Next-to-leading order QCD and electroweak corrections to Higgs-strahlung processes at the LHC

In this paper we calculate the total and fiducial cross sections as well as differential distributions for the Higgs-strahlung or VH process $p p \to VH \to l\nu_l/l^- l^+ + H$, (V = W or Z, l=e,$\mu$) including QCD and electro-weak corrections up to next-to-leading order before and after reweighting photon PDFs of NNPDF2.3qed, NNPDF3.0qed, MRST2004qed, CT14QEDinc, and LUXqed at the LHC with 13 TeV and Higgs boson mass $\ M_{H}=125$ GeV. The predictions from the various photon PDFs before and after reweighting against each other are in good agreement. The photon PDF uncertainties of the photon-induced cross sections decrease significantly with the reweighting PDFs.


I. INTRODUCTION
The Standard Model (SM) predicts the Higgs boson, that is, the remnant of the electroweak (EW) symmetry-breaking mechanism that generates the gauge boson and fermion masses [1]. One of the most important Higgs boson production mechanisms at hadron colliders is the Higgs-strahlung process, i.e. the associated production of Higgs bosons and weak gauge bosons, pp → W H + X → H ν l l + X and pp → ZH + X → H l + l − + X In order to determine the Higgs boson properties with high precision at the LHC, it is necessary to calculate the higher order quantum chromodynamics (QCD) and EW corrections.
One particular EW correction of interest is that due to photons coming from the proton in the initial state. It is thus necessary to use photon parton distribution functions (PDFs) both for consistency when EW corrections are included, and because photon-induced processes can become important at high energies. So far, a number of PDF groups, namely, the MRST, NNPDF and CTEQ collaborations, have introduced photon PDFs along with PDF evolution at leading order (LO) in QED and next-to-leading order (NLO) or next-to-next-to-leading order (NNLO) in QCD. Reference [2] calculated NLO QCD corrections to the pp → W ± H + X (sum of W + H and W − H) and pp → ZH + X total cross sections at the LHC at 16 TeV and 40 TeV, and discussed in detail the total cross section dependence on the choice of factorization scale and for three different sets of PDFs (MT, KMRS and DFLM) within the DIS scheme and MS scheme for m H = 100 GeV. References [3,6] presented the impact of the EW corrections on the cross section predictions for the processes pp/pp → W ± H + X and pp/pp → ZH + X at the Tevatron ( √ s = 1.96 T eV ) and LHC ( √ s = 14 T eV ) for the three different input parameter schemes (G µ , α(M Z ), and α(0)-schemes), and studied the theoretical uncertainties induced by factorization and renormalization scale dependence and by the PDFs by using the CTEQ6L1 and CTEQ6M [4] PDFs, respectively. The EW corrections in the G µ and α(M Z )-schemes are significant and reduce the cross section by 5-9% and by 10-15%, respectively. The EW corrections in the α(0)-scheme are rather small. References [5,6] [8] have discussed the impact of EW radiative corrections to the Higgs-strahlung processes at the Tevatron (1.96 TeV) and LHC (7 TeV and 14 TeV) within the G µ -scheme for M H = 120GeV using the HAWK Monte Carlo program. The LHC Higgs Cross Section Working Group [9] provided an update of the total and fiducial cross-section, together with theoretical uncertainties from renormalization and factorization scales, of the process pp → W ± /Z + H → lν l /l + l − + H including NNLO QCD and NLO EW corrections for different proton-proton collision energies ( √ s = 7T eV, 8T eV, 13T eV, 14T eV ) for a Higgs boson mass M H = 125GeV . Reference [13] studied the impact of the CMS measurements of W boson pair production at the LHC at 8 TeV on the NNPDF2.3qed, NNPDF3.0qed [14], CT14QEDinc [15], and LUXqed [16] photon PDFs, and found that the data strongly suggest that the NNPDF photon error band should be significantly reduced. Therefore, it is necessary to re-examine the physics effects induced by photon-initiated process. In this paper, by using various PDFs, we update the predictions for the processes pp → W ± /Z + H → lν l /l + l − + H at the LHC at 13 TeV and Higgs boson mass M H = 125 GeV, including the NLO QCD and EW corrections using the most recent PDFs. The paper is organized as follows. In Section II, first we give the Feynman diagrams for the Higgs-strahlung processes pp → W ± /Z + H → lν l /l + l − + H at LO and NLO. Then we provide our numerical results for these processes at the LHC with center-of-mass energy 13 TeV using the Monte Carlo program HAWK [17] by adopting the NNPDF2.3qed, NNPDF3.0qed, MRST2004qed, CT14QEDinc, and NNLO LUXqed PDFs, as well as their reweighting photon PDFs. We obtain reweighting photon PDFs by using the reweighting method [18][19][20][21] and the LHC CMS 8 TeV data [22] as in Ref. [13]. Our conclusion is given in Section III.

