Valence quark distributions in the soliton bag model

Volume 215, number 2

PHYSICSLETTERSB

15 December 1988

VALENCE QUARK DISTRIBUTIONS IN THE S O L I T O N BAG M O D E L C.J. BENESH l and G.A. MILLER

Institutefor Nuclear Theory, Departmentof Physics, FM-15, Universityof Washington, Seattle, WA 98195, USA Received 6 September 1988

Valence quark distributions are calculated in a momentum projected version of the soliton bag model. These distributions are evolved to Q2= 15 GeV2 using QCD perturbation theory, where they are compared with experiment. Substantial agreement is achieved.

1. Introduction In this letter, the authors continue the search for a model of hadrons that may be used for both high and low energy phenomenology. As in previous calculations [ 1 ], we use a Peierls-Yoccoz projected bag wavefunction and calculate the Bjorken limit of the current-current correlation function. These distributions are interpreted as the twist-two piece of the nucleon structure function evaluated at a low momentum scale Q2 =/z2 ,,~0.5 GeV 2. QCD perturbation theory is then used to evolve the distributions to Q2= 15 GeV 2, where higher twist effects are small, and comparison with experiment is made.

2. Interpretation of quark model calculations In the calculations that follow, we shall adopt a point of view suggested by Jaffe and Ross [ 2 ]. In particular, we shall assume that the calculations of structure functions in the bag represent the twist-two piece of the physical structure functions evaluated at a low value of Q:=/~2. The motivation for this is the observation from QCD that the quark structure of the nucleon changes with the scale Q2 at which one probes it. Thus, if the nucleon looks like three valence quarks in a confining interaction at some scale/Zo2, radiative QCD corrections will change its composition at higher Address after August 1, 1988:PhysicsDepartment, IowaState University, Ames, IA 50011, USA. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

Q2. Quarks will radiate gluons and these in turn will pair produce quarks until the nucleon becomes a very complicated object. Knowing the twist-two piece of the structure function at Q2=;tg, we may use QCD perturbation theory to determine it at higher Qe. At high enough Q2, higher twist effects will become negligible and we may compare the resulting distributions with data. Note that this prescription implies that the structure functions calculated at Qe=/z2 should not look like any data, since the physical structure function certainly has important contributions from all twists at this scale.

3. The model The calculations to follow are performed using the coherent state projection techniques developed by Wilets et al. [3] and the Friedberg-Lee [4] soliton model. The lagrangian for the model is given by

£#=itpyuOu~t-gatfiv/+ ½(Oua)2-U(a) ,

(3.1)

where U(a) = ½aa2+ (b/3!)a3 + (c/4!)a4+da+p. In ref. [ 3 ], the model is solved by making a trial wavefunction for the ground state of the form IN)= f d3ab*(a)bt(a)bt(a)lEB;a) ,

(3.2)

d

where the b*'s create quarks whose wavefunctions are g o ( r - a ) and 381

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Table 1 Low energy nucleon properties Low energy observables

Calculated

Experiment a)

EN

930 MeV 279 MeV -- 100 MeV 193 MeV 0.83 fm 2.67 1.27

939 MeV

% ~OOE ~o

(rE) 1/2

pp

gA

0.83 fm 2.76 1.26

a) From ref. [ 3 ].

(3.3)

the m e t h o d s ofref. [ 1 ] to the soliton model. This has three advantages over our previous calculations in the M I T bag. First, the m o m e n t u m projection techniques o f ref. [ 3 ] allow an u n a m b i g u o u s determination o f the mass o f the system including center-ofmass corrections which has not been available in our previous work. Second, in this m o d e l the confining forces are represented by a d y n a m i c a l field a n d therefore the valence quarks are not required to carry all the m o m e n t u m . Finally, the m o d e l p r o v i d e s a description o f the e m p t y bag state a n d therefore we m a y calculate the e m p t y b a g - e m p t y bag m a t r i x elements that appear in the expressions for quark distributions. In the calculations that follow, we shall use a covariantly n o r m a l i z e d m o m e n t u m eigenstate, i.e. IN p = 0 ) = f l f d3a

is a coherent state o f the tr field with m e a n value ao(r-a). The functional forms ofgto a n d ao are given by

(3.4)

where the functions #, t~and V(r) are the m e a n field solutions o f the model, a n d ~, a n d 2 are v a r i a t i o n a l parameters. The state described in eq. (3.2) is manifestly an eigenstate o f zero m o m e n t u m . A n a p p r o x i m a t e energy eigenstate is o b t a i n e d by m i n i m i z i n g the hamilt o n i a n d e r i v e d from (3.1) (H)=

(NIHIN) (NIN)

= f l f d3a

(3.5)

a n d identifying ( H ) as the ground state mass. Using these methods, Wilets et al. o b t a i n a very good fit to the low energy p r o p e r t i e s o f the nucleon which is s u m m a r i z e d in table 1. A similar calculation for mesons yields m ~ - 177 MeV, mp---7 35 M e V a n d f== 147 MeV. Thus, the m o d e l p r o v i d e s a very successful low energy phenomenology.

