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From QCD asymptotic predictions to few GeV experiments: The example of exclusive hard processes
223~
Nuclear Physics A497 (1989) 223~ -228~ North-Holland, Amsterdam
FROM QCD ASYMPTOTIC PREDICTIONS TO FEW GeV EXPERIMENTS: THE EXAh4PLEOF EXCLUSIVE HARD PROCESSES Bernard PIRE Centre de Physique Theorique
,Ecoie Polytechnique
,91128Palaiseau
Cedex France
.
1. IntroductlqB
Inclusive reactions at large momentum transfer are in general well understood in terms of hard scatterings between elementary constituents - quarks, gluons, photons, 2 and w’s - in the framework of perturbative quantum ch~m~ynamics.
These hard scattering models are however in
great difficulties when trying to describe reactions with typical transfer of a few GeV. Among these , the case of exclusive processes has been much studied recently [l].The no~ali~ation
and
energy dependence of these reactions cross sections indeed limit the available data to the few GeV range. It turns out that asymptotic predictions derived in the framework of perturbative Q.C.D. do not correctly describe actual data, apart from some qualitative features such as the power law energy dependence of cross sections at fixed angle. In particular, spin effects experimentally observed are at odds with helicity conservation rules characteristic of the asymptotic picture : it seems necessary to include some non perturbative effects in order to allow for helicity flips. One may think of the presence of some higher twist effect, an example of which is the role of diquarks-a system of two rather tightly bound quarks with a radius of say 0.1 - 0.3 fm, with acts as an elementary constituent of baryons probed with a few GeV transfer reaction. From different experimental
and theoretical approaches, there have indeed been many
indications suggesting the presence, inside baryons, of diquarks. They were introduced long time ago in hadron spectroscopy where they constitute an intermediate state in the building up of bound states of three quarks. They have also been advocated to explain inclusive baryon production at large transverse momenta and in deep inelastic lepton -hadron scattering [Z]. We have applied [3] the diquark model to exclusive reactions at large momentum transfer. Our basic assumption is the active presence of diquarks for intermediate regions (few GeV) of momentum transfer. The diquarks can act as real quasi-elementary constituents taking part in the hard scatterings. The basic ingredients of such a description is the hadronic wave function and the effective diquark-gluon (or photon) couplings. 0375-9474/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. Pire / QCD asymptotic predictions
224c 2. The
uroton
wave
function,
A way of writing the baryon wave function is first to couple two quarks together in a diquark and then couple the diquark to the third quark. Assuming zero orbital angular momentum between the two quarks forming the diquark, we have only spin 0 diquarks (S) and spin 1 diquarks (V). In terms of colour we have two types of diquarks, namely sextet (6) and anti triplet (3) states. Only the latter ones can form together with a quark an ordinary baryon. The (6) diquarks may play a role in multiquark exotics. The requirement of a totally antisymmetric baryon wave function then fixes the quantum numbers of the (3) diquarks. In the non-strange S=I=0
S(ud)
S=I= 1
V(W), V(ud), V(dd).
sector we therefore have the diquarks
In terms of diquarks the proton wave function reads (colour indices are suppressed).
VP+=& {‘[Vi,
(ud)u+- fi Vu (uu) d+l @,, (~1
+
[fl V, (ud) u+2 VI Wd. I qvl 6)
+ 3 SW)
u+$, (4
I /
The explicit value of the quark-diquark choices can be advocated
space wave functions is model dependent.
for. For instance, one may follow Chernyak and Zhitnitsky
QCD sum rules and get the following parametrization
Qs = 2.87 I$ as [I - 5.94x + 9.3x2] +v, = 5.58 Qas [l - 4.75x + 5.33x2]
qv,
= 4.81$,, [l - 4.72x + 5.47 x2]
with $,=20x(1
-x)3
[4].
Various [I] use of
225c
B. Pire / QCD asymptotic predictions
3. The diauark
couulinm,
How to treat the diquarks ? The coupling of photons to scalar and vector diquarks follows standard
description.
For the case of gluons we write in obious generalization
of the photon
coupling. SgS
vgv
iG, h%(qt
+ q2) P
-i Xa/2[G1(q1+ q#
gw - G2 (qK2 gkv + qpl g”‘9
where the h’s are the usual Gell-Mann
colour matrices. The four couplings
G,, G,, G2 and
G3 are in fact form factors depending on the momentum transfer squared Q2. For elementary vector particles G3 is zero and G, = G2 Moreover, because of the high powers of momenta multiplying G,, this contribution is suppressed at small and intermediate 42. For our Q2region
of interest,
therefore,
where we think of the vector diquark as a quasi-
elementary particle we neglect this type of coupling. Next we can further reduce the arbitrariness couplings perturbation
by requiring gauge invariance
introduced
in the model by the many possible
in reactions such as Sg+Sg
or Vg +Vg
to lowest order
theory. This leads to
G, = g, F, (Q2)
6
= G2= g, F, (Q2)
where g, = d4nas is the strong coupling constant. As usual the form factors are unknown and one has to assume a certain dependence Advice for a possible paramehization is obtained from the asymptotic behaviour. One gets
F, (Q3 =
h&Q’,
=
a, (Q?JQ: Q;+ Q2
a, (Q5 Q: Q:+Q2 2
F,,(Qz, =--% F,,(Q2) Q;+Q2
on Q2.
