LIGHT CONE AND SHORT DISTANCES
Yitzhak FRISHMAN
Trang 2LIGHT 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 hi d A A Juark g a 0 a multiplicity are analyzed Finally, we comment on various other problems and approaches Single orders for this issue PHYSICS REPORTS (Section C of PHYSICS LETTERS) 13, No 1 (1974) 1-52 ay be obtained at the price given below All orders should be sent directly to the Publisher Orders Single issue price Dfl 20.00, postage included (*)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
Trang 3Contents:
1 Introduction 3 4.4 Current conservation 32
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
2.2 Regge behaviour and sum rules 9 7 Other problems and approaches 42
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
3.2 The Thirring model and its generalization to 7.3 pp > u*u"X 42
U(n) symmetry 14 7.4 Can quarks escape? 44
3.3 Products of electromagnetic currents 17 7.5 Studies in field theory 44
3.4 Generalized Cornwall-Norton sum rules 18 7.6 Conformal symmetry 45
3.5 Fixed poles 20 7.7 Null plane 45
4 The Fritzsch-Gell-Mann algebra 24 7.8 Relation between scattering and annihilation
4.1 Quark algebra on the light cone 24 scaling functions 45
4.2, Results for deep inelastic scattering: structure 7.9 Early scaling 47
functions, relations and sum rules 25 7.10 Finite QED 47
4.3 Further implications — Non-forward matrix ele- Acknowledgements 47
ments 31 References 48
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- tion of his studies in model field theories These have the form Af + Df ` fol; 4X)5V/) ^* CC ` ` (xu > Yp) [a] rial tl Œ) onn — `x—+ (x_—ÿy)# 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 (x — y) it is assumed that only a finite number of terms appear in 1 the expansion eq (1) This ex- the 2 dynamical dimension governing short distance behaviour [1] Wilson | argues, that such di- *
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
alda dime c1 “ 3 o ^ “^^
Vy s Lj CŨ: 1O s 1C LJ LJ L C ° C CTICI#X Y= O U C OT d L
sion 4
@ ex ion is an expansion uc s when irs =ti is-
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
Trang 4A(@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
terms Summation of infinite sets of diagrams considered so far show no possibility of obtaining bilocal operators and scaling [12] Instead, even when one considers sets in which only power
singularities appear (where one considers neither self-energy nor vertex corrections), the singulari- ties near the light cone depend both on the spin » of the local operators and the coupling con-
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
Trang 5
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
light cone singularities, scaling and logarithmic multiplicities are consistent, as follows from the singularity structure of products of two electromagnetic currents and two hadronic sources at
short distances [24] (for the difference between the space—time points of the electromagnetic currents) In section 7 we mention other problems and approaches In particular, one photon amplitude processes (like form factors, exclusive electroproduction and pp> wv u 3), summations
schemes were ‘recently studied i in that limit), null plane quantization and s sum rules, and finite
Trang 6Finally, 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
d?ơ a? cos?+ 6
JQdE" AEM sintto (4° ») + 28220 Mu(4°, »)] 2 (4)
where F and E£’ are initial and final electron energies and Ø the scattering angle in the laboratory frame, p and M the four-momentum and mass of the target, g the virtual photon four momentum, and My=q-: p W, and W, are 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,
da Ana2—_{Fv 7 \
tay ` Can: 5 lÍ 20: 49) =3)* xa, 43132]
ashe W(x, a {a —y)+‡y? TRE (6) x = 1/w=(—q?)/2Mp, y=v/F, S = 2ME, » OL ewig WW Wi W, — Ww, cay R=== wr, ~ „Mộ (7) Bjorken’s scaling is [17] W:(q?, P) > F,(o), pW;(q?, P) |; F;(03) (8) B
where B is the limit of g? > —œ (space like) and vy > », with w = 2Mv/(—q’) fixed This is the re-
Trang 7transformations (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, and W, through 2 (=q”) p2 † 2»†z 38» W;—W=(-q Vy (1U) (—q’)
Note that V, and V, have no kinematic zeros at q? = 0, since W, and v*W,/(—q?) — W, have to
vanish only linearly at g? = 0 Bjorken scaling is obtained by [7] WrQ@G7, p- x) = —(21i)e(xa)ơ(x?)f, (p- x) V¿(x?, p- x) = (2m1)e(xa)0(x?)ƒ#;(p- x} as leading 8 Hightcone singularities W ~ M TZ V 23 (1) value of R, the ratio of longitudinal to transverse cross section, of 0.18 The data i is Shown in fig
fo get the structure functions for electron—neutron scattering one scatters electrons off deuterons The data shown in fig 2 is for R = 0.18 for the deuteron [29] (all points with (—q’)
Trang 90.00 -~ DEUTERIUM R=0.18 0.30 0.80}- 0.70 0.60 vWa 0.50 0.40 0.30 0.20 9.10 L J | Jt l jo 1 J 0.00 0.10 0.20 030 040 0.50 0.60 070 0.80 090 1.00 xtet @ Fig 2 „MỞ versus x' = 1/2 1 1 fii x)= f dre,Qdexplitp-x), ftp x) =f dd gaA)exp(idp- x) (14) -1 -1
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 ơ(x? — a’) singularity, and found that it has, relative to a 6(x?) contribution, an extra factor of „ 3/4 ime nm ¢) , k2 Œ 3 cy wv, cớ, 1/4 ˆ s “3 NY ed A 7 ee, rE ArA O =f} J dHAG (Ì 2 ' a we, was assumed In another calculation an extra factor of 1/v, with no extra oscillations, was found W
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
Trang 10be 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),
F1(@) —— 6209), F2(œ) ——> w* (15)
Ww > % Ww >
This implies
g(a) v IAI EO FIC [a(0)], ^>0 B20) IAIt =ô(đ], {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,
