Improved versions of the event generators NUNUGPV and WRAP are made .... matrix elements and pt-dependent QED Structure Functions (SF) to account for ...
May 6, 2013 - Center for Particle Physics, Prague, Czech Republic. 7Czech Technical University in Prague, Prague, ... 25Tata Institute of Fundamental Research, Mumbai, India. 26University College Dublin, Dublin, Ireland ... contribution of virtual he
Jan 11, 2017 - A polarized electron beam would provide suitable platform searching of the SM and for diagnosing new physics. Observation of even the tiniest signal which conflicts with the SM expectations would be a convincing evidence for physics be
Mar 24, 2004 - time is considered, fixing the others to zero. Limits from the most sensitive channels are shown in addition to the combined results. The region mH % âs â mZ is excluded by the e+eâ â HZ search for any value of the four couplin
Sep 25, 2013 - I. INTRODUCTION. In the next generation of e+eâ colliders, multiple vector-boson production will provide a ... give important clues about the existence of new particles and/or interactions beyond the. SM. ... The cross section for th
Our results show that although the analysis of. Tevatron Run I data can only provide limits on these quartic couplings which are worse than the existing bounds from. LEPII searches, the Tevatron Run II could yield bounds of the same order of magnitud
Jun 23, 1993 - forbids any contribution to the Ï parameter. We must also require that the phenomenological lagrangians are invariant under local U(1)em symmetry. The lowest order operators that comply with the above requirements and give genuinely q
Jun 23, 1993 - possible deviations of the quartic vector boson couplings from the ...... Angle with the beam pipe and rapidity distributions for Z and e in the ...
lous quartic gauge couplings (QGC) are constrained using production of WWÎ³, .... vertex. The signature is one electron lost in the beam pipe and missing trans-.
(c) Department of Physics, University of California at Davis, Davis CA 95616 .... use the âââ renormalization scheme of Kennedy and Lynn  which is defined ...
Mar 25, 2015 - Probe of anomalous quartic WWZÎ³ couplings in photon-photon ..... photons are scattered at very small angles from the beam pipe, so they have.
Aug 4, 2016 - at a center-of-mass energy of 8 TeV is pre- sented. Events consistent with the topology of associated VH production, where the. Higgs boson decays to a pair of bottom quarks and the vector boson decays lepton- ically, are analyzed. The
Apr 12, 2016 - resulting in a data sample corresponding to a total integrated luminosity of 4.59 Â± 0.08 fbâ1 . The .... event selection together with the top-quark mass and HT requirements are selected. ..... When reconstructing events in the
lous couplings on the cross section for Î½Â¯Î½Î³Î³ production via WW-fusion at LEP2. (âs = 200 ... Note that in practice the anomalous contributions arising from WW-.
Jan 26, 2015 - Analysis of anomalous quartic WWZÎ³ couplings in Î³p collision at ..... real photons are scattered at very small angles from the beam pipe, and ...
Aug 2, 2010 - ... de physique des particules, CEA/Saclay, 91191 Gif-sur-Yvette cedex, .....  ATLFast++ package for ROOT, http://root.cern.ch/root/Atlfast.html.
Apr 6, 2001 - Abstract: All lowest-order amplitudes for e+eâ â 4fÎ³ are calculated including five anomalous quartic gauge-boson couplings that are allowed by electromagnetic gauge invariance and the custodial SU(2)c symmetry. Three of these anoma
We analyze the potential of the e+eâ Linear Colliders, operating in the eÎ³ and Î³Î³ modes, to probe anomalous quartic vectorâboson interactions through the multiple production of W's and Z's. We examine all SU(2)LâU(1)Y chiral operators of ord
Dec 31, 2008 - Academy of Sciences of the Czech Republic, Prague, Czech Republic. 12Universidad San Francisco de Quito ... 56Purdue University Calumet, Hammond, Indiana 46323, USA. 57Iowa State University, Ames, Iowa .... are the s-channel production
Dec 14, 2011 - built from the Higgs field and the SM gauge fields only. We will ...... The reader can find in App. D the relevant interactions as well as their.
Nov 13, 2012 - We show that the bounds on these couplings are at the order of .... ATLAS and CMS collaborations have a program of forward physics with ...
