JHEP09(2012)005 Springer April 4, 2012 June 11, 2012 : : August 12, 2012 : September 4, 2012 : Revised Received 10.1007/JHEP09(2012)005 Accepted Published doi: , indicating a deviation from 13 θ Published for SISSA by . σ 2 . b [email protected] , and Michael Trott a 1203.4410 Neutrino Physics, Solar and Atmospheric Neutrinos We develop a formalism for constructing the Pontecorvo-Maki-Nakagawa- [email protected] Theory Division, Physics Department,CH-1211 CERN, Geneva 23, Switzerland E-mail: Department of Physics, UniversitySan of Diego, California, La Jolla, CA 92093 U.S.A. b a Open Access Keywords: ArXiv ePrint: Our approach does not dependstates on choosing of the the rotation charged betweento the lepton weak examine fields and the to mass eigen- oration be symmetry diagonal. reporting implications the of We discoverytri-bimaximal comment the form, of on recent with using a results a this non significance from expansion zero of the 5 value Daya-Bay for collab- Abstract: Sakata (PMNS) matrix andsequence neutrino of masses heavy using rightnism an handed for the expansion neutrinos origin that are of neutrinostructure originates integrated masses. of out, when the The assuming a PMNS expansion establishes matrix aimplications relationships and for seesaw between the measured the mecha- mass deviations differences from of tri-bimaximal neutrinos, form to and be allows symmetry studied systematically. Benjam´ınGrinstein An expansion for neutrino phenomenology JHEP09(2012)005 (1.1) (1.2) 3 ]. Taking and three P, 2 , 23 1    0 0 , θ 13 12 12 c s , θ 12 12 with the conventions θ 12 0 0 1 s c ij − θ 11    × 4 = cos form    8 9 iδ . ij ) c − 13 e 9 c 13 ν, L s ( and PMNS , the PMNS matrix is given by U 0 0 U ij ) θ iδ ···} − 6 e, L e ( 13 0 1 0 † c 12 13 – 1 – = sin s 3 U − ν, e, µ, ij is a Dirac phase. This latter phase can contribute in = { s form δ    = a general parameterization of this matrix is given by × 10 f TB π    2 PMNS 4 U 23 23 ≤ 1) is a function of the Majorana phases, present if the right c s , 2 , 2 1 . Defining / 23 2 , where 2 23 s , c 1 iα − δ, α L/R , e 1 00 0 0 f 2 δ, α ≤ / ) 1    1 iα 13 and flavour space = e f, L/R matrix can be parameterized in terms of three angles ( 2, and 0 U PMNS π/ PMNS = diag( 2.4.2 TheU expansion with flavour alignment 2.2.1 Model dependence of perturbing the seesaw 2.4.1 The expansion without flavour alignment = U ≤ P PMNS ij U 0 L/R θ 2.5 3.1 Perturbative breaking of 2.1 Review of2.2 seesaw the mechanism Perturbing the seesaw 2.3 Developing the2.4 seesaw perturbations Inverted and normal hierarchy and flavour space f ≤ where handed neutrinos are Majorana, while The CP violating phases 0 species is related to thenos, misalignment characterized of by the the (PMNS) Pontecorvo-Maki-Nakagawa-Sakata interaction matrix and [ the mass relation eigenstates between of the the interaction neutri- by (primed) and mass (unprimed) eigenstates to be given 1 Introduction The standard theory offrom neutrino their oscillations, interaction where eigenstates leading three towith mass the current eigenstate observed experimental neutrino neutrinos oscillations, data. differ is The consistent amplitude of the oscillations among various neutrino 4 Conclusions 3 (Un)relating flavour symmetries to the Contents 1 Introduction 2 Constructing a flavour space expansion for neutrino phenomenology JHEP09(2012)005 j < 13 ν ( θ ) = ] at , ij ] for 3 (1.5) (1.3) (1.4) 0 ) 8 e, L > ( 1). This , U 2 32 1 PMNS form of the , m in ref. [ U B σ 3 T = ( 0 i > ν 2 and ∆ ¯ / ) = diag(1 ) 2 2 , i.e. , m e, L ( , + ) , , U 2 1 ) PMNS     m 007 007 U 17 16 form is , , . . ( . . 2 2 2 0 0 005(syst) 1 2 . − √ +0 − B 1 +0 − 0 √ 0 2 3 eV eV ] for a coherent discussion on this T − 5 3 5 m ± 025 − − 312 3 3 3 . . 1 1 1 √ √ √ ≡ 10 10 (0 (0 form in light of this result, where again ] with a reported value of 6 6 2 32 2 3 × × 7 1 1 . It is reasonable to expect further specu- √ √ B ) ) m = 0 the . q 13 T − − 18 15 007 008 22 26 12 09 ...... θ 08 03 0 0 13 016(stat) 0 0 . . .     0 θ +0 − +0 − +0 − +0 − ), see ref. [ – 2 – 0 +0 − = 58 35 ± . . 