Higgs bosons at high $p_T$

We present a calculation of $H+j$ at next-to-leading order including the effect of a finite top mass. Where possible we include the complete dependence on $m_t$. This includes the leading order amplitude, the infrared poles of the two-loop amplitude and the real radiation amplitude. The remaining finite piece of the virtual correction is considered in an asymptotic expansion in $m_t$, which is accurate to $m_t^{-4}$. By successively including more $m_t$-exact pieces, the dependence on the asymptotic series diminishes and we find convergent behavior for $p_{T,H}>m_t$ for the first time. Our results justify rescaling by the $m_t$-exact leading-order (LO) cross section to model top-mass effects in effective field theory results up to $p_T$ of 250 to 300 GeV. We show that the error made by using the LO rescaling becomes comparable to the next-to-next-to-leading-order scale uncertainty for such large energies. We implement our results into the Monte Carlo code mcfm.

Figure 1

Representative Feynman diagrams for the production of a Higgs boson plus one jet at LO (left) and NLO (center, right). The NLO corrections include two-loop “virtual” topologies (center), and one-loop Higgs plus two parton “real” topologies (right).

Figure 2

Higgs inclusive $p_T$ spectrum for three different approximations, each taking into account higher orders of an asymptotic expansion in $1/m_t$. The upper panel shows the absolute distribution, while the lower two panels display the ratio to the LO distribution and the ${\mathrm{NLO}}^{*}$ $1/m_t^0$ approximation, respectively.

Figure 3

Invariant mass spectrum of the Higgs plus hardest jet system at LO. The upper part displays the absolute distribution, while the lower part displays the ratio to the EFT result.

Figure 4

Invariant mass spectrum of the Higgs plus hardest jet system at NLO. The upper part displays the absolute distribution, while the lower part displays the ratio to the EFT result.

Figure 5

Ratio of the ${\mathrm{NLO}}^{*}$ finite virtual piece ${\mathcal{F}}_{\mathrm{SI}}^{\mathrm{in}}$ in the asymptotic expansion as in Eq. (7) to the LO rescaled EFT virtual piece ${\mathcal{F}}_{\mathrm{EFT}\text{-resc}}^{\mathrm{in}}$ as in Eq. (5).

Figure 6

Rescaling ratios for the inclusion of mass effects in the NNLO Higgs inclusive $p_T$ spectrum. The upper panel shows ratios directly, while the lower panel displays the ratio to the LO ratio to give an estimation of the error made by using the LO rescaling scheme. The solid lines are locally weighted scatter plot smoothing curves to guide the eye.

Figure 7

${\mathrm{NLO}}^{*}$ transverse momentum distributions of the Higgs boson with minimum jet transverse momenta of 30, 70 and 110 GeV. The upper panel shows the absolute distribution, while the lower two panels display the ratio to the LO distribution and the ${\mathrm{NLO}}^{*}$ $1/m_t^0$ approximation, respectively.

Figure 8

Higgs rapidity distribution for different orders and approximations. The upper panel shows the absolute distribution, while the lower two panels display the ratio to the LO distribution and the ${\mathrm{NLO}}^{*}$ $1/m_t^0$ approximation, respectively.

Figure 9

Higgs inclusive transverse momentum distribution at LO and ${\mathrm{NLO}}^{*}$. The ribbon is obtained by varying the factorization and renormalization scale $\mu =\sqrt{p_{T,H}^2+m_H^2}$ by a factor of 2 and $1/2$. The middle panel shows the ratio to the LO cross section. The lower panel shows the mean of the upper and lower bound change with respect to the central value in percent, commonly known as the scale uncertainty.

Figure 10

Higgs rapidity distribution at LO and ${\mathrm{NLO}}^{*}$. The ribbon is obtained by varying the factorization and renormalization scale $\mu =\sqrt{p_{T,H}^2+m_H^2}$ by a factor of 2 and $1/2$. The middle panel shows the ratio to the LO cross section. The lower panel shows the mean of the upper and lower bound change with respect to the central value in percent, commonly known as the scale uncertainty.