SLAC-PUB-9116 hep-ph/0201105 February 1, 2008 Leptonic Unitarity Triangle and CP-violation Y. Farzan1,2 and A. Yu. Smirnov3,4 1 Scuola Internazionale Superiore di Studi Avanzati, SISSA, I-34014, Trieste, Italy 2 0 2 Stanford Linear Accelerator Center, Stanford University, Menlo Park, California 94025 0 2 3 The Abdus Salam International Centre for Theoretical Physics, I-34100 Trieste, Italy n 4 Institute for Nuclear Research, RAS, Moscow, Russia a J 8 2 Abstract 2 The area of the unitarity triangle is a measure of CP-violation. We introduce the v 5 leptonicunitaritytrianglesandstudytheirproperties. Weconsiderthepossibilityofre- 0 constructingtheunitaritytriangleinfutureoscillationandnon-oscillationexperiments. 1 A set of measurements is suggested which will, in principle, allow us to measure all 1 0 sidesof thetriangle, andconsequently toestablish CP-violation. For different values of 2 the CP-violating phase, δ , the required accuracy of measurements is estimated. The 0 D h/ key elements of the method include determination of |Ue3| and studies of the νµ −νµ p survival probability in oscillations driven by the solar mass splitting ∆m2sun. We sug- - gestadditionalastrophysicalmeasurementswhichmayhelptoreconstructthetriangle. p e The method of the unitarity triangle is complementary to the direct measurements of h CP-asymmetry. It requires mainly studies of the survival probabilities and processes : v where oscillations are averaged or the coherence of the state is lost. i X r a 1 Introduction Measurement of CP-violation in leptonic sector is one of the main challenges in particle physics, astrophysics and cosmology. For three neutrinos (similarly to the quark sector [1]) there is a unique complex phase in the lepton mixing matrix, δ , which produces observable CP-violating effects [2]. (If D neutrinos are Majorana particles, two additional CP-violating phases exist. These phases, the so-called Majorana phases, do not appear in the oscillation patterns.) The phase δ D leads to CP-asymmetry [3], P(ν ν ) = P(ν¯ ν¯ ), as well as T-asymmetry [4], P(ν α β α β α → 6 → → ν ) = P(ν ν ), of the oscillation probabilities (see also [5] and references therein). β β α 6 → 1 Measurements of the CP- and T- asymmetries provide a direct method of establishing CP-violation. There are a number of studies of experimental possibilities to measure the asymmetries. It was realized that in the 3ν-schemes of neutrino mass and mixing which explain the atmospheric and solar neutrino data, the CP-violation and T-violation effects are small and it will be difficult to detect them [6]. The smallness is due to small values of U (restricted by CHOOZ result) and ∆m2 (responsible for the solar neutrino conversion). e3 sun Still, the effect can be seen in the new generation of the long baseline (LBL) experiments provided that the LMA-MSW is the solution of the solar neutrino problem and that U > e3 0.05 [7, 8, 9]. Two types of LBL experiments sensitive to δ are under consideration [10]: the exper- D iments with superbeams [8, 9] and neutrino beams from muon storage rings (the neutrino factories) [11]. Analysis shows [8, 12] that for U > 0.05 and ∆m2 = 5 10−5 eV2, | e3| sun × neutrino factories can discriminate between δ = 0 and δ = π at the 3σ level [13] while D D 2 according to [8] superbeams are able to distinguish at the 3σ level δ = 0 from δ = π. In D D 9 these experiments, the sensitivity to δ decreases linearly with ∆m2 . So, the present un- D sun certainty in ∆m2 results in an order of magnitude uncertainty in evaluation of sensitivity sun to δ in the future neutrino factories and superbeam experiments. If ∆m2 is smaller than