A Rational Approach to Resonance Saturation
in large-Nc QCD
Pere Masjuan and Santiago Peris
Grup de F́ısica Teòrica and IFAE
Universitat Autònoma de Barcelona, 08193 Barcelona, Spain.
Abstract
We point out that resonance saturation in QCD can be understood in the
large-Nc limit from the mathematical theory of Pade Approximants to meromor-
phic functions. These approximants are rational functions which encompass any
saturation with a finite number of resonances as a particular example, explaining
several results which have appeared in the literature. We review the main prop-
erties of Pade Approximants with the help of a toy model for the 〈V V − AA〉
two-point correlator, paying particular attention to the relationship among the
Chiral Expansion, the Operator Product Expansion and the resonance spectrum.
In passing, we also comment on an old proposal made by Migdal in 1977 which
has recently attracted much attention in the context of AdS/QCD models. Fi-
nally, we apply the simplest Pade Approximant to the 〈V V − AA〉 correlator
in the real case of QCD. The general conclusion is that a rational approximant
may reliably describe a Green’s function in the Euclidean, but the same is not
true in the Minkowski regime due to the appearance of unphysical poles and/or
residues.
http://arxiv.org/abs/0704.1247v1
1 Introduction
The strong Chiral Lagrangian is a systematic organization of the physics in powers of
momenta and quark masses, but requires knowledge of the low-energy constants (LEC)
to make reliable phenomenological predictions. As with any other effective field theory,
these LECs play the role of coupling constants and contain the information which comes
from the integration of the heavy degrees of freedom not explicitly present in the Chiral
Lagrangian (e.g. meson resonances).
At O(p4) there are 10 of these constants [1]. Although at this order there is enough
independent information to extract the values for these constants from experiment,
this will hardly ever be possible at the next order, O(p6), because the number of
constants becomes more than a hundred [2]. In the electroweak sector the proliferation
of constants appears already at O(p4) [3]. Although in principle these low-energy
constants may be computed on the lattice, in practice this has only been accomplished
in a few cases for the strong Chiral Lagrangian at O(p4), and only recently [4].
The large Nc expansion [5] stands out as a very promising analytic approach capa-
ble of dealing with the complexities of nonperturbative QCD while, at the same time,
offering a relatively simple and manageable description of the physics. For instance,
mesons are qq states with no width, the OZI rule is exact and there is even a proof
of spontaneous chiral symmetry breaking [6]. Furthermore, its interest has recently
received a renewed boost indirectly through the connection of some highly supersym-
metric gauge theories to gravity [7], although the real relevance of this connection for
QCD still remains to be seen. However, in spite of all this, the fact that no solution to
large-Nc QCD has been found keeps posing a serious limitation to doing phenomenol-
ogy. For instance, in order to reproduce the parton model logarithm which is present
in QCD Green’s functions in perturbation theory, an infinity of resonances is necessary
whose masses and decay constants are in principle unknown.
On the other hand, QCD Green’s functions seem to be approximately saturated by
just a few resonances; a property which has a long-standing phenomenological support
going all the way back to vector meson dominance ideas [8], although it has never
been properly understood. In a modern incarnation, this fact translated into the very
successful observation [9] that the strong LECs at O(p4) seem to be well saturated by
the lowest meson in the relevant channel,1 after certain constraints are imposed on some
amplitudes at high-energy in order to match the expected behavior in QCD [11, 12].
It was then realized that all these successful results could be encompassed at once as
an approximation to large-Nc QCD consisting in keeping only a finite (as opposed to
the original infinite) set of resonances in Green’s functions. This approximation to
large-Nc QCD has been termed Minimal Hadronic Approximation (MHA) [13] because
it implements the minimal constraints which are necessary to secure the leading non-
trivial behavior at large energy of certain Green’s functions through the marriage of
the old resonance saturation and the large-Nc approximation of QCD. In recent years,
a large amount of work has been dedicated to studying the consequences of these ideas
[14].
1This is less clear in the scalar channel, however. See Ref. [10]
However, the high-energy matching with a finite set of resonances, first suggested in
[11], makes it clear that the treatment is not amenable to the methods of a conventional
effective field theory. An effective field theory is an approximation for energies smaller
than a heavy particle’s mass and, therefore, cannot deal with momentum expansions
at infinity as in the case of the Operator product Expansion (OPE). In other words,
the fact that the set of resonances in each channel is really infinite precludes the
naive expansion at large momentum because there is always a mass in the spectrum
which is even larger. The sum over an infinite set of resonances and the expansion
for large momentum are operations which do not commute [15]. In those Green’s
functions containing a contribution from the parton model logarithm, this is made
self-evident since a naive expansion at large momentum can only produce powers and
not a logarithm, which is why large-Nc QCD requires an infinity of resonances in the
first place.
The problem can be delayed one power of αs if one requires the use of the resonance
Lagrangian [9] to be limited only to Green’s functions which are order parameters of
spontaneous chiral symmetry breaking. These order parameters vanish to all orders in
αs in the chiral limit
2 and, therefore, avoid the presence of the parton model logarithm
which, otherwise, would preclude from the outset any matching to a finite number of
resonances. However, the concept of a Lagrangian whose validity is restricted only to a
certain class of Green’s functions has never been totally clear; and even if the resonance
Lagrangian is restricted by definition to order parameters, the problem surfaces again
in the presence of logarithmic corrections from nontrivial anomalous dimensions, which
make the exact matching at infinite momentum impossible.
In a slightly different context, a somewhat similar observation was also made in Ref.
[16]. In this paper it was observed that it is impossible to satisfy the large momentum
fall-off expected in large-Nc QCD for the form factors which can be defined through
a three-point Green’s function, if the sum over resonances in the Green’s function is
restricted to a finite set. Interestingly, this again pointed to an incompatibility of the
QCD short-distance behavior with an approximation to large Nc which only kept a
finite number of resonances.
A further piece of interesting evidence results from the comparison between the
analysis in Refs. [17] and [11]. After imposing some good high-energy behavior in
several Green’s functions and form factors including, in particular, the axial form
factor governing the decay π → γeν, Ref. [11] obtains, keeping only one vector state V
and one axial-vector state A, that their two masses must be constrained by the relation
2MV . The work in Ref. [17], on the contrary, does not use the axial form factor
and obtains, after performing a very good fit within the same set of approximations,
the precise values MV = 775.9 ± 0.5 MeV and MA = 938.7 ± 1.4 MeV. These values
for the masses, although close, are not entirely compatible with the previous relation.
In other words, the short-distance constraint from the axial form factor is not fully
compatible with the short-distance constraints used in [17] if restricted to only one
vector and one axial-vector states3.
2E.g., the two-point correlator 〈V V −AA〉.
3Adding one further state does not change the conclusion [17].
In this paper we would like to suggest that all the above properties can be un-
derstood if the approximation to large Nc QCD with a finite number of resonances is
reinterpreted within the mathematical Theory of Pade Approximants (PA) to mero-
morphic functions [18]. QCD Green’s functions in the large Nc and chiral limits have
an analytic structure in the complex momentum plane which consists of an infinity
of isolated poles but no cut, i.e. they become meromorphic functions [19]. As such,
they have a well-defined series expansion in powers of momentum around the origin
with a finite radius of convergence given by the first resonance mass4. This is all that
is needed to construct a Pade Approximant. A theorem by Pommerenke [20] assures
then convergence of any near diagonal PA to the true function for any finite momen-
tum, over the whole complex plane, except perhaps in a zero-area set. The poles of
the original Green’s function (i.e. the resonance masses) belong to this zero-area set
because not even the original function is defined there, but there are also extra poles.
These extra poles are called “defects” in the mathematical literature [18]. When the
Green’s function being approximated is of the Stieltjes type5, the poles of the PA are
always real and located on the Minkowski region Re(q2) = Re(−Q2) > 0, approaching
the physical poles as the order of the PA is increased [21]. However, this takes place
in a hierarchical way and, while the poles in the PA which are closest to the origin are
also very close to the physical masses, the agreement quickly deteriorates and one may
find that the last poles are several times bigger than their physical counterparts [22].
The same is true of the residues. In section 3, we will see with the help of a model
that the same properties are met in a meromorphic function whose spectral function
is not positive definite, except that some of the poles in the PA may even be complex.
This means that Minkowskian properties, such as masses and decay constants,
cannot be reliably determined from a PA except, perhaps, from the first poles which
are closest to the origin. If not all the residues and/or masses are physical, then there
is no reason why they should be the same in the form factor governing π → γeν and
in the Green’s function 〈V V − AA〉, explaining the different results found in [11] and
[17] we alluded to above. Furthermore, the form factors of all but the lightest mesons,
defined through the residues of the corresponding 3-point Green’s functions, will not
be reliably determined from a PA to that Green’s function, again in agreement with
the findings in [16].
The situation in the Euclidean is different. In general, PAs cannot be expanded at
infinite momentum to generate an OPE type expansion for the true function. Never-
theless, Pommerenke’s theorem assures a good approximation at any finite momentum,
no matter how large. Of course, the order of the PA will have to increase, the larger
the momentum region one wishes to approximate. For instance, in Ref. [21] it was
shown with the help of a simple model how, even in the case of the 〈V V 〉 correlator
which contains a logQ2 at large values of Q2 > 0, the PAs are capable of approximat-
ing the true function at any arbitrarily large (but finite) value of Q2 > 0, without the
need for a perturbative continuum. In section 3 we will show, again with the help of a
4The pion pole can always be eliminated multiplying by enough powers of momentum. We are
assuming here the existence of a nonvanishing gap in large-Nc QCD.
5Roughly this means that the associated spectral function is positive definite, like in the case of
the two-point correlator 〈V V 〉. See Ref. [18] (chapter 5) for a more precise definition.
model, how this is also true in the more general case of a non-positive definite spectral
function such as 〈V V −AA〉. This means that PAs are a reliable way to approximate
the original Green’s function in the Euclidean but not in the Minkowskian regime.
In 1977, A.A. Migdal [23] suggested PAs as a method to extract the spectrum of
large-Nc QCD from the leading term in the OPE of the 〈V V 〉 correlator, i.e. from
the parton model logarithm. However, nowadays this proposal should be considered
unsatisfactory for a number of reasons [24], the most simple of them being that different
spectra may lead to the same parton model logarithm [25]. In fact, the full OPE series
is expected to be only an asymptotic expansion at Q2 = ∞ (i.e. with zero radius
of convergence), and PAs constructed from this type of expansions cannot in general
reproduce the position of the physical poles [26]. For instance, we show this explicitly
with the help of a model for 〈V V − AA〉 in the Appendix. Migdal’s approach has
been recently adopted (in disguise) in some models exploiting the so-called AdS/QCD
correspondence [27] and, consequently, the same criticism also applies to them.
In Ref. [28] a model for the 〈V V − AA〉 two-point correlator with a spectrum
consisting of an infinity of resonances was suggested as a theoretical laboratory for
studying the relationship between the spectrum and the coefficients of the OPE. In
this paper several conventional methods usually employed in the literature were tested
against the exact result from the model. These included: Finite Energy Sum Rules as
in Ref. [33], pinched weights as in Ref. [34], Laplace transforms as in Ref. [35] and,
finally, also resonance saturation as in the MHA method. The bottom line was that
no method was able to produce very accurate predictions for the OPE coefficients. In
all the methods but the last one, the reason for this lack of accuracy was basically due
to the fact that the OPE requires an integral over the whole spectrum, whereas the
integral is actually cut off at an upper limit (in the real case, the upper limit is mτ ).
This is why even if one uses the real spectrum the result may be inaccurate [29]. In
the case of the MHA the reason was, as we will comment upon below, that the poles
were not allowed to be complex.
In section 3 we will revisit this 〈V V − AA〉 model, now from the point of view of
PAs. The model reproduces the power behavior of QCD at large Q2 > 0 except that
the model is simple enough not to have any logQ2 and, therefore, it cannot reproduce
the nonvanishing anomalous dimensions which exist in QCD. We do not think this is
a major drawback, however, because in QCD these logarithms are always screened by
at least one power of αs and, hence, in an approximate sense, it may be licit to ignore
them. In the model such an approximation becomes exact6. Will the PAs be able to
reproduce the large Q2 expansion of the 〈V V −AA〉 model? We will see that the answer
is affirmative. Therefore, the reason why the MHA method was not able to predict
accurately the OPE coefficients in Ref. [28] is because the lowest PA has complex
poles which were not allowed in [28]. When these complex poles are considered, the
accuracy achieved is better and, most importantly, improves for a higher PA. Since
the model allows the construction of PAs of a very high order, we have checked this
convergence up to the Pade P 5052 , which is able to reproduce the first non vanishing
coefficient of the OPE in the model with an accuracy of 52 decimal figures. Together
6For a model with a logQ2, the reader may consult Ref. [21].
with other numerical examples which will be discussed in section 3, we take this as a
clear evidence of the convergence of the method. This renders some confidence that
PAs may also do a good job in the real case of QCD.
