Study of weak corrections to Drell-Yan, top-quark pair, and dijet production at high energies with MCFM

Electroweak (EW) corrections can be enhanced at high energies due to the soft or collinear radiation of virtual and real $W$ and $Z$ bosons that result in Sudakov-like corrections of the form ${\alpha }_W^l{\log }^n(Q^2/M_{W,Z}^2)$, where ${\alpha }_W=\alpha /(4\pi {\sin }^2{\theta }_W)$ and $n\le 2l-1$. The inclusion of EW corrections in predictions for hadron colliders is, therefore, especially important when searching for signals of possible new physics in distributions probing the kinematic regime $Q^2\gg M_V^2$. Next-to-leading order (NLO) EW corrections should also be taken into account when their size [$\mathcal{O}(\alpha )$] is comparable to that of QCD corrections at next-to-next-to-leading order (NNLO) [$\mathcal{O}({\alpha }_s^2)$]. To this end, we have implemented the NLO weak corrections to the neutral-current Drell-Yan process, top-quark pair production and dijet production in the parton-level Monte Carlo program MCFM. This enables a combined study with the corresponding QCD corrections at NLO and NNLO. We provide both the full NLO weak corrections and their Sudakov approximation since the latter is often used for a fast evaluation of weak effects at high energies and can be extended to higher orders. With both the exact and approximate results at hand, the validity of the Sudakov approximation can be readily quantified.

Figure 1

Feynman diagrams for the NC DY process at LO.

Figure 2

Weak vertex corrections of $\mathcal{O}(\alpha )$ to the NC DY process $(V^a=Z,W)$.

Figure 3

Gauge-boson self-energy correction at $\mathcal{O}(\alpha )$, where ${\widehat{\mathrm{\Sigma }}}_T^{V^a{\overline{V}}^b}$ is the renormalized vector boson self-energy, and ${\mathrm{\Gamma }}_{V^a}$ the width of the gauge boson ($V^a=\gamma ,Z)$. The red dot denotes the tree-level coupling to the initial $q\overline{q}$ pair, $I_{q\overline{q}}^{V^a}$, and $I_{l\overline{l}}^{V^b}$ describes the coupling to the final lepton pair.

Figure 4

Feynman diagrams for weak one-loop box corrections to the NC DY process ($V^a=Z/W$).

Figure 5

Feynman diagrams for LO strong $t\overline{t}$ production at $\mathcal{O}({\alpha }_s^2)$.

Figure 6

Sample diagrams for one-loop weak virtual corrections to the quark-antiquark annihilation subprocess in strong $t\overline{t}$ production, which consist of vertex and box corrections, respectively. The $\widehat{u}$-channel box diagrams are not explicitly shown.

Figure 7

Sample diagrams for one-loop weak virtual corrections to the gluon fusion subprocess in strong $t\overline{t}$ production, which consist of vertex, self-energy, and box corrections, respectively. The ellipses represent the vertex, self-energy and box diagrams which are not explicitly shown.

Figure 8

The contribution of the $\widehat{t}$-channel two-gluon box diagram interfered with the $Z$-mediated Born diagram to the NLO cross section for $t\overline{t}$ production at $\mathcal{O}(\alpha {\alpha }_s^2)$. The contribution of the $\widehat{u}$-channel two-gluon box diagram is not explicitly shown.

Figure 9

Sample diagrams for real corrections to $t\overline{t}$ production in the $q\overline{q}$ annihilation channel contributing to the $\mathcal{O}(\alpha {\alpha }_s^2)$ cross section.

Figure 10

Sample tree-level Feynman diagrams for dijet production via QCD and EW interactions.

Figure 11

Sample Feynman diagrams for virtual corrections to four-quark subprocesses of category 2 of Table 3, $u_i{\overline{u}}_i\to d_j{\overline{d}}_j$, $d_i{\overline{d}}_i\to u_j{\overline{u}}_j$, where $i$, $j$ denote the $i$th- or $j$th-generation of the light (anti)quarks, i.e., $i,j\in (1,2)$. The diagrams above the double line contribute to $\delta \mathcal{M}({\alpha }_s\alpha )$ and below the double line to $\delta \mathcal{M}({\alpha }_s^2)$. Weak one-loop corrections similar to the ones shown in Fig. 6 are not explicitly shown.

Figure 12

Sample Feynman diagrams for virtual corrections to four-quark subprocesses of category 3 of Table 3, $q\overline{q}\to q\overline{q}$, $q\in \{u_i,d_i\}$. Contributions similar to the ones shown in Fig. 6 and 11 are not explicitly shown.

