Kinematics of deep inelastic scattering in leading order of the covariant approach

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by  Petr Zavada

Kinematics of deep inelastic scattering in leading order of the covariant approach
Petr Zavada∗
Institute of Physics AS CR, Na Slovance 2, CZ-182 21 Prague 8, Czech Republic
Westudythekinematicsofdeepinelasticscatteringcorrespondingtotherotationallysymmetric
distribution of quark momenta in the nucleon rest frame. It is shown that rotational symmetry
together with Lorentz invariance can in leading order impose constraints on the quark intrinsic
momenta. Obtained constraints are discussed and compared with the available experimental data.
PACSnumbers: 12.39.-x11.55.Hx13.60.-r13.88.+e
1. INTRODUCTION more consistent QCD treatment. In this sense it would
be interesting to compare our results with the leading
order of a real QCD approach, like e.g. the recent study
The motion of quarks inside the nucleons plays an im-
2 of perturbative QCD evolution of TMDs [11, 12]. How-
portant role in some effects which are at present inten-
1 ever such task would go beyond the scope of this short
0 sivelyinvestigatedbothexperimentallyandtheoretically.
report. Anyway, in general a comparison between the
2 Actual goal of this effort is to obtain a more consistent
experimental data and the leading order predictions can
3-Dpictureofthequark-gluonstructureofnucleons. For
n be important and instructive.
example the quark transversal momentum creates the
a
J asymmetries in particle production in polarized (SIDIS)

1 or in unpolarized (Cahn effect) experiments on deep in-
2. KINEMATIC VARIABLES
elastic scattering (DIS). Relevant tool for the study of

] these effects is the set of the transverse momentum de-
h 2.1. The Bjorken variable and light-cone
pendent distributions (TMDs). Apparently, a better un-
p coordinates
- derstanding of the quark intrinsic motion is also a neces-
p saryconditiontoclarifytheroleof quarkorbitalangular
e momenta in generating nucleon spin. First, let us shortly remind the properties of the
h
Bjorken variable
[ We have paid attention to these topics in our recent
2 studies, see [1–5] and citations therein. In particular Q2
we have shown that the requirements of Lorentz invari- x = , (1)
v B 2Pq
ance (LI) and the nucleon rotational symmetry in its
7
0 rest frame (RS), if applied in the framework of the 3- which plays a crucial role in phenomenology of lepton
6 D covariant quark-parton model (QPM), generate a set – nucleon scattering. Regardless of mechanism of the
5 of relations between parton distribution functions. Re-
process, this invariant parameter satisfies
. cently we obtained within this approach relations be-
6
0 tweenusualpartondistributionfunctionsandtheTMDs. 0≤x ≤1. (2)
B
1 The Wanzura-Wilczek approximate relation (WW) and
1 someotherknownrelationsbetweentheg andg struc- This is a very well-known textbook result. A possible
1 2
v: turefunctionsweresimilarlyobtainedinthesamemodel proof is suggested also in [10]. Now let us consider a
i before [7]. Let us remark that the WW relation has QPM approach, where the process of lepton – nucleon
X
been obtained independently also in another approaches scattering is initiated by the lepton interaction with a
r [14, 15] in which the LI represents a basic input. quark (see Fig. 1), which obeys
a
The aim of the present report is to consistently apply
the assumption LI&RS to the kinematics of DIS and to p(cid:48) =p+q, p(cid:48)2 =p2+2pq−Q2; Q2 =−q2. (3)
obtain the constraints on related kinematical variables.
The second equality implies
Thatisacomplementarytasktothestudyofabovemen-
tionedrelationsbetweendistributionfunctions,whichde-
Q2 =2pq−δm2; δm2 =p(cid:48)2−p2, (4)
pendonthesevariables. So,thereportcanbeconsidered
as an addendum to our former papers related to the co-
which with the use of relation (1) gives
variant QPM [1–9].
