NLO QCD corrections to Single Top and W associated production at the LHC with forward detector acceptances

In this paper we study the Single Top and W boson associated photoproduction via the main reaction $\rm pp\rightarrow p\gamma p\rightarrow pW^{\pm}t+Y$ at the 14 TeV Large Hadron Collider (LHC) up to next-to-leading order (NLO) QCD level assuming a typical LHC multipurpose forward detector. We use the Five-Flavor-Number Schemes (5FNS) with massless bottom quark assumption in the whole calculation. Our results show that the QCD NLO corrections can reduce the scale uncertainty. The typical K-factors are in the range of 1.15 to 1.2 which lead to the QCD NLO corrections of 15$%$ to 20$%$ correspond to the leading-order (LO) predictions with our chosen parameters.


Introduction
The Large Hadron Collider (LHC) at CERN generates high energetic proton-proton (pp) collisions with a luminosity of L = 10 34 cm −2 s −1 and provides the opportunity to study very high energy physics. After the discovery of Higgs boson [1,2], probing new physics beyond the Standard Model (BSM) turns to the main goal of the LHC. In such context, studying the heaviest elementary particle, the top quark, is particularly interesting since it is the only fermion with a natural Yukawa coupling to the Higgs boson of the order of unity. Its charged weak coupling might be sensitive to the existence of an additional heavy fermion. These couplings can be probed by measuring specific top quark production cross sections and branching ratios. However, these measurements will be challenging due to the composite internal structure of the colliding particles, i.e., the large QCD or electroweak (EW) backgrounds, the unknown precise centre-of-mass (c.m.s.) energy of the collisions occurring between the partons of protons, the complicate composition of underlying events within the central detector, etc. In this case, very high energy interactions involving quasireal incoming photons may provide a solution to some of these problems.
General diagrams for the photon induced interactions at the LHC is presented in to photoproduction or photon-proton (γp) production: a photon from a proton induces a deep inelastic scattering with the incoming proton and produces a proton remnant Y in addition to the centrally produced X system. Despite a lower available luminosity, photoproduction can occur under better known initial conditions, with fewer final states particles and at high energy scale (∼ TeV), thus can be studied as a complementary tool to normal pp collisions at the LHC. Indeed, the CDF collaboration has already observed such kinds of phenomenon including the exclusive dilepton [3,4], diphoton [5], dijet [6] production and charmonium (J/ψ) meson photoproduction [7], etc. Both the ATLAS and the CMS collaborations have programs of forward physics. They are devoted to studies of high rapidity regions with extra updated detectors located in a place nearly 100-400m close to the interaction point [8,9,10]. Technical details of the ATLAS Forward Physics (AFP) projects can be found, for example, in Refs. [11,12]. A brief review of experimental prospects for studying photon induced interactions are summarized in Ref. [13].
As previously mentioned, the top quark is the heaviest known elementary particle which makes it an excellent candidate for new physics searches. Among top quark production channels, single top production has some special features that top pair production can not achieve: it offers a unique possibility of the direct measurement of V tb , the Cabibbo-Kobayashi-Maskawa quark-mixing matrix (CKM), allowing non-trivial tests of the properties of this matrix in the SM [14,15,16]. In normal pp collision, single top produces mainly through two body s-channel (t-channel) single top in association with a b (light) quark, Wt channels and three body tbq ′ channel. Here we focus on the study of Wt channel. This channel is invisible at the Tevatron, however, it will be important at the LHC and even comparable to single top s-channel production. Even though, its cross section is almost a factor 100 smaller than the most dangerous background coming from the tt process. This makes the measurement error of Wt process as large as ∼ 41% for an integrated luminosity of 10 fb −1 through normal pp collision [17]. Single top production can also proceed through γp collisions mentioned above. This time through mainly Wt and tbq ′ channels. Compare these two channels, we