Anomalous quartic couplings in collision at the LHC
We investigate the constraints on the anomalous quartic couplings through the process at the LHC. Taking into consideration various forward detector acceptances and integrated LHC luminosities, we find 95% confidence level bounds on the anomalous coupling parameters. We show that the bounds on these couplings are at the order of which are about four orders of magnitude more restricted with respect to current experimental bounds.
Precision measurements of gauge boson self-interactions at the LHC will be the crucial test of the gauge structure of the standard model (SM). Any deviation of the couplings from the SM expectations would indicate the existence of new physics. It is very common to investigate the new physics via effective Lagrangian approach. The theoretical basis of such an approach rely on the assumption that at higher energies beyond the SM, there is a more fundamental theory which reduces to the SM at lower energies. Hence, SM is assumed to be an effective low-energy theory in which heavy fields have been integrated out. Such a procedure is quite general and independent of the details of the model. For this reason this approach is sometimes called model independent analysis.
In this paper we have analyzed genuine quartic couplings via single Z boson production in a -proton collision at the LHC. Genuine quartic couplings have different origins than anomalous trilinear couplings. They arise from effective operators that do not induce any trilinear gauge boson coupling. Hence, genuine quartic couplings are free from constraints on trilinear couplings. Imposing custodial symmetry and local symmetry, C and P conserving dimension 6 effective lagrangian for couplings are given by boudjema1 (); boudjema2 (); eboli1 (),
where is the triplet, and is the electromagnetic field strength. and are the dimensionless anomalous coupling constants and is the scale of new physics. The vertex functions generated from the effective lagrangians (2) and (3) are given respectively by eboli1 ()
where and are the momenta of photons and for a convention, we assume that all the momenta are incoming to the vertex.
The current best limits on couplings are provided by the OPAL Collaboration. These are
at 95% C.L. Abbiendi:2004bf ().
Studying photon-induced reactions in a hadron collider is not a very new phenomena. The reactions such as cdf1 (); cdf4 (), cdf3 (); cdf4 (), cdf3 () were verified experimentally by the CDF collaboration at the Fermilab Tevatron. These results raise interest on the potential of LHC as a photon-photon and photon-proton collider. Probing new physics via photon-induced reactions at the LHC is common in the literature. Phenomenological studies involve: supersymmetry, extradimensions, unparticle physics, gauge boson self-interactions, neutrino electromagnetic properties etc. fd11 (); fd12 (); fd13 (); lhc1 (); lhc2 (); lhc3 (); lhc4 (); lhc5 (); lhc6 (); lhc7 (); lhc8 (); lhc9 (); lhc10 (); lhc11 (); lhc12 ().
Ii Equivalent photon approximation and cross sections
A quasireal photon emitted from one proton beam can interact with the quarks of the other proton and the subprocess can occur at the LHC. Emitted quasireal photons are described by equivalent photon approximation (EPA) budnev (); Baur (); fd18 (). Their virtuality is very low and it is a good approximation to assume that they are on-mass-shell. Therefore to some extent it is possible to study -proton collision at the LHC. A schematic diagram describing this process is given in Fig.1.
Any process in a -proton collision can be discerned from pure deep inelastic scattering processes by means of two experimental signatures rouby (): First signature is the forward large-rapidity gap. Quasireal photons have a low virtuality and scattered with small angles from the beam pipe. Since the transverse momentum carried by a quasireal photon is small, photon emitting intact protons should also be scattered with small angles and exit the central detector without being detected. This causes a decrease in the energy deposit in the corresponding forward region compared to the case in which the proton remnants are detected by the calorimeters. As a result of this, one of the forward regions of the central detector has a significant lack of energy. This defines the forward large-rapidity gap and usual pp deep inelastic processes can be rejected by applying a selection cut on this quantity. Second experimental signature is provided by the forward detectors. Forward detectors are capable to detect particles with a large pseudorapidity. When a photon emitting intact proton is scattered with a large pseudorapidity, it exceeds the pseudorapidity coverage of the central detectors. The detection of this intact proton by the forward detectors provides a distinctive signal for the -proton collision.
