- Email: [email protected]
Composite nucleon approach to the deuteron problem
ANNALS
OF PHYSICS
95, 112-126 (1975)
Composite
Nucleon
Approach
to the Deuteron
Problem
B. K. AGARWAL* International
Centre for Theoretical Physics, Trieste, Italy Received April 9, 1975
A composite model is suggested for the nucleons by assuming a longrange strong gluon force between a diquark boson B and a quark A. In the proton, A is trapped inside B in an oscillator potential and, in the neutron, A is on the surface of B in a hydrogenlike state. Nucleon form factors are obtained in agreement with experiments. The model contains a mechanism for a large effective mass of the quark A. When B is identified with VTand A with CL,one can fix the gluon charge value and obtain the magnetic moments of the proton and neutron. The (TV) atomic model for the nucleon can be used to construct the deuteron on a hydrogen molecule model. It leads to values for the binding energy, electric quadrupole moment, and form factors of the deuteron that are in agreement with experiments.
1. INTRODUCTION
It is clear that we do not understand the deuteron problem as we do the hydrogen atom problem or even the hydrogen molecule problem. The main reason is that while the explicit nature of the Coulomb force is known, we have learned very little about the nuclear force. A basic limitation of the thought process is that we can think only in terms of what we know. A new idea is born when previously disconnected ideas are correlated to describe or meet a specific situation. To achieve this fusion one is forced to introduce some new assumptions. Then the theory becomes more than an analogy. The solar system and the atom were correlated in this sense by Bohr. In nuclear physics also one tries to combine classical with quantummechanical
considerations
to obtain
theoretical
expressions
that
are weighted
empirically to yield an agreement with experiments [l]. The analogy of the nuclear force with the chemical force was noted by Heisenberg [2] even before the advent of the Yukawa theory. The well-known common features are the saturation property and the exchange property [3]. Attempts * On leave of absence from the Physics Department, 112 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
Ailahabad
University, India.
COMPOSITE NUCLEON AND DEUTERON
113
were made to understand the nuclear force in terms of the exchange of fields or particles between the nucleons, in analogy to the photon exchange for the electromagnetic force. Heisenberg [4] and Ivanenko and Tamm [5] considered the exchange of an electron-neutrino pair, and Wentzel [6] and Gamow and Teller [7] added the exchange of e+-e- pairs. Finally, Yukawa [8] assumedthat the nuclear force is due to the exchange of pions. However, all attempts to understand the nuclear forces in terms of pion exchange, or exchange of other bosom, have run into practical difficulties. Barut [3] has asked the interesting question: Is it possible to develop a theory of nuclear forces that would revive the pre-Yukawa picture of chemical-type forces and yet explain the practical successof the Yukawa theory? We try to answer this question herein on the basis of a series of papers we have written on this subject [9-131.
2. PREVIOUS ATOMIC
MODELS
FOR NUCLEONS
Barut [3] has regarded the proton as a dyonium atom made out of spinless magnetic monopoles. In his model the neutron is made up of a proton and the negatively charged (e-v)-particle system, which effectively acts as a single charged spinlessparticle B-. It is an attractive model to consider, but there are several difficulties present from the very beginning. For example, like quarks, the magnetic monopoles have not shown up in the experiments designed to detect them. Also, if B- is identified as a valence rr-, then the explanation of beta decay is complicated by the possible emission of p-. Otherwise, one must introduce an unobserved boson B-. According to Goebel [14], such a model amounts to the identification of monopoles with quarks. However, since monopoles are spinless, this analogy seemsto be incorrect [15]. In the model of Drell and Johnson [16] a baryon is pictured as three valence quarks plus a “soul” whose existence is necessary to obtain a simple antisymmetric ground-state wavefunction. It also involves a short-range quark-quark repulsion to give the required l/t2 fall-off of the electromagnetic form. The concept of “soul” remains a mystery. Kuti and Weisskopf [17] consider the nucleon to be a three-valence quark structure accompanied by a core of virtual quark-antiquark pairs. As nothing is known about the mediation of the interaction between quarks, they assume it in terms of gluons. It is not clear how the valence quarks are prevented from coming out in a scattering experiment. If nuclear forces are of the same nature as chemical forces, then nucleons should be composite systems, like a hydrogen atom. A favorite model [18] is to treat them as atomlike systems consisting of a nucleonic core surrounded
114
B. K. AGARWAL
by a pionic cloud, as suggested by Yukawa’s theory. By assuming a pion-nucleon potential of the form
&
= f (a,v. MfN - C-Jv(r) W, 11,
W> < 0,
(1)
one obtains a strongly attractive state j = 4, t = 4, since
.I, -1 2Y;.t,I = +*
for for
j, t = 4 j, t = 8.
(2)
It is repulsive in the j = 4 , t = $ and j = $ , t = 4 states. An attractive state, but less strong than the ground state, is obtained for j = t = 4, which may be interpreted as a resonance state. The pion orbital momentum can be exploited to obtain values = f3 nm for the magnetic moments of the physical proton and neutron. However, despite its initial success, this model is also beset with some insurmountable difficulties. For example, the concept of the nucleonic core is not clear. Also, the dipole fit to the e-N scattering form factor implies that the nucieons do not have any charged core, but rather can be described in terms of a continuous (e.g., exponential) distribution of charge and magnetization. The equal and opposite values for the magnetic moments of p and n are also not in agreement with observations. In view of such shortcomings, we have looked for the various atomic models available in the old literature and not merely at the Bohr model. A well-known and useful model for our purpose is provided by the Thomson watermelon model [19] of the atom, which on analysis leads to a harmonic oscillator potential.
3. PION-MUON
MODEL
FOR NUCLEONS
We shall work with a diquark (44) called boson B and a quark (q) called fermion A. The Be is spinless and it carries electric charge e and a strong gluon charge +g. The Ae is a spin-4 quark, carries electric charge e, magnetic moment pa , and a strong gluon charge -g. The strong gluon potential has the long-range form -l/r. The possibility of this long-range form has been justified on various grounds [20]. The quark fields are assumed to obey (ia - m3 g(x) = c4 g(x), where m, is the bare quark mass and I(x) is the interaction term giving binding. For a sufficiently strong binding the gluon field can be treated classically and one can replace it by a c-number potential. Assuming V(r)
=
V(a)
-
b/r,
(3)
COMPOSITE
NUCLEON
AND
115
DEUTERON
Suura [20] obtains a wave equation for a hydrogen atom with corresponding hadronic mass spectrum. The presence of quark-antiquark pairs has been postulated [lo, 15-171 to explain the diffractive behavior in high-energy e-N scattering. From Lorentz covariance, charge conservation, and invariance under space reflections, we find that there exist two electromagnetic form factors for a &spin particle
where u(p) and u( p’) are the free Dirac spinors and y,, is the Dirac matrices; F,(q2) is called the Dirac form factor and F2(q2) is called the Pauli form factor. We shall refer to them as Ftcharse) and J’cmoment), respectively. We can assume, following Sachs [21], that a nonrelativistic calculation of the form factor is meaningful in the nucleon’s rest frame. Then it is possible to interpret the form factor in a classical fashion in terms of the appropriate spatial densities. We can identify B with rr and A with CL. Then the nucleon in our model is a (np) composite system. In particular, p ~= (TOP+) and n = (r+p-). There is an attractive gluon force, -g2/r2, between x and CL,where g is the gluon charge. The proton is assumed to consist of TO, smeared out in a uniform spherical distribution of positive strong gluon charge +g, with p+ of strong gluon charge -g embedded in it. The embedded TV+experiences a radial attraction -g2rlR3, where r is the distance of EL+from the center of 7~~of radius R. In other words, CL+ experiences the harmonic oscillator potential +kr2, k = g2/R3. The groundstate wavefunction of p+ is a 312 exp( - +a2r2), oscillator = - ,1/z
*
( 1
(4)
a2 = (m,k/fi2)1/2. The corresponding
charge density is poscillator
=
-$
exp(-a2r2),
leading to the form factors FWmxce) = F(moment)ppl = 9 9
(4747)
1
pOsmatO&)(sin
qr)
r
dr
(6)
= exp( - q2/4a2). This reproduces the form of the empirical gaussian model of the proton, which is in agreement with the data of Hofstadter et al. [22] at energies ~300 to 1000 MeV and four-momentum transfers -4 to 20 F-2.