find that cross section of Wt channel (order of ∼ 1 pb), is much larger than that of tbq ′ channel (order of ∼ 0.028 pb), and becomes the most important single top production channel at the γp collision at the LHC. This is different from normal pp collision that tbq ′ production channel accounts for 44.2% (39%) of the total single top quark production cross section, while Wt channel stands only 28% (5%) at LHC (Tevatron) [18]. In contrast, Wt channel stands over 40% of the top quark photoproductions, since the top pair photoproduction has a cross section of only ∼ 1.5 pb. Enhancement of the ratio σ Wt /σ tt might certainly be a good feature of related measurements through Wt channel.
First results on the measurement of the V tb matrix element using Wt photoproduction are presented in Refs. [19,20]. There comes the conclusion that the expected error on the measurement of V tb is 16.9% for the semi-leptonic channel and 10.1% for the leptonic one after 10 fb −1 of integrated luminosity, while the expected uncertainty from the equivalent study based on partonic interactions is 14% [21] using the same integrated luminosity, showing that photoproduction is at least competitive with partonic-based studies and that the combination of both studies could lead to significant improvement of the error.
In addition to the V tb measurement, Wt photoproduction can also be used to study the W-t-b vertex and test precisely the V-A structure of the charged current weak interaction of the top quark. Anomalous measurement of this vertex may lead direct evidence of new physics beyond the SM. It may manifest itself via either loop effects or inducing non-SM weak interactions to introduce new single top production channels. Typical studies involve, i.e., measuring anomalous W-t-b coupling in ep collision [22,23], in normal pp collision [24,25,26] as well as in γp collision at the LHC [27]. Ref. [27] studies pp → pγp → pW ± t + Y up to the leading order (LO) induced by anomalous W-t-b coupling. In this case, SM Wt photoproduction turns to its irreducible background. Moreover, a lot of studies are performed at the γp colliders, i.e., testing anomalous gauge boson couplings [28,29,30,31,32,33] or probing flavour changing neutral currents (FCNC) through single top photoproduction [34,35], etc. In cases like these, SM Wt photoproduction turns to be a most important reducible background that needs precise measurement and analysis. Furthermore, determining Wt production cross sections and once compared with experiments, will provide a direct access to the bottom quark parton density in nucleons and help understand the nature of the b quark parton distribution function (PDF).
As a result, accurate theoretical predictions including higher order QCD corrections for Wt photoproduction are needed. NLO QCD corrections for Wt production in the normal pp collisions have already been well studied [36,37,38,39,40,41,42,43]. In this paper, we present its production at the γp collision for the first time, assuming a typical LHC multipurpose forward detector, including the NLO QCD corrections via the main reaction pp → pγp → pW ± t+Y. Typically, we use the Five-Flavor-Number Schemes (5FNS) with massless b quark mass assumption through the whole calculation. Our paper is organized as follow: we build the calculation framework in Section 2 including brief introduction to the Equivalent Photon Approximation (EPA), general inelastic photoproduction cross section, LO and QCD NLO Wt photoproductions. Section 3 is arranged to present the input parameters, cross checks and numerical results of our study. Finally we summarize our conclusions in the last section. In our paper, we focus on the discussion of photoproductions pp → pγp → pXY through γp collisioins see Fig.1[right figure]. Photoproduction is a class of processes in which one of the two interacting protons is not destroyed during the collision but survive into the final state with additional particle (or particles) state(s). Protons of this kind are named intact or forward protons. The kinematics of a forward proton is often described by means of the reduced energy loss ξ, which is also defined as the forward detector 50, 51], unparticle physics [52], top triangle moose model [53], gauge boson self-interactions [28,29,30,31,32,33,54,55,56,57], neutrino electromagnetic properties [58,59,60], the top quark physics [19,20,27,34,35], dark matter searches [61] and triplet Higgs production [62], etc.

