Anomalous quartic gauge boson couplings at hadron colliders

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Anomalous quartic gauge boson couplings at hadron colliders

O. J. P. E´ boli,1,*M. C. Gonzalez-Garcı´a,2S. M. Lietti,1and S. F. Novaes1 1

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405–900, Sa˜o Paulo, Brazil 2_{Instituto de Fı´sica Corpuscular IFIC CSIC, Universidad de Valencia, Edificio Institutos de Paterna, Apartado 2085,}

46071 Valencia, Spain

共Received 22 September 2000; published 7 March 2001兲

We analyze the potential of the Fermilab Tevatron and CERN Large Hadron Collider 共LHC兲 to study anomalous quartic vector-boson interactions ␥␥ZZ and ␥␥W⫹W⫺. Working in the framework of SU(2)L

丢U(1)Ychiral Lagrangians, we study the production of photon pairs accompanied by l⫹l⫺, l⫾␯, 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 CERN 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.

DOI: 10.1103/PhysRevD.63.075008 PACS number共s兲: 12.60.Cn

I. INTRODUCTION AND FORMALISM

Within the framework of the standard model 共SM兲, 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 indi-cation of the existence of new phenomena at a higher scale

关1兴. Presently, the triple gauge-boson couplings are being

probed at the Fermilab Tevatron 关2兴 and CERN e⫹e⫺ col-lider LEP 关3兴 through the production of vector boson pairs; however, we have only started to study the quartic gauge-boson couplings关4,5兴.

It is important to independently measure the trilinear and quartic gauge-boson couplings because there are extensions of the SM 关6兴 that leave the trilinear couplings unchanged but modify the quartic vertices. A simple way to generate, at the 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

␥␥W⫹W⫺ and␥␥ZZ have already been carried out for␥␥ 关7,8兴, e␥ 关9兴, and e⫹_e⫺ _{关10兴 colliders. Some preliminary}

estimates of the potential of the Tevatron collider have also been presented in Ref.关11兴 where only the effect on the total cross section for ‘‘neutral’’ final states ␥W⫹W⫺ and ␥␥Z

were considered while the most promising charged final state

␥␥W⫾was not included. In this paper we analyze the poten-tial of hadron colliders to unravel deviations on the quartic vector boson couplings by examining the most relevant pro-cesses 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 Z0 in the anomalous contribution, i.e.,

p⫹p共p¯兲→␥⫹␥⫹共W*→兲l⫹␯, 共1兲

p⫹p共p¯兲→␥⫹␥⫹共Z*→兲l⫹l, 共2兲 as well as the production of photon pairs accompanied by jets

p⫹p→␥⫹␥⫹ j⫹ j 共3兲

for the CERN Large Hadron Collider 共LHC兲.

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 共form factor兲 of the anomalous cou-plings 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 considerably tighten 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␥␥W⫹W⫺

and␥␥ZZ interactions关12兴. Assuming that there is no Higgs

boson in the low energy spectrum we employ a nonlinear representation of the spontaneously broken SU(2)L 丢U(1)Ygauge symmetry. To construct such a lagrangian, it is useful to define the matrix-valued scalar field ␰(x)

⫽exp(2iXa␸a(x)/v), where Xa are the broken generators and

␸a_{are the Nambu-Goldstone bosons of the global} symmetry-breaking pattern SU(2)L丢U(1)Y→U(1)em. We denote the unbroken generator by Q and our conventions are such that Tr(XaXb)⫽

1

2␦ab and Tr(XaQ)⫽0.

The action of a transformation G of the gauge group

SU(2)L丢U(1)Y on␰ takes the form *Present address. Instituto de Fı´sica da USP, C.P. 66.318, Sa˜o

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␰→␰

⬘

, where G␰⫽␰

⬘

H†. 共4兲

H⫽exp(iQu) is defined requiring that ␰

⬘

contains only the broken generators. In order to write the effective Lagrangian for the gauge bosons, it is convenient to introduce the auxil-iary quantity

D␮共␰兲⬅␰†⳵␮␰⫺i␰†共gWa␮Ta⫹g

⬘

B␮Y兲␰, 共5兲 where Ta 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丢U(1)Y:

eA_␮⬅Tr关QD_␮共␰兲兴 eA_␮→eA_␮⫹⳵_␮u, 共6兲

冑

g2⫹g⬘2Z_␮⬅Tr关X3D␮共␰兲兴 Z␮→Z␮, 共7兲

gW_␮⫾⬅i冑2 Tr关T_⫿D_␮共␰兲兴 W_␮⫾→e⫾iuQW_␮⫾,

共8兲

with the standard definition T_⫾⫽T1⫾iT2. Notice that the fields A, Z, and W⫾ transform only electromagnetically under SU(2)_L丢U(1)_Y. Therefore, effective Lagrangians must be invariant exclusively under the unbroken U(1)em. Moreover, in the unitary gauge (␰⫽1) we have that A

→A, Z→Z, and W⫾_→W⫾_.

