UAB-FT-630
UB-ECM-PF-07-07
Constraints on Regge models from perturbation theory
Jorge Mondejara and Antonio Pinedab
a Dept. d’Estructura i Constituents de la Matèria
U. Barcelona, Diagonal 647, E-08028 Barcelona, Spain
b Grup de F́ısica Teòrica and IFAE, Universitat Autònoma de Barcelona, E-08193
Bellaterra, Barcelona, Spain
Abstract
We study the constraints that the operator product expansion imposes on large Nc inspired
QCD models for current-current correlators. We focus on the constraints obtained by going
beyond the leading-order parton computation. We explicitly show that, assumed a given
mass spectrum: linear Regge behavior in n (the principal quantum number) plus corrections
in 1/n, we can obtain the logarithmic (and constant) behavior in n of the decay constants
within a systematic expansion in 1/n. Our example shows that it is possible to have different
large n behavior for the vector and pseudo-vector mass spectrum and yet comply with all
the constraints from the operator product expansion.
http://arxiv.org/abs/0704.1417v3
1 Introduction
The operator product expansion (OPE) has been used since long in order to gain information
on the non-perturbative dynamics of the hadronic spectrum and decays [1, 2, 3, 4, 5, 6,
7, 8, 9]. In this article we revisit this problem. We want to obtain the constraints that
the knowledge of the perturbative expansion in αs(Q
2) of the current-current correlators
in the Euclidean regime poses on the relation between the decay constants and the mass
spectrum for excitations with a large quantum number n (where n is the the quantum
number of the bound state). We put special emphasis in going beyond the leading-order
parton computation. We will work with a specific model for the hadronic spectrum. This is
compulsory, since different spectral functions1 may yield the same OPE expression, yet we
believe some aspects of our discussion may hold beyond the assumptions of our model.
In order to have a well defined bound state it is crucial to consider the large Nc approx-
imation [10]. This ensures infinitely narrow resonances at arbitrarily large energies. We will
consider to be in the large Nc limit in what follows, as well as in the exact chiral (massless)
limit. We will then set a specific model for the hadronic spectrum, valid for large values of
n (we only need the behavior of the spectrum and decays for large n, we do not aim to get
any information from perturbation theory for low values of n). This model will be based
on the Regge behavior plus corrections in 1/n that will be included in a systematic way.
The model is based on the assumption that the Regge behavior is a good description of the
spectrum for large n (this can be explicitely seen in the ’t Hooft model [11] and it is also
consistent with phenomenology). Given the 1/n corrections to the mass spectrum, the ex-
pression of the correlator can also be written as a systematic expansion in 1/n, where higher
powers in 1/n are equivalent to higher orders in 1/Q2 in its OPE. By matching the OPE
and hadronic expressions order by order in 1/Q2, we will be able to predict the logarithmic
dependence on n of the decay constants (actually also the constant terms). This result can
also be systematically organized within an expansion in 1/n together with an expansion in
1/ lnn. We will give explicit expressions up to order 1/n2 and 1/ ln3 n. We will also make
some numerical estimates of the impact of these corrections. Finally, we would like to stress
that we are able to introduce power corrections in 1/n to the Regge behavior and yet comply
with the OPE. This is in contrast with Ref. [5], where, besides the Regge behavior, only
exponentially suppressed terms are introduced (parametrically smaller than any finite power
of 1/n for large n). This parameterization is however fine if considered as a fit not emanated
from the large n limit.
2 Correlators
For definiteness, we will consider the vector-vector correlator but most of the discussion
applies to any other current-current correlator (axial-vector, scalar, ....).
V (q) ≡ (q
µqν − gµνq2)ΠV (q) ≡ i
d4xeiqx〈vac|T {J
V (x)J
V (0)} |vac〉 , (1)
1In particular the one derived directly from perturbation theory, which we do not consider, since we will
work in the large N
limit with infinitely narrow resonances.
where J
Qf ψ̄fγ
µψf . In order to avoid divergences, we will consider the Adler function
A(Q2) ≡ −Q2
Π(Q2) = Q2
(t+Q2)2
ImΠV (t) , (2)
where Q2 = −q2 is the Euclidean momentum.
Since we are working in the large Nc limit, the spectrum consists of infinitely narrow
resonances, and the Adler function can be written in the following way
A(Q2) = Q2
F 2V (n)
(Q2 +M2V (n))
. (3)
On the other hand, for large positive Q2, one may try to approximate the Adler function
by its OPE, which reads
AOPE(Q
C(αs(Q
β(αs(ν))〈vac|G
2(ν)|vac〉+O
where αA(Q
2) admits an analytic expansion in terms of αs(Q
2) (computed in the MS scheme),
β(αs) = −β0
+ · · · , (5)
with β0 = 11/3Nc, β1 = 34/3N
c , β2 = 2857/54N
c , and [12]
C(αs(Q
2)) = −
+ · · ·
. (6)
3 Matching
High excitations of the QCD spectrum are believed to satisfy linear Regge trajectories:
M2V,n
= constant.
For generic current-current correlators, such behavior is consistent with perturbation
theory in the Euclidean region at leading order in αs if the decay constants are taken to be
“constants”, ie. independent of the principal quantum number n.
The inclusion of subleading effects in αs can be incorporated into this model by changing
the n dependence of the decay constants without changing the ansatz for the spectrum. The
inclusion of these effects has consequences on subleading sum-rules and the relation with the
non-perturbative condensates.
Here we would like to go beyond the analysis at leading order in αs, as well as to consider
power-like corrections in 1/n. We will consider that the large n expression for the mass
spectrum can be organized within a 1/n expansion in a systematic way starting from the
asymptotic linear Regge behavior. In order to fix (and simplify) the problem we will assume
that no lnn term appears in the mass spectrum2. Therefore, we write the mass spectrum in
the following way (for large n)
M2V (n) =
(−s) = B
V n+B
+ · · · , (7)
where B
V are constants. We will usually denote M
V,LO(n) = B
V n, M
V,NLO(n) = B
V and so on. To shorten the notation, we will denote B
V = BV , B
V = AV and B
For the decay constants, we will have a double expansion in 1/n and 1/ lnn.
F 2V (n) =
F 2V,s(n)
= F 2V,0(n) +
F 2V,1(n)
F 2V,2(n)
+ · · · , (8)
where the coefficients F 2V,s(n) have a logarithmic dependence on n:
F 2V,s(n) =
V,s(n)
lnr n
. (9)
As we did with the masses, we will define F 2V,LO(n) = F
V,0(n), F
V,NLO(n) = F
V,0(n) +
and so on. Note that in this case we also have an expansion in 1/ lnn.
We are now in position to start the computation. Our aim is to compare the hadronic
and OPE expressions of the Adler function within an expansion in 1/Q2, but keeping the
logarithms of Q. In order to do so we have to arrange the hadronic expression appropiately.
