Gluon operator coefficients in electroproduction

Nuclear Physics B152 (1979) 273-284 © North-Holland Publishing Company

GLUON OPERATOR COEFFICIENTS IN ELECTROPRODUCTION Jon SHEIMAN

Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Received 14 December 1978

We compute the Wilson coefficients of the twist-two gluon operators which are relevant to the W2 tensor structure in electroproduction. We find that the simple Green functions used to define the renormalization of the operators each have two independent tensor structures. This gives rise to varying renormalization conditions. We choose conditions which are different from those of other authors, and which simplify the computations. Taking these differences into account, we show that almost all of the published results are equivalent.

1. Introduction Recently there has been some interest in extending the twist-two operator product expansion (OPE) analysis o f electroproduction beyond the lowest order in ~ [1,2] (where ~ is the running coupling constant o f QCD). This analysis gives predictions for the moments of the structure functions F o f the form [3]

f dxxn-2 F(x, Q2) = ~] c(~n)[f(Q2), I] 0 a,b

Gexp

fM /3(~) -- 2)

ab

(l) where the C(an) andA(~n) are related respectively to the Wilson coefficients and the nucleon matrix elements of the twist-two operators o f spin n in the OPE of the product o f two electromagnetic currents; 3'n is the anomalous dimension matrix for these operators;/3(g) is the usual 13 function; the G symbol denotes g ordering in the exponential; M is the renormalization point; and x is the Bjorken scaling variable. Extending the analysis to next to leading order involves computing C(~n) to order ~2, 3'n to order ~4, and/3 to order ~ s * Research is supported in part by the National Science Foundation under grant no. PHY7722864. 273

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274

Differences in the renormalization of the operators give rise to ambiguities in the next to leading order corrections to the C (n), 7n, and A(bn) (which must cancel when these quantities are combined in eq. (1)). The twist-two operators of spin n are defined formally via

o(n) =_i n-I

[~,~lalDu:

... Dun ~ ] sym

o(n) -~ ~li"-2[FaUl D u2 ... Dlan-1

--

traces,

F ~ tan] sym - -

(2a) traces

0 (n) = i "-1 [~Xr~UlD v'2 ... DUn ~]sym - traces,

(2b) (2c)

where the subscript sym denotes symmetrization of the Lorentz indices, the Xr are the generators of the flavor group, D is the gauge covariant derivative, and sums over flavors and colors are implied. To completely define these operators, we must specify a renormalization condition which determines their multiplicative matrix renormalization. We use the following condition for the flavor singlet operators (which we call "g renormalization"):

arnp =6ab6KAga#PU'... P an + ....

(3a)

(0 IT(~j,(e)~o(n)(o)

(3b)

q,/f(-e))[0)am

p =

4 5 f f f K ~ pUl ... pUn + ....

where f denotes the flavor of the quark, p2 = _ M 2, K = A or qJ, amp denotes amputation of the Green functions, and ... denotes other tensor structures. Altarelli et al. [2] use a condition (which we shall call "t renormalization") in which eq. (3a) is replaced by:

(0 IT(A~(P)O(~)(O)A~b(-P))IO)amp = ~ab~K¢ 2(P ul ... Pun + ...).

(4)

In sect. 2 (eq. 15b), we show that the Green function of eq. (3a) has two independent tensor structures, so that the g and t schemes are not the same. We will also show that a similar problem occurs for the Green function of eq. (3b) (if we don't take the trace with ~). Finally, Bardeen et al. [1 ] use the minimal subtraction (MS) scheme in which the 1 PI Green functions involving these operators are computed in d dimensions, with the infinite parts subtracted as poles in d - 4. All of these renormalization conditions are chosen to be satisfied automatically to zeroth order, however they differ at order ~2. The operators O}~) (K = A, ~b) are a complete set of flavor singlet twist-two, spin-n operators occurring in the OPE of the product of two currents. Thus the O F) ~_determined by one renormalization condition must be a linear combination of the O ~ ) determined by another, i.e.,

J. Sheiman / Gluon operator coefficients

275

where (6) A similar equation can be written for the A (~)'s by making the replacement O F) ~ A ~ ) and O ~ ) ~ A'~) *. The product of two currents is invariant under these changes in normalization, thus the change in the O's must be compensated for by a change in the Wilson coefficients:

