The anomalous gluon contribution to polarized leptoproduction

Volume 212, number 3

PHYSICS LETTERS B

29 September 1988

THE ANOMALOUS GLUON CONTRIBUTION TO POLARIZED LEPTOPRODUCTION G. A L T A R E L L I and G.G. ROSS

CERN, CH-1211 Geneva23, Switzerland Received 29 June 1988

We show that, due to the anomaly, the gluon contribution to the first moment of the polarized proton structure function, as measured in deep inelastic scattering, is not suppressed by a power of the strong coupling evaluated at a large scale. As a result, the EMC result for the first moment of polarized proton electroproduction is consistent with a large quark spin component.

A measurement of the polarized proton structure function in deep inelastic leptoproduction was recently completed by the EMC Collaboration at C E R N [ 1 ]. In particular for the integral of the structure function g~ at ( Q 2 ) = 11 GeV 2 the measured asymmetry implies the relation

Here c-quarks are neglected and the following notation was adopted: 1

Aq(x)-

t dx [ q + ( x ) + ( t + ( x ) - q _ ( x ) - ( 7 _ ( x ) ] 0

~- (P, Sl ilYoYsqlP, s)

1

f dxgP(x, Q2) _ 0 . 1 1 4 + 0 . 0 2 9 ,

( 1)

0

where statistical and systematic errors have been added in quadrature. By combining the EMC results with previous data from SLAC [ 2 ] at ( Q 2) = 5 GeV 2 one obtains the improved determination [ 3 ] 1

f dxg~(x,

Q2) =0.115_+0.021

(2)

0

if the Q2 dependence is neglected in the relevant Q2 range. In the naive parton model [ 4 ]

with q_+ ( ~ ) being the densities of parton quarks (antiquarks) with helicities + ½ in a proton with helicity + ½Aq(x) is connected to the matrix element o f the quark axial vector current between polarized protons, as also shown in eq. (4). The flavour non-singlet components can be obtained from the measured values o f the axial-vector matrix elements by the relations A u - A d = g A / g v = F + D and A u + A d - 2 A s = 3 F - D [5]. From a recent fit o f hyperon decays F = 0 . 4 7 7 + 0 . 0 1 1 and D = 0 . 7 5 5 + 0 . 0 1 1 [6]. By combining the above results with the EMC experimental value in eq. ( 1 ) one finds [ 7 ] Au---0.74+0.08,

Ad~-0.51+0.08,

As= -0.23+0.08 ,

1

J

dx gP(x)

= ½

i~l p~ Aq,(x) = ½( ~ A u + { A d + ~As)

0

+ ~(Au+ Ad+ As).

(3)

On sabbatical leave from Department of Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK.

(5)

so that A S - A u + A d + As_~ 0.00 + 0.24.

_ ~(Au-Ad)+~6(Au+Ad--ZAs)

(4)

(6)

It has been suggested that the errors have been significantly underestimated [ 8 ] allowing for considerably larger values of AS. We do not pursue this possibility, but consider the theoretical implications o f a value for AS in the range given by eq. (6). These results are at first sight surprising because 391

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they appear to contradict the intuitive expectation that a substantial fraction of the proton spin is carried by parton quarks and antiquarks (AS= 1 if all the proton spin is carried by q and ~) and the guess that As should be very small [9 ]. This fact has produced many speculations (including claims of evidence against QCD [ 10 ] ). In ref. [ 11 ] it was shown that chiral symmetry and 1/N~ expansion imply the vanishing of the SU (3) singlet axial current matrix element. Also the corrections due to finite quark masses and Arc= 3 were argued to be small on the basis ofa skyrmion model of the proton. In this picture most of the proton spin is carried by orbital angular momentum. Explanations based on a rapid Qz dependence (the integral in eq. (1) is known [12] to change sign at Q2= 0 due to the Drell-Hearn sum rule [ 13 ] ) appear to be discarded by experiment. In fact data at different Q2 exist [ 14] down to Q2 as low as Q2 =0.5 GeV 2, and no appreciable Q2 dependence is visible. Indeed it is found that the asymmetry becomes negative at the N* resonance [14], however this fluctuation is very local. Actually, as shown in ref. [ 14], even at Q2=0.5 GeV 2 the scaling curve is a good average of the fluctuating asymmetry measured in the resonance region. Note that the position of the N* resonance is far outside the x range measured by EMC, near x = 1. A more physically significant form of Q2 dependence was advocated in ref. [ 15 ] where it was stressed that the conservation of the SU(3) singlet axial current, valid for massless quarks, is broken in QCD by the Adler-Bell-Jackiw anomaly [ 16 ]. In particular while the anomalous dimension of the non-singlet axial current vanishes at all orders [ 17], the singlet axial current anomalous dimension is different from zero at two loops and its value was computed by Kodaira [ 18 ]. In ref. [ 15 ] it is proposed that due to this non-conservation a rapid variation of AS takes place in the non-perturbative region of Q2. In the present letter we show that the effect of the anomaly in the singlet sector is actually more dramatic and interesting than envisaged in ref. [ 15 ]. The anomaly in fact induces a gluon contribution at all values of Q2 in the singlet part of the integral ofg~. This gluon term can in principle be large and can well explain the difference between parton and constituent quarks that the EMC result appears to imply. 392

