Impact of vacuum stability, perturbativity and XENON1T on global fits of $$\mathbb {Z}_2$$ and $$\mathbb {Z}_3$$ scalar singlet dark matter

Table 6 Individual contributions to the $\varDelta $ log-likelihood for the various best-fit points (see Tables 5 and 7) compared to an ‘ideal’ case. We take this to be the background-only likelihood for exclusions, and the central observed value for detections. Note that because each likelihood is dimensionful, the absolute values are less meaningful than the offset with respect to another point (see section 8.3 of Ref. [15] for more details on the normalisation). The best-fit points are labelled as follows: A represents a fit with the only constraint that $\varLambda _P>\max (\overline{m}_{\scriptscriptstyle S},m_t)$, B is a fit with the additional constraint of absolute vacuum stability, C includes the constraint of $\varLambda _P>10^{15}$ GeV and D also includes the requirement that $\varOmega _Sh^2$ be within 1$\sigma $ of the observed relic density

Likelihood contribution	Ideal	$\varDelta \ln \mathcal {L}$
		$\mathbb {Z}_2$A	$\mathbb {Z}_2$B	$\mathbb {Z}_2$C	$\mathbb {Z}_2$D	$\mathbb {Z}_3$A	$\mathbb {Z}_3$B	$\mathbb {Z}_3$C	$\mathbb {Z}_3$D
Relic density	5.989	0	0.001	0	0.120	0	0	0.034	0.142
LUX Run II 2016	$-1.467$	0.001	0.112	0.221	0.207	0.001	0.095	0.528	0.592
PandaX 2016	$-1.886$	0	0.071	0.140	0.131	0.001	0.059	0.339	0.380
PandaX 2017	$-1.550$	0.001	0.156	0.298	0.280	0.002	0.130	0.678	0.752
XENON1T 2018	$-3.440$	0.210	0.003	0.218	0.179	0.209	0.074	1.465	1.770
$\gamma $ rays (Fermi-LAT dwarfs)	$-33.244$	0.105	0.148	0.165	0.170	0.105	0.112	0.196	0.207
Higgs invisible width	0.000	0	0	0	0	0	0	0	0
Hadronic elements $\sigma _s$, $\sigma _l$	$-6.625$	0	0.001	0.016	0.019	0	0	0.099	0.043
Local DM density $\rho _0$	1.142	0	0.010	0.039	0.101	0	0.001	0.547	0.499
DM velocity $v_0$	$-2.998$	0	0	0	0.001	0	0	0.001	0.013
DM escape velocity $v_{esc}$	$-4.474$	0	0	0	0.005	0	0	0.002	0
$\alpha _s$	5.894	0	0	0	0.002	0	0.001	0.004	0.001
Higgs mass	0.508	0	0	0	0.043	0	0	0.082	0.004
Top quark mass	$-0.645$	0	0	0.022	0.196	0	0	0	0.041
Vacuum stability	0.000	0	0	0	0	0	0	0	0
Total		0.317	0.503	1.121	1.455	0.318	0.473	3.975	4.443

Likelihood contribution	Ideal	\(\varDelta \ln \mathcal {L}\)
		\(\mathbb {Z}_2\)A	\(\mathbb {Z}_2\)B	\(\mathbb {Z}_2\)C	\(\mathbb {Z}_2\)D	\(\mathbb {Z}_3\)A	\(\mathbb {Z}_3\)B	\(\mathbb {Z}_3\)C	\(\mathbb {Z}_3\)D
Relic density	5.989	0	0.001	0	0.120	0	0	0.034	0.142
LUX Run II 2016	\(-1.467\)	0.001	0.112	0.221	0.207	0.001	0.095	0.528	0.592
PandaX 2016	\(-1.886\)	0	0.071	0.140	0.131	0.001	0.059	0.339	0.380
PandaX 2017	\(-1.550\)	0.001	0.156	0.298	0.280	0.002	0.130	0.678	0.752
XENON1T 2018	\(-3.440\)	0.210	0.003	0.218	0.179	0.209	0.074	1.465	1.770
\(\gamma \) rays (Fermi-LAT dwarfs)	\(-33.244\)	0.105	0.148	0.165	0.170	0.105	0.112	0.196	0.207
Higgs invisible width	0.000	0	0	0	0	0	0	0	0
Hadronic elements \(\sigma _s\), \(\sigma _l\)	\(-6.625\)	0	0.001	0.016	0.019	0	0	0.099	0.043
Local DM density \(\rho _0\)	1.142	0	0.010	0.039	0.101	0	0.001	0.547	0.499
DM velocity \(v_0\)	\(-2.998\)	0	0	0	0.001	0	0	0.001	0.013
DM escape velocity \(v_{esc}\)	\(-4.474\)	0	0	0	0.005	0	0	0.002	0
\(\alpha _s\)	5.894	0	0	0	0.002	0	0.001	0.004	0.001
Higgs mass	0.508	0	0	0	0.043	0	0	0.082	0.004
Top quark mass	\(-0.645\)	0	0	0.022	0.196	0	0	0	0.041
Vacuum stability	0.000	0	0	0	0	0	0	0	0
Total		0.317	0.503	1.121	1.455	0.318	0.473	3.975	4.443