Mass corrections to scaling in deep inelastic processes

Nuclear Physics Bl17 (1976) 50-76 © North-Holland Publishing Company

MASS C O R R E C T I O N S TO S C A L I N G IN D E E P I N E L A S T I C P R O C E S S E S R. B A R B I E R I , J. ELLIS, M.K. G A I L L A R D * and G.G. ROSS

CERN, Geneva Received 30 September 1976

We study subasymptotic hadron target and quark-parton mass corrections to scaling in deep inelastic scattering, ignoring interactions. The results can be summarized using a modified scaling variable common to parton, light-cone and short-distance operator product expansion formalisms, but with model-dependent spectral conditions. The analysis is expected to break down near the kinematic boundaries because of the bound state nature of hadrons. Related effects probably also dominate mass corrections due to very light quarks, but the analysis should be applicable to the production of new heavy quarks in neutrino production. Experimental deviations from scaling in deep inelastic electroproduction do not seem to be describable in terms of mass corrections alone, suggesting that interaction effects may be important at large momentum transfers as suggested by the renormalization group.

1. Introduction Scaling in deep inelastic l e p t o p r o d u c t i o n [1] or in e÷e - annihilation is equivalent to the absence o f scale parameters with the dimensions o f masses, as in the quark-parton m o d e l [2] for example. Hadronic m a t t e r abounds in mass scale, and these have i m p o r t a n t threshold effects at low energies. In renormalizable field theories there are also deviations from scaling at high energies due to the persistent effects o f scales associated w i t h the interactions and expressible in terms o f the renormalization group [3]. Experimentally, scaling is not exact, and we want to k n o w what mass scales are doing the dirty work. A possibly i n c o m p l e t e list o f candidate masses includes. (i) The target hadron mass ** (ii) The masses o f hadronic constituents, assumed to be quark-partons. (iii) An interaction mass scale/a at which the strong interaction coupling constant has a fixed value, e.g., g2/4rr = 1 in coloured quark-vector gluon theory. (iv) Possible non-perturbative mass scales expressing the " s i z e " o f hadrons. * Laboratoire de Physique Th6orique et Particules El6mentaires (Laboratoire associ~ au CNRS) Orsay. ** Or produced particle mass in e+e - annihilation. 50

R. Barbieri et al. / Mass corrections to scaling

51

Examples include the pion decay constant f~, the Regge trajectory slope c( or the average transverse momentum (PT) typical of hadronic processes. In disentangling these effects we first note that the non-perturbative scales seem to be typically O(rn,r to 300 MeV), reflecting a hadron "size" of ~1 fro. Thus they are probably less important at moderate energies than the scale-breaking effects of target nucleon and heavy (e.g., charm) quark masses. On the other hand, they may not be negligible compared with light (up, down or strange) quark masses. In order to see the possible effects of an interaction mass scale/l, it is therefore desirable [4] to compute as completely as possible the effects of target and quarkparton masses, completely neglecting parton interactions. For consistency with approximate scaling it seems necessary that strong interactions be weak at short distances. One motivation for this is provided by asymptotically free gauge theories [3], which suggest that the neglect of quark-patton interactions may be a fair approximation over a range in Q2 (~ __q2) from 1 to 10 GeV 2, say. This would correspond to/~ < < 1, so that interaction effects would be small and only become clear over a larger range of Q2. The above strategy for distinguishing mass scales by computing exactly the effects of hadron and quark masses was initiated by Georgi and Politzer [4,5]. They found subasymptotic mass corrections could be summarized in a generalized scaling variable ~ which approaches x - Q2/2u in the scaling limit. Their analysis was made using short distance operator product expansions in the deep Euclidean region. Their original ~ variable differed from that derivable from the patton model [6] or light-cone expansions under equivalent assumptions on the neglect of interactions and has now been corrected. We show that the variable Q2 + mb2 _ m 2a + V ~ 2 + rn~ - rna2)2 + 4ma2Q2 , ~=

(1.1)

2(, + ~ / ~ +rn~tQ z )

where m i, mf and m H are the initial quark, final quark, and hadron target masses, respectively, can be derived from any of the parton, lightcone, or short-distance expansion techniques *. The only difference between the results is that the parton and light-cone techniques impose a spectral condition ~ ~> (mjmi4)2 which is absent in the operator product expansion approach. This reflects the fact that the parton and light-cone models work with the absorptive part of the current correlation function, which necessarily contains physical quarks if interactions are neglected, whereas the operator product expansion works with the real part of the amplitude in the deep Euclidean region. Another peculiarity of the parton and light-cone models is that they require the structure functions to have the symmetry F(~) = F ( m ~ / m ~ ) . We also show in appendix B how similar ideas can be applied to the production of hadrons through heavy quarks in e+e - annihilation.

;~ As far as we know, this variable was first written down by Frampton [6], see also ref. [7]. Some of our results were already presented in ref. [8]. For related analyses see ref. [9].

52

R. Barbieri et al. /Mass corrections to scaling

Unfortunately, the variable (1.1) is of limited application. One would expect, on general grounds, that the bound state nature of hadrons would be important near the kinematic boundaries ~ ~ 0, ~ ~ 1. In the low ~ region, Regge trajectories which are analytic continuations of crossed-channel bound states are expected to dominate, and it has been shown [9] that paradoxes arise in electroproduction from free field assumptions if the Regge intercept ~(0) > - 1. On the other hand, a threshold fixed in ~ near 1 is, in general, unphysical in the missing (mass) 2 W2 , usually being Q2 dependent, and we show that in a coloured quark-gluon theory the neglect of interactions is a non-uniform approximation which breaks down as ~ -+ 1. Also, as remarked above, we would expect other mass parameters to be as important as the masses of light quarks ~< 0(300 MeV). More problems arise if one applies renormalization group arguments: renormalizing on mass shell introduces effects due to the large infrared coupling, and is therefore of limited use. If one renormalizes at a Euclidean mass scale At one generally finds 0(//2/0 2) and/or O(m2/Q 2) terms arising from unknown operator matrix elements, and which are in fact necessary for maintaining gauge invariance in the case of electroproduction. Hence other m2/Q 2 effects can only be believed if m 2 > > At2;g2(At2)/47r <(<1. For example, only if the strong coupling constant is small at some mass scale At< < mH will target hadron mass corrections in (1.1) be believable. However (t.1)should be useful for the neutrinoproduction of new heavy quarks, such as charm [10, 11]. In an intermediate range o f x (or ~) applying the variable (1.1) is not obviously unjustified, and we compare it with data on electroproduction - including target mass effects, but neglecting light quark masses - to see if it reproduces the observed scaling violations. We find that it fails to reproduce the trends of scaling violation at large Q2 in W P, vW a and the ratio WN1/W P, and predicts too small a value for the quantity R a. We conclude that interaction effects are not negligible for Q2 > 10 GeV 2, and believe [7, 12] scaling deviations associated with an interaction scale Atas expressed via the renormalization group [3] are appearing. However, it will be a long time before data confirm asymptotically free gauge theory or any other renormalizable field theory. Sect. 2 of this paper discusses mass corrections in the free parton and light-cone approaches; sect. 3 uses free field theory short-distance operator product expansions working in momentum space. Sect. 4 discusses what free field theory results can be motivated from asymptotic freedom, while sect. 5 discusses generally limitations on the free field analyses. Sect. 6 contains a phenomenological analysis of deep inelastic electroproduction. There are appendices discussing the infinite momentum frame interpretation of the scaling variable and e+e - annihilation. 2. Deep inelastic scattering from the light-cone and the parton model The usual frameworks for discussing scaling in deep inelastic lepton-hadron scattering include:

