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Electroweak precision constraints on the electroweak-scale right-handed neutrino

model

Vinh Hoang, Pham Q. Hung, and Ajinkya Kamat

Department of Physics, University of Virginia, Charlottesville, VA 22904-4714, USA

(Dated: May 2, 2013)

A model of electroweak-scale right-handed neutrino (EWR ) model was constructed five years

ago in which the right-handed neutrinos are members of mirror fermion weak doublets and where

the Majorana masses of the right-handed neutrinos are found to be naturallyof the order of the

electroweak scale. These features facilitate their searches at the LHC through signals such as like-

sign dilepton events. This model contains, in addition to the mirror quarks and leptons, extra

scalars transforming as weak triplets. In this paper, we study the constraints imposed on these

additional particles by the electroweak precision parameters S, T, and U. These constraints are

crucial in determining the viability of the electroweak R model and the allowed parameter space

needed for a detailed phenomenology of the model.

I. INTRODUCTION

Two of the most pressing problems in particle physicsare, without any doubt, the nature of the spontaneous

breaking of the electroweak symmetry and the nature

of neutrino masses and mixings. It goes without saying

that the discovery of a Higgs-like particle with a mass

of 126 GeV at the LHC goes a long way in the attempt

to answer to the first question although much remains

to be determined if the 126 GeV object is truly a 0++

particle predicted by the Standard Model (SM) or it is

something else beyond the Standard Model. As to the

second question concerning neutrino masses, the general

consensus is that the discovery of neutrino oscillationsis best explained by neutrinos having a mass- albeit a

very tiny one. One might say that this is the first sign of

Physics Beyond the Standard Model since neutrinos are

massless in the SM. There has also been important ad-

vances in measuring mixing angles in the PMNS matrix

of the neutrino sector. In particular, the angle13 was

found by the Daya Bay experiment [1] to be quite large, a

number which was subsequently confirmed by the RENO

experiment [2].

In spite of these successes, we still do not know whether

the neutrinos are of the Dirac type or of the Majoranatype. In either case, the simplest approach is to add

right-handed neutrinos which are definitely particles be-

yond the SM spectrum. What is the nature of these

[email protected] [email protected] [email protected]

right-handed neutrinos? The standard assumption is one

in which they are SM singlets: the so-called sterile neu-

trinos. For this singlet assumption to become a physi-cal reality, one should be able to test it in order to ei-

ther prove or disprove it. Presently, there is no evidence

for these sterile neutrinos. Furthermore, since nothing is

known about the possible existence and associated prop-

erties of right-handed neutrinos, it is prudent to enter-

tain other logical possibilities. Why should right-handed

neutrinos be sterile? Would the assumption of SM-non

singletright-handed neutrinos be also reasonable? Could

one test it? This latter assumption is one that was pro-

posed by one of us in formulating the EWR model [4]

to which we will come back below.The most elegant mechanism for generating tiny neu-

trino masses is the quintessential seesaw mechanism in

which a large lepton-number-violating Majorana mass,

MR, typically of the order of some Grand Unified The-

ory mass scale, was given to the SM-singlet right-handed

neutrino and a Dirac mass, mD MR, was assumedto come form the electroweak sector, giving rise to a

mass m2D/MR mD which could be of the orderof O(< eV) [5]. How does one test this version of seesaw

mechanism? One could either look for the right-handed

neutrinos and/or search for lepton-number-violating pro-cesses. It is however practically impossible to directly

detect the SM-singlet right-handed neutrino unless ex-

treme fine-tuning is carried out to make the right-handed

neutrinos much lighter than the GUT scale[3]. The most

common way to test the seesaw mechanism is to look for

signals where a lepton-number violating process such as

the neutrino-less double beta decay is present. However,

such a process is extremely hard to detect and so far one

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has not had much luck with it.

In a generic seesaw scenario, one has two scales: MRMGUT 1016 GeV andmD EW 246 GeV. Out ofthose two scales, only EW is observable while MGUT is

a hypothetical scale that may or may not exist. With-

out fine-tuning, the fate of the SM-singlet (sterile) right-

handed neutrinos is linked to that of this hypothetical

scale.

The question that was asked in [4] was as follows: Is

it possible to naturallymake the Majorana mass of the

right-handed neutrinos of the order of the electroweak

scale? The answer is yes. One onlyneeds to extend the

SM in the fermion and scalar sectors. The gauge group is

still SU(3)CSU(2) U(1)Y, where the usual subscriptL for S U(2) is absent for reasons to be explained below.

IfMR EW and because ofm2D/MR, one would needmD O(keV ) in order to have neutrino masses of the

order of eV or less. This then requires the introductionof a hypothetical scale,mS, which, in contrast toMGUT ,

is of O(keV). This scale may be related to the physics of

dark matter [6]. This is the model of electroweak-scale

right-handed neutrinos presented in [4]. As one will see

in the brief review of this model, this necessitates the

introduction of mirror fermion doublets of the SM gauge

group, of which the right-handed neutrinos are members.

The right-handed neutrinos in the model of [4] acquire a

Majorana mass naturally ofO(EW). Furthermore, they

belong to weak doublets and couple to Ws and Z and

have electroweak production cross sections at colliders

such as the LHC. Some of the signals are described briefly

in [4]. In addition, the EWR model contains one Higgs

doublet and two Higgs triplets, one of which contains a

doubly-charged scalar. Some of the phenomenology of

this sector of the model was explored in [7]. We shall

come back to the implication of this scalar sector on the

126 GeV object in a separate paper.

The model of [4] contains mirror quarks and lep-

tons which are accessible at the LHC. The phenomenol-

ogy of these fermions will be presented in [8]. Since, for

every SM left-handed doublet, one has a right-handeddoublet (and similarly for the SM right-handed singlets),

the number of chiral doublets has increased by a factor of

two. This raises the obvious question of potential prob-

lems with electroweak precision data through the S, T

and U parameters. In particular, even if one artificially

makes the top and bottom members of these mirror dou-

blets degenerate, one is faced with a large contribution

to the S parameter. (In fact, this was a big problem

with Technicolor models [9].) These large contributions

from the extra chiral doublets would have to be offset by

contributions from other sectors with the opposite sign

in such a way that the sum falls within the experimen-

tal constraints. It was mentioned in [4] that such extra

contributions can be found in the scalar sector, in partic-

ular the Higgs triplet sector where its contribution can be

negative enough to offset the positive contribution fromthe mirror fermions to S. It is the purpose of the present

manuscript to examine in detail the contributions of the

mirror fermions and the extended Higgs sectors to the

electroweak precision parameters. As we shall see below,

the EWR model fits nicely with the electroweak preci-

sion constraints which, in turn, put limits on the mass

splittings within the multiplets of mirror fermions and

the Higgs multiplets and so on.

Finally, one should notice that there are aspects of the

SM which are intrinsically non-perturbative such as the

electroweak phase transition. The most common frame-work to study non-perturbative phenomena is through

lattice regularization. It is known that one cannot put a

chiral gauge theory such as the SM on the lattice without

violating gauge invariance. However, a gauge-invariant

formulation of the SM on the lattice is possible if one

introduces mirror fermions [10]. Is it possible that the

mirror fermions of the EWR model play such a role?

We end the Introduction by quoting part of a sentence

in the famous paper about parity violation by Lee and

Yang [11]: If such asymmetry is indeed found, the ques-

tion could still be raised whether there could not existcorresponding elementary particles exhibiting opposite

asymmetry such that in the broader sense there will still

be over-all right-left symmetry.. [4] is, in some sense, a

response to this famous quote.

The plan of the manuscript will be as follows. First, we

summarize the essential elements of the EWR model of

[4]. Second, we present calculations of the electroweak

precision parameters in the EWR model. Third, we

discuss the implications coming from the constraints on

the electroweak precision parameters on the various mass

splittings and parameters of the mirror sector as well asof the extended Higgs sector. We conclude with some

remarks concerning the 126 GeV boson.

II. THE EWR MODEL

[4] asked the following two questions: 1) Could one

obtain the right-handed neutrino Majorana mass strictly

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within the SM gauge group SU(3)c SU(2)L U(1)Yby just extending its particle content?; 2) If it is possible

to do so, what would be the constraints on the Dirac

mass scale? The answer to the first question lies in the

construction of the EWR model [4].

In a generic seesaw scenario, Rs are SM singlets and,

as a result, a right-handed neutrino mass term of theformMRTR 2R is also a singlet of the SM. As a result,

MR can take on any value and is usually assumed to be

of the order of some GUT scale if the SM is embedded in

a GUT group such asSO(10). To constrainMR, one has

to endow the right-handed neutrinos with some quantum

numbers. For example, ifR belongs to a 16ofSO(10) it

is natural forMR to be of the order of theSO(10) break-

ing scale. Another example is the left-right symmetric

extension of the SM [12] whereR belongs to a doublet of

SU(2)R. The aforementioned Majorana mass term would

still be a singlet underSU(2)Lbut it is no longer so under

SU(2)R. It is then natural thatMR MR ML, whereMR,L are the breaking scales ofS U(2)R andS U(2)L re-

spectively. In all of these scenarios, the value of the Dirac

mass mD in m2D/MR usually comes from the breaking

of the SM SU(2)L and is naturally proportional to the

electroweak breaking scale. The smallness of neutrino

masses gives rise, without fine-tuning, to an energy gap

mD O(EW) MR O(MGUT ) or O(MR). With-out fine-tuning, the large value ofMR O(MGUT ) wouldmake it practically impossible to detect the SM-singlet

right-handed neutrinos at machines such as the LHC and

to directly test the seesaw mechanism. However, in the L-

R model, the production ofR can proceed first through

the production ofWR as first shown in [13]. The feasibil-

ity of such a process was discussed in [14]. (Other mech-

anisms proposed to make the SM-singlet right-handed

neutrinos accessible at the LHC through SM Ware dis-

cussed in [15] although it might be very difficult to do so

due to the size of the Dirac Yukawa coupling.)

