JHEP01(2016)007 - Springer2016... · 2017-08-29 · JHEP01(2016)007 The idea of combining discrete ﬂavor symmetries and extra dimensions is quite at-tractive and has already been

JHEP01(2016)007

Published for SISSA by Springer

Received September 28 2015

Revised December 1 2015

Accepted December 19 2015

Published January 4 2016

Warped flavor symmetry predictions for neutrino

physics

Peng Chena Gui-Jun Dinga Alma D Rojasb CA Vaquera-Araujob and

JWF Valleb1

aDepartment of Modern Physics University of Science and Technology of China

Hefei Anhui 230026 ChinabAHEP Group Institut de Fısica Corpuscular mdash CSICUniversitat de Valencia

Parc Cientific de Paterna CCatedratico Jose Beltran 2 E-46980 Paterna (Valencia) mdash Spain

E-mail pchemailustceducn dinggjustceducn

almarojasificuves vaqueraificuves valleificuves

Abstract A realistic five-dimensional warped scenario with all standard model fields

propagating in the bulk is proposed Mass hierarchies would in principle be accounted for

by judicious choices of the bulk mass parameters while fermion mixing angles are restricted

by a ∆(27) flavor symmetry broken on the branes by flavon fieldsThe latter gives stringent

predictions for the neutrino mixing parameters and the Dirac CP violation phase all

described in terms of only two independent parameters at leading order The scheme also

gives an adequate CKM fit and should be testable within upcoming oscillation experiments

Keywords Quark Masses and SM Parameters Neutrino Physics Discrete and Finite

Symmetries

ArXiv ePrint 150906683

1URL httpastroparticleses

Open Access ccopy The Authors

Article funded by SCOAP3doi101007JHEP01(2016)007

JHEP01(2016)007

Contents

1 Introduction 1

2 Basic structure of the model 2

3 Lepton sector 4

31 Lepton masses and mixing 5

32 Phenomenological implications 8

4 Quark sector 10

5 High order corrections 14

6 Conclusions 16

A The profile of the Higgs zero mode 17

B Group theory of ∆(27) and its representation 18

C Vacuum alignment 21

1 Introduction

The understanding of flavor constitutes one of the most stubborn open challenges in particle

physics [1] Two aspects of the problem are the understanding of fermion mass hierarchies

as well as mixing parameters Various types of flavor symmetries have been invoked in this

context [2ndash9] These efforts have been partly motivated by the original success of the tri-

bimaximal mixing ansatz [10] The resulting non-Abelian flavor symmetries are typically

broken spontaneously down to two different residual subgroups in the neutrino and the

charged lepton sectors leading to zero reactor mixing parameter θ13 = 0 However the

measurement of a non-zero value for the reactor angle [11ndash14] implies the need to revamp

the original flavor symmetry-based approaches in order to generate θ13 6= 0 [15] or else look

for alternative possibilities such as bi-large neutrino mixing [16ndash18]

The existence of warped extra dimensions has been advocated by Randall amp Sun-

drum [19] as a way to address the hierarchy problem since the fundamental scale of grav-

ity is exponentially reduced from the Planck mass down to the TeV scale as a result of

having the Higgs sector localized near the boundary of the extra dimensions Moreover if

standard model fermions are allowed to propagate in the bulk and also become localized

towards either brane the scenario can also address the flavor problem possibly acting in

synergy with the flavor group predictions This is what we do in the present paper

ndash 1 ndash

JHEP01(2016)007

The idea of combining discrete flavor symmetries and extra dimensions is quite at-

tractive and has already been discussed in the literature within the context of large extra

dimensions [20ndash22] warped extra dimensions [23ndash31] and holographic composite Higgs

models [32ndash34] However such models try to generate tri-bimaximal neutrino mixing

which has been ruled out by the measurement of the reactor angle θ13 [11ndash14] and also

global fits of neutrino oscillation data [35] One of us has constructed a warped extra

dimension model with S4 flavor symmetry where democratic mixing is produced at leading

order and non-zero θ13 can arise from subleading corrections [36] In this work we shall

re-consider the issue of predicting flavor properties in particle physics by combining the

conventional predictive power inherent in the use of non-Abelian flavor symmetries with

the presence of warped extra dimensions We propose a warped five-dimensional scenario

in which all matter fields propagate in the bulk and neutrinos are treated as Dirac particles

Our model can accommodate all the strengths of the standard model Yukawa couplings

and resulting fermion mass hierarchies by making adequate choices of fermion bulk mass

parameters while the fermion mixing parameters can be restricted by means of the as-

sumed flavor symmetry We present a ∆(27) based flavor symmetry which nicely describes

the neutrino oscillation parameters in terms of just two independent parameters leading

to interesting correlations involving the neutrino mass hierarchy and the leptonic Dirac CP

phase not yet reliably determined by current global oscillation fit [35] Our predictions in-

clude a neat leading order relation between the solar and reactor mixing parameters which

should be tested at future oscillation experiments

2 Basic structure of the model

In this section we present the basic setup of a warped five-dimensional (5D) model for

fermions constructed under a ∆(27) otimes Z4 otimes Z prime4 flavor symmetry The 5D field theory is

defined on a slice of AdS5 where the bulk geometry is described by the metric

ds2 = eminus2kyηmicroνdxmicrodxν minus dy2 (21)

with ηmicroν = diag(1minus1minus1minus1) and k as the AdS5 curvature scale The fifth dimension

y is compactified on S1Z2 and two flat 3-branes of opposite tension are attached to the

orbifold fixed points located at y = 0 (UV brane) and y = L (IR brane)

Our framework is built upon the most minimal version of the RS model We adopt a

non-custodial GSM = SU(2)L otimesU(1)Y bulk electroweak symmetry where the 5D fermions

and the Higgs field are allowed to propagate into the bulk It is well known that models with

a brane-localized Higgs and no custodial symmetry suffer from large constraints imposed

by electroweak precision tests [37 38] However the tensions with electroweak precision

tests [39ndash42] and flavour physics [42ndash45] can be significantly reduced in the case of a 5D

Higgs field living in the bulk offering a new elegant explanation for the the tiny neutrino

masses [46 47] That is to say a SU(2)L doublet bulk Higgs field

) (22)

ndash 2 ndash

JHEP01(2016)007

is responsible for the spontaneous symmetry breaking (SSB) of GSM to U(1)EM through its

vacuum expectation value (VEV) where the superscripts ldquo+rdquo and ldquo0rdquo denote the electric

charge of the field The 5D Higgs field H(xmicro y) can be decomposed into Kaluza-Klein

(KK) modes as

H(xmicro y) = H(xmicro)fH(y)radic

L+ heavy KK Modes (23)

For an adequate choice of the Higgs potential (see appendix A for an explicit realization)

its zero mode profile fH(y) can be written as [48 49]

fH(y) =

radic2kL(1minus β)

1minus eminus2(1minusβ)kLekLe(2minusβ)k(yminusL) (24)

where we have introduced the Higgs localization parameter β =radic4 +m2

Hk2 in terms of

the Higgs field bulk mass parametermH Furthermore the Higgs zero mode obtains a VEV

