. Therefore, depending upon the region ofvalidity of the perturbative framework which is used to derive the correlation, the formof parameter correlation changes. We call all these features collectively as the “dynamicalnature” of the parameter correlation. We want to see explicitly whether a change in features of the correlation occurs when thevalues of the parameters involved are varied, or its effect is incorporated into the frameworkof perturbation theory. For this purpose let us go back to the collective variable correlationin (3.2). We know now the value of θ is larger than what was assumed at the time theCervera et. al. formula was derived [40]. The latest value from Daya Bay is s = 0 . [61], which is of the order of √ (cid:15) = 0 . . Then, we need higher order corrections of s upto the fourth order terms to match to the second order accuracy in (cid:15) [62, 63]. When it iscarried out with inclusion of NSI [63], it is seen that part of the additional terms generateddo not fit to the form of collective variables given in (3.2). Therefore, when we make θ larger, the parameter correlation which produced the collective variables (3.2) is started todissolve.Thus, the analysis of this particular example reveals the dynamical nature of the pa-rameter correlation in the neutrino propagation with NSI. We expect that overseeing theresults of computations of the oscillation probabilities in this and the previous papers [38]would reveal the similar dynamical behavior of the parameter correlation in the three-flavorneutrino evolution in matter with non-unitary mixing. Can we extract information of the δ - α parameter phase correlation from the collectivevariables (3.2)? The answer is Yes if we assume a “uniform chemical composition model”of the matter. As far as the propagation NSI is concerned there is a one to one mappingbetween NSI ε parameters and the UV α parameters, as noticed by Blennow et. al. [26]under the assumption N n = N e , an equal neutron and proton number densities in charge-neutral medium. Of course, an extension to the more generic case of N e = rN n [38] can beeasily done without altering the conclusion. For the purpose of the present discussion, onealso has to “approve” the procedure by which the e iδ dependence of the collective variables(3.2) is fixed. That is, removing an overall phase from the matrix element S eµ to make the s term δ free, as done in ref. [55].Assuming that the two conditions above are met, it leads to the collective variables in One must be aware that our terminology of “dynamical” correlation may be different from those usedin condensed matter physics or many body theory. In our case the correlated parameters are not thedynamical variables in quantum theory and there is no direct interactions between them. – 6 –3.2) written by the UV α parameters, Θ = s ∆ m a + 12 (cid:110) s (cid:16) ¯ α µe e − iδ (cid:17) ∗ + c (cid:16) ¯ α τe e − iδ (cid:17) ∗ (cid:111) , Θ = c s e iδ ∆ m a + 12 (cid:110) c (cid:16) ¯ α µe e − iδ (cid:17) ∗ − s (cid:16) ¯ α τe e − iδ (cid:17) ∗ (cid:111) , (3.3)where we have to use the α parameters defined in the PDG convention of U MNS . Theemerged correlation between δ and the α parameters is consistent with the canonical phasecombination obtained in ref. [38] in the PDG convention. For the relationships between the α parameters with the various U MNS conventions, see section 4.1. It is not unreasonablebecause the regions of validity of the perturbative frameworks in refs. [55] and [38] overlaps.
Physics discussion in this paper necessitates a new analytical framework to illuminate theeffect of non-unitary mixing matrix in region of the solar-scale enhanced oscillations, theUV extended version of the “solar-resonance perturbation theory” [41].
As is customary in our formulation of the three active neutrino evolution in matter withunitarity violation (UV) [38], we start from the evolution equation in the vacuum masseigenstate basis, whose justification is given in refs. [25, 26]. With use of the “check basis”for the vacuum mass eigenstate basis, it takes the form of Schrödinger equation i ddx ˇ ν = ˇ H ˇ ν (4.1)with Hamiltonian ˇ H ≡ E m
00 0 ∆ m + N † a − b − b
00 0 − b N (4.2)where E is neutrino energy and ∆ m ji ≡ m j − m i . A usual phase redefinition of neutrinowave function is done to leave only the mass squared differences. N denotes the non-unitaryflavor mixing matrix which relates the flavor neutrino states to the vacuum mass eigenstatesas ν β = N βi ˇ ν i . (4.3)where β (and the other Greek indices) runs over e, µ, τ , while the mass eigenstate index i (and the other Latin indices) runs over , , and . It must be noticed that the neutrino In a nutshell, the equation (4.1) with (4.2) describes evolution of the active three neutrinos in the × sub-space in the (3 + N s ) model (as a model for low-scale UV) [24, 25], or just the three neutrino systemin high-scale UV, see e.g., [26]. – 7 –volution described by eq. (4.1) is unitary, as is obvious from the hermitian Hamiltonian(4.2). How the apparent inconsistency between the unitary evolution and the non-unitarityof the flavor basis S matrix will be resolved in section 4.7, one of the points of emphasis inref. [38]. Notice that due to limited number of appropriate symbols the notations for thevarious basis may not be always the same in our series of papers.The functions a ( x ) and b ( x ) in (4.13) denote the Wolfenstein matter potential [8] dueto CC and NC reactions, respectively. a = 2 √ G F N e E ≈ . × − (cid:18) Y e ρ g cm − (cid:19) (cid:18) E GeV (cid:19) eV ,b = √ G F N n E = 12 (cid:18) N n N e (cid:19) a. (4.4)Here, G F is the Fermi constant, N e and N n are the electron and neutron number densitiesin matter. ρ and Y e denote, respectively, the matter density and number of electron pernucleon in matter. We define the following notations for simplicity to be used in thediscussions hereafter in this paper: ∆ ji ≡ ∆ m ji E , ∆ a ≡ a E , ∆ b ≡ b E . (4.5)For simplicity and clarity we will work with the uniform matter density approximationin this paper. But, it is not difficult to extend our treatment to varying matter density caseif adiabaticity holds.Throughout this paper, due to the reasoning mentioned in section 1, we use the SOLconvention of the U MNS matrix, the standard × unitary flavor mixing matrix U SOL = c s − s c c s − s c c s e iδ − s e − iδ c
00 0 1 ≡ U U U , (4.6)where we have used the obvious notations s ij ≡ sin θ ij etc. and δ denotes the lepton KMphase [6], or the ν SM CP violating phase. Our terminology “SOL” is because the phasefactor e ± iδ is attached to the “solar angle” s . It is physically equivalent to the commonlyused PDG convention [40] in which the phase factor is attached to s .We use the α parametrization of non-unitary mixing matrix [23] defined in the U SOL convention N = ( − ˜ α ) U SOL = − ˜ α ee α µe ˜ α µµ α τe ˜ α τµ ˜ α ττ U SOL (4.7)As seen in (4.7), and discussed in detail in ref. [38], the definition of the α matrix dependson the phase convention of the flavor mixing matrix U MNS . In consistent with the notationused in ref. [38], we denote the α matrix elements in the SOL convention as ˜ α βγ .– 8 –he other convention of the MNS matrix which is heavily used in ref. [38] is the “ATM”convention in which e ± iδ is attached to the “atmospheric angle” s : U ATM = c s e iδ − s e − iδ c c s − s c c s − s c
00 0 1 . (4.8)The α parameters defined in the ATM and PDG conventions of U MNS are denoted as α βγ and ¯ α βγ , respectively, in ref. [38], the notation we follow in this paper. Then, we recapitulatehere the relationships between the α parameters defined with the PDG ( ¯ α ), ATM ( α ) andthe SOL ( ˜ α ) conventions of U MNS : ˜ α µe = ¯ α µe e − iδ = α µe e − iδ , ˜ α τe = ¯ α τe e − iδ = α τe , ˜ α τµ = ¯ α τµ = α τµ e iδ . (4.9)where we note that the diagonal α parameters are equal among the three conventions. In this section, we aim at constructing the perturbative framework which is valid at aroundthe solar oscillation maximum, ∆ m L/ E ∼ O (1) . Given the formula ∆ m L E = 0 . (cid:18) ∆ m . × − eV (cid:19) (cid:18) L km (cid:19) (cid:18) E MeV (cid:19) − , (4.10)it implies neutrino energy E = (1 − × MeV and baseline L = (1 − × km. Inthis region, the matter potential is comparable in size to the vacuum effect represented by ∆ m , a ∆ m = 0 . (cid:18) ∆ m . × − eV (cid:19) − (cid:18) ρ . g/cm (cid:19) (cid:18) E MeV (cid:19) ∼ O (1) . (4.11)Hence, our perturbative framework must fully take into account the MSW effect caused bythe earth matter effect. A more detailed discussion of the region of validity without UVeffect is given in ref. [41].As in the solar resonance perturbation theory we will have the “effective” expansionparameter in the ν SM sector, A exp = c s (cid:0) a/ ∆ m (cid:1) ∼ − , as will be discussed insection 4.8. The reason for having such a very small expansion parameter is due to thespecial structure of our perturbative Hamiltonian.To formulate our perturbative framework with UV, we use ˜ α βγ defined in eq. (4.7) asthe extra expansion parameters. That is, we assume that deviation from unitarity is small.Therefore, ˜ α βγ (cid:28) holds for all β and γ . Though we follow basically the same procedure asin ref. [41], we give a step-by-step presentation of the formulation because of the additionalcomplexities associated with inclusion of UV, and to make this paper self-contained.– 9 –hat would be a reachable or a possible target sensitivity to ˜ α βγ in the context ofunitarity test? For the sake of rough estimation, we assume momentarily a perfect knowl-edges of the ν SM mixing parameters. Then, let us ask: Which level of sensitivity to UV ˜ α βγ parameters could one expect given the accuracy of measurement of ∆ P βα ≡ P ( ν β → ν α ) − P ( ν β → ν α ) ν SM (see eq. (7.1)) is of order, for example, − or − ? Notice that ∆ P βα is the non-unitary contribution to the oscillation probability. The former number ismore or less the situation at the current time or in the near future, while the latter is takenarbitrarily as an expectation in a foreseeable feature. Since ∆ P βα ∼ ˜ α βγ , we would expectthe constraints on ˜ α βγ parameters of the order of − , or − , respectively.We note that once the accuracy of measurement reached a “perturbative regime” thefirst-order UV correction is sufficient, as far as qualitative discussions are concerned. Thesecond order computation yields terms of the order of ˜ α βγ ∼ − , or ∼ − , respectively,far beyond the accuracy of ∆ P βα measurement in each era. This is the reason why werestrict ourselves to the first-order formulas in this and the companion papers [38].In low-scale UV scenarios, the probability leaking term as well as the flux “mis-normalization”term in the appearance channels are of order ∼ | W | where W denotes collectively theactive-sterile mixing matrix elements [25]. Due to unitarity in the whole N sterile space, ˜ α βγ must be of order (cid:39) W . Then, the leaking and the mis-normalization terms are oforder ˜ α βγ ∼ − or − in the above two regimes, respectively, which are far too smallcompared to the accuracy of ∆ P βα measurement in each era. It constitutes one of theserious problems in their determination. We transform to a different basis to formulate our perturbation theory for solar-scale en-hancement. It is the tilde basis ˜ ν i = ( U ) ij ˇ ν j (4.12)with Hamiltonian ˜ H = U ˇ HU † , or ˇ H = U † ˜ HU . (4.13)Notice that the term “tilde basis” has no connection to our notation of ˜ α parameters in theSOL convention. The Hamiltonian in the tilde basis is given by ˜ H = ˜ H ν SM + ˜ H (1) UV + ˜ H (2) UV (4.14)where each term of the right-hand side of (4.14) is given by ˜ H ν SM = s ∆ c s e iδ ∆ c s e − iδ ∆ c ∆