,

(7)

so that the Wilson coefficients differ at order ~ 2. Finally, to preserve eq. (1) at all Q2, the anomalous dimension matrix must change as follows:

rn

,

(8)

so that the 7's differ at order ~4. The g scheme has the following advantages. Firstly, in computing the anomalous dimensions, graphs in which an initial or final gluon comes from a gauge-covariant derivative (e.g., fig. 3b) cannot give rise to ga~ tensor structure, and thus do not contribute. Secondly, in computing the coefficient of the gC~pupv tensor structure of the current-current correlation o f fig. 2 (which is needed for the determination of the gluon coefficients C(An)) graphs 2c and 2d are infrared finite ~ . In the t scheme, we take a trace in the indices a and/3, so this is no longer true. In the MS scheme, no explicit reference is made to the tensor structure of gluon Green functions. Thus the anomalous dimensions and gluon coefficients can be calculated by extracting the ga~ tensor structure, with the advantages this implies *** This paper is organized as follows. In sect. 2 we present the algorithm for computing the order ~2 corrections to the Wilson coefficients. We will show that all of these are the same in the g and t schemes with the exception of the gluon operator coefficients relevant to the W2 tensor structure. We compute these coefficients in sect. 3. In sect. 4, we compare the published results for this quantity by using them to compute the vW2 structure function for on-shell gluons with massive quarks.

* In the language of the parton model, this implies that the quark and gluon distributions are ambiguous at orderg-2. ** To see this, note that in the light-cone limit, graphs 2c and 2d must reduce to 3b, 3b' or 1. The infrared divergence then comes from figs. 3b and 3b', but these cannot give rise to ga/3 tensor structure. (This comment is due to M. Peskin.) *** The authors of ref. [ 1 ] choose to take a trace in the indices a and/3.

J. Sheiman / Gluon operator coefficients

276

2. The algorithm for computing structure functions The Fourier transform of the twist-two OPE of the product of two electromagnetic currents is

i fd4y eiq'y T(JU(y) if(O)) =~n [~,gV2 _ (~u,qV +g~lqU)qu2 +g.uq.lqu2]

× 2- q . ~ q.. ~(.) ,~,,...,. + ~7) o2...,. 1 +

2

gUy qUg~12n qu,"" q"n It(n) O~ 1-..•n

+ C(,) O~l...#n]

(9)

where the subscripts 2 and L denote terms which contribute respectively to W2 and WL tensor structures in electroproduction, Q2 _ _q2, and we have written down only the contribution of flavor singlet operators [which are renormalized independently of the operators O(rn) of eq. (2c)]. The normalization of the C's is chosen so that

ci'~ : L E Q ~ + o(~:), nf /

(n even),

(10)

where the sum runs over flavors, Q/is the charge of flavor/" quarks, and nf is the number of flavors. All of the other coefficients vanish to zeroth order. Taking the spin-averaged nucleon matrix element of eq. (9) gives

TUV = i fd4y

e iq'y

1

(PITJU @):'(o) IP)

~

•

-- ~n X-~'~-2[PUP - (e"qV + p U q " ) ~

(eq?]

+gUU q-ff

"]

4 rp(")A(n) + C(n) A(n)] X -~ [~2,~ 2,A

1 [W

q~"]lc~ ")A<")+ ~<') A(")~+ flavor non-singlet

+ O(p2/q2), where P is the nucleon momentum,

(11) x -~ Q2/2P. q, and the A's are defined via

(PI O~ l"''"n I P ) = A ~ ) ( P ~' ... PUn - traces), where K = ~ or A.

(12)

3. Sheiman / Gluon operator coefficients

277

Thus the OPE gives a Laurent expansion of T uv in x valid for Ixl > 1. The coefficients in this expansion are related to integrals around the cut (from x = - 1 to x = 1), which are in turn related to W ~'v by the optical theorem, where _

i

f d'y

1 +--~n(PU-qU~-~)(PU-qV~)W2

= (--g'UV + ~ ) W

.

(13)

The connection is *" 1

f xn-ZvW2(Q2, x)dx

=

(14a)

½(A(n)C(2n) +A('Oc(2n) ) + ....

o 1

( x n - ' W I (Q 2, x ) d x

= 4t: ! r A ( "~) ( Cv 2,~ ('i -

C (L,q) ' ) ) + A (A " ) ( C ( " 2,A ) - C L,A ('))1

+ ""'

o

(14b)

where ... indicates the non-singlet contribution and terms of order p2/q2, v -

P. q/mn, m n is the nucleon's mass, and n is even. To find the C's, we form simple off-shell Green functions using eq. (9). First we define the relevant Green functions involving the O's: (01T(~(_p)jJO~I ...Un if(p))10)amp - 4nfQ~)(P ul ... pUn _ traces),

