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Strange form factors, the σ-term and strange quark distributions in the Gross-Neveu model
Physics Letters B 268 ( 1991 ) 419-423 North-Holland
PHYSICS LETTERS B
Strange form factors, the a-term and strange quark distributions in the Gross-Neveu model Matthias Burkardt 1
Stanford LinearAccelerator Center, Stanford University, Stanford, CA 94309, USA Received 12 July 1991
For N¢--,~ the structure function of physical fermions in the Gross-Neveu model is of pure valence type. Nevertheless several strange form factors of u-quarks, including the induced magnetic moment and the a-term acquire substantial ~ (N °) contributions from virtual gs pairs. Those arise from matrix elements which are off-diagonal in Fock space, i.e., they do not have a simple interpretation in terms of the quark distributions. It is shown that, under certain circumstances, a similar behavior in QCD3+ t is conceivable.
Recently, there has been a growing discussion about the "strangeness c o n t e n t " o f the nucleon (for a review, see e.g. ref. [ 1 ] ). Although gs pairs carry less than 5% o f the nucleon m o m e n t u m [2] it has been suggested that they contribute significantly to the nucleon spin ~l and almost cancel the spin carried by uand d-quarks. A n o t h e r surprising result that emerged from an analysis o f the low energy nN forward scattering a m p l i t u d e [3 ] seems to require a large value for the strange scalar "charge" or a-term ms ( N [~s IN ) o f the nucleon. In this note I will show that a large strange a-term together with small gs structure functions does not necessarily m e a n a contradiction. Although this will be d e m o n s t r a t e d on the basis o f the G r o s s - N e v e u m o d e l [4 ] in 1 + 1 dimensions, I will argue that a priori something similar might h a p p e n in QCD3+ ~. Based on the observation that ~s is twist-three whereas structure functions are m e a s u r e d by twist-two operators it has been already e m p h a s i z e d that such a behavior is conceivable [ 5 ]. The example to be studied here will help to u n d e r s t a n d the underlying physics o f large, strange a-terms c o m b i n e d with a suppression o f strange quarks in the structure function. Supported in part by a grant from Alexander von HumboldtStiftung and in part by the Department of Energy under contract DE-AC03-76SF00515. ~l This interpretation of the EMC II effect is still controversial.
In some simple field theories (like free fermions or Q C D ~ + I ) r n s ( N l Y s l N ) has a direct p a r t o n m o d e l interpretation, i.e., one can express the strange scalar form factor in the limit o f zero m o m e n t u m transfer (the scalar charge) in terms o f the strange quark distribution functions S(Xbj ) and g(xbj). In these theories the constraint equation for the left-handed fields takes the form J 2 i O+ ~/L = m~/R,
(1)
i.e., it is linear and thus
gs=s~ SR +S~SL =X/~ ms+ (i 0 + ) - ' S R .
(2)
F o r vanishing q÷ (q~ being the m o m e n t u m transferred by the gs insertion), where p a i r creation is suppressed at the vertex, the matrix elements o f ~s become diagonal in Fock space yielding [ 6 ] 1
m ( N l g s l N ) = x / ~ m2 f dx [ s ( x ) + ~ ( x ) l .
(3)
0
F o r these simple theories a large strange a-term will in general imply a large strange quark distribution function a n d vice versa ~2. In most field theories the simple constraint equation ( 1 ) will be replaced by ~2 Here possible divergences for x-~0 are neglected since, for m # 0, these do not occur in QCD~+ ~or for free fermions.