226c
4.
3. Pire f @CD asymptotic predict~~ns
Electromagnetic form factors
of the
Let us now look into the consequences consider elec~ma~etic nucleon form factors.
of introducing
diquarks as active constituent
and
Neglecting internal transverse momenta in the proton and using the wave functions given in section 2 , we get (quark masses neglected, collinear kinematics), the following expressions for the form factors of the proton
-Q’/gm”F,(Q3
.
jdxdy+:(y)
a,(&(1 - xl (1 -Y) F, (0;
4, (x>
XY
-‘IT CF
&Q5=
-F,(QZ)(dxdy$f Q2K
x, CQ’, F, (43 XY with 6”
=(I-x)(1-y)Qz
(~1
4%(xl
andQ2=xyQ2.
m is the mass of the vector diquark for with we
use the value 580 MeV. The colour factor C, is 4/3. The comparison measurements interpreted
to experimental
data requires some caution. In [5], the most recent and precise
of the elastic cross section ep -+ ep at relatively
in terms of the el~~magnetic
not measured in this experiment,
large 42 (3 I Q2 2 31 GeV2) are
form factors Gm (Q2) and GE (Q*). However, Gn (Q2) is
neither in other experiments
at comparable
Q2. In [5] the scaling
relation Gg (Q3 = G L (QT / l+, where pp is the proton magnetic moment (up = 2.79), is assumed to hold through the whole @ range although is has been checked only for small Q2 values. In our model, we definitely expect that such a scaling relation wilI break at moderately large 42, approaching quite quickly the asymptotic result
Gg (Q”, / G$ (Q3
--) I.
B. Pire / QCD asymptotic predictions
227c
We shall thus use the simple ansatz : G;(Q;j=Gh(Q2)
forQ2>3GeV2
which leads to the simple expression for the elastic cross section
(1+$a2 ;]
z=(z));,, The values for G L
found by us are compared with those given in (51 in Fig l.The differences
between the two sets of data may be considered as a systematic error. In Fig 1 WC.also show our results. Since, as we discus.. scalar diquark dominates, the 42 behaviour of GL enters in the definition
above, the contribution from the
strongly depends on the parameter Qi which
of F,. We see that we get a reasonable understanding
of GM for
Qf = 3.2GeV2.
1” c> * 0
0.4-
0 0 0 O? OO .-o-.-:--,-,~_+_.-
0.3 : 0.2 0.1 I
IO
20
30
I
Q* ( GeV? Fig. 1. Q4 G3t.tP
as a function of 6. Data taken from [5] using GE= GM (0) and C$ = G&
resp. The theoretical curve is normalized at Qa = 15 Gev(Qi
(0)
= Qt = 3.2 GeV’).A very simplified
wave function has been used [3].
5. Conclusions. Other reactions have been studied in this same framework [3,4,6]. In all cases, there seems to emerge a reasonable description of exclusive data in terms of quarks and diquarks acting both as quasi elementary constituents of baryons. The route from asymptotic QCD predictions to few GeV momentum uanfer reactions thus seems to pass through a quark-diquark description of the proton. It would be very helpful that non perturbative techniques applied to the QCD wave function found this fact on solid grounds.
228~
B. Pire / QCD asymptotic predictions
Reference
[l] S.J. Brodsky and G.P. Lepage : Phys Rev Letters 43 (1979) 545, 1625 (E); V.L. Chemyak and I.R. Zhitnitsky : Nucl.Phys B246 (1984) 52; G. Farrar, E. Maina, F. Neri Nucl.Phys B259 (1985) 702; B263 (1986) 746; N. Isgur, these proceedings.
[2] S. Ekelin et al : Phys Rev D30 (1984) 2310; Phys.Lett 162B (1985) 373 and references therein.
[3] M. Anseltnino,
P. Kroll and B. Pire : Z.Phys C36 (1987) 89.
[4] M. Anselmino,
F. Caruso, P. Kroll and W. Schweiger,
Cern preprint TH 4941 (1987).
[5] R.G. Arnold et al : Phys Rev Lett 57 (1986) 174
[6] P. Kroll and W. Schweiger, Nucl Phys A474 (1987) 608 M. Anselmino, F. Caruso, S. Forte and B. Pire, Torino preprint DFIT21/87.
