One-loop radiative corrections to the ρ parameter in the Left Right Twin Higgs model

Journal of the Korean Physical Society, Vol. 63, No. 6, September 2013, pp. 1114∼1127

One-loop Radiative Corrections to the ρ Parameter in the Left Right TwinHiggs Model

Jae Yong Lee∗

Department of Physics, Korea University, Seoul 136-701, Korea

Dong-Won Jung†

School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea

(Received 20 May 2013)

We implement a one-loop analysis of the ρ parameter in the Left Right Twin Higgs model, includ-ing the logarithmically enhanced contributions from both heavy fermion and scalar loops. Numericalanalysis indicates that the one-loop corrections are dominant over the tree-level contributions inmost regions of parameter space. The experimentally allowed values of the ρ-parameter divide theallowed parameter space into two regions; less than 670 GeV and larger than 1100 GeV roughly,for the symmetry breaking scale f . Therefore, our result significantly reduces the parameter spacethat is favorably accessible to the LHC.

PACS numbers: 12.15.Lk, 12.60.Cn, 12.60.FrKeywords: Extended Higgs sector, Electroweak precision testsDOI: 10.3938/jkps.63.1114

I. INTRODUCTION

The Standard Model (SM) has excellently describedhigh energy physics up to energies of O(100) GeV. Theonly undetected constituent of the SM to date is theHiggs boson, which is essential to generating fermionand gauge boson masses. Theoretically, the Higgs bo-son’s mass squared is quadratically sensitive to any newphysics scale beyond the Standard Model (BSM) andstabilization of the Higgs mass squared prefers an en-ergy scale below O(1) TeV. However, electroweak pre-cision measurements with naive naturalness assumptionraise the energy scale of the BSM up to 100 TeV or evenhigher. Hence, a tension associated with the stabiliza-tion of the SM Higgs mass, remains between theory andexperiment. With the latest data from the LHC the ten-sion has gotten stronger.

The basic idea of little Higgs is that the SM Higgs isa pseudo-Nambu-Goldstone boson (pNGB) [1–7]. Stabi-lization of the Higgs mass in the little Higgs theories isachieved by the “collective symmetry breaking”, whichnaturally renders the SM Higgs mass much smaller thanthe global symmetry breaking scale. The distinct ele-ments of little Higgs models are a vector-like heavy topquark, and various scalar and vector bosons. The for-mer is universal while the latter are model-dependent.

∗E-mail: [email protected]†E-mail: [email protected]

Both of them contribute significantly to one-loop pro-cesses and, hence, set strict constraints on the parame-ter space of little Higgs models. At worst, electroweakprecision tests push up the symmetry breaking scale to 5TeV or higher, and even revive the fine-tuning problemof the Higgs potential.

The idea of twin Higgs shares the same origin with thatof little Higgs. However, there is a difference between thetwo as for the stabilization of the Higgs mass squared.The twin Higgs mechanism introduces additional discretesymmetry to render no quadratic divergence in the Higgsmass squared. For instance, the mirror twin Higgs modelcontaining a complete copy of the SM identifies the dis-crete symmetry with mirror parity. The SM world andits mirror world communicate only through the Higgs sothat the mirror particles are very elusive in the SM worldand yield poor phenomenology at the LHC.

The twin Higgs mechanism can also be realized byidentifying the discrete symmetry with left-right symme-try in the left-right model [9]. The left-right twin Higgs(LRTH) model contains U(4)1 × U(4)2 global symme-try as well as SU(2)L × SU(2)R × U(1)B−L gauge sym-metry. The left-right symmetry acts on only the twoSU(2)’s gauge symmetry. A pair of vector-like heavytop quarks play a key role in triggering electroweak sym-metry breaking just as that of the little Higgs theories.On top of that, the non-SM Higgs particles acquire largemasses not only at the quantum level but also at thetree level, causing the model to deliver much richer phe-

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One-loop Radiative Corrections to the ρ Parameter· · · – Jae Yong Lee and Dong-Won Jung -1115-

nomenology at the LHC [10] compared with the mirrortwin Higgs model. Moreover, they lead to large radiativecorrections to one-loop processes, so the allowed param-eter space can be significantly reduced. In this paper,we perform a one-loop analysis of the ρ-parameter in theLRTH model to reduce the parameter space.

The paper is organized as follows: The LRTH model isbriefly reviewed in Section II. The renormalization proce-dure for the ρ-parameter is explained in Section III. Thenumerical analysis on the ρ-parameter is performed inSection IV. We present our conclusions in Section V. Thetechnical details on the computation of the ρ-parameterare reckoned in the Appendices.

II. LEFT RIGHT TWIN HIGGS MODEL IN ANUTSHELL

We review the LRTH model in Ref. [10]. The LRTHmodel is based on the global U(4)1 × U(4)2 symme-try, with a locally gauged subgroup SU(2)L ×SU(2)R ×U(1)B−L. A pair of Higgs fields, H and H, are intro-duced, and each transforms as (4, 1) and (1, 4), respec-tively, under the global symmetry. They are written as

H =(

HL

HR

), H =

(HL

HR

), (1)

where HL,R and HL,R are two component objects thatare charged under the SU(2)L × SU(2)R × U(1)B−L as

HL and HL : (2, 1, 1), HR and HR : (1, 2, 1). (2)

The global U(4)1 (U(4)2) symmetry is spontaneouslybroken down to its subgroup U(3)1 (U(3)2) with VEVs:

〈H〉T = (0, 0, 0, f), 〈H〉T = (0, 0, 0, f). (3)

The spontaneous symmetry breaking results in sevenNambu-Goldstone bosons (NGB), which are parameter-ized as

H = feπ/f

⎛⎜⎝

0001

⎞⎟⎠ , π =

⎛⎜⎜⎜⎝

− N2√

30 0 h1

0 − N2√

30 h2

0 0 − N2√

3C

h∗1 h∗

2 C∗√

3N2

⎞⎟⎟⎟⎠ ,

(4)

where π is the corresponding Goldstone field matrix. Nis a neutral real field, C and C∗ are a pair of charged com-plex scalar fields, and hSM = (h1, h2) is the SM SU(2)L

Higgs doublet. Note that the normalization of N is nat-urally altered when applying the unitary gauge. H isparameterized in the identical way by its own Goldstoneboson matrix, π, which contains N , C, and h = (h+

1 , h02).

