One-loop weak corrections to Higgs production

TL;DR

This paper calculates mixed QCD-weak corrections to Higgs production at the LHC and finds they are negligible within current uncertainties, supporting the factorization of QCD and weak effects.

Contribution

It provides the first numerical assessment of mixed QCD-weak corrections to inclusive Higgs production, confirming their insignificance compared to existing uncertainties.

Findings

Mixed QCD-weak corrections are negligible for current predictions.

The results support the factorization of QCD and weak corrections.

Contributions from $g q ightarrow H q$ are also insignificant.

Abstract

We compute mixed QCD-weak corrections to inclusive Higgs production at the LHC from the partonic process $g g \rightarrow H q \bar{q}$. We start from the UV- and IR-finite one-loop weak amplitude and consider its interference with the corresponding one-loop QCD amplitude. This contribution is a $\mathcal{O}(\alpha_s\alpha)$ correction to the leading-order gluon-fusion cross section, and was not numerically assessed in previous works. We also compute the cross section from the square of this weak amplitude, suppressed by $\mathcal{O}(\alpha^2)$. Finally, we consider contributions from the partonic process $g q \rightarrow H q$, which are one order lower in $\alpha_s$, as a reference for the size of terms which are not enhanced by the large gluon luminosity. We find that, given the magnitude of the uncertainties on current state-of-the-art predictions for Higgs production, all contributions computed in this work can be safely ignored, both fully inclusively and in the boosted Higgs regime. This result supports the approximate factorisation of QCD and weak corrections to that process.

Tables

Table 1. Table 1 : Summary of contributing amplitudes to the weak corrections to Higgs inclusive production involving one (bottom table) and two (top table) initial-state gluons, for various perturbative orders. The results reported in this work are highlighted with a green background, while those addressed in ref.  [ 7 , 9 ] are denoted in blue. Together, these form the complete σ g ​ g → H + X ( 1 , 1 ) subscript superscript 𝜎 1 1 → 𝑔 𝑔 𝐻 𝑋 \sigma^{(1,1)}_{gg\to H+X} weak correction.

$\times$	$A_{gg\to H}^{(0,0)\star }$	$A_{gg\to H}^{(2,0)\star }$	$A_{gg\to H}^{(0,2)\star }$	$A_{gg\to H}^{(2,2)\star }$	$A_{gg\to Hg}^{(1,0)\star }$	$A_{gg\to Hg}^{(1,2)\star }$	$A_{gg\to Hq\overline{q}}^{(2,0)\star }$	$A_{gg\to Hq\overline{q}}^{(0,2)\star }$
$A_{gg\to H}^{(0,2)}$	${\sigma }_{gg\to H}^{(0,1)}$	${\sigma }_{gg\to H}^{(1,1)}$	${\sigma }_{gg\to H}^{(0,2)}$	${\sigma }_{gg\to H}^{(1,2)}$
$A_{gg\to H}^{(2,2)}$	${\sigma }_{gg\to H}^{(1,1)}$	${\sigma }_{gg\to H}^{(2,1)}$	${\sigma }_{gg\to H}^{(1,2)}$	${\sigma }_{gg\to H}^{(2,2)}$
$A_{gg\to Hg}^{(1,2)}$					${\sigma }_{gg\to Hg}^{(1,1)}$	${\sigma }_{gg\to Hg}^{(1,2)}$
$A_{gg\to Hq\overline{q}}^{(0,2)}$							${\sigma }_{gg\to Hq\overline{q}}^{(1,1)}$	${\sigma }_{gg\to Hq\overline{q}}^{(0,2)}$

Table 2. Table 2 : SM parameters used for obtaining all numerical results presented in Table  4 . Dimensionful parameters are given in GeV. Lower-case mass parameters correspond to their complex-valued counterpart in the complex-mass scheme, i.e. m W = M ¯ W 2 − i ​ Γ ¯ W ​ M ¯ W subscript 𝑚 𝑊 superscript subscript ¯ 𝑀 𝑊 2 𝑖 subscript ¯ Γ 𝑊 subscript ¯ 𝑀 𝑊 m_{W}=\sqrt{\bar{M}_{W}^{2}-i\bar{\Gamma}_{W}\bar{M}_{W}} .

Parameter	value	Parameter	value	Parameter	value
PDF set	PDF4LHC15_nlo_30	${\mu }_R={\mu }_F$	$M_H/2,M_H$	$M_t$	174.3
${\alpha }_S\left(m_Z^2\right)$	from PDF set	$G_F$	$\frac{\pi \alpha }{\sqrt{2}{m}_W^2\left(1-m_W^2/m_Z^2\right)}$	${\Gamma }_t$	1.35408
$\sqrt{\hat{s}}$	13000	${\alpha }^{-1}$	132.507	$\frac{y_{t}v}{\sqrt{2}}$	$m_t$
${\overline{M}}_Z$	91.188	${\overline{\Gamma }}_Z$	2.42823	$M_b$	0.0
${\overline{M}}_W$	80.419	${\overline{\Gamma }}_W$	2.02844	$\frac{y_{b}v}{\sqrt{2}}$	0.0
$M_H$	125.0	${\Gamma }_H$	0.0	$V_{ij}^{CKM}$	${\delta }_{ij}$

