The limit mq→0 of perturbative QCD

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PHYSICS LETTERS

19 May 1983

THE LIMIT mq ~ 0 OF PERTURBATWE QCD F.V. TKACHOV Institute for Nuclear Research of the USSR Academy of Sciences, 60th OctoberProspect 7a, 11 7312Moscow, USSR Received 6 January 1983

The effect of light quark masses, as seen from short distances, is discussed. The large logs of mq/~ are shown to simulate "vacuum condensates".

1. The aim of this letter is to show that the role of light quark masses has heretofore been strongly underrated. Indeed, according to the popular philosophy the effect of light quark masses should be relatively unimportant, while the leading role is ascribed to the complex non-perturbative vacuum which is considered as only lightly affected by mq @ 0 [1]. Nobody, however, has as yet seriously studied the effect of large logs of mq/12, la being the renormalization parameter, which appear in perturbative calculations. Below we argue that these logs, when resummed, can drastically change the naive understanding of the limit mq -+ 0, so that the latter should rather be considered in the spirit of Bogoliubov's quasiaverages [2]. By this we mean that the real QCD near mq ~ 0 may differ qualitatively from the theory with massless quarks if such a theory exists (which is far from being obvious). To study the limit mq -+ 0 of the full theory one must first expand the full Green functions in mq and only then try, if one wishes, to calculate the resulting coefficients by expanding them in a s. For obvious reasons such a procedure is hardly possible. Instead one usually proceeds in a completely opposite direction: one takes the limit mq -+ 0, graphwise, after expanding in as, so that the log (mq//~)'s are always suppressed by powers of mq. It would be preferable to try to see the whole truth by resumming, as best we could, those logs prior to making conclusions. Needless to say, this is no substitute for a non-perturbative analysis, but it 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

is reasonable to expect that this way corresponds better to the original intention. 2. The Bogoliubov-inspired [3] extension principle (EP) for studying asymptotics of Feynman integrals [4] in conjunction with the R*-operation for removing a class of infrared divergences in the presence of ultraviolet ones [5], offer a means to study the structure of the m ~ 0 expansions of Feynman integrals straightforwardly and to all orders of perturbation theory (PT). To avoid irrelevant combinatorial complications we take a very simple example, but which retains all the essential features of the full proof *~ . Our example is similar to the one considered in ref. [4] and our reasoning closely follows [6]. Therefore only formal steps of the derivation will be indicated below. The MS-scheme [7] is used throughout the paper. Consider a free theory with a massless field, 9, and a massive one, • in four dimensions, and the current ](x) = : ~(x)¢(x) :. Consider 17(0 2, m 2) = i

fexp(iqx) dD x(RTj(x)j(O)) 0

[" dDp 1 c.t., = d (27r)D' (p _ Q)2(m2 + p2) -

(1)

where Q2 = _q2, and the rotation to euclidean re,1 The full and general proof will be given elsewhere; it repeats, in essence, the derivation below step by step. 85

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gion has been performed on the rhs of (1). The UV counterterm in (1) does not depend on m. A formal expansion gives:

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ogous to the mixing of operators under UV renormalization. A generalization of (6) reads:

i fexp (iqx) dx {A(x)B(O)} (1) ?= f(27r)Dd;P Q)2 ( ) 2

~ 2 ) -- c't" + °(m2)" (2)

~-- ~ ~y cif(Q,p)m 2i {O/.(0)}. Q2__.+~ i=0 ' The UV counterterm still subtracts the UV divergences from the first term in (2). The second term has no UV divergences but a logarithmic IR one. Following EP [4], to obtain a correct ,2 expansion in m, we must add a counterterm into (2):

- m 2/t74 _+- m 2/p4 + g(m) (2rr)o 6 (p).