Trang 11= 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 # (ĩ2), 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]
s=1+ f ale, — DS C,(ayiai-@r (18)
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
Trang 123 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(x)8Œœ) =3} Œl(x — y)Ƒ#\, y) (20) [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, Sl] (x) = (—x? + iex 9) Ir (+al*)), (22) The value of d!@! is given by the dimensions of the operators as 2dll = d(A) + d(B) — [a(F) 6) — 1 (23) where d(A), the dimension of the operator A, is defined by the transformation law under a scale change x > dx, namely
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
ail = na FE.) n 29
z between spin and dimension, of the operator [42] The targer d ity near the light cone
_— +1 the more stronger the singular- [a ta n When the operator product Biv) A(x) 1s considered, we have _ a : a Q, œ ~ “ œx On ˆ + ˆ
NOTE s An Om ne aa nange o neq OmMmnD ed neg eve ning 1s
Trang 13[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) which for d!*! + —n, n =0, 1, 2, is S'l¢) ——— + e(xa)Ø(x?)(x?)” alel_,_, n! (28) A useful formula is the Fourier transform of S,(x), namely, [d°x e'#%6 (x) = i(4m)?4-25;_¿(4) (29) For the time ordered product [7] the singularity is SISlx) = (~x? + ie)~##lr(+alsÐ, (30)
The simplest example of light cone expansions is provided within the framework of free field theory, and so far the only example with canonical singularities Thus taking J pix) =i: g*a uƯ °› where ở 1s a free scalar field, we have
LJ, (x), Jy) = [AG —y) 1823 Ay) ] — [AC —y) 182782) 6°06): +: 9° )GOD (31) where Ax) =~ 55 alate 7 *e(p9)6(p" — M*) ~ ~> 2) near x? = 0 (32a) and 1 l A,(x)= _!*ð(p?— M?)~ ——P—~ (32b) 21“ X The leadi ber singularities in eq (31 ‘onal 2x,x, 6'"'(x?) + 3g, 6'(x’) (33)
Note that the leading term, namely the first one, is not separately conserved The sum of the leading 6’’’ and next to leading 6” are conserved As for the operator term, the leading singularity is ax,x,6''(x’) with the scalar bilocal [:4*(x)6(y): +: 6"(v)¢():] The divergence for the leading singularity has no 8’’’(x*) term, and the 6'’(x”) term is cancelled by a corresponding term of a
next to leading singularity Terms of the order of 5'(x”) in the divergence are cancelled only after
Trang 14For 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
u=ftx, U=f—X
Since both the axial current in and the vector current jy are conserved, and since Th = €uuŸ” , it turns out that j, =/»9 +7, depends on u only and j_ =j) — /, depends on v only Since 7„ has no
divergence and no curl, we may be tempted to write j,, = 4,6, where ¢ is a free massless scalar field However, in one space dimension a massless scalar field does not exist, since the Fourier transform oft the propagator 1/p? does not exist One can introduce regularization procedures
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 1 âđfnrmm 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
Trang 15Y Frishman, Light cone and short distances
[/,(), 7,3] = 21c ơ (u — 1)
[_(0),7 (0)] =2ie ơ (0 ~ 0) (35)
[7,(z),/_(u)] = 0
c is a number, which serves to normalize the current Also [i,(u), w(w'v')] = —(a + @ys)Wu'v')b(u — u’)
[j_(0), j(/0')] = —(4 — y;)J( 0 )ỗ(0 — 0)
Equations (35) and (36) result from equal-time commutators and conservation of j,, However, a and a cannot be equal to their canonical value 1 unless g = 0 [45] The energy-momentum tensor can be written in terms of currents [46, 6]
(36)
l
yn = 52 (2 lye? Bun * fal] (37)
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
-exp|—L [(a +) ƒ 7.(#) đỹ + (a — 8) Í 7n) ani\ tụ (38)
L2c ⁄ ỳ U )
For ;Ú; replace a > —a4
Trang 16i[D, ÿŒ@)] =[x*ơ„ + $ + 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*?*/()ơ(w — +) + 21 c,82°5'(u — u') [/“(0), /'(0')] = 2/“?°//(u)ơ(0 — 0) + 2i c¡ơ"P8'(u — 0) (44) [//), /4u)] = 0 and also [„(x),/70)] = 0, u,U=0, 1 (45) T Buy ~ 2(n/2m+ei) L2:/5: —#8uy„://`:] _ L2:7„7,: —#uy:/4ƒ`:Ì (46)
where the coefficients are determined by the Lorentz group commutation laws As for the spinors,
——————— the groupstructure dictatesnoW_———————————————————————————————————————————=———
Trang 17When 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, d[w]=4 + 22c/4n for ø¡=0 (48) d[J]=+z tín — lịn+g?cl4m for gì= 4ml@+ l) and for the composite operators of scalar and pseudoscalar densities, l/a 1+ (—— 1) for g,=0 n'.a a yw) = đ(Ú+;) = (49) 1+2 n— | 1\/a 47 + ¬ i for g,= n H/\a n+]
Note that for g, = 42/(m + 1) the commutation law of the axial SU() charge density with the spinor fields has the opposite sign as compared with g, = 0
We should mention that for the case of the Thirring model the S-matrix is unity and the on shell structure trivial Such is the case also for the generalized model to include U(n) symmetry when we are on the eigenvalue for the coupling constant However, as we saw before, the struc- ture for the scalar and pseudoscalar densities is non-trivial and interesting 3.3 Products of electromagnetic currents In the case of electromagnetic currents, we write the light cone expansion as [5] [F.(<), J,0)1 = [8a — g, aga ]1C(x — ») FL, y)I + [8 ,.9O? 3 + 2,69998) — 8,58,,9 099 — 2, BaP Cate — vIF P(x, y)] +
The other terms do not contribute to forward spin averaged matrix elements Forward matrix elements are analyzed as in the previous section, with
[(œI\F?®%(x, 0)lp)] „2x ọ = p#p8ƒ;(p-x)+ (S1)
The other contributions are less leading in the scaling limit For the case of a canonical singularity a(x) © €(X%o)8(x*), the term p*p"f,(p- x) yields scaling for vpW, Note that when the longitudina cross section has no scaling contribution, namely C, has no 6(x’) and a leading 0(x’) singularity
