The spin content of the proton in a chiral quark model

Volume 232, number 2

PHYSICS LETTERS B

30 November 1989

T I l E S P I N C O N T E N T O F T H E P R O T O N IN A C H I R A L QUARK M O D E L M. W A K A M A T S U Department of Physics, Faculty of Science, Osaka University, Jbyonaka, Osaka 560, Japan Received 9 June 1989; revised manuscript received 31 July 1989

The spin content of the proton is investigated within the framework ofa chiral quark model. A main objective of the present study is to throw light on the underlying quark model foundation of the unique prediction of the Skyrmc model concerning the spin dependent structure function of the proton.

The Skyrme model has been providing one of the most exciting topics in the realm of hadron physics in recent years [ 1,2 ]. Although its quantitative predictions should not be taken too rigorously, its theoretical consistency has been tested in various ways. Among others, the underlying meaning of the identification of the topological charge as the baryon number has been made clear through the analysis of the gauged W e s s - Z u m i n o - W i t t e n action [3,4]. Another interesting observation beyond the single baryon problem is the correct reproduction of the two-body meson-exchange currents [5,6] as well as the longrange nuclear forces, based on the product ansatz of the two Skyrme solitons. All these facts as well as many other observations seem to support the correctness of the basic idea of the Skyrme model. An important questions remains, however. How can we test experimentally the fundamental idea of the Skyrme model? Unfortunately, low energy observables of the nucleon such as the charge radius, the magnetic moments etc. are reproduced by many other existing models of the nucleon even better than the Skyrme model. We must therefore admit that we could not so far find any positive reason that favors the Skyrme model to the others. The situation has been changed remarkably after the recent measurement of the spin dependent structure functions of the proton by high energy muons [ 7 ]. According to the analysis by the EMC group, the fraction of the proton spin carried by the quarks was found to be almost zero. (A more precise statement is that the matrix element of the flavor-singlet axial-

vector current between the proton states is nearly zero.) Interestingly enough, it was pointed out by Brodsky, Ellis and Karliner [ 8 ] that this observation is consistent with the prediction of the Skyrme model, whereas many other models of the nucleon such as the non-relativistic quark model and the MIT bag model etc. predict a much larger value (not extremely smaller than 1 ) for the same quantity. The EMC experiment can therefore be a touchstone of the idea of the Skyrme model. Here we do not immediately intend to explain the experimental data, but rathcr try to clari~ the underlying reason of this fairly unique prediction of the Skyrme model. (For the same reason wc concentrate on the two-flavor model. ) In other words, an attempt is made to translate the physical content of the Skyrme model into a quark language. A hint for carrying out this program lies in the work of Kahana and Ripka [ 9 ]. The startingpoint of their analysis is the linear sigma-quark model. After solving the Dirac equation for the quarks under the influence of the external background potential of hedgehog shape (which might be identified with the meson configuration in the Skyrme model), the valence and Dirac-sea quark contributions to the baryon density are calculated. They thus find that, for sufficiently large soliton sizes, the resultant baryon density is nearly identical to that obtained from the meson configuration as the topological current density of the Skyrme model. A prominent feature of their work is to have provided a point of contact between the quark model and the Skyrmelike topological soliton model in a most clear fashion.

0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

251

Volume 232, number 2

PHYSICS LETTERS B

As we shall see, however, there exists in the Skyrme model essential dissimilarity between the quantities like the bah'on number and those like the spin and/ or isospin. A peculiar feature of the Skyrme model will become more transparent through an analogous analysis below for the latter quantitics. Here, for pedagogical reason, we start with the Nambu-Jona-Lasinio model given as -~'NJt.= ~/)iTu0!, ~-/- 1G [ (~/ff)2-t- (/]/i75r/ff) 2 ] .

(1)

Using the well-known technique [10], this lagrangian density can be transformed into an equivalent form as follows:

- (g2/2G) (a2+n 2) .

