The precision predictions of Higgs-Boson production cross section become the most important topic after the discovery of the Higgs-Boson [1, 2] at the Large Hadron Collider (LHC). There were enormous amount of research works have been performed in providing higher-order corrections to the Higgs-Boson cross section and reducing the theoretical uncertainty of the Standard Model (SM) predictions via various production mechanism of the SM Higgs-Boson, namely, the gluon gluon fusion through a heavy-quark loop $p\,p(g\,g)\to H+X$, the Higgs-Strahlung or VH process $p\,p\to V\,H+X$, (V = W, Z), and vector-Boson-fusion (VBF) process—the production of a Higgs-Boson in association with two hard quark jets $p\,p\to H+{jj}+X$ [3-5]. References [6, 7] calculated next-to-leading order (NLO) QCD and electro-weak (EW) corrections to the ${pp}\to {W}^{\pm }H+X$ and ${pp}\to {ZH}+X$ at the LHC and Tevatron using Monte Carlo program HAWK [8]. References [9-11] studied the NLO QCD corrections for the VBF processes qQ → qQH for a Higgs-Boson in the mass range m_H = 115-200 GeV, and computed differential cross sections in rapidity and transverse momentum of the quark jet at a pp collider with $\sqrt{s}=14$ TeV by Monte Carlo program MCFM [12] with CTEQ6L1 at leading order (LO) and CTEQ6M at NLO [13]. By using structure function approach [14] presented total cross section at next-to-NLO (NNLO) in QCD as a function of m_H for a $\sqrt{s}=7$ TeV LHC for Higgs production via VBF by employing MSTW (parton distribution functions) PDFs [15]. References [16, 17] studied strong and EW correction to the production of Higgs+2 jets via weak interaction at the LHC at $\sqrt{s}=14$ TeV using MRST2004QED [18] for various M_H = 120, 150, 170, 200, 400, 700 GeV.

In this paper, we calculate the production of Higgs+2 jets at the $\sqrt{s}=7,8,13,14$ TeV for m_H = 125 GeV by using the Monte Carlo program HAWK [8] for Higgs production in VBF and Higgs-Strahlung, and it includes the complete NLO QCD corrections (real correction, virtual correction and incoming gluon correction) and EW corrections (real correction, virtual correction, and incoming photon correction) for MMHT2015qed [19], NNPDF3.1luxQED [20], LUXqed [21] and CT14QEDinc [22] photon parton distribution functions (PDFs).

The representative LO Feynman diagrams for the partonic processes qq → Hqq, $q\bar{q}\to {Hq}\bar{q}$, and $\bar{q}\bar{q}\to H\bar{q}\bar{q}$ are shown in figure 1, that include VBF weak-Boson fusion diagrams and quark-antiquark annihilation diagrams, i.e. t-channel, u-channel diagrams with VBF-like vector-Boson exchange and s-channel Higgs-Strahlung diagrams with hadronic decays of the weak-Boson. Representative Feynman diagrams corresponding to the NLO QCD and EW corrections to the LO processes are shown in figure 2. NLO QCD corrections to the Higgs+2 jets include incoming gluon corrections, outgoing gluon corrections, and the virtual NLO QCD corrections by adding a gluon loop to qqV vertex; and the EW corrections include incoming photon corrections, outgoing photon correction, virtual EW correction.

In the following, we study NLO total cross sections without any cuts and cross sections with VBF cuts of the production of Higgs+2 jets at the LHC. The details of the NLO calculations of the EW and QCD corrections to Higgs production via VBF at the LHC have already been described in [9-11, 17]. The NLO numerical calculation is embedded in the Monte Carlo program HAWK [8], which uses the $\overline{{MS}}$ factorization for QCD and DIS factorization for QED. We use only five quark flavors for the external partons, i.e. we have included the effect of initial- and final-state b quarks in the calculations. For the PDFs, we use MMHT2015qed [19], NNPDF3.1luxQED [20], LUXqed [21], CT14QEDinc [22] photon PDFS. We use the default set of parameters in [8], in which: Fermi constant G_F = 1.16637 × 10⁻⁵ Ge V⁻², the strong coupling constant ${\alpha }_{s}({M}_{Z})=0.1187\pm 0.0015$, the Higgs mass ${M}_{H}=125.0\,\mathrm{GeV}$, the W Boson width ${{\rm{\Gamma }}}_{W}=2.08872\,\mathrm{GeV}$, the W Boson mass ${M}_{W}=80.398\,\mathrm{GeV}$, the Z-Boson width ${{\rm{\Gamma }}}_{Z}=2.4952\,\mathrm{GeV}$, the Z-Boson mass ${M}_{Z}=91.1876\,\mathrm{GeV}$. For both the renormalization and factorization scales we use $\mu ={\mu }_{R}={\mu }_{F}={M}_{W}$.