D sun 2 10−5 eV2, the direct methods will not be sensitive to δ [7]. Moreover, neutrino factories D × and superbeams are very expensive and technically difficult, interpretation of their results can be rather complicated and ambiguous. In view of these difficulties, we need to explore any alternative way to search for CP-violation. Notice that apart from the asymmetries, the phase δ can be determined also from D measurements of CP-conserving quantities, the oscillation probabilities themselves, which depend on δ [14]. D The alternative method to establish CP-violation is to measure the area of unitarity triangle. This method iswell elaboratedinthe quarksector. Indeed, thearea ofthe unitarity triangle, S, isrelatedtotheJarlskog invariant, J , which isaparameterizationindependent CP measure of CP-violation, as 1 S = J . (1) CP 2 So, to establish CP-violation it is sufficient to show that the longest side of the triangle, is smaller than the sum of the other two. The problem is to measure lengths of the sides of the triangle. As we will see, the method of the unitarity triangle differs from measurements of asymmetries and may have certain advantages from the experimental point of view. 2 Previously, somegeneralpropertiesoftheunitaritytrianglesforleptonicsector(geometric features, test of unitarity) have been discused in [15], [16], [17]. In thispaper we will consider the possibility to reconstruct the leptonic unitaritytriangle. In sect. 2, we introduce the leptonic unitarity triangles and study their properties. We estimate the accuracy with which the sides of the triangle should be measured to establish CP-violation. In sect. 3, we describe a set of oscillation measurements which would in principle allow us to reconstruct the triangle. Additional astrophysical measurements which would allow us to realize the method are suggested in sect. 4. Discussions and conclusions are given in sect. 5. 2 Leptonic unitarity triangles In the three-neutrino schemes the flavor neutrino states, ν (ν ,ν ,ν ), and the mass f e µ τ ≡ eigenstates ν (ν ,ν ,ν ), are related by the unitary MNS (Maki-Nakagawa-Sakata mass 1 2 3 ≡ [18]) matrix ∗: U U U e1 e2 e3   U = U U U . (2) MNS µ1 µ2 µ3       U U U   τ1 τ2 τ3   The unitarity implies U U∗ +U U∗ +U U∗ = 0, e1 µ1 e2 µ2 e3 µ3 U U∗ +U U∗ +U U∗ = 0, (3) e1 τ1 e2 τ2 e3 τ3 U U∗ +U U∗ +U U∗ = 0. τ1 µ1 τ2 µ2 τ3 µ3 In the complex plane, each term from the sums in (3) determines a vector. So, the Eqs. (3) correspond to three unitarity triangles. The CP-violating phase, δ , vanishes if and only D if phases of all elements of matrix (2) are factorizable: U = ei(σα+γi) U . In this case αi αi | | U U∗ = ei(σα−σβ) U U , and therefore the unitarity triangles shrink to segments. αi βi | αi|| βi| To construct the unitarity triangle, one needs to measure the absolute values of the elements of two rows (or equivalently two columns) in the mixing matrix. The area of ∗Themixingofthreeflavorstates(twolightneutrinosandheavyneutralleptonfromthethirdgeneration) have been discussed in [19]. 3 the triangle is given by the Jarlskog invariant, J Eq. (1). The area is non-zero only if CP sinδ = 0. D 6 2.1 e µ triangle; properties − We will consider the triangle formed by the e- and µ-rows of the matrix (2) (see Eq. (3-a)). (Up to now, there is no direct information about the elements of the third row. Moreover, even in future, both creation of intense ν beams and detection of ν seem to be difficult.) τ τ To reconstruct the e µ triangle three quantities should be determined independently: − U U∗ , U U∗ , U U∗ . (4) | e1 µ1| | e2 µ2| | e3 µ3| The form of the triangle depends on the yet unknown value of