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
419
Volume 268, number 3,4
PHYSICSLETTERSB
more complicated relations. For example in gauge theories in more than two dimensions
x/~iO+~u_=[a±.(iO±-gAl)+y°m]~u+
(4)
or in Yukawa theories V/2 i O+~u_ =
[ia~ .0k +y°(m-&Os-iysgpq~p) ]~'+
(5)
(where ~u+ = A + ~ ' = (y°y+/v/2)~, and 0s/~p are sealar/pseudoscalar fields). Here ~ u contains interaction terms (cubic in the fields) which in general implies for the matrix elements - even for zero p + transfer - contributions from off-diagonal terms in Fock space [ 7 ]. This makes it rather difficult (if not impossible ) to relate ( N Igsl N ) to the strange quark distribution functions. As an example for the latter case (nonlinear relation between ~+ and ~/_ ) we will consider a generalized Gross-Neveu model [4] described in the lagrangian g2 ~=~(i~-mo)~U+ ~ [(~u)2-(~ysr~)2], (6) where ~uis an N~-spinor in color space and a two-spinor in flavor space. For m o = 0 this lagrangian is invariant under ~3 Uv ( 1 ) × SUL (2) × SUR (2) transformations - a symmetry which is, for N~--,~, spontaneously broken down to Uv (2). The model is asymptotically free (important if we want to "measure" deep inelastic structure functions) and describes nonconfined fermions. We will use the model to develop some intuitive understanding about the virtual qq cloud surrounding valence quarks in strong interactions. Besides being chiraUy symmetric one major advantage of the Gross-Neveu model, compared to conventional pion or kaon cloud models [ 8 ], is a consistent description of meson and quark degrees of freedom. In meson cloud models the mesons appear as elementary particles with substructure, i.e., one faces the possibility of double counting in higherloop corrections. Although there is a pion in the Gross-Neveu model it appears as a quark-antiquark bound state rather than an elementary particle. Since the Gross-Neveu model can be formulated entirely ~3 In addition to the SUv(Nc) symmetry. 420
17 October 1991
in terms of quark degrees of freedom the problem of double counting is avoided. It should be emphasized that the term "color" in the Gross-Neveu model has a priori little to do with the color in QCD. In particular, the 1/N¢ expansion used here might give rise to results which are totally different in a 1~No expansion in QCD. Here we consider the 1/Nc expansion only as an ordering principle for the perturbation series. The term "color" is used only for convenience. In the following we will label the two flavor states up ( u ) and strange (s); i.e., for the sake of simplicity, down quarks will be omitted from the discussion. An extension of the model to three or more flavors is possible and straightforward [4 ] though this would require the 't Hooft determinantal interaction [ 9 ] to break UA ( 1 ). Furthermore u- and s-quarks will be assumed to be degenerate. The main conclusion in this work, which are anyway qualitative, would be the same in a scenario with explicitly broken flavor symmetry and/or more flavors. All observables will be computed in the framework o f a 1/Nc expansion up to the first nontrivial contributions from s-quarks. Let us consider the structure function of a physical (=dressed) up quark. At (~(N °) the only modifications of the quark propagator arise from tadpole graphs (fig. 1 ). Those diagrams do not give any contribution to structure functions for Xbj~O, since in Pauli-Villars regularization [ 10 ]
f
1 dp-p2_m2=O
(p+¢0).
(7)
Thus the physical structure functions of an up quark read to (~(N~° )
U(X)=t~(X--1) , s(x) = e ( x ) = a(x) = o.
(8) U,S
ll,S U,S
u Fig. 1. Typical ¢ (N O) contributions to the propagator of a uquark.
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PHYSICS LETTERS B
To o r d e r N~- ~ the fermions are dressed by chains o f bubbles (fig. 2). Canonical q u a n t i z a t i o n on a x ÷ = c o n s t a n t surface (i.e., so-called light-cone or light-front q u a n t i z a t i o n ) yields to (9 ( N ~- t ) ( = leading o r d e r in N~- ~ ) for the structure functions
17 October 1991 1
D~-| (q 2 )
#2
1 j dz M2F _ z( 1 - z ) / z 2
f
0 1
-q2j dz" 0
I--X
2
1
MF--Z(1--z)q 2'
(12)
1
s(x)=N~ I dy ( l ~ } ~ ) ( x , y ) 1 2 + I~Jffu)(X,y)l 2) ,
1
Dz'(q2)=#~ f dZ M 2 - z ( 1 - z ) l t2
0
0 l--y
l
g(y)=N¢ f d x ( I q / J ) ~ ) ( x , y ) l 2
+(4M~-q (9)
where
~)2 (x, y) 1
-No {D"(q2)[1-1/(1-x-Y)](1/y+l/x)
X[1-1/x-1/y-1/(1-x-y)] with q 2 =
(1/y- l/x)} -1 ,
Here #, is the physical mass o f the pseudoscalar as meson (which is, besides MF, the only free p a r a m e t e r in the r e n o r m a l i z e d theory). N o t e that ( f o r / t 2 > 0) there is no b o u n d state in the scalar channel. The other type o f observables we are interested in are the various strange form factors [ 11 ]
(p'lgy~'slp ) =fi(p')(yUF~'( q 2) (10)
•
_ME(x.l_y)2/ ( 1 - x - y ) and
)
+ 2-~FaU~q~F~S(q2) u(p),
(2) (X, y ) q/se~
ths (P'lgslp) =a(p')u(p)F~S(q 2) ,
2
-- N~ { D " ( q 2 ) ( 1 - 1 / x ) t l / y + I / ( I - x - Y ) ] } X[1-1/x-l/y-1/(1-x-y)]
(13)
0
+ I ¢t~ff~)(x,y ) I 1 ) ,
+ D ~ ( q 2) [1 + 1 / ( l - x - y ) ]
d Z M 2 _ z ( 1l _ z ) q 2 .
