frishman. light cone and short distancecs

We review briefly the kinematics and light cone dominance, and then discuss the structure of operator products at nearly light like distances.. Matrix elements of the bilocal operators i

LIGHT CONE AND SHORT DISTANCES

Yitzhak FRISHMAN Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305, U.S.A

and Weizmann Institute of Science, Rehovot, Israel

LIGHT CONE AND SHORT DISTANCES *)(#*)

Yitzhak FRISHMAN

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305, U.S.A

and Weizmann Institute of Science, Rehovot, Israel

Received October 1973

Abstract:

In this article short distance and almost light-like distance behaviour of operator products are discussed In particular, products

of electromagnetic and weak currents are treated and applications made to the region of deep inelastic lepton—hadron scattering

This is motivated by the observed scaling behaviour in the deep inelastic region

We review briefly the kinematics and light cone dominance, and then discuss the structure of operator products at nearly light like distances Light cone expansions are postulated and bilocal operators are introduced These are a generalization of Wilson’s

short distance expansion and were abstracted from studies in model field theories Applications include, among others, treatment

of Regge behaviour in relation to sum rules and fixed poles implied by scaling

Regarding models, we discuss the Thirring model and its generalization to include U(m) symmetry The latter shows scale in- variance only for one value of the coupling constant In both cases anomalous dimensions occur Regarding field theories in four

dimensions, gauge theories are “nearest” to canonical scaling, for which only logarithmic violations occur

We review the quark algebra on the light cone and discuss the applications to structure functions and sum rules In e*e” anni-

ion i e revie e i Ẹ ingle icle i ive annihilation the singularit ucture and

(*)Work supported by the U.S Atomic Energy Commission

(**)This paper is based on the Rapporteur talk of the author at the XVI International Conference on High Energy Physics

(Chicago-NAL, September 1972, Vol 4), in which many changes and additions have been made for more completeness and

in view of developments since that Conference

Contents:

2 Deep inelastic electron—nucleon scattering 6 5 Total annihilation ete” — all, and 1° > 2y 34 2.1 Scaling behaviour and light cone dominance 6 6 Single particle inclusive e*e” annihilation 37

3 Light cone expansions of operators products 12 7.1 The parton model 42 3.1 General structure and examples 12 7.2 One photon amplitudes 42

4 The Fritzsch-Gell-Mann algebra 24 7.8 Relation between scattering and annihilation

functions, relations and sum rules 25 7.10 Finite QED 47

1 Introduction

The subject of singularity structure of operator products at almost light-like distances has re- ceived much attention in the last several years It is a generalization of earlier studies of short distances structure of operator products There has been much activity recently in the latter sub- ject too The idea of “‘scale invariance” at short and almost light-like distances, or generalizations

of this idea, are central to these approaches

Short distance expansions of operator products were introduced by Wilson [1] as an abstrac-

where A, B and F!@! are local operators and C!“l(x — y) singular c-number functions The index

[a] characterizes Lorentz as well as internal quantum numbers To any degree of accuracy in

œ ^ QS Q ate “em ìe HO1 fa 2 Ane 4a no 7 9 O 4

ry a iong in-g ry _ e i a $ 4 r1 đit Œ at Œ a s

Lagrangian consideration, unless there are special reasons against that Thus local current algebra

Vy s Lj CŨ: 1O s 1C LJ LJ L C ° C CTICI#X Y= O U C OT d L

sion 4

tance approaches light-like separations It was suggested as a generalization of the short distance

expansion, to study high virtual mass limits in deep inelastic lepton hadron scattering It has the

form [3—5]

A(@œ)8() ~_ DCI - y)F#l(x, y) (2)

(x — y)2 >0 [ø|

where #ƑÍ“Ì(x, y) are bilocal operafors, depending on the two points x and y and regular at

(x — y)?=0 In fact, it turns out that they are analytic in (vy — x), as follows from the spectral conditions in deep inelastic lepton—hadron scattering The expansions of the form eq (2) and the existence of bilocal operators were postulated to hold in nature, namely for the fully interact-

ing theories [3—5] It was an abstraction from Wick’s expansion for three fields and from the

existence of such a light cone expansion in the Thirring model [6] In the latter case the light cone singularities are not canonical, but rather depend on the coupling constant Matrix elements

of the bilocal operators in the expansion of products of currents are directly measurable in deep

inelastic scattering experiments, which exhibit simple scaling phenomena and therefore imply the

appearance of canonical light cone singularities [7—9]

Expanding the bilocal operators in a Taylor series,

Flix, yy = 2) Oe — vy) Oe — ym FN (y) (3a)

n=0

lf a 3

M Lax%1 axon x=y

We get, for each light cone singularity, an infinite number of local terms in Wilson’s expansion

eq (1) [10] Inversely, if we have a Wilson expansion, with an infinite number of terms of local operators with Increasing spin a and the same singularity function C Ie, we e may s sum them up to

ical) dimensions)

Wilson’s expansions have been demonstrated to hold in renormalizable quantum field theories

