On some properties of the fourth-rank hadronic vacuum polarization tensor and the anomalous magnetic moment of the muon

Some short-distance properties of the fourth-rank hadronic vacuum polarization tensor are re-examined.Their consequences are critically discussed in the context of the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon.

for HLxL. These different views then lead to quite different numerical evaluations of the pion-pole contribution, see for instance the discussion in Ref. [42]. Although the pion pole is only one contribution to HLxL among many, and what should actually matter in the end is the full contribution to a µ from HLxL, it is certainly of interest, given the importance of this contribution, to understand what are the whys and wherefores of this rather confusing situation.
Thus, the purpose of this note is therefore not to provide yet another new evaluation of the HLxL contribution. Rather, it was written with the aim of scrutinizing this particular issue in greater detail and, possibly, of providing some understanding that may contribute to settle it. The outline of the remaining part of the text is as follows. First, I recall, in Section 2, general properties of the four-and three-point functions relevant for this discussion, including the short-distance condition that relates them. I then describe in detail the implementation of this condition in Section 3 in general, before focusing on its implications for the contribution of the pseudoscalar poles. Finally, I give a summary and conclusions in Section 4. Some more technical aspects related to the short-distance expansion have been gathered in Appendix A for the interested reader. Appendix B illustrates the discussion from the perspective of the low-energy expansion.
2 Some hadronic four-and three-point functions and their properties As mentioned in the introduction, the central object of interest is the connected four-point QCD correlator where j µ (x) stands for the light-quark component of the hadronic part of the electromagnetic current, and |Ω denotes the QCD vacuum. For notational convenience, I have written this correlator as a function of four variables, but only three momenta are actually independent, since invariance under tranlations requires that the condition holds. Let me recall that a HLxL µ , the HLxL contribution to the anomalous magnetic moment of the muon, can be expressed in terms of this correlator in the following way [61] a HLxL µ ≡ 1 48m ℓ tr[( p + m ℓ )[γ σ , γ τ ]( p + m ℓ )Γ HLxL στ (p, p)], (2.4) where p stands for the momentum of the muon and Γ HLxL στ (p, p) is the limit of the vertex function Γ HLxL στ (p ′ , p), defined as u(p ′ )Γ HLxL στ (p ′ , p)u(p) = e 6 d 4 q 1 (2π) 4 ×W µνρστ (q 1 , q 2 , k − q 1 − q 2 , −k), (2.5) when the momentum difference k = p ′ − p vanishes. This definition involves the derivative of the four-point function, with respect to its fourth momentum k. Eq. (2.4) then requires to take the limit k → 0 of this derivative. Due to the conservation of the current j µ , the rank-four hadronic vacuum polarization tensor satisfies the Ward identities {q 1µ ; q 2ν ; q 3ρ ; q 4σ }W µνρσ (q 1 , q 2 , q 3 , q 4 ) = {0; 0; 0; 0}. (2.7) Based on these transversality properties combined with Bose symmetry, the authors of Ref. [22] have obtained a decomposition of the tensor W µνρτ , W µνρτ (q 1 , q 2 , q 3 , q 4 ) = 54 i=1 W i (q 1 , q 2 , q 3 , q 4 )T µνρτ i (q 1 , q 2 , q 3 , q 4 ). (2.8) in terms of invariant functions W i (q 1 , q 2 , q 3 , q 4 ) free from kinematic singularities and zeroes. These functions actually depend on the invariants that can be built with the products q i · q j , i, j = 1, 2, 3, 4, but for simplicity I write them as functions of the momenta for the time being. Not much is known about these functions beyond the kinematic properties mentioned above, and in order to estimate them, or at least the subset of them that contributes to a HLxL µ , it is important to make sure that they satisfy the few properties that can be deduced directly from QCD. One of these properties arises from the well-known behaviour [62,63] of the time-ordered product of two currents (2.2) at short distances, where it is understood here and in what follows that the limit holds when the momentum q belongs to the Euclidian region and when all its components become simultaneously large. The axial current appearing on the right-hand side of this relation is defined as (2.10) Then, writing q 1 =q +q, q 2 =q −q,q 2 = −Q 2 , Q 2 > 0, (2.11) one establishes [60] the following short-distance behaviour when the momenta carried by the first two currents