F.V. Tkachov, Phys. Lett. 125B (1983) 85-87 1



THE LIMIT m
q 0 OF PERTURBATIVE QCD

F.V. TKACHOV
Institute for Nuclear Research of the USSR Academy of Sciences, 60th October Prospect, 7a, 117312 Moscow, USSR

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


1. The aim of this Letter is to show that the role of light p 1 L 1 m2 O 2
quark masses has heretofore been strongly underrated. Indeed, Eq. (1) = - C.T. o(m ) . (2)
(2 ) ( p - Q)2 NM p2 p4 QP - +
z dD D
according to the popular philosophy the effect of light quark
masses should be relatively unimportant, while the leading role The UV counterterm still subtracts the UV divergences from
is ascribed to the complex non-perturbative vacuum which is the first term in (2). The second term has no UV divergences
considered as only lightly affected by m
q 0 [1]. Nobody, but a logarithmic IR one. Following EP [4], to obtain a correct 2
however, has as yet seriously studied the effect of large logs of expansion in m , we must add a counterterm into (2):
mq /  ,  being the renormalization parameter, which appear m2 m2 D
in perturbative calculations. Below we argue that these logs, - - + z(m) (2) ( p) . (3)
p4 p4
when resummed, can drastically change the nave understand-
ing of the limit m (NB Derivatives of ( p) need not and should not be added
q 0 , so that the latter should rather be
considered in the spirit of Bogoliubov's quasi-averages [2]. By [4], [6].) To fix the unknown z(m) we demand that
this we mean that the real QCD near m
q 0 may differ quali- p L 1 1 2
m O
tatively from the theory with massless quarks if such a theory 0 = z dD D
( p) z(m) ( ) ( p) (4)
D M - + - 2 P
2 2 2 4
( ) Mm +p p p P
exists (which is far from being obvious). 2 N Q
To study the limit m
q 0 of the full theory one must first for any smooth ( p) such that ( )
0 = 1 and lim ( p) < +
p
expand the full Green functions in mq and only then try, if one
wishes, to calculate the resulting coefficients by expanding [4], [6]. Choosing ( p) 1 we obtain:
them in D
S . For obvious reasons such a procedure is hardly p 1
z(m) = T 2
0
z d
possible. Instead one usually proceeds in a completely opposite ( ) 0 (5)
D
(2 ) m2+p2 0

direction: one takes the limit m
q 0 , graph-wise, after ex- (note the absence of normal ordering in 2 (0) ).
panding in 
S , so that log(mq / ) 's are always suppressed by
2 D-4 2 2

powers of m Using RT (0) = m Z +
0 T (0) to express
q . 0 0
It would be preferable to try to see the whole truth by re- (5) via convergent quantity RT 2 (0) we finally obtain:
0
summing, as best we could, those logs prior to making conclu- ( ,
Q )
m =
sions. Needless so say, this is no substitute for a non- (Q, )
0

perturbative analysis, but it is reasonable to expect that this m2C (Q, ) 2 
1 + R
T ( )
0 C ( ,
Q )
way corresponds better to the original intention. + 0 2 + (
o m2 ). (6)
Q2
2. The Bogoliubov-inspired [3] extension principle (EP) for
studying asymptotics of Feynman integrals [4] in conjunction Each quantity in (6) is finite by construction.
with the R -operation for removing a class of infrared diver- Interaction modifies (6) in three respects:
gences in the presence of ultraviolet ones [5], offer a means to (i) C1 and C2 become series in coupling constants and
study the structure of the m
q 0 expansion of Feynman inte- log(Q /) ;
grals straightforwardly and to all orders of perturbation theory
(PT). (ii) RT2 (0) goes over into 2
m (0)r RT 2(0)S 0 ,
0 0
 v2 11 Jan 1999 To avoid irrelevant combinatorial complications we take a S being the S-matrix;
very simple example, but which retains all the essential fea-
tures of the full proof1. Our example is similar to the one con- (iii) along with 2
k (0)p , there appear the vacuum averages
k p
sidered in [4] and our reasoning closely follows [6]. Therefore 2(0) and
{ ( )
0 ( )
0 } with their own coefficient functions,
only formal steps of the derivation will be indicated below. due to a mixing mechanism which is analogous to the mixing
The MS-scheme [7] is used throughout the paper. of operators under UV renormalization.
Consider a free theory with a massless field and a mas- A generalization of (6) reads:
sive one in 4 dimensions, and the current
i qx i
0 2
l q 
j(x) = : ( x) ( x): . Consider i e dx A( x) B( ) C (Q, ) m O
o ( )
0 t.
z (7)
2 ij j
Q + i=0 j
(Q2,m2) = i eiqx D
d x RTj(x) j( )
0
z k p
0 Oj ( )
0 are vacuum averages of local composite MS-