Our strategy is to split the sum over hadronic resonances into two pieces, for n above or
below some arbitrary but formally large n∗, such that ΛQCDn
∗ ≪ Q. The sum up to n∗
can be analytically expanded in 1/Q2 and will not generate lnQ2 terms (neither a constant
term at leading order in 1/Q2). For the sum from n∗ up to ∞, we can use Eqs. (7) and (8)
and the Euler-MacLaurin formula to transform the sum in an integral plus corrections in
1/Q2. Whereas the latter do not produce logarithms, the integral does. These logarithms of
Q are generated by the large n behavior of the bound states and the introduction of powers
of 1/n is equivalent (once introduced in the integral representation, and for large n) to the
introduction of (logarithmically modulated) 1/Q2 corrections in the OPE expression.
Therefore, by using the Euler-MacLaurin formula, we write the Adler function in the
following way (B2 = 1/6, B4 = −1/30, ...)
A(Q2) = Q2
F 2V (n)
(Q2 +M2V (n))
F 2V (n)
(Q2 +M2V (n))
F 2V (n)
(Q2 +M2V (n))
F 2V (n
(Q2 +M2V (n
(−1)k
|B2k|
(2k)!
d(2k−1)
dn(2k−1)
F 2V (n)
(Q2 +M2V (n))
, (10)
2This is a simplification. If one considers, for instance, the ’t Hooft model [11], lnn terms do indeed
appear.
where n∗ stands for the subtraction point we mentioned above, such that for n larger than
n∗ one can use the asymptotic expressions (7) and (8). This allows us to eliminate terms
that vanish when n → ∞. Note that the last sum in Eq. (10) is an asymptotic series, and
in this sense the equality should be understood.
Note also that for n below n∗, we will not distinguish between LO, NLO, etc... in masses
or decay constants, since for those states we will not assume that one can do an expansion
in 1/n and use Eqs. (7) and (8).
Finally, note that the expressions we have for the masses and decay constants become
more and more infrared singular as we go to higher and higher orders in the 1/n expansion.
This is not a problem, since we always cut off the integral for n smaller than n∗. Either way,
the major problems would come from the decay constants, since, in the case of the mass, Q2
effectively acts as an infrared regulator.
3.1 LO Matching
We want to match the hadronic, Eq. (10), and OPE, Eq. (4), expressions for the Adler
function at the lowest order in 1/Q2. This means that we have to consider the lowest order
expressions in 1/n for the masses and decay constants, i.e. F 2V,LO(n) and M
V,LO(n), since
the corrections in 1/n give contributions suppressed by powers of 1/Q2.
Only the first term in Eq. (10) can generate logarithms or terms that are not suppressed
by powers of 1/Q2. Therefore we obtain the following equality,
Apt.(Q2) ≡ Q2
F 2V,LO(n)
(Q2 +M2V,LO)
. (11)
This equation can be fulfilled by demanding that
F 2V,LO(n)
|dM2V,LO(n)/dn|
pert.
V,LO(n)) . (12)
By using the perturbative expression for ImΠ
pert.
V (see [13]), we obtain
F 2V,LO(n) = BV
Ncαs(nBV ) +
243− 176 ζ(3)
128π2
N2c α
s (nBV ) (13)
346201− 2904π2 − 324528 ζ(3) + 63360 ζ(5)
27648π3
N3c α
s (nBV ) +O
α4s (nBV )
where αs(nBV ) should actually be understood as a function of αs(BV ) and lnn. Therefore,
it is obvious that the above expression is resumming powers of αs(BV ) lnn:
F 2V,LO(n) = BV
1 + 11
αs(BV )
ln(n)
αs(BV )
2673− 1936 ζ(3)− 408 ln
1 + 11
Ncαs(BV ) ln(n)
88(1 + 11
αs(BV )
ln(n))2
N2c α
s (BV )
(4π)2
N3c α
s (BV )
(4π)3
52272π(1 + 11
αs(BV )
ln(n))3
[−350427Ncαs(BV ) ln(n)
+121π
346201− 2904π2 − 324528 ζ(3) + 63360 ζ(5)
−3672π(2877− 1936 ζ(3)) ln
αs(BV ) ln(n)
+749088π ln2
αs(BV ) ln(n)
α4s (BV )
Doing so we see that we are able to obtain the dependence of the decay constant in lnn
(somewhat we are assuming that αs(BV ) is an small parameter, BV ∼ 1 GeV).
We can also rewrite the decay constant as an expansion in 1/ lnn by using the equality
ln ñ =
αs(nBV )
αs(nBV )
αs(nBV )
, (15)
where ñ = nBV /ΛMS. We then obtain
F 2V,LO(n) = BV
ln ñ
ln2 ñ
ln ln ñ +
ln3 ñ
46818
161051
ln2 ln ñ +
322102
(−2877 + 1936ζ(3)) ln ln ñ +
42272605
2576816
20283 ζ(3)
360 ζ(5)
ln4 n
. (16)
Note that the lowest contribution in 1/ lnn for the decay constant, BV
Q2f ,
which, usually, is the only one considered, reproduces the leading-order partonic prediction
for the Adler function.
Note also that there is no problem with the Landau pole, even if the result is written in
the form of Eq. (16), since it holds only for n larger than an n∗ such that ΛMS ≪ n
∗BV (the
integral has an infrared cutoff at n∗).
Finally, we remind that, strictly speaking, we can only fix the ratio between the decay
constant and the derivative of the mass. We have fixed this ambiguity by arbitrarily imposing
the n dependence of the mass spectrum.
3.2 NLO matching
We now want to obtain extra information on the decay constant by demanding the validity
of the OPE at O(1/Q2), in particular the absence of condensates at this order. We then
have to use the NLO expressions for M2V (n) and F
V (n). With the ansatz we are using for
the mass at NLO, it is compulsory to introduce (logarithmically modulated) 1/n corrections
to the decay constant if we want this constraint to hold. Note that it is possible to shift all
the perturbative corrections to the decay constant.
Imposing that the 1/Q2 condensate vanishes produces the following sum rule:
Apt. −
Apt. +
F 2V (n)−
dnF 2V,LO(n)
F 2V (n
(−1)k
|B2k|
(2k)!
d(2k−1)
dn(2k−1)
F 2V (n)
F 2V,1(n)/n
(Q2 +M2V,LO(n))
F 2V,1(n)/n
(Q2 +M2V,LO(n))
= 0 .
This equality should hold independently of the value of n∗, which formally should be taken
large enough so that αs(n
∗BV ) ≪ 1, i.e. the limit ΛMS ≪ n
∗BV ≪ Q
2. Again, the meaning
of the asymptotic series appearing in Eq. (17) should be taken with care. If we forget about
this potential problem, only a few terms in Eq. (17) can generate lnQ2 terms, which should
cancel at any order. Those are the first two and the last two terms. Actually, the next to last
term does not generate logarithms, but it allows to regulate possible infrared divergences
appearing in the calculation. Therefore, asking for the cancellation of the 1/Q2 suppressed
logarithmic terms produced by the first two and the last term in Eq. (17) fixes F 2V,1. The
non-logarithmic terms should also be cancelled but they cannot be fixed from perturbation
theory.