[11] to any order in the coupling constant To any finite order, scaling is violated by logarithmic

stant [12] So far there is no nontrivial model of quantum field theory in which canonical light

cone singularities are exhibited (Note that scaling behaviour holds for ladder graphs in ¢° theory [13] However, this theory does not have a positive definite Hamiltonian )

for the moments of the structure functions [15] (see se sec 3 4) The scaling law for total ere”

nnih Cl ion Ci non a Q he 4 or C4 Q G non Cl a `7 AAaceA heorie Cl e “ta JINDLO 7 ~ free” The nofion of Wilson’ S dimensionality of operators does not hold here, since the former

DbHe DOWC Ingula C Te TOT 5 d CU CAPa UIT C aDppiOd s YL asy tr So,

is here also rogarithmis Present day experiments show a faster approach (81

scaling at SLAC L8], a phenomena remarkably predicted by Bjorken [17] An earlier approach

that emerged from scaling is the parton model [18], followed by cutoff field theory calculations

[19] Later, “‘soft field theory” [20] calculations were developed, and duality ideas were also in- corporated [21]

A very important step in the development of the light cone approach is the quark algebra structure suggested by Fritzsch and Gell-Mann [22], which is to assume free quark field algebra for the SU(3) X SU(3) structure on the light cone This made clear the connection with the par- ton approach in deep inelastic scattering, and shed light on which results of the parton model are

of a general nature and which dependent on specific assumptions peculiar to that model

This paper is organized as follows In section 2 we discuss deep inelastic electron—nucleon

scattering In 2.1 we review briefly the experimental situation and discuss the light cone domi- nance analysis, and in 2.2 discuss the Regge behaviour in the deep inelastic limit and the relation with equal-time commutator sum rules In section 3 light cone expansions of operator products are considered We review the general structure in 3.1 In 3.2 we discuss the Thirring model, where anomalous dimensions appear It also exhibits the phenomena of “‘softening”’ for compo- site operators, namely that their dimensionality may be less than the sum of the dimensionalities

of the constituent fields, and it may also be canonical (as is the case for the currents, but not the scalar and pseudoscalar densities) When the model is generalized to include interactions quadratic

in SU(m) currents, scale invariance (with anomalous dimensions) is obtained only for one (non- zero) value of the coupling constant, which is 4a/(n + 1) [23] We then review the deep inelastic

scattering (sec 3.3) and the Cornwall-Norton sum rules (sec 3.4), the latter in relation with re-

sults from summations in field theory In section 3.5 we discuss the subject of fixed poles and

the polynomial residue in the mass variables of the “photons’’ In section 4 we discuss the quark algebra on the light cone as suggested by Fritzsch and Gell-Mann It is the light cone singularities that are exhibited in 4.1 as for currents constructed out of free fields The resulting bilocal operators ® have matrix elements which include all the complexity of strong 1 interactions, and have

dependent, in contrast to the leading singularity, the structure of which is interaction indepen-

de If1 odel dependent i e sense 0 e kind of constitue used) ectio we discu

total e*e” annihilation into hadrons and 7° > 2 The relation between the two following from

consistency considerations of operator product expansions and quark schemes are reviewed In section 6 we discuss single particle inclusive annihilation, namely e*e annihilation with the detec-

tion of the momentum of a given hadron in the final state The scaling properties are reviewed

Special attention is given to the question of asymptotic multiplicity It is shown that canonical

schemes were ‘recently studied i in that limit), null plane quantization and s sum rules, and finite

OED Zia,

Finally, we should emphasize that the most important issues ahead are:

(1) Checking scaling and relations among structure functions for higher virtual mass and energy carried by the currents Also checking of the various sum rules

(2) More studies, experimentally and theoretically, of details of final states: distributions, charge ratios, multiplicities, etc These in both the scattering and annihilation regions

2 Deep inelastic electron—nucleon scattering 2.1 Scaling behaviour and light cone dominance Consider deep inelastic electron-nucleon scattering For an unpolarized target and in the one photon exchange approximation, the differential cross section is given by

WAG P) = > d*x ei4x(psl[Z,(x), J,(0)] Ips)

= (gy 3") mrtat +S (0, 9 a.) (pp 24 a, \wata?n q mM? Ve qe “#})\ 4° '") (5)

where j,, is the electromagnetic current Also,

tay ` Can: 5 lÍ 20: 49) =3)* xa, 43132]

where B is the limit of g? > —œ (space like) and vy > », with w = 2Mv/(—q’) fixed This is the re-

sult obtained when one ignores mass parameters and assumes canonical invariance under scale

transformations (see our discussion in sec 3.4 for possible power deviations due to anomalous dimensions, or logarithmic deviations as in “‘asymptotically free” gauge theories) When one changes p > Ap and q > dq, and takes into account the variation of the matrix element of the current commutator through J,,(x) > (1/A°)/,,(x/A) and Ip) > Aldp), one then gets Bjorken’s scaling for W, and vpW, The condition |p) > AlAp) emerges from the invariant normalization (p|p’) = p, 6° (p — p’) and can be applied for p? = 0 As our discussion in 3.4 indicates, the latter step is justified in evaluating the matrix element of the current commutator only in the case of canonical dimension for an infinite set of local operators of increasing spin that appear in the short

distance expansion of two electromagnetic currents

In the Bjorken limit most contributions come from the singularities near the light cone of the current commutator in eq (5) [25, 26] This is easily seen in the nucleon rest frame Then, con- sidering also the more general case of the covariantized time ordered product, the invariant am-

plitudes of 2X <ps|T* x), J (0))|ps) depend on x? and p: x = Mx, only, in which case the

angular integration dQ, can be explicitly performed, and one encounters now exponentials of the form exp(ivxg)exp{tiv v? + g?lx1} We thus have, in the Bjorken limit, that (using