Search for anomalous quartic W W Zγ couplings at the future linear e+ e− collider M. K¨oksal∗ Department of Optical Engineering,
arXiv:1406.2496v3 [hep-ph] 3 Jul 2015
Cumhuriyet University, 58140, Sivas, Turkey A. Senol† Department of Physics, Abant Izzet Baysal University, 14280, Bolu, Turkey
Abstract In this paper, the potentials of two different processes e+ e− → W − W + γ and e+ e− → e+ γ ∗ e− → e+ W − Zνe at the Compact Linear Collider (CLIC) are examined to probe the anomalous quartic √ W W Zγ gauge couplings. For s = 0.5, 1.5 and 3 TeV energies at the CLIC, 95% confidence level limits on the anomalous coupling parameters defining the dimension-six operators are found via the effective Lagrangian approach in a model independent way. The best limits on the anomalous
an and Λ 2 which can be achieved with the integrated luminosity of Lint = 590 √ at the CLIC with s = 3 TeV are [−8.80; 8.73] × 10−8 GeV−2 , [−1.53; 1.51] × 10−7 GeV−2 ,
The Standard Model (SM) of particle physics has been demonstrated to be quite successful until now through very important experimental tests, particularly by the recent discovery of a new particle in the mass region around 125 GeV which is consistent with the SM Higgs boson [1, 2]. However, the SM does not fully answer some of the most fundamental questions such as the origin of mass, the large hierarchy between electroweak and Planck scale, the strong CP problem, and matter/antimatter asymmetry. To clarify these questions, new physics beyond the SM is needed. A simple way to discover new physics beyond SM is to probe anomalous gauge boson self-interactions. In the electroweak sector of SM, gauge boson self-interactions are completely determined by SUL (2) × UY (1) gauge invariance. Hence, the high precision measurements of gauge boson self-interactions are extremely important in the understanding of the gauge structure of the SM. Any deviation from the expected values of these couplings would imply the existence of new physics beyond the SM. Investigation of the new physics through effective Lagrangian method is a well known approach. The origin of this method is based on the assumption that at high energies above the SM, there is a grander theory which reduces to the SM at lower energies. Therefore, SM is supposed to be an effective low energy theory in which heavy fields have been integrated out. Since this fundamental method is independent of the details of the model, it is occasionally called model independent analysis. In this paper, we examine the anomalous quartic W W Zγ gauge boson couplings by analyzing two different processes e+ e− → W − W + γ and e+ e− → e+ γ ∗ e− → e+ W − Zνe at the CLIC. Genuine quartic couplings consisting of effective operators, have different origins than anomalous trilinear gauge boson couplings. Hence, we assume that genuine quartic gauge couplings can be independently analyzed from the effects arosen from any trilinear gauge couplings. In the literature, to examine genuine quartic W W Zγ couplings, there are usually two different dimension-six effective quartic Lagrangians that keep custodial SU(2)c symmetry and local U(1)QED symmetry. The first one, CP-violating effective Lagrangian is given as the following 
iπα (i) an ǫijk Wµα Wν(j) W (k)α F µν 2 4Λ
where α is the electroweak coupling constant, W (i) is the SU(2)c weak isospin triplet, Fµν , 2
which equals to ∂µ Aν − ∂ν Aµ , is the tensor for electromagnetic field strength, an represents the strength of anomalous coupling and Λ represents the energy scale of possible new physics. The anomalous vertex generated from the above effective Lagrangian is given in the Appendix. Additionally we perform the notation of Ref.  in the writing of CP-conserving effective operators. There are fourteen effective photonic operators associated with the anomalous quartic gauge couplings (as shown in Eq. (5) of Ref. ). They are determined by fourteen w,b,m w,m b independent couplings k0,c , k1,2,3 and k1,2 that parameterise the strength of the anomalous
quartic gauge couplings. These effective photonic operators can be described in terms of independent Lorentz structures. Among them, the lowest order effective W W γγ and ZZγγ interactions are expressed by four Lorentz invariant structures
−e2 g 2 Fµν F µν W +α Wα− , 2
−e2 g 2 Fµν F µα (W +ν Wα− + W −ν Wα+ ), 4
−e2 g 2 Fµν F µν Z α Zα , 4cos2 θW