Table 3. Table 3 : Fully and semi inclusive cross sections obtained with SM input parameters given in Table  2 for the processes g ​ g → H ​ q ​ q ¯ → 𝑔 𝑔 𝐻 𝑞 ¯ 𝑞 gg\to Hq\bar{q} and g ​ g → H ​ b ​ b ¯ → 𝑔 𝑔 𝐻 𝑏 ¯ 𝑏 gg\to Hb\bar{b} , as (partial) contributions to the corrections of order ( α s 3 ​ α 2 ) order superscript subscript 𝛼 𝑠 3 superscript 𝛼 2 \order{\alpha_{s}^{3}\alpha^{2}} and ( α s 2 ​ α 3 ) order superscript subscript 𝛼 𝑠 2 superscript 𝛼 3 \order{\alpha_{s}^{2}\alpha^{3}} to inclusive Higgs production. For the contributions of order ( α s 2 ​ α 3 ) order superscript subscript 𝛼 𝑠 2 superscript 𝛼 3 \order{\alpha_{s}^{2}\alpha^{3}} labelled “no-HZ”, the diagrams of the class  1(d) and  1(e) are ignored, as they are best accounted for in the narrow-width approximation as the LO contribution to the process g ​ g → H ​ Z → 𝑔 𝑔 𝐻 𝑍 gg\to HZ , which is also shown. We also report the ( α s 2 ​ α 2 ) order superscript subscript 𝛼 𝑠 2 superscript 𝛼 2 \order{\alpha_{s}^{2}\alpha^{2}} and ( α s ​ α 3 ) order subscript 𝛼 𝑠 superscript 𝛼 3 \order{\alpha_{s}\alpha^{3}} contributions from the quark-initiated processes q ​ g → H ​ q → 𝑞 𝑔 𝐻 𝑞 qg\to Hq and b ​ g → H ​ b → 𝑏 𝑔 𝐻 𝑏 bg\to Hb . Finally, we consider the boosted regime, in which the Higgs transverse momentum is required to be at least 400 400 400 GeV. In each bracket separated by a dashed line, the upper number corresponds to the scale choice μ R = μ F = m H / 2 subscript 𝜇 𝑅 subscript 𝜇 𝐹 subscript 𝑚 𝐻 2 \mu_{R}=\mu_{F}=m_{H}/2 while the lower one corresponds to μ R = μ F = m H subscript 𝜇 𝑅 subscript 𝜇 𝐹 subscript 𝑚 𝐻 \mu_{R}=\mu_{F}=m_{H} .

cross section	[fb]			cross section	[fb]
interferences				squared amplitudes
${\sigma }_{gg\to Hq\overline{q}}^{\left({\alpha }_s^3{\alpha }^2\right)}$	$11.93$	$\pm$	$0.04$	${\sigma }_{gg\to Hq\overline{q}}^{({\alpha }_s^2{\alpha }^3,\text{no-HZ})}$	$-0.260$	$\pm$	$0.004$
	$13.31$	$\pm$	$0.08$		$-2.135$	$\pm$	$0.003$
\cdashline2-4      \cdashline6-8   ${\sigma }_{gg\to Hb\overline{b}}^{\left({\alpha }_s^3{\alpha }^2\right)}$			$-5.94$	$\pm$	$0.03$	${\sigma }_{gg\to Hb\overline{b}}^{({\alpha }_s^2{\alpha }^3,\text{no-HZ})}$	$3.867$	$\pm$	$0.008$
	$-7.36$	$\pm$	$0.03$		$0.882$	$\pm$	$0.006$
\cdashline2-4      \cdashline6-8   ${\sigma }_{qg\to Hq}^{\left({\alpha }_s^2{\alpha }^2\right)}$ + ${\sigma }_{\overline{q}g\to H\overline{q}}^{\left({\alpha }_s^2{\alpha }^2\right)}$			$-163.9$	$\pm$	$0.1$	${\sigma }_{qg\to Hq}^{\left({\alpha }_s{\alpha }^3\right)}$ + ${\sigma }_{\overline{q}g\to H\overline{q}}^{\left({\alpha }_s{\alpha }^3\right)}$	$52.3$	$\pm$	$0.2$
	$-137.0$	$\pm$	$0.2$		$48.6$	$\pm$	$0.1$
\cdashline2-4      \cdashline6-8   ${\sigma }_{bg\to Hb}^{\left({\alpha }_s^2{\alpha }^2\right)}$ + ${\sigma }_{\overline{b}g\to H\overline{b}}^{\left({\alpha }_s^2{\alpha }^2\right)}$			$20.95$	$\pm$	$0.04$	${\sigma }_{bg\to Hb}^{\left({\alpha }_s{\alpha }^3\right)}$ + ${\sigma }_{\overline{b}g\to H\overline{b}}^{\left({\alpha }_s{\alpha }^3\right)}$	$13.78$	$\pm$	$0.05$
	$19.45$	$\pm$	$0.06$		$13.82$	$\pm$	$0.02$
\cdashline6-8 ${\sigma }_{\text{total}}^{\text{interf.+squared}}$	$30.9$	$\pm$	$0.2$	${\sigma }_{gg\to HZ}^{\left({\alpha }_s^2{\alpha }^2,{\overline{\Gamma }}_Z=0\right)}$	$98.17$	$\pm$	$0.05$
	$24.9$	$\pm$	$0.2$		$76.27$	$\pm$	$0.03$
$p_T\left(H\right)>400$ GeV
${\sigma }_{gg\to Hq\overline{q}}^{\left({\alpha }_s^3{\alpha }^2\right)}$	$-0.0054$	$\pm$	$0.0002$	${\sigma }_{gg\to Hq\overline{q}}^{({\alpha }_s^2{\alpha }^3,\text{no-HZ})}$	$0.00390$	$\pm$	$0.00003$
	$0.00674$	$\pm$	$0.00008$		$0.00154$	$\pm$	$0.00004$
\cdashline2-4      \cdashline6-8   ${\sigma }_{gg\to Hb\overline{b}}^{\left({\alpha }_s^3{\alpha }^2\right)}$			$-0.0093$	$\pm$	$0.0002$	${\sigma }_{gg\to Hb\overline{b}}^{({\alpha }_s^2{\alpha }^3,\text{no-HZ})}$	$0.0363$	$\pm$	$0.0003$
	$-0.00197$	$\pm$	$0.00009$		$0.0118$	$\pm$	$0.0002$
\cdashline2-4      \cdashline6-8   ${\sigma }_{qg\to qH}^{\left({\alpha }_s^2{\alpha }^2\right)}$ + ${\sigma }_{\overline{q}g\to H\overline{q}}^{\left({\alpha }_s^2{\alpha }^2\right)}$			$-1.005$	$\pm$	$0.003$	${\sigma }_{qg\to Hq}^{\left({\alpha }_s{\alpha }^3\right)}$ + ${\sigma }_{\overline{q}g\to H\overline{q}}^{\left({\alpha }_s{\alpha }^3\right)}$	$0.1019$	$\pm$	$0.0002$
	$-0.7486$	$\pm$	$0.0005$		$0.0841$	$\pm$	$0.0001$
\cdashline2-4      \cdashline6-8   ${\sigma }_{bg\to Hb}^{\left({\alpha }_s^2{\alpha }^2\right)}$ + ${\sigma }_{\overline{b}g\to H\overline{b}}^{\left({\alpha }_s^2{\alpha }^2\right)}$			$-0.0326$	$\pm$	$0.0001$	${\sigma }_{bg\to Hb}^{\left({\alpha }_s{\alpha }^3\right)}$ + ${\sigma }_{\overline{b}g\to H\overline{b}}^{\left({\alpha }_s{\alpha }^3\right)}$	$0.1033$	$\pm$	$0.0003$
	$-0.0268$	$\pm$	$0.00003$		$0.0950$	$\pm$	$0.0002$
\cdashline6-8 ${\sigma }_{\text{total}}^{\text{interf.+squared}}$	$-0.502$	$\pm$	$0.003$	${\sigma }_{gg\to HZ}^{\left({\alpha }_s^2{\alpha }^2,{\overline{\Gamma }}_Z=0\right)}$	$0.3049$	$\pm$	$0.0006$
	$-0.3615$	$\pm$	$0.0008$		$0.2159$	$\pm$	$0.0003$