Vv? +q? = v—-M/w),

Ilxal — lxllS 1/, lxl œ/M and thus |x?! < 4/lq?l Light cone analysis of W,,(q, p) then proceeds through the introduction of the casual functions V, (x’, p- x) and V2(x?, p- x), defined as

(ILI, (x), JON? = (By, ~ 8,9,) V2, p- x)

+[ P,P, — Py, + P,P, 9) + guy(p- 9)2]W;(x2, p- x) (9)

The Fourier transform defines functions V, (q?, gq: p) and V2(q?, q: p), which are related to W,

vanish only linearly at g? = 0 Bjorken scaling is obtained by [7]

WrQ@G7, p- x) = —(21i)e(xa)ô(x?)f, (p- x) V¿(x?, p- x) = (2m1)e(xa)0(x?)ƒ#;(p- x}

as leading 8 Hightcone singularities

0.00 -~ DEUTERIUM R=0.18 0.30

0.80}- 0.70 0.60 vWa 0.50

0.40 0.30

0.20 9.10

The A integration is limited to [Al < 1 by the spectral conditions, which also implies that f, and f,

are analytic in (p- x) In these considerations we assume that there are no strongly varying parts

to the commutator inside the light cone, which may contribute in the scaling limit This is cer- al ble physical ion M Jaffe [31] calculated tl but; F ô(x? — a’) singularity, and found that it has, relative to a 6(x?) contribution, an extra factor of

U d d U CU d UJ C C OU Of CValud ) Wad CQUIVaAIC ©O dvYCTlaE Py OVE

oscillations (One can understand the connection as follows From Jaffe’s calculation, eq (8) in ref [31], taking g(1) # 0, one gets a vy '* suppression as compared with 6(x*), times a factor (1 — 1/w)y 4 explia/ 2Mvr(1 — I/w)] for w # | and v> ~- Now, since So đdẸg(‡)(1 — Ey '4 X

ex plia/ 2Mv(1 — &)] yon vp °"* 9(1), the oscillator factor is equivalent to an extra v’* The result

can be obtained from eq (7) in ref [31] directly integrating as above.) Thus even a 6(x? — a?) singularity, which is certainly too singular for any realistic situation, is less important than a 6(x?)

in the Bjorken limit Another point of view is to consider the oscillator factor in the distribution

sense, which means that it vanishes faster than any power of v [33]

be given by nonleading light cone singularities [5, 7] (Thus, for example, in the combination a(v/(—q?))'4 + (b/(—q?))(v/(—q?))'” the first term dominates in the Bjorken limit while the second dominates in the Regge limit.) Adopting this unification of Regge and scaling limits, we get for the contribution of a Regge pole with intercept a(0),

Ww > % Ww >

This implies

g(a) v IAI EO FIC [a(0)], ^>0 B20) IAIt =«(®]Œ, {a(0)] (16)

For any a(0) > 0 eq (16) implies that the first Fourier transform in eq (14) does not exist as

a usual integral It has to be understood, of course, as a generalized function [5, 7,35] We re- place, for À > Oand 1 > a(0)> 0,

IÀAI- t()+1] > —_ 1t {(—-r + ie) [e(0)t1] " (—À —— ie)~a(0)#1]1

= f dAfø()— 22 €Œ(@)IÀITG?ĐJexp(iAp:x) + 22 Dy (a)l(p- x)! (17)

œ>0 a>o

For a = 1 we can take a limit a > 1, or take 5[1/(A + ie)? + 1/(A — ie)?], from the start in g, (A)

[5] As for a = 0, making C, (a) « « we see that in the limit a > 0 we get a constant contribution

to f, (0) and a part « 6(A) in g, (A) Such a part does not contribute to # (ó2), but contributes a subtraction term to 7) Since f, (0) = S gives the matrix element of the operator Schwinger term,

we have the sum rule [36, 37]

a>o

— co

excluding any other J = O singularity in g, ¢, is a kronecker delta singularity at J = 0 in the real

part If present, it will show up also in certain exclusive electroproduction processes (see discussion

in 3.5) Whenever spin 0 or field algebra spin-]1 couplings are present, we have a longitudinal cross section [38] In the first case we have a nonvanishing S, while in the latter S = 0 [39]

