306 J . P. H S U 9 -

where q = k -p' and E =Po. Because of the large- ness of m,, the "elastic" neutrino cross sections for v,n- pe' and Cap- ne' a r e essentially the same a s the usual resultsS for E , < 50 GeV.

4. Implications of the rules I = k and (AS ( # 2. These rules suggest4 that we may need at least two more neutral intermediate bosons So and So with zero spin in the model. In this case, we may consider

(S', So) and (S', So)

a s two isodoublets. We assume that &(AS = *I) conserves isospin and &(AS = 0) violates isospin by an amount /AT 1 = 3, where C(AS =*I) and &(AS = 0) denote the interactions between the inter- mediate bosons and, respectively, the AS = *l cur- rents and the AS = 0 currents. On the basis of symmetry considerations, we expect that the de- cay rates A - pn- and A - nnO would have the same

anomalous energy dependence. We may remark here that the coupling between the neutral inter- mediate bosons and the neutral leptonic currents may be forbidden by a symmetry principle.'

In conclusion, we know that i t i s almost impos- sible to construct a CPT-violating model that sat- isfies Lorentz invariance and the usual causality. This does not necessarily imply that the sugges- tions of a CPT-violating model not satisfying Lo- rentz invariance should not be taken seriously. It demonstrates the close relationship between CPT invariance and Lorentz invariance. Also, it is clear that new accelerators provide a unique pos- sibility of testing CPT and Lorentz invariances by measuring the lifetime of the particle in flight at very high energies.

The author would like to thank Professor R. Rockmore and Professor E. C. G. Sudarshan for useful comments.

*Work supported in part by NSF Grant and by the U. S. (1960); T. D. Lee and C. S. Wu, Annu. Rev. Nucl. Sci. Atomic Energy Commission. 15, 420 (1965).

?Present address. 4 ~ - ~ . Lee and C. N. Yang, Phys. Rev. 119, 1410 'J. P. Hsu, Phys. Rev. D 5, 981 (1972). (1960). '5. P. Hsu and M. Hongoh, Phys. Rev. D_6, 256 (1972). 5 ~ . P. Hsu, Phys. Rev. D 5, 1161 (1972). 'T. D. Lee and C. N. Yang, Phys. Rev. Lett. 4, 307

PHYSICAL REVIEW D VOLUME 9 , NUMBER 1 1 JANUARY 1974

Measuring anomalous dimensions at high energies*

R. Carlitz and Wu-Ki Tung The Enrico Fermi Institute and the Department of Physics, The University of Chicago, Chicago, Illinois 60637

(Received 6 August 1973)

A search for operators with anomalous dimensions is crucial in resolving the conflict of the light-cone algebra hypothesis with the results of explicit calculations in interacting-field theories. A likely place to find evidence for anomalous dimensions is in the large-w region of high-energy inelastic eN and piV scattering.

A great deal of attention has been focused in recent years on the apparent scaling behavior of structure functions in deep-inelastic eN scatter- ing.' These structure functions specify hadronic matrix elements of products of current operators and hence bear directly on the short-distance structure of hadrons. A general framework fo r investigations of this sor t is provided by Wilson's hypothesis of scale invariance for the short-dis- tance expansion of products of currents.' This hypothesis was motivated by and provenS to hold

in various field-theory models. In this context the bold assumption of scale invariance for the leading light-cone singularities in the operator product i s found to give a direct interpretation of the observed scaling behavior.' Furthermore, i t lends itself to even more specific predictions where light-cone algebras of various naive field- 'theory models a r e i n ~ o r p o r a t e d . ~ In particular, ,the predictions of quark models have met with qualitative success in comparison with existing experimental data. It i s also gratifying that most

9 - MEASURING ANOMALOUS DIMENSIONS A T HIGH E N E R G I E S 307

of these results can be given a rather physical interpretation in terms of the quark-parton model.'

Unfortunately, the simple and elegant ansatz of a light-cone expansion of operator products in terms of bilocal operators with canonical dimen- sions i s not substantiated by studies of interact- ing-field t h e ~ r i e s . ~ Indications a re that the vari- ous tensor operators which contribute to the terms most singular on the light cone acquire "anomalous dimensions" which render scale invariance for the light-cone expansion untenable in these t h e o r i e ~ . ~ A resolution to this serious dilemma can be sought from the experimental side in the measurement of these anomalous dimensions. ~ a c k ~ showed that the anomalous dimension of the Jth -rank tensor can be obtained from the q2 dependence of the J th- moment integral of the structure function over the scaling variable l / w (see below). To perform this test, one needs data on the structure functions over the entire range of w for a number of values of q2. At present, available data permit estimates yielding only weak bounds on the sizes of these anomalous dimensions.'O This i s a reflection of the often-neglected fact that existing data establish scaling for only a limited range of the variable w.

