Anomalous Quartic Gauge Boson Couplings at Hadron Colliders

We analyze the potential of the Fermilab Tevatron and CERN Large Hadron Collider (LHC) to study anomalous quartic vector--boson interactions (photon photon Z Z) and (photon photon W+ W-). Working in the framework of SU(2)_L X U(1)_Y chiral Lagrangians, we study the production of photons pairs accompanied by (e+e-), (e nu), and jet pairs to impose bounds on these new couplings, taking into account the unitarity constraints. We compare our findings with the indirect limits coming from precision electroweak measurements as well as with presently available direct searches at LEPII. We show that the Tevatron Run II can provide limits on these quartic limits which are of the same order of magnitude as the existing bounds from LEPII searches. LHC will be able to tighten considerably the direct constraints on these possible new interactions, leading to more stringent limits than the presently available indirect ones.

We analyze the potential of the Fermilab Tevatron and CERN Large Hadron Collider (LHC) to study anomalous quartic vector-boson interactions γγZZ and γγW + W − . Working in the framework of SU (2)L ⊗ U (1)Y chiral Lagrangians, we study the production of photons pairs accompanied by ℓ + ℓ − , ℓ ± ν, and jet pairs to impose bounds on these new couplings, taking into account the unitarity constraints. We compare our findings with the indirect limits coming from precision electroweak measurements as well as with presently available direct searches at LEPII. We show that the Tevatron Run II can provide limits on these quartic limits which are of the same order of magnitude as the existing bounds from LEPII searches. LHC will be able to tighten considerably the direct constraints on these possible new interactions, leading to more stringent limits than the presently available indirect ones.

I. INTRODUCTION AND FORMALISM
Within the framework of the Standard Model (SM), the structure of the trilinear and quartic vector boson couplings is completely determined by the SU (2) L × U (1) Y gauge symmetry. The study of these interactions can either lead to an additional confirmation of the model or give some hint on the existence of new phenomena at a higher scale [1]. Presently, the triple gauge-boson couplings are being probed at the Tevatron [2] and LEP [3] through the production of vector boson pairs, however, we have only started to study directly the quartic gauge-boson couplings [4,5].
It is important to independently measure the trilinear and quartic gauge boson couplings because there are extensions of the SM [6] that leave the trilinear couplings unchanged but do modify the quartic vertices. A simple way to generate, at tree level, new quartic gauge boson interactions is, for instance, by the exchange of a heavy boson between vector boson pairs.
The phenomenological studies of the anomalous vertices γγW + W − and γγZZ have already been carried out for γγ [7,8], eγ [9], and e + e − [10] colliders. Some preliminary estimates of the potential of the Tevatron collider have been also presented in Ref. [11] where only the effect on the total cross section for "neutral" final states γW + W − and γγZ were considered while the most promising charged final state γγW ± was not included. In this paper we analyze the potential of hadron colliders to unravel deviations on the quartic vector boson couplings by examining the most relevant processes which are the production of two photons accompanied by a lepton pair, where the fermions are produced by the decay of either a W ± or a Z 0 in the anomalous contribution, i.e.
as well as the production of photon pairs accompanied by jets for the LHC. We carry out a detailed analysis of these reactions taking into account the full SM background leading to the same final state. We introduce realistic cuts in order to reduce this background and we include the effect of detector efficiencies in the evaluation of the attainable limits. We further consider the energy dependence (form factor) of the anomalous couplings in order to comply with the unitarity bounds. 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 magnitude as the present LEPII limits. Moreover, the LHC will be able to tighten considerably the direct constraints on these possible new interactions, giving rise to limits more stringent than the presently available indirect bounds.
In order to perform a model independent analysis, we use a chiral lagrangian to parametrize the anomalous γγW + W − and γγZZ interactions [12]. Assuming that there is no Higgs boson in the low energy spectrum we employ a nonlinear representation of the spontaneously broken SU (2) L ⊗ U (1) Y gauge symmetry. To construct such lagrangian, it is useful to define the matrix-valued scalar field ξ(x) = exp(2iX a ϕ a (x)/v), where X a are the broken generators and ϕ a are the Nambu-Goldstone bosons of the global symmetry-breaking pattern SU (2) L ⊗U (1) Y → U (1) em . We denote the unbroken generator by Q and our conventions are such that Tr(X a X b ) = 1 2 δ ab and Tr(X a Q) = 0. The action of a transformation G of the gauge group SU (2) L ⊗ U (1) Y on ξ takes the form H = exp(iQu) is defined requiring that ξ ′ contains only the broken generators. In order to write the effective lagrangian for the gauge bosons, it is convenient to introduce the auxiliary quantity where T a and Y are the generators of SU (2) L and U (1) Y respectively. Now we can easily construct fields which have a simple transformation law under SU (2) L ⊗ U (1) Y : with the standard definition T ± = T 1 ± iT 2 . Notice that the fields A, Z, and W ± transform only electromagnetically under SU (2) L ⊗ U (1) Y . Therefore, effective lagrangians must be invariant exclusively under the unbroken U (1) em . Moreover, in the unitary gauge (ξ = 1) we have that A → A, Z → Z, and W ± → W ± . Requiring C and P invariance, the lowest order effective interactions involving photons is In order to avoid the strong low energy constraints coming from the ρ parameter we impose the custodial SU (2) symmetry which leads to β 1 = c 2 W β 2 = β 0 and β 3 = c 2 W β 4 = β c . With this choice L ef f reduces to the parametrization used in Ref. [7]. In the unitary gauge, (9) gives rise to anomalous γγZZ and γγW + W − vertices which are related by the custodial symmetry.