(3)

(NB: derivatives of 6(p) need not and should not be added [4] .) To fix the unknown Z(m) we demand that

dDp

0=d~

ff(p)[1/(p 2 + m 2) - lip 2 + m2/p 4

- (2rr) o Z(m) 6(p)]

(4)

for any one smooth ~(p) such that if(O) = 1 and lim if(p) < +oo [4]. Choosing i f ( p ) - 1 we obtain: P ~

[" dDp 1 .,-- (T42(0))0 4:0 Z(m) = d ( 2 ~ p2+ m 2

(5)

(note the absence of normal ordering in 42). Using (RT 42(0))0 = pD-am2Zo + (T 42(0))0 to express (5) via the convergent quantity (RT 42(0))0 we finally obtain: FI(Q, m) = H(Q,0) + Q -2 [m2Cl(Q,p) + (RT 4)2(0))0 C2(Q,p)] + o(m2).

(6)

Each quantity in (6) is f'mite by construction. Interaction modifies (6) in three respects (i) C I and C 2 become series in coupling constants and log (Q/p); (ii) (RT 42(0))0 goes over into {42(0)} --=(RT [42(0) S[ )0 ¢ 0, S being the S-matrix; (iii) along with {42(0)}, there appear the vacuum averages {¢2(0)} and {~(0)4(0)} with their own coefficient functions, due to a mixing mechanism which is anal*2 "Correct" means (i) "IR convergent" and (ii) "ensuring the o(rn2) estimate for the neglected terms in (2)". 86

(7)

{O/(0)} are vacuum averages of local composite MSrenormalized operators [8], inevitably non-vanishing at least while m 4= 0. The leading power behaviour of cij at Q -+ ~ is determined by power counting and is modified by p only via the log (Q/p) contributions. For explicit formulae for calculating cq(Q, p) see ref. [9]. 3. Several conclusions can be drawn from the above derivation: (i) The techniques of dimensional regutarisation and minimal subtraction [7] are crucial for achieving the neat separation of Q and m in (7) (the phenomenon of "perfect factorization" in terms of ref. [6], see also ref. [ 10] ). (ii) The quantities {Oj} in (7) summarize information on the small-p behaviour of the theory. So, in a way, the above derivation can be considered as a formalization of heuristic arguments of ref. [11 ] which led to the introduction of non-zero vacuum condensates into deep-euclidean QCD amplitudes (for a renormalization group analysis of {O/} see below). (iii) By reexpressing the MS-renormalized operators in (7) in terms of, say, Zimmennann's normal products [12] with zero vacuum averages, all the log (m/ll) terms could only be pumped into the coefficient functions but never got rid of them. It should be clearly understood that the vacuum condensates {Oj} are faetorization-prescription dependent quantities, and the virtue of the MS-scheme consists not in making them non-zero, but in collecting within {@} all the logs of m/p. (iv) It is clear from the derivation that if there are heavy particles in the theory and/or there are other large momenta (in the case of three- etc. -point functions), all the heavy, i.e. O(q2), parameters will enter only coefficient functions in (7), so that (7) represents a general form of how the world of "light" confined particles affects the "heavy" world [6]. That is to say, whenever the introduction of vacuum conden-

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sates is justified from the phenomenological viewpoint of ref. [11], they do appear as an effect of light quark masses already at the perturbative level. To make this remark more precise, it should be noted that if heavy particles are present then the parameters of the effective lagrangian of the "light" world to be used in calculating "vacuum condensates" are related to those of the full lagrangian via the well-understood decoupling mechanism. 4. Now, the question arises: does the perturbative analysis of the vacuum condensates {@} make sense? Consider the set 0 d of the aggregates m 2i (O/} with a given total dimensionality d in units of mass (they contribute to the same power of Q -2 and form a closed set with respect to renormalization). The RG equation for 0 d reads:

{m~/Om

13*(as)~/Oas

- [d - 7(as) ]/[1 -/3m (as) ] } 0 d = 0,

(8)

where 7 is the matrix of anomalous dimensions,/3*(a) = 13(a)[1 - 13m(a)] 1 and ~m (as) =/2 dm/d~ Im B,asB"

(9)

Eq. (8) is similar in form to the well-known evolution equations for moments of structure functions of DIS [13]. The solution to (8) reads:

