The anomalous dimensions of the anisotropic dimension four operators in SU(N) gauge theory coupled to fermions are calculated to lowest order. The possibility of comparison with numerical simulations is pointed out

Robertson, David G.

Wilczek, Frank

Physics Letters B

English

Otterbein University

Otterbein University Digital Commons @ Otterbein Physics Faculty Scholarship Physics 11-22-1990 Anomalous Dimensions of Anisotropic Gauge Theory Operators David G. Robertson Otterbein University Frank Wilczek Princeton University Follow this and additional works at: https://digitalcommons.otterbein.edu/phys_fac  Part of the Physics Commons Repository Citation Robertson, David G. and Wilczek, Frank, "Anomalous Dimensions of Anisotropic Gauge Theory Operators" (1990). Physics Faculty Scholarship. 20. https://digitalcommons.otterbein.edu/phys_fac/20 This Article is brought to you for free and open access by the Physics at Digital Commons @ Otterbein. It has been accepted for inclusion in Physics Faculty Scholarship by an authorized administrator of Digital Commons @ Otterbein. For more information, please contact digitalcommons07@otterbein.edu. Volume 251, number 3 PHYSICS LETTERS B 22 November 1990 Anomalous dimensions of anisotropic gauge theory operators David G. Robertson Department ofPhysics, University of California, Santa Barbara, CA 93106, USA and Frank Wilczek 1 School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA Received 1September 1990 The anomalous dimensions ofthe anisotropic dimension four operators in SU (N) gauge theory coupled to fermions are calcu- lated to lowest order. The possibility of comparison with numerical simulations is pointed out. 1. Introduction Although great effort has been put into the numer- ical simulation of discretized gauge theories, there has been a notable paucity of quantitative analytic re- sults with which to compare the simulations. The running of the coupling constant according to the re- normalization group coefficient r, and some clever generalizations i volving anisotropic cutoffs [1 ], are the only examples that come to mind. We have calculated the anomalous dimensions of certain gauge-invariant, generally anisotropic, di- mension four operators in SU(N) gauge theories coupled to fermions, using techniques of continuum perturbation theory. We shall argue that these anom- alous dimensions ought also to be readily calculable from the simulations. If such calculations were done and agreed with the predictions, they would supply highly nontrivial verification that renormalized quantum field theory at weak coupling does indeed describe the short-distance b havior of the (nonper- turbative) lattice theory. There has been some doubt expressed about this recently [2 ]. Even if one be- lieves that such checks are redundant in principle, Research supported in part by DOE grant DE-FG02- 90ER40542. their success would be welcome reassurance that in practice the lattice size has been taken sufficiently small to reproduce the continuum. We shall first give an account of the technique of calculation and of the results obtained, and then sketch ow a comparison with numerical lattice gauge theory results could be implemented. 2. The anisotropic anomalous dimensions Let us first consider the pure gluon theory, speci- fied by the lagrangian L~= - ~ ~ Fa~F~. + (gauge fixing) + (ghosts), a,~,v  (2.1) with the gluon field strength " ~ O~Au+g f ~4uA ~. (2.2) Fu.  = O~A. - a abc b c (We work in euclidean spacetime throughout. ) A ba- sis for the gauge-invariant dimension four aniso- tropic operators we wish to consider is provided by the six objects 0,2 -½ X F~2F~2, 013 -~1 X F~3F~3, .... (2.3) ~t a 434 0370-2693/90/$ 3.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) Volume 25 1. number 3 PHYSICS LETTERS B 22 November 1990 By reflection symmetry there is no mixing between these operators and e.g., 1, F’f2FY3, so that this set of operators is closed under renormalization #‘. Now, upon renormalization there are evidently three independent operator mixings which can occur: aP ,2-Yb(OL3+014+0*3+0*4) -YcO34. (2.4) It is worth noting, however, that there are certain identifiable linear combinations of the O,,, multipli- catively renormalized, with anomalous dimensions that are combinations of y=, yb and yC. These are GI = 012 - 034 , (2.5) GII,~o012+034-t(013+0,4+023+024), with aP (2.6) ~~G*I,=-(~,-~~,+~L)G,,I. There are two other operators of the same general form as Grr, and one of the same form as Gnr, which we do not display. Now, Gr is just F2, the lagrangian. Its anomalous dimension is closely related to the con- ventional beta function [ 31. The three G,r-type op- erators are linear combinations of the diagonal ele- ments of the energy-momentum tensor, and are therefore expected on general grounds to have van- ishing anomalous dimensions. The actual computation of the anomalous dimen- sions is straightforward, though