General γp Photoproduction Cross Section
We denote the general photoproduction processes at the LHC, no metter at LO or NLO level, as 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 are its lower (upper) limits which means that the forward detector acceptance satisfies ξ min ≤ ξ ≤ ξ max . x 2 is the momentum fraction of the proton momentum carried by the gluon (quark). The quantityŝ = z 2 s is the effective c.m.s. energy with z 2 = x 1 x 2 . s = 4E 2 mentioned above and 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. f = dN dEγ dQ 2 is the Q 2 dependent relative luminosity spectrum present in Eq.(2). Q 2 max = 2GeV 2 is the maximum virtuality.
1 avgfac is the product of the spinaverage factor, the color-average factor and the identical particle factor. |M n | 2 presents the squared n-particle matrix element and divided by the flux factor [2ŝ(2π) 3n−4 ]. The n-body phase space differential dΦ n and its integral Φ n depend only onŝ and particle masses m i due to Lorentz invariance: with i and j denoting the incident particles and k running over all outgoing particles. We denote the Wt photoproduction process as:

Wt Photoproduction at Leading Order
where p i are the particle four momentums. There are four LO Feynman diagrams for this partonic process as shown in Fig.2. There Fig.2(1) and Figs.2(2-4) are the s-channel and t-channel diagrams for the partonic process, respectively. Fig.2(3) include b-t-G vertex that can be safely omitted in the massless b quark assumption. We only consider the W − t production while its charge-conjugate contribution is the same [41]. In order to describe the process γ( Here notation of e i/ix equal p i/ix /( √ŝ /2) and is needed in our following description.
The LO cross section for the partonic process γb → W − t is obtained by using the following formulaσ where dΦ 2 is the two-body phase-space element, and p 1 is the momentum of the initial photon in the c.m.s.. The integration is performed over the two-body phase space of the final particles W − t. The summation is taken over the spins and colors of the initial and final states, and the bar over the summation indicates averaging over the intrinsic degrees of freedom of initial partons.
The LO total cross section for pp → pγp → pW − t can be expressed as There G i/P j , i=b, j = A, B represent the PDFs of parton i in proton P j , µ f and µ r are the factorization and renormalization scales separately.
Here we use f γ/P A (x 1 ) to take place of the dEγ dQ 2 in Eq.(5) for simplicity. And we address here that during calculation, we use the ξ, Q 2 dependent form of Eq.(5).

General Description
We use the Five-Flavor-Number Scheme (5FNS) in our whole LO and QCD NLO calculations. As we see, at tree level in the 5FNS scheme the Wt photoproduction process consists of only one partonic subprocess, namely γb → W − t, as illustrated in Fig [70] present that being often the effects of resummation very mild, 4FNS calculations can be put to use, on the other hand, for 5NFS schemes, i.e., can typically provide quite accurate predictions for total rates and being simpler, in some cases allow the calculations to be performed at NNLO. We address the interesting of considering both schemes while here we use 5FNS in our calculation. Even in 5FNS, it will be interesting to consider two schemes [71,72]: one is the massless b quark scheme where we drop the mass of b quark during calculation while the other is the massive b quark scheme where we retain it everywhere.
In our paper, we adopt the 5FNS scheme with the massless b quark assumption. In this case, the first order of QCD corrections to the pp → pγp → pW − t + Y consist of: • The QCD one-loop virtual corrections to the partonic process γb → W − t.
• The contribution of the real gluon radiation partonic process γb → W − t + g.
• The contribution of the real b-quark emission partonic process γg → W − t +b.
• The corresponding contributions of the PDF counterterms.
We use the dimensional regularization method in D = 4 − 2ǫ dimensions to isolate the ultraviolet (UV) and infrared (IR) singularities. In massless b quark scheme, we split each collinear counter-term of the PDF, δG b/P (x, µ f ) (P=proton), into two parts: the collinear gluon emission part δG gluon b/P (x, µ f ) and the collinear b quark emission part δG quark b/P (x, µ f ). The analytical expressions are presented as follows and the explicit expressions for the splitting functions P ij (z), (ij = bb, bg) can be found in Ref. [73].

Virtual
The amplitude at the QCD one-loop level for the partonic process γb → W − t in the SM contains the contributions of the self-energy, vertex, box and counter-term graphs which are shown in Fig.3(1)- (18). Same as the leading level that diagrams include b-t-G vertex that can be safely omitted in the massless b quark scheme. Even in the massive b quark scheme, their contributions are also quite small.
To remove the UV divergences, we need to renormalize the mass of the quarks and the wave function of the quark fields. In the massless b quark assumption we introduce Figure 3: QCD one loop Feynman diagrams for γb → W − t in the SM. the following renormalization constants: where m t are the top-quark mass. ψ L,R b(t) denote the fields of bottom (top) quark. For the masses and wave functions of the fields are renormalized in the the on-shell scheme and the relevant counter-terms are expressed as with refer to the UV(IR) divergences. For massless b quark, there is no need to renormalize the mass of bottom and we use modified minimal subtraction (MS) scheme for b field as: UV singularities are regulated by adding renormalization part to the virtual corrections only leaving IR singularities that will be removed by combining the real emission corrections. We calculate the virtual one-loop corrections (σ V ) using a Feynman diagram approach based on FeynArts, FormCalc and our modified LoopTools (FFL) [74,75,76] packages. Tensor one loop integrals are checked with OneLoop [77] and QCDLoop [78] packages. Figure 4: The tree parton level Feynman diagrams for the real gluon emission subprocess γb → W − tg related to Eq.(16).  The first order of QCD corrections also consist of the real corrections in the form of other two subprocesses with an additional radiated parton, namely gluon emission and real (anti) quark emission presented as