29 September 1988

We introduce the gluon helicity content in the polarized proton 1

A g - ~ dx [g+ (x,

QZ)-g_(x, QZ)] ,

(7)

0

with g+ being the gluon density with helicity + 1 in a proton with helicity + ½. At order o~s the QCD evolution of AS and Ag is well known [4 ], Ag = ~

2n

Ag~l,

AgOg l)

Ag

0 3CF ~(llCA--4T)

Ag

'

where t=ln QZ/lz2, AAB=f~ dx APAB(X), APAB(X) being the lowest order QCD splitting functions for polarized partons given in ref. [19]. (i.e. APA~ =PA+B+ PA-B+), cF =4, CA=3, T = -~f and f is the number of excited flavours (f= 3 if c-quarks are neglected). The crucial point is that the gg entry of the evolution matrix in eq. (8) is nothing else than bo~, where b is the first coefficient of the QCD fl function: fl(o~)=-bo~Z(l+b'o~+...) with b=(llca-4T)/ 12n. Then at this order Ag belongs to an eigenvector whose evolution is determined by the factor [c~s/ as(t) ]a with d = 1 (o~= o~s(0) ). This implies that if a mixing with Ag is present at order o~s(t), induced by the anomaly via the diagrams in fig. 1, the gluon component will not vanish at large Q2, because c~s(t) [Ag(t) +...] = ~s(t) [~/o~s(t) ] [Ag(0) +...], where the ellipses indicate the quark completion (at this order) of the relevant eigenvector. The contribution of Ag to g~ at order OLsis obtained from the diagrams in fig. 1 by computing the difference of cross sections 6a= a(yl.g+ ) - a(TRg+ ) ,

t

,/° _

(9)

,~

q

Fig. 1. Diagrams contributing to a finite mixing of order cq between g]' and the polarized gluon parton density.

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PHYSICS LETTERSB

where 7L or YRis a left- or right-handed photon of virtual mass - Q 2 . With a dimensional over-all factor removed, one obtains % T

ao~ ~ 7

2

Ix - ( l - x )

~1

1 × ( l _ f l c1 o s 0 + 1 +flcosO

1) ,

(10)

with 0 being the Y-g center of mass angle. Here the normalization is fixed by the correct leading logaritmic splitting function APqg, i.e. the coefficient of 1/ ( 1 - f l c o s 0 ) . Since the first moment of APqg vanishes, the corresponding finite coefficient is well defined, i.e. independent of the regularization procedure. In eq. (10) we have adopted the physical quark mass m as a regulator, so that fl=p/ E= 1 -2m2x/[Q2( 1 --X) ]. The finite coefficient is then found to be 1

o~Cg =

2 ~sTf r f J dx IX2-- ( l - - x ) 2] in -___. 1 fXx O

29 September 1988

the growth of Ag is compensated by the variation of a d o as discussed above. The Fq entry becomes O (o~) due to the factor of as in AF. This means that eq. ( 12 ) receives only small radiative corrections at o (c~2). For practical purposes we may use eq. (12) as it is and the reader not interested in theoretical niceties should proceed to the phenomenological discussion at the end of the paper. First, however, we consider the QCD evolution of &S and AF at second order in CXs.Now the renormalisation group equations read d

AS

"g

+o(~3)

v(2) Jgg

(14) The label (n) in y~n) denotes the number of loops. There is an obvious reshuffling of powers because of the as factor in the relation between AFand Ag. Quite in general the Q2 evolution of the singlet component of the first moment is given by M~ ( 0 2) = ½(e2)c(~x~(t) )E(o~s, as(t) )O (c~s),

o~ s T

--

2re f "

(15)

(11)

where M~ is defined in eq. (12), O ( a s ) = ( i f ) and to the relevant order in %