Equations

$${\sigma }_{pp\to H+X}=\sum _{m,n}{\sigma }_{pp\to H+X}^{(m,n)},\text{where}{\sigma }_{pp\to H+X}^{(m,n)}\propto {\alpha }_s^{m+2}{\alpha }^{n+1},$$

$$A_{ab\to H+X}^{(i,j)}\propto g_s^{i+2}{g}^{j+1},$$

$${\sigma }_{pp\to H+X}^{(m,n)}\sim {\sigma }_{pp\to H+X}^{(m,0)}\cdot {\sigma }_{pp\to H+X}^{(0,n)}.$$

$${\sigma }_{gg\to Hq\bar{q}}^{(m,n)}=$$

$$\displaystyle\qquad\int\differential{\Phi_{Hq\bar{q}}}\Big{[}\mathcal{M}^{(m,n)}_{gg\to Hq\bar{q}}\mathbf{J}(\phi_{Hq\bar{q}})-\sum_{\pi}\mathbf{C}_{gq}\otimes\mathcal{M}^{(m-1,n)}_{gq\to Hq}\mathbf{J}(\tilde{\phi}_{Hq})\Big{]}$$

$$\displaystyle\qquad+\sum_{\pi}\int\differential{\Phi_{Hq}}\int_{0}^{1}\differential{\xi}\left[\left\langle\mathbf{C}_{gq}\right\rangle(\xi)+\mathrm{\Delta}_{gq}(\xi)\right]\mathcal{M}^{(m-1,n)}_{gq\to Hq}\mathbf{J}(\phi_{Hq})\bigg{\}},$$

$$\begin{gathered} [{\mathbf{C}}_{g_1{q}_3}\otimes {\mathcal{M}}_{{\bar{q}}_{13}{g}_2\to {\bar{q}}_4{H}_5}]({\varphi }_{q_3{\bar{q}}_4{H}_5})\equiv (8\pi {\alpha }_s{\mu }^{2ϵ})\frac{1}{s_{g_1{q}_3}}{\hat{P}}_{g_1{q}_3}^{s{s}^{\prime }}(1/z)\frac{\omega ({\bar{q}}_{13})}{\omega (g_1)} \\ \times {\mathcal{M}}_{{\bar{q}}_{13}{g}_2\to {\bar{q}}_4{H}_5}^{s{s}^{\prime }}({\tilde{\varphi }}_{{\bar{q}}_4{H}_5})\theta (y_0-y), \end{gathered}$$

