TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING
Biswajoy Brahmachari and Probir Roy Dept of Physics, Vidyasagar Evening College, Kolkata and Saha Instutute of Nuclear Physics, Kolkata and CAPPS, Bose Institute, Kolkata
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING Plan
1. Neutrino mixing puzzle 2. TBM and additive perturbation 3. Lowest order results 4. Discussion 5. Bottom line
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING Neutrino mixing puzzle
Quark and charged lepton masses hierarchical. Neutrinos ? Quark mixing hierarchical:1-2 » 2-3 » 1-3.
0.9 0.2 0.004 (1) V 0.2 0.9 0.01 | CKM |∼ 0.008 0.04 0.9
Neutrino mixing pattern is different
0.8 0.5 0.2 (2) U 0.4 0.6 0.7 | PMNS|∼ 0.4 0.6 0.7
Either neutrinos are quasi-degenerate or the neutrino mass hierarchy is mild and does not affect neutrino mixing significantly. A non-hierarchical mixing principle seems operative in the neutrino sector.
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING Neutrino mixing puzzle
For fermion of type t (=u,d,l,ν)
t t 2 2 2 (3) U †M † M U = M †M = diag.( m , m , m ) tf tf t t | t1| | t2| | t3| Take neutrinos to be Majorana particles
iα21 iα32 (4) M = diag.( m , m e− , m e− ) ν | ν1| | ν2| | ν3| Majorana phase matrix is,
(5) K = diag.(1,eiα21/2,eiα32/2)
l ν (6) UPMNS = U †U K
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING Neutrino mixing puzzle
(7) UPMNS =
iδCP c12c13 s12c13 s13e− (8) s c c s s eiδCP c c s s s eiδCP s c K. − 12 23 − 12 23 13 12 23 − 12 23 13 23 13 s s c c s eiδCP c s s c s eiδCP c c 12 23 − 12 23 13 − 12 23 − 12 23 13 23 13 Here cij = cos θij and sij = sin θij.
Global fits (3σ)
31◦ < θ12 < 36◦
36◦ < θ23 < 55◦
7.2◦ < θ13 < 10◦
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING Neutrino mixing puzzle
2 5 2 7.00 <δ21(10 eV ) < 8.09 2 3 2 2.195 <δ32(10 eV ) < 2.625 Direct 2 3 2 2.649 <δ32(10 eV ) < 2.242 Inverted − −
We repeat our operative principle. Either neutrino masses are quasi degenerate or at least a non-hierarchical principle seems to be involved in neutrino mixing.
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING TBM and additive perturbations
Mass basis and flavor basis of neutrinos (ν) and charged leptons (l)
f (9) ψνl = Uνl ψνl flavor basis mass basis
Majorana mass matrix is complex symmetric Mνf in flavor basis.
T (10) diag (m1,m2,m3) = Mν = Uν Mf Uν
2 2 2 (11) diag ( m , m , m ) = M †M = U †M † M U | 1| | 2| | 3| ν ν ν νf νf ν
(12) Ul†Uν K = UPMNS
We will now examine Mνf and Uν in the limit of tribimaximal symmetry(superscript zero). Theoretically tribimaximal mixing can be related to A4, S3 ∆27 symmetries.
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING TBM and additive perturbations
X Y Y 0 − (13) M = Y X + Z Y + Z , νf − Y Y + Z X + Z − − Here X,Y,Z are arbitrary complex mass dimensional parameters.