The two U(4)/U(3)’s symmetry breakings yield fourteenNGBs in all.

The linear combination of C and C, and the linearcombination of N and N are eaten by the gauge bosons ofSU(2)R ×U(1)B−L, which is broken down to the U(1)Y .The orthogonal linear combinations, a charged complexscalar φ± and a neutral real pseudoscalar φ0, remain asNGBs. On top of that, the SM Higgs acquires a VEV,〈hSM 〉 = (0, v/

√2), so electroweak symmetry SU(2)L ×

U(1)Y is broken down to U(1)EM , but h’s do not get aVEV and remain as NGBs. At the end of the day, thetwo Higgs VEVs are given by

〈H〉 =

⎛⎜⎝

0if sin x

0f cos x

⎞⎟⎠ , 〈H〉 =

⎛⎜⎜⎝

000f

⎞⎟⎟⎠ , (5)

where x = v√2f

. The values of f and f will be boundedby electroweak precision measurements. In addition, f

and f are interconnected once we set v = 246 GeV.

1. Gauge Sector

The whole gauge symmetry of the model is SU(3)C ×SU(2)L × SU(2)R × U(1)B−L. However, SU(3)C gaugesymmetry is not taken into account in this paper becauseour interest lies in the electroweak symmetry breaking.The generators of the SU(2)L × SU(2)R × U(1)B−L aregiven, respectively, as(

12σi 00 0

),

(0 00 1

2σi

),

12

(12 00 12

), (6)

and the corresponding gauge fields are W±,0L ,W±,0 and

B. The covariant derivative is then given as

Dµ = ∂µ − igWµ − ig′qB−LBµ, (7)

where

W =12

⎛⎜⎜⎝

W 0L

√2W+

L 0 0√2W−

L −W 0L 0 0

0 0 W 0R

√2W+

R

0 0√

2W−R −W 0

R

⎞⎟⎟⎠ ,

B =12

⎛⎜⎝

B 0 0 00 B 0 00 0 B 00 0 0 B

⎞⎟⎠ , (8)

and g and g′ are the gauge couplings for SU(2)L,R andU(1)B−L, and qB−L is the charge of the field underU(1)B−L.

The kinetic term for the two Higgs fields can be writtenas

LH = (DµH)†DµH + (DµH)†DµH, (9)

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with qB−L = 1. The Higgs mechanism for both H and Hmakes the six gauge bosons massive, but one gauge bo-son, the photon, massless. For the charged gauge bosons,there is no mixing between the W±

L and W±R : W±

L isidentified with the SM weak gauge boson W± while W±

R

is much heavier than W± and is denoted as W±H . Their

masses are

M2W =

12g2f2 sin2 x, M2

WH=

12g2(f2 + f2 cos2 x). (10)

Note that M2W +M2

WH= g2

2 (f2 + f2). The linear combi-nations of the neutral gauge bosons W 0

L,W 0R and B yield

photon and two neutral massive gauge bosons Z,ZH withrespective masses:

M2A = 0, (11)

M2Z =

g2 + 2g′2

g2 + g′22M2

W M2WH

M2WH

+ M2W +

√(M2

WH− M2

W )2 + 4g′2g2+g′2 M2

WHM2

W

, (12)

M2ZH

=g2 + g′2

g2(M2

W + M2WH

) − M2Z . (13)

For later use, we define the Weinberg angle of the LRTHmodel:

sw = sin θw =g′√

g2 + 2g′2, (14)

cw = cos θw =

√g2 + g′2

g2 + 2g′2, (15)

c2w =√

cos 2θw =g√

g2 + 2g′2. (16)

The unit of the electric charge is then given by

e = gsw =gg′√

g2 + 2g′2. (17)

2. Fermion Sector

To cancel the quadratic sensitivity of the Higgs massto the top quark loops, we incorporate a pair of vector-like, charge 2/3 fermion (QL,QR) into the top Yukawasector:

LY uk = yLQL3τ2H∗LQR+yRQR3τ2H

∗RQL−MQLQR+h.c.,

(18)

where τ2 =(

0 −11 0

), QL3 = −i(uL3, dL3) and QR3 =

(uR3, dR3) are the third generation up- and down-typequarks, respectively. The left-right parity indicates yL =yR(≡ y). The mass parameter M is essential to the topmixing. The value of M is constrained by the Z → bbbranching ratio. It can also be constrained by the oblique

parameters, which we will do in the paper. Furthermore,it yields a large log divergence of the SM Higgs mass. Tocompensate for that, the non-SM gauge bosons also getlarge masses by increasing the value of f . Therefore, itis natural for us to take M � yf .

Expanding the HL,R field in terms of the NGB fields,we acquire the mass matrix of the fermions. By diago-nalizing it, we obtain not only the mass eigenstates forthe SM-like and heavy top quarks, but also the mixingangles for the left-handed and right-handed fermions:

m2t =

12(M2 + y2f2 − Nt),

m2T =

12(M2 + y2f2 + Nt), (19)

sin αL =1√2

√1 − (y2f2 cos 2x + M2)/Nt, (20)

sin αR =1√2

√1 − (y2f2 cos 2x − M2)/Nt, (21)

where Nt =√

(y2f2 + M2)2 − y4f4 sin2 2x.

3. Higgs Sector

Among the fourteen NGBs in both π and π, sixNGBs are eaten by the gauge bosons. The remain-ing eight NGBs get masses through quantum effectsand/or soft symmetry breaking terms, so called µ-term.The Coleman-Weinberg potential, obtained by integrat-ing out the heavy gauge bosons and top quarks, yieldsthe SM Higgs potential, which determines the SM HiggsVEV and its mass, as well as the masses for the other


Higgs, φ±, φ0, h±1 and h0

2. Moreover, the µ-term con-tributes to the Higgs masses at the tree level:

Vµ = −µ2r(H

†RHR + c.c.) + µ2H†

LHL. (22)

One may include −µ2l (H

†LHL + c.c.) to the Higgs poten-

tial, but here we choose µl = 0 so as to keep the original

motivation of the model and to preserve the stability ofh2 dark matter [10].