306 021 42 e, L . . . ( 6= 0 in global fits is reported to be TB † 3 and 092 ) / U . = (7 = (2 13 | ≈ U ) = 0 ) = 0 ) = 0 θ ], (with new reactor flux results in brackets): 2 21 ( 2 32 23 12 13 3 2 ) = 1 θ θ θ ) = 0 m m ( ( ( ) and 12 2 2 2 13 ∆ error. Note that ∆ ¯ ∆ ¯ θ θ | PMNS ( σ sin sin sin 2 U d, L (2 ( 2 † evidence for non zero U sin σ 2 . 3. There have been many attempts to link the approximate ]. Fixing sin , 4 2 ] for a review. Recent experimental results provide evidence for deviations from , 6 1) is frequently used. When this assumption is employed the relationship between , form. Evidence for sin , 5 1 = 1 B , ) form [ T There is no clear experimental support for assuming that The structure of this matrix could be fixed by underlying symmetries. In attempting i, j B T structure of thethe Cabibbo-Kobayashi-Maskawa relation (CKM) between matrix,approach. and This a choice can further alsoan ansatz be example. justified on Conversely, with in model grandinvolved building unified in in models the principle, frequently seesaw see associated mechanism, ref. with the [ the quark high and scale lepton mass matricies can be related in corresponding to 5 lation about the originthe of flavour the basis deviation will from be frequently assumed. choice can be motivated by an ansatz related to the origin of the approximately diagonal this this time. As thiswas announced paper by was the approaching Daya completion, Bay the Collaboration discovery [ of non-vanishing diag(1 the weak and mass neutrino eigenstates isfor identified with the neutrino mass matrix to symmetriessee of ref. the [ right handed neutrino interactions in this basis, for a particular phase convention. to determine such an origin of this matrix, the “flavour” basis where one assumes 0) corresponds to ais normal perhaps (inverted) suggestive mass of spectrum. a( PMNS This matrix pattern that of has experimental at least data an approximate “tri-bimaximal” The error is the reported 1 global fit results onreactor neutrino fluxes mass are differences given and in measured ref. mixing [ angles using old/new principle to neutrino oscillation measurements, while the Majorana phases cannot. Recent JHEP09(2012)005 We ) or ), so form (2.1) (2.2) 1 2 . m B We also Ri 2). Then T & N 3 . . √ f, L/R 2 h.c. , ( Rj for the SU(2) i, j e + U , v/ ij R > m 0 Lj ) e ` 3 = (0 † m ˜ i e, R H ( T ij U ) and also the index on the ) H 3 h 0 , ν = 2 y , 1 ( ) corresponding to 0 Ri m 0 Ri ¯ N e, L ( − U , e 0 Rj Lj e . We use this notation to distinguish these 0 ` i ij N ij ) ) and ) 0 E y ( ]. ν, L L, L ( 22 ( H U – ], with three right handed neutrinos U – 3 – 0 Li the charge conjugation matrix. There is freedom 20 ¯ ` 19 3. = , – C for the three 2 − ). In our initial discussion we will assume a normal , 0 Li 17 2 0 ), such as the flavour basis. In this paper, we develop a Rj m Using this approach we show a basis independent rela- with N a, b, c ]. 1 . 0 e, L 7 ij T Ri ( , ` 1 ¯ M N U Rj is the Higgs doublet of the SM, with C c ) for the left handed SU(2) doublet field and 0 R i N < m L H = N ij 3 , e ) 1 2 c L m Ri , which run over 1 0 ν i N − N N,R ( 0 = ( Ri , where ? ` U reported in ref. [ H ∂/N = ]. 2 13 0 Ri 16 θ 0 i τ Ri ¯ – N N i = 14 , ˜ 9 = H L It is of interest to have a formalism for neutrino phenomenology that is as robust and This is a modern implementation in the neutrino sector of an oldOur idea initial of discussion relating will theThese largely mixing flavour follow matricies indices ref. [ will run over 2 3 1 of the standard model (SM)see to ref. the [ measured quark or neutrino masses, for pioneering studies with this aim flavours from the measuredYukawa mass vectors differences of of each the physical eigenstates ( here we have defined to rotate to thethat mass basis by introducing the unitary rotation matricies define the lepton sector of the Lagrangian in the seesaw scenario is the following 2.1 Review ofRecall seesaw the the standard mechanism seesawuse scenario the [ notation singlet lepton field carrying