(14) (M 0 ) = (M 0 ) , νf 12 − νf 13 0 0 (15) (Mνf )22 = (Mνf )33, (16) (M 0 ) (M 0 ) = (M 0 ) (M 0 ) νf 11 − νf 13 νf 22 − νf 23
0 −1 1 ◦ 0 ◦ 0 (17) θ12 = sin 35.3 , θ23 = 45 , θ13 = 0 . √3 ∼ violated allowed 3σ level
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING TBM and additive perturbations
Chosen weak basis is where
0 0 0 0 (18) Mℓ = diag. (mτ ,m ,mτ )
In the TBM limit
(19) U 0† M 0 †M 0 U 0 = diag ( m0 2, m0 2, m0 2), ν νf νf ν | ν1| | ν2| | ν3| 2/3 1/3 0 0 (20) U = 1/6 1/3 1/2 . ν − 1/6 1/3 1/2 − 1 0 0 0 (21) Uℓ = 0 1 0 0 0 1
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING TBM and additive perturbations
0 (22) Mℓ = Mℓ + Mℓ′ TBM conserving TBM violating 0 (23) Mνf = Mνf + Mνf′
TBM conserving TBM violating
M ′ elements are of O(ǫ). ǫν s 0.16 , ǫℓ expected smaller than ǫν because they are related to | |∼ 13 ∼ | | | | inverse powers of charged lepton mass squares. Neglecting O(ǫ2) terms, we can write,
0 0 0 0 2 (24) M †M M †M + M †M ′ + M ′†M + O(ǫ ) ∼
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
Flavor Basis:
(25) ψν,ℓ = ψ0ν,ℓ + ǫν,ℓ ψ0ν,ℓ + O(ǫ2). | i f | i f ik | k f k=i In mass basis: (26) ψν,ℓ = ψ0ν,ℓ + ǫν,ℓ ψ0ν,ℓ + O(ǫ2). | i | i ik | k k=i
ν,ℓ ν,ℓ 0 2 0 2 1 ν,ℓ (27) ǫ = ǫ ∗ =( m m )− p , ik − ki | ν,ℓi| − | ν,ℓk| ki ν,ℓ 0ν,ℓ 0 0 0ν,ℓ (28) p = ψ M †M ′ + M ′ †M ψ . ik i | ν,ℓ ν,ℓ ν,ℓ ν,ℓ| k ν,ℓ ν,l One can explicitly check that ǫik and pik are identical in either basis.
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
Arranging first order flavor eigenstates column by column we get mixing matrices for charged leptons and neutrinos.
ℓ ℓ 1 ǫ12∗ ǫ13∗ ℓ ℓ (29) U † = ǫ ∗ 1 ǫ ∗ ℓ − 12 23 ǫℓ ǫℓ 1 − 13 − 23
Uν =
2 1 ν 1 2 ν∗ 2 ν∗ 1 ν∗ 3 + 3 ǫ12 3 3 ǫ 12 3 ǫ 13 3 ǫ 23 − − − 1 1 ν 1 ν 1 1 ν∗ 1 ν 1 1 ν∗ 1 ν∗ (30) 6 + 3 ǫ12 + 2 ǫ13 3 + 6 ǫ 12 + 2 ǫ23 2 + 6 ǫ13 3 ǫ 23 − − 1 1 ν 1 ν 1 1 ν∗ 1 ν 1 1 ν∗ 1 ν∗ 6 3 ǫ12 + 2 ǫ13 3 6 ǫ 12 + 2 ǫ23 2 6 ǫ 13 + 3 ǫ 23 − − − −
(31) UPMNS = Uℓ†Uν K
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
UPMNS given in PDG convention in Eqn. (7). In identification the matrix K cancels out.
νℓ Relations among ǫij ,cij, sij and δCP emerge.
iδCP cij and sij are real and e is complex.