It is clear as presented in [4] that one of the natural and

minimal ways (in terms of the gauge group) to test the

seesaw mechanism and to detect the right-handed neu-trinos at colliders such as the LHC is to make the right-

handed neutrinos non singletsunder the SM S U(2)L for

two reasons. The first reason has to do with the mass

scaleMR. IfRs are non-singlets underSU(2)LthenMRnecessarily comes from the breaking ofSU(2)Land there-

fore would naturally be of the order of the electroweak

scale. Energetically-speaking, it could be directly de-

tected at the LHC [4]. The second reason has to do

with the possible detection of Rs themselves. Being

SU(2)L non-singlets, they can couple to the SM elec-

troweak gauge bosons and the production cross sections

would be naturally of the order of the electroweak cross

sections [4].

The simplest way to make RsS U(2)L non-singlets is

to group them into SU(2)L right-handed doublets with

the right-handed charged partners which are new chargedleptons with opposite chirality to the SM charged lep-

tons. Anomaly freedom would necessitate the introduc-

tion ofSU(2)L doublets of right-handed quarks. These

new right-handed quarks and leptons are called mirror

fermions in [4]. The right-handed quarks and charged

leptons are accompanied by their left-handed partners

which are SU(2)L singlets, a complete mirror image of

the SM fermions, so mass terms can be formed by cou-

pling to the Higgs doublet. TheSU(2)L U(1)Y fermioncontent of the EWRmodel of [4] is given, for each family,

as follows.

SU(2)L lepton doublets:

SM :lL =

LeL

; Mirror: lMR =

ReMR

(1)

for the SM left-handed lepton doublet and for the

right-handed mirror lepton doublet respectively.

SU(2)L lepton singlets:

SM :eR; Mirror: eML , (2)

for the right-handed SM lepton singlet and left-

handed mirror lepton singlet respectively.

Similarly, for the quarks, we have

SU(2)L quark doublets:

SM :qL =

uLdL

; Mirror: qMR =

uMRdMR

(3)

for the SM left-handed quark doublet and for the

right-handed mirror quark doublet respectively.

SU(2)L quark singlets:

SM :uR, dR; Mirror: uML , d

ML (4)

for the right-handed SM quark singlets and left-

handed mirror quark singlets respectively.

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Apart from chiralities, the SU(2)L U(1)Y quantumnumbers of the mirror fermions are identical to those of

the SM fermions. A remark is in order at this point.

What we refer to as mirror fermions are the particles

listed above and they are not to be confused with parti-

cles in the literature which have similar names but which

are entirely of a different kind. As the above listing

shows, the mirror quarks and leptons are particles whichare differentfrom the SM ones. It is for this reason that

a superscript M was used in [4] and here in order to

avoid possible confusion. These chiral mirror fermions

will necessarily contribute to the precision electroweak

parameters and potentially could create disagreements

unless contributions from other sectors are taken into ac-

count. This will be the main focus of the next sections.

As with the SM leptons, the interaction of mirror lep-

tons with the SU(2)L U(1)Y gauge bosons are foundin the terms

lMR /DlMR ; e

ML

/DeML , (5)

where the covariant derivatives /D are the same as the

ones used for the SM leptons and are listed explicitly in

the Appendix C. The gauge interactions of the mirror

quarks can similarly be found.

We next review the salient point of the EWR model

of [4]: The electroweak seesaw mechanism. For the sake

of clarity, we repeat here the arguments given in [4].

As discussed in [4], a Majorana mass term of the type

MR

TR

2R

necessarily breaks the electroweak gauge

group. The reason is as follows. The bilinear l M,TR 2lMR

containsTR2R and transforms underSU(2)L U(1)Yas (1+3, Y /2 = 1). For obvious reasons, the Higgs fieldwhich couples to this bilinear cannotbe an S U(2)L sin-

glet with the quantum number (1, Y /2 = +1) since this

singlet charged scalar cannot develop a VEV. This leaves

the triplet Higgs field = (3, Y /2 = +1) as a suitable

scalar which can couple to the aforementioned bilinear

and whose neutral component can develop a VEV:

=

1

2 .= 12

+ ++

0 12

+ . (6)The Yukawa coupling of the bilinear to this Higgs field

was given in [4] and is written down again here

LM =gM(lM,TR 2) ( 2) lMR=gM(

TR2R

0 12

TR2eMR

+ 12

eM,TR 2R+

+eM,TR 2eMR

++) , (7)

From Eq.(7), one notices the Yukawa termgMTR2R

0

which upon having

0 =vM, (8)

gives rise to the right-handed Majorana mass

MR = gMvM. (9)

As it has been stressed in [4], MR is naturally of the

order of the electroweak scale since vM O(EW) andis constrained to be larger than MZ/2 46 GeV becauseof the constraint coming from the width of the Z boson

(no more than three light neutrinos). A triplet Higgs field

with such a large vacuum expectation value will destroy

the custodial symmetry value = 1 at tree level. A

nice remedy for this problem was given in [4] and will be

reviewed below.

A Dirac mass term of the type LR comes from lLlMRwhich is 1 + 3 under SU(2)L. It was argued in [4] why

a singlet scalar field is the appropriate choice and why

a triplet is phenomenologically ruled out [16]. As in [4],

the interaction with the singlet scalar is given as

LS=gSllLSlMR +H.c.=gSl (LR+ eLe

MR) S+ H.c. (10)

With

S =vS, (11)

the neutrino Dirac mass is given by

mD =gSlvS, (12)

If gSl O(1), this implies that vS O(105 eV). Ithas been discussed in [4] that this value for vS is six

orders of magnitude smaller then the electroweak scale

EW and this hierarchy requires a cross coupling be-

tween the singlet and the triplet scalars to be of or-

der 1012. To evade this fine tuning, it was pro-posed in [4] that the classical singlet scalar field takes

a value S(t0) (105 eV ) at the present time and itsvalue changes with time whose rate is dictated by a slow-

rolling effective potential. It goes without saying thatmuch remains to be worked out for this scenario. Al-

ternatively, one can assume that vS O(EW) and setgSl 107 as suggested in [6] to obtain a Dirac massof the desired order. (This is actually not so unnatural

as the example of the electron mass being 107EWillustrates.)

Eq.(7) gives a Majorana mass to the right-handed neu-

trinos but one could easily have from gauge invariance

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a term such as gL(lTL2) ( 2 ) lL which would yield a

large Majorana mass for the left-handed neutrinos unless

fine-tuning is carried out.

As discussed in [4], in order to guarantee that left-

handed neutrinos have vanishing Majorana masses at tree

level, a mirror global symmetry U(1)M was imposed:

(lM

R, eM

L ) ei M

(lM

R, eM

L ) , e2 i M

, S ei M

S,(13)

for the mirror leptons and triplet and singlet scalars and

(qMR , uML , d

ML ) ei M(qMR , uML , dML ) , (14)

for the mirror quarks. In [4], it was mentioned that the

left-handed neutrinos can acquire a Majorana mass at

one-loop of the type ML = 1

162mD 2MR

ln MRMS

, where

is theSquartic coupling. This is smaller than the light

neutrino mass by at least two orders of magnitude and

can be neglected.

Beside preventing the left-handed neutrinos from ac-quiring a large tree-level Majorana mass, this U(1)Msymmetry also prevents terms such as qLqMR , uRu

ML and

dRdML . Therefore, as stressed in [4], any bilinear mixing

between SM fermions and mirror fermions will have to

couple with the singlet scalar S in order to be U(1)M-

invariant at tree level, namely qLSqMR, uRSu

ML and

dRSdML . Because of these mixings between the two

sectors, the mass eigenstates are not pure left-handed

SM quarks or right-handed mirror quarks. This was dis-

cussed in [4]. However, the deviation from the pure

states, i.e. for example uL = uL + O(vS/EW)uM,cR , is

proportional to vS/EW 106 and, for most practicalpurposes, one can neglect this mixing.

To finish up with the review of the EWR model, we

review the triplet scalar sector of [4]. Let us recall that

the -parameter for arbitrary Higgs multiplets is given

by = (

i[T(T+ 1) T23 ]iv2i cT,Y)/(2

iT23iv

2i ), where

cT,Y = 1 for complex multiplet and cT,Y = 1/2 for real

multiplet [17]. If one just has the triplet and noth-

ing else, one would obtain = 1/2 in contradiction with

the fact that experimentally 1. Pure Higgs doubletswould give naturally = 1. A mixture with one tripletand one doublet would give 1 if the VEV of thetriplet,vM, is much less than that of the doublet, v2, i.e.

vM v2. But this is not what we want since we wouldlike to have vM and v2 of O(EW). To preserve the

custodial symmetry with a Higgs triplet, another triplet

Higgs scalar = (3, Y /2 = 0) is needed in addition to

the aforementioned (3, Y/2 = 1) and the usual doublet

= (2, Y /2 = 1/2). The potential for these three scalar

multiplets and its minimization is given in the Appendix.

This potential possesses a global SU(2)L SU(2)R sym-metry under which the two triplets are combined into the

following (3, 3) representation [7, 1820]:

=

0 + ++

0 +

0

. (15)

Similarly, and = 2 can be grouped into a (2, 2)

representation:

=

0 +

0

. (16)

Proper vacuum alignment so that SU(2)L U(1)Y U(1)em gives

=

vM 0 0

0 vM 0

0 0 vM , (17)and

=

v2/

2 0

0 v2/

2

. (18)

This breaks the global SU(2)L SU(2)R down to thecustodial S U(2)D. One obtains MW =g v/2 and MZ=

MW/ cos W, with v =

v22+ 8 v2M 246 GeV and, at

tree level, = MW/MZ cos W= 1 as desired.