Requiring C and P invariance, the lowest order effective interactions involving photons is

Leff⫽⫺ ␲␣␤1 2 F ␮␯_F ␮␯W⫹␣W␣⫺⫺␲␣␤₄ 2F␮␯F␮␯Z␣Z␣ ⫺␲␣␤3 4 F ␮␣_F ␮␤共W␣⫹W⫺␤⫹W␤⫹W⫺␣兲 ⫺␲␣␤4 4 F ␮␣_F ␮␤Z␣Z␤. 共9兲

In order to avoid the strong low energy constraints coming from the␳ parameter we impose the custodial SU(2) sym-metry which leads to ␤1⫽cW

2_␤

2⫽␤0 and ␤3⫽cW 2_␤

4⫽␤c. With this choice Leffreduces to the parametrization used in Ref.关7兴. In the unitary gauge, Eq. 共9兲 gives rise to anomalous

␥␥ZZ and ␥␥W⫹W⫺ vertices which are related by the cus-todial symmetry.

II. PRESENT CONSTRAINTS: PRECISION DATA, LEPII, AND UNITARITY BOUNDS

The couplings defined in the effective Lagrangian Eq.共9兲 contribute at the one-loop level to the Z physics关9兴 via ob-lique 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 ⌸VV(0,c) and here we take the opportunity to update the constraints on␤0and␤cderived in Ref. 关9兴.

It is easy to notice from the structure of the Lagrangians that the contributions to the W and Z self-energies are

con-stant, i.e., they do not depend on the external momentum. Moreover, due to the SU(2) custodial symmetry they are related by

⌸WW(0,c)⫽cw 2_⌸

ZZ(0,c). 共10兲 As a consequence the couplings␤0and␤cdo not contribute to T⫽⌬␳关13兴. Equivalently their contribution to sin␪W van-ishes. Moreover, the unbroken U(1)_emsymmetry constrains the photon self-energy contribution to be of the form

⌸␥␥(0,c)共q2兲⫽q2⌸␥␥(0,c)

⬘

, 共11兲

where for the anomalous interactions Eq. 共9兲 ⌸_␥␥(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 关13兴.

Following the standard procedure, we evaluated the vec-tor boson two-point functions using dimensional regulariza-tion and subsequently kept only the leading nonanalytic con-tributions 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

2 4⫺d →log

⌳2

⫹ 3 c_W4 ln

冉

⌳2_c W 2 M_W2

冊册

冎

, 共12兲 ␣U⫽sw 2 cW 2 S. 共13兲

The allowed ranges of S and U depend on the SM param-eters. As an illustration of the size of the bounds, we take that for the Higgs boson mass of MH⫽300 GeV, the 95% C.L. limits on S and U are 0.34⭐S⭐0.02 and ⫺0.13⭐U

⭐0.37 关14兴. These bounds can then be translated into the

95% C.L. limits on ␤₀ and␤_cpresented 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⫺␥,

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Z␥␥, and ␯␯¯␥␥, while the ALEPH collaboration has re-ported results only on the last reaction关4,5兴. The combined results for all these searches lead to the following 95% C.L. direct limits on the quartic vertices 关5兴

⫺4.9⫻10⫺3 _GeV⫺2_⬍_␤

0⬍5.6⫻10⫺3 GeV⫺2, 共14兲

⫺5.4⫻10⫺3 _GeV⫺2_⬍_␤

c⬍9.8⫻10⫺3 GeV⫺2. 共15兲 Another way to constrain the couplings in Eq. 共9兲 is to notice that this effective Lagrangian leads to tree-level uni-tarity 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 (a˜_␯␮j ) for the inelastic scattering ␥(␭1)␥(␭2)→V(␭3)V(␭4), with V

⫽Z and W⫾_{; see Table II. Unitarity requires that}_关15兴 ␤V

兺

␯ 兩a˜␯␮

j _兩2_⭐1

4, 共16兲

共19兲 is basically equivalent to a rescaling of the anomalous

couplings ␤0,c, therefore we should perform this rescaling when comparing results obtained at hadron and e⫹e⫺ collid-ers. For example, for our choice of n and ⌳ the LEP limits should be weakened by a factor ⯝1.6.