In this note we propose a simple alternate way of measuring anomalous dimensions." In contrast to the above-mentioned technique of taking moments to isolate a single tensor operator, we look a t the accumulated effect of the anomalous dimensions of all the tensor operators a t once. This can be done by examining the q2 dependence of the struc- ture functions in the very large-w region-a task that can be easily achieved in the forthcoming high-energy pN scattering experiments a t the National Accelerator Laboratory (NAL).

For clarity, let us consider the forward Compton amplitude for scalar currents $ on a spin-zero target with 4-momentum p,

where w = -2q.p /q2. The scale-invariant short- distance expansion of the product of two current operators is12

1

J= even

+ nonleading terms , (2)

where for simplicity we write down one (local) tensor of rank J for each J. The common factor (x')-~ with d = dim(3) - 1 is determined by counting the canonical dimensions of the various operators, and the factor (x2)'J i s added to compensate for

the "anomalous dimension" 2A3 of the operator 0 ,. For forward scattering matrix elements between spinless particle states, only totally symmetric and traceless tensors occur for each J, and we have

(PIOP~P~. - . I J~ I P ) = B ~ ( pPl pp2. . .PPJ- trace terms) . (3)

Substituting this in (1) and taking the Fourier transform, one gets

T(w, q2) =(q2)3-2 cJ(q2)-*J J= even

In Mack's approach, one makes use of the fact that T(w, q2) i s an analytic function of w at w = O . Thus after the series (4) is rearranged into

+

T(w, q2) = (q2)d-2 w ~ c ~ ( ~ ~ ) - * J [ ~ + 0(m2/q2)1 J = even

one can pick out the coefficient of wJ by forming the contour integral

This latter quantity can then be converted into a moment integral over W(w, q2) = ImT(w, q2) using the dispersion relation satisfied by T(w, qa), with the result

One observes that Eq. (4) i s nothing but the t - channel partial-wave expansion of the amplitude T(w, q2) at high energies. The summation over the infinite number of partial waves can be performed neatly if the amplitude is Regge-behaved.'' In this case, we have

Here a is presumably the intercept of the leading Regge trajectory a t t=O, 2A, is the analytically continued anomalous dimension 2h, for J = a , and C, is a constant, the "Regge residue" at J = a.

Clearly, if A, = 0 for all J then A, = 0. In that case Eq. (7) i s the familiar Regge asymptotic form of the scaling function. If hJ # 0 for J> 2 (as i s indicated in field-theoretic calculations), our re- sult (7) depends on the possibility of analytically

308 R . C A R L I T Z AND W U - K I T U N G

continuing the summand in Eq. (4) a s a function of J (Ref. 14) and performing a Sommerfield-Watson transformation a s is usually done in Regge theo- r ies. Alternatively, one can argue that given ce r - tain smoothness conditions on the J dependence of the summand, the s e r i e s (4) o r (5) can be summed analytically for small values of w and analytically continued to obtain the asymptotic form of (7). (This i s the Van Hove -Durandls view of the Regge formula.)

In the physical case of deep-inelastic scattering of electrons (or muons) off unpolarized nucleon targets, the relevant amplitudes can be defined a s

and an analysis entirely s imi lar to that just given yields the following behavior for the invariant am- plitudes:

1 + eiaa q2Ul(w, q2) 0 , - (q2)-' a wa,

large w,q2 s i n r a

1 + eiau (9)

(q2)2u2(w, q2)hgz ,c2 P2 (q2) -Aaw a - 2 .

In te rms of the conventional s tructure functions, we have the result

F o r large and fixed w the leading Regge trajec- tory [a = 1 for (ep+ en) and CY = & for (ep - en)] gives r i s e to a (q2)-', dependence of the structure functions-a violation of the scaling behavior ob- served for moderate values of w. Data on these structure functions for large w a r e presently available16 only for rather small values of q 2 (due to the limited range of v). Comparison with our Eq. (10) is probably without much rea l significance. It i s true, however, that for fixed large values of w, the data points r i se with q2 (see Fig. 1). The forthcoming experiments on p - N scattering a t NAL will reveal whether o r not this r i s e continues.

In comparing the proposed test for the existence of anomalous dimensions with the standard tech- nique, we note that the expression (10) represents the average effect of anomalous dimensions for al l the tensors which contribute to the structure function in the asymptotic region. By contrast,

t R=0.18 , W >2.0 GeV /c2

FIG. 1. The structure function vW2 as a function of q2 for three ranges of w'=w +m2/q2 . Data points are taken from H. Kendall's talk at the Cornell Conference, 1971, Ref. 1.

the moment integrals [ ~ q . (6)] pick out individual te rms in the short-distance expansion. It i s nat- ural, therefore, that l e s s extensive data suffice for comparing Eq. (10) with experiment. There i s no question, however, that in principle the moments contain more direct and complete in- formation on the anomalous dimensions.