In this this section we also calculate production cross section of Higgs+2 jets after applying typical VBF cuts in [9, 10, 16, 17] on the outgoing jets. To reconstruct jets from the final-state partons the k_T-algorithm [23] with the pseudo-rapidity cut $| \eta | \lt 5$ on partons and jet resolution parameter (i.e. the jet size) D = 0.8 are used. We calculate the cross sections by imposing the following cuts for events with at least two hard jets with transverse-momentum p_T^j and rapidity y_j,Furthermore, we apply the rapidity gap cut between the two tagging jets, that are defined as the two jets passing the cuts 1, with highest p_T and ${p}_{{{Tj}}_{1}}\gt {p}_{{{Tj}}_{2}}$,

In tables 1-8 we show results for the LO cross section σ_LO, NLO cross section σ_NLO, photon-induced cross section σ_γ, relative QCD corrections δ_QCD, relative EW corrections δ_EW, relative incoming gluon corrections δ_g, relative photon-induced corrections ${\delta }_{\gamma }$ and total corrections δ_total, respectively, at the $\sqrt{s}=7,8,13,14$ TeV for ${m}_{H}=125\,\mathrm{GeV}$ without any phase space cuts (tables 1, 3, 5 and 7 ) and with VBF cuts (tables 2, 4, 6 and 8). The complete EW corrections δ_EW includes virtual and real EW corrections.

In tables 1, 3, 5 and 7, we list our results without applying VBF cuts in equations (1) and (2) for ${M}_{H}=125\,\mathrm{GeV}$. Without cuts, the QCD corrections are about +6%, +5%, +4% and +3% for $\sqrt{s}=7,8,13$ and 14 TeV, respectively; and the EW corrections about −4% and −5% for $\sqrt{s}=7,8$ TeV and 13, 14 TeV, respectively.

When VBF cuts are imposed, s-channel diagrams and interferences are suppressed, Higgs-Boson is produced via pure EW processes at LO involving only quark and antiquark parton distributions. In tables 2, 4, 6 and 8, we list our results by applying VBF cuts in equations (1) and (2). With cuts the QCD corrections vary between 2% and −4.7% for $\sqrt{s}=7,8,13$ and 14 TeV, while the EW corrections are approximately −6.0%.

From tables 1-8, we see that the incoming photon correction ${\delta }_{\gamma }$ increases the EW corrections roughly 1% for various PDFs (note that there is only a small difference of about 0.1% or 0.2% between the various photon PDFs). The incoming-gluon correction reduce the cross section by −3% to −4% for the photon PDFs under consideration.

In this paper we presented complete NLO QCD and EW corrections for Higgs+2 quark jets production using various photon PDFs, namely, MMHT2015qed, LUXqed, NNPDF3.1luxQED, CT14QEDinc for M_H = 125 GeV at the LHC at $\sqrt{s}=7,8,13,14$ TeV by using the Monte Carlo program HAWK. The LO and NLO total cross sections and corresponding NLO QCD and EW corrections for the processes $p\,p\to H+2{jets}+X$ are shown in tables 1-8. We find that the NLO EW corrections without VBF cuts decreases the total cross section roughly about −4% to −5% for various photon PDFs; the EW correction with VBF cuts is of the order of −6%, which also include real corrections from incoming photons, for total cross section for various photon PDFs. We found that photon-induced processes increase LO results by roughly 1%. The NLO QCD corrections without VBF cuts are about +6%, +5%, +4% and +3% for $\sqrt{s}=7,8,13$ TeV and 14 TeV, respectively. With cuts the QCD corrections vary between 2% and −4.7% for $\sqrt{s}=7,8,13$ and 14 TeV. The incoming-gluon correction is of the order of −3% to −4% for various photon PDFs. When we apply cuts the full NLO QCD+EW corrections for total cross section for MMHT2015qed, LUXqed, NNPDF3.1luxQED, and CT14QEDinc decrease the total cross section by 4%, 6%, 10% and 11% for $\sqrt{s}=7,8,13,14$ TeV, respectively.