The dynamical effect of the above form factor can be seen in Fig. 1 where we present the normalized invariant mass distribution of the ␥␥ pair for the process Eq. 共1兲 at the Tevatron run II and LHC, assuming that only␤0contributes. As expected, the form factor reduces the number of photon pairs with high invariant mass. Similar behavior is obtained for reaction共2兲 and for the anomalous␤ccontribution.

III. SIGNALS AT HADRON COLLIDERS

In this work we studied reactions共1兲 and 共2兲 for the Teva-tron and LHC, that is, the associated production of a photon pair and a W*or Z*which decay leptonically, as well as the process Eq.共3兲 only for the LHC since the Tevatron center-of-mass energy is too low for this process to be of any sig-nificance. Process Eq. 共1兲 can be used to study the

␥␥W⫹W⫺vertex while the process Eq.共2兲 probes the␥␥ZZ

interaction and reaction 共3兲 receives contributions from

␥␥W⫹W⫺ and␥␥ZZ. We evaluated numerically the

helic-ity amplitudes of all the SM subprocesses leading to the

␥␥l⫾␯,␥␥l⫹l⫺, and␥␥j j final states where j can be either

a gluon, a quark, or an antiquark. The SM amplitudes were generated using Madgraph 关16兴 in the framework of Helas

关17兴 routines. The anomalous interactions arising from the

Lagrangian Eq. 共9兲 were implemented as subroutines and were included accordingly. We consistently took into ac-count 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 run I, i.e.,

冑

s⫽1.8 TeV and an integrated

TABLE II. a˜_␯␮0 for the reactions ␥(␭1)␥(␭2)→V(␭3)V(␭4), with V⫽Z and W⫾, where␮⫽␭1⫺␭2 and␯⫽␭3⫺␭4. ␤ stands for␤0or␤c, and nW⫾⫽1 (4) for␤0(␤c), and nZ⫽cW

冉

␣s 16nV

冊

␤ TABLE I. 95% C.L. limits on␤oand␤csteaming from oblique

parameters S and U.

⌳ 共TeV兲 Parameter ␤0 (GeV⫺2) ␤c(GeV⫺2) 0.5 S (⫺0.09,1.5)⫻10⫺4 (⫺0.29,4.9)⫻10⫺4

U (⫺5.4,1.9)⫻10⫺4 (⫺18.,6.2)⫻10⫺4 2.5 S (⫺0.04,0.69)⫻10⫺4 (⫺0.15,2.5)⫻10⫺4

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luminosity of 100 pb⫺1. We also investigated the reach of the Tevatron run II assuming

冑

s⫽2 TeV and an integrated

luminosity of 2⫻103 pb⫺1. For the LHC, we took a center-of-mass energy of 14 TeV and a luminosity of 105 pb⫺1. In our calculations we used the Martin-Roberts-Sterling set G

关MRS 共G兲兴关18兴 of proton structure functions with the

fac-torization scale Q2⫽sˆ.

We started our analysis of the processes, Eqs.共1兲 and 共2兲, imposing a minimal set of cuts to guarantee that the photons and charged leptons are detected and isolated from each other:

p_T(l,␯)⭓20 共25兲 GeV for l⫽e共␮兲, E_T␥⭓20 GeV,

兩␩␥,e兩⭐2.5, 共20兲

兩␩␮兩⭐1.0, ⌬Ri j⭓0.4,

where i and j stand for the final photons and charged leptons. For the␥␥l␯ final state, we also imposed a cut of the trans-verse mass of the l␯ pair ( M_Tl␯):

65 GeV⭐M_Tl␯⭐100 GeV. 共21兲 In the case of␥␥l⫹l⫺production, we required the tag of a Z decaying leptonically imposing that

75 GeV⭐Mll⭐105 GeV, 共22兲

where Mll 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 an 85% detection efficiency of isolated photons, electrons, and muons. Therefore, the efficiency for reconstructing the final state ␥␥l␯ is 61% while the efficiency for␥␥l⫹l⫺is 52%. Considering the cuts Eqs. 共20兲, 共21兲, and 共22兲, and the detection efficiencies discussed above, the SM prediction for

FIG. 1. Normalized invariant mass distribution of the␥␥ pair

for the reaction p⫹p(p¯)→␥⫹␥⫹(W*_→)l⫹␯ at Tevatron run II

共a兲 and LHC 共b兲. The solid histogram represents the SM

contribu-tion while dashed共dotted兲 histograms are the anomalous␤0

contri-bution with共without兲 unitarity form factor. We chose n⫽5 and ⌳

⫽0.5 共2.5兲 TeV for the Tevatron 共LHC兲.