D
d p 1 renormalized operators [8], inevitably non-vanishing at least
= - .
C T.
z while m 0 . The leading power behavior of C
(2 )D ( p - Q)2 (m2 + p2 ) (1) ij at Q is
determined by power counting and is modified by  only via
where Q2 q2
= - , and the rotation to Euclidean region has the log(Q /) contributions. For explicit formulae for calcu-
been performed on the r.h.s. of (1). The UV counterterm in (1) lating C 
ij (Q, ) see [9].
does not depend on m . Formal expansion gives:


2 "Correct" means (i) "IR convergent" and (ii) "ensuring the o(m2) esti-
1 The full and general proof will be given elsewhere; it repeats, in essence,
the derivation below step by step. mate for the neglected terms in (2)".


F.V. Tkachov, Phys. Lett. 125B (1983) 85 2



3. Several conclusions can be drawn from the above deriva- limit then the Gell-Mann-Okubo formula for the -meson3 is
tion: lost, while the masses of the quarks u and d are larger than in
(i) The techniques of dimensional regularization and mini- the standard framework, which gives one a possibility to ac-
mal subtraction [7] are crucial for achieving the neat separation count for the large deviations from the chiral-invariant picture
of Q and  in (7) (the phenomenon of "perfect factorization" observed in the pion-nucleon scattering.
in terms of [6]; see also [10]). However, theoretical estimates of the vacuum condensate
(ii) The quantities {O values from (10) are hardly feasible at present -- one of the
j} in (7) summarize information on
the small-p behavior of the theory. So the above derivation is, reasons is that the value of / is related to the strong-
in fact, a formalization of heuristic arguments of [1] which led coupling behavior of m , which is unknown.
to introduction of non-zero vacuum condensates into deep- 6. To conclude, we have demonstrated that upon resumming
Euclidean QCD amplitudes (for a renormalization group analy- large logs of the ratio mq/ , where mq represents light quark
sis of {Oj} see below). masses and  is the renormalization [=factorization] parame-
(iii) By re-expressing the MS-renormalized operators in (7) ter, within usual perturbation theory, a collection of "vacuum
in terms of, say, Zimmermann's normal products [11] with zero condensates", i.e. non-zero vacuum averages of composite op-
vacuum averages, all the log(Q /) terms could only be erators, emerges whenever they are liable to appear according
pumped into the coefficient functions but never got rid of. It to, and in exactly the same form as prescribed by, the phe-
should be clearly understood that the vacuum condensates nomenological rules of [1]. The renormalization group analysis
{O shows that with those condensates we are in a strong-coupling
j} are factorization-prescription dependent quantities, and
the virtue of the MS-scheme consists not in making them non- region, so that their behavior at m
q 0 is hardly predictable
zero, but in collecting within {O at present. However, the fact that the log(m 
q / ) 's fit into the
j} all the logs of m / .
(iv) It is clear from the derivation that if there are heavy par- pattern of vacuum condensates is in itself very nice, for it en-
ticles in the theory and/or there are other large momenta (in the sures that those uncalculable terms, being hidden within con-
case of three- etc. -point functions), all the heavy, i.e. densates, cannot modify the formulae of [1], whatever is the
O(q2 ) ,
interpretation.
parameters will enter only coefficient functions in (7), so that
(7) represents a general form of how the world of "light" con- 7. Acknowledgements. The author is grateful to
fined particles affects the "heavy" world. That is to say, when- S.G. Gorishny and V.F. Tokarev for discussions. The author is
also indebted to K.G. Chetyrkin for promptly turning the
ever the introduction of vacuum condensates is justified from
author's attention to diagrammatic interpretation of the con-
the phenomenological viewpoint of [1], they do appear as an sistency conditions, Eq. (5), and to J. Gasser (Bern Univ.) for
effect of light quark masses already at the perturbative level. an enlightening discussion of the chiral limit.
4. Now, the question arises: does the perturbative analysis
of the vacuum condensates {O References
j} make sense?
Consider the set O 2i
d of the aggregates m O
{ j } with a [1] M.A.Shifman, A.I.Vainshtein and V.I.Zakharov, Nucl. Phys. B147
given total dimensionality d in units of mass (they contribute (1979) 385.
[2] N.N. Bogoliubov, in: Selected papers on statistical physics, Mos-
to the same power of Q-2 and form a closed set with respect cow Univ. Press, 1979, p.193 (in Russian).
to renormalization). The RG equation for Od reads: [3] N.N. Bogoliubov, Doklady Acad. Sci. USSR 82 (1952) 217.
F [4] F.V. Tkachov, in Proc. Int. Seminar Quarks'82, Sukhumi, 5-6 May
L d -
( ) O
m S
-