After spontaneous breaking ofSU(2)L U(1)Y, besidethe three Nambu-Goldstone bosons which are absorbedby W and Z, there are ten physical scalars which are

grouped into 5 + 3 + 1 of the custodial SU(2)D. (In

fact there are two singlets ofS U(2)D.) These states are

discussed in the next section and in the Appendix B.

Last but not least in this mini review is the question

of charged fermion masses, in particular the top quark

and mirror fermion masses. This is a topic on its own

but a few words are in order here. It goes without say-

ing that this interesting topic deserves a detailed inves-

tigation but such endeavor is beyond the scope of this

manuscript. Since v = v22+ 8 v2M 246 GeV, it is ev-ident thatv2, vM MZ/2, it

follows thatvM 46 GeV for gM 1. This implies thatv2 209 GeV. Since naivelya charged fermion mass isgiven by (ignoring mixings in the mass matrix for now)

mf = gfv2/

2 gf 148 GeV, one expects a Yukawa

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coupling gt 1.2 for the top quark. This coupling canactually be even smaller if the SM quark mass matrix is

of the democratic type i.e. having all matrix elements

being equal to 1 [21]. (A more realistic version differs

slightly from this one.) The largest mass eigenvalue in

such a model is 3 gf148 GeV giving gt 0.4. Forvery heavy mirror quarks, the Yukawa couplings might

be larger but one may use a less restrictive definitionof strong coupling, namely f g2f/4 O(1). Fromthat point of view, a value of gf 2or3 might not beproblematic. There is also an interesting twist in the

situation when the Yukawa couplings become large: A

possibility that the electroweak symmetry can be broken

dynamically by condensates of heavy fermions through

the exchange of a fundamental scalar as it has been done

for a heavy fourth generation [22].

III. OBLIQUE PARAMETERS

As we have mentioned in the introduction, the contri-

butions to the S parameter coming from the extra mirror

fermions will be positive and will exceed the constraints

imposed by electroweak precision data. These contri-

butions will have to be cancelled by those coming from

another sector such as the triplet Higgs present in the

EWR model. This has been suggested in [4]. In this

section, we carry out a detailed calculation of the elec-

troweak precision parameters, the so-called oblique pa-

rameters, within the framework of the EWR model.In Appendix B, we summarize the discussion of the

minimization of the scalar potential given by Eq.(B1).

In what follows, we list the expressions for the physical

states and for the Nambu-Goldstone bosons in terms of

the original scalar fields.

Physical observables like the oblique parameters are

to be expressed using the masses of physical scalars. To

express these physical states we use the subsidiary fields

[7]:

0 12v2+0r + 0,

0 vM+ 12

0r + 0

; (19)

12

+

, 1

2

(20)

for the complex neutral and charged fields respectively.

With these fields the Nambu-Goldstone bosons are given

by

G3 =cH+sH,

G03=

cH0 +sH0

. (21)

The scalar potential in Eq. (B1) preserves the custodial

SU(2)D. Hence, the physical scalars can be grouped, as

stated in the previous section, based on their transfor-mation properties under S U(2)D as follows:

five-plet (quintet) H5 , H5 , H05 ;

triplet H3 , H03 ;

two singlets H01 , H01 , (22)where

H++5 =++, H+5 =

+, H+3 =cH+ sH+,

H0

5 =

1

620 20r, H03 = cH0 +sH0,H01 =

0r, H01 = 1

3

20r +0

, (23)

with H5 = (H++5 )

, H5 =(H+5), H3 =(H+3 ),and H03 =(H03 ). The oblique parameters, the Feyn-man rules and the loop diagrams will be expressed in

terms of these physical scalar five-plet, triplet, two scalars

and their masses, mH,,05

, mH,03

, mH1 , mH1 respec-

tively. We will also use

sH= sin H=2

2vMv , cH= cos H=

v2v (24)

The effects of vacuum polarization diagrams (oblique

corrections) on the electroweak-interaction observables

can be described by three finite parameters S, T and

U, known as the Oblique Parameters. Using these pa-

rameters one could probe the effects of new Physics on

the electroweak interactions at the one-loop level, if the

new Physics scale is much larger as compared to MZ[2325]. Hence, these parameters can be defined using

perturbative expansion as [23]:

S 4e2[33(0) 3Q(0)],

T e2

s2Wc2WM

2Z

[11(0) 33(0)],

U 4e2[11(0) 33(0)] , (25)where sW = sin W, cW = cos W are the functions

of the weak-mixing angle W. 11 and 33 are the

vacuum polarizations of the isospin currents and 3Q

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the vacuum polarization of one isospin and one elec-

tromagnetic current. The functions are defined as(0) =

(q2) (0) /q2 in general, and we will be

using q2 =M2Z.

These parameters can be expressed in terms of the self-

energies of the W, Z and bosons and the Z mixing

[23]. For the purpose of the EWR model the constraints

on the new Physics in EWR model from S,T,Ucan be

obtained by subtracting the the Standard Model (SM)

contributions to S, T, U from the corresponding total

contributions due to the EWR model. Hence, the new

Physics contributions to theS,T,Udue to EWRmodel

are denoted by

S,

T,

U respectively (following notation

used in [26]) and they can be expressed as

4s2Wc2W S= 1M2Z

ZZ(M

2Z)

c2Ws2Wc2Ws2W

Z(M2Z) (M2Z)

EWR

1M2Z

ZZ(M

2Z)

c2W

s2W

c2Ws

2W

Z(M

2Z) (M2Z)

SM(26)

T= 1M2W

11(0) 33(0)

EWR 1

M2W

11(0) 33(0)

SM(27)

4s2W U=

W W(M2W)

M2Wc2W ZZ(M2Z)M2Z 2sWcW Z(M

2Z)

M2Zs2W (M2Z)M2Z

EWR

W W(M2W)

M2W

c2W

ZZ(M2Z)

M2Z 2

sW

cW

Z(M2Z)

M2Z

s2W

(M2Z)

M2Z

SM, (28)

where all quantities with a hat on top ( ) i.e.sW,cW, g2s2W/(4) are defined in theM Sscheme evaluatedat MZ [24]. Hereafter, in this article the hats on top of

these and other quantities are omitted, but implied. The

notation (q2) = (q2) (0) [23] and the superscriptEWRdenotes the contribution due to EWRmodel. We

can see that Sis associated with the difference between

the Z self-energy at q2 = M2Z and q2 = 0. T is propor-

tional to the difference between W and Z self-energies

at q2 = 0. The new physics contribution to U in EWRmodel is small as compared to that to S and T. Also,

this contribution is constrained only by mass, MW, and

width, W, ofWboson. Thus, we can project the S T U

parameter space on 2-D ST parameter space in U = 0

plane. Hence, in the this paper our emphasis will be on

constraints on the S and Tparameters only. The steps

in the derivations of the new Physics contributions to S

andTare provided in the Appendix D. The new Physics

contributions to S, T from the scalar sector in EWRmodel (denoted bySscalar,Tscalar respectively) and thecontributions from the mirror fermion sector in EWRmodel (denoted bySfermion,Tfermion respectively) arecalculated separately and then added to obtain the to-

tal new Physics contributions in EWR model, contribu-

tionsS,T. Note that the scalar sector contributions andmirror fermion sector contributions in EWR model are

separately finite. Thus,

S=Sscalar+Sfermion (29)T =Tscalar+Tfermion. (30)The contribution to Sby the new Physics due to the

scalar sector of the EWR model is given by

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8

Sscalar = SEW Rscalar SSMscalar=

1

M2Z

4

3s2H

B22(M

2Z; M

2Z, m

2H05

) M2ZB0(M2Z; M2Z, m2H05

)

+ 2s2H

B22(M

2Z; M

2Z, m

2

H+5

)

M2W B0(M

2Z; M

2Z, m

2

H+5 ) +c2H B22(M2Z; M2Z, m2H1) M2ZB0(M2Z; M2Z, m2H1)

+ 8

3s2H

B22(M

2Z; M

2Z, m

2H1

) M2ZB0(M2Z; M2Z, m2H1

)

+4

3c2HB22(M

2Z; m

2H05

, m2H03

)

+ 2c2HB22(M2Z; m

2

H+5

, m2H+3

) +s2HB22(M2Z; m

2

H03

, m2H1) +8

3 c2HB22(M

2Z; m

2

H03

, m2H1

)

4B22(M2Z; m2H++5

, m2H++5

) B22(M2Z; m2H+5

, m2H+5

) B22(M2Z; m2H+3

, m2H+3

)

B22(M

2Z; M

2Z, m

2H) M2ZB0(M2Z; M2Z, m2H)

, (31)

where The functions B22 and B0 are as defined in [27].

The definitions in terms of two point integrals are given

in eqn. (A3). In dimensional regularization they can be

evaluated using eqns. (A7), (A6). These individual loop

integral functions do have divergences in their definitions,

but when all the contributing diagrams are added, the

divergences cancel as expected resulting in finiteSscalar.Similar cancellations ensure the finiteness of each term

on the right hand sides of eqns. (29), (30).

The new Physics contribution to T due to the scalar

sector in EWR model is given by

Tscalar = TEW Rscalar TSMscalar=

1

4s2WM2W

1

2F(m2

H++5

, m2H05

) +3

4F(m2

H+5

, m2H05

) +1

4F(m2

H+3

, m2H03

) +c2H

2F(m2

H++5

, m2H+3

)

+ c2H

4F(m2

H+5

, m2H03

) +c2H12

F(m2H05

, m2H+3

) c2H

2F(m2

H+5

, m2H+3

) c2H

3F(m2H0

5

, m2H03

)

+ s2H

4

F(m2

H+3

, m2H1) F(m2H03 , m2H1

)

+2

3 c2H

F(m2

H+3

, m2H1

) F(m2H03

, m2H1

)

+ s2H

2F(M2

W, m2

H++

5

)

s2H

4F(M2

W, m2

H+

5

)

s2H

4F(M2

W, m2

H0

5

)

+ M2Ws2HB0(0; M

2W, m

2

H05

) +M2Ws2HB0(0; M

2W, m

2

H+5

) M2Ws2HB0(0; M2W, m2H++5

)

, (32)

whereF(m21, m22) is defined in eqn. (A12).The corresponding new Physics contributions toSand

Tdue to the lepton sector in EWR model are given as

Slepton =SEW Rlepton SSMlepton=

(NC)lepton

6

3

i=1 2Ylepton xi

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9

and [25]

Tlepton = TEW Rlepton TSMlepton=

(NC)lepton8s2WM

2W

3i=1

F(m2i , m2ei) (34)

respectively. Here, because we have subtracted the con-tribution from three generations of SM leptons, the sum-

mation is over three generations of mirror leptons only.