TABLE III. SM cross sections after the cuts. We applied the cuts Eqs.共20兲–共22兲 to the l␯␥␥ and l⫹l⫺␥␥

processes while we used the cuts Eqs.共20兲 and Eqs. 共26兲, 共27兲 to the␥␥ j j final state. We present between

parenthesis the Tevatron II results after we included the additional cut Eq. 共23兲 for l␯␥␥ and l⫹l⫺␥␥

productions. In the case of j j␥␥ production at LHC, we exhibit between parenthesis 共brackets兲 the results

after cuts Eq.共28兲 for ⌳⫽0.5 共2.5兲 TeV.

Collider Process Cross section共pb兲 Number of events

Tevatron I p p¯→l⫾␯l⫾␥␥ 1.93⫻10 ⫺4 _1.93_⫻10⫺2 p p¯→l⫹l⫺␥␥ 1.58⫻10⫺4 1.58⫻10⫺2 Tevatron II p p¯→l⫾␯l⫾␥␥ 2.13⫻10 ⫺4_(7.89_⫻10⫺6₎ _{0.43 (1.58}_⫻10⫺2₎ p p¯→l⫹l⫺␥␥ 1.77⫻10⫺4(5.90⫻10⫺6) 0.35 (1.18⫻10⫺2) LHC p p→l⫾␯l⫾␥␥ 1.08⫻10⫺3(1.32⫻10⫺5) 108共1.3兲 p p→l⫹l⫺␥␥ 6.45⫻10⫺4(4.25⫻10⫺6) 65共0.43兲 p p→ j j␥␥ 3.19⫻10⫺2(6.28⫻10⫺3) 关1.12⫻103兴 3190共628兲关112兴

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the cross sections and expected number of events of the pro-cesses Eqs.共1兲 and 共2兲 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 anoma-lous predictions appears in the transverse energy of the pho-tons, which is shown in Fig. 2 for the reaction Eq. 共1兲 and

␤0⫽0. Similar behavior is obtained for the reaction 共2兲 and for the anomalous ␤c contribution. Therefore, we tightened the cut on the transverse energy of the final photons, as sug-gested by Fig. 2, to enhance the significance of the anoma-lous contribution.

E_T␥1(2)_{⭓75 共50兲 GeV for Tevatron run II}

and 共23兲

E_T␥1(2)⭓200 共100兲 GeV for LHC.

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 ex-pected at the Tevatron run II after this new cut, while very few events survive at the LHC.

We parametrized the cross sections for processes 共1兲 and

共2兲 after cuts Eqs. 共20兲–共23兲 as

␴⬅␴sm⫹␤␴inter⫹␤2␴ano, 共24兲 where ␴sm, ␴inter, and␴ano are, respectively, the SM cross section, interference between the SM and the anomalous contribution, and the pure anomalous cross section.␤stands for ␤0 or ␤c. The results for␴sm, ␴inter, and␴anoare pre-sented in Table IV.

Process 共3兲 receives contributions from W* and Z* pro-ductions and their subsequent decay into jets, as well as from vector boson fusion共VBF兲

p⫹p→q⫹q⫹共W*⫹W*or Z*⫹Z*兲→q⫹q⫹␥⫹␥.

