Eur. Phys. J. C (2009) 63: 305–315DOI 10.1140/epjc/s10052-009-1103-2

Regular Article - Theoretical Physics

New physics contributionto neutral trilinear gauge boson couplings

Sukanta Dutta1,a, Ashok Goyal2,b, Mamta1,c

1SGTB Khalsa College, University of Delhi, Delhi 110007, India2Department of Physics & Astrophysics, University of Delhi, Delhi 110007, India

Received: 13 May 2009 / Revised: 1 June 2009 / Published online: 31 July 2009© Springer-Verlag / Società Italiana di Fisica 2009

Abstract We study the one-loop new physics effects to theCP even triple neutral gauge boson vertices γ �γZ, γ �ZZ,Z�Zγ and Z�ZZ in the context of Little Higgs models. Wecompute the contribution of the additional fermions in Lit-tle Higgs models in the framework of direct product groupswhere [SU(2) × U(1)]2 gauge symmetry is embedded inSU(5) global symmetry and also in the framework of thesimple group where SU(N) × U(1) gauge symmetry breaksdown to SU(2)L × U(1). We calculate the contribution ofthe fermions to these couplings when T parity is invoked.In addition, we re-examine the MSSM contribution at thechosen point of SPS1a′ and compare with the SM and LittleHiggs models.

PACS 12.15.-y · 12.60.cn · 13.10.+q · 13.10.+q · 14.70.-c

1 Introduction

Multiple gauge boson production channels at the LargeHadron Collider (LHC) provide a novel opportunity to probethe trilinear and quartic gauge boson couplings [1]. Theproposed International Linear Collider (ILC) will be muchmore sensitive to these couplings due to its clean environ-ment and fixed center of mass energy [2]. The availabil-ity of a high luminosity in both these colliders gives us anunique facility to understand the non-Abelian gauge struc-ture of the Standard Model (SM) and confront the presenceof the new physics above the weak scale. The charged WWγ

and WWZ couplings have been extensively studied and the-oretical predictions in the context of SM and the minimalsupersymmetric standard model (MSSM) have been made

a e-mail: Sukanta.Dutta@fnal.govb e-mail: agoyal@iucaa.ernet.inc e-mail: mdahiya@physics.du.ac.in

[3–6]. The neutral gauge boson couplings Zγγ , ZZγ andZZZ, which can be studied in Zγ and ZZ pair productionin e+e− and in hadron colliders through e+e− → Zγ,ZZ

and qq̄ → Zγ,ZZ respectively, have been analyzed withinthe SM and MSSM [7, 8]. Recently Armillis et al. [9] per-formed a detailed study of the trilinear gauge boson interac-tions with additional anomalous U(1)’s, which arise in theconstruction of various string motivated and large extra di-mension models. A model independent analysis of the neu-tral gauge boson couplings for hadron colliders exists in theliterature [10–12] and has been also recently studied withthe Tevatron Data [13, 14]. Recent LEP studies on the triplegauge boson couplings have been made in [15, 16].

The forthcoming experiments at the LHC and the pro-posed ILC offer the exciting prospect of probing physics be-yond SM. In particular if supersymmetry exists in nature,there will be a real possibility of discovering supersymmet-ric partners. As is well known supersymmetry (SUSY) pro-vides an elegant solution to the hierarchy problem in the SMby canceling the quadratic divergence arising in the Higgsmass due to the contributions of SM particles by the contri-butions from their superpartners.

Recently there has been a proposal to consider Higgsfields as pseudo Nambu–Goldstone bosons of a global sym-metry [17–24] which is spontaneously broken at some highscale. The Higgs boson acquires mass through electroweaksymmetry breaking triggered by radiative corrections lead-ing to a Coleman–Weinberg type of potential. Since theHiggs boson is protected by approximate global symmetry,it remains light and the quadratic contributions to its massare canceled by the contributions of heavy gauge bosons andheavy fermionic states that are introduced in the model. Ifthis Little Higgs mechanism is realized in nature, it will beof great importance to verify it at the LHC. The realizationof Little Higgs mechanism discussed in the literature essen-tially fall into two classes [23]. The majority of implemen-tations rely on the product group for the structure of gauge

306 Eur. Phys. J. C (2009) 63: 305–315

symmetry, its breaking pattern and the treatment of the newheavy fermionic sector required to cancel the quadratic di-vergence contribution coming from the top quark. The sec-ond implementation has a simple group structure and has anadditional model parameter. The Littlest Higgs model (LH)[18] is a minimal model of the product group class whichaccomplishes this task to one-loop order within a minimalmatter content. The SU(3) simple group model [23, 24] is arepresentative model of the second class. The SU(3) gaugesymmetry in this model forces the introduction of heavypartners with each SU(2)L fermion doublets of the SM. Thismodel has an extra parameter. Both these models however,suffer from severe constraints [25–29] from precision elec-troweak measurements which can be satisfied only by tuningthe model parameters once again and thus introducing whatis called a Little Hierarchy problem.