somewhat laborious; it involves evaluating the diagrams shown in fig. 1. Before we present our results, however, a few re- marks about operator mixing in gauge theories are in order. As is well known, the renormalization of a compos- ite operator in general requires counter-terms propor- xl Other closed sets of anisotropic operators, containing FJ,,, etc., could also be considered. Fig. 1. tional to other composite operators of the same (or lower) mass dimension and carrying the same quan- tum numbers. Obviously it is very helpful if the num- ber of operators which mix together is fairly small, so that the calculation of the matrix of renormalization constants is not too difftcult. Life would be most pleasant in this regard if the counterterms needed for a gauge-invariant operator (i.e., one formally invar- iant under the usual gauge transformations of the el- ementary fields) were themselves gauge-invariant. Indeed it is somewhat frightening to contemplate the alternative, since the number of gauge-noninvariant operators which a priori might mix with a given gauge- invariant one is usually enormous. Actually, the truth is somewhere in between. We now briefly summarize the result of a rather involved analysis, which is fully presented in ref. [ 41. Define type I operators to be the formally gauge-invariant operators of interest. It can be shown that operators which mix with type I operators (and are not them- selves type I ) are of two special types: type IIa oper- ators, which are BRS variations of some other oper- ators, and type IIb operators, which formally vanish by the equations of motion. These three sets of oper- ators together form a closed set under renormaliza- 435 Volume 251, number 3 PHYSICS LETTERS B 22 November 1990 tion. In this basis of operators the renormalization matrix Zab , defined by obare  7 /~ren (2.7) a ~g-'abt-/b takes a simple form. The operators of type II mix only with each other. Thus Zab has the block triangular form g i i , i i  / " So if we are willing to work on-shell and to refrain from looking at BRS null states, the renormalization constants ZLI that connect ype I operators among themselves are adequate. They yield the physically correct, gauge-independent anomalous dimensions for the type I operators. The physical on-shell matrix elements, i.e. those with all external (momenta) 2 set to zero and with physical polarization vectors attached, of all type II operators vanish. Thus we can isolate the type I counterterms ZLI simply by ignoring the gauge-non- invariant operators, and evaluating all matrix ele- ments on-shell. Finally, from Z~,~ we compute the physically relevant gauge-independent anomalous dimensions. This is an eminently practical scheme, and is the one we employed. The calculation is slightly complicated by the fact that we must evaluate all matrix elements on-shell. Indeed, the two-gluon matrix elements of Ou~ at zero momentum transfer vanish on-shell, so to see this op- erator we must either consider three-gluon matrix elements or evaluate the operator insertions at non- zero momentum transfer. We chose the latter alternative. Our results are, for pure SU(N) gauge theory g2C h g2C a g2C h Ya =6(8n2) ,  Yb=-- 8~C ' y¢=6(8n2) ,  (2.9) where CA is the Casimir invariant of the adjoint rep- resentation of SU (N): CAaab~-facdf bed , (2.10) with the f~b~ the structure constants of the group. These results are consistent with the general expec- tations discussed above, namely y~=y¢ and ya+ 4yb + ~ = 2fl(g)/g. We also checked the results by calculating the y's in the general Fermi-type gauge, and finding that they are independent of the gauge parameter. To include massless fermions, we augment the la- grangian (2.1) with the term ~f=i ~ ~Tyu Duq/, (2.11) ,u where m • a a D u -Ou-lgAut (2.12) and the t a are generators of SU(N) in the fermion representation. The four new operators we must in- elude are Pl=i~yi DIY, Pz=-i~itTzDz~l . . . . .  (2.13) We define additional mixing coefficients by 0 ~ e~ = - r~e l  - ~o(e2-P3 +P4) - -  y f (012  +013 +014)  --) 'h (023  "1"024 "]'-034) , (2.14) O12= (old terms) --Yk(PI +P2) -- Y~(P3+P4) •/z Evaluating the additional graphs of fig. 2 then yields g2 1 4 z,~= ~ (gCA +~T.) , g2C A g2C A Yt, = 8~2 , Yc----- 6 (81~2)  , 2g2CR 2g2CR ~)d = 8~ 2 , ~)e-- 3 (8n2)  , 4g2TR g2CR (2.15) yf=- -Yh  = 3(87~2 ) , ~k = --Yl~-- 3 (87t2)  , where TR and CR are the index and Casimir invar- iant, respectively, of the fermion representation: Trt6ab=tr(tat b) , CR~ij= (t~t~)ij. (2.16) Again, we have a sum rule relating our anomalous di- mensions to the ordinary fl function: Y~+4?b+7¢= 2fl(g)/g. Also, there must he four operators with zero anomalous dimension, corresponding to the diago- nal components of the energy-momentum tensor. This last condition provides ahighly nontrivial check on the calculation, as it corresponds to the vanishing 436 Volume 251, number 3 PHYSICS LETTERS B 22 November 1990 /k Fig. 2. of the determinant of the 9 × 9 matrix of anomalous dimensions. As another check on the calculation we also com- puted the insertions of the Pu into the Green function with two external fermions and one gluon. This amounts to considering the pieces of the counter- terms proportional to qT~.Auq/rather than q?yu Ou~. The coefficients of these must come out the same, in order that they may be assembled into a gauge-invar- iant counterterm for OYu DuV. 3. Applications The operators Our appear prominently in lattice gauge theory, as the limit, when the lattice spacing is small, of the operators 1 a2g----- ~ tr ( Uu~ - 1 ) ( U~ - 1 ) ~ O,~. (3.1) Here Uu~ is the Wilson line integral around a pla- quette in the/tv plane, and a 2 is the area enclosed by such a plaquette. Similarly, the fermion operators we have calculated above are simple objects on the lattice. The correlation function between Qu,, at nearby points, and possibly with different indices, can be calculated using the operator product expansion. The leading singularity as x-~0 is of the form (GA(X)GB(O) > ..~ CgAa (X, g;/t) 1 +... (3.2) for any of the multiplicatively renormalized combi- nations GA appearing in (2.5). Here the dots repre- sent higher-dimension operators, and <gaa is a c- number function obeying the renormalization group equation ( °  ° ) +fl~g -YA--YB ~aB(X,g;/t)=0. (3.3) Eq. (3.3) may be solved to yield / . . . .  (r°+y°)/2bo C~AB().--Ix, g) =~,S I~-~ ) ~Aa(X, g(t) ) (3.4) to lowest nontrivial order, where 7g=gZy0A+ .... fl= -bog3+ .... t - ln2 ,  and g(t) is the running cou- pling as usually defined. For sufficiently short distances, uch that the effec- tive coupling becomes mall, we may calculate :gga approximately using perturbation theory. The corre- lation functions of the operators appearing in (2.5) may be obtained from the perturbative r sults for the Our. In lowest order perturbation theory the three distinct cases are, e.g., (x~ 2 2 ~ 2 "[- X  - -X  3 - -X4)  <012(X)012(0)> = 27~4(X2) 6 ' 2x~x 2< O12(X)013(0)>-  7~4(X2) 6 ' < 0~2(x)034(0) > =0,  (3.5) where we have neglected &function singularities. Combining eqs. (2.6), (3.4), and (3.5) we obtain predictions for the correlation functions of interest. Similarly, we find for the fermion operator prod- ucts in lowest order perturbation theory 16 < Pu(x)P.(O ) > - zt,,(x2)6 [ 2x2ux~ _X2XuXvg,uv l 2 2 2 1 2 2 -~x  (Xu+X,)+~(x ) 1; (3.6) no summation is implied in this expression. 437 Volume 251, number 3 PHYSICS LETTERS B 22 November 1990 The anisotropic operators we have been discussing are in principle readily accessible on the lattice, as we have stressed before. However, one subtlety ought to be mentioned. The correlation functions (3.5) and (3.6) vary rapidly with distance (as 1/x s) even in free field theory. The interesting QCD modif ications of this behavior are minor by comparison; only pow- ers of logarithms at small x. Since there is some am- biguity concerning the correct cont inuum interval x to associate with pairs of extended operators uch as our plaquette operators on the lattice, an accurate comparison between the analytic predictions and the numerical results for the change of the correlation function with distance would seem to be quite diffi- cult. Fortunately, there is an alternative procedure: to compare the correlation functions in a fixed ge- ometry, as a function of coupling. The variation in this case will come only from the anomalous dimensions of interest, which thereby become practically accessible. Comparison between the analytically calculated, continuum correlation functions and lattice simula- tions clearly is called for. Acknowledgement We would like to thank the Harvard Physics De- partment, where portions of this work were com- pleted, for its hospitality. D.G.R. thanks B. Hill for helpful discussions. This research was supported in part by the National Science Foundation under Grant No. PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration, at the University of California at Santa Barbara. References [ 1 ] F. Karsch, Nucl. Phys. B 205 [ FS5 ] (1982) 285; J.C. Sexton and H.B. Thacker, Phys. Rev. Lett. 57 (1986) 2131. [2] K. Cahill, M. Herbert and S. Prasad, Phys. Lett. B 210 (1988) 198; A. Patrascioiu, E. Seiler and I.O. Stamatescu, Phys. Lett. B 107 (1981) 364. [3] H. Kluberg-Stern and J.B. Zuber, Phys. Rev. D 12 (1975) 467; R. Tarrach, Nucl. Phys. B 196 (1982) 45; B. Grinstein and L. Randall, Phys. Len. B 217 (1989) 335; D.G. Robertson, UCSB preprint UCSBTH-89-57. [4] H. Kluberg-Stern and J.B. Zuber, Phys. Rev. D 12 (1975) 3159; S. Joglekar and B.W. Lee, Ann. Phys. 97 ( 1976 ) 160; J.C. Collins, Renormalization (Cambridge U.P., Cambridge, 1984). 438 

Anomalous Dimensions of Anisotropic Gauge Theory Operators

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Anomalous Dimensions of Anisotropic Gauge Theory Operators

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