Renormalization group improved perturbative QCD

Volume 95B, number 1

PHYSICS LETTERS

8 September 1980

RENORMALIZATION GROUP IMPROVED PERTURBATIVE QCD G. GRUNBERG 1

Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853, USA Received 5 June 1980

The results of perturbative QCD calculations are reformulated as renormalization-scheme independent predictions; in so doing, we obtain a renormarization group improvement of perturbation theory. As an application, we show that asymptotic freedom alone does not give the correct quantitative relation between pseudoscalar charmonium decay and the scaring violations in deep inelastic scattering.

One of the most attractive features of QCD is its property of asymptotic freedom [1 ], which allows perturbative calculations to be performed for large momentum transfer processes where the running coupling constant as(Q2 ) becomes small. In the last few years, these calculations have been pushed beyond the leading order [2], but have revealed in a number of cases the existence o f poorly convergent expansions in c~s. In this note, we suggest a simple method based on the renormalization group to deal with such expansions, which yield the same result in all renormalization schemes. To be precise, consider a dimensionless physical quantity o(Q 2) which has the expansion at large Q2 : o(Q 2) = a 0 + Olas(Q 2) + o2a2(Q 2) + ..., where all the coefficients except a 0 and o 1 depend on the definition of a s. We are concerned with the renormalization prescription dependent part of the expansion, i.e., with the series: [o(Q 2) - Oo]/O 1 = as(Q2)[1 +(o2/Ol)O~s(Q 2) + ...]. (1) Suppose that in some scheme the effective expansion parameter (o2/o 1) O~s(Q2) is large for the range o f Q2 of interest, making the expansion eq. (1) unreliable. i Permanent address: Laboratoire de Physique Mathematique, USTL, 34060 Montpcllier Cedex, France (Physique Mathematique et Th6orique, 6quipe de recherche associ6e au CNRS). 70

Since the coefficients in eq. (1) are scheme dependent, it is always possible to define a process-dependent c~s such that they all vanish. Typically, to the generic physical quantity o(Q 2) one can associate a "physical" c~s (Q2) by the identity: o(Q 2 ) - o 0 + alas(Q2),

(2)

where all the higher order corrections to the process considered are absorbed into the definition of c~s. Well known examples of this procedure are the coupling constants defined by the total e+e - annihilation cross section into hadrons:

(Qi are the quark charges), or by the scaling violation in some non-singlet structure function [3] : Q2 d lnMn/dO2 = - (7~/8n)as(Q2)

(4)

(7~ is a one loop anomalous dimension). The various a s so defined are renormalization scheme independent (and gauge independent) quantities which can be extracted unambiguously from experiment. Since they are coupling constants, each o f them is predicted in QCD in terms of a process dependent scale parameter A and dimensionless 13 function (if the quark mass can be neglected). The t3 function is defined as usual by the equation:

Q2 ~[/~Q2 =/3(p),

(5)

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where p - C~s/47r, and can in principle be evaluated from QCD. It is convenient to write the exact solution of eq. (5) as * 1. boln(Q2/A2) = p 1 + (bl/bo)In(bOp )

P + f dx Ix 2 _ (bl/bo)x-1 + bo//3(x) ] =-F(p), 0

(6)

where b 0 and b 1 are, respectively, the coefficients in the one loop [1] and two loop [5]/3 function: 3(x) = -box2 blX3 + O(x4), and are scheme and process independent. Higher order terms in/3 will depend upon the definition of p, and can be obtained from the perturbative expansion of p in any scheme. The integral in eq. (6) has been arranged to vanish at p = 0, using the expansion of the inverse/3 function:

bo//3(x ) = - x -2 + (b 1/bo) x -1 + O ( x ° ) .

(7)

By construction, dF/dp = bo//3(p ). The scale parameter A in eq. (6) is defined by the choice of the integration constant (i.e. the (b 1/bo) In b 0 term), fixed by the standard convention that, for t = Q2/A2 large, p(Q2) has the asymptotic expansion: 1

p ( Q 2 ) - bo l n t

way to the coupling/)MS of the MS scheme. As noted in ref. [8], eq. (8) is exact. We stress that both the/3 function and the scale A in eq. (6) are renormalization-scheme independent quantities, whose knowledge is equivalent to that of p (Q2). Before discussing the connection of the present method with perturbation theory, let us make the following remarks: (i) Eq. (6) predicts the Q2 evolution of p independently of any particular choice of A. Indeed, given the t3 function from QCD, the function F(p) is known. Plotting the experimental values of F(p) against In Q2/~2 (where/a is an arbitrary scale), one should get a straight line with slope b 0. The intercept is arbitrary, since it depends upon the choice of ~t, and in fact A as defined in eq. (6) can be measured by finding the value of/1 for which the intercept is zero. Note that eq. (6) need not be inverted to test QCD (in so doing, one would introduce a complicated A dependence into the prediction). (ii) Any/3 function in QCD is presently known only in weak coupling, and up to two loop. Correspondingly, dropping higher order terms in eq. (7), the integral in eq. (6) vanishes identically and we are left with the asymptotic freedom formula: b0 ln(Q2/A2) = p-1 + (bl/bo) ln(b0P ) - F o ( P ) .

