- Email: [email protected]
Structure of the constituent quark
Ptog.
Part.
Nucl. Phys.. Vol. 36, pp. 19-28, 1996 Copyright 0 1996 Elsevier Science Ltd
Printed in Oreat Britain. All rights reserved SO146-6410(96)00004-X
Structure
of the Constituent
0146-6410/96
i32.00
+ 0.00
Quark
H. J. PIRNER InstitutfiirThcorerische Philosophenwcg
Physik, Universitiit Heidelberg. 19. D-69120 Heidelberg, Germany
ABSTRACT We calculate the differential cross section for semi-inclusive pion production in electron proton reactions using a model where the physical quark fluctuates with some probability to quark plus pion. Diffractive vector meson production is analysed in terms of a gluon cloud of the quark.
KEYWORDS &CD; electroproduction;
constituent
quark.
INTRODUCTION Since Yukawa the nucleon has been perceived as an extended object with a pionic cloud around it. The one-pion exchange interaction bktween nucleons is a naturai consequence of pion emission and absomtion by nucleons. The extent of the nionic cloud of the nucleon has. however. been debated. Proponents df the N-N-one-boson exchange iode use form factors with cut-ok values h’ z 2 - 2.5 GeV, which give a large amount of pion admixture to the nucleon. In an analysis of pion-electroproduction data on the proton (Giittner et al., 1984) an estimate of the pion cloud of the nucleon has yielded a rather small value, namely a 3% admixture of mr + to the naked p state. This small probability would make it unrealistic to use coherent pion emission of the nucleon to create sea quarks in the nucleon. On the other hand, it is very natural to couple quarks to scalar (u) and pseudoscalar mesons (?) in a chirally invariant way. The first coupling leads to the constituent quark mass after spontaneous breaking of the (0, ?)-symmetry. The second coupling to pions produces a pion cloud of the constituent quark. There is a difference between a pion cloud of the n&eon and the pion cloud of the qua&. Emission and absorption of a pion by the same quark determines the pion cloud of the constituent quark. The intermediate states on the nucleon level would be I3q7r) states of arbitrary excitation energy. Allowed intermediate states of the nucleonic pion cloud are ) N’r) or ) Ar) states. In this sense the pion is part of the hadronic system. The analysis of electroproduction at small inclusive mass and correspondingly z z ET/u -+ 1 will give only the pion cloud of the nucleon, as described Gfittner et al. (1984). We suggest that the constituent quark structure is the more interesting quantity to study. The constituent quark picture haa been a simple and convincing concept to explain the static properties of baryons and mesons. It is surprising how well magnetic moments and masses of baryons are reproduced. In deep inelastic lepton nucleon scattering the quarks are visible with their momentumdistributions and charges, Yet the constituent quark model has never been able to explain the measured structure functions of the proton. Especially the transition with Q’ from an extended constituent quark with its own internal structure to a pointlike current quark haa not been investigated. At low Q’, the constituent quark has intrinsic structure, namely a pion and possibly also a gluon cloud. The baryon will be described by a light cone wave function of three constituent quarks. This wave
19
20
H. J. Pirner
function takes care of the relativistic motion of the quarks and can, in principle, be obtained from a suitable light cone Hamiltonian. We want to define a procedure to separate its internal structure from the bound state dynamics contained in the quark wave function. The contents of this paper on the pion cloud of the constituent quark are part of a longer article investigating also the experimental possibilities for semiinclusive pion production at Hermes (Baumgirtner et al., 1995). This article contains an extensive list of references. We will also give some preview of work in progress on diffractive production of vector mesons which may shed some light on the gluon cloud of the constituent quark (Dosch et al., 1995). THE PION CONTENT
OF THE CONSTITUENT
QUARK
Instead of presenting a detailed derivation of the properties of the constituent quark, which have been made with a Lagrangian (Gell-Mann et al., 1960), we want to employ a toy model (Eichten et al., 1992; Cheung et al., 1992) that already captures the essential features. In this simplified picture a nucleon consists of up and down quarks-only. In the chiral quark model a u-quark can emit a s+ (containing a valence u-quark and a valence d-antiquark) or a K’ as depicted in Fig. 1.
Figure 1. Fluctuation of a u-quark into quark and pion. The model parameter a is the probability an up-quark to turn into a down-quark with the emission of a B+ a = I( &+I u) 1’.
In this picture a physical constituent (naked) states:
T
ef N;/u
quark (U) after one such interaction
=
et [(l+
=
\/;Ela-~)+
ia)
is a superposition
N,/ut %Nqu] + 4 [;Nqu + $Nqu]
and analogously: ID)
‘j$
efNi,D
= ei
[(I + ia)
~/sod)+ Nd/D +
J((l-a-4)ld) ~NJ,~] t ei
[?N,,D
+
?N~,D]
for
of different
Structure of the Constituent The flavor content 7
efN/,
of the proton and neutron = ez [(z t $>
in this simplified
NulP t ($)
N,l,]
Quark
21
model hence is given by
t ei [(It
T)
&lP t ($)
NJ,,]
(3)
and 7
e:Ni,,=e:[(2+~)Nd/,f(~)~l,]te:[(l+~)~,l.’(~)~’l.]
.