Motivated by these considerations, an implementation ofa discrete symmetry called T-parity is proposed. T-parity ex-plicitly forbids any tree-level contribution from the heavymass states to observables involving only the SM particles.It also forbids the interactions that impart a VEV to tripletHiggs bosons, thereby generating the corrections to pre-cision electroweak observables only at the one-loop level.In this Little Higgs model with T-parity (LHT) [30], thereare heavy T-odd partners of the SM gauge bosons and SMfermions called mirror fermions. In the top quark sector,the model incorporates two heavy T-even and T-odd topquarks in addition to the T-even SM top quark, which arerequired for canceling the quadratically divergent contribu-tion of the SM top quark to the Higgs mass. The LHT hasa rich phenomenology and the LHC has great potential tounravel it by directly observing the T-partners of the SMparticles as well as by studying indirect phenomenologicalconsequences [31–39].

In this paper we study the CP conserving trilinear neutralgauge boson couplings in Little Higgs models and MSSMas discussed above. In Sect. 2, we evaluate the one-loopfermion contribution to these three point functions when oneof the gauge bosons is off-shell and in Sect. 3 we analyze andinterpret the numerical results.

2 Neutral gauge-boson couplings

Bose–Einstein statistics render the three neutral gauge bo-son couplings γ γZ, γZZ and ZZZ to vanish when all thethree vector bosons are on shell. The most general CP con-serving coupling of one off-shell boson V ≡ Z/γ to a pairof on-shell Zγ and ZZ gauge bosons (all incoming) can bewritten as (see [8])

Γμαβ

VZγ (Q,p1,p2)

= i

[HV

3 εμβαηp2η + HV4

M2Z

{εμβρηp2ρQηQ

α}]

(1a)

Γμαβ

VZZ(Q,p1,p2) = i[

F V5 εμαβσ (p1 − p2)η

](1b)

where the form factors HVi and F V

5 are related to those ofreference [40] by

Hγ

i = Q2

m2Z

hγ

i , HZi = m2

Z − Q2

m2Z

hZi and

F V5 = −Q2 − m2

V

m2Z

f V5

(1c)

Here Γμαβ

VV1V2(Q,p1,p2) represents the coupling of off-shell

neutral gauge boson V μ carrying momentum Q with thebosons V α

1 and Vβ

2 carrying momenta p1 and p2 respec-tively.

In the SM these couplings vanish at the tree level.These couplings can however, be generated at the loop level(Fig. 5). On account of the totally antisymmetric nature ofεμαβσ , these couplings can never be generated by scalarsand vector bosons running in the loop. Thus fermions run-ning in the loop with one axial and two vector couplings orall the three axial-couplings at the vertices can generate suchcouplings. Further at the one-loop level, the couplings Hγ

4and HZ

4 are not generated i.e.

Hγ

4 = HZ4 = 0 (2)

Thus the only couplings likely to appear at one loop areF γ,Z

5 and Hγ,Z

3 . These couplings can in general be complexquantities. However, they pick up an imaginary contributiononly when Q2 crosses the threshold for fermion pair produc-tion (i.e. Q2 > 4m2

f ) for time-like Q2 or when M2Z exceeds

this threshold (i.e. MZ > 2mf ) for space-like Q2.In order to evaluate these couplings, we write the inter-

actions of the vector boson V ≡ γ,Z with fermions in thestandard notation,

Lint = f̄iγμ

[gV

LijPL + gV

RijPR

]fj Vμ (3)

For V ≡ γ , we have gLij= gRij

= eqiδij , qi being thecharge of fermion fi . For V = Z, the couplings g

Land g

Rin

various models are listed in Table 4 of the Appendix. In theabsence of any CP violating interactions, all these couplingsare real and gL,Rij = gL,Rji because of hermiticity.

Using the notation of Passarino–Veltman (PV) functions,the contribution of the fermionic triangle graphs to the trilin-ear vector boson couplings F γ,Z

5 and Hγ,Z

3 can be expressedin terms of scalar PV functions as given in the Appendix.

2.1 SM contribution

The contribution to the trilinear neutral gauge couplings inthe SM arise from the three families of quarks and leptons.

Eur. Phys. J. C (2009) 63: 305–315 307

The anomaly cancellation ensures that all the couplings goto zero for Q2 much larger than the fermion pair produc-tion threshold. It is obvious that of all the thresholds (atQ2 = 4M2

f ), the largest contribution comes from the heavi-

est fermion loop. The couplings F γ

5 and Hγ,Z

3 get contribu-tions only from the charged fermions, whereas F Z

5 receivescontributions from the neutrinos as well. Note that in SM,the same fermion runs in all the three sides of triangle loopas there is no mixed coupling of neutral boson with differentfermions.