( ) - O 1982. INR, Moscow, 1982.
HG I
0 , (8)
m S d
NM1-
( ) QPKJ =
[5] K.G. Chetyrkin and F.V. Tkachov, Phys. Lett. 114B (1982) 340.
S m S [6] F.V. Tkachov, Phys. Lett. 124B (1983) 212.
where is the matrix of anomalous dimensions, [7] G. `t Hooft, Nucl. Phys. B61 (1973) 455.
1
( )
( ) 1
b g [8] J.C. Collins, Nucl. Phys. B92 (1975) 477.
m -
= - ( ) , and [9] S.G. Gorishny, S.A. Larin and F.V. Tkachov, Phys. Lett. 124B
(1983) 217.
d
( ) =  m (9)
m S [10] K.G. Chetyrkin, S.G. Gorishny and F.V. Tkachov, Phys. Lett. 119B
d  m ,
B SB (1982) 407.
Eq. (8) is similar in form to the well-known evolution equa- [11] W. Zimmermann, in: Lectures on elementary particles and QFT,
tions for moments of structure functions of DIS (for a review v.1, MIT, Cambridge (Mass.), 1970.
[12] A.V. Efremov and A.V. Radyushkin, Riv. Nuovo Cim. 6 (1981) n.5.
see e.g. [12]). The solution to (8) reads: [13] J. Gasser and H. Leutwyler, Phys. Reports 87 (1982) 77, and refs.
d -a
F m L
(m) O 1 therein.
O (m) = O (m ) 1 a (
m) (10)
K ,
d d 0 HG I
2
m KJ References added in 1998:
0 NM (m )0QP  + +
[14] F.V. Tkachov, preprints INR P-332 and P-358 (1984) [scanned
where ( )
m is governed by . Assuming that images of this and other INR preprints cited below are available
-1 from the KEK preprint library, nos. 8406265  at the
( )
m b
c log(m2/ 2
)h with ~ ~
MS 100 500 MeV , the URL http://ccdb1.kek.jp/KISS.v3/kiss_prepri.html].
conclusion is unavoidable that with O m
d ( ) and light quarks [15] F.V. Tkachov, Int. J. Mod. Phys. A8 (1993) 2047
we are in a deep non-perturbative region. .
[16] G.B. Pivovarov and F.V . Tkachov, preprint INR P-370 (1984)
5. The most important point perhaps is that even if the md [KEK no. 8502210 at the URL of ref. [14]].
factor in (10) prevails in the chiral (i.e. m
q 0 ) limit, e.g. if [17] G.B. Pivovarov and F.V . Tkachov, preprint INR -459 (1986)
q q [KEK no. 8610360 at the URL of ref. [14]].
0 in the above interpretation vanishes as mu,d 0 , such
[18] G.B. Pivovarov and F.V. Tkachov, Int. J. Mod. Phys. A8 (1993)
a possibility is by no means inconsistent with the current- 2241 .
algebra-PCAC phenomenology, as is explained in the appendix [19] F.V. Tkachov, Phys. Part. Nucl. 25 (1994) 649
E of [13]. As explained there, if q q .
0 vanishes in the chiral

