# Diphoton and digluon vertices with SPheno

For the calculation of the partial width of a neutral scalar Φ decaying into two gluons or two photons SPheno follows closely for the LO and NLO contributions. The partial widths at LO are given by

`\Gamma(\Phi \to \gamma \gamma)_{\rm LO} = \frac{G_F \alpha^2(0) m_\Phi^3}{128 \sqrt{2} \pi^3} \Bigg|\sum_f N^f_c Q_f^2 r^\Phi_f A_f(\tau_f) + \sum_s N^s_c r^\Phi_s Q_s^2 A_s(\tau_s) + \sum_v N^v_c r^\Phi_v Q_v^2 A_v(\tau_v) \Bigg|^2, \\ \Gamma(\Phi \to g g)_{\rm LO} = \frac{G_F \alpha_s^2(\mu) m_\Phi^3}{36 \sqrt{2} \pi^3} \Bigg|\sum_f \frac{3}{2} D_2^f r^\Phi_f A_f(\tau_f) + \sum_s \frac{3}{2} D_2^s r^\Phi_s A_s(\tau_s) + \sum_v \frac{3}{2} D_2^v r^\Phi_v A_v(\tau_v) \Bigg|^2.`

Here, the sums are over all fermions *f*, scalars *s* and vector bosons *v* which are charged or coloured and which couple to the scalar *Φ*. *Q* is the electromagnetic charges of the fields, *N*_{c} are the colour factors and *D*_{2} is the quadratic Dynkin index of the colour representation which is normalised to `\frac12`

for the fundamental representation. We note that the electromagnetic fine structure constant *α* must be taken at the scale *μ* = 0, since the final state photons are real . In contrast, *α*_{s} is evaluated at *μ* = *m*_{Φ} which, for the case of interest here, is *μ* = 750 GeV. *r*_{i}^{Φ} are the so-called reduced couplings, the ratios of the couplings of the scalar *Φ* to the particle *i* normalised to SM values. These are calculated as

`r^\Phi_f = \frac{v}{2 M_f} (C^L_{\bar f f \Phi}+C^R_{\bar f f \Phi}), \\ r^\Phi_s = \frac{v}{2 M^2_s} C_{s s^\* \Phi},\\ r^\Phi_v = -\frac{v}{2 M^2_v} C_{v v^\* \Phi}.`

Here, *v* is the electroweak VEV and *C* are the couplings between the scalar and the different fields with mass *M*_{i} (*i* = *f*, *s*, *v*). Furthermore,

`\tau_x = \frac{m_\Phi^2}{4 m_x^2}`

holds and the loop functions are given by

`A_f = 2 (\tau + (\tau -1) f(\tau))/\tau^2, \\ A_s = -(\tau-f(\tau))/\tau^2, \\ A_v = -(2 \tau^2 + 3\tau + 3 (2 \tau -1) f(\tau) )\tau^2,`

with

`f(\tau) = \begin{cases} \text{arcsin}^2 \sqrt{\tau} \hspace{1cm} \text{for} \,\, \tau \le 1,\\ -\frac{1}{4}\left(\log \frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}} -i\pi \right)^2 \text{for} \,\, \tau gt; 1. \end{cases}`

For a pure pseudo-scalar state only fermions contribute, i.e. the LO widths read

`\Gamma(A \to \gamma \gamma)_{\rm LO} = \frac{G_F \alpha^2 m_A^3}{32 \sqrt{2} \pi^3} \left|\sum_f N^f_c Q_f^2 r^A_f A^A_f(\tau_f) \right|^2, \\ \Gamma(A \to g g)_{\rm LO} = \frac{G_F \alpha_s^2 m_A^3}{36 \sqrt{2} \pi^3} \left|\sum_f 3 D_2^f r^A_f A^A_f(\tau_f) \right|^2,`

where

*A*_{f}^{A} = *f*(*τ*)/*τ* ,

and *r*_{f}^{A} takes the same form as *r*_{f}^{Φ} in , simply replacing *C*_{f̄fΦ}^{L, R} by *C*_{f̄fA}^{L, R}.

These formulae are used by SPheno to calculate the full LO contributions of any CP-even or odd scalar present in a model including all possible loop contributions at the scale *μ* = *m*_{Φ}. However, it is well known, that higher order corrections are important. Therefore, NLO, NNLO and even N^{3}LO corrections from the SM are adapted and used for any model under study. In case of heavy colour fermionic triplets, the included corrections for the diphoton decay are

`r^\Phi_f \to r_f \left(1 - \frac{\alpha_s}{\pi} \right), \\ r^\Phi_s \to r_s \left(1+\frac{8 \, \alpha_s}{3 \pi} \right).`

These expressions are obtained in the limit *τ*_{f} → 0 and thus applied only when *m*_{Φ} < *m*_{f}. *r*_{f}^{A} does not receive any corrections in this limit. For the case *m*_{Φ} > 100*m*_{f}, we have included the NLO corrections in the light quark limit given by

`r^X_f \to r^X_f \left(1+ \frac{\alpha_s}{\pi} \left[-\frac{2}{3} \log 4\tau + \frac{1}{18} \left(\pi^2 -\log^2 4\tau\right) + 2\log \left( \frac{\mu_{\text{NLO}}^2}{m_f^2} \right) + i \frac{\pi}{3}\left(\frac{1}{3} \log 4\tau +2 \right) \right] \right)`

for *X* = *Φ*, *A*. *μ*_{NLO} is the renormalisation scale used for these NLO corrections, chosen to be *μ*_{NLO} = *m*_{Φ}/2. In the intermediate range of 100*m*_{f} > *m*_{Φ} > 2*m*_{f}, no closed expressions for the NLO correction exist. Our approach in this range was to extract the numerical values of the corrections from HDECAY and to fit them. For the digluon decay rate, the corrections up to N^{3}LO are included and parametrised by

`\Gamma(X \to g g) = \Gamma(X \to g g)_{\rm LO} \left(1 + C_X^{\rm NLO} + C_X^{\rm NNLO} + C_X^{\rm \text{N}^3LO} \right)\,,`

with

`C_\Phi^{\rm NLO} = \left(\frac{95}{4} - \frac76 N_F \right) \frac{\alpha_s}{\pi}\,, \\ C_\Phi^{\rm NNLO} = \Bigg(370.196 + (-47.1864 + 0.90177 N_F) N_F + (2.375 + 0.666667 N_F)\log \frac{m_\Phi^2}{m_t^2}\Bigg) \frac{\alpha^2_s}{\pi^2}\,, \\\ C_\Phi^{\rm \text{N}^3LO} = \left(467.684 + 122.441 \log \frac{m_\Phi^2}{m_t^2} + 10.941 \left(\log \frac{m_\Phi^2}{m_t^2}\right)^2 \right) \frac{\alpha^3_s}{\pi^3}\,,`

and

`C_A^{\rm NLO} = \left(\frac{97}{4} - \frac76 N_F \right) \frac{\alpha_s}{\pi}\,, \\ C_A^{\rm NNLO} = \left(171.544 + 5 \log \frac{m_\Phi^2}{m_t^2}\right) \frac{\alpha^2_s}{\pi^2}`

For pseudoscalar we include only corrections up to NNLO as the `\rm \text{N}^3LO`

are not known for CP-odd scalars.