U and on the specific e3 | | solution of the solar neutrino problem. In what follows, we will consider mainly the LMA- MSW solution which provides the best fit for the solar neutrino data. In Figs. 1 and 2, we show examples of the unitarity triangles for different values of U e3 and δ . In these figures we have normalized the sides of the triangles in such a way that the D length of the first side equals one: U U∗ U U∗ x = 1, y = | e2 µ2| and z = | e3 µ3| . (5) U U∗ U U∗ | e1 µ1| | e1 µ1| WeusethestandardparameterizationoftheMNSmixing matrix[20]intermsoftherotation angles θ , θ , θ and the phase δ . We take values of θ and θ from the regions allowed 12 13 23 D 12 23 by the solar and atmospheric neutrino data. In Fig. 1 we present the triangles which correspond to sin22θ = 0.12 (the upper bound 13 from the CHOOZ experiment for ∆m2 = 3 10−3 eV2). The arcs show 10% uncertainty atm × in measurements of the sides y and z. From Fig. 1, one can conclude that for maximal CP-violation, δ = 90◦, the existence of CP-violation can be established at the 3σ-level or D even better if the sides of the triangle are measured with 10% accuracy. For δ = 60◦, the D confidence level is approximately 2σ. No statement can be made for δ 45◦ unless the D ≤ accuracy of measurements of the sides will be better. These estimates should be considered as tentative ones. In order to make precise statements one needs to perform careful analysis taking into account, in particular, correlations of the errors. The triangles shrink for smaller values of sin22θ (Fig. 2). According to Fig. 2 which 13 corresponds to sin22θ = 0.03, for δ = 90◦ CP-violation might be established at 2σ 13 D ∼ level. No conclusion can be made for δ < 70◦. D 4 The form of the triangle is also sensitive to variations of the angle θ within the allowed 12 LMA region. In Fig. 3, we have set θ = π and sin22θ = 0.18. As follows from this 12 4 13 figure with 10% uncertainty in determination of the sides, CP-violation can be established for δ = 90o and δ = 60◦. D D Note that y O(1) and z is the smallest side, although its length may not be much ∼ smaller than others. So, CP-violation implies that U U∗ < U U∗ + U U∗ . (6) | e1 µ1| | e2 µ2| | e3 µ3| Similar triangles can be obtained for the LOW and VAC solutions. The unitarity triangle is different in the case of the SMA-MSW solution. Taking tan2θ = tan2θ = 0.0016, 12 sun sinθ = 1/√2 and sinθ = 0.15, we find y = 0.25, z = 0.96. Now y is the smallest side, 23 13 are the two other sides have comparable lengths. Note that in spite of small mixing of the electron neutrino the smallest side is not very small. Even in this case a moderate accuracy in determination of the sides would allow us to establish CP-violation. In general, to establish CP-violation, one needs to construct the triangle without using the unitarity conditions. However, if we assume that only three neutrino species take part in the mixing and that there are no other sources of CP-violation apart from the MNS- matrix, we can use some equalities which follow from unitarity. In particular, we can use the independent normalization conditions: U 2 = 1 , U 2 = 1. (7) ei µi | | | | i=1,2,3 i=1,2,3 X X In this case, the CP-violation effect can be mimicked at some level by the 4th (sterile) neutrino. To eliminate such a possibility, one should check the normalization conditions experimentally. Thus, to find the sides of the triangle we should determine moduli of four mixing matrix elements: U , U , U , U . (8) e2 µ2 e3 µ3 | | | | | | | | They immediately determine the second and third sides. The two other elements, U and e1 | | U , and consequently the first