hypercharge. These fields carry flavour indicies eigenstates with the structure of theexpansion PMNS (FSE). matrix, There which is we will anthis refer experimental to time. ambiguity as in a The the flavour space an measured neutrino mass inverted mass hierarchy hierarchies spectrum at ( canhierarchy. The be formalism a can normal be hierarchy reinterpreted ( for an inverted hierarchy. of non zero 2 Constructing a flavourIn space this expansion section we for develop a neutrino formalism phenomenology linking the measured differences in the neutrino mass perturbative approach to the structurestudies of that the PMNS are matrix basis as dependent. tionship an between alternative the to eigenvectors symmetry thatat give leading order inthe our data expansion. in low scale We measurements alsobreaking, of illustrating examine neutrino how properties the our with prospects approach high can scale of be flavour relating symmetry used patterns to study in the recent reported discovery basis can be highly artificial. basis independent as possible. Clearlyrequires linking that symmetries the to physical thecan form consequences be of of arbitrarily the a choosen PMNS symmetry for matrix are not dependent on a basis that such a manner that the flavour basis cannot be chosen, or at least, the choice of the flavour JHEP09(2012)005 . . i c Ri N N (2.3) (2.4) ? i related η ν p , y + E Ri N ] i M, y η 23 √ = i a perturbative expansion N 4 , † is diagonal real and nonnegative. ) . , † . The key observation that we use ; see the next section for definitions and h.c. , † ) ) M ij ) a,b,c matrix and ν N,R ) + ~ρ y ( in sequence j L, L e, R ν ( ` 1 ( U y i † − U U are the Majorana phases. Working in this N ˜ ν M H E y ( ? y ηM ) ) ) ij a,b,c T ν effects the quality of the FSE. Experimentally, – 4 – η C y 5 ) i N,R L, L N,R defined by the set of all possible ` states, which are not mass diagonal in general. ( ( ( = ( † ν ˜ Ri ij H ( N ), here , y C → U → U → U 1 2 E 0 0 ν 0 E a,b,c y = ], however, the formalism we will develop is distinct from y M µ one obtains the dimension five operator [ 5 ( 13 M, y i L – N a,b,c 10 η = | ). Initially the Majorana mass matrix has three complex eigenvalues a,b,c can be constructed. Note that this is also the key point in the sequential M | N,R ij ( with respect to the leading order eigenvectors ] shifts any Majorana phases into the C U for such states, compared to a,b,c 22 i η – N x~,~z ~x,~y, = 20 We distinguish in this paper right handedBy neutrinos the that flavour are orientation in of a the mass Yukawa couplings, diagonal we basis, mean reserving the the orientation of the Yukawa coupling 4 5 a,b,c FSE would be of some utility. notation vectors further details. the low energy parameterswe in develop this allows effective perturbative theoryangles investigations in in of this many the paper. scenarios neutrinounknown and However, mass parameters the is conspired spectrum to formalism in and forbid factseek an mixing quite expansion to of general. illustrate this this form It point from would being in be present. this surprising We section, if by all demonstrating of a simple the scenario where the Also, in principle, theat flavour this orientation time, all offorced these to assume Lagrangian that parameters a are perturbativeis individually expansion of to unknown, the thus view form we we the willattempt employ be exists. seesaw to uniquely Lagrangian Our identify approach given and restrict in ourselves eq. to a (6) particular UV as physics an that dictates effective theory, and we do not 2.2 Perturbing the seesaw 2.2.1 Model dependenceThe of FSE perturbing we the develop seesaw depends on the mass spectrum and the Yukawa couplings of the with the matrix Wilson coefficient in this work isfor that neutrino phenomenology when that integratingbutions is out related to the to adominance hierarchy in idea the of magnitude refs.these of [ past the results. contri- basis [ Integrating out the heavy We choose to make the initialThis rotation to fixes a basis where M basis in the flavour spacethrough of The kinetic terms are unchanged by these rotations, and one is free to further rotate to a JHEP09(2012)005 6 ). ij  (2.5) (2.6) (2.7) ). can be 2 . (1+ 0 ) ij 0 2  ) in generic , ,  M M ( (1 O 2.1 ' 0 are not distin- 0 0 ij M 0 M Ri i, j. M ' N = . ) masses are the same, 3 3 L 2.4 , µ ν 2 , µ and the Yukawa couplings is naively expected to have , µ   1 0 ij ij µ no sum on  a,b,c M µ i m c β L + – 8 – | | ~x ~x ~x b 2 ` i | | | ~z ~z 3 m 2 a N ) ) ) | | ~x 2 | ? ? ? 