QCD relation between deep inelastic neutrino reactions and polarized electroproduction

Volume 89B, number 3,4

PHYSICS LETTERS

28 January 1980

QCD RELATION BETWEEN DEEP INELASTIC NEUTRINO REACTIONS AND POLARIZED ELECTROPRODUCTION T. MUTA

Research Institute for FundamentalPhysics, Kyoto University, Kyoto 606, Japan Received 29 October 1979

Within the framework of perturbation theory in quantum chromodynamics (QCD), a phenomenologically useful relation is derived between the moments of the structure functions in charged-current neutrino reactions and in polarized electroproduction (eq. (1)). The relation is illustrated by practical QCD results up to the next-to-leading order. The phenomenological consequences of the relation are analyzed with the help of existing data.

In the present communication we derive the following relation between the moments of the structure function F 3 in charged-current neutrino reactions [ 1] and those of the structure function gl in polarized electroproduction (or muoporduction) [2] :

Me(02)/~(02)

(1)

~ 2 = Myn(Q 2 )/an(Qo),

where 1

Me(Q 2)

= f dx xn-lgI~-n(x,Q2), 0

1

Mn(.Q2) = f dxxn-lF;+~(x, 0

Q2),

v + ~ = rt~u3N + F~N with N = (p + n)/2 the isospin-singlet target, Q2 is the momentum transana glp - n _- glep - glen ' -~3 fer squared of the leptons with Q2 the fixed reference point, and x the Bjorken scaling variable. In the derivation of eq. (1) we work in the framework of QCD perturbation theory and the quark masses are neglected relative to Q2. The structure functions in neutrino reactions and polarized electroproduction are defined in terms of the nucleon matrix elements of products o f the weak and electromagnetic currents, respectively. The moments of these structure functions are related to the coefficient functions appearing in the operator product expansions of the current products. The antisymmetric part of the T-product of the electromagnetic current.iu(x ) is relevant to polarized electroproduction and has the following operator product expansion [3,4] :

eiq'x ZO'g(X)fv(O)]A = 6tavhc~qx ~n Eln(Q

2 C~#l""gn-1 )R 1

(0)qut

...qun_l(Q2/2) -n

2 cvgul + (--eta~a'aqvqX + ev~'°cq~qX + e~vag3q2) ~n E2n (Q)R2 ""Un-2(0) qt~l ""qtsn_2(Q2/2)-n'

(2)

where the suffix A refers to the antisymmetric part, Q2 = _q2, Ein(Q2) (i = 1,2) are the Fo.urier transforms of the coefficient functions, singular on the light cone, and R~l""t~n(x) (i = 1,2) are regular composite operators. It should be noted that Ein(Q 2) and R~ l'''un(0) in eq. (2) are unrenormalized with respect to the operator renormalization. We shall discuss this point in more detail shortly. We are interested in flavor-non singlet combinations 413

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PHYSICS LETTERS

28 January 1980

of structure functions like gp-n (i = 1,2) and hence only the quark composite operators are relevant (there is no operator mixing between the quark and gluon composite operators). The operators responsible for the most singular coefficient functions are

Ra.u 1 l ...un_ 1t~'~ t~s = S'~(x)~ 'c~75 D'ul ...o#n - 1 ~ Xi ~(x) - trace,

(3)

R~.ul ...Un-2 (X) = S'~(x)7 ~ 7519~D uz ...D un-2 ½Xi ~(x) - trace, where ¢J(x) is the quark field, D u the covariant derivative, Xi the flavor generator matrix, and S and S' are the sylnmetry operations as defined in ref. [4]. The coefficient functions Ein(Q 2) may be made finite by operator renormalization [5,6]. The renormalized coefficient functions ~in(Q 2) are related to the moments of the structure functions by moment sum rules, e.g. [3,4,7], 1

Mne(Q2) = f dx x n- lgp-n(x, Q2) = al n ~ln(Q2), 0

(4)