The rest of the paper is organized as follows. In section 2 we review some generali-
ties of rational approximants, in section 3 we describe the 〈V V −AA〉 model and apply
different rational approximants to learn about the possible advantages and disadvan-
tages of them. In section 4 we apply the simplest PA to the case of the real 〈V V −AA〉
two-point function in QCD. Finally, we close with some conclusions.
2 Rational approximations: generalities
Let a function f(z) have an expansion around the origin of the complex plane of the
f(z) =
n , z → 0 . (1)
One defines a Pade Approximant (PA) to f(z) , denoted by PMN (z), as a ratio of two
polynomials QM(z), RN (z)
7, of order M and N (respectively) in the variable z, with
a contact of order M + N with the expansion of f(z) around z = 0. This means
that, when expanding PMN (z) around z = 0, one reproduces exactly the first M + N
coefficients of the expansion for f(z) in Eq. (1):
PMN (z) =
QM(z)
RN (z)
≈ f0 + f1 z + f2 z2 + ...+ fM+N zM+N +O(zN+M+1) . (2)
At finite z, the rational function PMN (z) constitutes a resummation of the series (1). Of
special interest for us will be the case when N =M +k, for a fixed k, because then the
function behaves like 1/zk at z = ∞. The corresponding PAs PMM+k(z) belong to what
is called the near-diagonal sequence for k 6= 0, with the case k = 0 being the diagonal
sequence.
The convergence properties of the PAs to a given function are much more difficult
than those of normal power series and this is an active field of research in Applied
Mathematics. In particular, those which concern meromorphic functions8 are rather
well-known and will be of particular interest for this work. The main result which
we will use is Pommerenke’s Theorem [20] which asserts that the sequence of (near)
diagonal PA’s to a meromorphic function is convergent everywhere in any compact set
of the complex plane except, perhaps, in a set of zero area. This set obviously includes
the set of poles where the original function f(z) is clearly ill-defined but there may
be some other extraneous poles as well. For a given compact region in the complex
plane, the previous theorem of convergence requires that, either these extraneous poles
move very far away from the region as the order of the Pade increases, or they pair
up with a close-by zero becoming what is called a defect in the mathematical jargon
7Without loss of generality we define, as it is usually done, RN (0) = 1.
8A function is said to be meromorphic when its singularities are only isolated poles.
[30]. These are to be considered artifacts of the approximation. Near the location of
these extraneous poles the PA approximation clearly breaks down but, away from these
poles, the approximation is safe.
In the physical case the original function f(z) will be a Green’s function G(Q2)
of the momentum variable Q2. In QCD in the large Nc limit this Green’s function
is meromorphic with all its poles located on the negative real axis in the complex Q2
plane. These poles are identified with the meson masses. On the other hand, the region
to be approximated by the PAs will be that of euclidean values for the momentum, i.e.
Q2 > 0. The expansion of G(Q2) for Q2 large and positive coincides with the Operator
Product Expansion.
In general a meromorphic function does not obey any positivity constraints and,
as we will see, this has as a consequence that some of the poles and residues of the
PAs may become complex 9. This clearly precludes any possibility that these poles and
residues may have anything to do with the physical meson masses and decay constants.
However, and this is very important to realize, this does not spoil the validity of the
rational approximation provided the poles, complex or not, are not in the region of
Q2 one is interested in. It is to be considered rather as the price to pay for using a
rational function, which has only a finite number of poles, as an approximation to a
meromorphic function with an infinite set of poles.
When the position of the poles in the original Green’s function is known, at least
for the lowest lying states, it is interesting to devise a rational approximation which has
this information already built in. The corresponding approximants are called Partial
Pade Approximants (PPAs) in the mathematical literature [31] and are given by a
rational function PMN,K(Q
N,K(Q
RN(Q2) TK(Q2)
, (3)
where QM(Q
2), RN(Q
2) and TK(Q
2) are polynomials of order M,N and K (respec-
tively) in the variable Q2. The polynomial TK(Q
2) is defined by having K zeros
precisely at the location of the lowest lying poles of the original Green’s function10
2) = (Q2 +M21 ) (Q
2 +M22 ) ... (Q
2 +M2K) . (4)
As before the polynomial RN (Q
2) is chosen so that RN (0) = 1 and, together with
2), they are defined so that the ratio PMN,K(Q
2) matches exactly the first M +N
terms in the expansion of the original function around Q2 = 0, i.e. :
N,K(Q
2) ≈ f0 + f1 Q2 + f2 Q4 + ... + fM+N Q2M+2N +O(Q2N+2M+2) . (5)
At infinity, the PPA in Eq. (3) obviously falls off like 1/Q2N+2K−2M . Exactly as
it happens in the case of PAs, also the PPAs will have complex poles for a general
9A special case which does obey positivity constraints is when the function is Stieltjes. In this
case the poles and residues of the PAs are purely real and with the same sign as those of the original
function [21].
10For simplicity, we will assume that all the poles are simple.
meromorphic function, which prevents it from any interpretation in terms of meson
states.
Finally, another rational approximant defined in mathematics is the so-called Pade
Type Approximant (PTA) [31] TMN (Q
QM (Q
TN (Q2)
, (6)
where TN (Q
2) is also given by the polynomial (4), now with N preassigned zeros at
the corresponding position of the poles of the original Green’s function, G(Q2). The
polynomial QM(Q
2) is defined so that the expansion of the PTA around Q2 = 0 agrees
with that of the original function up to and including terms of order M + 1, i.e.
2) ≈ f0 + f1 Q2 + f2 Q4 + ... + fM Q2M +O(Q2M+2) . (7)
At large values of Q2, one has that TMN (Q
2) falls off like 1/Q2N−2M . Clearly the PTAs
are a particular case of the PPAs, i.e. TMN (Q
2) = PM0,N(Q
2) and coincide with what has
been called the Hadronic Approximation to large-Nc QCD in the literature [13].
Let us summarize the mathematical jargon. A Pade Type Approximant (PTA)
is a rational function with all the poles chosen in advance precisely at the physical
masses. A Pade Approximant (PA) is when all the poles are left free. The intermediate
situation, with some poles fixed at the physical masses and some left free, corresponds
to what is called a Partial Pade Approximant (PPA).
3 Testing rational approximations: a model
Let us consider the two-point functions of vector and axial-vector currents in the chiral
limit
ΠV,Aµν (q) = i
d4x eiqx〈JV,Aµ (x)J† V,Aν (0)〉 =
qµqν − gµνq2
ΠV,A(q
2) , (8)
with J
V (x) = d(x)γ
µu(x) and J
A(x) = d(x)γ
µγ5u(x). As it is known, the difference
ΠV (q
2)− ΠA(q2) satisfies an unsubtracted dispersion relation11
ΠV−A(q
t− q2 − iǫ
ImΠV−A(t) . (9)
Following Refs. [32, 28], we define our model by giving the spectrum as
ImΠV (t) = 2F
ρ δ(t−M2ρ ) + 2
F 2V (n)δ(t−M2V (n)) ,
ImΠA(t) = 2F
0 δ(t) + 2
F 2A(n)δ(t−M2A(n)) . (10)
11The upper cutoff which is needed to render the dispersive integrals mathematically well defined
can be sent to infinity provided it respects chiral symmetry [15].
Here Fρ,Mρ are the electromagnetic decay constant and mass of the ρ meson and
FV,A(n) are the electromagnetic decay constants of the n− th resonance in the vector
(resp. axial) channels, while MV,A(n) are the corresponding masses. F0 is the pion
decay constant in the chiral limit. The dependence on the resonance excitation number
n is the following:
F 2V,A(n) = F
2 = constant , M2V,A(n) = m
V,A + n Λ
2 , (11)
in accord with known properties of the large-Nc limit of QCD [5] as well as alleged
properties of the associated Regge theory [37].
The combination
ΠLR(q
(ΠV (q
2)−ΠA(q2)) (12)
thus reads
ΠLR(q
−q2 +M2ρ
−q2 +M2V (n)
−q2 +M2A(n)
. (13)
This two-point function can be expressed in terms of the Digamma function ψ(z) =
log Γ(z) as [28]
ΠLR(q
−q2 +M2ρ
−q2 +m2A
−q2 +m2V
. (14)
To resemble the case of QCD, we will demand that the usual parton-model logarithm
is reproduced in both vector and axial-vector channels and that the difference (9) has
an operator product expansion which starts at dimension six. A set of parameters
satisfying these conditions is given by12
F0 = 85.8 MeV , Fρ = 133.884 MeV , F = 143.758 MeV , (15)
Mρ = 0.767 GeV, mA = 1.182 GeV, mV = 1.49371 GeV , Λ = 1.2774 GeV ,
and is the one we will use in this section. This set of parameters has been chosen to
resemble those of the real world, while keeping the model at a manageable level. For
instance, the values of Fρ and Mρ in (15) are chosen so that the function ΠLR in (14)
has vanishing 1/Q2 and 1/Q4 in the OPE at large Q2 > 0, as in real QCD. In fact, the
model admits the introduction of finite widths (which is a 1/Nc effect) in the manner
described in Ref. [32], after which the spectral function looks reasonably similar to
the experimental spectral function. This comparison can be found in Fig. 5 of Ref.
[28]. But this model is also interesting for a very different reason. In Ref. [28] several
attempts were made at determining the coefficients of the OPE by using the methods
which have become common practice in the literature. Among those we may list Finite
Energy Sum Rules [33], with pinched weights [34], Laplace sum rules [35] and Minimal
Hadronic Approximation [13]. As it turned out, when these methods were tested on
12These numbers have been rounded off for the purpose of presentation. Some of the exercises which
will follow require much more precision than the one shown here.
C0 C2 C4 C6 C−4 C−6 C−8
−7.362 21.01 −43.92 81.81 −2.592 1.674 −0.577
Table 1: Values of the coefficients C2k from the high- and low-Q
2 expansions of
Q2 ΠLR(−Q2) in Eq. (16) in units of 10−3 GeV 2−2k. Notice that C−2 = 0 and
C0 = −F 20 (the pion decay constant in the chiral limit), see text.
the model, none of them was able to produce very accurate results. We think that this
makes the model very interesting (and challenging !) as a way to assess systematic
errors [36].
Defining the expansion of the Green’s function (9) in Q2 = −q2 around Q2 = 0,∞
Q2 ΠLR(−Q2) ≈
C2k Q
2k , with k = 0,±1,±2,±3, . . . (16)
one obtains that the coefficients accompanying inverse powers of momentum, akin to
the Operator Product Expansion at large Q2 > 0, are given by (p = 1, 2, 3, ... with
k = 1− p):
C2k = −F 20 δp,1 +
(−1)p+1
F 2ρM
F 2Λ2p−2
, (17)
where Bp(x) are the Bernoulli polynomials [10]. As stated above, Fρ andMρ are defined
by the condition that the above expression (17) vanishes for k = 0,−1 enforcing that
Q2 ΠLR(−Q2) ∼ Q−4 at large momentum, as in QCD. We emphasize that the above
coefficients of the OPE in Eq. (17) can not be calculated by a naive expansion at large
Q2 of the Green’s function in Eq. (13). In other words, physical masses and decay
constants do not satisfy the Weinberg sum rules [15].
On the other hand, for the coefficients accompanying nonnegative powers of mo-
mentum, akin to the chiral expansion at small Q2, one has (k = 1, 2, 3, ...):
C0 = −F 20 , C2k = (−1)k+1
(k − 1)!
ψ(k−1)
− ψ(k−1)
where ψ(k−1)(z) = dk−1ψ(z)/dzk−1. In Table 1 we collect the values for the first few of
these coefficients C2k.
Let us start with the construction of the rational approximants to the function
Q2 ΠLR(−Q2). Since our original function (14) falls off at large Q2 as Q−4, this is a
constraint we will impose on all our approximants.
The simplest PA satisfying the right falloff at large momentum is P 02 (Q
2), so we
will begin with this case. In order to simplify the results, and unless explicitly stated
otherwise, we will assume that dimensionful quantities are expressed in units of GeV
to the appropriate power. Fixing the three unknowns with the first three coefficients
from the chiral expansion of (14) (i.e. C0,2,4) one gets the following rational function
P 02 (Q
− r2R
(Q2 + zR)(Q2 + z
, r2R = 3.379×10−3 , zR = 0.6550+ i 0.1732 . (19)
5 10 15 20 25
Im (q  )
Re (q  )
Figure 1: Location of the poles (dots) and zeros (squares) of the Pade Approximant P 5052 (−q2)
in the complex q2 plane. We recall that Q2 = −q2. Notice how zeros and poles approximately
coincide in the region which is farthest away from the origin. When the order of the Pade
is increased, the overall shape of the figure does not change but the two branches of complex
poles move towards the right, i.e. away from the origin.
