aa r X i v : . [ h e p - ph ] J a n DOI: 10.1103/PhysRevD.74.113010
CP nonconservation in the leptonic sector
Petre Dit¸˘a ∗ In this paper we use an exact method to impose unitarity on moduli of neutrino PMNS matrixrecently determined, and show how one could obtain information on CP nonconservation from alimited experimental information. One suggests a novel type of global fit by expressing all theoreticalquantities in terms of convention independent parameters: the Jarlskog invariant J and the moduli | U αi | , able to resolve the positivity problem of | U e | . In this way the fit will directly provide a valuefor J , and if it is different from zero it will prove the existence of CP violation in the availableexperimental data. If the best fit result, | U e | <
0, from M. Maltoni et al , [New J.Phys. (2004)122] is confirmed, it will imply a new physics in the leptonic sector. PACS numbers: 12.15.Hh, 12.15.-y, 14.60.Pq
I. INTRODUCTION
Neutrino experiments have shown that neutrinos havemass and oscillate, the last property suggesting a mixingof leptons similar to that of quarks [1]-[4]; for more ref-erences on experimental data see, e.g., [5]-[6]. The pres-ence of the lepton mixing opens the possibility to see aCP violation in the leptonic sector, phenomenon which isconsidered an indispensable ingredient that could gener-ate the excess of the baryon number of Universe [7], sincethe baryon number can be generated by leptogenesis [8].Excluding the LSND anomaly, see [9]-[10], all neutrinodata are explained by three flavor neutrino oscillations,and the determination of oscillation parameters is one ofthe main goals of phenomenological analyzes. The LSDNsignal for ¯ ν µ → ¯ ν e has been interpreted as a possible exs-tence of a fourth light neutrino, or as a signal of CP T violation. The current opinion is that neither explana-tion provides a good fit to all existing neutrino data, seee.g. [11]. Recently two detailed combined analyzes of allneutrino data have been published, see Refs. [5-6], thatmaterialized in upper and lower bounds on sin θ ij , thesebounds being converted in Ref. [12] into intervals for themoduli of the PMNS unitary matrix [13]. In both abovepapers the CP phase δ was not considered a free param-eter, e.g., only the particular cases cos δ = ± V = . ± .
045 0 . ± . < . . ± .
165 0 . ± .
155 0 . ± . . ± .
165 0 . ± .
15 0 . ± . , (1)where the central values are given by half the sums oflower and upper bounds entering [12], and the errors cor-respond to a three sigma level, excepting V e for whichonly an upper bound is given. Complementary infor-mation concerns the absolute value of the two neutrino ∗ Institute of Physics and Nuclear Engineering, P.O. Box MG6,Bucharest, Romania mass-squared differences [5], but we have no informationon: • the magnitude of U e -element of the leptonic mixingmatrix, or the value of the mixing angle θ ; • the existence of leptonic CP violation, or a value forthe Dirac phase δ = 0 , π entering the PMNS matrix [13]; • the order of the mass spectrum, or the sign of ∆ m ; • the magnitude of neutrino masses, etc.; see, e.g., [14]-[16].Although it is believed that only the next generationexperiments will give an answer to the first above items,[17], we consider that by using unitarity we can obtainsome information from experimental data as given in (1)for both the existence of CP violation in leptonic sector,and the parameter θ , all the more as measurement of θ is considered a first mandatory ingredient for inves-tigation of CP leptonic violation in ν µ → ν e transitionsand for the mass hierarchy determination. ConcerningCP violating phase, δ , the general opinion is that it willrequire a major experimental effort because of the intrin-sic difficulty to measure it, see e.g. Ref. [18].The current phenomenological analyzes of data aregiven in terms of mixing angles θ ij , see, e.g., [5]-[6], andphase δ which all depend on the chosen parametriza-tion of the unitary matrix one starts with. In the pa-per we propose another method for analyzing the databy expressing all measurable quantities directly relatedto oscillation amplitudes in terms of moduli, | U αi | , andJarlskog invariant J , [19]. From a theoretical point ofview this approach has the advantage that the recon-struction of a unitary matrix from knowledge of modulileads essentially to a unique solution, see, e.g., Ref. [20].From an experimental point of view the method allowsto get from nowadays limited information on neutrinophysics all moduli of the unitary matrix such that onecan predict the approximate form for the oscillation am-plitudes, for example ν τ → ν µ , which could be of greathelp for experimenters. It is easily seen that the experi-menters have measured quantities directly related to os-cillation amplitudes, which contain information on themoduli | U e | , | U e | , | U µ | , and | U τ | . The above mod-uli are correlated by unitarity to | U e || U e | = 1 − | U e | − | U e | = 1 − | U µ | − | U τ | (2)By using (2), from the first row and from the third col-umn in (1) one gets the bounds − . ≤ | U e | ≤ . , − . ≤ | U e | ≤ .
35 (3)The moduli of lower and upper bounds in the aboverelations are of the same order of magnitude, whichmeans that the possibility V e ≤ V e = − .