Note that Regge contributions influence the high (p- x) behaviour of matrix elements of bilocal operators As is clear from eq (17), the contribution of a Regge pole isa I(p- x)|* as l(?- x)l~> % (the integral on the right-hand side of eq (17) gives a vanishing contribution in that limit) Simi-

larly, ƒ(- x) > l- x)l%(8)~2,

Note that the Regge term 3X gС (ø)l(p- x)!* is not analytic at (x- p) = 0, but the whole ex-

pression on the right-hand side of eq (17) obviously must be, as we showed before To see this

we derive yet another expression for f, (p- x) Define f, =f, —SR38, =, — gf where gk, fP

3 Light cone expansions of operator products 3.1 General structure and examples

In the previous section we were concerned with the singularity structure for one matrix element

In order to get relations among various experiments we need an operator statement This is pro- vided, for deep inelastic processes, by the light cone expansion [4, 5] as in eq (2),

[a]

To be specific about the Lorentz structure, we write A(x) BQ”) = 27 S11 (x — y) F(x) ey )™ Fil (x, y) (21)

[a] n where S!¢l(x — y) is a scalar singular function,

A(x) > A,(x) = MAYA (AX)

This transformation is applicable in the short distance or light cone limit of operator products,

when mass parameters may be neglected Current algebra fixes the dimensions of the currents to

be 3 The energy-momentum tensor must have dimension 4 Other operators may have a dimen-

sion which is not the canonical value as derived from formal Lagrangian considerations (see also

our discussion of the Thirring model in sec 3.2) Thus the degree of singularity is determined by the difference

z between spin and dimension, of the operator [42] The targer d ity near the light cone

NOTE s An Om ne aa nange o neq OmMmnD ed neg eve ning 1s

same This follows from locality and analyticity of the bilocal operators Thus, in order for

[A(x), B(v)] to vanish at space-like separations, the bilocal operators in eq (21) and eq (25) have to be the same for (x — y) space-like, and by the assumed analyticity have to be the same operator every where

For the commutator the singularity is

Slel(x) = P(tdlel[(—x? + iex ya! — (—x? — iex,)- ty (27)

where ở 1s a free scalar field, we have

LJ, (x), Jy) = [AG —y) 1823 Ay) ] — [AC —y) 182782) 6°06): +: 9° )GOD

next to leading singularity Terms of the order of 5'(x”) in the divergence are cancelled only after

use of the - equations of motion for ó Since we are going to generalize to the case of the real cur-

For the case of J,,(x) = : W(x)%„ W(x):, W(x) a free Fermi field of mass M, we have [5]

1[Z,(x),/,@Œ)] = [2„A(x—y)][9,Ai(Œx—y)] + [2„A(x—y)][ð,Ai(Œœx—v)]

— #uy[2„A(x—y)][2#A¡(x—y)] + M ?ø„„A(x—y)A¡(@œ—y)

— i[2„A(x=y)][:@)Y„17+„UG): — :JŒ)y„Y +„():]

— MA(x—y)[:ÿ()„x„ÚÚ): + :JŒ)„y„(x):] (34)

The bilocal operators multiplying 2„A(x — y) can be expressed in terms of vector and axial vec-

tor bilocals by use of ¥, YoY, = Sua Vv + Sva%y — SuvVa ~ ieyavrg 87% The bilocals multiplying

A(x — v) can be expressed in terms of scalar and anti-symmetric tensor bilocals, since

Vp Vy =8, + 3 LY p> +,Ì `

It is interesting to comment, that anomalous dimensions appear in the study of solutions of the Dirac equation in a Coulomb potential Ze?/r Thus the behaviour near the origin, r > 0, of the wave function with a given angular momentum is

y, ~ pitt it 1/2)? — (Za)? ] 1/2

}

which depends on the coupling constant A similar situation appears in the Schrodinger equation with a 1/r? potential In both cases, anomalous dimensions appear when the potential has the same dimension as the kinetic energy term

3.2 The Thirring model and its generalization to U(n) symmetry

Recently, an operator solution of the Thirring model was exhibited i in terms oO the full light cone

^+^ a ato © ere e Are ic Dana va 7 mses cla Ln colpece eninge

Dđ11S1O OT proau O C 1O Gi yy: a d 5 Od ÄS d Di O

field in one space dimension interacting through L, = — 3Ø HP: Define

ducts 1 in equations of motion and i in defining currents [45] Our way of obtaining the solution i is

io r1 C ya TC UI a 3 Marry TUE Œ†1 ©®fnrmm vane ion q ate On > y ^/1 wu a He ry `} ¬ > wd h c3 he

currents ¢ are massless free fields, one can use a normal V ordering with Tespect to their creation and

for an interacting Fermi field, turn out to be very useful in the course of solving the model The

commutation rules are

Y Frishman, Light cone and short distances

(36)

l

where the normal ordering is with respect to the frequencies in the Fourier decomposition of the

current Our expression for Ouy and eqs (35) and (36) for the commutation rules, and the normal

ordering procedure are sufficient to solve the model without any problems of singular products at the same point The resulting operator product expansion for two spinor fields is