〈H(xmicro)〉 = vHradic2

) (25)

and it is peaked toward the IR brane allowing for electroweak SSB at the TeV scale The

fermion content is the same as in the standard model Three families of fermion fields are re-

quired to describe each generation (labeled by i = 1 2 3) of quarks and leptons All fermion

fields propagate into the bulk and transform under the minimal representation of the gauge

group SU(2)L otimesU(1)Y In the lepton sector the three multiplets of the model are given as

Ψℓi =

(ν[++]i

e[++]i

)sim (2minus1) Ψei = e

[minusminus]i sim (1minus2) Ψνi = ν

[minusminus]i sim (1 0) (26)

while for the quark sector we have

(u[++]i

d[++]i

)sim (2 13) Ψdi = d

[minusminus]i sim (1minus23) Ψui

= u[minusminus]i sim (1 43)

In the above equations each 5D fermion field is described by a 4-component Dirac spinor

field and fields with different sign assignments must be understood as independent The

bracketed signs indicate Neumann (+) or Dirichlet (minus) BCs for the left-handed component

of the corresponding field on both UV and IR branes The right-handed part of the field

satisfies opposite BCs Only fields with [++] BCs have left-handed zero modes whereas

right-handed zero modes exist solely for fields with [minusminus] BCs The KK decomposition for

such fields has the form

ψ[++](xmicro y) =e2kyradicL

micro)f(0)L (y cL) + heavy KK modes

ψ[minusminus](xmicro y) =e2kyradicL

micro)f(0)R (y cR) + heavy KK modes

ndash 3 ndash

JHEP01(2016)007

with ψ = νi ei ui di and zero mode profiles [50ndash52]

f(0)L (y cL) =

radic(1minus 2cL)kL

e(1minus2cL)kL minus 1eminuscLky f

(0)R (y cR) =

radic(1 + 2cR)kL

e(1+2cR)kL minus 1ecRky (29)

where cL and cR are the bulk mass parameters of the 5D fermion fields in units of the AdS5curvature k Thus the low energy spectrum contains left-handed doublets ℓiL = (νiL eiL)

QiL = (uiL diL) alongside right-handed singlets νiR eiR uiR diR In the following we

identify all standard model fields with this set of zero modes (ie the so called zero mode

approximation ZMA) For future convenience we denote the flavor components of charged

leptons and quarks as e123 = e micro τ Q123 = UC T u123 = u c t d123 = d s b

In the present work we choose the flavor symmetry to be ∆(27) augmented by the

auxiliary symmetry Z4 otimes Z prime4 The group ∆(27) was originally proposed to explain the

fermion masses and flavor mixing in refs [53 54] and has been used for Dirac neutrinos

in [55] by one of us Here we study its implementation in a warped extra dimensional theory

The flavor symmetry ∆(27)otimesZ4otimesZ prime4 is broken by brane localized flavons transforming as

singlets under GSM We introduce a set of flavons ξ σ1 σ2 localized on the IR brane and

a flavon ϕ localized on the UV brane Both ξ and ϕ are assigned to the three-dimensional

representation 3 of ∆(27) while σ1 and σ2 transform as inequivalent one-dimensional

representations 101 and 100 respectively A summary of the ∆(27) group properties and

its representations can be found in appendix B There are two different scenarios for the

model determined by the two possible VEV alignments for ξ namely

〈ξ〉 = (0 1 0)vξ Case I

〈ξ〉 = (1 ω 1)vξ Case II(210)

with ω = e2πi3 As indicated above we will denote the models described by each alignment

as cases I and II respectively Note that the case II vacuum pattern frequently appears in

the context of geometrical CP violation [56 57] The VEVs for the remaining flavon fields

〈ϕ〉 = (1 1 1)vϕ 〈σ1〉 = vσ1 〈σ2〉 = vσ2 (211)

Further details regarding this vacuum configuration are offered in appendix C

3 Lepton sector

Once the basic framework has been laid out we are in position to discuss the structure of

the lepton sector and its phenomenological implications As we will show below charged

lepton as well as Dirac neutrino masses are generated at leading order (LO) and non-zero

values for the ldquoreactor anglerdquo θ13 arise naturally The model is predictive in the sense that

the three mixing angles and the Dirac CP phase will ultimately be determined in terms of

only two parameters

ndash 4 ndash

JHEP01(2016)007

Field Ψℓ Ψe Ψmicro Ψτ Ψν1 Ψν2 Ψν3 H ϕ ξ σ1 σ2

∆(27) 3 100 110 120 100 100 100 100 3 3 101 100

Z4 1 1 1 1 minus1 i minus1 1 1 minus1 1 i

Z prime4 1 i i i minus1 minus1 minus1 1 minusi 1 minus1 minus1

Table 1 Particle content and transformation properties of the lepton and scalar sectors under the

flavor symmetry ∆(27)otimes Z4 otimes Z prime4

31 Lepton masses and mixing

The transformation properties of leptons and scalars under the family symmetry ∆(27)otimesZ4 otimes Z prime

4 are given in table 1 Note that the Higgs field is inert under the flavor symmetry

Since the three left-handed lepton doublets are unified into a faithful triplet 3 of ∆(27)

they will share one common bulk mass parameter cℓ On the other hand both right-

handed charged leptons and right-handed neutrinos are assigned to singlet representations

of ∆(27) Therefore there are six different bulk mass parameters cei and cνi (i = 1 2 3)

for these fields From the particle transformation properties we can write the most general

lepton Yukawa interactions that are both gauge and flavor invariant at LO1

radicG

ye(ϕΨℓ

HΨe + ymicro(ϕΨℓ

HΨmicro + yτ(ϕΨℓ

radicG

(Λprime)72

(ξσ1Ψℓ

HΨν1 + y31(ξσlowast

1Ψℓ

HΨν1 + y22(ξσ2Ψℓ

HΨν2

+y13(ξσ1Ψℓ

1Ψℓ

HΨν3

δ(y minus L) + hc (31)

with H equiv iτ2Hlowast and τi as the Pauli matrices After electroweak and flavor spontaneous

symmetry breaking all leptons develop masses dictated by the above Yukawa interactions

The generated masses are modulated by the overlap of the relevant zero mode fermion

profiles the VEV profile of the Higgs and the flavon VEVs given in eqs (210) (211)

From eq (31) The mass matrix ml for charged leptons is

(LΛ)32

vHradic2

radic3Ul

ye 0 0

0 ymicro 0

0 0 yτ

where Ul stands for the so-called magic matrix

Ul =1radic3

1 ω ω2

1 ω2 ω

1Notice that the Majorana coupling of the form ΨνiΨcνj

with i j = 1 3 can be forbidden by including

addtional auxiliary cyclic group such as Z3

ndash 5 ndash

JHEP01(2016)007

and yemicroτ are modified Yukawa couplings defined as

yemicroτ = yemicroτF (0 cℓ cei) (34)

in terms of the overlapping function

F (y cL cR) equiv f(0)L (y cL) f

(0)R (y cR)fH(y) (35)

radic2 (1minus βH) (1minus 2cL) (1 + 2cR) k3L3

[1minus eminus2(1minusβH)kL

] [e(1minus2cL)kL minus 1

] [e(1+2cR)kL minus 1

] eminus(1minusβH)kLe(2minusβHminuscL+cR)ky

Given that U daggerl Ul = 1 the diagonalization of the charged lepton mass matrix is straightfor-

ward leading to charged lepton masses of the form

memicroτ =

radic3 yemicroτ

(LΛ)32

vHradic2 (36)