become hard, while the other two remain soft (note that q 3 + q 4 = −q 1 − q 2 remains soft as well), where the three-point function W µνρ (q 1 , q 2 ) is defined as It satisfies the Ward identities where A stands for the anomalous contribution [64,65] These Ward identities feature yet another three-point function, where m q , q = u, d, s, denotes the masses of the three lightest quarks and G µν is the gluon field strength. The decomposition of W µν (q 1 ; q 2 ) is quite simple, since it involves a single function that is also free of kinematic singularities, This representation is entirely fixed by Lorents covariance, Bose symmetry, invariance under parity and conservation of the current j µ , which imposes transversality, Achieving a similar decomposition for the three-point function W µνρ (q 1 , q 2 ) is not quite as straightforward. Using only Lorentz covariance, invariance under parity, Bose symmetry and Schouten's identity to eliminate two additional possible structures, q ν 1 ǫ µραβ q 1α q 2β + q µ 2 ǫ νραβ q 1α q 2β and q ν 2 ǫ µραβ q 1α q 2β − q µ 1 ǫ νραβ q 1α q 2β , one obtains, to start with, the general decomposition in terms of six amplitudes that are free of kinematic singularities. The use of Schouten's identity, as well as Bose symmetry, may well introduce kinematic zeroes, but this issue is not relevant for our present purposes, so I will not take it into consideration. Bose symmetry further requires Conservation of the electromagnetic current implies These identities allow to eliminate W 4 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) and W 5 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) in terms of the remaining functions without introducing kinematic singularities. The result reads An alternative but equivalent decomposition in terms of four functions free of kinematical singularities can also be found in Eq. (4.9) of Ref. [66]. The condition (2.14) on (q 1 + q 2 ) ρ W a µνρ (q 1 ; q 2 ) further requires when combined with Eq. (2.18). Expressing W 0 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) in terms of the remaining functions through this relation leads to the decomposition given in Ref. [67], with a slightly different notation, (2.27) in terms of a set of three fully transverse tensors t µνρ i (q 1 , q 2 ), and with But this elimination is done at the expense of introducing kinematic singularities into the tensors t µνρ i (q 1 , q 2 ), and hence a kinematic constraint on the functions w i (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ). Indeed, Eq. (2.26) precisely materializes this constraint, since it states that the combination has to be equal to (q 1 + q 2 ) 2 times a function free of any kinematic singularity.
Since the authors of Ref. [42] use the notation of Ref. [67], let me, before closing this section, provide the connection between the two. It is straightforward to establish the relations one obtains and, making, for convenience, the change of notation W 0 −→ w 0 , It is clear from this relation that the function w L (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) does in general exhibit kinematic singularities. At this stage, let me formulate two remarks: • All the above properties still hold if instead of considering the correlators involving the current A µ , I had replaced the latter by one of its components A a µ defined in Eq. (2.10), with the proviso that each function like W i or w i is endowed with a corresponding superscript a, where a = 3, 8, 0, and that the anomaous contribution A is replaced by A a ≡ A tr(Q 2 λ a /2)/trQ 4 . Following common practice, I will refer to these three cases a = 3, 8, 0 as the iso-triplet, octet, and singlet channels, respectively.
• In the limit where q 2 vanishes, or equivalently in the combined limit q 2 Two observations can be made from this relation. The first is that the combination A + H(q 2 1 , 0, q 2 1 ) vanishes as O(q 2 1 ), a statement in which, when restricted to the iso-triplet channel and with the anomaly removed, one recognizes the Sutherland-Veltman theorem [68,69], see also Refs. [70] and [66]. The second observation is more relevant for the subject of this note: in the chiral limit, or in the combined chiral and large-N c limit in the case of the singlet channel, H(q 2 , 0, q 2 ) vanishes, and the relation (2.35) reduces to the usual expression w L (q 2 , 0, q 2 )/8π 2 = A/q 2 . Although the combination w 0 (q 2 , 0, q 2 ) + w 1 (q 2 , 0, q 2 ) tends to the same expression in this limit, the way it arises, and the physical content it conveys, is completely different. I will come back to this issue and its consequences later on.