Yiluv) Wilw'v') = folilu — u') + €]~@F 4x6 L(y — y') + e] “OAM x

i u! v

For ;Ú; replace a > —a4

Comparing with the equations of motion we get

i[D, ÿŒ@)] =[x*ô„ + $ + ø2e/4m)]0(x) (43)

The main conclusions we can draw from this model are:

(a) Currents are more regular than the respective products of fields and obey simple commuta- tion rules

(b) Scalar and pseudoscalar densities, which have no algebraic reason to have canonical dimen- sions, indeed have anomalous dimensions Their dimensions are canonical only for g = 0 [48]

When one generalizes the model to include U(7) symmetry, and introduces an extra interaction

— 581 :77* 72: (where j* are the SU(m) currents), one no more has scale invariance for arbitrary g,

Scale invariance, with anomalous dimensions for the Fermi fields, is obtained only when g, = 0 or when g, = 4m/(m + 1) [23] For these values of the coupling one can solve the model completely

Note that the currents 75 are now no more conserved From the Lagrangian it follows formally that 07 = 8/°59/2°j °#(ƒ“”° the structure constants of SUŒ)) However, one can show that

whenever a scale invariant solution exists, and the dimension of i? is canonical, namely 1 (as fol- lows from current algebra), then it has to be conserved [23] In such a case we can use the normal ordering procedure as for the usual Thirring model, since all currents are again massless free fields

The extra commutation rules are [/), 7/4 ')Ì = 2*?*/()ô(w — +) + 21 c,82°5'(u — u')

where the coefficients are determined by the Lorentz group commutation laws As for the spinors,

——————— the groupstructure dictatesnoW_———————————————————————————————————————————=———

[/(), @/u')] = —(1 + ô+y;)G X2)Q@¿0))ỗ(u — u’)

When one considers the four point function for the spinor fields, and imposes conformal invari- ance and isospin crossing structure, one gets that the only solutions obtained are whenever

c, = 1/27, the free field value, and then 6 = 1 yields g, = 0 and 6 = —1 yields g, = 4m/(n + 1) It can be verified that for those values one indeed obtains a quantum field theory [23]

As for the dimension of the spinor field,

only, then the extra terms in [51] of the form ø#xể + p®x® are as important in their contribution

to V, {7, 49] Here, although the tensor structure is that multiplying F$°(x, 0), since the matrix

element involves x* terms one has a part like a next to leading term in V, (this is clear when one uses x*elax = —ja' (e'%*) in the Fourier transform and integrates by parts), A x%x® term in eq

(51) has an extra f ) v suppression as compared with (p#xể + p#x*) (we exclude gø® terms, since

those are identical in eq (50) to the V, term)

3.4 Generalized Cornwall-Norton sum rules

The Cornwall-Norton sum rules [50] express integrals over moments of the scaling functions

in terms of commutators of corresponding numbers of time derivatives of a space component of the current with a space component, at infinite momentum They are,

as the coefficient of 2i(—)"*'/q?"*? in the expansion of T;,(q, p) = 2ifd*x exp(iq-x Kp ITJ,(x)J,(0) |p)

(only even powers appear because of crossing symmetry), and then taking py > © to isolate the

highest spin component, (2” + 2) Starting from an unsubtracted dispersion relation for 7},

If F,(w, g*) scales for g* > —~, then the nth term behaves like p$/q3"* ? However, a non-leading

term in F;(w, q’), like in F,(@, g*) = F,(@) + F 3(c2)/(—q?), would mean an extra part play r4

in the nøth term Thus the coefficient of 1/4?"*”? has a part like p2" from #;(œ›) and a part pr 2 from Fw) If one now lets pp > ©, one picks up only the contribution of the scaling limit F,(w)

Then, in 7,,(q, p) one would have a pint? term as the coefficient of 1/qg?"*?, which means a spin

(2n + 2) ovetator in the coefficient of 6@)(x) in [22”*'J,(x), J,(0)] (powers of g correspond to

gradients of delta) Thus the existence of the scaling limit implies the existence of the spin

—(2n + 2) operator coefficient of B(x) i in (pI [Bể i iO), J ;60)] p? However, if w we have a non-

lower spin operators [37] (This does not happen for half-odd integer powers For details see ref

h h ontain genera di IO'n O ne rela ion _be veen igh one ingula es ang

equal time commutators.) In this case an expansion in terms of integer powers of 1/q, is not justi- Fied-and the Wil oo vant

The general results for the moments of " and W, for Mã > coe can be > obtained from Wilson’ S

operators, and therefore write, instead of Cx - — WFC yi in eq (50), the form

VSF(X, OV ~ DE CO) ays Kevyn BeBe 8 2(0)5, Ce) = [( x? Hex 9) 8" (x? -i€x 9) W(4d,,)