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$°(x, 0), since the matrix
element involves x* terms one has a part like a next to leading term in V, (this is clear when one uses x*elax = —ja' (e'%*) in the Fourier transform and integrates by parts), A x%x® term in eq
(51) has an extra f ) v suppression as compared with (p#xể + p#x*) (we exclude gø® terms, since
Trang 183.4 Generalized Cornwall-Norton sum rules
The Cornwall-Norton sum rules [50] express integrals over moments of the scaling functions in terms of commutators of corresponding numbers of time derivatives of a space component of the current with a space component, at infinite momentum They are, _—M(_ z2322n & 2,.,! 12 lim <2- <4? ƒ _ im J q2>_—s= 271 M? œ2832 lp > = ni expQ4 - x){(I[92"*17 (x), J,(0)] Ip) 1 — (pl[ð2”*17 (x),J,(0)]Ip)}, (52a) 2n +2 Z M;(q?o›')du3'? lim (—)"*! f we 2 = lim f= x;9xpQ4 - xXpl[ð2"*17 (x), J (0)] Ip) q?> 27i œ2" 4 Ip | + 2% lp Ị281 2 1 (S2b) where we choose p in the z direction and g in the y direction They are obtained by using the
Bjorken-Johnson-Low [2] limit of expanding in powers of 1/qo, identifying (p!| [a2"* T(x), J,(0)] |p)
as the coefficient of 2i(—)"*'/q?"*? in the expansion of T;,(q, p) = 2ifd*x exp(iq-x Kp ITJ,(x)J,(0) |p)
(only even powers appear because of crossing symmetry), and then taking py > © to isolate the
highest spin component, (2” + 2) Starting from an unsubtracted dispersion relation for 7}, W2(w"', g?)dw'? T —————_—_—_—— 53 (oa m2 CIOS —1€E (63) (54) n=0 đ j G2 T n= 0 (—@q’),
If F,(w, g*) scales for g* > —~, then the nth term behaves like p$/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
A + a
3 `, a a ~ vid kỉ A c Tid CC c© 7 c
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
Trang 19
VSF(X, OV ~ DE CO) ays Kevyn BeBe 8 2(0)5, Ce) = [( x? Hex 9) 8" (x? -i€x 9) W(4d,,) 0
(55)
n=
The terms we omitted have less singular C,, Taking the-singularity structure of the time ordered product, and going to the limit of g,, > © for all components, with g? > — (namely qụ =À4,
with À > s and gq any fixed space- ‘like vector), and then taking po > », we obtain
Tạ> Ð; A,(p- q)?2( Ca?) 8n 2n~1, (56)
n=0
The first limit of \ > implies that for each n it is the largest d, that is leading, and the second limit of pạ > © justifies keeping only the spin (d,, + 2) part for the Ƒs88 %2n(0) We could have taken simultaneously À > œ and øạ > œ such that øạ/Ä > 0, to obtain the same result
Comparing eq (56) with eq (54), we get eels , w )dw?
af a TƯ mi (56a)
Scaling means that d,, = 0 for all n In that case the infinite sum of local operators in eq (55) de- fines a bilocal operator
Studies of infinite sums of ladder graphs in perturbation theory show that in general the right- hand side of eq (36 1S nof a power of q? [5 1] Thus the notion of dimensionality of operators a, = gs (57) ? 16m? (2n + 2)(2n +3) no King at the [unction (—g?)~zo W, instead of M;, this example serves to show that Bjorken and dy = 0 (53), * ® * ry ats ~ WG ệ O ale c c1 a nDỌtO ở wa 4 ee we Đa a ve C 1O wD eC ca ne Cc HE cả ep d aTe ° c7 OfLd a very &%
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
Trang 20have 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] 3.5 Fixed poles It was demonstrated in various works that a J = 0 fixed pole exists in the amplitude T2, either ° 66 +9 fi q > —œ and WwW > © we have a piece proportional to ql in nữ» Similarly, a J =0 fixed singularity
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
Trang 21Thus, for > œ, T(q?, Đ)——> 4T? p> Ẳœ 2 - đÀ „« <P ƒ x80) (60) —O
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
I [đ]
rial ~ (gr) f _ÿ (T4? — 2XMb + 2 + ie)? 8 210) (61)
where d > 0O and where we also introduce an ““effective”” uw”, which represents less leading contri- butions In the scaling limit
1 drg!l4l(n)
.—-ằ vo 1
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 > ~, THA) > (—q? v4 {vl(—q? + w?) "4 Vụ Thus considering (v?/(— q?))T> = R,, we see that the sum of contributions of the leading light cone singularity and a representative of the non-leading singularities is, = ad 2 đ† 6C) œŒ) =1 M ©
where we now look in a non-forward direction and allow for momentum transfer t dependence (the first term can also be rewritten as
~ dN wy
‘f enue
pearance of a fixed pole in i @))T at J = 0 with a residue that is independent of g* However,
en & is ve TO, a separate its contri e
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
Trang 22residue 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
and thus light cone dominates for v > © and fixed q’
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
t “ ° + + ° * + *
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 E1 d » đ av 7 a = › Œ¿ —= #8(1+—— 1+)” apf [ ptp’ _| { p—p' _| [(IƑ7#QGx, — ;x)lpÌ 2 =o =P pry dy data (Ar Ra Dex | id 2— x fexp ida = | 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-
Trang 23Finally, 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) ¬ Defining ^“[đl/»» [oles y `¬^ oy !* tít a) 8, (AD = 8A) — Ly Cy TAT 7 œ>(-đ) we get that the function ø1#!(A)/IAI#*? is integrable both at A = 0 and at À = + (remember that
Trang 244 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,
- l
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 74# — - (+32 (1 4 m)! Yay) | (62) Tu WAVY ET AVY ST QR WAT wee? we get + bt ~ : cab + + + [J2*(x), J3*0)] Af 8 LSE, V8 t SEO, W8ra — SupSa (% y) (x-—y)*® 0 : Fie *“*uUpe AStP (yy) Ys JF FS ba" Dix — vy) + đ2P°{S«< Ary /J 7 Cc A} J (69) ad F where 1 D(zÌÀ~= — c(z \Af{ 72) 27 and
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
Trang 25
