Hugonweb | BSM Effective Interactions for the Higgs Boson

Higgs-Fermion Effective Interactions

From¹², an effective $\psi^2\phi^3$ operator may modify the Higgs-fermion coupling:

$$g_{Hf} = \sqrt{2}\frac{m_f}{v} + \frac{\alpha v^2}{\Lambda^2}$$

where $m_f$ is the fermion mass, $\Lambda$ is the scale of the new physics, $v$ is the Higgs vacuum expectation value, ~246 GeV/c$^2$, and $\alpha$ is the "Wilson Coefficient" or effective coupling strength.

Coupling scale-factor:

$$k_f = 1 + \frac{\alpha v^3}{\sqrt{2}m_f\Lambda^2}$$

Scale Factor on production rate:

$$\mu = k_f^2$$

New physics mass scale for a given $k_f$:

$$\Lambda = \sqrt{\frac{\alpha v^3}{\sqrt{2} m_f ( k_f - 1 )}}$$

Higgs-Gluon Effective Interactions

From³,

Scale factor on the production rate or decay width:

$$\mu = \left\| 1 - \frac{8 \pi^2 v^2 c_G }{ \Lambda^2 I^g } \right\|^2 + \left\| \frac{8 \pi^2 v^2 \tilde{c}_G }{\Lambda^2 I^g } \right\|^2$$

where all variables are as in the above section, but $c_G$ and $\tilde{c}_G$ are the Wilson Coeffecients for the CP-conserving and CP-violating effective interactions respectively. $I^g$ is the amplitude for the top loop, which can be approximated for $m_t\rightarrow\infty$ as:

$$\frac{1}{I^g} = 2.76 - 6.37 \times 10^{-2} \left(\frac{m_H}{100 \mathrm{GeV}}\right)^2$$

Note that for the CP-conserving part, $c_G<0$ yields an increase in the rate, while $c_G>0$ decreases the rate.

The scale of new physics, $\Lambda$, can then be parameterized in terms of the coupling modifier $k_g = \sqrt{\mu}$. If only the CP-conserving interaction is involved:

$$\Lambda = \sqrt{\frac{8\pi^2 v^2 c_G}{I^g (k_g-1)}}$$

Higgs-Photon Effective Interactions

Similar to the Higgs-gluon case, and from the same reference:

$$k_\gamma^2 = \left\| 1 - \frac{4 \pi^2 v^2 c_{\gamma\gamma}}{ \Lambda^2 I^\gamma } \right\|^2 + \left\| \frac{4 \pi^2 v^2 \tilde{c}_{\gamma\gamma} }{ \Lambda^2 I^\gamma } \right\|^2$$

where $c_{\gamma\gamma}$ is the Wilson coefficient for the CP-conserving operator and $\tilde{c}_{\gamma\gamma}$ is the Wilson Coefficient for the CP-violating operator. $I^\gamma$ is the amplitude for the loop, in which both top and W are important, which can be approximated as:

$$\frac{1}{I^\gamma} = -0.85 + 0.16\left(\frac{m_H}{100 \mathrm{GeV}}\right)^2$$

The scale of new physics, $\Lambda$, can then be parameterized in terms of the coupling modifier $k_\gamma = \sqrt{\mu}$. If only the CP-conserving interaction is involved:

$$\Lambda = \sqrt{\frac{4\pi^2 v^2 c_{\gamma\gamma}}{I^\gamma (k_\gamma-1)}}$$

CMS Current Upper Limits on Higgs Couplings

From ⁴, reading off the 95% CL upper limit from the profile-likelihood scan plots at $-2\Delta\ln\mathcal{L}=2.7$. The $k_\mu$ values are my estimates, assuming the $H\rightarrow \mu\mu$ sensitivity to be similar to the $t\bar{t}H$ sensitivity.

Effective Guon and Photon Couplings

Assuming SM Loop Structure

CMS Projected Uncertainty on Higgs Couplings

From⁵, systematics scenario 1, $1\sigma$ uncertainties. The $k_\mu$ values are my estimates, assuming the $H\rightarrow \mu\mu$ sensitivity to be similar to the $t\bar{t}H$ sensitivity.

Expected 1-Sigma Uncertainties

Expected 95% CL Upper Limits

These are computed from the above $1\sigma$ uncertainties assuming Gaussian errors.

Results: Limits on Effective Mass Scale from CMS Coupling Modifiers

Results Assuming Effective Gluon and Photon Couplings

Assuming unit Wilson coefficients for fermions. Wilson coefficients for gluon effective interaction are assumed to be -1 and 0 for CP-conserving and 0 CP-violating interactions, respectively. For the photon effective interaction, the CP-conserving wilson coefficient is assumed to be 1, and the CP-violating coefficient 0.

Results Assuming SM Loop Structure

Assuming unit Wilson coefficients.

Projected Limits

Assuming unit Wilson coefficients for fermions. Wilson coefficients for gluon effective interaction are assumed to be -1 and 0 for CP-conserving and 0 CP-violating interactions, respectively. For the photon effective interaction, the CP-conserving wilson coefficient is assumed to be 1, and the CP-violating coefficient 0.

95% CL Lower Limit on Effective Mass Scale TeV/c$^2$

Results: Limits on Dimensionless Couplings from CMS Coupling Modifiers

Method

Instead of assuming unit couplings, one may assume the effective mass scale equal to $v=246$ GeV/c$^2$, the electroweak symmetry breaking scale, and then extract limits on the couplings. In this case:

$$c_f' = \sqrt{2}(k_f-1)\frac{m_f}{v}$$

$$c_g' = \frac{I^g(k_g-1)}{8 \pi^2}$$

$$c_{\gamma\gamma}' = \frac{I^\gamma(k_\gamma-1)}{4\pi^2}$$

Results with Effective Gluon and Photon Couplings

CP-Odd coefficients assumed zero.

Assuming SM Loop Structure

CP-Odd coefficients assumed zero.

Projected Limits

CP-Odd coefficients assumed zero.

95% CL Lower Limit on Effective Coupling