00 0 ∆ + c ∆ a c s ∆ a c s ∆ a s ∆ a , (4.15) ˜ H (1) UV = ∆ b U † U † α ee (cid:16) − ∆ a ∆ b (cid:17) ˜ α ∗ µe ˜ α ∗ τe ˜ α µe α µµ ˜ α ∗ τµ ˜ α τe ˜ α τµ α ττ U U , (4.16)– 10 – H (2) UV = − ∆ b U † U † ˜ α ee (cid:16) − ∆ a ∆ b (cid:17) + | ˜ α µe | + | ˜ α τe | ˜ α ∗ µe ˜ α µµ + ˜ α ∗ τe ˜ α τµ ˜ α ∗ τe ˜ α ττ ˜ α µe ˜ α µµ + ˜ α τe ˜ α ∗ τµ ˜ α µµ + | ˜ α τµ | ˜ α ∗ τµ ˜ α ττ ˜ α τe ˜ α ττ ˜ α τµ ˜ α ττ ˜ α ττ U U . (4.17)In this paper, we restrict ourselves to the perturbative correction to first order inthe expansion parameters. There is a number of reasons for this limitation. It certainlysimplifies our discussion of the δ − α parameter phase correlation, though we will make a briefcomment on effect of ˜ H (2) UV to the correlation in section 6.2. Unfortunately, the expressionof the first order UV correction to the oscillation probability is sufficiently complex atthis order, as we will see in sections 5.1, 5.2 and appendix D.1. We do not consider ourrestriction to first order in ˜ α βγ a serious limitation because the framework anticipates aprecision era of neutrino experiment for unitarity test in which the condition ˜ α βγ (cid:28) should be justified. F and K matrices To make expressions of the S matrix and the oscillation probability as compact as possible,it is important to introduce the new matrix notations F and K : F ≡ F F F F F F F F F = U † α ee (cid:16) − ∆ a ∆ b (cid:17) ˜ α ∗ µe ˜ α ∗ τe ˜ α µe α µµ ˜ α ∗ τµ ˜ α τe ˜ α τµ α ττ U , (4.18) K = U † F U ≡ K K K K K K K K K = c F + s F − c s ( F + F ) c F − s F c F − s F + c s ( F − F ) c F − s F F s F + c F c F − s F + c s ( F − F ) s F + c F s F + c F + c s ( F + F ) . (4.19)The explicit expressions of the elements F ij and K ij defined in eqs. (4.18) and (4.19),respectively, are given in appendix A. By using these notations the first order Hamiltonianin the tilde basis (4.16) can be written as ˜ H (1) UV = ∆ b K. (4.20) We use the “renormalized basis” such that the zeroth-order and the perturbed Hamiltoniantakes the form ˜ H = ˜ H + ˜ H . ˜ H is given by (we discuss ˜ H later) ˜ H = s ∆ + c ∆ a c s e iδ ∆ c s e − iδ ∆ c ∆
00 0 ∆ + s ∆ a . (4.21)– 11 –o formulate the solar-resonance perturbation theory with UV, we transform to the“hat basis”, which diagonalizes ˜ H : ˆ ν i = ( U † ϕ ) ij ˜ ν j (4.22)with Hamiltonian ˆ H = U † ϕ ˜ HU ϕ (4.23)where U ϕ is parametrized as U ϕ = cos ϕ sin ϕe iδ − sin ϕe − iδ cos ϕ
00 0 1 (4.24) U ϕ is determined such that ˆ H (0) is diagonal, which leads to cos 2 ϕ = cos 2 θ − c r a (cid:113)(cid:0) cos 2 θ − c r a (cid:1) + sin θ , sin 2 ϕ = sin 2 θ (cid:113)(cid:0) cos 2 θ − c r a (cid:1) + sin θ , (4.25)where r a ≡ a ∆ m = ∆ a ∆ . (4.26)The three eigenvalues of the zeroth order Hamiltonian ˜ H in (4.21) is given by h = sin ( ϕ − θ ) ∆ + cos ϕc ∆ a ,h = cos ( ϕ − θ ) ∆ + sin ϕc ∆ a ,h = ∆ + s ∆ a . (4.28)Then, the Hamiltonian in the hat basis is given by ˆ H = ˆ H + ˆ H ν SM + ˆ H UV where ˆ H = h h
00 0 h , ˆ H ν SM = c ϕ c s ∆ a s ϕ c s e − iδ ∆ a c ϕ c s ∆ a s ϕ c s e iδ ∆ a , ˆ H UV = ∆ b U † ϕ KU ϕ , (4.29)where the K matrix is defined in (4.19), and the simplified notations are hereafter used: c ϕ = cos ϕ and s ϕ = sin ϕ . Notice that we have omitted the second order ˆ H UV , though onecan easily compute it from (4.17) if necessary. Notice that one can show that h = ∆ (cid:20)(cid:0) c r a (cid:1) − (cid:113) (cos 2 θ − c r a ) + sin θ (cid:21) ,h = ∆ (cid:20)(cid:0) c r a (cid:1) + (cid:113) (cos 2 θ − c r a ) + sin θ (cid:21) . (4.27) – 12 – .6 Calculation of ˆ S and ˜ S matrices To calculate ˆ S ( x ) we define Ω( x ) as Ω( x ) = e i ˆ H x ˆ S ( x ) . (4.30)Then, Ω( x ) obeys the evolution equation i ddx Ω( x ) = H Ω( x ) (4.31)where H ≡ e i ˆ H x ˆ H e − i ˆ H x . (4.32)Notice that ˆ H = ˆ H ν SM + ˆ H UV as in (4.29). Then, Ω( x ) can be computed perturbativelyas Ω( x ) = 1 + ( − i ) (cid:90) x dx (cid:48) H ( x (cid:48) ) + ( − i ) (cid:90) x dx (cid:48) H ( x (cid:48) ) (cid:90) x (cid:48) dx (cid:48)(cid:48) H ( x (cid:48)(cid:48) ) + · · · , (4.33)and the ˆ S matrix is given by ˆ S ( x ) = e − i ˆ H x Ω( x ) . (4.34)Using ˆ H = ˆ H ν SM + ˆ H UV in (4.29), ˆ S matrix of the ν SM part is given to the zeroth andthe first orders in the effective expansion parameter s
13 ∆ a h − h by ˆ S (0+1) ν SM ( x ) = e − i ˆ H x Ω ν SM ( x )= e − ih x c ϕ c s
13 ∆ a h − h (cid:8) e − ih x − e − ih x (cid:9) e − ih x s ϕ c s e − iδ ∆ a h − h (cid:8) e − ih x − e − ih x (cid:9) c ϕ c s
13 ∆ a h − h (cid:8) e − ih x − e − ih x (cid:9) s ϕ c s e iδ ∆ a h − h (cid:8) e − ih x − e − ih x (cid:9) e − ih x , (4.35)where we have used the fact that ∆ b is spatially constant as a consequence of the uniformmatter density approximation.Then, ν SM part of the tilde basis ˜ S matrix is given by ˜ S (0+1) ν SM = U ϕ ˆ S (0+1) ν SM U † ϕ = ˜ S (0) ν SM + ˜ S (1) ν SM , (4.36)where ˜ S (0) ν SM = c ϕ e − ih x + s ϕ e − ih x c ϕ s ϕ e iδ (cid:0) e − ih x − e − ih x (cid:1) c ϕ s ϕ e − iδ (cid:0) e − ih x − e − ih x (cid:1) s ϕ e − ih x + c ϕ e − ih x
00 0 e − ih x . (4.37) ˜ S (1) ν SM can be written in the form ˜ S (1) ν SM = X Y e − iδ X Y e iδ , (4.38)– 13 –here X = c s (cid:26) ∆ a h − h c ϕ (cid:16) e − ih x − e − ih x (cid:17) + ∆ a h − h s ϕ (cid:16) e − ih x − e − ih x (cid:17)(cid:27) ,Y = c s c ϕ s ϕ (cid:26) − ∆ a h − h (cid:16) e − ih x − e − ih x (cid:17) + ∆ a h − h (cid:16) e − ih x − e − ih x (cid:17)(cid:27) . (4.39)Notice that ˜ S ν SM respects the generalized T invariance.Now, we must compute the UV parameter related part of ˆ S and ˜ S matrices. Byremembering ˆ H UV = ∆ b U † ϕ KU ϕ , the UV part of H in (4.32), is given by H UV = e i ˆ H x ˆ H UV e − i ˆ H x = ∆ b e i ˆ H x U † ϕ KU ϕ e − i ˆ H x = ∆ b U † ϕ (cid:16) U ϕ e i ˆ H x U † ϕ (cid:17) K (cid:16) U ϕ e − i ˆ H x U † ϕ (cid:17) U ϕ . (4.40)Due to frequent usage of the factors in the parenthesis above we give the formula for them S ( ± ) ϕ ≡ (cid:16) U ϕ e ± i ˆ H x U † ϕ (cid:17) = c ϕ e ± ih x + s ϕ e ± ih x c ϕ s ϕ e iδ (cid:0) e ± ih x − e ± ih x (cid:1) c ϕ s ϕ e − iδ (cid:0) e ± ih x − e ± ih x (cid:1) s ϕ e ± ih x + c ϕ e ± ih x
00 0 e ± ih x . (4.41)Notice that ˜ S (0) ν SM is nothing but S ( − ) ϕ . Then, H UV takes a simple form H UV = ∆ b U † ϕ S (+) ϕ KS ( − ) ϕ U ϕ ≡ ∆ b U † ϕ Φ U ϕ = ∆ b U † ϕ Φ Φ Φ Φ Φ Φ Φ Φ Φ U ϕ (4.42)where we have introduced another simplifying matrix notation Φ ≡ S (+) ϕ KS ( − ) ϕ and itselements Φ ij . The explicit expressions of Φ ij are given in appendix A.Since U ϕ rotation back to the tilde basis removes U † ϕ and U ϕ in (4.42), it is simpler togo directly to the calculation of the tilde basis ˜ S matrix ˜ S ( x ) (1) EV = U ϕ ˆ S ( x ) (1) EV U † ϕ = U ϕ e − i ˆ H x Ω( x ) (1) UV U † ϕ = ∆ b U ϕ e − i ˆ H x U † ϕ (cid:20) ( − i ) (cid:90) x dx (cid:48) Φ( x (cid:48) ) (cid:21) = ∆ b S ( − ) ϕ ( − i ) (cid:90) x dx (cid:48) Φ ( x (cid:48) ) Φ ( x (cid:48) ) Φ ( x (cid:48) )Φ ( x (cid:48) ) Φ ( x (cid:48) ) Φ ( x (cid:48) )Φ ( x (cid:48) ) Φ ( x (cid:48) ) Φ ( x (cid:48) ) . (4.43)Hereafter, the subscript “EV” is used for the ˜ S and ˆ S matrices to indicate that they describeunitary evolution. The computed results of the elements of ˜ S ( x ) (1) EV are given in appendix B.Notice that again ˜ S ( x ) (1) UV respects the generalized T invariance.Thus, we have computed all the tilde basis S matrix elements to first order as ˜ S = ˜ S (0) ν SM + ˜ S (1) ν SM + ˜ S (1) EV . (4.44)The first and the second terms are given, respectively, in (4.37) and (4.38) with (4.39), andthe third in appendix B. – 14 – .7 The relations between various bases and computation of the flavor basis S matrix We first summarize the relationship between the flavor basis, the check (vacuum masseigenstate) basis, the tilde, and the hat (zeroth order diagonalized hamiltonian) basis.Only the unitary transformations are involved in changing from the hat basis to the tildebasis, and from the tilde basis to the check basis: ˆ H = U † ϕ ˜ HU ϕ , or ˜ H = U ϕ ˆ HU † ϕ , ˜ H = U ˇ HU † , or ˇ H = U † ˜ HU = U † U ϕ ˆ HU † ϕ U (4.45)The non-unitary transformation is involved from the check basis to the flavor basis: ν β = N βi ˇ ν i = { (1 − ˜ α ) U } βi ˇ ν i . (4.46)The relationship between the flavor basis Hamiltonian H flavor and the hat basis one ˆ H is H flavor = { (1 − ˜ α ) U } ˇ H { (1 − ˜ α ) U } † = (1 − ˜ α ) U U † U ϕ ˆ HU † ϕ U U † (1 − ˜ α ) † = (1 − ˜ α ) U U U ϕ ˆ HU † ϕ U † U † (1 − ˜ α ) † . (4.47)Then, the flavor basis S matrix is related to ˆ S and ˜ S matrices as S flavor = (1 − ˜ α ) U U U ϕ ˆ SU † ϕ U † U † (1 − ˜ α ) † = (1 − ˜ α ) U U ˜ SU † U † (1 − ˜ α ) † . (4.48)Using the formula eq. (4.48), it is straightforward to compute the flavor basis S matrixelements. Notice, however, that U is free from CP phase δ due to our choice of the SOLconvention of the U MNS matrix in (4.6).The flavor basis S matrix has a structure S flavor = (1 − ˜ α ) S prop (1 − ˜ α ) † where S prop ≡ U U ˜ SU † U † describes the unitary evolution despite the presence of non-unitary mixing[38]. The factors (1 − ˜ α ) and (1 − ˜ α ) † , parts of the N matrix which project the flavor statesto the mass eigenstates and vice versa, may be interpreted as the ones analogous to the“production NSI” and “detection NSI” [64], but a very constrained one determined by the“propagation NSI”. As announced in section 4.2, the expression of ˜ S (1) ν SM in (4.38) with (4.39) tells us that wehave another expansion parameter [41] A exp ≡ c s (cid:12)(cid:12)(cid:12)(cid:12) a ∆ m (cid:12)(cid:12)(cid:12)(cid:12) = 2 . × − (cid:18) ∆ m . × − eV (cid:19) − (cid:18) ρ . g/cm (cid:19) (cid:18) E MeV (cid:19) , (4.49)which is very small. The reason for such a “generated by the framework” expansion param-eter is the special feature of the perturbed Hamiltonian in (4.29).