def vmin_migdal(w, erec, mw):
"""Return minimum WIMP velocity to make a Migdal signal with energy w,
given elastic recoil energy erec and WIMP mass mw.
"""
y = (wr.mn() * erec / (2 * wr.mu_nucleus(mw)**2))**0.5
y += w / (2 * wr.mn() * erec)**0.5
return np.maximum(0, y)
def vmin_w(w, mw):
"""Minimum wimp velocity to emit a Bremsstrahlung photon w
:param w: Bremsstrahlung photon energy
:param mw: WIMP mass
From Kouvaris/Pradler [arxiv:1607.01789v2], equation in text below eq. 10
"""
return (2 * w / wr.mu_nucleus(mw))**0.5
def erec_bound(sign, w, v, mw):
"""Bremsstrahlung scattering recoil energy kinematic limits
From Kouvaris/Pradler [arxiv:1607.01789v2], eq. between 8 and 9,
simplified by vmin (see above)
:param sign: +1 to get upper limit, -1 to get lower limit
:param w: Bremsstrahlung photon energy
:param mw: WIMP mass
:param v: WIMP speed (earth/detector frame)
"""
return (wr.mu_nucleus(mw)**2 * v**2 / wr.mn() *
(1 - vmin_w(w, mw)**2 / (2 * v**2) + sign *
(1 - vmin_w(w, mw)**2 / v**2)**0.5))