We provide a precise description of the Higgs boson transverse momentum distribution
including top and bottom quark contributions, that is valid for transverse momenta in the
range

Article funded by SCOAP3

(1-z_1^{(\ell_2)})$) \begin{equation} \label{eq:zlimit} z_1^{(\ell_1)} < 1-\frac{k_{\perp1}}{\mh}. \end{equation} An analogous limit on $z_1^{(\ell_2)}$ as in eq.~\eqref{eq:zlimit} holds when $k_1$ is collinear to $\tilde{p}_2$. Subsequent emissions off leg $\ell_1$ can be parametrized analogously to eq.~\eqref{eq:Sudakov}, replacing the reference momentum $\tilde{p}_1 $ with \begin{equation} \tilde{p}_1 \to \left(\prod_{\substack{i=1\\\ell_i = \ell_1}}^k z_i^{(\ell_i)}\right)\tilde{p}_1\simeq \tilde{p}_1, \end{equation} where the product runs over all emissions off leg $\ell_1$ that occur prior to the emission we are parametrizing, and we used the fact that in the soft limit $ z_i^{(\ell_i)}\simeq 1$. A similar parametrization holds for emissions off leg $\ell_2$. The transverse recoil of the radiation is absorbed entirely by the Higgs boson that acquires a transverse momentum \begin{align} \pt = |\sum_{i}\vec{k}_{\perp i}|. \end{align} In order to predict the $\pt\to 0$ limit, we need to sum emissions at all orders in the strong coupling. With LL accuracy, the squared amplitude for $n$ emissions can be approximated by a product of $n$ independent splitting kernels, as the soft correlation between emissions starts contributing at NLL order. The physical picture corresponding to this approximation is given by a set of independent emissions off legs $\ell_1$ and $\ell_2$. In this approximation, the differential partonic distribution can be written as \begin{align} \label{eq:eikonal_app} \frac{{\rm d}\hat \sigma}{{\rm d}\pt} \simeq [d p_H]\mathcal|\mathcal M(\tilde p_1 + \tilde p_2\to H)|^2 \delta^{(4)}(\tilde p_1+\tilde p_2 - p_H)\notag \\ \times \frac{1}{n!}\prod_{i=1}^{n} [dk_i]|M_{\rm soft}(k_i)|^2 \delta\left(\pt- |\sum_{i}\vec{k}_{\perp i}|\right), \end{align} where the eikonal squared amplitude for a single emission reads \begin{align} \label{eq:single-emsn} [dk] | M_{\rm soft}(k)|^2=\sum_{\ell=1,2} 2 C_A \frac{\alpha_s(k_{\perp})}{\pi}\frac{dk_{\perp}}{k_{\perp}} \frac{d z^{(\ell)}}{1-z^{(\ell)}}\, \Theta\left((1-z^{(\ell)}) - k_{\perp}/\mh\right) \frac{d\phi}{2\pi}. \end{align} In eq.~\eqref{eq:single-emsn}, the coupling is evaluated at $k_{\perp}$ to account for the leading-logarithmic contribution of the gluon branching into either a pair of soft quarks or gluons, see e.g.~\cite{Banfi-ml-2004yd} for a detailed explanation. The resummation is naturally performed at the level of the cumulative distribution, defined as \begin{equation} \Sigma(\pt) = \int_0^{\pt} d \pt' \frac{d\sigma}{d \pt'}. \end{equation} Indeed while the differential spectrum involves plus distributions in $\pt$, $\Sigma(\pt)$ is a regular function. From eq.~\eqref{eq:eikonal_app}, it follows that the cumulative distribution with LL accuracy can be written as \begin{equation} \Sigma(\pt) \simeq \left[f_g(\mu_F)\otimes f_g(\mu_F)\right](\mh^2/s)\times\int d\hat\sigma\,\Theta\left(\pt- |\sum_{i}\vec{k}_{\perp i}|\right), \end{equation} where $f_g(\mu_F)$ is the gluon parton density evaluated at the factorization scale $\mu_F$, and the convolution is defined as usual \begin{equation} \left[f\otimes g\right](x) \equiv \int_0^1 {\rm d}y \; {\rm d}z \; \delta(x-yz) f(y) g(z). \end{equation} Since $\pt$ only constrains the transverse momentum of the emissions, we can perform the integrals over the $z_i^{(\ell_i)}$ components