(2)

After neglecting the quantum fluctuation of the a and the n fields, the set of equations of motion reduces to a self-consistent Hartree problem: [ i7u0~, - g ( a + i751r•n) ] ~u= 0 , o=-

G - (~v), g

G 7 r = - -- (~75rqz) , g

(3)

where ( ) stands for the ground state expectation value and correspondingly the a and n in eq. (3) should be regarded as c-number fields. (To be strict, in the infinite cutoff limit which will be discussed later, the second and third equations need some modification, since the RHS are divergent in this case. But this is irrelevant for our discussion. ) The point we want to emphasize here is that the a and 7r fields acting as external potential for the quarks are themselves formed of the quark fields. This means that any physical quantities in the present model can be described in terms of the quark variables. The self-consistency problem is of course hard to handle in practice. Following Kahana and Ripka [9], we therefore assume an appropriate configuration for the a and ¢r fields, i.e. the hedgehog shape on the chiral circle as follows:

a=f=cosF(r),

n=J,~fsin

F(r) ,

252

().l,u) ----aaa,

where hD--

a~-V

i

+tiM(cos F + i 7 5 , . i s i n F) ,

(6)

with M=gf~. Because of the v.~ term in the potential, the above single-particle orbitals are neither the eigenstates of the total angular momentum J = L + ~r nor those of the isospin ½v, but they are those of the grand spin K = J + ½r. For a technical reason necessary for obtaining the proton state with the definite spin and isospin, we shall exclude all the states with K ¢ 0 from our model space. There is no physical justification for this approximation. Because of this restriction, we must say that our proton state defined below is a "schematic" proton rather than the "real" proton. This however simplifies the problem enormously and allows us to learn much about the spin content of the proton in a transparent fashion. Now we shall proceed as follows. First, it is convenient to introduce plane-wave basis states as follows. They are introduced as eigenstates of the free Dirac hamiltonian (hD with F(r) = 0 ) labelled by their momentum k = Ikl, their parity and their grand spin K. (In practice, we shall retain K = 0 states only.) With the fixed value of k, wc have four independent solutions. They arc two positive energy solutions,

ijo(kr)lO) I~0~-> =:% k ~j,(kr) 12) ij~ (kr)12 I¢o~) =Nk

] (7)

with the opposite parities, and two negative energy solutions,

(4)

with the boundary condition F(0)=zr, F ( o ¢ ) = 0 . The quark orbitals are therefore eigenstates of the Dirac hamiltonian hD hD I;t) = e a l 2 ) ,

30 November 1989

(5)

=~

i ~ j ok ( k r ) l O ) (8)

--Jl (kr)12)

Volume 232, number 2

PHYSICS LETTERS B

30 November 1989

I i ~ j ,k( k r ) 1 2 > ,

0-:M

O:M

(8 cont'd)

0":M-1

O':M

jo(kr) lO)

0*(valence) Here I 0 ) and 12 ) are angular wave functions given as

IO>=I(L=O)J=~;K=O) , 12>=I(L=I)J=½;K=O> ,

(9)

in an obvious notation. (The coupling scheme is such that 3 = L + ~a followed by K = J + ½~. ) According to Kahana and Ripka [9], the plane wave basis above is made discrete by imposing the boundary' condition j~=o(kD)=0 at a distance D which is well beyond the distance where the chiral angle F(r) vanishes. Furthermore, the basis is made finite by imposing a cutoff on the momenta k as k
IH>=l-[atat at Ivae> j=O

Jl

J2 "'"

)NC

(10)

Here the number of the color Nc is assumed to be an odd integcr. (Fig. 1 illustrates the " \ ~ = 3 c a s e . ) T h e

:::i ]07:M O:M

0-:M

::=

-:

g

:

:

:

=

0":M

(b)

(a)

Fig. 1. The schematic level structure of the K=0 orbitals (a) in the absence, and (b) in the presence of the chiral background potential. indexj runs ovcr one valence (0 + ), and M 0 ÷ and M 0 - negative-energy orbitals. The ajt are the creation operator of a quark in the single particle states (iu~o°(r) IO) "] Iq/~ =°+ ) = \ v2~*)(r)12) J '

Igt~=°-)=

{iui*)(r)12)~ \v~,)(r)10 >),

(11)

with i = 0 , ..., M for 0 + and i-- 1, ..., M for 0 - states. The radial wave functions are normalized as

f drrZ{[u~i)(r)

