New Higgs Couplings at e+ e- and Hadronic Colliders

We examine the potentiality of both CERN LEP and Fermilab Tevatron colliders to establish bounds on new couplings involving the bosonic sector of the standard model. We pay particular attention to the anomalous Higgs interactions with $\gamma$, $W^{\pm}$ and $Z^0$. A combined exclusion plot for the coefficients of different anomalous operators is presented. The sensitivity that can be achieved at the Next Linear Collider and at the upgraded Tevatron is briefly discussed.

We examine the potentiality of both CERN LEP and Fermilab Tevatron colliders to establish bounds on new couplings involving the bosonic sector of the standard model. We pay particular attention to the anomalous Higgs interactions with γ, W ± and Z 0 . A combined exclusion plot for the coefficients of different anomalous operators is presented. The sensitivity that can be achieved at the Next Linear Collider and at the upgraded Tevatron is briefly discussed. 14.80.Cp

I. INTRODUCTION: EFFECTIVE LAGRANGIANS FOR HIGGS INTERACTIONS
We certainly expect that the standard model (SM), despite its astonishing success in describing all the precision high energy experimental data [1], to be an incomplete picture of Nature at high energy scales. In particular, the Higgs sector of the model, responsible for the spontaneous electroweak symmetry breaking and for the mass generation, is introduced in an ad hoc way and has not yet been directly probed.
Although we do not know the specific theory which will eventually supersede the SM, we can always parametrize its effects by means of an effective Lagrangian [2] that contains operators with dimension higher than four and involves the fields and symmetries of the low energy theory. The effective Lagrangian approach is a modelindependent way to describe new physics that is expected to manifest itself directly at an energy scale Λ, larger than the scale where the experiments are performed.
The effective Lagrangian depends on the particle content at low energies. We consider here the possibility of having a light Higgs boson that should be present in the higher dimensional operators. Hence, we use a linearly realized SU L (2) × U Y (1) invariant effective Lagrangian [3,4] to describe the bosonic sector of the SM, keeping the fermionic sector unchanged.
A general set of dimension-6 operators that involve gauge bosons and the Higgs scalar field, respecting local SU L (2) × U Y (1) symmetry, and C and P conserving, contains eleven operators [3]. Some of these operators either affect only the Higgs self-interactions or contribute to the gauge boson two-point functions at tree level and is strongly constrained from low energy physics below the present sensitivity of high energy experiments [4]. The remaining five "blind" operators can be written as [3,4], where Φ is the Higgs field doublet,B µν = i(g ′ /2)B µν , andŴ µν = i(g/2)σ a W a µν with B µν and W a µν being the field strength tensors of the U (1) and SU (2) gauge fields respectively.
Anomalous Hγγ, HZγ, and HZZ and HW W and couplings are generated by (1), which modify the Higgs boson production and decay [5]. In the unitary gauge they are given by HZγ HA µν Z µν + g (1) HZZ HZ µν Z µν + g (2) HW W HW + µν W µν where A(Z) µν = ∂ µ A(Z) ν − ∂ ν A(Z) µ . The effective couplings g Hγγ , g (1,2) HZγ , and g (1,2) HZZ and g (1,2) HW W are related to the coefficients of the operators appearing in (1) and can be found elsewhere [5]. In particular the Higgs couplings to two photons is given by with g being the electroweak coupling constant, and s(c) ≡ sin(cos)θ W . Equation (1) also generates new contributions to the triple gauge boson vertex [3,4] [4,5]. Therefore, one cannot obtain any constraint on these couplings from the study of anomalous trilinear gauge boson couplings. Finally, we should point out that the dimensionsix operators (1) do not induce 4-point anomalous couplings like ZZγγ, Zγγγ, and γγγγ, being these terms generated only by dimension-eight and higher operators.
Anomalous Higgs boson couplings have been studied in Higgs and Z 0 boson decays [5], and in e + e − [6][7][8][9][10], pp [11][12][13] and γγ colliders [14]. In this work, we make a combined analysis, based on several experimental searches at the CERN LEP collider and at the Fermilab Tevatron collider, in order to establish the attainable bounds on the coefficient of the effective operators describing the anomalous bosonic sector. Our results are presented in Section II. In Section III we discuss the sensitivity that can be achieved at the Fermilab Tevatron upgrade and at the Next Linear Collider (NLC). Finally, in Section IV, we compare our results with existing limits on the coefficients of dimension-six operators based on searches for anomalous triple gauge boson couplings.