R. Barbieri et al. /Mass corrections to scaling

53

(i) The light-cone (LC) analysis of the singularities of current commutators [13]. (ii) The quark parton model (PM) [2, 14, 15]. (iii) Short-distance operator product expansions (OPE) [16]. Given the approximate validity of Bjorken scaling [1 ], it is worth-while to consider within the above frameworks possible corrections to the naive picture, in particular at moderate Q2, where mass effects are still expected to be important. In this section we analyze these corrections to scaling in the approximation of negligible interactions. This is motivated naively by the feeling that the observed precocious scaling requires interaction effects to be already small for Q2 ~ 1. A more sophisticated discussion is referred to the renormalization group analysis of sect. 4. In the light-cone analysis the starting point is the connection between the standard structure functions and the matrix element of the current commutator

Wuv(q,p) - ~--~f

d4y


guvWx(V, q2) +PuPv W20) ' q2) + ½ieuuox ~ m2

W3(v' q2) + ....

(2.1)

mH

where

S.0;) = "~f(F) (a')'# + b')'#'),5) @i(F)" -- :~f(F) P#~i(.v);

(2.2)

is a general vector current composed of two fermion fields, p is the momentum of the hadron target of mass mH, q is the spacelike momentum transfer to the hadrons (_q2 - Q2), v = p" q, spin averages have been taken, and we have suppressed in the right-hand side terms proportional to qu and qv which give terms proportional to the lepton mass when contracted with the leptonic currents. In the standard analysis one assumes canonical commutation relations of the fermion fields on the light cone. To incorporate the effect of masses we are going to assume here that even for subasymptotic distances the interactions between the quarks are negligible, so that the fields ~i, ~f can be regarded as free fields. Free field commutation relations give the following expression for the current commutator

IS.0,), : - 8 ~ A ( v , m 2) :

+ 8~A(v,m~): ~i(0)

+ imiA(y,m~): f f ( v ) F u I ' ~ f ( 0 ) : + imfA(y, m2): f i ( 0 ) pvPu~i(.v): , (2.3) where A(y, m2) = - l fd4k e_ikYe(ko ) 6(k2 _ m 2 ) . (2~) a~'

(2.4)

The right-hand side of eq. (2.3) corresponds to the handbag diagram of fig. lb. The

R. Barbieriet al. / Masscorrectionsto scaling

54

(a)

(b)

(c)

Fig. 1. The (a) disconnected (b) handbag, and (c) cat's ears terms in the operator product expansion of two currents. purely disconnected piece of fig. la does not contribute to deep inelastic scattering, while the cat's ears diagram of fig. 1 c does not contribute to the current commutator

if quark-parton interactions are neglected. By inserting eq. (2.3) in the left-hand side of eq. (2.1), after a straightforward reduction of the Dirac matrices, one easily expresses the matrix element of the current commutator, and therefore the structure functions themselves, in terms of the following matrix elements of bilocal operators

A ~ (p,y) = (p[ (;fi, f Qv) ').'u'),v~ i,f(~): + "~i,f (0) "),V'yU~./i,f(F)" }[p), Z~f(p,y) = (/9t (:fi, f(.v)')'~i,f(0): - :fi, f(0)')'#t~i,f(Y) : ) IP) , Ai, f(P,Y) = (Pl (:~i,f(.F) ~//i,f(0): + :~i,f(0)~i,f(Y): } IP) •

(2.5)

A full exploitation of the relationships implied for these matrix elements by the free field equation obeyed by the quark fields allows us to represent them in terms of only two scalar functions of one variable fi, f(x) - one for each fermion field as follows Ai, f(P,Y) = - m i f, ~ d4 k e_ikYfi,f ( mk"2 )p

6 ( k 2 - m2f) ,

A~f(p,y) = f d4k e_iKv kffif ( k. P t 6 ( k 2 - mi2'f) ' ~ k m2H /

A~(p,y) = _mi, f j v f d4ke-iky fi, e(k" P )8(k2 --mi,f) 2 . \m~-H

(2.6)

Notice the 6 functions in these representations which keep on their mass shells the momenta carried by the quark field, and which are a direct consequence of the free field equations for ~f and ~i- Combining eqs. (2.1), (2.3), (2.4), (2.5) and (2.6), we find

Wuv(P'q) c c f d 4 k f i ( k" m~P ) 5 ( k 2 - m~)6 [(k + q ) 2

m~] e(ko+qo)

X ((a 2 + b 2) [qukv * qvk u - 2kukv + ½ff,uv(Q~ + m~ + m~)]

R. Barbieri et al. / Mass corrections to scaling

55

+ 2iab euvxoqhk p - (a 2 - b2)g~vmimf} + (i ~ 0 -

(2.7)

This equation is just what we would have written down in a covariant formulation [15] of the naive patton model [2, 14], with both initial and final quark partons essentially free (and therefore on mass shell) - as in a very simple-minded impulse approximation. Thus tile light cone, under the additional assumptions stated above, and a naive parton formulation lead to the same effective result and are therefore equivalent. Notice the explicit gauge invariance of eq. (2.7) for b = 0 and m i mf. In this free quark model, gauge invariance is ensured by treating on the same footing the initial and the struck quark *. Out of eq. (2.7) the calculation of the various structure functions, in terms of the function fi, f(x), is straightforward. Disregarding for a moment the Lorentz structure, the typical expression for a corresponding scalar structure function is =

W(v, Q2)

'

-

x Iv2 + m~Q2

[Fii\mH! +i°+ Fi

-

~

+ (i ~

(2.8)

'

where Fi(z ) -= mHTrfmHz dk°fi(k°/mH), with k ° the parton energy in the target rest frame. The limits E+ are specified by the on-mass-shell conditions in eq. (2.7); they are

E+ -

1 [_u(Q2 + m~ - m ~ + Nip2 + QZm~ x/(Q 2 + m~ - m~) ~ + 4m~Q 2 ] 2Q2mH (2.9)

Fig. 2 illustrates the ranges of -+E+ for different values of v between ½Q2 and ~o. One sees that - E _ varies strictly between the kinematic limits indicated, whereas E+ can become as small as m i when u = mH(Q 2 + m ~ - m~)/2m i. The physical spectral condition u ~> ½Q2 requires Fi(E/mH) = 0 for E >~(m 2 + m~)/2mH;this boundary is also indicated in the figure. This condition corresponds to the recoil momentum being timelike: (p - k) 2 >~ O. We see that for Q2 greater than some limit value Q2o (= (m 2 - m 2)2/2(m~ + m 2) for m i = mf = m) the F i ( - E _ / m H ) term in eq. (2.8) is zero. This equation can then be written as

W(u, Q.2)

_

1

l(~+..q

x/v: + Q2m~ Fi ~ \

+ (i~ f),

* See sect. 4 for a discussion on this point.