We write the masses for the Higgs as

M2φ0 =

µ2rff

f2 + f2 cos2 x

[ f2(cos x + sin xx (4 + x2))

f2(cos x + 2 sin xx )2

+2 cos x(cos x + 4 sin x

x )3(cos x + 2 sin x

x )+

f2 cos2 x(4 + cos x)

9f2

], (23)

M2φ± =

316π2

g′2M2WH

M2ZH

− M2Z

[( M2W

M2ZH

− 1)Z(MZH

) −(M2

W

M2Z

− 1)Z(MZ)

]

+µ2

rff

f2 + f2 cos2 x

[ f2x2

f2 sin2 x+ 2 cos x +

f2 cos3 x

f2

], (24)

M2h2

=3

16π2

[g2

2(Z(MW ) −Z(MWH

)) +2g′2 + g2

4M2

WH− M2

W

M2ZH

− M2Z

(Z(MZ) −Z(MZH))

]+ µ2

r

f

fcos x + µ2, (25)

M2h1

= M2h2

+3

16π2

g′2M2W

M2ZH

− M2Z

[(M2WH

M2ZH

− 1)Z(MZH

) −(M2

WH

M2Z

− 1)Z(MZ)

]. (26)

where

Z(x) = −x2(lnΛ2

x2+ 1), (27)

with Λ being a UV cutoff. The SM Higgs potential arisesmainly from both the top and the gauge sectors. Thecontribution of the fermion loops to the SM Higgs masssquared is negative, and its dominance over the contri-bution of the gauge boson loops and the tree level massparameter µ2

rf2f triggers electroweak symmetry breaking.

We fix the SM Higgs VEV as v = 246 GeV.

III. RENORMALIZATION PROCEDURE

We follow the renormalization procedure in Ref. [11]to calculate the ρ-parameter at one-loop order. The Z-pole, W -mass, and neutral current data can be used tosearch for and set limits on the deviations from the SM.The the ρ-parameter is defined as

ρ ≡ M2W

M2Zc2

θ

. (28)

The electroweak mixing angle s2θ(≡ 1−c2

θ) in the effec-tive leptonic (electronic) vertex of the Z boson is definedas

s2θ ≡ 1

4

(1 + Re

geV

geA

)(29)

in terms of the effective vector and axial vector couplingsge

V,A of the Z to electrons;

L = ieγµ(gV + gAγ5)eZµ. (30)

The effective Lagrangian of the charged current interac-tion in the LRTH model is given by

Lcc =g√2

(W+

µLJµ−L + W−

µLJµ+L

)+ (L → R), (31)

where Jµ±L,R is the charged currents. For momenta quite

small compared to MW , this effective Lagrangian givesrise to the effective four-fermion interaction with theFermi coupling constant GF√

2= g2

8M2W

, and the vector andthe axial vector parts of the neutral current Zee couplingconstants are given to the order v2/f2 as

geV =

g

2cw

[( − 12

+ 2s2w

)+

v2

4(f2 + f2)

s2w(c22

w − 2)c4w

],

(32)

geA =

g

2cw

[12

+v2

4(f2 + f2)

s2wc22

w

c4w

]. (33)

The effective leptonic mixing angle s2θ in Eq. (29) is then

related to the mixing angle s2w as

s2θ = s2

w − v2

f2 + f2

s4w

c2w

, (34)

which can then be inverted to

s2w = s2

θ + ∆s2θ, (35)


where

∆s2θ

s2θ

= −ζ +12

c2θ

s2θ

−√−ζ +

(ζ − 1

2c2θ

s2θ

)2

, (36)

with

ζ ≡ v2

f2 + f2. (37)

The SM SU(2)L gauge coupling constant, g, is ex-pressed by using both the effective leptonic mixing angle,s2

θ, and the fine-structure constant, α, as

g2 =e2

s2w

=4πα

s2θ

(1 − ∆s2

θ

s2θ

). (38)

The ρ-parameter at tree level is

ρtree =πα√

2GF c2θs

2θM

2Z

(1 − ∆s2

θ

s2θ

). (39)

The ρ-parameter at tree level is different from unity, andits deviation from unity is of order v2/f2.

Because the loop factor arising from radiative correc-tions, 1/16π2, is similar in magnitude to v2/f2 (for f � 5TeV), the one-loop radiative corrections can be compa-rable in size to the next-to-leading order corrections attree level. At one-loop order, the mass relation reads [12]

s2θc

2θ =

πα(MZ)√2GF M2

Zρ

[1 − ∆s2

θ

s2θ

+ ∆rZ

], (40)

where ∆rZ includes radiative effects from varioussources:

∆rZ =δα

α− δGF

GF− δM2

Z

M2Z

−(c2

θ − s2θ

c2θ

)δs2θ

s2θ

. (41)

We should mention that ∆rZ in Eq. (41) differs from theusual ∆rZ defined in the SM by extra corrections due tothe renormalization of s2

θ. In general, the vertex and boxcontributions to the radiative effects are relatively smallcompared to the other corrections [12, 13]: hence, weconsider only the “oblique corrections”, i.e. the vacuumpolarization corrections to W , Z and the photon. Thecorrection due to the vacuum polarization of the photon,δα, is given by

δα

α= Πγγ′

(0) + 2(ge

V − geA

Qe

)ΠγZ(0)M2

Z

. (42)

Because we ignore the vertex and the box corrections, theelectroweak radiative correction to the Fermi constant,δGF , stems from the W -boson vacuum polarization:

δGF

GF= −ΠWW (0)

M2W

. (43)

The counterterms for the Z-boson mass, δM2Z , and the

leptonic mixing angle, δs2θ, are given, respectively, by [13]

δM2Z = Re[ΠZZ(M2

Z)], (44)

δs2θ

s2θ

= Re[( cθ

sθ

){ΠγZ(M2Z)

M2Z

+v2

e − a2e

aeΣe

A(m2e)

− ve

2sθcθ

(ΛZeeV (M2

Z)ve

− ΛZeeA (M2

Z)ae

)}], (45)

where ΛZeeV,A are the vector and the axial vector form fac-

tors of the unnormalized one-loop Zee vertex corrections,and Σe

A is the axial part of the electron self-energy. Onceagain, we ignore the non-oblique corrections.

