implications of cp 0 decay - hindawi publishing...

Research ArticleImplications of 120575CP119897 sim 270∘ and 12057923 ≳ 45∘ for TextureSpecific Lepton Mass Matrices and 0]120573120573 Decay

Rohit Verma

Chitkara University Barotiwala Himachal Pradesh 174103 India

Correspondence should be addressed to Rohit Verma rohitvermalivecom

Received 30 June 2016 Accepted 10 October 2016

Academic Editor David Latimer

Copyright copy 2016 Rohit Verma This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We study the phenomenological consequences of recent results from atmospheric and accelerator neutrino experiments favoringnormal neutrino mass ordering1198981 lt 1198982 lt 1198983 a near maximal lepton Dirac CP phase 120575119897 sim 270∘ along with 12057923 ≳ 45∘ for possiblerealization of natural structure in the lepton mass matrices characterized by (119872119894119895) sim 119874(radic119898119894119898119895) for 119894 119895 = 1 2 3 It is observed thatdeviations from parallel texture structures for119872119897 and119872] are essential for realizing such structures In particular such hierarchicalneutrino mass matrices are not supportive for a vanishing neutrino mass 119898]1 rarr 0 characterized by Det119872] = 0 and predict119898]1 ≃ (01ndash80)meV 119898]2 ≃ (80ndash130)meV 119898]3 ≃ (470ndash520)meV Σ ≃ (560ndash710)meV and ⟨119898119890119890⟩ ≃ (001ndash100)meVrespectively indicating that the task of observing a 0]120573120573 decay may be rather challenging for near future experiments

1 Introduction

One of the intriguing phenomena in particle physics is theorigin of fermionmasses which appear to span several ordersof magnitudes starting with neutrinos to the top quark Themasses and flavor mixing schemes of quarks and leptons aresignificantly different with the quark sector exhibiting strongmass hierarchy small mixing angles and relatively heaviermass spectrum whereas the neutrinos are extremely lightwhile two of their mixing angles are still large In the currentscenario there is also a lack of consensus on the nature ofneutrinos that is Dirac orMajorana alongwith doubts on thepossible ordering of neutrinomasses namely normal that is1198981 lt 1198982 lt 1198983 (NO) or inverted that is1198983 lt 1198981 lt 1198982 (IO)This nevertheless makes the task of constructing the fermionmass matrices nontrivial especially in the context of quark-lepton complementarity

The confirmation of Higgs Boson by the ATLAS andCMS collaborations [1] completes the Standard Model (SM)of particle physics Within this model the quark mass termsin the Lagrangian are expressible asminus119871quarksmass = 119902119906119871119872119906119902119906119877 + 119902119889119871119872119889119902119889119877 + hc (1)

where 119902119906119871 (119877) and 119902119889119871 (119877) are the left (right) handed quark fieldsand 119872119902 are the quark mass matrices with 119906 119889 for the ldquouprdquotype and ldquodownrdquo type quarks The resulting weak chargedcurrent quark interactions are given by

minus119871quarkscc = 119892radic2(119906 119888 119905)119871120574120583119881CKM(119889119904119887)

119871

119882+120583 + hc (2)

where 119881CKM = 119880119906dagger119871 119880119889

119871 is the Cabibbo-Kobayashi-Maskawa(CKM) matrix [2 3] or the quark mixing matrix measuringthe nontrivial mismatch between the flavor and mass eigen-states of quarks for example119880119906dagger

119871 119872119906119872dagger119906119880119906

119877 = Diag (1198982119906 1198982

119888 1198982119905 ) 119880119889dagger

119871 119872119889119872dagger119889119880119889

119877 = Diag (1198982119889 1198982

119904 1198982119887) (3)

where 119880119902 are unitary matricesInterestingly the quark masses as well as the elements of

CKM matrix observe a hierarchical pattern namely 1198981 ≪1198982 ≪ 1198983 and (119881119906119887 119881119905119889) lt (119881119888119887 119881119905119904) lt (119881119906119904 119881119888119889) lt(119881119906119889 119881119888119904 119881119905119887) It is natural to expect this hierarchy to be

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 2094323 11 pageshttpdxdoiorg10115520162094323

2 Advances in High Energy Physics

embedded within the corresponding quark mass matricesnamely (for 119902 = 119906 119889)11987211 lt 1198721221 ≲ 1198721331 lt 11987222 lt 1198722332 lt 11987233 (4)

with 11987222 ≪ 11987233 Recent investigations [4] in this regardindicate that the current quarkmixing data indeed permit thequark mass matrices to have such a natural and hierarchicalstructure provided (119872119894119895) sim 119874(radic119898119894119898119895) for 119894 119895 = 1 2 3 119894 = 119895and (119872119894119894) sim 119874(119898119894) Such hierarchical mass matrices havebeen referred to as natural mass matrices [5] In particularnaturalness provides a rationale framework to correlate theobserved fermionmass ratios the correspondingmassmatri-ces and observed mixing angles Specifically for (11987213) =(11987231) = 0 the observed strong hierarchy among the quarkmasses and CKM elements gets naturally translated onto thestructure of the corresponding mass matrices

A concomitant of such naturalness in mass matrices isthe absence of parallel texture structure for the ldquouprdquo andldquodownrdquo type quark mass matrices [4] A parallel texturestructure corresponds for example to the mass matrices119872119906

and119872119889 with texture zeros at identical positions in both themass matrices Hierarchical structures have fetched greaterimportance in the literature as these predict certain verysimple yet compelling relations among the CKM elementsand the quark mass ratios [4 6ndash14]

However the mass spectrum for leptons is quite distin-guished from the quark sector wherein the charged leptonsmasses are strongly hierarchical that is 119898119890 ≪ 119898120583 ≪ 119898120591while at least two of the neutrinos are allowed to have thesame order of mass It should be interesting to investigate ifnaturalness can provide a unique explanation for the fermionmassmatrices corresponding observedmass spectra and themixing angles both for the quark and the lepton sectors

Since the neutrinos are massless within the SM one hasto explore beyond the realms of SM to comprehend theorigin of neutrino masses and observe neutrino oscillationsphenomenon A simplistic way to achieve this is to extendthe SM theory by assuming neutrinos as Dirac-like particlesIn this case the neutrinos acquire mass through the Higgsmechanism in the similar way as quarks and charged leptonsdo within the SM through a Dirac mass term for exampleminus119871119897mass = 119897119871119872119897119897119877 + hcminus2119871Diracmass = ]119871119872]119863]119877 + hc (5)

where 119872119897 and 119872]119863 represent the charged-lepton and Diracneutrinomassmatrix respectively Indeed the current exper-iments have not ruled out such a possibility In this contextit is also observed that highly suppressed Yukawa couplingsfor Dirac neutrinos can naturally be achieved using modelswith extra spatial dimensions [15 16] or through radiativemechanisms [17ndash22] However such a possibility is perceivedto be highly unlikely due to several orders of magnitudedifference among119898120572 (120572 = 119890 120583 120591) and119898]119894 (119894 = 1 2 3)

A rather convincing and natural explanation of neutrinomasses can be obtained if neutrinos are assumed to beMajorana particles This usually involves adding the lepton

number (and flavor) violating Majorana mass terms forneutrinos in the Lagrangian for exampleminus2119871Majorana

mass = ]119871119872]119871]119888119877 + ]119888119871119872]119877]119877 (6)

where119872]119871 and119872]119877 correspond respectively to the left andright-handedMajorana neutrinomassmatrices and the latterusually has an extremely high mass scale This facilitatesgenerating the light neutrino masses through Type I or TypeII seesaw mechanisms namely119872] = minus119872119879

]119863119872minus1]119877119872]119863 119872] = 119872]119871 minus119872119879]119863119872minus1

]119877119872]119863 (7)

where119872] is usually a complex symmetric matrix for exam-ple

119872] = (119890] 119886] 119891]119886] 119889] 119887]119891] 119887] 119888]) (8)

This allows writing the corresponding charged weakcurrent term for leptons as

minus119871leptonscc = 119892radic2(]119890 ]120583 ]120591)119871120574120583119881(119890120583120591)119871

119882+120583 + hc (9)

where 119881 = 119881PMNS = 119880dagger119897119871119880]119871 is the Pontecorvo-Maki-

Nakagawa-Sakata (PMNS) mixing matrix [23] or the neu-trinomixingmatrix and emerges through the diagonalizationof the matrices119872119897 and119872] for example119880dagger