Since the present discussion is motivated and based pq (cid:18) δm2(cid:19)
=x 1+ . (5)
on our earlier study of a covariant version of QPM, ob- Pq B Q2
tained results correspond only to the leading order of a
The basic input for the construction of QPM is the as-
sumption
∗Electronicaddress: [email protected] Q2 (cid:29)δm2, (6)
2
Now one can easily check that Eq. (7) with the use of
this ratio and Eq. (13) imply
k k'
p q0−p q1 p −p (cid:18) (cid:18)4M2x2 (cid:19)(cid:19)
x = 0 1 = 0 1 1+O B .
q B P0q0−P1q1 P0−P1 Q2
(15)
In this way we have proved that replacement of Bjorken
p
variable by the invariant light-cone ratio in Eq. (9) is
p'
valid provided the inequality (8) is satisfied.
P
The relation (9) expressed in the nucleon rest frame
reads
FIG. 1: Diagram describing DIS as a one photon exchange
p −p
between the charged lepton and quark. Lepton and quark x= 0 1, (16)
momentaaredenotedbyk,p(k(cid:48),p(cid:48))ininitial(final)state,P M
is initial nucleon momentum.
which after inserting into (2) gives
p −p
which allows us to identify 0≤ 0 1 ≤1. (17)
M
Q2 pq
xB = 2Pq = Pq (7) However the most important reason why we require
largeQ2 isinphysics. Ifweacceptscenariowhenaprob-
and to directly relate the quark momentum to the pa- ing photon interact with a quark, we need sufficiently
rameters of scattered lepton. Moreover, if one assumes large momentum transfer Q2 at which the quarks can be
considered as effectively free due to asymptotic freedom.
Q2 (cid:29)4M2x2B, (8) At small Q2 the picture of quarks (with their momenta
and other quantum numbers) inside the nucleon disap-
where M is the nucleon mass, then one can identify
pear.
p −p
x =x≡ 0 1 (9)
B P −P
0 1
2.2. Rotational symmetry
in any reference frame in which the first axis orientation
is defined by the vector q. This relation can be proved
The RS means that the probability distribution of the
as follows. Let us consider Eq. (7) in the same frame:
quarkmomentap=(p ,p ,p )inthenucleonrestframe
1 2 3
p q0−p |q| depends, apart from Q2, on |p|. It follows that also −p
xB = P0q0−P1 |q|. (10) is allowed, so together with the inequality (17) we have
0 1
p +p
In the nucleon rest frame we denote the photon momen- 0≤ 0 1 ≤1. (18)
tum components by the subscript R and using the usual M
symbol ν =q0 we have
R The combinations of (17),(18) imply
|qR|2 =ν2+Q2, (11) M
0≤|p |≤p ≤M, |p |≤ . (19)
1 0 1 2
which with the use of Eq. (1) gives
Andifweagain refertoRS,thenfurtherinequalitiesare
|q |2 4M2x2
R =1+ B. (12) obtained:
ν2 Q2
M
It means that for Q2 (cid:29)4M2x2B we obtain 0≤|p|≤p0 ≤M, |p|≤ 2 , 0≤pT ≤p0 ≤M
(20)
|qR| =1+O(cid:18)4M2x2B(cid:19). (13) and
ν Q2
M
p ≤ , (21)
(Since for Q2 → ∞ we have also |q |,ν → ∞, so the T 2
R
ratio |q |/ν → 1 does not contradict Eq. (11).) In
R where
a reference frame connected with the rest frame by the
Lorentz boost in the direction opposite to q we have (cid:113) (cid:113)
R |p|= p2+p2+p2, p = p2+p2.