]2+ [v~O(r)]2}= 1 ,

f d r r 2 { [ u ~ O ( r ) ] 2 + [v~i)(r)]2}=l ,

(12)

for each i. The state IH ) cannot yet be identified with a physical nucleon, because it is neither an eigenstate of ar nor that of r. The physical proton state can be constructed by using the cranking method or the generator coordinate projection method. The latter method is adopted here, since it does not suffer from the restriction of adiabatic rotation. (An analytical treatment is possible for it, owing to the restriction to the K = 0 Hilbert space.) The "schematic" proton state in this framework is defined by [ 11-13 ]

[pT)=,V I d~2Vl//~*_~/2(12)l~(ff2)lH) ,

(13)

where D~.M(O) is a usual D-function and/~ (12) is a rotation matrix in the isospace. 253

Volume 232, number 2

PHYSICS LETTERSB

The expectation value of an operator 0 between the proton states (with up spin) is given by ---

(Pl"l Olpl") (PtlPt)

(r3)-

(a3) - 2:,+1 1 (~,=o

(14)

The standard angular-momentum algebra is sufficient for evaluating this matrix element. Wc first show the answer for the expectation value of the isospin operator r3 -- Z,~,p(f),~,aa~aa:

1 (~, fdrrZ{[u~i)(r)]2+[v~2')(r)]2} ,=o

30 November 1989

f d''={[ug'(r)]~-~[v~')(r)]=}

+ ,=~"f drr2{[v~°(r)]2-~[u~°(r)]2}).

Before discussing the implication of this result, it is instructive to sce the result for the orbital angularmomentum operator/53 for quarks. The answer is

2M+~

l (~fd,'r22 (L3)=2M+l i=0 3[v~i)(r)]2

+ i=, ~ ~ drr2{[u~O(r)]2+[v~O(')'2})

+ ,=,Zf drr22[ui')(r)]2). ~ 1 ( ~ 1+ ~ 1) 2 M + 1 /=o i=l

1.

(17)

(15,

Here we have used the normalization condition (12). The final answer itself is only natural, because the definition ( 13 ) just projects the 1"=½,7-3( = ½r3) = component out of the hedgehog state IH ). What is not trivial is the detailed content ofeq. ( 15 ). It means that the proton isospin is carried equally by all the quark orbitals including the Dirac-sea ones. In particular, we do not observe any dominant role played by the valence quark orbital. This is just a typical feature of collective models. Many degrees of freedom work cooperatively to produce the total isospin of the proton in the above model. We claim that this is an underlying quark model interpretation of the Skyrme model in which the proton isospin is supposed to be carried by the collective iso-rotation. We point out that the situation for the baryon number is completely different. If we evaluate the baryon number expectation value of the proton stale defined by (13), we would obtain ( B ) = 1+2M. Here the first term is the contribution of the valence orbital and the second is that of the Dirac-sea orbitals. The latter is of course to be eliminated by the subtraction of the (unperturbed) vacuum contribution. This reveals that the baryon number of the proton is essentially carried by the valence quarks. The Diracsea quarks, although they induce the local polarization of the baryon number density, give no effect on the integrated (or net) baryon number. Now we turn to a more interesting quantity, the proton spin. The expectation value of 63 is given by 254

(16)

By using eqs. (16) and (17) together with the normalization condition (12), it is straightforward to check that (2L3 + a3 ) -= ( 2J3 ) = 1 as it should be. Different from the isospin case, the numerical value of (a3) depends on the dynamical detail of the model such as the shape o f F ( r ) and the cutoffmomentum A. The limit A ,av has a special consequence, however. Wc can show that the value of ( a 3 ) given by eq. (16) approaches ~ as A goes to infinity. This can be seen as follows. For sufficiently large momentum, the single particle wave functions are expected to approach the negative-energy plane-wave solutions, i.e.

u~i)(r)~Nijo(kir), v~°(r)~--.V, jl(k,r), u~i)(r)~ "Vj,(kir), v~°(r)-,N, jo(kir) with Ni-~{~D 3 X [j~(k~r) ]2} -1/2. (Here we have used the fact that E, = k~>> M for large enough k~. ) Then, with use of the identities of the spherical Bessel functions, D

