The next-to-leading QCD approximation to the Ellis-Jaffe sum rule

11 August 1994 PHYSICS LETTERS B

Physics Letters B 334 (1994) 192-198

ELSEVIER

The next-to-leading QCD approximation to the Ellis-Jaffe sum rule S.A. L a r i n 1

TheoryDivision,CERN,CH-1211Geneva23,Switzerland Received 25 March 1994 Editor. R Gatto

Abstract

The~correctiontotheEllis-Jaffesumruleforthe structure functiongl ofpolarizeddeep inelastic lepton-nucleon scattenng is calculated.

The results of the EMC Collaboration at CERN [ 1 ] and the El30 Collaboration at SLAC [2] for the Ellis-Jaffe [3] sum rule S~ dxg~(x, Q2) attracted a lot of attention to this sum rule; see [4-12] and references therein. Recent data of the SMC Collaboration at CERN [ 13] on polarized scattering of muons off deuterium and of the E142 Collaboration at SLAC [ 14] on polarized scattering of electrons off helium 3He allowed the determination of the analogous sum rule S~ dxg~(x, Q2) for a neutron. This in turn allowed us to find a difference f~dx[gP(x, Q2) _ g~(x, Q2)] which is the Bjorken sum rule [15]. At present, the Bjorken sum rule is calculated within QCD with quite high accuracy. The as correction [ 16], c~2 correction [ 17] and cr3 correction [ 18] are calculated in the leading twist approximation. The higher twist corrections are also calculated [ 19]. For the Ellis-Jaffe sum rule only the as correction was calculated in the leading twist [20]. The power corrections were calculated in [21 ]. In the present paper we obtain the as2 correction to the Elhs-Jaffe sum rule in the leading twist massless quark approximation. All calculations are performed in dimensional regularizaUon [ 22 ]. Renormalizations are done within the MS-scheme [ 23 ], the standard modification of the Minimal Subtraction scheme [ 24 ]. Polarized deep inelastic electron-nucleon scattering is described by the hadronic tensor

Wg~(p, q) =

~

1I

d4zelqZ(p,

sJJ~,(z)J~(O)Jp,s)

= ( - g ~ + q~q~F,(x, Q2)+(p~-~2 q~)(p~- P'q q2 ] -~-q~ p_qF2(x, 0 2) • ( +le,,,,,oo'qp

s¢ gx(x, Q2)+s¢p'q-P-q's p.q (p.q)2 g2(x,Q2) ) ,

t Permanent address. INR, Moscow 117312, Russia Elsevier Science B.V

SSD10370-2693( 94 ) 00744-R

(1)

S.A. Larm / Physics Letters B 334 (1994) 192-198

193

of which we will consider the structure function gl. Here J~, = ~y~,/~$= "-" z.,,"*= ~e , ~~., y~, $, is the electromagnetic quark current and J~ = diag (2/3, - 1 / 3, - 1/3 . . . . ) is the quark electromagnetic charge matrix, x = Q 2/(2p. q) is the Bjorken variable, Q 2 = _ q 2 . The nucleon state IP, s) is covariantly normalized as (p, sip', s ' ) = 6,,,2p°(2~ -) 3 3 ( 3 ) ( p - p ' ) . s~ is the polarization vector of the nucleon: s,~ = U(p, s)y~ysU(p, s), where U is the nucleon spinor, U(p, s) U(p, s) = 2M. The moments of the deep inelastic structure functions are expressed [ 25 ] via quantities of the Wilson operator product expansion (OPE) of the corresponding currents. We need the OPE of two electromagnetic currents. The strict method of the OPE ensures [26,27] that the OPE of two gauge-invariant currents can contain only gaugeinvariant operators with their renormalization basis. Thus we have only contributions from the non-singlet and singlet axial currents in the OPE of electromagnetic currents in the leading twist for the considered structure: i J dz e l q z T { J ~ ( z ) J u ( O )

]

°2~ ,~t~qq-~2[ ~ C~ (log (~2), a~(tz2))J~,~(O) +C~ (log (~2), a~(tz2))j~(O) +higher twists] , (2) where the non-singlet contribution can be rewritten as E

a

caj~a(o)