m .2

))+(i~0

m~ H

~

1

x/u2 + Q2m~ ~ri(~)

(2.1o)

R. Barbieri et al. /Mass corrections to scaling

56 where

Q2 + m ~ - mi2 + ~/(Q2 + m~ - m~) 2 + 4m~Q 2 ~-

(2.11) 2(v + ~

+ Q2m~)

We observe that ~ -+x = Q2/2p as Q2 ~ oo *. The physical spectral condition E+ <~(m~ + m~)/2mr-i requires (m-]mH) 2 ~< ~ ~< 1, whereas the condition E + / > mi does not impose any constraint on ~ ** A point which is apparent from eq. (2.10) is the invariance o f the function ~ri(~) under the transformation ~ ~ m~/m~a~, which reflects the fact that the range of E+ turns around at m i when v = mH(Q 2 + m~ - m~)/2mi. This line in the (Q2, v) plane corresponds to the invariance point ~ = m j m H of the above transformation. The structure function has then a turning point at this value at ~. If you believed this analysis, which is very.questionable as we will see in sect. 5, this would be an easy way of finding in the data an indication on the value of the struck quark mass m i. The present data are consistent with rnjrnrl < < 1 in this framework. Coming back to eq. (2.7) with the proper Lorentz structure taken into account, one gets [8] for the structure functions defined in eq. (2.1)

W1 = (a 2 + b 2) 2X/v 2 + Q2m~i

-

-

(a 2 -- b 2)

14

Q2

cYi (~) + ~

mimf X/p2 + Q2m~ ~ri(~) ,

W 2 = ( a 2 + b 2)

Q4m~

i] (2.12a)

I(~)2cyi(~)+3~t

2(v 2 + Q2m~)3/2

(2.12b)

W3 = 2ab ( v2 + Q2rn~)

1 ( +X/v2+Q2m~t f

d~' 1

I m -~- '2] firi(~'

,

(2.12C)

* The variable ~ can in fact be interpreted as the fraction of longitudinal momentum carried by the parton in a target infinite momentum frame [7, 14], and 7i(~) is related to the probability of finding a quark with momentum fraction ~ as shown explicitly in appendix A. ** In a previous paper [8] we had the constraint ~ > mi/m H. This is because we were sticking to the definition ~ =E+ + ,JE2+- m 2, which is actually not necessary.

R. Barbieri et al. / Mass corrections to scaling

UJ

fi

~

57

Q2/2 boundary

Ill parton maximum

÷

E

I11

>~ =,o~2boundary for E+

E0

,,.

Q2

(m2_ma)2/2(m2 .m2 ) •

+

•

•

-

Fig. 2. The range of the variables _E+ m eq. (2.9) with the physical (u >~_ 1 ~ Q 2 ) and free parton [E <~(my + m~)/2mH] spectral conaition indicated. The curves corresponding to v = ~Q2 are appropriate to the case mf = m i.

with

emh ~i

t

( m~'21

=

1

~ 2 + Q2m~I /

d~'

(,

I

- - ~'21 /~, d,~" m2

R --- [(Q2 + m 2 _ m ] ) 2 + 4 m 2 Q 2 ] 1/2

(,

m? mH~

(2.12d)

(2.12e)

The single and double integrals o f the function 9"i which appear in eq. (2.12) correspond, respectively, to the terms linear and quadratic in k u in eq. (2.7); the factor

(1 - m ] / m ~ 2) is nothing but the Jacobian of the transformation E -+ ~(~ + m~/ m ~ ) *. A similar analysis can be carried out for e+e - annihilation, as shown in appendix B.

3. The operator product expansion In this section we discuss the q2 dependence o f m o m e n t s [7] o f the deep inelastic structure functions as deduced from free field o p e r a t o r product expansion * We do not need to assume the parton PT = 0, as suggested by Georgi and Politzer in sect. 8 of the revised version of ref. [ 5 ].

R. Barbieri et al. /Mass corrections to scaling

58

[16] (OPE) techniques. The essential difference from the LC and PM approaches is that the OPE is used for the time-ordered product of two currents rather than the comutator (see also ref. [17]). The connection with the deep inelastic structure functions is made via sum rules, and under suitable assumptions, results similar to (2.12) emerge. The structure of the OPE is abstracted from free (or asymptotically free) models, but since we do not deal directly with the absorptive part, we do not have to put the patton states on mass shell, and so avoid the patton spectral conditions. Deferring a justification in the context of asymptotically free gauge theories [3] to the next section, we now consider the OPE for a free field theory. For simplicity of illustration, we consider a model with scalar currents

J(x)

=

:el(X) ei(x):

Then for small relative distances the time-ordered product may be expanded [16] in a series of local operators

T(S(x) J(0)) =

~ J4n~)( - x 2 + ie) x ul"''un O la(c0 ~'05 1...lant J ,

X "~ 0

(3.1)

n,o~

where c~ labels the differe.t operators of spin n The O,(f) - are symmetric traceless tensor operators of dimension d(nc~) and twist t (c0 ~5~ ~'n- n. The Fourier transform of the matrix element of (3.1)between nucleon states gives the forward Compton amplitude

T(u, Q2) _ - i f d 4 x eiqX (p[T(J(x)J(0))lp) •

(3.2)

Inserting (3.1) into (3.2) we find matrix elements of the traceless operators which may be written as

(nlO(e0t~' #l"'Un IP) =- (p[lO~llp)

{Pul ""Pun

-

-

trace terms } .

(3.3)

The evaluation of (3.2), without neglecting any of the trace terms in (3.3), is complicated, but has been done by Nachtmann [7]. He finds

T(u, Q2) = ~

C1

sin ~ n(n + 1)

~tn(Q2),

(3.4)

n even

where

iQ 2n /'/n(Q2)

=

sin g,-r(n 1 + 1)"

(~)n fd4x eiqx ~f~)~

( - x 2 + ie)(pllO(nc0 liP),

(3.5)

and the Cn1(r/) are Gegenbauer polynomials. The deep inelastic structure funclion W(u, Q2) is given by

w(u, Q2)=2~lm T(~', Q2),

(3.6)

R. Barbieri et al. /Mass corrections to scaling

59

and the moments pn(Q 2) are related to the structure functions by the formulae

un(O2)=8£ doW(u,Q2)iQ212

02

Q2

1,,+,

(3.7)

r, + X/u2 + rn2 Q ~

In the limit m H ---*0, eq. (3.7) reduces to the more familiar sum rules

pn(Q 2) = 8 f dv Q2/:

W(u,

Qz) x n+l ,

(3.8)

Q2

where x --~Q2/2u. Eqs. (3.4) and (3.7) give the target mass corrections necessary to project on to the moments corresponding to traceless operators when m 2 / Q 2 is not negligible. In our free field scalar boson example the traceless operators Oua...Un will be constructed from the operators

Oul ...tan = q~i(0) ~ul "'" 0~un¢i(0) '

(3.9)

and their traces. The leading contributions to the pn(Q 2) will come from the 0ul ...Un themselves, and contributions O(m~/Q 2) will arise from the trace terms. They may then be calculable in free field theory using (plTr On[p) = --m 2 (plO n-2 tP),

(3.10)

in an obvious shorthand notation. However, the reader (and an author) may believe applying (3.10) to hadron targets involves making assumptions about relations between operator matrix elements which should not be abstracted from free field theory (see also sect. 4). An exhausting discussion of the decomposition into traceless operators and the resulting inclusion of initial quark-mass effects for the spin -1 case has been given by Georgi and Politzer [4,5] *. The efl~cts of the final quark mass arise from a factor of the scalar boson propagator AF(X , m] ) in the c number singular functions ~ ) ( - x 2 + ie) in eqs. (3.1) and (3.5). Fourier transforming the x space expressions via (3.5) yields the dependences Ofpn(Q 2) on m i and mf. An equivalent, but more direct, method of extracting the pn(Q 2) comes from applying the Nachtmann [7] projection (3.7) to the free field theory structure function

W(u, Q2) = 6(2u

Q2 _ rnf2 + m ~ ) ,

* See in ref. [5] eqs. (4.7), (4.8) and (6.5) to (6.10).