D

drr2jo(kir)jo(l%r)= f dr r2j'(k'r)j'(kfl ") 0

0

=(~ij'½O3[j] (ktD )12 ,

(18)

where k,D and kiD are two zero's of jo(z), it is straightforward to show that

f drr2[u~i)(r)]2-+½, f drr2[v(2i)(r)]2 ,12, f drr2[u~i'(r)]',½, f drr2[t,'~"(r)]2--+½, or equivalently'

(19)

Volume 232, number 2

f

PHYSICS LETTERS B

drr2{[u~i)(r)]2 -- ~[v~i)(r) ] 2,; ,!3,

f dr r2{ [v~')(r) ]2_ [u~O(r)]2}_.~

9

(20)

Since there are infinitely many such orbitals in the A_.cc limit, it is clear that given by eq. (16) approaches -~. It is also obvious that the expectation value = ~ and the prediction of the Skyrme model < a 3 ) = - 0 [8] ). This is of course due to the incompleteness of our treatment in which the effect of the K ¢ 0 Dirac-sea orbitals are neglected. It is naturally expected that, if we include such Diracsea orbitals with larger and larger grand spin K (largc K means large L), more and more total spin would be eaten up by the orbital part and the expectation value (a3> becomes much smaller than (possibly zero) in the A _ . ~ limit. Unfortunately we could not show this explicitly, because the projection method does not allow analytical treatment with inclusion of the K # 0 orbitals. From the analysis so far, the importance of the Dirac-sea orbitals seems self-evident. Without calling out these degrees of freedom, it would be difficult to obtain the spin expectation value much smaller than 1. This can be most directly seen by excluding those from the model space, i.e. by setting IH> = 4 nt tit I vac ), ( this is the valence quark approxa,,, - v_~........... imation ), and rcpeting the following analysis. In this case, we get (a3)=

fdrr2{[u~,°)(r)]Z-~[v$°)(r)]2},

(21/

where u~°)(r) and vl°)(r) are the radial wave functions of the valence orbital. Although the numerical value of the above matrix element depends on the wave function of the valence orbital, it would be

30 November 1989

highly accidental if this gives a value extremely smaller than 1. It should be noticed that the above result is entirely independent of the color number Nc of quarks, so that nothing is changed by taking the limit N c - * ~ . This throws a little doubt on the widespread view which seek for the point of contact between the quark model and the Skyrme model in the "Vc ~ limit within the valence quark approximation [ 16,17 ]. In our opinion, what gives a close contact bctween the quark model and the Skyrme model is the Dirac-sea quark degrees of freedom. (This appears quite clear from the viewpoint of the functional bosonization of chiral quark lagrangians, through which Skyrme-likc effective meson lagrangians can be derived.) By calling out the infinitely many degrees of freedom provided by the Dirac-sea quarks, the collective iso-rotation characteristic in the Skyrme model can be precisely mimicked by the quark system. It cannot be emphasized too much that the postulated formation of the mean field potential of the hedgehog type is crucial for the realization of such a scenario. Otherwise, the spin and isospin saturated Dirac sea of quarks would give no contribution to quantities like (z3) and (a3 >. The picture explained above, although it nicely matches the Skyrme model, looks pretty radical. It appears to absolutely contradict the picture of the successful SU (6) type (valence) quark model. How can we accept such a picture that the isospin of the proton is hardly carried by the valence quark? What we have shown here is, however, that it is likely to be a quark model counterpart of the Skyrme model. A lesson wc can learn from the present analysis is therefore that something might be wrong (or oversimplified) with the Skyrme model. How can we avoid such an extreme picture, then'? We recall that the above correspondence made between the Skyrme model and the chiral quark model is exact only in the A = oo limit. The above model with the infinite cutoff turns out to be equivalent to the model of Kahana and Ripka. (The underlying reason for it is that, under a particular renormalization condition, the NJL model with the infinite cutoff can be made exactly equivalent to the linear sigma-quark model [18,19].) Their model is howcvcr known to have a difficulty of a vacuum instability paradox [191. Alternatively, we could start with the NJL lagran255

