Probe Anomalous $tq\gamma$ couplings through Single Top Photoproduction at the LHC

In this work we study the constraints on the anomalous $tq\gamma$ ($q=u$, $c$) couplings by photon-produced leading single top production and single top jet associated production through the main reaction $pp\rightarrow p\gamma p\rightarrow pt\rightarrow pW(\rightarrow\ell \nu_\ell) b+X$ and $pp\rightarrow p\gamma p\rightarrow ptj\rightarrow pW(\rightarrow\ell \nu_\ell) bj+X$ assuming a typical LHC multipurpose forward detectors in a model independent effective lagrangian approach. Our results show that: for the typical detector acceptance $0.0015<\xi_1<0.5$, $0.1<\xi_2<0.5$ and $0.0015<\xi_3<0.15$ with a luminosity of 2 $\rm{fb}^{-1}$, the lower bounds of $\kappa_{tq\gamma}$ through leading single top channel (single top jet channel) are 0.0096 (0.0115), 0.0162 (0.0152) and 0.0098 (0.0122), respectively, correspond to $\rm{Br}(t\rightarrow q\gamma)\sim 3\times 10^{-5}$. With a luminosity of 200 $\rm{fb}^{-1}$, the lower bounds of $\kappa_{tq\gamma}$ are 0.0031 (0.0034), 0.0051 (0.0047) and 0.0031(0.0038), respectively, correspond to $\rm{Br}(t\rightarrow q\gamma)\sim 4\times 10^{-6}$. We conclude that both channels can be used to detect such anomalous $tq\gamma$ couplings and the detection sensitivity on $\kappa_{tq\gamma}$ can be improved.


Introduction
The top quark is the heaviest known elementary particle which makes it an excellent candidate for new physics searches. One possible manifestation of new interaction in the top quark sector is to alter its couplings to the gauge bosons. Such anomalous couplings would modify top production and decay at colliders. The most widely studied cases are the ttV, with V = γ, Z, g, and tbW three-point functions. In addition, the flavor change neutral current (FCNC) interactions tqV, with q=u, c, will also offer an ideal place to search for new physics. They are very small in the Standard Model (SM). For instance, while radiative B-menson decays have branching ratios of order Br(b → sγ) ∼ 10 −4 , typical FCNC top quark decays, such as t → cZ, t → cγ and t → cg, are highly suppressed by GIM mechanism with SM branching ratios of order at most 10 −14 , 10 −13 and 10 −12 [1,2], respectively, which in practice are impossible to be measured. In this instance any positive observation of these transitions would be signal presence of new physics. Actually, t → cV have been studied in various new physics models beyond the SM [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. There they often predict much larger FCNC top quark decay interactions which can be explored in future collider experiments.
In addition to the direct top quark decays, production of top quarks by FCNC interactions can also be used to probe such vertices. Studies have been presented at linear colliders [18,19,20,21,22,23,24], lepton-hadron colliders [25,26,27,28], as well as hadron colliders [24,29,30,31,32,33,34,35,36,37,38,39,40], see also reference there in. In this paper, we study the tqγ anomalous couplings through the leading single top photoproduction and single top jet associated photoproduction via the main reaction pp → pγp → pt → pW(→ ℓν ℓ )b + X and pp → pγp → ptj → pW(→ ℓν ℓ )bj + X assuming a typical LHC multipurpose forward detectors in a model independent effective lagrangian approach. Feynman diagrams for these processes present with anomalous tqγ couplings arise from the initial photon. Similar studies was presented in Ref. [25] and tried to study tqγ coupling through γb → Wb at CLIC+LHC ep colliders while recently moved to the photon-proton (γp) collision in Ref. [41]. In addition, feasibility studies of anomalous κ tqγ via single top photoproduction at the LHC have also been carried out in Ref. [42,43,44].
Typically, our study will also include the single top jet associated production channel.
The possibility of adding forward proton detectors to both the ATLAS and CMS experiments has received quite some attention since the possibility of forward proton tagging would provide a very clean environment for new physics searches. Our paper is organized as follow: we build the calculation framework in Section 2 include a brief introduction to the anomalous tqγ couplings, Equivalent Photon Approximation implementation, general photoproduction cross section. Section 3 is arranged to present the selected processes and numerical results as well as the signal and background analysis. In Section 4 we present the bounds on anomalous tqγ couplings at the future LHC. Finally we summarize our conclusions in the last section.