(*I
One sees: by assuming a pion cloud of the u- and d-constituent quarks (that is: a # 0) there arises a flavor asymmetry in the nucleon’s sea the extent of which depends on the value of the model parameter a. Hence, the Gottfried Sum Rule defect can be expressed in terms of a:
Of course, the flavor asymmetry properties of the nucleon. Let
represented
by the parameter
Aq, := jdr
a also affects other static and quasi-static
[q!(s) - q!(z)]
(6)
0
denote the integrated difference of quarks (and antiquarks) with helicity parallel (f) and antiparallel (1) to the nucIeon’s helicity. In the naive quark-parton model with SU(6)- wave functions for a polarized proton one has Au= Ad and gets the foilowing prediction
$ =
-f
(7)
for the axial vector coupling constant g,,=Au-Ad=?
gA (8)
3’
If we consider a pion cloud of the constituent quark in the form of the just explained toy model, the spin content of the constituent quark is changed, since the pion is emitted in a p-wave state. One gets AU
=
AD
=
4 7a 3-3
-i-2” 3
AS=
0”
(9)
and thus a value of gA gA=AU-AD=i(l-a), which is modified as compared to the naive expectations (a # 0) to the constituent quark.
(IO) in Eq. 8 by the existence
of a pionic admixture
Similarly, gi, defined as the proton matrix element of the flavor singlet axial current
(PI~~YS~IP)=: dlx+~x>
PPWP-B
(11)
22
H. I. Pirner
where 1c,= (u,d,s) denotes spinor y, can be expressed
the quark field and lp) is the wave function using Eq. 9:
of a proton
at rest with Pauli
(12) gi measures how much of the proton’s helicity is carried by constituent quarks. In the naive quark parton model a = 0 and the proton’s helicity is carried entirely by quarks (9: = 1). In the model of a pion cloud this value is reduced by the pions’ (or sea-quarks’) contribution and thus gi < 1. A rich source of insight into nucleon structure are structure functions for deep inelastic inclusive scattering on polarized nucleon targets. They can also be expressed using quark distribution functions: (13) For experimental
reasons,
instead
of the structure
functions
one rather considers
moments
of them:
(14) In the pion cloud model one obtains
for the proton and neutron
rp1 =
We will now analyse the available experimental IT’” to determine the value of a.
spin structure
functions
$l-2a)
data on the above mentioned
quantities
So, gA, gi and
Fig. 2 shows the various values for the model parameter a. We also plot as a thick vertical line the mean from the results of the three independent experimental determinations. The best known quantity is gA. It is this value which therefore essentially determines the weighted mean for a: a = 0.246 f 0.002
(17)
As one sees in Fig. 2 the various corrections are essential. By correcting the NMC result on So for shadowing effect and the El42 result for &‘-dependence the original values are shifted to within la of the weighted mean of a = 0.246. We take this as evidence that the employed toy model captures at least the essential features of the idea of a pion cloud of the constituent quark and that this idea is capable of explaining some properties of the nucleon. quark turns However, it might come as a surprise that the probability for a r + in an up-constituent out to be almost 25 %. A perturbative calculation in the linear a-model with pseudoscalar quark-pionparameter coupling (coupling constant grpq = 3.76) reproduces this value if one uses a regularization of A = 3.37 GeV. This seems to be very large for a hadronic cut-off. A possible explanation for both large values might be the non-relativistic character of the employed models. It should be appropriate to describe the nucleon as consisting of three relativistic quarks. Relativistic quark models with lightcone wave functions for the constituent quarks have already been used for a quantitative analysis of static and quasi-static properties of the nucleon (Yabu et al., 1993; Brodsky et al., 1994). While such calculations can predict static properties of the nucleon, like g,J, almost exactly without recurring to
Structure of the Constituent Quark
-
El42
: g; (moos ured El 22 (d-corrected) EMC
best fit of all spin
ds
tructure
23
SMC
: a
: g:
function experiments
_NMC 1 So (with shadow correction) -NMC
: S, (unc orrected)
Figure 2. The various results for a from different observables.
an effective description like a pion cloud! they nevertheless experiment as far as quasi-static properties are concerned.
The thick vertical line indicates
cannot give a satisfactory
a = 0.246.
agreement
with
We conclude that even in a relativistic description of what constitutes a nucleon the idea of a pion cloud of the constituent quark might be helpful. However, the above performed estimates of the pion probability a in the constituent quark, which are based on integral properties of the nucleon, are only of limited use. A direct and differential measurement of the pionic substructure of constituent quarks will offer more insight. To this purpose we turn in the next section to a differential description of the nucleon as seen in semi-inclusive electroproduction of pions on nucleon targets. SEMI-INCLUSIVE We consider electron
scattering
PION PRODUCTION on the proton with at least one pion a(k) in the final state: e+p+e’+7r+X.
(IS)
Semi-inclusive pion production can be looked at as a soft hadronization process. Data (Arnedo et al., 1985) at high energies agree rather well with LUND-simulations. However, in certain kinematic regions higher twist effects can visibly modify the leading gluon cascade. For example, one expects contributions (Brandenburg et al., 1995) from the hard fragmentation r*q + rq. We will describe the pion production in the naive parton model with the following assumptions. The nucleon consists of three valence constituent quarks, which are weakly bound. The square of their wave functions inte rated over transverse momentum P; up to the resolution Q2 gives the probability to find a quark with # avor q and light cone momentum fraction z in the proton:
(19)
24
H. J. Pirner
The constituents are massive Dirac particles (MQ = yKgp(cr)) with a pointlike coupling to the photon and a pseudoscalar coupling ig,, ($y&!~)? to the pion. The coupling constant grpp couples the quark equally to both fields of the (4, i) SU(2)~xSu(2) n multiplet (g,?). The coupling strength is determined from the empirical value of the constituent quark mass AJo = 350 MeV and the value of (u) = fir = 93.3 MeV: 9 “9,r = 3.76 . (20) ygally
one could have estimated
gnqq from quark additivity
. :
LhNN
g
In the parton model, with the distribution
= 3.57
- -
“”
-
3gA
pion production is calculated N,(s) of the quarks:
and the a-N-coupling
constant
gnNN =
*
(21)
as a folding of the partonic
cross section
eq -+ e’q’r
(2‘4 The kinematical y = u/E
variables
of this semi-inclusive
and on the pion side Ic” = (ET, i).