We can hardly overemphasize the striking appearance of a pair of complex-conjugate
poles on the Minkowski side of the complex Q2 plane. Obviously, this means that these
poles cannot be interpreted in any way as the meson states appearing in the physical
spectrum (10,13). In spite of this, if one expands (19) for large values of Q2 > 0, one
finds C−4 = −r2R = −3.379 × 10−3 which is not such a bad approximation for this
coefficient of the OPE, see Table 1. Even better is the prediction of the fourth term in
the chiral expansion, which is C6 = 79.58× 10−3.
This agreement is not a numerical coincidence and the approximation can be sys-
tematically improved if more terms of the chiral expansion are known. In order to
exemplify this, we have amused ourselves by constructing the high-order PA P 5052 (Q
This rational approximant correctly determines the values for C−4,−6,−8 with (respec-
tively) 52, 48 and 45 decimal figures. In the case of C103, which is the first predictable
term from the chiral expansion for this Pade, the accuracy reaches some staggering 192
decimal figures. This is all in agreement with Pommerenke’s theorem [20].
As it happens for the PA (19), also higher-order PAs may develop some artificial
poles. In particular, Figure 1 shows the location of the 52 poles of the PA P 5052 (Q
2) in
the complex q2 plane. Of these, the first 25 are purely real and the rest are complex-
conjugate pairs. A detailed numerical analysis reveals that the poles and residues
reproduce very well the value of the meson masses and decay constants for the lowest
part of the physical spectrum of the model given in (13-15), but the agreement deteri-
orates very quickly as one gets farther away from the origin, eventually becoming the
complex numbers seen in Fig. 1. It is by creating these analytic defects that rational
functions can effectively mimic with a finite number of poles the infinite tower of poles
present in the original function (14).
For instance the values of the first pole and residue in P 5052 (Q
2) reproduce those
of the ρ in (15) within 193 astonishing decimal places for both. However, in the case
of the 25th pole, which is the last one still purely real, its location agrees with the
physical mass only with 3 decimal figures. This is not to be considered as a success,
however, because after the previous accuracy, this is quite a dramatic drop. In fact,
the residue associated with this 25th pole comes out to be 29 times the true value.
The lesson we would like to draw from this exercise should be clear: the determination
of decay constants and masses extracted as the residues and poles of a PA deteriorate
very quickly as one moves away from the origin. There is no reason why the last
poles and residues in the PA are to be anywhere near their physical counterparts and
their identification with the particle’s mass and decay constant should be considered
unreliable. Clearly, this particularly affects low-order PAs.
A very good accuracy can also be obtained in the determination of global euclidean
observables such as integrals of the Green’s function over the interval 0 ≤ Q2 < ∞.
Notice that the region where one approximates the true function is far away from the
artificial poles in the PA. For instance, one may consider the value for the integral
Iπ = (−1)
dQ2 Q2ΠLR(Q
2) = 4.78719× 10−3, (20)
which, up to a constant, would yield the electromagetic pion mass difference in the
chiral limit [38] in the model (14). The PA P 5052 (Q
2) reproduces the value for this
integral with more than 42 decimal figures. This suggests that one may use the integral
(20) as a further input to construct a PA.
For example if we fix the three unknowns in the PA P 02 (Q
2) by matching the first
two terms from the chiral expansion but now we complete it with the pion mass dif-
ference (20) instead of a third term from the chiral expansion as we did in (19),13 the
approximant results to be
P̃ 02 (Q
− r2R
(Q2 + zR)(Q2 + z
, with r2R = 2.898× 10−3 , zR = 0.5618 + i 0.2795 .
This determines C−4 = −2.898×10−3 and C4 = −41.26×10−3, which shows that using
the pion mass difference is not a bad idea. Notice how the position of the artificial pole
has changed with respect to (19).
Artificial poles and analytic defects are transient in nature, i.e. they appear and
disappear from a point in the complex plane when the order of the Pade is changed. On
the contrary, the typical sign that a pole in a Pade is associated with a truly physical
pole is its stability under these changes in the order of the Pade. Of course, when the
order in the Pade increases there have to be new poles by definition, and it is natural
to expect that some of them will be defects. Pade Approximants place some effective
poles and residues in the complex Q2 plane in order to mimic the behavior of the true
Green’s function, but it can mimic the function only away from the poles, e.g. in the
Euclidean region. Obviously, PAs cannot converge at the poles, in agreement with
Pommerenke’s theorem [20], since not even the true function is well defined there. The
point is that what may look like a small correction in the Euclidean region may turn
out to be a large number in the Minknowski region. To exemplify this in simple terms,
13We remark that this procedure, although reasonable from the phenomenological point of view,
strictly speaking lies outside the standard mathematical theory of rational approximants [18, 31].
let us consider a very small parameter ǫ and imagine that a given Pade P (Q2) produces
the rational approximant to the true Green’s function G(Q2) given by
G(Q2) ≈ P (Q2) ≡ R(Q2) + ǫ
Q2 +M2
, (22)
where R(Q2) is the part of the Pade which is independent of ǫ. Although for Q2 > 0
there is a sense in which the last term is a small correction precisely because of the
smallness of ǫ, for Q2 < 0 this is no longer true because of the pole at Q2 = −M2.
This pole is in general a defect and may not represent any physical mass. In fact,
associated with this pole, there is a very close-by zero of the Pade P (Q2) at Q2 =
−M2 − ǫ R(−M2)−1, as can be immediately checked in (22). This is another way of
saying that a defect is characterized by having an abnormally small residue and is the
origin of the pairs of zeros and poles in the y-shaped branches of Fig. 1. Therefore,
not only are defects unavoidable but one could say they are even necessary for a Pade
Approximant to approximate a meromorphic function with an infinite set of poles.
Similarly to masses, also decay constants may be unreliable. To see this, imagine
now that our Pade is given by
P (Q2) =
Q2 +M2
(Q2 +M2) (Q2 +M2 + ǫ2)
, (23)
again for a very small ǫ. As before, the term proportional to ǫ may be considered
a small correction for Q2 > 0. However, at the pole Q2 = −M2 the decay constant
becomes F + ǫ−1 which, for ǫ small, may represent a huge correction. When the poles
are preassigned at the physical masses, like in the case of PTAs, it is the value of the
residues that compensates for the fact that the rational approximant lacks the infinite
tower of resonances. As we saw before, the residues of the poles in the Pade which lie
farthest away from the origin are the ones which get the largest distortion relative to
their physical counterparts.
In real life, the number of available terms from the chiral expansion for the con-
struction of a PA is very limited. Since the masses and decay constants of the first few
vector and axial-vector resonances are known, one may envisage the construction of a
rational approximant having some of its poles at the prescribed values given by the
known masses of these resonances. If all the poles in the approximant are prescribed
this way (as in the MHA), we have a PTA. On the contrary, when some of the poles are
prescribed but some are also left free, then we have a PPA (see the previous section).
Assuming that the first masses are known, let us proceed to constructing the PTAs
(6). The lowest such PTA is T02(Q
2), which contains two poles at the physical masses
of the ρ and the first A in the tower. Fixing the residue through the chiral expansion
to be C0 = −F 20 , one obtains
− F 20M2ρM2A
(Q2 +M2ρ )(Q
2 +M2A)
. (24)
Even though it has the same number of inputs (C0 and the two masses), this rational
approximant does not do such a good job as the PAs (19) or (21). For instance, C−4
is 2.3 times larger than the true value in Table 1. As we have already stated, one
way to intuitively understand this result is the following. The OPE is an expansion at
Q2 = ∞ and therefore knows about the whole spectrum because no resonance is heavy
enough with respect to Q2 to become negligible in the expansion, i.e., the infinite tower
of resonances does not decouple in the OPE. Chopping an infinite set of poles down
to a finite set may be a good approximation, but only at the expense of some changes.
These changes amount to the appearance of poles and residues in the PA which the
original function does not have. This is how the PA (19) manages to approximate
the true function (14). However, by construction, the PTA (24) does not allow the
presence of any artificial pole because, unlike in a PA, all its poles are fixed at the
physical values. Consequently, it only has its residues as a means to compensate for
the infinite tower of poles present in the true function and, hence, does a poorer job
than the PA (19), particularly in determining large-Q2 observables like C−4. Indeed,
the role played by the residues in the approximation can be appreciated by comparing
the true values of the decay constants to those extracted from (24). Although the one
of the ρ is within 30% of the true value, that of the A is off by 100%.
A different matter is the prediction of low-energy observables such as, e.g., the
chiral coefficients. In this case heavy resonances make a small contribution and this
means that the infinite tower of resonances does decouple.14 Truncating the infinite
tower down to a finite set of poles is not such a severe simplification in this case, which
helps understand why a PTA may do a good job predicting unknown chiral coefficients.
Indeed, (24) reproduces the value of C2 within an accuracy of 15%, growing to 22% in
the case of C4. A global observable like Iπ averages the low and the high Q
2 behaviors
and ends up differing from the true value (20) by 35%. This gives some confidence
that observables which are integrals over Euclidean momentum may be reasonably
estimated with MHA as, e.g., in the BK calculation of Ref. [39].
Improving on the PTA (24) by adding in the first resonance mass from the vector
tower produces the following approximant
a + b Q2
(Q2 +M2ρ )(Q
2 +M2A)(Q
2 +M2V )
, with
a = −13.5× 10−3,
b = +1.33× 10−4 , (25)
where the values of the chiral coefficients C0 and C2 have been used to determine
the parameters a and b. The prediction for C4 is much better now (only 2% off),
in agreement with our previous comments. The prediction for C−4 is still very bad,
becoming now 19 times smaller than the exact value. Nevertheless, it eventually gets
much better if PTAs of very high order are constructed. For instance, we have found
C−4 = −2.58 × 10−3 for the approximant T79 with 9 poles. Similarly, we have also
checked that the prediction of the chiral coefficients and the integral (20) improve with
higher-order PTAs.
However, another matter is the prediction of the residues. For instance, the predic-
tion for the decay constant of the state with mass MV in (25) is smaller than the exact
value in the model (15) by a factor of 2. In general, we have seen that the residues
14This is because the residues F 2 in the Green’s function (14) stay constant as the masses grow.
This behavior does not hold in the case of the scalar and pseudoscalar two-point functions [10].
of the poles always deteriorate very quickly so that the residue corresponding to the
pole which is at the greatest distance from the origin is nowhere near the exact value.
We again explicitly checked this up to the approximant T79, in which case the decay
constant for this pole is almost 5 times smaller than the exact value. The conclusion,
therefore, is that PTAs are able to approximate the exact function only at the expense
of changing the residues of the poles from their physical values. Identifying residues
with physical decay constants may be completely wrong in a PTA for the poles which
are farthest away from the origin.
As an intermediate approach between PAs and PTAs, there are the PPAs (3) where
some poles are fixed at their physical values while some others are left free. The simplest
of such rational approximants is P01,1(Q
2) (see the previous section for notation). Fixing
its 3 unknowns with M2ρ , C0 and C2, one obtains
1,1(Q
− r2R
(Q2 +M2ρ )(Q
2 + zR)
, with r2R = 3.75× 10−3 , zR = 0.8665 . (26)
As can be seen, the mass (squared) of the first A resonance is predicted to be at zR
which is sensibly smaller than the true value in (15)15. The rational function (26)
predicts C−4 = −r2R = −3.75 × 10−3 which is a better determination than that of the
PTA (24) with the same number of inputs, and C4 = −45.52× 10−3 which is not bad
either. Concerning the pion mass difference, one gets Iπ = 5.22 × 10−3. However, as
compared to the PAs (19) or (21), the PPA (26) does not represent a clear improvement.
In order to improve on accuracy of the PPA, one may try to use the mass and decay
constant of the first resonance, Mρ and Fρ, in addition to the pion mass difference and
the chiral coefficients C0, C2 and build the P
2,1(Q
2), which can be written as:
2,1(Q
F 2ρM
Q2 +M2ρ
a− F 2ρM2ρ Q2
(Q2 + zc) (Q2 + z∗c )
a = 17.43× 10−3,
zc = 1.24 + i 0.34 .
This PPA, upon expansion at large and small Q2, determines C−4 = −2.47× 10−3 and
C4 = −44.0× 10−3 to be compared with the corresponding coefficient in Table 1. The
accuracy obtained is better than that of (21), but this is probably to be expected since
(27) has more inputs.
