Do T asymmetries for neutrino oscillations in uniform matter have a CP-even component?
PPrepared for submission to JHEP
Do T asymmetries for neutrino oscillations in uniformmatter have a CP-even component?
Jos´e Bernab´eu and Alejandro Segarra
Departament de F´ısica Te`orica and IFIC, Universitat de Val`encia - CSIC, E-46100, Spain
E-mail:
[email protected] , [email protected]
Abstract:
Observables of neutrino oscillations in matter have, in general, contributionsfrom the effective matter potential. It contaminates the CP violation asymmetry addinga fake effect that has been recently disentangled from the genuine one by their differentbehavior under T and CPT. Is the genuine T-odd CPT-invariant component of the CPasymmetry coincident with the T asymmetry? Contrary to CP, matter effects in uniformmatter cannot induce by themselves a non-vanishing T asymmetry; however, the questionof the title remained open. We demonstrate that, in the presence of genuine CP violation,there is a new non-vanishing CP-even, and so CPT-odd, component in the T asymmetryin matter, which is of odd-parity in both the phase δ of the flavor mixing and the matterparameter a . The two disentangled components, genuine A T;CP αβ and fake A T;CPT αβ , couldbe experimentally separated by the measurement of the two T asymmetries in matter( ν α ↔ ν β ) and (¯ ν α ↔ ¯ ν β ). For the ( ν µ ↔ ν e ) transitions, the energy dependence of thenew A T;CPT µe component is like the matter-induced term A CP;CPT µe of the CP asymmetrywhich is odd under a change of the neutrino mass hierarchy. We have thus completed thephysics involved in all observable asymmetries in matter by means of their disentanglementinto the three independent components, genuine A CP;T αβ and fake A CP;CPT αβ and A T;CPT αβ . a r X i v : . [ h e p - ph ] F e b ontents Neutrino oscillations from a terrestrial accelerator source take place through their propaga-tion in matter of the Earth mantle. There are matter effects originated in the interferencebetween the forward scattering amplitude of electron neutrinos with matter electrons andthe free propagation. Matter being CP —and CPT— asymmetric leads to well knownfake contributions to the CP asymmetry A CP αβ between the α → β flavor transition prob-abilities for neutrinos and antineutrinos. Recently, the disentanglement of genuine andmatter-induced CP violation has been solved by means of a theorem [1] able to separatethe experimental CP asymmetry into two components with well defined behavior under thediscrete symmetries: a genuine term A CP;T αβ odd under T and CPT-invariant, and a faketerm A CP;CPT αβ odd under CPT and T-invariant. In addition, their peculiar different energydistributions at a fixed baseline provide signatures [2] for their separation in an actualexperiment. Their definite different parities under the baseline, the matter potential andthe genuine CPV phase allow the use of guiding simple and precise enough expressions interms of vacuum parameters.A question immediately arises: are these A CP;T αβ and A CP;CPT αβ components coincidentwith the T and CPT asymmetries, respectively? As the genuine T-odd A CP;T αβ componentis CPT-invariant and thus CP-odd, a positive answer to this question would be equivalentto claim the absence of fake matter-induced terms in the T asymmetries, and hence anegative answer to the title of this paper. In the literature on T asymmetries for