(3.11)

60

R. Barbieri et al. / Mass corrections to scaling

which yields

Q2

7 n+l + x/Q '2 + 4mi2Q2J

(3.12)

where Q,2 =Q2 + m ] - m i z .

(3.13)

Eq. (3.12) is the scalar analogue of eq. (6.9) in the analysis of Georgi and Politzer [4,5]. Note that the absorptive part (3.11) of the quark amplitude is only used to abstract the Q2 dependence of the moments in the deep Euclidean region. To get to the moments/an(Q 2) for a hadron target neglecting interactions, we need only multiply (3.12) by the reduced matrix element of the traceless operator On between hadron states: /an (Q2) =/aFF (Q2) -

(3.14)

Eqs. (3.12), (3.14) ~md (3.4) can then be used to recover the full.amplitude

T(v, Q2) and the structure function W(u, Q2). We represent the Gegenbauer polynomials in the form Z-n-1

C~(r/) = - ~ / f ( l -

(3.15)

dz

2rv +z2~ '

C

where C is a contour enclosing the origin. Then from eqs. (3.4) and (3.15) we have

1

T(v, Q 2 ) = - ~ 7

fdz c

. . (o

(-iozi

°

E ( 2{ iv ]z 2) nevenz 1 - I,Qml4] +z

(3.16)

By deforming the contour to enclose the poles arising from the factor [1 - 2(iv/

Qmu)z +z 2] we find T(v, Q2) =

Q2

. {~/an(Q2)[(_Zo)-n-1 + (-z,) -n-' ]} ,

x/p2 + mZ Q2

n

(3.17)

where Z0 = - -1[ - - l ) + ~ Q 2 1

Z 1 = _ _1

m~t

[P + NIp 2 +

m~tQ2].

(3.18)

R. Barbieriet al. / Masscorrections to scaling

61

Using the form for/ln(Q 2) given by (3.12) and (3.14) we see that

W(v, Q2)_

1

VlV2 + m2 Q2

F(~),

(3.19)

where ~ is given by eq. (1.1). The second term in (3.17) vanishes because of the physical spectral condition, as we discussed in sect• 2. • 1 The analysis of spm-~ partons proceeds as for the spin-0 case. But there are two series of operators which must now be considered 0 7 - ~ a ~ l ~ U 2 ... ~ n ~ , 0~ ~ 11/~1

(3.20)

... ~p, nt]./ .

Each moment ktn(Q2) will get contributions from both series of operators. In general their matrix elements are unrelated, each ~tn(Q2) having an unknown Q2 dependence, and no simple scaling variable will exist• But in free field theory

(plO~lp) = m i (plOTIp) ,

(3.21)

and the operator matrix elements are related *. The reader (and an author) may not believe this relation should be abstracted from free field theory. If we use (3.21), • 1 we then find for electroproduction with spln-~ partons that eq. (3.7) is replaced by the two moment equations [7] 2 (3WI(v ' Q2)

Ua (a2): f

m~tQ 2

Q2

×f

(v2 + m~lQ2) W2(v' Q2)]

I

21n+l

(3.22)

P + N/P2 + m~ta

m,n(O2) = fdv

~

{

v2 + m~Q 2 - wl(v, a2)4

m~a 2

W2(v, 0 2)

I

2(n+ l)v(~/v2 + r n ~ Q 2 _ v ) _ ( n + 2)Q2m~_l)[ Q2 21n+ × 1+ -(n+~(n+3)-(~+m~Q 2) V+~/v 2 +m~Q (3.23) To extract the blight:(Q 2) corresponding to (3.12), we insert the free field structure

* Recall that the free field equations of motion applied to the bilocal operators yielded a similar uniquenessin the LC approach•

62

R. Barbieri et aL/ Mass corrections to scaling

functions W1 = vS(2u-- Q2), W2 = 2m~ 6 ( 2 u -

Q2),

(3.24)

into eqs. (3.22) and (3.23). As in the spin zero case, the l~n(Q 2) are, neglecting interactions,

lai,n(Q2)__ la~n vv (Q2) (pltOntlp).

(3.25)

The resummation of the series (3.22) is identical to that of the scalar series (3.16), except that the left-hand side yields

U2 + m 2 Q 2 3WI(U, 02)

rn2 a 2

W2(v, O2),

instead of W(v, Qz) in the analogue ofeq. (3.19). The resummation of the series (3.23) is more complicated because of the n dependent terms. We do not give the details here as they are equivalent to the analysis of Georgi and Politzer [4,5]. The result of the resummation is the same as obtained in the LC and PM approaches eqs. (2.12). But, as emphasized above, we do not get the patton model spectral condition ( m i / m , ) 2 ~< ~ ~< 1. This is because the moments lai,n(Q 2) have no memory of the parton model spectral condition, which is a restriction on the operator matrix elements (PllOn liP) which need not be respected in the general OPE framework. The use of the partonian absorptive parts (3.24) was only a matter of calculational convenience for extracting the Q2 dependence of the moments in the deep Euclidean region. Unlike the LC and PM analysis of sect. 2, the OPE does not have unphysical features of the quark-parton model's absorptive part - i.e., physical quark partons, the relations (2.10) they imply between values of the structure functions at different values of ~, and the spectral condition ~ ~>(mi/mH) 2 .

