  Geminga is the first pulsar around which a remarkable TeV gamma-ray halo
extending over a few degrees was discovered by MILAGRO, HAWC and later by
H.E.S.S., and by Fermi-LAT in the GeV band. More middle-aged pulsars have
exhibited gamma-ray halos, and they are now recognized as an emerging class of
Galactic gamma-ray sources. The emission appears in the late evolution stage of
pulsars, and is most plausibly explained by inverse Compton scattering of CMB
and interstellar photons by relativistic electrons and positrons escaping from
the pulsar wind nebulae. These observations pose a number of theoretical
challenges. Tackling these questions requires constraining the ambient magnetic
field properties, which can be achieved through X-ray observations. If the
gamma-ray halos originate from a distribution of highly energetic electrons,
synchrotron losses in the ambient magnetic fields of the same particles are
expected to produce a diffuse X-ray emission with a similar spatial extension.
We present the most comprehensive X-ray study of the Geminga pulsar halo to
date, utilising archival data from XMM-Newton and NuSTAR. Our X-ray analysis
covers a broad bandwidth ($0.5\rm{-}79$ keV) and large field of view ($\sim
4^\circ$) for the first time. This is achieved by accurately measuring the
background over the entire field of view, and taking into account both focused
and stray-light X-ray photons with NuSTAR. We find no significant emission and
set robust constraints on the X-ray halo flux. These are translated to
stringent constraints on the ambient magnetic field strength and the diffusion
coefficient by using a physical model considering particle injection, diffusion
and cooling over the pulsar's lifetime, which is tuned by fitting
multi-wavelength data. Our novel methodology for modelling and searching for
synchrotron X-ray halos can be applied to other pulsar halo candidates.