= Cns E

a

Tr(E2ta)J~a(O)'

to introduce, as usual, the unique non-singlet coefficient function C "s not depending on the number a. Here J~(x) = ~(x)y~,yst~b(x) is the non-singlet axial current, where t ~ is a generator of a flavour group, Tr(t~t o) = 1~aa , and J~(x) -- Y'.':f =11 (b,(x)Y,~Y5~b,(x) is the singlet axial current. The known twist-two and spin-one __ 1 ~e abc,.A a~,. A b,~,. A C,~j"~ has also the necessary quantum numbers, but axial gluon current K~-4e,~,~ ,~3(A a,~ 0 ~2A% + ~gT it cannot contribute to the above operator product expansion because it is not gauge invariant. Here and further on (before presenting the final results), we use the most practical definition for the strong coupling constant from the calculational point of view:

as =

g2

O~s

16at 2

4~-

The Ellis-Jaffe sum rule is expressed as

1

f

dXglp(n) (x, Q2) = CnS( 1, a~(Q 2) ) ( ___~ l g A I + ~ a s ) + C~( 1, a~(Q 2) ) exp

0

as(Qf~) da' y ( ~)~ 1 ~ f l ' - ~ ~') a } 9 ]~

(

as(/z2)

"a

(/z2)'

(3) where some comments are in order. H e r e p ( n ) denotes a target: proton (or neutron). The plus (minus) before IgAI corresponds to the proton (neutron) target. The proton matrix elements of the axial currents are defined as follows:

[ga Iso,= 2(P, slJ~; 3 IP, s)= ( A u - Ad)s,,, where

gA/gV =

-- 1.2573 5:0.0028 [ 28 ] is the constant of the neutron beta-decay,

slJ~ 8 [p, s)= ( Au + Ad- 2As)s~, (P, slJ~lP, s ) = (Au+Ad+As)s,~,

asS,~ = 2V/'3(p, E(/x2)s~ =

and we use the standard notation Aq(/x2)s~=(,P,

slqy~ysqlp, s), q=u, d, s.

S.A, L a r m / Phystcs Letters B 334 (1994) 192-198

194

We omitted the contributions of the nucleon matrix elements for quarks heavier than the s-quark but it is straightforward to include them. We should stress here that gA and a8 do not depend on the renormalization point/~2 since the correspondmg non-singlet currents j~3 and j~;8 are conserved in the massless limit, and hence their renormalization constants are equal to one. On the contrary, the singlet axial current has a non-trivial renormalization constant. Hence the quantity ,~(/z 2) does depend on the renormalization point (that is why it is not a physical quantity). The coefficient functions CnS( 1, as(Q 2) ) and CS( 1, as(Q 2) ) are normalized in the standard way to unity at the tree level. The renormalization group technique was applied to the coefficient functions to kill logarithms log (/z2/Q 2). The singlet axial current has the non-zero anomalous dimension yS(as) due to the axial anomaly [29,30]. So one can say that the axial anomaly contributes to this sum rule through the renormalization group exponent of the singlet current contribution in Eq. (3). Let us define the functions y~(a~) and/3(as) in the renormalization group exponent of Eq. (3). The renormahzation group QCD/3-function is calculated [ 31,32 ] in the MS-scheme at the 3-100p level: /3(as) = / x 2 das d/~2

=_(ll_Z

=

-

floa~

-

-

flxas

3

-

-

¢]2as

4

~ne)asz - ( 1 0 2 - ~3 nf)a3s - ( z~27 - 5o33, 18 " f

~

2

"JP 54 n f ) a s

4•

(4)

To define the anomalous dimension of the singlet axial current we should first define the singlet axial current itself within dimensional regularization. To define the singlet axial current we will follow the lines of [33], where the 't Hooft-Veltman definition [22] of the ys-matrix is elaborated for the multiloop case. The singlet current j 5 is renormalized multiplicatively and is expressed via the bare one [ j s ] B as 5_

s s

5

J~--ZsZMs[J~r]B

(5)

.