What can we say about the signs and magnitudes of the A,; and, assuming they can be measured, what can one infer about the A,'s of the individual tensor operators? Our theoretical knowledge on anomalous dimensions is very limited. Positivity requirementsg' l7 on the structure functions demand that X, be a nondecreasing function of J, bounded for large J by a linear function of J . Calculations1" in 4 - E space-time dimensions yield the following expression for A,:

This result i s consistent with the general behavior mentioned above. In addition, the "constant" factor i s found to be very small (roughly - 0.01) resulting in rather small values of A, for J 2 2. Even so, an inspection of Eq. (11) shows that lXJ1 could be much bigger a t J = 1 o r *. One should quickly add, however, that these considerations should only be regarded a s illustrations of how A, might behave. After all, E - 0 in 4-dimensional space-time and the theory reduces to that of the free-field case. Another field-theoretic approach,'' based on sum- ming the leading logarithmic behavior of various (infinite) c lasses of Feynman diagrams, yields rather involved results concerning anomalous dimensions. One explicit result-for the lowest- order ladder graphs of the pseudoscalar theory-

9 - M E A S U R I N G A N O M A L O U S D I M E N S I O N S A T HIGH E N E R G I E S

where g i s the coupling constant. This seems to indicate a behavior similar to Eq. ( l l ) , although, again, i t s significance i s hard to e ~ t i m a t e . ~ "

What these examples do show is that the extrap- olated anomalous dimension A , (at a = 1 o r 3) could be rather different from the individual AJ for each tensor operator a s measured by the q2 dependence of the associated moment integrals. In particular, even if each individual A, i s small, the summation over the infinite tower of tensor opera- tors could result in a sizable effective;\., at J = a. In the case of the I = 0 amplitude, if one assumes that the only second-rank tensor contributing to the short-distance expansion i s the energy-momen- tum tensor, then hz=O. We expect, therefore, that A, will have the opposite sign a t J = a = 1 a s a t J = 4,6, . . . . This results in an increasing q2depen- dence in the large-w region of the structure func- tions [ ~ q . ( lo) ] a s opposed to a decreasing q2 de- pendence for the individual moments." This can be readily understood in a plot of vW,(w, q2) a s a function of l /w for various values of q2. The zeroth moment (corresponding to J = 2) i s constant in qZ. This means that the area under the curve remains constant when q2 varies. Since vWz i s positive-definite and since the higher moments a r e decreasing functions of q2, the curve must have a slow falloff with q2 in the moderate-w re- gion'' with a compensating r i se in q2 near 1/w = O . This type of behavior i s illustrated on a somewhat exaggerated scale in Fig. 2.

If this qualitative picture i s correct , then the total c ros s section for virtual-photon absorption - increases a s q2 increases for large w. This i s , a t f i r s t sight, a rather surprising result. Conven- tional arguments on the geometrical size of the virtual photon lead to a decreasing o r constantz2 function of q2. The total c ross section is not, however, a direct measure of the geometrical size (as, for instance, i s the slope of the diffrac- tion peak). When w becomes large, the usual coherent diffraction scattering mechanism becomes more important. The virtual photon behaves more and more like a superposition of (virtual) hadron states. In this context, the increase in the total c ross section (as a function of qz) could be inter- preted a s the effect of increasing participation of

FIG. 2. Possible dependence of vW* on 1/u for three different values of q2 . The solid, dashed, and dotted curves represent increasing values of q2.

higher -mass hadron states.23 To summarize, the conflict between a simple

scale-invariant light-cone structure (i.e., s tr ict scaling behavior for the structure functions) and the time-honored models of field theory calls for vigorous efforts to establish the existence (or absence) of anomalous dimensions. Besides the well-known method of forming moment integrals to detect the anomalous dimensions of individual tensor operators, an experimentally more feasible technique is to examine the q2 dependence of the structure functions a t very large w. This mea- sures the aggregate effect of the anomalous di- mensions of all the tensor operators contributing to the deep-inelastic scattering process. If A , i s found to be finite, then parton models and the naive abstraction of the light-cone behavior from quark field theories will have been shown to be invalid. If, on the other hand, A , i s zero, then the hope that these models provide a basis for a viable physical theory may well be justified.

Note added: Recent developments in field theoryz4 have revealed the possibility of "asymp- totically free" theories, which violate scaling only by terms logarithmic in q2. The breakdown of str ict scale invariafice in these theories should be, once again, most apparent in the large-w re- gion.

ACKNOWLEDGEMENT

We would like to thank G. F a r r a r and H. T. Nieh for useful discussions.