The tree-level and radiative corrections, except for theW-boson, are summed and expressed as

∆r = −∆s2θ

s2θ

− Re[ΠZZ(M2Z)]

M2Z

+ Πγγ′(0)

+ 2(ge

V − geA

Qe

)ΠγZ(0)M2

Z

− c2θ − s2

θ

cθsθ

Re[ΠγZ(M2Z)]

M2Z

,

(46)

and then, Eq. (40) is rewritten as,

s2θc

2θ =

πα(MZ)√2GF M2

Zρ

[1 +

ΠWW (0)M2

W

+ ∆r]. (47)

Solving Eqs. (28) and (47) for the W -boson mass, we get

M2W =

12

[a(1+∆r)+

√a2(1 + ∆r)2 + 4aΠWW (0)

], (48)

with a ≡ πα(MZ)√2GF s2

θ

. Finally, the ρ parameter is calcu-

lated using Eq. (28).

IV. NUMERICAL ANALYSIS

In order to take the precision measurements, we needthe standard experimental values as input parameters.These are the input parameters we take [14]:

GF = 1.16637(1) × 10−5 GeV−2, (49)MZ = 91.1876(21) GeV, (50)

α(MZ)−1 = 127.918(18), (51)

s2θ = 0.23153(16). (52)

We also take the top and the bottom quark masses as[14–16]

mt = 172.3 GeV, mb = 3 GeV, (53)

where mt is the central value of the electroweak fit andmb is the running mass at the MZ scale with the MS


Fig. 1. (Color online) Plots of f versus f for differentvalues of M .

scheme. The ρ-parameter itself is measured very accu-rately [14]:

ρ ≡ ρ0ρ ≡ M2W

M2Zc2

θ

, (54)

ρ0 = 1.0002+0.0007−0.0004, (55)

ρ = 1.01043 ± 0.00034. (56)

Including all the SM corrections (top quark loop, bosonicloops), we take the allowed range of the ρ parameter as

1.00989 ≤ ρexp ≤ 1.01026. (57)

The input parameters of the LRTH model aref, M, µr, and µ, where f is the Higgs VEV in Eq.(3),M is the heavy top quark mass scale, and both µr andµ are soft symmetry breaking terms. The masses of thetop and the heavy top quarks are determined mainly byf and M while those of the scalar particles h1, h2, φ±

and φ0 largely depend on µ, µr and f . Another scale f ,associated with the masses of the heavy gauge bosons,can be determined by the electroweak symmetry break-ing condition: there is a generic relation between f andf because the Coleman-Weinberg potential of the Higgsboson mostly depends on M,f and f . For the scalar po-tential, there is a tree-level mass term proportional to µ2

r,so we may not acquire the negative mass squared termthat is necessary for the electroweak symmetry breaking,which gives an upper bound for the value of µr.

Figure 1 shows f versus f for various values of theheavy top mass scale, M . For a given f , f becomeslarger as M increases. The heavy top loop through Mcontributes positively to the Higgs mass while the heavygauge boson loop through f contributes negatively to theHiggs mass. The two contributions cancel out in orderto retain v = 246 GeV. There is also a contribution fromthe tree-level mass term µ2

r, but in most cases, it makeslittle difference to the relation as long as µr is much

Fig. 2. (Color online) Plot of the ρ-parameter versus f fordifferent values of M .

smaller than the Higgs mass scale. This insensitivitycan be figured out with a simple evaluation. First, fromthe electroweak symmetry breaking condition, the mass-squared contribution from the soft symmetry-breakingterm µ2

rf2f should be smaller than that from the fermion

loop. This can be written approximately as

µ2r

f

2f<

38π2

(M2 + y2f2). (58)

In the above inequality, we ignore the gauge boson loopcontributions because they are small compared to thefermion loop contributions. In general, f is larger thanf by about 5 times or more and 3

8π2 is very small, sowe can see that µr should be very much smaller than f .To get the f that reproduces the electroweak symmetry-breaking scale v = 246 GeV, we should solve the equation

3g4

64π2ff2+µ2

r f+2λv2f− 34π2

f(M2+f2)+3g4

64π2= 0 (59)

for given f,M and µr. λ in the above equation is thecoefficient of the quartic term and is less than 1 in gen-eral. Note that we derive the above equation with somedegree of approximation. For example, we ignore the log-arithmic terms, but the crude behavior will be similar.In this equation, the coefficients of f2 and f are muchsmaller than constant term, so the solution f is almostinsensitive to the value of µr [17].

Plots in Fig. 2 illustrate the behavior of one-loop ρ-parameter for various values of M . At M = 0, wherethere is no mixing between the top and the heavy topquarks, ∆ρ increases monotonically with f . For nonzeroM where the mixing is turned on, the mass of the heavytop quark becomes large as M increases for a given f , andthe fermionic loop contributions tend to become large,too. The effects of mixing angles on the fermionic loopsbecome significant as either f or M increases while thecondition of electroweak symmetry breaking is retained.In other words, because the mixing angles are determined


Fig. 3. (Color online) Scatter plots (a) for the mass parameter M and (b) for the mass parameter µr with the horizontalaxis being the scale parameter f .

Fig. 4. (Color online) Scatter plots (a) for the heavy top mass, (b) for the heavy Z boson mass, and (c) the heavy W -bosonmass with the horizontal axis being the scale parameter f .

by f and M , the one-loop corrected ρ parameter be-gins to waver as f increases even with the fixed scalarmass parameters. Because of this, fine tuning in theρ-parameter is inevitable for large f , as will be shownlater.