Subscripts i and ei represent right-handed neutrino

(Ri) and mirror electron (eMi ) member ofi

th mirror lep-

ton generation respectively. (NC)lepton = 1 is the lepton

color factor and Ylepton =1 is hypercharge for mirrorleptons. xi, ei= (mi, ei/MZ)2, wheremi, eiare masses

ofRi and eMi respectively. And G(x) is given by eqn.

(A19)

Similarly, the new Physics contributions to S and T

due to the quarks in EWR model are given by,

Squark =SEW Rquark SSMquark=

(NC)quark6

3i=1

2

4

Yquark2

+ 3

xui

+ 2

4 Yquark

2 + 3

xdi Yquark ln

xuixdi

+

3

2+ Yquark

xui+

Yquark2

G(xui)

+3

2 Yquark

xdi Yquark

2

G(xdi)

(35)

Tquark = TEW Rquark TSMquark=

(NC)quark8s2WM

2W

3i=1

F(m2ui, m2di). (36)

Once again, because we have subtracted the contri-

bution from three generations of SM quarks, the sum-mation is over three generations of mirror quarks only.

Subscriptsui and di represent mirror up- (uMi ) and mir-

ror down- (dMi ) member of ith mirror-quark generation

respectively. (NC)quark = 3 is the quark color factor

and Yquark =1/3 is hypercharge for mirror quarks.xui, di = (mui, di/MZ)2, where mui, di are masses ofuMianddMi respectively. Refer to Appendix D 2 for the mir-

ror fermion loop diagrams contributing to SandT.

IV. NUMERICAL RESULTS

In this section we will study numerically the results

presented in Section III and is organized as follows. First,

we present unconstrained scatter plots for the S and T

parameters coming from the mirror fermion sector and

from the scalar sector. These scatter plots are given in

the T-Splane for the scalar and mirror fermion sectorsseparately. The main desire is to observe possible regions

where the two sectors can cancel each other. Second, we

generate the scatter plots for T and Sfor the scalar sec-

tor as a function of the mass splittings among the scalars.

In particular, we will notice below there is a significant

region in the parameter space where Scan be quite neg-

ative if the mass splitting between the doubly-charged

scalar with the other ones is large. Third, we combine two

sectors and plot the scatter points of the EWR model

in the T-S plane endowed with the 1 and 2 ellipses

coming from experiment. It is shown below that themodel is well consistent with precision electroweak data.

Fourth, as an example (and simply as an example), we

fix the values of some of the scalar masses and present

a 3-dimensional plot ofSscalar versus the mass splittings

among members of the quintet and among members of

the triplet.

A. Unconstrained S and T parameters for the

mirror fermion and scalar sectors

The S and T parameters as shown in Section III de-

pend on a number of parameters such as the masses of

the scalars as well as the mixing parameter sin Has de-

fined in Section III, and the masses of the new fermions

from the model. For simplicity, we allow for the scalar

masses to go from MZto 650 GeV and for sin H to go

from 0.1 to 0.89 as discussed in [7] (we stretch the lower

value to 0.1 for numerical purpose). The right handed

neutrino masses are taken from MZ/2 to 650 GeV, while

the mirror charged lepton and mirror quark masses vary

from MZto 650 GeV. The scatter plots are generated

with about 10,000 points. We first show these plots as

they are before discussing their implications. The elec-

troweak precision constraints which we will be using are

given as S=0.02 0.14; T = 0.06 0.14 [24] for SMHiggs mass of 126 GeV.

We will present the plots as follows:

Scatter plot ofT versus Sfor the scalar sector with-outthe 1 and 2 experimental contours (FIG. 1);

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10

SS~

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

S

T~

-5

0

5

10

15

20

25

30

FIG. 1. T versus Sfor the scalar sector

Scatter plot ofT versus S for the mirror fermionsectorwithoutthe 1 and 2 experimental contours

(FIG. 2);

MFS~

-1 -0.5 0 0.5 1 1.5 2

MF

T~

-5

0

5

10

15

20

25

30

35

40

FIG. 2. T versus S for the mirror fermion sector

Scatter plot ofT versus Sfor the scalar sector withthe 1 and 2 experimental contours (FIG. 3);

SS~

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

S

T~

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

FIG. 3. T versus S for the scalar sector with the 1 and 2

experimental contours

Scatter plot ofT versus S for the mirror fermionsector with the 1 and 2 experimental contours

(FIG. 4).

MFS~

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

MF

T~

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

FIG. 4. T versus

S for the mirror fermion sector with the 1and 2 experimental contours

Before showing the combined scalar and mirror fermion

contributions to T and S, a few remarks are in order at

this point. The plots FIG. 1 and FIG. 2 shows the re-

gions in the T-Splane which are unconstrained by the

electroweak precision data. Let us look at the Sparame-

ter. From FIG. 2, one can see that the contribution to S

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11

from the mirror fermion sector is almost always positive

and can be quite large. This is to be expected since the

addition of extra chiral doublets (the mirror fermions)

always leads to such a phenomenon- a well-known fact.

Cancellations from other contributions with the opposite

sign will be needed in order to agree with the electroweak

precision constraints. A look at FIG. 1 reveals that that

the contribution to S coming from the scalar sector, inparticular the Higgs triplet sectors, can be quite nega-

tive allowing for such cancellation to occur. This has

been anticipated in [4] but this is the first detailed calcu-

lations of such a contribution to the electroweak precision

parameters.

To see a little more explicitly why the two sectors com-

plement each other in such a way as to bring the EWRmodel to be in agreement with the electroweak precision

data, let us take a look at FIG. 3 and FIG. 4. From

FIG. 3, we can see that the 1 and 2 experimental con-

tours are well inside the region generated by the scalars

of the EWR model. However, these contours are way

outside the region generated by the mirror fermions of

the model. Again, one notices the importance of the

scalar sector in bringing the EWR into agreement with

the electroweak precision data.

B. Constrained S and T parameters

To compare the model with data, we, of course, con-

sider the total sum of the two contributions, namely

S= SS+ SMF and T= TS+ TMF . This is shown below

in a plot which also includes the 1 and 2 experimental

contours.

S~

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

T~

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

FIG. 5. Total T versus S with the 1 and 2 experimental

contours

One can now see from FIG. 5 that the sum of the scalar

and mirror fermion contributions to T and Sgenerates

data pointsinsidethe 1 and 2experimental contours.

It implies that there is a large region of parameter space

where the EWR model is consistent with electroweak

precision constraints. One notices again the crucial role

played by the triplet scalars.

To understand better the data points in FIG. 5, we

plot the constrained SS versus SMF as shown below.

MFS~

-0.2 0 0.2 0.4 0.6 0.8

S

S~

-0.6

-0.4

-0.2

0

0.2

0.4

0.6constraint1+

constraint2

FIG. 6. S versus SMF for S and T satisfying 1 and 2

constraints of FIG. 5

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12

In the plot below, we also show the constrained TSversus TMF .

MFT~

-1 0 1 2 3 4 5

S

T~

-5

-4

-3

-2

-1

0

1

constraint1+

constraint2

FIG. 7. T versus TMF for S and T satisfying 1 and 2


From FIG. 6 and FIG. 7, one can clearly see the can-

cellation among the scalar and mirror fermion sectors in

their contributions to T and S. At the 1 level, one can

see from FIG. 6 that the scalar contribution, SS, ranges

roughly from 0 to -0.3.

The last constrained plot we would like to show in this

section is the variation ofSand Tas a function of sin H.

Hsin

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S~

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6constraint1+

constraint2

FIG. 8. S versus sin H for S and T satisfying 1 and 2


Hsin

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

T~

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6constraint1+

constraint2

FIG. 9. T versus sin H for S and T satisfying 1 and 2


These plots should be considered, because in EWRmodel various couplings and, hence, branching ratios and

cross sections (e.g. Eq. (37)) depend onH[7, 28]. Thus,

it is necessary to take into account any constraint on

sin H from oblique parameters. Any restriction on the

allowed range of sin H, from constraints on S, T, would

surely affect the agreement of the model with experimen-

tal data and the searches for experimental signals of this

model. From FIG. 8 and FIG. 9, we notice that the

EWR model agrees with electroweak precision data for

the entire allowed range of sin H.

The next question that one might wish to ask is how

the above informations influence the masses and mass

splittings in the scalar sector. In the next section, we

will show some samples of three-dimensional plots ofSSand TS versus the mass splittings in the scalar quintet

and triplet (i.e. H5 and H3). Some specific mass values

are used in these plots for the purpose of illustration. An

exhaustive study of a large range of masses is beyond the

scope of this paper.

C. SS and TS versus scalar mass splittings

When the custodial symmetry is preserved at tree level,

all the masses of scalars within a multiplet (SU(2)Dquin-

tet, triplet in EWR model, refer to Appendix B) are

degenerate for the purpose of all tree level calculations.

However, at loop level the custodial symmetry can be

broken and the physical scalars within a multiplet can

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13

have non-degenerate masses. It means, when we con-

sider the radiative corrections to self energy diagrams of

gauge vector bosons from the scalar sector, it is allowed

to have non-degenerate masses of scalars within a mul-

tiplet. It not problematic, if these mass splittings result

in large|

SS|,|

TS| as long as the total

S,

T satisfy the

experimental constraints. The effect of mass splittings

within scalar and fermion multiplets on Sparameter wasdiscussed in [29]. And forTparameter, it can be realized

in the same manner.