Od(m ) = Od(mO) (m/mo)d [a*(m)/a*(mo) ] -aa X [1

+a2a*(m ) + ...1,

(10)

where a*(m) is governed by/3*. Assuming that a*(m) = [b log (m2/A*2)] -1 with A* ~ AW~s ~ 1 0 0 - 5 0 0 MeV, the conclusion is unavoidable that with Od(m ) and light quarks we are in a deeply non-perturbative region. 5. The most important point perhaps is that even if the m d factor in (10) prevails in the chiral (i.e. mq -+ 0) limit, e.g. if 0 in the above interpretation vanishes as mu, m d -+ 0, such a possibility is by no means inconsistent with the current-algebra-PCAC phenomenology, as is explained in the appendix E of ref. [1] (see also the references therein). As explained there, if 0 vanishes in the chiral limit then the Gell-Mann-Okubo formula for the r? meson ,3 is lost,

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while the masses of the u and d quarks are larger than in the standard framework, which gives one a possibility to account for the large deviations from the chiralinvariant picture, observed in pion-nucleon scattering. However, theoretical estimates of the vacuum condensates values from (10) are hardly feasible at present - one of the reasons is that the value of A/A* is related to the strong-coupling behaviour of/3m, which is unknown. 6. To conclude, we have demonstrated that upon resumming large logs of the ratio mq/ll, where mq represents light quark masses and/l is the renormalization parameter, within usual perturbation theory, a collection of "vacuum condensates", i.e. non-zero vacuum averages of composite operators, emerges whenever they are liable to appear according to, and in exactly the same form as prescribed by, the phenomenological rules of ref. [ 11 ]. The renormalization group analysis shows that with those condensates we are in a strong-coupling region, so that their behavior at mq ~ 0 is hardly predictable at present. However, the fact that the log (mq//~)'s fit into the pattern of vacuum condensates is in itself very nice, for it ensures that those uncalculable terms, being hidden within condensates, cannot modify the formulae of ref. [11 ], whatever the interpretation may be. The author is grateful to Professor V.A. Matveev and Professor A.N. Tavkhelidze, and to S.G. Gorishny and A.V. Radiushkin (JINR, Dubna), S.A. Latin and V.V. Srnirnov (Moscow University), and V.A. Kuz'min, V.A. Rubakov, M.E. Shaposhnikov and V.F. Tokarev for discussions. The author is also indebted to K.G. Chetyrkin for promptly turning the author's attention to a diagrammatic interpretation of the consistency conditions, eq. (5), and to J. Gasser (Bern University) for an enlightening discussion of the chiral limit.

References [1] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77, and references therein. [2] N.N. Bogolubov, in: Selected papers on statistical physics (Moscow U.P., Moscow, 1979) p. 193 [in Russian]. [3] N.N. Bogotubov, Dokl. Acad. Sci. USSR 82 (1952) 217.

,3 And nothing else, as was kindly explained to me by J. Gasser (Bern University). 87

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[4] F.V. Tkachov, in: Proc. Intern. Seminar Quarks-82 (Sukhumi, May 1982) (INR Press, Moscow, 1982). [5] K.G. Chetyrkin and F.V. Tkachov, Phys. Lett. l14B (1982) 340. [6] F.V. Tkachov, On the operator-product expansions in the MS-scheme (1982). [7] G. 't Hooft, Nucl. Phys. B61 (1973) 455. [8] J.C. Collins, Nucl. Phys. B92 (1975) 477. [9] S.G. Gorishny, S.A. Latin and F.V. Tkachov, The algorithm for OPE coefficient functions in the MS-scheme (1982).

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[10] K.G. Chetyrkin, S.G. Gorishny and F.V. Tkachov, Phys. Lett. l19B (1982) 407. [11] M.A. Shifman, A.I. Vainstein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385. [12] W. Zimmermann, in: Lectures on elementary particles and QFT, Vol. 1 (MIT Press, Cambridge, MA, 1970). [13[ A.V. Efremov and A.V. Radiushkin, Riv. Nuovo Cimento (1981) no. 5.