2.2 MSSM

The MSSM contribution to the trilinear neutral gauge cou-plings has been calculated in references [7, 8]. Charginoscontribute to all the four anomalous couplings whereasthe neutralinos contribute only to the F Z

5 . We re-calculatethe MSSM contribution in the light of the reference pointSPS1a′, which is defined at a characteristic scale of 1 TeVwith its origin in minimal supergravity (mSUGRA) [43].The root GUT scale mSUGRA parameters in this referencepoint SPS1a′ are the gaugino mass M1/2 = 250 GeV, theuniversal scalar mass M0 = 70 GeV, the trilinear couplingA0 = −300 GeV, tanβ(M̃) = 10 and sign(μ) = +1. Extrap-olating these parameters to M̃ = 1 TeV generates the MSSMLagrangian parameters. The relevant evolved MSSM para-meters for our calculations are the Higgs mixing parameterμ = 396 GeV and M2 = 193.2 GeV.

2.3 LH contribution

In the Little Higgs models, the Higgs bosons are realizedas pseudo-Goldstone bosons. The generic structure of Lit-tle Higgs models is a global symmetry broken at a high(TeV) scale f . At this scale there are new gauge bosons,scalars and fermions responsible for the cancellation of thequadratic divergent one-loop contributions to the Higgs bo-son mass from the SM gauge bosons, Higgs self interac-tions and from the top quark respectively. The Littlest Higgsmodel accomplishes this task with the minimal matter con-tent. In this model [SU(2) × U(1)]2 gauge symmetry is em-bedded in an SU(5) global symmetry. The gauge symmetryis broken down to the SM SU(2)×U(1) gauge symmetry bya single vacuum condensate at f ≈ 1 TeV. The new fermi-onic degrees of freedom in the Littlest Higgs model are inthe heavy quark sector and consist of a pair of vector-likeSU(2)-singlet quarks that couple to the top sector. The resul-tant top sector consists of a top quark t and its heavy partnerT whose masses and couplings are given in terms of modeldependent parameters by

mt = λ1λ2√λ2

1 + λ22

v (4)

MT =√

λ21 + λ2

2f = 1√XL(1 − XL)

mt

vf (5)

where XL = λ21/(λ

21 + λ2

2), λ1 and λ2 being the couplingsthat appear in the heavy quark sector of the interaction La-grangian. The interactions of the left and right handed fermi-ons with the Z boson in this model can be found in [41] andcan also be realized from the LHT couplings given in Table 4by retaining only the leading terms in v/f because of thelarge scale factor requirement from precision electroweakdata.

The second class of Little Higgs models feature a sim-ple group that contains an SU(N) × U(1) gauge symme-try that is broken down to SU(2)L × U(1), giving rise to aset of TeV scale gauge bosons. The two gauge couplingsof SU(N) × U(1) are fixed in terms of two SM gaugecouplings, leaving no free parameters in the gauge sector.Furthermore, due to enlarged SU(N) gauge symmetry, allfermionic SM representations are extended to transform asfundamental or conjugate representations of SU(N). Thisgives rise to additional heavy fermions in all the three quarkand lepton sectors. The simplest realization of this simplegroup class is the SU(3) simple gauge model [23, 24] withanomaly-free embedding of extra fermions. The expansionof the SU(2)L gauge group to SU(3) requires the introduc-tion of heavy fermions associated with each SU(2)L doubletof the SM. The first two generations of quarks are enlargedto contain the new TeV scale D and S quarks of charge −1/3that are 3̄ representations of SU(3). The quarks of the thirdgeneration and three generation of leptons are put in the 3representation of SU(3). The electric charge of the heavythird generation quark T is +2/3 and all the heavy leptonsNi of three generations are electrically neutral. The massesof these heavy fermions are given in terms the parameters ofthe model, namely,

MT =√

λ21c

2β + λ2

2s2βf = √

2t2β + X2

λ

(1 + t2β)Xλ

mt

vf

MD,S = sβλD,Sf (6)

MNi= sβλNi

f

where tβ ≡ tanβ = f2/f1 is an additional parameter in thesimple SU(3) model and XL = λ2

1/(λ21 +λ2

2). In SU(3) sim-

ple group, f is defined as f ≡√

f 21 + f 2

2 . In these expres-sions the effect of light quark masses is neglected and theneutrinos are taken to be massless. Constraints from elec-troweak precision measurements require the breaking scalef to be greater than 5 TeV in the Littlest Higgs Model. Thisconstraint can however be brought down to about 2–3 TeV(see for example [41]). In the anomaly-free SU(3) simplegroup [29], the constraint on the scale is f > 3.9 TeV fortβ = 3. The scale f can only be marginally brought down

308 Eur. Phys. J. C (2009) 63: 305–315

by a slightly different realization [42]. In these two classesof models the T mass has a lower bound given by

MT ≥ 2mt

vf ≈ √

2f for λ1 = λ2 (7)

in the Littlest Higgs model and

MT ≥ 2√

2sβcβ

mt

vf ≈ f sin(2β) for

λ1

λ2= tanβ (8)

in the SU(3) simple group model.We calculate the contributions to the trilinear gauge bo-

son couplings in the LH model. Unlike SM, here the con-tribution to these couplings also comes from the trianglegraph with both the SM top t and heavy top T simultane-ously present in the loop because of the presence of the T̄ Zt

coupling. The couplings relevant for our study can be readout from the Table 4 of the Appendix.