inclusively. It is therefore convenient to introduce the~functions \begin{equation} \label{eq:R'1,2} \begin{split} R'_1\left(\pt\right)&= \int [dk] |M_{\rm soft}(k)|^2 \,(2\pi) \delta(\phi-\bar \phi)\, \pt\delta\left(\pt-k_{\perp}\right)\Theta(z^{(2)} -z^{(1)}) \,, \\ R'_2\left(\pt\right)& = \int [dk] |M_{\rm soft}(k)|^2\,(2\pi) \delta(\phi-\bar \phi)\, \pt\delta\left(\pt-k_{\perp}\right)\Theta(z^{(1)} -z^{(2)})\,. \end{split} \end{equation} This notation allows us to parametrize the real-emission matrix element and phase space~as \begin{align} \label{eq:single-emsn-rp} [dk_i] |M_{\rm soft}(k_i)|^2 = \frac{dk_{\perp i}}{k_{\perp i}} \frac{d\phi_i}{2\pi}\sum_{\ell_i=1,2} R'_{\ell_i}\left(k_{\perp i}\right) &= \frac{d\zeta_i}{\zeta_i} \frac{d\phi_i}{2\pi} \sum_{\ell_i=1,2} R'_{\ell_i}\left(\zeta_i k_{\perp1}\right) \,, \end{align} where we defined $\zeta_i = k_{\perp i}/k_{\perp1}$. We now discuss the purely virtual corrections, which are encoded in the gluon form factor $|\mathcal M(\tilde p_1 + \tilde p_2\to H)|^2$. We write it as \begin{equation} |\mathcal M(\tilde p_1 + \tilde p_2\to H)|^2 = {\cal H}(\mh)|\mathcal M_B(\tilde p_1 + \tilde p_2\to H)|^2, \end{equation} where the function ${\cal H}$ contains all the IRC singularities and the constant finite corrections of the form factor, and ${\cal M}_B$ denotes the Born amplitude. Since we are working with LL accuracy, we are only interested in the leading singular term of ${\cal H}$ at all orders (while neglecting all finite terms) which can be written as \begin{equation} \label{eq:form_factor} {\cal H}(\mh) \simeq \exp\left\{-\int [dk] |M_{\rm soft}(k)|^2\right\}. \end{equation} Note that the integral in eq.~\eqref{eq:form_factor} is divergent and is to be considered as regularized. In order to cancel the IRC divergences of the real emissions~\eqref{eq:eikonal_app} against the ones in the virtual corrections~\eqref{eq:form_factor} at all orders, we introduce a small slicing parameter $\epsilon >0$ such that all emissions with a transverse momentum $k_{\perp i}$ smaller than $\epsilon k_{\perp1}$ can be ignored in the computation of the observable $\pt$, in the limit $\epsilon\to 0$. The real emissions with $k_{\perp i}<\epsilon k_{\perp1}$, hereby denoted as \emph{unresolved}, can be directly combined with the virtual corrections at all orders. Their combination gives rise to an exponential suppression factor of the type \begin{align} {\cal H}(\mh)&\sum_{m}^{\infty}\frac{1}{m!}\prod_{i=1}^m \bigg[\int\frac{dk_{\perp i}}{k_{\perp i}} \frac{d\phi_i}{2\pi}\sum_{\ell_i=1,2} R'_{\ell_i}\left(k_{\perp i}\right) \Theta\left(\epsilon k_{\perp1}-k_{\perp i}\right) \bigg] \notag\\ &= \exp\left\{ -\int\frac{dk_{\perp}}{k_{\perp}} \frac{d\phi}{2\pi}\sum_{\ell=1,2} R'_{\ell}\left(k_{\perp}\right) \Theta\left(k_{\perp} - \epsilon k_{\perp1}\right)\right\}\equiv e^{-R(\epsilon k_{\perp1})}. \end{align} On the other end, emissions with $k_{\perp i}>\epsilon k_{\perp1}$, that we denote as \emph{resolved}, are constrained by the observable's measurement function and therefore cannot be integrated over inclusively. The resummed LL cross section thus reads \begin{align} \label{eq:XS_LL} \Sigma(\pt) \simeq &\,\sigma_B \int \frac{dk_{\perp1}}{k_{\perp1}} \frac{d\phi_1}{2\pi} e^{-R(\epsilon k_{\perp1})}\sum_{\ell_1=1,2} R'_{\ell_1}\left(k_{\perp1}\right)\notag\\ &\times \sum_{n=0}^{\infty}\frac{1}{n!