The transverse momentum dependent distribution functions in the bag model Page: 12 of 26
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12
B. Transverse momentum dependence
In the context of TMDs the most interesting aspect is, of course, their transverse momentum dependence. In
principle, all information is contained in the two-dimensional functions j(x, kL) for a generic TMD, but here we shall
content ourselves to discuss 'one- or zero-dimensional' projections of that information.
The first point we address is: what are the typical transverse momenta of unpolarized quarks in the bag TMDs?
For that we define for a generic TMD jq(x, kL) the following quantities
f dx f d2kL kL j(x, kL) 2 f dx f d2kL k2 j(,kL)
(PT) f dx f d2kL j(x, kL) (PT) fdx f d2k lj(x, kl) (64)
Due to the simple spin flavor structure of the MIT bag model the (PT) and (p4) are flavor-independent for all TMDs.
The first observation is that depending on the TMD (PT) and (p) in Eq. (64) may not exist in the bag model,
because the momentum-space wave-function components t(k), Eq. (17), do not vanish sufficiently fast at large k.
This is the case especially for fl(x, kl).
For the same reason also the (1)-moment fil)q(x) does not exist. However, the (1/2)-moment f(1/2)q(x) defined
according to (12) exists, and can be used to introduce an x-dependent average transverse momentum (pT(x)) as
f(1/2)q(i)
(PT(x)) 2MN f . (65)
Fig. 2a shows the result for f l/2)q(x). (The divergence of (PT) from (64) emerges when one tries to integrate f 1/2)q(i)
over x, recalling that this integration extends to the entire x-axis, see Sec. II.)
Now the (1)-moment fii)q(x) is divergent, but its derivative with respect to x exists, see the dotted line in Fig. 2b.
Hereby it is understood that the (1)-moment is computed with a finite cutoff Acut MN, then the derivative is
taken, and only then the limit Acut -> zo is performed.
By integrating the well-defined a fi)q(x) we can compute a regularized (1)-moment fil)q(x)6eg. The result depends
on some arbitrary integration constant, which we fix such that the (1)-moment vanishes at x 1. This choice is
reasonable but not unique, if we recall that in the MIT bag model TMDs in general have a non-zero (though small)
support for ixl > 1, see Sec. II. Our main conclusions in this respect, to be presented below in this Section, depend
weakly on the chosen value of the integration constant, provided reasonable choices are made (such as, for example,
f~(x)Teg = fi1/2)q(x) at x 1). The result for ffl)q(1)eg defined in this way is shown as solid line in Fig. 2b.
With fil)q(x)6eg we are in the position to define an x-dependent average transverse momentum square (p4(x)) as
(p (x)) 2MN fl)q(i)T69 (66)
(1/2>l (x)r
f1r (x) (a) f1 )(x) (b) <PT(X)> (c)
( (122)u( (1)u x) () <p
0.5 0.5 . 0.5
0.4
,/
0 0 0.3
0.2
-0.5 0.5 0.1 bag model
0
0 0.2 0.4 0.6 0.8 x 0 0.2 0.4 0.6 0.8 x 0 0.2 0.4 0.6 x
FIG. 2: For the unpolarized TMD fq(x, kl) (a) the (1/2)-moment defined in Eq. (65), (b) the derivative of the (1)-moment
and the regularized (1)-moment as discussed in the text, and (c) (pT(x)) in comparison to (7r(pf(x))/4)1/2. In the Gauss-model
the two quantities would be equal. (The dotted marks the value (pT(x)) = 0.25 GeV.)
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Avakian, Harut; Efremov, Anatoly; Schweitzer, Peter & Yuan, Feng. The transverse momentum dependent distribution functions in the bag model, article, January 29, 2010; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc1015520/m1/12/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.