side, can be found from the normalization conditions (7). µ1 | | For the first side we have U∗ U = (1 U 2 U 2)(1 U 2 U 2). Taking into | e1 µ1| −| e2| −| e3| −| µ2| −| µ3| account this correlation in determinaqtion of the sides of the triangle one can estimate accu- racy of measurements of the elements (8) needed to establish CP-violation via the inequality (6). Let us introduce A U U + U U (1 U 2 U 2)(1 U 2 U 2) (9) e2 µ2 e3 µ3 e2 e3 µ2 µ3 ≡ | || | | || |− −| | −| | −| | −| | q 5 which is a measure of CP violation. CP is conserved if A = 0. For the most optimistic cases, where U is close to the CHOOZ bound and δ = 90◦, we find A = 0.10 0.13. e3 D − Suppose the elements U are measured with accuracies ∆ U . Assuming that the αi αi | | | | errors ∆U are uncorrelated, we can write the error in the determination of A as αi | | 2 dA ∆A = (∆ U )2, (10) v αi uuα=eX,µ,i=2,3 d|Uαi|! | | u t where dA U U dA U U e2 µ1 e3 µ1 = U + | || |, = U + | || |, (11) µ2 µ3 d U | | U d U | | U e2 e1 e3 e1 | | | | | | | | dA U U dA U U e1 µ2 e1 µ3 = U + | || |, = U + | || |. (12) e2 e3 d U | | U d U | | U µ2 µ1 µ3 µ1 | | | | | | | | As an example, let us choose the oscillation parameters used in Fig. 1 and δ = 90◦. Then D from Eqs. (11, 12) we find dA/d U = 0.82, dA/d U = 0.77, dA/d U = 2.0 and e2 e3 µ2 | | | | | | dA/d U = 1.9. Note that for muonic elements the derivatives are larger by factor of 2. µ3 | | This is a consequence of the appearance of the relatively small element U in denominators µ1 | | of (12). So, the muonic elements should be measured with the accuracy two times better than the electronic elements. For our example we find from Eq. (10) that ∆A < 0.065, which would allow to es- tablish deviation of A from zero at the 2σ level, if ∆ U = ∆ U < 0.03 and ∆ U = e2 e3 µ2 | | | | | | ∆ U < 0.02. This, in turn, requires the following upper bounds for relative accuracies µ3 | | of measurements of the matrix elements: 6% for U , 17% for U , and 3% for U and e2 e3 µ2 | | | | | | U . Since µ3 | | 2 ∆ U U ∆ U µ2 µ1 µ1 | | = | | | | Uµ2 Uµ2 ! Uµ1 | | | | | | and U 0.5U , the required 3% accuracy in U corresponds to 12 % uncertainty in U . µ1 µ2 µ2 µ1 ≃ | | If there are correlations between ∆ U , the situation may become better. So, the above αi | | estimations can be considered as the conservative ones. 2.2 Present status At present, we cannot reconstruct the triangle: knowledge of the mixing matrix is limited to the elements of the first row (from the solar neutrino data and CHOOZ/Palo Verde experiments) andthethirdcolumn(fromtheatmospheric neutrino data). To reconstruct the triangle one needs to know at least one element fromthe block U , where β = µ,τ, i = 1,2. βi 6 That is, one should measure the distribution of the ν (or/and ν ) in the mass eigenstates µ τ with split by the solar ∆m2. Using the unitarity condition we can estimate only the ranges for these matrix elements. Clearly, present data are consistent with any value of the CP- violating phase and, in particular, with zero value which corresponds to degenerate triangles. Let us summarize our present knowledge of the relevant matrix elements. 