Calculation Framework
The effective Lagrangian involving anomalous tqγ (q=u, c) couplings can be written as where Λ is an effective scale which we set equal to the top quark mass m t by convention.
Usually the value of Λ should be at the TeV scale. For the other choice of Λ the results can be rescaled by ( mt Λ ) 2 . e, e t are the electric charge of the electron and the top quark, respectively. σ µν is defined as (γ µ γ ν −γ ν γ µ )/2 with γ µ the Dirac matrices. q ν is the photon 4-vector momentum. κ tuγ and κ tcγ are real and positive anomalous FCNC couplings.
As the SM predictions for Γ(t → qγ) are exceedingly small, we need only consider t → qγ decays mediated by the anomalous tqγ interactions, which can be considered at the next-to-leading order (NLO) [76] and resulted for the final decay widths Γ(t → qγ): with the leading order (LO) decay width obtained from Eq.(1) as Γ 0 (t → qγ) = 2 9 α ew m 3 t κ 2 tqγ Λ 2 with α ew = 1 137 . It is convenient to relate the branching ratios Br(t → qγ) to the FCNC partial widths of the top-quark as .
The decay width of the dominant top-quark decay mode t → Wb at the LO and the NLO could be found in Ref. [77], and is given below 2 is the velocity of the W-boson in the top-quark rest frame.
Present constraints on the FCNC tqγ couplings come from the following experimental bounds: The CDF collaboration [78] has performed a direct search for FCNC top decays and has placed the 95% confidence level (C.L.) limits on the branching fractions Br(t → qγ) < 3.2% (q=u, c), which gives the constraint of κ tqγ ≤ 0.26. The ZEUS collaboration [79] provide at 95% C.L. the effective FCNC coupling κ tuγ < 0.174 with the assumption of m t = 175 GeV. The current limits from H1 collaboration are κ tqγ < 0.305 [80]. These constraints will be improved significantly by the large top quark sample to be available at the LHC. In particular, both the ATLAS [81] and CMS [82] collaborations have presented their sensitivity to these rare top quark decays induced by the anomalous FCNC interactions [83].
where α is the fine-structure constant, E is the energy of the incoming proton beam which is related to the quasi-real photon energy by E γ = ξE and M p is the mass of the proton. µ 2 p = 7.78 is the magnetic moment of the proton. F E and F M are functions of the electric and magnetic form factors. The intact protons with some momentum fraction loss is described by the formula ξ = (|p| − |p ′ |)/|p|, which is defined as the forward detector acceptances.
We denote the photoproduction processes as pp → pγp → p + γ + q/q/g → p + i + j + k + ... + X with q=u, d, c, s, b and i, j, k, ... the final state particles. The hadronic cross section at the LHC can be converted by integrating γ + q/q/g → i + j + k + ... over the photon (dN(x, Q 2 )), gluon and quark (G g,q/p (x 2 , µ f )) spectra: where x 1 is the ratio between scattered quasi-real photons and incoming proton energy x 1 = E γ /E and ξ min (ξ max ) are its lower (upper) limits. x 2 is the momentum fraction of the proton momentum carried by the gluon (quark). The quantityŝ = z 2 s is the effective center-of-mass system (c.m.s.) energy with z 2 = x 1 x 2 . M inv is the total mass of the related final states. 2z/x 1 is the Jacobian determinant when transform the differentials from dx 1 dx 2 into dx 1 dz. G g,q/p (x, µ f ) represent the gluon (quark) parton density functions, µ f is the factorization scale. 1 avgfac is the times of spin-average factor, color-average factor and identical particle factor. |M n | 2 presents the squared n-particle matrix element and divided by the flux factor [2ŝ(2π) 3n−4 ]. dΦ n and Φ n are the n-body phase space differential and its integral depending onŝ and particle masses.

The Processes and Numerical Results
We implement the anomalous interaction vertices deduced from the Lagrangian (see in Eq.(1)) into FeynArts and use FeynArts, FormCalc and LoopTools (FFL) packages [85,86,87] to create the amplitudes and perform the numerical calculation for both the signal and background. We adopt CT10 [88] PDF for the parton distributions for collider physics and BASES [89] to do the phase space integration while Kaleu [90]  • CMS − TOTEM forward detectors with 0.0015 < ξ 1 < 0.5 • CMS − TOTEM forward detectors with 0.1 < ξ 2 < 0.5 • AFP − ATLAS forward detectors with 0.0015 < ξ 3 < 0.15 which we simply refer to ξ 1 , ξ 2 and ξ 3 , respectively.

