The spins inside the constituent quarks

Nuclear Physics B (Proc. Suppl.) 23B (1991) 91-97 North-Holland

91

THE SPINS INSIDE THE CONSTITUENTQUARKSx Harald FRITZSCH Sektion Physik der U n i v e r s i t ~ t MUnchen and Max-Planck-lnstitut f u r Physik und Astrophysik -Werner Heisenberg I n s t i t u t f u r Physik MUnchen, Germany Constituent quarks are surrounded by a cloud of polarized (qq) pairs, due to the QCD anomaly. The "Zweig rule" is strongly violated. A large part, i f not a l l of the spin of the constituent quarks and of the nucleon is carried by the angular momentum of the cloud. Constituent quarks have a nont r i v i a l internal f l a v o r structure.

The

dynamical QCD

details of the

mechanism

in

understood,

especially the fact

are

still

not that

confinement completely to

"light" Goldstone bosons like the pions as (~[Q) bound states in the pseudoscalar channel.

a good

Usually it is assumed that the mechanism, which

approximation the baryons behave as loosely bound

leads to the

systems of three constituent quarks carrying a mass of

symmetry in QCD is solely the source of the effective

about one third of tile nucleon mass. In many applications (e.g. for the calculation of the magnetic

masses of the constituent quarks, and it does not alter

moments)

dynamical

breaking of the

chiral

their spin structure, nor their flavor structure.

the constituent quarks are treated as

In this note we shall demonstrate that this is not

structureless Fermi-Dirac particles. Here I would like

the case. Due to the QCD anomaly, which strongly

to point out that this procedure may be justified in certain cases (e.g. by calculating matrix elements of

influences the pattern of chiral symmetry breaking,

SU(3) octet operators), but fails in other cases, due to

structure. They are composed of the corresponding

the QCD anomaly.

"valence" quark, surrounded by a cloud of polarized (q-q)-pairs, which in addition modifies the internal

The hadrons can be viewed as bound states of the type (QQQ)

or (~Q),

where Q denotes a

"constituent quark". In the simplest models these objects carry the electric charge 2/3 or - 1 / 3 (in units of e), an effective mass, which for the "light" u - and

the constituent quarks acquire a particular internal

spin structure substantially. Before we start our discussion about the "constituent quarks", let us remind the reader that

d--quarks is of the order of 300 MeV, and the spin 1/2

the simple SU(6) type quark model describes rather well the static properties of the baryons like the

(see e.g. ref. (1)). In QCD the effective masses of the

magnetic moments or their SU(3) coupling properties,

constituent quarks of the u, d type are due to the

e.g.

mechanism of spontaneous breaking of the chiral

constants. The well-known prediction

symmetry in QCD, i.e. an effect of order A[QCD]. Note that the "Lagrangian" masses mu, m d for these flavors are estimated to be less than 10 MeV, i.e. negligible. Thus the constituent masses are due to the same mechanism which causes the appearance of

the

D/P

ratio for the axialvector coupling

IGA/GvI =

5/3 is violated in nature, however the departure of the observed value

[GA/@V[=

1.26 + 0.04 from the

SU(6) value can be understood as an effect due to orbital motion 2) and / or a gluonic effect 3). Thus there is no contradiction with the constituent quark

0920-5632/91/$03.50 @ 1991 - Elsevier Science Publishers B.V. All rights reserved.

92

H. Fritzsch / The spins inside the constituent quarks

scheme. On the other hand the axialvector coupling constants can be related, via the PCAC relations 4)," to the coupling constants for the corresponding pseudoscalar mesons and their decay constants. The PCAC relations for the octet of pseudoscalar mesons can be derived only in the limit of the chiral symmetry SU(3)L x SU(3)I3. The ninth one, i.e. the r/'-meson, acquires a mass of order A[QCD], due to the U(1) anomaly. The associated chiral symmetry of the axialsinglet charge is broken due to the QCD interaction. Thus the axialsinglet charge cannot be related to a pole in the formfactor of the corresponding current; its value remains undetermined by the chiral dynamics (see also ref. (5,6,7,8,9)). In the limit of chiral SU(3)L x SU(3)R the axial

aryon matrix elements of the axialvector currents. We shall see that this leads to interesting conclusions about the internal flavor and spin structure of the baryons and of the constituent quarks. Let us consider a simplified case, namely the one of QCD with the two flavors u and d only. The strange quarks and the "heavy" flavors c, b and t are disregarded. Furthermore we assume m u = m d = 0, i.e. the chiral symmetry SU(2)L x SU(2)R is exactly fulfilled. The pions are massless. Due to the QCD anomaly the singlet pseudoscalar r/(quark composition (uu + ~td) / ~-) has a mass of the order of the nucleon mass M. The Goldberger - Treiman relation is exactly valid: 2Mng A = 2 F

singlet charge is given by the nucleon matrix element of the anomalous divergence of the singtet current: 5'6)