119897119871119872119897119872dagger119897 119880119897119877 = Diag (1198982

119890 1198982120583 1198982

120591) 119880dagger]119871119872]119872dagger

]119880]119877 = Diag (1198982]1 1198982

]2 1198982]3) (10)

This mixing matrix relates the neutrino flavor states to theneutrino mass eigenstates through

]120572119871 = sum119894=123

119881120572119894]119894119871 (11)

In the standard parametrization [24] the PMNS matrix isexpressed as 119881 = 119880 sdot 119875119900 where 119875119900 equiv Diag119890119894120588 119890119894120590 1 with120588 120590 being two Majorana CP violating phases and 119880 can beparametrized in terms of three mixing angles 12057912 12057913 and 12057923and one Dirac CP violating phase 120575119897 namely119880= ( 1198881211988813 1199041211988813 11990413119890minus119894120575119897minus1199041211988823 minus 119888121199041311990423119890119894120575119897 1198881211988823 minus 119904121199041311990423119890119894120575119897 11990423119888131199041211990423 minus 119888121198882311990413119890119894120575119897 minus1198881211990423 minus 119904121199041311988823119890119894120575119897 1198882311988813 )

(12)

with 119904119894119895 = sin 120579119894119895 and 119888119894119895 = cos 120579119894119895 for 119894119895 = 12 13 23 Theneutrino oscillation experiments provide constraints on the

Advances in High Energy Physics 3

three mixing angles 12057912 12057913 and 12057923 along with the two masssquare differences namely 1205751198982 = 1198982

2 minus 11989821 and Δ1198982 =120578[1198982

3 minus (11989821 + 1198982

2)2] with 120578 = +1 for NO and 120578 = minus1 forIO cases

In the current scenario the global picture of neutrinooscillation parameters for NO at 3120590 suggests [25]1205751198982 = (699ndash818) times 10minus5 eV2Δ1198982 = (223ndash261) times 10minus3 eV2119904212 = 0259ndash0359119904213 = 00176ndash00295119904223 = 0374ndash0626120575119897 (1120590) = 201∘ndash239∘

(13)

Moreover the above data does not seem to forbid 119898]1 = 0for NO or 119898]3 = 0 for IO cases the signatures for whichare obtained through Det119872] = 0 The Planck collaborationmeasurements of the cosmic microwave background [26]provide further insight into the sum of absolute neutrinomasses for exampleΣ = 119898]1 + 119898]2 + 119898]3 lt 023 eV (14)

More recent results from long-baseline accelerator neutrinoexperiments T2K [27] and NO]A [28] are indicative of a nearmaximal Dirac CP phase that is120575119897 sim 270∘12057923 ≳ 45∘ (15)

along with preference for the normal ordering (NO) ofneutrino masses These results are also supported by the pre-liminary results from the atmospheric neutrino experiment atSuper-Kamiokande [28] In addition a statistical analysis ofthe cosmological data [29] also indicates preference for NOproviding maximum likelihood for Majorana effective massthat is ⟨119898119890119890⟩ lt 16meV (16)

in neutrinoless double beta decay at 1120590 where⟨119898119890119890⟩ = 10038161003816100381610038161003816119890119894120588 1003816100381610038161003816100381611988021198901

10038161003816100381610038161003816 119898]1 + 119890119894120590 1003816100381610038161003816100381611988021198902

10038161003816100381610038161003816 119898]2 + 1003816100381610038161003816100381611988021198903

10038161003816100381610038161003816 119898]310038161003816100381610038161003816 (17)

As the mixing angles are related to the correspondingmass matrices it therefore becomes desirable to study theimplications of a combination of NO 120575CP sim 270∘ alongwith 12057923 ≳ 45∘ for lepton mass matrices assuming quarksand leptonmass matrices have similar origins and investigatethe conditions affecting the possibility of obtaining naturallepton mass matrices synchronous with the quark sectorNevertheless from a top-down prospective it should bemoreeconomical to have a common framework explaining thefermion masses and mixing for the quark and lepton sectors

2 Lepton Mass Matrices

Phenomenologically the problem of constructing thefermion mass matrices has always been a difficult task withinthe framework of Standard Model (SM) and its possibleextensions wherein the flavor structure of these matricesis usually not constrained by the gauge symmetry As aresult the matrices 119872119897 and 119872] remain arbitrary 3 times 3complex matrices thereby involving several free parametersas compared to the number of physical observables namelysix lepton masses three mixing angles and one Dirac-likeCP phase 120575119897 along with two Majorana phases 120588 and 120590

In this regard the ldquotexture zerordquo ansatz initiated byWeinberg [30] and Fritzsch [31 32] has been quite successfulin explaining the fermion masses and mixing patterns [33ndash52] However one requires handling all possible texturestructures on a case to case basis In this context a commonframework allowing for the study of such possibilities is moredesirable This is addressed in the following section

3 Constructing the PMNS Matrix

In order to reconstruct the PMNS matrix one requiresobtaining the diagonalizing transformations for the corre-sponding mass matrices To start with for 119902 = 119897 ] weconsider the following two possibilities of texture one-zeromass matrices as

119872119902 = (119890119902119890119894120595119902 119886119902119890119894120572119902 0119886119902119890119894120572119902 119889119902119890119894120596119902 1198871199021198901198941205731199020 119887119902119890119894120573119902 119888119902119890119894120574119902)1198721015840

119902 = ( 0 1198861015840119902119890119894120572119902 1198911015840119902119890119894Δ 1199021198861015840119902119890119894120572119902 1198891015840119902119890119894120596119902 11988710158401199021198901198941205731199021198911015840

119902119890119894Δ 119902 1198871015840119902119890119894120573119902 1198881015840119902119890119894120574119902 )(18)

referred to as Type I and Type II structures respectively inthe following text

Onemay also consider thesematrices to be Hermitian forDirac neutrinos Using standard procedures it is not possibleto obtain the exact diagonalizing transformations for thelatter case In order to avoid a large number of free parametersin these matrices we assume that the phases are factorizablein these requiring

120595119902 = 2120572119902120596119902 = 0Δ 119902 = 120572119902 + 120573119902120574119902 = 2120573119902(19)


for symmetric119872119902 and1198721015840119902 and

120595119902 = 0120596119902 = 0Δ 119902 = 120572119902 + 120573119902120574119902 = 0(20)

for Hermitian119872119902 and1198721015840119902

The diagonalization of119872119902 above is realized using119872Diag119902 = 119874119879

119902 119902119874119902 = Diag (1198981 minus1198982 1198983) (21)

with 1 2 3 = 119890 120583 120591 for 119902 = 119897 and 1 2 3 = ]1 ]2 ]3 for 119902 = ]Here 119875119902 = Diag (119890minus119894120572119902 1 119890minus119894120581120573119902)

119902 = 119875119902119872119902119876119902 = (119890119902 119886119902 0119886119902 119889119902 1198871199020 119887119902 119888119902) (22)

and 119876 = 119875 (symmetric case) and 119876 = 119875dagger (Hermitian case)Considering 119890119902 and 119889119902 as free parameters one can write [45]

119874119902 =((((((((

radic (119890119902 + 1198982) (1198983 minus 119890119902) (119888119902 minus 1198981)(119888119902 minus 119890119902) (1198983 minus 1198981) (1198982 + 1198981) radic (1198981 minus 119890119902) (1198983 minus 119890119902) (119888119902 + 1198982)(119888119902 minus 119890119902) (1198983 + 1198982) (1198982 + 1198981) radic (1198981 minus 119890119902) (119890119902 + 1198982) (1198983 minus 119888119902)(119888119902 minus 119890119902) (1198983 + 1198982) (1198983 minus 1198981)radic (1198981 minus 119890119902) (119888119902 minus 1198981)(1198983 minus 1198981) (1198982 + 1198981) minusradic (119890119902 + 1198982) (119888119902 + 1198982)(1198983 + 1198982) (1198982 + 1198981) radic (1198983 minus 119890119902) (1198983 minus 119888119902)(1198983 + 1198982) (1198983 minus 1198981)minusradic (1198981 minus 119890119902) (1198983 minus 119888119902) (119888119902 + 1198982)(119888119902 minus 119890119902) (1198983 minus 1198981) (1198982 + 1198981) radic (119890119902 + 1198982) (119888119902 minus 1198981) (1198983 minus 119888119902)(119888119902 minus 119890119902) (1198983 + 1198982) (1198982 + 1198981) radic (1198983 minus 119890119902) (119888119902 minus 1198981) (119888119902 + 1198982)(119888119902 minus 119890119902) (1198983 + 1198982) (1198983 minus 1198981)))))))))