Parton Radiation
with Feynman diagrams depicted in Fig.4 and Fig.5, respectively.
In the massless b quark scheme, singularities associated with initial state collinear . If in this case, a resonant t propagator (with flowing momentum equal or close to p W + p b ) is encountered, a divergence will arise. Actually these diagrams can be interpreted as the production of a tt pair at LO, with subsequent decay of the top into a Wb system. This is the well known interference between Wt and tt production, namely doubly resonant.
Such resonant becomes extremely large in certain phase space region and renders the perturbative computation of the Wt cross section meaningless, thus should be preferable excluded from the NLO corrections to the Wt process.
Several approaches have been outlined in the literature, i.e., making a cut on the invariant mass of the Wb system to prevent the t propagator from becoming resonant [18], subtracting the contribution from the resonant diagrams so that no on-shell piece remains [79,80], bottom quark PDF method, technically, perform calculation of the Wt process by applying a veto on the p T of the additional b quark that appears at next-toleading order aids the separation of this process from doubly-resonant tt production [42], etc [41,81]. Here in our case we use the PROSPINO scheme [82] which is defined as a replacement of the Breit-Wigner propagator This subtraction scheme helps to avoid double counting and to not artificially ruin the convergence of the perturbative QCD description of these production channels with the remove of on-shell particle contributions from the associated production. This scheme has be done in some other refs like [83,84,85,86], etc.
Some attention should be paid to the light-(anti)quark emission subprocess, see Figs.5 (5) and Fig.5(10). Contributions from these two diagrams can be considered as part of the NLO EW corrections to the LO process pp → gb → Wt through normal pp collision.
Since we concentrate on the photoproduction of pp → pγp → pW − t where forward protons are considered, the pp → gb → Wt and its full NLO EW corrections are not taken into account. Then, the subprocesses in Figs. 5(5) and 5(10) are defined applying a small p T cut on the tagged b quark, which regularize the collinear splitting of the photon into a bb pair. Indeed, the choice of specific kinematical cuts to select events in the forward re-gion (small p T cut applied on the tagged b quark) forces the contributions from these two diagrams to be quite small. Thus, even when one is forced to consider them as part of the NLO QCD corrections to the pp → pγp → pWt production process, their contribution results for only but a tiny theoretical uncertainty.
Then the cross section for each of the real emission partonic processes can be written asσ R =σ S +σ H =σ S +σ HC +σ HC . After integrating over the photon and quark spectra, we get the real contributions as σ R = σ S + σ HC + σ HC .

Total QCD NLO Cross Section
After combining all the contributions that are mentioned before, the UV, IR singularities in our final total cross section are exactly cancelled. The logarithmic dependence on the arbitrary small cutoff parameters δ s and δ c are then cancelled (but power-like terms survive). These cancelations can be verified numerically in our numerical calculations. The final results of the total QCD NLO cross section in the 5FNS scheme can be expressed as: Here F soft and F hc are the factors contain soft and collinear singularities as well as finite terms. F 1,2 are the factors that finite. In the massless b quark assumption, there analytical expression are and Notations of e i/ix can be found in Eq. (8). The dilogarithm function Li 2 (x) is defined in Ref [87]. Technical details can be found in Ref [73,88].

Input Parameters
We take the input parameters as M p = 0.938272046 GeV, α ew (m Neglecting terms of order m 2 b /m 2 t , α s , and (α s /π)m 2 W /m 2 t , the width predicted in the SM at NLO is: By taking α ew (m 2 Z ) −1 | MS = 127.918 and α s (m 2 t ) = 0.1079, we obtain Γ t = 1.41595 GeV. Based on the forward proton detectors to be installed by the CMS-TOTEM and the ATLAS collaborations we choose the detected acceptances to be • 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. During calculation we use ξ 1 in default unless otherwise stated. Note here we do not consider the decay of the heavy final states as well as the survival probability in the γp collision or simply taken to be unit.