II. BOUNDS FROM THE RECENT LEP AND TEVATRON SEARCHES
In this section, we derive combined bounds on anomalous Higgs boson interactions taking into account both LEP [15] and Tevatron [16][17][18] data on the following signatures: Events containing two photons plus missing energy, additional photons or charged fermions represent a signature for several theories involving physics beyond the SM, such as some classes of supersymmetric models [19] and they have been extensively searched for [15][16][17][18]. In the framework of anomalous Higgs couplings presented before, they can also arise from the production of a Higgs boson which subsequently decays in two photons. In the SM, the decay width H → γγ is very small since it occurs just at one-loop level [20]. However, the existence of the new interactions (2) can enhance this width in a significant way. Recent analyses of these signatures presented a good agreement with the expectations from the SM. Thus we can employ these negative experimental results to constrain new anomalous couplings in the bosonic sector of the SM.
We have included in our calculations all SM (QCD plus electroweak), and anomalous contributions that lead to these final states. The SM one-loop contributions to the Hγγ and HZγ vertices were introduced through the use of the effective operators with the corresponding form factors in the coupling. Neither the narrow-width approximation for the Higgs boson contributions, nor the effective W boson approximation were employed. We consistently included the effect of all interferences between the anomalous signature and the SM background. The SM Feynman diagrams corresponding to the background subprocess were generated by Madgraph [21] in the framework of Helas [22]. The anomalous couplings arising from the Lagrangian (1) were implemented in Fortran routines and were included accordingly. For the pp processes, we have used the MRS (G) [23] set of proton structure functions with the scale Q 2 =ŝ.
All processes listed in (4) have been the object of direct experimental searches. In our analysis we have closely followed theses searches in order to make our study as realistic as possible. In order to establish bounds on the values of the anomalous coefficients f i , i = W W, BB, W, B, we have imposed an upper limit on the number of signal (anomalous) events based on Poisson statistics. In the absence of background this implies N signal < 1 (3) at 64% (95%) CL. In the presence of background events, we employed the modified Poisson analysis [24].
For events containing three photons in the final state at electron-positron collisions [8], we have used the recent OPAL data [15] where data taken at several energy points in the range √ s = 130 -172 GeV, were combined. They have established an upper limit at 95% CL for σ(e + e − → γ + X) × BR(X → γγ) where X is a scalar particle. These results were used to derive our limits.
The process can also be employed to further constrain the anomalous Higgs boson couplings described in (2) [11]. DØ Collaboration reported the results for the search of high invariant-mass photon pairs in pp → γγjj events [16] at √ s = 1.8 TeV and 100 pb −1 of integrated luminosity.
In our analysis, we applied the same cuts of Ref. [16] and included the particle identification and trigger efficiencies. We have searched for Higgs boson with mass in the range 100 < M H < ∼ 220, since after the W W (ZZ) threshold is reached the diphoton branching ratio of Higgs is quite reduced. Since no event with two-photon invariant mass in the range 100 < M γγ < ∼ 220 were observed, a 95% CL in the determination of the anomalous coefficient f i is attained requiring 3 events coming only from the anomalous contributions.
For events containing two photons plus large missing transverse energy (γγ E T ) [12] we have used the results from DØ collaborations [17]. Anomalous Higgs couplings can give rise to this final state via, where in the latter case the charged lepton (ℓ = e, µ) escapes undetected. In order to compare our predictions with the results of DØ Collaboration, we have applied the same cuts of last article in Ref. [17]. After these cuts we find that 80% to 90% of the signal comes from associated Higgs-Z 0 production while 10% to 20% arrises from Higgs-W . We also include in our analysis the particle identification and trigger efficiencies which vary from 40% to 70% per photon [25]. Since no event with two-photon invariant mass in the range 100 < M γγ < ∼ 2M W were observed, a 95% CL in the determination of the anomalous coefficient f i , i = W W, BB, W, B is attained requiring 3 events coming only from the anomalous contributions. Table I shows the 95% CL allowed region of the anomalous couplings in the above scenario. We exhibit in Fig. 1 the 95% CL exclusion region in the plane f BB × f W W obtained from the DØ data on γγ+ E T [17].
Finally we have also analyzed events with three photons in the final state [13] pp → γ + H(→ γγ) , and compare our results with the recent search reported by CDF Collaboration [18] for this signature. They looked for γγγ events requiring two photons in the central region of the detector, with a minimum transverse energy of 12 GeV, plus an additional photon with E T > 25 GeV. The photons were required to be separated by more than 15 • . In these conditions they were able to establish that the signal should have less than 3 events, in the 85 pb −1 collected data, at 95 % CL. We have used the results described above to constrain the value of the coefficients f i of (1). The coupling Hγγ (3) involves f W W and f BB [5], and in consequence, the anomalous signature ff γγ is only possible when those couplings are not vanishing. The couplings f B and f W , on the other hand, affect the production mechanisms for the Higgs boson. In Fig. 1(a) we present our results for the excluded region in the f W W , f BB plane from the different channels studied for M H = 100 GeV assuming that these are the only non-vanishing couplings. Since the anomalous contribution to Hγγ is zero for f BB = −f W W , the bounds become very weak close to this line, as is clearly shown in Fig. 1.
In order to reduce the number of free parameters one can make the assumption that all blind operators affecting the Higgs interactions have a common coupling f , [4,5,26]. We present in Table I the 95% CL allowed regions of the anomalous couplings in this scenario, for different Higgs boson mass.
These results obtained from the analysis of the four reactions (4) can be statistically combined in order to obtain a better bound on the coefficient of the effective operators (1). We exhibit in Fig. 1(b) the 95% CL exclusion region in the plane f BB × f W W obtained from combined results. In Fig. 2, we present the combined limits for the coupling constant f = f BB = f W W = f B = f W (upper scale) for Higgs boson masses in the range of 100 ≤ M H ≤ 220 GeV.