To extract meaningful information on the model pa-rameters from the ρ-parameter, we scan the parameterspace generally, i.e.,

500 GeV ≤ f ≤ 2500 GeV, 0 ≤ M, µr, µ ≤ f.

(60)

We take a rather large value of f , 2.5 TeV, as an upperlimit for completeness of scanning. As a result of theρ-parameter calculation, we can obtain the allowed re-gions of parameter space. As an example, Fig. 3 showsthe allowed regions of parameter space (a) for f versusM and (b) for f versus µr. Note that the allowed pa-rameter space for f is divided into two regions: less than670 GeV and larger than 1100 GeV roughly. This can befigured out as follows: The loop corrections tend to belarger as f increases because the masses of the particlesinvolved in the one-loop correction increase, in general,as f increases. However, at the same time, the mixingangles of the top-heavy top quarks also vary. Because themixing angles depend on not only f but also M , thesetwo effects compete during the increase of f . Because of

this interplay of top mixing angles and masses, we havetwo distinct allowed parameter spaces. For small f , so-lution points prefer very small values of M . This meansthere is no large mixing between the top and the heavytop quarks. In general, ΠWW (0) is large for small f anddecreases as f increases. Thus, for fitting the observedW-boson mass in the small f region, which is directly re-lated to the ρ-parameter, we restrict the ∆r to a rathersmall range. Because the ∆r is mostly determined byΠZZ(M2

Z), it should also be small. For doing that, weshould take a small value of M , which makes the massesand mixing angles of heavy top quark small. We findthat in the small f region, M should be smaller thanabout 22 GeV. The soft symmetry-breaking parameterµr is restricted to the values less than around 60 GeV.This bound arises mainly from the electroweak symmetrybreaking condition and is generically independent of theρ-parameter. Another free parameters µ is not restrictedby the one-loop corrected ρ-parameter. The reason isthat µ only contributes to the masses of h1 and h2, andtheir contributions are effectively cancelled among therelevant loop diagrams. This is pointed out in AppendixC.

This region of parameter space can provide constraintson the masses of the many particles that appear in thismodel. First, let us consider the masses of the heavy top


Fig. 5. (Color online) (a) Scatter plot for the mass of h1 with the horizontal axis being the scale parameter f . It is similar

to that of h2. (b) Scatter plot for the mass of φ0 with the horizontal axis being the scale parameter f . (c) Scatter plot for themass of φ± with the horizontal axis being the scale parameter f .

Fig. 6. (Color online) Scatter plot for the SM Higgs masswith the horizontal axis being the scale parameter f .

and the heavy gauge bosons. As shown in Fig. 4, theirmasses generically increase as f increases. The mass ofthe heavy top quark is uniquely determined when f, fand M are fixed, as is the top Yukawa coupling. Ba-sically, f is determined by the electroweak symmetrybreaking condition, but M and µr dependence of theheavy top and the heavy gauge bosons provoke ambigui-ties in their values. For the small f region, because M isalso very small, the MT is almost determined by f alone.This appears as straight line in Figure 4(a). For the largef region, M becomes spread due to the top mixing an-gles. The plots of the heavy Z and W boson massesversus f are quite similar to that of the heavy top massversus f . In the case of the heavy W boson, the strongestconstraint come from KL − KS mixing. The strongestbound ever known is mWH

> 1.6 TeV, with the assump-tion of gL = gR [18]. This can exclude some region fromFig. 4(c). In this case, small f region can be completelyexcluded, but the analysis of Ref. [18] did not include thehigher order QCD corrections and used vacuum insertionto obtain the matrix element. Thus, we will not considerthat bound seriously here. A detailed study includingQCD corrections and others is being done by the authorsof Ref. [10]. Once the lower bound for f is confirmed,

we can give the lower bound for f as 1.1 TeV from ourcalculation of the ρ-parameter and for the many parti-cles that appear in the model. Another constraint on themWH

from CDF and D0 is a lower bound of 650 ∼ 786GeV [19,20]. Our results remain safe based on these ex-perimental bounds. The heavy Z boson has also beenstudied in detail by many experimentalists. The currentexperimental bound is about 500 ∼ 800 GeV from preci-sion measurements [14] and ∼ 630 GeV from CDF [14].In this case, also safe is the mass of the heavy Z boson.

With the parameters allowed by the ρ-parameter, themasses of new scalar bosons h1,2, φ

0 and φ± can be con-strained. h1,2 have almost degenerate masses and dependon both µr and µ, unlike the φ0,±, which depend only onµr. Their masses are substantially constrained accordingto the value of f . Unfortunately, we cannot give a lowerbound on the mass of φ0. In fact, its mass, though itis quite small, arises from radiative corrections. For φ±,the loop effects are rather large, so they are heavier thanφ0 as shown in Fig. 5.

The distribution of the SM Higgs mass as a functionof f is shown in Fig. 6. As for the lower bound of the SMHiggs mass we adopt the LEP bound for the Higgs mass,114.4 GeV [21], because its structure is the same as thatof the SM. Its upper bound is approximately given as167 GeV.

We summarize the results of our analysis as follows:With the observed ρ-parameter, the allowed parameterspace is divided into two separate regions: f smaller thanabout 670 GeV and f larger than about 1.1 TeV. We cangive the mass bounds of the particles in the LRTH modelfor either region. However, the heavy gauge bosons re-main safe from the experimental constraints. Unlike theother particles, we cannot set a lower bound for the neu-tral φ0 scalar. The loop corrections play an importantrole in the charged φ± scalars, yielding a mass differ-ence between the charged and the neutral scalars. Fur-ther analysis is required to reduce the allowed region.If the small f region is excluded, for example by Ref.[10], we can provide exact lower bounds for the massesof T,ZH ,WH , h1,2, and φ± but even in that case, wecannot do so for φ0 and the SM Higgs boson.


V. CONCLUSION

The left-right twin Higgs model is a concrete realiza-tion of the twin Higgs mechanism. The model predicts aheavy top quark, heavy gauge bosons and various scalarbosons, along with a light SM Higgs boson, and will,in turn, yield rich phenomenology of the new particlesat the LHC. We have performed an indirect search forthe existence of the particles. The heavy top and newscalars contribute significantly to the isospin violatingthe ρ-parameter. One-loop radiative corrections to theρ-parameter reduce the parameter space of the modeland can set rough bounds for the masses of the heavyparticles. In particular, we demonstrated that the sym-metry breaking parameter f can be either smaller than660 GeV or larger than 1.1 TeV, which is a crucial regionin parameter space. More analysis on other one-loop pro-cesses, as well as study of collider physics, is mandatoryto further reduce the region of parameter space.