In Fig. 3 we give a definition inclusive
reaction
process
are on the electron
side xs = &‘/~MNv
It is useful to discuss the kinematics
of all momenta
in the target-at-rest
there are two planes of interest,
system,
the lepton scattering
P’“_=JMN, 0’). In the semi-
plane (f!, C) and the photon-pion
plane (<,i). They include an angle 4% between them. The energy of the incoming electron energy of the outgoing electron is E’. The photon energy is Y, the pion energy is E,. momentum
z is divided into a momentum
and transverse
to the photon
momentum
parallel
to the direction
and
in more detail.
of the photon
momentum
is E, the The pion $
9’ (24)
Unpolarized cross sections do not depend two invariants are sufficient to characterize
on the azimuthal angle of the scattered the virtual photon. These are
Q2 = -q2 v
=
4 EE’ sin2 @,I2
=
PC&P-
electron,
therefore
(25)
MN
or
xs=-= y=P.e
Q2
-q2
2P.
P.q -=--.
q
2Mf,r u v E
(26)
Structure of the Constituent
25
Quark pion scattering
Figure 3. The reaction
plane
ep + e’?rX in the target-rest-system.
The momentum of the pio_n has three components. We separate the azimuthal angle & from the two residual variables $1 and Ikll. Th e p arallel momentum /cl, can be replaced by the energy variable E, or the scaling variable
P.k
E,
(27)
z=G=u’ A
conventional set of variables thus (v/4 ICI,)dz d$: d& the five-fold differential da
=
dx, dy dz di$ d&
consists of (Q2, xer z, $11, &). cross section has the form
d3k/2ET
=
4m2MNE _ v
$
In this paper we will concentrate
With
Q4
{
GY2'H1+(1-y)'H2t
(2 - y) J1-y
cos $,?I3
(1 - y) co5 29%T-lH4. 1
on & and ci integrated
+
(28)
cross sections.
The five-fold differential cross section can be evaluated with the four different structure functions ‘H;. For a first comparison with future experimental data it will be necessary - for reasons of statistics - to
H. J. Pirner
26 reduce this five-dimensional
space to the 3 dimensions
xer Q’, and z by integrating
out &, and ii::
&=x H,(x=,Q’,z)
:=
; Td,
/
0
0
e
d!$ ‘?&(xB, Q2, z, if)
IL
P-9 The cross section
is then given by da
~
=
dx, dy dz da
dx,dQ2dz Of main functions
=
interest is the contribution we obtain:
&4
{X,Y2fJ1f(l-Y)Hz}
=
or
4ircXz -{xaYZ%+(l-y)Hz}. xa Q” to the cross section
da dx,dQ2dz
8affZMp4E
(39)
from longitudinal
and transverse
structure
+g{2(l-y)HL+(1+(1-Y)2)Hr}~
The two contributions from HL and HT are is dominated by the t-channel, whereas the of the longitudinal part of the cross section allow to isolate the presence and strength 1995) of hadronization effects of the struck in the longitudinal structure function.
(31)
plotted in Fig. 4. Clearly the contribution from HL at z + 1 s-channel contributes strongly to HT. Thus a determination and correspondingly the longitudinal structure function will of a pion cloud. A recent calculation (Brandenburg et al., quark also showed evidence for a peak at large values of .Z
It should be stressed, that we require a large invariant mass of the hadronic system excluding the pion, such that the quasi-elastic channel y’p -+ ~+n does not contribute. As an example, for xg = 0.15 at E, = 27GeV this hadronic mass is only 1 GeV (2.4 GeV) f or z -+ 1 (z --f 0), with the hadronic mass increasing with decreasing xg. Thus a measurement at larger center-of-mass energy is desirable. GLUON
CLOUD
OF THE
CONSTITUENT
QUARK
The chiral properties of the constituent quark determine a large part of low energy hadronic physics. This can also be seen e.g. from the contributions of L.Y. Glozmann, K. Goeke and D. Dyakonov in this meeting. In addition, QCD-sum rules show that the light hadron masses mainly depend on the value of the quark condensate ($$). I n my opinion, it is expected that a neglect of the long-range confining forces necessitates a too large constituent quark mass, which leads to a discrepancy e.g. of the magnetic moments with experiment. I therefore advocate to add to the effective Lagran ian of the linear a-model a part containing the long-range gluon fields. I prefer a parametrization o P the longrange gluon physics after the spontaneous breaking of chiral symmetry, since an inclusion of long-range confining gluon forces together with zero-range forces, e.g. in the Nambu-Jona-Lasinio model, does not seem easy and is probably not correct. Besides the lattice color-dielectric model I think the continuum It model of the stochastic vacuum (Dosch et al. 1988) is closest to lattice QCD at large distances. Note, because of includes nonvanishing gluon field strenghts &,l( x ) correlated over short distances. the Aharonov-Bohm
effect
that
the associated
gluon fields At)(r)
arising
e.g.
from a localized
color
Structure of the Constituent
da / dx, d@ dz
0.2
[ pb/GeV’
(H, only)
0.2
Quark
E
da / dx, dd
27
dz
(H, only)
I
0.175
r:; ,--.....p-channel
0.075
: ,i’
‘...