−e2 g 2 Fµν F µα Z ν Zα . 4cos2 θW
In addition, the lowest order effective ZZZγ operators are parameterized as
−e2 g 2 Fµν Z µν Z α Zα , 2cos2 θW
−e2 g 2 Fµν Z µα Z ν Zα . 2cos2 θW
There are only five basic Lorentz structures also related to anomalous quartic W W Zγ vertex as follows: 3
WZ0 = −e2 g 2Fµν Z µν W +α Wα− ,
e2 g 2 Fµν Z µα (W +ν Wα− + W −ν Wα+ ) 2
egz g 2 µν + − F (Wµν Wα− Z α + Wµν Wα+ Z α ) 2
egz g 2 µν + − F (Wµα W −α Zν + Wµα W +α Zν ) 2
egz g 2 µν + − F (Wµα Wν− Z α + Wµα Wν+ Z α ) 2
WZc = −
WZ1 = −
WZ2 = −
WZ3 = −
with g = e/sin θW , gz = e/sin θW cos θW and Vµν = ∂µ Vν − ∂ν Vµ where V = W ± , Z. The vertex functions for the anomalous quartic W W Zγ couplings generated from Eqs. (8)-(12) are given in Appendix. As a result, these fourteen effective operators can be written more simply as the following: kcγ γ k1γ γ k0γ γ γ γ (Z + W0 ) + 2 (Zc + Wc ) + 2 Z0 L= Λ2 0 Λ Λ γ Z Z X kW k23 k k i + 2 Zγc + 02 ZZ0 + c2 ZZc + WZi , 2 Λ Λ Λ Λ i (13) where kjγ = kjw + kjb + kjm
(j = 0, c, 1)
γ k23 = k2w + k2b + k2m + k3w + k3m
sin θW b cos2 θW − sin2 θW cos θW w (k0 + k1w ) − (k0 + k1b ) + ( )(k0m + k1m ), sin θW cos θW 2cos θW sin θW 4
cos θW w sin θW b cos2 θW − sin2 θW (kc + k2w + k3w ) − (kc + k2b ) + ( )(kcm + k2m + k3m ),(17) sin θW cos θW 2cos θW sin θW
cos θW w sin θW b cos2 θW − sin2 θW m k0 − k0 + ( )k0 , sin θW cos θW 2cos θW sin θW
cos θW w sin θW b cos2 θW − sin2 θW m kc − kc + ( )kc , sin θW cos θW 2cos θW sin θW
1 kjW = kjw + kjm 2
(j = 1, 2, 3).
In this work, we are only interested in the kiW (i = 0, c, 1, 2, 3) parameters given in Eqs. (18)-(20) related to the anomalous W W Zγ couplings. These kiW parameters are correlated with couplings defining anomalous W W γγ, ZZγγ and ZZZγ interactions . Hence, we need to separate the anomalous W W Zγ couplings from the other anomalous quartic couplings. This can be achieved by imposing additional restrictions on kij parameters . Thus, we set all kij parameters to zero except k2m and k3m in the anomalous W W Zγ couplings. Additionally, we require k2m = −k3m . Therefore, the effective interactions can be obtained below Lef f = In the literature, the
k2m (W Z − W3Z ). 2Λ2 2
couplings describing the anomalous quartic W W Zγ vertex are
examined by Refs. [4–6]. However, the
couplings obtained with the aid of Eqs.
(18)-(19) provide the current experimental limits related to the anomalous quartic W W Zγ couplings. In this paper, we analyze the limits on the CP-conserving parameters the CP-violating parameter
which are the current experimental limits on the anomalous
quartic W W Zγ gauge couplings, and compare our limits with the phenomenological studies on
k2m . Λ2
Anomalous quartic W W Zγ couplings at linear colliders and their eγ and γγ modes have been examined through the processes e+ e− → W + W − Z, W + W − γ, W + W − (γ) → 4f γ [7– 11], eγ → W + W − e, νe W − Z [3, 12] and γγ → W + W − Z [13, 14]. These couplings appear as 5
W + W − e and νe W − Z final state productions of eγ collision at linear colliders. νe ZW − production is more sensitive to anomalous quartic W W Zγ couplings with respect to eW − W + production . This production isolates the anomalous W W Zγ couplings from W W γγ couplings. These couplings have also been investigated at the Large Hadron Collider (LHC) via the processes pp → W (→ jj)γZ(→ ℓ+ ℓ− )  and pp → W (→ ℓνℓ )γZ(→ ℓ+ ℓ− ) . Although anomalous quartic W W Zγ couplings have been examined in many studies by analyzing either CP-violating or CP-conserving effective Lagrangians in the literature, these couplings have been investigated using two effective Lagrangians only by Ref. . On the other hand, the limits on
parameter of the anomalous quartic W W Zγ couplings
are constrained at the LEP by analysing the process e+ e− → W + W − γ [15–17]. This reaction is sensitive to both the anomalous W W γγ and W W Zγ couplings. The latest results obtained by L3, OPAL and DELPHI collaborations are given by −0.14 GeV−2 < an Λ2
an Λ2 −2
< 0.14 GeV
< 0.13 GeV−2 , −0.16 GeV−2 <
< 0.15 GeV−2 , and −0.18 GeV−2 <
at 95% confidence level (C. L.), respectively. However, the recent most
restrictive experimental limits on
parameters of the anomalous quartic W W Zγ
couplings are determined through the process qq ′ → W (→ ℓν)Z(→ jj)γ by CMS col-
laboration at the LHC . These are −1.2 × 10−5 GeV−2 < −1.8 × 10−5 GeV−2 <
< 1.7 × 10−5 GeV−2 at 95% C. L..
