ESTIMATION OF THE PROPERTIES OF QUARKS

1. VECTOR QUARK-QUARK INTERACTION

By G. B. SMITH*t and L. J. TASSIE*

[Manuscript received June 11, 1970]
Abatract
From the sum rules for hadron scattering, the nucleon form factors, and the
masses of the hadrons, the properties of quarks are estimated as: 2 Ge V Jc 2 ;:; quark
mass ;:; 30 Ge V Jc 2 , 0·1 fm ;:; range of the quark-quark interaction ;:; 0·25 fm.
The hadrons are described by a relativistic independent quark model using the
Dirac equation with a potential energy term for the effective interaction. This
model is compared with a rigid rotor model.

1. INTRODUCTION
This paper reports an attempt to estimate the properties of quarks, as indicated
by the successes of the quark model. The quark model (Kokkedee 1969), in which a
baryon consists of three quarks and a meson consists of a quark and anti quark,
cannot give a complete description of hadrons. For instance, a nucleon will contain
additional quark-antiquark pairs, some of which form mesons; because the binding
energy of a meson in the nucleon is very much smaller than that of the other con-
stituents of the nucleon, the density of mesons will extend further out of the nucleon.
The quark model can only be expected to apply to some core of a hadron, the
properties concerning the outer parts of the hadron being more appropriately de-
scribed by other means, such as a virtual meson cloud around a nucleon, or by more
complicated models including the additional quark-antiquark pairs.
In this paper we estimate the quark mass, and the root-mean-square separation
distance of the quarks in the core. By considering the scattering sum rules for the
total hadronic cross sections and the nucleon form factors we estimate the root-mean-
square separation distance of the quarks in the hadronic core.
We use the one-particle Dirac equation as the basis of the dynamical description
of the hadrons. There are two alternatives for introducing the quark-quark inter-
action. We may write the interaction as a potential energy (i.e. the fourth component
of a 4-vector), as would be expected if the quark-quark interaction is a vector inter-
action, or alternatively we may write the interaction as a scalar term, as would be
expected if the quark-quark interaction is a scalar interaction. In the present paper
we choose the former while in the following paper (Smith 1970; present issue, pp.
627-32) the latter interaction is used. .
Using this dynamical description, and the estimate of the root-mean-square
separation distance of the quarks in the cor~, the quark mass required to produce the
observed hadron spectra is estimated.
* Rese\lorch School of Physical Sciences, Australian National University, P.O. Box 4,
Canberra, A.C.T. 2600.
t Present address.: Canberra College of Advanced Education, P.O. Box 381, Canberra City,
A.C.T. 2601.

Aust. J. Phya., 1970. 23, 615-25

616 G. B. SMITH AND L. J. TASSIE

II. RANGE OF THE QUARK-QUARK INTERAOTION

Sum rules relating meson-baryon and baryon-baryon total cross sections

obtained in the quark model are based on the simple postulates (Levin and Frankfurt
1965; Lipkin and Scheck 1966) that the quark forward scattering amplitudes are
additive and isospin invariant. These assumptions led to the following sum rules for
total cross sections (Lipkin and Scheck 1966; James and Watson 1967)

a(pp)+a(pn) = 2a(1T-p) + a(1T+p) , (2)

a(pn)+a(pp) = a(1T-p)+2a(1T+p) . (3)

Since the quarks in the core of the hadron are very strongly bound, we expect

where rc is the radius of the core and b is the range of the quark-quark interaction.
We assume that the range for the quark-quark interaction is the same as for the
quark-antiquark interaction and that the size of the core is the same for mesons and
baryons, for the sake of simplicity and for the lack of any experimental evidence as
yet against these assumptions. The quark model should apply to hadron-hadron
scattering when the distance between the two hadrons is less than 2b. Since the
investigation of this region requires a wavelength ,\ smaller than or comparable with
the diameter of the region,
,\ ;s 4b, (4)