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 gluon interactions [42] Note that 1 ina theory of quarks interacting with neutral gluons, the force between two quarks duce a non- “Abelian group structure for the gluons and quarks such that the known hadrons and a ¬e c 99:
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 46 9? > namely only logarithmic deviations from scaling [14, 15] 4.2 Results for deep inelastic scattering: structure functions, relations and sum rules
The analysis of deep inelastic processes proceeds as in our discussion in section 2 In electron nucleon scattering, only the đ“?° coupling appears, the spin averaged matrix elements involve
only the anti-symmetric vector bilocals, and the spin dependent ones involve the symmetric axial
Trang 26where 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 scalar contribution For neutrino—nucleon scattering [69, 70], the matrix elements that enter are (v) =_L 4 igx WwW W+ bests 2 u ) đụ 1v | pq pq y -(-2,, TT \werg?, v) +—-(p, —— „2 a) (v, — x2 a.) WS (q?, ») \ qo iN q 7X ki , i GÀ — Sa Suuag "4W? (4), v) (73) 1 1 1 + we 449 ,WY(q?, v) + aM? (4„0„ + 4,p„)WŸ (42, v) + ap (4u„P„ — 4,P„)W‡ 42, v) and W€(q, p) =—W)(—a, p),
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,
Trang 27in the scaling limit The positivity conditions here are
y2 /p2 —_ „2
. g M | Mại (75)
We have, for the cross section,
d2g¥” =— G2 1 Fer = , 2 + — y? 2 Fur = 2 = ( y\2x ) Fur = , 2 | 7
ay „ S|d =1 2 (G2, đ”) 2 L2XF (3,4)]*yÄ1 5m (G3, Gg’) (76)
where x = 1/w, y = v/E and S = 2ME (terms of order M/E where ignored)
ES ai — Fe*(w, gq?) +3 fe orion i f Sgro a] 09
Thus if the integrals in this expression are g* independent (corresponding to dy = O in our discus- sion regarding eq (56)), the total neutrino and anti-neutrino cross sections will grow linearly with energy, 0ø = œÈE, (78) Experiments, which measure ø”P? + ø”" and ơ?? +ø”", vield [9] a” = 0.69 + 0.14, a? = 0.27 + 0.05 (79) for energies up to about 10 GeV Taking @, = 0, we have from isospin symmetry Fye= FP FPP = Frvn, i= 1, 2,3 (80) Thus F? + 7" = F?? + F?" = 2F Experimentally [9], g”N/g7N = 0.38 + 0.02 (81) do Tủ y f d9 2w (d9 2N Jun MỸ? mm “oJ 2 wM 3 ` Now, since F,(w, qg?)/M > 2F,(w, qg*)/w, it follows from the equality of the above integrals that 1 , 2 › positivity condition M F,(w, q’) > | F'3(w, qg’)| and the equality of the above integrals imply
F3(@, 7°) = —M F\(@, q’) We thus see that the data implies a negative F; and almost maximal in ab- solute value This means that anti-quarks almost do not contribute to the integrals appearing in
the expression for the total cross-section, which means that they do not play a role for low
Trang 28wF>(w) = 2M F,(w) (82) for all processes Setting @, = 0, we get [71] 6u( Fy ?(w) — Fy"(w)) = (FZP(w) — F3°(w)) (83) One obviously also has the Adler sum rule [72] in the scaling limit, r dw ƒ —~ tF??(6) — F?M(@)] = —2 (84) 1 One also gets the Gross-Llewellyn Smith sum rule [73] eo dw f — 2 LF 3"(co) + F3%(c0)] = —6 (85) 1
The first follows from the equal time commutation relations between time components while the second follows from the d coupling part of the commutator between space components of vector and axial-vector currents Note that the Adler sum rule can be derived by the p > ~ method, and holds for any g? < 0 fixed in the form [72, 74]
f avlw2P(v, 2) — W2%(y, q?)) = 2 (86)
0
Equation (84) is the g? > —= limit of it The Gross-Llewellyn Smith sum rule cannot be derived by the p > © method since it involves a commutator between space components where Z-diagram
contributions are important [74] It holds only in the g? ~ — limit However in some cases,
one can include z-diagram contributions in fixed g* sum rules using null-plane commutators [75] This is not the case for the sum rule eq (85) since in null plane commutators one deals with
Sdq-(W,(9 Pl g+= o> where q* = q° q°, and therefore the only combination that can come out is oo 2 oo { dv qi £ J —(OW3) ~~ J dw F3(wq’), 0 Pp Pp 1 where q, is the component of 4 in the x,y plane By considering the commutator T lý; (X),‹ J, (Ơi |x+= integrating dư (since one takes q* = 0) over the matrix element of the respective scalar bilocal
(multiplying €;,) one gets an extra I/p* times a number f[d(a- x)] F(p- x), with F(p- x) the matrix element of the scalar bilocal Thus all one gets is that f° dw F3(wq’) is q? independent This is not useful, since this integral is expected to diverge in the high w limit (F3(w, 9°) ~ w* forv7> ©
with fixed q?, and the leading trajectory, the omega, has a ~ 3) Also, when considering f dq {W,, A, P)Ì + g*=o» One integrates only over space like g? However, if one first starts with
ƒdq [W,„(4, p)] „+-„, one also enfers time-like g? for large enough 4", and one has‘to argue that these vanish as e > 0 [75]
Trang 29
We thus see that light cone expansions put all current components on the same footing as far as sum rules at g? > —- are concerned Bjorken’s sum rule for W, [76] coincides here with the Adler sum rule, since the longitudinal cross section vanishes
So far the scaling phenomena and all other relations following from the algebra eq (69) are consistent with experiments [8, 9] For spin dependent amplitudes and sum rules see ref [77]
One may obtain information about W, and W, by taking the divergence of the axial currents to be like in a model of quarks interacting with neutral gluons, namely
Dt = HIN =i WYs (2M, m}Ủ: (87)
where m is the quark mass matrix [67] When considering the light cone structure of the commu-
tator of DĐ“ with a weak current, one obtains
(pI[D%(x), F-0)] x2 = 0 Ip) ~ S*°(p- x)d,€(x 9)8(x”) (88)