< 1 × 10−5 GeV−2 and
The LHC which is the current most powerful particle collider, is able to carry out proton√ proton collisions at s = 14 TeV. It may generate large massive particles and allow us to reveal new physics effects beyond the SM. However, the analysis of the LHC data is quite difficult due to backgrounds from strong interactions. The linear e− e+ colliders generally provide clean environment with reference to hadron colliders and they can be used to determine new physics effects with high precision measurements. The Compact Linear Collider (CLIC) is one of the most popular linear colliders, planned to realize e− -e+ collisions in three energy stages of 0.5, 1.5, and 3 TeV . The CLIC’s first energy stage will provide an opportunity for the achievement of high precision measurements of various observables of the SM gauge bosons, top quark and Higgs boson. The second energy stage will allow the detection of theories that lie beyond the SM. Moreover, Higgs boson properties such as the Higgs self-coupling and rare Higgs decay modes will be investigated in this stage . √ CLIC’s operation at s = 3 TeV reaches a higher effective center-of-mass energy than the LHC for elementary particle collisions . This enables the determination of new parti6
cles and the testing of various models such as supersymmetry, extra dimensions, and so forth beyond the LHC’s capability. Besides, the linear colliders have eγ and γγ modes to probe the new physics beyond the SM. High energy real photons in the eγ and γγ processes can be produced by converting the original e− or e+ beam into a photon beam through the Compton back-scattering technique [22, 23]. In addition, eγ ∗ , γγ ∗ and γ ∗ γ ∗ collisions coming from quasireal photons at the linear colliders also are examined. eγ ∗ collision is the interaction of an incoming lepton beam and a quasireal γ ∗ photon associated with the other beam particle; γγ ∗ collision is the interaction of a real photon and a quasireal photon; and γ ∗ γ ∗ collision is the interaction between quasireal photons. The Weizsacker-Williams approach, known as the Equivalent Photon Approximation (EPA), can be applied to the photons in these processes [24–28]. In the framework of EPA, the virtuality of the quasireal γ ∗ photons is very low and they are assumed to be almost real. In EPA, these photons carry a small transverse momentum. Hence, they deviate at very small angles from the incoming lepton beam path. Moreover, eγ ∗ and γ ∗ γ ∗ processes are more realistic than eγ and γγ processes since they naturally occur spontaneously from the e− e+ process itself. In the literature, photon-induced reactions through the EPA have been extensively studied at the LEP, Tevatron, and LHC [29–57].