One can actually find an explicit solution to the above constraint for F 2V,1 by performing
some integration by parts. We obtain
F 2V,1(n)
F 2V,0(n) (18)
ln2 ñ
(1− 2 ln (ln ñ)) +
36 ζ(3)
ln3 ñ
2576816
−45794053 + 351384π2 + 41637552 ζ(3)− 7666560 ζ(5)
−3672 ln (ln ñ) (−3013 + 1936 ζ(3) + 204 ln (ln ñ))]
ln4 ñ
ln5 ñ
or in terms of αs(nBV ) or αs(BV ),
F 2V,1(n)
N2c α
s (nBV )−
2877− 1936 ζ(3)
768π3
N3c α
s (nBV ) (19)
11(376357− 2904π2 − 344112 ζ(3) + 63360 ζ(5))
110592π4
N4c α
s (nBV ) +O
α5s (nBV )
F 2V,1(n)
(1 + 11
αs(BV )
ln(n))2
α2s (BV )
(4π)2
2877− 1936 ζ(3)− 408 ln
1 + 11
Ncαs(BV ) ln(n)
12(1 + 11
αs(BV )
ln(n))3
α3s (BV )
(4π)3
4752π(1 + 11
αs(BV )
ln(n))4
α4s (BV )
(4π)4
[−233618Ncαs(BV ) ln(n)
+121π
376357− 2904π2 − 344112 ζ(3) + 63360 ζ(5)
−3672π(3013− 1936 ζ(3)) ln
αs(BV ) ln(n)
+749088π ln2
αs(BV ) ln(n)
α5s (BV )
Note that besides the 1/n suppression, we also have an extra α2s (nBV ) suppression.
In principle one could think of the existence of 1/n×constant terms in the decay constant,
i.e. without any associated logarithm. However, such terms produce ln(Q2) contributions in
the Euclidean regime that do not appear in the perturbative computation, so they can be
ruled out. This appears to be a general statement since 1/nm × constant for any m integer
also produces logarithms. Note that in order to give meaning to these integrals it is implicit
that the integral over n has an infrared cutoff at n∗. Nevertheless, the logarithm does not
appear to multiply powers of the infrared cutoff (as expected).
Finally, we would like to mention that, besides the constraints coming from the logarith-
mic related behavior of the OPE, there is also the constraint from its constant terms, which
should sum up to zero. Nevertheless, for this constraint we cannot give a closed expression.
This is due to the fact that the lnQ2 independent terms may receive contributions from
any subleading order in the 1/n expansion of the masses and decay constants. The reason
is that the decay constant at a given order in 1/n is obtained after performing some inte-
gration by parts, which generate new (lnQ2-independent) terms that can be Q2 enhanced.
This statement is general and also applies to any subleading power in the 1/Q2 matching
computation.
3.3 NNLO matching
We now consider expressions for the mass and decay constants at NNLO. For the first time
we have to consider condensates. Simplifying terms that do not produce logs, we obtain the
following equation,
β(αs(ν))〈vac|G
2(ν)|vac〉
(Q2 +BV n)2
F 2V,2(n)
V,0(n)
V,1(n)
where
= stands for the fact that we can only predict the lnQ2 dependence. Constant terms
are not fixed by this relation.
In order to get a more closed expression is convenient to use the following equality,
(Q2 +BV n)2
BV n2
α2s (nBV ) , (22)
valid up to terms that do not produce logarithms or those that are subleading.
We get then
F 2V,2(n) = −CV
Ncαs(nBV )
287− 176 ζ(3)
128π2
11A2V
64π2BVCV
β(αs(ν))〈vac|G
2(ν)|vac〉
BVCVN2c
N2c α
s (nBV ) (23)
α3s (nBV )
Note that in this case we only consider up to O(α2s (nBV )) corrections, since higher order
loops are unknown. The accuracy is set by our knowledge of the matching coefficient of the
gluon condensate. Note also that F 2V,2(n) does not have αs suppression. Therefore, for low n,
this contribution could be practically of the same size than, formally, more important terms.
4 Axial versus vector correlators
The above discussion has been performed for the vector-vector correlator Adler function.
It goes without saying that we could perform a similar analysis with axial-vector currents,
since the perturbative expansions for both correlators are equal. Here it comes an important
observation. We could change the coefficients for the mass spectrum BA 6= BV , AA 6= AV ,
· · · , yet we would obtain the same expression for the OPE (at the order we are working, the
first chiral breaking related effects are O(1/Q6)). Therefore, we conclude that the OPE does
not fix BA = BV as it is sometimes claimed in the literature [1, 3]
3. Our computation gives
a specific counter example. Moreover, it is nice to see what the role played by BA and BV
is in our case. When one goes to the Euclidean regime, BA and BV become renormalization
factorization scales and, obviously, the physical result does not depend on them (for large
Q2 in the Euclidean). On the other hand, it is evident that having different constants: BA,
BV , . . . produces different physical predictions for the masses and decay widhts for vector or
axial-vector channels. The point to be emphasized is that BA = BV cannot come from an
OPE analysis alone. This point has already been stressed in Refs. [4, 8], what we think is
novel in our analysis is that we have seen that the inclusion of corrections in αs does not
affect that conclusion, and that BA and BV play the role of the renormalization scale in the
analogous perturbative analysis in the Euclidean regime, and are therefore unobservable.
Finally, we cannot avoid mentioning the analysis of Ref. [14] where, using AdS/CFT, they
explicitly find Regge behavior with different slopes for vector and axial-vector channels.
3Another issue, on which we do not enter, is whether some other kind of arguments (relying on the specific
model used), like semiclassical arguments, may fix those parameters to be equal.
In any case, even though the constants that characterize the spectrum can be different
for the vector and axial-vector channel, they have to yield the same expressions for the OPE
when combined with the decay constants. This produces some relations that we list in what
follows. We first define t ≡ BV n = BAn
′ and take n and n′ as continuous variables. We then
obtain the following equalities
F 2V,LO(n)
F 2A,LO(n
pert.
V (t) ≡ f0(t) , (24)
F 2V,1(n)
F 2A,1(n
f0(t) , (25)
F 2V,2(n)
V,0(n)
V,1(n)
F 2A,2(n
A,0(n
A,1(n
β(αs(ν))〈vac|G
2(ν)|vac〉f1(t) , (26)
where
f1(t) =
α2s (t)
(4π)2
+ · · · . (27)
5 Numerical Analysis
n = 1 n = 2 n = 3 n = 4
Mρ(I) 781.3(775.5± 0.4) 1440.2(1459± 11) 1891.8(1870± 20) 2257(2265± 40)
Mρ(II) 771.5(775.5± 0.4) 1471.7(1459± 11) 1855(1870± 20) 2154.8(2149± 17)
Ma1 1235.6(1230± 40) 1621.7(1647± 22) 1962(1930
−70) 2257.8(2270
FV (I) 156(156± 1) 155 154 153
FV (II) 185(156± 1) 147 139 135
FA 123(122± 24) 137 139 139
Table 1: We give the experimental values of the masses (in MeV) and electromagnetic decay
constants (when available) for vector and axial vector particles (within parenthesis), compared
with the values obtained from the fit. For the vector states we consider two possible Regge
trajectories that we label I and II respectively. We take αs(1GeV) = 0.5 and β〈G
−(352MeV)4.
