Using an OPE modification, which takes non-perturbative non-locality into
account, the difference and sum of vector and axial-vector correlators are
evaluated in euclidean position space. For distances up to 0.8 fm the
calculated behavior is close to the instanton liquid result and the ALEPH data,
in contrast to the implications of the conventional OPE.Comment: 4 pages, 2 figure

Barate

Bode

D'Elia

deGrand

Di Giacomo

Dosch

Hofmann

Ralf Hofmann

Schäfer

Shifman

Physics Letters B

English

arXiv

Crossref

Non-perturbative non-locality: V±A correlators in euclidean position space

AbstractUsing an OPE modification, which takes non-perturbative non-locality into account, the difference and sum of vector and axial-vector correlators are evaluated in euclidean position space. For distances up to 0.8 fm the calculated behavior is close to the instanton liquid result and the ALEPH data, in contrast to the implications of the conventional OPE

Hofmann, Ralf

Elsevier - Publisher Connector 

15 November 2001
Physics Letters B 520 (2001) 254–256
www.elsevier.com/locate/npe
Non-perturbative non-locality:
V ±A correlators in euclidean position space
Ralf Hofmann
Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, 80805 München, Germany
Received 10 September 2001; accepted 25 September 2001
Editor: P.V. Landshoff
Abstract
Using an OPE modification, which takes non-perturbative non-locality into account, the difference and sum of vector and
axial-vector correlators are evaluated in euclidean position space. For distances up to 0.8 fm the calculated behavior is close to
the instanton liquid result and the ALEPH data, in contrast to the implications of the conventional OPE.
 2001 Published by Elsevier Science B.V.
The determination of the momentum or distance
range in which a truncated operator product expansion
(OPE) in QCD approximates well is very important
for many applications. Based on global quark–hadron
duality the low-energy precision measurements of the
spectral functions of vector and axial-vector channels
(ALEPH experiment τ → ντ + hadrons [1]) could
be confronted with the associated OPE’s [2] but
also with the instanton liquid model [3] and lattice
calculations [4].
In this Letter we investigate how OPE-based non-
perturbative non-locality affects the behavior of the
V ± A correlators in euclidean position space. The
results are compared to the instanton liquid data of [3]
and the conventional OPE approach.
Working in the chiral limit, in which both ΠV,Aµν are
transverse, and applying naive vacuum saturation at
dimension 6, we obtain the following truncated OPE’s
E-mail address: ralfh@mppmu.mpg.de (R. Hofmann).
[3] at a large normalization scale Q0
RV−A(x)|Q0 ≡
ΠV (x)−ΠA(x)
2Π0(x)
∣∣∣∣
Q0
= π
3
9
αs(Q0)〈q¯q〉2Q0 log
[
(xQ0)
2]x6,
RV+A(x)|Q0 ≡
ΠV (x)+ΠA(x)
2Π0(x)
∣∣∣∣
Q0
= 1− π
2
96
〈
αs
π
(
Faµν
)2〉
Q0
x4
(1)
− 2π
3
81
αs(Q0)〈q¯q〉2Q0 log
[
(xQ0)
2]x6.
Thereby, we have only taken into account the leading-
order αs corrections in each mass dimension. Π0
denotes the parton-model result which is equal for
V ±A. The V −A correlator is free of purely pertur-
bative contributions and only sensitive to chiral sym-
metry breaking. On the other hand, the V + A cor-
relator includes pure perturbation theory and more-
0370-2693  2001 Published by Elsevier Science B.V.
PII: S0370-2693(01) 01 16 7- 4
Open access under CC BY license.
Open access under CC BY license.
R. Hofmann / Physics Letters B 520 (2001) 254–256 255
over should directly reflect non-perturbative gluonic
dynamics. Since the running of αs almost cancels
the log-powers at dimension 6, which come from the
anomalous dimensions of the contributing operators,
we can consider αs to be fixed at scale Q0 [2].
A modification of the conventional OPE was pro-
posed in [5] and applied to light-quark channels in [6].
Since the idea is explained at length in these two pa-