III. FUTURE PERSPECTIVES
The effect of the anomalous operators becomes more evident with the increase of energy, and higher sensitivity to smaller values of the anomalous coefficients can be achieved by studying their contribution to different processes at the upgraded Tevatron collider or at new machines, like the Next Linear Collider.
We first extend our analysis of the pp → γγ E T and pp → γγjj reactions for the upgraded Tevatron collider. We have considered the Run II upgrade with a luminosity of 1 fb −1 , and for the TeV33 upgrade we assumed 10 fb −1 [27]. In our estimates we have taken the same cuts and detection efficiencies given in our previous analysis.
For the γγγ final state we have studied the improvement on the sensitivity to the anomalous coefficients by implementing additional kinematical cuts [13]. Best results are obtained for the following set of cuts: E T1 > 40 GeV, with E T2,3 > 12 GeV where we have ordered the three photons according to their transverse energy, i.e. E T1 > E T2 > E T3 . We always require the photons to be in the central region of the detector (|η i | < 1) where there is sensitivity for electromagnetic showering. In our estimates we assume the same detection efficiency for photons as considered by CDF Collaboration [18].
In Table II we present the 95% CL limit on the anomalous couplings for Tevatron Run II and for TeV33 for each individual process. All couplings are assumed equal (f = f BB = f W W = f B = f W ) and the Higgs boson mass is varied in the range 100 ≤ M H ≤ 220 GeV. Combination of the results obtained from the analysis of the three reactions (6,7,8) leads to the improved bounds given in the last column of Table II. Comparing these results with those in Table. I (or with the upper scale of Fig. 2) we observe an improvement of about a factor ∼ 2-3 [∼ 4-6] for the combined limits at RunII [TeV33].
The Next Linear electron-positron Collider will open an important opportunity to further improve the search for new physics. In particular, the anomalous Higgs boson couplings can be investigated in the processes [9,10]: We studied the sensitivity of NLC to these processes assuming a energy in the center-of-mass of √ s = 500 GeV and an integrated luminosity L = 50 fb −1 . We adopted a cut in the photon energy of E γ > 20 GeV and required the angle between any two particles to be larger than 15 • . We have analyzed these processes for different values of the Higgs boson mass. We have investigated different distributions of the final state particles in order to search for kinematical cuts that could improve the NLC sensitivity. The most promising variable is the photon transverse momentum. We observe that the contribution of the anomalous couplings is larger in the high p Tγ region. Since the anomalous signal is dominated by on-mass-shell Higgsγ production with the subsequent H → W + W − or Z 0 Z 0 decay, the photon transverse momentum is distributed around the monochromatic peak E mono γ = (s − M 2 H )/(2 √ s). In consequence for Higgs boson masses in the range where on-shell production is allowed, a cut of p Tγ > ∼ 100 drastically reduces the background. For lighter Higgs bosons, e.g. M H < 2M W,Z , the p Tγ cut is ineffective since the Higgs boson is off-mass-shell and the peak in the photon transverse momentum distribution disappears. This makes the bounds on the anomalous coefficients obtained from the W + W − (Z 0 Z 0 )γ production to be very loose.
In Fig. 3 we show the 95% CL exclusion region in the plane f BB × f W W for M H = 200 GeV from the study of reactions (9) and (10). Notice that for these two reactions the exclusion region closes the gap at f BB = −f W W since the anomalous decay widths H → W + W − (Z 0 Z 0 ) do not vanish along this axis [5].
We present in Table III the limits on the coefficient f /Λ 2 based on a 95% C. L. deviation in the total cross section for a Higgs mass in the range 170 ≤ M H ≤ 350 GeV. The results coming from the Z 0 Z 0 γ production are a little better than the ones obtained from W + W − γ production, and they are one order of magnitude better than the actual limits derived from LEP and Tevatron data analyses.