~'~.tt1 ...~tn_ 1 where aln is the nucleon matrix element of the renormalized composite operator..1 Neutrino reactions are described by the product of the weak currents J.u(x), the V - A interference term of which has the following operator product expansion:

e iq'x T[Ju(x)'~ J ( O ) I v A = e uvxaq X ~n C3n(Q2)O°~'ul ""tan-1 (O)q.u a ...q.un_ 1(Q2/2)-n '

(5)

where the suffix VA refers to the V - A interference term and

O~Ul ""Un-z (x) = S ~ ( x ) 7 C~19"z ...D #n-1 ~(x) - trace.

(6)

Here we consider only the flavor-singlet operator since we are interested in a flavor-singlet target. It should be noted that, although another singlet operator may be constructed with gluon fields, it does not contribute to the V - A interference term according to C-invariance [8]. Again the coefficient function C3n(Q 2) and the operator O~Ul ...Un-x (x) are unrenormalized with respect to the operator renormalization. The renormalized coefficient function C3n(Q 2) is directly connected to the moment o f F 3 [8,9] • 1

Mn(Q2 ) = f d x x.n-l-v+V,. (7) r 3 tx, Q Z ) = a 3 n ~3n(Q2), 0 where a3n is the nucleon matrix element of the renormalized composite operator ~a.ul .-..un_l, and only the odd-n moments are physically meaningful because of crossing relations. It is the standard method, in order to calculate the coefficient functions Eln , E2n and C3n , to sandwich eqs. (2) and (5) between off-shell quark states. Define unrenormalized amputated Green's functions by T u ( p, q) =

i fax

dy dz e iq "x+ip'(Y-Z)(o[ T[~(y)fu(x)jv(O ) 5(2)] A [0)amp'

A~I """n(p) = f dy d2 eiP'(Y-Z)(0[ T[~(y)R~ 1 ""t~n(0)~(z)] 10)amp, TSgv(p, q) = i fax dy dz eiq'x+ip'(Y-Z)(o[ T[ff(y) J~(x) J ( 0 ) ~(z)] vAl0)am p, .un(p)

_-f

dy a~ eip'(Y-Z)(ol T[ if(y) O m ....un(0) ~-(z)] [0)amp.

These Green's functions are assumed to be properly regularized by the common regularization scheme. For example, we choose the dimensional regularization scheme ,1. From eqs. (2), (5) and (8) we obtain ,1 As for the treatment of 3's we follow the prescription of Chanowitz et al. [10]. 414

(8)

Volume 89B, number 3,4

PHYSICS LETTERS

28 January 1980

Tv(p ' q) = ieuvxo~qXn ~odd Eln(Q 2)A1c~l...Un-1(P)qul ""qun-x (Q2/2)-n + i(_e#x~qvqX + euxocquq x + e TSuv(p,q)=ieu~,~q x ~

n odd

uccq2) ~ n odd

E

t'-~2~-~Zl'"gn-2" " 2n t~d )/~2 (P)qul

""qun : (Q2/2)-n'

C3n(a2)l'C'm'"Un-~(p)qm...qun_ (Q2/2)-n.

(9)

1

If we work in QCD perturbation theory and deal only with massless quarks, we immediately find the following simple relations to every order o f the perturbation series: r

:

--

(10)

with the normalization T2 = 1. In fact, as can be seen in the typical Feynman diagrams in fig. 1,75 appearing at a vertex may be shifted towards one o f the external quark lines by anticommuting with the 7u's. Thus the amplitude with 3'5 is simply related to the one without 3'5 by eq. (10). From eqs. (9) and (10) we find E I n ( Q 2) :

C3n(Q2).

(1 1)

We perform the operator renormalization for the operators R 1 and O:

Rl=z#'

1,

(12)

with the multiplicative renormalization constants Z R and Z O. Here we neglected the quark field renormalization constant as it is irrelevant to the present argument. By the renormalization (12) the coefficient functions are made finite and the renormalized coefficient functions are given by

fin -_ ZnR Eln'

~C3n =ZOnC3n

(13)

We have the freedom o f choosing different normalizations for the operators R 1 and O and hence, in general,

M~(Q2)xIOa 10.C

T.v

TSv

~p

~'~-5D'u !....