Based on the previous numerical experiments done on the model in Eq. (14,15) (and
many others), we now summarize the following conclusions. Although, in principle,
the PAs have the advantage of reaching the best precision by carefully adjusting the
polynomial in the denominator to have some effective poles which simulate the infinite
tower present in (14), they have the disadvantage that some of the terms in the low-
Q2 expansion are required precisely to construct this denominator. This hampers the
construction of high-order PAs and consequently limits the possible accuracy.
When the locations of the first poles in the true function are known, there is the
possibility to construct PTAs (with all the poles fixed at the true values) and PPA
(with some of the poles fixed and some left free). As we have seen, although the PTA
may approximate low-Q2 properties of the true function reasonably well, the large-Q2
15Intriguinly enough, this is also what happens in the real case of QCD [9, 17].
properties tend to be much worse, at least as long as they are not of unrealistically
high order. The PPAs, on the other hand, interpolate smoothly between the PAs
(only free poles) and the PTAs (no free pole). Depending on the case, one may choose
one or several of these rational approximants. However, common to all the rational
approximants constructed is the fact that the residues and/or poles which are farthest
away from the origin are in general unrelated to their physical counterparts.
4 The QCD case
Let us now discuss the real case of large-Nc QCD in the chiral limit. In contrast to the
case of the previous model, any analysis in this case is limited by two obvious facts.
First, any input value will have an error (from experiment and because of the chiral and
large-Nc limits), and this error will propagate through the rational approximant. And
second, it is not possible to go to high orders in the construction of rational approxi-
mants due to the rather sparse set of input data. In spite of these difficulties one may
feel encouraged by the phenomenological fact that resonance saturation approximates
meson physics rather well.
The simplest PA to the function Q2ΠLR(−Q2) with the right fall-off as Q−4 at large
Q2 is P 02 (Q
P 02 (Q
1 + A Q2 +B Q4
. (28)
The values of the three unknowns a, A and B may be fixed by requiring that this PA
reproduces the correct values for F0, L10 and Iπ
16 given by
F0 = 0.086± 0.001 GeV ,
δmπ = 4.5936± 0.0005 MeV =⇒ Iπ = (5.480± 0.006)× 10−3GeV4 ,
L10(0.5 GeV) ≤ L10 ≤ L10(1.1 GeV) =⇒ L10 = (−5.13± 0.6)× 10−3 . (29)
The low-energy constant L10 is related to the chiral coefficient C2, in the notation of
Eq. (16), by C2 = −4L10. Since L10 does not run in the large-Nc limit, it is not
clear at what scale to evaluate L10(µ) [40]. In Eq. (29) we have varied µ in the range
0.5 GeV ≤ µ ≤ 1.1 GeV as a way to estimate 1/Nc systematic effects. The central
value corresponds to the result for L10(Mρ) found in Ref. [42]. The other results in
(29) are extracted from Refs. [1, 41].
Obviously, the PA (28) can also be rewritten as
P 02 (Q
(Q2 + zV )(Q2 + zA)
, (30)
in terms of two poles zV,A. In order to discuss the nature of these poles, we will define
the dimensionless parameter ζ by the combination
ζ ≡ −4L10
= 2.06± 0.25 , (31)
16Recall that Iπ is, up to a constant, the electromagnetic pion mass difference δmπ [17] and is
defined in terms of ΠLR as in Eq. (20).
where the values in (29) above have been used in the last step. Imposing the constraints
(29) on the PA (30) one finds two types of solutions depending on the value of ζ : for
ζ > 2 the two poles zV,A are real, whereas for ζ < 2 the two poles are complex. At
ζ = 2, the two solutions coincide. To see this, let us write the set of equations satisfied
by the PA (30) as:
F 20 =
zV zA
−4L10 = F 20
Iπ = F
zV zA
zA − zV
. (32)
The first of these equations can be used to determine the value of the residue r2 in
terms of zV zA. In order to analyze the other two, let us first assume that both poles
zV,A are real. In this case, they also have to be positive or else the integral Iπ will not
exist because it runs over all positive values of Q2. Let us now make the change of
variables
zV = R (1− x) , zA = R (1 + x) . (33)
The condition zV,A > 0 translates into R > 0, |x| < 1. In terms of these new variables,
the second and third equations in (32) can be combined into
1 + x
, (34)
where the definition (31) for ζ has been used. With the help of the identity log(1 +
x/1− x) = 2 th−1x (for |x| < 1), one can finally rewrite this expression as
th−1x , (x real) (35)
which is an equation with a solution for x only if ζ ≥ 2. Once this value of x is found,
the value of R can always be obtained from one of the last two equations (32) and this
determines the two real poles zV,A from (33).
On the other hand, when ζ < 2, Eq. (35) does not have a solution. However,
according to (31), ζ can also be smaller than 2. In order to study this case, we may
use the identity th−1(i y) = i tan−1(y) to rewrite the above equation (35) in terms of
the variable x = i y (y real) as
tan−1y , (y real). (36)
One now finds that this equation has a solution for y when ζ ≤ 2. In this case
the poles of the PA (28) are complex-conjugate to each other and can be obtained as
zV,A = R(1±i y). These poles, obviously, cannot be associated with any resonance mass
and this is why this solution has been discarded in all resonance saturation schemes
up to now. However, from the point of view of the rational approximant (28) there is
C0 C2 C4 C6 C8 C−4 C−6 C−8
−F 20 −4L10 −43± 13 81± 53 −145± 120 −4.1± 0.5 6± 2 −7± 6
Table 2: Values of the coefficients C2k in the high- and low-Q
2 expansions of
Q2 ΠLR(−Q2) in Eq. (16) in units of 10−3 GeV 2−2k. Recall that C−2 = 0.
nothing wrong with this complex solution, as the approximant is real and well behaved.
From the lessons learned in the previous section with the model, there is no reason to
discard this solution since, as we saw, rational approximants may use complex poles
to produce accurate approximations. Therefore, we propose to use both the complex
as well as the real solution for the poles zV,A, at least insofar as the value for ζ ≷ 2. In
this case we obtain, using the values given in Eqs. (29),
(ζ ≥ 2) , r2 = −(4.1± 0.5)× 10−3 , zV = (0.77)2 ± 0.15 , zA = (0.96)2 ± 0.41 (37)
(ζ ≤ 2) , r2 = −(3.9± 0.1)× 10−3 , zV = z∗A = (0.66± 0.06) + i (0.25± 0.25) , (38)
in units of GeV6 for r2, and GeV2 for zV,A. The two solutions in Eqs. (37,38) have
been separated for illustrative purposes only. It is clear that they are continuously
connected through the boundary at ζ = 2, at which value the two poles coincide and
zV = zA ≃ 0.72. The errors quoted are the result of scanning the spread of values in
(29) through the equations (32).
With both set of values in (37,38), one can get to a prediction for the chiral and OPE
coefficients by expansion in Q2 and 1/Q2, respectively. These expansions of the PA can
be done entirely in the Euclidean region Q2 > 0, away from the position of the poles
zV,A, whether real or complex. Recalling the notation in Eq. (16), the above P
produces the coefficients for these expansions collected in Table 2. The values for the
OPE coefficients C−4,−6,−8 in this table are compatible with those of Ref. [17], after
multiplying by a factor of two in order to agree with the normalization used by these
authors. However, the spectrum in our case is different because of the complex solution
in (38). As we saw in the previous section with a model, this again shows that Euclidean
properties of a given Green’s function, such as the OPE and chiral expansions, or
integrals over Q2 > 0 are safer to approximate with a rational approximant than
Minkowskian quantities, such as resonance masses and decay constants.
5 Conclusions
In this article we pointed out that approximating large-Nc QCD with a finite number
of resonances may be reinterpreted within the mathematical Theory of Pade Approxi-
mants to meromorphic functions [18].
The main results of this theory may be summarized as follows. One may expect
convergence of a sequence of Pade Approximants to a QCD Green’s function in the
large-Nc limit in any compact region of the complex Q
2 plane except at most in a
zero-area set [20]. This set without convergence comprises the poles of the original
Green’s function together with some other artificial poles generated by the approximant
which the original function does not have. As the order of the PA grows, the previous
convergence property implies that any given artificial pole either goes to infinity, away
from the relevant region, or is almost compensated by a nearby zero. This symbiosis
between a pole and a zero is called a defect. Although close to a pole the rational
approximation breaks down, in a region which is far away from it the approximation
should work well.
We have reviewed the main results of this theory with the help of a model for the
two-point Green’s function 〈V V −AA〉. The simpler case of a Green’s function of the
Stieltjes type, such as the two-point correlator 〈V V 〉, was previously considered in Ref.
[21]. We have seen in the case of this particular model how rational approximants
create the expected artificial poles (and the corresponding residues) in the Minkowski
region Re(q2) > 0 while, at the same time, yielding an accurate description of the
Green’s function in the Euclidean region Re(q2) < 0. This happens in a hierarchical
way: although the first poles/residues in a PA may be used to describe the physical
masses/decay constants reasonably well, the last ones give only a very poor description.
Therefore, it is in general unreliable to extract properties of individual mesons, such
as masses and decay constants, from an approximation to large-Nc QCD with only a
finite number of states. Since a form factor, like a decay constant, is obtained as the
residue of a Green’s function at the corresponding pole(s), this also means that one
may not extract a meson form factor from a rational approximant to a 3-point Green’s
function, in agreement with [16]. This observation may explain why the analysis of
Ref. [43], which is based on an extraction of matrix elements such as 〈π|S|P 〉 and
〈π|P |S〉 from the 3-point function 〈SPP 〉, finds values for the Kℓ3 form factor which
are different from those obtained in other analyses [44].
In spite of all the above problems related to the Minkowski region, our model
shows how Pade Approximants may nevertheless be a useful tool in other regions of
momentum space. We think that this is also true in the real case of QCD in the large-
Nc limit. In this case one may use the first few terms of the chiral and operator product
expansions of a given Green’s function to construct a Pade Approximant which should
yield a reasonable description of this function in those regions of momentum space
which are free of poles. In this construction, Pade Approximants containing complex
poles, if they appear, should not be dismissed.
We have also reanalyzed the simplest approximation to the 〈V V − AA〉 Green’s
function in real QCD which consists of keeping only two poles, and we have found
that, depending on the value of the combination ζ in Eq. (31), these two poles may
actually be complex.
However, if not all the residues and masses in a rational approximant are physical,
this poses a challenge to any attempt to use a Lagrangian with a finite number of
resonances such as, for example, the ones in Ref. [9, 11], for describing Green’s functions
in the large-Nc limit of QCD. Even if these Lagrangians are interpreted in terms of
PTAs, with the poles fixed at the physical value of the meson masses, we have seen how
the residues then get very large corrections with respect to their physical counterparts.
Can these residues be efficiently incorporated in a Lagrangian framework? We hope to
be able to devote some work to answering this and related questions in the future.
Acknowledgements
S.P. is indebted to M. Golterman, M. Knecht and E. de Rafael for innumerable
discussions during the last years which have become crucial to shape his understanding
on these issues. He is also very grateful to C. Diaz-Mendoza, P. Gonzalez-Vera and R.
Orive, from the Dept. of Mathematical Analysis at La Laguna Univ., for invaluable
conversations on the properties of Pade Approximants as well as for hospitality. We
thank S. Friot, M. Golterman, M. Jamin, R. Kaiser, J. Portoles and E. de Rafael for
comments on the manuscript.
This work has been supported by CICYT-FEDER-FPA2005-02211, SGR2005-00916
and by the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”.
APPENDIX
Here we will show how the PAs constructed from the OPE do not in general re-
produce even the first resonances in the spectrum, unlike those constructed from the
chiral expansion. Again, we will use the model of section 3 as an example. Recalling
the definition of the OPE given in Eq. (16), with the corresponding coefficients (17), it
is straightforward to construct a PA in 1/Q2 around infinity, i.e. by matching powers
of the OPE in 1/Q2. The construction parallels that in Eq. (2) but with the replace-
ment z = 1/Q2. Since the function Q2ΠLR(−Q2) behaves like a constant for Q2 → 0,
we will consider diagonal Pade Approximants, i.e. of the form P nn (1/Q
2), in order to
reproduce this behavior. Figure 2 shows the position of the poles and zeros of the PA
P 5050 (−1/q2) in the complex q2 plane. As it is clear from this plot, the positions of the
poles have nothing to do with the physical masses in the model, given by Eqs. (11-15),
even for the lightest states. This is to be contrasted with what happens with the PA
constructed from the chiral expansion around Q2, which is shown in Fig. 1. The differ-
ence between the two behaviors is due to the fact that, while the chiral expansion has
a finite radius of convergence, the radius of convergence of the OPE vanishes because
this expansion is asymptotic.