116
B. K. AGARWAL
The properties of the proton and neutron are known to be different. Therefore, it would be reasonable to have a slightly different model for the neutron. We assume for the neutron a hydrogenlike composite system in which p- is the valence constituent accompanied by the central charge distribution of 7~+,under the strong gluon long-range force. The ground-state wavefunction of II- is ti hydrogen
=
($-)“*
t?Xp
(-
F),
(7)
a = 2m,g2/h2.
The corresponding
negative charge density is phydrogen
=
(8)
exp(--olr) = pn- .
c
The resulting form factors are $charge) n-
For simplicity,
=
#moment) R
p,~
= (1 + $,-‘.
we assume a gaussian distribution Pn = Pn+ pharge) n
(9)
of the 7r+ charge. Then
pn- = -$
exp(a2r2) - $- exp(--olr),
=
_
F(charge)
(W
FF(chawe)
= e1; (- &y--
(1 + $)-‘.
Thus, in our model, Eqs. (10) and (11) represent the neutron, with J p,(r) dr = 0. The Pharge)(q2) given by Eq. (11) is very similar in behavior to that derived by Schiff :23]. It will fit the electromagnetic neutron structure factor [22]. The scaling law in our model is &$harge)(qz) Fcmoment)(q2) 7%
= /L,Fy-)(q2) F;moment)(q2)
~ * )
(12)
which is in agreement with experiments for small four-momentum transfers. For a recent review of electromagnetic form factors of nucleons, see the article by Gourdin [24].
117
COMPOSITE NUCLEON AND DEIJTERON
4. MAGNETIC We assume that-the comes mainly from:the
MOMENTS
OF NUCLEONS
contribution to the magnetic moment of the nucleon $-spin constituent muons. Then we expect [25]
MO Pa = + __ nm, mcl+
pn = - arim mu+
>
(13)
where M, is the rest mass of the nucleon and m, is the relativistic mass of the muon in the model in the rest frame of the nucleon. For the observed value pn = -1.9 nm we need m,- = MO/l.9 = 4.7muo , where m,o is the rest mass of the muon. In other words, CL- is moving with a speed u = 0.98~. In the neutron the p-- is moving in a hydrogenlike orbit defined by v/c = g2/fic = 0.98. This fixes the value of strong gluon charge g. If we assume that p- behaves like a particle of effective mass 4.7mwo then 01 = 2g2m,-/iP = 4.92 F-l = 0.94 GeV. With this value of 01, Eq. (9) gives a reasonable fit to the experimental [22] Fimoment)(q2). To obtain the observed value of pD, we need m,+ = M,/2.79 = 3.21mGo . It implies that the TV+is moving with a speed of 0.95~. If we use the well-known relations AE = Am?,
Ax = c At = cfi/AE,
(14)
then for Am = 3.21m,o we obtain Ax = 0.58 F. Taking R = 0.58 F we find that k = g2/R3 = 9.9 x lo2 MeV F-2 and a2 = 2.93 F-2. If this value of a2 is substituted into Eq. (6), we obtain a good fit to the experimental FFharge)(q2) and Fmoment)(q2). Eq. (11) is now also in agreement with experiments. It may be meitioned that the known size of a proton and baryon level spacing requires [26] an effective quark mass around 300 MeV m 3mGo. Thus, we have shown that our model contains a possible mechanism for a large effective mass of muons, which fixes the single parameter g of the model, and which in a consistent way leads both to the observed magnetic moments and the electromagnetic form factors of the proton and neutron. This avoids the use of any unobserved constituent quarks or partons. Also, the model is more viable than the pion-atomic model [18] of nucleons. Since +iw = h(k/m,)1/2 * mrr and $iio = m, , the ground-state (or zero-point) energy will appear as a source for virtual pion cloud. The breakup of the proton into its constituents can be avoided by a new assumption: All the energy given to the system is absorbed to create pions from the single level ground state or muons from the two-level transitions, the former process being more probable.
118
B. K. AGARWAL 5. BETA DECAY
In the long-range potential model of Suura [20], the constant term V(co) in the potential (3) is related to a self-mass, which acts as a trapping barrier for quarks. Then the quarks cannot propagate out asymptotically. Our model for the proton has some features of the “bag model” [27]. In the neutron, the p-is on the surface of the bag, and thus, it is allowed to decay (forced decay of I”in the presence of n+) I*- -+ e- + Pe + v, . (15) The v, is assumed to carry the same gluon charge as p-, and consequently, it cannot leave the bag. However, it can fall back into the bag leading to the process v, + 7r+ + pf + 7To= p.
(16)
This will explain the beta decay of the neutron in terms of the muon decay. We can now say that the difference between leptons (p, v,) and (e, v,) is that the former set possesses a gluon charge while the latter does not. The emission and reabsorption of this gluon field will then account for the greater mass of ~(v,,) than of e(v,). 6. DEEP INELASTIC
SCATTERING
We have seen that the (np)-atomic model of nucleons accounts for the observed electromagnetic structure features up to about 1000 MeV. What will happen in our model when, at high energies, deep inelastic scattering sets in ? In e-N scattering at low energies, the incident electron primarily interacts with p+ in the proton (no is not coupled to the photon due to zero electric charge) and with the valence p- in the neutron. It is like electron-muon scattering with a cross section of the form (17)
The proton of mass AJo sees the electron of initial energy E by receiving from it a photon of energy v = E - E’ and q2 = -4EE sin2(8/2) = -Q2, where E’ is the final energy of the electron scattered at the laboratory angle 19.For the high-energy inelastic process e + p --t p + hadrons, the cross section can be written as
da0 domott -dQdE’ = ~ dL’ [ W2(v, q2)+ Wdv, 43 2tg2;I,
(18)
COMPOSITE
NUCLEON
where W, and W, are the unknown is assumed to be absent,
AND
structure functions.
W:)(v, (2") --f -&
S (& 0
Wf)(v,
119
DEUTERON
When the structure
- v), 0
Q*) + S 2MQ2 c 0
v . 1
When the energy increases, the electron begins to probe into the (qij) pair B” in the proton, resulting in the creation of more quarks and antiquarks, B = qq + qq@ = BB --t BBB, etc.
