Weinberg like sum rules revisited
© Afonin 2009
Received: 10 September 2008
Accepted: 08 January 2009
Published: 08 January 2009
The generalized Weinberg sum rules containing the difference of isovector vector and axial-vector spectral functions saturated by both finite and infinite number of narrow resonances are considered. We give a historical survey and summarize the status of these sum rules analyzing their overall agreement with phenomenological Lagrangians, low-energy relations, parity doubling, hadron string models, and experimental data.
PACS codes: 11.55.Hx, 11.30.Rd
The sum rules equating certain moments of the spectral weight functions of two-point current correlators turned out to be an extremely fruitful concept in the hadron physics. They were first proposed by Weinberg who considered the difference of vector (V) and axial-vector (A) correlators , later this difference was recognized to be an order parameter of spontaneous chiral symmetry breaking in QCD in the chiral limit. The original derivations of such sum rules suffered from a lack of rigour as long as they were based on ad hoc postulates about the high-energy behaviour of certain combinations of two-point correlators and about the possible nature of Schwinger terms of the current commutators. The original Weinberg's derivation rested on a proof of equality of V and A Schwinger terms and on the assumption of asymptotic chiral SU(2) × SU(2) symmetry at high momentum, thus providing for the first time a concrete realization of the notion of chiral symmetry at short distances. The latter proposal was immediately generalized in , where it was shown that the idea of asymptotic symmetries may serve as a powerful tool for deriving various interesting results in the pole approximation. These papers were followed by the SU(3) × SU(3) generalization of Weinberg sum rules  and subsequent early applications to hadron physics [4–8]. The Weinberg sum rules were exploited for derivation of electromagnetic mass difference  and Das-Mathur-Okubo sum rule , both results have been widely used up to now. The former one relies essentially on the validity of the second Weinberg sum rule, otherwise this mass difference is not finite (later it was demonstrated  that the use of the second spectral-function sum rule is not necessary if an exchange by intermediate weak gauge boson is taken into account). This and other phenomenological observations called for a rigorous justification of spectral-function sum rules. An important requirement was also the universality of such a justification, as, say, the original method used in  to derive the equality of V and A Schwinger terms does not work, generally speaking, in the scalar case.
The first attempt in this direction was the proposal to replace the current algebra by the algebra of gauge fields , the Schwinger terms in the latter are explicitly known c-numbers. Considerable progress in understanding the Weinberg sum rules was achieved due to the introduction of Wilson's Operator Product Expansion (OPE) at short distances  as a substitute of Lagrangian models, the new techniques was applied to the sum rules already in the original paper . The Wilson's proposal happened to be very useful tool for analysis of convergence of spectral-function sum rules. Wil-son proved, for instance, that if his OPE method is true, the necessary and sufficient condition for the validity of Weinberg sum rules is that the chiral SU(2) × SU(2) group be an exact invariance (see also  for a more general proof). The discovery of asymptotic freedom in non-Abelian gauge theories [15, 16] inspired to consider the status of spectral-function sum rules within the framework of asymptotically free theories , in particular, exploiting the techniques of Wilson's OPE . A general recipe for extracting exact spectral-function sum rules for asymptotically free theories was presented in .
Weisberger showed  that the Wilson's results can be reproduced in Lagrangian theory by means of the use of the renormalization-group equation. In particular, the general theoretical criterium for the validity of spectral sum rules was derived: the symmetry-breaking terms in a renormalizable Lagrangian must have canonical dimensions δ ≤ 3. Such a soft symmetry breaking can be implemented in practice by mass terms and by scalar fields with nonvanishing vacuum expectation value. Provided this condition is satisfied, the propagators approach their symmetric values in the asymptotic spacelike limit. Choosing then certain linear combinations of propagators whose asymptotics will be less singular than that of the individual terms and applying the spectral representations for those combinations, one obtains precisely the Weinberg sum rules. To realize the program completely, one needs to know nonleading asymptotic terms which depend on the symmetry breaking effects. The earlier studies of chiral symmetry breaking within various spectral-function sum rules can be found in [2, 17, 18, 21–25].
In summary, it became clear that the two Weinberg sum rules are very general and do not depend on the dynamics of chiral symmetry breaking in the vacuum for the asymptotically free theories, such as QCD, while the higher-order sum rules do depend on that dynamics, hence, on the specific details of the QCD Lagrangian.