We restrict ourselves to the SU(2) case (non-strange sector) and study the vector and
axial-vector channels. We would like here to assess the importance of including perturbative
corrections to a standard analysis based on the OPE. We do not aim to perform a full fledged
2 4 6 8 10
FV,LOHIL
2 4 6 8 10
FV,LOHIIL
Figure 1: In this plot we show FV,LO(I) and FV,LO(II) at different orders in αs.
analysis, but only to see the importance of the corrections. In table 1, we give the values
of the masses and decay constants. In Figure 1 we show the changes in both FV,LO(I) and
FV,LO(II) as we include higher orders in the expansion in αs, and in Figure 2 the changes
in the full FV (I) and FV (II) as we include higher orders in 1/n. In figure 3 we show the
same plots for the axial-vector case. We take the experimental values from Ref. [15]. In
principle there are more states in the particle data book, in particular in the vector channel.
Nevertheless, it is not clear whether they belong to the same Regge trajectory or whether
they belong to some daughter one, see, for instance, the discussion in Ref. [5]. For the time
being we will disregard the study of other possible (vector) Regge trajectories and restrict
the analysis to a single trajectory. We will consider the two possibilities listed in Table 1.
Our choice of states for the set (I) is motivated by the discussion of Ref. [16] on the possible
formation of multiplets in the case of chiral symmetry restoration. The set (II) is based on
the assignment of states made in Ref. [5] (based on the existence of S and D-wave daughter
trajectories) and in particular on the analysis of Ref. [17], where the state 2265 is argued to
belong to the D-wave Regge trajectory4.
In order to fix the parameters of the mass spectrum we use the experimental values of
the masses we list in the table. We obtain the values:
BV (I) = 1.525× 10
6MeV2 , AV (I) = −1.038× 10
6MeV2 , CV (I) = 0.123× 10
6MeV2 ,
BV (II) = 1.128× 10
6MeV2 , AV (II) = 0.353× 10
6MeV2 , CV (II) = −0.885× 10
6MeV2 ,
BA = 1.278× 10
6MeV2 , AA = −0.100× 10
6MeV2 , CA = 0.349× 10
6MeV2 .
We should mention that the values obtained for these parameters are not very stable
under the change of number of data points, except for BV and BA, which are roughly stable,
although with quite sizeable uncertanties. For the subleading terms A and C, their values
are basically random with the fit. We roughly find BV ≃ BA within the uncertainties. The
n dependence of the axial and vector (model II) decay constants is small but sizeable (and it
goes in the right direction for low n). The 1/n corrections are always corrections compared
with the leading order terms. Nevertheless, the 1/n2 correction is much larger than the 1/n
one for the range of values of n that we explore. This appears to be due to the α2s/(4π)
suppression of the 1/n term, as well as to the difference in size between the constants A
4We also thank S. Afonin for discussions on this point.
2 4 6 8 10
FVHIL
2 4 6 8 10
132.5
137.5
142.5
147.5
FVHIIL
Figure 2: In this plot we show FV (I) and FV (II) at different orders in the 1/n expansion.
2 4 6 8 10
FA,LO
2 4 6 8 10
Figure 3: In this plot we show FA,LO and FA at different orders in αs and in the 1/n
expansion, respectively.
and C. This is so for the axial and vector (model II) decay constants. Nevertheless, for the
vector (model I) decay constants the n dependence appears to be quite small also at NNLO.
This appears to be due to the small value of the coefficient CV (I). The gluon condensate
contribution is a small correction to the total NNLO term. Either way, our predictions
compare favorably with experiment when this comparison is possible.
We should keep in mind that these results have been obtained for a specific model, so
we are testing the impact of the perturbative corrections for this specific model. On the
other hand, if one believes that the large n behavior of the spectrum is dictated by the
Regge behavior and that the corrections can be obtained as an expansion in 1/n, the set
up is general. The only ambiguity comes from where the logarithms should be introduced
(masses or decays). At this respect it is worth mentioning that, as a matter of principle, this
ambiguity could be fixed if enough experimental information were available for the masses
and decays.
6 Conclusions
We have studied the constraints that the OPE imposes on large Nc inspired QCD models for
current-current correlators. We have focused on the constraints obtained by going beyond the
leading-order parton computation. We have explicitly showed that, assumed a given mass
spectrum (Regge plus corrections in 1/n), we can obtain the logarithmic (and constant)
behavior in n of the decay constants within a systematic expansion in 1/n. More than that,
power-like 1/n corrections can only be incorporated in the analysis if full consideration to the
perturbative corrections in the Euclidean regime is made. This is due to the fact that these
type of contributions produce logarithms of Q in the Euclidean (this is one of the reasons
why this sort of corrections are not usually considered in quark-hadron duality analysis). On
the other hand, the existence of lnn in the decay constants may point to the existence of
two scales in the problem, ΛQCD and nΛQCD, in the Minkowski regime.
We have also performed some numerical estimates of the importance of these corrections.
The n dependence of the decay constants is small but sizeable for the axial and vector (model
II) channel, for the vector (model I) one this dependence is small. On the other hand the
uncertainties of the calculation are large. Either way, our predictions compare favorably
with experiment when this comparison is possible.
Our example shows that it is possible to have different large n behavior for the vector
and pseudo-vector mass spectrum and yet comply with all the constraints from the OPE.
An important caveat of our analysis is that we have not considered what the effect of
renormalons could be. We have focused on the effect of low orders in perturbation theory
to the decay constants. It would be interesting to see whether the knowledge of the higher
order behavior of perturbation theory may give some extra constraints on the values of these
constants and the mass spectrum. At this respect we have to say that we have obtained
approximated expressions for the decay constants as an expansion in αs(nBV ), with just
the low order contributions in αs. It is quite likely that this expansion is asymptotic and
that different orders in 1/n are related in a similar way to the one found in the renormalon
analysis for the OPE expansion for different orders in 1/Q2. Therefore, the results obtained
for the 1/n corrections could be affected as well by the asymptotic behavior of the 1/ lnn
expansion in the leading-order term. This is obviously related with renormalons. We expect
to come back to this issue in the future.
Acknowledgments. We thank S. Afonin, A. Andrianov, D. Espriu, and S. Peris for
discussions and L. Glozman for correspondence. This work is partially supported by the
network Flavianet MRTN-CT-2006-035482, by the spanish grant FPA2004-04582-C02-01,
by the catalan grant SGR2005-00564 and by a Distinció from the Generalitat de Catalunya.