ACKNOWLEDGMENTS

Both D. W. Jung and J. Y. Lee are supported inpart by a National Research Foundation (NRF) ResearchGrant 2012R1A2A1A01006053. J. Y. Lee is also sup-ported in part by the Basic Science Research Programthrough the NRF of Korea funded by the Ministry ofEducation, Science and Technology (2011-0003974).

APPENDIX A: COUPLING CONSTANTS OFTHE LRTH MODEL

We summarize the relevant coupling constants rele-vant to our calculation. The gauge-fermion interactionis given by

L = iψ1γµ(gV + gAγ5)ψ2Xµ

= iψ1γµ(cLPL + cRPR)ψ2Xµ, (A1)

where PL,R = 12 (1 ∓ γ5) are the projection operators.

We make a list of the gauge coupling constants of thefermions in Table 1.

The other gauge-scalar interactions are also taken intoaccount. We choose the unitary gauge in which all gauge-scalar mixing terms vanish. The various gauge couplingconstants of the scalar fields are given in Table 2, 3,and 4.

APPENDIX B: ONE-LOOP INTEGRALS

We list scalar integrals relevant for one-loop Feynmandiagrams. The one-loop scalar integrals are decomposedin terms of Passarino-Veltman functions [23], which aredefined in d = 4 − 2ε dimensions:

Q4−d

∫ddk

(2π)d

1k2 − m2 + iε

≡ i

16π2A0(m2), (B1)

Q4−d

∫ddk

(2π)d

1(k2 − m2

1 + iε)((k − p)2 − m22 + iε)

≡ i

16π2B0(p2,m2

1,m22), (B2)

Q4−d

∫ddk

(2π)d

kµ

(k2 − m21 + iε)((k − p)2 − m2

2 + iε)≡ i

16π2pµB1(p2,m2

1,m22), (B3)

Q4−d

∫ddk

(2π)d

kµkν

(k2 − m21 + iε)((k − p)2 − m2

2 + iε)≡ i

16π2[gµνB22(p2,m2

1,m22) + pµpνB11(p2,m2

1,m22)], (B4)

where Q is the renormalization scale and 1/ε =(4π)εΓ(1 + ε)/ε. We also define the following integrals:

I1(a) ≡∫ 1

0

dx ln[1 − ax(1 − x)], (B5)

I3(a) ≡∫ 1

0

dx x(1 − x) ln[1 − ax(1 − x)], (B6)

I4(a, b) ≡∫ 1

0

dx ln[1 − x + ax − bx(1 − x)], (B7)

I5(a, b) ≡∫ 1

0

dx x ln[1 − x + ax − bx(1 − x)],

(B8)

I6(a, b) ≡∫ 1

0

dx (1 − x) ln[1 − x + ax − bx(1 − x)],

(B9)

I7(a, b) ≡∫ 1

0

dx x(1 − x) ln[1 − x + ax − bx(1 − x)].

(B10)


Table 1. Relevant coupling constants Xψψ. The mixing angles CL = cos αL, CR = cos αR, etc. are given in Eq. (19).

Xψψ

Wtb cL = eCL/(√

2sw) cR = 0

WTb cL = eSL/(√

2sw) cR = 0

Ztt gV = e( 14C2

L − 23s2

w)/(cwsw) gA = − 14eC2

L/(cwsw)

Zbb gV = e(− 12

+ 23s2

w)/(2cwsw) gA = e/(4cwsw)

ZTT gV = e( 14S2

L − 23s2

w)/(cwsw) gA = − 14eS2

L/(cwsw)

ZT t cL = eCLSL/(2cwsw) cR = ef2x2swCRSR/(2f2c3w)

Aff gV = eQf gA = 0

Fig. 7. One-loop corrections to the self-energy Πγγ .

Fig. 8. One-loop corrections to the self-energy ΠγZ .

APPENDIX C: GAUGE BOSONSELF-ENERGIES IN THE LRTH MODEL

We calculate the four gauge boson self-energies,Πγγ′

(0), ΠγZ(M2Z), ΠWW (0) and ΠZZ(M2

Z). In gen-eral, the gauge independence in the bosonic sector canbe retained by using the pinch technique [24, 25] or byusing the background field method [26]. In our calcu-

lations, three one-loop diagrams are involved with aninternal gauge boson propagator. In these diagrams, wetake only gauge-invariant parts that are proportional toln(M2

S)/16π2.

1. Contributions to Πγγ′(0)

The one-loop corrections to the self-energy Πγγ of theLRTH model are shown in Fig. 7. The total contributionto the self-energy is

Πγγ′(0) =

α

4π

[169

lnQ2

m2t

+49

lnQ2

m2b

+169

lnQ2

m2T

+13

lnQ2

m2φ+

+13

lnQ2

m2h1

+143ε

]. (C1)

2. Contributions to ΠγZ(M2Z)

The one-loop corrections to the self-energy ΠγZ(M2Z)

are shown in Fig. 8. These are (i) fermionic loops hav-ing (tt), (T T ) and (bb), (ii) the scalar loops due to SSV

coupling, (φ+φ−), (h+1 h−

1 ), and (iii) φ+ and h+1 scalar

loops due to SSV V quartic couplings. The contribu-tions to ΠγZ(M2

Z) due to the fermion loops through thecouplings in Table 1 are

ΠγZtt (M2

Z) =Ncα

π

23swcw

(14C2

L − 23s2

w

)M2

Z

[13

(ln

Q2

m2t

+1ε

)− 2I3

(M2Z

m2t

)], (C2)

ΠγZTT

(M2Z) =

Ncα

π

23swcw

(14S2

L − 23s2

w

)M2

Z

[13

(ln

Q2

m2T

+1ε

)− 2I3

(M2Z

m2T

)], (C3)

ΠγZ

bb(M2

Z) =Ncα

4π

13swcw

(12− 2

3s2

w

)M2

Z

[13

(ln

Q2

m2b

+1ε

)− 2I3

(M2Z

m2b

)]. (C4)


Table 2. Relevant gauge coupling of the scalar fields, CXSS . p1, p2 and p3 refer to the incoming momenta of the first, secondand third particle, respectively [22].