1). The values of the mixing parameters U and U can be obtained from studies of e1 e2 | | | | solar neutrinos. Neglecting small effect due to U , for the LMA-MSW solution we obtain e3 U e2 | | = tanθ = 0.39 0.77, (95% C.L.) (13) sun U | | − e1 | | and then using the normalization condition: U [1+tan2θ ]−1/2 = 0.79 0.93, (95% C.L.). (14) e1 sun | | ∼ − 2). The absolute value of U is restricted from above by the CHOOZ[21] and Palo e3 | | Verde [22] experiments. The 2ν analysis of the CHOOZ data [21], gives for the best fit value of ∆m2 atm U < 0.20, (90% C.L.). (15) e3 | | For lower values of ∆m2 , the bound is weaker: U < 0.22. atm | e3| 3). The admixture ofthe muonneutrino inthethird mass eigenstate, U , is determined µ3 | | bytheatmosphericneutrino data. Again, neglecting effects duetonon-zeroU , wecanwrite e3 4 U 2(1 U 2) = sin22θ , (16) µ3 µ3 atm | | −| | where sin22θ can be extracted, e.g., from analysis of the zenith angle distribution of the atm µ-like events in terms of the ν ν oscillations. Using the Super-Kamiokande data, we find µ τ − U = 0.707+0.12, (90%C.L.). (17) | µ3| −0.14 4). At present, there is no direct information about U and U . To measure these µ1 µ2 | | | | elements, one needs to study the oscillations of muon neutrinos driven by ∆m2 . The sun normalization condition allows us to impose a bound on a combination of these elements: U 2 + U 2 = 1 U 2 = (0.33 0.67). (18) µ1 µ2 µ3 | | | | −| | − So, to determine U and U separately we need to measure a combination of these µ1 µ2 | | | | elements which differs from the normalization condition (18). 7 3 Reconstructing the unitarity triangle Let us consider the possibility to determine the triangle in the forthcoming and future oscil- lation experiments. We suggest a set of oscillation measurements with certain configurations (base-lines, neutrino energies and features of detection) which will allow us to measure the moduli of the relevant matrix elements (see Eqs. (4, 5)). In general, for 3ν-system the oscillation probabilities depend not only on moduli of the mixing matrixelements, (8) we areinterested in, but also onother mixing parameters includ- ing the unknown relative phases of the mixing matrix elements, δ . Therefore, the problem x is to select configurations of oscillation measurements for which the dominant effect is deter- mined by relevant moduli and corrections which depend on unknown elements and phases are negligible or sufficiently small. The hierarchy of mass splittings: ∆m2 ∆m2 helps to solve the problem. We use atm ≫ sun ∆m2 ǫ sun 0.02 (19) ≡ ∆m2 ∼ atm as an expansion parameter, where the estimation corresponds to the best fit values of the mass squired differences. Another small parameter in the problem is U . e3 | | In what follows, we suggest a set of measurements for which the oscillation probabilities depend mainly on the relevant moduli: P = P ( U , U )+∆P (δ ), α,β = e,µ , (20) αβ αβ ei µi αβ x | | | | where ∆P P. We estimate corrections, ∆P (δ ), due to unknown mixing elements and αβ x ≪ phases. It is convenient to study the dynamics of oscillations in the basis of states obtained through rotation by the atmospheric mixing angle: (ν ,ν′,ν′), where e µ τ 1 1 ν′ = (U ν U ν ), ν′ = (U∗ ν +U∗ ν ). (21) µ 1 U 2 τ3 µ − µ3 τ τ 1 U 2 τ3 τ µ3 µ e3 e3 −| | −| | q q Projections of these states onto the mass eigenstates equal U∗ U U∗ ν ν = U∗ , ν′ ν = e2 , ν′ ν = e1 e3 (22) h e| 1i e1 h µ| 1i − 1 U 2 h τ| 1i − 1 U 2 e3 e3 −| | −| | q q and ν ν = U∗ , ν′ ν = 0, ν′ ν = 1 U 2. (23) h e| 3i e3 h µ| 3i h τ| 3i −| e3| q 8 Note that in the limit U = 0, the state ν′ coincides with mass eigenstate ν , whereas e3 τ 3 ν′ = sinθ ν +cosθ ν . µ − 12 1 12 2 In matter, the system of three neutrinos (ν ,ν′,ν′) has two resonances associated with e µ τ the two different ∆m2. The corresponding resonance energies for the typical density in the mantle of the Earth, are ER = 7 GeV, ER = 0.15 GeV. (24) 23 12 These energies determine the typical energy scales of the problem as well as the energies of possible experiments. Also there are two length scales in the problem which correspond to the oscillation lengths: 4πE E 4πE E l = 5 104 km , l = 103 km . (25) 