for the contribution of the leading light cone singularity However, a non-leading light cone sin- gularity like e(x9)65(x?) multiplying a vector bilocal is as important in its contribution to W, as the leading one [67] This is so since the contribution of the leading singularity to the coefficient of p, in the Fourier transforms of the above matrix element is (by integrating by parts) the same
like we had —[8,,S*?(p- x)] €(x0)5(x?) = —p, Ss? (p- x)e(X9)5(x?), which is like the contribution of
a vector bilocal multiplying a 6(x?) singularity In fact, if we take into account the mass term contribution of {W(x), (0) } in calculating the above commutator of a divergence and a current,
the resulting W; is identically zero as a result of an exact cancellation between the leading and
next to leading singularities What makes W, non-vanishing are extra terms involving the curl of the gluon field and map ying the © non-leading singularity | However, the ve eadng TH TỦ in leading light cone > behaviour, (pl[D“(x), D°(0)] lp) ~ A°Œœ': x)ð“e(xu)ồ(x?) (89) where Az°(- x) is an antisymmetric vector bilocal This now determines the scaling function Note that another s sum rule, derived within the parton model with extra specific assumptions regarding the 66 +3 f [78] [—TFP(@)— FR@=3 edu : (90) 1
cannot be derived here It is not related to any local commutator; rather, the left-hand side is proportional to f° {dp x)/(p x) }f4 (px), namely an integration over a line on the light cone
running to infinity [79] In the parton model, one has
4 f 80 | (63) — F?P(o›)]T— 3 | de | mep, )— F2(w)] = 2 | dx Lf ( }— f( )]
Ww (o>)
i 1 0
Trang 30where ƒ;(x), fs (x) are the momentum distribution functions for n and p partons, respectively
The sum rule eq (89) is obtained from K dx[ƒg(x) — ƒs(x)] = = 0 and the Adler sum rule One can argue that ƒ; ~ iE for small x (x < 1/100 ?) from exchange degeneracy, and that f ~ 0, i, ~ Oforx > 0.1 from neutrino scattering data The latter, however, has very large errors (see ref [9]) Recent experimental evaluation [30], using latest data [29], gives 20 dw ƒ —= [FƑ£P(œ) — F£"(œ)] = 0.18 + 0.04 (91) 1 and using a Regge extrapolation from w = 20 to œ, oo dw f ——?(@) ~ F‡"(@)] = 0.27 + Œ) (92) 1
where [Ƒ2P(œ›)T— F7"(w)],, = 5) = 0.045 was taken When considering the combination
3) Í d°x(pl[a,Jƒ(œ), 7#(0)]|p),
one gets that the leading light cone singularity contributes 2 a term which is the part of the kinetic energy carried by the quark fields, namely «(p! ÿ(,Š); + tổ; )Wip) Since only the kinetic energy part of @,,, contributes to the p,,p,, part of the matrix element, we get a sum rule [22],
6 f LFS (co) + FEISS = f LEI) + FMW =3 (1 — e) i G2 i G2 (93)
re € is the fraction of rgy_ carried constituen at do not couple to the currents, like
neutral gluons Recent experiments indicate that e = 0.46 + 0.21 [9], namely about half of the
energy is carried by neutrals
All the results of the light cone algebra depend on the hypothesis that the current constituents
C c Vit cá “uid CS = vid Go oo CU Cid 5 ki 3 Bo AGED,
he latter appearing in n quark model spectroscopy considerations, is a subject of recent activity
Of Q1)
OU, OL]
Other implications result from the internal group structure and positivity, and are in the form
O ir equd itie WÏ NM Noid OT da G) ese WeTeE [ di OVE ed wi [ ie Dd O Ode S
and then shown to hold from general light cone considerations [83, 84] We mention here the
Trang 311.00 - | 0.90F i 0.80 F 070+ Pu oof oly | | | en ep Fo /Fo 0.40 - —>— 030P 9.20 9.IoF 0.00 O10 0-20-03 0-04 0-050 -6-66-8:78-6-88-8:9E—-+00 Fig 3 F§"/FS§P versus X' = 1/w’
The latter being severe for those w where the ratio in eq (94) is close to the lower limit Experi- mentally, the ratio Y = F$"(w)/F5°(w) approaches ~ | at large w, and is essentially decreasing
with decreasing w to about ~0.4 at w ~ 1.2 [29] The data points, in fig 3, have all (—qg?) 2 1 GeV? « ve-peer AAs V-đd1SCUSSIOTIS ©eØardĩne a LJ na A đđl¬a+e is very slow (w ~ L002) or that F7"/FTP is large It is argued that [85] either the convergenc for V,1 T IVI T TEge T to; [86] Thus, for $ > Y >3, Z=F/"/F;??">$ (l — Y)/(Y — 3) (96)
Forš > Y > 3 we have Z> 5 only (for4> Y> 2 we have Z > (Y — 1)/(1T—z Y)) It may be pos-
i | ~ 30 — 40 itt 2 ratio Eˆyn/Ƒ'vP of 3 — 4 for ¿›< 9 [86] See also [87] 4.3 Further implications — Non-forward matrix elements Since the singularity structure near the light cone is a c-number, the scaling laws will be the ˆ
U q L1 DO C C w TC JIIUOCdl1S O c iICqaC L RUTd VY lIIdVC 21! "Vda
elements In particular, varying the momenta of the states in the matrix elements of bilocal Operators constitute a severe test of the idea of c-number singularities Such matrix elements oc- cur in amplitudes with two currents, and to get to the light cone we need both ““masses”” of the
two currents to become large, in either space like or time like directions
One can consider e*e annihilation into a u*y pair and a given hadronic state [88] We are in- terested in the part of the amplitude which is the diagram of fig 4
Trang 32Sw bw QE
Fig 4 e*e => w°w` X(p) [C = +1] Fig 5 ete > uty X(p) [(C = —1]
matrix element is
A„„ = [d*x el9X(XIT*[7,( x)J„(—š x)] I0) (97)
One can consider here the BJL limit of Qy > » with Q fixed, which is in the physical region The scaling limit here is v= Q- P> © with w = 2Q: P/Q? fixed (w < 1) In this limit one can use the light cone expansion for the time ordered product in eq (97) Moreover, by squaring the matrix element and summing over X, and then letting M,, > © (first v > © with P and w fixed), one can check the assumption that the bilocals obey a closed algebra when all distances are light like [22, 42] If correct, one obtains an explicit expression for the cross section as a function of w in the above limit One has to separate the contribution of the diagrams where the hadrons are in C = — Ì