2)
0
-~ ,
(14) (15)
where th~ is the current quark mass defined via
qU~z~yu75~= - 2rh~(tz~y5~ #5. (11)
with q2= _M2( 1- x ) 2 / x are the two color c o m p o nents o f the sYu states. Here ME is the physical ferm i o n mass #4. The functions D , a n d Do act like form factors in the pseudoscalar a n d scalar channel. They arise from s u m m i n g up the infinite chain o f qq bubbles a n d can be interpreted as effective meson propagators. Explicitly one finds [4 ] ~4 Note that we have already made use of the fact that s(x) and g(x) are independent of the total p+ of the fermion when we expressed the results in terms of wavefunctions which have total p + = 1.
Most surprisingly the Pauli and scalar form factors have nonzero (9 ( N O) contributions
F~(q 2) = 2M2FD~(q2),
( 16 )
1 .¢s
2
2
f
F~ ( q ) = / z , ' ½ M r a
1 dXM2_x(l_x)lt2. O~(q 2)
0
= ½ M F D ~ ~( 0 ) O ~ ( q z) .
( 17 )
There are no contributions to F~ (q2) from gs pairs to this order in N~F~(q2)=0.
(18)
N o t e that F2 a n d F~ have a nonzero (9 ( N O) contri-
Fig. 2. Typical ¢ (N ~-l ) contribution to the u-quark propagator.
~5 Note that the matrix elements of qT~k,as well as those of flY5~' are cutoff dependent as is ths. The leading cutoff dependence of ~g and ~Ysg are the same. Since the matrix elements of rhs~OzzTsq/areby definition finite, the same will be true for the matrix elements of ths~ . 421
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PHYSICS LETTERS B
bution even for q2 _.-2q +q - = 0. Thus, even though ~s pairs are not present in the structure function to (~(Nc°) [see eqs. ( 9 ) , ( 1 0 ) ] , neither the strange aterm nor the induced a u : c o u p l i n g show this 1/Nc suppression. In order to understand the different powers of 1 ~No we will evaluate the Arc dependence o f form factors and structure functions in ordinary p e r t u r b a t i o n theory (fig. 3). A m e a s u r e m e n t o f the quark distribution always involves only such operators which are diagonal in Fock space (no pair creation at the insertion o f the external probe = " c r o s s " ) . Thus the gs pair in fig. 3a cannot be annihilated by the m e a s u r e m e n t but only through a subsequent interaction. Therefore ~(x) and s ( x ) are o f order a2N¢ (the factor N~ arises from the quark l o o p ) where a = g 2 / N ¢ ~6; i.e., or order 1/Nc. F o r m factor measurements are quite different in this respect. If the o p e r a t o r is inserted with q + ~ 0 (which is necessary in 1 + 1 d i m e n s i o n s if one wants to have q Z = 2 q + q - ~ O ) terms diagonal and off-diagonal in Fock space contribute (fig. 3b). Although the latter usually vanish for F~ for q+ ~ 0 in general they do not vanish for F2 or F~ because these involve so-called b a d currents [ 12 ]. The diagonal terms (fig. 3c) are also o f o r d e r 1/Nc (like the quark d i s t r i b u t i o n ) but the off-diagonal terms do not need another interaction to create the gs pair (it is created at the insertion). Thus the form factors are (only the leading contribution is conside r e d ) o f the order ozNc, i.e., o f C ( N ° ) . One can express this result also in a different way.
17 October 1991
The gs c o m p o n e n t o f a u-quark wavefunction is o f the o r d e r 1/Nc. Thus operators which are diagonal in Fock space and measure the gs content will be o f order Ncl ~Ge, I2 ~ 1/N¢ (there is always an extra Nc multiplying the matrix element because o f the sum over the colors o f the s-quarks). In contrast off-diagonal operators are linear in q/se,, i.e., o f the o r d e r Nc~ss, ~ 1 #,7 The m a i n question arising at this point is whether there is a similar effect possible in QCD3+I ~8. The answer is in principle yes although the i m p o r t a n c e o f such an effect is not obvious. A light-cone expansion o f the matrix elements o f the o p e r a t o r gs, very similar to the expansion o f the matrix element o f g2(x) in ref. [13], will (even for q + = 0 ) contain o p e r a t o r products o f the form bsd~a ÷ (fig. 4) which annihilate an s g p a i r at the Ys insertion and replace it by a gluon. N o w let us imagine a situation where there is a large gluonic a d m i x t u r e to the p r o t o n wavefunction but a small sg a d m i x t u r e ~9
I ~qqq I "~ I qlqqqg[
>>
I ~qqqes I
(19)
•
#7 A quite similar pattern emerges in QCDI + 1(No--'oo) where to (5(N ° ) the structure function of mesons is pure valence qq while form factors are nevertheless modified by "vector meson dorninance"-type corrections of order N O [10]. Intuitively one attributes such kinds of corrections with a qq-cloud around the valence quark which is in this case a questionable interpretation. However, there are no flavor mixing effects to leading order 1/Nc in QCDI + i. ~8 Here we neither assume Nc~oo nor do we associate Nc in QCD3÷ l with the value of Noin the Gross-Neveu model above. ,9 This picture is probably not too far reality.