3 And nothing else, as was kindly explained to me by J.Gasser.


F.V. Tkachov, Phys. Lett. 125B (1983) 85 3



[20] A.N. Kuznetsov, F.V. Tkachov, and V.V. Vlasov, preprint PSU- be published without reference to any derivation method (as a
92/108-T (1992) ; A.N. Kuznetsov and F.V. calculational trick that just worked), which gave me a possi-
Tkachov, preprint INR-809/93 (1993) . bility to "bootstrap" the publications of the theory of AO --
[21] A.H. Mueller, Nucl. Phys. B250 (1985) 327.
[22] J.C. Collins, D.E. Soper, and G. Sterman, in: Perturbative QCD, a non-trivial task given a complexity and utter novelty of the
A.H. Mueller, ed., World Scientific, 1989. theory (it deviated from the then prevailing BPHZ paradigm
[23] S.A. Larin, F.V. Tkachov, and J.A.M. Vermaseren, Phys. Rev. Lett. from the ground up; for more on this see [19]). In the final ver-
66 (1991) 862. sion of the theory, no reference to the R -operation is neces-
[24] A.N. Kuznetsov and F.V. Tkachov, in Renormalization Group '91. sary, of course, and the latter remains just an exotic applica-
II Int. Conf. Dubna, USSR. 3-6 Sept., 1991. World Scientific, tion. In particular, no knowledge of the R -operation is re-
1992.
[25] F.V. Tkachov, Phys. Lett. B412 (1997) 350 . quired to understand the example.
[26] N.A. Sveshnikov and F.V. Tkachov, Phys. Lett. B382 (1996) 403 Unfortunately, I had to rely on the method of dimensional
. regularization (regularization-independent understanding of the
[27] A.A. Slavnov, Phys. Lett. 98B (1981) 57. method was achieved after this Letter went to press). The use
[28] J.C.Collins and F.V.Tkachov, Phys. Lett. 294B (1992) 403 [hep- of dimensional regularization somewhat obscured the underly-
.
[29] J.M. Cornwall, Phys. Rev. D26 (1982) 1453. ing elementary analytical mechanism.
[30] J.H. Khn, in: Cracow International Symposium on Radiative Cor- The same example without dimensional regularization
rections, Cracow, Poland, 1996. .
[31] F.V. Tkachov, "Perturbation theory with unstable fundamental A regularization-independent treatment of AO to all orders
fields" . was accomplished between 1984 [14] and 1990 (see the sum-
[32] G.P. Korchemsky and A.V. Radyushkin, Sov. J. Nucl. Phys. 45 mary publications [20]). This allows me to explain the example
(1987) 910.
of the Letter without the somewhat confusing use of dimen-
Comments (December, 1998) sional regularization which nullifies certain terms that play an
important role in the mechanism of factorization; the same
The subject of the comments below is to explain the sys- mechanism works to all orders [20].
tematic approach to studying the structure of power corrections The starting point is now the following 4-dimensional ana-
which was demonstrated in this Letter and which remains the log of the integral in (1):
only valid and completely meaningful scenario for studying the
4
structure (not necessarily numerical values) of power- d p
L 1 O
F = lim - z UV 0.1
z ln( / ) ,
suppressed corrections.4 NM ( )4 ( p - Q)2 (m2 + p2
2 ) QP