The invention of famous ITEP sum rules  was a significant progress in development of OPE-based sum rules. Within that method, one improves the convergence and suppresses the contribution of higher excitations by performing the Borel transformation in Euclidean space, and then one parametrizes all non-perturbative effects by means of a few condensates entering the numerators of OPE. As a result, the perturbative and non-perturbative contributions become effectively factorized, which permits to make numerous predictions having at hand only several inputs – the phenomenological values of condensates. There is still no complete understanding why this method works so well in the phenomenology. In particular, the application of the same method to a solvable quantum-mechanical problem  shows that when the hadron continuum is not known and is modelled by an effective continuum threshold (this very situation one has usually in practice), the systematic uncertainties of the method cannot be controlled. The assumption of dominance of the lowest-lying resonances is sometimes also in doubt . A possibility to take into account the higher excitations appeared within the Finite Energy Sum Rules (FESR) [29–31] (see, e.g.,  for references), which represent a kind of extension of the Weinberg sum rules based on analytic properties of correlation functions and on quark-hadron duality.
About twenty years ago it was realized that the sum rules can be directly confronted with experiment through semi-leptonic τ-lepton decays, namely the V and A spectral functions can be reconstructed in the kinematical range limited by the τ-lepton mass. The first attempt was undertaken in  using the ARGUS data . This analysis was followed by improved versions [35–37]. Subsequently, much more precise data of the ALEPH and OPAL collaborations on the V and A spectral functions [38, 39] gave rise to a large series of papers devoted to the extraction of hadronic parameters, such as condensates, with the help of FESR and other methods [40–52].
In the last decade the sum rules saturated by narrow resonances have found numerous applications in the phenomenology, it would require a long paper to survey this activity. In this respect, a question might even appear whether new papers on such well known sum rules are really needed. We believe, however, that some new trends in the phenomenology invite to return to foundations of resonance sum rules and to revise them. Each scheme of resonance saturation implies a certain pattern for the chiral symmetry breaking at low energies, the most known pattern is based on the conception that the a1-meson is the chiral partner of the ρ-meson and the σ-meson is that of pion. If the Wigner-Weyl realization of chiral symmetry was somehow restored maintaining confinement in QCD, these chiral partners would be degenerate. The same pattern is used for construction of many effective quark models, equal resonance content provides a possibility to match these models to QCD sum rules establishing thereby a correspondence of effective models to the fundamental theory (see, e.g., [53–67] and references therein). There is, however, an alternative possibility  where the ρ-meson is a "would-be" chiral partner of pion. It is not excluded that this pattern would be preserved even if the chiral symmetry was restored – the so-called vector manifestation scenario . To a certain extent, the recent phenomenological observations yield an unexpected support for this scenario – the ρ-meson belongs to the leading Regge trajectory, the states lying on such trajectories, probably, do not have parity partners [70–72] and if the chiral symmetry is effectively restored above the chiral symmetry breaking scale, the chiral partner of the a1-meson seems to be the ρ(1450)-resonance, the first "radial" excitation of the ρ-meson (such a possibility was explored in ). This example shows that further systematic studies of both saturation schemes and relations between the QCD sum rules and phenomenological Lagrangians are needed, we will address to these subjects.
Recently the resonance sum rules were employed to demonstrate that the chiral symmetry is realized in the Wigner-Weil mode in the upper part of meson spectrum [73–75], the first attempt of this kind seems to go back to , where the baryon sector was analyzed. These attempts boiled down to justification of parity doubling among the highly excited states, the procedure turned out to be model-dependent, thus not replying unambiguously whether the chiral symmetry is restored or not. Another approach was put forward in , where the spectrum was split into the "chirally symmetric" and "nonsymmetric" parts. The first part, after summation over resonances and comparison with the OPE, yields no contribution to the condensates responsible for the chiral symmetry breaking. If the chiral symmetry gets restored, the second part has to represent the asymptotically vanishing corrections to the first part. Technically, one should fix an ansatz for mass spectrum, e.g. take the linear one, calculate its input parameters from the imposed constraints, and compare with the experimental data and known theoretical relations to check whether this works. Unfortunately, the existing uncertainties both in the experimental data and in the OPE condensates do not permit to perform this reliably.
The chiral symmetry restoration still remains a rather iffy concept [72, 78] in spite of all efforts to justify it , for this reason we would prefer to use the term "parity doubling", the latter is easier to compare with the actual spectroscopy. As parity doubling seems to be an important observable phenomenon , it is interesting to consider to what extent the approximate parity degeneracy can take place in the sum rules saturated by finite number of resonances (leaving aside the case of trivial degeneracy)  and check, if possible, whether such mass spectra are more preferable from the phenomenological point of view in comparison with mass spectra without approximate parity doubling. This subject will be also addressed in the present work.