(23)

such that 119888119902 = 1198981 minus 1198982 + 1198983 minus 119889119902 minus 119890119902119886119902 = radic (1198981 minus 119890119902) (1198982 + 119890119902) (1198983 minus 119890119902)(119888119902 minus 119890119902) 119887119902 = radic (119888119902 minus 1198981) (1198983 minus 119888119902) (119888119902 + 1198982)(119888119902 minus 119890119902) 1198981 gt 119890119902 gt minus1198982(1198983 minus 1198982 minus 119890119902) gt 119889119902 gt (1198981 minus 1198982 minus 119890119902)

(24)

The above constraints on the parameters 119890119902 and 119889119902 neverthe-less allow hierarchical mass matrices that is 119890119902 lt 119886119902 lt 119889119902 lt119887119902 lt 119888119902 Texture rotation from (13) and (31) positions in119872119902

to (11) position in 1198721015840119902 is realized by rotating (11) element

in 119872119902 to (13) and (31) positions in 1198721015840119902 through a unitary

transformation 119877119902 on119872119902 using119872119902 997888rarr 1198721015840119902 = 119877119879

119902119872119902119877119902 (25)

for symmetric mass matrices and119872119902 997888rarr 1198721015840119902 = 119877dagger

119902119872119902119877119902 (26)

for Hermitian case where 119877119902 is a complex rotation matrix inthe 1ndash3-generation plane for example

119877119902 = ( cos 12057813119902 0 minus119890minus119894(120572119902minus120581120573119902) sin 120578131199020 1 0119890119894(120572119902minus120581120573119902) sin 12057813119902 0 cos 12057813119902 ) (27)

where 120581 = +1 for symmetric matrices and 120581 = minus1 forHermitian matrices

The condition of a texture zero rotation from (13 31)positions in119872119902 to (11) position in1198721015840

119902 requires0 = 119890119902cos212057813119902 + 119888119902sin212057813119902 (28)

which can be translated to

tan212057813119902 = minus119890119902119888119902 997904rArrtan 12057813119902 = 120591119902radicminus119890119902119888119902 (29)

where 120591119902 = plusmn1 and 119890119902 is always negativeNote that the rotationangle 12057813119902 is not a free parameter and is completely fixedthrough 119890119902 and 119888119902 due to repositioning of texture zeros asa result of the rotation 119877119902 One can now relate the matrix


elements in1198721015840119902 with the corresponding elements in119872119902 for

example 1198861015840119902 = 100381610038161003816100381610038161003816119886119902 cos 12057813119902 + 120591119902119887119902 sin 12057813119902 100381610038161003816100381610038161003816 1198871015840119902 = 100381610038161003816100381610038161003816119887119902 cos 12057813119902 minus 120591119902119886119902 sin 12057813119902 100381610038161003816100381610038161003816 1198881015840119902 = 119888119902cos212057813119902 + 119890119902sin212057813119902 1198891015840119902 = 1198891199021198911015840119902 = 1003816100381610038161003816100381610038161003816radicminus1198901199021198881199021003816100381610038161003816100381610038161003816

(30)

The texture rotation in 1ndash3-generation plane allows 1198891015840119902 =119889119902 Note that 1198911015840119902 prop radicminus119890119902 while the other off-diagonal

elements essentially get rescaled due to texture rotationFurthermore for 119890119902 sim minus1198981 one expects 1198911015840

119902 sim 119874(radic11989811198983)allowing hierarchical structures inType II possibility namely1198861015840119897 lt 1198911015840

119897 lt 1198891015840119897 lt 1198871015840119897 lt 1198881015840119897 along with 1198861015840] sim 1198911015840] sim 1198891015840] ≲ 1198871015840] ≲1198881015840] since 119874(radic119898]1119898]2) sim 119874(radic119898]1119898]3) sim 119874(119898]2) are allowed

by oscillation data Henceforth it is trivial to obtain theorthogonal transformation 1198741015840

119902 for1198721015840119902 (symmetric case) as1198741015840

119902 = 119875119902119877119879119902119875dagger119902119874119902 = 119879

119902119874119902 (31)

and (Hermitian case) as1198741015840119902 = 119875dagger119902119877dagger

119902119875119902119874119902 = 119879

119902119874119902 (32)

with 1198721015840Diag119902 = 1198741015840119879

119902 1015840

1199021198741015840119902 = 119872Diag

119902 (33)

with 1015840

119902 = 1198751199021198721015840119876119902 Note that in the absence of texturerotation 119902 = 119868 (unit matrix) for119872119902 while

119902 = (cos 12057813119902 0 minus sin 120578131199020 1 0sin 12057813119902 0 cos 12057813119902 ) (34)

for1198721015840119902 signifying the corresponding effect of such rotation on

real diagonalizing transformation 1198741015840119902 The resulting mixing

matrix for119872119902 andor1198721015840119902 may be constructed as119881 = 119874119879

119897 119897119875119897119875dagger] 119879

]119874] (35)

Also 119875119897119875dagger] = Diag(119890minus1198941206011 1 1198901198941206012) 1206011 = 120572119897 minus 120572] and 1206012 = 120573] minus 120573119897(symmetrics case) or 1206012 = 120573119897 minus 120573] (Hermitian case) Notethat a change in sign for 1198861015840119902 and 1198911015840

119902 can always be accom-modated in the redefinition of phases 120572119902 and 120573119902 whichonly appear implicitly in the PMNS matrix through 1206011 and1206012 Considering the six lepton masses 1206011 1206012 119889119902 and 119890119902as free parameters one can reconstruct the unitary mixingmatrix 119881 using the above procedure and confront it with

the current oscillation data In lieu of this we restrictour investigation to only texture four-zero mass matricesinvolving ten free parameters Furthermore the condition ofnaturalness forbids a texture zero at (33)matrix elements

Recent works [53ndash56] in this regard suggest that thereexist several viable texture structures of leptonmassmatricesMost of these investigationswork in the flavor basiswith diag-onal charged-lepton mass matrix or enforce parallel texturestructures for lepton mass matrices119872119897 and119872] In this letterwe investigate all possible structures for four-zero leptonmass matrices both symmetric andor Hermitian assumingfactorizable phases (for simplicity) in these The resultingstructures are summarized in Tables 1 and 2 wherein we enlistall texture five and four zeros in agreement with current dataat 3120590119883119897 and119883] in the tables represent the position of texturezeros in the corresponding mass matrices It is observed thatthe constraints of naturalness near maximal 120575119897 119904223 ≳ 050and normal ordering for neutrino masses taken togethergreatly reduce the number of possible viable structures andonly a few possibilities seem to survive the testThepossibilityof a vanishing neutrino mass is also studied for these texturestructures

4 Fritzsch-Like Four Zeros

It has been observed [36 47] that in the absence of 120575119897 sim 270∘constraint the Fritzsch-like texture four-zero mass matricesare physically equivalent to the generic lepton mass matricesInterestingly these matrices can be obtained from the abovestructures using the assumption of 119890119902 = 0 and 1198911015840

119902 = 0 Inparticular 119877119902 = 119902 = 119868 where 119868 is a unit matrix for this caseThe predictions from these matrices and their experimentaltests can be found in previous works To start with using (13)(35) and allowing free variations to the parameters119898]1 119889119897 119889]1206011 and 1206012 we first reconstruct the viable structures for 119897 (inunits of GeV) and ] (in units of eV) for 119889] sim 119898]2 using theavailable oscillation data and obtain the following best-fits

119897 = ( 0 0007 ndash 0010 00007ndash0010 0ndash0822 0423ndash09240 0423ndash0924 0822ndash1644) GeV]

= ( 0 00066ndash00104 000066ndash00104 00076ndash00115 00223ndash002600 00223ndash00260 00302ndash00383) eV(36)

along with 1206011 = 0∘ndash50∘ 267∘ndash360∘ and 1206012 = 180∘ndash285∘ Thecorresponding predictions for the absolute neutrinomassesΣand ⟨119898119890119890⟩ read 119898]1 = (296ndash670)meV 119898]2 = (905ndash1150)meV119898]3 = (477ndash519)meV Σ = 602ndash696meV and ⟨119898119890119890⟩= 0008ndash900meV respectively In the context of agreementwith 120575119897 sim 270∘ along with 12057923 ≳ 45∘ it is observed thatnaturalness is allowed in119872] independent of 11990423 octant Thisis depicted in Figure 1 where one observes that 119889] ≲ 119898]2 isstill consistent with 119904223 ≳ 05 However one finds that thenear maximal constraint of 120575119897 ≃ 270∘ requires large deviation