4. The renormalization group and asymptotic freedom In the previous sections we used free field theory models for the structure of the operator product expansion of two currents, and the resulting form of the deep inelastic structure functions. The basic idea was that there may be a regime of Q2 where the effective strong interaction coupling constant ~(Q2) is small, while O(rn2/Q 2) effects are still substantial. We now discuss the justification for these analyses in the framework of an asymptotically free colour gauge theory [3]. We must first specify a renormalization prescription. One possibility is to renorrealize Green's functions and operators on mass shell. This will yield Callan-Symanzik equations [181, where mass terms enter via inhomogeneous terms which should vanish asymptotically in the deep Euclidean region by Weinberg's theorem [19]. Un-

R. Barbieriet al. / Masscorrectionsto scaling

63

fortunately the operator product expansion will then involve the strong coupling constant evaluated on mass shell where infrared slavery presumably makes it nonnegligible. We therefore renormalize at spacelike momentum: p2 = _/x2. To define quark masses we follow Georgi and Politzer [4,5] and specify mq(/a) by the identification S-I(P)[

2 2 p =-g

=~

mq(U)

(4.1)

of the inverse propagator at the renormalization momentum. The resulting renormalization group equations for Green's functions in the Landau gauge are m

(4.2)

The corresponding equations for the moments ~(~) (Q2) corresponding to operators O (c~) of definite spin n are

tx-~+3~+%nm

~' + 3'~

=0,

(4.3)

where -')'c~,~' n is given by the transpose of the matrix of anomalous dimensions of the set of operators of spin n. These equations have the familiar solution /j(~)(Q2, g0-0, rn(u)) ~"(nC~')(/~2'gc~' ( Q ) ' m ( Q ) ) e x p { T ; y n @

-(~')'~')ldlft/.t 7~7J '

(4.4)

where T denotes that the matrices ~,n are ordered in/1' in the integral. In AFGT g ( Q ) -+ 0 as Q -+ ~ and we believe that for large enough Q2/l(n~) i/j2, g (Q), re(Q)] may be reliably estimated by perturbation series expansion in ~(Q). Of course, as Q -+ ,,% N(Q) contributes to/J(n~) in O [N=(Q)/Q=] which also -+ 0. The analyses of the previous sections assumed there was some range of Q= where m2(Q)/Q 2 was O(1) while g(Q) was small, so that /j(c0(/~2, ~ (Q), ~t (Q)) -~ bL(c0(/a2 , 0, ~ ( Q ) ) .

(4.5)

It was also assumed that the logarithmic scaling violations generated by the exponential in (4.4) are negligible over the range of Q2 where N2(Q)/Q2 effects are appreciable. There are other places where the interactions affect the discussion of the OPE as given in sect. 3. First, we observe that the operators On1,2 must be made gauge invariant by replacing 4->

4+

Ou ~ D " - o u-igTA",a a (4.6) where T a is a colour group representation matrix and A~ : a = 1 ..... 8 are the

64

R. Barbieri et al. / Mass corrections to scaling

gluons. Then since the operators are normalized at p2 = _//2, the mass terms on the right-hand side of the spin-½ analogues of the trace eqs. (3.10) get altered: m ] ~ _//2. Also the interactions generate new terms in the trace equations: for example (plTr O~lp) =/a2 (pl0~-2lp) + (n - 2) g(u) (pl4~n-2lP),

(4.7)

where

Ot~ - S~Du2 ... Dun q2 , and

42t~- S ~ T

DvDul . . . . ~n_l I ~n ~

(4.8)

is a non-leading twist operator, and S denotes symmetrization in the indices/a~ .../an and division by n!. Eq. (4.7) means that the O(m~/Q 2) effects in (2.12) generated by the construction of traceless operators described in sect. 3 are replaced by O ~ 2 / Q 2) effects. On the other hand, the mi effects from the free field eqs. (3.21), and the mr" effects from the propagator in fig. lb, remain because of the renormalization prescription (4.1). Explicit gauge invariance will then be violated to the extent that _112 @ mi2 (m 2 = m ] for electroproduction), but this is not unreasonable as the hadronic matrix elements are unknown to the same extent, even if the hadronic wave function is constructed from effectively free quarks. The analysis of the previous sections also requires the ~bn (and other operator towers) contributions, as seen in (4.7), to be small. We must therefore choose/a so that g(/a) is not too big and the matrix elements (p[~)n IP) are not badly normalized (do not blow up). Since the ~bn are normalized at p2 = _/a2, the (p[Onlp} will probably be well behaved if the average quark momenta in the proton are 0(/2). Thus/a = 0(typical hadronic size) = O((PT) ) = 0(300 MeV) might be preferred as a renormalization point. But it then obvious that g(/a) is small: probably it is not. Therefore, because of the terms on the right-hand side of the trace eq. (4.7), the results (2.12) will only be true to O ~ 2 / Q 2) and the m~/Q 2 pieces are generally unreliable. Also, we will want to choose/a as small as possible so that g(g) is still negligible. Even then we are reduced to piously hoping the matrix elements , etc., are not too large. Ifg(/a) is small, then N(Q) can be replaced by m(/a) in eq. (4.5) for/a(n~) (Q2). Only the effects of masses m > > / a : g(/a) is small can be believed in eqs. (2.12). Thus the m2/Q 2 effects may be believable for heavy quarks [11 ], as in charm production in neutrino interactions [10]. Whether the O(m2/Q 2) effects are believable, e.g., for a nucleon target, depends crucially on the value ofg(/a ~ 1 GeV). But independent of the approximation of neglecting interactitms or the formulae (2.12), the moments (3.22) and (3.23) are the theoretically correct way of extracting the Q2 dependence of specific spins, and would be ideal quantities to compute from the experimental data.

R. Barbieriet al. / Mass corrections to scaling

65

5. Limitations of the analysis In the previous sections, while deriving the wholly scaling variable formulae (2.12), we have emphasized some of the limitations on their applicability. In this section we will try to discuss them all systematically: for another discussion, see ref. [20].

5.1. ~-+1 This is the threshold region where you might naively expect direct channel bound state [21 ] effects to be important (see fig. 3a). They are indeed reflected in our analysis in several ways. First we noticed a negative patton energy contribution F i ( - E _ ) to the structure function in (2.8). If one seeks to apply (2.12) out 1 2 to ~ = 1, then v must be restricted to > 5mH. More importantly, the formulae (2.12) only admits a threshold fixed in ~, which means a threshold dependent on Q2 in general and usually unphysical. The physical condition Q2 ~< 2v corresponds to ~<

QZ+rn]_m

r + (x/~+m]-m~)

z+4m]Q 2

Q2 + x / ~ + 4m~O 2 and in the case m i = mf = 0 the physical asymptotic spectral condition ~ ~< 1 corresponds to W2 = ( p + q)2 = m ~ + 2 v -

Q2 ~>0

for all Q2 , . Furthermore, the Adler sum rule

A=

-yvw2

(5.1)

v0

should be a constant independent of Q2 since it is derivable from a current algebra equal time commutator. In quark-parton language it just counts the number of quarks in the target. We have verified that A is a constant only if the threshold v0 in (5.1) is taken to be fixed in ~ (and so varying with Q2), and that then it indeed counts the number of quarks given by the density fi(k" pimP)) in eq. (2.7) (see appendix A). We expect these threshold anomalies to be cured by interactions, and asymptotically free gauge theory [3] indicates at least two ways in which the neglect of interactions becomes a worse approximation as x(~) -~ 1. First it is known [3, 17, 22] that the structure function at a given x is dominated by moments lan(Q2) of order * These problems are removed by Derman's [10] choice of variable which has a less direct field theoretical significance than does ~.