~1 n>,

axialvector coupling constant gA is given by the

while the octet charges are determined by the residua of the poles of the associated Goldstone bosons. On the other hand within the SU(6) type constituent quark model the axial singlet charge can easily be determined by noting that strange quarks are absent in the wavefunction of the nucleon, and one has therefore: 10)

=
(2)

In the SU(6) type constituent quark model the

Ot S

gTrNN"

(1)

~'rff-r~sIn>

(In>: nucleonstate). This relation can be derived on the basis of the "Zweig rule". In an approach based on the chira] symmetry there is no reason why eq. (1) should hold. Thus the chiral dynamics contradicts the "Zweig rule" and the SU(6) type quark model. This is well known for the spectrum of the pseudoscalar mesons (for an early discussion see ref. (11)). Here we should like to point out that a similar contradiction exists for the

nucleon expectation value of the quark spin operator 1

2%:

gA = <°z(U)> - < ° z ( d ) > = 5/3

(3)

where one has:

1/2 az(U) = 2/3

1/2 #z(d) = - 1/6,

(4)

1/2 (¢z(U) + %(d)) = 1/2 (= nucleon spin).

In reality gA is not equal to 5/3, but about 1.27, i.e. the prediction of the "constituent model" is violated by about 24 %. This violation can be understood, as mentioned above, without giving up the simple ideas of the constituent quark model. Thus in the isovector channel both the chiral dynamics and the constituent quark model do no__Atcontradict, but rather supplement each other. This observation encourages us to consider the "constituent quarks" as

H. Fritzsch / The spins inside the constituent quarks

93

separate entities. In a "Gedankenexperiment" we

success of the "Zweig" rule relies on the assumption

could consider a polarized "constituent quark" Q (Q = U,D) and study its coupling constants. Such a

that (qq)-pairs contribute very little to the hadronic wave functions. Correspondingly we could consider a

"constituent quark" would be surrounded by a cloud

limit in which the (qq)-pairs are neglected ("valence

of "Ooldstone" pions, just like an ordinary hadron. It

quark dominance"). In this limit we find for a

would also obey a Ooldberger-Treiman relation:

U-quark:

:2 Mq ~A = 2 FTr g~.QQ

(5)

u+=g_

(r)

=u_ =0

d+ =d_=~[+ = 0 . (Mq: constituent quark mass, gA axialvector coupling constant of the constituent quark, g~rQQ: pion -- quark

Only the density function u+is different from zero.

coupling constant).

This is easily understood if we consider the free quark

Suppose we consider the corresponding matrix

model, in which the "constituent quarks" and the

elements of the vector and axialvector currents and

"current quarks" are identical and we have not only

relate them to the various moments of the quark

the relations (7), but in addition the function u+ is

density functions. One finds naively:

known: u+ = ~x-1).

15.~#uIu>


= P#/Mu =

(6)

Thus

the

essential

difference

between

a

"constituent quark" inside a hadron and a free quark "~[U o

+ u

- 5"+ - ~ _ )

lies in the shape of the density function u+. The

dx

confinement forces merely cause this function to depart from a g-function and to spread out over the

= o

available x-range.



It turns out that the picture of a constituent quark described above is noA consistent with the

I

=s

t

•

Au=s t

• !