CROSS SECTIONS AND NUMERICAL ANALYSIS
In this work, we obtain limits on the CP-conserving parameters violating parameter
and the CP-
which are the current experimental limits on the anomalous quartic
W W Zγ gauge couplings, and also compare our limits with phenomenological studies on
derived in Refs. [3, 4, 6]. In order to examine our numerical calculations, we have used the W W Zγ vertex in CompHEP . The general form of the total cross sections for two processes e+ e− → W − W + γ and e+ e− → e+ γ ∗ e− → e+ W − Zνe including CP-conserving
anomalous quartic couplings kiW (i = 0, c) can be written as σtot = σSM +
X kW i
σi + 2 int
X kiW kjW i,j
where σSM is the SM cross section, σint is the interference terms between SM and the anomalous contribution, and σano is the pure anomalous contribution. The contributions of the interference terms to total cross section for both processes are negligibly small comparing 7
to pure anomalous terms. But in this study, the small contributions of the interference terms are taken into account in the numerical calculations. Moreover, the general expression of the cross section including CP-violating anomalous quartic coupling is derived by replacing kiW = kjW with an in Eq. (23). But this anomalous coupling (an ) does not interfere with the SM amplitude in all processes . Therefore the total cross section depends only on the quadratic function of anomalous coupling an . The total cross sections of the process kW
e+ e− → W − W + γ are presented in Figs. 1-4 as functions of anomalous Λ02 , kΛc2 , Λ22 and √ an couplings with s = 0.5, 1.5 and 3 TeV. In Figs. 1-4, we consider that only one of the 2 Λ anomalous quartic gauge coupling parameters is non-zero at any given time, while the other couplings are fixed at zero. We can see from Figs. 1-3 that the value of the anomalous cross section including on
is larger than the value of
couplings. Hence, the limits
coupling are expected to be more sensitive according to the limits on
couplings. Similarly, the total cross sections of the process e+ e− → e+ γ ∗ e− → e+ W − Zνe are presented in Figs. 5-8 as functions of anomalous √ s = 0.5, 1.5 and 3 TeV.
The pT distribution of the final state photon in e+ e− → W − W + γ process with the √ W km kW anomalous W W Zγ couplings Λ02 , kΛc2 , Λ22 and Λan2 , together with SM backgrounds at s=0.5, 1.5 and 3 TeV are given in Figs. 9-11, respectively. From these figures, the final state photon in the e+ e− → W − W + γ process is radiated from massless fermion-photon, W W γ and W W Zγ vertices. The massless fermion-photon vertex causes infrared singularities in the cross section. Therefore, the strong peak arises at the low pT region of the photons. Above pT of 20 GeV we see an obvious splitting and enhancement of the signal from SM background. The effects of infrared singularities which diminish the contribution of anomalous couplings to SM cross section become dominant for the high pT region, as shown in Fig. 9-11. It is km
clear from Fig. 9 that the distributions are more sensitive to Λ22 than to Λan2 . On the other √ hand, at s = 1.5 and 3 TeV, it shows exactly the opposite behavior. In addition, the momentum dependence of
for all center of mass energies is bigger than
between four different anomalous couplings is highest at the momentum dependence of √ s = 3 TeV. Consequently, we impose a pT > 20 GeV cut to reduce the SM background without affecting the signal cross sections due to anomalous quartic couplings. In the course of statistical analysis, the limits of anomalous
at 95% C.L. are obtained by using χ2 test since the number of SM background events of the 8
examined processes is greater than 10. The χ2 function is defined as follows
σSM − σN P σSM δstat
where σN P is the total cross section in the existence of anomalous gauge couplings, δstat =
is the statistical error in which N is the number of events. The number of expected events of the process e+ e− → W − W + γ, N is obtained by N = Lint ×σSM ×BR(W → ℓνℓ )×BR(W →
q q¯′ ) where Lint is the integrated luminosity, σSM is the SM cross section and ℓ = e− or µ− .
Similarly, the number of expected events of the process e+ e− → e+ γ ∗ e− → e+ W − Zνe is calculated as N = Lint × σSM × BR(W → ℓνℓ ) × BR(Z → q q¯). In addition, we impose the acceptance cuts on the pseudorapidity |η γ | < 2.5 and the transverse momentum pTγ > 20
GeV for photons in the process e+ e− → W − W + γ. After applying these cuts, the SM √ background cross sections for the process e+ e− → W − W + γ are 1.65 × 10−1 pb at s = 0.5 √ √ TeV, 6.00 × 10−2 pb at s = 1.5 TeV, and 2.63 × 10−2 pb at s = 3 TeV. They are √ √ 3.58 × 10−3 pb at s = 0.5 TeV, 5.92 × 10−2 pb at s = 1.5 TeV, and 1.61 × 10−1 pb at √ s = 3 TeV for the process e+ e− → e+ γ ∗ e− → e+ W − Ze− . The one-dimensional limits on anomalous couplings
at 95% C.L.
