Higgs couplings and BSM

• Let each visible SM partial width scale as \(\Gamma_i = \kappa_i^2\,\Gamma_i^{\rm SM}\) .
• Define the visible width–scaling factor \(\kappa_H^2 \;\equiv\; \sum_i \kappa_i^2\,{\rm BR}i^{\rm SM}\),

so that the sum of visible partial widths is \(\kappa_H^2\,\Gamma_{\rm SM}\) (because \(\sum_i \Gamma_i^{\rm SM} = \Gamma_{\rm SM}\)).

• Add BSM (invisible/undetected) partial widths \(\Gamma_{\rm BSM}\equiv \Gamma_{\rm inv}+\Gamma_{\rm undet}\). It is convenient to describe them by branching fractions

\(B_{\rm inv}=\frac{\Gamma_{\rm inv}}{\Gamma_{\rm tot}},\qquad\) \(B_{\rm undet}=\frac{\Gamma_{\rm undet}}{\Gamma_{\rm tot}},\qquad\) \(B_{\rm BSM}\equiv B_{\rm inv}+B_{\rm undet}\)

Then the total width satisfies \[ \Gamma_{\rm tot} =\kappa_H^2\,\Gamma_{\rm SM} + \Gamma_{\rm BSM} =\kappa_H^2\,\Gamma_{\rm SM} + B_{\rm BSM}\,\Gamma_{\rm tot}. \]

Solve for \(\Gamma_{\rm tot}:\)

\[ \Gamma_{\rm tot}\,(1-B_{\rm BSM})=\kappa_H^2\,\Gamma_{\rm SM} \;\;\Rightarrow\;\; \frac{\Gamma_{\rm tot}}{\Gamma_{\rm SM}}=\frac{\kappa_H^2}{1-B_{\rm BSM}}\;. \tag{1} \]

For production mode p (coupling \(\kappa_p\)) and decay to final state f (coupling \(\kappa_f\)), the rate scales as \[ \mu_{p,f} \equiv \frac{\sigma_p\times{\rm BR}f}{(\sigma_p\times{\rm BR}f)_{\rm SM}} = \frac{\kappa_p^2\,\kappa_f^2}{\displaystyle \frac{\Gamma{\rm tot}}{\Gamma_{\rm SM}}} = \frac{\kappa_p^2\,\kappa_f^2}{\kappa_H^2/(1-B_{\rm BSM})} = \frac{\kappa_p^2\,\kappa_f^2}{\kappa_H^2}\,\bigl(1-B_{\rm BSM}\bigr). \tag{2} \] This is the standard compact form you’ll often see:

\[ \boxed{\;\mu_{p,f} \;=\; \frac{\kappa_p^2\,\kappa_f^2}{\kappa_H^2}\,(1-B_{\rm BSM})\;} \]

Consider a common rescaling of all visible couplings:

\[ \kappa_i \;\to\; c\,\kappa_i \quad \text{for all visible } i. \]

Then

\[ \kappa_p^2 \to c^2\kappa_p^2,\quad \kappa_f^2 \to c^2\kappa_f^2,\quad \kappa_H^2 \to c^2\kappa_H^2. \]

Plug into box above: \[ \mu_{p,f}\;\to\; \frac{(c^2\kappa_p^2)(c^2\kappa_f^2)}{c^2\kappa_H^2}\,(1-B_{\rm BSM}^\prime) = c^2\,\frac{\kappa_p^2\kappa_f^2}{\kappa_H^2}\,(1-B_{\rm BSM}^\prime). \]

To keep \(\mu_{p,f}\) unchanged for every channel, choose a new BSM width such that \[ c^2\,(1-B_{\rm BSM}^\prime) = (1-B_{\rm BSM}) \;\;\Rightarrow\;\; 1-B_{\rm BSM}^\prime = \frac{1-B_{\rm BSM}}{c^2}. \tag{3} \] Equivalently, \[ B_{\rm BSM}^\prime = 1 - \frac{1-B_{\rm BSM}}{c^2}. \] As you increase c>1 (make all visible κ’s larger), you can increase the BSM branching fraction

\(B_{\rm BSM}\) per (3) so that all measured \(\mu_{p,f}\) stay exactly the same.

That one-parameter family \(\{c,\,B_{\rm BSM}^\prime\}\) is the flat direction—the experimental likelihood is nearly unchanged along it.

Two sanity checks: • If B_BSM=0 initially, then after rescaling by c, you need \(B_{\rm BSM}^\prime = 1-1/c^2\) to keep the same \(\mu\)’s. • Physicality requires \(0\le B_{\rm BSM}^\prime\le 1\), which merely bounds how far you can push c for a fixed starting point.

Experimental Challenges Higgs EWK Fall 2025 Physics