Higgs boson production at large transverse momentum within the SMEFT: analytical results

Abstract

We consider Higgs boson production through gluon fusion at large transverse momentum in hadronic collisions. We present the analytic expressions of the relevant one-loop QCD amplitudes including the effects of the complete set of dimension-six operators. The latter correspond to modifications of the top and bottom Yukawa couplings, to an effective point-like Higgs coupling to gluons and to the chromomagnetic operator of the top quark. The quantitative impact of the chromomagnetic operator is also studied. Our results confirm previous findings that the effect of the chromomagnetic operator at high $p_T$ can be large and should not be neglected.

1 Introduction

After the discovery of the scalar resonance of mass 125 GeV [1, 2] the measurement of its properties is one of the main activities of the LHC program. The Run I measurements [3] showed that the new resonance is compatible with the Standard Model Higgs boson. There is, however, still the possibility that more precise measurements will uncover small deviations from the Standard Model (SM) predictions. These might be the long lasting legacy of the LHC, which will encompass the searches for New Physics. The need of a consistent framework to capture small deviations from the SM is reflected in the wide discussions in Refs. [4,5,6]. The Standard Model Effective Field Theory (SMEFT) is a promising and theory motivated approach, in which the deviations from the SM are parametrised with higher-dimension operators, in the first approximation dimension six [7, 8].

Next to the inclusive quantities, differential Higgs observables were measured in Run I [9,10,11,12,13,14,15] and with a partial data set of Run II [16,17,18], although still with relatively large uncertainties. With the increasing amount of collected data, the statistical accuracy will improve, thereby allowing us to put stringent constraints on the SMEFT parameters. One of the observables which is able to shed light on the structure of the Higgs sector is the transverse momentum spectrum ($p_T$) of the Higgs particle. For example, a measurement of the $p_T$ spectrum could give insight on the nature of the Higgs boson coupling to gluons (see e.g. Refs. [19, 20]).

Recent years have seen considerable progress in the theoretical control of the $p_T$ spectrum in the SM [21,22,23,24,25,26,27]. In particular, the NLO QCD corrections are now available including the exact dependence on the top-quark mass [28].

Dedicated calculations and tools are needed to enable the experimental analyses to set bounds on the SMEFT operators. Approximate results for the total gluon fusion Higgs production cross section including modified top and bottom Yukawa couplings and an additional direct Hgg interaction have been obtained at NNLO in QCD perturbation theory in Ref. [29] and at N$^3$LO in Refs. [30, 31]. As far as gluon fusion is concerned, the inclusion of dimension-six and dimension-eight operators in the Higgs $p_T$-spectrum also has been considered in Refs. [32,33,34] and [35, 36], respectively. Strategies for extracting information on the Higgs-gluon couplings from the measurements were studied in Ref. [34], and the study the low-$p_T$ range therein was made possible by using Monte Carlo Parton Shower. Also in Ref. [20] the prospects of the determination of the Wilson coefficients in the high-luminosity LHC and future colliders were considered. The mentioned studies usually omitted the effects of the chromomagnetic operator, but a dedicated work analysed its effect on the LO Higgs production [37]. This was followed by a LO study [38] on the interplay of the SMEFT operators entering top-induced Higgs production channels, with the chromomagnetic operator treated in the heavy-top limit (HTL). Recently, the program of the SMEFT at NLO QCD was started by the MadGraph5_aMC@NLO group [39] and led to the calculation of $t\bar{t}H, tH$ [40] and recently also of Higgs production through gluon fusion [41].

In this work we recall the results for the LO Higgs production via gluon fusion and we extend our study [42] of the Higgs $p_T$ spectrum to include the effects of the chromomagnetic operator. More precisely, we present the analytic expressions of the relevant one-loop QCD amplitudes including the effects of the complete set of dimension-six operators and we shortly illustrate the impact of the chromomagnetic operator on the high-$p_T$ tail of the spectrum. Note that, due to the automated character of the calculations in the MadGraph5_aMC@NLO framework [40, 41], they can be considered complementary to the analytic calculations presented here.

The paper is organised as follows. In Sect. 1 we review the LO results and set up our notation. In Sect. 2 we present the analytical results for the SMEFT one-loop QCD amplitudes in all partonic channels, and we briefly discuss the impact of the chromomagnetic operator at high $p_T$. In Sect. 4 we draw our conclusions.

2 Framework and LO results

We consider the effective Lagrangian

$$\begin{aligned} \mathcal{L}=\mathcal{L}_{SM}+\sum _i \frac{c_i}{\Lambda ^2}\mathcal{O}_i, \end{aligned}$$

(1)

where the SM is supplemented by the inclusion of a set of dimension-six operators describing new physics effects at a scale $\Lambda$ well above the EW scale. We focus on the following three operators

$$\begin{aligned}&\mathcal{O}_1 = |H|^2 \bar{q}_L H^c t_R + h.c.\quad \mathcal{O}_2 = |H|^2 G^a_{\mu \nu }G^{a,\mu \nu }\quad \quad \quad \nonumber \\&\mathcal{O}_3 = \bar{Q}_L H \sigma ^{\mu \nu }T^a t_R G_{\mu \nu }^a + h.c. \end{aligned}$$

(2)