In Table 1, we give the values of real and imaginary partsof all four trilinear neutral gauge couplings as a functionof symmetry breaking scale f for fixed ratio r = λ1

λ2= 1.

The values are given for some typical values of√

Q2 wheretheoretical peaks are expected. It is worthwhile to mentiononce again that the entries corresponding to the large valuesfor the scale f say, ∼2–3 TeV corresponds to both the Lit-tle Higgs Model with and without T-parity. However, sincethe measured value of precision observables forbids a lowervalue of f in the Littlest Higgs model, the results derivedfor the lower values of the scale f correspond only to theLittle Higgs model with T-parity (discussed in the follow-ing section). It is to be noted that up to the leading order in

x = O(v/f ), the couplings in both the models are the sameand for large f the higher order terms in x may be easilyneglected.

We also calculate the contributions to the trilinear gaugeboson couplings in the anomaly-free SU(3) simple groupmodel. A new feature in this model is the contribution frommixed t and T , mixed SM and TeV range quarks of thefirst two generations and the mixed neutrino and TeV massheavy neutrinos (Ni) of all the three generations in the tri-angle loop. However, the pure T quark loop, pure TeV massquark loop of the first two generations and TeV mass heavyneutrinos of three generations do not contribute to F Z

5 inthe model. This is clear from the couplings of Z-boson tovarious new fermions (Table 4).

Table 2 lists the values of the couplings for the anomaly-free SU(3) simple group model. All values correspond totanβ = r = 3, scale f = 3 TeV and mt = 175 GeV. At thesevalues of the parameters, the mass of heavy top is MT =1.8 TeV and masses of all other heavy fermions have beentaken to be Mi = 3 TeV. As expected, the threshold valuesat 2mt has roughly the same magnitude as that in the SM.At higher

√Q2, the effect of new heavy fermions shows up

but the threshold values are an order of magnitude lowerthan that at the 2mt threshold. However, at these

√Q2, the

SM contribution is negligible. In Fig. 1 we have shown thisbehavior of couplings as a function of

√Q2. The values of

various parameters are the same as given in Table 2 but themass of the heavy fermions U and N are taken to be 2 TeVeach in the figure.

Table 1 The values of various triple neutral gauge boson couplings inLH model (written as complex numbers) at some typical

√Q2 (where

peaks are expected) for different values of symmetry breaking scale f .All values correspond to r = λ1/λ2 = 1 and mt = 175 GeV. The values

are written in units of 10−4. Note that the lower values of the scale cor-respond only to the Littlest Higgs model with T-parity (the couplingsin the two models being the same up to O(v/f )) with mT+ = mT

Scale mT

√Q2 Hγ

3 HZ3 F γ

5 F Z5

(in TeV) (in GeV) (10−4) (10−4) (10−4) (10−4)

0.5 711.4 2mt −89.34 − ι0.0130 25.49 + 0ι −27.65 + 0ι −18.06 + 0ι

mt + MT 0.9154 − ι28.02 5.388 + ι7.730 −1.313 − ι9.229 −8.534 − ι11.50

2MT 3.710 − ι14.24 −0.3113 + ι6.901 −0.0809 − ι5.481 1.717 − ι10.66

1.0 1422.8 2mt −92.97 − ι0.0147 28.46 + ι0.0003 −30.40 + 0ι −19.98 + ι0.0002

mt + MT 4.6901 − ι12.51 −0.1816 + ι3.951 1.152 − ι4.267 −0.5294 − ι6.851

2MT 2.976 − ι5.004 −0.842 + ι2.255 0.6727 − ι1.942 1.710 − ι3.926

2.0 2845.5 2mt −93.87 − ι0.0152 29.26 + ι0.0008 −31.22 + 0ι −20.57 + ι0.0002

mt + MT 3.039 − ι4.590 −0.7061 + ι1.501 0.9257 − ι1.573 0.9056 − ι2.891

2MT 1.396 − ι1.587 −0.4402 + ι0.6802 0.3927 − ι0.6076 0.8645 − ι1.2824

3.0 4268.3 2mt −94.04 − ι0.0152 29.41 + 0ι −31.37 + 0ι −20.68 + 0ι

mt + MT 0.8188 − ι0.7906 0.2639 + ι0.3314 0.2422 − ι0.2998 0.5222 − ι0.6467

2MT 1.9801 − ι2.4200 −0.5342 + ι0.7975 0.6259 − ι0.8290 0.7985 − ι1.6052

Eur. Phys. J. C (2009) 63: 305–315 309

Table 2 The values of various couplings (written as complex num-bers) at some typical