$${\hat{P}}_{gq}^{s{s}^{\prime }}(z)\equiv {\delta }^{s{s}^{\prime }}{C}_F\left[\frac{1+(1-z{)}^2}{z}-ϵz\right],$$

$$z\equiv \frac{Q\cdot (p_1-p_3)}{Q\cdot p_1},$$

$$\left\langle\mathbf{C}_{gq}\right\rangle(\xi)=\frac{\alpha_{s}}{2\pi}S_{\epsilon}\left[\frac{\mu_{R}^{2}}{Q^{2}}\right]^{\epsilon}\frac{T_{R}}{C_{F}}\bigg{\{}[\xi^{2}+(1-\xi)^{2}]\bigg{[}-\frac{1}{\epsilon}+\ln(1-\xi)(1+\theta[\xi-(1-y_{0})])\\ +\ln(y_{0})\theta[(1-y_{0})-\xi]\bigg{]}+2\xi(1-\xi)\bigg{\}}+\order{\epsilon},$$

$$S_{ϵ}\equiv \frac{(4\pi {)}^{ϵ}}{\Gamma (1-ϵ)}.$$

$${\mathrm{\Delta }}_{gq}(\xi )=\frac{{\alpha }_s}{2\pi }{S}_{ϵ}\frac{1}{ϵ}{\left[\frac{{\mu }_R^2}{{\mu }_F^2}\right]}^{ϵ}{P}_{gq}(\xi ),$$

$$P_{gq}(\xi )=T_R[{\xi }^2+(1-\xi {)}^2].$$

$$A_{gg\to q\bar{q}H}^{(0,2)}$$

$$={\epsilon }_{{\mu }_1}(p_1){\epsilon }_{{\mu }_2}(p_2){\bar{u}}^{s_3}(p_3)v^{s_4}(p_4){\mathcal{F}}_{s_3{s}_4}^{{\mu }_1{\mu }_2},$$

$$ud{W}^{+}\propto g_{\mathrm{V}}{\gamma }^{\mu }+g_{\mathrm{A}}{\gamma }^{\mu }{\gamma }^5,ud{W}^{-}\propto g_{\mathrm{V}}^{\ast }{\gamma }^{\mu }+g_{\mathrm{A}}^{\ast }{\gamma }^{\mu }{\gamma }^5\text{and}$$

$$qqZ\propto g_{\mathrm{V},Z}{\gamma }^{\mu }+g_{\mathrm{A},Z}{\gamma }^{\mu }{\gamma }^5,$$

$$p_1\cdot {\epsilon }_1(p_1)=0$$

$$\sum _{\text{polarization}}{\epsilon }_{\mu }^{\ast }(p_i){\epsilon }_{\nu }(p_i)=-g_{\mu \nu }+\frac{p_{1\mu }{p}_{2\nu }+p_{2\mu }{p}_{1\nu }}{p_1\cdot p_2}\text{for }i=1,2.$$

$${\mathcal{F}}^{{\mu }_1{\mu }_2}=\sum _i{\alpha }_i{\mathcal{S}}_i^{{\mu }_1{\mu }_2}=\sum _i{\alpha }_i\int \frac{N^{{\mu }_1{\mu }_2}(k)}{D_1\dots D_{m_i}}\text{differential}[D]k,$$

$${\mathcal{S}}_i^{{\mu }_1{\mu }_2}=\sum _j{T}_j^{{\mu }_1{\mu }_2}(p_1,p_2,p_3,p_4)S_j.$$

$$N^{{\mu }_1{\mu }_2}(k)=c_0^{{\mu }_1{\mu }_2}+c_1^{{\mu }_1{\mu }_2;{\alpha }_1}{k}_{{\alpha }_1}+c_2^{{\mu }_1{\mu }_2;{\alpha }_1{\alpha }_2}{k}_{{\alpha }_1}{k}_{{\alpha }_2}+c_3^{{\mu }_1{\mu }_2;{\alpha }_1{\alpha }_2{\alpha }_3}{k}_{{\alpha }_1}{k}_{{\alpha }_2}{k}_{{\alpha }_3}+\dots ,$$

$${\mathcal{S}}_i^{({\alpha }_1{\alpha }_2\dots {\alpha }_n)}=\int \frac{{\tilde{N}}^{({\alpha }_1{\alpha }_2\dots {\alpha }_n)}(k)}{D_1\dots D_{m_i}}\text{differential}[D]k=\sum _j{t}_j^{({\alpha }_1{\alpha }_2\dots {\alpha }_n)}(g,p_1,p_2,p_3,p_4)c_j,$$

$$t^{({\alpha }_1\dots {\alpha }_n)}=\frac{1}{n!}\sum _{\sigma \in {\Sigma }_n}{t}^{{\alpha }_{\sigma (1)}\dots {\alpha }_{\sigma (n)}},$$

$$t_{p_1,p_2}^{({\alpha }_1{\alpha }_2)}=p_1^{({\alpha }_1}{p}_2^{{\alpha }_2)}=\frac{1}{2}\left(p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}+p_1^{{\alpha }_2}{p}_2^{{\alpha }_1}\right).$$