These operators, in the case of single Higgs production, may be rewritten as:

$$\begin{aligned} \frac{c_1}{\Lambda ^2}\,\mathcal{O}_1&\rightarrow c_1 \frac{m_t}{v} h \bar{t} t\,, \end{aligned}$$

(3)

$$\begin{aligned} \frac{c_2}{\Lambda ^2}\,\mathcal{O}_2&\rightarrow c_2 \frac{\alpha _{\mathrm {S}}}{\pi v} h G^a_{\mu \nu }G^{a,\mu \nu }\,, \end{aligned}$$

(4)

$$\begin{aligned} \frac{c_3}{\Lambda ^2}\,\mathcal{O}_3&\rightarrow c_{3}\frac{g_{\mathrm {S}}m_t}{2v^3} (v+h)G_{\mu \nu }^a({\bar{t}}_L\sigma ^{\mu \nu }T^a t_R+h.c), \end{aligned}$$

(5)

where $\alpha _{\mathrm {S}}$ is the QCD coupling ($\alpha _{\mathrm {S}}=g_{\mathrm {S}}^2/(4\pi )$), $m_t$ is the (pole) mass of the top quark, v is the expectation value of the Higgs field, $v=(\sqrt{2}G_F)^{-1/2}\sim 246$ GeV and $\sigma ^{\mu \nu }=\frac{i}{2}\left[ \gamma ^\mu ,\gamma ^\nu \right]$. The operator $\mathcal{O}_1$ is the Yukawa operator, and describes modifications of the $t\bar{t}H$ coupling. The operator $\mathcal{O}_2$ provides a contact interaction of the Higgs boson and gluons with the same structure as in the heavy-top limit of the SM. The operator $\mathcal{O}_3$ is the chromomagnetic dipole moment operator, which modifies the interactions between gluons and the top quark. In our convention, based on the SILH basis [43, 44], we express the Wilson coefficients as factors in the canonically normalized Lagrangian.

To set up our convention we reproduce the results for the LO inclusive cross section for $gg\rightarrow H$ as exemplified in Refs. [42, 45, 46]. After renormalizing the point-like Higgs-gluon coupling $c_2$ in the ${\overline{\mathrm{MS}}}$ scheme the LO matrix element can be decomposed as

$$\begin{aligned} \mathcal{T}_{gg\rightarrow H}(p_1,p_2)= & {} i\frac{\alpha _{\mathrm {S}}}{3\pi v}\epsilon _{\mu }(p_1)\epsilon _{\nu }(p_2)\nonumber \\&\times \left[ p_1^\nu p_2^\mu -(p_1p_2)g^{\mu \nu }\right] F(\tau _H), \end{aligned}$$

(6)

where $p_1$, $p_2$ are the gluon momenta, $\epsilon (p_1)$, $\epsilon (p_2)$ their polarisations and $\tau _H=4m_t^2/m_H^2$, $m_H$ being the Higgs boson mass. The form factor $F(\tau )$ is defined as

$$\begin{aligned} F(\tau )=c_1 F_1(\tau )+c_2(\mu _{R}) F_2(\tau )+Re(c_3)\frac{m_t^2}{v^2} F_{3}(\tau ), \end{aligned}$$

(7)

with¹

$$\begin{aligned} F_1(\tau )&=\frac{3}{2}\tau \left[ 1+(1-\tau )f(\tau )\right] \,, \end{aligned}$$

(8)

$$\begin{aligned} F_2(\tau )&=12,\end{aligned}$$

(9)

$$\begin{aligned} F_3(\tau )&=3\left( \tau f(\tau )+2g(\tau )-1-2\ln \frac{\mu _{R}^2}{m_t^2}\right) \, \end{aligned}$$

(10)

and the functions f, g are defined in “Appendix”. In the HTL the form factors approach the simple expressions

$$\begin{aligned} F_1(\tau )&\rightarrow 1\,, \end{aligned}$$

(11)

$$\begin{aligned} F_2(\tau )&\rightarrow 12,\end{aligned}$$

(12)

$$\begin{aligned} F_3(\tau )&\rightarrow 6\left( 1-\ln \frac{\mu _{R}^2}{m_t^2}\right) \, . \end{aligned}$$

(13)

3 Higgs plus jet production

Higgs boson production in association with a jet is the LO contribution to Higgs boson production at finite transverse momenta. This process is mediated by gg, gq and $q\bar{q}$ initial states. We start the presentation of our results for the gg channel,

$$\begin{aligned} g(p_1)+g(p_2)\rightarrow g(p_3)+H(q) \end{aligned}$$

and the Mandelstam variables are defined as

$$\begin{aligned}&s=2p_1\cdot p_2\quad \quad \quad t=-2p_1\cdot p_3\quad \quad \quad \nonumber \\&u=-2p_2\cdot p_3\quad \quad \quad \mathrm{with}\quad \quad \quad s+t+u=m_H^2\, . \end{aligned}$$

(14)

The contributing generic SM diagrams are shown in Fig. 1.