The experimental searches for the scalars presented in

[4, 7] will be guided partly by the mass splittings among

the scalars. These, in turn, are dictated by the elec-

troweak precision constraints discussed above. In fact,

the amount of mass splittings is constrained by e.g. the

allowed ranges ofSS, TS which, at 1 level, range ap-

proximately from 0 to -0.3 and from 0 to 4, respectively.We will present a few of these plots and will comment on

their implications.

Plots: The 3-D plots shown below will be for twodefinite values of sin H, namely 0.1 and 0.89, in

order to illustrate also the dependance ofSS and

TSon sin H. As mentioned above, an exhaustive

analysis for arbitrary scalar masses is beyond the

scope of this paper. As a consequence, we will fix

the values ofmH1 ,mH1

,mH03

,mH05

andmH++5

and

will vary mH+3

and mH+5

(two of the three axes in

the plots).

X0 0.5

11.5 2

2.5 33.5 4

Y1

2

3

4

5

S

S

-0.02

-0.01

0

0.01

0.02

0.03

= 0.1H= 600 GeV; sin

1H

= 100 GeV; m,1

Hm

= 200 GeV++5

H= 100 GeV; mo

5H

= 126 GeV; mo3

Hm

FIG. 10. SS versus Y mH+5

mH05

1 and XmH+3

mH03

1 forsin H= 0.1 and mH1 = 600 GeV

X00.5 1

1.5 22.5

33.5 4Y

1

2

3

4

5

S

S~

-0.22

-0.21

-0.2

-0.19

-0.18

-0.17

-0.16

= 0.1H= 600 GeV; sin

1H

= 100 GeV; m,1

Hm

= 500 GeV++5

H= 100 GeV; m

o

5H

= 126 GeV; mo3

Hm


mH05

1 and XmH+3

mH03

1 for

sin H= 0.1 and mH1 = 600 GeV

X00.5

11.5 2

2.53

3.5 4

Y1

2

3

4

5

S

S~

-0.35

-0.3

-0.25

-0.2

-0.15

= 0.89H

= 600 GeV; sin1

H= 100 GeV; m,

1Hm

= 200 GeV++5H

=100 GeV; mo

5H= 126 GeV; mo

3H

m


mH05

1 and XmH+3

mH03


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14

X00.5

11.5 2

2.53

3.5 4

Y1

2

3

4

5

S

S

-0.55

-0.5

-0.45

-0.4

-0.35

= 0.89H= 600 GeV; sin

1H

= 100 GeV; m,1

Hm

= 500 GeV++5

H=100 GeV; mo

5H

= 126 GeV; mo3

Hm


mH05

1 and XmH+3

mH03

1 for


X

00.5

11.5

22.5

33.5

4

Y0

1

2

3

4

5

S

T

-2

0

2

4

6

8

10

12

14

16

= 0.1H

= 600 GeV; sin1

H= 500 GeV; m,

1Hm

= 200 GeV++5

H= 100 GeV; m

o

5H

= 126 GeV; mo3

Hm

FIG. 14. TS versus Y mH+5

mH05

1 and XmH+3

mH03


X

00.5

11.5

22.5

33.5

4Y1

2

3

4

5

S

T~

-2

0

2

4

6

8

10

= 0.1H

= 600 GeV; sin1

H= 500 GeV; m,

1Hm

= 500 GeV++5

H= 100 GeV; mo

5H

= 126 GeV; mo3

Hm


mH05

1 and XmH+3

mH03

1 for


X

00.5

1 1.52 2.5

33.5

4Y0

0.51

1.52

S

T~

-1

0

1

2

3

4

5

6

7

= 0.89H

= 600 GeV; sin1

H= 650 GeV; m,

1Hm

= 200 GeV++5

H= 200 GeV; mo

5H

= 126 GeV; mo3

Hm


mH05

1 and XmH+3

mH03


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X

00.5

1 1.52

2.53

3.54

Y

-0.5

0

0.5

1

1.5

2

S

T

0

1

2

3

4

5

6

= 0.89H= 600 GeV; sin

1H

= 650 GeV; m,1

Hm

= 350 GeV++5H

=200 GeV; mo5H

= 126 GeV; mo3

Hm


mH05

1 and XmH+3

mH03

1 for


Remarks :

In the above figures, arbitrary values are cho-

sen (for the purpose of illustration) for the two

scalars H1 and H

1 , namely mH1 = 600 GeV

and mH1

= 100, 500, 650 GeV. The reasons

for these particular- albeit arbitrary- values

will be given below.

The value of 126 GeV was set for H03 in the

plots also for illustrative purpose. In the nextsection, we will discuss what the most recent

LHC result on the spin-parity of the 126 GeV

object implies on the minimal EWR model

and what extension is needed in the scalar sec-

tor.

A look at FIG. 10 - FIG. 13 reveals the follow-

ing pattern. SSbecomes more negative as the

mass of the doubly-charged scalar, H++5 , goes

from 200 GeV to 500 GeV. It also becomes

more negative as one increases sin Hfrom 0.1

to 0.89.

For sin H = 0.1 (FIG. 10 and FIG. 11) , we

notice that SSdecreases, becoming more neg-

ative, when mass splittings between H+3 and

H03 AND between H+5 and H

05 become simi-

lar. This features persists until SSreaches its

lowest value where it stays stable along the

linemH+5

mH05

1 mH+3

mH03

1. This implies that

SS reaches its most negative value when the

mass splittings among H5s and among H3s

are similar.

For sin H = 0.89, SS decreases, becoming

more negative, whenH+5 andH05 become more

and more degenerate whileH+3 becomes heav-

ier and heavier compared withH03 . This is in

marked contrast with the sin H= 0.1 case.

If one restricts oneself to the 1 constraint,

namely SS 0.0 to 0.3, then FIGs. 10, 11and 12 seem to be favored.

FIG. 14 - FIG. 17 reveal that the dependence

ofTSon the mass splittings is different with

that ofSS.

For sin H= 0.1 (FIG. 14 and FIG. 15), with

particular choice of mH1

= 500 GeV, TS be-

comes more negative for large mass splitting

within the triplet, and small mass splitting be-

tweenH+5 andH05 .

For sin H = 0.89 (FIG. 16 and FIG. 17), we

choose mH1

= 650 GeV. The low value re-

gions ofTS correspond to 1 < X < 4 and

0 < Y < 1. And, noticeably, TStends to be

more negative as mH++5

approaches mH05

.

The above observations are very useful in the search

for signals of the EWR model, in particular its

scalar sector in light of recent results from the LHC.

We shall discuss this aspect in the next section deal-

ing with experimental implications.

V. SOME EXPERIMENTAL IMPLICATIONS

As we have seen in the above discussions, the EWRmodel contains a large region of parameter space which

agreeswith the electroweak precision data and thus has

passed the first (indirect) test. The next test would be

direct observations of the signatures coming from the

new particles of the model: The mirror quarks and lep-tons and the scalars. Some of such signatures have been

suggested in [4] such as like-sign dileptons as a sign of

lepton number violation coming from the decay of the

Majorana right-handed neutrinos. As mentioned in [4],

this would be the high-energy equivalent of neutrino-less

double beta decay. This signal and those of other mir-

ror quarks and leptons will be presented in a separate

publication.

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16

The scalar sector of the EWR model has been studied

in some details in [7]. In light of the new LHC results, it

is timely to update the status of this sector. In particular,

the question that one may ask is the following: What are

the implications of the above analysis on the masses of

the scalars and their couplings to fermions? Several of

these issues will be presented in a follow-up paper on the

126 GeV scalar but it is important to set the foundationfor that paper here.

In terms of the minimal EWR model discussed in this

manuscript, one is most interested at this point in the

neutral scalars given in Eq. 23. Furthermore, the 126-

GeV object appears to be consistent, in terms produc-

tion and decays, with the SM Higgs boson which is a 0+

particle [30]. Although recent data on the spin-parity

[31] seemed to disfavor the 126-GeV object as a 0 par-ticle and is more consistent with the 0+ interpretation,

it did not completely rule out the 0 possibility. In con-sequence, we will keep an open mind. As seen in Eq. 23,

there are four neutral states: H05 ,H03 ,H

01 andH

01 . Since

the triplet scalars and do not couple to SM and mir-

ror quarks while the doublet does, one can see from

Eq. 23 that onlyH03 andH01 could be candidates for the

126 GeV object. However, a close look at the production

cross section reveals that, parity aside, only H03 fits the

bill. We summarize here some of the details which will

be given in full in [28].

The dominant production mechanism for the afore-

mentioned scalars is through gluon fusion. As a result,

one should know the couplings ofH03 andH01 to the SM

and mirror quarks. For H01 , the coupling to SM and

mirror quarks is given generically as gH01

q q = mqg2mWcHwhere qrepresents SM and mirror quarks and cH is an

abbreviation for cos H. This is greater than the SM

coupling by the factor 1/ cos H. Furthermore, the gluon-

fusion cross section is now proportional the square of the

number of heavy quarks, namely (7)2 = 49 where we

count the top quark and the six mirror quarks. As a re-

sult, EW R 49 SM. This is evidently not acceptable.As a consequence, to be consistent with the LHC data,

H01 will have to be heavier than 600GeV.We are left with H03 . From [ 7, 32], one c an

find its coupling to the SM and mirror quarks

as follows: gH03

tt = + mtg2mW

tan H5, gH03

bb =

mbg2mW

tan H5, gH03

uMuM = muMg2mW tan H5,gH0

3dMdM = +

mdM

g

2mWtan H5. Notice that the pseudo

scalar H03 contains the imaginary part of0 which cou-

ples to the up and down quarks with opposite signs. This

fact reflects in the above sign differences in the couplings.