Finally, let me mention that the first equality in Eq. (2.34) also appears as Eq. (B13) of Ref. [40], but its implications have not been discussed by the authors.

Implementing the short-distance constraints
Coming back to the short-distance behaviour given in Eq. (2.12), it may now be rewritten as The task that needs to be done next is to work out the consequences of the short-distance constraint (3.1) on the invariant functions W i (q 1 , q 2 , q 3 , q 4 ) that describe the fourth-rank vacuum polarization tensor as shown in Eq. (2.8). A procedure through which this can be achieved is described in Appendix A. Here I will merely discuss, through one example, some of the consequences that follow from the condition (3.1). The example I wish to consider involves, following Ref. [23], the combination For this combination, the short-distance constraint (3.1) requires the condition to hold. Before considering some specific aspects of this relation, a few general statements may be useful: • This condition holds as it stands, i.e. for all values of the invariants q 2 3 , q 2 4 and (q 3 + q 4 ) 2 . • Both sides are free from kinematic singularities. Since such singularities are absent on the left-hand side by construction, none should show up on the right-hand side, which is the case.
• Dynamical singularities in the variables q 2 3 , q 2 4 and (q 3 +q 4 ) 2 , i.e. poles due to single-particle exchanges or cuts due to multi-particle exchanges, have to match on both sides; those present in the functions w 0 and w 1 must correspond to singularities also present inŴ 1 and that moreover survive in the limit under consideration; likewise, singularities inŴ 1 that have no counterpart in w 0 or w 1 must fall into the subleading contributions to the short-distance expansion.
• Since the momenta q 3 and q 4 are generic (i.e. non-exceptional in the sense of Weinberg's theorem [71]), the chiral limit can be taken on both sides; the same holds for the large-N c limit, or for the combination of both limits.
• The limit where in addition q 2 3 becomes large in the Euclidian region can also be taken on both sides, as long as the condition −q 2 ≫ −q 2 3 remains satisfied.

Pion pole
Let us now consider the contribution coming from the exchange of a single neutral pion. It produces inŴ 1 a pole in the variable (q 3 + q 4 ) 2 ,Ŵ involving the pion transition form factor F πγ * γ * defined as and where Bose symmetry means that the form factor is unchanged upon replacing q by −q. Notice that the above definition implies that It differs of course from A pole singularity similar to the one inŴ 1 also shows up on the right-hand side of Eq. (3.4), since where the ellipsis stands for terms that are regular at (q 3 + q 4 ) 2 = M 2 π and F π denotes the pion decay constants defined as That the pion pole is located in the function w 0 and that it takes the form given above follows directly from the structure of the three-point function W µνρ as given in Eq. (2.24), and from the structure of the two matrix elements in Eqs. (3.6) and (3.10). The interested reader may actually check this property explicitly on the calculation of the functions w i at next-to-leading order in the low-emergy expansion presented in Appendix B. According to the third item in the list that follows Eq. (3.4), this same pole singularity in w 0 has to be recovered in the asymptotic limit ofŴ 1 (q +q,q −q, q 3 , q 4 ). This requires a property that is known to hold [72,73], and that also follows from the result given in Eq. (2.9). Furthermore, the compatibility, via the short-distance constraint (3.4), between the two expressions (3.5) and (3.9) manifestly continues to hold in the chiral limit.