Bibliographic comments where: z is a finite constant which can be chosen independent
This Letter was in the first burst of publications ([4], [6], of m (this is important; see below); is an intermediate upper
[9]) on what eventually evolved into the theory of Asymptotic cutoff; UV parameterizes the arbitrariness of UV subtraction
Operation (hereafter AO; see [19] and refs. therein, and [25] (the subscript is introduced to explicitly distinguish it from the
for the more recent ultimate extension to arbitrary non- factorization parameter).
Euclidean situations). The formulas discovered within the My purpose here is to simply exhibit the factorization
framework of this vast and powerful theory formed the basis of mechanism for power correction in the final answer, not ex-
an entire industry of multiloop calculations (NNL moments of plain how such a final answer is arrived at via a deterministic
DIS structure functions, [23] and many subsequent publica- procedure of AO (for that, see [15]). So I am going to simply
tions; mass expansions, [30] and refs. therein5). present the final answer in parallel with the formulas in the
main text -- an expansion of F including one power term in
Remarks on the general method which the dependence on the large parameter, Q , is cleanly
The derivation presented in sec. 2 follows the pattern of AO separated ("factorized") from the dependence on m .
explicitly worked out to all orders in [14], [15] (for systematic Since the dependence on UV is logarithmic, it is irrele-
discussions see [15] and the review [19]). At the time of writ- vant whether one considers Q as large or m as a small pa-
ing of this Letter, the notion of AO had not yet crystallized out rameter. As usual in the theory of AO, it is more convenient to
from concrete calculations, instead I emphasized the "extension work with small parameters, i.e. expand in m 0 .
principle" which is a central but somewhat abstract element in Instead of the integral (2), we now write:
prescriptions for writing out formulas of AO for concrete cases.
As to the R -operation mentioned in the first paragraph of 4
L d p 1 1
lim  
sec. 2, it was actually obtained with an essential use of the F z
z ln( / )
0 4
NM - O
UV
(2 ) ( p - Q 2
) p2 QP
method of AO (as explained in [19]) but final formulas could
4
d p R 1 ( p2 2
 ) UR m2 U
+ z S| - < V|S V . 0.2
4 2 2 4
4 ( ) ( ) ( ) T W
Concerning the discussion of the chiral limit in sec. 5 of the main text 2 p - Q 0 - Q p
T| W| -
I only add that, given how nearly impossible it is to do and publish work
that goes against a prevailing dogma created by many fine experts, I would To understand the structure of this expression, compare it with
not be surprised if the possibility of quark condensates vanishing in the the formal expansion of the m -dependent propagator:
chiral limit (although at a slower than canonical rate; cf. (10)) would turn 1 1 2
m
out to be correct in the end. But that is only a possibility which one can = - + 0.3
K
neither prove nor completely exclude. 2 2 2 4
m + p p p
5 Unfortunately, the publications by Khn et al. systematically fail to give In 0.2, the UV subtraction is inherited by the integral corre-
proper credit to the original publications in which the mass expansion for-
mulas they used were discovered, namely, [14][18]. This is the more de- sponding to the first term on the r.h.s. of 0.3, p-2 , whereas
plorable that the resulting misinformation of the research community as the second term, m2 p 4
- , is accompanied by another subtrac-
regards the true source of the underlying mathematical expertise proves to tion and the cutoff ( p2 > ) , which play a role similar to that
be detrimental to the progress of theoretical high energy physics because
the full potential of the theory of AO is far from being exhausted (cf. [31]).