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	Introduction
	Rational approximations: generalities
	Testing rational approximations: a model
	The QCD case
	Conclusions

  We point out that resonance saturation in QCD can be understood in the
large-Nc limit from the mathematical theory of Pade Approximants to meromorphic
functions. These approximants are rational functions which encompass any
saturation with a finite number of resonances as a particular example,
explaining several results which have appeared in the literature. We review the
main properties of Pade Approximants with the help of a toy model for the
<VV-AA> two-point correlator, paying particular attention to the relationship
among the Chiral Expansion, the Operator Product Expansion and the resonance
spectrum. In passing, we also comment on an old proposal made by Migdal in 1977
which has recently attracted much attention in the context of AdS/QCD models.
Finally, we apply the simplest Pade Approximant to the <VV-AA> correlator in
the real case of QCD. The general conclusion is that a rational approximant may
reliably describe a Green's function in the Euclidean, but the same is not true
in the Minkowski regime due to the appearance of unphysical poles and/or
residues.

Introduction
The strong Chiral Lagrangian is a systematic organization of the physics in powers of
momenta and quark masses, but requires knowledge of the low-energy constants (LEC)
to make reliable phenomenological predictions. As with any other effective field theory,
these LECs play the role of coupling constants and contain the information which comes
from the integration of the heavy degrees of freedom not explicitly present in the Chiral
Lagrangian (e.g. meson resonances).
At O(p4) there are 10 of these constants [1]. Although at this order there is enough
independent information to extract the values for these constants from experiment,
this will hardly ever be possible at the next order, O(p6), because the number of
constants becomes more than a hundred [2]. In the electroweak sector the proliferation
of constants appears already at O(p4) [3]. Although in principle these low-energy
constants may be computed on the lattice, in practice this has only been accomplished
in a few cases for the strong Chiral Lagrangian at O(p4), and only recently [4].
The large Nc expansion [5] stands out as a very promising analytic approach capa-
ble of dealing with the complexities of nonperturbative QCD while, at the same time,
offering a relatively simple and manageable description of the physics. For instance,
mesons are qq states with no width, the OZI rule is exact and there is even a proof
of spontaneous chiral symmetry breaking [6]. Furthermore, its interest has recently
received a renewed boost indirectly through the connection of some highly supersym-
metric gauge theories to gravity [7], although the real relevance of this connection for
QCD still remains to be seen. However, in spite of all this, the fact that no solution to
large-Nc QCD has been found keeps posing a serious limitation to doing phenomenol-
ogy. For instance, in order to reproduce the parton model logarithm which is present
in QCD Green’s functions in perturbation theory, an infinity of resonances is necessary
whose masses and decay constants are in principle unknown.
On the other hand, QCD Green’s functions seem to be approximately saturated by
just a few resonances; a property which has a long-standing phenomenological support
going all the way back to vector meson dominance ideas [8], although it has never
been properly understood. In a modern incarnation, this fact translated into the very
successful observation [9] that the strong LECs at O(p4) seem to be well saturated by
the lowest meson in the relevant channel,1 after certain constraints are imposed on some
amplitudes at high-energy in order to match the expected behavior in QCD [11, 12].
It was then realized that all these successful results could be encompassed at once as
an approximation to large-Nc QCD consisting in keeping only a finite (as opposed to
the original infinite) set of resonances in Green’s functions. This approximation to
large-Nc QCD has been termed Minimal Hadronic Approximation (MHA) [13] because
it implements the minimal constraints which are necessary to secure the leading non-
trivial behavior at large energy of certain Green’s functions through the marriage of
the old resonance saturation and the large-Nc approximation of QCD. In recent years,
a large amount of work has been dedicated to studying the consequences of these ideas
[14].
1This is less clear in the scalar channel, however. See Ref. [10]
However, the high-energy matching with a finite set of resonances, first suggested in
[11], makes it clear that the treatment is not amenable to the methods of a conventional
effective field theory. An effective field theory is an approximation for energies smaller
than a heavy particle’s mass and, therefore, cannot deal with momentum expansions
at infinity as in the case of the Operator product Expansion (OPE). In other words,
the fact that the set of resonances in each channel is really infinite precludes the
naive expansion at large momentum because there is always a mass in the spectrum
which is even larger. The sum over an infinite set of resonances and the expansion
for large momentum are operations which do not commute [15]. In those Green’s
functions containing a contribution from the parton model logarithm, this is made
self-evident since a naive expansion at large momentum can only produce powers and
not a logarithm, which is why large-Nc QCD requires an infinity of resonances in the
first place.
The problem can be delayed one power of αs if one requires the use of the resonance
Lagrangian [9] to be limited only to Green’s functions which are order parameters of
spontaneous chiral symmetry breaking. These order parameters vanish to all orders in
αs in the chiral limit
2 and, therefore, avoid the presence of the parton model logarithm
which, otherwise, would preclude from the outset any matching to a finite number of
resonances. However, the concept of a Lagrangian whose validity is restricted only to a
certain class of Green’s functions has never been totally clear; and even if the resonance
Lagrangian is restricted by definition to order parameters, the problem surfaces again
in the presence of logarithmic corrections from nontrivial anomalous dimensions, which
make the exact matching at infinite momentum impossible.
In a slightly different context, a somewhat similar observation was also made in Ref.
[16]. In this paper it was observed that it is impossible to satisfy the large momentum
fall-off expected in large-Nc QCD for the form factors which can be defined through
a three-point Green’s function, if the sum over resonances in the Green’s function is
restricted to a finite set. Interestingly, this again pointed to an incompatibility of the
QCD short-distance behavior with an approximation to large Nc which only kept a
finite number of resonances.
A further piece of interesting evidence results from the comparison between the
analysis in Refs. [17] and [11]. After imposing some good high-energy behavior in
several Green’s functions and form factors including, in particular, the axial form
factor governing the decay π → γeν, Ref. [11] obtains, keeping only one vector state V
and one axial-vector state A, that their two masses must be constrained by the relation
2MV . The work in Ref. [17], on the contrary, does not use the axial form factor
and obtains, after performing a very good fit within the same set of approximations,
the precise values MV = 775.9 ± 0.5 MeV and MA = 938.7 ± 1.4 MeV. These values
for the masses, although close, are not entirely compatible with the previous relation.
In other words, the short-distance constraint from the axial form factor is not fully
compatible with the short-distance constraints used in [17] if restricted to only one
vector and one axial-vector states3.
2E.g., the two-point correlator 〈V V −AA〉.
3Adding one further state does not change the conclusion [17].
In this paper we would like to suggest that all the above properties can be un-
derstood if the approximation to large Nc QCD with a finite number of resonances is
reinterpreted within the mathematical Theory of Pade Approximants (PA) to mero-
morphic functions [18]. QCD Green’s functions in the large Nc and chiral limits have
an analytic structure in the complex momentum plane which consists of an infinity
of isolated poles but no cut, i.e. they become meromorphic functions [19]. As such,
they have a well-defined series expansion in powers of momentum around the origin
with a finite radius of convergence given by the first resonance mass4. This is all that
is needed to construct a Pade Approximant. A theorem by Pommerenke [20] assures
then convergence of any near diagonal PA to the true function for any finite momen-
tum, over the whole complex plane, except perhaps in a zero-area set. The poles of
the original Green’s function (i.e. the resonance masses) belong to this zero-area set
because not even the original function is defined there, but there are also extra poles.
These extra poles are called “defects” in the mathematical literature [18]. When the
Green’s function being approximated is of the Stieltjes type5, the poles of the PA are
always real and located on the Minkowski region Re(q2) = Re(−Q2) > 0, approaching
the physical poles as the order of the PA is increased [21]. However, this takes place
in a hierarchical way and, while the poles in the PA which are closest to the origin are
also very close to the physical masses, the agreement quickly deteriorates and one may
find that the last poles are several times bigger than their physical counterparts [22].
The same is true of the residues. In section 3, we will see with the help of a model
that the same properties are met in a meromorphic function whose spectral function
is not positive definite, except that some of the poles in the PA may even be complex.
This means that Minkowskian properties, such as masses and decay constants,
cannot be reliably determined from a PA except, perhaps, from the first poles which
are closest to the origin. If not all the residues and/or masses are physical, then there
is no reason why they should be the same in the form factor governing π → γeν and
in the Green’s function 〈V V − AA〉, explaining the different results found in [11] and
[17] we alluded to above. Furthermore, the form factors of all but the lightest mesons,
defined through the residues of the corresponding 3-point Green’s functions, will not
be reliably determined from a PA to that Green’s function, again in agreement with
the findings in [16].
The situation in the Euclidean is different. In general, PAs cannot be expanded at
infinite momentum to generate an OPE type expansion for the true function. Never-
theless, Pommerenke’s theorem assures a good approximation at any finite momentum,
no matter how large. Of course, the order of the PA will have to increase, the larger
the momentum region one wishes to approximate. For instance, in Ref. [21] it was
shown with the help of a simple model how, even in the case of the 〈V V 〉 correlator
which contains a logQ2 at large values of Q2 > 0, the PAs are capable of approximat-
ing the true function at any arbitrarily large (but finite) value of Q2 > 0, without the
need for a perturbative continuum. In section 3 we will show, again with the help of a
4The pion pole can always be eliminated multiplying by enough powers of momentum. We are
assuming here the existence of a nonvanishing gap in large-Nc QCD.
5Roughly this means that the associated spectral function is positive definite, like in the case of
the two-point correlator 〈V V 〉. See Ref. [18] (chapter 5) for a more precise definition.
model, how this is also true in the more general case of a non-positive definite spectral
function such as 〈V V −AA〉. This means that PAs are a reliable way to approximate
the original Green’s function in the Euclidean but not in the Minkowskian regime.
In 1977, A.A. Migdal [23] suggested PAs as a method to extract the spectrum of
large-Nc QCD from the leading term in the OPE of the 〈V V 〉 correlator, i.e. from
the parton model logarithm. However, nowadays this proposal should be considered
unsatisfactory for a number of reasons [24], the most simple of them being that different
spectra may lead to the same parton model logarithm [25]. In fact, the full OPE series
is expected to be only an asymptotic expansion at Q2 = ∞ (i.e. with zero radius
of convergence), and PAs constructed from this type of expansions cannot in general
reproduce the position of the physical poles [26]. For instance, we show this explicitly
with the help of a model for 〈V V − AA〉 in the Appendix. Migdal’s approach has
been recently adopted (in disguise) in some models exploiting the so-called AdS/QCD
correspondence [27] and, consequently, the same criticism also applies to them.
In Ref. [28] a model for the 〈V V − AA〉 two-point correlator with a spectrum
consisting of an infinity of resonances was suggested as a theoretical laboratory for
studying the relationship between the spectrum and the coefficients of the OPE. In
this paper several conventional methods usually employed in the literature were tested
against the exact result from the model. These included: Finite Energy Sum Rules as
in Ref. [33], pinched weights as in Ref. [34], Laplace transforms as in Ref. [35] and,
finally, also resonance saturation as in the MHA method. The bottom line was that
no method was able to produce very accurate predictions for the OPE coefficients. In
all the methods but the last one, the reason for this lack of accuracy was basically due
to the fact that the OPE requires an integral over the whole spectrum, whereas the
integral is actually cut off at an upper limit (in the real case, the upper limit is mτ ).
This is why even if one uses the real spectrum the result may be inaccurate [29]. In
the case of the MHA the reason was, as we will comment upon below, that the poles
were not allowed to be complex.
In section 3 we will revisit this 〈V V − AA〉 model, now from the point of view of
PAs. The model reproduces the power behavior of QCD at large Q2 > 0 except that
the model is simple enough not to have any logQ2 and, therefore, it cannot reproduce
the nonvanishing anomalous dimensions which exist in QCD. We do not think this is
a major drawback, however, because in QCD these logarithms are always screened by
at least one power of αs and, hence, in an approximate sense, it may be licit to ignore
them. In the model such an approximation becomes exact6. Will the PAs be able to
reproduce the large Q2 expansion of the 〈V V −AA〉 model? We will see that the answer
is affirmative. Therefore, the reason why the MHA method was not able to predict
accurately the OPE coefficients in Ref. [28] is because the lowest PA has complex
poles which were not allowed in [28]. When these complex poles are considered, the
accuracy achieved is better and, most importantly, improves for a higher PA. Since
the model allows the construction of PAs of a very high order, we have checked this
convergence up to the Pade P 5052 , which is able to reproduce the first non vanishing
coefficient of the OPE in the model with an accuracy of 52 decimal figures. Together
6For a model with a logQ2, the reader may consult Ref. [21].
with other numerical examples which will be discussed in section 3, we take this as a
clear evidence of the convergence of the method. This renders some confidence that
PAs may also do a good job in the real case of QCD.