states, fig 5 These can be calculated in terms of e*e > all For more details see ref [88]
Other processes which involve two high off shell currents are inclusive electroproduction of pop pairs [89] and ete” > e*e X [90] In the former one can relate the inclusive cross section, again assuming the algebra of bilocals, to total electroproduction In the case of e*e” > e*e X one has the advantage that in certain regions of variables the amplitude is dominated by the diagram of fig 6 Here both exchanged photons are space like, which simplifies the analysis of connection
with experiments [90] (In the case ee > ee X these are, of course, the only contributions.) In
Trang 33sistent with current conservation to leading order, namely when applying 04“) we do not get 5''(x?) terms (because D(x) = 0) However, we do get terms with 5'(x”) singularities in general These should be cancelled by the corresponding contributions of the next to leading singularities near the light cone, which in the current commutators involve 6(x*) singularities, and therefore 6'(x?) terms when a divergence is taken
Note that if we try and write the general terms contributing to W?? and W2? in an explicitly conserved way like for the case of the electromagnetic currents eq (50), we get that this gives
ƒ đx(pl[7Z(x), 72(0)] Ip› = 0 for local functions Vÿ” and V7? of eq (9), a result known long ago
[91] Thus insisting on current algebra and local invariant functions, we cannot have an exactly current conserving expression for the leading singularity, and next to leading terms have to com-
pensate
We can write the next term in eq (69) as Cae (x, y)D(x — y) Also, we do not want (pIŒ (+, y)lp)
to have a p,p, term, since such a term contributes to the leading scaling behaviour Let us first demonstrate our results for the f?°° part of the commutator [92] Introducing Z = x + y and A=x — y, we get from the conservation conditions
(8° S JA, + (OES, — 82S q + i€yyoq dt AB )A* + (Cy, — Ci )A% = A’e, (98a) (AA SA, + (ABS, — AOSy + 1€, 543" AR JAX + (Cy, + C,)A* = A*h, (98b) where g,, and h,, are new bilocals, which do not contribute to the next to leading light cone sin-
gularity Since [S.(x, y)], , = 2/,(x) which is conserved, it follows that
a°2 S$ (x, vy) = AS (x, y) (99) Therefore, (a°25,)A, = A°A,S, = AXA,S, — A, 5,) + AS, and eq (98a) implies Cay — Cyq = AgS, — AYSy — ZS, — 82S q + i€ jy pq3"8A2) + (Cay — Cy) (100a) where A*(Ở„„T— Ở „) = O(A?) (100b)
As for solving (98b), we observe that
O28, — 905 t+ 1€y 559" A8 = Fray, d”* (101)
since each term on the left-hand side vanishes for A > 0 Thus, observing that Fry ,Ja, 1S anti-
symmetric in the first two indices,
Trang 34Note that only in the free field case, neglecting masses, we can have C,,, = 0 This is so since in
any interacting theory S, # 0, since S,, = 0 means an infinite number of local conserved quanti-
ties through 2*Z,$ (x, y) = 0 [93] The same is true for the left-hand side of eq (101), which vanishes only for free fields
When considering the d?"° part of the commutator, one gets again that a non-leading light cone
singularity can be added to yield current conservation to next to leading order, but now when a
local constraint is satisfied by the bilocals
11(94Ag) 4-9 — FA ga =o] = i€apupd4(S3)4 = o- (103)
Or, in local form
WO) Ye Dg — YgDa IW) : = 1€ ag 08 Ls DO M57? Wx) :] (104)
where
DyW= (@, +igV,)V, Dy =D, —Dy (105)
One may also introduce a mass term (M,) and couplings to scalar (a) or pseudoscalar (7) SU(3)
singlet fields, thus yielding non-conserved axial currents One may then inquire about the next to leading singularity such that the expansion near the light cone be consistent with the divergence
equations
187 “(x) = 2: Ủ(x)}:š X“[Ma + gơ(x) + iy¿Gm(x)lử(x) : (106)
One then gets, that a non-leading light cone singularity as above may be found, but from the đ“?° coupling one now gets a new local condition,
1((aAsaaag — (OFAsadg aol=: VO) sy, AD, W : — : VOD ENAD, WY :
= 1€ pu pd97(S)a = ti: WOM Yes Vg] 3 A(My + B0(x) + IGygm(x)) W(x): (107)
For details see ref [92] 2 a(s) = ama p(s) 3s (108) where gq is the total momentum and s = q’, and the function p(s) is related to the commutator of Ss electromagnetic_currents by 1 ub xúp ~ 3CTƑF uty App 0
Here we are dealing with a vacuum expectation value of a commutator, and therefore the short
distance and light-cone structure coincide The asymptotic behaviour of o(s) is therefore given by
Trang 35
the short distance structure of the left-hand side of eq (109) Using free-field singularities near the light cone, we obtain that o(s) « 1/s fors > - [94, 95] However, the coefficient cannot be
determined unless we also assume that the unrenormalized fields from which the current is con- structed satisfy canonical commutation relations In such a case, we obtain
Oeste > a (s)
p(s) = -|2 Q?+2 Dai) (110)
Ceres uty (5) $“5
However, we may avoid discussing Fermi field propagator and obtain the coefficient of total annihilation by relating it to 7° > 2y decay and the connected part of a space—space commutator
(see below)
Note that when one calculates the short distance structure for free fields using j,, =: Vy, one obtains for the vacuum expectation value near x? = 0,
(0117), 7„(0)]109~ {Trly, yy") }{[8,4 (x) ] [8,4°-e)] — [8,4°()] [8,A°) 1}
~ Fy SONG, 2? — 2X XY) q11
and thus one obtains for the time—space commutators, for xạ> 0,
(0Ia(),7g(0)10) ——» —z— 88 9œ) +7 9, ABH) xo>0 Ố7?lXg | TT (112)