Analogously taking into account the two distinct VEV alignments for the flavon triplet ξ

in eq (210) the neutrino mass matrix for each respective case can be written as

mIν =

(LΛprime)32

vξΛprime

vHradic2

y11vσ1Λprime 0 y13

vσ1Λprime

0 y22vσ2Λprime 0

y31vlowastσ1Λprime 0 y33

vlowastσ1Λprime

mIIν =

(LΛprime)32

vξΛprime

vHradic2

radic3V0

vσ1Λprime

0 y22vσ2Λprime 0

vlowastσ1Λprime

yij = yijF (L cℓ cνj ) (39)

V0 equiv1radic3

ω 1 1

1 ω 1

1 1 ω

Thus the diagonalizing matrix for the neutrino sector can be parameterized as

U Iν =

cos θν 0 sin θνeiϕν

minus sin θνeminusiϕν 0 cos θν

U IIν = V0

In terms of the auxiliary functions

Xplusmnν = |y31|2+|y33|2plusmn|y11|2plusmn|y13|2 Yν = y11y33minusylowast13y

lowast31 Zν = y11y

lowast31+y13y

lowast33 (313)

ndash 6 ndash

JHEP01(2016)007

the relevant parameters of the model θν and ϕν are given by

tan 2θν = 2|Zν |Xminusν ϕν = arg

(v2σ1

) (314)

and the neutrino mass eigenvalues for both NH and IH are determined as

bull Case I

NH m1 =v1radic2Mminus

ν Yν

) m2 = v2 |y22| m3 =

v1radic2M+

ν Yν

) for Xminus

ν cos 2θν gt 0

IH m1 =v1radic2M+

ν Yν

) m2 = v2 |y22| m3 =

v1radic2Mminus

ν Yν

) for Xminus

ν cos 2θν lt 0

bull Case II

NH m1 =

radic3

minus(X

+ν Yν

) m2 =

radic3v2 |y22| m3 =

radic3

+ν Yν

) for Xminus

ν cos 2θν gt 0

IH m1 =

radic3

+ν Yν

) m2 =

radic3v2 |y22| m3 =

radic3

minus(X

+ν Yν

) for Xminus

ν cos 2θν lt 0

where we have defined

Mplusmn(x y) =

radicxplusmn

radicx2 minus 4|y|2 (319)

∣∣∣∣∣1

(LΛprime)32

vξΛprime

Λprime

vHradic2

∣∣∣∣∣ α = 1 2 (320)

Without loss of generality the angle θν is restricted to the interval [0 π] Notice

that Xminusν cos 2θν = 2|Zν | cos2 2θν sin 2θν As a result for non-vanishing values of Zν the

neutrino mass spectrum displays Normal Hierarchy (NH) provided 0 lt θν lt π2 whereas

Inverted Hierarchy (IH) is realized for π2 lt θν lt π The angle ϕν on the other hand

can take any value in the interval [0 2π]

At leading order the lepton mixing matrix UPMNS = U daggerl Uν becomes

U IPMNS =

1radic3

cos θν minus eminusiϕν sin θν 1 cos θν + eiϕν sin θν

cos θν minus ωeminusiϕν sin θν ω2 ω cos θν + eiϕν sin θν

cos θν minus ω2eminusiϕν sin θν ω ω2 cos θν + eiϕν sin θν

U IIPMNS =

minusiωradic3

ω cos θν minus ω2eminusiϕν sin θν 1 ω2 cos θν + ωeiϕν sin θν

ω cos θν minus eminusiϕν sin θν ω2 cos θν + ωeiϕν sin θν

In both cases the solar atmospheric and reactor angles can be written in terms of θν and

ϕν as

sin2 θ12 =1

2minus sin 2θν cosϕν

ndash 7 ndash

JHEP01(2016)007

sin2 θ13

sin2 θ23

sin2 θ12

010 015 020 025 030 035 040

θνπ

ϕνπ

sin2 θ13

sin2 θ23

sin2 θ12

060 065 070 075 080 085 090

θνπ

ϕνπ

Figure 1 1σ 2σ and 3σ ranges of sin2 θ12 (green) sin2 θ23 (blue) and sin2 θ13 (red) for normal (left

panel) and inverted (right panel) neutrino mass hierarchies Best-fit contours for sin2 θ13 (sin2 θ23)

are indicated by dotted (short-dashed) lines The long-dashed contour in the left panel represents

the local minimum in the first octant of θ23

sin2 θ23 =1minus sin 2θν sin(π6minus ϕν)

sin2 θ13 =1

3(1 + sin 2θν cosϕν) (323)

A convenient description for the CP violating phase in this sector is the Jarlskog invariant

JCP = Im[Ulowaste1U

lowastmicro3Umicro1Ue3] [58] which in this parameterization takes the compact form

JCP = minus 1

6radic3cos 2θν (324)

It is worthy of attention the independence of JCP upon ϕν and the simple predicted relation

between the solar and reactor angles θ12 and θ13

sin2 θ12 cos2 θ13 =

3 (325)

32 Phenomenological implications

As shown above only two parameters are required to generate the three angles and the

Dirac CP violating phase characterizing the lepton mixing matrix making this model

highly predictive In the remaining part of this section we explore in detail the predictions

for the lepton mixing parameters and the neutrino mass spectrum

In figure 1 the θν ndash ϕν parameter region compatible with experimental data is delimited

using the global fit of neutrino oscillations given in [35] for each mass ordering shown as

the left and right hand panel The model can reproduce successfully the best-fit values

for the atmospheric and reactor angles reaching simultaneously the 2σ region for the solar

ndash 8 ndash

JHEP01(2016)007

NH1 NH2 IH

sin2 θ2310minus1 567 473 573

sin2 θ1310minus2 226 226 229

sin2 θ1210minus1 341 341 341

JCP10minus2 minus(+)271 minus(+)337 +(minus)257

Table 2 Central predictions for sin2 θ12 and JCP obtained from the central values of the atmo-

spheric and reactor angles reported in ref [35] The sign of JCP in the parentheses corresponds to

the bracketed prediction for θν in eq (326)

angle The intersecting points of the ldquocentralrdquo or best fit curve in the sin2 θ13 contour and

the corresponding ones in the sin2 θ23 contour are located at

NH1 θνπ = 0204(0296) ϕνπ = 0924

NH2 θνπ = 0193(0307) ϕνπ = 1031

IH θνπ = 0707(0793) ϕνπ = 1917

where NH1 denotes the best-fit contour of sin2 θ23 and NH2 corresponds to its local min-

imum in the first octant Notice that the numbers in parenthesis denote the intersection

values within the range θν isin [π4 π2] cup [3π4 π]