$$\begin{array}{rlrlrlrl}{T}_1^{{\mu }_1{\mu }_2}& =/p{}_1{\gamma }^{{\mu }_1}{\gamma }^{{\mu }_2},& T_2^{{\mu }_1{\mu }_2}& =/p{}_2{\gamma }^{{\mu }_1}{\gamma }^{{\mu }_2},& T_3^{{\mu }_1{\mu }_2}& =g^{{\mu }_1{\mu }_2}/p{}_1,& T_4^{{\mu }_1{\mu }_2}& =g^{{\mu }_1{\mu }_2}/p{}_2,\\ T_5^{{\mu }_1{\mu }_2}& ={\gamma }^{{\mu }_1}{p}_3^{{\mu }_2},& T_6^{{\mu }_1{\mu }_2}& ={\gamma }^{{\mu }_1}{p}_4^{{\mu }_2},& T_7^{{\mu }_1{\mu }_2}& =/p{}_1/p{}_2{\gamma }^{{\mu }_1}{p}_3^{{\mu }_2},& T_8^{{\mu }_1{\mu }_2}& =/p{}_1/p{}_2{\gamma }^{{\mu }_1}{p}_4^{{\mu }_2},\\ T_9^{{\mu }_1{\mu }_2}& ={\gamma }^{{\mu }_2}{p}_3^{{\mu }_1},& T_{10}^{{\mu }_1{\mu }_2}& =/p{}_1/p{}_2{\gamma }^{{\mu }_2}{p}_3^{{\mu }_1},& T_{11}^{{\mu }_1{\mu }_2}& ={\gamma }^{{\mu }_2}{p}_4^{{\mu }_1},& T_{12}^{{\mu }_1{\mu }_2}& =/p{}_1/p{}_2{\gamma }^{{\mu }_2}{p}_4^{{\mu }_1},\\ T_{13}^{{\mu }_1{\mu }_2}& =/p{}_1{p}_3^{{\mu }_1}{p}_3^{{\mu }_2},& T_{14}^{{\mu }_1{\mu }_2}& =/p{}_2{p}_3^{{\mu }_1}{p}_3^{{\mu }_2},& T_{15}^{{\mu }_1{\mu }_2}& =/p{}_1{p}_3^{{\mu }_1}{p}_4^{{\mu }_2},& T_{16}^{{\mu }_1{\mu }_2}& =/p{}_2{p}_3^{{\mu }_1}{p}_4^{{\mu }_2},\\ T_{17}^{{\mu }_1{\mu }_2}& =/p{}_1{p}_3^{{\mu }_2}{p}_4^{{\mu }_1},& T_{18}^{{\mu }_1{\mu }_2}& =/p{}_2{p}_3^{{\mu }_2}{p}_4^{{\mu }_1},& T_{19}^{{\mu }_1{\mu }_2}& =/p{}_1{p}_4^{{\mu }_1}{p}_4^{{\mu }_2},& T_{20}^{{\mu }_1{\mu }_2}& =/p{}_2{p}_4^{{\mu }_1}{p}_4^{{\mu }_2}.\end{array}$$

$$b_0(s;m^2,0)=h(ϵ)\int \frac{1}{(k^2-m^2)(k-p{)}^2}\text{differential}[D]k$$

$$={\left(\frac{{\mu }^2}{m^2}\right)}^{ϵ}[\frac{1}{ϵ}+2-\frac{(x-1)\ln (1-x)}{x}$$

$$+\left(4+\frac{{\pi }^2}{6}+\frac{(1-x)}{2x}\left(2{\mathrm{Li}}_2\left(-\frac{x}{1-x}\right)-{\ln }^2(1-x)+4\ln (1-x)\right)\right)ϵ+\text{order}{ϵ}^2],$$

$$b_0(s;0,0)=h(ϵ)\int \frac{1}{k^2(k-p{)}^2}\text{differential}[D]k={\left(-\frac{s}{{\mu }^2}\right)}^{-ϵ}\left[\frac{1}{ϵ}+2+4ϵ+\text{order}{ϵ}^2\right],$$

$$a_0(m)=h(ϵ)\int \frac{1}{k^2-m^2}\text{differential}[D]k=m^2{\left(\frac{{\mu }^2}{m^2}\right)}^{ϵ}\left[\frac{1}{ϵ}+1+\left(1+\frac{{\pi }^2}{6}\right)ϵ+\text{order}{ϵ}^2\right],$$

$$h(ϵ)\equiv \frac{{\mu }^{2ϵ}}{i{\pi }^{2-ϵ}}\frac{\Gamma (1-2ϵ)}{\Gamma (1-ϵ{)}^2\Gamma (ϵ+1)},$$

$$\underset{\eta \downarrow 0}{\lim }\ln \left(1-\frac{s+i\eta }{m^2}\right)=\{\begin{array}{ll}\ln (1-x)& s<m^2,\\ \ln (1-x)-2i\pi & s>m^2,\end{array}$$

$$A_{gg\to d\bar{d}H}^{(0,2),\mathrm{AA}}=\frac{|g_{\mathrm{A}}{|}^2}{|g_{\mathrm{V}}{|}^2}{A}_{gg\to d\bar{d}H}^{(0,2),\mathrm{VV}}.$$

Full text

1. 1 Introduction

2. 2 Classification of contributions

3. 3 Initial-state collinear singularities

4. 4 Setup of the computation and numerical results

5. 5 Conclusion

6. A Initial-collinear counterterms

7. B Analytical computation of the amplitudes $A^{(2,0)}_{gg\to Hq\bar{q}}$ and $A^{(0,2)}_{gg\to Hq\bar{q}}$

8. C Validation material

1 Introduction

In the quest towards an ever more accurate prediction for the inclusive Higgs production cross section at hadron colliders, one of the major tasks is the computation of fixed-order corrections in the perturbative expansion in powers of the Standard Model (SM) couplings. Our understanding of pure QCD corrections, which are known to be very important for this process, has reached an unprecedented level of accuracy in recent times. A milestone in this programme was achieved with the computation of the third correction term in the expansion in the strong coupling $\alpha_{s}$ of the cross section for Higgs production via gluon fusion in the infinite top mass limit [1, 2]. In a typical setup for the LHC running at a centre-of-mass energy of 13 TeV, this contribution shifts the prediction for the total cross section upwards by roughly 3% [3].

