  We present a lattice determination of the leading-order hadronic vacuum
polarization (HVP) contribution to the muon anomalous magnetic moment,
$a_{\mu}^{\rm HVP}$, in the so-called short and intermediate time-distance
windows, $a_{\mu}^{\rm SD}$ and $a_{\mu}^{\rm W}$, defined by the RBC/UKQCD
Collaboration [1]. We employ gauge ensembles produced by the Extended Twisted
Mass Collaboration (ETMC) with $N_f = 2 + 1 + 1$ flavors of Wilson-clover
twisted-mass quarks with masses of all the dynamical quark flavors tuned close
to their physical values. The simulations are carried out at three values of
the lattice spacing equal to $\simeq 0.057, 0.068$ and $0.080$ fm with spatial
lattice sizes up to $L \simeq 7.6$~fm. For the short distance window we obtain
$a_\mu^{\rm SD}({\rm ETMC}) = 69.27\,(34) \cdot 10^{-10}$, which is consistent
with the recent dispersive value of $a_\mu^{\rm SD}(e^+ e^-) = 68.4\,(5) \cdot
10^{-10}$ [2]. In the case of the intermediate window we get the value
$a_\mu^{\rm W}({\rm ETMC}) = 236.3\,(1.3) \cdot 10^{-10}$, which is consistent
with the result $a_\mu^{\rm W}({\rm BMW}) = 236.7\,(1.4) \cdot 10^{-10}$ [3] by
the BMW collaboration as well as with the recent determination by the CLS/Mainz
group of $a_\mu^{\rm W}({\rm CLS}) = 237.30\,(1.46) \cdot 10^{-10}$ [4].
However, it is larger than the dispersive result of $a_\mu^{\rm W}(e^+ e^-) =
229.4\,(1.4) \cdot 10^{-10}$ [2] by approximately $3.6$ standard deviations.
The tension increases to approximately $4.5$ standard deviations if we average
our ETMC result with those by BMW and CLS/Mainz. Our accurate lattice results
in the short and intermediate windows point to a possible deviation of the $e^+
e^-$ cross section data with respect to Standard Model predictions in the low
and intermediate energy regions, but not in the high energy region.