C-21)
The diffractive scattering is now dominated by these newly created quarks. The Bjorken scaling law [28] suggests that the inelastic scattering is just the incoherent sum of the elastic scattering from the constituent pointlike quarks (or partons) [29, 301. Then Wl,, = 1 J@iYm,>,
(22)
where m, is the parton mass. Also, we can write
Wz(v,
Qz>= 2 s1 d’(x) S(x - co-‘) dx - g%rge 72 0
v W*(w),
where x = m,/M, , w = 2m,v/Q2, and f(x) dx is the probability that the parton mass is found to be between xMo and (x + dx) A4, . This is the scale invariance. If we regard the proton as moving with momentum P, then for the longitudinal direction one can write gpn=p
or
11, n
= 1,
where P, is the longitudinal momentum of the nth parton and 1, = P,/P. If L(ln, P) gives the probability distribution of the created partons, then for N created partons, d-L(x, ,..., x,> = AS(l
- ~lr,)&LP)dl,.
(2%
where A is a normalization constant. This can be integrated to yield the parton contribution to F, , provided one can guess a suitable form for L(& , P). For the proton in our model, the quark A, being embedded in the B “sea,” will also
120
B. K. AGARWAL
participate in the diffractive scattering. Therefore, we expect W, > Wn in our model when the number of created quarks N is not high. For large N, W, will approach W,, . Thus for nondiffractive scattering W, m W, , for low w diffractive scattering W, > W, , and for high w diffractive scattering again W, w W, . This is in agreement with experiments [31]. 7.
THE
DEUTERON
We have found that the (~11~)modified atomic model can explain many highand low-energy features of nucleons in a consistent way with only one parameter g. We can now form the deuteron on the molecular model. A simple calculation for the binding energy of the deuteron can be performed by considering the interaction Hamiltonian (26)
where +g is the gluon charge on pions, -g is the gluon charge on muons, rQZ is the distance between rr” and p-, r bl is the distance between rrf and $-, r12 is the distance between the two muons, R, is the distance between two pions, and g2/R, is regarded as constant. To the first order of approximation, the binding energy is given by EB = j @‘*(r,, , rb2) fWr,,
, rb2) d71 dT2 9
(27)
where @ is the normalized wavefunction of the deuteron, ral is the distance between no and p+, rb2 is the distance between yr+ and TV-, and drl and dr2 are the volume elements. If &(r,,) and &,(r,,) are the normalized wavefunctions of the proton and the neutron, respectively, we have @@k , rb2) = h&J
Mrb2).
P-9
It is well known that very little error is introduced by replacing the harmonic oscillator wavefunction by the hydrogenlike wavefunction for the ground state. Then 4dr> = Mr) A straightforward
calculation EB=
+(A
ew (-
F)
-
(29)
from Eq. (27) gives a
2y = R,a.
It has a minimum
= &
at y = 1.88.
-;y+~y2+;y3)e-2y,
(30)
121
COMPOSITE NUCLEON AND DEUTERON
In our model the spin dependence comes from the two constituent muons. Since (CT,+ * o,-) = + 1 for parallel spins, a positive contribution is obtained for only parallel spins. As a result, for the deuteron we obtain an attractive potential for the 3S, state alone. We also can calculate the quadrupole moment. In our model it is given by Qe = (32” - r2)a2,= +R,2,
(31)
because the net contribution due to muons is zero, and therefore, the entire contribution comes from 7~+ situated at a distance &R, from the origin. So far we have taken g2/?ic = 0.98. This may be compared with the pionic strong coupling gr2/hc = 0.3 and the electromagnetic coupling e2/fic = l/137. We know that in treating hydrogen molecule formation one must use an effective value for the electronic charge to obtain an agreement with the experiment. Analogous calculation for the deuteron formation will need an effective value of the gluon charge g, especially when the coupling involved is so large. However, we must require that both Es and Qe are simultaneously in agreement with observations for the same effective value gefP . With this requirement in mind, we find g&/!ic = 0.48. Then aeff = 2.51 F-l and EB = -2.22 MeV
(-2.2246 MeV),
Q
2J - 1 Qe = 2.8 x 1O-27 cm2 max= 2(J+ 1)
(32)
(2.796 x 1O-27 cm2),
where quantities in parentheses are the corresponding Thus, the composite model of nucleons leads to an for the deuteron. It is able to account for (i) parallel energy, and (iii) electric quadrupole moment of the any admixture of D states.
(33)
experimental values [32, 331. acceptable molecular model spins of n and p, (ii) binding deuteron, without requiring
8. FORM FACTORS FOR THE DEUTERON
Following Hofstadter
[34], we can express the form factors for the deuteron as
pharge)(q2) D p-#jmomeW(q2)
= [~~hawe)(q2j =
[F;moment)(q2)
+ F;hawe)(qpl +
FfmmeW(q2j]
fD , fD
(34) ,
(35)
122
B. K. AGARWAL
where fD(q2) is the form factor of the deuteron when p and n are point particles and FDsn are given by Eqs. (6) and (11). We can writef, as fD(q2)
= T J”om exp(p2hr)
sin qr dr, (36)
x = ( -yny2.
Calculations
for the deuteron form factors give, for q2 = 1, 3, 4 F-2, FC$%w) = 0.48 (0.55), 0.26 (0.26), 0.21 (O.lS),
pilFtymw
= 0.41 (0.41), 0.25 (0.26), 0.18 (O.lS),
where the quantities in parentheses are the experimental the agreement is reasonable.
(37) (38)
values 135, 361. Again,
9. VAN DER WAALS FORCEAND NEUTRONMATTER When the interneutron distance is large, the exchange phenomenon is unimportant and even in this case we can write H’ as in (26). We can expand H’ in a Taylor series in inverse powers of R, . With the two neutrons located on the z-axis,
H’ = &
+ y, y, - 2~~2,)
(x1x2
+ l+$ n lr12z2- r22z1+
+
(2x92
where x1 , y1 , z1 are coordinates of the are coordinates of the second p- relative the dipole moments of the two neutrons. easily in the second-order perturbation W(R,)
>
-
F
zz
-
- 3z,z,)(z, - z2)> + ***, (39)
first EL- relative to its n+, and x2 , yz , z2 to its 7~f. The first term in (39) represents The dipole-dipole interaction is evaluated theory to give 1371
0.02 R6’ 77
__
2Yo2
(A = c = in,, = 1).
(40)
Recently, a Van der Waals force has been employed to fit the Reid potential (soft core) between two nucleons and to develop a corresponding state approach to nuclear and neutron-star matter [38]. In view of the result (40), our model can provide a suitable basis for such an approach. Recently, Sawada [39] has
COMPOSITE
NUCLEON
AND
123
DEUTERON
shown that we should add a long-range Van der Waals potential, -C/R6, to the nucleon-nucleon potential. He needs C = 0.12, while Barut [40] gets C = 1O-g - lo-lo for his monopole model. Our value is close to Sawada’s prediction.
10.
CONCLUDING
REMARKS
We have shown that a boson B and fermion A bound in an atomlike state under a strong long-range gluon force provide a suitable model for nucleons. This composite model is suggested by the high-energy e-N scattering experiments and the resulting form factors. The form factors of nucleons and the scaling behavior of the structure factor W, are obtained easily in this model. The reason for the scaling behavior also can be seen as follows. The virtual photoproduction cross section is the imaginary part of the forward Compton scattering amplitude. In our model it looks like Fig. 1, and the scaling behavior of the structure function arises due to the appearance of the proton as core B” plus quark A+, with A+ propagating between the photons. If the four-momenta are labeled as in Fig. 1, proJon
1; I PII J-4 I I
I 01
Bl
FIG.
1.
Forward
q
XP
; I A+
Compton scattering in a (PA ‘1
model
of the proton.
then apart from the spin-dependent parts, which are immaterial here, and neglecting any momenta transverse to the proton-photon axis [41], we have for the quark propagator [(xp + q)2 - n1,$2- k-1.