The rest of the paper is organized as follows. In section 2 we review some generali-
ties of rational approximants, in section 3 we describe the 〈V V −AA〉 model and apply
different rational approximants to learn about the possible advantages and disadvan-
tages of them. In section 4 we apply the simplest PA to the case of the real 〈V V −AA〉
two-point function in QCD. Finally, we close with some conclusions.
2 Rational approximations: generalities
Let a function f(z) have an expansion around the origin of the complex plane of the
f(z) =
n , z → 0 . (1)
One defines a Pade Approximant (PA) to f(z) , denoted by PMN (z), as a ratio of two
polynomials QM(z), RN (z)
7, of order M and N (respectively) in the variable z, with
a contact of order M + N with the expansion of f(z) around z = 0. This means
that, when expanding PMN (z) around z = 0, one reproduces exactly the first M + N
coefficients of the expansion for f(z) in Eq. (1):
PMN (z) =
QM(z)
RN (z)
≈ f0 + f1 z + f2 z2 + ...+ fM+N zM+N +O(zN+M+1) . (2)
At finite z, the rational function PMN (z) constitutes a resummation of the series (1). Of
special interest for us will be the case when N =M +k, for a fixed k, because then the
function behaves like 1/zk at z = ∞. The corresponding PAs PMM+k(z) belong to what
is called the near-diagonal sequence for k 6= 0, with the case k = 0 being the diagonal
sequence.
The convergence properties of the PAs to a given function are much more difficult
than those of normal power series and this is an active field of research in Applied
Mathematics. In particular, those which concern meromorphic functions8 are rather
well-known and will be of particular interest for this work. The main result which
we will use is Pommerenke’s Theorem [20] which asserts that the sequence of (near)
diagonal PA’s to a meromorphic function is convergent everywhere in any compact set
of the complex plane except, perhaps, in a set of zero area. This set obviously includes
the set of poles where the original function f(z) is clearly ill-defined but there may
be some other extraneous poles as well. For a given compact region in the complex
plane, the previous theorem of convergence requires that, either these extraneous poles
move very far away from the region as the order of the Pade increases, or they pair
up with a close-by zero becoming what is called a defect in the mathematical jargon
7Without loss of generality we define, as it is usually done, RN (0) = 1.
8A function is said to be meromorphic when its singularities are only isolated poles.
[30]. These are to be considered artifacts of the approximation. Near the location of
these extraneous poles the PA approximation clearly breaks down but, away from these
poles, the approximation is safe.
In the physical case the original function f(z) will be a Green’s function G(Q2)
of the momentum variable Q2. In QCD in the large Nc limit this Green’s function
is meromorphic with all its poles located on the negative real axis in the complex Q2
plane. These poles are identified with the meson masses. On the other hand, the region
to be approximated by the PAs will be that of euclidean values for the momentum, i.e.
Q2 > 0. The expansion of G(Q2) for Q2 large and positive coincides with the Operator
Product Expansion.
In general a meromorphic function does not obey any positivity constraints and,
as we will see, this has as a consequence that some of the poles and residues of the
PAs may become complex 9. This clearly precludes any possibility that these poles and
residues may have anything to do with the physical meson masses and decay constants.
However, and this is very important to realize, this does not spoil the validity of the
rational approximation provided the poles, complex or not, are not in the region of
Q2 one is interested in. It is to be considered rather as the price to pay for using a
rational function, which has only a finite number of poles, as an approximation to a
meromorphic function with an infinite set of poles.
When the position of the poles in the original Green’s function is known, at least
for the lowest lying states, it is interesting to devise a rational approximation which has
this information already built in. The corresponding approximants are called Partial
Pade Approximants (PPAs) in the mathematical literature [31] and are given by a
rational function PMN,K(Q
N,K(Q
RN(Q2) TK(Q2)
, (3)
where QM(Q
2), RN(Q
2) and TK(Q
2) are polynomials of order M,N and K (respec-
tively) in the variable Q2. The polynomial TK(Q
2) is defined by having K zeros
precisely at the location of the lowest lying poles of the original Green’s function10
2) = (Q2 +M21 ) (Q
2 +M22 ) ... (Q
2 +M2K) . (4)
As before the polynomial RN (Q
2) is chosen so that RN (0) = 1 and, together with
2), they are defined so that the ratio PMN,K(Q
2) matches exactly the first M +N
terms in the expansion of the original function around Q2 = 0, i.e. :
N,K(Q
2) ≈ f0 + f1 Q2 + f2 Q4 + ... + fM+N Q2M+2N +O(Q2N+2M+2) . (5)
At infinity, the PPA in Eq. (3) obviously falls off like 1/Q2N+2K−2M . Exactly as
it happens in the case of PAs, also the PPAs will have complex poles for a general
9A special case which does obey positivity constraints is when the function is Stieltjes. In this
case the poles and residues of the PAs are purely real and with the same sign as those of the original
function [21].
10For simplicity, we will assume that all the poles are simple.
meromorphic function, which prevents it from any interpretation in terms of meson
states.
Finally, another rational approximant defined in mathematics is the so-called Pade
Type Approximant (PTA) [31] TMN (Q
QM (Q
TN (Q2)
, (6)
where TN (Q
2) is also given by the polynomial (4), now with N preassigned zeros at
the corresponding position of the poles of the original Green’s function, G(Q2). The
polynomial QM(Q
2) is defined so that the expansion of the PTA around Q2 = 0 agrees
with that of the original function up to and including terms of order M + 1, i.e.
2) ≈ f0 + f1 Q2 + f2 Q4 + ... + fM Q2M +O(Q2M+2) . (7)
At large values of Q2, one has that TMN (Q
2) falls off like 1/Q2N−2M . Clearly the PTAs
are a particular case of the PPAs, i.e. TMN (Q
2) = PM0,N(Q
2) and coincide with what has
been called the Hadronic Approximation to large-Nc QCD in the literature [13].
Let us summarize the mathematical jargon. A Pade Type Approximant (PTA)
is a rational function with all the poles chosen in advance precisely at the physical
masses. A Pade Approximant (PA) is when all the poles are left free. The intermediate
situation, with some poles fixed at the physical masses and some left free, corresponds
to what is called a Partial Pade Approximant (PPA).
3 Testing rational approximations: a model
Let us consider the two-point functions of vector and axial-vector currents in the chiral
limit
ΠV,Aµν (q) = i
d4x eiqx〈JV,Aµ (x)J† V,Aν (0)〉 =
qµqν − gµνq2
ΠV,A(q
2) , (8)
with J
V (x) = d(x)γ
µu(x) and J
A(x) = d(x)γ
µγ5u(x). As it is known, the difference
ΠV (q
2)− ΠA(q2) satisfies an unsubtracted dispersion relation11
ΠV−A(q
t− q2 − iǫ
ImΠV−A(t) . (9)
Following Refs. [32, 28], we define our model by giving the spectrum as
ImΠV (t) = 2F
ρ δ(t−M2ρ ) + 2
F 2V (n)δ(t−M2V (n)) ,
ImΠA(t) = 2F
0 δ(t) + 2
F 2A(n)δ(t−M2A(n)) . (10)
11The upper cutoff which is needed to render the dispersive integrals mathematically well defined
can be sent to infinity provided it respects chiral symmetry [15].
Here Fρ,Mρ are the electromagnetic decay constant and mass of the ρ meson and
FV,A(n) are the electromagnetic decay constants of the n− th resonance in the vector
(resp. axial) channels, while MV,A(n) are the corresponding masses. F0 is the pion
decay constant in the chiral limit. The dependence on the resonance excitation number
n is the following:
F 2V,A(n) = F
2 = constant , M2V,A(n) = m
V,A + n Λ
2 , (11)
in accord with known properties of the large-Nc limit of QCD [5] as well as alleged
properties of the associated Regge theory [37].
The combination
ΠLR(q
(ΠV (q
2)−ΠA(q2)) (12)
thus reads
ΠLR(q
−q2 +M2ρ
−q2 +M2V (n)
−q2 +M2A(n)
. (13)
This two-point function can be expressed in terms of the Digamma function ψ(z) =
log Γ(z) as [28]
ΠLR(q
−q2 +M2ρ
−q2 +m2A
−q2 +m2V
. (14)
To resemble the case of QCD, we will demand that the usual parton-model logarithm
is reproduced in both vector and axial-vector channels and that the difference (9) has
an operator product expansion which starts at dimension six. A set of parameters
satisfying these conditions is given by12
F0 = 85.8 MeV , Fρ = 133.884 MeV , F = 143.758 MeV , (15)
Mρ = 0.767 GeV, mA = 1.182 GeV, mV = 1.49371 GeV , Λ = 1.2774 GeV ,
and is the one we will use in this section. This set of parameters has been chosen to
resemble those of the real world, while keeping the model at a manageable level. For
instance, the values of Fρ and Mρ in (15) are chosen so that the function ΠLR in (14)
has vanishing 1/Q2 and 1/Q4 in the OPE at large Q2 > 0, as in real QCD. In fact, the
model admits the introduction of finite widths (which is a 1/Nc effect) in the manner
described in Ref. [32], after which the spectral function looks reasonably similar to
the experimental spectral function. This comparison can be found in Fig. 5 of Ref.
[28]. But this model is also interesting for a very different reason. In Ref. [28] several
attempts were made at determining the coefficients of the OPE by using the methods
which have become common practice in the literature. Among those we may list Finite
Energy Sum Rules [33], with pinched weights [34], Laplace sum rules [35] and Minimal
Hadronic Approximation [13]. As it turned out, when these methods were tested on
12These numbers have been rounded off for the purpose of presentation. Some of the exercises which
will follow require much more precision than the one shown here.
C0 C2 C4 C6 C−4 C−6 C−8
−7.362 21.01 −43.92 81.81 −2.592 1.674 −0.577
Table 1: Values of the coefficients C2k from the high- and low-Q
2 expansions of
Q2 ΠLR(−Q2) in Eq. (16) in units of 10−3 GeV 2−2k. Notice that C−2 = 0 and
C0 = −F 20 (the pion decay constant in the chiral limit), see text.
the model, none of them was able to produce very accurate results. We think that this
makes the model very interesting (and challenging !) as a way to assess systematic
errors [36].
Defining the expansion of the Green’s function (9) in Q2 = −q2 around Q2 = 0,∞
Q2 ΠLR(−Q2) ≈
C2k Q
2k , with k = 0,±1,±2,±3, . . . (16)
one obtains that the coefficients accompanying inverse powers of momentum, akin to
the Operator Product Expansion at large Q2 > 0, are given by (p = 1, 2, 3, ... with
k = 1− p):
C2k = −F 20 δp,1 +
(−1)p+1
F 2ρM
F 2Λ2p−2
, (17)
where Bp(x) are the Bernoulli polynomials [10]. As stated above, Fρ andMρ are defined
by the condition that the above expression (17) vanishes for k = 0,−1 enforcing that
Q2 ΠLR(−Q2) ∼ Q−4 at large momentum, as in QCD. We emphasize that the above
coefficients of the OPE in Eq. (17) can not be calculated by a naive expansion at large
Q2 of the Green’s function in Eq. (13). In other words, physical masses and decay
constants do not satisfy the Weinberg sum rules [15].
On the other hand, for the coefficients accompanying nonnegative powers of mo-
mentum, akin to the chiral expansion at small Q2, one has (k = 1, 2, 3, ...):
C0 = −F 20 , C2k = (−1)k+1
(k − 1)!
ψ(k−1)
− ψ(k−1)
where ψ(k−1)(z) = dk−1ψ(z)/dzk−1. In Table 1 we collect the values for the first few of
these coefficients C2k.
Let us start with the construction of the rational approximants to the function
Q2 ΠLR(−Q2). Since our original function (14) falls off at large Q2 as Q−4, this is a
constraint we will impose on all our approximants.
The simplest PA satisfying the right falloff at large momentum is P 02 (Q
2), so we
will begin with this case. In order to simplify the results, and unless explicitly stated
otherwise, we will assume that dimensionful quantities are expressed in units of GeV
to the appropriate power. Fixing the three unknowns with the first three coefficients
from the chiral expansion of (14) (i.e. C0,2,4) one gets the following rational function
P 02 (Q
− r2R
(Q2 + zR)(Q2 + z
, r2R = 3.379×10−3 , zR = 0.6550+ i 0.1732 . (19)
5 10 15 20 25
Im (q  )
Re (q  )
Figure 1: Location of the poles (dots) and zeros (squares) of the Pade Approximant P 5052 (−q2)
in the complex q2 plane. We recall that Q2 = −q2. Notice how zeros and poles approximately
coincide in the region which is farthest away from the origin. When the order of the Pade
is increased, the overall shape of the figure does not change but the two branches of complex
poles move towards the right, i.e. away from the origin.
