Attilio Cucchieri, David Dudal, Nele Vandersickel
We study the Landau-gauge Gribov ghost form-factor sigma(p^2) for SU(N)
Yang-Mills theories in the d-dimensional case. We find a qualitatively
different behavior for d=3,4 w.r.t. d=2. In particular, considering any
(sufficiently regular) gluon propagator D(p^2) and the one-loop-corrected ghost
propagator G(p^2), we prove in the 2d case that sigma(p^2) blows up in the
infrared limit p -> 0 as -D(0)\ln(p^2). Thus, for d=2, the no-pole condition
\sigma(p^2) < 1 (for p^2 > 0) can be satisfied only if D(0) = 0. On the
contrary, in d=3 and 4, sigma(p^2) is finite also if D(0) > 0. The same results
are obtained by evaluating G(p^2) explicitly at one loop, using fitting forms
for D(p^2) that describe well the numerical data of D(p^2) in d=2,3,4 in the
SU(2) case. These evaluations also show that, if one considers the coupling
constant g^2 as a free parameter, G(p^2) admits a one-parameter family of
behaviors (labelled by g^2), in agreement with Boucaud et al. In this case the
condition sigma(0) <= 1 implies g^2 <= g^2_c, where g^2_c is a 'critical'
value. Moreover, a free-like G(p^2) in the infrared limit is obtained for any
value of g^2 < g^2_c, while for g^2 = g^2_c one finds an infrared-enhanced
G(p^2). Finally, we analyze the Dyson-Schwinger equation (DSE) for sigma(p^2)
and show that, for infrared-finite ghost-gluon vertices, one can bound
sigma(p^2). Using these bounds we find again that only in the d=2 case does one
need to impose D(0) = 0 in order to satisfy the no-pole condition. The d=2
result is also supported by an analysis of the DSE using a spectral
representation for G(p^2). Thus, if the no-pole condition is imposed, solving
the d=2 DSE cannot lead to a massive behavior for D(p^2). These results apply
to any Gribov copy inside the so-called first Gribov horizon, i.e. the 2d
result D(0) = 0 is not affected by Gribov noise. These findings are also in
agreement with lattice data.
View original:
http://arxiv.org/abs/1202.1912
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