共25兲

TABLE IV. Results for␴sm,␴inter, and␴ano; see Eq.共24兲.␴interand␴anoare obtained for the anomalous coupling␤0(␤c) in units of GeV⫺2. We considered n⫽5 and different values of ⌳; see Eq. 共19兲. Collider Process ␴sm共pb兲 ␴inter (pb⫻GeV2) for␤0(␤c) ␴ano (pb⫻GeV4) for␤0(␤c) Tevatron I _{p p}¯_→l⫾_␯ l⫾␥␥ 1.93⫻10 ⫺4 _5.09(2.58)_⫻10⫺3 _15.0_共5.50兲 ⌳⫽0.5 TeV pp¯→l⫹_l⫺_{␥␥ 1.58⫻10}⫺4 _7.18(1.22)_⫻10⫺3 _3.63_共1.37兲 Tevatron II p p¯→l⫾␯l⫾␥␥ 7.89⫻10 ⫺6 _1.20(1.03)_⫻10⫺3 _6.21_共2.92兲 ⌳⫽0.5 TeV pp¯→l⫹_l⫺_{␥␥ 5.90⫻10}⫺6 1.38(0.36)⫻10⫺3 1.78共0.86兲 LHC p p→l⫾␯l⫾␥␥ 1.32⫻10⫺5 3.13(3.97)⫻10⫺4 6.79共59.2兲 ⌳⫽0.5 TeV pp→l⫹_l⫺_{␥␥ 4.25⫻10}⫺6 _6.06(0.49)_⫻10⫺4 _4.82_共18.5兲 p p→ j j␥␥ 6.28⫻10⫺3 - 1.02⫻104_(7.56_⫻102₎ LHC p p→l⫾␯l⫾␥␥ 1.32⫻10⫺5 1.17(22.4)⫻10⫺3 5570共2900兲 ⌳⫽2.5 TeV pp→l⫹_l⫺_{␥␥ 4.25⫻10}⫺6 _1.15(1.08)_⫻10⫺2 ₃₉₈₀_共1390兲 p p→ j j␥␥ 1.12⫻10⫺3 - 1.07⫻107(7.34⫻105)

FIG. 2. Normalized transverse energy distribution of the most

energetic photon for the reaction Eq.共1兲 at Tevatron run II 共a兲 and

LHC共b兲. The solid histogram represents the SM contribution while

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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

E_T␥1(2)⬎50 共25兲 GeV, 共26兲兩␩␥(1,2)兩⬍5.0,

while the jets should comply with

p_Tj1(2)⬎40 共20兲 GeV, 兩␩j_(1,2)兩⬍5.0, 兩␩j₁⫺␩j₂兩⬎4.4, ␩j₁•␩j₂⬍0, 共27兲 min兵␩_j 1,␩j1其⫹0.7⬍␩␥(1,2)⬍max兵␩j1,␩j1其⫺0.7, ⌬Rj j⬎0.7, ⌬Rj␥⬎0.7.

Assuming an 85% detection efficiency of isolated photons, the efficiency for reconstructing the final state jet⫹jet⫹␥

⫹␥ is 72%. Table III also contains the SM cross section for the VBF production of photon pairs taking into account the above cuts. As we can see, the VBF reaction possesses much higher statistics than the production of photon pairs associ-ated to leptons. In order to enhance the VBF signal for the anomalous couplings we studied a few kinematical distribu-tions and found that the most significant difference between the signal and SM background occurs in the diphoton invari-ant mass spectrum; see Fig. 3. Thusly, we imposed the fol-lowing additional cuts:

200 共400兲 GeV⭐M_␥␥⭐700 共2500兲 GeV for

⌳⫽500 共2500兲 GeV. 共28兲

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 ⌳⫽500 and 2500 GeV. The re-sults for␴smand␴anoof Eq.共24兲 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␴inter.

Taking into account the integrated luminosities of the Tevatron and LHC and the results shown in Table IV, we evaluated the potential 95% C.L. limits on␤0 and␤c in the case where there is no deviation from the SM predictions; see Table V. We also exhibit in this table our choice for the scale⌳ appearing in the form factor. Therefore, at the Teva-tron, the most restrictive constraints are obtained from the reaction Eq.共1兲 for␤0and␤c. Combining both reactions we

are able to impose a 95% C.L. limit 兩␤_0,c兩ⱗ1.5

⫻10⫺2 _GeV⫺2_{at the Tevatron run II, which is of the same}

order as 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 reac-tions Eqs.共1兲 and 共2兲. This general statement does not seem to apply for the limits on ␤_c with⌳⫽500 GeV, which is more strongly constrained by the process Eq.共1兲. This is not surprising because, for the reactions Eqs.共1兲 and 共2兲, the set of cuts Eq. 共23兲 leave the ␤c signal practically unaffected, i.e., this set of general cuts is particularly optimum for this coupling and reactions. This is also the reason why the de-rived limits on␤care better than the limits for ␤0 only for this case.