We can hardly overemphasize the striking appearance of a pair of complex-conjugate
poles on the Minkowski side of the complex Q2 plane. Obviously, this means that these
poles cannot be interpreted in any way as the meson states appearing in the physical
spectrum (10,13). In spite of this, if one expands (19) for large values of Q2 > 0, one
finds C−4 = −r2R = −3.379 × 10−3 which is not such a bad approximation for this
coefficient of the OPE, see Table 1. Even better is the prediction of the fourth term in
the chiral expansion, which is C6 = 79.58× 10−3.
This agreement is not a numerical coincidence and the approximation can be sys-
tematically improved if more terms of the chiral expansion are known. In order to
exemplify this, we have amused ourselves by constructing the high-order PA P 5052 (Q
This rational approximant correctly determines the values for C−4,−6,−8 with (respec-
tively) 52, 48 and 45 decimal figures. In the case of C103, which is the first predictable
term from the chiral expansion for this Pade, the accuracy reaches some staggering 192
decimal figures. This is all in agreement with Pommerenke’s theorem [20].
As it happens for the PA (19), also higher-order PAs may develop some artificial
poles. In particular, Figure 1 shows the location of the 52 poles of the PA P 5052 (Q
2) in
the complex q2 plane. Of these, the first 25 are purely real and the rest are complex-
conjugate pairs. A detailed numerical analysis reveals that the poles and residues
reproduce very well the value of the meson masses and decay constants for the lowest
part of the physical spectrum of the model given in (13-15), but the agreement deteri-
orates very quickly as one gets farther away from the origin, eventually becoming the
complex numbers seen in Fig. 1. It is by creating these analytic defects that rational
functions can effectively mimic with a finite number of poles the infinite tower of poles
present in the original function (14).
For instance the values of the first pole and residue in P 5052 (Q
2) reproduce those
of the ρ in (15) within 193 astonishing decimal places for both. However, in the case
of the 25th pole, which is the last one still purely real, its location agrees with the
physical mass only with 3 decimal figures. This is not to be considered as a success,
however, because after the previous accuracy, this is quite a dramatic drop. In fact,
the residue associated with this 25th pole comes out to be 29 times the true value.
The lesson we would like to draw from this exercise should be clear: the determination
of decay constants and masses extracted as the residues and poles of a PA deteriorate
very quickly as one moves away from the origin. There is no reason why the last
poles and residues in the PA are to be anywhere near their physical counterparts and
their identification with the particle’s mass and decay constant should be considered
unreliable. Clearly, this particularly affects low-order PAs.
A very good accuracy can also be obtained in the determination of global euclidean
observables such as integrals of the Green’s function over the interval 0 ≤ Q2 < ∞.
Notice that the region where one approximates the true function is far away from the
artificial poles in the PA. For instance, one may consider the value for the integral
Iπ = (−1)
dQ2 Q2ΠLR(Q
2) = 4.78719× 10−3, (20)
which, up to a constant, would yield the electromagetic pion mass difference in the
chiral limit [38] in the model (14). The PA P 5052 (Q
2) reproduces the value for this
integral with more than 42 decimal figures. This suggests that one may use the integral
(20) as a further input to construct a PA.
For example if we fix the three unknowns in the PA P 02 (Q
2) by matching the first
two terms from the chiral expansion but now we complete it with the pion mass dif-
ference (20) instead of a third term from the chiral expansion as we did in (19),13 the
approximant results to be
P̃ 02 (Q
− r2R
(Q2 + zR)(Q2 + z
, with r2R = 2.898× 10−3 , zR = 0.5618 + i 0.2795 .
This determines C−4 = −2.898×10−3 and C4 = −41.26×10−3, which shows that using
the pion mass difference is not a bad idea. Notice how the position of the artificial pole
has changed with respect to (19).
Artificial poles and analytic defects are transient in nature, i.e. they appear and
disappear from a point in the complex plane when the order of the Pade is changed. On
the contrary, the typical sign that a pole in a Pade is associated with a truly physical
pole is its stability under these changes in the order of the Pade. Of course, when the
order in the Pade increases there have to be new poles by definition, and it is natural
to expect that some of them will be defects. Pade Approximants place some effective
poles and residues in the complex Q2 plane in order to mimic the behavior of the true
Green’s function, but it can mimic the function only away from the poles, e.g. in the
Euclidean region. Obviously, PAs cannot converge at the poles, in agreement with
Pommerenke’s theorem [20], since not even the true function is well defined there. The
point is that what may look like a small correction in the Euclidean region may turn
out to be a large number in the Minknowski region. To exemplify this in simple terms,
13We remark that this procedure, although reasonable from the phenomenological point of view,
strictly speaking lies outside the standard mathematical theory of rational approximants [18, 31].
let us consider a very small parameter ǫ and imagine that a given Pade P (Q2) produces
the rational approximant to the true Green’s function G(Q2) given by
G(Q2) ≈ P (Q2) ≡ R(Q2) + ǫ
Q2 +M2
, (22)
where R(Q2) is the part of the Pade which is independent of ǫ. Although for Q2 > 0
there is a sense in which the last term is a small correction precisely because of the
smallness of ǫ, for Q2 < 0 this is no longer true because of the pole at Q2 = −M2.
This pole is in general a defect and may not represent any physical mass. In fact,
associated with this pole, there is a very close-by zero of the Pade P (Q2) at Q2 =
−M2 − ǫ R(−M2)−1, as can be immediately checked in (22). This is another way of
saying that a defect is characterized by having an abnormally small residue and is the
origin of the pairs of zeros and poles in the y-shaped branches of Fig. 1. Therefore,
not only are defects unavoidable but one could say they are even necessary for a Pade
Approximant to approximate a meromorphic function with an infinite set of poles.
Similarly to masses, also decay constants may be unreliable. To see this, imagine
now that our Pade is given by
P (Q2) =
Q2 +M2
(Q2 +M2) (Q2 +M2 + ǫ2)
, (23)
again for a very small ǫ. As before, the term proportional to ǫ may be considered
a small correction for Q2 > 0. However, at the pole Q2 = −M2 the decay constant
becomes F + ǫ−1 which, for ǫ small, may represent a huge correction. When the poles
are preassigned at the physical masses, like in the case of PTAs, it is the value of the
residues that compensates for the fact that the rational approximant lacks the infinite
tower of resonances. As we saw before, the residues of the poles in the Pade which lie
farthest away from the origin are the ones which get the largest distortion relative to
their physical counterparts.
In real life, the number of available terms from the chiral expansion for the con-
struction of a PA is very limited. Since the masses and decay constants of the first few
vector and axial-vector resonances are known, one may envisage the construction of a
rational approximant having some of its poles at the prescribed values given by the
known masses of these resonances. If all the poles in the approximant are prescribed
this way (as in the MHA), we have a PTA. On the contrary, when some of the poles are
prescribed but some are also left free, then we have a PPA (see the previous section).
Assuming that the first masses are known, let us proceed to constructing the PTAs
(6). The lowest such PTA is T02(Q
2), which contains two poles at the physical masses
of the ρ and the first A in the tower. Fixing the residue through the chiral expansion
to be C0 = −F 20 , one obtains
− F 20M2ρM2A
(Q2 +M2ρ )(Q
2 +M2A)
. (24)
Even though it has the same number of inputs (C0 and the two masses), this rational
approximant does not do such a good job as the PAs (19) or (21). For instance, C−4
is 2.3 times larger than the true value in Table 1. As we have already stated, one
way to intuitively understand this result is the following. The OPE is an expansion at
Q2 = ∞ and therefore knows about the whole spectrum because no resonance is heavy
enough with respect to Q2 to become negligible in the expansion, i.e., the infinite tower
of resonances does not decouple in the OPE. Chopping an infinite set of poles down
to a finite set may be a good approximation, but only at the expense of some changes.
These changes amount to the appearance of poles and residues in the PA which the
original function does not have. This is how the PA (19) manages to approximate
the true function (14). However, by construction, the PTA (24) does not allow the
presence of any artificial pole because, unlike in a PA, all its poles are fixed at the
physical values. Consequently, it only has its residues as a means to compensate for
the infinite tower of poles present in the true function and, hence, does a poorer job
than the PA (19), particularly in determining large-Q2 observables like C−4. Indeed,
the role played by the residues in the approximation can be appreciated by comparing
the true values of the decay constants to those extracted from (24). Although the one
of the ρ is within 30% of the true value, that of the A is off by 100%.
A different matter is the prediction of low-energy observables such as, e.g., the
chiral coefficients. In this case heavy resonances make a small contribution and this
means that the infinite tower of resonances does decouple.14 Truncating the infinite
tower down to a finite set of poles is not such a severe simplification in this case, which
helps understand why a PTA may do a good job predicting unknown chiral coefficients.
Indeed, (24) reproduces the value of C2 within an accuracy of 15%, growing to 22% in
the case of C4. A global observable like Iπ averages the low and the high Q
2 behaviors
and ends up differing from the true value (20) by 35%. This gives some confidence
that observables which are integrals over Euclidean momentum may be reasonably
estimated with MHA as, e.g., in the BK calculation of Ref. [39].
Improving on the PTA (24) by adding in the first resonance mass from the vector
tower produces the following approximant
a + b Q2
(Q2 +M2ρ )(Q
2 +M2A)(Q
2 +M2V )
, with
a = −13.5× 10−3,
b = +1.33× 10−4 , (25)
where the values of the chiral coefficients C0 and C2 have been used to determine
the parameters a and b. The prediction for C4 is much better now (only 2% off),
in agreement with our previous comments. The prediction for C−4 is still very bad,
becoming now 19 times smaller than the exact value. Nevertheless, it eventually gets
much better if PTAs of very high order are constructed. For instance, we have found
C−4 = −2.58 × 10−3 for the approximant T79 with 9 poles. Similarly, we have also
checked that the prediction of the chiral coefficients and the integral (20) improve with
higher-order PTAs.
However, another matter is the prediction of the residues. For instance, the predic-
tion for the decay constant of the state with mass MV in (25) is smaller than the exact
value in the model (15) by a factor of 2. In general, we have seen that the residues
14This is because the residues F 2 in the Green’s function (14) stay constant as the masses grow.
This behavior does not hold in the case of the scalar and pseudoscalar two-point functions [10].
of the poles always deteriorate very quickly so that the residue corresponding to the
pole which is at the greatest distance from the origin is nowhere near the exact value.
We again explicitly checked this up to the approximant T79, in which case the decay
constant for this pole is almost 5 times smaller than the exact value. The conclusion,
therefore, is that PTAs are able to approximate the exact function only at the expense
of changing the residues of the poles from their physical values. Identifying residues
with physical decay constants may be completely wrong in a PTA for the poles which
are farthest away from the origin.
As an intermediate approach between PAs and PTAs, there are the PPAs (3) where
some poles are fixed at their physical values while some others are left free. The simplest
of such rational approximants is P01,1(Q
2) (see the previous section for notation). Fixing
its 3 unknowns with M2ρ , C0 and C2, one obtains
1,1(Q
− r2R
(Q2 +M2ρ )(Q
2 + zR)
, with r2R = 3.75× 10−3 , zR = 0.8665 . (26)
As can be seen, the mass (squared) of the first A resonance is predicted to be at zR
which is sensibly smaller than the true value in (15)15. The rational function (26)
predicts C−4 = −r2R = −3.75 × 10−3 which is a better determination than that of the
PTA (24) with the same number of inputs, and C4 = −45.52× 10−3 which is not bad
either. Concerning the pion mass difference, one gets Iπ = 5.22 × 10−3. However, as
compared to the PAs (19) or (21), the PPA (26) does not represent a clear improvement.
In order to improve on accuracy of the PPA, one may try to use the mass and decay
constant of the first resonance, Mρ and Fρ, in addition to the pion mass difference and
the chiral coefficients C0, C2 and build the P
2,1(Q
2), which can be written as:
2,1(Q
F 2ρM
Q2 +M2ρ
a− F 2ρM2ρ Q2
(Q2 + zc) (Q2 + z∗c )
a = 17.43× 10−3,
zc = 1.24 + i 0.34 .
This PPA, upon expansion at large and small Q2, determines C−4 = −2.47× 10−3 and
C4 = −44.0× 10−3 to be compared with the corresponding coefficient in Table 1. The
accuracy obtained is better than that of (21), but this is probably to be expected since
(27) has more inputs.