Nuclear Physics A497 (1989) 223~ -228~ North-Holland, Amsterdam
FROM QCD ASYMPTOTIC PREDICTIONS TO FEW GeV EXPERIMENTS: THE EXAh4PLEOF EXCLUSIVE HARD PROCESSES Bernard PIRE Centre de Physique Theorique
,Ecoie Polytechnique
,91128Palaiseau
Cedex France
.
1. IntroductlqB
Inclusive reactions at large momentum transfer are in general well understood in terms of hard scatterings between elementary constituents - quarks, gluons, photons, 2 and w’s - in the framework of perturbative quantum ch~m~ynamics.
These hard scattering models are however in
great difficulties when trying to describe reactions with typical transfer of a few GeV. Among these , the case of exclusive processes has been much studied recently [l].The no~ali~ation
and
energy dependence of these reactions cross sections indeed limit the available data to the few GeV range. It turns out that asymptotic predictions derived in the framework of perturbative Q.C.D. do not correctly describe actual data, apart from some qualitative features such as the power law energy dependence of cross sections at fixed angle. In particular, spin effects experimentally observed are at odds with helicity conservation rules characteristic of the asymptotic picture : it seems necessary to include some non perturbative effects in order to allow for helicity flips. One may think of the presence of some higher twist effect, an example of which is the role of diquarks-a system of two rather tightly bound quarks with a radius of say 0.1 - 0.3 fm, with acts as an elementary constituent of baryons probed with a few GeV transfer reaction. From different experimental
and theoretical approaches, there have indeed been many
indications suggesting the presence, inside baryons, of diquarks. They were introduced long time ago in hadron spectroscopy where they constitute an intermediate state in the building up of bound states of three quarks. They have also been advocated to explain inclusive baryon production at large transverse momenta and in deep inelastic lepton -hadron scattering [Z]. We have applied [3] the diquark model to exclusive reactions at large momentum transfer. Our basic assumption is the active presence of diquarks for intermediate regions (few GeV) of momentum transfer. The diquarks can act as real quasi-elementary constituents taking part in the hard scatterings. The basic ingredients of such a description is the hadronic wave function and the effective diquark-gluon (or photon) couplings. 0375-9474/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
B. Pire / QCD asymptotic predictions
224c 2. The
uroton
wave
function,
A way of writing the baryon wave function is first to couple two quarks together in a diquark and then couple the diquark to the third quark. Assuming zero orbital angular momentum between the two quarks forming the diquark, we have only spin 0 diquarks (S) and spin 1 diquarks (V). In terms of colour we have two types of diquarks, namely sextet (6) and anti triplet (3) states. Only the latter ones can form together with a quark an ordinary baryon. The (6) diquarks may play a role in multiquark exotics. The requirement of a totally antisymmetric baryon wave function then fixes the quantum numbers of the (3) diquarks. In the non-strange S=I=0
S(ud)
S=I= 1
V(W), V(ud), V(dd).
sector we therefore have the diquarks
In terms of diquarks the proton wave function reads (colour indices are suppressed).
VP+=& {‘[Vi,
(ud)u+- fi Vu (uu) d+l @,, (~1
+
[fl V, (ud) u+2 VI Wd. I qvl 6)
+ 3 SW)
u+$, (4
I /
The explicit value of the quark-diquark choices can be advocated
space wave functions is model dependent.
for. For instance, one may follow Chernyak and Zhitnitsky
QCD sum rules and get the following parametrization
Qs = 2.87 I$ as [I - 5.94x + 9.3x2] +v, = 5.58 Qas [l - 4.75x + 5.33x2]
qv,
= 4.81$,, [l - 4.72x + 5.47 x2]
with $,=20x(1
-x)3
[4].
Various [I] use of
225c
B. Pire / QCD asymptotic predictions
3. The diauark
couulinm,
How to treat the diquarks ? The coupling of photons to scalar and vector diquarks follows standard
description.
For the case of gluons we write in obious generalization
of the photon
coupling. SgS
vgv
iG, h%(qt
+ q2) P
-i Xa/2[G1(q1+ q#
gw - G2 (qK2 gkv + qpl g”‘9
where the h’s are the usual Gell-Mann
colour matrices. The four couplings
G,, G,, G2 and
G3 are in fact form factors depending on the momentum transfer squared Q2. For elementary vector particles G3 is zero and G, = G2 Moreover, because of the high powers of momenta multiplying G,, this contribution is suppressed at small and intermediate 42. For our Q2region
of interest,
therefore,
where we think of the vector diquark as a quasi-
elementary particle we neglect this type of coupling. Next we can further reduce the arbitrariness couplings perturbation
by requiring gauge invariance
introduced
in the model by the many possible
in reactions such as Sg+Sg
or Vg +Vg
to lowest order
theory. This leads to
G, = g, F, (Q2)
6
= G2= g, F, (Q2)
where g, = d4nas is the strong coupling constant. As usual the form factors are unknown and one has to assume a certain dependence Advice for a possible paramehization is obtained from the asymptotic behaviour. One gets
F, (Q3 =
h&Q’,
=
a, (Q?JQ: Q;+ Q2
a, (Q5 Q: Q:+Q2 2
F,,(Qz, =--% F,,(Q2) Q;+Q2
on Q2.