A. Feynman Diagrams
The LO Feynman diagrams for Higgs boson production in association with weak gauge bosons V = W ± , Z, at the proton-antiproton collider Tevatron and proton-proton collider LHC, are shown in Fig. 1. The corresponding partonic processes are q i,uqj,d → HW Hl + ν l and q i,dqj,u → HW − → Hl −ν l , where q i,u and q i,d denote up-and down-type quarks of the ith generation, as well as q iqi → HZ → Hl + l − , where q i denotes any light quark of the ith generation.
At NLO the corrections to the Higgs-strahlung process includes QCD corrections (virtual corrections with gluon exchange in the qq vertex, gluon-induced processes, and the emission of an additional gluon) and complete EW corrections (incoming photon corrections, virtual Vertex diagrams: Box diagrams: In this section, we present numerical results for total and differential cross sections for associated W ± H (W + → l + ν l and W − → l −ν l , where l = e, µ) and ZH (Z → l + l − ) production with NNPDF2.3qed, NNPDF3.0qed, MRS2004qed, CT14QEDinc, and LUXqed photon PDFs at the LHC at 13 TeV including the NLO QCD and EW corrections.

B. Input Parameters and Cuts
In our numerical analysis of the VH processes (V = W ± , Z) we use the same VH input parameters and cuts as in Ref. [9], The electromagnetic coupling is fixed in the G F -scheme, Feynman diagrams corresponding to the EW real corrections to the LO processes. and the weak mixing angle is defined in the on-shell scheme, Both the factorization and the renormalization scales are set to the value of the invariant mass of the VH system, We also provide results for the total cross section and differential cross section in the same kinematic region as in Ref. [9], where p T l is the transverse momentum of the lepton and y l is its rapidity. For Z(→ l + l − )H (a)Virtual diagrams: production the invariant mass of the two leptons is in the range 75GeV < M ll < 105GeV. (6)

C. Total Cross Sections
First we study the effect of the NLO corrections on the LO cross sections without any cuts on the final-state charged leptons using the SM input parameters given in Section II B. The total NLO VH cross sections σ V H N LO and relative corrections δ QCD/EW are calculated according to and For the LO calculations, strictly speaking, one should use LO PDFs, but for convenience, we used NLO PDFs. In Tables I, II, and III we list the LO and NLO total cross sections (σ LO , σ NLO ) including NLO QCD and EW corrections (δ QCD , δ EW ), and photoninduced cross sections (σ γ , σ γ P R ) with PDF uncertainties for the Higgs-strahlung processes pp → W ± /Z + H → lν l /l + l − + H with NNPDF2.3qed, NNPDF3.0qed, MRS2004qed, CT14QEDinc, and LUXqed photon PDFs and Higgs-boson mass M H =125 GeV at the LHC at √ s=13 TeV. The photon-induced cross section is only significant for WH production.
Note that we did not apply reweighting to the LUXqed photon PDFs since the PDF uncertainty of the photon-induced cross section is already small. We see from Tables I and  II that the photon-induced cross section σ γ and uncertainties depend on the choice of the photon PDFs. Especially, the photon PDF uncertainties decrease significantly as we change from NNPDF2.3qed and NNPDF3.0qed to MRST2004qed, CT14QEDinc, and to LUXqed. The NLO QCD corrections for various photon PDFs are of the order of +17.0% for total cross sections. The EW corrections are insensitive to the choice of the photon PDFs, and are about −7.0%. The reweighting PDF uncertainties of the photon-induced cross section σ γ P R are reduced significantly after including the CMS data to reweight the NNPDF2.3qed, NNPDF3.0qed, MRS2004qed, and CT14QEDinc photon PDFs, while the corresponding central predicted cross sections are unchanged. From Table I (pp → H + W + (→ l + ν l )), the relative errors ∆σ γ /σ γ for the photon-induced cross sections reduce from 88.7% and 89.5% in the NNPDF2.3qed and NNPDF3.0qed predictions to 5.5% and 8.2% in the reweighting NNPDF2.3qed and reweighting NNPDF3.0qed predictions; the relative errors ∆σ γ /σ γ reduce from 23.9% in the MRST2004qed predictions to 7.6% in the reweighting MRST2004qed predictions; the relative errors ∆σ γ /σ γ reduce from 13.4% in the CT14QEDinc predictions to 7.1% in the reweighting CT14QEDinc predictions. From Table II    we see that the relative errors ∆σ γ /σ γ of the photon-induced cross sections reduce from 96.6% and 95.4% in the NNPDF2.3qed and NNPDF3.0qed predictions to 5.7% and 9.5% in the reweighting NNPDF2.3qed and reweighting NNPDF3.0qed predictions; the relative errors ∆σ γ /σ γ reduce from 24.5% in the MRST2004qed predictions to 7.6% in the reweighting MRST2004qed predictions; and the relative errors ∆σ γ /σ γ reduce from 11.8% in the CT14QEDinc predictions to 6.3% in the reweighting CT14QEDinc predictions. From Table  III (pp → H + Z(→ l + l − )) we see that the photon-induced cross section contributions is small for various photon PDFs; the NLO QCD corrections are of the order of +17% for total cross sections; the EW corrections are insensitive to the choice of the PDF set, and are about −5%; the PDF uncertainties of the photon-induced cross sections σ γ P R are reduced