neutrinooscillations [3–15] one does not find definite claims on this question, even for a T-symmetricmatter between the source and the detector (and, a fortiori, for uniform matter) neither inone nor the other sense of the response. Contrary to the fake A CP;CPT αβ component of theCP asymmetry, which exists even in the absence of true CP violation, if the medium is T-symmetric it, by itself , cannot generate a T asymmetry in neutrino oscillations. However,in the presence of genuine CP violation, this last reasoning does not lead to a definiteconclusion whether the entire T asymmetry is also CP-odd and CPT-invariant. In this last– 1 –ase, the genuine A CP;T αβ component in the disentangled CP asymmetry of Ref. [1] couldbe separately measured by the T asymmetry. On the contrary, a positive answer to thetitle of this paper would mean that the medium generates an additional A T;CPT αβ componentwhich is CP-even by the combined effect of genuine and matter amplitudes. This CPT-oddcomponent would then be a fake effect even for a T-symmetric medium.We plan to solve this open question in this paper and, anticipating a possible presenceof A T;CPT αβ , prove an analogous T asymmetries Disentanglement Theorem into two com-ponents: a CP-odd CPT-invariant A T;CP αβ component which is identified with the knowngenuine A CP;T αβ and a new CPT-odd CP-invariant A T;CPT αβ component. As a lemma for thefake CPT asymmetries, written as a combination of CP and T asymmetries, they wouldbe disentangled into the two components A CPT;CP αβ = A CP;CPT αβ of the CP asymmetry and A CPT;T αβ = A T;CPT αβ of the T asymmetry. No experimental asymmetry would be genuine byitself: only the A CP;T αβ = A T;CP αβ component is so, whereas A CP;CPT αβ = A CPT;CP αβ is purely amedium effect and A T;CPT αβ = A CPT;T αβ combines genuine and matter-induced amplitudes.The paper is organized as follows. In Section 2 we prove the second DisentanglementTheorem, this time for the T asymmetries in matter, and identify the relevant rephasing-invariant mixings and oscillation factors appearing separately in each of the two components A T;CPT αβ and A T;CP αβ , which are both odd functions of the baseline L . For the three-familyHamiltonian in terms of vacuum parameters and the matter potential, we establish inSection 3 the definite parities of the two components under the phase δ of the flavor mixingmatrix and the matter parameter a . We calculate their peculiar energy dependencies inthe energy region between the two MSW [16, 17] resonances, both numerically and usingthe analytic perturbation expansion performed in Ref. [2], for the golden ν µ → ν e channel.Section 4 discusses the specific combinations which appear in CPT asymmetries, as a lemmaof the two Disentanglement Theorems. The main conclusions are given in Section 5. In this Section we study the time evolution of neutrinos propagating in matter in terms oftheir effective masses ˜ m and mixings ˜ U . Notice that both ingredients are energy dependent.The flavor oscillation probability is P ( ν α → ν β ) = δ αβ − (cid:88) j