IV. SUMMARY AND CONCLUSIONS

We are just beginning to test the SM predictions for the quartic vector boson interactions. Because of the limited

FIG. 3. Normalized invariant mass distribution of the␥␥ pair

for the reaction p⫹p→␥⫹␥⫹jet⫹jet at LHC. The solid histogram

represents the SM contribution while dotted 共dashed兲 histograms

are the anomalous␤0(␤c) contribution with unitarity form factor.

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available center-of-mass energy, the first couplings to be studied contain two photons, and just at the LHC and the Next Linear Collider共NLC兲 we will be able to probe VVVV (V⫽W or Z) vertices 关19兴. In this work we analyzed the production of photon pairs in association with l⫾␯, l⫹l⫺, or

j j in hadron colliders. These processes violate unitarity at

high energy; therefore, we cut off the growth of the subpro-cess cross section via the introduction of form factors which enforce unitarity and render the calculation meaningful.

We showed that the study of the processes Eqs. 共1兲 and

共2兲 at Tevatron run I lead to constraints on the quartic

anomalous couplings that are a factor of 4 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 ⌳⫽2.5 TeV, a limit of 兩␤_0,c兩ⱗ10⫺5 GeV⫺2 will be reached. Therefore, the direct limits on the anomalous

inter-action steaming from LHC will be stronger than the ones coming from the precise measurements at the Z pole. It is interesting to note that the LHC will lead to limits that are similar to the ones attainable at an e⫹e⫺collider operating at

冑

s⫽500 GeV with a luminosity of 300 pb⫺1, which are

兩␤0,c兩ⱗ3⫻10⫺5 GeV⫺2关10兴.

In conclusion, the LHC will be able to impose quite im-portant limits on genuine quartic couplings studying the

␥␥l⫹l⫺, ␥␥l␯, and␥␥j j productions. ACKNOWLEDGMENTS

This work was supported by Conselho Nacional de De-senvolvimento Cientı´fico e Tecnolo´gico 共CNPq兲, by Fun-daçaõ de Amparo a` Pesquisa do Estado de Saõ Paulo

共FAPESP兲, by Programa de Apoio a Nu´cleos de Exceleˆncia 共PRONEX兲, by the Spanish DGICYT under Grant Nos.

PB98-0693 and PB97-1261, by the Generalitat Valenciana under Grant No. GV99-3-1-01, and by the TMR network Grant No. ERBFMRXCT960090 of the European Union.

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Tevatron I _{p p}¯_→l⫾_␯ l⫾␥␥ (⫺4.5,4.4)⫻10 ⫺2 ₍_{⫺7.4,7.4)⫻10}⫺2 p p¯→l⫹l⫺␥␥ (⫺9.2,9.0)⫻10⫺2 (⫺15.,15.)⫻10⫺2 ⌳⫽0.5 TeV Combined (⫺4.0,4.0)⫻10⫺2 (⫺6.6,6.5)⫻10⫺2 Tevatron II _{p p}¯_→l⫾_␯ l⫾␥␥ (⫺1.6,1.5)⫻10 ⫺2 ₍_{⫺2.3,2.2)⫻10}⫺2 p p¯→l⫹l⫺␥␥ (⫺2.9,2.9)⫻10⫺2 (⫺4.2,4.1)⫻10⫺2 ⌳⫽0.5 TeV Combined (⫺1.4,1.3)⫻10⫺2 (⫺2.0,2.0)⫻10⫺2 LHC p p→l⫾␯l⫾␥␥ (⫺2.2,2.1)⫻10⫺3 (⫺7.4,7.3)⫻10⫺4 p p→l⫹l⫺␥␥ (⫺2.4,2.3)⫻10⫺3 (⫺12.,12.)⫻10⫺4 ⌳⫽0.5 TeV p p→ j j␥␥ (⫺2.2,2.2)⫻10⫺4 (⫺8.0,8.0)⫻10⫺4 LHC p p→l⫾␯l⫾␥␥ (⫺7.6,7.6)⫻10⫺5 (⫺11.,10.)⫻10⫺5 p p→l⫹l⫺␥␥ (⫺8.2,7.9)⫻10⫺5 (⫺14.,13.)⫻10⫺5 ⌳⫽2.5 TeV p p→ j j␥␥ (⫺4.4,4.4)⫻10⫺6 (⫺1.7,1.7)⫻10⫺5

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