Comparing eq (56) with eq (54), we get

since ` LF" ape 4n] = 4+ 2n — 24, and since Fy af “presumably has a part which is the energy-

nsi no, we expec T e data, since

fi (doo/w?)F>"(w, q?) s seems to be about 2 £ of the value of fy (deo eo?) F2 (we, q?) [8] (the ter being 0.16 = 0.02, and the former 0, 12 = 0.02, where the error includes also Regge extra-

polation for w > 20), F; is about 2 isoscalar and about 4 ~ isovector Thus if the above of = would

persist for (—q*) > ~, this would mean the existence of: an isovector tensor with dimension 4

(Note that for the “asymptotically free” gauge theories the difference f*(dw/w?)[F3?(w, q?) — Fy"(w, q*)] goes to zero as a power of (lg q”), [15], thus implying that F$° is pure isoscalar, and that at present we have not reached yet the “‘asymptotic limit’’.) If d, is anomalous, d, < 0, we

have a situation in which pW, goes to zero for gq? > — at each fixed w The decrease is at least

as fast as (—g?)#= (for the ‘asymptotically free” gauge theories it is at least (lg(—gq?))?@= }

Since the n = 0 integral is g? independent, this means that for low w the function F, de- creases with q’, for high w increases with q?, and the point w = w,,, at which the function does not change with g7, must itself move to higher values with increasing g? Thus measuring a de- crease with qg* for fixed w would show such deviations from scaling Low w are preferable ex-

perimentally, since then one can have higher q’

It can be shown that the d, are convex from above inn, namely (d/dn)d, < 0 and (d?/dnw?) d, 20

as follows from eq (56) and the positivity of W, (this fact was first discussed by Nachtman [55])

Thus if d; = d;,, =, it follows that d, = d for any n > 7 Now, if d, > d for any n < n, there

would not be an unsubtracted dispersion relation for 7, This is so since when considering eq

(55), modified for the time ordered product, its matrix elements between single particle states correspond to expansion of 7, in powers of (w7) for w* > 0 If only a finite number of d,, have

different values, we have a situation in which we have a bilocal and several local terms, with the

local terms multiplying different x? singularities than the one for the bilocal Now, since any

given term in eq (55) does not contribute to W, for g? < 0 (it is only the infinite series, first summed, that yields a finite contribution to W,; see ref [5S] for details), the local terms yield

subtractions for 7,, which we do not want

The conclusion is that either all d,, are equal, in which case we get Bjorken scaling, or that all are different, convex from above This also shows that one cannot prove Bjorken scaling from Wilson’s expansion, dy = 0 and unsubtracted dispersion relation for 72 [53] All one can show is

that it is impossible to have all but a certain finite number of d,, equal A recent estimate from experiment, taking dy = 0, yields (—d,)< +, [56]

limit in both cases, vamely ˆ qƒM = = —3 ub G GeV for the proton and zero for the neutron 159] (in

C C [ CTTOIT are darge 14a d O DE dTRUCC 1a 2/G d e lo C C idue ©

the fixed singularities are q7 independent for all g? [60]

Let us return to V; of eq (9) For the leading light cone contribution we have

ˆ dÀg;(À)

T¿ ~ 16ỀMÊ(-đ)_ | (C2Z— 2AMp+iof ` lâu

Subtracting all Regge contributions with a > 0 (assume no contributions to g, at a = 0), then

; M4) ƒ đÀøz(A) HS

and the integral is convergent since g,(A) vanishes faster than |A! at AX > 0 (see eq (16) for œ(0) <

0) But are we justified in taking only the leading singularity near the light cone, if we deal with fixed qg?? It turns out that this is allright for 7, For suppose we take a less leading singularity

Then its contribution to 7, is of the form

A Regge pole with intercept a Will be generated by a term IAI! ~® in gl@l(n) for À > 0, as in eq

(16) This is so since Regge behaviour is obtained from the small A or high (p- x) behaviour of the matrix elements of the bilocal operators, and is therefore independent of the type of singularity near the light cone [5, 7] Thus, for > ~,

pearance of a fixed pole in i @))T at J = 0 with a residue that is independent of g* However,

through finite energy sum rules Since we expect d = 1 for the next to leading singularity (as mass term corrections, for example), we see that the second term is especially important near 4? = 0, while for g? > — only the first term obviously survives An effective change in the value of the

residue of the fixed pole around (—g?) ~ yp? in a phenomenological analysis may therefore not

be surprising The value of 4? should be around that value where scaling begins in (—g?) When (d + a) < 0, the integral f°, dd &!7!(4)/A2*4 converges and thus

7a > (—q?)v 2-4

which is a fixed pole at a = —d

Since T, = v’T,/(—q’) — T, vanishes at g? = 0, it follows that the J = 0 fixed pole in T, im- plies a fixed singularity at J = 0 for gq? = 0 in 7, The latter is a Kronecker delta singularity in the J-plane (since J = 0 is a physical partial wave in the t-channel for 7,) One can thus separate the first term in eq (62) by looking at Compton scattering for t # 0 If we assume that R, , the residue of the J = 0 singularity in 7, , is g? independent (see reasoning below), then it must vanish everywhere Thus 7, has aJ = 0 fixed singularity for all g? One may also detect the f-de- pendence in amplitudes with one real photon and one off-shell, like bremsstrahlung in electron—

nucleon scattering [61], since the residue of the fixed singularity does not depend on the photon masses (see below) Our assumption is that a(t) changes with f, since it comes from the matrix

element of the bilocal, and there is no reason for that to be fixed (Anything that can move —

moves!) Moreover, it can be argued that the decrease of the first term in (62) with ¢ is like that of

a form factor [61], and therefore for not small ¢ would dominate over usual Regge contributions (the latter, of the form B(t)y%0*%*, have a fall-off from B(t) and also exponentially like

exp{—a, (Inv) It! })