The rest of review is organized as follows. In Sect. 2 we concern a relation between the generalized Weinberg sum rules and the phenomenological Lagrangians of effective field theory. Sect. 3 deals with the same sum rules in the correlator approach. Sect. 4 is devoted to solutions of sum rule equations at different saturation schemes and additional assumptions, with the main emphasis being placed on the possibility for parity doubling. In Sect. 5 we comment on some problems emerging in the sum rules with infinite number of states. Our conclusions are summarized in Sect. 6.
2. Sum rules: Lagrangian approach
Deriving the higher-order Weinberg sum rules from the OPE for correlation functions, one typically encounters the following problem: From the OPE side, the condensate terms have anomalous dimensions starting from dimension 6, while from the resonance side, the anomalous dimensions are absent by construction, thus the question arises about the correctness of equating both sides. In this section, we will argue that the sum rules are closely related to the Lagrangian approach, in fact they can be derived from this approach assuming the asymptotic chiral symmetry restoration at large momentum transfer. The problem with anomalous dimensions can be escaped in this case.
The decay constant Fφ, nmay be related to observable quantities. For instance, if we associate V1 with the ρ-meson, matrix element (5) can be estimated from the decay ρ0 → e+e- (see Eq. (84)).
in the expressions above. Here f π is the weak pion decay constant, f π = 92.4 MeV, in the chiral limit f π ≈ 87 MeV. The corresponding axial current is then conserved in the chiral limit, m π = 0, which we shall adopt. Furthermore, we will make use of quite common assumption of generalized PCAC, i.e. the covariant derivative of pion field is mixed with the axial field only, the axial fields , n > 1, are supposed to correspond to heavier states ("radial" excitations).
The exact chiral symmetry implies that the expressions in the braces of Eqs. (11) and (12) have to be equal at all p. This is possible only if mV, n= mA, n, FV, n= FA, n, f π = 0, which is far from reality. Instead of exact chiral symmetry, one usually imposes an asymptotic chiral symmetry at large energies.
Sum rules (14) and (15) are the Weinberg sum rules  generalized to the case of arbitrary number of states. Some general properties of set of equations (14)-(16) were studied in , where these sum rules were derived from the OPE of two-point quark current correlators . It should be emphasized the distinction of the OPE-based ideology from the one adopted here. The quantities Δ k are usual finite numbers in our approach, while in the OPE they are related to the condensates of appropriate dimension multiplied by the coefficient functions calculated from QCD by means of the perturbation theory (the ALEPH/OPAL data on the V and A spectral functions from τ decays was used for numerical estimations of quantities Δ k up to dimension 18, see  for a review). The relation with the fundamental theory is an advantage of the OPE-based methods, in particular, the restrictions Δ0 = 0 and Δ1 = 0 follow automatically in the chiral limit. However, from the point of view of sum rules (16), the OPE has two shortcomings. First, the OPE represents, at best, an asymptotic expansion with zero radius of convergence, therefore the calculation of Δ k at large k (in practice, at k ≳ 4) is not reliable because the divergence sets in. Second, as mentioned above the condensate terms have an anomalous dimension starting from dimension 6 while the l.h.s. of Eq. (16) does not have, this circumstance caused a critics of such sum rules recently .
Another derivation of generalized Weinberg sum rules, Eqs. (14) and (15), which does not use the correlation functions, was proposed in , where the method consisted in analysis of certain Current Algebra commutation relations in infinite momentum frame.
3. Sum rules: correlator approach
Originally the sum rules under consideration were derived in  from some asymptotic restrictions on correlation functions with subsequent saturation by narrow states (an earlier application of asymptotic equality for the V and A correlators exists in the literature , where the rate for ω0 → π0 + γ was calculated from one of sum rules emerging in the infinite energy limit; we are grateful to Prof. S. Gasiorowicz for this remark). We will consider a generalization of this method to the case of arbitrary number of narrow states and discuss the underlying physics.
and get the sum rules considered above, Eqs. (14)–(16).