66

R. Barbieri et al. / Mass corrections to scaling

(a)

(b)

Fig. 3. Possible interaction corrections to the free field approximation.

n ~ (4G In txs/ln x), where ixs = g2/4rc and G is determined by the colour group and quark representations. It can be shown that interaction corrections go like [as(ln n)3] i i = 1,2,3 ..., and so are important unless ~[ln~--~

! <<1,

(5.2)

which excludes the region x(~) ~ 1. There are also the g(p)¢n-2 interaction corrections in the trace relation (4.7) cc(n - 2), which indicate that the perturbation series in ~-(Q) is only reasonable for %(n - 2) 2 < < 1. We, in fact, find that for a structure function cc(l - ~)a the corrections from this source are (1 - ~)a X O [ax/oq/ (1 - ~)] as ~ -+ 1. These are obviously not negligible as ~ -+ l, and not obviously negligible anywhere. The problem is more acute for higher twist operators contributing in higher order in (1/Q2). 5.2. ~ - * 0 In this region, usually expected to be related to Regge behaviour, you might naively expect crossed channel bound state effects to be important (see fig. 3b). We saw this directly in the LC and PM approaches where the formulae (2.12) were restricted to ~/> (m.JmH) 2 because the quark-parton being struck was real. The OPE approach allowed ~ < (mffmH) 2, but in apicture based on physical partons this possibility requires interaction effects. In appendix A of his paper, Broadhurst [9] has used equivalent free field assumptions to ours to show that if WL(V' Q2)__-

1+

W2(v ' Q2)_

Q2m

WI(V ' Q2)

(5.3)

and if l i m v - ~ W2(v, Q2) ccv ~-2 in electroproduction, then

R(v, Q 2 ) =

WL(~Q2)

OL(~Q2)

WI(v, Q2)

OT(V,Q2)

has the limiting behaviour lim

lira

Q2_+ 0 v'-*

R(v, Q 2 ) = - ( a + 1),

(5.4)

R. Barbieriet al. /Mass corrections to scaling

67

which is absurd if ct > - 1 since R must be ~> 0. Thus using free field assumptions near v -+ oo (~ -+ 0) is inconsistent with the structure functions having Regge behaviour with any intercept c~ > - 1 * In A F G T it is well known [23] that the perturbation expansion of/~n(Q 2) in a s breaks down in the Regge limit. For SU(flavour) singlet structure functions the O(as) contributions to the anomalous dimensions of operators of spin n have a pole at n = - 1 and it is found that interaction terms of order a s and above also have singularities at n = - 1 proportional to

.....

For small x the effective value o f n is o f order n+l

Jinx

(5.5)

and so interaction corrections will be important unless [

[lnxl

'~3/2

aS~gG i n ( l / a s ) ]

< < 1.

(5.6)

The condition (5.6) excludes the Regge region x(~) ~ 0: similar considerations apply to tile non-singlet structure functions.

5. 3. m i effects We saw in sect. 2 that ~ ~> (mjmH) 2 in the PM and LC approaches, implying that these analyses were totally inapplicable when mi > m~t. Also we saw in sect. 4 that in an asymptotically free gauge theory O(m~/Q 2) pieces in (2.12) are suspect, many being replaced by 0(122/Q 2) effects where presumably/2 > mi.

5.4. Small mi, mf In all our analyses we have neglected other strong interaction mass scales, some probably non-perturbative like f~, a', and possibly (PT). Effects due to these scales and the mass scale/l probably dominate mass corrections due to very light quarks such as up, down, and probably also strange quarks. For this reason details of the formulae ( 2 . 1 2 ) s h o u l d not be taken very seriously in this case.

~' The paper of Dash [9] implicitly contains a variable similar to ~ in the Regge limit. He did not encounter the problems found by Broadhurst because he did not use free field theory for the spin kinematics.

68

R. Barbieri et al. /Mass corrections to scaling

6. Phenomenological applications Having calculated hadronic and quark-parton mass corrections to scaling as best we can, we will now make some comparisons with the observed [ 2 4 - 2 7 ] patterns of deviations from scaling in deep inelastic scattering. We will ignore other corrections to scaling, wanting to know whether the trivial mass factors of (2.12) can explain all or some of the scale breaking. We discuss mainly electroproduction and restrict ourselves to moderate values ofx(~) because of the arguments of Section 5. We will mainly be concerned with data taken at SLAC, hence only need the formulae (2.12) when m i ~ mf ~ 0. We then have

WI(v' Q2) =.

02 2~/v 2 + m~Q 2

[F(~) + G(~)] ,

Q4m~ W2(v' Q2) _

2(v 2 + m~a2) 3/2

(6.1)

[F(~) + 3G(~)],

where 1

1

-

1

f

G(~) x/v2 + m2 Q2 f d ~ ' F ( ~ ' ) + v 2 + m 2 Q 2 f~ d~' ~,

d'-"F ......

~

(~ ) ,

(6.2)

02

V + X/V2 + m2Q 2 in this case. In the range -13~
m2 02(1 - ~)n ['1 + .. l - G + 2x/v-~+m~%2[" x/v2 +m~qQ2n+ l

. . Q4m2(1 . . . . _ ~)n W2 2(v 2 + m 2 Q 2 ) 3 / 2

I1

3m~

m4

( 1 - - ~ ) 2_

_']

v2 +m~qQ2 ( n + l ) ( n + Z ) J '

1- ~+

~x/v 2 +m~Q2n+ l

3m~l

¥(1- 1-) ~~)2 - + 2)]]

v2 +m~Q2(n (6.3)

The most precise data in the suitable range o f x comes from MIT-SLAC [24] and SLAC [25] groups. They report that their data are best fitted by n ~- 4 in polynomial expansions with various different scaling variables. We have checked that n = 4 in eqs. (6.3) also fits best t h e x dependence of the data at Q2 ~ 6 GeV 2 for ½ ~
69

R. Barbieri et aL/ Mass corrections to scaling /

0.42 % ~-

~,

0.18

x = 0.50 -1

0.26

x=050 ee

0.34

e

~N0.~

E ~ 0.10

"

0.08

-~.~._..

> 006

''t!

0.05

0.08

2 I

I

I

I

4

6

8

[

I

]

I

10 12 l& 16

i

0

I 4

i 6

I I 1 8 10 12 14 16

QZ(GeV 2)

(a)

i

i 2

004

x=0.67

i

i

i

i

0.12 0.5 < x < 0.7 0.10 o_ ~-

008

+

E= ¢M

v

0.06 0.04 0

c~

I

I

4

8

I

I

L

12 16 20 (b) Q2 (GeV 2)

0/-,f

I

24

28

0.3Xl3<
O2 0 2

, , ,,,,1 10 (c)

? ,, IO0

Q2 (GeVZ)

Fig. 4. Comparisons of the formulae (6.3) for n = 4 with (a) MIT-SLACdata on 14/1P and uW2P [23]; (b) SLAC data on W1P [24]; (c) the data of Anderson et al. [26] (solid dots) on vWD. trend of scale breaking is not well reproduced by the mass corrections to scaling in (6.3), particularly in the case of uW2p. In fig. 4b we have compared SLAC [25] data on W~ for 0.5 < x < 0.7 for Q2 ~< 24 to the Q2 dependence contained in (6.3) with an arbitrary normalisation, and again we find that (6.3) does not fit the data well. In fig. 4c we have compared the FNAL data of Anderson et al. [26] with (6.3) in the kinematic range ~ < x < 1. On the scale the data is presented, the variation in the theoretical formula is negligible, whereas the experimental value of uW~ falls by a factor ~2. Another application of the formula (6.3) is to measurements of [4/2(' V2

+ 1~ -- W1

\Q2m~ R =

I411

/ (6.4)

In fig. 5 we have compared SLAC data on R P with the formulae (6.3). In the rele-

R. Barbieri et al. /Mass corrections to sealing

70

0.4

0.3

.