Here Z h s is the MS renormahzation constant which contains only poles in the regularization parameter e, the dimension of the space-time being D = 4 - 2E. The extra finite renormalization constant Z[ is introduced to keep the exact l-loop Adler-Bardeen form [ 34] for the operator anomaly equation within dimensional regularization in all orders in a s : O~,JSu = a s ~ ( G G ) ,

(6)

0 , , Aa ~4. where all quantities are renormalized ones. GG=eu,,apG%,,Gaao and G u ~ -_ d u A l _ _ ~ofabcAbAc , - u - ~ is the gluonic field strength tensor. In fact the full physical quantity, the Ellis-Jaffe sum rule, does not depend on the choice of the normalization constant Z[. But to be definite we adopt the normalization of [33] to keep in the MS-scheme the singlet axial current satisfying Eq. (6). The anomalous dimension of the singlet axial current is zero at the l-loop level and starts from the 2-loop level. To have the next-to-leading approximation we need two non-zero terms, i.e. the 3-loop approximation. The 3-loop approximation for the anomalous dimension of the smglet axial current was calculated in [ 33 ] and confirmed in [ 35 ]. The result in the adopted normalization reads YS(as) = / z 2 d log(Z~Z~ts) d tz 2

y(O)a~ + y(1)aZ + y(2)a3

=aZ(_6CFnf)+a~[(18C~_~ZCFCA)nr+4

~CFnr]2 .

(7)

Here CF = 4 and CA = 3 are the Casimir operators of the fundamental and adjoint representation of the colour group SU(3). The a 2 term agrees with the calculation [20] after multiplication of our result by the factor ( - 2) due to different normalizations. The non-singlet coefficient function C"S( 1, as(Q2) ) was calculated in the a~ approximation in [ 18] where the

S A. Larm ~Physics Letters B 334 (1994) 192-198

195

Bjorken sum rule was calculated in this approximation. We want to obtain the a 2s correction to the singlet contribution to the Ellis-Jaffe sum rule (3): (as(f 2) CS( 1, as(QZ)) exp

yS(a's)/ I

da's fl(a's)"~]-9 E(/z2)

as(/~2)

~ CS(1,

as(Q2))

(1 - a s ( Q 2) To) y(l)/31 ~ o +as(Q2)2

" lt -

(y(l)) 2--'y(2)]~0] 1 "2--flo Y "J 9 ~ ....

(8)

where we introduced the notation

(I Elnv = exp --

Sa, oas ~ 1

fl( a~) )

•(/z 2)

(9)

for the renormalization group-invariant (i.e./z2-independent) nucleon matrix element of the single axial current. Beside the 3-loop (the order a 3) approximation of the anomalous dimension, we need also the 2-loop (the order a 2) approximation for the singlet coefficient function C2( 1, as) which has already been calculated in [ 36]. Here we present the calculation of C ~ with another method to confirm the validity of the result obtained in [36]. We use the "method of projectors" [ 37 ]. To project out the coefficient function C s from the OPE of Eq. (2), one should sandwich this equation between quark states and nullify the quark momentum p. To be more precise, one should consider the following Green function:

i f dz e'qz(oIZ~(p) Y~Y5~b(p)J~(z)J~(O) = e,~p~,q~ C s log

as(tz:)

I0) v=oamputated

(OIT~p(p)'Y,~ys~P(p)ZSsZhs[JS(O)]B 10)

p=oamputated,

(10)

where some remarks are in order. ~b(p) is the Fourier transform of the quark field carrying the momentum p. Quark legs are amputated. The essence of the method [ 37 ] is the nullification of the quark momentum p. In the dimensional regularization scheme all massless vacuum diagrams are equal to zero. So on the r.h.s, only the tree graphs survive after the nullification ofp. In our case the only operator which produces a tree graph is j s . The non-singlet axial current does not contribute because of the nullification of the flavour trace: Tr(t ~) = 0. It is interesting to note that at p = 0 we have infrared divergences in the diagrams of the l.h.s, of l~q. (10). But these divergences are cancelled by the ultraviolet poles of the renormahzatlon constant Z~as of the singlet current. Thus to calculate C s we need to calculate the diagrams contributing to the 1.h.s. of Eq. (10). These are the diagrams of the forward scattering of a photon off quarks with photon momentum q and zero quark momentum. In comparison with the calculation of the non-singlet coefficient function C as [ 17,18 ], we have at the 2-loop level two extra diagrams where both electromagnetic vertices are inside a closed quark loop. The analytic calculation of the diagrams has been done with the symbolic manipulation program FORM [ 38 ] by means of the package MINCER [39]. This package is based on algorithms of [40]. The result is CS( 1, as(Q 2) ) = 1 + as( - 3CF) + a2[ ~ C ~ - 23CF CA + (8~3 +