13; see also Figure A.2 in Ref. [5] and, respec-tively Figure 14 in Ref. [6]. Hence the phenomenologistshave to take into account this possibility, which, if con-firmed, could be interpreted as a breaking of the estab-lished rules, signaling really a new physics. A slight mod-ification of the theory can take it properly into account.After nonvanishing masses, such a surprise coming fromneutrino physics is not a priori excluded.As we will show in the paper our approach makes fulluse of unitarity property, which implies many correla-tions between all moduli of a unitary matrix. For acomplete determination of an (exact) unitary matrix theknowledge of four independent moduli is necessary, see, e.g., Ref. [20]. Taking into account that a 3 × CP violation in the leptonic sec-tor starting with the current knowledge on moduli matrix(1). We find that within the moduli intervals given by(1) there are values corridors that are compatible to theexistence of at least one unitary matrix, in fact one canconstruct a continuum of solutions, a result which couldbe interpreted as a signal of CP violation in the leptonicsector. A clear signal of CP violation will be a value J = 0 for the Jarlskog invariant, a value which can beprovided by a global fit suggested in the paper. If thenew fit will confirm the result, | U e | <
0, by Maltoni etal. , [5], this will be equivalent with the existence of newphysics in the leptonic sector.The theoretical tool that will be used in the followingis unitarity of the PMNS matrix, and the constraintson physical observables generated by this property. Wemake full use of phase invariance property to write thematrix under the form U = c c c s s − c s e iδ − c s s c c e iδ − s s s s c s s e iδ − c c s − c s e iδ − s c s c c (4)and the notation is c ij = cos θ ij and s ij = sin θ ij , where θ ij are the mixing angles with ij = 12 , ,
23, and δ isthe Dirac phase which encodes CP violation.The matrix U is assumed and built as a unitary matrix,but nobody assures that the moduli | U ij | , the angles θ ij ,or the phase δ extracted from experiments come from aunitary matrix. Usually it is assumed that irrespectiveof how the measured data are, they are compatible tothe existence of a unitary matrix. A problem which wasnot considered in physical literature is the property ofunitary matrices to be naturally embedded in a largerclass of matrices, the set of double stochastic matrices.A 3 × M is said to be double stochastic if itsentries satisfy the relations [21] m ij ≥ , P i =1 m ij = 1 , P j =1 m ij = 1 , i, j = 1 , , M and M are double stochastic, matrices from the set M = x M + (1 − x ) M , with 0 ≤ x ≤ m ij of thedouble stochastic matrix by relations m ij = | U ij | (7)One easily sees that relations (5) are satisfied because ofthe unitarity property U U † = U † U = I (8)where † denotes the hermitian conjugate matrix, and I is the 3-dimensional unit matrix. The subset of dou-ble stochastic matrices coming from unitary matrices isknown as unistochastic matrices. It is well known in themathematical literature that there are double stochasticmatrices that do not come from unitary matrices, see,e.g., [21] - [24]. This unpleasant feature rises a few prob-lems which have to be solved before trying to do a fit ofthe experimental data. The first one is to find the neces-sary and sufficient conditions the data have to satisfy inorder they should come from a unitary matrix. In otherwords we have to find the separation criteria between thetwo sets. After that we have to provide an algorithm forthe reconstruction of U from the given data. With theseaims in view we have developed a formalism for express-ing the necessary and sufficient conditions for a 3 × × δ as the function of modulimust take physical values, i.e. − ≤ cos δ ≤
1. The useof moduli is also justified by the fact that mixing anglesand CP phase are not well defined objects, they dependon the form of the unitary matrix. In this respect thereare at least two popular forms: the PDG proposal [25],and the classical proposal [26] by Kobayashi-Maskawa,and the physics has to be invariant, i.e. not dependentof one or another choice. The moduli are such invari-ants. The above statements can be formalized by thefollowing (obvious) axiom: irrespective of the physicalprocesses where the moduli appear, or the phenomeno-logical models used in analyzing the data, the numericalmoduli values should be the same.We used the novel formalism to develop an algorithmfor the reconstruction of a unitary matrix from a doublestochastic one when it is compatible to a unistochasticone. In this sense we modified the standard χ -test toallow the implementation of all unitarity constraints, fora better processing of experimental error affected data.To see that the found unitarity constraints are nec-essary to be used in any phenomenological analysis weconstructed an exact double stochastic matrix, from themoduli (1), which is perfectly acceptable from an experi-mental point of view, i.e. all the moduli are very close tothe central values in (1) being within the experimentalerror bars, but which is not compatible with a unitarymatrix.Last but not least we used the convexity property ofdouble stochastic matrices to develop a method for do-ing statistics on