226c
4.
3. Pire f @CD asymptotic predict~~ns
Electromagnetic form factors
of the
Let us now look into the consequences consider elec~ma~etic nucleon form factors.
of introducing
diquarks as active constituent
and
Neglecting internal transverse momenta in the proton and using the wave functions given in section 2 , we get (quark masses neglected, collinear kinematics), the following expressions for the form factors of the proton
-Q’/gm”F,(Q3
.
jdxdy+:(y)
a,(&(1 - xl (1 -Y) F, (0;
4, (x>
XY
-‘IT CF
&Q5=
-F,(QZ)(dxdy$f Q2K
x, CQ’, F, (43 XY with 6”
=(I-x)(1-y)Qz
(~1
4%(xl
andQ2=xyQ2.
m is the mass of the vector diquark for with we
use the value 580 MeV. The colour factor C, is 4/3. The comparison measurements interpreted
to experimental
data requires some caution. In [5], the most recent and precise
of the elastic cross section ep -+ ep at relatively
in terms of the el~~magnetic
not measured in this experiment,
large 42 (3 I Q2 2 31 GeV2) are
form factors Gm (Q2) and GE (Q*). However, Gn (Q2) is
neither in other experiments
at comparable
Q2. In [5] the scaling
relation Gg (Q3 = G L (QT / l+, where pp is the proton magnetic moment (up = 2.79), is assumed to hold through the whole @ range although is has been checked only for small Q2 values. In our model, we definitely expect that such a scaling relation wilI break at moderately large 42, approaching quite quickly the asymptotic result
Gg (Q”, / G$ (Q3
--) I.
B. Pire / QCD asymptotic predictions
227c
We shall thus use the simple ansatz : G;(Q;j=Gh(Q2)
forQ2>3GeV2
which leads to the simple expression for the elastic cross section
(1+$a2 ;]
z=(z));,, The values for G L
found by us are compared with those given in (51 in Fig l.The differences
between the two sets of data may be considered as a systematic error. In Fig 1 WC.also show our results. Since, as we discus.. scalar diquark dominates, the 42 behaviour of GL enters in the definition
above, the contribution from the
strongly depends on the parameter Qi which
of F,. We see that we get a reasonable understanding
of GM for
Qf = 3.2GeV2.
1” c> * 0
0.4-
0 0 0 O? OO .-o-.-:--,-,~_+_.-
0.3 : 0.2 0.1 I
IO
20
30
I
Q* ( GeV? Fig. 1. Q4 G3t.tP
as a function of 6. Data taken from [5] using GE= GM (0) and C$ = G&
resp. The theoretical curve is normalized at Qa = 15 Gev(Qi
(0)
= Qt = 3.2 GeV’).A very simplified
wave function has been used [3].
5. Conclusions. Other reactions have been studied in this same framework [3,4,6]. In all cases, there seems to emerge a reasonable description of exclusive data in terms of quarks and diquarks acting both as quasi elementary constituents of baryons. The route from asymptotic QCD predictions to few GeV momentum uanfer reactions thus seems to pass through a quark-diquark description of the proton. It would be very helpful that non perturbative techniques applied to the QCD wave function found this fact on solid grounds.
228~
B. Pire / QCD asymptotic predictions
Reference
[l] S.J. Brodsky and G.P. Lepage : Phys Rev Letters 43 (1979) 545, 1625 (E); V.L. Chemyak and I.R. Zhitnitsky : Nucl.Phys B246 (1984) 52; G. Farrar, E. Maina, F. Neri Nucl.Phys B259 (1985) 702; B263 (1986) 746; N. Isgur, these proceedings.
[2] S. Ekelin et al : Phys Rev D30 (1984) 2310; Phys.Lett 162B (1985) 373 and references therein.
[3] M. Anseltnino,
P. Kroll and B. Pire : Z.Phys C36 (1987) 89.
[4] M. Anselmino,
F. Caruso, P. Kroll and W. Schweiger,
Cern preprint TH 4941 (1987).
[5] R.G. Arnold et al : Phys Rev Lett 57 (1986) 174
[6] P. Kroll and W. Schweiger, Nucl Phys A474 (1987) 608 M. Anselmino, F. Caruso, S. Forte and B. Pire, Torino preprint DFIT21/87.