The fact, that the fixed pole term is dominated by the light cone singularity even at low (q”) follows from the standard phase variation argument in eq (5) defining W,,,- Our arguments in Sec 2 1 led to exponentials I like ©eXP{if(xo = ae) FexpGM ix i/o) Thus [lxo! — Ix] < lịp as before,

damped for high values such that only Ixạl< < a are | important, we have

Ix?1< 2a/p

In the analysis for non-forward direction and different photon “‘masses” in Compton scattering one has matrix elements of the bilocal operator between different momentum states,

[(DiF oS? (x, OVP 12-9 = P°P* fda dp exp{ilap- x + 8pˆ- x)}ø;(œ, ổ, f) r¬œ8 At-v¬ als sa ps at sy As (63) (ens

PH L7 = {7 1 L7s H 1e avs > q Anna 7 oI a a ars 4 =) On alF= ae; 1 A PION 7 C7

in the variables a, =a + 8 and a, =a — 8, turns out to be the area within the lines connecting the

with ns, À bounded by four lines connecting t the poin's (+1, 0) and (O, +1) Thus i in eq we

d O sees the °

analysis i in the non-forward direction shows that there i is also an additional fixed ole i in a spin-

flip ampliti ude {61]

Finally, we would like to comment that there may be a fixed pole at J = 0 in the 7, amplitude coming from non-leading singularities, of the form

u?q?/(u? _ q?)v?

which is non-polynomial in qg? This is a result of an infinite number of light cone singularities, since no finite number can produce such a term However, such a term will show up also as a

fixed pole in electroproduction of the hadronic states with the mass yw’, that give rise to the dis-

continuity at g? = u? One may of course replace 1/(qg? — w?) by fdm?p(m?)/(q? — m?), with

Sp(m’)dm? = 1, to get the same effect as before for the J = 0 singularity, but now with a con- tinuum contribution for the discontinuity in g* The quantum numbers of these hadronic states are those of the electromagnetic current One can of course also have a fixed singularity

(Kronecker delta) at / = O in the 7, amplitude, which from the leading light cone singularity im- plies af, term in the Schwinger term sum rule eq (18) Such a singularity also implies J = 0 fixed singularities in electroproduction of hadronic states, and also changes the relation between 7, and 7, fixed singularities It also ruins the polynomial dependence, since it may be of the form q?/(q? — w?) It may be argued in this case that such terms are absent, since they are not produced

by dispersion integrals over the imaginary part, but by real subtractions only (The dispersion integral has no J = O fixed behaviour once the a > O Regge contributions are subtracted.) How- ever, for R,; = 0 such terms appear in 7;

Thus, if there are no fixed singularities in one photon amplitudes, namely in photoproduction

or electroproduction of hadronic states (according to the rule: anything that can move — moves!), then the J = 0 fixed pole in 7,/(—q’) has a residue independent of g? and equal to that of the

J = 0 fixed singularity in 7, Both arise from the fact that one has a canonical light cone singularity

in two current amplitudes

We saw before that from a light cone singularity (x7)? in V, we get a fixed pole at a = —đ in T;,

ra oS C LÀ Đ G z2 O ở dl 2? 5 Note > a d cy “7 now Ne ger Q L] r1 ed pete k7 ha ry LÝ Cả a d ( `

to the imaginary part W, This is so since the contribution of the above singularity to W, is

1 Wl4l(g?, v) ~ e(p)(—4?) Jdà øl2A)JI(—4? + 2XMb + ie)~(4+2) — (Ta? + 2XMb — ie)~3-?]| (64)

¬

4 The Fritzsch-Gell-Mann algebra

4.1 Quark algebra on the light cone

A very important step forward in the study of deep inelastic processes was the hypothesis of Fritzsch and Gell-Mann [22], that not only is the leading light cone singularity given by a free field of spin-) constituents, but that so is also the whole SU(3) X SU(3) structure of the bilocals

of the leading singularity [64] This implied many relations, and it thus became clear which re- sults of the parton model are a consequence of the SU(3) X SU(3) structure on the light cone and which depend on specific assumptions of that model

To obtain the commutation relations, one writes the electromagnetic and weak currents in terms of quark fields,

JEM = 23 py,p —3ny,n—3 yd =F: vt (As +a} v:, (66)