An interesting consequence of the correlator formalism is that the first Weinberg sum rule follows also from the requirement of cancellation of Schwinger terms in the Π V - Π A difference , this fact is evident from Eqs. (21) and (22). The cancellation of Schwinger terms is quite natural as long as the Π V - Π A difference is related to observable quantities. Thus, one arrives at a kind of equivalence between the asymptotic chiral symmetry and the cancellation of Schwinger terms. A question emerges, which principle is more fundamental?
where jΓ is the current corresponding to the gamma-matrix structure Γ. These terms can be both operators and c-numbers, generally speaking, their form is model-dependent. The appearance of Schwinger terms is an inescapable consequence of Lorentz invariance and of positive definiteness for probability, otherwise the theory is trivial [88, 89]. This observation suggests that the cancellation of Schwinger terms is likely more fundamental requirement than the cancellation of the longitudinal parts at large momentum. In addition, the former cancellation should take place at all momenta as it is momentum-independent. The given property explains why the first Weinberg sum rule is much better fulfilled in the phenomenology than the second one, the reason seems to be that it is not only asymptotic sum rule – its validity extends beyond the high-energy domain. Indeed, provided the equality of Schwinger terms in spectral representation (20) and in its axial-vector analogue, the first Weinberg sum rule equally emerges if one takes the limit |p| → 0 in condition (23). On the other hand, it should be mentioned that the problem with the Schwinger terms can be escaped from the very beginning if we subtract automatically the contribution due to contact terms in definitions (17) and (18) performing thereby an additive renormalization. Such an operation gives the so-called T*-product which is gauge-invariant and Lorentz covariant. This could be a better starting point, but our aim was to demonstrate some delicate points with the Weinberg sum rules.
where is the electromagnetic mean mass squared radius of the charged pions and is the axial-vector coupling. Relation (33) is nothing but the Das-Mathur-Okubo low-energy theorem .
The constant turns out to be related with a remarkable physical observable, the electromagnetic pion mass difference , , we will exploit this relation later.
4. Weinberg-like sum rule equations: solutions for particular cases
We are going to solve the system of equations (14)–(16) for some typical cases. Partly, our analysis may be regarded as a revision and extension of results obtained in . We, however, place quite different emphasis – we are interested in maximally degenerate solutions (aside from the trivial ones) for excited V and A mesons and in overall correspondence of different possibilities to the actual experimental data.
4.1 One Vector and One Axial-vector
In this subsection we revisit the standard Weinberg ansatz and clarify its modern status.
Experimentally  (in MeV),
F V = 154 ± 8, F A = 123 ± 25, M V = 776, M A = 1230 ± 40,
The phenomenological and experimental values above yield the following estimate,
Δ OPE ≈ -0.3.
where the first and the second Weinberg sum rules have been used.
X ρ = 2,
X A = 1, Y A = 2.
Identifying the A state with the a1-meson, the Weinberg ansatz was widely used in the literature for matching conditions and other purposes.
We would like to note, however, that the status of the KSFR relation caused much discussions in the literature. In particular, its original derivation was revised in [96–101], the main lesson was that it required more ad hoc assumptions in comparison with the ones made in [94, 95]. Notably, the modern experimental data (37) favors rather to the ansatz
X ρ = 3,
Comparing the predictions with the corresponding experimental and phenomenological values given above, one can see immediately that the experimental ansatz works better substantially than the Weinberg one: Lowering M A by 15% (which is within the large-N c accuracy), one amends noticeably all other five quantities.
Finally, it should be mentioned that the degenerate case, which would mean here Y A = 1, cannot be obtained within the considered ansatz.
4.2 One Vector and No Axial-vector
depends explicitly on the cutoff μ. Treating the cutoff as an input parameter, one can achieve any value for , in particular, the experimental one, 4.6 MeV, is implemented at ≈ 1435 MeV. In this respect, it is not surprising that can be calculated without use of the a1-meson, see e.g. . On the other hand, the cutoff should not exceed the mass of the higher resonance, otherwise the latter has to be included explicitly. Since the higher resonances give a sizeable contribution to , the physical interpretation for seems to be the following: It indicates indirectly on the typical scale of the higher meson excitations such as a1(1230) and ρ(1450). The result of the previous subsection is reproduced if we identify the cutoff with the mass of the lowest A-meson.
In summary, the ρ- π ansatz is not senseless within the sum rules, it seems to be indeed the simplest ansatz consistent, to a certain extent, with the phenomenology.
4.3 Two Vectors and One Axial-vector
It is natural to associate the second vector state with the ρ (1450)-meson, whose mass is 
M V = 1459 ± 11 MeV,
with the inequality,
Y V ≥ Y A ,
Physically we expect that the ρ-meson dominates over heavier resonances, this leads to the inequalities
X A ≤ X ρ , X V ≤ X ρ .