Fig. 1

Generic diagrams originating from the $\mathcal{O}_1$ operator that also provide the SM contribution

Fig. 2

Generic diagrams originating from the $\mathcal{O}_2$ operator. These correspond also to the effective SM diagrams in the HTL

Fig. 3

Generic diagrams induced by the chromomagnetic operator $\mathcal{O}_3$

The contribution from the modified Yukawa coupling can be straightforwardly obtained by rescaling the SM result. The effective Higgs-gluon coupling gives rise to the diagrams in Fig. 2. When considering the insertion of the chromomagnetic operator we obtain 54 additional diagrams (see Fig. 3) out of which just 2 types are topologically equivalent to the SM ones.

The amplitude can be cast into the form

$$\begin{aligned}&\mathcal {T}_{gg\rightarrow gH}(p_1,p_2,p_3)\nonumber \\&\qquad = \frac{ig_S^3 m_H^4}{12 \pi ^2 v} f_{abc} {\mathcal {M}}_{gg\rightarrow gH}^{\mu \nu \rho } \epsilon _{\mu }(p_1) \epsilon _{\nu }(p_2) \epsilon ^*_{\rho }(p_3), \end{aligned}$$

(15)

where $\epsilon (p_i)$ are the gluon polarisation vectors. The amplitude can be decomposed into four independent tensor structures [47]

$$\begin{aligned} \begin{aligned}&{\mathcal {M}}_{gg\rightarrow gH}^{\mu \nu \rho }(p_1,p_2,p_3) = F_1(p_1,p_2,p_3) \mathcal {Q}^{\mu \nu \rho }_1\\&\qquad + F_2(p_1,p_2,p_3) \mathcal {Q}^{\mu \nu \rho }_2 \\&\qquad + F_3(p_1,p_2,p_3) \mathcal {Q}^{\mu \nu \rho }_3 + F_4(p_1,p_2,p_3) \mathcal {Q}^{\mu \nu \rho }_4 \end{aligned} \end{aligned}$$

(16)

where

$$\begin{aligned} \mathcal {Q}^{\mu \nu \rho }_1&= p_1^{\rho } p_2^{\mu } p_3^{\nu } - p_1^{\nu } p_2^{\rho } p_3^{\mu }\nonumber \\&\quad + g^{\mu \nu } [ (p_1 \cdot p_3) p_2^{\rho } - (p_2 \cdot p_3) p_1^{\rho } ] \nonumber \\&\quad + g^{\mu \rho } [ (p_2 \cdot p_3) p_1^{\nu } - (p_1 \cdot p_2) p_3^{\nu } ] \nonumber \\&\quad + g^{\nu \rho } [ (p_1 \cdot p_2) p_3^{\mu } - (p_1 \cdot p_3) p_2^{\mu } ] \nonumber \\ \mathcal {Q}^{\mu \nu \rho }_2&= [ (p_2 \cdot p_3) p_1^{\rho } - (p_1 \cdot p_3) p_2^{\rho } ] \frac{p_1^{\nu }p_2^{\mu } - (p_1 \cdot p_2) g^{\mu \nu }}{(p_1 \cdot p_2)} \nonumber \\ \mathcal {Q}^{\mu \nu \rho }_3&= [ (p_2 \cdot p_3) p_1^{\nu } - (p_1 \cdot p_2) p_3^{\nu } ] \frac{p_1^{\rho }p_3^{\mu } - (p_1 \cdot p_3) g^{\mu \rho }}{(p_1 \cdot p_3)} \nonumber \\ \mathcal {Q}^{\mu \nu \rho }_4&= [ (p_1 \cdot p_3) p_2^{\mu } - (p_1 \cdot p_2) p_3^{\mu } ] \frac{p_2^{\rho }p_3^{\nu } - (p_2 \cdot p_3) g^{\nu \rho }}{(p_2 \cdot p_3)}\, .\nonumber \\ \end{aligned}$$

(17)

It is useful to note that based on the definitions of the tensor structures the form factors have the following properties due to Bose symmetry,

$$\begin{aligned} F_2(p_1,p_2,p_3)= & {} F_2(p_2,p_1,p_3)\nonumber \\= & {} - F_3(p_1,p_3,p_2) = F_4(p_3,p_2,p_1) \end{aligned}$$

(18)

and $F_1$ is totally symmetric. When squaring the amplitude, the decomposition in Eq. (16) leads to mixed terms in the form factors $F_i$. If we rearrange the form factors as

$$\begin{aligned} C_1(s,t,u;m_t)&= \frac{stu}{2}~[2 F_1(p_1,p_2,p_3) + F_2(p_1,p_2,p_3)\nonumber \\&\quad - F_3(p_1,p_2,p_3) + F_4(p_1,p_2,p_3)]\nonumber \\ C_2(s,t,u;m_t)&= \frac{stu}{2}~F_2(p_1,p_2,p_3)\nonumber \\ C_3(s,t,u;m_t)&= \frac{stu}{2}~F_3(p_1,p_2,p_3)\nonumber \\ C_4(s,t,u;m_t)&= \frac{stu}{2}~F_4(p_1,p_2,p_3), \end{aligned}$$

(19)

where the $C_i$ develop the analogous Bose-symmetry properties as the previous form factors

$$\begin{aligned} \begin{aligned} C_3(s,t,u;m_t)&= -C_2(t,s,u;m_t)\\ C_4(s,t,u;m_t)&= C_2(u,t,s;m_t) \end{aligned} \end{aligned}$$