we expect the sum rules (1), (2), and (3) to apply only at energies high enough to
satifyequation (4) for the wavelength of the projectile in the centre-of-mass system.
Thus it is possible to estimate b from the energy at which the sum rules (1), (2), and
(3) fail. At wavelengths larger than 4b we expect the hadron scattering cross section
to be determined by the properties of the outer regions of the hadron, as calculated
using such models as the peripheral model (Jackson 1965):
Assuming that the internal velocity of the quarks inside each hadron is small
compared with the relative velocity of the two hadrons, identical kinematical condi-
tions are provided if the different cross sections are compared at the same centre-of-
mass energy for the quark-quark or quark-antiquark system. However, in relating
the total centre-of-mass energy of the hadrons to the total centre-of-mass energy of
the constituent quarks the mass of the quark is an independent variable and this
mass may be interpreted as either the bare quark mass (James and Watson 1967)
or an effective quark mass (Kokkedee and Van Hove 1966). In the former case the
sum rules are graphed as functions of the relative velocity of the two hadrons; this
is easily accomplished since the laboratory energy E lab of the projectile divided by its
rest mass m is dependent only on the projectile's velocity in the laboratory frame;
or in the latter case the sum rules are graphed as functions of the laboratory momentum
Plab of the projectile since, in the spirit of additivity, the hadronic momentum is
assumed to be the sum of the momenta of the constituent quarks. For sum rules
 PROPERTIES OF QUARKS. I 617

involving both meson-baryon and baryon-baryon total cross sections we must

compare them at laboratory momentum in the ratio 2 : 3 (Kokkedee and Van Hove
1966).
In Figure 1 the Johnson-Treiman (1965) relations

Ha{K+p)-a{K-p)} = a{rr+p)-a{rr-p) = a{K+n)-a{K-n) (5)

are shown as functions of (a) Plab and (b) E lab /mC 2 . The data used in this paper are
taken from Lindenbaum et al. (1961), Diddens et al. (1963), Citron et al. (1964),
Galbraith et al. (1965), Foley et al. (1967), and Allaby et al. (1969). Figure 1 shows

-4

-3

1- ~~:::~c"
2
Fig. l.-Jolmson-Treiman
KP relations (equations (5)) plotted
:g (a) fIP against (a) P1ab and (b) E 1ab/mc 2 :
.~ -4 10 15 20
KP = Ha{K+p)-a{K-p)}
P1ab (GeV/cl
U
~ ,--\ lIP = a{ 7T+p) - a{ 7T-p)
-3 KN = a{K+n)-a{K-n)
fIP
KN
-2
(b) KP
10 20 30 40
E 1ab /mc 2

that this comparison does not resolve the question whether cross sections should be
compared at the same Plab or the same E lab /mC 2 . Other sum rules involving only
meson-baryon total cross sections, namely (Lipkin and Scheck 1966) the symmetric
sum rule

and the antisymmetric sum rule

a{K+p)-a{K-p)-{a{K+n)-a{K-n)} = a{rr+p)-a{rr-p) ,

have been examined but they do not resolve the question either. In Figure 2 (see
also James and Watson 1967) sum rule (2) is plotted for the same abscissae as in
Figure 1 except that the meson-baryon and baryon-baryon total cross sections are
compared at Plab in the ratio of 2 : 3. Figure 2 indicates that E lab /mC 2 should be used
to compare sum rules, and this conclusion is supported by the results of sum rule
(3) also. This comparison implies that the bare quark mass should be used as the mass
variable in relating the total centre-of-mass energy of the hadrons to the total centre-
of-mass energy of the constituent quarks.
618 G. B. SMITH AND L. J. TA$SIE

In Figure 3 the sum role (1) is plotted as a function of Elab/mc2; this relation
was selected because accurate experimental data are available. By extrapolation
sum rule (1) appears to hold for
E/mc 2 ~ 45
corresporiding to a laboratory momentum;::: 6 GeV/c for the pion. The analysis of

\MB
100 '
\ ..,' .
,~,

100 :;;
..s
~
.j 90
90 ~
~ ...
1: ITP
U
80
-.2
c8
80
(a) (a)
~ ~_---'1~0----~15~~~----
~ 100 --_ .. _-\ Plab(nucleon) (GeV/cl
1·0
u