‘..,
‘\.
‘..,
0.05
!
‘..,
t
‘.._ Y,
0.025
_,_ ;‘,,,_ --
Oo
_..I
.. -\
j
0.2
--
-.. _. ---fJ_. 0.4
,f”
,’
x,,
‘\
‘... __, _.:.
II.6
_ ._
0.8
Figure 4. Contribution from HL and HT to the cross section for semi-inclusive values of Q2 and ~a are fixed at 1 GeV’ and 0.10, respectively.
magnetic field strength & = F,$ 1 ) s p rea d over large distances the Wilson area law for heavy quarks.
pion production.
and satisfy the confinement
The
criteria
of
(32) The string tension
u = 1 GeV/fm
is obtained
as in the stochastic
0 w o.o2(g~F~“F~Ja~
vacuum model (33)
with a = 0.35 fm as correlation length and the gluon condensate (gzFfiyFfiy) = 2.4 GeV4 from QCDsum rules. In soft high-energy scattering (Dosch et al., 1992), the hadrons interact via the correlations induced upon the respective Wilson loops formed by the paths of the quarks and antiquarks (or diquarks) inside the hadrons. Naively, we can consider the vacuum as made up from domains (or bubbles) with constant gluon field strength. Then two quarks can only interact if they hit the same bubble. In the rest system of the hadron, the target quark can be pictured to be surrounded by a bubble of constant gluon field, with which the scattering quark interacts. Consequently, we expect the size of the gluon cloud of the constituent quark to be identical with the correlation length of the gluon fluctuations in the stochastic vacuum. Indeed the differential cross section for TN + pN scattering has an exponential slope whose size is mostly given by the correlation size a and not by the size of the hadronic wave functions of the p-meson and the nucleon. Diffractive constituent
electroproduction of p-mesons is a very good instrument to investigate the gluonic size of the quark since the transverse size of the incoming qq state can be varied with the virtuality of
28
H. J. Pirner
the incoming photon Q2. Until now we have investigated the low Q2 region, where the cross section can be modelled in a Generalized Vector-Meson Dominance Model with three p-resonances, the ~(770 MeV), p’( 1450 MeV) and p”( 1770 MeV). The first excited state is mostly a radial excitation (2s,3 Sr), whereas the second one corresponds to a 3D1 state. In principle the coupling of these vector mesons to the photon is calculable in the constituent quark model. The nonrelativistic wave function of the p-meson unfortunately reproduces very badly the width of the 3Si state Ip + e+e-. Relativistic corrections are the only reason for a nonvanishing coupling of the 3Di-state, since its nonrelativistic wavefunction is zero at the origin. So we have used a semiempirical way to include the Ip, -+ e+e- widths and the signs The yN production amplitude for the vector-meson pi by transverse of the frpj coupling constants. photons is then given as:
The stochastic vacuum model allows us to @-scattering. In general, the off-diagonal with the size of the integrated p-production B of the differential cross section:
calculate diagonal and off-diagonal T-matrix elements for matrix elements are important. Preliminary results agree cross section in the low Q2-region and the slope parameter
, da = ao~e-Bl*l _ dt
with Bezp N 8 GeV-‘.
(36)
The size of the gluon cloud of the constituent quark contributes about the same amount as the size of the wave functions of the hadrons to the theoretical slope parameter Btheor E 1.6~~ + O.lS((pilr’lp,) + (NIr’IN)) CONCLUSION Employing a model where the physical quark fluctuates with some probability to a quark plus a pion we have calculated the differential cross section for semi-inclusive pion production in electron proton reactions. As expected, the biggest contribution from the pion cloud to the cross section is at small to the values of I~ N 0.1 and large values of z = ET/v + 1. In particular, the sole contribution longitudinal structure function will allow an experimental determination of the magnitude of the pion cloud of the quark. The gluon cloud of the constituent quark can be investigated in diffractive vector meson production. REFERENCES Arnedo, M. et al. (NMC) (1985). Phys. Lett. 150, 458. Baumgartner, S., H.J. Pirner, K. Kijnigsmann and B. Povh (1995). How to determine the pion cloud of the constituent quark. Preprint hep-ph/9507309, to be published in Z. Phys. A. Brandenburg, A., V.V. Khoze and D. Miiller (1995). Phys. Lett. B347, 413-418. Brodsky, S.J. and F. Schlumpf (1994). Phys. Lett. B329, 111. Cheung, C.Y., C.F. Wai and H.L. Yu (1992). Phys. Lett. B279, 373. Dosch, H.G. and Y.A. Simonov (1988). Phys. Lett. B205, 339. Dosch, H.G., E. Ferrera and A. Kramer (1994). Phys. Rev. D50, 1992. Dosch, H.G., E. Gubankova, G. Kulzinger and H.J. Pirner. Diffractive vectormeson production in the model of the stochastic vacuum. Work in progress. Eichten, E.J., I. Hinchliffe and C. Quigg (1992). Phys. Rev. D45, 2269. Gell-Mann, M. and M. Levy (1960). Nuovo Cimento 16, 705. Giittner, F., G. Chanfray, H. J. Pirner and B. Povh (1984). Nucl. Phys. A429, 389. Yabu, H., M. Takizawa and W. Weise (1993). 2. Phys. A345, 193.