<0 IT(Aaa ( - P ) O ~ ' ""UriAh(P))[O)am p

--= (~ab G(~)[g°~P"l? ~2

-- g°t~l : P

(1 5a) ~2

_g#Ulp°~pU2+p2g~ig#U2]e"3 pUn+6ab"~(n)[ga# _ Pae"TpUl • .. p: ] . . pUn . . (15b) where the right-hand side of eq. (15b)is understood to be made symmetric and traceless in the indices ~1 " " /2n, K =A o r 4 , p2 < 0, and a sum over flavors is simplified in eq. (15a) **. The two-tensor structures in eq. (15b) are the only ones consistent with color gauge invariance. Referring back to eq. (3a), we see that for the g scheme, at p2 = _ M 2, the renormalization condition is

O(~ ) + "G(ff) = 6 KA •

(16a)

* Eq. (14) is in disagreement with eqs. (2.11) and (2.12) of ref. [1], where the same definitions are used. ** The flavor singlet nature of these Green functions guarantees that the operators 0 (n) will not contribute.

J. Sheiman / Gluon operator coefficients

278

Similarly for the t scheme, we have (16b) We now form the analagous Green functions, from eq. (9)

T~" --ifd4y eiq'Y ( OIT(~(-P)PJU(y)ff = ~.

1

4

v

(O) t~(P))10)amp

"n" Q(n) + C(n) Q(n)] 2,A

x-~_2-~Pue 4nf[C~,i 1 qUqV

+ ~n ~h ~-4nf[C(Ln, )

Q(n) +C(.) Q(n)] + ... L,A

(17a)

,

T~V~~ - i f d4y e iq'y ( 0l T(A~ (- P)J• (y)JU(O)A~ (P))] 0)amp =6ab ~n

+

1

2 ~-dg l

4 ~

v

n

Q2 'C(') t L,+ (C(') • +

~(en)

c(n) (c(n)+~(n))]

+ C(") L,A (G? + Cq"))] + .... (17b)

Tp.V;c~ Gabot

=6abOrtxn-2 I Q2 4 pupv[c(2n,)t2a(n '. # )+ 3~(n)~+p(n)QG(n) # " ~2,A ~. A +3~(n))] qUqV ~ + 6ab n V1 __[c(n)Q2 L,~ (2a(n) + 3G(~n) ) + c(n)L,A (2G(A n)'+3G("))]+ "'",

07c) where + ... denotes other tensor structure + O(P2/q2). The left-hand sides of eqs. (17) and the G's and Q's can be computed in perturbation theory, resulting in an expansion of the C's in powers of ~. For the g scheme, eq. (17b)~ more useful than eq. (17c) because it involves precisely the combination of G and G appearing in the renormalization condition (eq. 16a). Similarly, in the t scheme, eq. (17c) is more useful than eq. (17b). We now discuss how the choice of renormalization scheme affects the C's in second-order perturbation theory. First, consider CL,~ and CL,A, both of which are order ~2. The qUqU tensor structure of eqs. (17) to order ~ 2 involves these coeffidents and the zeroth-order operator Green functions (which are of course the same in all schemes). Thus to order ~2, the CL,~ and CL,A are scheme independent. Consider now the order ~ 2 contribution to the pup, tensor structure of eq. (17a). The c(2n,) Q(n) term is order ~4, and thus does not contribute. Thus to determine ¢-(n) ~2,~ to order ~-2 from T~2U,we need only ~/3(n)to order ~2. However the graphs involved do not make use of eq. (3a), eq. (4a), or their MS analogue. Thus the g and t schemes give the same C2(n ,) to order ~2; MS gives a different result.

J. Sheiman / Gluon operator coefficients

279

The difference b e t w e e n the g and t schemes arises from the fact that there are two distinct tensor structures occurring in eq. (15b). The same is true for quark-quark Green f u n c t i o n s * (0 1 T f ( - e ) o ~

1 "'~tn ~(e)10)am p =R(Kn)')'~IP#2 ... P un + R ~ ) e

~

-,

(18) where the fight-hand side is u n d e r s t o o d to be made symmetfized and traceless in the indices/a 1 .../a n . The Dirac m a t r i x structure is limited to 7 matrices b y chirality conservation. The renormalization c o n d i t i o n of eq. (3b) is R ~ ) + R ~ ) = 6K~ ,

(19a)

at p2 = - M 2. De Rfijula et al. [4] c o m p u t e the C (n)2,~ and C(Ln.~ to second order using a different renormalization, which in our language is (at p2 -' _ M 2) ** R(~ ) = 6K~

(19b)

.