To estimate the ensuing restriction on the mass M V we can admit Δ = 0. Then one has M V ≥ 1550 MeV for X ρ = 2 and M V ≥ 1340 MeV for X ρ = 3. Thus, the physical value of ρ(1450)-meson (60) mass is incompatible with the KSFR relation within the ansatz under consideration.
Δ + Y A = 0.
The impossibility to provide Δ = 0 for degenerate case in realistic situations recurs at introducing more resonances. For instance, if we add a pair of resonances (ρ" and mesons) with equal masses then one can show that the condition Δ = 0 may be adjusted only when X A > 2(X ρ + 1) while we expect X A ≤ X ρ .
The conclusion made in  was that when relation (69) takes place then X V = 0 and the highest vector state decouples. Evidently, this is not true if Y V = Y A , i.e. when the V and A mesons are exactly degenerate.
4.4 Arbitrary Finite Number of States
In this example the ρ-meson is singled out, its residue is in accord with the KSFR relation, the universal slope agrees with the phenomenology and some models (see  for discussions) as well as the universal residue . We have made here a minimal manipulation with residues – the residue of the highest vector state is two times less than the universal one. The physical interpretation could be given the following: The resonance of mass is the heaviest in the system, if one cuts off at μcut = mV, Nthen the half of its decay width (namely, the right half from the position of resonance) is thrown away, in the narrow-width approximation this loss of information can be mimicked by halfing the residue.
We could not achieve the completely degenerate case, at least in asymptotics, a removal of degeneracy seems to be unavoidable if one likes to get rid of nonlinear corrections. It is interesting to note that the mass spectrum of the ansatz above resembles that of old dual models  (a somewhat similar model, but for infinite number of states, was considered in ).
we arrive at the standard pattern of resonance saturation, so the usual formulae remain formally valid as if the D-wave states were decoupled. A qualitative argument in favour of D-wave decoupling presented in  relied, in essence, on the fact that the vector interpolating current (19) couples to the e+e- annihilation, which is a point-like process, hence, the extended objects like the D-wave states should decouple, i.e. their residues vanish rapidly. However, quasiclassical arguments (see, e.g., discussions in ) tell us that the orbitally and radially excited mesons are equally extended objects, at least in the hadron string picture, in which the size of meson is defined by its mass only. Thus, once we adopt the large-N c limit and introduce thereby the high radial excitations regarding them as coupled to a certain local current, we inevitably should encounter the orbital excitations coupled to the same current, under the vector mesons we should then understand the mixture defined above.
5. Infinite number of states
The sum rules dealing with infinite number of V and A states are largely covered in the literature, see e.g. [73, 74, 104, 112–115] and references therein. In this section we give some relevant comments.
where we neglect (α s ) correction to the partonic logarithm (the impact
here S.C. means "subtraction constant". It should be noted that the perturbative continuum is not present any more in Eq. (80), this circumstance expresses the quark-hadron duality.
which should hold at least at n → ∞. To advance further one needs some ansatz for or . There is no experimental information on the residues of highly excited states, but fortunately experiment  indicates on the Regge behaviour of masses, ~n. From the theoretical side, the same behaviour is suggested by quantization of quasiclassical meson string (see, e.g., [71, 72, 121]). These arguments justify the standard use of linear ansätze (up to corrections vanishing at n → ∞) in the sum rules under consideration. Relation (82) gives then constant residues. To the best of our knowledge, this is the only reason why one believes that residues are independent of n, at least at large n. The question is whether it is possible to justify qualitatively the constant behaviour of residues independently?
Comparing relations (83) and (84) we conclude that the residues should not depend on n, i.e. they are constant.
Finally, we arrive at a quite unexpected result: Assuming a well-motivated string picture for excited mesons, one is able to argue for the constant residues, then relation (82), which is a consequence of quark-hadron duality, yields the linear mass spectrum, ~n, i.e. this linearity may be deduced from the sum rules without quantization of hadron string.
Another comment concerns derivation of the KSFR relation from the sum rules. This relation is believed to hold in the phenomenology, in the sum rules, however, it is usually imposed from outside. As was argued above, it is natural to expect that V and A meson residues – the electromagnetic decay constants – are a universal constant, = F2. Assuming that this feature extends up to the ρ-meson, the KSFR relation reads F2 = . In principle, we may choose the slope of the linear mass spectrum such that the KSFR relation is reproduced due to Eq. (82), this trick is consistent with the phenomenology . An interesting question rising here is whether it is possible for the KSFR relation to infer from the sum rules in a way weakly dependent on a concrete ansatz for the mass spectrum? We will show that under assumptions above this indeed can be done, the only assumption about the mass spectrum we need is that going up in energy one encounters the resonances in the order V -A -V -A -... . It is a rather weak assumption and consistent with the phenomenology.