Direct Single Top Photoproduction
The first single top photoproduction with anomalous tqγ interactions we consider is the direct leading single top production via the process where q=u,ū, c,c. The Feynman diagrams for the subprocess is presented in Fig.1(2) corresponds to the signal and Fig In this case, the studied topology is simply one of a tagged b-jet, one isolated, either positive or negative, lepton ℓ ± , and a missing transverse momentum from the undetected neutrino. In addition to the irreducible background from Fig.1(1,3,4), the main background comes from associated production of W boson and the light jets with jet faking a b-jet. Though jet charge can be a possibility for labeling jets, it is not well measured experimentally, we can not use charge to separate them. In our analysis, we assume a b-jet tagging efficiency of ǫ b = 60% and a corresponding mistagging rate of ǫ light = 1% for light jets (u, d, s quark or gluon) and ǫ c = 10% for a c-jet, consistent with typical values assumed by the LHC experiments [93].
For the direct leading single top production, we impose a cut of pseudorapidity |η| < 2.5 for the final state particles since central detectors of the ATLAS and CMS have a pseudorapidity coverage 2.5. The general acceptance cuts for both the signal and background events are: where ∆R = ∆Φ 2 + ∆η 2 is the separation in the rapidity-azimuth plane. p jet,ℓ dσ dp jet
The transverse momentum differential cross sections of the final state jets (p jet T ) are given in Fig.2. The anomalous coupling is chosen to be κ tuγ = κ tcγ = κ = 0.01(0.03) and the forward detector acceptance is chosen to be 0.0015 < ξ 1 < 0. dσ dp jet  taken into account. In the p T distribution we can clearly see a resonance in the signal which correspond to the top quark. In order to improve the signal to background ratio we can apply an invariant mass cut on the W-jet system around the top quark mass. To determine the invariant mass of the W-jet system, we follow Ref. [25,32] and reconstruct p t = p ℓ + p ν + p b−jet . The transverse momentum of the neutrino can be deduced from the missing transverse momentum. The longitudinal component of the neutrino momentum is given by  (10). We see that the invariant mass cut can reduce the W-jet background obviously while make the signal reduce slightly. To see how the cross sections for signal and background depend on the final jet (p jet T ) cuts, we present this dependence in Tab.1 with the invariant mass cut taken to be 160 GeV < M Wj < 180 GeV. We find that for small value of κ, for example, κ = 0.01, larger p jet T cut can reduce the background cross section essentially while make the signal reduce slightly. This can be seen directly by comparing differential cross sections in Fig.2 and Fig.3. We also calculate the statistical significance (SS) for the signal and background on different values of p jet T cuts in Tab.2 with the following formula [94]: In the following calculation, we apply p jet T > 35 GeV and 160 GeV < M Wj < 180 GeV. In Fig.5, we present the signal cross sections of pp → pγp → pW(→ ℓν ℓ )b (ℓ = e, µ) as functions of the anomalous κ tqγ couplings and three forward detector acceptance: 0.0015 < ξ 1 < 0.5, 0.1 < ξ 2 < 0.5 and 0.0015 < ξ 3 < 0.15. Compare different acceptance regions we see that although lines correspond to ξ 1 , ξ 2 and ξ 3 have almost the same features, ξ 1 and ξ 3 do not differ much from each other while both of them are much larger than cross section of ξ 2 . We observe from these figures that cross sections are large for high values of κ tqγ and are sensitive to the anomalous couplings as expected.

Single Top Jet Associated Photoproduction
Black blobs represent the anomalous tqγ couplings parameterized by Eq.(1).
The second single top photoproduction with the anomalous tqγ interactions we examined via the main processes with q=u, c, where we simply refer to these processes as tj productions. The main reactions include parton level photon-quark collision γq → tg and photon-gluon collision γg → tq. The motivation for the study of tj process is that: first, tj associated production is another interesting single top photoproduction through γp collision at the LHC in addition to the direct single top photoproduction, both study on them would provide complementary information from one to the other; second, although an additional particle appear in the final state, another γg collision mode may also appear. Since the larger value of gluon parton distribution function, it will be interesting to find out how this tj channel works to detect the anomalous tqγ couplings. Some Feynman diagrams are shown in Fig.6(1-4). Same as before that the black blobs in these figures represent the anomalous tqγ couplings. Still we concentrate on the semi-leptonic decay of the single top quark, taking ℓ = e, µ. Both the process and its charge-conjugate state are implied.
As can be seen, the studied topology of our signal in this case therefore give rise to the where a 0 is the SM prediction, the term a 1 linear in κ tqγ arises from the interference between SM and the anomalous amplitudes, whereas the quadratic term a 2 is the selfinterference of the anomalous amplitudes. Here we still assume κ tuγ = κ tcγ = κ.   Tab.3 summarises the signal and background cross sections after the application of the basic cuts in Eq. (10). However, for the tj production, since the extra jet in this case will be in forward region already, we do not impose the |η| < 2.5 in this calculation.
In addition to the invariant mass cut of the b-jet, the charged lepton and the neutrino system (m(ℓν ℓ j)) close to the top mass has also been included. So that we can require the final state to be consistent with the tj(tj) production. Since we have two jets in the final state, we require a random one satisfy 150 GeV < m(ℓν ℓ j) < 200 GeV will pass into our selections. During calculation, we consider all the backgrounds listed in Tab.3 except the ones that can be safely omitted. We can notice in Tab To see how the cross sections and statistical significance depend on the m(ℓν ℓ j) = M Wj cut, we also require 160 GeV < m(ℓν ℓ j) < 180 GeV and compare it with the former case (150 GeV < m(ℓν ℓ j) < 200 GeV) in Tab.4. We see by applying the invariant mass of 160 GeV < m(ℓν ℓ j) < 180 GeV, the signal is reduced slightly while the backgrounds can be reduced obviously thus leading to a better signal over background ratio and higher statistical significance. The statistical significance for different values of L is presented in the Tab.4. In the following calculation we apply 160 GeV < m(ℓν ℓ j) < 180 GeV.