– 15 –n fact, our perturbative framework is peculiar from the beginning in the sense that thekey non-perturbed part of the Hamiltonian (4.21), its top-left × sub-matrix, is smallerin size than the 33 element by a factor of ∼ , and is comparable with ˆ H ν SM in (4.29).The secret for emergence of the very small effective expansion parameter (4.49) is that the33 element decouples in the leading order and appear in the perturbative corrections onlyin the energy denominator, making them smaller for the larger ratio of ∆ m / ∆ m . Thelatter property holds because of the special structure of perturbative Hamiltonian ˆ H ν SM with non-vanishing elements only in the third row and third column.With inclusion of the UV Hamiltonian (4.20), however, the size of the first order correc-tion is controlled not only by A exp in (4.49) but also the magnitudes of ˜ α β,γ . In computingthe higher order corrections the energy denominator suppression does not work for all theterms because the last property, “non-vanishing elements in the third row and third columnonly”, ceases to hold in the first order Hamiltonian. It can be confirmed in looking into theformulas of the oscillation probabilities in section 5.1, appendix D.1, and section 5.2. The oscillation probability can be calculated by using the formula P ( ν β → ν α ) = | ( S flavor ) αβ | . (5.1)We denote the flavor basis S matrices corresponding to ˜ S (0) ν SM , ˜ S (1) ν SM , and ˜ S (1) UV as S (0) ν SM , S (1) ν SM , and S (1) UV , respectively, as they are related through (4.48). To first order we have S flavor = S (0) ν SM + S (1) ν SM + S (1) EV − ˜ αS (0) ν SM − S (0) ν SM ˜ α † . (5.2)Then, we are ready to calculate the expressions of the oscillation probabilities using theformula (5.1) to first order in the expansion parameters. Following ref. [38], we categorize P ( ν β → ν α ) into the three types of terms: P ( ν β → ν α ) = P ( ν β → ν α ) (0+1) ν SM + P ( ν β → ν α ) (1) EV + P ( ν β → ν α ) (1) UV , (5.3)where P ( ν β → ν α ) (0+1) ν SM = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) S (0) ν SM (cid:17) αβ (cid:12)(cid:12)(cid:12)(cid:12) + 2 Re (cid:20)(cid:16) S (0) ν SM (cid:17) ∗ αβ (cid:16) S (1) ν SM (cid:17) αβ (cid:21) ,P ( ν β → ν α ) (1) EV = 2 Re (cid:20)(cid:16) S (0) ν SM (cid:17) ∗ αβ (cid:16) S (1) EV (cid:17) αβ (cid:21) ,P ( ν β → ν α ) (1) UV = − Re (cid:20)(cid:16) S (0) ν SM (cid:17) ∗ αβ (cid:16) ˜ αS (0) ν SM + S (0) ν SM ˜ α † (cid:17) αβ (cid:21) . (5.4)The subscripts “EV” and “UV” refer the unitary evolution part and the genuine non-unitarycontribution, terminology defined in ref. [38].The first term in eq. (5.4), P ( ν β → ν α ) (0+1) ν SM , is already computed in ref. [41]. Hence,we do not repeat the calculation, but urge the readers to go to this reference. Notice– 16 –hat use of the SOL convention of U MNS does not alter the expression of the oscillationprobabilities. The rest of the terms in eq. (5.4) can be computed straightforwardly by usingthe expressions of the tilde basis S matrix which are given explicitly in appendix B, andthe ˜ α matrix defined in (4.7).Unfortunately, the resulting expression of the oscillation probability even at first orderis far from simple. Therefore, to give a feeling to the readers, we will show in the next twosubsections a part of the first order probability in the ν µ → ν e channel. In section 5.1, oneof the five terms in P ( ν µ → ν e ) (1) EV is given, and the whole expression of P ( ν µ → ν e ) (1) UV in section 5.2. We leave the rest of the terms of P ( ν µ → ν e ) (1) EV to appendix D.1. Inthis appendix, we also give a practical suggestion to the readers on how to compute theoscillation probabilities in the ν µ − ν τ sector.For notational simplicity, we define, following ref. [41], the reduced Jarlskog factor inmatter as J mr ≡ c s c s c ϕ s ϕ = J r (cid:104)(cid:0) cos 2 θ − c r a (cid:1) + sin θ (cid:105) − / , (5.5)which is proportional to the reduced Jarlskog factor in vacuum, J r ≡ c s c s c s [65]. We have used eq. (4.25) in the second equality in (5.5).In this paper, we do not discuss numerical accuracy of the first order oscillation prob-ability because (1) the ν SM part, which is controlled by A exp ∼ − , is known to be veryaccurate already in first order [41], and (2) accuracy of the UV related part is trivial, thesmaller the α βγ , the better the accuracy. P ( ν µ → ν e ) (1) EV We first introduce the decomposition of P ( ν µ → ν e ) (1) EV . After computation of all the terms,we assemble them according to the types of the K ij variables involved. See eq. (4.19) and(A.2) in appendix A for the definitions and the explicit expressions of the K ij , respectively.For bookkeeping purpose we decompose P ( ν µ → ν e ) (1) EV into the following four terms: P ( ν µ → ν e ) (1) EV = P ( ν µ → ν e ) (1) EV | D-OD + P ( ν µ → ν e ) (1) EV | OD1 P ( ν µ → ν e ) (1) EV | OD2 + P ( ν µ → ν e ) (1) EV | OD3 , (5.6)where the subscripts “D” and “OD” refer to the diagonal and the off-diagonal K ij variables.The organization inside each term is largely determined such that the symmetry under thetransformation ϕ → ϕ + π is manifest. See section 5.3 for the ϕ symmetry. If we set the target accuracy of unitarity test at a % level, α βγ < ∼ − . Then, within the accuracy ofthe ν SM part, − < ∼ α βγ < ∼ − , the second order UV corrections could play a role. But, it is of order of ∼ α βγ < ∼ − , and hence it is negligible. – 17 –ere, we only present the first term in (5.6), P ( ν µ → ν e ) (1) EV | D-OD , leaving the others toappendix D.1: P ( ν µ → ν e ) (1) EV | D-OD = 4 J mr sin δ cos 2 ϕ ( K − K ) (∆ b x ) sin ( h − h ) x
2+ 2 ( K − K ) (∆ b x ) (cid:20) J mr cos δ sin( h − h ) x − s c s (cid:8) c ϕ sin( h − h ) x + s ϕ sin( h − h ) x (cid:9)(cid:21) + 4 J mr ( K − K ) (∆ b x ) × (cid:20) δ sin ( h − h ) x h − h ) x h − h ) x δ (cid:26) sin ( h − h ) x − sin ( h − h ) x (cid:27)(cid:21) + (cid:104) cos 2 ϕ ( K − K ) + sin 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:105) (∆ b x ) × (cid:20) c c ϕ s ϕ (cid:26) c c sin( h − h ) x − s s sin ( h − h ) x h − h ) x h − h ) x (cid:27) + 2 J mr cos δ (cid:2) c ϕ sin( h − h ) x + s ϕ sin( h − h ) x (cid:3) − J mr sin δ (cid:26) c ϕ sin ( h − h ) x s ϕ sin ( h − h ) x − c ϕ s ϕ sin ( h − h ) x (cid:27)(cid:21) + 2 (cid:104) sin 2 ϕ ( K − K ) − cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:105) × (cid:20) s c s c ϕ s ϕ (cid:26) (∆ b x ) (cid:2) c ϕ sin( h − h ) x + s ϕ sin( h − h ) x (cid:3) + 2 ∆ b h − h (cid:20) sin ( h − h ) x − sin ( h − h ) x − cos 2 ϕ sin ( h − h ) x (cid:21)(cid:27) + 4 J mr cos δc ϕ s ϕ ∆ b h − h sin ( h − h ) x (cid:21) − c c c ϕ s ϕ (cid:104) cos 2 ϕ ( K − K ) + sin 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:105) × ∆ b h − h (cid:20) − s cos δ (cid:26) sin ( h − h ) x − sin ( h − h ) x − cos 2 ϕ sin ( h − h ) x (cid:27) + c sin 2 ϕ sin ( h − h ) x s sin δ sin ( h − h ) x h − h ) x h − h ) x (cid:21) + 2 J mr c ϕ s ϕ (cid:104) sin 2 ϕ ( K − K ) − cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:105) (∆ b x ) × (cid:20) − (cid:26) cos δ sin( h − h ) x + 2 sin δ cos 2 ϕ sin ( h − h ) x (cid:27) + 2 sin δ (cid:26) sin ( h − h ) x − sin ( h − h ) x − cos 2 ϕ sin ( h − h ) x (cid:27) + 4 cos δ sin ( h − h ) x h − h ) x h − h ) x (cid:21) . (5.7)We first note that the α parameter dependence is expressed through the K jj elements,see its definition and the expression eq. (4.19) and (A.2), respectively. It is noticeable thatthe diagonal K jj elements organize themselves into the form of difference, K − K type– 18 –ombinations, as it should be, because it comes from the rephasing invariance. It leadsto the similar structure expressed by the diagonal α parameters, see section 6.1. P ( ν µ → ν e ) (1) UV To calculate P ( ν µ → ν e ) (1) UV defined in the last line in eq. (5.4), we need the expressionsof zeroth-order elements of ν SM matrix S (0) ν SM , which are given in appendix C. They canbe easily obtained from the tilde basis S matrix in (4.37). Using the S (0) matrix elements P ( ν µ → ν e ) (1) UV can be readily calculated as P ( ν µ → ν e ) (1) UV = −
2( ˜ α ee + ˜ α µµ ) | S (0) eµ | − Re (cid:104) ˜ α µe ( S (0) ee ) ∗ S (0) eµ (cid:105) = −