(41)
The imaginary part of the amplitude includes the delta function S(x2MBz + q2 + 2xp . q - mA2), which comes from the quark propagator.
124
B. K. AGARWAL
Neglecting x2MP2 and mA2for very large v, q2, we find that the amplitude contains the delta function constraint
so that the structure function is dependent only on the ratio v/Q,. The strength of our model is that, unlike other parton models, Fig. 1 is the only box diagram possible for Compton scattering (B” has zero charge), yp -+ yp, and therefore the structure function will scale. If we identify B with rr and A with CL,then our model can be used to calculate the magnetic moments of the proton and neutron by fixing the single parameter of the theory, g. One can form the deuteron on a molecular model and correctly predict (i) the binding energy, (ii) electric quadrupole moment, and (iii) form factors of the deuteron. Despite this phenomenal success of the model, some questions remain unanswered. The mass of the composite nucleons has not been calculated [42]. If all the zero-point energy is assumed to be stored by the constituent particles in the form of effective mass, one can think of a dynamical equilibrium, say in the proton, via the following sequence of processes: p = rr” + p+ + Pi -
y + y + pf -
E.L+ + I-L--+ CL++ p-
p++y+y+p++~“=P.
To conclude, our work shows that once the high-energy experiments have suggested a composite model for the nucleon, it follows that a strong long-range potential can account for the short-range nature of the n-p potential by regarding the deuteron on a molecular model requiring hybridization of muon clouds. This picture involves no unobserved quarks and only one parameter, g, is sufficient to understand the basic features of nucleons and deuteron. Thus, both the high-energy electron scattering experiments and low-energy nuclear phenomena can be described in terms of an explicit force, bringing it in line with gravitational and electromagnetic forces in nature. This has more esthetic appeal. ACKNOWLEDGMENTS The author is grateful to Prof. K. M. Khanna and Drs. S. C. K. Nair and S. B. Khadkikar for helpful discussions. He is also grateful to Prof. Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, Italy.
COMPOSITE NUCLEON AND DEUTERON
125
Added in Proof. In view of Eq. (14), m, N 3.2 rn&,, and m, N 3.2 m,, . In a simple way, N m, + m, + @J - $.(g2/R) N 930 MeV. Note that the x0 lifetime is much smaller than the p+ lifetime. Note
M,,
REFERENCES 1. P. MARMIER AND E. SHELDON, “Physics of Nuclei and Particles,” Vol. I, p. 53, Academic Press, 1969. 2. W. HEISENBERG,“Nuclear Physics,” Methuen, London, 1953. 3. See, for example, A. 0. Barut, Nature of nuclear bond and hadron structure, in “The Structure of Matter” (B. G. Wybourne, Ed.), University of Canterbury, New Zealand, 1972. 4. W. HEISENBERG,Solvay Conference, 1933. 5. D. IVANENKO AND I. TAMM, Nature (London) 133 (1934). 981. 6. G. WENTZEL, Heft. Phys. Acfa 10 (1936), 107. 7. G. GAMOW AND E. TELLER, Phys. Rev. 51 (1937), 289. 8. H. YUKAWA, Proc. Phys. Math. Sot. (Japan) 17 (1935), 48. 9. B. K. AGARWAL, Indian J. Phys. 34 (1971), 406. 10. B. K. AGARWAL, Lettere Nuovo Cim. 9 (1974), 185. 11. B. K. AGARWAL, Lettere Nuovo Cim. 9 (1974), 294. 12. B. K. AGARWAL, Y. M. GUPTA, AND C. WATAL, “Some Implications of the Composite Model of Nucleon,” Allahabad University Preprint BKA/74-3 (1974). 13. B. K. AGARWAL, Y. M. GUPTA, AND C. WATAL, “A Composite Model for Nucleon and Deuteron,” Allahabad University Preprint BKA/75-1 (1975). 14. C. J. GOEBEL, in “Quanta” (P. G. 0. Freund, C. J. Goebel, and Y. Nambu, Eds.), p. 338, New York, 1970. 15. A. 0. BARUT, private communication. 16. S. D. DRELL AND K. JOHNSON, in “Proceedings of the XVI International Conference on High Energy Physics” (J. D. Jackson and A. Roberts, Eds.), p. 801, Chicago, Ill, 1972. 17. J. KUTI AND V. F. WEISSKOPF,Phys. Rev. D4 (1971), 3418. 18. B. T. FELD, Ann. Pfzys. (N.Y.) 1 (1957), 58. 19. J. J. THOMSON, Philos. Mag. 7 (1904), 237. 20. H. SUURA, Preprints DESY 72/7 (1972) and DESY 72/47 (1972). 21. R. G. SACHS, Phys. Rev. 126 (1962), 2256. 22. R. HOFSTADTER, F. BUMILLER, AND Y. YEARIAN, Rev. Mod. Phys. 30 (1958), 482. T. A. GRIFFY, R. HOFSTADTER, E. B. HUGHES, T. JANSSENS,AND M. R. YEARIAN, in “Nuclear Structure” (R. Hofstadter and L. I. Schiff, Eds.), p. 61, Stanford, Calif., 1964. 23. L. I. SCHIFF,Rev. Mod. Phys. 30 (1958), 462. 24. M. GOURDIN, Phys. Rept. 99C (1974), 30. 25. E. M. HENLEY AND W. THIRRING, “Elementary Quantum Field Theory,” p. 230, McGrawHill, New York, 1962. 26. See, for example, G. MORPURGO, The quark model, in “Proceedings of the XIV International Conference on High Energy Physics,” Vienna, 1968. 27. A. CHODOS, R. L. JAFFE, K. JOHNSON,C. B. THORN, AND V. F. WEISSKOPF,Phys. Rev. 9D (1974), 3471. 28. J. D. BJORKEN, Phys. Rev. 179 (1969), 1547. 29. R. P. FEYNMAN, Phys. Rev. Letters 23 (1969), 1415. 30. J. D. BJORKEN AND E. A. PAXHOS, Phys. Rev. 185 (1969), 1975.
126
B.
K.
AGARWAL
R. SOGARD, in “Proceedings of the XVI International Conference on High Energy Physics,” (J. D. Jackson and A. Roberts, Eds.), Vol. 2, p. 98, Chicago, Ill., 1972. 32. W. V. PRESTWICK, Can. J. Phys. 43 (1965), 2086. 33. H. NAMURI AND T. WATANABE, Progr. Theoret. Phys. (Kyoto) 35 (1966), 1154. 34. R. HOFSTADTER, An. Rev. Nucl. Sci. 7 (1957), 231. 35. N. K. GLENDENNING AND G. KRAMER, Phys. Rev. 126 (1962), 2159. 36. J. I. FRIEDMAN, H. W. KENDALL, AND P. A. M. GRAM, Phys. Rev. 120 (1960), 992. 37. See, for example, L. I. SCHIFF, “Quantum Mechanics,” 2nd ed., p. 179, McGraw-Hill, New York, 1955. 38. R. G. PALMER AND P. W. ANDERSON, Phys. Rev. 9D (1974), 3281. 39. T. SAWADA, Tokyo University of Education, Preprint TUETP-74-9 (1974). 40. A. 0. BARUT AND J. NAGEL, Phys. Letters 55B (1975), 147. 41. F. E. CLOSE, Daresbury Laboratory Preprints DNPL/P177 (1973) and DNPL/R31 (1973). 31. M.
