[Paper Review] Analytic Virtual Corrections for Higgs Transverse Momentum Spectrum at $O(α_s^2/m_t^3)$ via Unitarity Methods

Duff Neill|ArXiv.org|Nov 16, 2009

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper presents analytic one-loop virtual corrections for Higgs transverse momentum production in gluon fusion and quark-antiquark annihilation at $O(\alpha_s^2/m_t^3)$ using unitarity methods and Higgs Effective Field Theory with dimension-eight operators. It derives exact helicity amplitudes for $HFF$, $HFFF$, and $HFDJ$ operators, enabling precise two-loop matching for the Higgs $p_T$ spectrum with full analytic control over kinematic singularities and anomalous dimensions.

ABSTRACT

Utilizing the Higgs Effective Theory, including dimension eight operators, in conjuction with spinor-helicity and unitarity methods, I present analytic expressions for the virtual corrections for $gg o Hg$ and $q\overline{q} o Hg$ amplitudes at order $α_s^2/m_t^3$. These contributions become enhanced as the invariant final state mass increases.

Motivation & Objective

• To compute virtual corrections to Higgs production with a hard jet at $O(\alpha_s^2/m_t^3)$ in the heavy top quark limit.
• To extend the Higgs Effective Theory to include dimension-eight operators for improved matching at two-loop order.
• To apply unitarity and spinor-helicity methods to compute analytic amplitudes for higher-dimensional operators with non-renormalizable power counting.
• To provide a systematic framework for computing subleading corrections in the heavy mass expansion for the Higgs transverse momentum spectrum.
• To enable accurate phenomenological predictions for Higgs $p_T$ distributions at the LHC by capturing dominant $m_t^{-3}$-suppressed effects.

Proposed method

• Utilizes Higgs Effective Field Theory with dimension-eight operators $O_1$ to $O_5$ to capture $m_t^{-3}$ corrections from top quark loops.
• Applies generalized unitarity and spinor-helicity techniques to compute one-loop amplitudes for $gg \to Hg$ and $q\bar{q} \to Hg$ processes.
• Employs dimensional regularization with $\epsilon$-dimensional cuts to handle infrared and ultraviolet divergences in loop integrals.
• Derives rational terms via auxiliary shifts and on-shell recursion, addressing large-$z$ behavior issues in loop-level amplitudes.
• Decomposes higher-dimensional operators using equations of motion and Bianchi identities to reduce the basis to $O_3$ and $O_5$, simplifying computation.
• Uses master integrals and $\overline{\text{MS}}$ renormalization to extract anomalous dimensions and match to full QCD at two loops.

Experimental results

Research questions

• RQ1 How can one systematically compute virtual corrections to the Higgs transverse momentum spectrum at $O(\alpha_s^2/m_t^3)$ using effective field theory?
• RQ2 What are the analytic one-loop amplitudes for $HFF$, $HFFF$, and $HFDJ$ operators in the context of Higgs EFT with dimension-eight operators?
• RQ3 How do unitarity and spinor-helicity methods handle the rational terms and infrared structure in amplitudes involving non-renormalizable operators?
• RQ4 What is the role of the $HFDJ$ operator in generating $m_t^{-3}$-suppressed corrections to the $Hgg$ vertex at one loop?
• RQ5 How do the derived amplitudes reproduce known anomalous dimensions and scheme dependence in the literature?

Key findings

• The one-loop amplitude for the $HFFF$ operator is found to be $m^1_{HFFF}(1^+,2^+,3^+) = m^0_{HFFF}(1^+,2^+,3^+) r_\Gamma N_c \frac{\alpha_s}{4\pi} \left( \frac{1-4\epsilon}{\epsilon^2} \left[ \left(\frac{-m_h^2}{-S_{12}}\right)^\epsilon + \cdots \right] \right) + O(\epsilon)$, capturing infrared and collinear singularities.
• The $HFDJ$ operator contributes to the amplitude only at one loop, with $m^1_{HFDJ}(1^+,2^+,3^+) = -m^0_{HFFF}(1^+,2^+,3^+) n_f \frac{\alpha_s}{4\pi} + O(\epsilon)$, showing its role in $m_t^{-3}$ corrections.
• For quark-gluon final states, the $HFDJ$ amplitude is $m^1_{HFDJ}(1^{-}_{\bar{q}},2^{+}_q,3^{+}) = m^0_{HFDJ}(1^{-}_{\bar{q}},2^{+}_q,3^{+}) \left( \frac{\alpha_s}{4\pi} r_\Gamma \right) \left( N_c U_1 + \frac{1}{N_c} U_2 \right)$, with $U_1$ and $U_2$ encoding kinematic dependence and $\epsilon$-poles.
• The $HFFF$ amplitude vanishes for helicity configuration $(1^+,2^+,3^-)$ at one loop: $m^1_{HFFF}(1^+,2^+,3^-) = 0 + O(\epsilon)$, consistent with helicity conservation and gauge invariance.
• The derived amplitudes correctly reproduce known anomalous dimensions for the $HFFF$ operator as shown in Gracey (2002rf) and scheme dependence from Catani (1996pk).
• The method successfully handles rational terms and large-$z$ behavior in loop amplitudes for non-renormalizable operators, extending unitarity techniques beyond renormalizable theories.