unitary matrices through their squaredmoduli, problem which was an open one in the literatureuntil now, see, e.g., Refs. [27]-[28] .The paper has the following structure. In Sec. II wepresent our phenomenological model that makes a noveluse of the unitarity property of the PMNS matrix, themain point being the focus on the condition that sepa-rates double stochastic matrices from unitary ones. Thiscondition is, in the same time, the consistency conditionbetween the theoretical model, represented by the PMNSmatrix, and the experimental data, as those appearing in(1). In Sec. III we present the reconstruction algorithmof unitary matrices from experimental data, when theyare compatible, and show how errors affected data can be consistently included in the formalism. We propose atwo terms χ -test which has to contain a piece enforcingthe fulfillment of unitarity constraints, and another piecewhich will properly take into account experimental data.In Sec. IV we propose a new type of global fit, done interms of invariant parameters, J and moduli | U ij | , whichcan resolve the existence of CP nonconservation in theleptonic sector by using the current experimental data,or, alternatively, prove the existence of new physics in theleptonic sector. In particular we show that around centralvalues from data (1) there is a continuous (approximate)unitary set of matrices, all of them being consistent with δ ≈ π/
2, for a broad range of numerical V e values. Bychanging the moduli values around one looks for unitarymatrices one gets other values for δ and V e . The paperends by conclusions. II. PHENOMENOLOGICAL MODEL
By phenomenological model we will understand in thefollowing a relationship between the entries of a doublestochastic matrix, provided in general by experiments,and the entries of a unitary matrix. The experimentaldata will be the numbers affected by errors entering (1).We remind that in order to find allowed ranges for themoduli | U ij | , as those given in (1), the solar neutrino andatmospheric neutrino measurements, as well as the reac-tor neutrino experiments have been used, see [5, 6, 12].The analyses lead to numbers for all moduli, but | U e | ,from the first row and the last column. Unfortunatelyonly three moduli are independent, because the unitar-ity gives | U e | + | U e | = | U µ | + | U τ | (9)Taking into account the embedding (7), which does notimply the unitarity fulfillment by experimental data, wewill use different notation for the theoretical quantities,and for experimental data. Hence in the following we as-sume that the data are contained in a numerical matrix, V, whose entries are positive V = V e V e V e V µ V µ V µ V τ V τ V τ (10)and the current experimental data on the moduli V αi , α = e, µ, τ, i = 1 , , , are summarized in matrix (1). From aphenomenological point of view the main problem to besolved is to see if from a numerical matrix as (1) one canreconstruct a unitary matrix as (4). For that we define aphenomenological model as follows V = | U | (11)relation that has to be understood as working entrywiseleading to the following equations V e = c c , V e = s c , V e = s V µ = s c , V τ = c c ,V µ = s c + s s c + 2 s s s c c cos δ,V µ = c c + s s s − s s s c c cos δ, (12) V τ = s c c + s s − s s s c c cos δ,V τ = s s c + c s + 2 s s s c c cos δ together with the double stochasticity relations X i =1 , , | U αi | − X i =1 , , V αi − , α = e, µ, ν X α = e,µ,ν | U αj | − X α = e,µ,ν V αj − , j = 1 , , The above phenomenological model is similar to thatproposed by Wolfenstein, [29], i.e. it is a direct relation-ship between the measured values V ij and the theoreti-cal parameters s ij and δ entering U . The difference be-tween the two approaches is the explicit use of the doublestochasticity properties (13), and making no approxima-tions on right hand side Eqs. (12). Also important isthe fact that on left hand side of Eqs. (12) we have setsof nine numbers, V ij , which all could be obtained fromexperiments, and on right hand side there are four inde-pendent parameters. Hence checking the consistency ofthe system of Eqs. (12) is a natural problem and it hasto be resolved. Since the matrix U is parameterized byfour independent parameters, e.g. s ij and δ , it is not atall obvious that the nine equations (12) have a physicalsolution for all experimental values allowed in (1). Onthe other hand we have to be aware that the parameters s ij and δ are convention dependent and have no intrinsictheoretical significance, and this fact rise the problem ofusing only invariant functions generated by the entriesof the matrix (4). Such functions are the Jarlskog in-variant J and the moduli V αi , see e.g. [30], and this isthe main reason for starting with moduli as independentparameters in our approach.We said before that for n ≥ s ij , ij = 12 , , V e + V e + V e = 1 , V e + V µ + V τ = 1 (14)we get always a physical solution for s ij which is uniqueunder the condition 0 ≤ s ij ≤