II. PRESENT CONSTRAINTS: PRECISION DATA, LEPII, AND UNITARITY BOUNDS
The couplings defined in the effective lagrangian (9) contribute at the one-loop level to the Z physics [9] via oblique corrections as they modify the W , Z, and photon two-point functions, and consequently they can be constrained by precision electroweak data. We denote the new contribution to the two-point functions as Π V V (0,c) and we take here the opportunity to update the constraints on β 0 and β c derived in Ref. [9].
It is easy to notice from the structure of the lagrangians that the contributions to the W and Z self-energies are constant, i.e. they do not depend on the external momentum. Moreover, due to the SU (2) custodial symmetry they are related by As a consequence the couplings β 0 and β c do not contribute to T = ∆ρ [13]. Equivalently their contribution to sin θ W vanishes. Moreover, the unbroken U (1) em symmetry constrains the photon self-energy contribution to be of the form where for the anomalous interactions (9) Π ′ γγ(0,c) is a constant. This also implies that these anomalous interactions do not modify the running of the electromagnetic coupling. However, both interactions give rise to corrections to ∆r or, equivalently, to the S and U parameters [13].
Following the standard procedure, we evaluated the vector boson two-point functions using dimensional regularization and subsequently kept only the leading non-analytic contributions from the loop diagrams to constrain the new interactions -that is, we maintained only the logarithmic terms, dropping all others. The contributions that are relevant for our analysis are easily obtained by the substitution where Λ is the energy scale which characterizes the appearance of new physics, and µ is the scale in the process, which we take to be M W . After this procedure we obtain that The allowed ranges of S and U depend on the SM parameters. As an illustration of the size of the bounds, we take that for the Higgs boson mass of M H = 300 GeV, the 95%CL limits on S and U are 0.34 ≤ S ≤ 0.02 and −0.13 ≤ U ≤ 0.37 [14]. These bounds can then be translated into the 95% CL limits on β 0 and β c presented in Table  I. The LEP collaborations have directly probed anomalous quartic couplings involving photons. L3 and OPAL have searched for their effects in the reactions e + e − → W + W − γ, Zγγ, and ννγγ, while the ALEPH collaboration has reported results only on the last reaction [4,5]. The combined results for all these searches lead to the following 95% CL direct limits on the quartic vertices [5] Another way to constrain the couplings in (9) is to notice that this effective lagrangian leads to tree-level unitarity violation in 2 → 2 processes at high energies. In order to extract the unitarity bounds on the anomalous interactions we evaluated the partial wave helicity amplitudes (ã j νµ ) for the inelastic scattering γ(λ 1 )γ(λ 2 ) → V (λ 3 )V (λ 4 ), with V = Z and W ± ; see Table II. Unitarity requires that [15] where β V is the velocity of the final state boson in the center-of-mass frame. For the anomalous interactions (9), the most restrictive bounds come from the J = 0 partial wave, which read where β = β 0 or β c and N = 1/4 (4) for β 0 (β c ). For instance, unitarity is violated for γγ invariant masses above 240 GeV for β 0 = 5.6 × 10 −3 GeV −2 (one of the present LEP bounds). These unitarity constraints are of relevance when extracting the bounds on the anomalous couplings at hadron colliders since it is possible to obtain large parton-parton center-of-mass energies, and consequently have a large unitarity violation. The standard procedure to avoid this unphysical behavior of the subprocess cross section and to obtain meaningful limits is to multiply the anomalous couplings by a form factor where M γγ is the invariant mass of the photon pair. Of course using this procedure the limits become dependent on the exponent n and the scale Λ which is not longer factorizable. In our calculations, we conservatively choose n = 5 and Λ = 0.5 TeV for the Tevatron and Λ = 0.5 (2.5) TeV for the LHC. In the case of e + e − colliders the center-of-mass energy is fixed and the introduction of the form factor (19) is basically equivalent to a rescaling of the anomalous couplings β 0,c , therefore we should perform this rescaling when comparing results obtained at hadron and e + e − colliders. For example, for our choice of n and Λ the LEP limits should be weakened by a factor ≃ 1.6. The dynamical effect of the above form factor can be seen in Figure 1 where we present the normalized invariant mass distribution of the γγ pair for the process (1) at the Tevatron Run II and LHC, assuming that only β 0 contributes. As expected, the form factor reduces the number of photon pairs with high invariant mass. Similar behavior is obtained for reaction (2) and for the anomalous β c contribution.