bl l n l n t b 0 (b 0 lnt)2

E(lnln 1)2 1

+°L OntV J ' with no 1/ln2t term. The value of A depends of course upon the definition of p ~ : , and can be obtained in any scheme by expanding O to second order; following an argument due to Celmaster and Gonsalves [8], and choosing for definiteness the MS scheme, we get the relation (also rederived below): b0 ln(A2/A2s) = d 2 ,

8 September 1980

(8)

where o(Q2) = PMs(Q2)[1 + d2PMs(Q2) + ...], and AMS is the scale parameter associated in the standard

+1 For a nonperturbative application of eq. (6) to the heavy QC~potential, see ref. [4]. :F2 The definition of A used here is very close to those suggested in the case of moments of structure functions by Bace [6] and Para and Sachrajda [7].

(9)

We deduce that the Q2 evolution (as opposed to normalization) of p depends only upon the universal constants b 0 and b 1 in the perturbative regime. This is the case, in particular, for the scaling violations in deep inelastic scattering * 3 [see eq. (4)]. (iii) We need a criterion for the validity of perturbative QCD; it is natural to choose as a necessary condition:

P/Pmax ~ (bl/bo)P < 1 ,

(10)

since (b l/b0) p is the effective expansion parameter for the 13function. This condition is renormalizationscheme independent and testable experimentally, but may not be sufficient, due to nonperturbative effects. Furthermore, we note that the approximation eq. (7) for the/3 function introduces a pathology (the analogue) of the Landau ghost) in the function Fo(P), which has a minimum at p = Pm~x (= 0.16 for 4 flavors), whereas +3 Eq. (9) automatically includes the "higher order" corrections discussed by Moshe [9], which are essentially higher powers of 1/In Q2 in the asymptotic expansion of p (Q2), coming from inverting eq. (9). 71

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the exact F(p) must decrease from +oo to - ~ . This fact clearly shows that perturbation theory breaks down for p >/9ma x. Alternatively, in terms of Q2, one must require Q2 > A 2 exp [bo1 F 0 (Pmax)] - Q 2 i n ,

(11)

i.e. Q > 1.62 A (for 4 flavors). The scale Qmin does not depend on the particular definition of A used in eq. (6), and in fact could be substituted to A as the natural short-distance scale associated to p. (iv) In the two loop approximation for the/3 function, the results of the present method are equivalent to those obtained in any scheme, to order %2, by the choice of renormalization point which makes the order a s2 correction vanish. Indeed, consider the expansion of p (Q2) in, say, the MS scheme, with an arbitrary renormalization point p2 : p(Q2) = PMS(/.t2)

8 September 1980

we get: ln(A2/A 2) = - b o 1 [ff(~) - F(p)]

=-

If

o

dx

f

dx I

/3-(x) 0 e % 5

'

(13)

where a limit process is understood at the lower ends of the integrals in eq. (13), and the universality of/3 and ~ u p to two loop has been used to obtain a finite result. Eq. (13) is exact, and gives the relation between p and ~ once the corresponding/3 functions and ln(A2/A 2) are known ,4. [Differentiating eq. (13) with respect to Q2, we recover the well-known relation d ~ / ~ ( ~ - do//3(0) = 0.] Solving for ~ as a power series in p in eq. (13), one finds: p = p [ 1 +c2P+C3 p2 + c 4 p 3 +...] , with

X { 1 + [d 2 - b 0 ln(Q2/p2)] PMS(P 2) + ...) .

c 2 = b 0 ln(A2/A 2) ,

To minimize the 2nd order correction, let us choose /a2 such that [10]

c3 = c2 + (bl/bo)c2 + (b2 - b2)/bo ,

d 2 - b 0 ln(Q2/p 2) = 0 .

c 4 = 2C32+ ~(bl/bo)c 2

(12)