F.V. Tkachov, Phys. Lett. 125B (1983) 85 4



of dimensional regularization and minimal subtraction in the mation in an integral comes not only from the region but first
main text.  will be seen to be the factorization parameter. and foremost from the integrand. This flaw manifests itself at
Eq. (5) is now to be replaced by the following expression: the level of non-leading powers:

p L 1 1 m2 (iv) Indeed, if one starts with splitting the initial integral into
2 2 O
z(m) = - + ( p >  ) ,
z d4 0.4 two regions: | p | < | | and | p| > | | (recall that in this example
(2 )4 NM m2+p2 p2 p4 QP
we work with a Euclidean integral), and performs the corre-
which is finite both at small and large p . One easily finds by a sponding expansions ( Q and m 0 , respectively) then
rescaling p mp that the results would contain various (negative and positive) pow-
z(m) = m2 c ln(m/) ers of  , and keeping track of their eventual cancellations as
1 + c ,
0 0.5
well as reassembling the correct factorization result 0.6, is hard
where both ci are numerical constants. already in one loop and is a hopeless endeavor in higher orders
A similar power-and-log dependence on Q2 (with an over- of PT where the overall mechanism becomes rather delicate;
all Q-2 ) is obtained for the second integral in 0.2 via rescaling cf. [20].
p |Q| p . (v) Plainly put, the mechanism of factorization based on hard
Now the "factorized" expansion is as follows: cutoffs fails with power-suppressed corrections and -- unlike
4
d p 1 an implicit belief behind the "theory of renormalons" [21] --
F = F 2
l ( )( )q ( )
0 + z m p + o m
z 4 cannot be a basis for a meaningful theory of such corrections.
(2 ) ( p - Q 2
)

4 Discussion of Eq. 0.6
d p
= 1
F 2

l q
0 + z(m)  ( p) + o(m
z ) . 0.6
4
(2 ) ( p - Q 2
) (i) Eq. 0.6 is in exact correspondence with Eq. (6): (Q, )
0

(In fact, the remainder is not just o(m2 ) [i.e. vanishes faster is exactly the first integral in the expression for F0 , the r.h.s.
than m2 ] but is actually O(m4 ln m) .) of 0.2; m2C 2
 -
1(Q, )Q corresponds to the second integral on
Correctness of the result 0.6 (i.e. the indicated smallness of the r.h.s. of 0.2; C 2
 -
2 (Q, ) Q is exactly the integral with -
the remainder) is easily checked by combining the r.h.s. and function on the r.h.s. of 0.6; and the aggregate RT2 (0) is
0
l.h.s. and simple algebraic rearrangements of various terms. identified with the z(m) , Eq. 0.4.
The r.h.s. of 0.6 together with 0.4 and the r.h.s. of 0.2 rep- (ii) The integral with the -function exactly corresponds to
resents a decomposition of the initial integral 0.1 into pieces the much used formulas for coefficient functions of OPE found
(without explicit integrations) in such a way as to obtain an in a companion Letter [6] and worked out in [9] (cf. the 3-loop
expansion in powers and logarithms of the expansion parame- calculation [23] that defined the state of the art in DIS OPE
ter (small m or, equivalently, large Q ). Prior to discussing the calculations).
interpretation of various terms in 0.6, let us focus on the pat-
tern of subtractions in the above formulas. (iii) The correspondence RT2 (0) z(m) is a special
0