Once we have determined θν and ϕν from the central values of the atmospheric and

reactor oscillation global fits the predictions for the solar angle and the Jarlskog invariant

can be straightforwardly obtained using eqs (323) (324) For completeness in table 2

we present the full set of mixing parameters derived from the points defined in eq (326)

Remarkably the central prediction for sin2 θ12 falls very close to its 1σ boundary

In addition notice that the 1σ range of JCP is entirely contained in the region θν isin[0 π4] cup [3π4 π]

We conclude this section bringing forth a consistent realization of lepton masses and

mixing angles In the numerical analysis we assume that the fundamental 5D scale is

k ≃ Λ ≃ MPl withMPl ≃ 244times1018GeV as the reduced Planck mass We also set the scale

Λprime ≃ kprime = keminuskL ≃ 15TeV in order to account for the hierarchy between the Planck and the

electroweak scales allowing for the lowest KK gauge boson resonances (with massesmKK =

3 sim 4TeV) to be within the reach of the LHC experiments The Higgs VEV is identified

with its standard model value vH ≃ 246GeV and the ratios vϕΛ vξΛprime vσ1Λ

prime vσ2Λprime are

all fixed to 01 (thus considering real-valued flavon VEVs) The Higgs localization param-

eter β common to all mass matrix elements is chosen as 095 in the following discussion

As an as illustrative example we can choose cℓ = 185 ce = minus027 cmicro = minus044

cτ = minus071 |ye| = 0861 |ymicro| = 0898 |yτ | = 0994 to generate the charged lepton

masses me = 0511MeV mmicro = 1057MeV mτ = 1777GeV For the neutrino sector

benchmark points (BPs) in parameter space are given in table 3 There the four BPs are

labeled according to their hierarchy scheme and case as NH-I NH-II IH-I IH-II One sees

that indeed the large disparity between charged lepton masses is reproduced for Yukawa

couplings of the same order of magnitude

ndash 9 ndash

JHEP01(2016)007

NH-I NH-II IH-I IH-II

cν1 minus140 minus141 minus139 minus140

y11 minus1000minus 0307i 0282 + 1166i 0752 + 0096i minus0674 + 0520i

y13 minus0451 + 0631i 0031minus 0880i 0919minus 0432i 1026minus 0542i

y22 0860 + 0353i 0097minus 1088i minus0905minus 0194i 0974 + 0431i

y31 0667 + 0397i 0001minus 0881i 0941 + 0383i minus1070 + 0450i

y33 0792minus 0683i minus0324 + 1154i 0746minus 0136i 0829minus 0191i

Table 3 Benchmark points for the neutrino sector featuring both NH and IH in Cases I and II

m1 [eV] 180times 10minus3 259times 10minus3 488times 10minus2 489times 10minus2

∆m221 [10

minus5eV2] 760 760 750 748

|∆m231| [10minus3eV2] 248 248 238 238

sin2 θ1210minus1 341 341 341 341

sin2 θ2310minus1 567 567 573 573

sin2 θ1310minus2 226 226 229 229

JCP10minus2 minus271 minus271 minus258 minus257

Table 4 Neutrino masses and oscillation parameters associated to the four chosen benchmark

points

The neutrino masses splittings and mixing angles associated to each BP are displayed

in table 4 All the obtained neutrino oscillation parameters are consistent with the global

fit in ref [35] In particular the reproduced atmospheric and reactor angles lie comfortably

in their respective 1σ region whereas the solar angle values are contained in the 2σ range

very close to the 1σ boundary

4 Quark sector

The quark transformation properties under the family group ∆(27)otimesZ4 otimesZ prime4 are given in

table 5 At leading order the most general invariant Yukawa interactions can be written as

radicG

(Λprime)52

lowast2ΨU HΨu + yctσ

lowast1ΨCHΨt + ytcσ

lowast2ΨT HΨc + yttσ1ΨT HΨt

+ydsσlowast1ΨUHΨs + ydbσ1ΨUHΨb + ysdσ

lowast2ΨCHΨd

ndash 10 ndash

JHEP01(2016)007

Field ΨU ΨC ΨT Ψu Ψc Ψt Ψd Ψs Ψb

∆(27) 102 101 100 102 100 102 101 100 101

Z4 minusi minusi minusi 1 1 minusi 1 minusi minusi

Z prime4 1 1 1 minus1 minus1 minus1 minus1 minus1 minus1

Table 5 Particle content and transformation properties of the quark sector under the flavor

symmetry ∆(27)otimes Z4 otimes Z prime4

+yssσ1ΨCHΨs + ybbσlowast1ΨTHΨb

Again after spontaneous electroweak and flavor symmetry breaking the mass matrices for

the up and down quark sectors read

(LΛprime)32

vHradic2

yuuvlowastσ2Λprime 0 0

0 0 yctvlowastσ1Λprime

0 ytcvlowastσ2Λprime yttvσ1Λ

(LΛprime)32

vHradic2

0 ydsvlowastσ1Λprime ydbvσ1Λ

ysdvlowastσ2Λprime yssvσ1Λ

prime 0

0 0 ybbvlowastσ1Λprime

yuiuj= yuiuj

F (L cQi cuj

ydidj = ydidjF (L cQi cdj ) (43)

The up-type quark mass matrix is already block-diagonal The diagonalization of the

down-type mass matrix md requires a more careful treatment For the sake of simplicity

in the following analysis we denote the ij element of mu (md) as muij (md

ij) The product

of the down-type mass matrix and its adjoint

mdmddagger =

|md12|2 + |md

13|2 md12m

dlowast22 md

13mdlowast33

mdlowast12m

d22 |md

21|2 + |md22|2 0

mdlowast13m

d33 0 |md

can be diagonalized in two steps in first place an approximate block diagonalization

UdprimedaggermdmddaggerUdprime ≃

|md12|2 md

12mdlowast22 0

mdlowast12m

d22 |md

21|2 + |md22|2 0

0 0 |md33|2

is accomplished with the aid of the transformation matrix

Udprime ≃

1 0 ǫ

minusǫlowast 0 1

ndash 11 ndash

JHEP01(2016)007

and subsequently the diagonalization is completed through a unitary rotation of the upper

block This approximation is consistent provided |md33| ≫ |md

12| |md13| |md

22| and |ǫ| ≪ 1

The resulting diagonalization matrices for the up and down sectors can be parameterized as