ν,ℓ Conditions emerge on Im ǫij
ν 2 (32) Im ǫ12 = O(ǫ ), (33) Im (ǫν √2ǫν ) = O(ǫ2), 13 − 23 l 2 (34) Im ǫ23 = O(ǫ ), (35) Im (ǫl ǫl ) = O(ǫ2). 12 − 13
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
Three measurable deviants from tribimaximal mixing:
2 ν l l 2 c12 2/3= 1/2 1/3 s12 + O(ǫ )= 1/3 ǫ12 1/6 ǫ12 ǫ13 + O(ǫ ), − − − − (36) ν ν l 2 c23 s23 = 2/3 ǫ13 √2 ǫ23 √2 ǫ23 + O(ǫ ), − − − − (37)
iδCP ν ν l l 2 s13 e = 1/3 √2 ǫ13 + ǫ23 + 1/2 ǫ12 + ǫ13 + O(ǫ ). − (38)
† † † ∗ † ∗ The basis independent Jarlskog invariant J = Im[(Uℓ Uν )e1(Uℓ Uν ) 2(Uℓ Uν )e2(Uℓ Uν ) 1] now turns out to be 1 1 (39) J = Im [ǫν (ǫl + ǫl )] + O(ǫ2). −√6 23 − √6 12 13
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
Let us now take the perturbing mass matrices for neutrinos and charged leptons, with respective complex mass dimensional parameters ij = ji and λij, as
11 12 13 λ11 λ12 λ13 ′ ′ ′ 2 (40) Mνf = 12 22 23 ,Mℓf = λ21 λ22 λ23 = Mℓ + O(ǫ ) 13 23 33 λ31 λ32 λ33 Off diagonal entries of neutrino perturbation in mass basis: 1 (M ′ )12 = (2 11 + 12 13 22 + 2 23 33) ν 3√2 − − − 1 (M ′ )13 = (2 12 + 2 13 22 + 33) ν 2√3 − 1 (M ′ )23 = ( 12 + 13 + 22 + 33) ν √6
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
Explicit substitution in Eqn. (25) yields
l 02 0 2 1 0 0 (41) ǫ = (m m )− (m λ + m λ∗ ), 12 e − 21 e 12 l 0 2 02 1 0 0 (42) ǫ = (m m )− (m λ + m λ∗ ), 23 − τ τ 32 23 l 02 02 1 0 0 (43) ǫ = (m m )− (m λ + m λ∗ ). 13 e − τ τ 31 e 13 Because of the hierarchical nature of charged lepton masses,(34) and (35) can only be satisfied, without unnatural cancellations, by
λ12,λ21,λ13,λ31,λ32 and λ23 all being real to order ǫ. That immediately implies l 2 l (44) Im ǫ12 = O(ǫ )=Im ǫ13.
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
Using (44)
ν 2 3 Im ǫ23 + O(ǫ ) (45) tan δCP = . Re [√2 ǫν + ǫν ] 3/2Re[ǫl + ǫl ] + O(ǫ2) 13 23 − 12 13 Also, 1 (46) J = Im ǫν + O(ǫ2). −√6 23
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
0 0 2 0 2 (47) ∆ij mνi mνj , ≡| 0 | −|0 | (48) a∓ m m . ij ≡ νi ∓ νj
i Im ǫν √ 0 12 ∓∗ 6 2∆12 ν = a 21 (2 11 + 12 13 22 + 2 23 33) c.c., Re ǫ12 − − − ∓ (49)
i Im ǫν √ 0 23 ∓∗ 2 6∆23 ν = a 32( 12 + 13 + 22 33) c.c., Re ǫ23 − ∓ (50)
i Im ǫν 1 1 √ 0 13 ∓∗ 2 3∆13 ν = a 31( 12 + 13 22 + 33) c.c., Re ǫ13 − 2 2 ∓ (51)
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
In the absence of unnatural cancellations, (32) and (49) would require 0 2 11 + 12 13 22 + 2 23 33 and mν2 to be real; the latter − − − 0 constrains the Majorana phase α21 to equal 0 or π in the TBM limit. These statements are valid neglecting O(ǫ2) terms. Furthermore, (33), (50) and (51) would require the equality.