2 2 m δ m δm ~z z z and x is required. Expanding on this important point in h i| i| · · 2 ab 2 ba 2 ca · i ~z ~z = = = 2 a | | x z x c b ( ( ( m δm δm δm ~ρ ~ρ 2 A 2 C Re [ 2 B δm i i i b c ~z |h |h m ~ρ ~ρ m | m 2 ac receives corrections at linear order to its mass, while the b | |  µ 2 cc 2 cc ~ρ ~z ~z a 2 i h h h µ µ m 2 ac 2 ac 2 ac = = 2 = depends on the hierarchy in µ µ µ , δm 2 c 2 b i 2 a 2 i = = = m m m m m 2 2 2 2 b c a 2 δ δ δ ~ρ ~ρ δ ~ρ δ 2 2 2 δ δ δ and i , only 2 i δm δm  It is also important to note that this formalism does not require a hierarchy of the form Finally integrate out the third right handed neutrino 2 i m 2 for an inverted hierarchy. AnFSE, inverted or hierarchy one requires can non triviallyThe generic modify size perturbations the of in expansion the tovectors, only and perturb also when the orientationwill in discuss flavour the space of size the of vectors the perturbations of the FSE in light of neutrino mass data in a 2.4 Inverted and normalFor a hierarchy normal and hierarchy flavour (withthe space notation equations above. The difference betweenappears the in normal the and inverted relative ( size of the more detail, it is not requiredthe that perturbation the perturbation due due to toof integrating integrating these out out right handed neutrinosintegrating perturbs out the the initial initial mass rightare matrix handed not — neutrino which necessarily is direct dominateddetail statements by in on the the next section. relative size of the It is interesting tothe note that leading a neutrino normal mass remaining hierarchy emerges masses only quite receive naturally corrections from at the second FSE order as inδ the perturbations. The measured masses of the neutrino’s are related to these perturbations as The mass perturbations are couples into a lineareigenvector combination perturbations are of the JHEP09(2012)005 . , 2 21 xz θ m 2 (2.20) (2.21) (2.19) ∆ 2 eV at , yields a . . Gener- cos 2 32 c pl < . We will 4 ij | so that the µ ) m M ~z O | e | a,b,c ~z ν ∆ ' ∼ ( 2 cc µ b integrated out, µ 0 < m µ c + 2 c | ∼ | M ~y N ' | m xy , 2 a θ requires more precise δ 2 a & µ 2 | | m ~z ~x 05 eV. This condition is | . cos ≈ 0 4 | number, and this condition ) ~y 2 b | ∼ | ∼ | L ~y . follows, m ] which quote 2 bb 2 2 32 µ R    δ 25 m  + N | ∼ | + ∆ xz ~x 2 b | θ 1 + p 2 (1) with the orientation in flavour space δm   . Consider the generic case in a normal ( < c O cos ) − c 2 1 + | 2 a µ ~z < µ    | m 2 b (2 | 2    – 9 – / δ ~x 2 | 1 + and the expansion requires | , + < µ ~z 2 ac z |    b 2 21 2 a ], assuming ΛCDM cosmology. It is also consistent µ 2 a ρ decay experiments [ 0 v θ µ m are similar in magnitude and 24 β 2 δm +2 M ∆ 2 ij xy + µ ' cos ∼ θ 2 a , i.e. i 2 0 4 2 b ij | m µ ~z | M cos δm = 2 cc 2 | µ ~y 2 32 | + 2 . m | y z ~x b c ∆ | ρ ρ θ θ 2 ab 2 2 µ GeV. Generically one expects the mass scale of the right handed neutrino vectors primarily. An example where this is the case is when the threshold 28 eV quoted in ref. [ cos cos +2 . 