0 cos θu sin θueiϕu

0 minus sin θueminusiϕu cos θu

Ud ≃

cos θd sin θdeiϕd ǫ

minus sin θdeminusiϕd cos θd 0

minusǫlowast cos θd minusǫlowast sin θdeiϕd 1

withtan 2θu = 2|Zu|Xminus

u ϕu = argZu

tan 2θd = 2|Zd|Xminusd ϕd = argZd ǫ = BdAd

andXplusmn

u = |mu33|2 + |mu

32|2 plusmn |mu23|2 Yu = mu

23mulowast32 Zu = mu

23mulowast33

Xplusmnd = |md

22|2 + |md21|2 plusmn |md

12|2 Yd = md12m

dlowast21 Zd = md

12mdlowast22

Ad = |md33|2 minus |md

12|2 minus |md13|2 Bd = md

13mdlowast33

Correspondingly the quark mass eigenvalues can be expressed in terms of Mplusmn defined in

eq (319) as

mu = |mu11| mc =

1radic2Mminus

u Yu) mt =

1radic2M+

md =1radic2Mminus

d Yd) ms =

1radic2M+

d Yd) mb = |mb

33| (410)

so that the CKM matrix is given by

VCKM = UdaggeruUd (411)

cos θd eiϕd sin θd ǫ

minuseminusiϕd cos θu sin θd minus eiϕu sin θu cos θdǫlowast cos θd cos θu minus ei(ϕu+ϕd) sin θu sin θdǫ

lowast minuseiϕu sin θu

minuseminusi(ϕd+ϕu) sin θd sin θu minus cos θu cos θdǫlowast eminusiϕu cos θd sin θu minus eiϕd cos θu sin θdǫ

lowast cos θu

Hence the quark sector Dirac CP phase (in PDG convention) and the Jarlskog invariant

take the form

δqCP = π minus arg(ǫ) + ϕd + ϕu (412)

JqCP ≃ 1

4|ǫ| sin 2θd sin 2θu sin δqCP (413)

According to eq (43) the size of up and down mass matrix elements is determined by

the overlap of the 5D quark field zero mode profiles ie muij prop f

(0)L (L cQi

)f(0)R (L cuj

mdij prop f

(0)L (L cQi

)f(0)R (L cdj ) If the wave function localization parameters cQi

cui cdi are

chosen such that the quark zero mode profiles obey

f(0)L (L cU ) ≪ f

(0)L (L cC) ≪ f

(0)L (L cT )

ndash 12 ndash

JHEP01(2016)007

f(0)R (L cu) ≪ f

(0)R (L cc) ≪ f

(0)R (L ct)

f(0)R (L cd) ≪ f

(0)R (L cs) ≪ f

(0)R (L cb) (414)

then the elements of mu and md approximately satisfy

mu11 ≪ mu

23 sim mu32 ≪ mu

33 md12 sim md

21 ≪ md22 ≪ md

33 md13 ≪ md

33 (415)

justifying the perturbative diagonalization performed on mdmddagger These relations imply

that X+ud ≫ |Yud| holds and therefore a rough estimate for the mixing parameters and

quark mass spectrum is

θu sim∣∣∣∣mu

∣∣∣∣ simf(0)L (L cC)

f(0)L (L cT )

θd sim∣∣∣∣md

∣∣∣∣ simf(0)L (L cU )

f(0)L (L cC)

|ǫ| sim∣∣∣∣md

∣∣∣∣ simf(0)L (L cU )

f(0)L (L cT )

mu sim |mu11| mc sim

∣∣∣∣mu

23mu32

∣∣∣∣ mt sim |mu33|

md sim∣∣∣∣md

12md21

∣∣∣∣ ms sim∣∣∣md

∣∣∣ mb sim∣∣∣md

∣∣∣ (416)

Thus in order to reproduce plausible quark masses and mixings namely

θu sim 10minus1 θd sim 10minus2 |ǫ| sim 10minus3

mu mc mt sim 10minus5 10minus2 1

md ms mb sim 10minus3 10minus2 1

the quark zero mode profiles must observe the following hierarchy

f(0)L (L cU ) f

(0)L (L cC) f

(0)L (L cT ) sim 10minus3 10minus1 1

f(0)R (L cu) f

(0)R (L cc) f

(0)R (L ct) sim 10minus2 10minus1 1

f(0)R (L cd) f

(0)R (L cs) f

(0)R (L cb) sim 10minus1 10minus1 1 (418)

To conclude this section an explicit realization of quark masses and mixings is pre-

sented The choice cU = 197 cC = 192 cT = 183 cu = minus076 cc = minus062 ct = minus056

cd = minus074 cs = minus069 cb = minus068 yuu = minus0438 minus 0954i yct = minus0360 minus 1038i

ytc = 1147 minus 0273i ytt = minus0372 minus 1073i yds = minus0966 minus 0285i ydb = 0290 + 0400i

ysd = 0838minus 0226i yss = minus0703minus 0207i ybb = 0637minus 0879i generates the quark mass

spectrum

mu = 230MeV mc = 1275GeV mt = 173GeV

md = 480MeV ms = 950MeV mb = 418GeV (419)

and fixes the magnitude of VCKM elements at

|VCKM| =

0974 0225 00035

0225 0973 00414

00089 0041 0999

ndash 13 ndash

JHEP01(2016)007

Finally the obtained values for the Dirac CP phase and the Jarlskog invariant are

δqCP = 125 JqCP = 306times 10minus5 (421)

The resulting quark masses and mixings are consistent with the current experimental

data [1] and the precision of the results can be improved by incorporating high order

corrections addressed in the next section

5 High order corrections

From the particle content and above transformation properties one finds that nontrivial

high order corrections to the charged lepton sector are absent in the present model The

next-to-leading order (NLO) corrections to the neutrino Yukawa interactions are given by

δLνY =

radicG

(Λprime)92

[(ξlowastξlowast)3σ

lowast2Ψl

HΨν2δ(y minus L) + hc (51)

However the contribution of these terms to the neutrino masses and mixing parameters can

be absorbed by a proper redefinition of the parameter y22 after SSB Hence in order to esti-

mate the effects of higher order corrections in this sector we need to investigate the Yukawa

terms involving an additional (vIRΛprime)2 suppression with respect to the lowest order terms

in eq (31) where we have introduced vIR to characterize the magnitude of vξ sim vσ1 sim vσ2

The contraction of the field products ΨlHΨν1 ΨlHΨν3 transforming as (3minus1minus1)

under ∆(27)otimes Z4 otimes Z prime4 with the flavon operators

(Λprime)112

(ξξlowast)1a2ξσ1 1

(Λprime)112

(ξξlowast)1a1ξσlowast1

(Λprime)112

ξσ31

(Λprime)112

ξσlowast31

as well as the combination of ΨlHΨν2 sim (3 iminus1) and

(Λprime)112

(ξξlowast)1abξσ2

(Λprime)112

ξσ21σ2

(Λprime)112

ξσlowast21 σ2 (53)

provide the desired high order corrections to the neutrino Yukawa interactions In the

above expressions the indices a b = 0 1 2 label the different singlets of ∆(27) Additional

terms that can be absorbed into y11 y13 y22 y31 and y33 have been omitted Taking into

consideration these corrections the neutrino mass matrix mν can be roughly written as

mν ≃ 1

(LΛprime)32

vHradic2

vξΛprime

vσ1Λprime

0 y22vσ2Λprime 0

vlowastσ1Λprime

(vIRΛprime

0 x12 0

x21 0 x23

0 x32 0

with xij = xijF (L cl cνj ) and xij as dimensionless parameters of order O(1)