~2

I EB; a)

[R=a) ,

(4.1)

4. Calculation of structure functions In this section, we seek to extend the success o f Wilets et al. to the deep inelastic regime by applying

2M f d3a

(R=aIR=O)

We wish to calculate the Bjorken limit o f

l;

W ul,----~

d4~e iq~

× (Np=OI[Ja(¢),J,(O)IINp=O) ,

(4.2)

where Ju(~) = ~(~)~'u~(~) is the usual vector current operator. M a k i n g the usual p a r t o n m o d e l assumptions ~1,2, eq. (4.2) reduces to ( x - -qZ/2p.q)

Wu,, -\-{ quq,,q2 -guy )El (x) +(Pu

#2

382

b~(a)b~(a)b~(a)

where

ao(r)-Crv = [~o(r/2)-avl,

{ O(r12) 9'o(r) =~ itr.Py~'(r/2 ) } '

15 December 1988

P'qqu'~{

f.

p'qqv.'~F2(x)

q2 ) ;.q

For a derivation of this see ref. [ 5 ]. One may wonder about the appropriateness of the parton model here, since the lagrangian ( 3.1 ) is not asymptotically free. The tr field is, however, a low energy effective field presumably representing a gluon condensate and therefore it is inconsistent to consider loops involving the a field. It is not unreasonable to expect that the high momentum quarks will not see the a field at all.

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15 December 1988

@

where

Fl(x) = ~x Fz(x),

F2(x)= Z x[qi(x)+qi(x)],

i=u,d

If d~-e iq+~-

Fig. 1. Bubble diagram.

qi(x)= ~

~

X ¢ ¢+=¢~=o'

P(K)O°(K)7+eo(K),

K+

(4.4 cont'd ) q~(x) = ~

d ~ - e ~+¢-

where

P(K) =

X ~ ~+=¢1=o'

f d3ze-iXzA2(z) (EB; zlEB; 0 ) ,

~o(K) = ~ d3ze-i~'zVo(l~) ,

~+= ~o+~3

q+_ -Mx

7+= Y°+73

(4.3) K± = IMx+ eq [ ,

The matrix elements may be evaluated in exactly the same fashion as in ref. [ 1 ] yielding ~/-2'b'2 i

q(x)= ~

KdK

K-

~

-

A(Z):{bt(z), b(0)}=

'

'

I

'

I

'

./..""..,,. /:....

.8

(4.5)

and eq is the single quark energy calculated in ref. [ 3 ]. A problem that is apparent in the analytic expressions is that q ( x ) < 0 . This is a result of our neglect

+

(-'~n)aF(K)Oo(K)?,~o(K), (4.4) '

~ d3r~t(r-z)gt(r),

'.

'

'

I

'

--Soliton Bag ................ MIT Bag

'

'

-~ _J

.

H N .~.4 ~ f X X

~"~"~"....,.........

I

... ...... i

o

0

i

I .2

.4

Bjorken x

.6

Fig. 2. Valence quark distributions at the bag scale for MIT and soliton bag

.8

I

(MN=1030 MeV). 383

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'

'

I

'

'

'

I

'

'

'

15 December 1988

I

I

I

I

I

............... p . o z = . 7 ........

•

i

Gev 2

--#ore=.6

.8

I

G e v ~'

#0~=.8 Gev 8 CDHS

Data

tt) II

.6

X I

.4

N

"-.. ,;

.2

0

0

.2

.4

.~.,"

Bjorken

x

.6

.8

Fig. 3. Results of QCD evolution to Q2= 15 GeV2 for MN= 1030 MeV. o f the bubble graph shown in fig. 1. According to Jaffe [ 5 ], quark distributions are not required to be positive definite if the bubble graph is neglected. This graph will not contribute to the valence quark distributions we are calculating a n d hence we will not concern ourselves with it further. Neglecting the one-gluon-exchange c o n t r i b u t i o n to the nucleon mass ~3, we o b t a i n the valence quark dist r i b u t i o n shown in fig. 2. F o r c o m p a r i s o n , we have also shown the valence quark distribution for the M I T bag calculated in ref. [ 1 ]. The smaller area u n d e r the soliton curve is a result o f the m o m e n t u m carried b y the cr degrees o f freedom, which we find to be about 30%. The spreading o f the curve is a result o f the fact that the quark wavefunctions are not a p p r o x i m a t e eigenstates o f IP[, as in the M I T bag. The valence distribution is evolved to Q2 = 15 G e V 2 using next to leading o r d e r Q C D p e r t u r b a t i o n [6] theory for p ~ = 0 . 6 , 0.7, 0.8 G e V 2. The results are shown in fig. 3, along with d a t a for XUv- xdv from the C D H S Collaboration [ 7 ]. The calculated curves track the d a t a for x>~0.3 a n d are too large in the smaller x ~3 To be completely consistent, one should calculate all the order txs corrections, which we have not done. 384

region. In this region one expects the effects o f virtual mesons in the nucleon wavefunction to become nonnegligible. In particular, these mesons will create a n o n p e r t u r b a t i v e sea ~4, which will cause the valence quarks to lose m o m e n t u m and hopefully lower the curve at small x. In fig. 3, we show the evolved curves for p~ = 0 . 4 , 0.5, 0.6 G e V 2 including one-gluon-exchange corrections to the nucleon mass. The theoretical curves are closer to the d a t a for the entire range o f x . Again, one m a y hope that the inclusion o f one-meson-exchange corrections will improve the description even further.