AI

F'

Fig. 1. Typical examples of Feynman diagrams in QCD for ~.,T'S'v, A~#1 """n-1 and r aM1 """n-1.

1D

10 100 Fig. 2. The QCD prediction normalized by eq. (19) at Qo2 = 2.2 GeV 2 including the next-to-leading-order correction together with the rescaled CDHS data.

415

Volume 89B, number 3,4

PHYSICS LETTERS

28 January 1980

ZRn 4: Z 0 . The difference may be factored out as a finite multiplicative factor b n ,

R_

O

(14)

Z n - bn Z n •

Combining eqs. (11), (13) and (14) we have Eln(Q 2) = b n C3n(Q2).

(15)

Eq. (15) is a finite well-defined relation where the space-time dimension may be set equal to 4. Since b n does not depend on Q2, we immediately obtain eq. (1) from eq. (15) by using eqs. (4) and (7). This completes our argument. Note that it is always possible to adjust the normalization condition so that b n = 1. In the above derivation of eq. (1) only the following three conditions were employed: (1) choosing the special structure functions to which only quark composite operators contribute, (2) working in perturbation theory and (3) assuming massless quarks. Thus relation (1) may hold for a wider class of field theories in which gluons are flavor-singlet and perturbative calculation makes sense. However, in order for perturbation theory to work at short distances the theory must be asymptotically free and the only such theory available is QCD. One of the important consequences of eq. (15) is that the QCD correction to the Gross-Llewellyn Smith sum rule should be the same as that to the Bjorken sum rule in polarized electroproduction. The QCD calculations up to the next-to-leading order confirm this statement [6,11,12]. We can also check eq. (15) for arbitrary n by comparing the lower-order calculations in QCD. For this purpose we use the following expression of the solution of the renormalization-group equations for ~ln and C3n : g ff~ln(Q2//22, g) = Eln ( 1 ,~) exp f dX 3'ln(X)//3(X), (16) g C3n(Q2//22, g) = C3n (1,~) exp f

dX 73n0Q//3(X),

(17)

g where we explicitly wrote the functional dependence offfln and C3n on/22 and g, t3 is the Gell-Mann-Low function, and 3'1n and 3'3n are the anomalous dimensions of the operators R 1 and O. In the leading order of the QCD calculation one readily finds Eln(1 ,~) = C3n(1 ,~) = 1 and 3'In = 3'3n [3,4,7,9]. Hence eq. (15) is trivially satisfied with b n = 1. Relation (15) turns out to be nontrivial if one includes the next-to-leading order. For example, if we apply different normalization conditions to the renormalized operators R1 and O, we find in general b n 4 : 1 and thus in a perturbative calculation b n = 1 + Cng 2 + ... with c n a calculable number. Hence the relation between Eln(1,~) and C3n(1 ,~) at order g 2 is distorted by the amount Cn, and, at first sight, we may be lost in proving eq. (15). In the minimal subtraction scheme of 't Hooft, the operators are not normalized and the renormalization constants do not include a finite part. Hence b n = 1. We may argue that 3'ln = 3'3n at the two-loop level in the minimal subtraction scheme. In fact, 3'5 appearing at the vertex in the calculation of 3'ln in two loops can always be shifted to the external quark line by anticommutation and so the net result for ")'In is the same as that for 3'3n [5]. The next-to-leading-order result offfTln(1 ,~) and "C3n(1 ,~) in the minimal subtraction scheme is [6, 11,12] ~2 4 ( l 1 , ~ ) = 1 + - -3 1 627 r - 9 + - +n n