(20)

the amplitude squared can now be expressed as a sum of absolute squares

$$\begin{aligned} |{\mathcal {T}}_{gg\rightarrow gH}|^2 = \frac{32\alpha _S^3}{3 \pi v^2} \frac{m_H^8}{s t u} (|C_1|^2+|C_2|^2+|C_3|^2+|C_4|^2). \end{aligned}$$

(21)

Here we present the results for $C_1(s,t,u;m_t)$ and $C_2(s,t,u;m_t)$ ($C_3$ and $C_4$ can be obtained from $C_2$ following Eq. (20)) in terms of the scalar integrals:

$$\begin{aligned} C_i^{(\alpha )} = \frac{1}{32}\tau _H^2\sum _{j=1}^{12} P^{(\alpha )}_{i,j} T_j, \end{aligned}$$

(22)

where the $\alpha$ index runs over the operators $\mathcal{O}_\alpha$ in Eq. (2) and

$$\begin{aligned} T_1&= 1&T_2&= 2\left[ 1-g\left( \tau _s\right) \right] \end{aligned}$$

(23)

$$\begin{aligned} T_3&= 2\left[ 1-g\left( \tau _t\right) \right]&T_4&= 2\left[ 1-g\left( \tau _u\right) \right] \end{aligned}$$

(24)

$$\begin{aligned} T_5&= 2[1-g(\tau _H)]&T_6&= 2 f(\tau _s) \end{aligned}$$

(25)

$$\begin{aligned} T_7&= 2 f(\tau _t)&T_8&= 2 f(\tau _u) \end{aligned}$$

(26)

$$\begin{aligned} T_9&= 2 f(\tau _H)&T_{10}&= J(\tilde{s},\tilde{t},\tilde{u}) \end{aligned}$$

(27)

$$\begin{aligned} T_{11}&= J(\tilde{s},\tilde{u},\tilde{t})&T_{12}&= J(\tilde{u},\tilde{s},\tilde{t})\, . \end{aligned}$$

(28)

In order to present our results in a compact form we have defined dimensionless quantities $\tilde{x} = x/m_t^2$ and $\tau _x=4/\tilde{x}$ for $x=s,t,u$ and $\rho =m_H^2/m_t^2$. The scalar functions f(x), g(x) and J(x, y, z) are given in “Appendix”. The $P_{i,j}^{(1)}$ coefficients, corresponding to the SM contribution, read

$$\begin{aligned}&\begin{array}{ll} P_{1,1}^{(1)} = 12 \rho &{} P_{1,2}^{(1)} = 0 \\ P_{1,3}^{(1)} = 0 &{} P_{1,4}^{(1)} = 0 \\ P_{1,5}^{(1)} = 0 &{} P_{1,6}^{(1)} = 3 (4 - \rho ) \\ P_{1,7}^{(1)} = 3 (4 - \rho ) &{} P_{1,8}^{(1)} = 3 (4 - \rho ) \\ P_{1,9}^{(1)} = -9 (4 - \rho ) &{} P_{1,10}^{(1)} = \frac{3}{2} \tilde{s} \tilde{u} (4 - \rho ) \\ P_{1,11}^{(1)} = \frac{3}{2} \tilde{s} \tilde{t} (4 - \rho ) &{} P_{1,12}^{(1)} = \frac{3}{2} \tilde{t} \tilde{u} (4 - \rho ) \end{array} \end{aligned}$$

(29)

$$\begin{aligned}&P_{2,1}^{(1)} = - 12 \tilde{s} \frac{\tilde{u}\tilde{t} - \tilde{s}^2}{(\tilde{s}+\tilde{u})(\tilde{s}+\tilde{t})} \nonumber \\&P_{2,2}^{(1)} = 0 \nonumber \\&P_{2,3}^{(1)} = \frac{12 \tilde{t} \tilde{u} \left( \tilde{u} + 2\tilde{s}\right) }{(\tilde{s}+\tilde{u})^2} \nonumber \\&P_{2,4}^{(1)} = \frac{12 \tilde{t} \tilde{u} \left( \tilde{t} + 2\tilde{s}\right) }{(\tilde{s}+\tilde{t})^2} \nonumber \\&P_{2,5}^{(1)} = -\frac{12 \tilde{t}\tilde{u}}{(\tilde{s}+\tilde{t})^2 (\tilde{s}+\tilde{u})^2}\left[ (3\tilde{s}^2+\tilde{t}\tilde{u} +2\rho \tilde{s})(\rho -\tilde{s})+4\tilde{s}^3\right] \nonumber \\&P_{2,6}^{(1)} = -3 (\tilde{s} - 4)\nonumber \\&P_{2,7}^{(1)} = -\frac{3 \left( 4\tilde{t}\tilde{u}(\tilde{s}+\tilde{u})^2 +\tilde{s}(4-\tilde{s})(\tilde{u}^2-\tilde{s}^2)+8\tilde{s}^2\tilde{u}\right) }{\tilde{s}(\tilde{s}+\tilde{u})^2} \nonumber \\&P_{2,8}^{(1)} = -\frac{3 \left( 4\tilde{t}\tilde{u}(\tilde{s}+\tilde{t})^2 +\tilde{s}(4-\tilde{s})(\tilde{t}^2-\tilde{s}^2)+8\tilde{s}^2\tilde{t}\right) }{\tilde{s}(\tilde{s}+\tilde{t})^2} \nonumber \\&P_{2,9}^{(1)} = \frac{3}{ \tilde{s} (\tilde{s}+\tilde{t})^2 (\tilde{s}+\tilde{u})^2} \left\{ 4\tilde{u}(\tilde{s}+\tilde{t})^2 [2\tilde{s}^2+\tilde{t}(\tilde{s}+\tilde{u})^2]\right. \nonumber \\&\quad + 8\tilde{s}^4\tilde{t}+4\tilde{s}^2\tilde{t}\tilde{u} (\tilde{s}^2+2\tilde{u}) \nonumber \\&\quad \left. + \tilde{s} (\tilde{s}-4) [\tilde{s}^2[(\tilde{s}+\tilde{t})^2+(\tilde{s}+\tilde{u})^2] + \tilde{s}^3 (2\rho -\tilde{s}) - \tilde{t}^2\tilde{u}^2] \right\} \nonumber \\&P_{2,10}^{(1)} = \frac{3 \tilde{s} \tilde{u} (4 - \tilde{s})}{2} \nonumber \\&P_{2,11}^{(1)} = \frac{3 \tilde{s} \tilde{t} (4 - \tilde{s})}{2} \nonumber \\&P_{2,12}^{(1)} = -\frac{3 \tilde{t} \tilde{u} \left( 4\tilde{t}\tilde{u} -\tilde{s}^2 + 12\tilde{s} \right) }{2 \tilde{s}}\, . \end{aligned}$$