IV. DISCUSSION
So far we have estimated the limits on anomalous dimension-six Higgs boson interactions that can be derived from the study of several signatures at LEP and Tevatron colliders. Combined results from the different reactions were established. We compare now these results with existing limits on the coefficients of other dimensionsix operators.
As discussed in Section I, for linearly realized effective Lagrangians, the modifications introduced in the Higgs and in the vector boson sector are related to each other. In consequence, the bounds on the new Higgs couplings should also restrict the anomalous gauge-boson self interactions. Under the assumption of equal coefficients for all anomalous Higgs operators, we can relate the common Higgs boson anomalous coupling f with the conventional parametrization of the vertex W W V (V = Z 0 , γ) [28], A different set of three independent couplings has been also used by the LEP Collaborations [29]: α BΦ , α W Φ , and α W . These parameters are related to the parametrization of Ref. [28] through α BΦ ≡ ∆κ γ − ∆g Z 1 c 2 W , α W Φ ≡ ∆g Z 1 c 2 W , α W ≡ λ γ , or in terms of the anomalous Higgs boson coupling f by, The current experimental limit on these couplings from combined results on double gauge boson production at Tevatron and LEP II [30] is: at 95 % CL. This limit is derived under the relations given in Eq. (11) [4].
In Table IV, we present the 95% CL limit of the anomalous coupling ∆κ γ using the limits on f /Λ 2 obtained through the analysis of the processes considered in Section II. We also present the expected bounds that will be reachable at the upgraded Tevatron and at the NLC. Our results show that the present combined limit from the Higgs production analysis obtained in this paper is comparable with the existing bound from gauge boson production (13) for M H ≤ 170 GeV, as can be seen in Fig. 2 (lower scale).
Summarizing, we have estimated the limits on anomalous dimension-six Higgs boson interactions that can be derived from the investigation of three photon events at LEP2 and Tevatron and diphoton plus missing transverse energy events or dijets at Tevatron. Under the assumption that the coefficients of the four "blind" effective operators contributing to Higgs-vector boson couplings are of the same magnitude, the study can give rise to a significant indirect limit on anomalous W W V couplings. We have also studied the expected improvement on the sensitivity to Higgs anomalous couplings at the Fermilab Tevatron upgrades and at the Next Linear Collider. ACKNOWLEDGMENTS M.C. G-G is very grateful to the Instituto de Física Teórica of Universidade Estadual Paulista for their kind hospitality. We would like to thank Alexander Belyaev for very useful discussions. This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), by Fundação de Amparoà Pesquisa do Estado de São Paulo (FAPESP), and by Programa de Apoio a Núcleos de Excelência (PRONEX).  1. (a) Exclusion region outside the curves in the fBB × fW W plane, in TeV −2 , based on the CDF analysis [18] of γγγ production (most external black lines), on the DØ analysis [16] of γγjj production (most internal black lines), on the DØ analysis [17] of γγ ET (blue lines), and on the OPAL analysis [15] of γγγ production (red lines) , always assuming MH = 100 GeV. The curves show the 95% CL deviations from the SM total cross section. (b) Same as a for the combined analysis.

MH(GeV)
TABLE III. 95% CL allowed range for f /Λ 2 , from W + W − γ and Z 0 Z 0 γ production at NLC, assuming all fi to be equal.