1, and depends on thethree chosen independent moduli. Substituting this so-lution in the last equations one gets four equations forcos δ , that lead to a unique solution for it. But nobodyguarantees that the solution will satisfy the physical con-straint − ≤ cos δ ≤ ≤ s ij ≤ , ij = 12 , , , − ≤ cos δ ≤ together with relations (12)-(13)prove the unitarity property of data, and provide an al-gorithm for recovery of unitary matrices from their ex-perimental moduli.The embedding relations (7) of unistochastic matricesinto the double stochastic set show that the number ofindependent parameters is the same for the two sets, i.e.in our case it is equal to four. Now we construct a doublestochastic matrix by using the constraints (13) and theentries from the data set, (10). In order to simplify theformulas form we make the notation V e = a, V e = b, V µ = c, V µ = d,V µ = e, V τ = g, V τ = h, V τ = f (17)As we already remarked the measured parameters a, b, e, and f are not independent since unitarity implies therelation a + b = e + f . Thus in principle we haveto chose only three of them. Our choice for the first twomatrices is, a and b , since they have the smallest errors,and c and e , respectively, d and f . The third matrix willbe generated by c, d, g, h , i.e. by moduli that are not yetmeasured. If the independent parameters are a, b, c, e ,the matrix S = ( S ij ), where S = ( S ij ) = a b √ − a − b c √ − c − e e √ − a − c √ c + e − b √ a + b − e (18)is double stochastic. Using the relations V αj = S αj in Eqs.(12) we find values for s ij and cos δ , e.g. s (1)12 = b √ a + b , s (1)13 = p − a − b , and s (1)23 = e √ a + b (19)cos δ (1) = − b ( a + b ) + c ( a + b ) + e ( b ( a + 1) + a ( a − a b e √ − a − b √ a + b − e In the second case when the independent parameters are a, b, d, f one gets s (2)12 = b √ a + b , s (2)13 = p − a − b , and s (2)23 = p a + b − f √ a + b (20)cos δ (2) = ( a + b )( b (1 − a − b ) − d ( a + b )) + f ( a + b ( a + b − a b f √ − a − b p a + b − f and, similarly, in the third case when used parameters are c, d, g, hs (3)12 = √ − d − h √ − c − d − h , s (3)13 = p c + d + g + h − , s (3)23 = √ − c − d √ − c − d − h (21)cos δ (3) = (cid:0) c (1 − c ) − d (1 − d ) − g (1 − g ) + h (1 − h ) + ( d g − c h )( c + d + g + h − (cid:1) / (22) (cid:16) p − c − d p − c − g p − d − h p − g − h p c + d + g + h − (cid:17) In the first two cases the mixing angles s ij depend only on three parameters, such that the only constraint on all fourparameters is given by − ≤ cos δ ≤ − cos δ = sin δ ≥ δ the above condition is equivalent to sin δ = − ( a + b ) (cid:0) b ( b − c ( a + b )) + c ( a + b ) + e (1 − a ) + 2 e b ( a −
1) + 2 c e ( a ( a −
1) + b (1 + a ) (cid:1) a b e (1 − a − b )( a + b − e ) ≥ CP violation. This area has to be a positive number if the CP symmetry is violated.If l i , i=1,2,3, denote the lengths of the triangle sides, from (18) one gets l = a b, l = c p − c − e , l = p − a − c p c + e − b (24)One makes use of Heron’s formula A = p p ( p − l )( p − l )( p − l ), where p = ( l + l + l ) / A , and the reality condition A ≥ A = − ( b ( b − c ( a + b )) + c ( a + b ) + e (1 − a ) + 2 e b ( a −
1) + 2 c e ( a ( a −
1) + b (1 + a ))) ≥ Looking at equations (23) and (25) we see that they areequivalent, the supplementary factors appearing in (23)being positive for a double stochastic matrix. Hence thecondition (16) on cos δ is a novel separation criterion be-tween the unistochastic matrices and the double stochas-tic ones. The separation border is given equivalently by A = 0, or cos δ = 1. Equivalent conditions to (23) were given in [31]-[33],where positivity of sin δ ≥ s ij and δ are quite different. The interesting thing is thatall the possible forms for s ij and cos δ take the samenumerical values when the numerical matrix is doublestochastic. Let us consider the following matrix S = √ √ q √ √ (26) ≈ .
835 0 .
55 0 . .
32 0 . . . . . whose entries are within the error bars from (1), andthe matrix S = ( S ij ) is a double stochastic matrix.The numbers entering it have been chosen such that themeasured parameters, V e , V e , V µ and V τ , should bearound the central values from (1). For this matrix weget s ( i )12 = 110 √ ≈ . , s ( i )13 = √ ≈ . ,s ( i )23 = 144 √ ≈ . , i = 1 , , δ ( i ) = − √ ≈ − . , i = 1 , , s ij are physical, but δ is not, i.e. the data matrix S , Eq.(26),does not come from a unitary matrix. Computing thetriangle area we find A = 0 . i (29)i.e. an imaginary number which sends the same signal,namely, the above data (26) are not compatible to theexistence of a unitary matrix.Thus the new unitarity constraint − ≤ cos δ ≤ S , s ij and cos δ do notdepend on the chosen four independent moduli used todefine it. If the modulus of cos δ is outside the inter-val ( − ,
1) the data are not compatible to the theoreti-cal model and the story ends here. If cos δ takes values within the interval ( − , δ and s ij nec-essary for the recovery of the unitary matrix are easilyobtained, e.g. from the relations (19), and by introducingthem in Eq.(4) we explicitly recover the unitary matrix.From almost any numerical experimental data as (1)we can form a doubly stochastic one by choosing at ourwill four independent parameters, the other five beingcompletely determined by equations similar to (18). Inthis way we obtained the equations (19)-(21) for cos δ .By using numerical information from matrix (1) for a, b, c, d, e, g, h and f , we getcos δ (1)+ = − . i, cos δ (1) c = − . ,cosδ (1) − = − . , cos δ (2)+ = 0 . i cos δ (2) c = 0 . , cos δ (2) − = 0 .