collision can also be studied in the framework of EPA. In fact, these two-photon processes are much more studied at the LHC. Two-photon processes provide a more clean environment with respect to -proton processes due to absence of the remnants of both proton beams. Moreover two-photon processes are generally electroweak in nature and they are mostly free from backgrounds coming from strong interactions. It is more probable for strong interactions that take part in a -proton process rather than in a two-photon process. Hence, two-photon processes generally have less backgrounds compared to -proton processes. On the other hand, for -proton processes energy reach and effective luminosity are much higher than for two-photon processes rouby (); deFavereaudeJeneret:2009db (). This feature might be important in probing new physics especially when the energy dependences of the anomalous cross sections are very high. For instance, anomalous couplings are described by effective lagrangians (2) and (3) which have an energy dimension of 6. Therefore anomalous cross section containing the vertex has a higher momentum dependence than the SM cross section. From a simple dimensional analysis we deduce that its momentum dependence is higher up to a factor of than the SM cross section where is the momentum incoming (or outgoing) to the vertex. Thus, new physics contribution to the cross section rapidly increases when the center-of-mass energy increases and the processes which have a higher energy reach are expected to have a high sensitivity to new physics. Another factor which has to be considered in a -proton or a collision is the survival probability. It is basically defined as the probability of the scattered protons not to dissociate due to the secondary soft interactions deFavereaudeJeneret:2009db (). In the -proton collision with quasireal photons virtuality of photons is very small. In the EPA that we have considered typical photon virtuality is fd18 (). Therefore proton impact parameter is much bigger than the range of strong interactions and proton survival probability is expected to be large. On the other hand, the survival probability for a -proton process is usually smaller than for a two-photon process deFavereaudeJeneret:2009db ().
ATLAS and CMS collaborations have a program of forward physics with extra detectors located at distances of 220m and 420m from the interaction point royon (); albrow (). These forward detectors have a capability to detect intact scattered protons with momentum fraction loss in the interval which is called the acceptance of the forward detectors. The acceptance proposed by the ATLAS Forward Physics (AFP) Collaboration is royon (); albrow (). The acceptance of the forward detectors in CMS is similar. There are also other scenarios with different acceptances. CMS-TOTEM forward detector scenario spans avati (); fd11 ().
The process takes part as a subprocess in the main reaction . We consider totaly 10 subprocesses for different type of quarks and anti-quarks:
Each of the subprocesses is described by seven tree-level diagrams(Fig.2). We see from Fig.2 that one of them contains anomalous vertex and others are SM contributions. The cross section for the main process can be obtained by integrating the cross sections for the subprocesses over the photon and quark spectra:
where is the fraction which represents the ratio between the scattered equivalent photon and initial proton energy and is the momentum fraction of the proton’s momentum carried by the quark. is the equivalent photon spectrum (see Appendix) and is the quark distribution function of the proton. The summations in (II) are performed over subprocesses in (II). The second integral in (II) is obtained by transforming the differentials into with a Jacobian determinant where . is the total mass of the final particles of the subprocess . is evaluated at . At high energies greater than proton mass, holds. Therefore it is a good approximation to assume that and in the first integral in (II). During calculations, the virtuality of the quark is taken to be where is the mass of the Z boson. In our calculations parton distribution functions of Martin, Stirling, Thorne and Watt pdf () have been used.
In Figs.3 and 4 we plot the total cross section of the process as a function of anomalous couplings and for the acceptances of and . We observe from these figures that cross sections are large for compared with as expected. We also see from these figures that, deviation of the anomalous cross section from its SM value is larger for the coupling than . Therefore sensitivity limits on the coupling are expected to be more restricted than the limits on .
In all results presented in this paper we assume that center-of-mass energy of the proton-proton system is TeV and cross sections have evaluated numerically by a computer code GRACE grace ().
Iii Limits on the anomalous couplings
During statistical analysis we use two different method. We employ a simple one-parameter test when the number of SM events is greater than 10. On the other hand, we employ a Poisson distribution when the number of SM events is less than or equal to 10. For AFP and CMS-TOTEM scenarios SM cross sections for the main reaction are 0.0046 pb and 0.0047 pb respectively. Hence, the number of SM events exceeds 10 for integrated luminosities which are equal or greater than . Therefore for AFP and CMS-TOTEM scenarios we employ both type of the statistical analysis depending on the luminosity. Forward detectors have a capability to detect protons in a continuous range of . Therefore one can impose some cuts and choose to work in a subinterval of the whole acceptance region. Imposing such cuts on forward detector acceptance is useful in suppressing the SM contribution. In addition to AFP and CMS-TOTEM scenarios we will consider and subintervals of the whole AFP and CMS-TOTEM acceptance regions. For these acceptances the number of SM events is less than 10. Therefore it is very appropriate to set bounds on the couplings using a Poisson distribution.