(30)

This result agrees with the one presented in Refs. [47,48,49].

The coefficients for the contribution arising from the chromomagnetic operator read

$$\begin{aligned}&\begin{array}{ll} P_{1,1}^{(3)} = 6 \rho ^2 \left( 1 - 2 \log \frac{\mu _R^2}{m_t^2}\right) &{} P_{1,2}^{(3)} = -6 \tilde{t} \tilde{u} \\ P_{1,3}^{(3)} = -6 \tilde{s} \tilde{u} &{} P_{1,4}^{(3)} = -6 \tilde{s} \tilde{t} \\ P_{1,5}^{(3)} = -6 \rho ^2 &{} P_{1,6}^{(3)} = -3 \left( 2 \rho - \tilde{t} \tilde{u}\right) \\ P_{1,7}^{(3)} = -3 \left( 2 \rho - \tilde{s} \tilde{u}\right) &{} P_{1,8}^{(3)} = -3 \left( 2 \rho - \tilde{s} \tilde{t}\right) \\ P_{1,9}^{(3)} = 18 \rho &{} P_{1,10}^{(3)} = -3 \tilde{s} \tilde{u} (\rho +\tilde{t}) \\ P_{1,11}^{(3)} = -3 \tilde{s} \tilde{t} (\rho + \tilde{u})&{} P_{1,12}^{(3)} = -3 \tilde{t} \tilde{u} (\rho + \tilde{s}) \end{array} \end{aligned}$$

(31)

$$\begin{aligned}&P_{2,1}^{(3)} = \frac{6 \rho \tilde{s} \left( \tilde{s}^2 - \tilde{t}\tilde{u}\right) }{(\tilde{s}+\tilde{t}) (\tilde{s}+\tilde{u}) } - 12 \tilde{s}^2 \log \frac{\mu _R^2}{m_t^2} \nonumber \\&P_{2,2}^{(3)} = 0 \nonumber \\&P_{2,3}^{(3)} = \frac{6 \tilde{t} \tilde{u} \left( \rho ^2-\tilde{t} (\tilde{t}+\tilde{u})\right) }{(\tilde{s}+\tilde{u})^2} \nonumber \\&P_{2,4}^{(3)} = \frac{6 \tilde{t} \tilde{u} \left( \rho ^2-\tilde{u} (\tilde{t}+\tilde{u})\right) }{(\tilde{s}+\tilde{t})^2} \nonumber \\&P_{2,5}^{(3)} = -\frac{6\rho }{(\tilde{s}+\tilde{t})^2 (\tilde{s}+\tilde{u})^2} (\rho \tilde{s}^2 (\tilde{s}^2+4\tilde{t}\tilde{u})\nonumber \\&\quad +2\tilde{s}\tilde{t}\tilde{u} (\rho -\tilde{s})^2-\tilde{t}^2\tilde{u}^2(2\tilde{s}-\rho )) \nonumber \\&P_{2,6}^{(3)} = -6 (2\tilde{s}-\rho )\nonumber \\&P_{2,7}^{(3)} = -\frac{3 \left\{ (\tilde{s}+\tilde{u})^2 \left[ 2\rho (\tilde{s}+\tilde{t}\tilde{u})-\tilde{s}\tilde{t}\tilde{u}\right] -4\tilde{s}^2[\tilde{s}(\rho -\tilde{s})+\tilde{u}^2]\right\} }{\tilde{s} (\tilde{s}+\tilde{u})^2} \nonumber \\&P_{2,8}^{(3)} = -\frac{3 \left\{ (\tilde{s}+\tilde{t})^2 \left[ 2\rho (\tilde{s}+\tilde{t}\tilde{u})-\tilde{s}\tilde{t}\tilde{u}\right] -4\tilde{s}^2[\tilde{s}(\rho -\tilde{s})+\tilde{t}^2]\right\} }{\tilde{s} (\tilde{s}+\tilde{t})^2} \nonumber \\&P_{2,9}^{(3)} = \frac{3}{ \tilde{s} (\tilde{s}+\tilde{t})^2 (\tilde{s}+\tilde{u})^2}\nonumber \\&\quad \qquad \times \left\{ 2 \tilde{s} \left[ 3 \rho \tilde{s}^4+ \tilde{s} (2\tilde{t}\tilde{u}-\rho \tilde{s})(\rho -\tilde{s})^2 + 2\tilde{s}^2\tilde{t}\tilde{u}(\tilde{s}+4\rho ) \right. \right. \nonumber \\&\quad \qquad \left. \left. +\,\tilde{t}^2\tilde{u}^2 (\rho -2\tilde{s})\right] +\tilde{t} \tilde{u} (\tilde{s}+\tilde{t})^2 (\tilde{s}+\tilde{u})^2 (2\rho - \tilde{s})\right\} \nonumber \\&P_{2,10}^{(3)} = -3 \tilde{s} \tilde{u} (\tilde{s}-\tilde{u}) \nonumber \\&P_{2,11}^{(3)} = -3 \tilde{s} \tilde{t} (\tilde{s}-\tilde{t}) \nonumber \\&P_{2,12}^{(3)} = -\frac{3 \tilde{t} \tilde{u} \left[ 2 (\rho -\tilde{s}) (3\tilde{s}+\tilde{t}\tilde{u}) + \tilde{s} \tilde{t}\tilde{u}\right] }{2 \tilde{s}}\, . \end{aligned}$$