Part.
Nucl. Phys.. Vol. 36, pp. 19-28, 1996 Copyright 0 1996 Elsevier Science Ltd
Printed in Oreat Britain. All rights reserved SO146-6410(96)00004-X
Structure
of the Constituent
0146-6410/96
i32.00
+ 0.00
Quark
H. J. PIRNER InstitutfiirThcorerische Philosophenwcg
Physik, Universitiit Heidelberg. 19. D-69120 Heidelberg, Germany
ABSTRACT We calculate the differential cross section for semi-inclusive pion production in electron proton reactions using a model where the physical quark fluctuates with some probability to quark plus pion. Diffractive vector meson production is analysed in terms of a gluon cloud of the quark.
KEYWORDS &CD; electroproduction;
constituent
quark.
INTRODUCTION Since Yukawa the nucleon has been perceived as an extended object with a pionic cloud around it. The one-pion exchange interaction bktween nucleons is a naturai consequence of pion emission and absomtion by nucleons. The extent of the nionic cloud of the nucleon has. however. been debated. Proponents df the N-N-one-boson exchange iode use form factors with cut-ok values h’ z 2 - 2.5 GeV, which give a large amount of pion admixture to the nucleon. In an analysis of pion-electroproduction data on the proton (Giittner et al., 1984) an estimate of the pion cloud of the nucleon has yielded a rather small value, namely a 3% admixture of mr + to the naked p state. This small probability would make it unrealistic to use coherent pion emission of the nucleon to create sea quarks in the nucleon. On the other hand, it is very natural to couple quarks to scalar (u) and pseudoscalar mesons (?) in a chirally invariant way. The first coupling leads to the constituent quark mass after spontaneous breaking of the (0, ?)-symmetry. The second coupling to pions produces a pion cloud of the constituent quark. There is a difference between a pion cloud of the n&eon and the pion cloud of the qua&. Emission and absorption of a pion by the same quark determines the pion cloud of the constituent quark. The intermediate states on the nucleon level would be I3q7r) states of arbitrary excitation energy. Allowed intermediate states of the nucleonic pion cloud are ) N’r) or ) Ar) states. In this sense the pion is part of the hadronic system. The analysis of electroproduction at small inclusive mass and correspondingly z z ET/u -+ 1 will give only the pion cloud of the nucleon, as described Gfittner et al. (1984). We suggest that the constituent quark structure is the more interesting quantity to study. The constituent quark picture haa been a simple and convincing concept to explain the static properties of baryons and mesons. It is surprising how well magnetic moments and masses of baryons are reproduced. In deep inelastic lepton nucleon scattering the quarks are visible with their momentumdistributions and charges, Yet the constituent quark model has never been able to explain the measured structure functions of the proton. Especially the transition with Q’ from an extended constituent quark with its own internal structure to a pointlike current quark haa not been investigated. At low Q’, the constituent quark has intrinsic structure, namely a pion and possibly also a gluon cloud. The baryon will be described by a light cone wave function of three constituent quarks. This wave
19
20
H. J. Pirner
function takes care of the relativistic motion of the quarks and can, in principle, be obtained from a suitable light cone Hamiltonian. We want to define a procedure to separate its internal structure from the bound state dynamics contained in the quark wave function. The contents of this paper on the pion cloud of the constituent quark are part of a longer article investigating also the experimental possibilities for semiinclusive pion production at Hermes (Baumgirtner et al., 1995). This article contains an extensive list of references. We will also give some preview of work in progress on diffractive production of vector mesons which may shed some light on the gluon cloud of the constituent quark (Dosch et al., 1995). THE PION CONTENT
OF THE CONSTITUENT
QUARK
Instead of presenting a detailed derivation of the properties of the constituent quark, which have been made with a Lagrangian (Gell-Mann et al., 1960), we want to employ a toy model (Eichten et al., 1992; Cheung et al., 1992) that already captures the essential features. In this simplified picture a nucleon consists of up and down quarks-only. In the chiral quark model a u-quark can emit a s+ (containing a valence u-quark and a valence d-antiquark) or a K’ as depicted in Fig. 1.
Figure 1. Fluctuation of a u-quark into quark and pion. The model parameter a is the probability an up-quark to turn into a down-quark with the emission of a B+ a = I( &+I u) 1’.
In this picture a physical constituent (naked) states:
T
ef N;/u
quark (U) after one such interaction
=
et [(l+
=
\/;Ela-~)+
ia)
is a superposition
N,/ut %Nqu] + 4 [;Nqu + $Nqu]
and analogously: ID)
‘j$
efNi,D
= ei
[(I + ia)
~/sod)+ Nd/D +
J((l-a-4)ld) ~NJ,~] t ei
[?N,,D
+
?N~,D]
for
of different
Structure of the Constituent The flavor content 7
efN/,
of the proton and neutron = ez [(z t $>
in this simplified
NulP t ($)
N,l,]
Quark
21
model hence is given by
t ei [(It
T)
&lP t ($)
NJ,,]
(3)
and 7
e:Ni,,=e:[(2+~)Nd/,f(~)~l,]te:[(l+~)~,l.’(~)~’l.]
.