310 R , C A R L I T Z A N D W U - K I T U N G

*Work supported in part by the National Science Foun- dation under Contract No. NSF-GP-32904X.

'~eferences to the extensive literature on this subject can be traced through the review papers by: K. Wilson, in Proceedings of 1971 International Symposium on Electron and Photon Interactions at High Energies, edited by N. B. Mistry (Laboratory of Nuclear Studies, Cornell University, Ithaca, N. Y., 1972); Y. Frishman, in Proceedings of the XVI International Conference on High Energy Physics, Chicago-Batavia, Ill., 1972, edited by J. D. Jackson and A. Roberts (NAL, Batavia, Ill., 1973), Vol. 4, p. 189; and E. Bloom, in Proceedings of the International Symposium on Electron and Photon Interactions at High Energies, Bonn, 1973 (to be pub- lished).

'K. Wilson, Phys. Rev. 179, 1499 (1969). 3 ~ . Zimmermann, in Lectures in Elementary Particles

and Quantum Field Theory (MIT Press, Cambridge, Mass., 1971).

4 ~ . A. Brandt, Phys. Rev. Lett. 22, 1149 (1969); 2, 1260 (1969); Y. Frishman, ibid. 25, 966 (1970); Ann. Phys. (N.Y.) 66, 373 (1971).

5 ~ . Fritzsch and M. Gell-Mann, in Broken Scale In- variance and the Light Cone, 1971 Coral Gables Con- ference on Fundamental Interactions at High Energy, edited by M. Dal Cin, G. J. Iverson, and A. Perlmutter (Gordon and Breach, New York, 1971), Vol. 2, p. 1; R. Jackiw, R. Van Royen, and G. B. West, Phys. Rev. D 2 , 2473 (1970).

6 ~ . Feynman, Lepton-Hadron Interactions (Benjamin, New York, 1973).

?See Y. Frishman, footnote 1, for detailed references. h he infinite number of tensor operators which add up

to a bilocal on the light cone in general acquire differ- ent anomalous dimensions. Hence the bilocal operators do not have well-defined dimension.

'G. Mack, Nucl. Phys. E, 592 (1971) (the result con- tained in the last section of this paper has been recog- nized to be wrong).

' '~stimates of the first two moments based on present data and extrapolations thereof typically yield bounds of -4 for the anomalous dimension. (Private communi- cation from G. Farrar . See also E. Bloom, Ref. 1.)

"see also G. Parisi, Phys. Lett. s, 207 (1973).

he summation is over J=even only because of cross- ing symmetry.

lS~tr ict ly speaking, the validity of Eq. (7) is independent of the Regge asymptotic behavior of the usual hadronic amplitudes. It only depends on the summability of the series (4) or (5) as discussed in the next paragraph. It is, however, entirely natural to assume that a is the intercept of the usual Regge trajectories since they do arise from a similar mechanism.

1 4 ~ o be more precise, in order for the continuation to be unique, the conditions of Carlson's theorem have to be satisfied. See, for instance, E. J. Squires, Complex Angular Momenta and Particle Physics (Benjamin, New York, 1963).

1 5 ~ . Van Hove, Phys. Lett. z, 183 (1967); L. Durand, m, Phys. Rev. 161, 1610 (1967).

"HOW large is 'large" w ? "Large" w should start a t the region where the relative contributions due to nonleading trajectories have dropped off. An indication of such behavior can be obtained by looking at data for wP2 - WZ . [See A. Bodek et a1 ., Phys. Rev. Lett. 30, 1087 (1973).1 This suggests that for small q 2 , w 2 10 is 'large". "0. Nachtmann, Institute for Advanced Study report

(unpublished). 'k. Wilson and J. Kogut, Phys. Rep. (to be published). "s. -J. Changand P. Fishbane, Phys. Rev. D 2, 1084

(1970); N. Christ, B. Hasslacher, and A. H. Mueller, ibid. 2, 3543 (1972).

%ne may notice that A,--- as J- 0 in both Eqs. (11) and (12). Whether this i s an intrinsic property of A, or an accident is not clear to us.

' l ~ h e higher moments are not sensitive to the behavior of vW2 nsar l / w = O .

2 2 ~ . L. Ioffe, Phys. Lett. z, 123 (1969); J. D. Bjorken, in Hadronic Interactions of Electrons and Photons, edited by J. Cummings and H. Osborn (Academic, New York, 1971); H. T. Nieh, Phys. Lett. E, 100 (1972); Phys. Rev. D 1, 3401 (1973).

2 3 ~ . J. Sakurai and D. Schildknecht, Phys. Lett. e, 121 (1972); H. T. Nieh, Ref. 22.

2 4 ~ . J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973); H. Politzer, ibid. 30, 1346 (1973).