the corresponding ratio 1 2 3 T 2 3
q1 |q |+βν Obviouslyinequality(21)issatisfiedalsoinanyreference
= R . (14)
q0 ν+β|q | frame boosted in the directions ±q. Further, the above
R
3
inequalities are apparently valid also for average values neglected and x depends on the both, longitudinal and
(cid:104)p (cid:105),(cid:104)p (cid:105),(cid:104)|p|(cid:105) and (cid:104)p (cid:105). In addition, if one assumes transversal quark momenta components. In the limit of
0 1 T
that p − distribution is a decreasing function, then nec- masslessquarkstheconnectionbetweenthevariablexin
T
essarily (16) and the quark momenta components is given by the
relations:
M
(cid:104)p (cid:105)≤ . (22) p −p (cid:113)
T 4 x= 0 1; p = p2+p2, (25)
M 0 1 T
The above relations are valid for sufficiently Mx(cid:18) p2 (cid:19) Mx(cid:18) p2 (cid:19)
high Q2 suggested by Eqs. (6) and (8). Let us note p1 =− 2 1− M2Tx2 , p0 = 2 1+ M2Tx2 .
that the on-mass-shell assumption has not been applied
for obtaining these relations. ThesevariableswereusedinourrecentpapersonTMDs
These inequalities can be compared with relations ob- [1, 2]. Simply the value of invariant variable x does not
tained in [9], where the additional on-mass-shell condi- dependonthereferenceframe,butitsinterpretatione.g.
tionm2 =p2 =p2−p2 hadbeenapplied. Corresponding intherestframediffersfromthatintheinfinitemomen-
0
relations are more strict: tum frame.
ii) The relations (20), (21), which follow from RS,
m2 M2+m2 M2−m2 can be confronted with the experimental data on (cid:104)p (cid:105)
≤x≤1, p ≤ , |p|≤ T
M2 0 2M 2M or (cid:104)|p|(cid:105). We have discussed the available data in [1, 2]
(23)
and apparently relation (21) is compatible with the set
and
oflowervalues(cid:104)p (cid:105)correspondingtothe’leptonicdata’.
T
(cid:18) m2(cid:19) On the other hand the second set giving substantially
p2 ≤M2 x− (1−x). (24) greater (cid:104)p (cid:105) and denoted as the ’hadronic data’, seems
T M2 T
tocontradictthisrelation. Actuallytheconflictwiththe
relation (21) would mean either the conflict with some
However, it is clear that in general the on-mass-shell as-
of the assumptions a)–c) or simply it can be due to ab-
sumptionisnotrealistic. Inthenextwewillassumeonly
sence of the higher order QCD corrections. However,
the off-mass-shell approach.
for possible comparison with the perturbative QCD ap-
proach [11] let us remark the following. This approach
generate evolved TMDs using as input the existing phe-
3. DISCUSSION
nomenological parametrizations extracted from the ex-
perimental data. For example, one of the inputs is the
First let us summarize more accurately what we have
scale-independent Gaussian fit [13]
done in the previous section. We assumed:
a) Lorentz invariance exp(cid:2)−p2/(cid:10)p2(cid:11)(cid:3)
It means that the theoretical description in terms of the Ff/P(x,pT)=ff/P(x) π(cid:104)Tp2(cid:105) T , (26)
standard kinematical variables (see Fig. 1) T
where (cid:10)p2(cid:11) = (0.38±0.06)GeV2. Obviously our con-
q, x , x, p=(p ,p ,p ,p ), P =(P ,P ,P ,P ) T
B 0 1 2 3 0 1 2 3 ceptRSdefinedaboveishardlycompatiblewiththisdis-
tribution. Infactintherestframethisdistributiongives
can be boosted also to the nucleon rest frame. much greater (cid:10)p2(cid:11) than the corresponding longitudinal
b) Inequality 0≤x≤1 T
term (cid:10)p2(cid:11). However RS requires (cid:10)p2(cid:11)=2(cid:10)p2(cid:11) only. Let
It means that the light-cone ratio x satisfies the same 1 T 1
us remark that this imbalance is of the same order as a
bound (2) as the Bjorken variable x .