(u++

5+

-u

_ -u _ )dx = s

#

• 1

constraints given by the chiral symmetry. In the constituent model we have CFi3 = 0. This implies for

= 0

a U-quark that both the isoscalar and the isovector the quark

combinations of the spin density moments are equal to

density functions refer to the U-quark and should

one: Au - Ad = Au + k d = 1. The isovector part is

(s#: spin vector, p#: four-momentum,

carry an index u, which is not explicitly denoted here.) These relations reflect the expectation that in a constituent U-quark the quark density functions must be arranged such that the correct flavor structure is obtained and that its total spin is carried by the u-flavor. The d-flavor is not supposed to contribute to the spin. We could go further and be more specific about the structure of the quark density functions. The

determined by the pion pole. If the isosinglet r/meson would also be a Goldstone particle, the associated coupling constants would conspire such that

the

isovector and isoscalar spin density moments would be equal, and the results of the "naive" constituent model would be obtained. However due to the QCD anomaly the isosinglet spin density function does not receive a @oldstone pole contribution. Instead it is given by the constituent quark matrix element of the anomalous

94

H. Fritzsch / The spins inside the constituent quarks

particular by the Gotdberger - Treiman relations for

divergence:

the axialvector matrix elements. Their appearance is a Zxu+Ad=A

(8)

nonperturbative phenomenon just like the generation of the r/mass due to the gluonic anomaly.

O~

G~l u>=2M

Au =(1+ A)/2

In ref. (5) the author has argued that the

uA~i'vSu

nucleon matrix element of the anomalous divergence should be very small. The argument is based on the observation that the "constituent quark model"

2xd=(A-1)[2

requires the quarks in a nucleon to be in an s-wave. In There is no dynamical reason why A should be

particular

the

gluonic

components

of the

wave

equal to one. If it were, the spin density moments

function should also be dominantly in an s-wave. This

would indeed reproduce the constituent quark model

implies that a pseudoscalar density like a

result. In particular Ad would vanish. This is not ruled out a prori, but if it were, it would be a miraculous coincidence. For all other values of A Ad does not vanish. We conclude that for A tt 1 the constituent U quark contains (~q)-palrs. Thus a violation of the "Zweig rule" is automatically implied. It is interesting that these pairs axe generated by the same nonperturbative mechanism due to the gtuon anomaly which causes the r/meson to acquire a mass and not to act as a Goldstone particle in the chiral limit of SU(2)L x

SU(2)R.

s

•G

#u

~#u

is not expected to have a sizable matrix element. This is in accordance with the experimental results, which give: 12) A 2 = Au + A d + A s = 0 . 0 1 + 0 . 2 9

(9)

since in the chiral limit of SU(3) x SU(3) the quantity AE ("naively the portion of the nucleon spin due to quarks") is given by the m.e. of the anomalous divergence. It is easy to apply similar considerations to the constituent quarks. Also for them we should have

the "Zweig rule" discussed above as follows. The chiral

A = 0, at least to a. good approximation. Thus we find for a constituent U quark that the "Zweig rule" for the

dynamics of a "constituent quark" would obey the

density moments is maximally violated:

Intuitively one can understand the violation of

"Zweig rule" if it were surrounded by a cloud of ~rand 7/ Ooldstone bosons. The @oldstone poles of the

a u = 1/2

&d = - 1[2.

(10)

axialvector current matrix elements would imply, via the Goldberger - T r e i m a n relations in the isovector

We can go further and specify the various density

and isoscalar channel, that the matrix elements obey

moments. If the "Zweig rule" were valid (both r and 7/

the

Ooldstone modes present), we would have

constraints

given by the

"Zweig rule"

(in

particular Ad = 0 for a u--quark etc.). However the QCD anomaly causes the r/pole at q2 = 0 to

f u+dx = 1,

(11)

disappear. As a result the "@oldstone cloud" of an U-quaxk

consists

dynamical structure

only

of

r-mesons.

Thus

of the constituent quark

the

u_=~+=K_

=0

d+=d_

=8+=8_=0

is

drastically changed. In particular (T~u) and (~ld) pairs are generated. We note that these pairs cannot simply be regarded as the pairs inside virtual ~-mesons. Their presence is caused by the chiral dynamics, in

Such a constraint which is not invariant under the renormalization group can only be imposed for a particular value of the energy scale #, which is

95

H. Fritzsch / The spins inside the constituent quarks

expected to be the characteristic hadronic energy

the U-quark, is unaffected by the QCD anomaly. The

scale. The removal of the ~/ Goldstone pole causes a

latter causes a large violation of the "Zweig rule" in

shift

the sense that (qq)-pairs are generated. We shall refer

in

the

density

moments,

which

we

can

parametrize by two functions h+ and h : u + = u + +V h +

7+ = d + = a + = h +

to this "cloud" of (~q)-pairs as the "anomaly cloud". The density functions C73 and C8-3 are different from (12)

zero, i.e. the pairs are polarized oppositely to the original constituent quark. The sum of all (anti) quark