14 4 4 4 | | | ~x 10 ~y ~z | = 0 | | 2 bb 2 cc 2 aa & ν x~,~z ~x,~y, µ µ µ 2 m | Expressing this condition in terms of the high mass scale of the = = = ~z | P / 2 A 2 C 2 B c m m m Consider the case that allmass of the spectrum is primarilyspace. dictated An by example the of an orientation inverted of or the normal Yukawa hierarchy vectors is in shown flavour in figure 1 in this case. ating a normal oralignments inverted in hierarchy flavour space if and one allowsmass also a spectrum. geometric has In interpretation of the the FSE, measured the neutrino tree level masses of the SM neutrinos are mass matrix with nearly degenerate eigenvalues. This occurs for example when In this case, a nearly degenerate mass spectrum of the 2.4.2 The expansionNow with consider the flavour case alignment where theof perturbative expansion the follows from the flavour orientation matching onto the UV physics is such that the Wilson coefficient matrix of with current bounds from95% Tritium C.L. µ operator to be thefor largest the scale integrated lightest out neutrino that is violated clearly consistent with naive expectations of while in the sameDue manner to this, theconsistent expansion with current requires bounds on theof absolute neutrino mass scale, with a 95% C.L. bound that the Yukawa coupling vectors arenot taken to significantly be effecting the qualityfrom of the the relative expansion. sizehierarchy. In of Using this the the case, FSE the and expansion retaining follows the dominant term then discuss the caseYukawa coupling where vectors the in expansion flavour space. arises primarily2.4.1 due to the The orientation expansionWe of can without examine the flavour the quality alignment of thein expansion by neutrinos. comparing to Consider the measured the mass differences case that the size of the perturbations is generic in the sense scenario where the perturbations are dictated primarily by a hierarchy in JHEP09(2012)005 ~x ) = and two c (2.22) (2.23) µ ~z ) in the ν, L ( ' U e, L b ( µ lightest U ' ~y ). As such we do a µ e, R a ( . U ~ρ is given by ) i ? ~ρ ) 2 i ) when a seesaw mechanism is ~ρ , δ . matrix is ensured when an 2 i ) δ τ ν, L δ~ρ ( + ( U i , m O µ δ~ρ PMNS b c + , U ~ρ ~ρ , m + e       i . In a mass degenerate case, they are − 3 2 1 ~ρ 3 2 1 m L ~σ ~σ ~σ ~σ ~σ ~σ ν · · · · · · = ( 1 1 1 1 1 1 ? i ~ρ ~ρ ~ρ ~v ~v ~v ~v 3 2 1 3 2 1 ~σ ~σ ~σ . As we have discussed, these neutrinos can be ~σ ~σ ~σ the orthonormal column eigenvectors diagonal- ) = diag( · · · · · · – 10 – i 2 2 2 2 2 2 ~σ ~x,~y ~ρ ~ρ ~ρ ~v ~v ~v 11 ) with e, L ( 3 2 1 3 2 1 ~z is of the form † ~σ ~σ ~σ ~σ ~σ ~σ ~x ~y · · · · · · U ν, L ( e 3 3 3 , with 3 3 3 in principle. Conversely, in principle, one could employ an ansatz ~ρ ~ρ ~ρ ~v ~v ~v  . The unitarity of the U † e M ? 1 PMNS Lj       ) ` U M , ~σ † e = = † 2 e, R M . This ambiguity also limits the application of a FSE to the CKM matrix. ( † ) , ~σ U † 3 ~σ e, L ? PMNS ( a  U U ~ρ = 2 and diagonalized by † √ ) and flavour space / . The expanded E . Normal hierarchy on the left, inverted on the right. † e | e, L ~z ( . The rotation matrix v y ). The charged lepton mass matrix after electroweak symmetry breaking is given M eigenvectors only depended on . A geometric interpretation of an inverted or normal hierarchy when U ? 1 e i = PMNS | ∼ | ~ρ ,~v U e ~y b M ? c 2 | ~ρ PMNS ~ρ The hierarchy of the charged lepton masses can be used to organize an expansion of can lead to the largest mass eigenvalues of the M ,~v & U ? 3 11 i | − ~v ~x same manner by constructing that the hierarchy innot these employ mass a eigenvalues FSE couldIn on be this manner, related the to expansionthe an we origin employ expansion of is of the most smallness useful of for the expanding neutrino masses. the FSE contribute to theof leading the order structure ofintegrated the out PMNS in matrix. sequence, The andN leading for order Yukawa couplings thatmore are strongly degenerate coupled the to the This result makes clear that the first two right handed neutrinos that were integrated out in We take izing ( by | 2.5 Now, assuming that a FSE expansionof exists, let us consider its utility in examining the form Figure 1 JHEP09(2012)005 . = =     1 (3.2) (3.3) (3.4) (3.1) being ~σ 12 2 2 c i 2 12 2 12 √ √ c c 0). Now σ PMNS 12 , s /v. U 0 b , , ) µ 12 , 3 2 2 c b ) 2 12 2 12 ~ρ √ √ c c 3 12 ~ρ δm 12 s and the ) = (0 1) and s c 12 , p 1 2 12 12 12 s 1 − ~ρ s c c  , , δ ~ρ 2 − 2 b 12 ~ρ 12 12 . 