"Lazy cutoffs" case of the general results on the connection of diagram-wise
UV renormalization (the R-operation) and the subtraction of
(i) Note how the cutoffs like ( p2 2
> ) enter the various asymptotics at large loop momenta (properly interpreted) di-
terms: ( p2 2
> ) is only introduced precisely into those sub- rectly from non-integrated integrands [24], [20].
tracted terms that correspond to the singularities of the formal (iv) It is a characteristic feature of the method of AO that the
expansion, whereas the non-singular term p-2 carries no cut- integrals one obtains in the "factorization" formulas allows a
off. For instance, the expression for z(m) should be regarded direct diagrammatic interpretation in terms of some operators
(this interpretation remains valid in all orders of perturbation -- one never needs any matching conditions.
theory) as obtained by subtracting from the integrand all those (v) The all-orders generalization of the discussed example as
and only those asymptotic terms for p that are responsi- described in the main text after (6) was achieved in [14][18]
ble for UV divergence, whereas the cutoff is introduced only where, moreover, the results were extended to arbitrary models
into the term corresponding to the logarithmic singularity and arbitrary Euclidean asymptotic regimes, as promised in
which necessarily also has a logarithmic singularity at p = 0 , remark (iv) in sec. 3 of the main text.
is why the cutoff is needed in the first place. In other words, (vi) An important update to the remark (i) in sec. 3 of the
the finiteness of the corresponding integrals is ensured by sub- main text is that the dimensional regularization is not crucial
traction of power-behaved terms rather than a hard cutoff. This for achieving the perfect factorization: the analysis of [20]
mechanism can be conveniently called lazy cutoff. shows how the required subtractions (with lazy cutoffs) are
(ii) Recall how factorization is discussed in the language of rather naturally implemented in a regularization-independent
"factorization theorems" (cf. [21], [22]). There, one discourses manner.
(cf. [21]) upon contributions of various integration regions, and (vii) In cases such as gluon condensates, the masslessness of
the factorization parameter  is interpreted as the boundary of perturbative gluons prevents non-zero condensates from ap-
various integration regions. However, a precise mechanism of pearance in lowest orders (due to absence of a mass scale).
factorization as exhibited in our example6 is such that the However, non-zero contributions re-emerge in higher orders
"factorized" integrals collect contributions from all p . (massive quark self-energy insertions).

(iii) So, the interpretation of factorization in terms of (viii) A theoretical estimate of the aggregates such as vacuum
"regions" is fundamentally flawed in one trivial respect: infor- condensates comprising the large logarithms in power-
suppressed corrections requires non-perturbative methods. For

6 Euclidean type objects (e.g. local vacuum condensates), the
Expansions in perfectly factorized form are essentially unique [15], so
one cannot talk about different equivalent representations here.