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	Introduction
	Correlators
	Matching
	LO Matching
	NLO matching
	NNLO matching
	Axial versus vector correlators
	Numerical Analysis
	Conclusions

  We study the constraints that the operator product expansion imposes on large
$N_c$ inspired QCD models for current-current correlators. We focus on the
constraints obtained by going beyond the leading-order parton computation. We
explicitly show that, assumed a given mass spectrum: linear Regge behavior in
$n$ (the principal quantum number) plus corrections in $1/n$, we can obtain the
logarithmic (and constant) behavior in $n$ of the decay constants within a
systematic expansion in $1/n$. Our example shows that it is possible to have
different large $n$ behavior for the vector and pseudo-vector mass spectrum and
yet comply with all the constraints from the operator product expansion.

Introduction
The operator product expansion (OPE) has been used since long in order to gain information
on the non-perturbative dynamics of the hadronic spectrum and decays [1, 2, 3, 4, 5, 6,
7, 8, 9]. In this article we revisit this problem. We want to obtain the constraints that
the knowledge of the perturbative expansion in αs(Q
2) of the current-current correlators
in the Euclidean regime poses on the relation between the decay constants and the mass
spectrum for excitations with a large quantum number n (where n is the the quantum
number of the bound state). We put special emphasis in going beyond the leading-order
parton computation. We will work with a specific model for the hadronic spectrum. This is
compulsory, since different spectral functions1 may yield the same OPE expression, yet we
believe some aspects of our discussion may hold beyond the assumptions of our model.
In order to have a well defined bound state it is crucial to consider the large Nc approx-
imation [10]. This ensures infinitely narrow resonances at arbitrarily large energies. We will
consider to be in the large Nc limit in what follows, as well as in the exact chiral (massless)
limit. We will then set a specific model for the hadronic spectrum, valid for large values of
n (we only need the behavior of the spectrum and decays for large n, we do not aim to get
any information from perturbation theory for low values of n). This model will be based
on the Regge behavior plus corrections in 1/n that will be included in a systematic way.
The model is based on the assumption that the Regge behavior is a good description of the
spectrum for large n (this can be explicitely seen in the ’t Hooft model [11] and it is also
consistent with phenomenology). Given the 1/n corrections to the mass spectrum, the ex-
pression of the correlator can also be written as a systematic expansion in 1/n, where higher
powers in 1/n are equivalent to higher orders in 1/Q2 in its OPE. By matching the OPE
and hadronic expressions order by order in 1/Q2, we will be able to predict the logarithmic
dependence on n of the decay constants (actually also the constant terms). This result can
also be systematically organized within an expansion in 1/n together with an expansion in
1/ lnn. We will give explicit expressions up to order 1/n2 and 1/ ln3 n. We will also make
some numerical estimates of the impact of these corrections. Finally, we would like to stress
that we are able to introduce power corrections in 1/n to the Regge behavior and yet comply
with the OPE. This is in contrast with Ref. [5], where, besides the Regge behavior, only
exponentially suppressed terms are introduced (parametrically smaller than any finite power
of 1/n for large n). This parameterization is however fine if considered as a fit not emanated
from the large n limit.
2 Correlators
For definiteness, we will consider the vector-vector correlator but most of the discussion
applies to any other current-current correlator (axial-vector, scalar, ....).
V (q) ≡ (q
µqν − gµνq2)ΠV (q) ≡ i
d4xeiqx〈vac|T {J
V (x)J
V (0)} |vac〉 , (1)
1In particular the one derived directly from perturbation theory, which we do not consider, since we will
work in the large N
limit with infinitely narrow resonances.
where J
Qf ψ̄fγ
µψf . In order to avoid divergences, we will consider the Adler function
A(Q2) ≡ −Q2
Π(Q2) = Q2
(t+Q2)2
ImΠV (t) , (2)
where Q2 = −q2 is the Euclidean momentum.
Since we are working in the large Nc limit, the spectrum consists of infinitely narrow
resonances, and the Adler function can be written in the following way
A(Q2) = Q2
F 2V (n)
(Q2 +M2V (n))
. (3)
On the other hand, for large positive Q2, one may try to approximate the Adler function
by its OPE, which reads
AOPE(Q
C(αs(Q
β(αs(ν))〈vac|G
2(ν)|vac〉+O
where αA(Q
2) admits an analytic expansion in terms of αs(Q
2) (computed in the MS scheme),
β(αs) = −β0
+ · · · , (5)
with β0 = 11/3Nc, β1 = 34/3N
c , β2 = 2857/54N
c , and [12]
C(αs(Q
2)) = −
+ · · ·
. (6)
3 Matching
High excitations of the QCD spectrum are believed to satisfy linear Regge trajectories:
M2V,n
= constant.
For generic current-current correlators, such behavior is consistent with perturbation
theory in the Euclidean region at leading order in αs if the decay constants are taken to be
“constants”, ie. independent of the principal quantum number n.
The inclusion of subleading effects in αs can be incorporated into this model by changing
the n dependence of the decay constants without changing the ansatz for the spectrum. The
inclusion of these effects has consequences on subleading sum-rules and the relation with the
non-perturbative condensates.
Here we would like to go beyond the analysis at leading order in αs, as well as to consider
power-like corrections in 1/n. We will consider that the large n expression for the mass
spectrum can be organized within a 1/n expansion in a systematic way starting from the
asymptotic linear Regge behavior. In order to fix (and simplify) the problem we will assume
that no lnn term appears in the mass spectrum2. Therefore, we write the mass spectrum in
the following way (for large n)
M2V (n) =
(−s) = B
V n+B
+ · · · , (7)
where B
V are constants. We will usually denote M
V,LO(n) = B
V n, M
V,NLO(n) = B
V and so on. To shorten the notation, we will denote B
V = BV , B
V = AV and B
For the decay constants, we will have a double expansion in 1/n and 1/ lnn.
F 2V (n) =
F 2V,s(n)
= F 2V,0(n) +
F 2V,1(n)
F 2V,2(n)
+ · · · , (8)
where the coefficients F 2V,s(n) have a logarithmic dependence on n:
F 2V,s(n) =
V,s(n)
lnr n
. (9)
As we did with the masses, we will define F 2V,LO(n) = F
V,0(n), F
V,NLO(n) = F
V,0(n) +
and so on. Note that in this case we also have an expansion in 1/ lnn.
We are now in position to start the computation. Our aim is to compare the hadronic
and OPE expressions of the Adler function within an expansion in 1/Q2, but keeping the
logarithms of Q. In order to do so we have to arrange the hadronic expression appropiately.