(*I
One sees: by assuming a pion cloud of the u- and d-constituent quarks (that is: a # 0) there arises a flavor asymmetry in the nucleon’s sea the extent of which depends on the value of the model parameter a. Hence, the Gottfried Sum Rule defect can be expressed in terms of a:
Of course, the flavor asymmetry properties of the nucleon. Let
represented
by the parameter
Aq, := jdr
a also affects other static and quasi-static
[q!(s) - q!(z)]
(6)
0
denote the integrated difference of quarks (and antiquarks) with helicity parallel (f) and antiparallel (1) to the nucIeon’s helicity. In the naive quark-parton model with SU(6)- wave functions for a polarized proton one has Au= Ad and gets the foilowing prediction
$ =
-f
(7)
for the axial vector coupling constant g,,=Au-Ad=?
gA (8)
3’
If we consider a pion cloud of the constituent quark in the form of the just explained toy model, the spin content of the constituent quark is changed, since the pion is emitted in a p-wave state. One gets AU
=
AD
=
4 7a 3-3
-i-2” 3
AS=
0”
(9)
and thus a value of gA gA=AU-AD=i(l-a), which is modified as compared to the naive expectations (a # 0) to the constituent quark.
(IO) in Eq. 8 by the existence
of a pionic admixture
Similarly, gi, defined as the proton matrix element of the flavor singlet axial current
(PI~~YS~IP)=: dlx+~x>
PPWP-B
(11)
22
H. I. Pirner
where 1c,= (u,d,s) denotes spinor y, can be expressed
the quark field and lp) is the wave function using Eq. 9:
of a proton
at rest with Pauli
(12) gi measures how much of the proton’s helicity is carried by constituent quarks. In the naive quark parton model a = 0 and the proton’s helicity is carried entirely by quarks (9: = 1). In the model of a pion cloud this value is reduced by the pions’ (or sea-quarks’) contribution and thus gi < 1. A rich source of insight into nucleon structure are structure functions for deep inelastic inclusive scattering on polarized nucleon targets. They can also be expressed using quark distribution functions: (13) For experimental
reasons,
instead
of the structure
functions
one rather considers
moments
of them:
(14) In the pion cloud model one obtains
for the proton and neutron
rp1 =
We will now analyse the available experimental IT’” to determine the value of a.
spin structure
functions
$l-2a)
data on the above mentioned
quantities
So, gA, gi and
Fig. 2 shows the various values for the model parameter a. We also plot as a thick vertical line the mean from the results of the three independent experimental determinations. The best known quantity is gA. It is this value which therefore essentially determines the weighted mean for a: a = 0.246 f 0.002
(17)
As one sees in Fig. 2 the various corrections are essential. By correcting the NMC result on So for shadowing effect and the El42 result for &‘-dependence the original values are shifted to within la of the weighted mean of a = 0.246. We take this as evidence that the employed toy model captures at least the essential features of the idea of a pion cloud of the constituent quark and that this idea is capable of explaining some properties of the nucleon. quark turns However, it might come as a surprise that the probability for a r + in an up-constituent out to be almost 25 %. A perturbative calculation in the linear a-model with pseudoscalar quark-pionparameter coupling (coupling constant grpq = 3.76) reproduces this value if one uses a regularization of A = 3.37 GeV. This seems to be very large for a hadronic cut-off. A possible explanation for both large values might be the non-relativistic character of the employed models. It should be appropriate to describe the nucleon as consisting of three relativistic quarks. Relativistic quark models with lightcone wave functions for the constituent quarks have already been used for a quantitative analysis of static and quasi-static properties of the nucleon (Yabu et al., 1993; Brodsky et al., 1994). While such calculations can predict static properties of the nucleon, like g,J, almost exactly without recurring to
Structure of the Constituent Quark
-
El42
: g; (moos ured El 22 (d-corrected) EMC
best fit of all spin
ds
tructure
23
SMC
: a
: g:
function experiments
_NMC 1 So (with shadow correction) -NMC
: S, (unc orrected)
Figure 2. The various results for a from different observables.
an effective description like a pion cloud! they nevertheless experiment as far as quasi-static properties are concerned.
The thick vertical line indicates
cannot give a satisfactory
a = 0.246.
agreement
with
We conclude that even in a relativistic description of what constitutes a nucleon the idea of a pion cloud of the constituent quark might be helpful. However, the above performed estimates of the pion probability a in the constituent quark, which are based on integral properties of the nucleon, are only of limited use. A direct and differential measurement of the pionic substructure of constituent quarks will offer more insight. To this purpose we turn in the next section to a differential description of the nucleon as seen in semi-inclusive electroproduction of pions on nucleon targets. SEMI-INCLUSIVE We consider electron
scattering
PION PRODUCTION on the proton with at least one pion a(k) in the final state: e+p+e’+7r+X.
(IS)
Semi-inclusive pion production can be looked at as a soft hadronization process. Data (Arnedo et al., 1985) at high energies agree rather well with LUND-simulations. However, in certain kinematic regions higher twist effects can visibly modify the leading gluon cascade. For example, one expects contributions (Brandenburg et al., 1995) from the hard fragmentation r*q + rq. We will describe the pion production in the naive parton model with the following assumptions. The nucleon consists of three valence constituent quarks, which are weakly bound. The square of their wave functions inte rated over transverse momentum P; up to the resolution Q2 gives the probability to find a quark with # avor q and light cone momentum fraction z in the proton:
(19)
24
H. J. Pirner
The constituents are massive Dirac particles (MQ = yKgp(cr)) with a pointlike coupling to the photon and a pseudoscalar coupling ig,, ($y&!~)? to the pion. The coupling constant grpp couples the quark equally to both fields of the (4, i) SU(2)~xSu(2) n multiplet (g,?). The coupling strength is determined from the empirical value of the constituent quark mass AJo = 350 MeV and the value of (u) = fir = 93.3 MeV: 9 “9,r = 3.76 . (20) ygally
one could have estimated
gnqq from quark additivity
. :
LhNN
g
In the parton model, with the distribution
= 3.57
- -
“”
-
3gA
pion production is calculated N,(s) of the quarks:
and the a-N-coupling
constant
gnNN =
*
(21)
as a folding of the partonic
cross section
eq -+ e’q’r
(2‘4 The kinematical y = u/E
variables
of this semi-inclusive
and on the pion side Ic” = (ET, i).