pers we can be short here. For dimension 4 it was
argued in [5] that a finite correlation in the non-
perturbative piece of the associated gauge invariant
correlation function must decrease the relevance of
fundamental field operators at low resolution. Coarse
graining the vacuum expectation values (VEV’s) of di-
mension 4 operators, an evolution equation was ob-
tained which has the following solution in euclidean
momentum space
(2)A(Q)=A(Q0) exp
[
− 4
5λ
(
1
Q
− 1
Q0
)]
.
Thereby, A(Q0) is the VEV of a fundamental, gauge
invariant composite and Q0 the “fundamentality”
scale down to which the description in terms of funda-
mental fields is sufficiently accurate. The (resolution
independent) correlation length is denoted by λ.
Assuming naive vacuum saturation [2] in the sense
of (1) in [6], the relevant 2 point functions to evalu-
ate the modified version of Eq. (1) are the gauge in-
variant scalar quark and field strength correlators [7]
which have been measured on the lattice [8]. The fol-
lowing parameter values (q = quark, g = gluon) were
obtained for NF = 4 (staggered) and a quark mass
am= 0.01
λq = 3.1 GeV−1,
Aq
(
a−1 =Q0 ∼ 2 GeV
)= 0.212 (GeV)3,
λg = 1.7 GeV−1,
(3)Ag
(
a−1 =Q0 ∼ 2 GeV
)= 0.015 (GeV)4.
Using (3) and αs(2 GeV) = 0.2 [9] as inputs, ap-
pealing to Eqs. (1), (2) (substituting Q = 1/x in (2)),
we are now in a position to calculate RV±A. Figs. 1
and 2 show the results. For a comparison with the
ALEPH data see Ref. [3] where a spectral threshold
s0 = 2.5 GeV2 for the onset of perturbation theory
is used in the x-space dispersion relation. Due to the
moderation by exponential factors the power-like blow
up of the non-perturbative corrections in conventional
Fig. 1.RV−A as a function of distance x in the chiral limit. The solid
line corresponds to the non-perturbatively coarse grained OPE, the
dashed line to the conventional OPE. Crosses depict the result of the
instanton liquid calculation of [3] which is taken from [4].
Fig. 2. Same as in Fig. 1, but now for RV+A.
OPE’s is not observed, and the modified OPE seems to
converge also at larger distances than the 0.3 fm deter-
mined in [3]. Extrapolating the behavior of the lowest
dimensions under non-perturbative coarse graining to
higher dimensions d < dc, each correction developes
a maximum at xd = d × λd . Thereby, dc denotes the
critical dimension of the asymptotic expansion which
even the modified OPE is believed to be [5]. It is
a rather striking observation that our solid lines in
Figs. 1 and 2 match the lattice results of Ref. [4] al-
most perfectly. In quenched QCD this lattice calcula-
tion is based on the evaluation of the (truncated) over-
lap quark propagator which subsequently is used to
build the correlators.
256 R. Hofmann / Physics Letters B 520 (2001) 254–256
To summarize we have shown that the concept of
a non-perturbative component of OPE-based coarse
graining leads in a wider range of distances to more
realistic results for the euclidean V ±A correlators in
position space than the conventional OPE’s do.
Acknowledgements
The author would like to thank Uli Nierste for
a stimulating conversation. Financial support from
CERN’s theory group during a research stay in June
are gratefully acknowledged. The author is indebted
to V.I. Zakharov for numerous useful discussions,
valuable comments, and the encouragement to write
this Letter.
References
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[3] T. Schäfer, E.V. Shuryak, Phys. Rev. Lett. 86 (2001) 3973.
[4] T. deGrand, hep-lat/0106001.
[5] R. Hofmann, hep-ph/0109007.
[6] R. Hofmann, hep-ph/0109008.
[7] H.G. Dosch, Phys. Lett. B 190 (1987) 177;
H.G. Dosch, Yu.A. Simonov, Phys. Lett. B 205 (1988) 339.
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Non-perturbative non-locality: V±A correlators in euclidean position space 

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