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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 expected [4] that the effects of operators occurring in the operator product expansion with twists greater than two become very important for 
, corresponding to 
(terms of higher order in 
also diverge in the limit 
at fixed 
, but this is only significant at much larger values of 
). Starting distributions need only be rejected, therefore, if they lead to negative distributions for 
(
). Nevertheless, as explained in sect. 1, it is interesting to consider the limit , where we find
where
For 
, the factor 
is negative. The distributions therefore become negative unless the first terms in eq. (10a) dominate as . Making the assumption of eq. (2) for the behaviour of the starting distributions as 
and 
behave like 
and 
, respectively, for large . Eq. (3) is then the condition that 
and 
remain positive for 
. If the inequalities of eq. (3) are satisfied, remains positive for all 
but still becomes negative for sufficiently small. For this occurs at
As stated in sect. 1, it is clear that distributions for 
generated from any starting distributions must obey eq. (3). When 
at fixed ,
so that eq. (10a) shows that 
: but in general this will only become true at extremely high when the structure functions will all be infinitesimal for 
close to one. To make more useful statements about the form of the structure functions for , we must assume a particular form for the starting distributions. If, for example, following ref. [5], we take 
for 
, then, ignoring 
terms, eqs. (10) and (4) give 
for .
Next we consider the 
moments. The energy-momentum sum rule implies 
, so that for the singlet structure functions
Hence The value of Figs. 1 and 2 show that the observed values of 
will inevitably become negative as 
, unless 
, in which case 
will become negative. In fig. 1 we have plotted the value of below which or becomes negative (which we call 
as a function of 
) for 
and 
(the results are very insensitive to the number of flavours). The very reasonable demand that 
requires 
) 
, (experimentally 
) 
and if 
(by which point higher-twist operators should be becoming significant for ), 
.
[ Figure 1 ] The value 
of below which the predicted total
momenta carried by the gluons (lower line) or quarks (upper line) becomes negative, as a function of the momentum
carried by the gluons at 
( and 
).
[ Figure 2 ] The value of below which the predicted total
momentum carried by one flavour of antiquark 
becomes negative, as a
function of the momentum it carried at for various starting gluon
momenta, 
) (
and ).
at which 
becomes negative is easily calculated from eq. (4) as a function of 
and 
using the fact that 
and 
(which has already been used in deriving eq. (11)). The results are shown in fig. 2 for and (they are insensitive to 
). The reasonable demand that 
gives 
) 
for 
and implies 
. Neutrino data suggest that 
not greatly in excess of this bound. For strange quarks the bound probably fails with 
but larger is needed for it to be credible in this case. However, our bounds show that the distributions of new flavours must be non-zero in regions of where the application of asymptotic freedom formulae to them is at all reasonable.
and 
are obtained if 
and 
vanish at 
. This was originally pointed out in ref [5].
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