|
Articles Particle Physics Positivity Constraints on Quark and Gluon Distributions in QCD (1978)
Positivity Constraints on Quark and Gluon Distributions in QCD (1978) |
|
We now consider the constraints which arise from demanding that 
and 
be non-negative (2). We begin with the necessary condition that their moments 
and 
be non-negative. If the starting distributions are positive it is easy to show that 
(eqs. (5), (6)), but one must consider two possibilities for 
.
If the starting distributions are positive, this holds for 
, and since 
it continues to hold for 
. For 
, however, it will eventually fail.
Eq. (3) shows that 
is non-negative as 
or 
since it is dominated by the positive terms 
or 
in these limits, but it has one minimum at
The validity of eq. (8) at 
implies that 
. Hence 
decreases between 
and 
but is positive throughout this region. For 
, however, 
may become negative, depending on the starting distributions.
We have now shown that the moments of the (anti)quark and gluon distributions remain positive for 
if they are positive at 
. In order that the distributions themselves remain positive, their moments must satisfy eq. (7) for all 
. For the large class of starting distributions whose moments behave like inverse powers of 
for large , the asymptotic freedom formulae predict that their moments continue to behave like inverse powers of for (apart from 
factors which will not affect our argument). In this case 
and 
will be positive for large since 
for 
(where 
is the th-order finite difference operator defined in eq. (7)). A detailed numerical investigation of 
and 
with a wide variety of positive starting distributions suggests that in all cases they indeed remain positive for 
for all .
We showed above that for 
some of the moments inevitably become negative. This is also true of their finite differences. For ``reasonable'' starting distributions, however, we find (sect. 5) that the 
conditions only very rarely fail before the 
conditions as 
is decreased below 
. This is to be expected since for a large class of distributions (including all those which decrease monotonically with 
) the finite differences of the moments cannot be negative unless the moments themselves are negative.