}\prod_{i=2}^{n+1}\int_{\epsilon}^1 \frac{d\zeta_i}{\zeta_i} \frac{d\phi_i}{2\pi} \sum_{\ell_i=1,2} R'_{\ell_i}\left(\zeta_i k_{\perp1}\right) \Theta\left(\pt- |\sum_{i}\vec{k}_{\perp i}|\right), \end{align} where $\sigma_B$ is the Born cross section. The above formula, in the limit $\epsilon\to 0$ exactly reproduces the LL corrections to the $\pt$ distribution, see ref.~\cite{Bizon-ml-2017rah} for a formal proof. Eq~\eqref{eq:XS_LL} can be further simplified by observing that in the resolved radiation one always has $\zeta_i\sim 1$, since configurations in which $\zeta_i\ll 1$ are automatically canceled against the exponential Sudakov factor $e^{-R(\epsilon k_{\perp1})}$. Therefore, one can expand the functions $R'\left(\zeta_i k_{\perp1}\right)$ in powers of $\ln(1/\zeta_i)$ as \begin{align} \label{eq:expansion} R'_{\ell_i}\left(\zeta_i k_{\perp1}\right) = R'_{\ell_i}\left(k_{\perp1}\right) +R''_{\ell_i}\left(k_{\perp1}\right) \ln\frac{1}{\zeta_i} + \dots, \end{align} and retain terms that contribute at a given logarithmic order. In particular, at LL, only the first term in this expansion contributes, and higher-order terms matter at higher logarithmic orders (see refs.~\cite{Bizon-ml-2017rah} for details). Similarly, we can consistently expand out the $\epsilon$ dependence of the exponential Sudakov~as \begin{equation} e^{-R(\epsilon k_{\perp1})}= e^{-R(k_{\perp1})} e^{-R'\left(k_{\perp1}\right) \ln\frac{1}{\epsilon} + \dots }, \end{equation} where the $\epsilon$ dependence manifestly cancels against the one in the resolved contribution, and we defined \begin{equation} R'\left(k_{\perp1}\right)\equiv \sum_{\ell_1=1,2} R'_{\ell_1}\left(k_{\perp1}\right). \end{equation} Therefore, with LL accuracy, eq.~\eqref{eq:XS_LL} becomes \begin{align} \label{eq:XS_LL_expanded} \Sigma(\pt) \simeq &\,\sigma_B \int \frac{dk_{\perp1}}{k_{\perp1}} \frac{d\phi_1}{2\pi} e^{-R(k_{\perp1})}\epsilon^{R'\left(k_{\perp1}\right)}\sum_{\ell_1=1,2} R'_{\ell_1}\left(k_{\perp1}\right)\notag\\ &\times \sum_{n=0}^{\infty}\frac{1}{n!}\prod_{i=2}^{n+1}\int_{\epsilon}^1 \frac{d\zeta_i}{\zeta_i} \frac{d\phi_i}{2\pi} \sum_{\ell_i=1,2} R'_{\ell_i}\left(k_{\perp1}\right) \Theta\left(\pt- |\sum_{i}\vec{k}_{\perp i}|\right). \end{align} Equation~\eqref{eq:XS_LL_expanded} is suitable for a numerical implementation, as explained in ref.~\cite{Bizon-ml-2017rah} in detail. The dependence on $\epsilon$ is at most power suppressed (i.e.\ ${\cal O}(\epsilon \pt)$) and it vanishes in the limit $\epsilon\to 0$. This limit can therefore be taken safely numerically, and the result is absolutely stable for very small values of $\epsilon$.\footnote{In our implementation we use $\epsilon = e^{-20}$, although any value below $\epsilon = e^{-6}$ does not lead to any appreciable differences.} We now introduce the resummation scale $Q$ as a possible way to switch off the resummation at large transverse momentum. This is defined with a procedure similar to the one discussed in the text. We first break the logarithm as follows \be L\equiv\ln \frac{\mh}{k_{\perp1}} = \ln\frac{\mh}{Q}+\ln\frac{Q}{k_{\perp1}}. \ee The above operation will allow us (as explained shortly) to have an additional handle (namely the scale $Q$) to estimate the size of subleading logarithmic terms. Moreover, we also slightly modify the phase space available for the radiation, by introducing power-suppressed contributions that ensure that at large $\pt$ the resummation effects completely vanish. This can be done, as a first step, by modifying the resummed logarithms as follows \be \ln\frac{Q}{k_{\perp1}} \to \frac{1}{p}\ln \left(\frac{Q^p}{k_{\perp1}^p}+1\right) \equiv \tilde L, \label{eq:modlog_kt1} \ee where $p$ is a positive real parameter which is chosen