2( ˜ α ee + ˜ α µµ ) (cid:20) c c sin ϕ sin ( h − h ) x s sin θ (cid:26) c ϕ sin ( h − h ) x s ϕ sin ( h − h ) x − c ϕ s ϕ sin ( h − h ) x (cid:27) + 4 J mr cos δ (cid:26) cos 2 ϕ sin ( h − h ) x − sin ( h − h ) x ( h − h ) x (cid:27) + 8 J mr sin δ sin ( h − h ) x h − h ) x h − h ) x (cid:21) + Re ( ˜ α µe ) (cid:20) c c sin 2 ϕ cos δ (cid:18) c cos 2 ϕ sin ( h − h ) x s (cid:26) sin ( h − h ) x − sin ( h − h ) x (cid:27)(cid:19) − c c sin 2 ϕ sin δ (cid:18) c sin( h − h ) x − s { sin( h − h ) x − sin( h − h ) x } (cid:19) − s sin 2 θ (cid:18) c sin ϕ sin ( h − h ) x − θ (cid:26) c ϕ sin ( h − h ) x s ϕ sin ( h − h ) x (cid:27)(cid:19)(cid:21) − Im ( ˜ α µe ) (cid:20) c c sin 2 ϕ cos δ (cid:18) c sin( h − h ) x − s { sin( h − h ) x − sin( h − h ) x } (cid:19) + 2 c c sin 2 ϕ sin δ (cid:18) c cos 2 ϕ sin ( h − h ) x s (cid:26) sin ( h − h ) x − sin ( h − h ) x (cid:27)(cid:19) + s sin 2 θ (cid:8) c ϕ sin( h − h ) x + s ϕ sin( h − h ) x (cid:9)(cid:21) . (5.8)Here, the dependence on the α βγ is manifest. The feature of the diagonal α parametercorrelation is vastly different from that of the unitary evolution part P ( ν µ → ν e ) (1) EV , aswill be discussed in section 6.1. It is observed in ref. [41] that for each matter-dressed mixing angle φ there is an invarianceunder the transformation φ → φ + π . φ can be θ or θ in matter. In our system in this The fact is well known in the systems with the NSI parameters ε βγ . For a demonstration of the ε ββ − ε γγ structure to third order in the NSI parameters, see ref. [55], in particular its arXiv v1 for theexplicit form. The θ counterpart is previously noticed in ref. [66]. – 19 –aper, the oscillation probability is invariant under the transformation ϕ → ϕ + π , (5.9)which induces the following transformations simultaneously h → h , h → h ,c ϕ → − s ϕ , s ϕ → + c ϕ , cos 2 ϕ → − cos 2 ϕ, sin 2 ϕ → − sin 2 ϕ. (5.10)Hence, J mr → − J mr under the transformation.It is interesting to observe explicitly that the symmetry is respected by P ( ν µ → ν e ) (1) EV and P ( ν µ → ν e ) (1) UV , whose former is given in section 5.1 and appendix D.1, and the latterin section 5.2. The nature of the symmetry is identified as the “dynamical symmetry”, nota symmetry in the Hamiltonian [41]. Yet, it serves for a powerful consistency check of thecalculation. ν SM and the UV α parameters In this section, we discuss correlations between ν SM and the UV α parameters, includingthe clustering of the latter, which are manifested in the oscillation probabilities calculatedin sections 5.1, 5.2, and appendix D.1. α parameter correlation As discussed in ref. [38], the diagonal α parameters have the particular types of correlationsin the evolution part of the probability (cid:18) ∆ a ∆ b − (cid:19) α ee + α µµ , and α µµ − α ττ , (6.1)which arises due to the rephasing invariance. It becomes manifest in the would-be flavorbasis H wb-flavor ≡ U ˇ HU † = U ˜ HU † . Of course, it must hold in regions of the solar-scaleenhanced oscillation. In our expressions of P ( ν µ → ν e ) (1) EV given in section 5.1 and inappendix D.1, it is hidden in the diagonal K jj parameters in the form of K jj − K ii : K − K = 2 c (cid:20) ˜ α ee (cid:18) ∆ a ∆ b − (cid:19) + ˜ α µµ (cid:21) − s − c s )( ˜ α µµ − ˜ α ττ ) − s ) c s Re ( ˜ α τµ ) + 2 c s Re ( s ˜ α µe + c ˜ α τe ) K − K = − s (cid:20) ˜ α ee (cid:18) ∆ a ∆ b − (cid:19) + ˜ α µµ (cid:21) + 2( s − c c )( ˜ α µµ − ˜ α ττ )+ 2(1 + c ) c s Re ( ˜ α τµ ) + 2 c s Re ( s ˜ α µe + c ˜ α τe ) . (6.2) For most of our purposes the expressions of the flavor basis S matrix, S flavor , are sufficiently informative,but not on the diagonal α parameter correlation, whose discussion requires rephasing invariant quantities. We refer the UV paramerters, in generic contexts, as the “ α parameters”, but use the notation “ ˜ α ” inmaking the statements about the formulas and the results obtained by using the SOL convention of U MNS . – 20 –e note that K − K is not independent of the above two as it is obtained by adding them.See (4.19) for definition of K ij , and appendix A for their explicit expressions. Though thediagonal α parameter correlation is written in terms of the SOL convention ˜ α jj variables,it is independent of the convention of U MNS because the variables do not depend on theconvention. ν SM phase δ and the α parameters In view of the expressions of the first order probability, its UV related but unitary part insection 5.1 and appendix D.1, we identify the following correlated pairs consisting of the δ − α parameters, K e − iδ and K e iδ , where the blobs of the α parameters K and K can be written as K e − iδ = c (cid:110) c (cid:16) ˜ α µe e iδ (cid:17) ∗ − s (cid:16) ˜ α τe e iδ (cid:17) ∗ (cid:111) − s e − iδ (cid:2) c s ( ˜ α µµ − ˜ α ττ ) + c ˜ α τµ − s ˜ α ∗ τµ (cid:3) ,K e iδ = s (cid:110) c (cid:16) ˜ α µe e iδ (cid:17) − s (cid:16) ˜ α τe e iδ (cid:17)(cid:111) + c e iδ (cid:2) c s ( ˜ α µµ − ˜ α ττ ) + c ˜ α ∗ τµ − s ˜ α τµ (cid:3) . (6.3)Therefore, the δ - complex α parameter correlation does exist in the SOL convention of U MNS , which is in marked contrast to the feature of no δ − α parameter phase correlation inregion of the atmospheric scale enhanced oscillation [38]. Notice that K e iδ = (cid:0) K e − iδ (cid:1) ∗ and K e − iδ = (cid:0) K e iδ (cid:1) ∗ , and therefore they do not introduce correlations independent ofthose in (6.3). In fact, the feature of the e ± iδ - K blob correlation can be traced back to theform of Φ ij given in appendix A.One can also conclude from the features of ˜ α µe vs e ± iδ correlation seen in (6.2) and (6.3)there is no definite “chiral” combination ˜ α µe e iδ and/or ˜ α τe e iδ , nor ˜ α τµ e ± iδ . Considerationof the non-unitary part of the probability (5.8) does not change the conclusion.To summarize, the feature of δ - α parameter correlation at around the solar scaleenhanced oscillation is different from the one in region of the atmospheric scale oscillationdiscussed in ref. [38], most notably, on the following two aspects: • The correlation between the ν SM phase δ and the α parameters does exist in the SOLconvention of U MNS in region of the solar scale enhanced oscillation. • But, the correlation does not have the “chiral” form, ˜ α βγ e ± iδ . Rather it takes theform of correlation between e ± iδ and the blobs composed of the α parameters.Since the correlation between δ and the K - K cluster variables lives in Φ matrixelements, which are the building block of the perturbation series, it is obvious that thecorrelation prevails to higher order in perturbation theory in the unitary evolution part. δ − α parameter correlation: Are they real? The result in ref. [38] shows that the SOL convention of U MNS is the unique case in theatmospheric-scale enhanced oscillation in which the δ - α parameter correlation is absent.Then, the first itemized statement above indicates that there is no U MNS convention inwhich the phase correlation is absent both at around the atmospheric- and the solar-scaleenhanced oscillations. Then, we can now conclude that the δ − α parameter correlations– 21 –een in this and the previous paper [38] are all physical. That is, it cannot be wiped awayby a U MNS convention choice.In fact, it is very likely that, in the solar-scale enhanced region, the phase correlationexists with all the three conventions of U MNS . The oscillation probability in the other U MNS conventions can be obtained simply by using the translation rule, eq. (4.9). Then, weobserve in the ATM and PDG conventions, even more complicated correlations between e ± iδ and the blobs composed of the α parameters inside which some of the α parametersare attached with e ± iδ .One may wonder why the features of the correlation between α and the α parame-ters are so different between the regions of the atmospheric- and the solar-scale enhancedoscillations. But, this is entirely normal. As we have learned in section 3, the nature ofthe parameter correlation in neutrino evolution with inclusion of outside- ν SM ingredientsare dynamical. Their features depend on the values of the relevant parameters as wellas the kinematical regions where different degrees of freedom play the dominant role. Thedynamical nature of the phase correlation will be demonstrated in a visible way in section 7. α parameters In addition to the δ − (blob of the α parameters) correlation, we observe a feature which maybe called the “clustering of the α parameters” in the first order unitary evolution part of thefirst order oscillation probability P ( ν µ → ν e ) calculated in sections 5.1 and appendix D.1.We can identify the following “clustering variables” at the level of the ˜ S (1) EV matrix elements: K e − iδ + K e iδ , c ϕ K − c ϕ s ϕ K e iδ , c ϕ s ϕ K + c ϕ K e iδ , (6.4)where we have not listed ( s ϕ K + c ϕ s ϕ K e iδ ) and ( c ϕ s ϕ K − s ϕ K e iδ ) . They are notdynamically independent from the ones in (6.4) because they can be generated by thesymmetry transformation (5.9) from the second and the third in (6.4). Also there existsthe exceptional, isolated one K e − iδ in eq. (D.2).In (6.4), we did not quote the diagonal variables which come as a form of the difference,for example ( K − K ) , because these combinations are enforced by rephasing invariance.But, these diagonal α parameter differences often come with the particular combinationwith the other cluster variables, e.g., as (cid:2) cos 2 ϕ ( K − K ) + sin 2 ϕ (cid:0) K e − iδ + K e iδ (cid:1)(cid:3) ,or (cid:2) sin 2 ϕ ( K − K ) − cos 2 ϕ (cid:0) K e − iδ + K e iδ (cid:1)(cid:3) . Moreover, the other blobs of variables (cid:0) c K − s K (cid:1) , and (cid:0) c K e iδ − s K e − iδ (cid:1) , which are not visible at the level of the ˜ S (1) EV matrix, shows up in the oscillation probability. See eqs. (5.7), (D.2) - (D.4) for all theabove examples of blobs.It appears that appearance of such cluster variables as well as the correlation between δ and the α parameter blobs are worth attention though we do not quite understand thecause of this phenomenon. To close possible loophole in this statement, we performed an explicit construction of the solar resonanceperturbation theory extended with the UV effect using the ATM convention of U MNS . A preliminaryinvestigation reveals that the same δ − (cluster of the α parameters) correlation as in (6.3) survives, but inside K ij ˜ α βγ must be transformed to α βγ ( α matrix elements in the ATM convention) by the transformationrule (4.9). It is the expected result and apparently there is no loophole in our prescription. – 22 – Physics of neutrino flavor transformation with non-unitary mixing ma-trix
Up to this section, we aimed at analytical understanding of the system around the regionof solar-scale enhanced oscillation, for short, the “solar region”. Likewise, we use belowthe simplified terminology “atmospheric region” for a region of enhanced atmospheric-scaleoscillation. Now, we discuss physics of neutrino flavor transformation in the solar region.But, we do it in comparison with that of atmospheric region as it proves to be morerevealing. We try to illuminate some new aspects of the system of the three-flavor activeneutrinos with non-unitary mixing matrix by using the numerical method together withour first order formula.We use the PDG convention of U MNS in all the computations in this section, because itis used in most of the analyses of neutrino flavor transformations. We also depart from our“official” notations ¯ α βγ of the α parameters in the PDG convention defined in section 4.1,and simply denote them as α βγ in this section beyond the next subsection 7.1. U MNS