Based on the previous numerical experiments done on the model in Eq. (14,15) (and
many others), we now summarize the following conclusions. Although, in principle,
the PAs have the advantage of reaching the best precision by carefully adjusting the
polynomial in the denominator to have some effective poles which simulate the infinite
tower present in (14), they have the disadvantage that some of the terms in the low-
Q2 expansion are required precisely to construct this denominator. This hampers the
construction of high-order PAs and consequently limits the possible accuracy.
When the locations of the first poles in the true function are known, there is the
possibility to construct PTAs (with all the poles fixed at the true values) and PPA
(with some of the poles fixed and some left free). As we have seen, although the PTA
may approximate low-Q2 properties of the true function reasonably well, the large-Q2
15Intriguinly enough, this is also what happens in the real case of QCD [9, 17].
properties tend to be much worse, at least as long as they are not of unrealistically
high order. The PPAs, on the other hand, interpolate smoothly between the PAs
(only free poles) and the PTAs (no free pole). Depending on the case, one may choose
one or several of these rational approximants. However, common to all the rational
approximants constructed is the fact that the residues and/or poles which are farthest
away from the origin are in general unrelated to their physical counterparts.
4 The QCD case
Let us now discuss the real case of large-Nc QCD in the chiral limit. In contrast to the
case of the previous model, any analysis in this case is limited by two obvious facts.
First, any input value will have an error (from experiment and because of the chiral and
large-Nc limits), and this error will propagate through the rational approximant. And
second, it is not possible to go to high orders in the construction of rational approxi-
mants due to the rather sparse set of input data. In spite of these difficulties one may
feel encouraged by the phenomenological fact that resonance saturation approximates
meson physics rather well.
The simplest PA to the function Q2ΠLR(−Q2) with the right fall-off as Q−4 at large
Q2 is P 02 (Q
P 02 (Q
1 + A Q2 +B Q4
. (28)
The values of the three unknowns a, A and B may be fixed by requiring that this PA
reproduces the correct values for F0, L10 and Iπ
16 given by
F0 = 0.086± 0.001 GeV ,
δmπ = 4.5936± 0.0005 MeV =⇒ Iπ = (5.480± 0.006)× 10−3GeV4 ,
L10(0.5 GeV) ≤ L10 ≤ L10(1.1 GeV) =⇒ L10 = (−5.13± 0.6)× 10−3 . (29)
The low-energy constant L10 is related to the chiral coefficient C2, in the notation of
Eq. (16), by C2 = −4L10. Since L10 does not run in the large-Nc limit, it is not
clear at what scale to evaluate L10(µ) [40]. In Eq. (29) we have varied µ in the range
0.5 GeV ≤ µ ≤ 1.1 GeV as a way to estimate 1/Nc systematic effects. The central
value corresponds to the result for L10(Mρ) found in Ref. [42]. The other results in
(29) are extracted from Refs. [1, 41].
Obviously, the PA (28) can also be rewritten as
P 02 (Q
(Q2 + zV )(Q2 + zA)
, (30)
in terms of two poles zV,A. In order to discuss the nature of these poles, we will define
the dimensionless parameter ζ by the combination
ζ ≡ −4L10
= 2.06± 0.25 , (31)
16Recall that Iπ is, up to a constant, the electromagnetic pion mass difference δmπ [17] and is
defined in terms of ΠLR as in Eq. (20).
where the values in (29) above have been used in the last step. Imposing the constraints
(29) on the PA (30) one finds two types of solutions depending on the value of ζ : for
ζ > 2 the two poles zV,A are real, whereas for ζ < 2 the two poles are complex. At
ζ = 2, the two solutions coincide. To see this, let us write the set of equations satisfied
by the PA (30) as:
F 20 =
zV zA
−4L10 = F 20
Iπ = F
zV zA
zA − zV
. (32)
The first of these equations can be used to determine the value of the residue r2 in
terms of zV zA. In order to analyze the other two, let us first assume that both poles
zV,A are real. In this case, they also have to be positive or else the integral Iπ will not
exist because it runs over all positive values of Q2. Let us now make the change of
variables
zV = R (1− x) , zA = R (1 + x) . (33)
The condition zV,A > 0 translates into R > 0, |x| < 1. In terms of these new variables,
the second and third equations in (32) can be combined into
1 + x
, (34)
where the definition (31) for ζ has been used. With the help of the identity log(1 +
x/1− x) = 2 th−1x (for |x| < 1), one can finally rewrite this expression as
th−1x , (x real) (35)
which is an equation with a solution for x only if ζ ≥ 2. Once this value of x is found,
the value of R can always be obtained from one of the last two equations (32) and this
determines the two real poles zV,A from (33).
On the other hand, when ζ < 2, Eq. (35) does not have a solution. However,
according to (31), ζ can also be smaller than 2. In order to study this case, we may
use the identity th−1(i y) = i tan−1(y) to rewrite the above equation (35) in terms of
the variable x = i y (y real) as
tan−1y , (y real). (36)
One now finds that this equation has a solution for y when ζ ≤ 2. In this case
the poles of the PA (28) are complex-conjugate to each other and can be obtained as
zV,A = R(1±i y). These poles, obviously, cannot be associated with any resonance mass
and this is why this solution has been discarded in all resonance saturation schemes
up to now. However, from the point of view of the rational approximant (28) there is
C0 C2 C4 C6 C8 C−4 C−6 C−8
−F 20 −4L10 −43± 13 81± 53 −145± 120 −4.1± 0.5 6± 2 −7± 6
Table 2: Values of the coefficients C2k in the high- and low-Q
2 expansions of
Q2 ΠLR(−Q2) in Eq. (16) in units of 10−3 GeV 2−2k. Recall that C−2 = 0.
nothing wrong with this complex solution, as the approximant is real and well behaved.
From the lessons learned in the previous section with the model, there is no reason to
discard this solution since, as we saw, rational approximants may use complex poles
to produce accurate approximations. Therefore, we propose to use both the complex
as well as the real solution for the poles zV,A, at least insofar as the value for ζ ≷ 2. In
this case we obtain, using the values given in Eqs. (29),
(ζ ≥ 2) , r2 = −(4.1± 0.5)× 10−3 , zV = (0.77)2 ± 0.15 , zA = (0.96)2 ± 0.41 (37)
(ζ ≤ 2) , r2 = −(3.9± 0.1)× 10−3 , zV = z∗A = (0.66± 0.06) + i (0.25± 0.25) , (38)
in units of GeV6 for r2, and GeV2 for zV,A. The two solutions in Eqs. (37,38) have
been separated for illustrative purposes only. It is clear that they are continuously
connected through the boundary at ζ = 2, at which value the two poles coincide and
zV = zA ≃ 0.72. The errors quoted are the result of scanning the spread of values in
(29) through the equations (32).
With both set of values in (37,38), one can get to a prediction for the chiral and OPE
coefficients by expansion in Q2 and 1/Q2, respectively. These expansions of the PA can
be done entirely in the Euclidean region Q2 > 0, away from the position of the poles
zV,A, whether real or complex. Recalling the notation in Eq. (16), the above P
produces the coefficients for these expansions collected in Table 2. The values for the
OPE coefficients C−4,−6,−8 in this table are compatible with those of Ref. [17], after
multiplying by a factor of two in order to agree with the normalization used by these
authors. However, the spectrum in our case is different because of the complex solution
in (38). As we saw in the previous section with a model, this again shows that Euclidean
properties of a given Green’s function, such as the OPE and chiral expansions, or
integrals over Q2 > 0 are safer to approximate with a rational approximant than
Minkowskian quantities, such as resonance masses and decay constants.
5 Conclusions
In this article we pointed out that approximating large-Nc QCD with a finite number
of resonances may be reinterpreted within the mathematical Theory of Pade Approxi-
mants to meromorphic functions [18].
The main results of this theory may be summarized as follows. One may expect
convergence of a sequence of Pade Approximants to a QCD Green’s function in the
large-Nc limit in any compact region of the complex Q
2 plane except at most in a
zero-area set [20]. This set without convergence comprises the poles of the original
Green’s function together with some other artificial poles generated by the approximant
which the original function does not have. As the order of the PA grows, the previous
convergence property implies that any given artificial pole either goes to infinity, away
from the relevant region, or is almost compensated by a nearby zero. This symbiosis
between a pole and a zero is called a defect. Although close to a pole the rational
approximation breaks down, in a region which is far away from it the approximation
should work well.
We have reviewed the main results of this theory with the help of a model for the
two-point Green’s function 〈V V −AA〉. The simpler case of a Green’s function of the
Stieltjes type, such as the two-point correlator 〈V V 〉, was previously considered in Ref.
[21]. We have seen in the case of this particular model how rational approximants
create the expected artificial poles (and the corresponding residues) in the Minkowski
region Re(q2) > 0 while, at the same time, yielding an accurate description of the
Green’s function in the Euclidean region Re(q2) < 0. This happens in a hierarchical
way: although the first poles/residues in a PA may be used to describe the physical
masses/decay constants reasonably well, the last ones give only a very poor description.
Therefore, it is in general unreliable to extract properties of individual mesons, such
as masses and decay constants, from an approximation to large-Nc QCD with only a
finite number of states. Since a form factor, like a decay constant, is obtained as the
residue of a Green’s function at the corresponding pole(s), this also means that one
may not extract a meson form factor from a rational approximant to a 3-point Green’s
function, in agreement with [16]. This observation may explain why the analysis of
Ref. [43], which is based on an extraction of matrix elements such as 〈π|S|P 〉 and
〈π|P |S〉 from the 3-point function 〈SPP 〉, finds values for the Kℓ3 form factor which
are different from those obtained in other analyses [44].
In spite of all the above problems related to the Minkowski region, our model
shows how Pade Approximants may nevertheless be a useful tool in other regions of
momentum space. We think that this is also true in the real case of QCD in the large-
Nc limit. In this case one may use the first few terms of the chiral and operator product
expansions of a given Green’s function to construct a Pade Approximant which should
yield a reasonable description of this function in those regions of momentum space
which are free of poles. In this construction, Pade Approximants containing complex
poles, if they appear, should not be dismissed.
We have also reanalyzed the simplest approximation to the 〈V V − AA〉 Green’s
function in real QCD which consists of keeping only two poles, and we have found
that, depending on the value of the combination ζ in Eq. (31), these two poles may
actually be complex.
However, if not all the residues and masses in a rational approximant are physical,
this poses a challenge to any attempt to use a Lagrangian with a finite number of
resonances such as, for example, the ones in Ref. [9, 11], for describing Green’s functions
in the large-Nc limit of QCD. Even if these Lagrangians are interpreted in terms of
PTAs, with the poles fixed at the physical value of the meson masses, we have seen how
the residues then get very large corrections with respect to their physical counterparts.
Can these residues be efficiently incorporated in a Lagrangian framework? We hope to
be able to devote some work to answering this and related questions in the future.
Acknowledgements
S.P. is indebted to M. Golterman, M. Knecht and E. de Rafael for innumerable
discussions during the last years which have become crucial to shape his understanding
on these issues. He is also very grateful to C. Diaz-Mendoza, P. Gonzalez-Vera and R.
Orive, from the Dept. of Mathematical Analysis at La Laguna Univ., for invaluable
conversations on the properties of Pade Approximants as well as for hospitality. We
thank S. Friot, M. Golterman, M. Jamin, R. Kaiser, J. Portoles and E. de Rafael for
comments on the manuscript.
This work has been supported by CICYT-FEDER-FPA2005-02211, SGR2005-00916
and by the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”.
APPENDIX
Here we will show how the PAs constructed from the OPE do not in general re-
produce even the first resonances in the spectrum, unlike those constructed from the
chiral expansion. Again, we will use the model of section 3 as an example. Recalling
the definition of the OPE given in Eq. (16), with the corresponding coefficients (17), it
is straightforward to construct a PA in 1/Q2 around infinity, i.e. by matching powers
of the OPE in 1/Q2. The construction parallels that in Eq. (2) but with the replace-
ment z = 1/Q2. Since the function Q2ΠLR(−Q2) behaves like a constant for Q2 → 0,
we will consider diagonal Pade Approximants, i.e. of the form P nn (1/Q
2), in order to
reproduce this behavior. Figure 2 shows the position of the poles and zeros of the PA
P 5050 (−1/q2) in the complex q2 plane. As it is clear from this plot, the positions of the
poles have nothing to do with the physical masses in the model, given by Eqs. (11-15),
even for the lightest states. This is to be contrasted with what happens with the PA
constructed from the chiral expansion around Q2, which is shown in Fig. 1. The differ-
ence between the two behaviors is due to the fact that, while the chiral expansion has
a finite radius of convergence, the radius of convergence of the OPE vanishes because
this expansion is asymptotic.
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	Introduction
	Rational approximations: generalities
	Testing rational approximations: a model
	The QCD case
	Conclusions
