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Articles Particle Physics Positivity Constraints on Quark and Gluon Distributions in QCD (1978)
Positivity Constraints on Quark and Gluon Distributions in QCD (1978) |
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It is necessary to resort to numerical techniques to make a complete investigation of the positivity of distributions for 
. As an example, we consider first the following simple parametrization of the starting distributions for the case of four flavours:
We take
but will allow 
and 
to vary. The moments of these 
are given by
where 
is the Beta function.
We consider the 
and 
generated from the starting distributions of eq. (12) at 
, which corresponds to 
for 
and 
. We know that these quantities are positive for 
(see figs. 1 and 2) but they become negative for large 
unless and satisfy eq. (3). Table 1 gives the minimum (which we call 
) at which this occurs for various and . The contributions of higher-twist operators probably invalidate the use of the asymptotic freedom formulae for 
(
), or 
in this case. We can therefore only exclude values of and which give 
. The outlined central area in table 1 (in which 
) corresponds to the allowed values of and ; it is considerably larger than the region allowed by the 
bound of eq. (3) but has similar features. For 
, it is which becomes negative for 
while for 
it is 
.
In this case, it turns out that the 
positivity constraints are at least as strong as the 
ones for a given . If we consider values of 
larger than 1.2, the maximum allowed value of tends to grow very slowly with 
, and the constraints begin to become stricter than the ones for small and (although this never happens in a region which is not seriously hampered by higher-twist effects). An increase (decrease) of 
increases (decreases) throughout the domain in which becomes negative for 
(this domain, which was originally is itself decreased (increased)). Likewise, an increase (decrease) of 
increases (decreases) in the region where becomes negative.
0f course, actual quark and gluon momentum distributions will not be of the simple form (12). Some idea of the results of using more complicated forms may be obtained by admixing a small 
component into the 
and 
of eq. (12). Very small contamination of 
is sufficient to prevent it from becoming negative for values of 
and 
at which asymptotic freedom formulae should apply (ignoring higher-twist operators). It would also, however, ruin the agreement between theory and experiment for the Drell-Yan process. On the other hand, even 10% admixtures into 
(changing it drastically for large ) do not appreciably change the point at which it becomes negative (although for small 
this occurs at larger 
.
for which the moment of one of the momentum distributions becomes negative at starting from distributions at 
(with ). The starting gluon and
antiquark momentum distributions are taken to behave like 
and 
, respectively (and the valence quark distributions like ). The lines enclose the region in which 
where the higher-twist operations are important.
If we use the phenomenological fits of Field and Feynman [6] or Barger and
Phillips [7] as the starting quark distributions, then we find that positivity is only respected for a small distance below 
if falls less rapidly with than about 
. Steeper starting gluon distributions (e.g. [8]) should therefore probably be abandoned.