737 (30)cos δ (3)+ = 0 . i, cos δ (3) c = 0 . i, cos δ (3) − = 0 . i, where + , c, − denote values of cos δ function obtainedfrom central values+ σ , central values, and, respectively,central values − σ . The corresponding values for cos δ onthe three rows are different because the numerical val-ues for parameters a, b, c, e , respectively, d, f, g, h donot come from the same doubly stochastic matrix, e.g. e = √ − c − d , or numerically 0 . = 0 . V ij , and this unitarity feature has to be wellunderstood by people doing phenomenology. The borderof the physical region, A = 0, or cos δ = ±
1, can beused to find maximal intervals for the unmeasured mod-uli. If we consider V e , V e , and V µ or V τ fixed to thecentral values one gets V µ ∈ (0 . , .
45) which represent40% from the 3 σ interval given by (1), and respectively, V µ ∈ (0 . , . σ inter-val. If one computes now the same quantities by usingthe non unitary double stochastic matrix (26) one finds V µ ∈ (0 . , . V µ ∈ (0 . , . χ -test with aseparate piece that should implement the fulfillment ofunitarity constraints, if we want to get reliable values forthe interesting physical parameters. III. RECONSTRUCTION OF UNITARYMATRICES FROM DATA
We have seen in the previous section that by usingnumerical values from neutrino data, Eq.(1), one findsvalues for cos δ outside the physical range [ − , δ are provided by the last four relations(12), and these formulas have to give the same numberwhen comparing theory with experiment, by supposingthe data come from a unitary matrix. Their explicit formdepends on the four independent moduli used to parame-terize the data, see e.g. (19)-(21), and since there are 57such independent groups one gets 165 different expres-sions for cos δ .Hence if data are compatible to the existence of a uni-tary matrix the mixing angles s ( m ) ij coming from all 57groups, and all 165 phases δ ( m ) , expressed as functionsof V αi , have to be (approximately) equal, and these arethe most general necessary conditions for unitarity; theycan be written as0 ≤ s ( m ) ij ≤ , s ( m ) ij = s ( n ) ij , m, n = 1 , . . . , , (31)cos δ ( i ) = cos δ ( j ) , i, j = 1 , . . . , − ≤ cos δ ( i ) ≤ V αj .Now we present the necessary input that have to betaken into account when doing a fit, the experimentaldata are considered moduli of a unitary matrix. It has toinclude as a separate piece the conditions for a completeimplementation of unitarity. For that we define a testfunction that has to take into account the double stochas-ticity property expressed by the conditions (13) and thefact that in general the numerical values of data are suchthat cos δ depends on the choice of the four independentmoduli and could take values outside the physical range.The proposal for the first piece is χ = X i Collaboration [27], the difference being the term χ thatimplements the theoretical model (4).The above form can be easily modified if we have otherexperimental information for some functions dependingon s ij and δ , or on V αi . The primary data in neutrinophysics are the transition probabilities between two neu-trino flavors ν α → ν β , α, β = e, µ, τ at a distance L ,whose generic form is [35] P ( ν α → ν β ; E, L ) = X i j U βi U ∗ βj U ∗ αi U αj exp( − i ∆ ij )= X i, j ℜ ( W jiαβ ) cos ∆ ij + X i, j ℑ ( W jiαβ ) sin ∆ ij (35)where W jiαβ = U αj U ∗ βj U ∗ αi U βi , ∆ ij = ( m i − m j ) L/ E ν (36)and α = e, µ, ν, i, j = 1 , , 3. From [32], see also [36],we know that the imaginary part has the form ℑ ( U αj U ∗ βj U ∗ αi U βi ) = J X γ, l ǫ αβγ ǫ jil (37)with J the Jarlskog invariant. On sees that the parameterinvolving the CP violation appears in transition prob-abilities, hence it can be directly determined from theexperimental data. By using the above relation one gets X i, j ℑ ( U αj U ∗ βj U ∗ αi U βi ) sin ∆ ji = ± J (sin ∆ + sin ∆ + sin ∆ ) (38)The term coming from the real part ℜ ( W jiαβ ) dependsexplicitly on the flavors α and β . When α = µ and β = e one gets X i, j ℜ ( W jiµe ) cos ∆ jk = (39) R cos ∆ + R cos ∆ + R cos ∆ where R = U e U e ℜ ( U µ U ∗ µ )= − ( c c s ( c − s s )+ c c c s s s ( c − s cos δ ) R = U e U e U µ ℜ ( U µ )= − c c s s ( c s s + c s cos δR = U e U e U µ ℜ ( U µ )= c s s s ( c c cos δ − s s s ) (40)The above formulas have been obtained by using thestandard form (4) of the PMNS matrix. We use nowthe phenomenological model (40) to write R ij in termsof invariant quantities, i.e. moduli. One gets R = 12 ( − b − a c + b c + e − a e ) R = 12 ( b − a c − b c − e + a e ) (41) R = 12 ( − b + a c + b c − e + a e + 2 b e ) Similar formulas have to be obtained for all amplitudesentering the experimental data. For example the three-flavor survival probability P ee in vacuum [5] is given by P ee = 1 − 12 sin θ − cos θ sin θ sin ∆ (42)= 1 − a + b ) + 2( a + b ) − a b sin ∆ Hence the transition probabilites, (40)-(42), andthose similar to them, depends on eight parameters: a, b, c, e, J, ∆ , ∆ , ∆ which can be used to fit thedata. In