0.2

0, 0

+++t 02

0.~ 0.4 x = Q2/2v

0.8

10

Fig. 5. Comparison ofR P from eq. (6.4) with SLAC data [24]. vant range ~ ~ x ~< ~ the experimental [24, 25] average value o f R ~ is ~ 10 - 15%. The mass corrections to scaling just give R P ~ 1-3%. A final application of these results is to the ratio of neutron and proton structure functions. Since mp ~ m N, the pattern of scale breaking for neutron and proton would be identical if WN(x)/WP(x) were constant at any Q2. In fact, the N/P ratio varies between ½ and ~1 in . the x range measured, so that the neutron and proton structure functions have different functional forms, and a variation in the N/P ratio with Q2 is, in'principle, possible, The data [24] on WN are reasonably described by a combination of (1 - x ) 4 and (1 - x ) s. Even if the (1 - x ) s term were dominant, which it is not, the variation in the N/P ratio would be

WP (02

6)/

W1P(O2 = 2 0

-1"04"

(6.5)

The experimental data [24] show a variation by ~30% in this ratio between Q2 = 6 and 20 GeV 2. The conclusion o f these phenomenological comparisons is that simple target and patton mass effects do not adequately describe scaling breakdowns in the electroproduction data. Since (pT)2/Q 2 ~- ~o~ when Q2 _~ 10, we do not believe hadron "size" effects are likely to make up the discrepancies at larger Q2. We conclude that renormalization group effects are probably significant in the observed scaling deviations, and possibly dominant in the quantities R p and wN/wPl. These conclusions are not novel: Nachtmann [7] has been saying it for two years. More recently, Baluni and Eichten and Gliick and Reya [12] have made more sophisticated analyses, and found evidence for interaction effects redolent of an asymptotically free gauge theory. In the case of neutrinoproduction o f new heavy quarks from light initial quarks

R. Barbieri et al. / Mass corrections to scaling

71

in a target of negligible mass, our fornmlae reduce to those of Barnett [10]:

1411 = 2~ F(~), P~I]2 4 m ~ 2 F ( ~ ) '

~-

=

Q2 + m] 2v

(6.6)

vW 3 = +4m~F(~), As observed by Barnett, the Callan-Gross relation becomes vW2 = 2m~Wl subasymptotically. We have not attempted detailed comparisons with neutrinoproduction data because a large fraction of the new particle production comes from small values of x where the neglect of parton interactions is dubious, and renormalizati0n group (asymptotic freedom) effects are likely to be dominant.

We would like to thank C. Bouchiat, R. Crewther, R. Gatto, R. Jaffe, G. Parisi and L. Susskind for frank and comradely discussions. Appendix A In order to establish contact with the more familiar infinite momentum frame parton model [2,14], consider an invariant distribution of quark momenta k in a proton of momentum p,

d4kg

6(k 2

m2)=½dtpdcosOkdkog-~ H ,

(A.1)

\rn H where the right-hand side is evaluated in the target rest frame, p = (mH, 0). The longitudinal fraction of patton momentum in an infinite momentum frame P -+ (3'mH, ")'13mH,0, 0), /3 -+ 1, is given in terms of the rest frame variables by x = (k o + k cos

O)/mrt •

Then the x distribution corresponding to the invariant distribution (A. 1) is

F(x)=n ['jkdk o d c o s 0 8

Ymax =~m~f

x

(1 dy

( o+ cos0) 1 x

g

m2 ) g(-~(y + m2 }); m~--v2

.

1

m~y..

k o = ~m H

m2y.. (A.2)

Eq. (A.2) is the defining integral for the function appearing in eq. (2.10) (~) - F(~)

(A.3)

72

R. Barbieri et al. /Mass corrections to scaling

if the distribution g(p. q/m 2) in (A.1) is identified with that in (2.7). The normalization is determined by the definition (p] [J#(x), J~(0)] [p) = f d 4 k g ( k" p~ 6(k2 _ m2 ) (kl [Ju(x), JJ(0)] [k), tm~ ] and the covariant state normalization (Pip') = 2E 63(p - p') such that (for unit charges)

p] k" p

fd4k_ik,

J

-

m:) : l

(A 4)

\ m~l l m~t

Using the spectral condition

m2 < ~ < 1

o<

2mH

'

we may use the definition (A.2) to express (A.4) in the form

l=--~1 f 1

d~( ~2

(m imH) 2

mm~H42 )

F'(~)=fJ (m /mH)2

2m4~ d~( ~+m~3 ]

F(~). (A.5)

Therefore the function p ( ~ ) = ( ~ + 2m4 )c_'J(~) m4~ 3 may be interpreted as the probability of finding a parton with momentum fraction in the infinite momentum frame. We find no restriction on the PT of the parton necessary for the validity of this interpretation of ~. One may verify that in the limits m2/Q 2 , rn~i/Q 2, m/mH ~ O, the expressions (2.12) reduce to the usual ones, e.g., Q2

~x-

2u '

u14I2 ~ m~tx 2 9r(x) = m~xp(x)

and that the right-hand side of the Adler sum rule is proportional to (A.5) as required by current algebra. In the case of e+e - annihilation, the distribution (A.2) applies for the momentum fraction of the decay hadron of mass m from a quark of infinite momentum and mass mlt withy = (Po + p)/mH defined in terms of the decay hadron energy in the quark rest frame. If k o, k are the quark energy and momentum in the hadron rest frame, we have: ko Po

k P

mH m

R. Barbieri et al. / Mass corrections to scaling

73

Defining ko - k -7

--

mH --

m

1 --

Po +P

Y'

we may rewrite (B.2) as t.o

F(x) -~ G ( c o )

=

½ranI f CO m in

dz[ m~ km2z

- 1)g(l(z+

m~)) m2z// '

(A.6)

co= I/x . Again eq. (A.6) defines the function ~ (~) appearing in eq. (B.7) with the roles of m n and m exchanged.

Appendix B e+e - annihilation In the context of the free covariant quark-parton model, the discussion of oneparticle inclusive e+e - annihilation is very similar to that given in sect. 2 for deep inelastic scattering. The counterpart of eq. (2.7) is here

q2 }

--fd4kg ( k" p )

q2 }

q2 ]

6(k2 _ m2) 6((q _ k) 2 _ mZ) (qukv + qvku - 2kukv

- l qguv) "

(B.1)

The function g(k. p / m y ) in eq. (B.1) can be interpreted as the spin-averaged probability for the fragmentation of an on-shell quark of mass rn and momentum k into a badron of momentum p (again p2 = m y ) + anything. Clearly the picture of the process, as expressed in eq. (B. i), imposes the constraints q2 >/4m 2 ~> 4m~. We also have the general kinematic limits q2 1> 2v, v ~> m H x/q 2. There is also the condition (k - p)2 ~> 0 which ensures that the "anything" produced by the quark decay has timelike momentum. The k integral in eq. (B.1) is most conveniently performed in the rest frame of the timelike photon. Discarding again for a moment the Lorentz structure, one finds for the corresponding scalar structure function

W(t', q l ) = X/v2 _ qZm2

G E

\mH]]

(u.2)

R. Barbieri et al. / Mass corrections to sealing

74

where

G(~H)=m~vrf

(B.3)

dzg(z),

E/m H

E+_-

1 [vq2 +_~q2(q2 _ 4m 2-) x/~ - q2m~l. 2rnHq2

(B.4)

The spectral boundary condition v = V~5-mH corresponds to E_+ = ½V/@-, while v = ½q2 corresponds to

= m2/mH E+

when

q2 = 4m 2 (threshold),

[~q2/2mH m~

+ m 2

(B.5) when

q2

~

eo .