In Fig. 3 we give a definition inclusive
reaction
process
are on the electron
side xs = &‘/~MNv
It is useful to discuss the kinematics
of all momenta
in the target-at-rest
there are two planes of interest,
system,
the lepton scattering
P’“_=JMN, 0’). In the semi-
plane (f!, C) and the photon-pion
plane (<,i). They include an angle 4% between them. The energy of the incoming electron energy of the outgoing electron is E’. The photon energy is Y, the pion energy is E,. momentum
z is divided into a momentum
and transverse
to the photon
momentum
parallel
to the direction
and
in more detail.
of the photon
momentum
is E, the The pion $
9’ (24)
Unpolarized cross sections do not depend two invariants are sufficient to characterize
on the azimuthal angle of the scattered the virtual photon. These are
Q2 = -q2 v
=
4 EE’ sin2 @,I2
=
PC&P-
electron,
therefore
(25)
MN
or
xs=-= y=P.e
Q2
-q2
2P.
P.q -=--.
q
2Mf,r u v E
(26)
Structure of the Constituent
25
Quark pion scattering
Figure 3. The reaction
plane
ep + e’?rX in the target-rest-system.
The momentum of the pio_n has three components. We separate the azimuthal angle & from the two residual variables $1 and Ikll. Th e p arallel momentum /cl, can be replaced by the energy variable E, or the scaling variable
P.k
E,
(27)
z=G=u’ A
conventional set of variables thus (v/4 ICI,)dz d$: d& the five-fold differential da
=
dx, dy dz di$ d&
consists of (Q2, xer z, $11, &). cross section has the form
d3k/2ET
=
4m2MNE _ v
$
In this paper we will concentrate
With
Q4
{
GY2'H1+(1-y)'H2t
(2 - y) J1-y
cos $,?I3
(1 - y) co5 29%T-lH4. 1
on & and ci integrated
+
(28)
cross sections.
The five-fold differential cross section can be evaluated with the four different structure functions ‘H;. For a first comparison with future experimental data it will be necessary - for reasons of statistics - to
H. J. Pirner
26 reduce this five-dimensional
space to the 3 dimensions
xer Q’, and z by integrating
out &, and ii::
&=x H,(x=,Q’,z)
:=
; Td,
/
0
0
e
d!$ ‘?&(xB, Q2, z, if)
IL
P-9 The cross section
is then given by da
~
=
dx, dy dz da
dx,dQ2dz Of main functions
=
interest is the contribution we obtain:
&4
{X,Y2fJ1f(l-Y)Hz}
=
or
4ircXz -{xaYZ%+(l-y)Hz}. xa Q” to the cross section
da dx,dQ2dz
8affZMp4E
(39)
from longitudinal
and transverse
structure
+g{2(l-y)HL+(1+(1-Y)2)Hr}~
The two contributions from HL and HT are is dominated by the t-channel, whereas the of the longitudinal part of the cross section allow to isolate the presence and strength 1995) of hadronization effects of the struck in the longitudinal structure function.
(31)
plotted in Fig. 4. Clearly the contribution from HL at z + 1 s-channel contributes strongly to HT. Thus a determination and correspondingly the longitudinal structure function will of a pion cloud. A recent calculation (Brandenburg et al., quark also showed evidence for a peak at large values of .Z
It should be stressed, that we require a large invariant mass of the hadronic system excluding the pion, such that the quasi-elastic channel y’p -+ ~+n does not contribute. As an example, for xg = 0.15 at E, = 27GeV this hadronic mass is only 1 GeV (2.4 GeV) f or z -+ 1 (z --f 0), with the hadronic mass increasing with decreasing xg. Thus a measurement at larger center-of-mass energy is desirable. GLUON
CLOUD
OF THE
CONSTITUENT
QUARK
The chiral properties of the constituent quark determine a large part of low energy hadronic physics. This can also be seen e.g. from the contributions of L.Y. Glozmann, K. Goeke and D. Dyakonov in this meeting. In addition, QCD-sum rules show that the light hadron masses mainly depend on the value of the quark condensate ($$). I n my opinion, it is expected that a neglect of the long-range confining forces necessitates a too large constituent quark mass, which leads to a discrepancy e.g. of the magnetic moments with experiment. I therefore advocate to add to the effective Lagran ian of the linear a-model a part containing the long-range gluon fields. I prefer a parametrization o P the longrange gluon physics after the spontaneous breaking of chiral symmetry, since an inclusion of long-range confining gluon forces together with zero-range forces, e.g. in the Nambu-Jona-Lasinio model, does not seem easy and is probably not correct. Besides the lattice color-dielectric model I think the continuum It model of the stochastic vacuum (Dosch et al. 1988) is closest to lattice QCD at large distances. Note, because of includes nonvanishing gluon field strenghts &,l( x ) correlated over short distances. the Aharonov-Bohm
effect
that
the associated
gluon fields At)(r)
arising
e.g.
from a localized
color
Structure of the Constituent
da / dx, d@ dz
0.2
[ pb/GeV’
(H, only)
0.2
Quark
E
da / dx, dd
27
dz
(H, only)
I
0.175
r:; ,--.....p-channel
0.075
: ,i’
‘...
