  We present a complete and independent set of dimension-7 operators in the low
energy effective field theory (LEFT) where the dynamical degrees of freedom are
the standard model five quarks and all of the neutral and charged leptons. All
operators are non-Hermitian and are classified according to their baryon
($\Delta B$) and lepton ($\Delta L$) numbers violated. Including
Hermitian-conjugated operators, there are in total $3168$, $750$, $588$, $712$
operators with $(\Delta B,\Delta L)=(0,0)$, $(0,\pm 2)$, $(\pm 1,\mp 1)$, $(\pm
1,\pm 1)$ respectively. We perform the tree-level matching with the standard
model effective field theory (SMEFT) up to dimension-7 (dim-7) operators in
both LEFT and SMEFT. As a phenomenological application we study the effective
neutrino-photon interactions due to dim-7 lepton number violating operators
that are induced and much enhanced at one loop from dim-6 operators that in
turn are matched from dim-7 SMEFT operators. We compare the cross sections of
various neutrino-photon scattering with their counterparts in the standard
model and highlight the new features. Finally we illustrate how these effective
interactions could arise from ultraviolet completion.