12 ≡ ∆m2 · GeV 23 ≡ ∆m2 GeV sun (cid:18) (cid:19) atm (cid:18) (cid:19) These numbers have been obtained for the best fit values of the mass squared differences. Letusconsiderpossibilities todeterminethemoduliofrelevantelements ofmixingmatrix (8) in turn. 3.1 U U e∗3 µ3 | | In principle, this product can bedirectly measured in studies of the ν ν oscillations driven µ e − by ∆m2 . Let us consider a relatively short baseline experiment in vacuum. The transition atm probability can be written as ∆m2 L P = 4 U∗ U 2sin2 atm +∆P , (26) µe | e3 µ3| 4E µe where ∆P is the correction due to existence of the ∆m2 splitting : ∆P 0 when µe sun µe → ∆m2 0. Thus, if the original flux is composed of pure ν (or pure ν ), detecting the sun → µ e appearance of ν (or ν ), one can measure immediately U∗ U provided that ∆P is small e µ | e3 µ3| µe enough. Note that ∆P depends on mixing matrix elements U , U , (α = e,µ), both on µe α1 α2 their absolute values and on phases which are unknown. So, we cannot predict ∆P and µe the only way to proceed is to find conditions for experiment at which this value is small. An alternative method would be independent measurement of U and U . e3 µ3 | | | | Forneutrinoenergies, E > 100MeV(whichareofpracticalinterest) theoscillationlength in vacuum, l , is more than several hundred kilometers. This means that the experiment 23 should bea long-baseline one, andtherefore oscillations will occur inthe matter of theEarth. 9 In a medium with constant density† the probability can be written as P = (Um)∗Um eiΦm32 1 +(Um)∗Um eiΦm12 1 2, (27) µe e3 µ3 − e1 µ1 − (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)(cid:12) (cid:12) (cid:12) where Um arethemixing(cid:12)matrixelements inmatter andΦm istheoscill(cid:12)ationphasedifference αi ij of i and j eigenstates. − − In the vacuum limit (one may consider a hypothetical configuration of experiment where neutrino beam propagates mainly in atmosphere or in a tunnel), Um = U and Φm = Φ . αi αi i i The first term in (27) corresponds to the mode of oscillation we are interested in, and the second term is due to the ∆m2 splitting. The main correction follows from the interference sun of these two terms. For the correction we find ∆P 2ǫ U∗ U U U∗ Φ [sin(δ Φ ) sinδ ], (28) µe ≈ − e1 µ1 e3 µ3 32 x − 32 − x (cid:12) (cid:12) (cid:12) (cid:12) where δ is the unknown phase(cid:12)of the product(cid:12)of four mixing matrix elements. In derivation x of (28), we have used the smallness of the phase Φ : 12 Φ = ǫΦ , (29) 12 32 assuming that Φ = O(1) (which maximizes the effect of oscillations). Then the relative 32 correction is of the order of ∆P sin2θ µe sun ǫ . (30) P ∼ U µe e3 | | Forthebest fitvaluesofthesolar oscillationparameters (LMA-MSWsolution) andU = 0.2 e3 we get ∆P /P 0.1. That is, the product U∗ U 2 can be measured with accuracy not µe µe ∼ | e3 µ3| better than 10% for maximal possible U . Consequently, the accuracy in the determination e3 of U∗ U cannot be better than 5%. | e3 µ3| There are two possibilities to improve the accuracy: 1) The main oscillation term and the interference term have different dependences on Φ and therefore on E/L. So, in principle 32 one can disentangle these terms by studying the energy dependence of the effect. 2) The sign of the interference term can be changed varying E/L. Therefore, the correction can be suppressed by averaging over energy, especially if δ is small. x Note that for other solutions of the solar neutrino problem (LOW, SMA, VAC), ∆m2 sun is much smaller and the correction is negligible. †Forsimplicitywewillconsidermatterwithconstantdensity. Densityvariationeffectsdonotchangeour conclusions. 10