Working under the same numerical framework established in section 3 one can readily

estimate the shift in the neutrino oscillation parameters induced by high order corrections

of the Yukawa interaction Particularly in Case I taking xij as random complex numbers

ndash 14 ndash

JHEP01(2016)007

with magnitudes ranging from 2 to 6 and vIR = 01 the resulting deviations in the neutrino

mixing parameters with respect to their LO values can be estimated as

δs212 sim 001 δs223 sim 001 δs213 sim 0001 δJCP sim 0001 (55)

On the other hand the corrections to the neutrino mass splittings are negligible

δ(∆m2

)sim 10minus7 eV2 δ

(∆|m2

31|)sim 10minus6 eV2 (56)

From eq (55) it is clear that high order corrections can easily drive s212 into its 1σ region

while keeping the remaining parameters optimal

Turning to the quark sector every bilinear formed by ΨQiand Ψui

or Ψdi can produce a

high order correction to the Yukawa interaction whenever it is contracted with the adequate

cubic flavon operator Beside terms that can be absorbed by a redefinition of yuiujor ydidj

in eq (41) all the NLO contributions can be classified into three categories

bull Invariant products of ΨUHΨc ΨCHΨu ΨTHΨd sim (101 iminus1) with

(Λprime)92

(ξξlowast)102σlowast2

(Λprime)92

σ21σ

lowast2 (57)

bull Invariant products of ΨUHΨt ΨCHΨb ΨTHΨs sim (100 1minus1) with

(Λprime)92

(ξξlowast)102σ1 1

(Λprime)92

σlowast31 (58)

bull Invariant products of ΨCHΨc ΨT HΨu and ΨUHΨd sim (102 iminus1) with

(Λprime)92

σlowast21 σlowast

2 (59)

Again after symmetry breaking the quark mass matrices mu and md can be approx-

imately written as

(LΛprime)32

vHradic2

(vIRΛprime

0 xuc xut

xcu xcc 0

xtu 0 0

(LΛprime)32

vHradic2

prime 0

0 0 ybbvσ1Λprime

(vIRΛprime

xdd 0 0

0 0 xsb

xbd xbs 0

Here we have defined xuiuj= xuiuj

F (L cQi cuj

) and xdidj = xdidjF (L cQi cdj ) where the

couplings xuiujand xdidj represent dimensionless parameters of order O(1) As a numerical

example taking xuiuj xdidj as random complex numbers with magnitudes ranging from

1 to 4 for xuc xcu xbd xcc xtu xdd and from 2 to 6 for xut xsb xbs while keeping the

ndash 15 ndash

JHEP01(2016)007

ds2 = eminus2kyηmicroνdxmicrodxν minus dy2 S1Z2

∆(27)otimes Z4 otimes Z prime4

ξ σ1 σ2

ℓL QL τR νR uR dR H eR microR

SU(2)L otimes U(1)Y

Figure 2 Pictorial description of the basic warped model structure showing the UV (IR) peaked

nature of the standard model fields

values of cQi cui

cdi yuiujand ydidj reported in section 4 the order of deviation with

respect to the LO values of the quark masses is

δmu sim 0001MeV δmc sim 10MeV δmt sim 01MeV

δmd sim 01MeV δms sim 01MeV δmb sim 05MeV (511)

The corresponding correction to the first order CKM matrix is of order

δ|VCKM| sim

0001 0005 00001

0005 0001 0001

00005 0001 000005

and the values for the quark CP violating phase and the Jarlskog invariant are displaced by

δ(δqCP) sim 01 δJqCP sim 10minus6 (513)

As for the lepton sector it is not difficult to find parameter values reproducing the quark

mass and mixing parameters required to fit the current experimentally observed values

6 Conclusions

We have proposed a five-dimensional warped model in which all standard model fields

propagate into the bulk Its structure is summarized in the ldquocartoonrdquo depicted in figure 2

Mass hierarchies in principle arise from an adequate choice of the bulk shape parameters

while fermion mixing angles are constrained by relations which follow from the postulated

∆(27) flavor symmetry group broken on the branes by a set of flavon fields The neutrino

ndash 16 ndash

JHEP01(2016)007

mixing parameters and the Dirac CP violation phase are described in terms of just two

independent parameters at leading order This leads to stringent predictions for the lepton

mixing matrix which should be tested in future neutrino oscillation experiments Likewise

the scheme also includes the quark sector providing an adequate description of the quark

mixing matrix The effect of next-to-leading order contributions is estimated to be fully

consistent with the experimental requirements

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant

Nos 11275188 11179007 and 11522546 by the Spanish grants FPA2014-58183-P Multi-

dark CSD2009-00064 SEV-2014-0398 (MINECO) and PROMETEOII2014084 (Gener-

alitat Valenciana) ADR and CAV-A acknowledge support from CONACyT (Mexico)

grants 250610 and 251357

A The profile of the Higgs zero mode

In this appendix we shall show that the Higgs zero mode profile in eq (24) can be achieved

for a proper choice of boundary conditions Following the original idea proposed in the

appendix A of ref [48] the action of the Higgs field reads as

SbulkH =

intd5x

radicG

[(DMH)

daggerDNH

]minusm2

HTr(HdaggerH)minus VUV(H)δ(y)minus VIR(H)δ(y minus L)

In order to generate a VEV close to the IR brane the potential VIR(H) is assumed to have

the usual ldquomexican hatrdquo form [48]

VIR(H) =λ

[Tr(HdaggerH)minus v2TeV

]2(A2)

On the UV brane only a mass term is added [42 47 48]

VUV(H) = minusmUV Tr(HdaggerH) (A3)

The variation of the action yields both the equations of motion as well as boundary condi-

tions for the bulk Higgs field as follows

(e2kypartye

minus4kyparty minus partmicropartmicro minus eminus2kym2H

)H = 0 (A4)

partyH + λH[Tr(HdaggerH)minus v2TeV2

]∣∣∣y=L

= 0 (A5)

partyH minusmUVH|y=0 = 0 (A6)

The Higgs potential in eq (A1) leads to non-zero VEV of the Higgs doublet H with

〈H〉 = 1radic2

1radic2

)fH(y)radic

L (A7)

ndash 17 ndash

JHEP01(2016)007

where v(y) equiv vHfH(y) and vH denotes the effective 4D VEV of the zero mode of φ0

Obviously fH(y) fulfills the following set of conditions

(e2kypartye

minus4kyparty minus eminus2kym2H

)fH(y) = 0 (A8)

partyfH(y) + λfH(y)[v2Hf2

H(y)minus v2TeV2]∣∣

y=L= 0 (A9)

partyfH(y)minusmUVfH(y)|y=0 = 0 (A10)

The most general solution of eq (A8) is given by

fH(y) = ae(2+β)ky + be(2minusβ)ky (A11)

where β =radic4 +m2

Hk2 a and b are determined by the boundary conditions in eq (A9)

and (A10) From eq (A10) we have

b= minus(2minus β)k minusmUV

(2 + β)k minusmUV (A12)