sensitivity at various integrated luminosities and center-of-mass energies are given in Tables I-VI. As can be seen in Tables I and II, the limits on
are approximately several
orders of magnitude more restrictive than those obtained from the LHC  while the best limits obtained on
for the process e+ e− → W − W + γ is five orders of magnitude more
restrictive than those obtained from the LEP . In addition, as shown in Table III, we improve sensitivity to
coupling with respect to limits derived by Ref. , in which
the best limits on this coupling in the literature are obtained. An important advantage of the examined e+ e− → e+ γ ∗ e− → e+ W − Zνe process is that it isolates the anomalous W W Zγ couplings, and therefore it enables us to examine W W Zγ couplings independently from W W γγ couplings. In Table IV, the limits on the anomalous couplings
are obtained as [−3.24; 3.24] × 10−7 and [−4.71; 4.70] × 10−7 which can almost improve the sensitivities up to 37 times for
with respect to LHC’s results. We show in
Table V that the best limits on the anomalous coupling
through the process e+ e− →
e+ γ ∗ e− → e+ W − Zνe are calculated as [−1.17; 1.17] × 10−6 GeV−2 which are more stringent than LEP’s results by almost five orders of magnitude. The best limits on 9
process e+ e− → e+ γ ∗ e− → e+ W − Zνe are 10 times than the process e+ e− → W − W + γ which improves the current experimental limits by a factor of 1.1. In addition, we compare our limits with phenomenological studies on the anomalous couplings on
an . Λ2
obtained from eγ ∗ collision are 11 times more restrictive than the best limits obtained
with the integrated luminosity of 200 fb−1 corresponding to W ± Zγ production at the 14 TeV LHC . These limits are almost of the same order with our result obtained through the √ process e+ e− → e+ γ ∗ e− → e+ W − Zνe at the CLIC with Lint = 100 fb−1 and s = 1.5 TeV. However, Ref.  has considered incoming beam polarizations as well as the final state polarizations of the gauge bosons in the cross-section calculations to improve the bounds on an Λ2
coupling. We can see that the limits expected to be obtained for the future √ γγ colliders with Lint = 500 fb−1 and s = 1.5 TeV are 5 times worse than our best limits √ when comparing to the unpolarized case. At the CLIC with s = 3 TeV for Lint = 590 anomalous
fb−1 , we can set more stringent limit by two orders of magnitude comparing to the limits on
in Ref.. k0W kcW Λ2 Λ2
plane for the e+ e− → W − W + γ process in √ Figs. 12-14 for various integrated luminosity at s = 0.5 , 1 and 3 TeV, respectively. We show 95% C.L. contours in the
Similarly, the same contours for the process e+ e− → e+ γ ∗ e− → e+ W − Zνe are depicted in Figs. 15-17. As we can see from Fig. 14, the best limits on anomalous couplings kcW Λ2
are [−1.90; 1.92] × 10−7 GeV−2 and [−3.34; 3.29] × 10−7 GeV−2 , respectively at
TeV for Lint = 590 fb−1 . According to Fig. 17, the attainable limits on
[−3.86; 3.85] × 10−7 GeV−2 and [−5.62; 5.60] × 10−7 GeV−2 , respectively. III.
The CLIC is an proposed collider with energies on the TeV scale and extremely high luminosity. Particularly, operating with its high energy and luminosity is extremely important in order to investigate geniue anomalous W W Zγ quartic gauge couplings that are described by dimension-six effective Lagrangians. Since energy dependences of the anomalous couplings are very high, the anomalous cross sections containing these couplings would have a higher momentum dependence than the SM cross section. We can easily understand that the contribution to the cross section of anomalous quartic couplings rapidly increases when the center-of-mass energy increases. Moreover, the geniue anomalous couplings can obtain 10
higher sensitivity via analyzed reactions in the linear colliders due to very clean experimental conditions and being free from strong interactions with respect to LHC. Thus in this paper, we have examined CP-violating and CP-conserving Lagrangians for the anomalous W W Zγ couplings in the processes e+ e− → W − W + γ and e+ e− → e+ γ ∗ e− → e+ W − Zνe at the CLIC. Appendix: The anomalous vertex functions for W W Zγ
The anomalous vertex for W + (pα1 )W − (pβ2 )Z(k2ν )γ(k1µ ) with the help of effective Lagrangian Eq. (1) is generated as follows
This work partially supported by the Abant Izzet Baysal University Scientific Research Projects under the Project no: 2015.03.02.867.
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