OF PHYSICS
95, 112-126 (1975)
Composite
Nucleon
Approach
to the Deuteron
Problem
B. K. AGARWAL* International
Centre for Theoretical Physics, Trieste, Italy Received April 9, 1975
A composite model is suggested for the nucleons by assuming a longrange strong gluon force between a diquark boson B and a quark A. In the proton, A is trapped inside B in an oscillator potential and, in the neutron, A is on the surface of B in a hydrogenlike state. Nucleon form factors are obtained in agreement with experiments. The model contains a mechanism for a large effective mass of the quark A. When B is identified with VTand A with CL,one can fix the gluon charge value and obtain the magnetic moments of the proton and neutron. The (TV) atomic model for the nucleon can be used to construct the deuteron on a hydrogen molecule model. It leads to values for the binding energy, electric quadrupole moment, and form factors of the deuteron that are in agreement with experiments.
1. INTRODUCTION
It is clear that we do not understand the deuteron problem as we do the hydrogen atom problem or even the hydrogen molecule problem. The main reason is that while the explicit nature of the Coulomb force is known, we have learned very little about the nuclear force. A basic limitation of the thought process is that we can think only in terms of what we know. A new idea is born when previously disconnected ideas are correlated to describe or meet a specific situation. To achieve this fusion one is forced to introduce some new assumptions. Then the theory becomes more than an analogy. The solar system and the atom were correlated in this sense by Bohr. In nuclear physics also one tries to combine classical with quantummechanical
considerations
to obtain
theoretical
expressions
that
are weighted
empirically to yield an agreement with experiments [l]. The analogy of the nuclear force with the chemical force was noted by Heisenberg [2] even before the advent of the Yukawa theory. The well-known common features are the saturation property and the exchange property [3]. Attempts * On leave of absence from the Physics Department, 112 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
Ailahabad
University, India.
COMPOSITE NUCLEON AND DEUTERON
113
were made to understand the nuclear force in terms of the exchange of fields or particles between the nucleons, in analogy to the photon exchange for the electromagnetic force. Heisenberg [4] and Ivanenko and Tamm [5] considered the exchange of an electron-neutrino pair, and Wentzel [6] and Gamow and Teller [7] added the exchange of e+-e- pairs. Finally, Yukawa [8] assumedthat the nuclear force is due to the exchange of pions. However, all attempts to understand the nuclear forces in terms of pion exchange, or exchange of other bosom, have run into practical difficulties. Barut [3] has asked the interesting question: Is it possible to develop a theory of nuclear forces that would revive the pre-Yukawa picture of chemical-type forces and yet explain the practical successof the Yukawa theory? We try to answer this question herein on the basis of a series of papers we have written on this subject [9-131.
2. PREVIOUS ATOMIC
MODELS
FOR NUCLEONS
Barut [3] has regarded the proton as a dyonium atom made out of spinless magnetic monopoles. In his model the neutron is made up of a proton and the negatively charged (e-v)-particle system, which effectively acts as a single charged spinlessparticle B-. It is an attractive model to consider, but there are several difficulties present from the very beginning. For example, like quarks, the magnetic monopoles have not shown up in the experiments designed to detect them. Also, if B- is identified as a valence rr-, then the explanation of beta decay is complicated by the possible emission of p-. Otherwise, one must introduce an unobserved boson B-. According to Goebel [14], such a model amounts to the identification of monopoles with quarks. However, since monopoles are spinless, this analogy seemsto be incorrect [15]. In the model of Drell and Johnson [16] a baryon is pictured as three valence quarks plus a “soul” whose existence is necessary to obtain a simple antisymmetric ground-state wavefunction. It also involves a short-range quark-quark repulsion to give the required l/t2 fall-off of the electromagnetic form. The concept of “soul” remains a mystery. Kuti and Weisskopf [17] consider the nucleon to be a three-valence quark structure accompanied by a core of virtual quark-antiquark pairs. As nothing is known about the mediation of the interaction between quarks, they assume it in terms of gluons. It is not clear how the valence quarks are prevented from coming out in a scattering experiment. If nuclear forces are of the same nature as chemical forces, then nucleons should be composite systems, like a hydrogen atom. A favorite model [18] is to treat them as atomlike systems consisting of a nucleonic core surrounded
114
B. K. AGARWAL
by a pionic cloud, as suggested by Yukawa’s theory. By assuming a pion-nucleon potential of the form
&
= f (a,v. MfN - C-Jv(r) W, 11,
W> < 0,
(1)
one obtains a strongly attractive state j = 4, t = 4, since
.I, -1 2Y;.t,I = +*
for for
j, t = 4 j, t = 8.
(2)
It is repulsive in the j = 4 , t = $ and j = $ , t = 4 states. An attractive state, but less strong than the ground state, is obtained for j = t = 4, which may be interpreted as a resonance state. The pion orbital momentum can be exploited to obtain values = f3 nm for the magnetic moments of the physical proton and neutron. However, despite its initial success, this model is also beset with some insurmountable difficulties. For example, the concept of the nucleonic core is not clear. Also, the dipole fit to the e-N scattering form factor implies that the nucieons do not have any charged core, but rather can be described in terms of a continuous (e.g., exponential) distribution of charge and magnetization. The equal and opposite values for the magnetic moments of p and n are also not in agreement with observations. In view of such shortcomings, we have looked for the various atomic models available in the old literature and not merely at the Bohr model. A well-known and useful model for our purpose is provided by the Thomson watermelon model [19] of the atom, which on analysis leads to a harmonic oscillator potential.
3. PION-MUON
MODEL
FOR NUCLEONS
We shall work with a diquark (44) called boson B and a quark (q) called fermion A. The Be is spinless and it carries electric charge e and a strong gluon charge +g. The Ae is a spin-4 quark, carries electric charge e, magnetic moment pa , and a strong gluon charge -g. The strong gluon potential has the long-range form -l/r. The possibility of this long-range form has been justified on various grounds [20]. The quark fields are assumed to obey (ia - m3 g(x) = c4 g(x), where m, is the bare quark mass and I(x) is the interaction term giving binding. For a sufficiently strong binding the gluon field can be treated classically and one can replace it by a c-number potential. Assuming V(r)
=
V(a)
-
b/r,
(3)
COMPOSITE
NUCLEON
AND
115
DEUTERON
Suura [20] obtains a wave equation for a hydrogen atom with corresponding hadronic mass spectrum. The presence of quark-antiquark pairs has been postulated [lo, 15-171 to explain the diffractive behavior in high-energy e-N scattering. From Lorentz covariance, charge conservation, and invariance under space reflections, we find that there exist two electromagnetic form factors for a &spin particle
where u(p) and u( p’) are the free Dirac spinors and y,, is the Dirac matrices; F,(q2) is called the Dirac form factor and F2(q2) is called the Pauli form factor. We shall refer to them as Ftcharse) and J’cmoment), respectively. We can assume, following Sachs [21], that a nonrelativistic calculation of the form factor is meaningful in the nucleon’s rest frame. Then it is possible to interpret the form factor in a classical fashion in terms of the appropriate spatial densities. We can identify B with rr and A with CL. Then the nucleon in our model is a (np) composite system. In particular, p ~= (TOP+) and n = (r+p-). There is an attractive gluon force, -g2/r2, between x and CL,where g is the gluon charge. The proton is assumed to consist of TO, smeared out in a uniform spherical distribution of positive strong gluon charge +g, with p+ of strong gluon charge -g embedded in it. The embedded TV+experiences a radial attraction -g2rlR3, where r is the distance of EL+from the center of 7~~of radius R. In other words, CL+ experiences the harmonic oscillator potential +kr2, k = g2/R3. The groundstate wavefunction of p+ is a 312 exp( - +a2r2), oscillator = - ,1/z
*
( 1
(4)
a2 = (m,k/fi2)1/2. The corresponding
charge density is poscillator
=
-$
exp(-a2r2),
leading to the form factors FWmxce) = F(moment)ppl = 9 9
(4747)
1
pOsmatO&)(sin
qr)
r
dr
(6)
= exp( - q2/4a2). This reproduces the form of the empirical gaussian model of the proton, which is in agreement with the data of Hofstadter et al. [22] at energies ~300 to 1000 MeV and four-momentum transfers -4 to 20 F-2.