the usual assumption that U is unitary only sixare independent, for example a, b, c, J, ∆ , ∆ . Thatmeans that we may eliminate modulus e . For that we usethe relation J = 2 A where, J is the Jarlskog invariant,and A is the area of (any) unitarity triangle. One gets e = b (1 − a − c ) + a c (1 − a − b )(1 − a ) + 2 ǫ p a b c (1 − a − c )(1 − a − b ) − (1 − a ) J (1 − a ) (43)where ǫ = ± 1. If the matrix is unitary then the followingcondition has to be satisfied 0 ≤ e ≤ 1. The secondparameter is eliminated from the relation∆ + ∆ + ∆ = 0If we use the the phenomenological models (20) and (21),i.e. if we take as parameters four independent moduli, wefind form different terms for all R ij appearing in formulae(40), see Eqs. (41) and (42). Thus χ could have the form χ = X α X i X j P i ( ν α → ν β ) − ^ P j ( ν α → ν β ) σ P j (44)where α runs over all transitions probabilities involvedin the fit, i runs over the sets of four independent mod-uli, such as those given by relations (19)-(21), which giveequivalent but form different expressions for the theoret-ical probabilities P ( ν α → ν β ), and j runs over the number of experimental points for each transition probability. Bydoing a fit with the form (34) where χ is given by (44)one gets values for moduli V αi , ∆ ij and J .The fit could provide a few (approximate) unitary ma-trices compatible with the data. In such a case the dou-ble stochastic matrices obtained from fit could be usedto do statistics on unitary matrices. At this point theembedding property of unitary matrices into the convexset of double stochastic matrices is essential. Indeed if U , U , . . . U n are unitary matrices, then M = i = n X i =1 x i · | U i | , i = n X i =1 x i = 1 , (45)0 ≤ x i ≤ , i = 1 , . . . , n is double stochastic. The above relation shows that thestatistics on unitary matrices has to be done through thestatistics on double stochastic matrices, i.e. the relevantquantities for doing statistics on unitary matrices are thesquared moduli, | U αj | . This property allows us to definecorrectly the mean value, h M i , and the error matrix, σ M ,for a set of doubly stochastic matrices, coming from a setof approximate unitary matrices, as follows h M i = vuut k = n X k =1 | U k | ! /n (46) σ M = vuut k = n X k =1 | U k | ! /n − h M i (47)If the moduli of the mean value matrix h M i obtained inthis way are compatible with a unitary matrix, or not toofar from moduli coming from a unitary one, one can re-construct from it an (approximate) unitary matrix, andin the following we use this technique to reconstruct uni-tary matrices compatible to the data, and for obtaininginformation on relevant physical parameters. IV. GLOBAL FITS By global fit we understand a fit that takes into ac-count all available experimental data, and (all) unitarity constraints expressed by invariant functions; on this lastrequirement see, e.g., Ref. [37], the aim in view being agood estimation of mixing and CP violation in neutrinophysics. For doing that we use the simple ansatz (32)-(34) in conjunction with data (1) to obtain qualitativeresults about J , δ and/or e V e . The last parameter wasconsidered free within the interval e V e ∈ (0 . − . 4) andits error was taken σ e = e V e . Around the central valuesfrom (1) we obtained many approximate double stochas-tic matrices, i.e. values for all moduli V αi , and by doingstatistics with formulae (47) the following mean valuematrix and associated error matrix were gotten: M = . . . . . . . . . (48) σ M = . . . . . . . . . (49)and the unitary matrix corresponding to M is U = . . . − . − . i − . . i . − . . i − . − . i . (50)If we define two matrices, let us say, by the relation M ± = q M ± σ M (51)we get the following prediction for CP violating phaseand unitarity triangle area δ = (89 . +0 . − . ) ◦ , A = (127 . ± . × − corresponding to a 3 σ level, and so on.The results (48) and (49) strongly depend on the mod-uli values around one looks for a minimum. Changingthese values within the error bars in (1) one gets othermatrices and other values for δ . For example the matrix M = . . . . . . . . . (52)leads to a value δ = (120 . ± . ◦ , J ≈ . × − ,and so on. This happens since the errors entering (1)are too large, such that for many choices of fixed three moduli, δ is practically unconstrained, i.e. there is acontinuum of unitary matrices with cos δ varying in manycases between ( − , θ ij and δ . The term containing e V e was not introduced in the fit. One gets easily θ = 0 . , θ = 0 . , θ = 0 . ,δ = 90 . ◦ , A = 9 . × − (53)where the values for θ ij are given in radians, and χ =8 . × − , where from one gets the following doublestochastic matrix M = . . . . . . . . . (54)The corresponding unitary matrix is given by0 U = . . . − . − . i − . . i . − . . i − . − . i . (55)The remarkable fact is that the fit provides a numericalvalue for V e which could not be too far from the true one,although the global fit [5] failed to find it. To estimatethe errors to the above parameters we used the modulimatrices √ V − σ and √ V + 4 σ , where V and σ are those given by (1), and doing again the fit we got χ σ = 1 . 