Table 1 Viable texture five zeros in relation to 119904223 ≳ 05 120575119897 sim 270∘ naturalness and119898]1 = 0Sr 119883119897 119883] (a) 119904223 ≳ 05 (b) 120575119897 sim 270∘ (c) natural (a + c) (b + c) Det119872] = 01 11 22 13 31 11 13 31 radic times radic radic times times2 11 13 31 11 22 13 31 radic times times times times times3 11 13 31 23 32 11 13 31 radic times times times times times4 12 21 22 13 31 11 13 31 radic times times times times times5 11 13 31 23 32 11 12 21 radic times times times times times6 11 22 13 31 11 12 21 radic times radic radic times times7 12 21 22 13 31 11 12 21 radic times radic radic times times8 11 12 21 23 32 11 13 31 radic times times times times times

Table 2 Viable texture four zeros in relation to 119904223 ≳ 05 120575119897 sim 270∘ naturalness and Det119872] = 0Sr 119883119897 119883] (a) 119904223 ≳ 05 (b) 120575119897 sim 270∘ (c) natural (a + c) (b + c) Det119872] = 01 11 13 31 11 13 31 radic radic radic radic times times2 13 31 23 32 11 13 31 radic radic times times times times3 11 22 11 13 31 radic radic radic radic radic times4 11 13 31 11 22 radic radic radic times times radic5 13 31 11 22 13 31 radic radic radic times radic times6 11 22 13 31 13 31 radic times radic radic times radic7 13 31 11 13 31 23 32 radic radic times times times times8 11 13 31 23 32 13 31 radic times times times times radic9 11 11 22 13 31 radic radic radic times radic times10 11 22 13 31 11 radic times radic radic times radic11 11 11 13 31 23 32 radic radic times times times times12 11 13 31 23 32 11 radic times times times times times13 22 13 31 11 13 31 radic radic radic radic radic times14 11 12 21 11 13 31 radic radic radic radic radic times15 11 13 31 11 12 21 radic radic radic radic times times16 13 31 23 32 11 12 21 radic radic times times times times17 22 13 31 11 12 21 radic radic radic radic times times18 11 12 21 11 22 radic radic times times times radic19 11 22 11 12 21 radic radic radic radic radic times20 12 21 13 31 11 13 31 radic radic times times times times21 12 21 13 31 11 22 radic radic times times times times22 22 12 21 13 31 13 31 radic times times times times radic23 12 21 22 13 31 11 radic times radic radic times times24 11 23 32 11 13 31 radic times times times times times25 11 12 21 23 32 11 radic times times times times times26 11 12 21 23 32 13 31 radic times radic radic times times27 13 31 11 12 21 23 32 radic radic times times times times28 11 11 12 21 23 32 radic radic times times times timesof 119872119897 from a possible natural structure In particular weidentify three vital sources for CP violation in these matricesnamely the two nontrivial phases 1206011 1206012 along with thefree parameter 119889119897 as elaborated in Figure 2 indicating 119889119897 gt06GeV sim 1198981205913 ≫ 119898120583 is required to obtain 120575119897 ≃ 270∘This also implies that Fritzsch-like texture five-zero matrices(119889119897 = 0) should be ruled out by 120575119897 ≃ 270∘ Our studyreveals this conclusion to hold true for all possible texturefive-zero structures all of which seem to be ruled out by anear maximal 120575119897 see Table 1 This calls upon investigating

alternate texture structures which on the one hand accountfor near maximal 120575119897 and at the same time allow possiblenatural structures for119872119897 and119872] (ie119872119895119896 sim 119874(119898119895119898119896))5 Natural Lepton Mass Matrices

In this context 119890119902 = 0 andor 1198911015840119902 = 0 in the correspond-

ing mass matrices provide greater possibility of realizingnaturalness in corresponding mass matrices as comparedto the Fritzsch-like structures wherein interactions between


0008 0009 0010 0011 00120007035

040

045

050

055

060

065

s2 23

d] (eV)

Figure 1 119904223 versus 119889] for Fritzsch-like four zeros

02 04 06 08 1000dl (GeV)

180

200

220

240

260

280

300

320

340

360

l(∘)

Figure 2 120575119897 versus 119889119897 for Fritzsch-like four zerosthe first and third generations of leptons are suppressed due totexture zeros invoked at (11) and (13 31)matrix elements Atleast for the quark sector nonvanishing (13 31) elements areobserved to be crucial in effectuating the natural structuresof corresponding mass matrices A careful analysis of allpossible texture four-zero structures reveals that only fourpossibilities for natural structures are allowed by recent datasee Table 2 We categorize these as Type I and Type II basedon the texture structure of119872]

51 Type I119872](11) = 119872](13 31) = 0Case A (119872119897(22) = 119872119897(13 31) = 0) The viable best-fitstructures of the leptonmassmatrices are summarized below

119897 = ( minus0003ndash0 0007ndash0019 00007ndash0019 0 0416ndash04260 0416ndash0426 1644ndash1647) GeV

0002 0003 0004 0005 0006 0007 00080001024

026

028

030

032

034

036

038

s2 12

m]1 (eV)

Figure 3 119904212 versus119898]1 for Case A

0006 0007 0008 0009 0010 0011 001200050016

0018

0020

0022

0024

0026

0028

0030

0032

s2 13

d] (eV)

Figure 4 119904213 versus 119889] for Case A]


with 1206011 = 0∘ndash340∘ 1206012 = 98∘ndash265∘ and 119898]1 = (199ndash701)meV 119898]2 = (865ndash113)meV 119898]3 = (477ndash519)meV Σ =(587ndash700)meV and ⟨119898119890119890⟩ = (001ndash923)meV respectivelyLike the Fritzsch-like texture four zeros 119904212 prop 119898]1 [36 3847 57] as depicted in Figure 3 However the other twomixingangles are fixed by the free parameter 119889] illustrated in Figures4 and 5 The latter also indicates that natural structure for119872] is allowed independent of 11990423 octant with 119889] ≲ 119898]2also accounting for 119904223 gt 05 Finally the parameter 119890119897 ≪ 119898120583

accounts for near maximal 120575119897 as shown in Figure 6 In par-ticular a small deviation of 120575119897 rarr 270∘ plusmn 30∘ provides greateragreement of 119890119897 sim 5MeV with the notion of naturalness inthe corresponding mass matrix


036038040042044046048050052054056058

0006 0007 0008 0009 0010 0011 00120005

s2 23

d] (eV)

Figure 5 119904223 versus 119889] for Case A

180

200

220

240

260

280

300

320

340

360

00000minus00005minus00010minus00015minus00020minus00025minus00030

el (GeV)

l(∘)

Figure 6 120575119897 versus 119890119897 for Case ACase B (119872119897(11) = 119872119897(22) = 0)Weobtain the following viablebest-fit structures for these lepton mass matrices namely1015840

119897

= ( 0 0001ndash0007 00003ndash00890001ndash0007 0 0413ndash042300003ndash0089 0413ndash0423 1644 ) GeV]


wherein 1206011 = 0∘ndash11∘ 251∘ndash360∘ 1206012 = 89∘ndash268∘ and119898]1 = (226ndash753)meV 119898]2 = (873ndash116)meV119898]3 = (477ndash520)meV Σ = (590ndash710)meV and⟨119898119890119890⟩ = (001ndash100)meV respectively 119898]1 dependence for119904212 remains the same as before while the other two mixingangles are fixed by the parameter 119889] Furthermore apart fromphases 1206011 and 1206012 120575119897 is now fixed by the parameter119891119897 = radicminus119890119897119888119897

002 004 006 008000fl (GeV)

180

200

220

240

260

280

300

320

340

360

l(∘)

Figure 7 120575119897 versus 1198911015840119897 for Case B

as shown in Figure 7 Naturalness in1198721015840119897 and119872] seems to be

in good agreement with 120575119897 sim 270∘ and 119904223 ≳ 05 compatiblewith 1198911015840

119897 sim 0075GeV sim 119874(radic119898119890119898120583) lt 119898120583 and 119889] ≲ 119898]2 respectively A greater agreement with naturalness in 119872119897 isachieved for 120575119897 rarr 270∘ plusmn 30∘ up to 1198911015840

119897 sim 005GeVCase C (119872119897(11) = 119872119897(12 21) = 0) The viable best-fitstructures so obtained for these lepton mass matrices areshown below