F.V. Tkachov, Phys. Lett. 125B (1983) 85 5



lattice QCD may provide such estimates. In non-Euclidean them in any way. Their regrouping in a gauge invariant way
situations, no meaningful method exists. may be necessary though (e.g. via studying dependence on the
gauge parameter).
Scenario for non-Euclidean regimes  Note that for identifying the contributing operators, match-
The scheme used in the main text of the Letter can be sum- ing conditions are never needed. Note also that if one follows
marized as follows: the prescriptions of [25] carefully, one never has to invent a
-- perform "perfect factorization" of small parameters such as priori prescriptions for renormalization of the resulting non-
quark masses; local operators (cf. [32]) -- the pattern of necessary subtrac-
-- identify contributions which give rise to large non- tions is dictated by the method of AO unambiguously (apart
perturbative logarithms; from inessential finite ambiguities similar to the difference
-- treat such PT-uncalculable contributions as phenomenologi- between MS and MS-bar renormalization schemes).
cal parameters.  Upon identification and classifications of independent "non-
In other words, the small masses etc. are used as a test of sta- perturbative" integrals, one simply treats them as phenomenol-
bility of perturbation theory in order to identify the structure of ogical constants.
potentially large non-perturbative corrections.  One should replace such integrals with constants as a
whole, i.e. without attempting to scale out powers of small
The argument that the condensates one thus obtains are not masses because such a possibility is a special feature of pertur-
"the" condensates of [1] is beyond the point for two reasons: bation theory. That a typical scale in such integrals should be
1) the large ln m contributions cannot be phenomenologically rather than, say, quark masses was clear since at least the
distinguished from contributions of, say, instantons; studies of QCD via Dyson equations in the early 80s [29]. Arti-
2) the usual analyses such as [1] also rely on PT results in re- ficial arguments such as based on Borel resummation of Borel-
gard of the structure of power-suppressed corrections (the non-resummable QCD PT series [21] cannot add anything new
OPE used in [1] was the perturbative OPE somewhat dramati- here as a matter of principle.
cally misinterpreted due to ignoring the non-zero large ln m  If one has trouble identifying independent integrals, the best
contributions which in Zimmermann's version of OPE are hid- solution is to consider the next order of PT (one does not need
den in coefficient functions); in this respect our scenario is only to do the integrals explicitly, only write out the prescriptions of
a straightforward clarification of the reasoning behind [1]. AO [25] explicitly, which requires only patience) where radia-
tive corrections would allow one to distinguish different op-
This scenario is applied to non-Euclidean situations in a erator objects.
straightforward manner (as was mentioned in [26] in connec-  To facilitate investigation of contributions involving gluon
tion with the physics of jets). All one needs is a generalization operators (which to lowest order may be zero due to perturba-
of the method of AO that allows treatment of arbitrary asymp- tive masslessness of the gluon), one may force an early appear-
totic regimes and phase space integrals along with loops; such ance of non-zero gluon operators by introducing a gluon mass
a generalization was found in [25]. One proceeds as follows:
 (concerning gauge invariance of such a procedure consult [27]).
One starts from complete diagrams with non-zero masses This is just a trick because quark self-energies would give non-
for quarks, ..., non-zero p , etc. zero contributions from higher orders anyway.
 One performs diagram-by-diagram factorization of the small  The examples I studied show that the breakdown of dimen-
parameters (say, < 1 GeV ) using the algorithms of non- sional regularization observed in [28] in a semi-artificial ex-
Euclidean AO [25]. Note that the method of AO yields ample, is quite typical in the case of power corrections to, say,
"perfect" expansions in which the large logarithms of small pa- jet observables, so one must exercise care with intermediate
rameters are localized in certain integrals. This is in contrast regularizations.
with the conventional methods; e.g. the version of OPE due to  The success of the QCD sum rules method [1] where one
Zimmermann had such logarithms hidden in coefficient func- dealt with essentially Euclidean regimes and was able to
tions, which phenomenologically is incorrect (see [19]). parameterize many observables with a few phenomenological
 Among various pieces of the resulting factorized expression, constants, is not likely to be repeated in non-Euclidean situa-
one identifies (using scalings like those used to obtain 0.5) the tions where one encounters a wider variety of non-perturbative
integrals which comprise the large logarithms of small masses. operator objects in addition to local operators and has to deal
The simple argument ln Q / m ln Q / - ln m / shows that if with unknown phenomenological functions of kinematic argu-
one normalizes ments, sometimes in infinite number.
S at  of order Q , a typical scale of the proc-  Moreover, whereas vacuum condensates (that are the only
ess being studied, then the logarithms of masses are automati- type of non-perturbative coefficients in Euclidean situations)
cally large, so neither analysis, nor even knowledge of the cor- can be estimated from, say, the lattice QCD, for the non-
responding evolution equations is really necessary (the data on Euclidean situations no useful and meaningful method exists.
power corrections are never precise enough to warrant that).
  Finally, a critical examination of renormalons-based results
One classifies independent "non-perturbative" integrals (i.e. demonstrates that their concrete content is always traced back
ones that give rise to large logarithms) which would then cor- to (rather imperfect) analysis of diagrams, whereas the renor-
respond to independent operator objects (being lower-orders malon phraseology effectively reduces to the unoriginal propo-
contributions to matrix elements of the latter). Note that inter- sition that the corresponding aggregates are non-perturbative
preting such objects in, say, coordinate representation as local without providing any clue as to how one could compute them.
etc. operators serves no useful purpose except, perhaps, as an
inessential notational convenience. Anyway, even if one de-
cides that one needs such an operator interpretation, one only
needs to recognize the emerging integrals (by simply looking at
them) as corresponding to some or other operator objects (not
necessarily local operators) -- one does not need to massage