Bounds for future LHC and the Conclusion
We follow Ref. [41] exactly to obtain the sensitivity limits. Typically, the limits are achieved by assuming the number of observed events equal to the SM background prediction, N obs = σ B × L × ǫ, with L for a given integrated luminosity and ǫ the detection efficiency. σ B is the cross section of SM background prediction. As can be seen, the SM background events can be less or larger than 10 for different values of the luminosity and different types of the detector acceptances. We thus estimate the sensitivity limits on the anomalous tqγ coupling through these two single top photoproduction channels by using two different statistical analysis methods depending on the number of observed events N obs . For N obs ≤ 10, we employ a Poisson distribution method. In this case, the upper limits of number of events N up at the 95% C.L. can be calculated from the formula Values for limits candidate N up can be found in Ref. [91]. The expected 95% C.L. limits on κ tqγ can then been calculated by the limits of the observed cross section. The integrated luminosity L will be taken as a running parameter. For N obs > 10, a chi-square (χ 2 ) analysis is performed with the definition where σ tot is the cross section containing new physics effects and δ = 1 √ N is the statistical error with N = σ B × L × ǫ. The parameter sensitivity limits on anomalous tqγ coupling as a function of the integrated luminosity can then be obtained. pp → pγp → pℓν ℓ j a 14 TeV LHC 0.0015 < ξ 1 < 0.5 0.1 < ξ 2 < 0.5 0.0015 < ξ 3 < 0.15 Figure 9: 95% C.L. lower bounds for the anomalous tqγ couplings as functions of various integrated luminosity and forward detector acceptances of 0.0015 < ξ 1 < 0.5, 0.1 < ξ 2 < 0.5 and 0.0015 < ξ 3 < 0.15. Bounds obtained by using channel pp → pγp → pW(→ ℓν ℓ )b.
photoproduction processes pp → pγp → pW(→ ℓν ℓ )b and pp → pγp → pW(→ ℓν ℓ )bj, both channels can be used to test the anomalous tqγ couplings. These parameter limits (bounds) are also comparable with the other phenomenological studies [26,27,28,41,95,96] and much better than the constraints from experiments [78,79,80,83]. Notice that in Fig.9 and Fig.10, we also present the bounds obtained when the luminosity become larger than 200fb −1 , see, up to 1000fb −1 . However, we should mention here that as the luminosity become larger, identify the signal under the high pileup running conditions will be challenge: the hadronic background of multiple pp interactions will be so large that any γp process will be completely swamped. This can be a drawback of γp productions.
In this case a more detailed study on the experimental effects, i.e., pileup rejection factors, should also be considered. These will and might significantly reduce the constraints on the bounds obtained. However, in our phenomenological study, we keep all the results up to high luminosity with the discussion been focused only up to 200fb −1 . Full detector simulation is beyond the scope of this analysis.

Summary
In this work, we examine the anomalous tqγ (q=u, c) coupling through photon-produced leading single top production and single top jet associated production through the main reaction pp → pγp → pt → pW(→ ℓν ℓ )b + X and pp → pγp → ptj → pW(→ ℓν ℓ )bj + X assuming a typical LHC multipurpose forward detectors in a model independent effective lagrangian approach. Full effects of the top quark leptonic decay modes (t → Wb → Br(t → qγ) ∼ 4 × 10 −6 . We see that for the typical detector acceptance 0.0015 < ξ 1 < 0.5 and 0.0015 < ξ 3 < 0.15, leading single top photoproduction is the better channel to test anomalous tqγ couplings than single top jet channel. While for 0.1 < ξ 2 < 0.5, single top jet channel becomes better. We conclude that both channels can be used to detect such anomalous tqγ couplings and the detection sensitivity on κ tqγ is obtained.