Toward the goal, we utilize the perturbative oscillation probability derived in section 5, aswell as the exact formula for the probability based on numerical integration of the evolutionequation whose latter is valid even for a varied matter density. It was pointed out in ref. [38] that the α matrix depends on convention of U MNS . Byusing the property, one can derive probability formula in the PDG or ATM conventions byusing the substitution rule from the ˜ α parameter in the SOL convention to the ¯ α parameterin the PDG, or the α parameter in the ATM conventions. See eq. (4.9) in section 4.1. Onecan also transform to a general U MNS convention by using the phase redefinition U ( β, γ ) defined in ref. [38]. Notice that the translation rule applies not only in the perturbativeformulas but also in the exact formulas. The first step to understand the effect of non-unitarity that brought into the ν SM threeneutrino system is to know where and how strongly the UV α parameters affect the neutrinoflavor transformation. For this purpose, we turn on each α βγ parameter one by one andcalculate the non-unitary contribution to the appearance probability ∆ P µe defined by ∆ P µe ≡ P ( ν µ → ν e ) − P ( ν µ → ν e ) ν SM = P ( ν µ → ν e ) EV + P ( ν µ → ν e ) UV (7.1)where P ( ν µ → ν e ) in eq. (7.1) denotes the appearance probability in the ν µ → ν e channelwith the UV effect fully implemented. Both P ( ν µ → ν e ) and P ( ν µ → ν e ) ν SM are computednumerically. In all the calculations in this section, the matter density is taken to be ρ =3 . − over the entire baseline. If the uniform matter density approximation applies one can also use the exact analytic formula forthe probability derived in ref. [25]. We note that the expression is reasonably simple despite its exactitude. – 23 – igure 1 . Plotted is ∆ P µe ≡ P ( ν µ → ν e ) − P ( ν µ → ν e ) ν SM by turning on one α βγ at one time, inorder from the top-left to the bottom-right panels, α ee , α µµ , α ττ , α µe , α τe , and α τµ . We take thevalue of each α βγ as half of the bound obtained by Blennow et al. [26] given in Table 2 in appendixA: α ee = 0 . , α µµ = 0 . , α ττ = 0 . , α µe = 0 . , α τe = 0 . , and α τµ = 0 . . The matterdensity is taken to be ρ = 3 . − over the entire baseline. – 24 –n figure 1 we show ∆ P µe by using color grading guided by the contour lines. In eachpanel we turn on one of α βγ , from the top-left to the bottom-right panels in order, α ee , α µµ , α ττ , α µe , α τe , and α τµ . In this section, we turn on only one of the α βγ parameters ineach panel, except for the top-right and bottom two panels in figure 2. To have an insightinto the required accuracy of the P ( ν µ → ν e ) measurement to improve the current boundsby a factor of 2, we take the value of each α βγ as half of the bound obtained by Blennow et al. [26] with the positive sign. Figure 1 as a whole, displays how large is the UV effectdepending upon the energy E and the baseline L . The “mountain ridges” roughly followthe line of L/E = constant. The atmospheric and the solar MSW enhancements are visible,respectively, at around E ∼ GeV and near the upper end of L = 10 km and E (cid:39) several × MeV and E (cid:39) several × km.We observe the two salient features: • ∆ P µe is at most (cid:39) ±
1% level in all the panels in figure 1, which means a 1% levelmeasurement of the probability is necessary for a factor 2 improvement of the bounds. • ∆ P µe changes sign depending upon which α βγ is turned on, and on region of kine-matical phase space, e.g., in the atmospheric region, or the solar region.The 1% accuracy measurement of the probability is mentioned at the end of section 4.2 inrelationship with the possible target accuracy of constraining UV α parameters.For the second point above, we notice in figure 1 that with turning on α ττ (middle-leftpanel) ∆ P µe is positive in the solar region and negative in the atmospheric region. Onthe other hand, this tendency is reversed completely with α τe (bottom-left panel), and lesscompletely with α µe (middle-right panel). In the other cases, ∆ P µe is negative in the bothregions of the atmospheric-scale and solar-scale enhancement. It means that if we turnon all α βγ at once, the effect of each element may cancel with each other at least partly.One must also take into account the fact that since we do not know a priori the sign ofthe α parameters, the pattern of the cancellation can be more complicated when all theparameters are turned on with arbitrary signs, or if phases are attached to the off-diagonal α parameters. It implies that (1) determination of the UV α βγ parameters (assuming itsexistence) could have additional difficulties due to confusion and degeneracy caused by thecancellation between the effect of different α parameters, (2) the bound on UV obtainedby using “one α βγ turned at one time” procedure could have made the bound artificiallystronger than the one obtained with the proper procedure of “all α βγ turned on but the restof them marginalized”.The features of possible cancellation between the effect of α βγ parameters may addanother difficulty to the task of identifying their effects, an already highly nontrivial one dueto high precision required to measurement of the probability. Therefore, further discussionof the question of how to disentangle the effects of different alpha parameters is called for. Notice that if the diagonal α parameters enter into the probability in the form α ββ − α γγ , only twoof the three diagonal α parameters are independent. But, since this subtractive dependence holds only infirst order in UV expansion [38], the independent bounds exist for all three of them. It is in sharp contrastto the situation of the NSI parameters. – 25 – .3 Unitary vs. non-unitary pieces of the UV related oscillation probability The UV α parameter related part of the probability ∆ P µe decomposes into the two parts,the unitary evolution part P ( ν µ → ν e ) EV and the genuine non-unitary part P ( ν µ → ν e ) UV [38], see eq. (5.3). Then, a natural question is which part is larger or dominating, andwhether they mutually tend to add up or cancel with each other.These questions are answered by figure 2. In the top two panels the whole UV effects, ∆ P µe = P ( ν µ → ν e ) UV + P ( ν µ → ν e ) EV are presented, with α µe = 0 . only in theleft panel, and with α ee = 0 . and α µµ = 0 . in the right panel. The values of the α parameters are the same as used in figure 1, and hence the left panel overlaps with a partof the middle-right panel of figure 1.The decomposition of ∆ P µe into P ( ν µ → ν e ) UV and P ( ν µ → ν e ) EV is displayed inthe middle ( α µe = 0 . case) and bottom ( α ee = 0 . and α µµ = 0 . case) panels offigure 2, respectively. We restrict ourselves into the two choices of the α parameters because P ( ν µ → ν e ) UV in first order depends only on the two combinations, α µe and α ee + α µµ . P ( ν µ → ν e ) EV is computed by using the formula P ( ν µ → ν e ) − P ( ν µ → ν e ) ν SM − P ( ν µ → ν e ) (1) UV with the first-order expression of P ( ν µ → ν e ) UV , and hence P ( ν µ → ν e ) EV is accurateonly to first order.An overall feature is that in wide areas in figure 2 P ( ν µ → ν e ) UV and P ( ν µ → ν e ) EV tend to cancel with each other. In looking into the figure more closely, however, we observea little more intricate features. In the α µe = 0 . case (middle panels), above L/E =10 km / MeV line, P ( ν µ → ν e ) UV contributes to lift up the probability, enhancing theyellow regions of P ( ν µ → ν e ) EV into the thinker ones in ∆ P µe . Below the line, P ( ν µ → ν e ) UV is more dominating in the blue solar resonance region, but partially cancelled by P ( ν µ → ν e ) EV . The cancellation is even more prominent in the bottom panels, the casewith α ee + α µµ turned on. The overall feature of the color-graded contour of ∆ P µe is similarto that of P ( ν µ → ν e ) UV , but P ( ν µ → ν e ) EV over-cancels the peaks of P ( ν µ → ν e ) UV abovethe L/E = 10 km / MeV line.The feature of cancellation is akin to, but is much more prominent compared to the oneobserved in the “atmospheric region” in ref. [38]. Unfortunately, we cannot offer physicalexplanation on why the cancellation between P ( ν µ → ν e ) UV and P ( ν µ → ν e ) EV takes place,and why the feature is common to both the atmospheric and the solar regions. In most ofthe regions it acts as a partial “hiding mechanism” of non-unitarity since a less prominenteffect is left in the observable, the appearance probability P ( ν µ → ν e ) . To obtain theinformation of the genuine non-unitary part P ( ν µ → ν e ) UV , it must be complemented bymeasurement of departure from unitarity, P ( ν µ → ν e ) + P ( ν µ → ν µ ) + P ( ν µ → ν τ ) (cid:54) = 1 . ν SM - α parameter phase correlation: The atmospheric vs. solar regions We have learned in the previous section 6 that the features of the parameter correlationbetween the ν SM and the UV new physics parameters in the solar region is different fromthe ones in the atmospheric region. A new δ - (blobs of the α parameters) correlationis observed. Then, it is natural to ask the question: What is the feature of ν SM - UV– 26 – igure 2 . In the top two panels, ∆ P µe = P ( ν µ → ν e ) UV + P ( ν µ → ν e ) EV are presented by thecolor grading, with α µe = 0 . (left panel), and with α ee = 0 . and α µµ = 0 . (right panel).In the middle and bottom panels ∆ P µe is decomposed to P ( ν µ → ν e ) UV and P ( ν µ → ν e ) EV in theleft and right panels, respectively. parameter CP phase correlation in the solar region, and which characteristic difference ithas from those in the atmospheric region?To discuss correlation between δ and phases of the off-diagonal α parameters, we– 27 –arametrize the latter as α βγ = | α βγ | e iφ βγ , (7.2)where βγ = µe, τ e, τ µ . To make the