On the other hand, weak corrections to the leading-order (LO) inclusive Higgs cross section also need to be considered. In the same setup mentioned before, the first weak term turns out to increase the total gluon fusion cross section by a significant 5% [4, 5, 6]. Since next-to-leading-order (NLO) QCD corrections can be as large as the leading contribution, the motivation to investigate mixed first-order QCD and first-order weak corrections is very strong. Although the exact size of this term is at present unknown, various approximations have been considered in the literature. The first estimate to appear was based on the argument that mixed QCD-weak effects on the inclusive Higgs production cross section are well approximated by combining the purely weak term and the full QCD series in a multiplicative fashion [7]. Following this factorisation approach, the authors of ref. [3] reported the mixed QCD-weak corrections to be approximately 3% of the full result, and conservatively estimated the uncertainty stemming from non-factorisable contributions to be 1% of the total. The estimates of [7, 3] are obtained by considering the unphysical limit $m_{H}\ll m_{W},m_{Z}$. The gluon induced interference contributions discussed in our work are suppressed in this limit by two powers of the weak boson masses with respect to the leading order $\mathcal{O}(\alpha_{S}^{2}\alpha)$ cross section, which we verified by explicit calculation. The theoretical uncertainty associated to each of the other main error sources (determination of parton distribution functions, truncation of the QCD perturbative series, and missing quark-mass effects) is currently of the same order. It is therefore highly desirable to remove the ambiguity due to the factorisation approximation.

Important steps have recently been made in this direction. Thanks to the calculation of the three-loop mixed QCD-weak correction to Higgs boson gluon fusion for arbitrary masses of the $W$, $Z$, and Higgs bosons [8], an estimate of the cross section in the soft-virtual approximation was obtained [9]. An independent work considered three-loop matrix elements in the limit of massless vector bosons instead, and combined them with a different class of two-loop real-emission contributions [10]. The estimates obtained using these approximations support the validity of the factorisation approach, since they include some non-factorisable effects and find that these are numerically small.

In order for the full mixed QCD-weak term to become available, however, two pieces of the puzzle are still missing. On the one hand there is the formidable challenge of computing two-loop matrix elements with an extra real emission for arbitrary $W$, $Z$, and Higgs masses. On the other hand, there are UV- and IR-finite one-loop weak contributions to the production of the Higgs in association with two partons, which feature more complicated kinematics but whose one-loop integrals are well understood. Although in general corrections with fewer or soft real emissions are expected to dominate within the inclusive cross section [9], the contributions with two extra hard partons are formally of the same order and may disrupt the approximate factorisation of weak and QCD corrections because of their final-state kinematic structure.

In the present paper, we address this issue by carrying out the exact inclusive computation of the contribution to mixed QCD-weak corrections from the one-loop partonic subprocess $gg\to Hq\bar{q}$. We stress that this contribution features one-loop pentagon topologies which appear only in matrix elements with (at least) two real emissions, that do not fit in a factorised picture and that have not been assessed before.

The paper proceeds as follows. In Section 2, we discuss the different contributions that enter our computation, we categorise them and identify potential competing mechanisms which are formally of the same order or slightly higher. Although the computation of the required matrix elements is straightforward using standard public codes for one-loop calculations, the computation of the pieces of cross sections we are interested in requires the renormalisation of parton distributions and the subtraction of initial-state collinear singularities. Given the very special features of the process examined, these steps require some care and are thus described in Section 3. Finally, we report and discuss numerical results.

2 Classification of contributions

In order to classify contributions to the Higgs inclusive cross section, it is useful to write its mixed QCD and weak expansion as

[TABLE]

where the prefactor $\alpha_{s}^{2}\alpha^{1}$ is chosen so as to match the couplings factorised by the leading-order loop-induced gluon-fusion contribution to inclusive Higgs production. Notice that we group all squared couplings that are not strong, including the Yukawa of the top quark, under the label $\alpha$, in view of their comparable strength and of the electroweak gauge relations often rendering their separate factorisation ambiguous. The corrections often labelled “QCD NmLO” and “(electro)weak NnLO” are then denoted by $\sigma^{(m,0)}_{pp\to H+X}$ and $\sigma^{(0,n)}_{pp\to H+X}$, as they become impractical when addressing the mixed cases $\sigma^{(m,n)}_{pp\to H+X}$. With such a notation in mind, the expected naive parametric suppression from the couplings, which counts $\alpha_{s}\sim 10^{-1}$ and $\alpha\sim 10^{-2}$, simply reads $\sigma^{(m,n)}_{pp\to H+X}\sim 10^{-m-2n}$. In order to discuss interference terms, we also find it useful to introduce a similar notation for amplitudes:

[TABLE]

where we denote by $g$ all couplings that are not $g_{s}$.