such that the resummed differential distribution vanishes as $1/\pt^{p+1}$ at large $\pt$. The above prescription essentially amounts to the following \begin{enumerate} \item First, we split the resummed logarithm $L$ into the sum of a \emph{small} logarithm $\ln(\mh/Q)$ (with $Q\sim \mh$) and a \emph{large} one $\ln(Q/k_{\perp1})$. This operation allows one to introduce a generic scale $Q$ which appears in the resummed logarithms. One can now expand $L$ about $\ln(Q/k_{\perp1})$, retaining all terms with the desired logarithmic accuracy. Effectively, this implies that $\ln(\mh/Q)$ is treated perturbatively at fixed order. Moreover, we replace $\ln(Q/k_{\perp1})$ by the modified logarithm $\tilde{L}$. In our LL example this means \begin{align} R(k_{\perp1}) \to \tilde{R}(k_{\perp1}) + {\cal O}(\ln \mh/Q);\qquad R(k_{\perp1}) \to \tilde{R}'(k_{\perp1}) + {\cal O}(\ln \mh/Q), \end{align} where $\tilde{R}$ and $\tilde{R}'$ are functions of the modified logarithm $\tilde{L}$ only. \item eq.~\eqref{eq:modlog_kt1} comes together with the following prefactor $\mathcal J$ in eq.~\eqref{eq:XS_LL_expanded} \begin{equation} \label{eq:jakobian} {\cal J}(k_{\perp1}) = \left(\frac{Q}{k_{\perp1}} \right)^p \left(1+\left(\frac{Q}{k_{\perp1}} \right)^p\right)^{-1}. \end{equation} This corresponds to the Jacobian for the transformation~\eqref{eq:modlog_kt1}, and ensures the absence of fractional (although power suppressed) $\alpha_s$ powers in the final distribution~\cite{Bizon-ml-2017rah}. This factor, once again, leaves the small $k_{\perp1}$ region untouched, and only modifies the large $\pt$ region by power-suppressed effects. This is effectively mapping the limit $k_{\perp1}\to Q$ onto $k_{\perp1}\to \infty$. Although this procedure seems a simple change of variables, we stress that the observable's measurement function (i.e.\ the $\Theta$ function in eq.~\eqref{eq:XS_LL_expanded}) is not affected by this prescription. As a consequence, the final result will depend on the parameter $p$ through power-suppressed terms. \end{enumerate} The difference between the above prescription and what was introduced in the text is that the argument of the (modified) logarithms is now $k_{\perp1}$ instead of $\pt$. This prescription is technically more correct, since in the small $k_{\perp1}$ region, which governs the $\pt\to 0$ limit, the modified logarithms leave eq.~\eqref{eq:XS_LL_expanded} untouched. Conversely, at large $k_{\perp1}$, where one has $k_{\perp1}\sim \pt$, the above prescription reduces to what was defined in the text, i.e.\ the modified logarithms of $k_{\perp1}$ in this region are formally equivalent to modified logarithms in $\pt$. To see this, we observe that when $k_{\perp1}\gg Q$ the function $R'(k_{\perp1})\ll 1$. Therefore, the probability of having any emission after the first one in eq.~\eqref{eq:XS_LL_expanded} is strongly suppressed. As a consequence, at large $k_{\perp1}$, the only relevant event is the one that involves a single emission $k_1$, for which the cross section reads \begin{align} \label{eq:XS_LL_asympt} \Sigma(\pt) \sim &\,\sigma_B \int \frac{dk_{\perp1}}{k_{\perp1}} \frac{d\phi_1}{2\pi} {\cal J}(k_{\perp1}) e^{-\tilde{R}(k_{\perp1})}\sum_{\ell_1=1,2} \tilde{R}'_{\ell_1}\left(k_{\perp1}\right)\Theta\left(\pt- |\vec{k}_{\perp1}|\right) = e^{-\tilde{R}(\pt)}. \end{align} It is easy to see that, if eq.~\eqref{eq:XS_LL_asympt} were evaluated without the factor ${\cal J}$, it would lead to additional power-suppressed terms with fractional power of the coupling, which are clearly~spurious. ]]>