1015840

119897 = ( 0 0 0029ndash01670 0003ndash0103 0395ndash05800029ndash0167 0395ndash0580 154ndash164 ) GeV]


wherein 1206011 = 0∘ndash36∘ 175∘ndash360∘ 1206012 = 97∘ndash265∘ and119898]1 = (04ndash78)meV 119898]2 = (84ndash117)meV 119898]3 =(476ndash520)meV Σ = (569ndash712)meV and ⟨119898119890119890⟩ =(002ndash96)meV respectively It is noteworthy that the con-dition of texture zero at1198721015840

119897 (12 21) that is 1198861015840119897 = 0 fixes theparameter 119890119897 and hence 1198911015840

119897 = radicminus119890119897119888119897 (40)

through (30) with 119890119897 = minus119898119890119898120583119898120591119889119897119888119897 (41)

This results in only one free parameter 119889119897 = 1198891015840119897 in 1198721015840119897 This

is depicted in Figure 8 This parameter also determines theDirac-like CP phase as shown in Figure 9 Other observationspertaining to the dependence of mixing angles remain thesame as in previous cases It is clear that naturalness in 1198721015840

119897

and119872] is in good agreement with 120575119897 sim 270∘ and 119904223 ≳ 05compatible with 1198911015840

119897 sim 0064GeV sim 119874(radic119898119890119898120583) lt 119898120583 and119889] ≲ 119898]2 respectively


002 004 006 008 010 012000dl (GeV)

002

004

006

008

010

012

014

016

018

f l(G

eV)

Figure 8 1198911015840119897 versus 1198891015840119897 for Case C

180

200

220

240

260

280

300

320

340

360

002 004 006 008 010 012000dl (GeV)

l(∘)

Figure 9 120575119897 versus 1198891015840119897 for Case C52 Type II119872](11) = 119872](12 21) = 0Case D (119872119897(11) = 119872](22) = 0) The best-fit values obtainedfor this possibility are summarized below1015840

119897


= ( 0 0 0006ndash00190 00041ndash00120 00178ndash002690006ndash0019 00178ndash00269 00282ndash00393) eV(42)

wherein 1206011 = 0∘ndash23∘ 256∘ndash360∘ 1206012 = 98∘ndash261∘ and119898]1 = (11ndash79)meV 119898]2 = (84ndash125)meV 119898]3 =(476ndash519)meV Σ = (575ndash708)meV and ⟨119898119890119890⟩ =(001ndash956)meV respectively It is observed that naturalnessis in good agreement with 120575119897 sim 270∘ and 119904223 ≳ 05 compatiblewith |1198911015840

119897 | sim 0088GeV sim 119874(radic119898119890119898120583) lt 119898120583 see Figure 10

005 010 015 020 025000fl (GeV)

180

200

220

240

260

280

300

320

340

360

l(∘)

Figure 10 120575119897 versus 1198911015840119897 for Case D

and 119889] ≲ 119898]2 respectively Again a greater agreement withnaturalness in119872119897 can be achieved for 120575119897 rarr 270∘ plusmn 30∘ up to|1198911015840

119897 | sim 005GeV6 Conclusions

Assuming factorizable phases in lepton mass matrices weshow that natural mass matrices characterized by (119872119894119895) sim119874(radic119898119894119898119895) for 119894 119895 = 1 2 3 119894 = 119895 and (119872119894119894) sim 119874(119898119894) provide areasonable explanation for the observed fermion masses andflavor mixing patterns in the quark as well as the lepton sec-tors It is also observed that deviations from parallel texturestructures for119872119897119889 and119872]119906 are essential for establishing suchnatural structures Such phenomenological textures have alsobeen observed to be stable under the renormalization grouprunning from the lightest right-handed neutrino mass scaleto the electroweak scale [39 40 53 58 59]

Interestingly naturalness in the lepton sector implies11990412 prop 119874(radic119898]1119898]2) and 119904223 prop 119889]119888] or 11990423 prop 119874(radic119898]2119898]3)such that the observed large values of these mixing angles areperhaps indicative of the possible realization of the neutrinomass ratios as obtained above that is 119898]1 ≃ (01ndash80)meV119898]2 ≃ (80ndash130)meV 119898]3 ≃ (470ndash520)meV Σ ≃(560ndash710)meV and ⟨119898119890119890⟩ ≃ (001ndash100)meV respectivelyIn particular the possibility of a vanishing neutrino massthat is 119898]1 = 0 is not supported by natural lepton matricesFrom the point of view of 0]120573120573 decays these results seemto indicate that multiton scale detectors may be required topossibly observe signals for such processes

Competing Interests

The author declares that they have no competing interests

Acknowledgments

The author would like to thank Shun Zhou IHEP Beijingfor discussions and valuable suggestions This work was sup-ported in part by the Department of Science and Technologyunder SERB Research Grant no SBFTPPS-1402013


References

[1] G Aad B Abbott J Abdallah et al ldquoCombined measurementof the higgs bosonmass in pp collisions atradic119904 = 7 and 8 TeVwiththe ATLAS and CMS experimentsrdquo Physical Review Letters vol114 no 19 Article ID 191803 33 pages 2015

[2] N Cabibbo ldquoUnitary symmetry and leptonic decaysrdquo PhysicalReview Letters vol 10 no 12 pp 531ndash533 1963

[3] MKobayashi andTMaskawa ldquoCP-violation in the renormaliz-able theory of weak interactionrdquo Progress of Theoretical Physicsvol 49 no 2 pp 652ndash657 1973

[4] R Verma and S Zhou ldquoQuark flavor mixings from hierarchicalmass matricesrdquoThe European Physical Journal C vol 76 no 5article 272 2016

[5] R Peccei andKWang ldquoNaturalmassmatricesrdquoPhysical ReviewD vol 53 pp 2712ndash2723 1996

[6] P Ramond RG Roberts andGG Ross ldquoStitching theYukawaquiltrdquo Nuclear Physics B vol 406 no 1-2 pp 19ndash42 1993

[7] H Fritzsch and Z-Z Xing ldquoFour-zero texture of Hermitianquark mass matrices and current experimental testsrdquo PhysicsLetters Section B Nuclear Elementary Particle and High-EnergyPhysics vol 555 no 1-2 pp 63ndash70 2003

[8] L J Hall and A Rasin ldquoOn the generality of certain predictionsfor quarkmixingrdquo Physics Letters B vol 315 no 1-2 pp 164ndash1691993

[9] G C Branco M N Rebelo and J I Silva-Marcos ldquoYukawatextures new physics and nondecouplingrdquo Physical Review Dvol 76 Article ID 033008 2007

[10] H Fritzsch ldquoQuark masses and flavor mixingrdquo Nuclear PhysicsB vol 155 no 1 pp 189ndash207 1979

[11] Z Z Xing and H Zhang ldquoComplete parameter space of quarkmass matrices with four texture zerosrdquo Journal of Physics GNuclear and Particle Physics vol 30 no 2 p 129 2004

[12] R Barbieri L J Hall andA Romanino ldquoPrecise tests of a quarkmass texturerdquo Nuclear Physics B vol 551 no 1-2 pp 93ndash1011999

[13] H D Kim S Raby and L Schradin ldquoQuark mass textures andsin2120573rdquo Physical Review D vol 69 Article ID 092002 2004

[14] R G Roberts A Romanino G G Ross and L Velasco-SevillaldquoPrecision test of a Fermion mass texturerdquo Nuclear Physics Bvol 615 no 1-3 pp 358ndash384 2001

[15] N Arkani-Hamed S Dimopoulos G Dvali and J March-Russell ldquoNeutrino masses from large extra dimensionsrdquo Physi-cal Review D Third Series vol 65 Article ID 024032 2001

[16] K R Dienes E Dudas and T Gherghetta ldquoGrand unificationat intermediate mass scales through extra dimensionsrdquo NuclearPhysics B vol 537 no 1ndash3 pp 47ndash108 1999

[17] P Q Hung ldquoNeutrino masses and family replicationrdquo PhysicalReview D vol 59 no 11 Article ID 113008 5 pages 1999

[18] P-H Gu ldquoFrom Dirac neutrino masses to baryonic and darkmatter asymmetriesrdquo Nuclear Physics B vol 872 no 1 pp 38ndash61 2013

[19] G C Branco and G Senjanovic ldquoThe question of neutrinomassrdquo Physical Review D vol 18 no 5 pp 1621ndash1625 1978

[20] D Chang and R N Mohapatra ldquoSmall and calculable Diracneutrinomassrdquo Physical Review Letters vol 58 no 16 pp 1600ndash1603 1987