00 0 m U † + a = 12 E ˜ U ˜ M ˜ U † , (3.1)where the only difference between neutrinos and antineutrinos comes from the sign of (i)the CPV phase δ in the unitary PMNS matrix U and (ii) the matter parameter a = 2 V E ,with V the matter potential.In this case, the functional form of the T asymmetry is necessarily A T αβ ≡ f ( a ) sin δ ,which leads to the antineutrino asymmetry ¯ A T αβ = − f ( − a ) sin δ . A power expansion of f ( a ) around the vacuum a = 0 clearly shows that all even powers of a lead to CPT-invariant CP-odd terms, whereas all odd powers lead to CPT-odd CP-invariant terms. Asa consequence, we discover that, even for a uniform (T-symmetric) matter, the answer tothe title of this paper is positive through terms of odd-parity in the matter parameter a .An exact numerical result of the energy dependencies of the two components A T;CP µe and A T;CPT µe is given in Fig. 1 for the two baselines L = 295 , A CP;CPT µe component of Ref. [2]. Their direct observabilitywould require terrestrial sources of electron neutrinos with equal energies to those for muonneutrino sources, not accessible at present. However, these A T µe and ¯ A T µe asymmetrieswould be natural for a neutrino factory [25] and they are ingredients for an analysis usingatmospheric neutrinos.The vacuum parameters used are those obtained by the global fit in Ref. [26] θ = 34 . ◦ , θ = 8 . ◦ , θ = 47 . ◦ , ∆ m = 7 . × − eV , ∆ m = 2 . × − eV , (3.2)assuming Normal Hierarchy. We consider a change of Hierarchy for the same physicalconfiguration, which corresponds to compute the Inverted Hierarchy case with the condition∆ m | IH = − ∆ m | NH , keeping the same values for all other quantities. In this way, one isconsistent in isolating the observable effects due to the change in the ordering of the sameneutrino mass states. – 4 – = 295 kmNH CP;CPTT;CPTT;CP L = 1300 kmNH CP;CPTT;CPTT;CP L = 295 kmIH CP;CPTT;CPTT;CP L = 1300 kmIH CP;CPTT;CPTT;CP
Figure 1 : Comparison of the CP-odd CPT-invariant (blue) and the CP-invariant CPT-odd (red) components of the ν µ ↔ ν e T asymmetry, together with the matter-inducedCPT-odd T-invariant (green) component of the CP asymmetry. Each label near the linesindicates the two symmetries under which the corresponding asymmetry component isodd. Lines show A T;CP µe / sin δ , A T;CPT µe / sin δ and the maximal-CPV (cos δ = 0) value of A CP;CPT µe , from both the exact numerical calculation (dashed) and analytical Eqs. (3.9a,3.9b, 3.4a) (solid). The shaded regions show all possible values of the components takinginto account the unknown value of sin δ . Normal Hierarchy assumed in the upper figures,Inverted Hierarchy in the lower ones; as seen, A T;CP µe is Hierarchy-independent, whereas thematter-induced A T;CPT µe is Hierarchy-odd and A CP;CPT µe is almost [2] Hierarchy-odd. Thematter-induced CP-even A T;CPT µe is the same in the antineutrino asymmetry case, whereasthe other two components change sign. Notice the different scales for the two baselines.As expected, the Figure shows that matter effects become relevant at longer baselines.Since matter effects cannot generate a T asymmetry if there is no genuine CP violation,both T-odd components vanish when sin δ = 0. We also find that, as happens in the CPasymmetry, there are special configurations at longer baselines where the matter-inducedterm vanishes, for any value of δ , near a maximal value of the genuine one.In order to characterize these points analytically, we consider a perturbative expansion– 5 –n the small quantities ∆ m ∆ m ∼ . , | U e | ∼ . , (3.3) (cid:18) ∆ m a (cid:19) ∼ . E/ GeV) , (cid:18) a ∆ m (cid:19) ∼ . E/ GeV) , ∆ a ∼ .