9O~.~
~ --.
80 (b)
0'8 (b)
10 40 10 40

Fig. 2.-Sum rule (2): Fig. 3.-Sum rule (1):

PP = a(pp)+a(pn), BB = a(pp)+a(pp),
lIP = 2a(1T-p)+a(1T+p). MB = !{a(1T+p)+a(1T-p)}+Ha(K+p)+a(K-p)}
In (a) the meson-baryon and baryon-baryon -Ha(K+n)+ a(K-n)},
cross sections are compared at Plab in the shown in (a) with the ratio BB/MB plotted
ratio 2: 3. In (b) all cross sections are com- in (b). The dashed lines are extrapolations.
pared at Elab/mc2.

James and Watson (1967) shows that sum roles (2) and (3) are consistent with experi-
ment for pion laboratory momenta ;::: 6 GeV/c. Then equation (4) yields a lower
limit of 0·2 fm for the range of the quark-quark interaction.
Further information about the quark interaction is available from the electro-
magnetic form factors of the nucleon (Ishida, Konno, and Shimodaira 1966). If the
nucleon consists of a core of three quarks acting as a source for a meson cloud, the
nucleon form factor is

where ~(k2) is the form factor of the meson cloud about a point source, Gc(k 2) is the
form factor of the core, and k 2 is the square of the 4-momeritum transfer. The form
factor of the meson cloud about the core is taken from dispersion theory (Gasiorowicz
 PROPERTIES OF QUARKS. I 619

1966); for the proton the electric form factor of the meson cloud is
222
(k 2 ) 0 mp 0 m", 0 m",
-2--2 + '" -2--2 + ", -2--2
.'>2 _
"3p - P
mp+k m",+k m",+k
and for the neutron
2 2 2
~a 0
"3n
(k 2 ) _
- -
mp
P-2--2
mp+k
+0 '" m",+k
m",
2
0 m",
2 + "'-2--2'
m",+k
where 0 P' 0"" and 0", are constants. Taking the electric form factor of the neutron as
G:(k2) =0
and neglecting the mass difference between the p and the w mesons, the electric form
factor relevant to the proton is
tm~
E
'§p(k)
2
= -2--2 + - 2tm~
--2' (6)
mp+k m",+k
The experimental nucleon form factor agrees (see e.g. Islam and Vasavada 1969)
with the dipole fit for 0 < k 2 < 25 (GeV/c)2,

G:(k2 )
k2
= ( 1+ 0.71 (GeV/c)2
)-2 (7)
Using
<r2) = -6dG(k 2)/dk2 at k2 = 0,

the root-mean-square radius of the core obtained from equations (6) and (7) is

(8)

Since other processes may contribute to the size of the nucleon, equation (8) is an
upper limit to the size of the core.
For a flat-bottomed well <R2)! is approximately half the range of the potential.
From the scattering sum rules we find a lower limit of 0 ·2 fm for the range of the
quark-quark interaction, which implies

From the nucleon form factors

Consequently the root-mean-square separation distance of the quarks in the core is

bound by

III. DYNAMICAL DESCRIPTION

To obtain further information about quarks by fitting the observed masses of

the hadrons, it is necessary to calculate energy levels according to the quark model.
While this should be done by solving the relativistic two-body and three-body
620 G. B. SMITH AND L. J. TASSIE

problems, it seems better first to extract as much information about quarks as possible
by simpler means. We use an independent quark model of both mesons and baryons,
and consider only states which differ from the ground state by a change in quantum
numbers of one quark. The quark wavefunction rpnl} is an eigenfunction of the Dirac
equation
(9)

'where m is the reduced mass of the quark; the spherically symmetric potential V
is taken to have the same form (although not necessarily the same strength) for both
baryons and mesons. The discrete eigenvalues of (9) are bounded as (Rose 1961)