Note that the infinite > Schwinger | term 1 [96] is here obtained without any point: splitting 1 in de-
equal -time commutators Mass corrections introduce terms œ ma ð9(x) 7 fiction for tt io in eq-(110) d 7 , , FE
o, ~ Oin deep inelastic electron scattering, we assume no spin-zero constituents For the -Mann-Zweig raCtional charge quarks, One obtains p(s) > R = ; When an extra quantum number is introduced [81, 97], “‘color’’, then R = 2 ‘Color’ quarks (“‘red’’, “‘white”’,
and “blue’) obey ordinary Fermi-Dirac statistics All physical states are postulated to be singlets
under the “color” group Thus mesons are constructed as 6,,q/q/, and baryons as €,,, giqi.q*,
where abc are “‘color” indices and ijk usual SU(3) ones
The latter scheme is also in agreement with the observed 7° > 2y decay rate, as given by the Adler-Bell-Jackiw anomaly [97], while the GMZ quarks give a value smaller by a factor of 9 For a discussion of the various quark schemes, see ref [97] Note that the Hahn-Nambu quarks [98] of integral charge yield the same value for the 7 719 > oY decay as “color” quarks while they pre-
J, = 22, + id wi (113)
where Q! = (4, —4, —+) as in GMZ and Q, = 4,4, —2) The first term, J(), is a singlet under
Trang 369r 8r 7F et ° ACO A NOVOSIBIRSK , —CEA ST FRASCATI GROUPS R °o YY 4L © BOSON HT 3r 2l -|- ì -|Ì0-|-Ì- _ _ ch 90 ị QUARK MODE LS Ir + CỐ TT QRDINARY i t L 1 1 L1 1 1 te — | 2 3 4 5678910 I6 25 s(6ev)Ê Tiere” — HADRONS) R= Flere —— ut”) Fig 7 R = ơ(e*e" > HADRONS)/o(e*e' > uty) then when considering matrix elements between usual hadrons, the relations are as for GMZ 1 en/“e 2?
which j is excluded by present experiments [8] °
Recent experiments at CEA yield R = 4.7 = I.1 at g* = 16 GeV? [101], and & = 5.8 + 2.0 at q? = 25 GeV? [102] Looking at the experimental results for R, [102], fig 7, we see that the two last experimental points indicate the scaling limit has not been reached yet If we are approaching a situation as predicted by the Hahn-Nambu model with R = 4, we must be also producing “charm’”’
states (Note that the Hahn-Nambu model is not suitable as an “‘asymptotically free’? gauge theory [14], since the photon is not a singlet under the gauge group.) Another possibility is that we are seeing an intrinsic breaking of scale invariance [103]
Crewther [104] showed that a relation exists between total annihilation, 7° decay, and a
Trang 37Also, only the isovector contribution of total annihilation enters Denoting the coefficient of the isovector part of A, by K (which can be measured by the difference between proton and neutron for polarized electron on polarized target scattering), and R, the isovector contribution to total annihilation, the relation is
S=KR,, (114)
5 = — 1 2 euvas f fas a4 12 x YXuVy (OIT*J,(x)J,(0)817 œ\X 8 5 (v) 10) 1 (115)
and, by use of PCAC, S is related to the 7° > 2y amplitude [106, 107] S is the coefficient of the anomaly in the PCAC equation as determined from the triangle diagram in renormalizable field theories
For GMZ quarks, S = 3, Ri = + , and K = For the “color” quarks, S and R, are multiplied by 3 From experiment, S ~ $4, [108]
Crewther [104] derives this relation by a consistency consideration, first using a short-distance
expansion in x > 0 and then in y > 0 in eq (75), and the free field form for the three-point func-
tion at short distances [109] This relation is of great importance, since it connects 7° decay to
other processes so that we can get the decay amplitude without any need for renormalized per-
turbation theory methods (In this case, however, one has to take the coefficient of 1/S in the total
e*e” annihilation into hadrons from considerations of a canonical quark field propagator.) This is relevant since out light-cone expansions do not hold in the latter approaches In fact, the form for the three-point function for all points near one light ray was demonstrated by Bardeen,
Fritzsch and Gell-Mann [97] to follow from consistency considerations in comparing the different
An interesting problem j is that of constraints imposed 0 on operator product expansions from the
1eld K `7 a ‘Yr a k rìe a eA ĐO 1 h O a 3 r^ d rr v5 C7 Yyrr a ry k7 a ry d aFataa > a Lì
out it that one Bets constraints on Wilson’ s short-distance expansion, namely, that line c integrals of with bilocal operators are not implied 6 Single particle inclusive e‘e’ annihilation
Light-cone expansions were generalized to include products of more than two operators [111, 112] to discuss single-particle inclusive experiments in ete” annihilation and eN scattering Re- cently, it was pointed out that certain regularity assumptions of the terms multiplying the light-
A careful examination of the singularity structure reveals that ¢ one can get a consistent formula-
iOn k2 A} Fy h vield a DOtRh 1Ø a ry ki ave Rd ahaa aa U r^ ⁄ oO AF) } ms eading au igh Ana L7
[24] The ¢ logarithmic multiplicity is obtained by a certain singularity structure at short distance
° U LY k C = =CO s bUuld Y QGOỌ€C FO q C Ie q Ee CC đUaA€C It LCid“
Trang 38that in ref [113]) It should be noted that our discussion, carried out here for the case of
Bjorken scaling, is essentially the same for any scaling law Namely, similar considerations to ours apply if vy W, (q*v) (see eq (116)) scales in the Bjorken limit rather than W,
We consider ete” > H + X, where the four-momentum p of H is observed, and p? = M? Define
[115]
W,(4, P) = = fatx explig: x) 2) (OW, (x)IH@)XXH@)X (0) 10)
X
= WiŒ, 4?) (-„„ + Wy 1 Mu Mv
nh wp Wale (0, a ano — a.) (116)
where we also sum over the spin of particle H and Mv = gp Assuming a one-photon exchange amplitude, we get
d?ơ Tro?