By arguments similar to those above, o n l y the C2(,n) are affected to order ~ 2. This m a y explain the differences in the results given in refs. [2,10] ***

3. C o m p u t a t i o n o f C~n,) in the g scheme We c o m p u t e -fur;a# ~Gab to order ~-2 and express the result as a power series in 1/x for

Ix > 1;i.e., =

g%°.e

28o

+ ....

(2o)

* This was pointed out by H. Georgi. ** The authors of ref. [4] falsely assert that the J~ terms are associated with higher-twist operators, *** The authors of ref. [2] claim that their result for C~I) is consistent with the Adler sum rule, while the result of ref. [4] is not (the c~nr) are coefficients of the non-singlet operators which to order ff 2 are trivially related to the c~h.~). Their argument hinges on the assumption that the anomalous dimension of the operator'O (1) vanishes, by virtue of flavor charge conservation. Charge conservation, however, dictates that the unrenorrnalized flavor current Jr~ defined via

= ~uXr*U~u, (where u denotes unrenormafized fields) has a vanishing anomalous dimension. This operator may not satisfy the renormalization condition placed on O~, so that in general

O~r = F(~ (M2))Jr~ , where F depends on the renormalization scheme. Thus Or~ need not have a vanishing anomalous dimension.

J. Sheiman / Gluon operator coefficients

280

Fig. 1. Zeroth-order Feynman diagrams for eq. (15b). The symbol ® denotes a gluon operator vertex; the curly lines are gluons.

where ... indicates other tensor structures + O(~ 4) + O(p2/qZ). T h e n , from eq. (17b)

a.~ 2 =c~"~ ~c~"~, + ~ " ~ +c ~n~2,(c~"~A ~ +'g~"b +o(~ 4)

(21)

.

Using eq. (16a), we see that in the g scheme

G(n) + ~(n) = b-gZ ln(_p2/M 2) + 0 ( ~ 4 ) .

(22)

The constant b is related to an element o f the zeroth-order anomalous dimension

matrix [5] b

=

1

nf n 2 +n + 2 47r2 n ( n + l ) ( n + 2 ) "

_

~7~A

(23)

Also, to zeroth order, the graph of fig. 1 gives

G(n) = 1,

G(An) = 0 .

(24)

P

P

(o)

(c) -q

-q

P

q

P (a')

q

(b) q

q

-

P

P (b')

P

-p (d)

Fig. 2. Second-order Feynman diagrams for the current-current correlations of eqs. (17b) and (17c). The wavy lines are photons.

J. Sheiman / Gluon operator coefficients

281

Combining eqs. ( 2 1 ) - ( 2 4 ) and eq. (10) gives C

:an~2_~Q j

~(M2)~

2 1 ~21n ~2 M 2 n(n + 1) (n + 2)

+ O(~4)

"

(25) We now proceed to compute the an. The relevant graphs are shown in fig. 2. Figs. 2a' and 2b' are equal respectively to 2a and 2b. Fig. 2b is related to 2a by x ~ - x and similarly for 2c'and 2d. The g~(Jpupv tensor structure of figs. 2c and 2d are finite as p2/q2 ~ 0; so we may take this limit directly. For the other graphs, we must keep p2/q2 4=0 and extract the ln(p2/q 2) and constant terms when doing the Feynman parameter integrations. This is a tricky procedure. In particular, there are p2/q2 terms in the numerator which give rise to non-vanishing results as p2/q2 _+O. The result is T~a~

2rr2Q2 ~Q~6ab PuPv l n ~ - ~ [ x 2 + ( - 2 x 4 + 2x 3 -

+(-8x4+8x3-2x2)ln

+~x 2 + x ~ - x } +

-x

+(4x3-4x2+2x)

_

o dalna_i~a~x

....

(26)

where ... indicates other tensor structures + O(~ 4) + O(p2/q2). Expanding this for Ixl > 1 gives * 1 ( an

p2

n2+n+2 n(n + 1)@ + 2)

÷

8 n+2

8 n+l

+

2 n

4 (n+2) 2

4 2 ) + (n ~ + 1) - n 2 ] '

(27)

for even n (an vanishes for odd n). Note that in eq. (25), the l n ( - P 2) pieces cancel, as they must if C2(,n) is to be p2 independent. Thus, taking M 2 = Q2, we find

8 + _ _4 + ~F ____.2 n+l n (n+2) 2

~2 8 C'~"~(g-'l)=~2-n2 ")"{(" ~s' Q Ln+2

4 @+1) 2

n~]+O(~4). (28)

* The expansion of the integral in eq. (26) is 1 0

1

iUdlx

,,=o x n _ 2 (n

l?