Our strategy is to split the sum over hadronic resonances into two pieces, for n above or
below some arbitrary but formally large n∗, such that ΛQCDn
∗ ≪ Q. The sum up to n∗
can be analytically expanded in 1/Q2 and will not generate lnQ2 terms (neither a constant
term at leading order in 1/Q2). For the sum from n∗ up to ∞, we can use Eqs. (7) and (8)
and the Euler-MacLaurin formula to transform the sum in an integral plus corrections in
1/Q2. Whereas the latter do not produce logarithms, the integral does. These logarithms of
Q are generated by the large n behavior of the bound states and the introduction of powers
of 1/n is equivalent (once introduced in the integral representation, and for large n) to the
introduction of (logarithmically modulated) 1/Q2 corrections in the OPE expression.
Therefore, by using the Euler-MacLaurin formula, we write the Adler function in the
following way (B2 = 1/6, B4 = −1/30, ...)
A(Q2) = Q2
F 2V (n)
(Q2 +M2V (n))
F 2V (n)
(Q2 +M2V (n))
F 2V (n)
(Q2 +M2V (n))
F 2V (n
(Q2 +M2V (n
(−1)k
|B2k|
(2k)!
d(2k−1)
dn(2k−1)
F 2V (n)
(Q2 +M2V (n))
, (10)
2This is a simplification. If one considers, for instance, the ’t Hooft model [11], lnn terms do indeed
appear.
where n∗ stands for the subtraction point we mentioned above, such that for n larger than
n∗ one can use the asymptotic expressions (7) and (8). This allows us to eliminate terms
that vanish when n → ∞. Note that the last sum in Eq. (10) is an asymptotic series, and
in this sense the equality should be understood.
Note also that for n below n∗, we will not distinguish between LO, NLO, etc... in masses
or decay constants, since for those states we will not assume that one can do an expansion
in 1/n and use Eqs. (7) and (8).
Finally, note that the expressions we have for the masses and decay constants become
more and more infrared singular as we go to higher and higher orders in the 1/n expansion.
This is not a problem, since we always cut off the integral for n smaller than n∗. Either way,
the major problems would come from the decay constants, since, in the case of the mass, Q2
effectively acts as an infrared regulator.
3.1 LO Matching
We want to match the hadronic, Eq. (10), and OPE, Eq. (4), expressions for the Adler
function at the lowest order in 1/Q2. This means that we have to consider the lowest order
expressions in 1/n for the masses and decay constants, i.e. F 2V,LO(n) and M
V,LO(n), since
the corrections in 1/n give contributions suppressed by powers of 1/Q2.
Only the first term in Eq. (10) can generate logarithms or terms that are not suppressed
by powers of 1/Q2. Therefore we obtain the following equality,
Apt.(Q2) ≡ Q2
F 2V,LO(n)
(Q2 +M2V,LO)
. (11)
This equation can be fulfilled by demanding that
F 2V,LO(n)
|dM2V,LO(n)/dn|
pert.
V,LO(n)) . (12)
By using the perturbative expression for ImΠ
pert.
V (see [13]), we obtain
F 2V,LO(n) = BV
Ncαs(nBV ) +
243− 176 ζ(3)
128π2
N2c α
s (nBV ) (13)
346201− 2904π2 − 324528 ζ(3) + 63360 ζ(5)
27648π3
N3c α
s (nBV ) +O
α4s (nBV )
where αs(nBV ) should actually be understood as a function of αs(BV ) and lnn. Therefore,
it is obvious that the above expression is resumming powers of αs(BV ) lnn:
F 2V,LO(n) = BV
1 + 11
αs(BV )
ln(n)
αs(BV )
2673− 1936 ζ(3)− 408 ln
1 + 11
Ncαs(BV ) ln(n)
88(1 + 11
αs(BV )
ln(n))2
N2c α
s (BV )
(4π)2
N3c α
s (BV )
(4π)3
52272π(1 + 11
αs(BV )
ln(n))3
[−350427Ncαs(BV ) ln(n)
+121π
346201− 2904π2 − 324528 ζ(3) + 63360 ζ(5)
−3672π(2877− 1936 ζ(3)) ln
αs(BV ) ln(n)
+749088π ln2
αs(BV ) ln(n)
α4s (BV )
Doing so we see that we are able to obtain the dependence of the decay constant in lnn
(somewhat we are assuming that αs(BV ) is an small parameter, BV ∼ 1 GeV).
We can also rewrite the decay constant as an expansion in 1/ lnn by using the equality
ln ñ =
αs(nBV )
αs(nBV )
αs(nBV )
, (15)
where ñ = nBV /ΛMS. We then obtain
F 2V,LO(n) = BV
ln ñ
ln2 ñ
ln ln ñ +
ln3 ñ
46818
161051
ln2 ln ñ +
322102
(−2877 + 1936ζ(3)) ln ln ñ +
42272605
2576816
20283 ζ(3)
360 ζ(5)
ln4 n
. (16)
Note that the lowest contribution in 1/ lnn for the decay constant, BV
Q2f ,
which, usually, is the only one considered, reproduces the leading-order partonic prediction
for the Adler function.
Note also that there is no problem with the Landau pole, even if the result is written in
the form of Eq. (16), since it holds only for n larger than an n∗ such that ΛMS ≪ n
∗BV (the
integral has an infrared cutoff at n∗).
Finally, we remind that, strictly speaking, we can only fix the ratio between the decay
constant and the derivative of the mass. We have fixed this ambiguity by arbitrarily imposing
the n dependence of the mass spectrum.
3.2 NLO matching
We now want to obtain extra information on the decay constant by demanding the validity
of the OPE at O(1/Q2), in particular the absence of condensates at this order. We then
have to use the NLO expressions for M2V (n) and F
V (n). With the ansatz we are using for
the mass at NLO, it is compulsory to introduce (logarithmically modulated) 1/n corrections
to the decay constant if we want this constraint to hold. Note that it is possible to shift all
the perturbative corrections to the decay constant.
Imposing that the 1/Q2 condensate vanishes produces the following sum rule:
Apt. −
Apt. +
F 2V (n)−
dnF 2V,LO(n)
F 2V (n
(−1)k
|B2k|
(2k)!
d(2k−1)
dn(2k−1)
F 2V (n)
F 2V,1(n)/n
(Q2 +M2V,LO(n))
F 2V,1(n)/n
(Q2 +M2V,LO(n))
= 0 .
This equality should hold independently of the value of n∗, which formally should be taken
large enough so that αs(n
∗BV ) ≪ 1, i.e. the limit ΛMS ≪ n
∗BV ≪ Q
2. Again, the meaning
of the asymptotic series appearing in Eq. (17) should be taken with care. If we forget about
this potential problem, only a few terms in Eq. (17) can generate lnQ2 terms, which should
cancel at any order. Those are the first two and the last two terms. Actually, the next to last
term does not generate logarithms, but it allows to regulate possible infrared divergences
appearing in the calculation. Therefore, asking for the cancellation of the 1/Q2 suppressed
logarithmic terms produced by the first two and the last term in Eq. (17) fixes F 2V,1. The
non-logarithmic terms should also be cancelled but they cannot be fixed from perturbation
theory.