III. SIGNALS AT HADRON COLLIDERS
In this work we studied the reactions (1) and (2) for the Tevatron and LHC, that is, the associated production of a photon pair and a W * or Z * which decay leptonically, as well as the process (3) only for the LHC since the Tevatron center-of-mass energy is too low for this process to be of any significance. Process (1) can be used to study the γγW + W − vertex while process (2) probes the γγZZ interaction and the reaction (3) receives contributions from γγW + W − and γγZZ. We evaluated numerically the helicity amplitudes of all the SM subprocesses leading to the γγℓ ± ν, γγℓ + ℓ − , and γγjj final states where j can be either a gluon, a quark or an antiquark. The SM amplitudes were generated using Madgraph [16] in the framework of Helas [17] routines. The anomalous interactions arising from the Lagrangian (9) were implemented as subroutines and were included accordingly. We consistently took into account the effect of all interferences between the anomalous and the SM amplitudes, and did not use the narrow-width approximation for the vector boson propagators.
In the case of the Tevatron collider, we considered the parameters of the Run I, i.e. √ s = 1.8 TeV and an integrated luminosity of 100 pb −1 . We also investigated the reach of the Tevatron Run II assuming √ s = 2 TeV and an integrated luminosity of 2 × 10 3 pb −1 . For the LHC, we took a center-of-mass energy of 14 TeV and a luminosity of 10 5 pb −1 .
In our calculations we used the MRS (G) [18] set of proton structure functions with the factorization scale Q 2 =ŝ. We started our analysis of the processes (1) and (2) imposing a minimal set of cuts to guarantee that the photons and charged leptons are detected and isolated from each other: In the case of γγℓ + ℓ − production, we required the tag of a Z decaying leptonically imposing that where M ℓℓ is the invariant mass of the lepton pair. In our calculations, we have also taken into account the detection efficiency of the final state particles. We assumed a 85% detection efficiency of isolated photons, electrons, and muons. Therefore, the efficiency for reconstructing the final state γγℓν is 61% while the efficiency for γγℓ + ℓ − is 52%. Considering the cuts (20), (21), and (22), and the detection efficiencies discussed above, the SM prediction for the cross sections and expected number of events of the processes (1) and (2) are presented in Table III. As we can see, the above basic cuts are enough to eliminate the SM background at the Tevatron Run I, however, further cuts are needed to control the background at the Tevatron Run II and LHC.
In order to reduce the SM background for the Tevatron Run II and LHC, we analyzed a few kinematical distributions. The most significant difference between the SM and anomalous predictions appears in the transverse energy of the photons, which is shown in Figure 2 for the reaction (1) and β 0 = 0. Similar behavior is obtained for the reaction (2) and for the anomalous β c contribution. Therefore, we tightened the cut on the transverse energy of the final photons, as suggested by Fig. 2, to enhance the significance of the anomalous contribution. The effect of these cuts can be seen in Table III where we display the new cross sections and expected number of events in parenthesis. As we can see, no SM event is expected at the Tevatron Run II after this new cut, while very few events survive at the LHC. We parametrized the cross sections for processes (1) and (2) after cuts (20)-(23) as where σ sm , σ inter , and σ ano are, respectively, the SM cross section, interference between the SM and the anomalous contribution and the pure anomalous cross section. β stands for β 0 or β c . The results for σ sm , σ inter , and σ ano are presented in Table IV.
Process (3) receives contributions from W * and Z * productions and their subsequent decay into jets, as well as from vector boson fusion (VBF) The signal for hadronic decays of W 's and Z's is immersed in a huge QCD background. Therefore, we tuned our cuts in order to extract the VBF production of photon pairs since it presents two very energetic forward jets that can be used to efficiently tag the events. In our analyses, we required that the photons satisfy while the jets should comply with Assuming a 85% detection efficiency of isolated photons, the efficiency for reconstructing the final state jet+jet+γ +γ is 72%. Table III also contains the SM cross section for the VBF production of photon pair taking into account the above cuts. As we can see, the VBF reaction possesses a much higher statistics than the production of photon pairs associated to leptons. In order to enhance the VBF signal for the anomalous couplings we studied a few kinematical distributions and found out that the most significant difference between the signal and SM background occurs in the diphoton invariant mass spectrum; see Fig. 3. Thusly, we imposed the following additional cuts 200 (400) GeV ≤ M γγ ≤ 700 (2500) GeV for Λ = 500 (2500) GeV.
This cut reduces the SM background cross section by a factor of at least 5; see Table III where we also present the signal cross section after cuts for Λ = 500 and 2500 GeV. The results for σ sm and σ ano of Eq. (24) are presented in Table IV. Since the interference between the SM and the anomalous contribution is negligible in this case, we do not present the results for σ inter . Taking into account the integrated luminosities of the Tevatron and LHC and the results shown in Table IV, we evaluated the potential 95% CL limits on β 0 and β c in the case that there is no deviation from the SM predictions; see Table V. We also exhibit in this table our choice for the scale Λ appearing in the form factor. Therefore, at the Tevatron, the most restrictive constraints are obtained from the reaction (1) for β 0 and β c . Combining both reactions we are able to impose a 95% CL limit |β 0,c | < ∼ 1.5 × 10 −2 GeV −2 at the Tevatron Run II, which is of the same order of the direct bounds coming from LEPII. On the other hand, the most stringent limits at the LHC will come from the photon pair production via VBF, whose bounds are a factor of 5-10 stronger than the ones coming from the reactions (1) and (2). This general statement does not seem to apply for the limits on β c with Λ = 500 GeV, which is more strongly constrained by the process (1). This is not surprising because, for the reactions (1) and (2), the set of cuts (23) leave the β c signal practically unaffected, i.e. this set of general cuts is particularly optimum for this coupling and reactions. This is also there reason why the derived limits on β c are better than the limits for β 0 only for this case.