D. Fiducial Cross Section
Apart from the total cross section without any cuts, in this subsection we consider the total cross section after applying cuts to the charged leptons. In Tables IV, V, and VI we list the LO and NLO total cross sections (σ LO , σ NLO ) including NLO QCD and EW corrections (δ QCD , δ EW ), and photon-induced cross sections (σ γ , σ γ P R ) with PDF uncertainties for the Higgs-strahlung processes pp → W ± /Z + H → lν l /l + l − + H with kinematical cuts in Section II B with NNPDF2.3qed, NNPDF3.0qed, MRS2004qed, CT14QEDinc, and LUXqed photon PDFs and Higgs boson mass M H =125 GeV at the LHC with √ s=13 TeV. The LO and NLO total cross sections are reduced by the kinematical cuts. The NLO QCD corrections for VH do not depend on the cuts on the final state charged lepton, but the EW corrections and the photon-induced cross section do depend on them. The photon-induced cross section is only significant for WH production. It is about 10 −3 fb for the pp → H +Z(→ l + l − ) process, as we can see from The reweighting PDF uncertainties of the photon-induced cross sections σ γ P R are reduced significantly after including the CMS data to reweight the NNPDF2.3qed, NNPDF3.0qed, MRS2004qed, and CT14QEDinc photon PDFs, while the corresponding central predicted cross sections are unchanged. From Table IV (pp → H + W + (→ l + ν l )) we see that the relative errors ∆σ γ /σ γ for the photon-induced cross sections reduce from 55.5% and 58.7% in the NNPDF2.3qed and NNPDF3.0qed predictions to 10.5% and 7.2% in the reweighting NNPDF2.3qed and reweighting NNPDF3.0qed predictions; the relative errors ∆σ γ /σ γ reduce from 21.6% in the MRST2004qed predictions to 6.6% in the reweighting MRST2004qed predictions; the relative errors ∆σ γ /σ γ reduce from 10.6% in the CT14QEDinc predictions to 5.5% in the reweighting CT14QEDinc predictions. From Table V (pp → H + W − (→ l −ν l )) we see that the relative errors ∆σ γ /σ γ for the photon-induced cross section reduce from 57.6% and 62.4% in the NNPDF2.3qed and NNPDF3.0qed predictions to 10.3% and 7.4% in the reweighting NNPDF2.3qed and reweighting NNPDF3.0qed predictions; the relative errors ∆σ γ /σ γ reduce from 22.2% in the MRST2004qed predictions to 6.9% in the reweighting MRST2004qed predictions; and the relative errors ∆σ γ /σ γ reduce from 11.3% in the CT14QEDinc predictions to 5.9% in the reweighting CT14QEDinc predictions.   GeV without kinematic cuts. The last row is the photon-induced process fiducial cross sections and

E. Fiducial Differential Cross Section
In this subsection we consider the differential cross sections after applying cuts to the charged leptons. We plot the transverse-momentum distributions for Higgs production in pp → W ± /Z + H → lν l /l + l − + H at LO and NLO in QCD for Higgs boson mass M H = 125 GeV with the kinematical cuts and set up described in Section II B. The p T (H) distributions of the Higgs boson are shown in Figs. 5, 6 and 7 for various PDFs. From the NLO (upper) and LO (lower) p T (H) distribution curves we see that the NLO correction is small at small p T (H), and increases slightly with p T (H) as the cross section falls rapidly. The upper left plot is the transverse-momentum distributions of Higgs boson production at LO (lower curves) and NLO (upper curves) including NLO QCD and EW corrections. The lower left plot is the photon correction δ γ = dσ γ /dσ LO   The bottom right panel is the ratio of δ γ /δ γLU Xqed .