5. Discussion

We mention some problems with our approach that should be m e n t i o n e d p r i o r to our final assessment o f the results shown in figs. 3 a n d 4. First, we observe that the soliton bag structure functions have a tail o f small m a g n i t u d e that extends b e y o n d x = 1. (The support is not perfect.) We believe this is due to our use o f a p r o t o n state that is not an exact eigenstate o f #4 For a discussion of this see ref. [ 8 ].

Volume 215, n u m b e r 2

P H Y S I C S LETTERS B

'

'

'

I

'

'

'

I

'

'

'

I

15 December 1988

I

}

I

I

,

i

--~u02=.4 Gee ~ .............. /.t,o2~,5 Gev 2 . . . . . . . . ./~o2=.6 Gev 2

.8

•

m t,

CDHS

Data

to ii

.6

x i

.4

"'.-.....~

x

.2

.2

.4

Fig. 4. Results of Q C D evolution to

the model hamiltonian. Our proton has small admixtures of excited states of mass higher than the proton, and this causes values of x higher than unity to be allowed. A related question concerns the sum rule on the number of valence quarks. Our calculation respects this sum rule if one integrates over the entire region, x>_-0. I f one integrates from 0 to 1, there is a small error (about 1%) caused by the extended tail of the distribution. We expect both problems (at high x) to disappear as the ansatz for the nucleon wavefunction is improved. Thomas [ 9 ] has discussed another approach which successfully deals with these support problems. Note also that figs. 3 and 4 show some oscillations in the quark distributions for values o f x near 1. This is discussed in ref. [ 1 ]. The oscillations at high x are a result of truncating the Legendre expansion we use [ 1 ] for m o m e n t u m evolution at ten terms. This is sufficient for convergence of the distribution for values o f x up to 0.7, where the experimental values of the distributions are large. These oscillations would become progressively smaller with the inclusion of more terms. We did not incorporate such terms because of the support problems at high x. We conclude from figs. 3 and 4 that the momen-

Bjorken x

Q2=

.6

15 GeV 2 for M

.8

N=

930 MeV.

tum projected soliton bag model of ref. [ 3 ] provides us with a good description of both the high and low energy scattering properties of the nucleon, except at high x. As we have stressed, the model ofref. [ 3 ] does not incorporate chiral symmetry and consequently cannot describe the low x region where meson exchange effects are expected to be important. Recently, several authors [ 10 ] have considered soliton models that have chiral symmetry. Perhaps the combination of the ideas ofref. [ 3 ] and chiral symmetry will yield improved results for the valence distributions. Another possibility is to consider other distributions measurable in deep inelastic scattering. O f particular interest are the spin dependent structure functions gt and g2. It would also be interesting to extend our ideas to the valence distributions of the pion, and to sea distributions in general. This would require a better understanding of one gluon effects than we presently have. Finally, we believe the model used here provides a sufficiently good description of deep inelastic scattering off a free nucleon that we may apply it to calculate deep inelastic scattering from the nucleus, and hopefully shed new light on the origin of the EMC effect. 385

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References [ 1 ] C.J. Benesh and G.A. Miller, Phys. Rev. D 36 ( 1987 ) 1344; Phys. Rev. D, to be published. [2] R.L. Jaffe and G.G. Ross, Phys. Lett. B 93 (1980) 313. [3] M.C. Birse, E.M. Henley, G. Liibeck and L. Wilets, in: Solitons in nuclear and elementary particle physics, Proc. 1984 Lewes Workshop, eds. A. Chodos, E. Hadjimichael and H.C. Tze (World Scientific, Singapore, 1984) p. 189; E.G. Liibeck, M.C. Birse, E.M. Henley and L. Wilets, Phys. Rev. D 33 (1986) 234. [4 ] R. Friedberg and T.D. Lee, Phys. Rev. D 15 (1977) 1694; D 16 1977) 1096;D 18 (1978)2623.

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15 December 1988

[5] R.L. Jaffe, in: Relativistic dynamics and quark nuclear physics, Proc. Los Alamos School ( 1985 ), eds. M.B. Johnson and A. Picklesimer (Wiley, New York, 1986). [6] A.J. Buras, Rev. Mod. Phys. 52 (1980) 199. [7] CDHS Collab., H. Abramowicz et al., Z. Phys. C 17 (1983) 283. [8] N.N. Nikolaev, Tokyo Report No. INS-Rep-539 ( 1985 ). [ 9 ] A.W. Thomas, at Topical Conf. on Nuclear chromodynamics (Argonne National Laboratory, May 1988 ), to be published. [10] A.G. Williams and L.R. Dodd, Phys. Rev. D 37 (1988) 1971; G. Fai, R.J. Perry and L. Wilets, Phys. Lett. B 208 (1988) 1.