084 ( L/ , where ∆ a ≡ aL/ E is the energy-independent matter-induced phase shift. Such an expan-sion is valid in the energy region between the two MSW resonances, ∆ m (cid:28) a (cid:28) (cid:12)(cid:12) ∆ m (cid:12)(cid:12) .The relevant ratios of the second line of Eq. (3.3) are quadratic due to the definite a -parityof each component of A T µe , so even expanding in them ensures that errors are well kept atthe few percent level.As thoroughly discussed in Ref. [2], this treatment leads to the disentanglement of theCP asymmetry into A CP µe = A CP;T µe + A CP;CPT µe by A CP;CPT µe = 16 ∆ a (cid:20) sin ∆ ∆ − cos ∆ (cid:21) ( S sin ∆ + J r cos δ ∆ cos ∆ ) + O (∆ a ) , (3.4a) A CP;T µe = − J r sin δ ∆ sin ∆ + O (∆ a ) , (3.4b)where S ≡ c s s , J r ≡ c c c s s s , ∆ a ≡ aL E ∝ L and ∆ ij ≡ ∆ m ij L E ∝ L/E .When the analogous treatment is applied to the T asymmetries, we find that the relevantmixings are Im ˜ J eµ = − Im ˜ J eµ = Im ˜ J eµ = (cid:18) ∆ m a + ∆ m ∆ m (cid:19) J r sin δ , (3.5)and the same for antineutrinos changing the sign of a and δ . This result is our perturbativeapproximation of the vacuum-matter invariance identity [8]∆ ˜ m ∆ ˜ m ∆ ˜ m Im ˜ J ijαβ = ∆ m ∆ m ∆ m Im J ijαβ . (3.6)Together with the mass differences∆ ˜ m ≈ a , ∆ ˜ m ≈ ∆ m , ∆ ˜ m ≈ ∆ m − a , ∆ ˜¯ m ≈ | a | , ∆ ˜¯ m ≈ ∆ m + | a | , ∆ ˜¯ m ≈ ∆ m , (3.7)we discover that, besides the global sin δ factor and the different a -parities, the samecombinations of vacuum parameters as for the CP asymmetry appear. We obtain A T µe = − J r sin δ ∆ sin ∆ (cid:20) sin ∆ + ∆ a (cid:18) sin ∆ ∆ − cos ∆ (cid:19)(cid:21) + O (∆ a ) , (3.8)which is disentangled into A T;CP µe = − J r sin δ ∆ sin ∆ + O (∆ a ) , (3.9a) A T;CPT µe = − a J r sin δ ∆ sin ∆ (cid:20) sin ∆ ∆ − cos ∆ (cid:21) + O (∆ a ) . (3.9b)– 6 –s expected, the leading term in the genuine component in the T asymmetry A T µe is its vac-uum limit A T;CP µe . It equals the genuine component A CP;T µe (3.4b) of the CP asymmetry, asrequired by the discussion in the previous Section. The leading term of the matter-inducedcomponent A T;CPT µe of the T asymmetry explicitly shows the combined proportionality to a sin δ and, consequently, it is CP-even. Whereas A CP µe contains either sin δ or a in itscomponents, A T;CPT µe needs both sin δ and a .The energy dependence of A T;CPT µe present in the oscillating phases is exactly the sameas the odd term under the change of hierarchy in A CP;CPT µe . Therefore, it vanishes in thesame set of δ -independent zeros as the matter-induced term A CP;CPT µe of the CP asymmetrydiscussed in Ref. [2]. These first-rank zeros correspond to the solutions oftan ∆ = ∆ , (3.10)whose highest-energy solution happens at E = 0 .
92 GeV L (cid:12)(cid:12) ∆ m (cid:12)(cid:12) . × − eV (3.11)near the second oscillation maximum ∆ = 3 π/ δ — genuine A T;CP µe at tan ∆ = − . (3.12)These analytical approximations are compared with the exact numerical results inFig. 1 for the two chosen baselines of T2HK and DUNE. We observe that the agreementis excellent. As is now understood, the T asymmetry is purely genuine at the same magicenergy (3.11) near the second oscillation maximum at which the experimental CP asym-metry is independent of matter effects. Even though a non-vanishing T asymmetry isa proof of genuine CP violation even in matter, this magic configuration provides a cleanmatter-independent point where sin δ could be measured without hierarchy ambiguities. Asshown in Ref. [2], a modest energy resolution would allow the separation of the