Up to 2nd order, this choice gives p(Q2) = PMS(#2). On the other hand, combining eq. (9) and its analogue for PMS with eqs. (8) and (12), we get: F 0 [PMS(/-*2)] = b 0 ln(/-12/A2Ms) = b 0 ln(Q2/A 2) = F 0 [p(Q2)] , hence p(Q 2) = PMS(p2), which reproduces the previous result. The same argument would go through to all orders (replacing F 0 by F) if the/3 functions for p and PMS were identical. Let us now show for completeness that the present method represents a renormalization group improvement of perturbation theory. Let p and ~ be two "physical" couplings (one of them may as well be the coupling constant in any renormalization scheme). From eq. (6) and its analogue for ~: b0 l n ( Q 2 / ~ 2 ) = ~ - 1 + (bilbo) ln(bo~ )

+ c2 (3b2/b0 - 2b2/bo) + ½(b3 - b3)/bo ,

(14)

etc., where the b i and bi's are (minus) the coefficients in the expansions of the/3 and ~ functions. In particular [8], we recover eq. (8) (with obvious substitutions). Eqs. (13) and (14) are straightforward generalizations of the well-known renormalization group relations between the couplings defined at two different scales in the same renormalization scheme; we then have p = p(O2), ~ = ~(O2) -- p ( a ' 2 ) , c2 = _bo In (Q'2/Q2) and/3 = ~. The only difference in the more general case discussed here is therefore the non-universality of the 13functions beyond two loop. In the weak coupling regime, we can replace F and P by FO, and thus get from eqs. (13) and (14): c 2 = _ IF 0 (P) - F 0 (/9)] = _ [~-1 _ p - I + (bl/bo) ln(~/p)] .

(15)

In practice, the b 1 term is often negligible [consistent+f

0 72

dx [x -2 - ( b l / b o ) x - 1 + bo/~(x)] ~ f f ' ( ~ ) ,

(6')

-~4 Choosing ~(x)~ -bo x2 - b ix 3, we obtain a nonperturbative construction of the 't Hooft transformation [ 11 ].

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ly with eq. (10)], giving the formula: ~ = p/(1-c2P ). We note that: (i) Ic21 can be arbitrarily large, whereas the corrections to the right-hand side of eq. (15) are 0 ( 0 , p-), and therefore vanish in weak coupling. In particular, if (b 1/bo)P and (bl/bo)~ are smaller than [c2Pl and Ic2~1, eq. (15) may improve perturbation theory, by resumming its most important terms [see eq. (14)]. (ii) There is no correlation a priori between the magnitude of c 2 and that of the coefficients b i and bi of the/3 functions, since c 2 expresses the relation between two coupling constants, whereas b i and bi are intrinsic to each coupling. For instance, in the example discussed above of two couplings with the same/3 function, one can make c 2 = -ln(Q'Z/Q 2) arbitrarily large without changing the/3 function. Turning to applications, we first note that if [c2l ~ b 1/bo, and condition eq. (10) is satisfied, eq. (15) reduces to the usual perturbation formula ~ = p X (1 + c2o). Important examples are predictions for R (Q2) in terms of the scaling violations in deep inelastic scattering [see eqs. (3) and (4)], or relations between various non-singlet moments. The results so obtained are well known [3,12,7]. The present method is more relevant to the case where Ic21 >> bl/b O. Only one interesting example has been calculated so far, that of pseudoscalar quarkonium decay. It has been found [13]: F(QQ -+ "had")/l~(QQ -+ 77) = x [1 +

(2/9e~)(as/a)2

ln(/a/Zm) - ~2 ln2

where we have written the result of ref. [13], for convenience, in the MS scheme [14], and 4 flavors are assumed; m and eQ are the mass and charge of the quark Q. We deduce F(QQ -+ "had")/F(QQ -+ 77) =

(2/9e~)(4nP/a) 2 , (16)

where 0 is given in the two loop approximation by eq. (9) with ' Q2 = 4m 2 ,

A = 5.38 A~S-

is clearly not applicable. At the bottonium level, m 5 GeV and eq. (11) is well satisfied. We obtain p 0.028 and I'(QQ -+ "had")/I~(OQ ~ ")'7) ~ 42 X 10 3 . For comparison, perturbation theory in the MS scheme with/a = 2m predicts: F(QQ ~ "had")/I'(QQ ~ 77) ~ 22 X 103 . No experimental number is yet available for this ratio, but I'(r/c -+ "had") and P ( ~ ~ e+e - ) have been measured. Combining the O(C~s) corrections to the leptonic width [15] with the result ofref. [13] for P(QQ -+ "had"), we obtain (in the MS scheme for 4 flavors, and assuming the wave functions at the origin cancel in the ratio):

r'(QQ --, "had")/P(QQ

(17)