-mc2 < E nl} < +mc 2 •

TABLE 1
j-j STATES DECOUPLED INTO L-S STATES

j-j State JPC of Equivalent L-S States

10-+> or 11--)
S.,P. 10++) or v'iI1++)-v'111+->
12++> or v'lll++>+v'tll+->

We interpret the state with the lowest possible eigenvalue E nlj = -mc 2 as a hadron
of zero mass, so that the mass of a hadron corresponding to a bound state at energy
E nl} will be taken as
(10)

For comparison, the Schrodinger equation was used with the same potential V,
and we selected the zero of Mc 2 at 2mc 2 below the lower bound of the energy con-
tinuum to correspond to the relativistic treatment. Disagreement between the results
of the Schrodinger equation and the Dirac equation would show that the motion of
a quark inside a hadron is not nonrelativistic.
In a model of elementary particles in which we describe the low lying hadrons
by ground state wavefunctions and the higher hadronic multiplets as angular
momentum excitations of these ground states, we are faced with the problem of the
relationship between the bound states of the model and the observed hadrons.
Experimentally the baryons are known to have half-integer total angular momentum
and good parity. The mesons have integer total angular momentum and good
parity and charge parity. L-S coupling of three spin t quarks to form a baryon,
and a quark and an antiquark to form a meson, have the experimentally required
properties. However, if we wish to use the Dirac equation (9) to describe these
hadrons then j-j coupling is appropriate.
For j-j coupling of a quark and antiquark to form a meson the parity and
charge parity of a j-j state is most readily found by decoupling it into L-S states.
In Table 1 the resultant j-j states of an s! quark (or antiquark) and another anti-
quark (or quark) is given with the decomposition of these states into L-S states.
 PROPERTIES OF QUARKS. I 621

Taking the splitting of the j-j states to form JP states as proportional to h .h the
energy of the j-j state can be written as the weighted average of the JP states it
forms. For example, for the Y = 0, I = 1 mesons the mass of the 1st state is

For either assignment of the compound states, to fit the J P = 0+, 1+, 1-, and 2+
mesons requires that
600 MeV(c 2 :s M(ls t ) :s 800 MeV(c 2 ,

400 MeV(c 2 :s M(IPt)-M(ls t ) :s 450 MeV(c2 ,

580 MeV(c 2 :s M(lp3/2)-M(ls1) :s 680 MeV(c 2 .

Using a model of the baryons in which we assume two quarks are coupled to
zero total spin, and assuming that the low lying baryons are angular momentum
excitations of the third quark, indicates that the sl state should describe the t+
octet, the Pi state the t-octet, and the P3/2 state the !- octet.
This extends the bound state energy ranges to

600 MeV(c 2 :s M(ls!) :s 1320 MeV(c 2 , (l1a)

(lIb)

470 MeV(c 2 :s M(lp3/2)-M(ls1) :s 680 MeV(c 2 . (l1c)

We require that the eigenvalues of (9) satisfy (11).

IV. NUMERICAL PROCEDURE

For five forms of the potential V: (1) square well; (2) cut-off harmonic
oscillator, namely,
V(r) = -A(I-r2 (f3), r :S:.; f3,

=0, r > f3;

(3) Woods-Saxon potential (roughly intermediate between forms (1) and (2)), namely,

V(r) = -B[I+exp{13'2(r-ro)(ro}]-1;

(4) Gaussian potential; and (5) exponential potential; eigenvalues of (9) were found
numerically by a trial-and-error procedure. For a spherically symmetric potential
the solution of (9) is (Rose 1961)

"'nlj = (g(r) X~k),

if(r) X~k

where xtio k are two component spinors and f and g (the radial wavefunctions) are the
622 G. B. SMITH AND L . .T. TASSIE

solutions of

(12)

where {tl = rg, {t2 = rj, x = rmejh, 10 = Ejme 2 , and v = Vjme 2• Equations (12)
were numerically integrated "out" from x = 0 to an arbitrary matching point x = a
and "in" from a point where the potential was assumed to be zero to x = a. The
difference between the "out" and "in" solutions were used to alter 10 until this
difference was zero. This was accomplished using the result (see Appendix)