One can actually find an explicit solution to the above constraint for F 2V,1 by performing
some integration by parts. We obtain
F 2V,1(n)
F 2V,0(n) (18)
ln2 ñ
(1− 2 ln (ln ñ)) +
36 ζ(3)
ln3 ñ
2576816
−45794053 + 351384π2 + 41637552 ζ(3)− 7666560 ζ(5)
−3672 ln (ln ñ) (−3013 + 1936 ζ(3) + 204 ln (ln ñ))]
ln4 ñ
ln5 ñ
or in terms of αs(nBV ) or αs(BV ),
F 2V,1(n)
N2c α
s (nBV )−
2877− 1936 ζ(3)
768π3
N3c α
s (nBV ) (19)
11(376357− 2904π2 − 344112 ζ(3) + 63360 ζ(5))
110592π4
N4c α
s (nBV ) +O
α5s (nBV )
F 2V,1(n)
(1 + 11
αs(BV )
ln(n))2
α2s (BV )
(4π)2
2877− 1936 ζ(3)− 408 ln
1 + 11
Ncαs(BV ) ln(n)
12(1 + 11
αs(BV )
ln(n))3
α3s (BV )
(4π)3
4752π(1 + 11
αs(BV )
ln(n))4
α4s (BV )
(4π)4
[−233618Ncαs(BV ) ln(n)
+121π
376357− 2904π2 − 344112 ζ(3) + 63360 ζ(5)
−3672π(3013− 1936 ζ(3)) ln
αs(BV ) ln(n)
+749088π ln2
αs(BV ) ln(n)
α5s (BV )
Note that besides the 1/n suppression, we also have an extra α2s (nBV ) suppression.
In principle one could think of the existence of 1/n×constant terms in the decay constant,
i.e. without any associated logarithm. However, such terms produce ln(Q2) contributions in
the Euclidean regime that do not appear in the perturbative computation, so they can be
ruled out. This appears to be a general statement since 1/nm × constant for any m integer
also produces logarithms. Note that in order to give meaning to these integrals it is implicit
that the integral over n has an infrared cutoff at n∗. Nevertheless, the logarithm does not
appear to multiply powers of the infrared cutoff (as expected).
Finally, we would like to mention that, besides the constraints coming from the logarith-
mic related behavior of the OPE, there is also the constraint from its constant terms, which
should sum up to zero. Nevertheless, for this constraint we cannot give a closed expression.
This is due to the fact that the lnQ2 independent terms may receive contributions from
any subleading order in the 1/n expansion of the masses and decay constants. The reason
is that the decay constant at a given order in 1/n is obtained after performing some inte-
gration by parts, which generate new (lnQ2-independent) terms that can be Q2 enhanced.
This statement is general and also applies to any subleading power in the 1/Q2 matching
computation.
3.3 NNLO matching
We now consider expressions for the mass and decay constants at NNLO. For the first time
we have to consider condensates. Simplifying terms that do not produce logs, we obtain the
following equation,
β(αs(ν))〈vac|G
2(ν)|vac〉
(Q2 +BV n)2
F 2V,2(n)
V,0(n)
V,1(n)
where
= stands for the fact that we can only predict the lnQ2 dependence. Constant terms
are not fixed by this relation.
In order to get a more closed expression is convenient to use the following equality,
(Q2 +BV n)2
BV n2
α2s (nBV ) , (22)
valid up to terms that do not produce logarithms or those that are subleading.
We get then
F 2V,2(n) = −CV
Ncαs(nBV )
287− 176 ζ(3)
128π2
11A2V
64π2BVCV
β(αs(ν))〈vac|G
2(ν)|vac〉
BVCVN2c
N2c α
s (nBV ) (23)
α3s (nBV )
Note that in this case we only consider up to O(α2s (nBV )) corrections, since higher order
loops are unknown. The accuracy is set by our knowledge of the matching coefficient of the
gluon condensate. Note also that F 2V,2(n) does not have αs suppression. Therefore, for low n,
this contribution could be practically of the same size than, formally, more important terms.
4 Axial versus vector correlators
The above discussion has been performed for the vector-vector correlator Adler function.
It goes without saying that we could perform a similar analysis with axial-vector currents,
since the perturbative expansions for both correlators are equal. Here it comes an important
observation. We could change the coefficients for the mass spectrum BA 6= BV , AA 6= AV ,
· · · , yet we would obtain the same expression for the OPE (at the order we are working, the
first chiral breaking related effects are O(1/Q6)). Therefore, we conclude that the OPE does
not fix BA = BV as it is sometimes claimed in the literature [1, 3]
3. Our computation gives
a specific counter example. Moreover, it is nice to see what the role played by BA and BV
is in our case. When one goes to the Euclidean regime, BA and BV become renormalization
factorization scales and, obviously, the physical result does not depend on them (for large
Q2 in the Euclidean). On the other hand, it is evident that having different constants: BA,
BV , . . . produces different physical predictions for the masses and decay widhts for vector or
axial-vector channels. The point to be emphasized is that BA = BV cannot come from an
OPE analysis alone. This point has already been stressed in Refs. [4, 8], what we think is
novel in our analysis is that we have seen that the inclusion of corrections in αs does not
affect that conclusion, and that BA and BV play the role of the renormalization scale in the
analogous perturbative analysis in the Euclidean regime, and are therefore unobservable.
Finally, we cannot avoid mentioning the analysis of Ref. [14] where, using AdS/CFT, they
explicitly find Regge behavior with different slopes for vector and axial-vector channels.
3Another issue, on which we do not enter, is whether some other kind of arguments (relying on the specific
model used), like semiclassical arguments, may fix those parameters to be equal.
In any case, even though the constants that characterize the spectrum can be different
for the vector and axial-vector channel, they have to yield the same expressions for the OPE
when combined with the decay constants. This produces some relations that we list in what
follows. We first define t ≡ BV n = BAn
′ and take n and n′ as continuous variables. We then
obtain the following equalities
F 2V,LO(n)
F 2A,LO(n
pert.
V (t) ≡ f0(t) , (24)
F 2V,1(n)
F 2A,1(n
f0(t) , (25)
F 2V,2(n)
V,0(n)
V,1(n)
F 2A,2(n
A,0(n
A,1(n
β(αs(ν))〈vac|G
2(ν)|vac〉f1(t) , (26)
where
f1(t) =
α2s (t)
(4π)2
+ · · · . (27)
5 Numerical Analysis
n = 1 n = 2 n = 3 n = 4
Mρ(I) 781.3(775.5± 0.4) 1440.2(1459± 11) 1891.8(1870± 20) 2257(2265± 40)
Mρ(II) 771.5(775.5± 0.4) 1471.7(1459± 11) 1855(1870± 20) 2154.8(2149± 17)
Ma1 1235.6(1230± 40) 1621.7(1647± 22) 1962(1930
−70) 2257.8(2270
FV (I) 156(156± 1) 155 154 153
FV (II) 185(156± 1) 147 139 135
FA 123(122± 24) 137 139 139
Table 1: We give the experimental values of the masses (in MeV) and electromagnetic decay
constants (when available) for vector and axial vector particles (within parenthesis), compared
with the values obtained from the fit. For the vector states we consider two possible Regge
trajectories that we label I and II respectively. We take αs(1GeV) = 0.5 and β〈G
−(352MeV)4.