XSS CXSS XSS CXSS

W+h†1h2 −e(p2 − p3)µ/(

√2sw) Ah†

1h1 −e(p2 − p3)µ

Zh†1h1 −e(c2

w − s2w)(p2 − p3)µ/(2cwsw) Zh†

2h2 e(p2 − p3)µ/(2cwsw)

Zφ−φ+ e(p2 − p3)µsw/cw Aφ−φ+ −e(p2 − p3)µ

Zhφ0 iexp1µ/(6cwsw)

The contributions to ΠγZ(M2Z) from the scalar loops through the couplings in Table 2 are

ΠγZ

h1h1(M2

Z) = − α

4π

c22w

cwsw

[(M2

h1− 1

6M2

Z

)(ln

Q2

M2h1

+1ε

)+

(16M2

Z − 23M2

h1

)I1

( M2Z

M2h1

)+ M2

h1− 1

9M2

Z

], (C5)

ΠγZφ+φ−(M2

Z) =α

2π

sw

cw

[(M2

φ+ − 16M2

Z

)(ln

Q2

M2φ+

+1ε

)+

(16M2

Z − 23M2

φ+

)I1

( M2Z

M2φ+

)+ M2

φ+ − 19M2

Z

]. (C6)

The contributions to ΠγZ(M2Z) from the scalar loops

through the couplings in Table 3 are

ΠγZ

h1h1(M2

Z) =α

4π

c22w

cwsw

[1 + ln

Q2

M2h1

+1ε

]M2

h1,

(C7)

ΠγZφ+φ−(M2

Z) = − α

2π

sw

cw

[1 + ln

Q2

M2φ+

+1ε

]M2

φ+ .

(C8)

The terms proportional to M2h1

and M2h1

ln(Q2/M2h1

) inEqs. (C5) and (C7) cancel between themselves, as dothe terms proportional to M2

φ+ and M2φ+ ln(Q2/M2

φ+) inEqs. (C6) and (C8). For p2 = 0, it can be easily checkedthat the total fermionic and scalar contributions vanishindividually. As expected in the unitary gauge, no mix-ing between the two gauge bosons takes place at one-loopdue to

ΠγZ(0) = 0. (C9)

3. Contributions to ΠWW (0)

The contributions to ΠWW (0) from the fermion loopsthrough the couplings in Table 1 are given as follows,

ΠWWtb (0) =

Ncα

4π

C2L

2s2w

f1(m2t ,m

2b), (C10)

ΠWWTb (0) =

Ncα

4π

S2L

2s2w

f1(m2T ,m2

b), (C11)

where 1/ε terms are omitted, and f1(m21,m

22) is defined

as

f1(m21,m

22) =

12(m2

1 + m22)

+m4

1

m21 − m2

2

lnQ2

m21

− m42

m21 − m2

2

lnQ2

m22

.

(C12)

The contributions to ΠWW (0) from the scalar loopsthrough the couplings in Table 3 are given as,

ΠWWh (0) =

α

16π

1s2

w

[1 + ln

Q2

M2h

+1ε

]M2

h , (C13)

ΠWWh+1

(0) =α

8π

1s2

w

[1 + ln

Q2

M2h1

+1ε

]M2

h1, (C14)

ΠWWh02

(0) =α

16π

1s2

w

[1 + ln

Q2

M2h2

+1ε

]M2

h2, (C15)

ΠWWφ+ (0) = − α

24π

x2

s2w

[1 + ln

Q2

M2φ+

+1ε

]M2

φ+ ,

(C16)

ΠWWφ0 (0) = − α

432π

x2

s2w

[1 + ln

Q2

M2φ0

+1ε

]M2

φ0 .

(C17)

Note that ΠWWφ+ and ΠWW

φ0 are much smaller than theother contributions due to the suppression factor x2.

The contribution to ΠWW (0) from the scalar loopsthrough the couplings in Table 2 has the following form

ΠWWh1h2

(0) = − α

2π

1s2

w

g1(M2h1

,M2h2

), (C18)


Table 3. Relevant gauge coupling constants of the scalarfields, CXXSS [22].

XXSS CXXSS XXSS CXXSS

W+W−hh e2/(2s2w) ZZhh e2/(2c2

ws22)

W+W−φ0φ0 −e2x2/(54s2w) ZZφ0φ0 −e2x2/(54c2

ws2w)

W+W−φ+φ− −e2x2/(6s2w) ZZφ+φ− 2e2s2

w/c2w

W+W−h†1h1 e2/(2s2

w) ZZh†1h1 e2c24

w/(2c2ws2

w)

W+W−h†2h2 e2/(2s2

w) ZZh†2h2 e2/(2c2

ws2w)

AAh†1h1 2e2 AAφ+φ− 2e2

ZAφ+φ− −2e2sw/cw ZAh†1h1 e2c22

w/(cwsw)

ZW+h†1h2 −e2/(

√2cw) ZW+h†

1h2 e2/(√

2sw)

where g1(m21,m

22) is defined as

g1(m21,m

22) =

38(m2

1 + m22)

+1

4(m21 − m2

2)[m4

1 lnQ2

m21

− m42 ln

Q2

m22

].

(C19)


and M2h1

ln(Q2/M2h1

)in Eqs. (C14) and (C18) cancel partially between them-selves and so do the terms proportional to M2

h2and

M2h2

ln(Q2/M2h2

) in Eqs. (C15) and (C18). Although theterms proportional to M2

φ+,0 and M2φ+,0 ln(Q2/M2

φ+,0) in

Eqs. (C16) and (C17) do not cancel out, their coefficientsare significantly small and so are their contributions toΠWW (0).

The contribution to ΠWW (0) from the SM Higgs-Wboson loops has the following form:

ΠWWhW (0) =

α

4π

M2W

s2w

[58− 3

8M2

h

M2W

+34

M2h

M2W − M2

h

lnQ2

M2W

+M2

h

M2W − M2

h

(−1 +M2

h

M2W

) lnQ2

M2h

+ (1 − M2W + M2

h

4M2W

)1ε

].

(C20)

We take only the contribution proportional toln(M2

S)/16π2, which is gauge invariant:

ΠWWhW (0) =

α

4π

M2W

s2w

lnQ2

M2h

[ M2h

M2W − M2

h

(−1+M2

h

M2W

)].