S

+ 4 ~s=l 1 ~ 1S j+=[l 1J

n

2 2 1 + 3 ~/'=1 I n + 1+

~

1 4~[~ 1'=1 j2

2

n

~-'~l n ( n + 1) I

t/

2 1) + 4 /~.21 n(n+ "= t (ln47r--3'E) } "

(18)

Thus we confirm eq. (15) with b n = 1 in the minimal subtraction scheme up to the next-to-leading order of QCD calculation. 416

Volume 89B, number 3,4

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28 January 1980

Finally we would like to test our relation (1) by existing data. Statistically enough data on M~(Q 2) have been provided by BEBC [13] and CDHS [14]. Unfortunately, however, only crude data on Mne(Q2) exist at one value of Q2 ((Q2) = 2.2 GeV 2) [15]. Hence a direct test o f e q . (1) cannot be made at present. Instead we use the data on Me(Q 2) in ref. [15] to normalize the QCD prediction, i.e. with 0 2 = 2.2 GeV2: e 2 )--- 0. 00 26, M~(Q2) = 0.0076, M~(Qo

(19)

where we used ordinary moments to deduce these numbers ,2 since the data are too crude to take into account the target-mass effect. The QCD prediction normalized by eq. (19) including the next-to-leading-order correction is plotted in fig. 2. Here we used the scale parameter A defined in the MS scheme [6] and took A~--S = 0.4 GeV. We then took the CDHS data [14] on M~(Q 2) (ordinary moments) and rescaled them by 0.050 for M~(Q 2) and 0.070 for M~(Q 2) to coincide with the curves in fig. 2. The data plotted in fig. 2 are 0.0503~3(Q2 ) and 0.070 M~(Q 2) for Q2 = 6.5, 10, 2 1 , 4 5 , 75 GeV 2, which fit the QCD prediction very well. New precise measurements ofglp-n in polarized electroproduction (muoproduction) at higher Q2 are definitely needed to really test relation (1). I would like to thank Bill Bardeen for enlightening discussions and J. Kodaira, S. Matsuda, K. Sasaki and T. Uematsu for useful discussions in the early stage of the present work. Thanks are also due to K. Kondo and N. Sasao for information on the experimental data. ,2 In order to get the above numbers for Me(Q~)we modified the original parametrization of the asymmetry parameter A 1 [15].. . . m. the. original form the Regge behavior - . - ofg p-n . . ofg p-n since 1 for small x was incorrect. The correct Regge behavxor I is x -~A1to) with aAl(0) -~ 0. Taking into account this fact we found the best fit to the data to be given byA 1 = 1.55x, which differs from the original one A1 = 0.78,¢rx, and which fits the data much better.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

C.H. Llewellyn Smith, Phys. Rep. 3C (1972) 261. A.J.G. Hey and J.E. Mandula, Phys. Rev. D5 (1972) 2610. K. Sasaki, Prog. Theor. Phys. 54 (1975) 1816. M.A. Ahmed and G.G. Ross, Nucl. Phys. B i l l (1976) 441. E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B129 (1977) 66. W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Phys. Rev. D18 (1978) 3998; E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B152 (1979) 493. H. lto, Prog. Theor. Phys. 54 (1975) 555. D. Bailin, A. Love and D.V. Nanopoulos, Nuovo Cimento Lett. 9 (1974) 501. D.J. Gross and F. Wilczek, Phys. Rev. D9 (1974) 980; H.D. Politzer, Phys. Rep. 14C (1974) 129. M. Chanowitz, M. Furman and I. Hinchliffe, preprint LBL-8855 (1979). J. Kodaira, S. Matsuda, T. Muta, K. Sasaki and T. Uematsu, Phys. Rev. D20 (1979) 627. V. Gupta, S.M. Paranjape and H.S. Mani, preprint TIFR/TH/79-37(1979). P.C. Bosetti et al., Nucl. Phys. B142 (1978) 1. J.G.H. de Groot et al., Phys. Lett. 82B (1979) 292. M.J. Alguard et al., Phys. Rev. Lett. 41 (1978) 70.

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