Thus we obtain 1

M](Q2)= j

dxg~(x,

c(a~) = ( 1 + czc~s, cv+c~-%),

Q2)sing,e t

0

E(a~, a s ( t ) ) = 1 +

(16)

a~-as(t) b Y"

(17)

-- (e2) ( z ~ - - 2 - ~ f Here c z = - 3 c v / 4 n was computed in ref. [17], c r = - f i s obtained from eq. ( 12 ) and y is the matrix in eq. (14). The explicit result is then given by where ( e 2) = E i e ~ = ~ (see eq. ( 3 ) ) and the factor of 2 in front of cgA, takes into account that there is a term - % / 4 1 r for each quark and each antiquark of a given flavour. Since o~sAgis conserved by the leading order QCD evolution it proves convenient to define the new density A/" by OLs

A F - -~-~Ag.

(13)

Now the renormalisation group equation (8) takes a very simple form as the variations of z ~ and AFboth vanish at order ols (t). The FF entry vanishes because

M~(Q 2) ~o,~l) = ½ ( e 2) { ( a1~-I-- ° ~ ( t b) , , , ~ 2I )' r ± qq TCF/gq

--

bcz)

+ as Cz)ZkS

+ l+aS-~s(t) b °tscl- ]cvAF~ . cr d

yg,~.q~ cr

bct + cr / (18)

3

It is instructive at this stage to establish contact with 393

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the analysis using the short-distance operator expansion (OPE). There are only two possible twist-two operators which may contribute to the first moment o f g L They are

29 September1988

The the first moment of the polarized structure function has the form

M](Q2)= ~e-~e2-~)2)IEl(~2,ods(t))al(#2 )

f

J~ = ~ q,7,75qi,

o~(t) 7(od)

"

2

Xexp( f fl--~-~dod)+E2(#,

i=1

od(t))a2(#2)]

ot ods

k~ = ~ e,~xo Tr[A ~(F a"- ]A~A ~) l ,

(19)

with A "= Z~4"Bt~ and Tr tAtB= IgAB. In general k, is gauge dependent. However the diagonal matrix element which is relevant for the first moment ofg~ is gauge invariant, being related to the topological charge. This is why it makes sense to use it to fill the gap between the parton language (where the first moment is as well defined as the other ones) and the conventional OPE formalism [ 18 ] where the first moment is left without gluons. The renormalization group equations for the operatorsj~ and k~ take the form [20]

~ \ G / = [oddt) \~,4. I )

\GJ

(2o)

Here the vanishing of 712 and ~22 results because the divergence o f k . is (Ods/2x) Tr (Fu.ff"~), i.e. is proportional to the U( 1 ) anomaly which is not renormalized in any order [21]. The non-zero entries 712) and 7~I ) are related because j . and k. are connected by the equations of motion ~:5

gOds

_1, =j~-~ Tr [F, vPU~l =f7'~k,.

(21)

Since this must be true for any renormalization scale #2, we have 712) =fT~l, =

3fCF 87~2

(22)

in agreement with the explicit two-loop calculation due to Kodaira [ 18 ]. Using these anomalous dimensions we may diagonalize eq. (20) to find the eigenvectors Oi and eigenvalues 2,. They are Oi,o~j

5 ,

__~,(2)~2 2 1 - - / ' 1 1 t,~s,

02,. = j ~ - f k . , 394

22 = 0 .

(23)

J (24)

where the quantities a~ are related to the operator matrix elements between protons of helicity s by

~ai(#2)Sot .

(25)

Taking matrix elements between quark and gluon states one obtains {EL (#2, Ods(#2) ) (&S-fA/-') M](#2)= --2-+ E 2 ( # 2, Ods(# 2 ) ) ~ } .

(26)

The coefficients E~ can then be derived by comparison with eq. (12) which gives El = 1, E2 = 0 at lowest order in Ods.Evaluating the exponential factor in eq. (24) and including the o (Od~)contribution to E~ then gives m ] (Q2) --

< e 2 > ( 1 - od~(t) ) 2 ~ (7~2)-bc~) ( A S - f A F ) o 2 ~ , (1 2

od~(t) 3 3 - - 8 f ) rc 3 3 - 2 f J ( Z k S - f A F ) ° ~ " (27)