=g =h

u =Cj = d

spins is zero. Thus for A = 0 the quarks do not

.

contribute to the spin of the constituent quark. The (u;:

intrinsic density function of U-quark in the

absence of the anomaly, J'u+Vdx = 1). We find:

latter is provided by the orbital angular momentum of the pairs. This can be seen as follows. If we would turn off the QCD anomaly (e.g. formally by setting n c = m), the "naive" picture should hold, i.e. the spin

1

AG= Au+Ad=l+4j(h+

-h_)dx = 0

(13)

O

Au - Ad = 1.

of the U--quark is carried by the valence quark u . v Once the anomaly is introduced, the u valence quark continues to contribute its spin. But the (~q) pairs cancel the latter. But their total angular momentum

It follows:

must be zero. Otherwise the introduction of the anomaly would violate the conservation of angular

1

[ (h+-h)

d+ = - 1/4.

(14)

momentum. Thus we have:

0

We observe that AZ vanishes because the constituent U quark contribution to A2 is cancelled by the pairs. A cancellation is only possible, if the density function h

Jz(U)= + 1/2 = Jz(Uv) + Jz(Cloud) + Lz(Cloud ) (16) = + 1/2 + ( - 1 / 2 )

+ (+ 1/2)

is different from zero. On the other hand

h+ can be zero, in accordance with the sum rule (14).

In the case A :~ 0 the cancellation between the spin of the valence quark and the spins of the

The simplest model obeying the constraints discussed

"anomaly cloud" would not be complete, but the sum

above is one in which we have

of the spins and of the orbited angular momenta of the pairs in the "anomaly cloud" would still be zero.

h+ = 0, [ h_dx = 1/4,

(15)

V

u+ = u+, j u_ dx = f C dx [

d_dx = I 3"dx =1/4

Thus far we have disregarded the polarization effects due to gluons. We find it unlikely, but not impossible that polarized gluons contribute also to the total angular momentum of the anomaly cloud, and the gluonic term would appear also in eq. (16).

d+ = ;3+ = u+ = 0 We obtain in the case A = 0 the following picture of a

Finally we consider the case of the three light flavors u,d,s. In the chiral limit of SU(3)L x SU(3)F/.

polarized constituent U quark in the SU(2)L x SU(2)R

we obtain for a constituent U quark in analogy to

limit: The density function u+, which describes the

eq. (15):

density of u-quarks polarized in the same direction as

96

H. Fritzsch / The spins inside the constituent quarks

u+(x) = uV(x) u =~

=d

g+ = d+ = a + = 0 =~t

=s

=s

(17)

=h

their contribution to the singlet charge could be calculated perturbatively(3'4).In our approach we see no reason for a large gluonic polarization. Thus the

1

effect discussed in ref. (14, 15) would be negligible in

J h_ dx = 1/6

comparison

O

Au = 2/3

kd =-

1/3

As = - 1/3.

to

the

nonperturbative

phenomenon

discussed here. The

In the symmetry limit the "anomaly cloud" is, of course, SU(3) symmetric. In reality symmetry

smallness

of

the

axialsinglet

charge,

parametrized above by the parameter A, follows also within the Skyrme type model, discussed in ref. (16).

breaking will be present. The result will be that the

However the connection of this model to the scheme

effects of the (ss) pairs are somewhat

discussed here remains unclear.

reduced

compared to those of the (~'u) and (a'd) pairs. For

The picture of "constituent quarks", carrying a

example, in a U---quark we expect: Ad > As. The

polarized "anomaly cloud", described here implies that

actual spin density momenta of the U,D constituent

many aspects of hadronic physics, especially *hose in

quarks will lie between the extreme case of SU(2) x

which polarization and spin aspects are relevant, must

SU(2) (Ad = - 1/2 for a U-quark) and of SU(.3) x SU(3) (Ad = - 1/3 for a U-quark). However we note

moments of the baryons, the polarization phenomena

that for A = 0 the limit of 8U(2)L x SU(2)lq, is not ruled out experimentally. It would imply for a proton:

be reconsidered. Among them

are the magnetic

of A hyperons in hadronic processes and the spin asymmetries observed in strong interaction processes. Many further tests of the ideas presented here can be

ku + kd = 0

(18)

envisaged, once spin asymmetries can be measured in electroweak lepton-hadron reactions at high energies.