√Q2 (where peaks are expected) in the SU(3)

simple model with anomaly-free embedding. All values correspond to

tanβ = r = 3, scale f = 3 TeV and mt = 175 GeV. At these values ofparameters, the mass of the heavy top is MT = 1.8 TeV and the massesof all other heavy fermions have been taken to be Mi = 3 TeV

√Q2 Hγ

3 HZ3 F γ

5 F Z5

(in TeV) (10−4) (10−4) (10−4) (10−4)

2mt −94.17 − ι0.0158 29.53 + 0ι −31.50 + ι0.0149 −22, .42 + ι0.0254

mt + MT 4.533 − ι9.136 −1.487 + ι3.008 1.757 − ι2.802 −0.1062 − ι4.751

2MT 2.582 − ι3.448 −2.417 − ι0.0712 0.5535 − ι0.677 0.9483 + ι0.9624

MU 3.146 − ι4.660 −2.503 + ι2.036 1.146 − ι1.152 1.699 − ι2.449

2MU 1.372 − ι1.455 0.1424 − ι1.523 2.947 − ι0.191 −3.151 + ι2.236

Fig. 1√

Q2-variation in the range 1–5 TeV of the real (left) and imag-inary (right) parts of the couplings in SU(3) simple Little Higgs Modelwith anomaly-free embedding for tanβ = r = 3, f = 3 TeV. With this

choice of parameters the mass of the heavy top MT = 1.8 TeV whilethe masses of all other heavy fermions are fixed at 2 TeV

2.4 LHT contribution

In LHT, as discussed in the Introduction, the T-odd heavy(TeV mass) fermions called mirror fermions couple vecto-rially to Zμ. Further, because of T-parity conservation thereis no coupling between Zμ and T-odd and T-even fermionsi.e. the coupling Zf̄+f− = 0. Thus the mirror fermions donot contribute to the trilinear neutral gauge boson couplings.The T-even partner of the top quark T+, however, has bothaxial and vector couplings with Zμ and hence contributesto the triangle loop. The top quark masses in this model aregiven by

mt = λ1λ2√λ2

1 + λ22

v

{1 + v2

f 2

(−1

3+ 1

2XL(1 − XL)

)}(9a)

MT = mt√XL(1 − XL)

f

v

×{

1 + v2

f 2

(1

3− XL(1 − XL)

)}(9b)

In addition to the contribution from SM fermions in theloop, we now get additional contributions from the heavyT-even partner and the top quark as well as contributionsfrom the triangle loops with mixed contributions from t and

310 Eur. Phys. J. C (2009) 63: 305–315

Table 3 The values of various couplings (written as complex numbers) at some typical√

Q2 (where peaks are expected) for different values ofr = λ1/λ2 in the LHT model. All values correspond to the symmetry breaking scale f = 500 GeV and mt = 175 GeV

Ratio MT+ (in GeV)√

Q2 Hγ

3 (10−4) HZ3 (10−4) F γ

5 (10−4) F Z5 (10−4)

