  Pulsar wind nebula (PWN) Boomerang and the associated supernova remnant (SNR)
G106.3+2.7 are among candidates for the ultra-high-energy (UHE) gamma-ray
counterparts published by LHAASO. Although the centroid of the extended source,
LHAASO J2226+6057, deviates from the pulsar's position by about $0.3^\circ$,
the source partially covers the PWN. Therefore, we cannot totally exclude the
possibility that a part of the UHE emission comes from the PWN. Previous
studies mainly focus on whether the SNR is a PeVatron, while neglecting the
energetic PWN. Here, we explore the possibility of the Boomerang Nebula being a
PeVatron candidate by studying its X-ray radiation. By modelling the diffusion
of relativistic electrons injected in the PWN, we fit the radial profiles of
the X-ray surface brightness and the photon index. The solution with a magnetic
field $B=140\mu$G can well reproduce the observed profiles and implies a severe
suppression of IC scattering of electrons. Therefore, a proton component need
be introduced to account for the UHE emission, in light of recent LHAASO's
measurement on Crab Nebula, if future observations reveal part of the UHE
emission originating from the PWN. In this sense, Boomerang Nebula would be a
hadronic PeVatron.