2 s s √ 3 ~ρ , c 12 , δ ~ρ ~ρ c     12 a form 12 12 c a + , c c δ ~ρ µ 1 + 12 b ~ρ + s ) = diag(1 1 2 2 . − ~ρ this gives (in a particular phase 2 eigenvectors order by order in the v ( ~ρ ( δm i PMNS √    2 2 e, L 2 ~v 12 12 1 1 (  at leading order in the FSE. This is U 2 s , √ √ s √ 1 2 U √ 1 0 = 0 and ( = √ + = † = = TB a − T 3 U 2 1    ) 2 2 12 12 12 a 1 δm PMNS √ √ c c s U − 2 2 δm 12 12 1 1 √ √ s s 12 – 11 – + c − a − − = 0 so 0 00 0 1 0 , ~σ matrix to proceed simply through solving 3    /v, ~y ~y , m , ~σ form at leading order in the FSE has a simple inter- ~σ    a b · 3 = µ ~σ ) form of ~x a 12 0). In this case, ) c , µ B a 12 TB , δm ) 0 PMNS TB s a , T U U , ~σ ) + δm maximal. We restrict the Yukawa vectors in this discussion 2 2 ) δm ) + v 2 + ~σ 2 23 ~σ = (1 + diag(0 a ~σ θ + a matrix is not necessarily unitary. 3 . Exact − m 1 ? + TB m ( 1 TB ~σ ( 1 U , ~σ ( ~σ U p ~σ ( 2 0) ( 12 = = 0 and , = 2 2 1) s √ PMNS PMNS 1 12 2 in our chosen phase convention. 1 , T T U , c √ √ U − 1 ≡ √ ~y~y ~y~y / − = = = b ) b 13 , a 2 2 θ µ µ µ PMNS = (0 3 1 2 v ~σ 2 ~ρ ~ρ ~ρ U 2 + eigenvectors and the = (0 − + T i 1 , ~σ T T ~ρ ). The flavour basis is appealing in that it allows the solution for the perturbations ~σ ~x~x ~x 1) form. Consider an exact ( ~x~x Consider the “flavour” basis as an illustrative example of the FSE, where It is important to note that, in general, it is the relationship between the eigenvectors , a 0 = ( ν, L 2 , µ ( v 1 2 ~ρ Trivially one finds It follows that in theconsider flavour including basis a third neutrino eigenvalue, retaining the required perturbation. For a (0 U to the Assuming dictated by to real values forconvention) simplicity. Including an unfixed 3 (Un)relating flavour symmetriesThis to formalism the can be usedTB to systematically explore symmetries that are consistent with expansion. Conversely, if thethe FSE constructed is used when the expansion parametersthat are determines not small, pretation. It follows from the relationship between the eigenvector expansion of this form is employed by normalizing the JHEP09(2012)005 = TB TB 0 . (3.7) (3.5) (3.6) T . ~y ! 36 2 12 s symmetric ± + 0 2] at leading ± . , 2 12 so that 5 . c = 2 b b c ], as such, we use ) ) N m m 3 form of the PMNS in the flavour basis, 2 2 12 23 δ δ c = 0. As a result the δθ δθ ( ( r TB ~z 2 2 ,N flavour symmetry in its form. It is instructive to · b 2 ] and taken a 1 and treat this breaking as τ ~x N 3 √ tan tan , 2 , z z TB measured by the DAYA-Bay c c ↔ 2 12 ) is given by [0 ρ ρ the eigenvectors determine the s θ θ 13 13 6= ∆ µ 2 2 assuming a normal hierarchy. At θ + δθ 1 10 = 0 as . ( 2 12 2 0 2 AC 2 a c but deviating from the flavour basis, + cos b c ± m m form can be falsified. One can perform sin between ~z 2 z + 2 cos b m m ∆ / δ 01 ρ ) 2 2 z . is consistent with b θ δ δ retains a TB ' ρ 23 ) with ∆ 2 has the pattern and ~z θ 2 b r δθ = 0 2 ∆ ( ~y N 2 AB 2 ) , 2 ) 1 – 12 – , . TB 1 m √ 2 cos cos form can be studied using the FSE and compared breaks such a symmetry so that deviations in and 23 13 , form ∆ c ~y , δθ δθ = 0 = = 0 2 12 ( N ( TB s 2 2 ) ) ) ) ) ) form. The value of b TB c 12 23 12 23 + (0 13 13 sin m m tan 2 2 T δθ δθ δθ δθ δθ δθ TB δ δ ( ( ( ( ~z ( ( form is broken in the flavour basis by 2 2 2 2 2 2 + = sin sin 0 tan tan tan tan 2 12 TB 32 T . c form are also correlated with mass perturbations in each case ~z 0 r is unperturbed by integrating out both ±

a TB ~ρ symmetry imposed on c 02 . is required, and consequently m τ 2 3 δ ). Using the expansion one finds the pattern of deviations are distinct. z = 0 v 2 c ↔ ) µ ) ∆ = µ 2 , 12 13 1 2 symmetry implemented on √ z δθ δθ ∆ τ ( ( ± , 2 2 = ↔ We emphasize however that the relationships sin tan T + (0 µ ~z T These breakings of which are trivial to determine using this formalism. form of the PMNS matrixretaining in general. a This is easy to demonstrate in more detail. Consider As the range oforder the predicted in value the of FSE,way, tan particular this mechanisms pattern of of thethe flavour breaking of exercise breaking does of not assuming reproduce the data. In this couplings to the charged leptons,form but are expected.a Fix perturbation using theleading FSE. order We in use the ∆ FSE the breaking of Here we have used the oneone sigma sigma new error. reactor flux Various values breaking ofto of ref. [ these results. Consider the case that collaboration is in agreementthe with fit the results global toconstruct find fit the the values following measured given ratios pattern inform of of ref. in experimental deviations each [ values in mixing characterizing angle. the Using deviations the of small angle approximation A matrix, as expected. 3.1 Perturbative breaking of Now consider the breaking of leading eigenvector one finds solution, ~y JHEP09(2012)005 . i 13 θ N TB TB one (3.8) (3.9) .    PMNS