The amplitude for the gluon fusion production ofH03 in-

volves a triangle loop denoted by I which depends on

rq = m2q /m

2H03

(q stands for any of the quarks). I 1when rq 1 and I 0 when rq 1. It is well-knownfrom the behavior ofIthat the gluon fusion production

of the SM Higgs boson is dominated by the top quark

loop. In our case, in addition to the top quark, we havethe mirror quarks which are assumed to be heavier than

H03 and hence rt, rqM 1. However, H03 is a pseudoscalar and, as we have seen above, the mirror up-quark

loop cancels that of the mirror down-quark loop because

both quarks are heavy so that rqM 1 and becausetheir couplings to H03 have opposite signs. This means

that IuM +IdM 0. As a consequence, again only thetop quark loop contributes. Details will be given in

[28]. Here we just quote the result. Because of the afore-

mentioned cancellation in the mirror quark sector, the

production cross section for H03 is

H03

= tan2 HHSM (37)

If we assume that the various branching ratios for the

H03 decays are comparable to those of the SM, one can

see from Eq. 37 that H03

HSM if tan H 1 orsin H 0.707 which is well inside our allowed rangeas shown in FIG. 8. The only hitch is that the par-

ity measurement seems to disfavor this interpretation ata 2 3 level but does not rule out completely thepseudo scalar interpretation. As a consequence, we will

keep an open mind regarding this possibility. In [28] we

also present a simple extension of the EWRmodel which

can accommodate the SM-like 0+ scalar as an interpre-

tation of the 126 GeV object.

Last but not least are the direct searches for mir-

ror fermions. In [4], it was mentioned that one of the

most tell-tale signs of the EWR is the production at the

LHC and the decays of Rs which are Majorana par-ticles and are their own antiparticles through the sub-

process q+ q R + R eM+W+ +eM+W+ e+ S+W+ +e+ S+W+, where e stands for ageneric charged lepton. These like-sign dileptons events

would be the high-energy equivalent of the low-energy

neutrino less double beta decay as emphasized in [4]. A

detailed study of this and other processes involving mir-

ror fermions is under investigation [8].

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17

VI. CONCLUSION

The assumption that right-handed neutrinos are non-

singletsunder S U(2) U(1) as proposed in [4] is a veryreasonable one which can be tested experimentally. The

EWR model preserves the gauge structure of the SM

but enriches it with mirror fermions and Higgs triplets.

The price paid might be considered to be minimal con-

sidering the fact that the EWR of [4] links the nature

of right-handed neutrinos -and hence the energy scale of

its Majorana mass- to details of the electroweak symme-

try breaking. In addition, these aspects can be tested

experimentally.

The first of such tests is the electroweak precision con-

straints. We have shown in this paper how the EWRmodel has a non-negligible range of parameter space to

fit the constraints ( see Fig. 5) on the oblique parame-

ters S and T despite the presence of right-handed mir-ror quarks and leptons which by themselves alone would

make a large positive contribution to the S parameter.

We have shown in details how the scalar sector, in partic-

ular the Higgs triplet fields, dramatically avoids this po-

tential disaster by making negative contributions which

offset those of the mirror fermions and thus bringing the

EWR model in agreement with the electroweak preci-

sion data. We have shown also how mass splittings, in

particular those of the scalar sector, affect the values

of the oblique parameters such as S whose constraints

in turn have interesting implications of those splittingsthemselves. This aspect would eventually be very use-

ful in the search for the scalars of the model. The mass

splittings of the mirror fermion sector can be straightfor-

wardly computed as a function of the mass splittings in

the scalar sector.

The next test of the model would be signatures and

searches for the mirror quarks and leptons and for the

additional scalars. Of immediate interest for the EWR(and for other BSM models as well) is the discovery of

the SM-like boson with a mass of 126 GeV. This discov-

ery puts a very strong constraint on any BSM model.What the EWR model has to say about this 126 GeV

object has been briefly discussed above and will be pre-

sented in detail in [28]. Basically, the minimal EWRmodel contains a pseudo-scalar,H03 , which could in prin-

ciple be a candidate whose production cross section can

be comparable to that for a SM Higgs boson with the

same mass with a choice of the angle Hwell within the

allowed range discussed above. However, the spin-parity

measurement [31] seemed to disfavor, but not completely

ruling out, the interpretation of the 126 GeV object as a

0 particle while the SM-like 0+ seems to be favored.Until more data come out to completely rule out the

pseudo-scalar interpretation, we will keep an open mind

however. Nevertheless, [28] presents a minimal extension

of the EWR model where the presence of an additional

0+ state can act like a SM Higgs boson. Needless to say,one expects several scalars beyond the 126 GeV boson to

be present in the model. A phenomenological study of

the scalar sector of the EWR model has been performed

[7] and it goes without saying that more studies of this

sector are needed. The input from the electroweak pre-

cision constraints will be valuable in a new study of this

sector.

One of the key points of the EWRmodel was the pro-

duction and detection of electroweak-scale right-handed

neutrinos through lepton-number violating signals such

as like-sign dilepton events at the LHC [4] which repre-sent the high-energy equivalent of the low-energy neu-

trino less double-beta decay. One could imagine that,

after taking care of the SM background, it would be a

much easier process to detect. This signal and others

related to searches for mirror quarks and leptons will be

presented in [8].

VII. ACKNOWLEDGEMENTS

We would like to thank Goran Senjanovic for illumi-nating discussions. This work was supported by US DOE

grant DE-FG02-97ER41027.

Appendix A: Loop Integrals and Functions

Different contributions to the oblique parameters are

expressed using loop integrals like A0, B0, B22, B1, B2.

and functions like F, G, etc. Therefore, it is very impor-tant to define all the loop integrals and functions we have

used in the calculations of different loop diagrams before

listing contributions from loop diagrams and details of

the calculations of the oblique parameters.

For calculation of oblique parameters we need the loop

diagrams with two external vector bosons. These dia-

grams have a general form

= Ag+ B qq (A1)

For the purpose of oblique parameters we only need the

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18

g term in this equation. Hence, hereafter in this paper

denotes only the first term on RHS above.

Loop diagrams involving one or two internal scalars or

one or two internal fermions appear in the calculation of

one-loop vector boson self-energy diagrams and Z - dia-

grams. Following loop integrals appear in the calculation

of loops with scalar particles [27]:

One-point integral: d4k

(2)41

(k2 m2)

162A0(m

2) (A2)

Two-point integrals: d4k

(2)41

(k2 m21)((k+q)2 m22)

162

B0(q2; m21, m

22),

(A3)

d4k(2)4

kk(k2 m21)((k+q)2 m22)

162

gB22(q2; m21, m

22)

(A4)

The expansion of LHS in the latter equation also has

term with qq[27], but this term is omitted as it does

not contribute to the oblique parameters [23].

Following [27], in the dimensional regularization these

integrals can be simplified to

A0(m2) = m2 + 1 ln(m2) (A5)B0(q

2; m21, m22) =

10

dxln(X ) (A6)

B22(q2; m21, m

22) =

1

4( + 1)

m21+m

22

q2

3

1

2

10

dxX ln(X ) (A7)

where

X

m21x +m

22(1

x)

q2x(1

x), (A8)

24 d + ln(4) . (A9)

ind space-time dimensions with = 0.577216..., the Eu-

lers constant [33]. The integrals in eqns. (A6), (A7) can

be calculated numerically up to desired accuracy. Note

that these equations involve the logarithm of a dimen-

sionful quantity,Xand the scale of this logarithm is hid-

den in the 2/(4 d) term in (refer to section 7.5 of

[33]). It is useful, especially in derivingTscalar in eqn.(32), to note that [34]

B0(0; m21, m

22) =

A0(m21) A0(m22)

m21 m22, (A10)

4B22(0; m21, m

22) = F(m21, m22) +A0(m21) +A0(m22),

(A11)

where

F(m21, m22) =m12 +m22

2 m

21m

22

m21 m2

2

ln

m21m2

2

,

ifm1=m2,

= 0 if m1= m2. (A12)

Note that

F(m21, m22) = F(m22, m21) . (A13)

Also notice that

B22(q2; m21, m

22) = B22(q

2; m22, m21)

B0(q2; m21, m

22) = B0(q

2; m22, m21). (A14)

While evaluating the fermion loops which contribute to

the oblique parameters following two-point loop integrals

are useful (refer section 21.3 of [33]):

B1(q2; m21, m

22) =

10

dx (1 x)lnX

M2

, (A15)

B2(q2; m21, m22) = 10

dx x(1 x)lnX M2 , (A16)where X is as defined in eqn. (A8). The logarithms in

these integrals involve a mass scale M. All the terms,

which depend on this scale cancel while evaluating the

final expressions for oblique parameters. For m1= m2=

mand q2 =M2Z,

B1(M2Z; m

2, m2) = 1 G(x)4

+lnm2

M2

, (A17)

B2(M2

Z

; m2, m2) = 1

18 32 G(x)2x+ 1+

12x 5 + 3 ln

m2M2

,

(A18)

where

G(x) = 4 4x 1Arctan 1

4x 1

. (A19)

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19

While derivingTfermion in eqn.(30) we need to evaluateintegrals in eqn. (A15) forq= 0 and m1= m2. One ofthe integrals, which appear in this calculation is 10

dx

m21x+m22(1 x)

lnm21x+m22(1 x)

M2

= m42 m41+ 2m41lnm21

M2 2m42lnm

21

M2

4

m21 m22 .

(A20)

Using the loop integrals and functions defined and en-

listed in this appendix we can derive the expressions for

the oblique parameters, which are suitable for the numer-

ical analysis.

Appendix B: Gauge Couplings of Higgs in EWR

model

In this appendix we derive the cubic and quartic cou-

plings of the Higgs in EWRmodel with the electroweak

gauge bosons. We start with the scalar fields and

in the EWR model, get the physical scalar states from

a generic potential with a global S U(2)L SU(2)R sym-metry, and which after spontaneous symmetry breaking

preservesSU(2)D custodial symmetry. Then we derive

the gauge couplings of the physical scalar states from the

kinetic part of the scalar Lagrangian in EWR model.