Here the Q2 dependent factor is just that found in ref. [ 18 ]. Note that the entire contribution to eq. [27] arises from the operator j,; the gluon component is introduced because the anomaly gives a finite gluon matrix element to j,. We may now compare the patton model analysis of eq. [ 18 ] with this result. Since it is only the combination multiplying od~(t) which has a physical meaning independent of all conventions, we can identify ,qq~,~2)+Cr.7~g~)-bcz with the corresponding quantity obtained in eq. (27). This gives 7q~2) = 0. In other words the quantity o' ~ ~2) of eq. (20), computed by Kodaira [ 18 ], is what here appears as t'F/' " o'~gql). Note that in fact the leading singularity of the first moment of the diagram in fig. 2 is precisely given by the prob-

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PHYSICS LETTERS B

i

-q q Fig. 2. The typical diagram contributing to the leading logarithms in the Q2 dependence ofgln at next-to-leading order. ability of finding a gluon in a quark times the finite cross section o f fig. 1, i.e. exactly by CF~gq (l) • Thus it follows that in parton terms AS is conserved, being idenfied with the combination ( j , - J k , ) which has no gluon matrix element. This combination is conserved (the anomaly cancels) and hence (see eqs. (20), (22) has vanishing anomalous dimensions. Finally, comparison o f the A F term in eq. (18) with that in eq. (27) shows that A F a n d AXhave the same Q2 dependence. We now consider the implications of our final result in eq. (27) which includes the effects o f the anomaly and the Q C D corrections to second order. As anticipated the Q2 evolution is expected to play no important role - as it arises from second-order effects which should be smaller than the moderate scaling violations observed in non-polarized deep inelastic scattering processes. However the crucial result is that the anomalous gluon contribution is not expected to be small as it is not suppressed by a power of oq(Q2). This means that the experimental value attributed to A S in the naive parton model should be interpreted as the quantity zLS-fAF. This separately applies to each flavour; A u - , A u - AF, Ad-, A d - Aft, As--, A s - AF. There is no reason in principle w h y f A F should be small, so it can well nearly compensate AS. Note that ifAg is positive, i.e. ifgluons give a positive contribution to the proton helicity, then the sign is correct for this compensation to occur. The rather large and negative value observed for the strange component A s - A F , eq. (5), can be attributed to gluons coupled by the anomaly. If one makes the assumption, following Ellis and Jaffe [ 9 ] that As is small, which we do not especially recommend, then A F = + 0 . 2 3 and A u + A d = 0 . 6 9 (with large errors). It is true that one can argue that one could take polar-

29 September 1988

ized deep inelastic scattering to define Aq, so that the anomalous term could be reabsorbed in a redefinition of Aq, but then one should not be surprised if e.g. the strange component is not small. Also our analysis goes beyond mere definitions for a sizeable gluon component can be tested in other moments (via the x distribution) and also in other processes. To conclude we have found it is possible to have a consistent picture o f the proton with a large component o f the spin being carried by valence quarks balanced by a sizeable gluon contribution arising from the anomaly. While this work was in preparation we received the paper in ref. [ 22 ]. We agree with these authors on the existence of a large mixing between A S and AF, but we differ on a n u m b e r of important points. First we have directly computed the crucial coefficient Cg in eq. ( 11 ) and obtain a different result than quoted in ref. [22]. Then we reach completely different conclusions on the Q C D evolution. In particular for us A S is conserved while in ref. [22] the conserved quantity is A S + 2fAF. Finally in ref. [22 ] this conserved quantity is identified with twice the total proton helicity, i.e. the authors o f ref. [22] set A S + 2 f A F = 1. We consider this statement not justified in any case, because in patton terms the gluon contribution to the proton helicity is by definition Ag (see eq. (7) ). Also the total helicity carried by the partons is not conserved because of the orbital angular momentum. We gratefully acknowledge useful discussions with C. Becchi and J. Ellis.

References

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[ 10] G. Preparata and J. Softer, Milan preprint ( 1988); A. Giannelli et al., Phys. Lett. B 150 (1985) 214. [ 11 ] S.J. Brodsky, J. Ellis and M. Karliner, Phys. Lett. B 206 (1988) 309. [ 12] B.L. Ioffe, V.A Khoze and L.N. Lipatov, Hard processes, Vol. 1 (North-Holland, Amsterdam, 1984); V.M. Belyaev, B.L. Ioffe and Ya.L. Kogan, Phys. Lett. B 151 (1985) 290. [ 13 ] S.D. Drell and A.C. Hearn, Phys. Rev. Lett. 16 (1966) 908; S.B. Gerasimov, Yad. Fiz. 2 (1966) 598. [ 14] G. Baum el al., Phys. Rev. Lett. 45 (1980) 2000. [ 15 ] R.L. Jaffe, Phys. Lett. B 193 ( 1987 ) 101. [ 16] S.L. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento A 51 (1969) 47.

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