0.5 889.2 2mt −93.40 − ι0.0149 28.85 + 0ι −30.83 + 0ι −20.29 + 0ι

mt + MT+ 4.068 − ι22.963 −0.705 + ι7.341 1.413 − ι7.612 −1.601 − ι10.82

2MT+ 4.595 − ι10.66 −1.499 + ι3.875 1.441 − ι3.666 1.574 − ι6.475

1.0 711.4 2mt −89.34 − ι0.0130 25.49 + 0ι −27.65 + 0ι −18.06 + 0ι

mt + MT+ 0.9154 − ι28.02 5.388 + ι7.730 −1.313 − ι9.229 −8.534 − ι11.50

2MT+ 3.710 − ι14.24 −0.3113 + ι6.901 −0.0809 − ι5.481 1.717 − ι10.66

2.0 889.2 2mt −81.80 − ι0.0094 19.84 − ι0.0054 22.00 + 0ι −14.31 + ι0.0002

mt + MT+ 0.7583 − ι19.59 14.88 + ι3.813 −4.727 − ι7.226 −15.09 − ι8.566

2MT+ 1.116 − ι9.099 1.424 + ι9.069 −2.866 − ι5.549 −5.549 − ι13.21

3.0 1185.6 2mt −78.51 − ι0.0079 17.58 − ι0.0068 −19.40 + 0ι −12.69 + ι0.0002

mt + MT+ 0.4505 − ι13.03 18.70 + ι2.076 −6.278 − ι6.100 −16.71 − ι6.937

2MT+ −0.623 − ι5.49 1.759 + ι9.726 −3.972 − ι5.508 4.953 − ι13.55

4.0 1511.7 2mt −77.05 − ι0.0072 16.60 − ι0.0074 −18.16 + 0ι −11.95 + ι0.0002

mt + MT+ 0.2019 − ι9.182 20.18 + ι1.320 −7.086 − ι5.541 −17.0 − ι6.011

2MT+ −1.712 − ι3.614 1.824 + ι9.939 −4.461 − ι5.508 5.479 − ι13.48

5.0 1849.6 2mt −76.30 − ι − 0.0068 16.10 − ι0.0077 −17.49 + 0ι −11.56 + ι0.0002

mt + MT+ −0.843 − ι6.819 20.78 + ι0.9261 −7.562 − ι5.220 −17.14 − ι5.422

2MT+ −2.416 − ι2.558 1.841 + ι10.01 −4.717 − ι5.511 5.727 − ι13.34

Fig. 2√

Q2-variation of real and imaginary parts of HZ3 and F Z

5 inthe range 0–3 TeV in the Little Higgs model with T-parity for r = 3 andf = 0.5 TeV. With this choice of parameters the mass of the T-even topis MT+ = 1.186 TeV

T+ quarks. The relevant fermion couplings with Z boson aregiven in Table 4 of the Appendix.

The Table 3 lists the values of the couplings in the Lit-tle Higgs Model for different values of the r ratio and atsome typical values of

√Q2. All the values are for symme-

try breaking scale f = 500 GeV and are given in units of10−4. This table also gives the value of heavy T-even topmass for different ratios. As expected for the same value off , the mass mT+ is the same for the ratio r and 1/r . It maybe mentioned that for higher ratios, a very interesting be-havior is shown by the couplings HZ

3 and F Z5 . Not only the

imaginary part becomes appreciable at high√

Q2 but also

the threshold values of the couplings at√

Q2 = mt + mT+are higher than those at

√Q2 = 2mt and are comparable to

the SM values. This is clearly brought about in Fig. 2 wherewe have plotted the real and imaginary parts of the couplingsHZ

3 and F Z5 for

√Q2 up to 3 TeV, scale f = 0.5 TeV and

the ratio r = 3.

3 Results and discussion

We calculate the one-loop contribution to the CP-conservingtrilinear neutral gauge boson couplings in SM, MSSM andthe two classes of Little Higgs Models for various para-

meters of the models. The√

Q2 variation of the real andimaginary parts of the couplings in all the four models is

Eur. Phys. J. C (2009) 63: 305–315 311

shown in Figs. 3 and 4. Values at some typical√

Q2 arealso given in Tables 1, 2 and 3 for different values of pa-rameters of the model for the Little Higgs Model with T-parity and for the SU(3) Model with anomaly-free embed-ding. Certain features are common to all these graphs whichwe note here. All couplings vanish asymptotically for large√

Q2 compared to the highest fermion mass in the theory.This is ensured by the anomaly cancellation in the models

considered. The relative importance of the real and imagi-

nary parts of the couplings is strongly energy dependent. As

expected and explained in Sect. 2, below the 2mt threshold,

the imaginary parts of all the couplings are negligible. At

and above this threshold the imaginary parts become com-

parable or even dominant in comparison to the real parts.

This behavior is shared by the couplings in all the models

considered.

Fig. 3√

Q2-variation of the real parts of the couplings in various models for the model parameter values f = 500 GeV, r = 1 for LHT andf = 3 TeV, tβ = 3 and the masses of all heavy fermions, Mi = 2 TeV for the SU(3) model. MSSM parameters are as discussed in the text

312 Eur. Phys. J. C (2009) 63: 305–315

Fig. 4√

Q2-variation of the imaginary parts of the couplings in various models for model parameter values f = 500 GeV, r = 1 for LHT andf = 3 TeV, tβ = 3 and masses of all heavy fermions, Mi = 2 TeV for the SU(3) model. MSSM parameters are as discussed in the text

As discussed in Sect. 1, the triple gauge boson couplings

in the SM and MSSM have already been studied by various

groups [7, 8]. Our results agree with the earlier results. How-

ever we have chosen a parameter space defined by the refer-

ence point SPS1a′ as mentioned in Sect. 1. For this chosen

point in the parameter space, the chargino masses are calcu-

lated to be mχ+1

= 183.7 GeV, mχ+2

= 415.4 GeV and the

neutralino masses are mχ01

= 94.8 GeV, mχ02

= 180.3 GeV,

mχ03

= 401.9 GeV and mχ04

= 411.8 GeV. In the MSSM

there is a peak at√

Q2 = 2mχ+1

, which is very close to the2mt SM peak for the special point chosen here. This results

in the enhancement of the couplings at this point as can be

seen from Figs. 3 and 4. This effect is more pronounced

in the imaginary parts of the couplings. All couplings in

MSSM show a threshold effect at√

Q2 = 2mχ+2

800 GeV

which is more pronounced in the real parts of fγ

5 and hZ3

Eur. Phys. J. C (2009) 63: 305–315 313

whereas in SM and Little Higgs model there is no such ef-fect up to 1 TeV. Besides, in the MSSM new peaks appear atmχ+

1+mχ+

2and 2mχ+

2. As mentioned before the neutralinos

contribute only to F Z5 .