We work in the t Hooft Feynman gauge (gauge param-

eter, gauge = 1) throughout the calculations in this ap-pendix and all the appendices, which follow. To calcu-

late the new Physics contributions due to EWR model

to the oblique parameters we also need the correspond-

ing contributions from SM (refer to equations (26, 27,

28)). Therefore, in this section we also list the related

SM couplings.

The most general scalar potential for and that

preserves global S U(2)L SU(2)R is given by [7, 18]:

V(, ) = 1T r v22

2

+2T r 3v2M

2

+3

T r v22+ T r 3v2M2

+4

(T r) (T r)

2 (T r a

2

b

2) (T rTaTb)

+5

3T r (T r)2

(B1)

where repeated indices a, b are summed over. Note that

this potential is invariant under so that thecubic terms in the potential are eliminated. In order for

this potential to be positive semidefinite the following

conditions must be imposed: 1+2+ 23> 0, 12+

13+23> 0, 4> 0, 5> 0.

At the minimum of this potential, the vacuum align-

ment in eqs. (17, 18) breaks the global SU(2)L SU(2)Rdown to the custodial SU(2)D. One obtainsMW =g v/2

and MZ = MW/ cos W, with v =

v22+ 8 v

2M

246 GeVand, at tree level, = MW/MZ cos W= 1 as

desired (this confirms that SU(2)D custodial is, indeed,

preserved at the tree level).

After spontaneous breaking ofSU(2)LU(1)Y, besidesthe three Nambu-Goldstone bosons which are absorbed

by W and Z, there are ten physical scalars which are

grouped into 5 + 3 + 1 (2 singlets) of the custodial

SU(2)D. These Nambu-Goldstone bosons and physical

scalars are given in eqns. (21), (23) respectively. Themasses of the physical scalars are given as:

m2H,,05

=m25= 3 (4c2H+ 5s

2H)

m2H,05

=m23= 4v2 . (B2)

The two singlets H01 and H01 can mix according to the

mass-squared matrix given as:

M2H01

, H01

=v2

8c2H(1+3) 2

6sHcH3

2

6sHcH3 3s2H(2+3)

(B3)

The oblique parameters, the Feynman rules and the loop

diagrams are expressed in terms of the VEVs of the dou-

blet and triplets, and the masses of the physical scalars-

mH,,05

, mH,03

, m1, mH1

.

The gauge couplings of the physical scalars can be ob-

tained from the kinetic part of the scalar Lagrangian in

EWR model [7, 19, 32]:

(LSEWR )kin =1

2T r

(D)

(D)

(B4)

+1

2T r

(D)

(D)

+ |S|2

The notation (LSEWR )kin is used to denote the kineticpart (denoted by subscript kin) of the Higgs Lagrangian

(denoted by subscript S for Scalar) in EWR model

(denoted by EW R in the subscript). Here, andare

used in their 2 2 and 3 3 representations respectively,as given in equations (15, 16); S is the neutral SU(2)

singlet scalar in EWR model and

D +12

g(W ) 12

gB3; (B5)

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D + g(W t) gBt3 (B6)

The i/2 and ti are the 2 2 and 3 3 representationmatrices of the SU(2) generators respectively, following

reference [32].

We work under the premise that the hierarchy in neu-

trino masses in EWR model comes from the VEV ofS.

Thus, vS 105

eV and as a result the mixing betweenS and other scalars in negligible. Hence, hereafter in

the related calculations we neglect this mixing, unless

otherwise is stated. After the spontaneous breaking of

SU(2)L U(1)Y to U(1)EM, expanding the Lagrangianin equation (B4), one can find the Feynman rules for the

three point and four point interactions between physical

scalars, Nambu-Goldstone bosons and electroweak gauge

bosonsW,Zand. For the corresponding SM Feynman

rules it is useful to recall the kinetic part of the SM-Higgs

Lagrangian:

(LSSM)kin = 12 T r

(D)(D)

(B7)

The resulting Feynman rules in EWR model and SM are

listed in tables (I, II, III and IV) below.

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TABLE I. S1S2V type couplings(V is a vector gauge boson and S1, S2 are Higgs/ Goldstone bosons), which contribute to

Oblique Corrections. Common factor: g(p p), where p(p) is the incomingmomentum of the S1(S2).

gH05H5W+

32

gH++5 H

5 Z (12s

2W)

2cW

gH+5H5 W+

12

gH+3H3 Z

(12s2W)2cW

gH03H3W+

12

gH+3H5 Z

12cW

gH+3H5 W+

12

cH gH03H05Z

13

cHcW

gH03H5W+

12

cH gG+3G3Z

(12s2W)2cW

gH05H3W+

123

cH gG03H05Z

13

sHcW

gG03G3W+

12

gG+3H5 Z

12cW

sH

gG+3H5 W+

12

sH gH01G03Z

cHcW

gG03H5W+

12

sH gH01G0

3Z

23sHcW

gH05G3W+

123

sH gH01H03Z sH2cW

gH01G3W+

12

cH gH01H0

3Z

23cHcW

gH01 G3W+

23

sH gH+5H5

sW

gH01H3W+

12

sH gH++5 H

5 2sW

gH01H

3W+ 23cH gH+

3H

3 sW

gH+5H

5 Z

(12s2W)2cW

gG+3G

3

sW

TABLE II. SV1V2 type couplings(V1 and V2 are vector gauge bosons and S is a Higgs boson), which contribute to Oblique

Corrections. Common factor: gMWg

gH05W+WsH3 gH05ZZ

23

sHc2W

gH++5 WW

2sH gH+

5WZ

sHcW

gH01W+W cH gH0

1ZZ

cHc2W

gH01W+W

223

sH gH01 ZZ

223

sHc2W

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TABLE III. H1H2V1V2 type couplings, which contribute to Oblique Corrections. Common factor: g2g

gH05H05W+W

53

gH05H05ZZ

23

1c2W

gH+5H

5W+W

32 gH+5H

5 ZZ

(c4

W+s4

W)c2W

gH++5 H

5 W+W

1 gH++5 H

5 ZZ

2(12s2W)2

c2W

gH03H03W+W (c2H+

s2H2

) gH03H03ZZ

(1+c2H)

2c2W

gH+3H

3W+W

( 12

+ c2H) gH+3H

3 ZZ

s2H2

(1s2W)2c2W

+ c2H(c4W+s

4W)

c2W

gG0

3G03W+W

(1+s2H )

2 gG0

3G03ZZ 12c2

W

(1 + 3s2H)

gG+3G

3W+W

( 12

+ s2H) gG+3G3ZZ

12

c2H(1 2s2W)2 + s2H(c4W+ s4W)

gH01H01W+W12 gH01H01ZZ

12c2W

gH01H0

1W+W

43

gH01H0

1 ZZ

43c2W

gH+5H

5

2s2W gH+5H

5 Z

sWcW

(1 2s2W)

gH++5 H

5

8s2W gH++5 H

5 Z

4 sWcW

(1 2s2W)

gH+3H

3

2s2W gH+3H

3 Z

sWcW

(1 2s2W)

gG+3G

3

2s2W gG+3G3Z

sWcW

(1 2s2W)

TABLE IV.H1H2V1V2 type couplings, which do notcontribute to Oblique Corrections. Common factor: g2g

gH01H0

5W+W

23

gH01H0

5ZZ 2

2

3c2W

gH+3H5W+W

cH2

gH+3H5 ZZ

cH(12s2W)

c2W

gH03

G0

3

W+W

cHsH

2

gH03

G0

3

ZZ

3

2

cHsH

c2

W

gH+3G3W+W

cHsH gH+3G3ZZ

cHsH2c2W

gH+5G3W+W

sH2

gH+5G3ZZ

sH(12s2W)

c2W

gH+3H5 Z

cHsWcW

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Appendix C: Gauge Couplings of mirror fermions in

EWR model

In this appendix we start with the fermion content

of the EWR model and derive the electroweak gauge

couplings of these fermions from the Lagrangian. The

SU(2)L U(1)Y fermion content of the EWR model of[4] is given in eqns. (1), (2), (3), (4).

The interaction of mirror leptons with the SU(2)LU(1)Ygauge bosons are found in the terms

lMR /DlMR ; e

ML

/DeML , (C1)

where

/DlMR ( 1

2 g(W ) +1

2 gB)lMR ,

/DeMe (+ gB)eML . (C2)

The gauge interactions for mirror quarks can similarly be

found. Thus, the Feynman rules for the gauge interac-

tions for fermions (Sm fermions and mirror fermions) inthe EWR model can be evaluated from

(LFEWR )int = (LFSM)int+ (LFM)int , (C3)

where (LFSM)int comes from the fermion-sector in theStandard Model (and is well known) and (LFM)intincludes interaction terms arising due to the mirror

fermion-sector in EWR model. (LFSM)int is well knownto be [33] To write the mirror fermions part (LFM)intremember that the W bosons couple only to SU(2)

doublets of fermions. Thus only right-handed mirror

fermions couple to the W, as opposed to (LFSM)int,where only left-handed SM fermions interact with the

W bosons. Similarly the three-point couplings of theright-handed mirror fermions with Z and bosons at

the tree-level are same as those for the left-handed SM

fermions. Hence, (LFM)int, is given by

(LFM)int = g

2

uMiR

dMRi+iR

eMRi

W+

+

dM i

R uMR i+e

M iR

MR i

W

+ gcW

fM= uM,dM,M,eM

TfM3 s2WQfM fM iR fMR i

fM= uM,dM,eM

s2WQfM fM i

L fML i

Z+e

fM= uM,dM,eM

QfM

fM i

R fMR i+f

M i

L fML i

A

(C4)

In equation (C4) i, j = 1, 2, 3, where i denotes fermions

in the ith mirror-quark or mirror-lepton generation.Sums over i are implicit, when an index is contracted.