#6 In perturbation theory one chooses g independent of No, in order to have finite qq scattering amplitudes for Nc-,~,
SS
I1 (a)
U (b)
(c)
Fig. 3. Perturbative (light-cone time ordered) diagrams contributing to (a) the strange structure function (only the contribution to the Edistribution is shown), and (b), (c) the formfactors. In (c) the external photon could also be attached to the s quark.
422
Fig. 4. Typical diagram yielding a contribution to the strange sigma term in a nucleon via terms which are off-diagonal in Fock space.
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PHYSICS LETTERS B
In this case the m a i n contributions to as= rhs (NIYsl N ) will come from the off-diagonal terms, i.e.,
a s Of. I lffqqqg'lffqqqgs I >> S( X ) of. I lffqqqgs 12 . ms
(20)
The explicit Fock-space expansion of these matrix elements is quite lengthy, since this involves twistthree operators and will be omitted here. It could be obtained by differentiating the light-cone hamilton i a n of Q C D [ 14 ]. The typical structure of such an expansion would be similar to the expansion of g2 (x) in ref. [ 13 ]. In the above discussion we concentrated on the scalar form factor ( 15 ). In the G r o s s - N e v e u model similar results a n d conclusion hold for the magnetic form factor F ~s [eq. ( 1 4 ) ] . However, for the magnetic m o m e n t of quarks in QCD3+I such contributions from off-diagonal Fock space seem to vanish for q+--*0 if one uses a y+ insertion, i.e., a good current [ 12 ], to the measure F2. F r o m the example studied here one should learn several lessons. First of all the question "is there a lot of strangeness in the proton" is a priori not well posed. The operator used to measure the content of strangeness might be crucial. Although our N ¢ ~ limit is somewhat extreme, a similar behavior is certainly possible in QCD3+ I. Secondly, although m a n y operators simplify in the light-cone analysis for q+--,0 there can still be a complicated operator left over in this limit. In particular, pair creation a n d / o r manybody effects might be i m p o r t a n t - or even d o m i n a n t . It is, however, not yet clear how i m p o r t a n t such effects are in QCD3+ i. If they are i m p o r t a n t this would mean that m a n y intuitive pictures, like the a-term of
17 October 1991
a h a d r o n originating from an intrinsic admixture of
gs pairs, have to be modified. I would like to acknowledge Brian Warr a n d Stan Brodsky for m a n y helpful discussions.
References
[ 1] R.L. Jaffe and A. Manohar, Nucl. Phys. B 337 (1990) 509. [2 ] H. Abramowiczet al., Z. Phys. C 15 (1982) 19. [3] R. Koch, Z. Phys. C 15 (1982) 161. [4] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235; for an excellentreview,see e.g.B. Rosenstein,B.J. Wart and S.H. Park, preprint SLAC-PUB-5349 (1990) submitted to Phys. Rep. [5] V. Bernard, R.L. Jaffe and U.-G. MeiBner, Nucl. Phys. B 308 (1988) 753. [6] M. Burkardt, Ph.D. Thesis (Erlangen, 1989). [7] J.F. Donoghue and C.R. Nappi, Phys. Lett. B 168 (1986) 105. [8 ] A.I. Signaland A.W. Thomas, Phys. Lett. B 191 ( 1987) 205; A.W. Thomas, Nucl. Phys. A 518 (1990) 186. [9] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. [ 10] T.-M. Yan, Phys. Rev. D 7 ( 1973 ) 1780. [ 11 ] D.B. Kaplan and A. Manohar, Nucl. Phys. B 310 (1988) 527; R.D. McKeown, Phys. Lett. B 219 (1989) 140; R.L. Jaffe, Phys. Lett. B 229 (1989) 275; E.J. Beise and R.D. McKeown,preprint OAP-707 (1990). [ 12] S.D. Drell, D.J. Levyand T.-M. Yan, Phys. Rev. 187 (1969) 2159; Phys. Rev. D 1 (1970) 1035. [ 13] L. Mankiewiczand Z. Ryzak, Phys. Rev. D 43 ( 1991 ) 733. [14]S.J. Brodsky, Invited Lectures Stellenbosch Advanced Course in Theoretical physics, preprint SLAC-PUB-3747 (1985).