[21] P-H Gu and U Sarkar ldquoRadiative neutrino mass dark matterand leptogenesisrdquo Physical Review D vol 77 Article ID 1050312008

[22] P-H Gu and H-J He ldquoNeutrino mass and baryon asymmetryfrom Dirac seesawrdquo Journal of Cosmology and AstroparticlePhysics vol 2006 no 10 2006

[23] Z Maki M Nakagawa and S Sakata ldquoRemarks on the unifiedmodel of elementary particlesrdquo Progress of Theoretical Physicsvol 28 no 5 pp 870ndash880 1962

[24] K A Olive ldquoReview of particle physicsrdquo Chinese Physics C vol38 no 9 Article ID 090001 2014

[25] G L Fogli E Lisi A Marrone D Montanino A Palazzo andA M Rotunno ldquoGlobal analysis of neutrino masses mixingsand phases Entering the era of leptonic CP violation searchesrdquoPhysical Review D vol 86 no 1 Article ID 013012 2012

[26] P A R Ade N AghanimM Arnaud et al ldquoPlanck 2015 resultsXIII Cosmological parametersrdquoAstronomyampAstrophysics vol594 article A13 2016

[27] K Abe J Adam H Aihara et al ldquoMeasurements of neutrinooscillation in appearance and disappearance channels by theT2K experiment with 66 times 1020 protons on targetrdquo PhysicalReview D vol 91 Article ID 072010 2015

[28] S Zhou ldquoUpdate on two-zero textures of theMajorana neutrinomass matrix in light of recent T2K Super-Kamiokande andNO]A resultsrdquo Chinese Physics C vol 40 no 3 Article ID033102 2016

[29] S DellrsquoOro S Marcocci M Viel and F Vissani ldquoThe contri-bution of light Majorana neutrinos to neutrinoless double betadecay and cosmologyrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 12 p 23 2015

[30] S Weinberg ldquoThe problem of massrdquo Transactions of the NewYork Academy of Sciences vol 38 no 1 pp 185ndash201 1977

[31] H Fritzsch ldquoCalculating the Cabibbo anglerdquo Physics Letters Bvol 70 no 4 pp 436ndash440 1977

[32] H Fritzsch ldquoWeak-interaction mixing in the six-quark theoryrdquoPhysics Letters B vol 73 no 3 pp 317ndash322 1978

[33] M Gupta G Ahuja and R Verma ldquoParticle physics astro-physics and quantum field theory 75 years since Solvayrdquo inProceedings of the Particle Physics Astrophysics and QuantumField Theory (PAQFT rsquo08) Singapore Novembre 2008

[34] H Fritzsch and S Zhou ldquoNeutrino mixing angles from texturezeros of the leptonmassmatricesrdquo Physics Letters B vol 718 no4-5 pp 1457ndash1464 2013

[35] Z-Z Xing and H Zhang ldquoLepton mass matrices with fourtexture zerosrdquo Physics Letters B vol 569 no 1-2 pp 30ndash402003

[36] R Verma ldquoLower bound on neutrino mass and possible CPviolation in neutrino oscillationsrdquo Physical Review D vol 88no 11 Article ID 111301 6 pages 2013

[37] P Fakay S Sharma R Verma G Ahuja and M GuptaldquoImplications of 12057913 on Fritzsch-like lepton mass matricesrdquoPhysics Letters B vol 720 no 4-5 pp 366ndash372 2013

[38] H Fritzsch and Z-Z Xing ldquoRelating the neutrino mixingangles to a lepton mass hierarchyrdquo Physics Letters B vol 682no 2 pp 220ndash224 2009

[39] M Fukugita M Tanimoto and T Yanagida ldquoPredictions fromthe Fritzsch-type lepton mass matricesrdquo Physics Letters SectionB Nuclear Elementary Particle and High-Energy Physics vol562 no 3-4 pp 273ndash278 2003

[40] M Fukugita Y Shimizu M Tanimoto and T T Yanagida ldquo12057913 in neutrino mass matrix with the minimal texturerdquo PhysicsLetters Section B Nuclear Elementary Particle and High-EnergyPhysics vol 716 no 2 pp 294ndash297 2012


[41] X-W Liu and S Zhou ldquoTexture zeros for Dirac neutrinos andcurrent experimental testsrdquo International Journal of ModernPhysics A vol 28 no 12 Article ID 1350040 28 pages 2013

[42] H Fritzsch ldquoTexture zero mass matrices and flavor mixing ofquarks and leptonsrdquo Modern Physics Letters A vol 30 no 28Article ID 1550138 2015

[43] H Fritzsch Z Z Xing and S Zhou ldquoTwo-zero textures ofthe Majorana neutrino mass matrix and current experimentaltestsrdquo Journal of High Energy Physics vol 2011 article 83 2011

[44] N Mahajan R Verma and M Gupta ldquoInvestigating non-Fritzsch-like texture specific quark mass matricesrdquo Interna-tional Journal of Modern Physics A vol 25 no 10 pp 2037ndash2048 2010

[45] RVerma ldquoMinimalweak basis textures and quarkmixing datardquoJournal of Physics G Nuclear and Particle Physics vol 40 no 12Article ID 125003 2013

[46] W Wang ldquoParallel texture structures with cofactor zeros inlepton sectorrdquo Physics Letters B vol 733 pp 320ndash327 2014Corrigendum to Physics Letters B vol 738 pp 524-525 2014

[47] R Verma ldquoGeneric lepton mass matrices and neutrino oscil-lationsrdquo Physical Review D vol 89 no 5 Article ID 053007 8pages 2014

[48] R Verma ldquoLepton textures and neutrino oscillationsrdquo Interna-tional Journal of Modern Physics A vol 29 no 21 Article ID1444009 2014

[49] GAhujaMGuptaM Randhawa andRVerma ldquoTexture spe-cificmassmatriceswithDirac neutrinos and their implicationsrdquoPhysical Review D vol 79 Article ID 093006 2009

[50] R Verma G Ahuja and M Gupta ldquoImplications of CPasymmetry parameter sin2120573 on structural features of texturespecificmassmatricesrdquo Physics Letters B vol 681 no 4 pp 330ndash335 2009

[51] R Verma G Ahuja N Mahajan M Gupta andM RandhawaldquoExploring the parameter space of texture four-zero quarkmassmatricesrdquo Journal of Physics G Nuclear and Particle Physics vol37 no 7 Article ID 075020 2010

[52] S-H Chiu T-K Kuo and G-H Wu ldquoHermitian quark massmatrices with four texture zerosrdquo Physical Review D vol 62 no5 Article ID 053014 2000

[53] J Liao D Marfatia and K Whisnant ldquoNeutrino seesawmechanism with texture zerosrdquo Nuclear Physics B vol 900 pp449ndash476 2015

[54] G Ahuja S Sharma P Fakay and M Gupta ldquoGeneral leptontextures and their implicationsrdquo Modern Physics Letters A vol30 no 34 Article ID 1530025 2015

[55] P O Ludl and W Grimus ldquoA complete survey of texture zerosin the leptonmassmatricesrdquo Journal of High Energy Physics vol2014 no 7 article 90 2014 Erratum to Journal of High EnergyPhysics vol 2014 no 10 article 126 2014

[56] P O Ludl and W Grimus ldquoA complete survey of texture zerosin general and symmetric quark mass matricesrdquo Physics LettersB vol 744 pp 38ndash42 2015

[57] H Fritzsch ldquoNeutrino masses and flavor mixingrdquo ModernPhysics Letters A vol 30 no 16 Article ID 1530012 2015

[58] C Hagedorn J Kersten and M Lindner ldquoStability of texturezeros under radiative corrections in see-saw modelsrdquo PhysicsLetters B vol 597 no 1 pp 63ndash72 2004

[59] M Fukugita M Tanimoto and T Yanagida ldquoPhenomenologi-cal lepton mass matrixrdquo Progress of Theoretical Physics vol 89no 1 pp 263ndash267 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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embedded within the corresponding quark mass matricesnamely (for 119902 = 119906 119889)11987211 lt 1198721221 ≲ 1198721331 lt 11987222 lt 1198722332 lt 11987233 (4)

with 11987222 ≪ 11987233 Recent investigations [4] in this regardindicate that the current quarkmixing data indeed permit thequark mass matrices to have such a natural and hierarchicalstructure provided (119872119894119895) sim 119874(radic119898119894119898119895) for 119894 119895 = 1 2 3 119894 = 119895and (119872119894119894) sim 119874(119898119894) Such hierarchical mass matrices havebeen referred to as natural mass matrices [5] In particularnaturalness provides a rationale framework to correlate theobserved fermionmass ratios the correspondingmassmatri-ces and observed mixing angles Specifically for (11987213) =(11987231) = 0 the observed strong hierarchy among the quarkmasses and CKM elements gets naturally translated onto thestructure of the corresponding mass matrices