(uMi and dMi ) denote the (up- and the down-) members

of a mirror-quark generation respectively. Following a

similar notation (Ri and eMi ) denote (the neutrino and

the electron) members of a mirror-lepton generation re-

spectively.

All the tree-level interactions calculated from equation

(C4) can be tabulated in a compact form as given in ta-

ble (V). Corresponding SM interactions can be similarly

calculated from equation (LFSM)int.In this paper we will not consider the mixings between

different quark and lepton generations or mirror-quarks

and mirror-lepton generations. Thus, the CKM, PMNS

matrices and also mirror-CKM and mirror-PMNS ma-

trices are identity. Since we are looking at wide ranges

of mirror fermions masses, which satisfy the experimen-

tal constraints from the oblique parameters, slight devi-

ations of mirror-CKM and mirror-PMNS matrices from

identity matrix will only move individual points in the

available paramter space, but will not affect significantly

the total available parameter space (refer to the plots in

section IV).

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TABLE V. fM1 fM2 V type couplings, which contribute to the Oblique Corrections. For each Feynman rule the charge conservation

is implicit. fM1R and fM2R are members of the same mirror fermion doublet with isospins

1

2 and1

2respectively (ref. [4], [33] &

[35]). Common factor for all couplings: g

gfM1Rf

M2R

W+12

gfMR fMR Z1cW

(Tf3 s2WQf)

gfMR f

MR

sWQf

gfML f

ML Z

s2WcW

Qf

gfML f

ML

sWQf

Appendix D: Calculation of One Loop Contributions

to Oblique Parameters in EWR model

The one loop contributions to the oblique parameters

in EWR model can be calculated from the cubic and

quartic couplings listed in Appendix B and C and using

the loop integral functions illustrated in Appendix A.

The SM loop diagrams contributing to S, T, U can be

similarly obtained from the SM cubic and quartic cou-

plings in equations (B7), (LFSM)int and using loop in-tegrals from Appendix A. Hereafter, the focus of cal-

culations will be on S and T parameters. The new

Physics contributions to S from the scalar sector andmirror fermion sector in EWR model will be calculated

separately and then added to find the total contributionS (eqn. (26)). Similary procedure will be followed tocalculateT (eqn. (27)). Thus, as in eqns. (29), (30),

S=Sscalar+Sfermion,T=Tscalar+Tfermion.Recall (eqn. (26)) that the contributions toScome fromZ and self-energies, Zmixing, each calculated up to

one-loop level. To evaluateT using equation (27) theisospin current 11 and electromagnetic current 33 areused. The W and Z self-energies are related to these

isospin currents by [23],

W W = e2

s2W11;

ZZ= e2

s2Wc2W

(33 2s2W3Q+s4WQQ) (D1)

Using these relations 11 can be obtained from the loop

contributions to W W listed in tables VI, VIII, IX. From

equation (D1) the one-loop contributions to 33 can be

obtained using limg0

(ZZ). These contributions are listed

below, separately from ZZfor scalar as well as fermion

sectors in EWR model.

1. One Loop Contributions toSscalar andTscalar

In this subsection the one-loop contributions to

Sscalar

andTscalar are listed. In every table the loop contribu-tions in EWR model are listed first and then the corre-

sponding contributions in SM are also listed. The one-

loop diagrams, which contribute toSscalar can be foundin tables VI, VIII, IX, XI, VII, X below. To calculateTscalar, 11 contributions from scalar sector in EWRmodel can be obtained from contributions to W W listed

in tables VI, VIII, IX. The scalar-loop diagrams con-

tributing to 33 are listed in tables XII, XIV, XIII.

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TABLE VI: One-loop diagrams with two internal scalar (S) (Higgs or

Goldstone boson) lines, which contribute to W+ and Z self-energies.

Common factor: g2

162

Contributions to WW(q2) Contributions to ZZ(q

2)

W+

i

Sj W+ Z

i

Sj Z

Si Sj Si Sj

H+5 H05 3B22(q

2; m2H+5

, m2H05

) H+5 H+5

c22Wc2W

B22(q2; m2

H+5

, m2H+5

)

H++5 H+5 2B22(q

2; m2H++5

, m2H+5

) H++5 H++5 4

c22Wc2W

B22(q2; m2

H++5

, m2H++5

)

H+3 H03 B22(q

2; m2H+3

, m2H03

) H+3 H+3

c22Wc2W

B22(q2; m2

H+3

, m2H+3

)

H++5 H+3 2c

2HB22(q

2; m2H++5

, m2H+3

) H+5 H+3

c2Hc2W

B22(q2; m2

H+5

, m2H+3

)

H+5 H03 c

2HB22(q

2; m2H+5

, m2H03

) H5 H3

c2Hc2W

B22(q2; m2

H+5

, m2H+3

)

H05 H+3

c2H3

B22(q2; m2

H05

, m2H+3

) H05 H03

4

3

c2Hc2W

B22(q2; m2

H05

, m2H03

)

H+3 H01 s

2HB22(q

2; m2H+3

, m2H1) H03 H

01

s2Hc2W

B22(q2; m2

H03

, m2H1)

H+3 H01

8

3 c2HB22(q

2; m2H+3

, m2H1

) H03 H01

8

3

c2Hc2W

B22(q2; m2

H03

, m2H1

)

G+3 H++5 2s

2HB22(q

2; M2W, m2H++5

) G+3 H+5

s2Hc2W

B22(q2; M2W, m

2H+5

)

G03 H+5 s

2HB22(q

2; M2Z, m2H+5

) G3 H5

s2Hc2W

B22(q2; M2W, m

2H+5

)

G+3 H05

s2H3

B22(q2; M2W, m

2H05

) G03 H05

4

3

s2Hc2W

B22(q2; M2Z, m

2H05

)

Continued on next page...

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TABLE VI continued from previous page

Si Sj Si Sj

G+3 H

01 c

2HB22(q

2

; M2W, m

2H1) G

03 H

01

c2Hc2W B

22(q2

; M2Z, m

2H1)

G+3 H01

8

3 s2HB22(q

2; M2W, m2H1

) G03 H01

8

3

s2Hc2W

B22(q2; M2Z, m

2H1

)

G+3 G03 B22(q

2; M2W, M2Z) G

+3 G

+3

c22Wc2W

B22(q2; M2W, M

2W)

Standard Model contributions

H G+

SM B22(q2

; M2W, m

2H) H G

0SM

1

c2W B22(q2

; M2Z, m

2H)

G+SM G0SM B22(q

2; M2W, M2Z) G

+SM G

+SM

c22Wc2W

B22(q2; M2W, M

2W)

TABLE VII: Tadpole diagrams with one internal scalar (S) (Higgs or

Goldstone boson) line, which contribute to W+ and Z self-energies.

Common factor: g2

162


2)

W+

i

W+

Z

i

Z

Si Si

H05

5

6 A0(m2H05 ) H

05

2

6c2WA0(m

2H05 )

H+5 3

2A0(m

2H+5

) H+5 c4W+ s

4W

c2WA0(m

2H+5

)

Continued on next page...

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TABLE VII continued from previous page

Si Si

H++5 A0(m

2H++5 ) H

++5 2

c22Wc2W A

0(m2H++5 )

H03 1

4(1 + c2H)A0(m

2H03

) H03 1

4c2W(1 + 3c2H)A0(m

2H03

)

H+3 1

2(1 + 2c2H)A0(m

2H+3

) H+3 c22W2c2W

(1 + c2H)A0(m2H+3

)

H01 1

4A0(m

2H1

) H01 1

4c2WA0(m

2H1

)

H

0

1 2

3 A0(m2

H1) H0

1 2

3c2WA0(m2

H1)

G03 1

4(1 + s2H)A0(M

2Z) G

03

1

4c2W(1 + 3s2H)A0(M

2Z)

G+3 1

2(1 + 2s2H)A0(M

2W) G

+3

c22W2c2W

(1 + s2H)A0(M2W)


H 1

4 A0(m

2

H) H 1

4c2WA0(m

2

H)

G0SM 1

4A0(M

2Z) G

0SM

1

4c2WA0(M

2Z)

G+SM 1

2A0(M

2W) G

+SM

c22W2c2W

A0(M2W)

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TABLE VIII. One-loop diagrams with one internal scalar (S) (Higgs or Goldstone boson) line and one internal vector boson

line, which contribute to W+ and Z self-energies. Common factor: g2

162


2)

W+

i

Vj W+ Z

i

Vj Z

Si Vj Si Vj

H05 W+ s

2H

3 M2WB0(q

2; M2W, m2H05

) H05 Z 4

3

s2Hc2W

M2ZB0(q2; M2Z, m

2H05

)

H01 W+ c2HM2WB0(q2; M2W, m2H1) H

01 Z

c2Hc2W

M2ZB0(q2; M2Z, m

2H1

)

H01 W+ 8

3 s2HM

2WB0(q

2; M2W, m2H1

) H01 Z 8

3

s2Hc2W

M2ZB0(q2; M2Z, m

2H1

)

H+5 Z s2Hc2W

M2WB0(q2; M2Z, m

2H+5

) H+5 W s

2H

c2WM2WB0(q

2; M2W, m2H+5

)

H++5 W 2s2HM2WB0(q2; M2W, m2H++

5

) H5 W+ s

2H

c2WM2WB0(q

2; M2W, m2H+5

)


H W+ M2WB0(q2; M2W, m2H) H Z M2Zc2W

B0(q2; M2Z, m

2H)

G+SM Z s4Wc2W

M2WB0(q2; M2Z, M

2W) G

+SM W

2 s4W

c2WM2WB0(q

2; M2W, M2W)

G+SM s2WM2WB0(q2; 0, M2W) GSM W+ 2s4Wc2W

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Nurstu 05-01-13 Revised 2