IV. SUMMARY AND CONCLUSIONS
We are just beginning to test the SM predictions for the quartic vector boson interactions. Due to the limited available center-of-mass energy, the first couplings to be studied contain two photons, and just at the LHC and the NLC we will be able to probe V V V V (V = W or Z) vertices [19]. In this work we analyzed the production of photon pairs in association with ℓ ± ν, ℓ + ℓ − , or jj in hadron colliders. These processes violate unitarity at high energy, therefore, we cutoff the growth of the subprocess cross section via the introduction of form factors which enforce unitarity and render the calculation meaningful.
We showed that the study of the processes (1) and (2) at Tevatron Run I lead to constraints on the quartic anomalous couplings that are a factor of four weaker than the presently available bounds derived from LEPII data. On the other hand, the Tevatron Run II has the potential to probe the quartic anomalous interactions at the same level of LEPII. An important improvement on the bounds on the genuine quartic couplings will be obtained at the LHC collider where, for Λ = 2.5 TeV, a limit of |β 0,c | < ∼ 10 −5 GeV −2 will be reached. Therefore, the direct limits on the anomalous interaction steaming from LHC will be stronger than the ones coming from the precise measurements at the Z pole. It is interesting to notice that the LHC will lead to limits that are similar to the ones attainable at an e + e − collider operating at √ s = 500 GeV with a luminosity of 300 pb −1 , which are |β 0,c | < ∼ 3 × 10 −5 GeV −2 [10]. In conclusion, the LHC will be able to impose quite important limits on genuine quartic couplings studying the γγℓ + ℓ − , γγℓν, and γγjj productions.

Collider Process Cross Section (pb) Number of Events
TABLE III. SM cross sections after the cuts. We applied the cuts (20)-(22) to the ℓνγγ and ℓ + ℓ − γγ processes while we used the cuts (20) and (26,27) to the γγjj final state. We present between parenthesis the Tevatron II results after we included the additional cut (23) for ℓνγγ and ℓ + ℓ − γγ productions. In the case of jjγγ production at LHC, we exhibit between parenthesis (brackets) the results after cuts (28) for Λ = 0.5 (2.5) TeV.