(32)

The $\log \frac{\mu _R^2}{m_t^2}$ terms arise from the absorption of the $1/\epsilon$ divergence in the renormalisation of the $c_g$ coupling. For the effective Higgs coupling to gluons we directly present the expressions of the $C_1$ and $C_2$ form factors, which read

$$\begin{aligned} \begin{aligned} C_1^{(2)}(s,t,u;m_t)&= 12\\ C_2^{(2)}(s,t,u;m_t)&= \frac{12 s^2}{m_H^4}, \end{aligned} \end{aligned}$$

(33)

which correspond also to the HTL of the SM result multiplied by a factor of 12. For completeness we report the HTL also for the contribution of the chromomagnetic operator

$$\begin{aligned} C_i^{(3)}\rightarrow \frac{1}{2}\left( 1-\ln \frac{\mu _R^2}{m_t^2}\right) \,C_i^{(2)} \quad i=1,2. \end{aligned}$$

The final results for the form factors $C_i$ read

$$\begin{aligned}&C_i(s,t,u;m_t)= c_1 C^{(1)}_i(s,t,u;m_t)\nonumber \\&\quad + c_{2}(\mu _R) C^{(2)}_i(s,t,u;m_t) +Re(c_{3})\frac{m_t^2}{v^2} C^{(3)}_i(s,t,u;m_t).\nonumber \\ \end{aligned}$$

(34)

We now move to the $q{\bar{q}}$ channel:

$$\begin{aligned} q(p_1)+{\bar{q}}(p_2)\rightarrow g(p_3)+H(q)\, . \end{aligned}$$

The contributing generic Feynman diagrams are depicted in Fig. 4.

Fig. 4

Generic diagrams of the $q\bar{q}$-channel diagrams in the SMEFT. The color coding is the same as in Figs. 1–3

The corresponding amplitude can be decomposed as

$$\begin{aligned}&\mathcal{M}_{q{\bar{q}}\rightarrow gH}(p_1,p_2,p_3) =\frac{ig_S^3}{16 \pi ^2 vs} {\bar{v}}(p_2)\gamma ^\mu T^a\,u(p_1)\nonumber \\&\quad \times \left[ g_{\mu \nu }(p_1+p_2)\cdot p_3 - p_{3\mu } (p_1+p_2)_\nu \right] \nonumber \\&\quad \epsilon ^\nu (p_3)\,D(p_1,p_2,p_3)\, . \end{aligned}$$

(35)

We again present the results in the form

$$\begin{aligned} D^{(\alpha )}(p_1,p_2,p_3)= \sum _{j=1}^5 P^{(\alpha )}_j T_j~~~~~~~~~~\alpha =1,2,3 \end{aligned}$$

(36)

with the basis of the scalar integrals:

$$\begin{aligned} \begin{array}{llll} T_1 &{}= 1 &{} &{} \\ T_2 &{}= 2[1-g(\tau _s)] &{} T_3 &{}= 2[1-g(\tau _H)] \\ T_4 &{}= 2 f(\tau _s) &{} T_5 &{}= 2 f(\tau _H)\, . \\ \end{array} \end{aligned}$$

(37)