116
B. K. AGARWAL
The properties of the proton and neutron are known to be different. Therefore, it would be reasonable to have a slightly different model for the neutron. We assume for the neutron a hydrogenlike composite system in which p- is the valence constituent accompanied by the central charge distribution of 7~+,under the strong gluon long-range force. The ground-state wavefunction of II- is ti hydrogen
=
($-)“*
t?Xp
(-
F),
(7)
a = 2m,g2/h2.
The corresponding
negative charge density is phydrogen
=
(8)
exp(--olr) = pn- .
c
The resulting form factors are $charge) n-
For simplicity,
=
#moment) R
p,~
= (1 + $,-‘.
we assume a gaussian distribution Pn = Pn+ pharge) n
(9)
of the 7r+ charge. Then
pn- = -$
exp(a2r2) - $- exp(--olr),
=
_
F(charge)
(W
FF(chawe)
= e1; (- &y--
(1 + $)-‘.
Thus, in our model, Eqs. (10) and (11) represent the neutron, with J p,(r) dr = 0. The Pharge)(q2) given by Eq. (11) is very similar in behavior to that derived by Schiff :23]. It will fit the electromagnetic neutron structure factor [22]. The scaling law in our model is &$harge)(qz) Fcmoment)(q2) 7%
= /L,Fy-)(q2) F;moment)(q2)
~ * )
(12)
which is in agreement with experiments for small four-momentum transfers. For a recent review of electromagnetic form factors of nucleons, see the article by Gourdin [24].
117
COMPOSITE NUCLEON AND DEIJTERON
4. MAGNETIC We assume that-the comes mainly from:the
MOMENTS
OF NUCLEONS
contribution to the magnetic moment of the nucleon $-spin constituent muons. Then we expect [25]
MO Pa = + __ nm, mcl+
pn = - arim mu+
>
(13)
where M, is the rest mass of the nucleon and m, is the relativistic mass of the muon in the model in the rest frame of the nucleon. For the observed value pn = -1.9 nm we need m,- = MO/l.9 = 4.7muo , where m,o is the rest mass of the muon. In other words, CL- is moving with a speed u = 0.98~. In the neutron the p-- is moving in a hydrogenlike orbit defined by v/c = g2/fic = 0.98. This fixes the value of strong gluon charge g. If we assume that p- behaves like a particle of effective mass 4.7mwo then 01 = 2g2m,-/iP = 4.92 F-l = 0.94 GeV. With this value of 01, Eq. (9) gives a reasonable fit to the experimental [22] Fimoment)(q2). To obtain the observed value of pD, we need m,+ = M,/2.79 = 3.21mGo . It implies that the TV+is moving with a speed of 0.95~. If we use the well-known relations AE = Am?,
Ax = c At = cfi/AE,
(14)
then for Am = 3.21m,o we obtain Ax = 0.58 F. Taking R = 0.58 F we find that k = g2/R3 = 9.9 x lo2 MeV F-2 and a2 = 2.93 F-2. If this value of a2 is substituted into Eq. (6), we obtain a good fit to the experimental FFharge)(q2) and Fmoment)(q2). Eq. (11) is now also in agreement with experiments. It may be meitioned that the known size of a proton and baryon level spacing requires [26] an effective quark mass around 300 MeV m 3mGo. Thus, we have shown that our model contains a possible mechanism for a large effective mass of muons, which fixes the single parameter g of the model, and which in a consistent way leads both to the observed magnetic moments and the electromagnetic form factors of the proton and neutron. This avoids the use of any unobserved constituent quarks or partons. Also, the model is more viable than the pion-atomic model [18] of nucleons. Since +iw = h(k/m,)1/2 * mrr and $iio = m, , the ground-state (or zero-point) energy will appear as a source for virtual pion cloud. The breakup of the proton into its constituents can be avoided by a new assumption: All the energy given to the system is absorbed to create pions from the single level ground state or muons from the two-level transitions, the former process being more probable.
118
B. K. AGARWAL 5. BETA DECAY
In the long-range potential model of Suura [20], the constant term V(co) in the potential (3) is related to a self-mass, which acts as a trapping barrier for quarks. Then the quarks cannot propagate out asymptotically. Our model for the proton has some features of the “bag model” [27]. In the neutron, the p-is on the surface of the bag, and thus, it is allowed to decay (forced decay of I”in the presence of n+) I*- -+ e- + Pe + v, . (15) The v, is assumed to carry the same gluon charge as p-, and consequently, it cannot leave the bag. However, it can fall back into the bag leading to the process v, + 7r+ + pf + 7To= p.
(16)
This will explain the beta decay of the neutron in terms of the muon decay. We can now say that the difference between leptons (p, v,) and (e, v,) is that the former set possesses a gluon charge while the latter does not. The emission and reabsorption of this gluon field will then account for the greater mass of ~(v,,) than of e(v,). 6. DEEP INELASTIC
SCATTERING
We have seen that the (np)-atomic model of nucleons accounts for the observed electromagnetic structure features up to about 1000 MeV. What will happen in our model when, at high energies, deep inelastic scattering sets in ? In e-N scattering at low energies, the incident electron primarily interacts with p+ in the proton (no is not coupled to the photon due to zero electric charge) and with the valence p- in the neutron. It is like electron-muon scattering with a cross section of the form (17)
The proton of mass AJo sees the electron of initial energy E by receiving from it a photon of energy v = E - E’ and q2 = -4EE sin2(8/2) = -Q2, where E’ is the final energy of the electron scattered at the laboratory angle 19.For the high-energy inelastic process e + p --t p + hadrons, the cross section can be written as
da0 domott -dQdE’ = ~ dL’ [ W2(v, q2)+ Wdv, 43 2tg2;I,
(18)
COMPOSITE
NUCLEON
where W, and W, are the unknown is assumed to be absent,
AND
structure functions.
W:)(v, (2") --f -&
S (& 0
Wf)(v,
119
DEUTERON
When the structure
- v), 0
Q*) + S 2MQ2 c 0
v . 1
When the energy increases, the electron begins to probe into the (qij) pair B” in the proton, resulting in the creation of more quarks and antiquarks, B = qq + qq@ = BB --t BBB, etc.