A concomitant of such naturalness in mass matrices isthe absence of parallel texture structure for the ldquouprdquo andldquodownrdquo type quark mass matrices [4] A parallel texturestructure corresponds for example to the mass matrices119872119906

and119872119889 with texture zeros at identical positions in both themass matrices Hierarchical structures have fetched greaterimportance in the literature as these predict certain verysimple yet compelling relations among the CKM elementsand the quark mass ratios [4 6ndash14]

However the mass spectrum for leptons is quite distin-guished from the quark sector wherein the charged leptonsmasses are strongly hierarchical that is 119898119890 ≪ 119898120583 ≪ 119898120591while at least two of the neutrinos are allowed to have thesame order of mass It should be interesting to investigate ifnaturalness can provide a unique explanation for the fermionmassmatrices corresponding observedmass spectra and themixing angles both for the quark and the lepton sectors

Since the neutrinos are massless within the SM one hasto explore beyond the realms of SM to comprehend theorigin of neutrino masses and observe neutrino oscillationsphenomenon A simplistic way to achieve this is to extendthe SM theory by assuming neutrinos as Dirac-like particlesIn this case the neutrinos acquire mass through the Higgsmechanism in the similar way as quarks and charged leptonsdo within the SM through a Dirac mass term for exampleminus119871119897mass = 119897119871119872119897119897119877 + hcminus2119871Diracmass = ]119871119872]119863]119877 + hc (5)

where 119872119897 and 119872]119863 represent the charged-lepton and Diracneutrinomassmatrix respectively Indeed the current exper-iments have not ruled out such a possibility In this contextit is also observed that highly suppressed Yukawa couplingsfor Dirac neutrinos can naturally be achieved using modelswith extra spatial dimensions [15 16] or through radiativemechanisms [17ndash22] However such a possibility is perceivedto be highly unlikely due to several orders of magnitudedifference among119898120572 (120572 = 119890 120583 120591) and119898]119894 (119894 = 1 2 3)

A rather convincing and natural explanation of neutrinomasses can be obtained if neutrinos are assumed to beMajorana particles This usually involves adding the lepton

number (and flavor) violating Majorana mass terms forneutrinos in the Lagrangian for exampleminus2119871Majorana

mass = ]119871119872]119871]119888119877 + ]119888119871119872]119877]119877 (6)

where119872]119871 and119872]119877 correspond respectively to the left andright-handedMajorana neutrinomassmatrices and the latterusually has an extremely high mass scale This facilitatesgenerating the light neutrino masses through Type I or TypeII seesaw mechanisms namely119872] = minus119872119879

]119863119872minus1]119877119872]119863 119872] = 119872]119871 minus119872119879]119863119872minus1

]119877119872]119863 (7)

where119872] is usually a complex symmetric matrix for exam-ple

119872] = (119890] 119886] 119891]119886] 119889] 119887]119891] 119887] 119888]) (8)

This allows writing the corresponding charged weakcurrent term for leptons as

minus119871leptonscc = 119892radic2(]119890 ]120583 ]120591)119871120574120583119881(119890120583120591)119871

119882+120583 + hc (9)

where 119881 = 119881PMNS = 119880dagger119897119871119880]119871 is the Pontecorvo-Maki-

Nakagawa-Sakata (PMNS) mixing matrix [23] or the neu-trinomixingmatrix and emerges through the diagonalizationof the matrices119872119897 and119872] for example119880dagger

119897119871119872119897119872dagger119897 119880119897119877 = Diag (1198982

119890 1198982120583 1198982

120591) 119880dagger]119871119872]119872dagger

]119880]119877 = Diag (1198982]1 1198982

]2 1198982]3) (10)

This mixing matrix relates the neutrino flavor states to theneutrino mass eigenstates through

]120572119871 = sum119894=123

119881120572119894]119894119871 (11)

In the standard parametrization [24] the PMNS matrix isexpressed as 119881 = 119880 sdot 119875119900 where 119875119900 equiv Diag119890119894120588 119890119894120590 1 with120588 120590 being two Majorana CP violating phases and 119880 can beparametrized in terms of three mixing angles 12057912 12057913 and 12057923and one Dirac CP violating phase 120575119897 namely119880= ( 1198881211988813 1199041211988813 11990413119890minus119894120575119897minus1199041211988823 minus 119888121199041311990423119890119894120575119897 1198881211988823 minus 119904121199041311990423119890119894120575119897 11990423119888131199041211990423 minus 119888121198882311990413119890119894120575119897 minus1198881211990423 minus 119904121199041311988823119890119894120575119897 1198882311988813 )

(12)

with 119904119894119895 = sin 120579119894119895 and 119888119894119895 = cos 120579119894119895 for 119894119895 = 12 13 23 Theneutrino oscillation experiments provide constraints on the



2 minus 11989821 and Δ1198982 =120578[1198982

3 minus (11989821 + 1198982



(13)





10038161003816100381610038161003816 119898]1 + 119890119894120590 1003816100381610038161003816100381611988021198902

10038161003816100381610038161003816 119898]2 + 1003816100381610038161003816100381611988021198903

10038161003816100381610038161003816 119898]310038161003816100381610038161003816 (17)







119872119902 = (119890119902119890119894120595119902 119886119902119890119894120572119902 0119886119902119890119894120572119902 119889119902119890119894120596119902 1198871199021198901198941205731199020 119887119902119890119894120573119902 119888119902119890119894120574119902)1198721015840

119902 = ( 0 1198861015840119902119890119894120572119902 1198911015840119902119890119894Δ 1199021198861015840119902119890119894120572119902 1198891015840119902119890119894120596119902 11988710158401199021198901198941205731199021198911015840

119902119890119894Δ 119902 1198871015840119902119890119894120573119902 1198881015840119902119890119894120574119902 )(18)



120595119902 = 2120572119902120596119902 = 0Δ 119902 = 120572119902 + 120573119902120574119902 = 2120573119902(19)



120595119902 = 0120596119902 = 0Δ 119902 = 120572119902 + 120573119902120574119902 = 0(20)



119902 119902119874119902 = Diag (1198981 minus1198982 1198983) (21)


119902 = 119875119902119872119902119876119902 = (119890119902 119886119902 0119886119902 119889119902 1198871199020 119887119902 119888119902) (22)


119874119902 =((((((((


(23)


(24)





119902119872119902119877119902 (25)


119902119872119902119877119902 (26)





119902 requires0 = 119890119902cos212057813119902 + 119888119902sin212057813119902 (28)







(30)







119902 = 119875119902119877119879119902119875dagger119902119874119902 = 119879

119902119874119902 (31)


119902119875119902119874119902 = 119879

119902119874119902 (32)

with 1198721015840Diag119902 = 1198741015840119879

119902 1015840

1199021198741015840119902 = 119872Diag

119902 (33)

with 1015840






119897 119897119875119897119875dagger] 119879

]119874] (35)


















0008 0009 0010 0011 00120007035

040

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s2 23

d] (eV)


02 04 06 08 1000dl (GeV)

180

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l(∘)




0002 0003 0004 0005 0006 0007 00080001024

026

028

030

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s2 12

m]1 (eV)


0006 0007 0008 0009 0010 0011 001200050016

0018

0020

0022

0024

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s2 13

d] (eV)






036038040042044046048050052054056058

0006 0007 0008 0009 0010 0011 00120005

s2 23

d] (eV)


180

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el (GeV)

l(∘)


119897




002 004 006 008000fl (GeV)

180

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l(∘)






1015840





119897 = radicminus119890119897119888119897 (40)




119897




002 004 006 008 010 012000dl (GeV)

002

004

006

008

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f l(G

eV)


180

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002 004 006 008 010 012000dl (GeV)

l(∘)


119897





005 010 015 020 025000fl (GeV)

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l(∘)






Competing Interests


Acknowledgments



References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2 minus 11989821 and Δ1198982 =120578[1198982

3 minus (11989821 + 1198982



(13)





10038161003816100381610038161003816 119898]1 + 119890119894120590 1003816100381610038161003816100381611988021198902