In the case of mUV = (2 minus β)k we have a = 0 such that only the term e(2minusβ)ky is picked

out Considering further the normalization conditionint L0 eminus2kyf2

H(y) = 1 we obtain

fH(y) =

radic2kL(1minus β)

1minus eminus2(1minusβ)kLekLe(2minusβ)k(yminusL) (A13)

It is exactly the desired Higgs profile zero mode Notice that the boundary conditions of

the Higgs field is of the ldquomixedrdquo type as shown in eqs (A5) (A6) and that a Higgs zero

mode can not be obtained in the absence of VIR(H) and VUV(H)

B Group theory of ∆(27) and its representation

The ∆(27) group is isomorphic to (Z3 otimes Z3) ⋊ Z3 It can be conveniently expressed in

terms of three generators a aprime and b which satisfy the following relations

a3 = aprime3= b3 = 1 aaprime = aprimea

babminus1 = aminus1aprimeminus1

baprimebminus1 = a (B1)

All ∆(27) elements can be written into the form bkamaprimen with km n = 0 1 2 The group

has 11 conjugacy classes given by

1C1 = 11C

(1)1 = aaprime2

1C(2)1 = a2aprime

3C(01)3 = a aprime a2aprime2

3C(02)3 = a2 aprime2 aaprime

3C(10)3 = b baaprime2 ba2aprime

ndash 18 ndash

JHEP01(2016)007

χ100 χ101 χ102 χ110 χ111 χ112 χ120 χ121 χ122 χ3 χ3

1C1 1 1 1 1 1 1 1 1 1 3 3

1C(1)1 1 1 1 1 1 1 1 1 1 3ω2 3ω

1C(2)1 1 1 1 1 1 1 1 1 1 3ω 3ω2

3C(01)3 1 ω ω2 1 ω ω2 1 ω ω2 0 0

3C(02)3 1 ω2 ω 1 ω2 ω 1 ω2 ω 0 0

3C(10)3 1 1 1 ω ω ω ω2 ω2 ω2 0 0

3C(11)3 1 ω ω2 ω ω2 1 ω2 1 ω 0 0

3C(12)3 1 ω2 ω ω 1 ω2 ω2 ω 1 0 0

3C(20)3 1 1 1 ω2 ω2 ω2 ω ω ω 0 0

3C(21)3 1 ω ω2 ω2 1 ω ω ω2 1 0 0

3C(22)3 1 ω2 ω ω2 ω 1 ω 1 ω2 0 0

Table 6 Character table of ∆(27)

3C(11)3 = ba baprime ba2aprime2

3C(12)3 = ba2 baaprime baprime2

3C(20)3 = b2 b2aaprime2 b2a2aprime

3C(21)3 = b2a b2aprime b2a2aprime2

3C(22)3 = b2a2 b2aaprime b2aprime2 (B2)

The ∆(27) has nine one-dimensional representations which we denote as 1kr (k r =

0 1 2) and two three-dimensional irreducible representations 3 and 3 The explicit form

of the group generators in each irreducible representation is

1kr a = ωr aprime = ωr b = ωk

ω 0 0

0 0 ω2

aprime =

ω2 0 0

0 ω 0

ω2 0 0

0 0 ω

aprime =

ω 0 0

0 ω2 0

where ω = e2πi3 is the cube root of unity Notice that 3 and 3 are complex representations

dual to each other From the character table of the group shown in table 6 we can

straightforwardly obtain the Kronecker products between the various representations

1kr otimes 1kprimerprime = 1[k+kprime][r+rprime] 3otimes 1kr = 3 3otimes 1kr = 3

3otimes 3 =2sum

1kr 3otimes 3 = 3oplus 3oplus 3 3otimes 3 = 3oplus 3oplus 3 (B4)

ndash 19 ndash

JHEP01(2016)007

where [n] stands for nmod 3 whenever n is an integer Starting from the representation ma-

trices of the generators in different irreducible representations we can calculate the Clebsch-

Gordan (CG) coefficients for the Kronecker products listed above All CG coefficients are

presented in the form α otimes β where αi stands for the elements of the first representation

and βj those of the second one In the following we adopt the convention α[3] = α0 equiv α3

bull 1kr otimes 1kprimerprime = 1[k+kprime][r+rprime]

otimes(β1

1kprimerprime

=(α1β1

1[k+kprime][r+rprime]

bull 3otimes 1kr = 3

otimes(β1

α[1+r]β1

ωkα[2+r]β1

ω2kα[3+r]β1

otimes(β1

α[1minusr]β1

ωkα[2minusr]β1

ω2kα[3minusr]β1

bull 3otimes 3 =sum2

kr=0 1kr

otimes

= (α1β1+α2β2+α3β3)100oplus (α1β1+ω2α2β2+ωα3β3)110

oplus (α1β1+ωα2β2+ω2α3β3)120

oplus(α3β1+α1β2+α2β3)101oplus (α3β1+ω2α1β2+ωα2β3)111

bull 3otimes 3 = 3S1 oplus 3S2 oplus 3A

otimes

α1β1

α2β2

α3β3

oplus 1

α2β3 + α3β2

α3β1 + α1β3

α1β2 + α2β1

oplus 1

α2β3 minus α3β2

α3β1 minus α1β3

α1β2 minus α2β1

where the subscripts ldquoSrdquo and ldquoArdquo denote symmetric and anti-symmetric combina-

tions respectively

ndash 20 ndash

JHEP01(2016)007

otimes

α1β1

α2β2

α3β3

oplus 1

α2β3 + α3β2

α3β1 + α1β3

α1β2 + α2β1

oplus 1

α2β3 minus α3β2

α3β1 minus α1β3

α1β2 minus α2β1

C Vacuum alignment

In this appendix we shall investigate the problem of achieving the vacuum configuration

in eq (210) and eq (211) For self-consistency all flavon fields ϕ ξ σ1 and σ2 are treated

as complex given the form of the ∆(27) representation matrices and the fact that the

Z4 charge of σ2 is purely imaginary Since the flavons ϕ and ξ σ1 σ2 are assumed to

be localized at y = 0 and y = L respectively the vacuum alignment problem is greatly

simplified At the UV brane y = 0 the flavon ϕ transforms in the manner listed in table 1

The scalar potential invariant under the flavor symmetry ∆(27)otimesZ4otimesZ prime4 can be written as

VUV(ϕ) = M2ϕ(ϕϕ

lowast)100 + f1

[(ϕϕ)

3S1(ϕlowastϕlowast)3S1

[(ϕϕ)