The effect of extra heavy fermions in the LHT Model isto decrease the threshold effects of the SM whereas the par-ticles in MSSM enhance it. The new threshold in the LHTat

√Q2 = mt + MT+ and in the MSSM as mentioned above

are opposite to each other but the magnitudes are compa-rable. As expected, the anomaly-free SU(3) simple modeldoes not show any appreciably different behavior than theSM up to

√Q2 = 1 TeV.

We have studied the trilinear neutral gauge boson cou-plings γ �γZ, Z�Zγ , γ �ZZ and Z�ZZ involving one off-shell vector boson as a function of Q2 in SM, MSSM andLittle Higgs models. We have made theoretical predictionof these couplings for the model parameters which are con-strained by the electroweak precision measurements. Thelarge s-channel contributions in the ZZ and Zγ produc-tion at the LHC due to the anomalous triple gauge bosoncouplings, could be the first indirect manifestation of newphysics. The ZZ cross section will be measured at the startup of LHC with a significance of 4.8σ at the 1 fb−1 in-tegrated luminosity and expected to suffer only a total of12.9% uncertainties which include the PDF and QCD un-certainties [1]. However, a the precise measurement of thetriple gauge boson couplings will only be possible with a10 fb−1 luminosity. Our analysis presented above allows usto confront and discriminate among various models consid-ered here on the basis of these couplings.

The complementary study of the one-loop contributionto the triple charge–neutral gauge boson vertices W+W−γ

and W+W−Z in the context of various Little Higgs Modelswith and without T-parity is in process.

Acknowledgements Authors would like to thank Prof. S. RaiChoudhury and Prof. Debajyoti Choudhury for fruitful discussions.The authors acknowledge the partial support from the Department ofScience and Technology, India under grant SR/S2/HEP-12/2006 andthe infra-structural support from the IUCAA Reference Center, Delhi.

Appendix: Fermion one loop contributionto CP even Couplings

Let us consider the vertex: V1(p1α)V2(p2β)V3(p3μ) whereVi ≡ γ,Z and p1 + p2 + p3 = 0. As for the CP conservingones, to one-loop order, only the fermions in the theory con-tribute. In Fig. 5, we draw a generic direct and a exchangediagram contributing to this process. Denote the fermion–gauge coupling by

L = f̄iγμ

[gV

LijPL + gV

RijPR

]fj Vμ (A.1)

with PL,R = (1 ∓ γ5)/2. Throughout our analysis the mo-menta p1 and p2 denote same kind of bosons, say Vα

1 and

Vβ

2 , and momenta p3 denotes Vμ3 , which is of the second

kind of vector boson unlike the case of ZZZ, where all arevector boson; they are the same. It is useful to define thecombinations in notation where V ≡ V1 ≡ V2.

gV,V31 = (gL

VabgL

VbcgL

V3ca − gR

VabgR

VbcgR

V3ca ),

gV,V32 = mamc(gL

VabgL

VbcgR

V3ca − gR

VabgR

VbcgL

V3ca ),

gV,V33 = mamb(gL

VabgR

VbcgR

V3ca − gR

VabgL

VbcgL

V3ca ),

gV,V34 = mbmc(gL

VabgR

VbcgL

V3ca − gR

VabgL

VbcgR

V3ca ).

(A.2)

Here mi , i = a, b, c, are the masses of the internal fermi-ons fi . The loop contribution with three distinct internalfermions fa , fb , fc having masses ma , mb and mc , respec-tively, corresponding to the direct diagram of Fig. 5(a) isproportional to εαβμη and can be parameterized as1

A Da,b,c(V,V3;p1,p2,p3)

= gV,V31

[p2

3(C11 + C21) − p22(C12 + C22)

− 2p1 · p2(C12 + C23)]

+ (m2

agV,V31 + g

V,V33 − g

V,V32

)(C11 + C0)

− gV,V34 C11. (A.3)

The contribution of exchange diagram with three distinctfermions fa , fb, fc as shown in Fig. 5(b) having massesma , mb and mc, respectively and proportional to the εαβμη

is parameterized in terms of

A Ea,b,c(V,V3;p1,p2,p3)

= −(m2

agV,V31 − g

V,V34 − g

V,V32

)(C11 − C12)

− gV,V33 (C11 + C0 − C12)

− gV,V31

[B023 + B123 − 2C24 + p2

1(C12 + C21)

+ (p2

2 + 2p1 · p2)(C22 − 2C23 + C21)

]. (A.4)

In (A.3) and (A.4), Bμjk and Cμ,Cμν are respectively thetwo-point and three-point Passarino–Veltman functions [44]defined as

B023 = B0(p2

2,m2b,m

2c

); B123 = B1(p2

2,m2b,m

2c

);C0;Cμ;Cμν = C0;Cμ;Cμν

(p2

3,p22,m

2a,m

2c,m

2b

) (A.5)

1We adopt the convention ε0123 = 1. Also we follow the notation ofPV functions as given in [44].