10038161003816100381610038161003816 119898]2 + 1003816100381610038161003816100381611988021198903

10038161003816100381610038161003816 119898]310038161003816100381610038161003816 (17)







119872119902 = (119890119902119890119894120595119902 119886119902119890119894120572119902 0119886119902119890119894120572119902 119889119902119890119894120596119902 1198871199021198901198941205731199020 119887119902119890119894120573119902 119888119902119890119894120574119902)1198721015840

119902 = ( 0 1198861015840119902119890119894120572119902 1198911015840119902119890119894Δ 1199021198861015840119902119890119894120572119902 1198891015840119902119890119894120596119902 11988710158401199021198901198941205731199021198911015840

119902119890119894Δ 119902 1198871015840119902119890119894120573119902 1198881015840119902119890119894120574119902 )(18)



120595119902 = 2120572119902120596119902 = 0Δ 119902 = 120572119902 + 120573119902120574119902 = 2120573119902(19)



120595119902 = 0120596119902 = 0Δ 119902 = 120572119902 + 120573119902120574119902 = 0(20)



119902 119902119874119902 = Diag (1198981 minus1198982 1198983) (21)


119902 = 119875119902119872119902119876119902 = (119890119902 119886119902 0119886119902 119889119902 1198871199020 119887119902 119888119902) (22)


119874119902 =((((((((


(23)


(24)





119902119872119902119877119902 (25)


119902119872119902119877119902 (26)





119902 requires0 = 119890119902cos212057813119902 + 119888119902sin212057813119902 (28)







(30)







119902 = 119875119902119877119879119902119875dagger119902119874119902 = 119879

119902119874119902 (31)


119902119875119902119874119902 = 119879

119902119874119902 (32)

with 1198721015840Diag119902 = 1198741015840119879

119902 1015840

1199021198741015840119902 = 119872Diag

119902 (33)

with 1015840






119897 119897119875119897119875dagger] 119879

]119874] (35)


















0008 0009 0010 0011 00120007035

040

045

050

055

060

065

s2 23

d] (eV)


02 04 06 08 1000dl (GeV)

180

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l(∘)




0002 0003 0004 0005 0006 0007 00080001024

026

028

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s2 12

m]1 (eV)


0006 0007 0008 0009 0010 0011 001200050016

0018

0020

0022

0024

0026

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0032

s2 13

d] (eV)






036038040042044046048050052054056058

0006 0007 0008 0009 0010 0011 00120005

s2 23

d] (eV)


180

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el (GeV)

l(∘)


119897




002 004 006 008000fl (GeV)

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l(∘)






1015840





119897 = radicminus119890119897119888119897 (40)




119897




002 004 006 008 010 012000dl (GeV)

002

004

006

008

010

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016

018

f l(G

eV)


180

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002 004 006 008 010 012000dl (GeV)

l(∘)


119897





005 010 015 020 025000fl (GeV)

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l(∘)






Competing Interests


Acknowledgments



References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120595119902 = 0120596119902 = 0Δ 119902 = 120572119902 + 120573119902120574119902 = 0(20)



119902 119902119874119902 = Diag (1198981 minus1198982 1198983) (21)


119902 = 119875119902119872119902119876119902 = (119890119902 119886119902 0119886119902 119889119902 1198871199020 119887119902 119888119902) (22)


119874119902 =((((((((


(23)


(24)





119902119872119902119877119902 (25)


119902119872119902119877119902 (26)





119902 requires0 = 119890119902cos212057813119902 + 119888119902sin212057813119902 (28)







(30)







119902 = 119875119902119877119879119902119875dagger119902119874119902 = 119879

119902119874119902 (31)


119902119875119902119874119902 = 119879

119902119874119902 (32)

with 1198721015840Diag119902 = 1198741015840119879

119902 1015840

1199021198741015840119902 = 119872Diag

119902 (33)

with 1015840






119897 119897119875119897119875dagger] 119879

]119874] (35)


















0008 0009 0010 0011 00120007035

040

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s2 23

d] (eV)


02 04 06 08 1000dl (GeV)

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l(∘)




0002 0003 0004 0005 0006 0007 00080001024

026

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s2 12

m]1 (eV)


0006 0007 0008 0009 0010 0011 001200050016

0018

0020

0022

0024

0026

0028

0030

0032

s2 13

d] (eV)






036038040042044046048050052054056058

0006 0007 0008 0009 0010 0011 00120005

s2 23

d] (eV)


180

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el (GeV)

l(∘)


119897




002 004 006 008000fl (GeV)

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1015840





119897 = radicminus119890119897119888119897 (40)




119897




002 004 006 008 010 012000dl (GeV)

002

004

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f l(G

eV)


180

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l(∘)


119897





005 010 015 020 025000fl (GeV)

180

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l(∘)






Competing Interests


Acknowledgments



References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






(30)







119902 = 119875119902119877119879119902119875dagger119902119874119902 = 119879

119902119874119902 (31)


119902119875119902119874119902 = 119879

119902119874119902 (32)

with 1198721015840Diag119902 = 1198741015840119879

119902 1015840

1199021198741015840119902 = 119872Diag

119902 (33)

with 1015840






119897 119897119875119897119875dagger] 119879

]119874] (35)


















0008 0009 0010 0011 00120007035

040

045

050

055

060

065

s2 23

d] (eV)


02 04 06 08 1000dl (GeV)

180

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l(∘)




0002 0003 0004 0005 0006 0007 00080001024

026

028

030

032

034

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038

s2 12

m]1 (eV)


0006 0007 0008 0009 0010 0011 001200050016

0018

0020

0022

0024

0026

0028

0030

0032

s2 13

d] (eV)






036038040042044046048050052054056058

0006 0007 0008 0009 0010 0011 00120005

s2 23

d] (eV)


180

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el (GeV)

l(∘)


119897




002 004 006 008000fl (GeV)

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l(∘)






1015840





119897 = radicminus119890119897119888119897 (40)




119897




002 004 006 008 010 012000dl (GeV)

002

004

006

008

010

012

014

016

018

f l(G

eV)


180

200

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002 004 006 008 010 012000dl (GeV)

l(∘)


119897





005 010 015 020 025000fl (GeV)

180

200

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260

280

300

320

340

360

l(∘)






Competing Interests


Acknowledgments



References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










0008 0009 0010 0011 00120007035

040

045

050

055

060

065

s2 23

d] (eV)


02 04 06 08 1000dl (GeV)

180

200

220

240

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300

320

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360

l(∘)




0002 0003 0004 0005 0006 0007 00080001024

026

028

030

032

034

036

038

s2 12

m]1 (eV)


0006 0007 0008 0009 0010 0011 001200050016

0018

0020

0022

0024

0026

0028

0030

0032

s2 13

d] (eV)






036038040042044046048050052054056058

0006 0007 0008 0009 0010 0011 00120005

s2 23

d] (eV)


180

200

220

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360


el (GeV)

l(∘)


119897




002 004 006 008000fl (GeV)

180

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l(∘)






1015840





119897 = radicminus119890119897119888119897 (40)




119897




002 004 006 008 010 012000dl (GeV)

002

004

006

008

010

012

014

016

018

f l(G

eV)


180

200

220

240

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280

300

320

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002 004 006 008 010 012000dl (GeV)

l(∘)


119897





005 010 015 020 025000fl (GeV)

180

200

220

240

260

280

300

320

340

360

l(∘)






Competing Interests


Acknowledgments



References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




036038040042044046048050052054056058

0006 0007 0008 0009 0010 0011 00120005

s2 23

d] (eV)


180

200

220

240

260

280

300

320

340

360


el (GeV)

l(∘)


119897




002 004 006 008000fl (GeV)

180

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300

320

340

360

l(∘)






1015840





119897 = radicminus119890119897119888119897 (40)




119897




002 004 006 008 010 012000dl (GeV)

002

004

006

008

010

012

014

016

018

f l(G

eV)


180

200

220

240

260

280

300

320

340

360

002 004 006 008 010 012000dl (GeV)

l(∘)


119897





005 010 015 020 025000fl (GeV)

180

200

220

240

260

280

300

320

340

360

l(∘)






Competing Interests


Acknowledgments



References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




002 004 006 008 010 012000dl (GeV)

002

004

006

008

010

012

014

016

018

f l(G

eV)


180

200

220

240

260

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320

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360

002 004 006 008 010 012000dl (GeV)

l(∘)


119897





005 010 015 020 025000fl (GeV)

180

200

220

240

260

280

300

320

340

360

l(∘)






Competing Interests


Acknowledgments



References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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