423
PHYSICS LETTERS B
Strange form factors, the a-term and strange quark distributions in the Gross-Neveu model Matthias Burkardt 1
Stanford LinearAccelerator Center, Stanford University, Stanford, CA 94309, USA Received 12 July 1991
For N¢--,~ the structure function of physical fermions in the Gross-Neveu model is of pure valence type. Nevertheless several strange form factors of u-quarks, including the induced magnetic moment and the a-term acquire substantial ~ (N °) contributions from virtual gs pairs. Those arise from matrix elements which are off-diagonal in Fock space, i.e., they do not have a simple interpretation in terms of the quark distributions. It is shown that, under certain circumstances, a similar behavior in QCD3+ t is conceivable.
Recently, there has been a growing discussion about the "strangeness c o n t e n t " o f the nucleon (for a review, see e.g. ref. [ 1 ] ). Although gs pairs carry less than 5% o f the nucleon m o m e n t u m [2] it has been suggested that they contribute significantly to the nucleon spin ~l and almost cancel the spin carried by uand d-quarks. A n o t h e r surprising result that emerged from an analysis o f the low energy nN forward scattering a m p l i t u d e [3 ] seems to require a large value for the strange scalar "charge" or a-term ms ( N [~s IN ) o f the nucleon. In this note I will show that a large strange a-term together with small gs structure functions does not necessarily m e a n a contradiction. Although this will be d e m o n s t r a t e d on the basis o f the G r o s s - N e v e u m o d e l [4 ] in 1 + 1 dimensions, I will argue that a priori something similar might h a p p e n in QCD3+ ~. Based on the observation that ~s is twist-three whereas structure functions are m e a s u r e d by twist-two operators it has been already e m p h a s i z e d that such a behavior is conceivable [ 5 ]. The example to be studied here will help to u n d e r s t a n d the underlying physics o f large, strange a-terms c o m b i n e d with a suppression o f strange quarks in the structure function. Supported in part by a grant from Alexander von HumboldtStiftung and in part by the Department of Energy under contract DE-AC03-76SF00515. ~l This interpretation of the EMC II effect is still controversial.
In some simple field theories (like free fermions or Q C D ~ + I ) r n s ( N l Y s l N ) has a direct p a r t o n m o d e l interpretation, i.e., one can express the strange scalar form factor in the limit o f zero m o m e n t u m transfer (the scalar charge) in terms o f the strange quark distribution functions S(Xbj ) and g(xbj). In these theories the constraint equation for the left-handed fields takes the form J 2 i O+ ~/L = m~/R,
(1)
i.e., it is linear and thus
gs=s~ SR +S~SL =X/~ ms+ (i 0 + ) - ' S R .
(2)
F o r vanishing q÷ (q~ being the m o m e n t u m transferred by the gs insertion), where p a i r creation is suppressed at the vertex, the matrix elements o f ~s become diagonal in Fock space yielding [ 6 ] 1
m ( N l g s l N ) = x / ~ m2 f dx [ s ( x ) + ~ ( x ) l .
(3)
0
F o r these simple theories a large strange a-term will in general imply a large strange quark distribution function a n d vice versa ~2. In most field theories the simple constraint equation ( 1 ) will be replaced by ~2 Here possible divergences for x-~0 are neglected since, for m # 0, these do not occur in QCD~+ ~or for free fermions.