Au

-

Ad

=

Au = 0.63

[gA/gVI

=

1.26 l:{ecently l~]lwanger and Stech 8) have proposed

Ad = - 0 . 6 3

an approach to the "constituent quark problem" based on

1

/ g, dx = 1/2 (4/9 Au + 1/9 ad) = 0.10,~,

effective

Lagrangian

methods.

Although

our

approach and the one of ref. (8) differ substantially,

O

the consequences seem to be similar, if one identifies the constituent quark fields used in ref. (8) with the

while the experimental value for this integral is 12)

constituent

quarks

discussed

here.

It

would be

interesting to explore this analogy, further.

1

Jgl dx = 0.114 i 0.038

(19)

O

In In this limiting case the SU(3) symmetry for the

axialvector currents would be very bad]y broken. It is known that the SU(3) breaking in the axialvector channel is sizeable 13), but certainly not as large as to

this

nonperturbative

paper

we

have

mechanism

which

described leads

to

a the

appearance of polarized qq-pairs. For example, in the SU(4)L x SU(4)R--]imit we would obtain (see eq. 17)) Ad = As = Ac= - 1/4 for a U-quark, i.e. the

allow the case As = 0. Details of the symmetry

U-quark would be surrounded by a cloud of polarized

breaking cannot be presented here. Recently it was argued that the anomaly could

7c-pairs, as well as of ad and ss-pairs. Once the

contribute to the axial singtet charge if gluons are

value of about 1,4 GeV as observed in nature, Ac will

highly polarized in a polarized nucleon. In this case

symmetry breaking is turned on and m c takes its

H. Fritzsch / The spins inside the constituent quarks

be reduced. A rough estimate of this nonperturbative Y_c--cloud in the proton could be ubtained by taking As z 0.20 as observed 12) and use the scaling relation

97

7.

A. Efremov, J. Softer and N. Tgrnqvist, Phys. IKev. Lett. 64 (1990) 1495

8.

U. Ellwanger and B. Stech, Phys. Lett. B 241 (1990) 409

9.

H. Fritzsch, preprint CE1KN-TH 5676/90 (March 1990)

10.

J. Ellis and Ft. Jaffe, Phys. Ftev. D 9 (1974) 1444

11.

H. Pritzsch and P. Minkowski, Nnovo Cimento 30 A, 393, 1975

12.

G. Baum et ai., Phys. Rev. Lett. 51 (1983) 1135 I. kshman et a.l., Phys. Lett. B 206 (1988) 364 V.W. Hughes et at., Phys. Left. B 212 (1988) 511

13.

J.P. Donoghue, B.R. Holstein and S.W. IKlimt, Phys. 1Key. D 35 (1986) 934

At/As = (ms/mc)~, which gives Ac z . - 0.4 %. This nonperturbative charm content of the nucleon can be identified with the "intrinsic charm" of the nucleon (for a recent discussion see ref. (17)). The order of magnitude estimated above is in agreem~.nt with other estimates of the "intrinsic charm". We emphasize that in our approach the "intrinsic charm" would be polarized. 'rhe generation of a cloud of (qq)-pairs by the QCD anomaly reminds us of the "Cooper pairs" in the BCS-theory of superconductivity. Indeed there are some analogies between superconductivity and hadronic physics in the chiral limit, e.g. the appearance of the mass gap, which in QCD is related to the anomaly and to the dynamical breaking of scale invariance and the chiral symmetry, and the presence of pairing forces, which in QCD are responsible for the removal of the Goldstone pole in the sing!et axialvector channel.

Acknowledgement: I would like to thank Prof. Narison for arranging this meeting in this wonderful city in Southern Prance.

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1.

P. (;lose, An introduction to Quarks and Patrons, Academic Press, Lond,,n !979

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L. Sehgal, Phys. D 10 (1974) 1663

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t1. Pritzsch, Mod. Phys. Lett. A 5 (t990) 625

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M. Ooldberger and S.B. Treiman, Phys. R~ev. 110 (1958) 1178

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H. Pritzsch, Phys. Lett. 2o9 B (1989) 122

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G. Veneziano, Mod. Phys. Left. A 17 (1989) !605

M. Ftoos, preprint HU-TFT-90-27 (1990) 14.

G. Altarelli and @. Ross, Phys. Lett. B 212 (1988) 391

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Ft. Carlitz, J. Collins and A. Mueller, Phys. Left. B 214 (1988) 229

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