Figure 13

Feynman diagrams representing soft-collinear contributions resulting in double logarithms (left), collinear contributions (middle) and contributions from wave function renormalization (right) both resulting in single logarithms.

Figure 14

Comparison of MCFM (red, solid) and ZGRAD2 (green, dashed) predictions for the weak one-loop relative correction ${\delta }_{\mathrm{wk}}$ to the invariant mass of the lepton-pair ($M(l^{+}{l}^{-})$, top), lepton transverse momentum ($p_T(l^{+})$, middle), and lepton rapidities ($\eta (l^{\pm })$, bottom) distributions in NC DY production at the 13 TeV LHC. The correction is expressed as a percentage of the LO result in each bin, according to Eq. (35).

Figure 15

LO predictions for the invariant lepton-pair mass distribution for the photon-induced process and the $q\overline{q}$-initiated (pink, long-dashed-dotted) NC DY process at the 13 TeV LHC. The photon-induced LO prediction is obtained with different photon PDFs as provided by CT14QED (red, solid), MRST2004QED (green, dashed), and NNPDF3.0QED (blue, dotted).

Figure 16

Comparison of relative corrections ${\delta }_{\mathrm{wk}}$ to the parton-level cross sections for $u\overline{u}\to t\overline{t}$ (left) and $gg\to t\overline{t}$ (right) from MCFM and the results from Fig. 8 of Ref. [36] (KSU) (green, dashed). The correction is expressed as a percentage of the LO $\mathcal{O}({\alpha }_s^2)$ cross section according to Eq. (35). In case of the $u\overline{u}\to t\overline{t}$ process the KSU results shown here only include weak one-loop vertex corrections, while MCFM results are provided for both the weak one-loop vertex corrections (blue, dotted) and the full $\mathcal{O}({\alpha }_s^2\alpha )$ contribution (red, solid).

Figure 17

Comparison of relative corrections ${\delta }_{\mathrm{wk}}$ from MCFM (red, solid) and the results from Fig. 20 and Fig. 22 of Ref. [36] (KSU) (green, dashed) to the invariant mass distribution of the top-antitop pair [$M(t\overline{t})$] (top center), the transverse momentum of the top quark [$p_T(t)$] (left), and the rapidity difference between the top and antitop quark [$\mathrm{\Delta }y(t\overline{t})$] (right), in $t\overline{t}$ production at the 13 TeV LHC. In the calculation of the $\mathrm{\Delta }y(t\overline{t})$ distribution a cut of $M(t\overline{t})>2\mathrm{TeV}$ is applied. The correction is expressed as a percentage of the LO $\mathcal{O}({\alpha }_s^2)$ cross section according to Eq. (35).

Figure 18

Comparison of relative corrections ${\delta }_{\mathrm{wk}}$ of Eq. (35) (blue, dashed and light blue, long-dashed-dotted), ${\delta }_{\mathrm{EW}}^{\mathrm{tree}}$ of Eq. (42) (green, short-dashed and yellow, dotted-long-dashed), and their sum (red, solid and pink, dotted) from MCFM and the results of Figs. 9 and 12 of Ref. [41] (DHS). Results are shown for the invariant dijet mass [$M(j_1{j}_2)$ (left)] and transverse jet momentum distributions for the leading jet [$k_T(j_1)$] (right) in dijet production at the 14 TeV LHC.

Figure 19

Relative weak one-loop corrections from MCFM to the invariant lepton-pair mass [$M(l^{+}{l}^{-},l=e,\mu )$] (left) and lepton transverse momentum [$p_T(l^{+})$] (right) distributions in the NC DY process at the 13 TeV LHC. The correction is expressed as a percentage of the LO result and results are shown for both the exact [${\delta }_{\mathrm{wk}}$ of Eq. (35)] (red, solid) and approximate Sudakov [${\delta }_{\text{Sudakov}}$ of Eq. (43)] calculation, the latter with (blue, dotted) and without (pink, long-dashed-dotted) the $\gamma -Z$ box contribution of Eq. (44).

Figure 20

Relative weak one-loop corrections to the pseudorapidity distributions of the positively (left) and negatively (right) charged leptons in the NC DY process at the 13 TeV LHC. The correction is expressed as a percentage of the LO result and results are shown for both the exact [${\delta }_{\mathrm{wk}}$ of Eq. (35)] (red, solid) and approximate Sudakov [${\delta }_{\text{Sudakov}}$ of Eq. (43)] calculation, the latter with (blue, dotted) and without (pink, long-dashed-dotted) the $\gamma -Z$ box contribution of Eq. (44).