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
419
Volume 268, number 3,4
PHYSICSLETTERSB
more complicated relations. For example in gauge theories in more than two dimensions
x/~iO+~u_=[a±.(iO±-gAl)+y°m]~u+
(4)
or in Yukawa theories V/2 i O+~u_ =
[ia~ .0k +y°(m-&Os-iysgpq~p) ]~'+
(5)
(where ~u+ = A + ~ ' = (y°y+/v/2)~, and 0s/~p are sealar/pseudoscalar fields). Here ~ u contains interaction terms (cubic in the fields) which in general implies for the matrix elements - even for zero p + transfer - contributions from off-diagonal terms in Fock space [ 7 ]. This makes it rather difficult (if not impossible ) to relate ( N Igsl N ) to the strange quark distribution functions. As an example for the latter case (nonlinear relation between ~+ and ~/_ ) we will consider a generalized Gross-Neveu model [4] described in the lagrangian g2 ~=~(i~-mo)~U+ ~ [(~u)2-(~ysr~)2], (6) where ~uis an N~-spinor in color space and a two-spinor in flavor space. For m o = 0 this lagrangian is invariant under ~3 Uv ( 1 ) × SUL (2) × SUR (2) transformations - a symmetry which is, for N~--,~, spontaneously broken down to Uv (2). The model is asymptotically free (important if we want to "measure" deep inelastic structure functions) and describes nonconfined fermions. We will use the model to develop some intuitive understanding about the virtual qq cloud surrounding valence quarks in strong interactions. Besides being chiraUy symmetric one major advantage of the Gross-Neveu model, compared to conventional pion or kaon cloud models [ 8 ], is a consistent description of meson and quark degrees of freedom. In meson cloud models the mesons appear as elementary particles with substructure, i.e., one faces the possibility of double counting in higherloop corrections. Although there is a pion in the Gross-Neveu model it appears as a quark-antiquark bound state rather than an elementary particle. Since the Gross-Neveu model can be formulated entirely ~3 In addition to the SUv(Nc) symmetry. 420
17 October 1991
in terms of quark degrees of freedom the problem of double counting is avoided. It should be emphasized that the term "color" in the Gross-Neveu model has a priori little to do with the color in QCD. In particular, the 1/N¢ expansion used here might give rise to results which are totally different in a 1~No expansion in QCD. Here we consider the 1/Nc expansion only as an ordering principle for the perturbation series. The term "color" is used only for convenience. In the following we will label the two flavor states up ( u ) and strange (s); i.e., for the sake of simplicity, down quarks will be omitted from the discussion. An extension of the model to three or more flavors is possible and straightforward [4 ] though this would require the 't Hooft determinantal interaction [ 9 ] to break UA ( 1 ). Furthermore u- and s-quarks will be assumed to be degenerate. The main conclusion in this work, which are anyway qualitative, would be the same in a scenario with explicitly broken flavor symmetry and/or more flavors. All observables will be computed in the framework o f a 1/Nc expansion up to the first nontrivial contributions from s-quarks. Let us consider the structure function of a physical (=dressed) up quark. At (~(N °) the only modifications of the quark propagator arise from tadpole graphs (fig. 1 ). Those diagrams do not give any contribution to structure functions for Xbj~O, since in Pauli-Villars regularization [ 10 ]
f
1 dp-p2_m2=O
(p+¢0).
(7)
Thus the physical structure functions of an up quark read to (~(N~° )
U(X)=t~(X--1) , s(x) = e ( x ) = a(x) = o.
(8) U,S
ll,S U,S
u Fig. 1. Typical ¢ (N O) contributions to the propagator of a uquark.
Volume 268, number 3,4
PHYSICS LETTERS B
To o r d e r N~- ~ the fermions are dressed by chains o f bubbles (fig. 2). Canonical q u a n t i z a t i o n on a x ÷ = c o n s t a n t surface (i.e., so-called light-cone or light-front q u a n t i z a t i o n ) yields to (9 ( N ~- t ) ( = leading o r d e r in N~- ~ ) for the structure functions
17 October 1991 1
D~-| (q 2 )
#2
1 j dz M2F _ z( 1 - z ) / z 2
f
0 1
-q2j dz" 0
I--X
2
1
MF--Z(1--z)q 2'
(12)
1
s(x)=N~ I dy ( l ~ } ~ ) ( x , y ) 1 2 + I~Jffu)(X,y)l 2) ,
1
Dz'(q2)=#~ f dZ M 2 - z ( 1 - z ) l t2
0
0 l--y
l
g(y)=N¢ f d x ( I q / J ) ~ ) ( x , y ) l 2
+(4M~-q (9)
where
~)2 (x, y) 1
-No {D"(q2)[1-1/(1-x-Y)](1/y+l/x)
X[1-1/x-1/y-1/(1-x-y)] with q 2 =
(1/y- l/x)} -1 ,
Here #, is the physical mass o f the pseudoscalar as meson (which is, besides MF, the only free p a r a m e t e r in the r e n o r m a l i z e d theory). N o t e that ( f o r / t 2 > 0) there is no b o u n d state in the scalar channel. The other type o f observables we are interested in are the various strange form factors [ 11 ]
(p'lgy~'slp ) =fi(p')(yUF~'( q 2) (10)
•
_ME(x.l_y)2/ ( 1 - x - y ) and
)
+ 2-~FaU~q~F~S(q2) u(p),
(2) (X, y ) q/se~
ths (P'lgslp) =a(p')u(p)F~S(q 2) ,
2
-- N~ { D " ( q 2 ) ( 1 - 1 / x ) t l / y + I / ( I - x - Y ) ] } X[1-1/x-l/y-1/(1-x-y)]
(13)
0
+ I ¢t~ff~)(x,y ) I 1 ) ,
+ D ~ ( q 2) [1 + 1 / ( l - x - y ) ]
d Z M 2 _ z ( 1l _ z ) q 2 .