+ flowast3

[(ϕϕ)

with real couplings M2ϕ f1 and f2 Note that ϕ = (ϕ1 ϕ2 ϕ3) is a ∆(27) triplet 3 and its

complex conjugate ϕlowast = (ϕlowast1 ϕ

lowast2 ϕ

lowast3) transforms consequently as 3 Focusing on the field

configuration

〈ϕ〉 = (1 1 1)vϕ (C2)

the minimum conditions for the UV potential read

partVUV(ϕ)

partϕlowast1

=partVUV(ϕ)

partϕlowast2

=partVUV(ϕ)

partϕlowast3

ϕ + 2(f1 + f2 + f3 + flowast3 )∣∣vϕ

∣∣2]= 0 (C3)

leading to a non zero solution

∣∣vϕ∣∣2 = minus

2(f1 + f2 + f3 + flowast3 )

that holds in a finite portion of parameter space with f1 + f2 + f3 + flowast3 lt 0

Similarly at the IR brane y = L the most general renormalizable scalar potential VIR

involving the flavon fields ξ σ1 σ2 is

VIR(ξ σ1 σ2) = M2ξ (ξξ

lowast)100+M2

σ1(σ1σ

lowast1)100

+M2σ2(σ2σ

lowast2)100

[(ξξ)

(ξlowastξlowast)3S1

[(ξξ)

+ glowast3

[(ξξ)

+g4σ21σ

lowast21 + g5σ

lowast22 + g6|σ1|2|σ2|2 + g7(ξξ

lowast)100|σ1|2 + g8(ξξ

lowast)100|σ2|2

+g9(ξξlowast)101

σ21 + glowast9(ξξ

lowast)102σlowast21 (C5)

ndash 21 ndash

JHEP01(2016)007

where all couplings excluding g3 and g9 are real For this potential the Case I alignment

〈ξ〉 = (0 vξ 0) 〈σ1〉 = vσ1 〈σ2〉 = vσ2 (C6)

determines the minimization conditions

partVIR(ξ σ1 σ2)

partξlowast1= glowast9vξv

lowastσ1

partVIR(ξ σ1 σ2)

partξlowast2= vξ

ξ + 2g1∣∣vξ

∣∣2 + g7∣∣vσ1

∣∣2 + g8∣∣vσ2

∣∣2)= 0

partVIR(ξ σ1 σ2)

partξlowast3= g9vξv

partVIR(ξ σ1 σ2)

partσlowast1

= vσ1

σ1+ 2g4

∣∣vσ1

∣∣2 + g7∣∣vξ

∣∣2 + g6∣∣vσ2

∣∣2)= 0

partVIR(ξ σ1 σ2)

partσlowast2

= vσ2

σ2+ 2g5

∣∣vσ2

∣∣2 + g8∣∣vξ

∣∣2 + g6∣∣vσ1

∣∣2)= 0 (C7)

From the above equations it is clear that non-trivial solutions in this sector are only achiev-

able by fine tuning the g9 parameter to satisfy g9 = 0 This choice can be enforced by an

additional dynamical mechanism capable of switching off the (ξξlowast)101σ21 and (ξξlowast)102σ

lowast21

terms in the potential Such scenario could be naturally realized in a supersymmetric exten-

sion [20 59] As this possibility lies beyond the scope or the present work we simply impose

the condition g9 = 0 in the general potential Then the obtained solutions are given by

|vξ|2 =(g26 minus 4g4g5)M

2ξ + (2g5g7 minus g6g8)M

+ (2g4g8 minus g6g7)M2σ2

2(4g1g4g5 + g6g7g8 minus g1g26 minus g4g28 minus g5g27)

|vσ1 |2 =(2g5g7 minus g6g8)M

2ξ + (g28 minus 4g1g5)M

+ (g27 minus 4g1g4)M2σ2

2(4g1g4g5 + g6g7g8 minus g1g26 minus g4g28 minus g5g27) (C8)

The right-handed side of these expressions can be positive in a finite region of parameter

space Analogously for the Case II vacuum configuration

〈ξ〉 = (1 ω 1)vξ 〈σ1〉 = vσ1 〈σ2〉 = vσ2 (C9)

the minimization conditions are

partVIR(ξ σ1 σ2)

partξlowast1= vξ

2ξ + 2

(g1 + g2 + ω

2g3 + ωg

lowast3

)∣∣vξ∣∣2 + g7

∣∣vσ1

∣∣2 + g8∣∣vσ2

∣∣2 + g9v2σ1

+ ωglowast9v

lowast2σ1

partVIR(ξ σ1 σ2)

partξlowast2= ωvξ

2ξ +2

(g1+g2+ω

2g3+ωg

lowast3

)∣∣vξ∣∣2+g7

∣∣vσ1

∣∣2+g8∣∣vσ2

∣∣2+ω2g9v

+ω2glowast9v

lowast2σ1

partVIR(ξ σ1 σ2)

partξlowast3= vξ

2ξ + 2

(g1 + g2 + ω

2g3 + ωg

lowast3

)∣∣vξ∣∣2 + g7

∣∣vσ1

∣∣2 + g8∣∣vσ2

∣∣2 + ωg9v2σ1

+ glowast9v

lowast2σ1

partVIR(ξ σ1 σ2)

partσlowast1

= vσ1

+ 2g4∣∣vσ1

∣∣2 + g6∣∣vσ2

∣∣2 + 3g7∣∣vξ

∣∣2]= 0

partVIR(ξ σ1 σ2)

partσlowast2

= vσ2

+ 2g5∣∣vσ2

∣∣2 + g6∣∣vσ1

∣∣2 + 3g8∣∣vξ

∣∣2]= 0 (C10)

ndash 22 ndash

JHEP01(2016)007

Again these equations are incompatible unless g9 = 0 Once the coupling g9 is enforced to

vanish we are left with three independent linear equations for the three unknown variables

vξ vσ1 and vσ2 The solutions can be easily found as

|vξ|2 =(4g4g5 minus g26)M

2ξ + (g6g8 minus 2g5g7)M

+ (g6g7 minus 2g4g8)M2σ2

2g(g26 minus 4g4g5) + 6(g4g28 + g5g27 minus g6g7g8)

|vσ1 |2 =(3g6g8 minus 6g5g7)M

2ξ + (4gg5 minus 3g28)M

+ (3g7g8 minus 2gg6)M2σ2

2ξ + (3g7g8 minus 2gg6)M

+ (4gg4 minus 3g27)M2σ2

2g(g26 minus 4g4g5) + 6(g4g28 + g5g27 minus g6g7g8) (C11)

where we have defined g equiv g1 + g2 + ω2g3 + ωglowast3 Therefore both 〈ξ〉 = (0 vξ 0) and

〈ξ〉 = (1 ω 1)vξ alignments can describe the local minimum of VIR(ξ σ1 σ2) depending

on the parameter values In the case of g1 ≪ g the VEV 〈ξ〉 = (0 vξ 0) is preferred over

〈ξ〉 = (1 ω 1)vξ while VIR(ξ σ1 σ2) is minimized by 〈ξ〉 = (1 ω 1)vξ for g1 ≫ g

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ndash 26 ndash

Introduction

Basic structure of the model

Lepton sector

Lepton masses and mixing

Phenomenological implications

Quark sector

High order corrections

Conclusions

The profile of the Higgs zero mode

Group theory of Delta(27) and its representation

Vacuum alignment