~CFnf],

( 11 )

where ~'3is the Riemann zeta-function (~'3= 1.202056903... ). This result of ours agrees with the calculation in [ 36 ] if one takes into account the fact that another finite constant Z~s (relevant for the non-smglet axial current, see [ 18,33] ) was taken in [36] for the normalization of the singlet axial current instead of our Z[. Multiplying our result ( 11 ) by the factor Z~ -~- 1 +as(--4CF) + a 2 ( 2 2 C 2 - -~-CACF 107 q- .31 18CFnf) + O ( a 3) Z~s 1 + a ( - - 4 C F ) + a 2 ( 2 2 C 2 ~CFCA + ~9CFnf) --

,

S.A Larin/ Physics Letters B 334 (1994) 192-198

196

one can reproduce the result of [36]. So we have a strong check of CL In principle our technique allows us to compute also the a s3 correction to the singlet coefficient function C s. But we need then also the 4-loop anomalous dimension of the singlet axial current in order to take into account this correction self-consistently in the next-to-leading approximation for the Ellis-Jaffe sum rule. The calculation of the 4-loop anomalous dimension is very time-consuming at present, although all the necessary techniques are available. Collecting together all the relevant results for the coefficient functions and the anomalous dimension, we obtain finally the next-to-leading approximation for the Ellis-Jaffe sum rule: 1

~dxgp(n)(x, Q2)= [1- (°ts(Q2-.-----~)) + ~s~)2(-~+~nf) 0 3 (1296

216

×

6n,

1+

2n-"-'~ 33 + as(,~2)

Xexp(-

f

'[-5~4~3--3~'5)nf

- 1 1648t~ 5 " 2 1 f-I

(+~lgAI +3~a8)

(1029/4)nf +(3(23/2)n23 - 2nf) 2 + (1/3)nf3'

]

y~(a'~)~ da's--fi~-~a,)]l~](/z2).

(12)

oq/'tr=g2/4"rr2

Here we use for the strong coupling constant. We keep the known (extra for the next-to-leading approximation) as3 term [ 18] for the non-singlet part, since this part determines the Bjorken sum rule. For the singlet part we factorize the QZ-dependent factors coming from the coefficient function (the first factor in the singlet part) and from the renormalization group exponent (the second factor). The leading as term agrees with [ 20]. For the case nf = 3 the sum rule reads

' f dxgp(n)(x,Q2) = [ 1 -

3] -3.5833

-20.2153

(___~lgA[ + ~ a 8 )

0

+ [1-(@)-1.0959 a,(iz2)

(a~)

2] [1 +0.6666 (as(Q2)~)+ 1.2130 ( a ~ )

2]

s a t

~--~-)'a ~) ] -~E(/x2) "

(13)

We can see that the coefficients of the second factor in the singlet part (which is due to the non-zero anomalous dimension of the singlet current) are approximately of the same magnitude and of opposite sign in comparison to the coefficients of the first factor which is due to the coefficient function. Thus one can not neglect the contribuUon of the anomalous dimension m comparison to the contribution of the coefficient function in this case. Opening the brackets in the singlet part we obtain our final representation for the Ellis-Jaffe sum rule for nr = 3:

, j dxgP("'(x,Q2) = [ 1 -

-3.5833

-20.2153

(___~[ghJ + ~ a 8 )

0

[

+ I_1-0.3333

-0.5495

+ O ( a s3)

]

l .... ~E

(14)

S.A Larm / Phystcs Letters B 334 (1994) 192-198

197

where E,nv is the renormalization group-invariant nucleon matrix element of the singlet axial current which includes the renormalization group exponent of Eq. (13) and is defined in Eq. (9). To estimate the size of the calculated a 2 correction we take e.g. Q2 = 10.7 GeV 2 which is the value of the EMC Collaboration [ I ]. Substituting as = 0.25 for this value of Q2 we find that the contribution of the as2 correction to the smglet part of the Ellis-Jaffe sum rule is of the order of 0.5% in the MS-scheme which is beyond the present accuracy of the experiments. I am grateful to K.G. Chetyrkin, J. Ellis, B.L. Ioffe, M. Karliner, W.L. van Neerven and J.A.M. Vermaseren for helpful discussions. I would hke to thank the Theory Dwision of CERN for warm hospitality. The work is supported in part by the Russmn Fund of the Fundamental Research, Grant N 94-02-04548-a.

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S A. Larm /Phystcs Letters B 334 (1994) 192-198

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