2)
0
-~ ,
(14) (15)
where th~ is the current quark mass defined via
qU~z~yu75~= - 2rh~(tz~y5~ #5. (11)
with q2= _M2( 1- x ) 2 / x are the two color c o m p o nents o f the sYu states. Here ME is the physical ferm i o n mass #4. The functions D , a n d Do act like form factors in the pseudoscalar a n d scalar channel. They arise from s u m m i n g up the infinite chain o f qq bubbles a n d can be interpreted as effective meson propagators. Explicitly one finds [4 ] ~4 Note that we have already made use of the fact that s(x) and g(x) are independent of the total p+ of the fermion when we expressed the results in terms of wavefunctions which have total p + = 1.
Most surprisingly the Pauli and scalar form factors have nonzero (9 ( N O) contributions
F~(q 2) = 2M2FD~(q2),
( 16 )
1 .¢s
2
2
f
F~ ( q ) = / z , ' ½ M r a
1 dXM2_x(l_x)lt2. O~(q 2)
0
= ½ M F D ~ ~( 0 ) O ~ ( q z) .
( 17 )
There are no contributions to F~ (q2) from gs pairs to this order in N~F~(q2)=0.
(18)
N o t e that F2 a n d F~ have a nonzero (9 ( N O) contri-
Fig. 2. Typical ¢ (N ~-l ) contribution to the u-quark propagator.
~5 Note that the matrix elements of qT~k,as well as those of flY5~' are cutoff dependent as is ths. The leading cutoff dependence of ~g and ~Ysg are the same. Since the matrix elements of rhs~OzzTsq/areby definition finite, the same will be true for the matrix elements of ths~ . 421
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bution even for q2 _.-2q +q - = 0. Thus, even though ~s pairs are not present in the structure function to (~(Nc°) [see eqs. ( 9 ) , ( 1 0 ) ] , neither the strange aterm nor the induced a u : c o u p l i n g show this 1/Nc suppression. In order to understand the different powers of 1 ~No we will evaluate the Arc dependence o f form factors and structure functions in ordinary p e r t u r b a t i o n theory (fig. 3). A m e a s u r e m e n t o f the quark distribution always involves only such operators which are diagonal in Fock space (no pair creation at the insertion o f the external probe = " c r o s s " ) . Thus the gs pair in fig. 3a cannot be annihilated by the m e a s u r e m e n t but only through a subsequent interaction. Therefore ~(x) and s ( x ) are o f order a2N¢ (the factor N~ arises from the quark l o o p ) where a = g 2 / N ¢ ~6; i.e., or order 1/Nc. F o r m factor measurements are quite different in this respect. If the o p e r a t o r is inserted with q + ~ 0 (which is necessary in 1 + 1 d i m e n s i o n s if one wants to have q Z = 2 q + q - ~ O ) terms diagonal and off-diagonal in Fock space contribute (fig. 3b). Although the latter usually vanish for F~ for q+ ~ 0 in general they do not vanish for F2 or F~ because these involve so-called b a d currents [ 12 ]. The diagonal terms (fig. 3c) are also o f o r d e r 1/Nc (like the quark d i s t r i b u t i o n ) but the off-diagonal terms do not need another interaction to create the gs pair (it is created at the insertion). Thus the form factors are (only the leading contribution is conside r e d ) o f the order ozNc, i.e., o f C ( N ° ) . One can express this result also in a different way.
17 October 1991
The gs c o m p o n e n t o f a u-quark wavefunction is o f the o r d e r 1/Nc. Thus operators which are diagonal in Fock space and measure the gs content will be o f order Ncl ~Ge, I2 ~ 1/N¢ (there is always an extra Nc multiplying the matrix element because o f the sum over the colors o f the s-quarks). In contrast off-diagonal operators are linear in q/se,, i.e., o f the o r d e r Nc~ss, ~ 1 #,7 The m a i n question arising at this point is whether there is a similar effect possible in QCD3+I ~8. The answer is in principle yes although the i m p o r t a n c e o f such an effect is not obvious. A light-cone expansion o f the matrix elements o f the o p e r a t o r gs, very similar to the expansion o f the matrix element o f g2(x) in ref. [13], will (even for q + = 0 ) contain o p e r a t o r products o f the form bsd~a ÷ (fig. 4) which annihilate an s g p a i r at the Ys insertion and replace it by a gluon. N o w let us imagine a situation where there is a large gluonic a d m i x t u r e to the p r o t o n wavefunction but a small sg a d m i x t u r e ~9
I ~qqq I "~ I qlqqqg[
>>
I ~qqqes I
(19)
•
#7 A quite similar pattern emerges in QCDI + 1(No--'oo) where to (5(N ° ) the structure function of mesons is pure valence qq while form factors are nevertheless modified by "vector meson dorninance"-type corrections of order N O [10]. Intuitively one attributes such kinds of corrections with a qq-cloud around the valence quark which is in this case a questionable interpretation. However, there are no flavor mixing effects to leading order 1/Nc in QCDI + i. ~8 Here we neither assume Nc~oo nor do we associate Nc in QCD3÷ l with the value of Noin the Gross-Neveu model above. ,9 This picture is probably not too far reality.