phase correlation visible clearly, we use ∆ P µe ≡ P ( ν µ → ν e ) − P ( ν µ → ν e ) ν SM defined in eq. (7.1), not the probability itself. Figure 3 . ∆ P µe ≡ P ( ν µ → ν e ) − P ( ν µ → ν e ) ν SM is presented in φ µe - δ plane by color graduation,which is calculated by turning on α µe = 0 . only. The top two panels are in the atmospheric regionwith energy E = 10 GeV, and the middle two panels are in the solar region with energy E = 200 MeV. The baseline is taken as L = 3000 km (left panel) and L = 12000 km (right panel), in both thetop and middle panels. The bottom panel is in the solar region with E = 300 MeV and L = 5000 km. In figures 3, the non-unitary contribution to the appearance probability ∆ P µe computedby turning on α µe only, is presented on φ µe - δ plane by showing the equi-contours of ∆ P µe with the color grading. In figures 4 and 5, the results of the similar exercises are presented,– 28 – igure 4 . ∆ P µe ≡ P ( ν µ → ν e ) − P ( ν µ → ν e ) ν SM is presented in φ µe - δ plane by color graduation,which is calculated by turning on α τe = 0 . only. The upper two panels are in the atmosphericregion with energy E = 10 GeV, and the lower two panels are in the solar region with energy E = 200 MeV. The baseline is taken as L = 3000 km (left panel) and L = 12000 km (right panel),in both the upper and lower panels. Figure 5 . The same as in figure 4 but with only α τµ = 0 . is turned on. – 29 –he case with α τe turned on (figure 4), and the one with α τµ (figure 5). In figures 3, 4,and 5, we use a large value α βγ = 0 . to enhance effects of the phase correlation, whichmerits higher visibility. The global features of the δ - α parameter phase correlation shownin figures 3, 4 and 5 are: • The linear, oblique correlation seen in the case of α µe (cid:54) = 0 (figure 3) and α τe (cid:54) = 0 (figure 4) both in the atmospheric region shown in the upper panels, but no clearlyvisible correlation in all the other panels. • The absolute value of | ∆ P µe | is larger in the panels with baseline L = 12000 km thanthose with L = 3000 km by a factor of ∼ . The statement applies to all the panelsincluding both the atmospheric and solar regions.Let us start from discussion of the phase correlation seen in the atmospheric region, thetop two panels in figures 3, 4 and 5. The linear, oblique correlation seen in figures 3 and 4, φ µe - δ and φ τe - δ correlation, respectively, and no visible correlation between φ τµ and δ shown in figure 5 (all in the upper two panels) is perfectly consistent with the “canonicalphase combination” [38] e − iδ α µe , e − iδ α τe , α τµ , (7.3)which holds under the PDG convention of U MNS . One should note the nontrivial U MNS convention dependence: In the ATM phase convention of U MNS (in which e ± iδ is attachedto s ), the phase correlation takes the form [ e − iδ α µe , α τe , e iδ α τµ ] [38].On the other hand, the features of the phase correlation in the solar region shown inthe lower panels in figures 3, 4, and 5 are more subtle and not easy to understand. In somepanels, the equal- ∆ P µe contours are vertical, which may imply that there is no significantcorrelation between δ and α parameter phases. In the other, there exists “circular-shapedcorrelation” with positive and negative signs of ∆ P µe in the two-dimensional phase space.Notice that in the panels with vertical correlation and with “circular correlation”, the δ (in-)dependence cannot be understood as a remnant of insufficient subtraction of the ν SM part.It is because the values of ∆ P µe and its variation in φ or δ directions can be as large as ∼ . , of the order of the α parameter that is turned on. The feature of the φ - δ phasecorrelation, in particular, coexistence of the vertical and circular shaped correlations is notunderstood, regrettably, by our analytic framework. If the probability calculated by first-order helio-UV perturbation theory [38] is sufficiently accurate,there should be no δ dependence in the upper two panels in figure 5, because the δ dependence would havebeen eliminated by the subtraction of P ( ν µ → ν e ) ν SM . Obviously, it is not the case. Notice that the resultspresented in figures 3, 4 and 5 are accurate as they do not rely on perturbative treatment. It means thatthe perturbative treatment fails to provide accurate description of the probability, which is natural dueto the large value 0.1 taken for α τµ . In fact, the remaining δ dependence is up to a few × − level for L = 3000 km, and is of order ∼ ± . for L = 12000 km, so that our interpretation may be valid. It appears that there are some regularities which may be relevant for understanding of the phasecorrelation in the solar region. That is, we often observe “red” ( ∆ P µe > ) and “blue” ( ∆ P µe < ) verticalcontours in central region, φ ∼ π . It is likely that the central “red” vertical correlation corresponds to – 30 –ith regard to the baseline dependence of the strength of the correlation, it might bethat | ∆ P µe | itself is larger at the longer baseline of L = 12000 km among the two baselineswe have chosen to display in figures 3, 4 and 5.The features of the α parameter phase - ν SM δ correlation in the atmospheric and thesolar regions presented in figures 3, 4 and 5 testify that the nature of the correlation isquite dynamical, confirming our view that stated in section 3. Unfortunately, physical un-derstanding of the features of the phase correlation in the solar region are not yet achieved,which calls for further studies. In this paper, we have attempted to achieve physics understanding of the three-flavor neu-trino system with non-unitary mixing matrix. We have focused our discussion on elucidatingthe nature of parameter correlations in such system, in particular the one between the ν SMand the UV new physics parameters. We do it in region of the solar-scale oscillations, forshort the “solar region”, in this paper. It nicely complements the one given in our previouspaper [38] which dealt with the region of atmospheric-scale oscillations, the “atmosphericregion”.Toward the goal, we have formulated a new perturbative framework to discuss effectof non-unitary mixing matrix in the solar region, the UV extended version of the “solar-resonance perturbation theory” [41]. It was necessary to resolve the question raised inref. [38] which casts doubt physical reality of the correlation between the ν SM δ and thephases of UV α parameters. But, in turn, the framework serves as a powerful analyticmachinery of analyzing the features of parameter correlation in the solar region. Theskepticism about the reality of the phase correlation, which is described in detail in section 1,is cleared up by showing that the phase correlation does exist in the solar region with theSOL ( e ± iδ attached to s ) convention of U MNS . See section 6.In fact, we have uncovered that the features of the ν SM - UV parameter correlationsare much more profound than we thought. This point can be illuminated most clearlyby contrasting the atmospheric region to the solar one. In the atmospheric region, themost notable feature is the ν SM δ - UV α parameter phase correlation of the “chiral type”, [ e − iδ α µe , e − iδ α τe , α τµ ] in the PDG convention of U MNS [38]. The picture no more holds in thesolar region, and the correlation takes the form of δ - (blobs of the α parameters) correlationas we saw in section 6.2. Another interesting observation in this context is that when wemove the kinematical region from E/L = 200
MeV / km to E/L = 300
MeV / km,the δ - φ µe correlation takes vastly different forms as shown in figure 3, where φ µe denotesthe phase of α µe .We have utilized the analytic framework developed in this paper as well as the numericalmethod to reveal more generic feature of the effects of the UV α parameters. In addition tothe above mentioned ones, we have observed that the effect of non-unitarity tends to cancel the region of negative ∆ P µe in figure 1. Whereas the central “blue” vertical correlation corresponds to theregion of positive ∆ P µe in figure 1. For the latter, we refer the lower-right panel of figure 4, and the caseof α τe = 0 . , E = 200 MeV and L = 5000 km. – 31 –etween the unitary evolution part (denoted as “EV”) and the non-unitary part (denotedas “UV”) of the probability, and between the different α βγ parameters. See section 7.One of the most intriguing features of the parameter correlation is that the form ofthe correlation depends also on the values of the mixing parameters. The phenomenon isbriefly mentioned in section 3.2 that as θ becomes larger, the correlation seen at smaller θ starts to dissolve. Since we cannot control the values of the mixing angles or ∆ m by ourselves, the discussion might look as appealing only to an academic interest. But,we believe that it merits deepening our understanding on the mechanism and the cause ofparameter correlation. We were not able to explore this point further in this paper, and afocused investigation on this issue is called for.All these features of the parameter correlation may be summarized by the term “dy-namical nature of the parameter correlation” .Finally, we remark that occurrence of dynamical correlations between the parametersin systems with the many degree of freedom is very common, as discussed in section 3. Arich variety of correlations we encountered in our system with non-unitarity adds anotherexample to this list. If one chooses the way of testing leptonic unitarity by setting up aclass of models with UV and confrontation of them with experimental data, understandingthe system with UV would be indispensable step to carry out this task. Yet we mustemphasize that our understanding on the system, e.g., on the parameter correlation, is farfrom sufficient, generically in the system with new physics beyond the ν SM.On the experimental side, if we want to utilize the low energy region with the solar-scale enhanced oscillation, in the context of precision unitarity test, possible advantageof the Kamioka-Korea identical two-detector setup [15, 67] may worth renewed attention.Fortunately, the construction of Hyper-K has been started, which may act as the Kamiokasite detector in an extended plan of the two-detector complex [16, 49].