As mentioned above, weak and QCD corrections are expected to factorise to a certain degree, such that

[TABLE]

This approximation is valid under the assumption that the main contributions to the mixed QCD-weak cross section are to be attributed either to soft gluons or Sudakov weak logarithms. If one is to assess violations of this factorisation, the expansion term $\sigma^{(1,1)}_{pp\to H+X}$ must be computed exactly. We now set out to discuss the many contributions this term receives.

In this work, we only consider weak corrections involving the $W$ and $Z$ bosons, as these dominate over the genuine electroweak corrections (i.e. unresolved photon exchange or emission) to contributions where the Higgs is produced from massive quark loop lines that are not the top-quark.

Also, the gluon initiated processes are expected to be the dominant contributions at the LHC, where quark parton distribution functions (PDFs) are small in comparison to the gluon one for the typical values of the Bjorken $x$’s probed by the kinematics involved. We therefore neglect all contributions to $\sigma^{(1,1)}_{pp\to H+X}$ that factorise parton luminosities with at least one quark. To get a reference for the size of these terms that we do not compute, we report numerical results also for $\sigma^{(0,1)}_{gq\to Hq}$. 111Note that our initial-state notation $gq$ encompasses in this context both permutations $gq$ and $qg$.

Weak corrections stemming from the interference with leading QCD production modes are often subject to kinematic suppressions that renders them smaller than what is naively expected from their factorised couplings. For this reason, we also report the pieces of the cross sections $\sigma^{(0,2)}_{gg\to Hq\bar{q}}$ and $\sigma^{(-1,2)}_{gq\to Hq}$ built from the square of the amplitudes $A^{(0,2)}_{gg\to Hq\bar{q}}$ and $A^{(-1,2)}_{gq\to Hq}$. These form a gauge-invariant subset of higher-order contributions.

Our work reports on the contribution $\sigma^{(1,1)}_{gg\to Hq\bar{q}}$ for the first time and, together with the results from refs. [7, 9], it completes the computation of $\sigma^{(1,1)}_{gg\to H+X}$. We now proceed to list in Table 1 all amplitudes building $\sigma^{(1,1)}_{gp\to H+X}$.

We now turn to discussing the Feynman diagrams building the amplitudes $A^{(2,0)}_{gg\to Hq\bar{q}}$, $A^{(0,2)}_{gg\to Hq\bar{q}}$, $A^{(1,0)}_{gq\to Hq}$ and $A^{(-1,2)}_{gq\to Hq}$ that contribute to the cross sections presented in this work.

The amplitude $A^{(2,0)}_{gg\to Hq\bar{q}}$ is built from the diagrams depicted in Fig. 1 where the Higgs is produced via weak vector boson fusion and interfered with the leading QCD gluon-fusion diagrams shown in Fig. 3.

Diagrams of the class 1(d) and 1(e), where the Higgs is produced via gluon-fusion, feature a Z-boson propagator222The diagram analogous to 1(e) with a photon instead of the Z-boson is exactly zero in virtue of Furry’s theorem. which however does not yield any Breit-Wigner resonance as they are interfered against the QCD diagrams of Fig. 3. We must nonetheless regulate the Z-boson propagator pole, which motivates our use in this computation of the complex-mass scheme [11, 12] with finite widths for the internal top quark and unstable weak gauge bosons. These diagrams 1(d) and 1(e) are however ignored when considering their squared contribution to $\sigma^{(1,1)}_{pp\to H+X}$, since in this case they are best accounted for in the narrow-width approximation as the LO prediction for associated Higgs production, i.e. $\sigma^{(1,1)}_{gg\to HZ}$ (also reported in this work).

Finally, diagrams of the class 1(f) are specific to the third-generation quarks where the Higgs can also be emitted from the top-quark running in the loop. This contribution is analogous to that of the heavy quarks in the two-loop electroweak corrections to Higgs production investigated in ref. [13] and, for this reason, we found it interesting to report our results separately for the processes $gg\to Hq\bar{q}$, with $q\equiv u,d,c,s$, and $gg\to b\bar{b}H$.

3 Initial-state collinear singularities

All of the one-loop amplitudes considered in this paper are free of explicit ultraviolet and infrared divergences that can arise from the integration over the loop momenta. In other words, working in dimensional regularisation with $D\equiv 4-2\epsilon$, their analytic expressions do not contain explicit poles in the dimensional regulator $\epsilon$. However, matrix elements may feature non-integrable infrared divergences in regions of the phase space which correspond to unresolved configurations. In order to discuss this issue, we concentrate on the amplitude $A^{(0,2)}_{gg\to Hq\bar{q}}$ as it constitutes the main focus of the present work.

In principle, the process $gg\to Hq\bar{q}$ presents infrared divergences when the quark-antiquark pair in the final state is collectively soft, and/or when one or both of the quarks are collinear to the direction of an incoming gluon. However, thanks to the factorisation properties of QCD, in double-unresolved configurations the amplitude $A^{(0,2)}_{gg\to Hq\bar{q}}$ can be approximated by universal factors times the reduced amplitude $A^{(-2,2)}_{q\bar{q}\to H}$ (that is, of order $g^{3}$) which is identically zero. Indeed, the triangle one-loop diagrams for $q\bar{q}\to H$ require a mass insertion for the chirality flip and therefore vanishes for massless onshell quarks. This explains why the interference involving the amplitude $A^{(0,2)}_{gg\to Hq\bar{q}}$ only requires the subtraction of single-unresolved infrared limits, while the interference built upon the amplitude $A^{(-1,2)}_{gq\to Hq}$ does not require IR subtraction at all.