C-21)
The diffractive scattering is now dominated by these newly created quarks. The Bjorken scaling law [28] suggests that the inelastic scattering is just the incoherent sum of the elastic scattering from the constituent pointlike quarks (or partons) [29, 301. Then Wl,, = 1 J@iYm,>,
(22)
where m, is the parton mass. Also, we can write
Wz(v,
Qz>= 2 s1 d’(x) S(x - co-‘) dx - g%rge 72 0
v W*(w),
where x = m,/M, , w = 2m,v/Q2, and f(x) dx is the probability that the parton mass is found to be between xMo and (x + dx) A4, . This is the scale invariance. If we regard the proton as moving with momentum P, then for the longitudinal direction one can write gpn=p
or
11, n
= 1,
where P, is the longitudinal momentum of the nth parton and 1, = P,/P. If L(ln, P) gives the probability distribution of the created partons, then for N created partons, d-L(x, ,..., x,> = AS(l
- ~lr,)&LP)dl,.
(2%
where A is a normalization constant. This can be integrated to yield the parton contribution to F, , provided one can guess a suitable form for L(& , P). For the proton in our model, the quark A, being embedded in the B “sea,” will also
120
B. K. AGARWAL
participate in the diffractive scattering. Therefore, we expect W, > Wn in our model when the number of created quarks N is not high. For large N, W, will approach W,, . Thus for nondiffractive scattering W, m W, , for low w diffractive scattering W, > W, , and for high w diffractive scattering again W, w W, . This is in agreement with experiments [31]. 7.
THE
DEUTERON
We have found that the (~11~)modified atomic model can explain many highand low-energy features of nucleons in a consistent way with only one parameter g. We can now form the deuteron on the molecular model. A simple calculation for the binding energy of the deuteron can be performed by considering the interaction Hamiltonian (26)
where +g is the gluon charge on pions, -g is the gluon charge on muons, rQZ is the distance between rr” and p-, r bl is the distance between rrf and $-, r12 is the distance between the two muons, R, is the distance between two pions, and g2/R, is regarded as constant. To the first order of approximation, the binding energy is given by EB = j @‘*(r,, , rb2) fWr,,
, rb2) d71 dT2 9
(27)
where @ is the normalized wavefunction of the deuteron, ral is the distance between no and p+, rb2 is the distance between yr+ and TV-, and drl and dr2 are the volume elements. If &(r,,) and &,(r,,) are the normalized wavefunctions of the proton and the neutron, respectively, we have @@k , rb2) = h&J
Mrb2).
P-9
It is well known that very little error is introduced by replacing the harmonic oscillator wavefunction by the hydrogenlike wavefunction for the ground state. Then 4dr> = Mr) A straightforward
calculation EB=
+(A
ew (-
F)
-
(29)
from Eq. (27) gives a
2y = R,a.
It has a minimum
= &
at y = 1.88.
-;y+~y2+;y3)e-2y,
(30)
121
COMPOSITE NUCLEON AND DEUTERON
In our model the spin dependence comes from the two constituent muons. Since (CT,+ * o,-) = + 1 for parallel spins, a positive contribution is obtained for only parallel spins. As a result, for the deuteron we obtain an attractive potential for the 3S, state alone. We also can calculate the quadrupole moment. In our model it is given by Qe = (32” - r2)a2,= +R,2,
(31)
because the net contribution due to muons is zero, and therefore, the entire contribution comes from 7~+ situated at a distance &R, from the origin. So far we have taken g2/?ic = 0.98. This may be compared with the pionic strong coupling gr2/hc = 0.3 and the electromagnetic coupling e2/fic = l/137. We know that in treating hydrogen molecule formation one must use an effective value for the electronic charge to obtain an agreement with the experiment. Analogous calculation for the deuteron formation will need an effective value of the gluon charge g, especially when the coupling involved is so large. However, we must require that both Es and Qe are simultaneously in agreement with observations for the same effective value gefP . With this requirement in mind, we find g&/!ic = 0.48. Then aeff = 2.51 F-l and EB = -2.22 MeV
(-2.2246 MeV),
Q
2J - 1 Qe = 2.8 x 1O-27 cm2 max= 2(J+ 1)
(32)
(2.796 x 1O-27 cm2),
where quantities in parentheses are the corresponding Thus, the composite model of nucleons leads to an for the deuteron. It is able to account for (i) parallel energy, and (iii) electric quadrupole moment of the any admixture of D states.
(33)
experimental values [32, 331. acceptable molecular model spins of n and p, (ii) binding deuteron, without requiring
8. FORM FACTORS FOR THE DEUTERON
Following Hofstadter
[34], we can express the form factors for the deuteron as
pharge)(q2) D p-#jmomeW(q2)
= [~~hawe)(q2j =
[F;moment)(q2)
+ F;hawe)(qpl +
FfmmeW(q2j]
fD , fD
(34) ,
(35)
122
B. K. AGARWAL
where fD(q2) is the form factor of the deuteron when p and n are point particles and FDsn are given by Eqs. (6) and (11). We can writef, as fD(q2)
= T J”om exp(p2hr)
sin qr dr, (36)
x = ( -yny2.
Calculations
for the deuteron form factors give, for q2 = 1, 3, 4 F-2, FC$%w) = 0.48 (0.55), 0.26 (0.26), 0.21 (O.lS),
pilFtymw
= 0.41 (0.41), 0.25 (0.26), 0.18 (O.lS),
where the quantities in parentheses are the experimental the agreement is reasonable.
(37) (38)
values 135, 361. Again,
9. VAN DER WAALS FORCEAND NEUTRONMATTER When the interneutron distance is large, the exchange phenomenon is unimportant and even in this case we can write H’ as in (26). We can expand H’ in a Taylor series in inverse powers of R, . With the two neutrons located on the z-axis,
H’ = &
+ y, y, - 2~~2,)
(x1x2
+ l+$ n lr12z2- r22z1+
+
(2x92
where x1 , y1 , z1 are coordinates of the are coordinates of the second p- relative the dipole moments of the two neutrons. easily in the second-order perturbation W(R,)
>
-
F
zz
-
- 3z,z,)(z, - z2)> + ***, (39)
first EL- relative to its n+, and x2 , yz , z2 to its 7~f. The first term in (39) represents The dipole-dipole interaction is evaluated theory to give 1371
0.02 R6’ 77
__
2Yo2
(A = c = in,, = 1).
(40)
Recently, a Van der Waals force has been employed to fit the Reid potential (soft core) between two nucleons and to develop a corresponding state approach to nuclear and neutron-star matter [38]. In view of the result (40), our model can provide a suitable basis for such an approach. Recently, Sawada [39] has
COMPOSITE
NUCLEON
AND
123
DEUTERON
shown that we should add a long-range Van der Waals potential, -C/R6, to the nucleon-nucleon potential. He needs C = 0.12, while Barut [40] gets C = 1O-g - lo-lo for his monopole model. Our value is close to Sawada’s prediction.
10.
CONCLUDING
REMARKS
We have shown that a boson B and fermion A bound in an atomlike state under a strong long-range gluon force provide a suitable model for nucleons. This composite model is suggested by the high-energy e-N scattering experiments and the resulting form factors. The form factors of nucleons and the scaling behavior of the structure factor W, are obtained easily in this model. The reason for the scaling behavior also can be seen as follows. The virtual photoproduction cross section is the imaginary part of the forward Compton scattering amplitude. In our model it looks like Fig. 1, and the scaling behavior of the structure function arises due to the appearance of the proton as core B” plus quark A+, with A+ propagating between the photons. If the four-momenta are labeled as in Fig. 1, proJon
1; I PII J-4 I I
I 01
Bl
FIG.
1.
Forward
q
XP
; I A+
Compton scattering in a (PA ‘1
model
of the proton.
then apart from the spin-dependent parts, which are immaterial here, and neglecting any momenta transverse to the proton-photon axis [41], we have for the quark propagator [(xp + q)2 - n1,$2- k-1.