‘..,
‘\.
‘..,
0.05
!
‘..,
t
‘.._ Y,
0.025
_,_ ;‘,,,_ --
Oo
_..I
.. -\
j
0.2
--
-.. _. ---fJ_. 0.4
,f”
,’
x,,
‘\
‘... __, _.:.
II.6
_ ._
0.8
Figure 4. Contribution from HL and HT to the cross section for semi-inclusive values of Q2 and ~a are fixed at 1 GeV’ and 0.10, respectively.
magnetic field strength & = F,$ 1 ) s p rea d over large distances the Wilson area law for heavy quarks.
pion production.
and satisfy the confinement
The
criteria
of
(32) The string tension
u = 1 GeV/fm
is obtained
as in the stochastic
0 w o.o2(g~F~“F~Ja~
vacuum model (33)
with a = 0.35 fm as correlation length and the gluon condensate (gzFfiyFfiy) = 2.4 GeV4 from QCDsum rules. In soft high-energy scattering (Dosch et al., 1992), the hadrons interact via the correlations induced upon the respective Wilson loops formed by the paths of the quarks and antiquarks (or diquarks) inside the hadrons. Naively, we can consider the vacuum as made up from domains (or bubbles) with constant gluon field strength. Then two quarks can only interact if they hit the same bubble. In the rest system of the hadron, the target quark can be pictured to be surrounded by a bubble of constant gluon field, with which the scattering quark interacts. Consequently, we expect the size of the gluon cloud of the constituent quark to be identical with the correlation length of the gluon fluctuations in the stochastic vacuum. Indeed the differential cross section for TN + pN scattering has an exponential slope whose size is mostly given by the correlation size a and not by the size of the hadronic wave functions of the p-meson and the nucleon. Diffractive constituent
electroproduction of p-mesons is a very good instrument to investigate the gluonic size of the quark since the transverse size of the incoming qq state can be varied with the virtuality of
28
H. J. Pirner
the incoming photon Q2. Until now we have investigated the low Q2 region, where the cross section can be modelled in a Generalized Vector-Meson Dominance Model with three p-resonances, the ~(770 MeV), p’( 1450 MeV) and p”( 1770 MeV). The first excited state is mostly a radial excitation (2s,3 Sr), whereas the second one corresponds to a 3D1 state. In principle the coupling of these vector mesons to the photon is calculable in the constituent quark model. The nonrelativistic wave function of the p-meson unfortunately reproduces very badly the width of the 3Si state Ip + e+e-. Relativistic corrections are the only reason for a nonvanishing coupling of the 3Di-state, since its nonrelativistic wavefunction is zero at the origin. So we have used a semiempirical way to include the Ip, -+ e+e- widths and the signs The yN production amplitude for the vector-meson pi by transverse of the frpj coupling constants. photons is then given as:
The stochastic vacuum model allows us to @-scattering. In general, the off-diagonal with the size of the integrated p-production B of the differential cross section:
calculate diagonal and off-diagonal T-matrix elements for matrix elements are important. Preliminary results agree cross section in the low Q2-region and the slope parameter
, da = ao~e-Bl*l _ dt
with Bezp N 8 GeV-‘.
(36)
The size of the gluon cloud of the constituent quark contributes about the same amount as the size of the wave functions of the hadrons to the theoretical slope parameter Btheor E 1.6~~ + O.lS((pilr’lp,) + (NIr’IN)) CONCLUSION Employing a model where the physical quark fluctuates with some probability to a quark plus a pion we have calculated the differential cross section for semi-inclusive pion production in electron proton reactions. As expected, the biggest contribution from the pion cloud to the cross section is at small to the values of I~ N 0.1 and large values of z = ET/v + 1. In particular, the sole contribution longitudinal structure function will allow an experimental determination of the magnitude of the pion cloud of the quark. The gluon cloud of the constituent quark can be investigated in diffractive vector meson production. REFERENCES Arnedo, M. et al. (NMC) (1985). Phys. Lett. 150, 458. Baumgartner, S., H.J. Pirner, K. Kijnigsmann and B. Povh (1995). How to determine the pion cloud of the constituent quark. Preprint hep-ph/9507309, to be published in Z. Phys. A. Brandenburg, A., V.V. Khoze and D. Miiller (1995). Phys. Lett. B347, 413-418. Brodsky, S.J. and F. Schlumpf (1994). Phys. Lett. B329, 111. Cheung, C.Y., C.F. Wai and H.L. Yu (1992). Phys. Lett. B279, 373. Dosch, H.G. and Y.A. Simonov (1988). Phys. Lett. B205, 339. Dosch, H.G., E. Ferrera and A. Kramer (1994). Phys. Rev. D50, 1992. Dosch, H.G., E. Gubankova, G. Kulzinger and H.J. Pirner. Diffractive vectormeson production in the model of the stochastic vacuum. Work in progress. Eichten, E.J., I. Hinchliffe and C. Quigg (1992). Phys. Rev. D45, 2269. Gell-Mann, M. and M. Levy (1960). Nuovo Cimento 16, 705. Giittner, F., G. Chanfray, H. J. Pirner and B. Povh (1984). Nucl. Phys. A429, 389. Yabu, H., M. Takizawa and W. Weise (1993). 2. Phys. A345, 193.