Consider the first Weinberg sum rule (14) with arbitrary number N of states and take the limit N → ∞. We should postulate the pattern of pairing for the V and A states. The most frequent assumptions which one commonly uses are: (i) the chiral symmetry of QCD provides equal number of V and A mesons (with the reservation on D-wave vectors above), so the n-th A-meson is paired with the n-th V-meson and the ordering V - A - V - A - ... is thus satisfied; (ii) the ρ-meson is singled out because of the chiral symmetry breaking at low energies, so the n-th A-meson is paired with the (n + 1)-th V-meson.
Here we encounter a usual problem – the limit N → ∞ leads to ill-defined sums and a generalized method of summation is needed in order that such sums make a definite sense. Fortunately, the sum in Eq. (85) is well known in mathematical analysis, there are many ways of generalized summation and practically all of them yield the same result for the sum in question.
Combining relations (85) and (86) we get the KSFR relation.
where we have used the assumption of V -A-V -A-... ordering of states in masses. Substituting relation (86) to sum rule (87) we again arrive at the KSFR relation, this universality of result is quite remarkable.
The operations with the divergent sums call for another one comment as they are often present in the sum rules in the large-N c limit, sometimes this circumstance causes criticism. Naively, such operations should indeed lead to very ambiguous results as long as these sums are ill-defined. This is correct if one throws away any physical content and considers the matter mathematically only. The situation then looks as if we worked with differential equations without boundary conditions, this would be useless for physics. The role of additional assumptions in the sum rules is somewhat similar to that of physical boundary conditions. When imposed correctly, the additional assumptions should always remove ambiguities.
In this regard, the sum Σ(-1)n+1 is a simple educative example. The answer for this sum depends on a way of grouping the terms, for this reason the problem is not well defined mathematically, but in physics the pattern of grouping is fixed by additional assumption(s). For instance, let us assume that the highly excited V and A states become exactly degenerate, mV, n= mA, n, FV, n= FA, nat n ≥ N. This means that going up in energy, since the scale mV, Nthe V and A resonances will be coming in pairs. This feature provides the physical pattern of grouping the individual contributions in that part of the spectrum,
1 - 1 + 1 - 1 + ⋯ = (1 - 1) + (1 - 1) + ⋯ = 0 + 0 + ⋯ = 0,
the same pattern will hold in the higher-order sum rules. Such a local conspiracy results in cancellations of the same type as cancellation of V and A perturbative continuums in the difference Π V (Q2)- Π A (Q2), in this sense one may think of a certain duality between that part of spectrum and perturbative continuum. It should be noted incidentally that assuming some pattern of pairing of states, any regularization of divergent sums has to respect this pattern, otherwise result will be senseless.
Let us assume now a more realistic case – the V and A mesons are not exactly degenerate at any finite N. This means that going up in energy we will be encountering the resonances consecutively state by state, hence, the summation has to be performed in the same way. For validity of this statement it is not necessary to require the V - A - V - A - ... recurrence, if the number of V and A contributions is almost equal (say, if the V and A mesons form approximately degenerate parity doublets), the permutation of some V and A contributions does not change the result. The partial sums of Σ(-1)n+1 will be then either 0 or 1. The generalized summations yield the averaged value . This result is in one-to-one correspondence with the fact that the Euclidean behaviour of correlators is sensitive only to averaged features of the spectral densities, for this reason the generalized summations are usually quite effective tools in the Euclidean domain.
We have considered various aspects of generalized Weinberg sum rules. It was argued that when one imposes the asymptotic chiral symmetry on phenomenological Lagrangians, the Weinberg like sum rules follow naturally. The difficulties related to the operator product expansion for correlation functions are problems of a specific derivation of the sum rules rather than problems of the sum rules themselves. In principle, one may use such sum rules including up to infinite number of narrow resonances having forgotten about those problems. In conclusion, we believe that the potential of considered QCD sum rules is not exhausted, further investigations in this direction may lead to interesting results and applications. Hopefully, the present revision of these known sum rules will be useful in this respect.
I am grateful to Prof. A. A. Andrianov for his critical comments and discussions. The work was supported by the Alexander von Humboldt Foundation.
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