U 6 6 eigenvectors), √ √ i σ − . One finds that i 6 3 3 + 1 ~σ √ √ √ in the FSE for the matrix only in a par- + − 2 2 2 + 3 √ ~x,~y √ √ − PMNS 1 6 1 6 U .     form is obtained, one finds 3 3 3

1 1 1 6 6 √ √ √ matrix. TB √ √ 2 3 form to the particular Yukawa coupling 6 6 2 as a simple interaction eigenvector for ) symmetry on the first interaction eigen- − − 1 1 q √ √ √ 3 3 − / TB 3 2 √ √ PMNS 2 √ 0) √ 1 2 U e, µ, τ , √ − 1 3 0 2+ √ 1 – 13 – 2 2 + − , − . At leading order in the expansion of √ √ i     symmetry is also not unique or of particular physi- ~σ form is still obtained for general 1 6 1 6 = (1 τ ) = T ↔ ~y TB form of the form (with an appropriate choice on the µ ν, L

( 3 3 U gives a valid solution at leading order in the expansion. This ) can lead to suspect physical conclusions. As an alternative to TB TB √ √ i 2 3, and ) so that at leading order ) basis independent manner. This formalism can be employed in 2 ~σ 3 symmetry imposed on the Lagrangian alone is not related to form at leading order in the FSE. Further, as the breaking of √ √ e, L − ( / e, L τ ν, L 2 + 2 2 − U ( ( 1) √ √ TB , U U ↔ 1 , 1 6 1 6 µ as above but unfixing    = (1 T ) = ~x x~,~z ~x,~y, matrix in a e, L form being broken. However, the approach we outline can accommodate any basis ( the flavour basis for U We have illustrated our approach in an example where the flavour basis was chosen, We also note that the impact of sterile neutrinos weakly coupled to the SM on neutrino As a further example that leading to an orthonormal eigenbasis at leading order, finding TB b PMNS of choice. Indeed, it is theU relationships between the eigenvectors thatmodel dictate building the to form attempt of tothat the determine corresponds a to compelling origin of the eigenvector relationship basis dependent symmetry studies, we havemeasured developed a masses perturbative of expansion relating the the neutrinosa to promising the framework form for ofimproving broadly the experimental PMNS neutrino understanding matrix. data, the particularly This implications in expansion of a offers normal the hierarchy. for systematically the sake of familiarity, and then shown how the expansion can control the predictions 4 Conclusions Flavour symmetries that are relatedticular basis to choice the of structure of the phenomenology can be systematicallyrelate studied any measured with value this ofvector approach. a of deviation For a from example, single onereported sterile by can neutrino, the Daya-Bay which collaboration canthe while SM the be three fields shown right give to handed an neutrinos accommodate exact partners the of value of This procedure can be used for any flavour symmetry chosen to fix Solving directly for cal significance in allowing consider the following procedure. Choosevector, a ( ie N can solve for theonly condition that makes clear that form in a basis independent manner. choosing JHEP09(2012)005 , ], 7 , ]. (2004) ]. 