J. Sheiman / Gluonoperatorcoefficients

282

4. Comparison of published results for C (n) We use the C2,A (n) to compute a quantity which is independent of the renormalization scheme used: the uW2 structure function for on-shell gluons in a theory with quarks of mass m.

4~ f eiq'y d4y( G, PIJU(y)ff(O)lG, P)

Here, IG, P) is a gluon state with momentum P, and an average over gluon helicities is taken. Optical theorem-OPE arguments analogous to those used for the proton structure functions give 1 f 0

=_

+

+

+

.

(30)

The helicity average has induced a trace in the gluon Lorentz indices, making the Green function of eq. (17c) relevant. The C's and G's of eq. (30) are those of the massive theory. Suppose, however, that we throw out all terms which vanish as m ~ 0. The Wilson coefficients have a smooth m ~ 0 limit; thus we may use the massless C's in eq. (30). Further, to zeroth order G(n) + 3~(n)= 1, independent ofm. Finally, the quark mass regulates all p2 ~ 0 singularities. From eq. (15b), we see that if(n) must vanish like p2 as p2 _+ 0 in order to avoid having a p2 _+ 0 pole; thus G(~~) Ip2=o = O. Eq. (29) becomes 1

1

~

1

mh^\

f xn- 2pWGdx =_ ~ [~] a]2(G~(n) +G(n))__ + C(n) ]2,A"It - O ( g 4 ) ' t ' O l ' - ~ - )

(31)

~v ~ H f

0

where we have used the zeroth-order c~(n~. We compute G(~n) +G!rn~) in the massive theory as the coefficient of gafF al ... PU~"in the appropriate Greer~ function (eq. (15b)), subtracting the result at p 2 = _M 2, as required by the g scheme. The graphs • . are shown in fig. 3. Figs. 3b and 3b t must have a g cO~i or a g#Uidue to the gluon attached to the operator vertex and are therefore irrelevant. Figs. 3a and 3a' give

G(n) ~(n) nfg 2[- n 2 + n +2

m2+~;) + G ~ = - ~ 2 Ln~ + ])(n + 2i(ln ~ /'=1

--

n

+

n+l

n+2

÷

n2

(n+

Combining eqs. (28), (31) and (32) gives, at M 2

= 0 2

+ O

(32)

J. Sheiman / Gluon operator coefficients

(a)

283

(b)

(b') Cb') Fig. 3. Second-order Feynman diagrams for eq. (15b). The symbol ® denotes a quark operator vertex.

f dx xn-2(vW2) G --

g2 ~ _ 2 ( n2 +n+2 + 1 _6 ~--6 n n+l n+2

lnm2 n

nl_~}

(33)

or equivalently

vW~2 = ~

Q2x 1 - 2 x + 2 x

2) ln

Q2(1 - x)

m2 x

7 - 1 + 8x(1 - x ) J .

(34)

This quantity has been computed by several authors. Our result is in agreement with those of Kingsley [6] and Witten [7] (if we multiply Witten's result by z~), but does not agree with that of Hinchliffe and Llewellyn-Smith [8] (in which the " 8 " of eq. (34) is replaced by a "6"). Bardeen et al. [1 ] have similarly converted their result for C (n) 2,A in the MS scheme into vW6; their result agrees with ours. Finally, both Altarelli et al. [2] and Bardeen et al. [1 ] compute the Green function of eq. (17c), and their results agree. Extracting c(2n,)Ain the t scheme from there is trivial. I would like to thank H. Georgi and M. Peskin for their useful suggestions.

References [1] W. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Fermilab publication, 78/42THY (1978). [2] G. AltareUi, R.K. Ellis and G. Martinelli, Nucl. Phys. B143 (1978) 521. [3] H. Georgi and H.D. Politzer, Phys. Rev. D14 (1976) 1829.

284 [4] [5] [6] [7] [8]

J. Shehnan / Gluon operator coefficients A. De Rfijula, H. Georgi and H.D. Politzer, Ann. of Phys. 103 (1977) 315. H.D. Politzer, Phys. Reports 14 (1974) 129. R.L. Kingsley, Nucl. Phys. B60 (1973) 45. E. Witten, Nucl. Phys. B104 (1976) 445. I. Hinehliffe and C.H. LleweUyn Smith, Nucl. Phys. B128 (1977) 93.