d£d(Gos8) ` m= [ốØ+(1 + cos20) + ø; (1 — cos26)] (117) where £ = q?/2Mp, 6 is the scattering angle of H in the e*e center-of-mass frame, and
ðy = Mù, ð, =M, +(/2MẸ — 1)M; (118)
where in eq (117) terms of order m?2/q? or Mé/q? were ignored (Note that here M, need not be
positive It is, in fact, negative for g, = 0.) do 47a? - Ama? ee dé 342‡? 0 (E, g+ FE, QDIE— MEG) 34'£” (119) We have q/2M da fo de HA) (120a) 1 pee ae do ——=g (q?) (120B) ì 2E dé tot g
Trang 39A is related to the rate of decrease of 0,,,(q7) In coordinate space
Wild, P) ~ fdtx d*y d4zexp{iq(x—y)}e!??(01[7%2,Gœn@0))] [T*(J,0 J y(z))] 10) (122)
where J, is the source of H (suppressing spin indices) and T* denotes a covariant anti-time- ordered product (operators with earlier times are to the left) Defining
F,,(u?, p- u) = f d*y d’zel?*(Ol[T*J, (u + Yi, (OM IT*I, Wy (2)] 100 (123)
we have
W,(q p) ~ [dtu elf, (u?, p- u) (124)
and standard arguments imply light-cone dominance |u?| < 1/q? for gq? > ~ and fixed & These arguments hold also for — > ~» as long as ME/q > O Thus we may get light-cone dominance terms also for large £, where one may get an increase of # with q? For f(é, qg?) > F() as q* > &, with F{‡) ~ & for large &, we get a logarithmic multiplicity Since from g/2u < & (with u > M) the contribution to fi is finite, most of the contribution comes from M€/q —> 0, in which case light- cone dominance applies We should remember, however, that a logarithmic increase in 7 may come from a non-scaling term altogether, like for example, 1 RE, q?) ~ ah (E) + ƒz() (125) with P( ~ Ệ? asÿ > and One then gets that only f, contributes to the energy sum rule, eq (120b), and only f, to the logarithmic increase in 7, as g? > ~, [116] We define, in analogy with eq (9), Wav (Bur? * WG V4", 9) + [(4- p(4uPy + Py) —PyPoT — Bud PIV 2G", ») (126) tensor structure as the product of two currents; when the space—time distance between the coor-
dinates of the two currents in the above two cases approaches the light cone A discussion similar
to what follows can be applied to V, # 0 We expect xz V2(q’, Đ)~ [d*u e'4*[Inu°(= u2 + ieua)]ƒP- u) (127) +
where Ci wie ome C m D2ramete Vt hẹ & noe Ata ỳ a LAN
Trang 40This is dictated by the fact that in eq (123), / u(x) is always to the left of /,(v), and for scaling the singularity in V, in ? is of zero order Writing
ƒ-u)= fda &(a) exp(—iap- u) (129)
we obtain
V,(q?, v) ~ fdog(a) {atu exp{i(g — ap) }In w?(—u? + ieug)
« fdag(a)O(qo — apo) (8((q — ap)? — 43) — ơ((4 — ập)?)]
~ Í da§(a)8(» — aM)8'((q — ap)?) (130)
Note that 6(k,)8'(k?) has no Fourier ‘transform due to an infrared divergence, as is obvious from eq (128) However, for calculating V, for g? > ~, the last two expressions in eq (130) are equiv- alent We obtain
V (q?, v) ~ v?B'(&) (131)
The other root a, ~ 2v/M of (q — ap)? = 0 does not contribute due to the @(v — aM) factor The spectral conditions also imply 2(£) = 0 for — < 1 To get logarithmic increase in multiplicity, we need g'(&) ~ & for large &, which means f(p- u) ~ 1/(p- u)* for small p- u
If one starts with the Fourier transform of a commutator, one gets an e(x9)0(x”) light-cone singularity for V, The procedure in ref [113] is to take over this singularity but modify the bi- local such as to pick up the part relevant to the annihilation process However, as pointed out in ref [114], in such a case the root a, ~ 2v/M also contributes, and in case g(&) grows as Đ > â, we get a violation of scaling Moreover, the spectral conditions are not maintained One can argue that these g(2v/M) terms are cancelled by less leading singularities with more singular p- u behav- iour However, this cannot happen, as can be shown [117] _
If one starts with an €(x)6(x’) singularity, its contribution to V(q?, v) is, V;(q°, ») ~ [dag(a)e(v — aM)5'(q? — 2a + a7M?) d 1 — fy? = 92 Ly a2 “ly l” _ #) (22) (132) If we take 2(a) ~ a, we get a contribution to V, like V(q?, v) ~ (const.) (133)