Acknowledgments
One of the authors (I.M.S.) acknowledges travel support from the Colegio de Física Fun-damental e Interdisciplinaria de las Américas (COFI). Fermilab is operated by the FermiResearch Alliance, LLC under contract No. DE-AC02-07CH11359 with the United StatesDepartment of Energy. The other (H.M.) thanks Center for Neutrino Physics, Departmentof Physics, Virginia Tech for hospitality and support.– 32 –
Explicit expressions of F ij , K ij and Φ ij The explicit expressions of the elements F ij , K ij and Φ ij defined, respectively, in eqs. (4.18),(4.19) and (4.42) are given as follows: F = 2 ˜ α ee (cid:18) − ∆ a ∆ b (cid:19) ,F = c ˜ α ∗ µe − s ˜ α ∗ τe ,F = s ˜ α ∗ µe + c ˜ α ∗ τe ,F = c ˜ α µe − s ˜ α τe = ( F ) ∗ F = 2 (cid:2) c ˜ α µµ + s ˜ α ττ − c s Re ( ˜ α τµ ) (cid:3) ,F = (cid:2) c s ( ˜ α µµ − ˜ α ττ ) + c ˜ α ∗ τµ − s ˜ α τµ (cid:3) ,F = s ˜ α µe + c ˜ α τe = ( F ) ∗ ,F = (cid:2) c s ( ˜ α µµ − ˜ α ττ ) + c ˜ α τµ − s ˜ α ∗ τµ (cid:3) = ( F ) ∗ ,F = 2 (cid:2) s ˜ α µµ + c ˜ α ττ + c s Re ( ˜ α τµ ) (cid:3) . (A.1) K = 2 c ˜ α ee (cid:18) − ∆ a ∆ b (cid:19) + 2 s (cid:2) s ˜ α µµ + c ˜ α ττ + c s Re ( ˜ α τµ ) (cid:3) , − c s Re ( s ˜ α µe + c ˜ α τe ) K = c (cid:0) c ˜ α ∗ µe − s ˜ α ∗ τe (cid:1) − s (cid:2) c s ( ˜ α µµ − ˜ α ττ ) + c ˜ α τµ − s ˜ α ∗ τµ (cid:3) = ( K ) ∗ ,K = 2 c s (cid:20) ˜ α ee (cid:18) − ∆ a ∆ b (cid:19) − (cid:0) s ˜ α µµ + c ˜ α ττ (cid:1)(cid:21) + c (cid:0) s ˜ α ∗ µe + c ˜ α ∗ τe (cid:1) − s ( s ˜ α µe + c ˜ α τe ) − c s c s Re ( ˜ α τµ ) = ( K ) ∗ ,K = 2 (cid:2) c ˜ α µµ + s ˜ α ττ − c s Re ( ˜ α τµ ) (cid:3) ,K = s ( c ˜ α µe − s ˜ α τe ) + c (cid:2) c s ( ˜ α µµ − ˜ α ττ ) + c ˜ α ∗ τµ − s ˜ α τµ (cid:3) = ( K ) ∗ ,K = 2 s ˜ α ee (cid:18) − ∆ a ∆ b (cid:19) + 2 c (cid:2) s ˜ α µµ + c ˜ α ττ + c s Re ( ˜ α τµ ) (cid:3) + 2 c s Re ( s ˜ α µe + c ˜ α τe ) . (A.2) Φ = K + 2 c ϕ s ϕ ( K − K ) − c ϕ s ϕ ( K − K ) (cid:110) e i ( h − h ) x + e − i ( h − h ) x (cid:111) − c ϕ s ϕ cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17) + c ϕ s ϕ (cid:110) − (cid:16) s ϕ K e − iδ − c ϕ K e iδ (cid:17) e i ( h − h ) x + (cid:16) c ϕ K e − iδ − s ϕ K e iδ (cid:17) e − i ( h − h ) x (cid:111) , Φ = e iδ (cid:20) c ϕ s ϕ cos 2 ϕ ( K − K ) + c ϕ s ϕ (cid:110) s ϕ e i ( h − h ) x − c ϕ e − i ( h − h ) x (cid:111) ( K − K )+ 2 c ϕ s ϕ (cid:16) K e − iδ + K e iδ (cid:17) + s ϕ (cid:16) s ϕ K e − iδ − c ϕ K e iδ (cid:17) e i ( h − h ) x + c ϕ (cid:16) c ϕ K e − iδ − s ϕ K e iδ (cid:17) e − i ( h − h ) x (cid:21) , Φ = (cid:16) s ϕ K + c ϕ s ϕ K e iδ (cid:17) e − i ( h − h ) x + (cid:16) c ϕ K − c ϕ s ϕ K e iδ (cid:17) e − i ( h − h ) x , – 33 – = e − iδ (cid:26) c ϕ s ϕ cos 2 ϕ ( K − K ) − c ϕ s ϕ (cid:110) c ϕ e i ( h − h ) x − s ϕ e − i ( h − h ) x (cid:111) ( K − K )+ 2 c ϕ s ϕ (cid:16) K e − iδ + K e iδ (cid:17) + c ϕ (cid:16) c ϕ K e iδ − s ϕ K e − iδ (cid:17) e i ( h − h ) x + s ϕ (cid:16) s ϕ K e iδ − c ϕ K e − iδ (cid:17) e − i ( h − h ) x (cid:27) , Φ = K − c ϕ s ϕ ( K − K ) + c ϕ s ϕ ( K − K ) (cid:110) e i ( h − h ) x + e − i ( h − h ) x (cid:111) + c ϕ s ϕ (cid:20) cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17) + (cid:16) s ϕ K e − iδ − c ϕ K e iδ (cid:17) e i ( h − h ) x − (cid:16) c ϕ K e − iδ − s ϕ K e iδ (cid:17) e − i ( h − h ) x (cid:21) , Φ = e − iδ (cid:20)(cid:16) c ϕ s ϕ K + c ϕ K e iδ (cid:17) e − i ( h − h ) x − (cid:16) c ϕ s ϕ K − s ϕ K e iδ (cid:17) e − i ( h − h ) x (cid:21) , Φ = (cid:16) s ϕ K + c ϕ s ϕ K e − iδ (cid:17) e i ( h − h ) x + (cid:16) c ϕ K − c ϕ s ϕ K e − iδ (cid:17) e i ( h − h ) x , Φ = e iδ (cid:20)(cid:16) c ϕ s ϕ K + c ϕ K e − iδ (cid:17) e i ( h − h ) x − (cid:16) c ϕ s ϕ K − s ϕ K e − iδ (cid:17) e i ( h − h ) x (cid:21) , Φ = K . (A.3) B The first order tilde basis unitary evolution ˜ S matrix elements Here, we present the result of unitary ˜ S matrix elements which come from first order UVparameter related part of the Hamiltonian. ˜ S ( x ) EV = ∆ b (cid:110) K + 2 c ϕ s ϕ ( K − K ) − c ϕ s ϕ cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111) ( − ix ) (cid:16) c ϕ e − ih x + s ϕ e − ih x (cid:17) + c ϕ s ϕ (cid:110) c ϕ s ϕ cos 2 ϕ ( K − K ) + 2 c ϕ s ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111) ( − ix ) (cid:16) e − ih x − e − ih x (cid:17) + c ϕ s ϕ (cid:104) − c ϕ s ϕ ( K − K ) + cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:105) h − h (cid:16) e − ih x − e − ih x (cid:17) . (B.1) ˜ S ( x ) EV = e iδ ∆ b (cid:20)(cid:110) c ϕ s ϕ cos 2 ϕ ( K − K ) + 2 c ϕ s ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111) ( − ix ) (cid:16) c ϕ e − ih x + s ϕ e − ih x (cid:17) + c ϕ s ϕ (cid:110) K − c ϕ s ϕ ( K − K ) + c ϕ s ϕ cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111) ( − ix ) (cid:16) e − ih x − e − ih x (cid:17) + (cid:26) − c ϕ s ϕ cos 2 ϕ ( K − K ) + (cid:110) K e − iδ − c ϕ s ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111)(cid:27) h − h (cid:16) e − ih x − e − ih x (cid:17)(cid:21) . (B.2) ˜ S ( x ) EV = ∆ b (cid:20)(cid:16) s ϕ K + c ϕ s ϕ K e iδ (cid:17) h − h (cid:16) e − ih x − e − ih x (cid:17) + (cid:16) c ϕ K − c ϕ s ϕ K e iδ (cid:17) h − h (cid:16) e − ih x − e − ih x (cid:17)(cid:21) . (B.3)– 34 – S ( x ) EV = e − iδ ∆ b (cid:20)(cid:110) c ϕ s ϕ cos 2 ϕ ( K − K ) + 2 c ϕ s ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111) ( − ix ) (cid:16) s ϕ e − ih x + c ϕ e − ih x (cid:17) + c ϕ s ϕ (cid:110) K + 2 c ϕ s ϕ ( K − K ) − c ϕ s ϕ cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111) ( − ix ) (cid:16) e − ih x − e − ih x (cid:17) + (cid:26) − c ϕ s ϕ cos 2 ϕ ( K − K ) + (cid:110) K e iδ − c ϕ s ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111)(cid:27) h − h (cid:16) e − ih x − e − ih x (cid:17)(cid:21) . (B.4) ˜ S ( x ) EV = ∆ b (cid:20) c ϕ s ϕ (cid:110) c ϕ s ϕ cos 2 ϕ ( K − K ) + 2 c ϕ s ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111) ( − ix ) (cid:16) e − ih x − e − ih x (cid:17) + (cid:110) K − c ϕ s ϕ ( K − K ) + c ϕ s ϕ cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:111) ( − ix ) (cid:16) s ϕ e − ih x + c ϕ e − ih x (cid:17) + (cid:26) c ϕ s ϕ ( K − K ) − c ϕ s ϕ cos 2 ϕ (cid:16) K e − iδ + K e iδ (cid:17)(cid:27) h − h (cid:16) e − ih x − e − ih x (cid:17)(cid:21) . (B.5) ˜ S ( x ) EV = e − iδ ∆ b (cid:26)(cid:104) c ϕ s ϕ K + c ϕ K e iδ (cid:105) h − h (cid:16) e − ih x − e − ih x (cid:17) − (cid:104) c ϕ s ϕ K − s ϕ K e iδ (cid:105) h − h (cid:16) e − ih x − e − ih x (cid:17)(cid:27) . (B.6) ˜ S ( x ) EV = ∆ b (cid:20)(cid:16) s ϕ K + c ϕ s ϕ K e − iδ (cid:17) h − h (cid:16) e − ih x − e − ih x (cid:17) + (cid:16) c ϕ K − c ϕ s ϕ K e − iδ (cid:17) h − h (cid:16) e − ih x − e − ih x (cid:17)(cid:21) . (B.7) ˜ S ( x ) EV = e iδ ∆ b (cid:20)(cid:16) c ϕ s ϕ K + c ϕ K e − iδ (cid:17) h − h (cid:110) e − ih x − e − ih x (cid:111) − (cid:16) c ϕ s ϕ K − s ϕ K e − iδ (cid:17) h − h (cid:110) e − ih x − e − ih x (cid:111)(cid:21) . (B.8) ˜ S ( x ) EV = ( − ix ∆ b ) e − ih x K . (B.9)– 35 – The zeroth-order ν SM S matrix elements
