A lattice formulation of chiral gauge theories Page: 8 of 69
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iSFN(p) = [(1/a) >j ,i sin(pua)]-, (3.5)
where p is the incoming fermion momentum. The order g and order g2 gauge-field vertices
that arise from the gauging of the naive action are
VuiN(pl) = TaV(1)N(P ,)PL = -igTayPLcos[(p, + lly)a], (3.6a)
Vi ,b(p,11, 2) = TaTbV(2)N(p, 11, l2)PL = ia92TaTbbv7APL sin[(p., + }ll, + }l2,)a], (3.6b)
where the VN's are the vertices that arise from the gauging of the naive lattice action for a
theory of fermions with vector-like couplings to an Abelian gauge field. Here Ta, Tb,... are
the gauge-group matrices, a, b,... are the gauge-field indices, I, v... are the polarization
indices, and l, l2,... are the incoming momenta, all of which are associated respectively
with the gauge fields. The incoming fermion momentum is p. The vertices of higher order '
in g can be obtained conveniently from the recursion relation
Al..11+1 (P, l1, ... , ln+l)
- -98n1Jn1 IV "..A(P + In+l, l1, .. - , ln) - V l..(P, li, ..- , ln) (37
= -6.+ d, '1l , , (3.7)
d, = (2/a) sin }pa. (3.8)
In addition to the usual pole at p = 0, the naive propagator (3.5) has extra poles when
one or more momentum components are equal to 7r/a. It can be seen that half of the poles
have positive chiral charge and half have negative chiral charge . Thus, this doubling
phenomenon leads to gauge-field couplings to both left- and right-handed species; the theory,
at this stage, is not chiral.
We follow the standard approach of eliminating the doublers by including a Wilson mass
term  in the action:
Sw = ad 7(x) 2a[2V(x) - V(x + a,) - 4(x - a,)]. (3.9)
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Bodwin, G.T. A lattice formulation of chiral gauge theories, report, December 1, 1995; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc693548/m1/8/: accessed June 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.