Higgs Pseudo-Observables, Second Riemann Sheet and All That

TL;DR

This paper develops a gauge-invariant framework to relate LHC/Tevatron observables to Higgs pseudo-observables by exploiting the Higgs complex pole on the second Riemann sheet. It provides a comprehensive method for analytic continuation of loop integrals with complex masses and invariants, introducing ln±, Li2±, contour deformation, and an extended BST approach to ensure correct high-order behavior. Three schemes (RMRP, CMRP, CMCP) are defined to study scheme dependence, with CMCP emerging as the consistent external-particle treatment, and extensive numerical results demonstrate significant effects at high Higgs masses. The work has practical implications for interpreting data and setting exclusions up to about 600 GeV, offering a concrete algorithmic path to compute Higgs production and decay pseudo-observables across perturbative orders.

Abstract

The relation between physical observables measured at LHC and Tevatron and standard model Higgs pseudo-observables (production cross section and partial decay widths) is revised by extensively using the notion of the Higgs complex pole on the second Riemann sheet of the $S $-matrix. The extension of their definition to higher orders is considered, confronting the problems that arise when QED(QCD) corrections are included in computing realistic observables. Numerical results are presented for pseudo-observables related to the standard model Higgs boson decay and production. The relevance of the result for exclusion plots of the standard model Higgs boson for high masses (up to $600 $GeV) is discussed. Furthermore, a recipe for the analytical continuation of Feynman loop integrals from real to complex internal masses and complex Mandelstam invariants is thoroughly discussed.

Figures (19)

• Figure 1: Gauge-invariant breakdown of the triply-resonant $g g \to 4\,$f signal into $gg \to H$ production, $H \to W^+ W^-$ decay and subsequent $W \to {\bar{f}} f$ decays.
• Figure 2: Analytical continuation from real $p^2$ to complex $p^2$ as seen in the $\chi\,$-plane with $\chi(x)= - s_{{{P}}}\,x\,(1-x) +\mu^2 - i\,\mu\,\gamma$, with $s_{{{P}}} = M^2 - i\,M\,\Gamma$ and $x\in [0,1]$. Solid lines represent the continuation for a low value of $M$ with a very small value for $\Gamma$. With increasing values for $M$ we reach the situation illustrated by the dot-lines, $\chi$ moving into the second quadrant, i.e. $\chi$ on the second Riemann sheet. Case $1$ holds for $\Gamma < (M/\mu)\,\gamma$ whereas case $2$ holds for $\Gamma > (M/\mu)\,\gamma$. Black circles correspond to $x=0, x=1$ whereas white circles correspond to $x= 1/2$.
• Figure 3: Analytical continuation of a $B_0\,$-function as seen in the $z\,$-plane with a cut along the positive real axis between $R_-$ and $R_+$ (Eq.(\ref{['RpmIpm']})). In the first part the integration path reaches the point $I_-$ (Eq.(\ref{['RpmIpm']})) and continuation after $z = I_-$ is in the second Riemann sheet. In the second part, where $I_- < R_-$ continuation must be, once again, in the second Riemann sheet; therefore the integration path which has moved into the lower half-plane must be deformed to cross the cut before moving once more into the lower half-plane (but on the second Riemann sheet).
• Figure 4: Example of contour deformations in computing a scalar two-point functions with equal (complex) internal masses and complex $p^2$.
• Figure 5: Examples of deformation in the $x$-complex plane ($x=u+iv$) of the integration contour $[0,1]$ for integral of $\ln^-V=\ln^-(ax^2+bx+c)$.