J =:[py, (1 — ys)n}: cosé, +:[py,(1 — ¥s)A}: sind,

=: Wy, (1 — ¥5)5 [Ou + idz)cosO, + (Ag + ïÀ;)sin6, ] Ủ: (67)

and then computes the commutators as for free fields One then postulates that the type of singularities and the SU(3) X SU(3) structure are the same for nature The space dependence of the bilocal operators is unknown — it is measured in deep-inelastic electron and neutrino scatter- ing experiments One should emphasize, that only the light cone singularities are of a free field nature The matrix elements include all the complications of strong interactions and may not

have any resemblance with a scale invariant limit of setting all mass parameters to zero Defining

A#*Œ, y) =: (x)y„(1 + +;)( Á2)): — (x s y) (70)

The commutators [7 a (xX), J >-(y)] are less singular near the light cone by one power of x”, namely

have a leading 5((x — y)’) singularity rather than the 6'((x — y)*) as in eq (69) They are also proportional to mass terms

We now adopt the structure of eq (69) to hold in nature for the leading light cone singularity

Note that we took over from free field theory the canonical leading light cone singularity and

the Lorentz and SU(3) X SU(3) structure of the bilocal operators However, matrix elements of

the bilocals will not be given by considerations of scale invariance, since they involve on mass- shell states and mass parameters are important there (for example, in non-forward direction Regge trajectory slopes enter)

One can try and argue that the leading light cone singularity is not going to be modified in re- normalizable field theories, proceeding as if canonical considerations are valid and “subtleties” of renormalization of infinities can be ignored [42, 65] One then discovers, that the leading bilocal

is not changed for interactions with scalars or pseudoscalars, while for neutral vector mesons,

“gluons’’, it gets multiplied by a line integral

: WOe)P Wy): > : Wor) exp{—ig S* V,, (&)dé" I y(y):

where V,,(x) is the gluon field and g is the gluon—quark coupling constant One does not have to worry about ordering problems in the definition of the exponential since when (x — y) is almost light-like and the Gupta-Bleuler commutation rules are taken for the gluon field, any two parts

of the line integral commute (Explicitly [J dé V,,(&), S52 dn’V,(n)) ~ ƒ dệ" /s2dn, X

€(Eo — No) 5((E — )”), and when Xi; X27; tend to points on one light ray, d&* - dn, becomes propor-

tional to (£ — 7)’, thus yielding (£ — n)?6((E — 7)”) = 0.)

One can go further and postulate closed commutation rules among bilocal operators [22], which yield light cone singularities multiplying the same set of bilocal operators For two bilocals F)(x¡y¡) and F,(x2y2), this is assumed to hold when all four points are near to one light ray (all

six distances are almost light like), as indicated from canonical considerations of quarks with

H Đ]"]OtOtra 1£ § espe O C O” di L IO'-O OTO H LÌ Gre d °

the virtue of explaining why non-zero triality states are much higher in mass [66] Also, taken as

only the anti-symmetric vector bilocals, and the spin dependent ones involve the symmetric axial

bilocal In neutrino—nucleon scattering both the f and d couplings appear One takes

¢c "¬ Cc we? x*=0 uA uŠS

that go with x„ do not contribute to the leading terms which give scaling However, in an approach

where the charged constituents are Fermi fields, the longitudinal combination W, = (1 — v’/q?)W,— W, vanishes in the scaling limit [38] The combination of the p,f and x,g are

then of the same order to W, , namely order 1/v The contribution of p,f to W, isW, = W2= F,/v, while the contribution of X,8 is 1 /v times a new scaling function, the latter related to g [49]

When considering non-leading contributions to the scaling limit, one has also to take into account 5(Z*) singularities in the current commutator However, those contribute of order 1/y* to W, and W, This is easy to see from eq (9), since a 6(Z’) in the current commutator means x76(x7”) sin- gularities in V, and V, of eq (9), which means [v*W,/(—q?) — W,]~ 1/v? since V, is two powers

of x* softer than canonical, and W, ~ 1/v? since V2 is one power of x? softer than canonical Thus,

if the leading light cone singularity 5'(Z*) appears with the structure like in eq (69), the leading contribution to the longitudinal cross section is from the x, terms also

For sum rules involving longitudinal cross-sections in electron and neutrino scattering in rela- tion to low energy parameters (sigma terms, octet masses and chiral symmetry breaking) see refs

[67, 68] (polynomial residues for J = O fixed singularities are assumed there)

When scalar constituents are present, the leading light cone singularity is a 6’’(x’) one [5], as follows from eq (31) One then has scaling for W, Now, a 6(x’) singularity in V, means W, ~ 1/p and a contribution 6'(Z7) in [/,,(x), J,)) with a tensor structure of scalar constituents Thus if scalar constituents are absent in their leading contribution but contribute in their next to leading singularity, we get another contribution to W, of order 1/v In that case, the 6’(Z?) in the current commutator comes partly from a leading Fermi contribution and partly from a next to leading

T invariance sets W = 0 Since we have SU(3) X SU(3) symmetry on the light cone, with all cur-

rents conserved, the scating limit for pW, 5 iS ZeTO

given by a structure like mass term corrections, then it is rather v?W, and v’W, that scale The

latter also do not contribute to the scattering cross sections in the limit of zero lepton masses As

for Ws,