hierarchy-independent genuine A T;CP µe component. A decomposition of the CPT asymmetry into T-odd CP-invariant and CP-odd T-invariantcomponents, A CPT αβ ≡ P αβ − ¯ P βα = A CPT;CP αβ + A CPT;T αβ , is constrained by the CP and Tasymmetries decomposition and the asymmetry sum rule (2.6), A CPT;CP αβ + A CPT;T αβ = A CP;CPT αβ + A CP;T αβ + A T;CPT αβ − A T;CP αβ . (4.1)This expression implies that, due to the separate invariance under the three discrete symme-tries, the following three equalities must be satisfied: the T-invariant A CPT;CP αβ = A CP;CPT αβ ,the CP-invariant A CPT;T αβ = A T;CPT αβ , and the CPT-invariant A CP;T αβ = A T;CP αβ .– 7 –herefore, the decomposition of all asymmetries, A T αβ = A T;CPT αβ + A T;CP αβ , A CPT αβ = A CPT;T αβ + A CPT;CP αβ , A CP αβ = A CP;T αβ + A CP;CPT αβ , ¯ A T αβ = A T;CPT αβ − A T;CP αβ , ¯ A CPT αβ = A CPT;T αβ − A CPT;CP αβ , (4.2)is written in terms of only three (per flavor channel) independent components, since the twosuperindices of all components commute. These three independent components correspondto (i) the genuine component of the CP asymmetry A CP;T αβ (2.2a), (ii) the matter-inducedcomponent in the CP asymmetry A CP;CPT αβ (2.2b) and (iii) the component A T;CPT αβ (2.5b),induced by matter in presence of genuine CP violation, breaking the vacuum identity ¯ A T αβ = −A T αβ . Notice that the interchange of flavor indices corresponds to a T transformation,so all T-odd components will be odd under α ↔ β , whereas T-invariant components willremain unchanged.A calculation of the CPT asymmetry following the considerations of the previousSection results in A CPT µe = 16∆ a (cid:20) sin ∆ ∆ − cos ∆ (cid:21) [∆ J r cos( δ + ∆ ) + S sin ∆ ] + O (∆ a ) , (4.3)which is consistent with the decomposition A CPT αβ = A CPT;CP αβ + A CPT;T αβ and the two com-ponents A CPT;CP µe = A CP;CPT µe (3.4a) and A CPT;T µe = A T;CPT µe (3.9b), the first one even inboth L and sin δ , the second one odd.Notice that the whole CPT asymmetry is odd in a , as expected, and vanishes at thepoints tan ∆ = ∆ . This vanishing condition is consistent with previous analyticalstudies on the CPT asymmetry [27], where the ( L, E ) values in which A CPT µe = 0 weresuggested as a clean configuration on which a non-vanishing CPT asymmetry is a proof ofintrinsic CPT violation. It is also consistent with the exact calculation [28] of the matter-induced CPT and CP asymmetry in the limit ∆ m = 0. We extend this result to thematter-induced components of the CP and T asymmetries: the disentanglement (4.2) ofall three asymmetries shows that A CPT µe vanishes at tan ∆ = ∆ because both CPT-oddcomponents separately vanish. This ensures that A CP µe and A T µe probe genuine CP and Tviolation, respectively, free from matter effects. This last statement cannot be ensured onlyby finding a vanishing CPT asymmetry, since a configuration in which the two CPT-oddcomponents cancel each other in A CPT µe = A CPT;CP µe + A CPT;T µe would still present fake termsin the CP and T asymmetries.This discussion clearly shows that, for neutrinos traveling through the Earth mantleat energies between the two MSW resonances, ∆ m (cid:28) a (cid:28) (cid:12)(cid:12) ∆ m (cid:12)(cid:12) , there are magic L/E values that satisfy tan ∆ = ∆ . At these points, of which L/E = 1420 km/GeV is thehighest-energy one, all three asymmetries probe genuine violation of its associated discretesymmetry.