which states that, between its discontinuities,

is a monotonically decreasing function of 10. This allows 10 to be increased or decreased

depending on whether D was positive or negative in the previous trial solution.
The number of nodes in the eigenfunction determines if the eigenvalue is the lowest
one.
V. MAss OF THE QUARK
The strength of the potential in (9) was adjusted to fit the 1st eigenvalue to the
values of M(ls.} given in (11). In Figure 4 the energy spacings E(lp3/2}-E(ls.}
and E(lP.}-E(ls t } are shown as functions of m for a fixed value of the root-mean-
square radius of the Is, state, <R2>!
= 0·2 fm. For <R2>~ ~ Mjme we found that
E(lp3/2}-E(ls t } and E(lP.}-E(ls t } were almost independent of M(ls.) for M(ls t )
as in (11). For m <: 3 GeVje 2 all potentials gave essentially the same energy spacing.
For m ;S 3 GeVje 2 the variation of the energy spacing for different potentials was
less than the variation for different M(ls t ). In Figure 4, curve A represents the
maximum and curve B the minimum for the various potentials with M(ls!) as in (11).
As <R2>"is increased the energy spacing decreases so that for <R2>!
= 1·0 fm the
maximum of E(lp3/2)-E(ls t ) is ,...." 100 MeV for M(ls.) as in (11). To fit the
JP = 0+, 1+, 1-, and 2+ mesons and the t+, t-, and !- baryons as set out in (11)
requires
<R2>!
;S 0·25 fm.

Using <R2>! = 0·1 fm, the reduced maSS of the quark is

8 GeVje 2 ;S m;S 15 GeVje 2

and, for <R2>! = 0·25 fm,
1 GeVje 2 ;S m ;S 2 GeV/e 2

if the p-wave, s-wave energy spacing is to be fitted. Consequently the root-mean-

.---- ---.~----

PROPERTIES OF QUARKS. I 623

square separation distance of the quarks in the core is

O·lfm ~ <R2)1 ~ 0·25fm

and the quark mass vi( is
2 GeVlc 2 ~ vi( ~ 30 GeVlc 2 •

For these parameter ranges the point at which the strength of the exponential
potential is lie that at the origin occurs at a radius greater than 1·5 fm, and on
these grounds the exponential well can be excluded as unrealistic.

---- Rigid rotor approXimation A

750
.8

500

250

(a) (b)
o 15 30 45 o 15 30 45
m (GeV/c 2 )
Fig. 4.-Energy spacings between the p-wave and s-wave eigenvalues for a fixed root·
mean-square radius of the s-state wavefunction: (a) Ipl/2-ls1/2 spacing, (b) IpS/2-1s1/2
spacing. The eigenvalues of the Schriidinger equation are well represented by the rigid
rotor approximation, while the eigenvalues of the Dirac equation are distinctly different
for a reduced mass;:; 6 Compton wavelengths (see text for details).

VI. RIGID ROTOR MODEL

The nonrelativistic results obtained from the Schrodinger equation are re-
produced remarkably well by a rigid rotor approximation. The energy E of rotation
of a rigid rotor is given by
E = L2/21 = /i 2l(l+l)/21 , -

where L is the total angular momentum and I the moment of inertia of the rotor.
For a spherically symmetric mass distribution

1= im<R2) ,
where m is the total mass and <R2) the mean square radius of the mass distribution.
We could fit the Ip-ls energy spacing with

E(lp)-E(ls) = 3/i2/2m<R2)lB

to within", 5% for m ;:::: 1·5/iI<R2)ic 2. For m ;:::: 6/iI<R2)lc 2 the nonrelativistic

and relativistic p-wave, s-wave energy spacings are almost identical. Consequently
the rigid rotor approximation fits the results obtained by solving the Dirac equation
for m ;:::: 6/iI<R2)lc 2. For m ~ 6/iI<R2)ic 2 the relativistic results are distinctly
different from the nonrelativistic results, and hence, for the stated parameter range
of the quark, relativistic effects are not negligible.
624 G. B. SMJTH AND L. J. TASSJE

VII. CONCLUSIONS

For quarks whose interaction is described by a non-singular potential energy,

the hadron spectrum and the scattering relations imply a quark mass J( in the range

2 GeV/c 2 :5 J( :5 30 GeV/c2

and a root-mean-square separation distance of the quarks in the core given by

0·1 fm :5 <R2)i :5 0·25 fm.