Cross Checks
Before presenting the numerical predictions, several cross checks should be done.
• First, during the calculation of the tensor one loop integrals, we use our modified LoopTools and cross check with OneLoop [77] and QCDLoop [78] packages. We can get exactly the same results in each phase space point.
• Second, when do the phase space integration we use BASES [92] and cross check independently with Kaleu [93] especially for the hard emission contributions. We can get the same integrated results within the error.
• Third, the UV and IR safeties should be verified numerically after combining all the contributions at the NLO QCD loop level. To check this, we display enough random phase space point as well as the cancellation for different divergent parameters, see in Table.1 corresponding to 5FNS massless b quark scheme. One thing that should be emphasized isσ V should include the counter-term contributions as well as the soft and collinear singularity terms coming from the real emissions. We implement this into our monte carlo codes which provide an automatic check of the dependence on these divergence parameters. We can see the UV and IR divergence can be canceled at high precision level in all the phase space thus leading the continuance of our following calculation.
• Fourth, since the total cross section is independent of the soft cutoff δ s (= ∆E g /E b , E b = √ŝ /2) and the collinear cutoff δ c , we display their values for pp → pγp → pWt versus the cutoff δ s , where we take δ c = δ s /100. Both δ s and δ c dependence should be  Table.2. The detector acceptance here is chosen to be 0.0015 < ξ 1 < 0.5. It is shown clearly that the NLO QCD correction does not depend on the arbitrarily chosen values of δ s and δ c within the calculation errors. In the further numerical calculations, we fix δ s = 10 −4 and δ c = δ s /100.

Scale dependence for different forward detector acceptances
We present the scale-µ dependence of the LO and QCD NLO corrected cross sections for pp → pγp → pWt + Y for the CMS-TOTEM forward detectors with 0.0015 < ξ 1 < 0.  Table.3 for different forward detector acceptances. In our further calculations we fix µ = µ 0 = (m W + m t )/2.

Distribution and Cross Section
In Fig.9, we show the LO (solid curves) and NLO (dotted curves) transverse momentum      Rapidity distributions for the W boson and top quark have been presented in Fig.10 and Fig.11. As can be seen the NLO corrections can shift the LO rapidity but in different ways for both W boson and top quark. For the W boson the distribution y W , QCD NLO corrections shift the LO peak range into different y and enhance them. For the top quark rapidity distributions there is not much enhancement can be found, instead, the rapidity values where they peaked shifts. Same behaviors but different values can be found for the other choices of ξ 2 and ξ 3 as can be see in Fig.10(a-c) and Fig.11(a-c),respectively. Their corresponding K-factors are present in the right panels. The reduction can be found for the K-factor when y W increase from -4 to -1. With y W < −0.8 NLO correction enhance the LO predictions while reverse in the range y W ≥ −0.8. This behavior is the same for all three value of forward detector acceptances ξ 1 , ξ 2 and ξ 3 . For y top , this value is around 0.2. That is to say, for y top < 0.2 NLO correction enhance the LO predictions while reverse in the range y top ≥ 0.2.
In Fig.12 we fix ξ min = 0.0015 and take ξ max as a running parameter from 0.15 to 1. The LO and NLO cross sections as well as the the K-factor defined as σ NLO /σ LO are presented as functions of different values of ξ max . The dotted, dashed and solid lines correspond to the LO, NLO predictions and K-factor, respectively. We can find that in the range 0.0015 < ξ max < 0.5, both LO and NLO predictions rely on the detector acceptances while in the region ξ max > 0.5, little contributions will shift the LO and NLO cross sections. No matter how the detector acceptances changes, the ratio of σ NLO to σ LO does not change much where a typical value of K-factor equal 1.1808 in the massless assumption, leading the QCD NLO corrections up to 18.08% related to the LO predictions with our chosen parameters.

Summary
In this work, we present the precise production of Single Top and W boson associated photoproduction up to NLO QCD level through the main reaction pp → pγp → pW ± t + Y at the future 14 TeV Large Hadron Collider (LHC) for the first time, assuming a typical LHC multipurpose forward detector. We use the Five-Flavor-Number Schemes (5FNS) through the whole calculation while treat the initial state b quark as massless.
This is the most important two body final state single top production channel at the