(41)
The imaginary part of the amplitude includes the delta function S(x2MBz + q2 + 2xp . q - mA2), which comes from the quark propagator.
124
B. K. AGARWAL
Neglecting x2MP2 and mA2for very large v, q2, we find that the amplitude contains the delta function constraint
so that the structure function is dependent only on the ratio v/Q,. The strength of our model is that, unlike other parton models, Fig. 1 is the only box diagram possible for Compton scattering (B” has zero charge), yp -+ yp, and therefore the structure function will scale. If we identify B with rr and A with CL,then our model can be used to calculate the magnetic moments of the proton and neutron by fixing the single parameter of the theory, g. One can form the deuteron on a molecular model and correctly predict (i) the binding energy, (ii) electric quadrupole moment, and (iii) form factors of the deuteron. Despite this phenomenal success of the model, some questions remain unanswered. The mass of the composite nucleons has not been calculated [42]. If all the zero-point energy is assumed to be stored by the constituent particles in the form of effective mass, one can think of a dynamical equilibrium, say in the proton, via the following sequence of processes: p = rr” + p+ + Pi -
y + y + pf -
E.L+ + I-L--+ CL++ p-
p++y+y+p++~“=P.
To conclude, our work shows that once the high-energy experiments have suggested a composite model for the nucleon, it follows that a strong long-range potential can account for the short-range nature of the n-p potential by regarding the deuteron on a molecular model requiring hybridization of muon clouds. This picture involves no unobserved quarks and only one parameter, g, is sufficient to understand the basic features of nucleons and deuteron. Thus, both the high-energy electron scattering experiments and low-energy nuclear phenomena can be described in terms of an explicit force, bringing it in line with gravitational and electromagnetic forces in nature. This has more esthetic appeal. ACKNOWLEDGMENTS The author is grateful to Prof. K. M. Khanna and Drs. S. C. K. Nair and S. B. Khadkikar for helpful discussions. He is also grateful to Prof. Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, Italy.
COMPOSITE NUCLEON AND DEUTERON
125
Added in Proof. In view of Eq. (14), m, N 3.2 rn&,, and m, N 3.2 m,, . In a simple way, N m, + m, + @J - $.(g2/R) N 930 MeV. Note that the x0 lifetime is much smaller than the p+ lifetime. Note
M,,
REFERENCES 1. P. MARMIER AND E. SHELDON, “Physics of Nuclei and Particles,” Vol. I, p. 53, Academic Press, 1969. 2. W. HEISENBERG,“Nuclear Physics,” Methuen, London, 1953. 3. See, for example, A. 0. Barut, Nature of nuclear bond and hadron structure, in “The Structure of Matter” (B. G. Wybourne, Ed.), University of Canterbury, New Zealand, 1972. 4. W. HEISENBERG,Solvay Conference, 1933. 5. D. IVANENKO AND I. TAMM, Nature (London) 133 (1934). 981. 6. G. WENTZEL, Heft. Phys. Acfa 10 (1936), 107. 7. G. GAMOW AND E. TELLER, Phys. Rev. 51 (1937), 289. 8. H. YUKAWA, Proc. Phys. Math. Sot. (Japan) 17 (1935), 48. 9. B. K. AGARWAL, Indian J. Phys. 34 (1971), 406. 10. B. K. AGARWAL, Lettere Nuovo Cim. 9 (1974), 185. 11. B. K. AGARWAL, Lettere Nuovo Cim. 9 (1974), 294. 12. B. K. AGARWAL, Y. M. GUPTA, AND C. WATAL, “Some Implications of the Composite Model of Nucleon,” Allahabad University Preprint BKA/74-3 (1974). 13. B. K. AGARWAL, Y. M. GUPTA, AND C. WATAL, “A Composite Model for Nucleon and Deuteron,” Allahabad University Preprint BKA/75-1 (1975). 14. C. J. GOEBEL, in “Quanta” (P. G. 0. Freund, C. J. Goebel, and Y. Nambu, Eds.), p. 338, New York, 1970. 15. A. 0. BARUT, private communication. 16. S. D. DRELL AND K. JOHNSON, in “Proceedings of the XVI International Conference on High Energy Physics” (J. D. Jackson and A. Roberts, Eds.), p. 801, Chicago, Ill, 1972. 17. J. KUTI AND V. F. WEISSKOPF,Phys. Rev. D4 (1971), 3418. 18. B. T. FELD, Ann. Pfzys. (N.Y.) 1 (1957), 58. 19. J. J. THOMSON, Philos. Mag. 7 (1904), 237. 20. H. SUURA, Preprints DESY 72/7 (1972) and DESY 72/47 (1972). 21. R. G. SACHS, Phys. Rev. 126 (1962), 2256. 22. R. HOFSTADTER, F. BUMILLER, AND Y. YEARIAN, Rev. Mod. Phys. 30 (1958), 482. T. A. GRIFFY, R. HOFSTADTER, E. B. HUGHES, T. JANSSENS,AND M. R. YEARIAN, in “Nuclear Structure” (R. Hofstadter and L. I. Schiff, Eds.), p. 61, Stanford, Calif., 1964. 23. L. I. SCHIFF,Rev. Mod. Phys. 30 (1958), 462. 24. M. GOURDIN, Phys. Rept. 99C (1974), 30. 25. E. M. HENLEY AND W. THIRRING, “Elementary Quantum Field Theory,” p. 230, McGrawHill, New York, 1962. 26. See, for example, G. MORPURGO, The quark model, in “Proceedings of the XIV International Conference on High Energy Physics,” Vienna, 1968. 27. A. CHODOS, R. L. JAFFE, K. JOHNSON,C. B. THORN, AND V. F. WEISSKOPF,Phys. Rev. 9D (1974), 3471. 28. J. D. BJORKEN, Phys. Rev. 179 (1969), 1547. 29. R. P. FEYNMAN, Phys. Rev. Letters 23 (1969), 1415. 30. J. D. BJORKEN AND E. A. PAXHOS, Phys. Rev. 185 (1969), 1975.
126
B.
K.
AGARWAL
R. SOGARD, in “Proceedings of the XVI International Conference on High Energy Physics,” (J. D. Jackson and A. Roberts, Eds.), Vol. 2, p. 98, Chicago, Ill., 1972. 32. W. V. PRESTWICK, Can. J. Phys. 43 (1965), 2086. 33. H. NAMURI AND T. WATANABE, Progr. Theoret. Phys. (Kyoto) 35 (1966), 1154. 34. R. HOFSTADTER, An. Rev. Nucl. Sci. 7 (1957), 231. 35. N. K. GLENDENNING AND G. KRAMER, Phys. Rev. 126 (1962), 2159. 36. J. I. FRIEDMAN, H. W. KENDALL, AND P. A. M. GRAM, Phys. Rev. 120 (1960), 992. 37. See, for example, L. I. SCHIFF, “Quantum Mechanics,” 2nd ed., p. 179, McGraw-Hill, New York, 1955. 38. R. G. PALMER AND P. W. ANDERSON, Phys. Rev. 9D (1974), 3281. 39. T. SAWADA, Tokyo University of Education, Preprint TUETP-74-9 (1974). 40. A. 0. BARUT AND J. NAGEL, Phys. Letters 55B (1975), 147. 41. F. E. CLOSE, Daresbury Laboratory Preprints DNPL/P177 (1973) and DNPL/R31 (1973). 31. M.