This paper represents the culmination of the study of the separate physics involved in thegenuine and fake matter-induced effects for all observable CP, T and CPT asymmetries– 8 –n neutrino oscillations. The starting point was the proof of Disentanglement Theoremsallowing each asymmetry to be written as the sum of two components with definite trans-formation properties under the other two symmetries A CP αβ = A CP;T αβ + A CP;CPT αβ , A T αβ = A T;CP αβ + A T;CPT αβ , A CPT αβ = A CPT;CP αβ + A CPT;T αβ . (5.1)Due to the proved symmetric character of superindices in the components, only three ofthem are independent. In going from neutrino to antineutrino asymmetries, the compo-nents with CP superindex change sign, whereas the others remain invariant. In addition,T-invariant components are symmetric under exchange of flavor indices, whereas T-oddones change sign under α ↔ β . The genuine CP, and T, asymmetry is given by the A CP;T αβ component, which is CPT-invariant as in vacuum; the other two independent components, A CP;CPT αβ and A T;CPT αβ , are induced by matter in the neutrino propagation and thus theyare odd in the matter potential. However, there is a very interesting distinction betweenthese two fake components: whereas A CP;CPT αβ can be induced by matter alone without anyfundamental CP violation for neutrinos, this paper has demonstrated a positive answer tothe title question on the existence of the matter-induced A T;CPT αβ component, even for T-symmetric matter. The possible surprise should disappear when realizing that the quantumlogic is not disjunctive —either free or matter— since interference terms are also relevant.In fact, the analytic solution (3.9b) for the fake A T;CPT µe component is proportional to both a and sin δ .Using the new T Asymmetry Disentanglement (2.5), the two observable T asymmetriesfor neutrinos and antineutrinos allow an experimental separation of genuine and fake effectsunder the same conditions of baseline and energy, A T;CP αβ = 12 (cid:0) A T αβ − ¯ A T αβ (cid:1) , A T;CPT αβ = 12 (cid:0) A T αβ + ¯ A T αβ (cid:1) , (5.2)the first component being odd in L and sin δ , even in a ; the second component beingodd in L , sin δ and a . The genuine component A T;CP µe is blind to the neutrino mass hier-archy and the fake component A T;CPT µe is odd under the change of hierarchy. The otherindependent fake component A CP;CPT µe can then be obtained from the CP Asymmetry Dis-entanglement (2.2) —or, equivalently, from the CPT asymmetry sum rule (2.6)— as A CP;CPT αβ = A CP αβ − A CP;T αβ , (5.3)being even in L and sin δ and odd in a .It is now clear that the CP asymmetry in matter cannot be written as the T asymmetryin matter plus a CP-odd fake term, since matter induces a CP-invariant term in the Tasymmetry. Our matter-induced CP-odd component (5.3) quantifies the fake effects in theCP asymmetry with the appropriate behavior under T and CPT.A remarkable result from our precise enough analytic calculation of the three com-ponents in terms of the vacuum parameters is that the two fake components A CP;CPT µe and A T;CPT µe vanish, with first-rank zeros, at the same set of magic L/E values satisfyingtan ∆ = ∆ , independent of the hierarchy, δ and a : only (cid:12)(cid:12) ∆ m (cid:12)(cid:12) matters. These– 9 –olutions appear near the values in which the genuine component A CP;T µe is maximal,tan ∆ = − . The joint fulfillment of these two equations occurs around the solu-tions for the even oscillation maxima in the transition probabilities, sin ∆ = 1.All in all, the conditions for a direct evidence of genuine CP and T violation for neutrinooscillations in matter by means of the measurement of an observable component odd underthe symmetry are now met. One may proceed by (a) measuring the genuine CP or Tasymmetry at the magic energies where matter effects vanish, or (b) follow the programoutlined in Eq. (5.2) at any chosen ( L ; E ) configuration to extract it. We emphasizethat this genuine component is independent of the hierarchy. So global fits in which theresulting δ phase is dependent on the assumed hierarchy say that true CP violating terms,proportional to sin δ and even in a , play a minor role in the intervening observables. On thecontrary, the two fake components have clear sign information on the hierarchy, being odd in a and containing δ -independent, cos δ and sin δ terms. The relative signs of genuine A T;CP µe and fake A T;CPT µe components are equal for normal hierarchy, the last one being smallerin magnitude at all energies and baselines. This is in contrast with the fake A CP;CPT µe component, which is larger in magnitude than the genuine A CP;T µe at large energies for longbaselines. Acknowledgments
This research has been supported by MINECO Project FPA 2017-84543-P, Generalitat Va-lenciana Project GV PROMETEO 2017-033 and Severo Ochoa Excellence Centre ProjectSEV 2014-0398. A.S. acknowledges the MECD support through the FPU14/04678 grant,and he is indebted to the hospitality of the Durham IPPP, where this work was finished.
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