(C21)

4. Contributions to ΠZZ(M2Z)

The one-loop corrections to the self-energy functionΠZZ(p2) are shown in Fig. 10. The complete list offermionic contributions to the self-energy function aregiven below:

ΠZZ(T t)(M

2Z) =

Ncα

4π

1c2w

[C2LS2

L

s2w

+C2

RS2Rf4x4

f4c4w

][(1ε

+ lnQ2

m2t

)(M2Z

6− m2

t + m2T

4

)

− M2ZI7

(m2T

m2t

,M2

Z

m2t

)− m2

T

2I5

(m2T

m2t

,M2

Z

m2t

)− m2

t

2I6

(m2T

m2t

,M2

Z

m2t

)]

+Ncα

8π

CLSLCRSR

c4w

x2f2

f2mtmT

[1ε

+ lnQ2

m2t

− I4(m2

T

m2t

,M2

Z

m2t

)],

ΠZZ(tT )(M

2Z) = ΠZZ

(T t)(M2Z)(mt ↔ mT ), (C22)

ΠZZ(tt)(M

2Z) =

Ncα

π

1c2ws2

w

[(12C2

L − 23s2

w

)2

+49s4

w

][(1ε

+ lnQ2

m2t

)(M2Z

6− m2

t

2

)− M2

ZI3

(M2Z

m2t

)− m2

T

2I1

(M2Z

m2t

)]

− 2Ncα

3π

1c2w

(12C2

L − 23s2

w

)m2

t

[1ε

+ lnQ2

m2t

− I1

(M2Z

m2t

)], (C23)

ΠZZ(T T )(M

2Z) =

Ncα

π

1c2ws2

w

[(12S2

L − 23s2

w

)2

+49s4

w

][(1ε

+ lnQ2

m2T

)(M2Z

6− m2

T

2

)− M2

ZI3

(M2Z

m2T

)− m2

T

2I1

(M2Z

m2T

)]

− 2Ncα

3π

1c2w

(12S2

L − 23s2

w

)m2

T

[1ε

+ lnQ2

m2T

− I1

(M2Z

m2T

)], (C24)

ΠZZ(bb)(M

2Z) =

Ncα

4π

1c2ws2

w

[(− 1 +

23s2

w

)2

+49s4

w

][(1ε

+ lnQ2

m2b

)(M2Z

6− m2

b

2

)− M2

ZI3

(M2Z

m2b

)− m2

b

2I1

(M2Z

m2b

)]

+Ncα

6π

1c2w

(− 1 +

23s2

w

)m2

b

[1ε

+ lnQ2

m2b

− I1

(M2Z

m2b

)]. (C25)


Table 4. Relevant gauge coupling constants of the scalarfields, CXXS [22].

X1X2S CX1X2S X1X2S CX1X2S

W+W−h eMW /sw ZZh eMW /(c2wsw)

ZZHh e2fx/(√

2c2wc2w)

Fig. 9. One-loop corrections to the self-energy ΠWW .

The contributions to ΠZZ(M2Z) from the scalar loops

through the couplings in Table 3 have the following form,

ΠZZ(h) (M

2Z) =

α

16π

1c2ws2

w

M2h

[1 + ln

Q2

M2h

+1ε

],

(C26)

ΠZZ(h1)

(M2Z) =

α

8π

c24w

c2ws2

w

M2h1

[1 + ln

Q2

M2h1

+1ε

],

(C27)

ΠZZ(h2)

(M2Z) =

α

8π

1c2ws2

w

M2h2

[1 + ln

Q2

M2h2

+1ε

],

(C28)

ΠZZ(φ+)(M

2Z) =

α

2π

s2w

c2w

M2φ+

[1 + ln

Q2

M2φ+

+1ε

],

(C29)

ΠZZ(φ0)(M

2Z) = − α

8π

x2

54c2ws2

w

M2φ0

[1 + ln

Q2

M2φ0

+1ε

].

(C30)

The contributions to ΠZZ(M2Z) from the scalar loops

through the couplings in Table 2 have the followingforms:

ΠZZ(h†

1h1)(M2

Z) = − α

8π

c24w

c2ws2

w

[(M2

h1− 1

6M2

Z

)(ln

Q2

M2h1

+1ε

)

+(1

6M2

Z − 23M2

h1

)I1

( M2Z

M2h1

)+ M2

h1− 1

9M2

Z

],

(C31)

Fig. 10. One-loop corrections to the self-energy functionΠZZ .

ΠZZ(h†

2h2)(M2

Z) = − α

8π

1c2ws2

w

[(M2

h2− 1

6M2

Z

)(ln

Q2

M2h2

+1ε

)

+(1

6M2

Z − 23M2

h2

)I1

( M2Z

M2h2

)+ M2

h2− 1

9M2

Z

],

(C32)

ΠZZ(φ+φ−)(M

2Z) = − α

2π

s2w

c2w

[(M2

φ+ − 16M2

Z

)(ln

Q2

M2φ+

+1ε

)

+(1

6M2

Z − 23M2

φ+

)I1

( M2Z

M2φ+

)+ M2

φ+ − 19M2

Z

],

(C33)

ΠZZ(hφ0)(M

2Z) = 0. (C34)


and M2h1

ln(Q2/M2h1

)in Eqs. (C27) and (C31) cancel between themselves, asdo the terms proportional to M2

h2and M2

h2ln(Q2/M2

h2)

in Eqs. (C28) and (C32). The terms proportional toM2

φ+ and M2φ+ ln(Q2/M2

φ+) in Eqs. (C29) and (C33) alsocancel between themselves. There are contributions ofscalar-gauge boson loops to ΠZZ(M2

Z). We take onlythe contribution proportional to ln(M2

S)/16π2, which isgauge invariant:

ΠZZ(Zh)(M

2Z) =

α

8π

M2W

c4ws2

w

lnQ2

M2h

[1− 3M2

h + 2M2Z

12M2Z

], (C35)

ΠZZ(ZHh)(M

2Z) =

α

16π

f2x2

c4wc22

w

lnQ2

M2h

[1−3M2

h + 3M2ZH

− M2Z

12M2ZH

](C36)

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One-loop radiative corrections to the ρ parameter in the Left Right Twin Higgs model