For the acceptances of and with a high luminosity, analysis is performed. The function is defined by
where is the cross section containing new physics effects and is the statistical error. The expected number of events has been calculated considering the leptonic decay channel of the Z boson as the signal , where or , is the integrated luminosity, is the jet reconstruction efficiency and is the survival probability factor. We have taken into account a jet reconstruction efficiency of and survival probability factor of . This survival probability factor was proposed for the single W boson photoproduction deFavereaudeJeneret:2009db (); survivalfactor (). We assume that same survival factor is valid for our process. ATLAS and CMS have central detectors with a pseudorapidity coverage . Therefore we place a cut of for final state particles. Moreover, we also demand that the transverse momenta of the final state photon and quark are greater than 15 GeV. For the acceptances of and , we employ a Poisson distribution. Sensitivity limits are obtained assuming the number of observed events equal to the SM prediction, i.e., . Upper limits of number of events at the 95% C.L. can be calculated from the formula deFavereaudeJeneret:2009db (); Pierzchala ()
Depending on the number of observed events, values for upper limits can be found in Table 33.3 in Ref.Nakamura (). In Table 1 we present number of observed events and upper limits of number of events for the cases in which the Poisson distribution has been used. In Table 1 the calculated values are rounded to the nearest integer. For instance, for forward detector acceptance of , and 1.38 for and respectively. Both of the values have been rounded to 1. The upper limits of number of events can be directly converted to the limits of anomalous couplings and . In Tables 2 and 3, we show 95% C.L. sensitivity limits on the anomalous couplings and for various integrated luminosities and forward detector acceptances of , , and . We see from Tables 2 and 3 that our limits are at the order of and limits on the coupling are more restricted than the limits on .
In this paper we considered all tree-level SM contributions for the subprocess (Fig.2). These constitute major SM contributions. Any other SM contribution coming to this subprocess is at the loop level and can be neglected compared to tree-level SM contributions. The leading order background process might be the pomeron exchange. A pomeron emitted from one proton beam can interact with the quarks of the other proton and same final state can occur. But when we analyze in detail we see that this background process is expected to has a minor influence on sensitivity bounds. In the deep inelastic scattering the virtuality of the struck quark is very high. During calculations in this paper, the virtuality of the struck quark is taken to be where is the mass of the Z boson. Therefore, when a pomeron strikes a quark it probably dissociates into partons. These pomeron remnants can be detected by the calorimeters and background from pomeron exchange can be eliminated. Furthermore, survival probability for a pomeron exchange is considerably smaller than that for a photon exchange. Hence, even if the background from pomeron exchange can not be eliminated, it can not be much bigger (probably smaller) than the tree-level SM contributions for the photon exchange. Finally we would like to stress that our bounds are not very sensitive to backgrounds. For instance, if we assume that background cross section is 2 times bigger than the tree-level SM contributions, our limits with a 200 luminosity are spoiled approximately a factor of 1.5 for and a factor of 1.3 for .
The process at the LHC with a center-of-mass energy of 14 TeV probes anomalous quartic couplings with a far better sensitivity than the current experimental bounds. It allows to improve the current sensitivity by almost four orders of magnitude. The potential of LHC to probe anomalous quartic couplings was examined via weak boson fusion processes and lietti () and photon-photon fusion process lhc2 (); lhc11 (); deFavereaudeJeneret:2009db (); Pierzchala (). In papers deFavereaudeJeneret:2009db (); Pierzchala () authors considered semi-leptonic decay channel of the final Z bosons ,i.e., where refers to jets. On the other hand in papers lhc2 (); lhc11 () authors considered fully-leptonic decay channel of the final Z bosons. The bounds obtained in lhc2 (); lhc11 () are considerably weaker than the bounds obtained in papers deFavereaudeJeneret:2009db (); Pierzchala (). This probably originates from the fact that semi-leptonic decay channel of the final Z bosons has a large branching ratio compared to fully-leptonic decay channel. In our paper we have considered leptonic decay channel of final Z. Therefore it is more appropriate to compare our bounds with the bounds obtained in lhc2 (); lhc11 (). The bounds obtained in lietti (); lhc2 (); lhc11 () are of the same order as our bounds. Anomalous quartic couplings were also studied for future International Linear Collider (ILC) and its operating modes of and . The limits expected to be obtained for such a machine are comparable with the LHC bounds Atag:2007ct ().
Appendix A Equivalent Photon Spectrum
In the above formula, E is the energy of the incoming proton beam and is the mass of the proton. and are functions of the electric and magnetic form factors. is the magnetic moment of the proton. It is taken to be . integration in (12) can be easily performed analytically. After integration over , (12) takes the form of fd11 ()
where the function is defined by
Here and are the upper and lower bounds of the integration. The contribution to the integral above is negligible. Therefore during calculations we set .
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