0 0 Im[(m ∗ m ∗)( + + )] ν3 − ν2 12 13 22 − 33 0 0 1 1 (52) = Im[(m ∗ m ∗)( + + )] ν2 − ν1 12 13 2 22 − 2 33
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
0 0 0 0 Either, one must have condition 1: mν1 = mν2, meaning mν1 = mν2 0 ν ν | | 2 plus α21 = 0, and 22 = 33 in which case, √2 Re ǫ23 = Re ǫ13 + O(ǫ ) l 2 and then, from (37), c23 s23 = √2 ǫ23 + O(ǫ ),i.e, l − 2 − s23 = (1/√2)(1 + ǫ23) + O(ǫ ); the latter implies via (37) and (42) that any deviation from maximality in the atmospheric neutrino mixing angle θ23 must come solely from the 2-3 off-diagonal element in the charged lepton mass perturbation Ml′ and is expected to be small l 1 since ǫ23 is scaled by (mτ )− , cf. (42) .
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Lowest order results
Or, what becomes necessary is condition 2: m0 m0 , m0 m0 as ν3 − ν1 ν3 − ν2 well as 12 + 13 and 22 33 have to be real; this means that the 0 −0 ν Majorana phase α21 and α32 in the TBM limit are 0 or π and ǫ23 is real, 2 in which case, by virtue of (45) as well as (46), s13 sin δCP = O(ǫ ) and J = O(ǫ2) so that, any observable CP-violation in neutrino oscillation experiments would vanish to the lowest order of TBM violating perturbations.
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Forthcoming experiments
Let us finally remark on the relevance of our results to planned neutrino experiments at the proton beam intensity frontier. A combination of data from the ongoing and upcoming runs of the T2K and NOvA experiments will sensitively probe the existence of any possible deviation from 1 in from converison probability 2 s23 P (ν ν ). Suppose such a deviation is discovered at a significant −→ e level, say comparable in percentage terms to (100 s13)% of the maximal value, then within our framework we would expect a non-observation of CP-violation in neutrino oscillations in the above data
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Forthcoming experiments
Contrariwise, the failure to measure any such deviation outside error bars would bolster the hope of detecting CP-nonconservation for oscillating neutrinos which would be allowed to order ǫ. The latter would be good news not only for a combined analysis of data from forthcoming runs of T2K and NOvA, but also for future experiments with superbeams, such as LBNE, LBNO or a neutrino factory at 10 GeV, aiming to observe CP-violation from the difference in oscillation probabilities, P (ν ν ) P (¯ν ν¯ ). → e − → e
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Bottom line
In summary, linear expressions have been analytically derived, valid with any possible neutrino mass hierarchy, for the three measurable TBM-variants in terms of lowest order perturbation parameters. Constrained interrelations among those parameters have been obtained from the requirement that the perturbed charged lepton and neutrino flavor eigenstate vectors have to constitute the columns of the respective unitary matrices Uℓ and Uν .
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – Bottom line
With the plausible assumptions of the mixing caused by charged leptons being significantly smaller than that due to neutrinos and no unnatural cancellations, we have obtained a result, forcing one of two possibilities, which should be testable in the foreseeable future. Our bottom line is that if both a deviation from maximality in θ23 and a non-zero amount of CP-violation are discovered in forthcoming neutrino oscillation experiments, perturbed tribimaximality will cease to be an attractive theoretical option.
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – References
1. Constrained analytical interrelations in neutrino mixing Biswajoy Brahmachari, Probir Roy; e-Print: arXiv:1403.2038 [hep-ph]
2. Perturbative generation of θ13 from tribimaximal neutrino mixing Biswajoy Brahmachari, Amitava Raychaudhuri; Phys.Rev. D86 (2012) 051302; e-Print: arXiv:1204.5619 [hep-ph]
3. Smallness of θ13 and the size of the Solar Mass Splitting: Are they related? Soumita Pramanick, Amitava Raychaudhuri; Phys.Rev. D88 (2013) 093009; e-Print: arXiv:1308.1445 [hep-ph]
4. Discrete Symmetries and Neutrino Mass Perturbations for θ13 L.J. Hall, G.G. Ross; JHEP 1311 (2013) 091; e-Print: arXiv:1303.6962 [hep-ph]
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING – THANK YOU
TESTABLE CONSTRAINT ON NEAR-TRIBIMAXIMAL NEUTRINO MIXING –