Figure 21

Differential lepton forward-backward asymmetry $A_{\mathrm{FB}}$ with respect to the invariant mass of the charged lepton pair in the NC DY process at the 13 TeV LHC. Results are shown for the LO (green, dashed) prediction and for both the exact (red, solid) and approximate Sudakov calculation, the latter with (blue, dotted) and without (pink, long-dashed-dotted) the $\gamma -Z$ box contribution of Eq. (44).

Figure 22

Relative weak corrections to the $p_T(t)$ (left) and $\mathrm{\Delta }y(t\overline{t})$ (right) distributions in $t\overline{t}$ production at the 13 TeV LHC. The correction is expressed as a percentage of the LO result and is shown for the exact weak relative correction ${\delta }_{\mathrm{wk}}$ of Eq. (35) (red, solid) and the Sudakov approximation of ${\delta }_{\text{Sudakov}}$ of Eq. (43) (blue, dotted).

Figure 23

Relative weak corrections to $M(t\overline{t})$ in $t\overline{t}$ production at the 13 TeV LHC with no cuts applied (top left), after application of a moderate rapidity cut (top right) and for central top quarks (center). The correction is expressed as a percentage of the LO result and is shown for the exact weak relative correction ${\delta }_{\mathrm{wk}}$ of Eq. (35) (red, solid) and the Sudakov approximation ${\delta }_{\text{Sudakov}}$ of Eq. (43) (blue, dotted).

Figure 24

Relative weak corrections to the invariant mass (upper left), the absolute rapidity difference between the two leading jets (upper right), and the transverse momentum distributions of leading (lower left) and subleading (lower right) jets in dijet production at the 13 TeV LHC. The correction is expressed as a percentage of the LO $\mathcal{O}({\alpha }_s^2)$ cross section and is shown for the exact weak relative corrections of Eq. (35) (red, solid) and the Sudakov approximation of Eq. (43) (blue, dotted).

Figure 25

Left: comparison of the effect of NLO weak (red, solid) and NNLO QCD corrections (blue, dotted) on the invariant mass distribution of the dilepton pair in the NC DY process at 13 TeV. The LO distribution is also shown (pink, long-dashed-dotted). Right: a comparison of the two procedures [${\delta }_{\text{add}}$ of Eq. (51) (green, dashed) and ${\delta }_{\mathrm{prod}}$ of Eq. (52) (red, solid)] used to combine NNLO QCD and NLO weak effects, together with the scale uncertainty in the pure NNLO QCD calculation (blue band). The NNLO QCD results have been obtained with MCFM [15].

Figure 26

Comparison of NNLO QCD (blue, solid) and combined NNLO $\mathrm{QCD}+\mathrm{NLO}$ weak (green, dashed) predictions for the $p_T(t)$ distribution in $t\overline{t}$ production with 8 TeV CMS data [138]. The data are divided by the theoretical prediction in each bin of the top quark $p_T$. The NNLO QCD predictions are taken from Ref. [48].

Figure 27

Left: comparison of the effect of NLO weak (red, solid) and NLO QCD (blue, dotted) corrections on the invariant mass distribution of the dijet pair in dijet production at the 13 TeV LHC. The LO QCD distribution at $\mathcal{O}({\alpha }_s^2)$ is also shown (green, dashed). Right: a comparison of the two procedures [${\delta }_{\text{add}}$ of Eq. (51) (green, dashed) and ${\delta }_{\mathrm{prod}}$ of Eq. (52) (red, solid)] used to combine NLO QCD and NLO weak effects. The NLO QCD results have been obtained with MEKS (version 1.0) [71].

Figure 28

Normalized ${\chi }_{\text{dijet}}$ distributions in dijet production at the 13 TeV LHC in a low (left) and high (right) invariant dijet mass bin at NLO QCD (red, solid) and when adding the $\mathcal{O}({\alpha }_s^2\alpha )$ (green, dashed) and LO EW (blue, dotted) contributions. The NLO QCD results have been obtained with MEKS (version 1.0) [71].

Figure 29

Relative weak corrections ${\delta }_{\mathrm{wk}}$ of Eq. (35) to the total cross sections in the high-energy region of the NC DY process (red, solid), $t\overline{t}$ (green, dashed) and dijet (blue, dotted) production, as a function of the proton-proton collision energy. The high-energy region is defined by a cut on the invariant mass of the leading pair of final-state particles, $M_{\text{final}}>\sqrt{S}/4$.