Here, we give the expressions of the flavor basis S matrix elements of ν SM part at zerothorder. The superscript “ ν SM” is abbreviated. S (0) ee = c (cid:16) c ϕ e − ih x + s ϕ e − ih x (cid:17) + s e − ih x ,S (0) eµ = c c c ϕ s ϕ e iδ (cid:16) e − ih x − e − ih x (cid:17) − s c s (cid:16) c ϕ e − ih x + s ϕ e − ih x − e − ih x (cid:17) ,S (0) eτ = − c c s (cid:16) c ϕ e − ih x + s ϕ e − ih x − e − ih x (cid:17) − s c c ϕ s ϕ e iδ (cid:16) e − ih x − e − ih x (cid:17) ,S (0) µe = c c c ϕ s ϕ e − iδ (cid:16) e − ih x − e − ih x (cid:17) − s c s (cid:16) c ϕ e − ih x + s ϕ e − ih x − e − ih x (cid:17) = S eµ ( − δ ) ,S (0) µµ = c (cid:16) s ϕ e − ih x + c ϕ e − ih x (cid:17) + s (cid:110) s (cid:16) c ϕ e − ih x + s ϕ e − ih x (cid:17) + c e − ih x (cid:111) − c s s c ϕ s ϕ cos δ (cid:16) e − ih x − e − ih x (cid:17) ,S (0) µτ = s c ϕ s ϕ (cid:16) s e iδ − c e − iδ (cid:17) (cid:16) e − ih x − e − ih x (cid:17) + c s (cid:104) s (cid:16) c ϕ e − ih x + s ϕ e − ih x (cid:17) + c e − ih x − (cid:16) s ϕ e − ih x + c ϕ e − ih x (cid:17)(cid:105) ,S (0) τe = − c c s (cid:16) c ϕ e − ih x + s ϕ e − ih x − e − ih x (cid:17) − s c c ϕ s ϕ e − iδ (cid:16) e − ih x − e − ih x (cid:17) = S eτ ( − δ ) ,S (0) τµ = s c ϕ s ϕ (cid:16) s e − iδ − c e iδ (cid:17) (cid:16) e − ih x − e − ih x (cid:17) + c s (cid:104) s (cid:16) c ϕ e − ih x + s ϕ e − ih x (cid:17) + c e − ih x − (cid:16) s ϕ e − ih x + c ϕ e − ih x (cid:17)(cid:105) = S µτ ( − δ ) ,S (0) ττ = s (cid:16) s ϕ e − ih x + c ϕ e − ih x (cid:17) + c (cid:110) s (cid:16) c ϕ e − ih x + s ϕ e − ih x (cid:17) + c e − ih x (cid:111) + 2 c s s c ϕ s ϕ cos δ (cid:16) e − ih x − e − ih x (cid:17) . (C.1) D The neutrino oscillation probability in the ν µ → ν e and the other chan-nels In this appendix, we give the expressions of the rest of the terms of P ( ν µ → ν e ) (1) EV whichare not presented in section 5. We also briefly mention how to compute the neutrinooscillation probability in the ν µ − ν τ sector. D.1 The neutrino oscillation probability in the ν µ → ν e channel: Rest of theunitary evolution part We recapitulate the definition (5.6) of the four terms of P ( ν µ → ν e ) (1) EV again for conve-nience: P ( ν µ → ν e ) (1) EV = P ( ν µ → ν e ) (1) EV | D-OD + P ( ν µ → ν e ) (1) int-UV | OD1 + P ( ν µ → ν e ) (1) int-UV | OD2 + P ( ν µ → ν e ) (1) int-UV | OD3 , (D.1)where the subscripts “D” and “OD” refer to the diagonal and the off-diagonal K ij variables.– 36 –he first term of eq. (D.1) is given in eq. (5.7). Now, we present the remaining three“OD” terms: P ( ν µ → ν e ) (1) EV | OD1 = 4 c c Re (cid:16) K e − iδ (cid:17) × ∆ b h − h (cid:20) − s cos δ (cid:26) sin ( h − h ) x − sin ( h − h ) x − cos 2 ϕ sin ( h − h ) x (cid:27) + c sin 2 ϕ sin ( h − h ) x s sin δ sin ( h − h ) x h − h ) x h − h ) x (cid:21) + 4 c s c Im (cid:16) K e − iδ (cid:17) × ∆ b h − h (cid:20) sin δ (cid:26) sin ( h − h ) x − sin ( h − h ) x − cos 2 ϕ sin ( h − h ) x (cid:27) + 2 cos δ sin ( h − h ) x h − h ) x h − h ) x (cid:21) . (D.2) P ( ν µ → ν e ) (1) EV | OD2 = − c c s c ϕ s ϕ Re (cid:16) c ϕ s ϕ K + c ϕ K e − iδ (cid:17) × ∆ b h − h (cid:26) sin ( h − h ) x − sin ( h − h ) x ( h − h ) x (cid:27) + 4 c c s c ϕ s ϕ Re (cid:16) c ϕ s ϕ K − s ϕ K e − iδ (cid:17) × ∆ b h − h (cid:26) sin ( h − h ) x − sin ( h − h ) x − sin ( h − h ) x (cid:27) + 4 c s c s (cid:26) cos δ Re (cid:16) c ϕ s ϕ K + c ϕ K e − iδ (cid:17) − sin δ Im (cid:16) c ϕ s ϕ K + c ϕ K e − iδ (cid:17)(cid:27) × ∆ b h − h (cid:20) c ϕ (cid:26) sin ( h − h ) x − sin ( h − h ) x (cid:27) + (1 + s ϕ ) sin ( h − h ) x (cid:21) − c s c s (cid:26) cos δ Re (cid:16) c ϕ s ϕ K − s ϕ K e − iδ (cid:17) − sin δ Im (cid:16) c ϕ s ϕ K − s ϕ K e − iδ (cid:17)(cid:27) × ∆ b h − h (cid:20) s ϕ (cid:26) sin ( h − h ) x − sin ( h − h ) x (cid:27) + (1 + c ϕ ) sin ( h − h ) x (cid:21) − c c s (cid:20)(cid:0) s s c ϕ cos δ + c c ϕ s ϕ (cid:1) Im (cid:16) c ϕ s ϕ K + c ϕ K e − iδ (cid:17) + s s c ϕ sin δ Re (cid:16) c ϕ s ϕ K + c ϕ K e − iδ (cid:17)(cid:21) ∆ b h − h sin ( h − h ) x h − h ) x h − h ) x − c c s (cid:20)(cid:0) s s s ϕ cos δ − c c ϕ s ϕ (cid:1) Im (cid:16) c ϕ s ϕ K − s ϕ K e − iδ (cid:17) + s s s ϕ sin δ Re (cid:16) c ϕ s ϕ K − s ϕ K e − iδ (cid:17)(cid:21) ∆ b h − h sin ( h − h ) x h − h ) x h − h ) x . (D.3)– 37 – ( ν µ → ν e ) (1) EV | OD3 = − c s c c ϕ s ϕ (cid:26) cos δ Re (cid:104) s ϕ (cid:0) c K − s K (cid:1) + c ϕ s ϕ (cid:16) c K e iδ − s K e − iδ (cid:17)(cid:105) + sin δ Im (cid:104) s ϕ (cid:0) c K − s K (cid:1) + c ϕ s ϕ (cid:16) c K e iδ − s K e − iδ (cid:17)(cid:105)(cid:27) × ∆ b h − h (cid:26) sin ( h − h ) x − sin ( h − h ) x ( h − h ) x (cid:27) − c s c c ϕ s ϕ (cid:26) cos δ Re (cid:104) c ϕ (cid:0) c K − s K (cid:1) − c ϕ s ϕ (cid:16) c K e iδ − s K e − iδ (cid:17)(cid:105) + sin δ Im (cid:104) c ϕ (cid:0) c K − s K (cid:1) − c ϕ s ϕ (cid:16) c K e iδ − s K e − iδ (cid:17)(cid:105)(cid:27) × ∆ b h − h (cid:26) sin ( h − h ) x − sin ( h − h ) x − sin ( h − h ) x (cid:27) − s c s Re (cid:104) s ϕ (cid:0) c K − s K (cid:1) + c ϕ s ϕ (cid:16) c K e iδ − s K e − iδ (cid:17)(cid:105) × ∆ b h − h (cid:20) c ϕ (cid:26) − sin ( h − h ) x ( h − h ) x (cid:27) − (1 + s ϕ ) sin ( h − h ) x (cid:21) − s c s Re (cid:104) c ϕ (cid:0) c K − s K (cid:1) − c ϕ s ϕ (cid:16) c K e iδ − s K e − iδ (cid:17)(cid:105) × ∆ b h − h (cid:20) s ϕ (cid:26) − sin ( h − h ) x ( h − h ) x (cid:27) − (1 + c ϕ ) sin ( h − h ) x (cid:21) + 8 s c (cid:26)(cid:0) c c ϕ s ϕ cos δ − s s c ϕ (cid:1) Im (cid:104) s ϕ (cid:0) c K − s K (cid:1) + c ϕ s ϕ (cid:16) c K e iδ − s K e − iδ (cid:17)(cid:105) + c c ϕ s ϕ sin δ Re (cid:104) s ϕ (cid:0) c K − s K (cid:1) + c ϕ s ϕ (cid:16) c K e iδ − s K e − iδ (cid:17)(cid:105)(cid:27) × ∆ b h − h sin ( h − h ) x h − h ) x h − h ) x
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