Volume 232, number 2

PHYSICS LETTERS B

gian with a finite cutoff. (This is certainly a more conventional usage of the NJL type lagrangian.) Of entirely similar nature is the chiral quark model of Diakonov, Petrov and Pobylitsa [ 20 ], which was derived from the instanton picture of the Q C D vacuum. It has a physical cutoff of about 600 MeV, which is only twice as large as the typical value of the constituent quark mass. Although we do not have enough space to explain this model, it is quite clear that the resultant physical picture is crucially dependent on the choice of the cutoff parameter, A---600 MeV or A = ~ . It is an inleresting challenge to investigate the spin a n d / o r the isospin content of the proton in such a model, without resorting to the K = 0 approximation. In summary, we have clarified the underlying quark model implication of the Skyrme model through the study of the spin and isospin content of the proton. We believe that the present analysis, although highly schematic in nature, exposed a fatal aspect of the Skyrmc model in an unprecedented form. Will such an extreme model survive in its original form? Or will partial modification be requircd for it? Or will it be completely rejected? Undoubtedly, further systematic study of low energy observables including the spin d e p e n d e n t structure function of the proton [ 21-23 ] would offer valuable information for answering these questions. The author would like to express his gratitude to W. Weise and A. Hosaka at the University of Regensburg for calling attention to the proton spin problem and for useful discussions.

256

30 November 1989

References [ 1] T.H.R. Skyrme, Proc. R. Soc. (London) A 260 ( 1961 ) 127. [2] G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. B 228 (1983) 552; G.S. Adkinsand C.R. Nappi, Nucl. Phys. B 233 (1984) 109. [3] E. Witten, Nucl. Phys. B 223 (1983) 422, 433. [41A.P. Balachandran, V.P. Nair, S.G. Rajeev and A. Stern, Phys. Rev. D 27 (1983) 1153. [5] E.M. Nyman and D.O. Riska, Nucl. Phys. A 454 (1986) 498;A 468 (1987) 473; U. Blom and D.O. Riska, Nucl. Phys. A 476 (1988) 603. [6] M. Wakamatsu and W. Weise, Nucl. Phys. A 477 (1988) 559. [ 7 ] European Muon Collab.,J. Ashman et al., Phys. Left. B 206 (1988) 364. [8] S.J. Brodsky, J. Ellis and M. Karliner, Phys. Lett. B 206 (1988) 309. [9] S. Kahana and G. Ripka, Nucl. Phys. A 429 (1984) 462. [ 10] T. Eguchiand H. Sugawara, Phys. Rev. D 10 (1974) 4275. [11 ] M.C. Birse, Phys. Rev. D 33 (1986) 1934. [12] A. Hosaka and H. Toki, Phys. Len. B 188 (1987) 301. [ 13] M. Fiolhais,IC Goeke, F. Grrimmerand J.N. Urbano, Nucl. Phys. A 481 (1988) 727. [ 14] A.D. Jackson, D.E. Kahana, L. Vepstas, H. Verschelde and E. Wrist, Nucl. Phys. A 462 (1987) 661. [ 15] B.-Y. Park and M. Rho, Z. Phys. A 331 (1988) 151. [ 16] K. Bardakci, Nucl. Phys. B 243 (1984) 197. [ 17] A.V. Manohar, Nucl. Phys. B 248 (1984) 19. [ 181T. Eguchi, Phys. Rev. D 14 (1976) 2755. [ 191 G. Ripka and S. Kahana, Phys. Rev. D 36 (1987) 1233. [ 20 ] D.I. Diakonov,V.Yu. Petrovand P.V. Pobylitsa,Nucl. Phys. B 306 (1988) 809; B 272 (1986)457. [21 ] R.L. Jaffe, Phys. Left. B 193 (1987) 101. [22 ] G. AltareUiand G.G. Ross, Phys. Lett. B 212 ( 1988) 391. [23] F.E. Close and R.G. Roberts, Phys. Rev. Lett. 60 (1988) 1471.