#6 In perturbation theory one chooses g independent of No, in order to have finite qq scattering amplitudes for Nc-,~,
SS
I1 (a)
U (b)
(c)
Fig. 3. Perturbative (light-cone time ordered) diagrams contributing to (a) the strange structure function (only the contribution to the Edistribution is shown), and (b), (c) the formfactors. In (c) the external photon could also be attached to the s quark.
422
Fig. 4. Typical diagram yielding a contribution to the strange sigma term in a nucleon via terms which are off-diagonal in Fock space.
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In this case the m a i n contributions to as= rhs (NIYsl N ) will come from the off-diagonal terms, i.e.,
a s Of. I lffqqqg'lffqqqgs I >> S( X ) of. I lffqqqgs 12 . ms
(20)
The explicit Fock-space expansion of these matrix elements is quite lengthy, since this involves twistthree operators and will be omitted here. It could be obtained by differentiating the light-cone hamilton i a n of Q C D [ 14 ]. The typical structure of such an expansion would be similar to the expansion of g2 (x) in ref. [ 13 ]. In the above discussion we concentrated on the scalar form factor ( 15 ). In the G r o s s - N e v e u model similar results a n d conclusion hold for the magnetic form factor F ~s [eq. ( 1 4 ) ] . However, for the magnetic m o m e n t of quarks in QCD3+I such contributions from off-diagonal Fock space seem to vanish for q+--*0 if one uses a y+ insertion, i.e., a good current [ 12 ], to the measure F2. F r o m the example studied here one should learn several lessons. First of all the question "is there a lot of strangeness in the proton" is a priori not well posed. The operator used to measure the content of strangeness might be crucial. Although our N ¢ ~ limit is somewhat extreme, a similar behavior is certainly possible in QCD3+ I. Secondly, although m a n y operators simplify in the light-cone analysis for q+--,0 there can still be a complicated operator left over in this limit. In particular, pair creation a n d / o r manybody effects might be i m p o r t a n t - or even d o m i n a n t . It is, however, not yet clear how i m p o r t a n t such effects are in QCD3+ i. If they are i m p o r t a n t this would mean that m a n y intuitive pictures, like the a-term of
17 October 1991
a h a d r o n originating from an intrinsic admixture of
gs pairs, have to be modified. I would like to acknowledge Brian Warr a n d Stan Brodsky for m a n y helpful discussions.
References
[ 1] R.L. Jaffe and A. Manohar, Nucl. Phys. B 337 (1990) 509. [2 ] H. Abramowiczet al., Z. Phys. C 15 (1982) 19. [3] R. Koch, Z. Phys. C 15 (1982) 161. [4] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235; for an excellentreview,see e.g.B. Rosenstein,B.J. Wart and S.H. Park, preprint SLAC-PUB-5349 (1990) submitted to Phys. Rep. [5] V. Bernard, R.L. Jaffe and U.-G. MeiBner, Nucl. Phys. B 308 (1988) 753. [6] M. Burkardt, Ph.D. Thesis (Erlangen, 1989). [7] J.F. Donoghue and C.R. Nappi, Phys. Lett. B 168 (1986) 105. [8 ] A.I. Signaland A.W. Thomas, Phys. Lett. B 191 ( 1987) 205; A.W. Thomas, Nucl. Phys. A 518 (1990) 186. [9] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. [ 10] T.-M. Yan, Phys. Rev. D 7 ( 1973 ) 1780. [ 11 ] D.B. Kaplan and A. Manohar, Nucl. Phys. B 310 (1988) 527; R.D. McKeown, Phys. Lett. B 219 (1989) 140; R.L. Jaffe, Phys. Lett. B 229 (1989) 275; E.J. Beise and R.D. McKeown,preprint OAP-707 (1990). [ 12] S.D. Drell, D.J. Levyand T.-M. Yan, Phys. Rev. 187 (1969) 2159; Phys. Rev. D 1 (1970) 1035. [ 13] L. Mankiewiczand Z. Ryzak, Phys. Rev. D 43 ( 1991 ) 733. [14]S.J. Brodsky, Invited Lectures Stellenbosch Advanced Course in Theoretical physics, preprint SLAC-PUB-3747 (1985).
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