We restrict ourselves to the SU(2) case (non-strange sector) and study the vector and
axial-vector channels. We would like here to assess the importance of including perturbative
corrections to a standard analysis based on the OPE. We do not aim to perform a full fledged
2 4 6 8 10
FV,LOHIL
2 4 6 8 10
FV,LOHIIL
Figure 1: In this plot we show FV,LO(I) and FV,LO(II) at different orders in αs.
analysis, but only to see the importance of the corrections. In table 1, we give the values
of the masses and decay constants. In Figure 1 we show the changes in both FV,LO(I) and
FV,LO(II) as we include higher orders in the expansion in αs, and in Figure 2 the changes
in the full FV (I) and FV (II) as we include higher orders in 1/n. In figure 3 we show the
same plots for the axial-vector case. We take the experimental values from Ref. [15]. In
principle there are more states in the particle data book, in particular in the vector channel.
Nevertheless, it is not clear whether they belong to the same Regge trajectory or whether
they belong to some daughter one, see, for instance, the discussion in Ref. [5]. For the time
being we will disregard the study of other possible (vector) Regge trajectories and restrict
the analysis to a single trajectory. We will consider the two possibilities listed in Table 1.
Our choice of states for the set (I) is motivated by the discussion of Ref. [16] on the possible
formation of multiplets in the case of chiral symmetry restoration. The set (II) is based on
the assignment of states made in Ref. [5] (based on the existence of S and D-wave daughter
trajectories) and in particular on the analysis of Ref. [17], where the state 2265 is argued to
belong to the D-wave Regge trajectory4.
In order to fix the parameters of the mass spectrum we use the experimental values of
the masses we list in the table. We obtain the values:
BV (I) = 1.525× 10
6MeV2 , AV (I) = −1.038× 10
6MeV2 , CV (I) = 0.123× 10
6MeV2 ,
BV (II) = 1.128× 10
6MeV2 , AV (II) = 0.353× 10
6MeV2 , CV (II) = −0.885× 10
6MeV2 ,
BA = 1.278× 10
6MeV2 , AA = −0.100× 10
6MeV2 , CA = 0.349× 10
6MeV2 .
We should mention that the values obtained for these parameters are not very stable
under the change of number of data points, except for BV and BA, which are roughly stable,
although with quite sizeable uncertanties. For the subleading terms A and C, their values
are basically random with the fit. We roughly find BV ≃ BA within the uncertainties. The
n dependence of the axial and vector (model II) decay constants is small but sizeable (and it
goes in the right direction for low n). The 1/n corrections are always corrections compared
with the leading order terms. Nevertheless, the 1/n2 correction is much larger than the 1/n
one for the range of values of n that we explore. This appears to be due to the α2s/(4π)
suppression of the 1/n term, as well as to the difference in size between the constants A
4We also thank S. Afonin for discussions on this point.
2 4 6 8 10
FVHIL
2 4 6 8 10
132.5
137.5
142.5
147.5
FVHIIL
Figure 2: In this plot we show FV (I) and FV (II) at different orders in the 1/n expansion.
2 4 6 8 10
FA,LO
2 4 6 8 10
Figure 3: In this plot we show FA,LO and FA at different orders in αs and in the 1/n
expansion, respectively.
and C. This is so for the axial and vector (model II) decay constants. Nevertheless, for the
vector (model I) decay constants the n dependence appears to be quite small also at NNLO.
This appears to be due to the small value of the coefficient CV (I). The gluon condensate
contribution is a small correction to the total NNLO term. Either way, our predictions
compare favorably with experiment when this comparison is possible.
We should keep in mind that these results have been obtained for a specific model, so
we are testing the impact of the perturbative corrections for this specific model. On the
other hand, if one believes that the large n behavior of the spectrum is dictated by the
Regge behavior and that the corrections can be obtained as an expansion in 1/n, the set
up is general. The only ambiguity comes from where the logarithms should be introduced
(masses or decays). At this respect it is worth mentioning that, as a matter of principle, this
ambiguity could be fixed if enough experimental information were available for the masses
and decays.
6 Conclusions
We have studied the constraints that the OPE imposes on large Nc inspired QCD models for
current-current correlators. We have focused on the constraints obtained by going beyond the
leading-order parton computation. We have explicitly showed that, assumed a given mass
spectrum (Regge plus corrections in 1/n), we can obtain the logarithmic (and constant)
behavior in n of the decay constants within a systematic expansion in 1/n. More than that,
power-like 1/n corrections can only be incorporated in the analysis if full consideration to the
perturbative corrections in the Euclidean regime is made. This is due to the fact that these
type of contributions produce logarithms of Q in the Euclidean (this is one of the reasons
why this sort of corrections are not usually considered in quark-hadron duality analysis). On
the other hand, the existence of lnn in the decay constants may point to the existence of
two scales in the problem, ΛQCD and nΛQCD, in the Minkowski regime.
We have also performed some numerical estimates of the importance of these corrections.
The n dependence of the decay constants is small but sizeable for the axial and vector (model
II) channel, for the vector (model I) one this dependence is small. On the other hand the
uncertainties of the calculation are large. Either way, our predictions compare favorably
with experiment when this comparison is possible.
Our example shows that it is possible to have different large n behavior for the vector
and pseudo-vector mass spectrum and yet comply with all the constraints from the OPE.
An important caveat of our analysis is that we have not considered what the effect of
renormalons could be. We have focused on the effect of low orders in perturbation theory
to the decay constants. It would be interesting to see whether the knowledge of the higher
order behavior of perturbation theory may give some extra constraints on the values of these
constants and the mass spectrum. At this respect we have to say that we have obtained
approximated expressions for the decay constants as an expansion in αs(nBV ), with just
the low order contributions in αs. It is quite likely that this expansion is asymptotic and
that different orders in 1/n are related in a similar way to the one found in the renormalon
analysis for the OPE expansion for different orders in 1/Q2. Therefore, the results obtained
for the 1/n corrections could be affected as well by the asymptotic behavior of the 1/ lnn
expansion in the leading-order term. This is obviously related with renormalons. We expect
to come back to this issue in the future.
Acknowledgments. We thank S. Afonin, A. Andrianov, D. Espriu, and S. Peris for
discussions and L. Glozman for correspondence. This work is partially supported by the
network Flavianet MRTN-CT-2006-035482, by the spanish grant FPA2004-04582-C02-01,
by the catalan grant SGR2005-00564 and by a Distinció from the Generalitat de Catalunya.
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	Introduction
	Correlators
	Matching
	LO Matching
	NLO matching
	NNLO matching
	Axial versus vector correlators
	Numerical Analysis
	Conclusions
