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G106.3$+$2.7, commonly considered a composite supernova remnant (SNR), ischaracterized by a boomerang-shaped pulsar wind nebula (PWN) and two distinct("head" & "tail") regions in the radio band. A discovery of very-high-energy(VHE) gamma-ray emission ($E_\gamma > 100$ GeV) followed by the recentdetection of ultra-high-energy (UHE) gamma-ray emission ($E_\gamma > 100$ TeV)from the tail region suggests that G106.3$+$2.7 is a PeVatron candidate. Wepresent a comprehensive multi-wavelength study of the Boomerang PWN (100"around PSR J2229+6114) using archival radio and Chandra data obtained from twodecades ago, a new NuSTAR X-ray observation from 2020, and upper limits ongamma-ray fluxes obtained by Fermi and VERITAS observatories. The NuSTARobservation allowed us to detect a 51.67 ms spin period from the pulsar PSRJ2229+6114 and the PWN emission characterized by a power-law model with $\Gamma= 1.52\pm0.06$ up to 20 keV. Contrary to the previous radio study by Kothes etal. 2006, we prefer a much lower PWN B-field ($B\sim3$ $\mu$G) and largerdistance ($d \sim 8$ kpc) based on (1) the non-varying X-ray flux over the lasttwo decades, (2) the energy-dependent X-ray PWN size resulting from synchrotronburn-off and (3) the multi-wavelength spectral energy distribution (SED) data.Our SED model suggests that the PWN is currently re-expanding after beingcompressed by the SNR reverse shock $\sim 1000$ years ago. In this case, thehead region should be formed by GeV--TeV electrons injected earlier by thepulsar propagating into the low density environment.

A multi-wavelength investigation of PSR J2229+6114 and its pulsar wind nebula in the radio, X-ray, and gamma-ray bands