The same observations can be made, perhaps more intuitively, by inspecting the representative Feynman diagrams depicted in Fig. 1. It is straightforward to see that propagators of massless partons which do not belong to closed loops can go on-shell only in the graphs of type 1(b) and 1(c). In the case of the diagram 1(b), this happens when antiquark $\bar{d}_{5}$ becomes collinear to gluon $g_{2}$ such that the hard scattering subgraph corresponds to diagram 2(a). By contrast, in the kinematic limit where quark $d_{4}$ is collinear to gluon $g_{1}$ and quark $\bar{d}_{5}$ is collinear to gluon $g_{2}$, both non-loop propagators of graph 1(c) are singular. The subgraph that describes the hard scattering process, however, evaluates to zero for massless quarks as explained before, thus avoiding the singularity. In the limit where only one of the quarks is collinear to an incoming gluon, the hard part of diagram 1(c) matches that of graph 2(b).

From the observations drawn so far, we conclude that for the local subtraction of implicit singularities it is sufficient to consider standard NLO initial-collinear counterterms. These subtraction terms are to be added back, analytically integrated over the unresolved degrees of freedom yielding explicit poles in the dimensional regulator $\epsilon$. These poles cancel against those part of the PDF renormalisation counterterms, as guaranteed by collinear beam factorisation, thus rendering the complete computation finite.

The formal expression which describes this subtraction procedure and the combination with PDF renormalisation counterterms reads:

[TABLE]

where the dependences on the factorisation and renormalisation scales $\mu_{F}$ and $\mu_{R}$ as well as on the kinematic inputs for the matrix elements have been suppressed for brevity. The sums run over the four permutations $\pi$ that are obtained exchanging the quark and the antiquark in the final state and/or the two initial-state gluons among themselves. The symbol $\mathbf{C}_{ij}$ denotes the local counterterm for particles $i$ and $j$ going collinear and $\left\langle\mathbf{C}_{ij}\right\rangle$ its counterpart analytically integrated over the unresolved degrees of freedom. The observable functions are indicated with $\mathbf{J}$, and $\mathrm{\Delta}_{ik}$ is the PDF renormalisation kernel for parton with flavour $i$ to change into species $k$ before entering the hard process. The notation $\tilde{\phi}_{Hq}$ indicates reduced kinematics of lower multiplicity which are obtained by mapping a pair of collinear partons to a massless parent. The concrete expressions of all subtraction ingredients closely follow ref. [14] and are presented more explicitly in appendix A, where we also explicitly show that our subtraction counterterms correctly regulate the relevant collinear singularities.

4 Setup of the computation and numerical results

The amplitudes $A^{(0,2)}_{gg\to Hq\bar{q}}$ and $A^{(-1,2)}_{gq\to Hq}$ that factorise a Higgs coupling to weak bosons were first computed analytically (for massless quarks only) in ref. [15], in the different context of NLO QCD corrections to weak vector-boson fusion. In the present case and as indicated in Table 1, in order to obtain contributions to $\sigma^{(1,1)}_{gg\to Hq\bar{q}}$ and $\sigma^{(0,1)}_{gq\to Hq}$, these amplitudes must be interfered against their corresponding QCD analog.

Nowadays such one-loop amplitudes are readily available from many automated one-loop matrix-element generators. However, a high degree of flexibility is necessary in order to be able to select the relevant diagrams and interferences, and to construct the appropriate subtraction terms. This motivates our choice of generating the relevant one-loop squared amplitudes using MadLoop [16], part of MadGraph5_aMC@NLO [17] (henceforth abbreviated MG5aMC), as it can efficiently generate and interfere [18] arbitrary one-loop amplitudes in the SM and beyond. MadLoop uses Ninja [19, 20] and OneLOop [21], or alternatively COLLIER [22], for performing one-loop reductions and for the evaluation of the scalar one-loop master integrals. We present in appendix C some details about the generation procedure as well as benchmark numbers in order to facilitate the reproduction of our results. Moreover, we have cross-checked MadLoop’s numerical implementation of the amplitudes $A^{(2,0)}_{gg\to Hq\bar{q}}$ and $A^{(0,2)}_{gg\to Hq\bar{q}}$ against a completely independent and analytical computation described in appendix B.

As already mentioned, we choose to renormalise all unstable particles in the complex-mass scheme [11, 12] and consider the SM input parameters given in Table 2.

The numerical Monte-Carlo integration as well as the necessary IR subtraction procedure, presented in Eqs. 3.1 and 3.2 as well as in appendix A, have been implemented in a private extension of MG5aMC currently under development. The poles in the dimensional regulator $\epsilon$ have been checked to cancel as expected. 333This check of course only considers the convoluted term of Eq. 3.2 as our computation involves no virtual contribution. Also, for the pole cancellation to occur, it is important to restrict the initial state contributions to gluons only, as poles from the beam factorisation terms $\mathrm{\Delta}_{qg}$ and $\mathrm{\Delta}_{qq}$ remain uncanceled given that we ignore the corresponding real-emission subprocesses. Moreover, we have validated our code by comparing NLO QCD cross sections against results from MG5aMC for the processes $pp\to Z$ and $pp\to H$, the latter in the Higgs Effective Theory.