314 Eur. Phys. J. C (2009) 63: 305–315

Fig. 5 Generic one-loop diagrams contributing to CP conserving trilinear neutral gauge boson vertices

Table 4 Relevant couplings of fermions with Z-boson in units ofg/2cw in the Littlest Higgs Model with T-parity (LHT) and SU(3)

simple group model with anomaly-free embedding. Note that T de-

notes the T-even heavy top quark in case of LHT and is the heavy topin case of SU(3) Model. Here xL = 1

1+r−2 with r = λ1/λ2

Vertex LHT Model SU(3) simple group

gL gR gL gR

q̄iZqi 2(T i3 − Qis2

w) −2Qis2w 2(T i

3 − Qis2w) −2Qis2

w

(for i = 1–5)

i.e. all SM quarks except top

l̄iZli 2(T i3 − Qis2

w) −2Qis2w 2(T i

3 − Qis2w) −2Qis2

w

(for i = 1–6)

i.e. all SM leptons

t̄Zt 1 − 43 s2

w − x2L

v2

f 2 − 43 s2

w 1 − 43 s2

w − x2L

v2

f 2 − 43 s2

w

T̄ Zt xLvf

0 12√

2(r2+t2β )

s2β(1 + t2β)(r2 − 1) v

f0

T̄ ZT x2L

v2

f 2 − 43 s2

w − 43 s2

w 0 0

D̄iZdi ×× ×× 1√2tβ

vf

0

(i = 1,2)

N̄iZνi ×× ×× − 1√2tβ

vf

0

(i = 1,2,3)

with

B0;Bμ;Bμν

(p2,m2

i ,m2j

)

= 1

ιπ2

∫d4k

1; kμ; kμν

(k2 + m2i )[(k + p)2 + m2

j ]C0;Cμ;Cμν

(p2

1,p22,m

2i ,m

2j ,m

2k

)

= 1

ιπ2

∫d4k

× 1; kμ; kμν

(k2 + m2i )[(k + p1)2 + m2

j ][(k + p1 + p2)2 + m2k]

(A.6)

1. The loop amplitude for γ �γZ is

κ

Nf∑a=1

Nf∑b=1

Nf∑c=1

Cf

[A D

a,b,c

(γ,Z;0,

√Q2,mZ

)

+ A Ea,b,c

(γ,Z;

√Q2,0,mZ

)], (A.7)

where Nf is the total number of flavors, Cf is the colorfactor of the fermion in the loop and κ is the overall loopfactor. Since electromagnetic interactions at two verticesforbids any flavor mixing, the above summation can bere-written as

Hγ

3 = −(

4παem

(2 cos θW sin θW )(16π2)

)

⊗Nf∑a=1

Cf

[A D

a,a,a

(γ,Z;0,

√Q2,mZ

)

+ A Ea,a,a

(γ,Z;

√Q2,0,mZ

)]. (A.8)

2. The loop amplitude for γ �ZZ has one e.m. vertex whichrenders the mixing among the weak interaction eigen-

Eur. Phys. J. C (2009) 63: 305–315 315

states at the other two weak vertices giving

F γ

5 =(

4παem

(2 cos θW sin θW )(16π2)

)

⊗Nf∑a=1

Nf∑b=1

Cf

[A D

a,b,a

(Z,γ ;mZ,mZ,

√Q2

)

+ A Ea,b,a

(Z,γ ;mZ,mZ,

√Q2

)](A.9)

3. The loop amplitude for Z�γZ follows the same suit asthe previous one with an additional change in the mo-mentum assignment.

HZ3 =

(4παem

(2 cos θW sin θW )2(16π2)

)

⊗Nf∑a=1

Nf∑b=1

Cf

[A D

a,b,a

(Z,γ ;mZ,

√Q2,0

)

+ A Ea,b,a

(Z,γ ;

√Q2,mZ,0

)]. (A.10)

4. The loop amplitude for Z�ZZ allows for weak mixing atall vertices.

F Z5 =

(4παem

(2 sin θW cos θW )3(16π2)

)

⊗Nf∑a=1

Nf∑b=1

Nf∑c=1

Cf

[A D

a,b,c

(Z,Z;mZ,mZ,

√Q2

)

+ A Ea,b,c

(Z,Z;mZ,mZ,

√Q2

)](A.11)

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