, 2m H For intermediate values of v, E+ varies strictly between these kinematic limits, while E_ can become as small as m when v = mH q2/2m. The timelike recoil condition (k - p)2 >~ 0 ensures G(E/mH) = 0 for E>/

m~ +

m2 (B.6)

2mH

In fig. 6 we havee plotted, the ranges of E+ corresponding to the physical spectral mHN/q2 K P K q2, and indicated the on-shell parton model spectral conditions m <. E <<.(rn~ + m2)/2mH , We see that for q2 ~ (m~ + rn2)/rn~ the G(E+/mH) term in (B.2) is zero. This reflects the physical fact that the decay products of a free massive particle can only populate the hemisphere into which the parent is moving, if it is travelling fast enough. The formula (B.2) can then be written as

conditions

W-(v'q2)-xf~-:m~q2G

~-+ m-Ztlm~/!

(B.7)

where ~- _q2 + v/~Z(q2 _ 4m 2) +

-

2)

is the e+e - analogue of the deep inelastic ~ variable (2.11). We observe that

(B.8)

R. Barbieri et al. /Mass corrections to scaling

75

.z_y =o2~2.ounda!y_ for E+

x%..~~~lllll~ n<,n,

'

m~.m212mH ~

~

m/

',or

._

parton maximum for E._

! minimuml

value of 014m2 1\(m2H+m2)21 _ _ _ m

~ p a r t o n minimum for E. Qz

E_

Fig. 6. The ranges of the variables E_+ in eq. (B.2), with the physical (B.5) and free patton (B.6) spectral conditions indicated.

-~--->~ =-qZ/2u as qe ~ ~, and that eqs. (B.7) and (B.8) are restricted to 1 ~<~- ~< (m/mH) z. As in deep inelastic scattering the structure function has the invariance property G(~) = G(rn2/m~). Just like the corresponding electroproduction prediction, this result should not be taken too seriously. The appropriate extension of (B.7) to the spin ½ case (B.1) is found to be q2

~)1 -

[q(~-) + ~ ] ,

2V/~

(B.9a)

m~q 2 mZq2) a/2

2( u2 _

~(~-) + 3 ~

,

(B.9b)

with mR ~=

_jdf'

- -

q2x/U2 - m 2 q 2 1

+

rn4 ---

d~' 1

R = x/~q2(q 2

4m2).

1

1

m2 m ~ g '2

m2

~(g') d~-"

1

~(f")

,

(B.9c)

mH~ H2

(B.9d)

76

R. Barbieri et al. / M a s s corrections to scaling

References [1] J.D. Bjorken, Phys. Rev. 179 (1969) 1547. [2] R.P. Feynman, Photon-hadron interactions (Benjamin, NY, 1972); J.D. Bjorken and E.A. Paschos, Phys. Rev. 158 (1969) 1975. [3] H.D. Politzer, Phys. Reports 14 (1974) 129. [4] H. Georgi and H.D. Politzer, Phys. Rev. Letters 36 (1976) 1281 and Erratum 37 (1976) 68. [5] tt. Georgi and H.D. Politzer, Harvard Preprint, Freedom at moderate energies: masses in colour dynamics (1976), revised version. [6] P.H. Frampton, UCLA Preprint UCLA/76/TEP/6 (1976). [7] O. Nachtmann, Nucl. Phys. B63 (1973) 237; B78 (1974) 455. [8] R. Barbieri, J. Ellis, M.K. Gaillard and G.G. Ross, Phys. Letters 64B (1976) 171. [9] D.J. Broadhurst, Open University, England, Preprint OUT-4102-1, Static properties and structure functions of a bag of finite-mass quarks (1975); H.J. Schnitzer, Phys. Rev. D4 (1971) 1429; J. Dash, Nucl. Phys. B47 (1972) 269; U.T. Cobley et al., Widdicomble Institute of Technology Preprint WIT-TY-1 (1976). [10] R.M. Barnett, Phys. Rev. Letters 36 (1976) 1163; E. Derman, Nucl. Phys. B110 (1976) 40. [11] S.L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285; M.K. Gaillard, B.W. Lee and J.L. Rosner, Rev. Mod. Phys. 47 (1975) 277; A. Benvenuti et al., Phys. Rev. Letters 36 (1976) 1478. [12] M. Glfick and E. Reya, Mainz Preprint MZ-TH 76/2 (1976); V. Baluni and E. Eichten, Inst. Adv. Study Princeton Preprint, COO2270 (1976). [13] Y. Frishman, Phys. Rev. Letters 25 (1970) 966; R.A. Brandt and G. Preparata, Nucl. Phys. B27 (1971) 541. [14] S.D. Drell, D.J. Levy and T.-M. Yah, Phys. Rev. D1 (1970) 1035. [15] P.V. Landshoff, J.C. Polkinghorne and R.D. Short, Nucl. Phys. B20 (1971) 225. [16] K.G. Wilson, Phys. Rev. 179 (1969) 1499. [17] N. Christ, B. Hasslacher and A.H. Mueller, Phys. Rev. D6 (1972) 3543. [18] C.G. Callan, Phys. Rev. D2 (1970) 1541; K. Symanzik, Comm. Math. Phys. 18 (1970) 227. [19] S. Weinberg, Phys. Rev. 118 (1960) 838. [20] R.K. Ellis, R. Petronzio and G. Parisi, Rome Preprint, Mass dependent corrections to the Bjorken scaling law (1976). [21] E.D. Bloom and F. Gilman, Phys. Rev. D4 (1971) 2901. 122] G. Parisi, Phys. Letters 43B (1974) 207; 50B (1974) 367; D. Gross, Phys. Rev. Letters 32 (1974) 1071. [23] A. de Rfijula et al., Phys. Rev. D10 (1974) 1649. [24] R.E. Taylor, Proc. 1975 Int. Symp. on lepton and photon interactions at high energies, ed. W.T. Kirk, SLAC Stanford, 1975, p. 679. [25] W. Atwood et al., SLAC Preprint, SLAC-PUB-1758 (1976). [26] L.W. Mo, Proc. 1975 Int. Symp. on lepton and photon interactions at high energies, ed. W.T. Kirk, SLAC Stanford, 1975, p. 651. [27] Y. Walanabe ct al., Phys. Rev. Letters 35 (1975) 898; C. Chang et al., Phys. Rcv. kctters 35 (1975) 901; tI.L. Anderson ct al., Phys. Rev. Letters 37 (1976) 4.