B
difference between the two data sets mentioned above.
c) Rotational symmetry
The kinematical region R3 of the quark intrinsic mo- iii) The first relation (20) apparently contradicts an
intuitive, Lorentz invariant condition
menta p = (p ,p ,p ) in the nucleon rest frame has
1 2 3
rotational symmetry (i.e. p∈R3 ⇒ p(cid:48) = Rp ∈ R3, |p|2 >p2 (27)
whereRisanyrotationinR3). Forexample,intermsof 0
the covariant QPM it means that probabilistic distribu- corresponding to bound, space-like quarks. Such conflict
tionofthequarkmomentaiscontrolledbysomefunction doesnottakeplacefortheleadingorderwithquarkson-
G(cid:0)pP/M,Q2(cid:1). mass-shell. However, for off-shell quarks the condition
Weprovedtheseassumptionsimplybounds(17)–(21). (27) is incompatible with the assumptions a)–c), which
Now we will shortly comment on the obtained results: imply (20). For example it means that any approach,
i) The ratio x of light-cone variables (9) has a simple whichisbasedprimarilyontheassumption (27)insome
interpretation in a frame, where the proton momentum starting reference frame (including the infinite momen-
is large - x is the fraction of this momentum carried by tum frame) cannot simultaneously satisfy the conditions
the quark. However an interpretation of the same vari- a)–c).
able in the nucleon rest frame is more complicated. In iv) In [2] we explained why the RS, if applied on the
this frame the quark transversal momentum cannot be level of QPM, follows from the covariant description. In
4
pT,maxM 2 which is equivalent to the on-mass-shell relation (24) for
0.30 m = 0. This general upper limit for p2 depending on
T
0.25 x is displayed in Fig. 2. Let us remark that results on
(cid:72) (cid:144) (cid:76) (cid:10)p2(x)(cid:11)obtainedin[14,15]arecompatiblealsowiththe
0.20 T
bound (29). An equivalent form of this inequality was
0.15
probably for the first time presented in [18].
0.10
0.05
x
0.0 0.2 0.4 0.6 0.8 1.0
4. SUMMARY
FIG. 2: Upper limit of the quark transversal momentum as
a function of x for µ = 0 (solid line), 0.1 (dashed line), 0.2 In the present report we studied the kinematic con-
(dotted line) and 0.3 (dash-dotted line). straints due to the rotational symmetry of the quark
momenta distribution inside the nucleon in the leading
order approach. In particular, we have shown that the
fact it means the assumptions a)–c) are common for our
light-cone formalism combined with the assumption on
QPM and for the approaches like [14, 15] where only
the rotational symmetry in the nucleon rest frame im-
Lorentzinvarianceisexplicitlyrequired. Thepredictions
ply p ≤ M/2. Only part of existing experimental data
T
of all these models are compatible with the bound (21).
on (cid:104)p (cid:105) satisfies this bound, but the another part does
T
This is just a consequence of the fact that general con-
not. In general, the existing methods for reconstruction
ditions a)–c) are satisfied in these approaches. Another
of (cid:104)p (cid:105) from the DIS data are rather model-dependent.
T
theoreticalreasonsforRShavebeensuggestedin[8]. Let
These are the reasons why more studies are needed to
usremarkthattherotational-symmetrypropertiesofthe
clarify these inconsistencies, since the phenomenological
nucleonstateinitsrestframeplayanimportantrolealso
distributions in the x−p plane at present serve also as
T
in the recent studies [17].
an input for a more fundamental calculation of the QCD
v)Therelation(24)isobtainedforthequarkson-mass-
evolution and other effects.
shell. Inamoregeneralcase,whereonlyinequalities(20),
(21) hold, this relation is replaced by
(cid:18) µ2 (cid:19)
p2 ≤M2 x− (1−x); µ2 ≡p2−p2, (28)
T M2 0
Acknowledgments
where the term µ2 is not a parameter corresponding to
the fixed mass, but it is only a number varying in the This work was supported by the project
limits defined by (20). The last relation implies for any AV0Z10100502 of the Academy of Sciences of the
µ2: CzechRepublic. IamgratefultoAnatoliEfremov,Peter
SchweitzerandOlegTeryaevformanyusefuldiscussions
p2 ≤M2x(1−x), (29) and valuable comments.
T
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