32, and, respectively, χ σ = 1 . 22, where thefollowing prediction was obtained δ = (90 . +2 . − . ) ◦ , A = (9 . +0 . − . ) × − (56)To see the difference between statistics done by using(47) and (51) and the naive one, we used the central ma-trix (1), and the corresponding error matrix σ obtaining V c + σ , to find θ = 0 . , θ = 0 . ,θ = 0 . , δ = 79 . ◦ (57)results which are equivalent with the moduli matrix W = . . . . . . . . . (58)This matrix is not compatible with a unitary matrix ifwe use our criteria. For example one gets cos δ (3) = − . i , and A = 9 . × − i . In fact there are 20 cos δ from a total of 165 which take imaginary values, so, inprinciple, the problem cannot be neglected. On the otherhand one finds χ = 6 . δ ≥ ◦ . Doing a similar fit for V − σ one gets χ = 8 . δ = 91 . ◦ .Of course in true fits one does not work with quanti-ties such as V ± σ , and they were given as pedagogicalexamples. They show that it is possible when using asindependent parameters θ ij and δ to obtain physical val-ues for all of them, although the area of unitarity trianglecould take imaginary values. The above computationsshow that s ij and δ are convention dependent, even iftheir numerical values are physical, and in order to min-imize this bias we expressed these parameters in termsof the moduli V αi which have to be the same irrespec-tive of the used convention for the unitary matrix (4),and we required the fulfillment of all constraints (31).The second lesson is that one could get values for δ evenif moduli where δ enters are not yet measured, and thishappens since unitarity over constrains the data implyingmany relationships between the moduli. Of course a neg-ative attitude to such results is compliant so long as thesubtleties implied by unitarity are not fully understood. In the following we want to stress the non uniqueness ofunitary matrices compatible with the error bars in (1).For example, one can find also a unitary matrix whichlead to δ = (75 . ± . ◦ ; its moduli are M = . . . . . . . . . (59)With the matrices M i , i = 2 , , 4, by using the prop-erty of double stochastic matrices we find a continuumof (approximate) unitary matrices whose squared moduliare given by M = x M + y M + (1 − x − y ) M , ≤ x, y ≤ M i , i = 2 , , 4, with formulae (47) onegets M = . . . . . . . . . (61) σ M = . . . . . . . . . (62)With matrix M one can obtain any value for δ ∈ (75 ◦ − ◦ ), and, respectively, A ∈ (2 × − − . × − ). Theprediction at the symmetric point x = y = 1 / M is δ = (88 . +14 . − . ) ◦ , A = (11 . ± . × − at one sigma level given by σ M .In the following we want to test the dependence ofmoduli matrix on the form of the unitary matrix U . Forthat we use the Kobayashi-Maskawa form [26] U = c c s s s c s − c c c + s s e iϕ − c c s − c s e iϕ s s − c s c − c s e iϕ − c s s + c c e iϕ (63) where c i = cos θ i and s i = sin θ i , and ϕ is the Diracphase. Of course selecting one or another form has notheoretical significance because all choices are mathemat-ically equivalent; however a clever choice may shed somelight on important issues. Such issues could be the | U e | magnitude and the CP phase δ, ( ϕ ), and the above form(63) could be more helpful.1Doing a fit by using the relations (12) and the newparameters θ i one gets the moduli matrix (54) with thesame digits. In fact we get θ = 0 . , θ = 0 . , (64) θ = 0 . , ϕ = 90 . ◦ that confirm the previous results, the only change be-ing the value for θ which is brought, in principle, to ameasurable value.A few conclusions of the above numerical fits could be:(a) there are large corridors for moduli around the centralvalues in (1) where data are compatible to the existenceof a unitary matrix with δ = 0 , π , and by consequencecompatible with a C P violation; (b) the real parts of thecomplex entries in unitary matrices are smaller by an or-der of magnitude than the imaginary parts, see e.g. ma-trix U , but their neglect completely spoils the unitarityproperty; (c) by consequence, reliable results can be ob-tained then and only then when all theoretical formulasused in fits are exact, not approximate .We are aware that the “experimental” data (1) are notnumbers obtained directly from experiments. In the fol-lowing we suggest a new kind of global fit able to resolve the existence or the lack of the CP nonconservation in theleptonic sector by measuring directly the Jarlskog invari-ant J , and by testing also the possible new physics whichcould explain the negative value for | U e | obtained in [5],by using the presentday experimental data. For doingthis we suggest the use of moduli, a, b, c, d, e, f, g, h and J, ∆ , ∆ , as parameters, although not all of them aretruly independent. However the moduli are correlatedthrough different forms for J and/or cos δ , and relationsas (31), or J ( a, b, c, e ) = J ( c, d, g, h ) have to be satisfiedto a great accuracy. And such a fit could provide val-ues for all moduli. If the new fit gives a positive valuefor J one can say that the fit confirms the CP violationin the leptonic sector. If the fit will confirm the resultsfrom Ref. [5], see its Appendix A, concerning the best fitobtained for negative values of | U e | , which imply that a and b entering the formula (42) for P ee are such that1 − a − b < 0, one can use the unitarity constraintsto obtain other correlations. For example the relation(43) implies also that 1 − a − c = | U τ | < 0, becauseotherwise e gets complex. If this is true then by usingas independent parameters a, c, e, and f , and eliminatingagain the parameter e , one gets e = ( a + c )( a − c f ) − ( a − c ) f ( a + c ) + 2 ǫ p a c f (1 − a − c )( a + c − f ) − ( a + c ) J ( a + c ) (65)Because | U τ | = 1 − a − c < 0, the preceding relationtell us that | U τ | = a + c − f < unitarity . In any case byusing the proposed fit there is a bigger flexibility to testvarious consequences of the theoretical model (4).Of course results such as | U τi | < i = 1 , 2, or | U e | < s ij and δ , which show easily how one can walk intoa trap. Such a fit will provide values for all moduli evenif the nowadays data come only from three sources, andit deserves to be done since it will give estimates for alltransition probabilities necessary for the design of futureneutrino factories. However, if such a result is confirmedthis will imply a new physics which could be interpretedas the first sign showing an evidence for a new generationof leptons, or the change of the theoretical frame (4), bytaking the matrix U as being unitary, for example, in a non-Euclidean metric. V. CONCLUSIONS In this work we have discussed the question of the rel-evance of the nowadays experimental neutrino data onthe determination of CP nonconservation in the leptonicsector, as well as on solving the negativity problem of | U e | < 0, raised by the best fit in Ref. [5]. For doingthat we used another consequence of unitarity propertyand defined a phenomenological model by taking as freeparameters the moduli of the unitary matrix, which areinvariant quantities, and which, naturally, leads to theseparation criterion between the double stochastic ma-trices and those arising from unitary ones. This crite-rion provided the necessary and sufficient conditions forthe consistency of experimental data with the theoreti-cal model encoded by the PMNS matrix, the strongestcondition being − ≤ cos δ ≤ χ -test and have two separatepieces: one which implements the unitarity constraints,and the other which properly takes into account the er-ror affected data. We have also shown how this algorithmcan be modified to take into account the oscillation prob-abilities that are the primary data measured in neutrinophysics.We have used the χ -test to obtain information on thefour parameters entering the standard parametrizationof the PMNS matrix. The test works very well, and byapplying it we obtained a continuum of (approximate)unitary matrices compatible with the nowadays availableexperimental data summarized in the numerical matrix(1). Their explicit construction shows that the modulidata (1) are compatible with the existence of CP viola-tion in the leptonic sector, even if one cannot yet find aprecise value for phase δ .The numerical results show that unitarity is a verystrong property, and the tuning of moduli implied byit is given at a higher level than expected, the statisticalerrors generated by the approximate character of data areat least an order of magnitude less than the experimentalerrors. It was also shown that the fitting method, using asindependent parameters moduli and Jarlskog invariant,can directly check the existence of CP violation in thepresentday neutrino data. On the other hand, the samemethod can resolve the positivity problem for | U e | . Ifthe results by Maltoni et al . [5] are confirmed, they willbe the first signal for a new physics in leptonic sector.Last but not least we used the natural embedding (7)of unitary matrices into the double stochastic set for de-vising a method for doing statistics on unitary matrices,problem that was an open one until now. The methodallowed us to find a procedure to obtain mean values,and, correspondingly, error matrices starting with a setof (approximate) unitary ones.As a final conclusion, a satisfactory solution of CP nonconservation can be obtained by diminishing the er-rors in all running experiments, as well as in all futureones, the present theoretical approach being able to testthe validity of the nowadays unitary model, or the neces-sity to modify it. Acknowledgments I thank also L Lavoura and C Hamzaoui for pointingout their papers to me. [1] B. Aharmim et al.(SNO) (2005), nucl-ex/0502021[2] Y.Ashie et al. (Super-Kamiokande), Phys.Rev. D 71 (2005) 112005[3] K. Eguchi et al.(KamLAND), Phys.Rev.Lett. 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