The coefficients corresponding to the SM and the Yukawa modifying operator in the SMEFT case read:

$$\begin{aligned} P_{1}^{(1)}&= \frac{8}{(\tilde{s}-\rho )} \nonumber \\ P_{2}^{(1)}&= \frac{8\tilde{s}}{(\tilde{s}-\rho )^2}\nonumber \\ P_{3}^{(1)}&= -\frac{8\tilde{s}}{(\tilde{s}-\rho )^2} \nonumber \\ P_{4}^{(1)}&= -\frac{4 \left( 4+\tilde{s}-\rho \right) }{(\tilde{s}-\rho )^2} \nonumber \\ P_{5}^{(1)}&= \frac{4 \left( 4+\tilde{s}-\rho \right) }{(\tilde{s}-\rho )^2}\, . \end{aligned}$$

(38)

The contribution of the chromomagnetic operator reads:

$$\begin{aligned} P_{1}^{(3)}&= \frac{4 \left[ \rho + 2 (\tilde{s}-\rho ) \ln \frac{\mu _R^2}{m_t^2}\right] }{(\tilde{s}-\rho )} \nonumber \\ P_{2} ^{(3)}&= \frac{2 \left[ 2 \tilde{s}^2+(\tilde{s}-\rho )^2\right] }{(\tilde{s}-\rho )^2}\nonumber \\ P_{3}^{(3)}&= -\frac{4\rho \left( 2\tilde{s}-\rho \right) }{(\tilde{s}-\rho )^2} \nonumber \\ P_{4}^{(3)}&= -\frac{8 (2\tilde{s}-\rho )}{(\tilde{s}-\rho )^2} \nonumber \\ P_{5}^{(3)}&= \frac{8 (2\tilde{s}-\rho )}{(\tilde{s}-\rho )^2}. \end{aligned}$$

(39)

Again, we finalise the results by presenting the amplitude for the point-like Higgs coupling to gluons which corresponds to the HTL of the SM:

$$\begin{aligned} D^{(2)}(p_1,p_2,p_3) = -16\, . \end{aligned}$$

(40)

The HTL of the relevant operators reads:

$$\begin{aligned}&D^{(1)}\rightarrow \frac{1}{12}D^{(2)}= -\frac{4}{3}~~~~~~~\\&D^{(3)}\rightarrow \frac{1}{2}(1-\ln \frac{\mu _R^2}{m_t^2})D^{(2)} = - 8 \left( 1-\ln \frac{\mu _R^2}{m_t^2}\right) \, . \end{aligned}$$

The final expression for the form factor is given by

$$\begin{aligned}&D(p_1,p_2,p_3)\nonumber \\&\quad = c_1 D^{(1)}(p_1,p_2,p_3) + c_2(\mu _R) D^{(2)}(p_1,p_2,p_3)\nonumber \\&\qquad +\,Re(c_3)\frac{m_t^2}{v^2} D^{(3)}(p_1,p_2,p_3)\, . \end{aligned}$$

(41)

The result for the qg channel can be obtained by crossing.

Fig. 5

Impact of the chromomagnetic operator on the $p_T$ spectrum of the Higgs boson in the region allowed by the current experimental constraints

The above results allow us to obtain complete predictions for Higgs boson production at high $p_T$ in the SMEFT. In Ref. [42] the effects of the $\mathcal{O}_1$ and $\mathcal{O}_2$ were studied, including the resummation of the large logarithmic contributions at small $p_T$, but neglecting the contribution of the chromomagnetic operator. We thus focus here on the effect of the chromomagnetic operator at high $p_T$. We consider pp collisions at $\sqrt{s} = 13$ TeV and use PDF4LHC2015 NLO parton distributions [50,51,52,53,54,55]. The central value of the renormalization and factorization scales is fixed to $\mu _F=\mu _R=0.5\times \sqrt{m_H^2+p_T^2}$.

In Fig. 5 we show the impact of the operator $\mathcal{O}_3$, by considering a variation of the coefficient $c_3$ within the range suggested by the study of Ref. [56]. The $p_T$ spectrum including the impact of the chromomagnetic operator is normalised to the SM result. Perturbative uncertainties are estimated with the usual 7-point scale variations. The numerical results, obtained with a modified version of the program Higlu [57], show that the chromomagnetic operator can significantly affect the $p_T$ spectrum, and the effects start to exceed the scale uncertainty of the SM result around $p_T \approx 400$–500 GeV. Considering also the improvements in the theoretical control of the SM prediction [21,22,23, 28], this means that at large transverse momenta the effect of the chromomagnetic operator could be clearly discriminated with respect to the SM prediction. Our numerical results agree with those of Ref. [40]. A more detailed study will be presented elsewhere.

4 Conclusions

In this work we studied Higgs boson production at large transverse momentum in gluon fusion within the SMEFT. We provided explicit and compact analytical results for the one-loop matrix elements of the corresponding partonic processes $gg,q\bar{q}\rightarrow Hg$ and $gq\rightarrow Hq$. The results, which are expressed in terms of standard one-loop scalar integrals, require the renormalization of the dimension-six point-like Higgs coupling to gluons in accordance with the related renormalization of the inclusive cross section [38, 42]. We studied the quantitative impact of the chromomagnetic operator and found that it can significantly distort the transverse-momentum spectrum of the Higgs boson at large $p_T$. Depending on the actual size of the corresponding Wilson coefficient, this contribution has to be taken into account for a solid study of relevant dimension-six operators within the SMEFT. Turning this argument around, the Higgs transverse-momentum distribution will provide a relevant observable to constrain the Wilson coefficient of the chromomagnetic operator when significant statistics is accumulated.

