The polarized TMDs in the covariant parton model approach

0.38 MB·English

by  A. V. Efremov

The polarized TMDs in the covariant
parton model approach1
A.V. Efremov1, P. Schweitzer2, O. V. Teryaev1, P. Zavada3
1
1 1 Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
0 2 Department of Physics, University of Connecticut, Storrs, CT 06269, U.S.A.
2 3 Institute of Physics AS CR, Na Slovance 2, CZ-182 21 Prague 8, Czech Rep.

n
E-mail: [email protected]
a
J

0 Abstract. We derive relations between polarized transverse momentum dependent distribu-
2 tionfunctions(TMDs)andtheusualpartondistributionfunctions(PDFs)inthe3Dcovariant
parton model, which follow from Lorentz invariance and the assumption of a rotationally sym-

] metric distribution of parton momenta in the nucleon rest frame. Using the known PDF gq(x)
h 1
asinputwepredictthex-andp -dependenceofallpolarizedtwist-2naivelytime-reversaleven
p T
- (T-even) TMDs.
p
e
h
[ TMDs [1, 2] open a new way to a more complete understanding of the quark-gluon structure

1 of the nucleon. Indeed, some experimental observations can hardly be explained without a more
v accurateandrealistic3Dpictureofthenucleon,whichnaturallyincludestransversemotion. The
5
azimuthal asymmetry in the distribution of hadrons produced in deep-inelastic lepton-nucleon
3
0 scattering (DIS), known as the Cahn effect [3], is a classical example. The intrinsic (transversal)
4 parton motion is also crucial for the explanation of some spin effects [4]–[16].
.
1 Inpreviousstudieswediscussedthecovariantpartonmodel, whichisbasedonthe3Dpicture
0 of parton momenta with rotational symmetry in the nucleon rest frame [17]–[26].
1
In this model we studied all T-even TMDs and derived a set of relations among them [23].
1
It should be remarked that some of the relations among different TMDs were found (sometimes
:
v
before) also in other models [27]–[34].
i
X In the recent paper [35] we further develop and broadly extend our studies [24]–[25] of
r the relations between TMDs and PDFs. The formulation of the model in terms of the light-
a
cone formalism [23] allows us to compute the leading-twist TMDs by means of the light-front
1 Contribution to the Proceedings of the 19th International Spin Physics Symposium (SPIN2010), Ju¨lich,
Germany, September 27 - October 2, 2010
correlators φ(x,p ) [2] as:
T ij
1 tr(cid:2)γ+ φ(x,p )(cid:3) = fq(x,p )− εjkpjTSTk f⊥q(x,p ), (1)
2 T 1 T M 1T T
1 tr(cid:2)γ+γ φ(x,p )(cid:3) = S gq(x,p )+ pTSTg⊥q(x,p ), (2)
2 5 T L 1 T M 1T T
1 tr(cid:2)iσj+γ φ(x,p )(cid:3) = Sj hq(x,p )+S pjT h⊥q(x,p ) (3)
2 5 T T 1 T L M 1L T
(pj pk − 1 p2δjk)Sk εjkpk
+ T T 2 T T h⊥q(x,p )+ T h⊥q(x,p ).
M2 1T T M 1 T
In the present contribution we report about new results related to the polarized distributions
[35].
In our approach all polarized leading-twist T-even TMDs are described in terms of the same
polarized covariant 3D distribution H(p0). This follows from the compliance of the approach
with relations following from QCD equations of motion [23]. As a consequence all polarized
TMDs can be expressed in terms a single “generating function” Kq(x,p ) as follows
T
1 (cid:18)(cid:16) m(cid:17)2 p2 (cid:19)
gq(x,p ) = x+ − T × Kq(x,p ) ,
1 T 2x M M2 T
1 (cid:16) m(cid:17)2
hq(x,p ) = x+ × Kq(x,p ) ,
1 T 2x M T
1 (cid:16) m(cid:17)
g⊥q(x,p ) = x+ × Kq(x,p ) , (4)
1T T x M T
1 (cid:16) m(cid:17)
h⊥q(x,p ) = − x+ × Kq(x,p ) ,
1L T x M T
1
h⊥q(x,p ) = − × Kq(x,p ) .
1T T x T
with the “generating function” Kq(x,p ) defined (in the compact notation of [23]) by
T
(cid:90) dp1 Hq(p0) (cid:18)p0−p1 (cid:19)
Kq(x,p ) = M2x d{p1} , d{p1} ≡ δ −x . (5)
T p0 p0+m M
We have shown that due to rotational symmetry the following relations hold:
Hq(p¯0) 1 (cid:18) p2 +m2(cid:19)
Kq(x,p ) = M2 , p¯0 = xM 1+ T , (6)
T p¯0+m 2 x2M2
(cid:18)M (cid:19) (cid:90) 1 dy dgq(x)
πx2M3Hq x = 2 gq(y)+3gq(x)−x 1 , (7)
2 y 1 1 dx
x
where we took the limit m → 0 in (7). In that limit we obtain for the generating function (6)
the result
Hq(Mξ) 2 (cid:18) (cid:90) 1 dy dgq(ξ)(cid:19) (cid:18) p2 (cid:19)