Relativistic effects are not negligible for these values of the quark parameters.
For larger values of the quark mass, the spacing of quark energy levels according to
the Dirac equation agrees with that of the Schrodinger equation and with the rigid
rotor model.

VIII. REFERENCES

ALLABY, J. V., et al. (1969).-Phys. Lett. B 30, 500.

CITRON, A., GALBRAITH, W., KYCIA, T. F., LEONTI6, B. A., PHILLIPS, R. H., and ROUSSE, A.
(1964).-Phys. Rev. Lett. 13, 205.
DIDDENS, A. N., JENKINS, E. W., KYCIA, T. F., and RILEY, K. F. (1963).-Phys. Rev. Lett. 10, 262.
FOLEY, K. J., et aT. (1967).-Phys. Ret). Lett. 19, 857.
GALBRAITH, W., et al. (1965).-Phys. Rev. 138, B9I3.
GASIOROWICZ, S. G. (1966).-"Elementary Particle Physics." Ch. 26, p. 444. (Wiley: New York.)
ISHIDA, S., KONNO, K., and SHIMODAIRA (1966).-Prog. theor. Phys., Kyoto 36, 1243.
ISLAM, M. M., and VASAVADA, K. V. (1969).-Phys. Rev. 178, 2140.
JACKSON, J. D. (1965).-Rev. mod. PhY8. 37, 484.
JAMES, P. B., and WATSON, H. D. D. (1967).-Phys. Rev. Lett. 18, 179.
JOHNSON, K., and TREIMAN, S. B. (1965).-PhY8. Rev. Lett. 14, 189.
KOKKEDEE, J. J. J. (1969).-"The Quark Model." (W. A. Benjamin: New York.)
KOKKEDEE, J. J. J., and VAN HOVE, L. (1966).-Nuovo. Oim. A 42,711.
LEVIN, E. M., and FRANKFURT, L. L. (1965).-Zh. elcsp. teor. Fiz. 2, 1'05; English translation in
Soviet Phys. JETP Lett. 2, 65.
LINDENBAUM, S. J., LOVE, W. A., NIEDERER, J. A., OZAKI, S., RUSSELL, J. J., and YUAN, L. C. L.
(1961).-PhY8. Rev. Lett. 7, 185.
LIPKIN, H. J., and SCHECK, F. (1966).-Phys. Rev. Lett. 16, 71.
ROSE, M. E. (1961).-"Relativistic Electron Theory." Ch. 5, pp. 159, 166. (Wiley: New York.)
SMITH, G. B. (1970).-Aust. J. Phys. 23, 627.

APPENDIX

Taking equations (12) for two values of £, namely £' and £1, we get

(AI)

(A2)
and

(A3)
 PROPERTIES OF QUARKS. I 625

(A4)

Multiplying (AI) by fL2 i and (A3) by fL2 i and substracting gives

.
and similarly from (A2) and (A4) we get

Substracting (A6) from (A5) gives

(A7)

Applying the boundary conditions fLl and fL2 tend to zero as x tends to zero or
infinity, integration of (A7) leads to

[ fLl1 1 1 1 ] out i1
fL2 -fLl fL2 x=a = (€ -€ )
[fa0 (fLl fLl i11j
+fL2 fL2 ) dr
]out
,

[ fLl
ij ji
fL2 -fLl fL2 x=a = ]il ij
(€ -€ )
[fa (fLl
00
ij ij
fLl +fL2 fL2 ) dr
]il
Hence dividing by fLl i fLl j and substracting we get

([ fL2
1
j-i
fL2 1]Out - [j
fL2
j-i
/12 1] ill)
fLl fLl fLl fLl x=a

Letting €i -+ €1 leads to
L
which is the required result.