Appendix: Scalar integrals

In this appendix we present the definitions of the functions f, g, J used for our analytical results and their relation to the scalar one-loop integrals.

$$\begin{aligned}&g(\tau ) = {\left\{ \begin{array}{ll} \displaystyle \sqrt{\tau -1}\arcsin \frac{1}{\sqrt{\tau }} &{} \tau \ge 1 \\ \displaystyle \frac{\sqrt{1-\tau }}{2}\left[ \ln \frac{1+\sqrt{1-\tau }}{1-\sqrt{1-\tau }} - i\pi \right] &{} \tau < 1 \end{array}\right. } \;. \end{aligned}$$

(42)

$$\begin{aligned}&f(\tau ) = {\left\{ \begin{array}{ll} \displaystyle \arcsin ^2 \frac{1}{\sqrt{\tau }} &{} \tau \ge 1 \\ \displaystyle - \frac{1}{4} \left[ \ln \frac{1+\sqrt{1-\tau }}{1-\sqrt{1-\tau }} - i\pi \right] ^2 &{} \tau < 1 \end{array}\right. } \; \end{aligned}$$

(43)

$$\begin{aligned}&J(\tilde{s},\tilde{t},\tilde{u}) = I_3(\tilde{s},\tilde{t},\tilde{u},\tilde{s}) + I_3(\tilde{s},\tilde{t},\tilde{u},\tilde{u}) - I_3(\tilde{s},\tilde{t},\tilde{u}, \rho )\nonumber \\&I_3(\tilde{s},\tilde{t},\tilde{u},\tilde{x}) = \frac{1}{\tilde{s} \tilde{u}} \frac{2}{\beta _+ - \beta _-} \bigg ( \text {Li}_2\left( \frac{\beta _-}{\beta _- - \alpha _-}\right) \nonumber \\&\quad - \text {Li}_2\left( \frac{\beta _+}{\beta _+ - \alpha _+}\right) + \text {Li}_2\left( \frac{\beta _-}{\beta _- - \alpha _+}\right) \nonumber \\&\quad - \text {Li}_2\left( \frac{\beta _+}{\beta _+ - \alpha _-}\right) + \log \left( -\frac{\beta _+}{\beta _-}\right) \log \left( 1+ \frac{\tilde{x} \tilde{t}}{\tilde{s} \tilde{u}}\right) \bigg ) ,\nonumber \\ \end{aligned}$$

(44)

with $\alpha _{\pm } = \frac{1}{2}(1 \pm \sqrt{1 - \frac{4}{\tilde{x}}})$ and $\beta _{\pm } = \frac{1}{2}(1 \pm \sqrt{1 - \frac{4\tilde{t}}{\tilde{s} \tilde{u}}})$. The function $\text {Li}_2$ denotes the Spence function. The functions f, g, J are related to the corresponding scalar one-loop integrals as ($n=4-2\epsilon$)

$$\begin{aligned}&B_0(p;m_t,m_t) = \int \frac{d^nk}{(2\pi )^n} \frac{\mu ^{2\epsilon }}{(k^2-m_t^2) [(k+p)^2-m_t^2]} \nonumber \\&\quad = i \frac{\Gamma (1+\epsilon )}{(4\pi )^2} \left( \frac{4\pi \mu ^2}{m_t^2}\right) ^\epsilon \left[ \frac{1}{\epsilon } + 2 - 2g(\tau )\right] + \mathcal{O}(\epsilon ) \quad \qquad \nonumber \\&\qquad (\tau =4 m_t^2/p^2) \nonumber \\&\quad C_0(p_1,p_2;m_t,m_t,m_t)\nonumber \\&\quad = \int \frac{d^nk}{(2\pi )^n} \frac{\mu ^{2\epsilon }}{(k^2-m_t^2) [(k-p_1)^2-m_t^2] [(k+p_2)^2-m_t^2]} \nonumber \\&\quad = \frac{i}{(4\pi )^2 m_t^2} \left[ -\frac{\tau }{2} f(\tau )\right] + \mathcal{O}(\epsilon ) \quad \qquad (\tau =4 m_t^2/(2 p_1p_2)) \nonumber \\&\quad D_0(p_1,p_2,p_3;m_t,m_t,m_t,m_t)\nonumber \\&\quad = \int \frac{d^nk}{(2\pi )^n}\nonumber \\&\qquad \times \frac{\mu ^{2\epsilon }}{(k^2-m_t^2) [(k+p_1)^2-m_t^2] [(k+p_{12})^2-m_t^2] [(k+p_{123})^2-m_t^2]} \nonumber \\&\quad = \frac{i}{(4\pi )^2 m_t^4} J\left( \frac{s}{m_t^2},\frac{t}{m_t^2},\frac{u}{m_t^2}\right) + \mathcal{O}(\epsilon ) \end{aligned}$$

(45)

where $p_{12} = p_1+p_2, p_{123} = p_1+p_2+p_3$ and $s=2 p_1p_2, t = 2p_1p_3, u=2p_2p_3$ where all $p_i~(i=1,2,3)$ correspond to incoming massless external particles ($p_i^2=0$).