6 (1998) 350 in ref. [ SPIRE ]. from global IN 13 ]. SPIRE θ 0 ]. IN > B 439 SPIRE ][ IN 13 SPIRE θ IN ][ SPIRE ]. New J. Phys. IN (1977) 185. ][ (2000) 85 , (1957) 429 [ ][ 38 6 SPIRE ]. Phys. Lett. IN , ]. B 576 ][ Evidence of SPIRE arXiv:1106.6028 SPIRE IN [ A Model of Lepton Masses from a hep-ph/9904210 IN ][ [ ][ arXiv:1203.1669 [ arXiv:0806.0356 Nucl. Phys. [ , Sov. Phys. JETP , hep-ph/0204360 (1999) 57 ]. [ Observation of electron-antineutrino disappearance (2011) 053007 – 14 – matrix, as the pattern of this breaking is further Tri-bimaximal mixing and the neutrino oscillation Remarks on the unified model of elementary particles (2008) 055 Trans. New York Acad. Sci. SPIRE hep-ph/0202074 B 562 (2012) 171803 , [ D 84 IN 10 arXiv:1002.0211 [ [ PMNS ]. (2002) 011 108 U Models of neutrino masses and mixings Discrete Flavor Symmetries and Models of Neutrino Mixing 09 ]. ]. JHEP SPIRE , (2002) 167 IN Phys. Rev. Nucl. Phys. (1962) 870 , , ][ SPIRE SPIRE JHEP (2010) 2701 , 28 IN IN ][ ][ B 530 82 form of the Mesonium and anti-mesonium This article is distributed under the terms of the Creative Commons Phys. Rev. Lett. collaboration, F. An et al., The Problem Of Mass , TB Atmospheric and solar neutrinos from single right-handed neutrino dominance and Large mixing angle MSW and atmospheric neutrinos from single right-handed Atmospheric and solar neutrinos with a heavy singlet Constructing the large mixing angle MNS matrix in seesaw models with right-handed Phys. Lett. hep-ph/0405048 , [ hep-ph/9912492 hep-ph/9806440 neutrino dominance and U(1)[ family symmetry neutrino dominance [ U(1) family symmetry Rev. Mod. Phys. DAYA-BAY at Daya Bay Warped Extra Dimension data 106 Prog. Theor. Phys. neutrino data analysis S. Weinberg, C. Cs´aki,C. Delaunay, C. Grojean and Y. Grossman, P. Harrison, D. Perkins and W. Scott, G. Altarelli and F. Feruglio, G. Altarelli and F. Feruglio, B. Pontecorvo, Z. Maki, M. Nakagawa and S. Sakata, G. Fogli, E. Lisi, A. Marrone, A. Palazzo and A. Rotunno, S. King, S. King, S. King, S. King, [9] [7] [8] [4] [5] [6] [1] [2] [3] [12] [13] [10] [11] Open Access. Attribution License which permits anyprovided use, the distribution original and author(s) reproduction and in source any are medium, credited. References We thank Mark WiseEnrique for Martinez for collaboration useful in conversations and theThe C. work Grojean initial of for stages comments B.G. on of wasDOE-FG03-97ER40546. the this supported manuscript. We thank in work. the part Aspen We Centre by for also the Theoretical Physics US thank for Department hospitality. of Energy under contract we expect the FSE tobreaking be of of the some phenomenologicalresolved utility experimentally in in falsifying the mechanisms years of ahead. the Acknowledgments form is now experimentally established due to the discovery of a non zero JHEP09(2012)005 , B Phys. , (2006) Nucl. Phys. ]. , (1979) 1566 Phys. Lett. B 636 , 43 Rept. Prog. Phys. SPIRE 13 , θ IN [ (2010) 031301 Proc. of the workshop on the 105 Erratum ibid. [ (1977) 436 , in Phys. Rev. Lett. , , Amsterdam, North Holland (1979), B 70 , KEK, Tsukuba (1979) pg. 95. (2003) 177 ]. Phys. Rev. Lett. ]. ]. , ]. B 552 Supergravity Phys. Lett. The Effective Lagrangian for the seesaw model of Upper Bound of 0.28eV on the Neutrino Masses Neutrino physics in the seesaw model , SPIRE – 15 – SPIRE SPIRE ]. IN IN SPIRE IN [ Neutrino Mass and Spontaneous Parity Violation ][ IN ][ ][ ]. Tribimaximal Mixing, Leptogenesis and SPIRE ]. Phys. Lett. IN , Hierarchy of Quark Masses, Cabibbo Angles and CP-violation [ Neutrino mass limit from tritium beta decay Mixing and Relations Among Quark Masses, Angles and ]. SPIRE ¯ (1987) 586 B SPIRE IN − [ IN SPIRE B ][ IN B 195 (1979) 277 hep-ph/0307058 arXiv:0909.2104 ][ [ [ arXiv:0807.4176 [ (1980) 912 Horizontal Symmetry and Masses of Neutrinos Baryon and Lepton Nonconserving Processes Calculating the Cabibbo Angle B 147 44 ]. Phys. Lett. (2003) 163 , hep-ph/0210271 (2008) 210 (2008) 086201 SPIRE arXiv:0911.5291 IN from the Largest Photometric Redshift[ Survey 71 668 [ neutrino mass and leptogenesis 330] [ B 672 pg. 315. unified Theory and Baryon Number in the Universe Rev. Lett. Phases Nucl. Phys. E. Otten and C. Weinheimer, E.E. Jenkins and A.V. Manohar, S. Weinberg, S.A. Thomas, F.B. Abdalla and O. Lahav, A. Broncano, M. Gavela and E.E. Jenkins, A. Broncano, M. Gavela and E.E. Jenkins, T. Yanagida, R.N. Mohapatra and G. Senjanovi´c, H. Harari and Y. Nir, C. Froggatt and H.B. Nielsen, M. Gell-Mann, P. Ramond and R. Slansky, H. Fritzsch, [25] [22] [23] [24] [20] [21] [18] [19] [15] [16] [17] [14]