Kq(x,p ) = 2 = 2 gq(y)+3gq(ξ)−x 1 , ξ = x 1+ T .
T Mξ πξ3M4 y 1 1 dξ x2M2
2 ξ
(8)
and from (4) we obtain
2x−ξ (cid:18) (cid:90) 1 dy dgq(ξ)(cid:19)
gq(x,p ) = 2 gq(y)+3gq(ξ)−ξ 1 . (9)
1 T πξ3M3 y 1 1 dξ
ξ
Figure 1. The TMD gq(x,p ) for u- (upper panel) and d-quarks (lower panel). Left panel:
1 T
gq(x,p )asfunctionofxforp /M = 0.10(dashed), 0.13(dotted), 0.20(dash-dottedline). The
1 T T
solid line corresponds to the input distribution gq(x). Right panel: gq(x,p ) as function of
1 1 T
p /M for x = 0.15 (solid), 0.18 (dashed), 0.22 (dotted), 0.30 (dash-dotted line).
T
This relation yields for gq(x,p ), with the LO parameterization of [36] for gq(x) at 4GeV2, the
1 T 1
results shown in Fig. 1.
The remarkable observation is that gq(x,p ) changes sign at the point p = Mx, which is
1 T T
due to the prefactor (this is the definition of the variable p¯1 in the limit m → 0)
(cid:18) (cid:19)
(cid:16) p (cid:17)2
2x−ξ = x 1− T = −2p¯1/M (10)
Mx
in (9). The expression in (10) is proportional to the quark longitudinal momentum p¯1 in the
protonrestframe,whichisdeterminedbyxandp [35]. Thismeans,thatthesignofgq(x,p )is
T 1 T
controlled by sign of p¯1. To observe these dramatic sign changes one may look for multi-hadron
jet-like final states in SIDIS. Performing the cutoff for transverse momenta from below and from
above, respectively, should affect the sign of asymmetry.
There is some similarity to gq(x) which also changes sign, and is given in the model by [21]
2
1 (cid:90) (cid:32) (cid:0)p1(cid:1)2−p2/2(cid:33) (cid:18)p0−p1 (cid:19) d3p
gq(x) = Hq(p0) p1− T δ −x . (11)
2 2 p0+m M p0
The δ−function implies that, for our choice of the light-cone direction, large x are correlated
with large and negative p1, while low x are correlated with large and positive p1. Thus, g (x)
2
changes sign, because the integrand in (11) changes sign between the extreme values of p1. Let
us remark, that the calculation of g (x) based on the relation (11) well agrees [19] with the
2
experimental data.
The other TMDs (4) can be calculated similarly and differ, in the limit m → 0, by simple
x-dependent prefactors
x 1
hq(x,p ) = Kq(x,p ), g⊥q(x,p ) = Kq(x,p ), h⊥q(x,p ) = − Kq(x,p ). (12)
1 T 2 T 1T T T 1T T x T
Figure 2. The TMDs hq(x,p ), g⊥q(x,p ), h⊥q(x,p ) for u- and d-quarks. Left panel: The
1 T 1T T 1T T
TMDsasfunctionsofxforp /M = 0.10(dashed),0.13(dotted),0.20(dash-dottedlines). Right
T
panel: The TMDs as functions of p /M for x = 0.15 (solid), 0.18 (dashed), 0.22 (dotted), 0.30
T
(dash-dotted lines).
The resulting plots are shown in Fig. 2. We do not plot h⊥q since this TMD is equal to
1L
−g⊥q in our approach [23]. Let us remark, that gq(x,p ) is the only TMD which can change
1T 1 T
sign. The other TMDs have all definite signs, which follows from (4, 12). Note also that
pretzelosity h⊥q(x,p ), due to the prefactor 1/x, has the largest absolute value among all
1T T
TMDs. Noteworthy, pretzelosity is related to quark orbital angular momentum in some quark
models [32, 33], including the present approach [26].
To conclude, let us remark that an experimental check of the predicted TMDs requires care.
In fact, TMDs are not directly measurable quantities unlike structure functions. What one
can measure for instance in semi-inclusive DIS is a convolution with a quark fragmentation
function. This naturally “dilutes” the effects of TMDs, and makes it difficult to observe for
instance the prominent sign change in the helicity distribution, see Fig. 1. A dedicated study
of the phenomenological implications of our results is in progress.
Acknowledgements. A.E.andO.T.aresupportedbytheGrantsRFBR09-02-01149and09-
02-00732,and(alsoP.Z.)Votruba-BlokhitsevProgramsofJINR.P.Z.issupportedbytheproject
AV0Z10100502 of the Academy of Sciences of the Czech Republic. The work was supported in
part by DOE contract DE-AC05-06OR23177. We would like to thank also Jacques Soffer and
Claude Bourrely for helpful comments on an earlier stage of this study.
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