Eqs. (16) and (17) are equivalent to a result given by Billoire [3 ]. Assuming the value A ~ ~ 0.4 GeV obtained from deep inelastic scattering [2] is correct, we get A ~ 2 GeV. At the charmonium level, m ~ 1.5 GeV, Q = 2m does not satisfy condition eq. (11), and perturbation theory, both improved and nonimproved,

-+ e+e - ) =

(2/3e~)(asfiX)2

X {1 + (as/Tr)[ ~ ln6u/2m) + ~-Tr2 - ~0 _ 2~ In 2

We deduce

r'(QO--, "had")/I'(Q0 ~

e+e - ) =

(2/3e~)(4np/a) 2 , (18)

where p is given by eq. (9) with: Q2 = 4m 2 ,

~r 2 + ~)] ,

8 September 1980

A = 6.785 AM~.

(19)

Again, perturbation theory breaks down at the charmonium level, assuming A ~ = 0.4 GeV. However, from the experimental values F(~ -+ e+e - ) ~ 4.8 keV and [16] F(r/c -~ "had") < 35 MeV we obtain p < 0.04 in eq. (18). For such values of p, condition eq. (10) is well satisfied, and we would expect perturbation theory to be valid. Eq. (9) then yields A < 1 GeV, which corresponds to A ~ < 0.15 GeV from eq. (19). Such a small value of A ~ may easily be accounted for by higher twist effects [2] ; in any case, we conclude that asymptotic freedom alone is not in quantitative agreement with the data, and that some other physics must be at work, either in the quarkonium system or in deep inelastic scattering. As a final comment, we note that the much discussed "large" corrections to the leptonic width of quarkonium [15], or to the Drell-Yan cross section ,s ,5 See ref. [2] for a list of references. 73

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are of the generic f o r m o f eq. (2). Therefore, according to the point o f view presented in this paper, t h e y cannot yet be considered as a threat to the applicability o f perturbative QCD to these processes, since the magnitude o f the effective a s in eq. (2) is predicted only by the O(~ 2) corrections. F u r t h e r m o r e , the relevant criterion for perturbative QCD is given by eqs. (10) and (11), and not by the c o n d i t i o n (Ol/gO)as(Q2) 1. Experimentally [17], c o n d i t i o n eq. (10) seems to be well satisfied for the D r e l l - Y a n cross section. I thank P. Lepage, S.-H.H. Tye and T.M. Yan for valuable discussions.

References [1] H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346; D.J. Gross and F. Wilzeck, Phys. Rev. Lett. 30 (1973) 1343. [2] For a review, see e.g., A.J. Buras, Rev. Mod. Phys. 52 (1980) 199; J. Ellis, CERN preprint TH 2744-CERN (1979). [3] R. Barbieri, L. Caneschi, G. Curci and E. d'Emilio, Phys. Lett. 81B (1979) 207. [4] W. Buchmiiller, G. Grunberg and S.-H.H. Tye, Cornell preprint CLNS 80/453 (1980).

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[5] W.E. Caswcll, Phys. Rev. Lett. 33 (1974) 244; D.R.T. Jones, Nucl. Phys. B75 (1974) 531. [6] M. Bace, Phys. Lett. 78B (1978) 132. [7] A. Para and C.T. Saehrajda, Phys. Lett. 86B (1979) 331. [8] W. Celmaster and R.J. Gonsalves, Phys. Rev. Lett. 42 (1979) 1435; Phys. Rev. D20 (1979) 1420. [9] M. Moshe, Phys. Rev. Lett. 43 (1979) 1851. [10] S. Wolfram, C~tech preprint CALT 68-690 (1978); for an application to the heavy QC~ potential, see also A. Billoire, Saclay preprint DPh-T/79/152. [11] G. 't Hooft, Lectures given at the Ettore Majorana Intern. School of Subnuclear physics (Erice, Sicily, 1977). [12] M.R. Pennington and G.G. Ross, Phys. Lett. 86B (1979) 371. [13] R. Barbieri, G. Curci, E. d'Emilio and E. Remiddi, Nucl. Phys. B154 (1979) 535. [14] W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Phys. Rev. D18 (1978) 3998. [15] R. Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Lett. 57B (1975) 455; W. Celmaster, Phys. Rev. D19 (1979) 1517; E.C. Poggio and H.J. Schnitzer, Phys. Rev. 20 (1979) 1175. [16] D. Coyne, talk given at the Intern. Symp. on High energy e+e- interactions (Vanderbilt Univ., May 1980). [17] J. Badier et al., Phys. Lett. 89B (1979) 145.