We may now consider the kinematic regime relevant for the evaluation of a HLxL µ . According to the formulas given in Eqs. (2.4), (2.5), and (2.6), this involves taking the derivative of the rank-four vacuum polarization tensor with respect to q 4 , and then letting q 4 → 0, taking the constraint (2.3) into account. Since the tensors T µνρσ i (q 1 , q 2 , q 3 , q 4 ) are all at least linear in the momentum q 4 , this limit can be rewritten as As far as the short-distance constraint (3.4) is concerned, this means that we need to compare the leading term in the short-distance expansion of (3.14) The two expressions clearly match, and keep on doing so if one further takes the chiral limit, where one gains the additional information that In the dispersive approach of Refs. [22,23,41,42], the invariant functions W i (q 1 , q 2 , q 3 , q 4 ) are first expressed in terms of a set of appropriate kinematic variables, namely Accordingly, the functions describing the three-point function W µνρ are to be written as w i (q 2 3 , q 2 4 , s). This rewriting in terms of the variables (3.16) does not change the short-distance condition (3.1), and the right-hand side involves the same sum, w 0 (q 2 3 , q 2 4 , s) + w 1 (q 2 3 , q 2 4 , s), as before. The pion-pole contributions in Eqs. (3.5) and (3.9) also remain the same, up to the denominators that are now rewritten as s − M 2 π . As long as we give the different variables in Eq. (3.16) generic values, the whole discussion leading to the condition (3.11) can be repeated again, mutatis mutandis. So let us therefore turn to the kinematic regime relevant for the evaluation of a HLxL µ . Here the dispersive approach requires to consider the reduced kinematics defined in Ref. [22], so that Eq. (3.12) is replaced by (3.17) For the contribution from the pion pole to the left-hand side of Eq. (3.4) we find (the pion-pole contribution does not depend on the variable t) whereas in the same limit its contribution to the right-hand side reads The results for the two sides of the condition (3.4) differ from the previous case, since the second transition form factor now retains a dependence on q 3 3 , but what matters is that they perfectly match, and this matching persists in the chiral limit, which can be taken without problem.
To summarize this discussion of the pion pole, I find that, in the chiral limit, the short-distance constraint (3.4) leads to . (3.20) in the case where the kinematic configuration corresponding to q 4 → 0, and considered by the authors of Refs. [60,43], is taken. In the kinematic configuration corresponding to the dispersive treatment of the pion pole advocated by the authors of Refs. [41,42], it instead leads to The second equality in this last equation holds when q 2 3 becomes large in the Euclidian region (but with −q 2 ≫ −q 2 3 ), where the result [74,75] lim can be used. Both limits are, as far as I can see, legitimate, in the sense that none reveals any incoherence. However, they will most likely lead to different numerical outcomes as far as the contribution of the pion pole to a HLxL µ is concerned. But this needs not be a problem per se since what matters in the end is the comparison of the results obtained once all contributions to a HLxL µ have been added up.

An apparent paradox and its solution
The debate in the literature on a HLxL µ that has resurfaced recently [43,42] takes its origin in the fact that Eq. (3.4) is usually written in terms of the function w L , Refs. [43,42]. (3.23) As discussed after Eq. (2.34), this is quite legitimate in the limit appropriate for the discussion of a HLxL µ , whether one considers it in the form (3.12) or in the form (3.17). But although w L (q 2 3 , 0, q 2 3 )/(8π 2 ) and w 0 (q 2 3 , 0, q 2 3 ) + w 1 (q 2 3 , 0, q 2 3 ) are the same functions, they differ by their physical content, and this difference lies at the heart of the debate. In order to explain this point, let me consider the chiral limit and consider the iso-triplet channel, see the first remark after Eq.
(2.34) for the explanation of the nomenclature and the notation. The discussion in the octet channel is exactly the same, with the η meson playing the role of the pion, and extends to the singlet channel and the η ′ meson if in addition the large-N c limit is taken as well. In the chiral limit, the function w 3 L is known exactly in QCD, for arbitrary kinematics, 24) and this single contribution is entirely produced by a dynamical pion pole. Comparing the pion pole inŴ 1 with the one in w 3 L /(8π 2 ) would lead to compare, in the "dispersive" limit (3.17) Clearly, the two expressions cannot match as such for all values of q 2 3 , and this mismatch is at the origin of the debate between the authors of Refs. [60,43] on the one hand, and the authors of Refs. [41,42] on the other hand, the former seeing "the dependence on this form factor [i.e. o F πγ * γ * (q 2 3 , 0)] on q 2 3 " as "ambiguous within the dispersive approach", whereas for the latter the model based on a constant form factor [i.e. o F πγ * γ * (0, 0) in the chiral limit] represents a "distorsion" of the low-energy behaviour of the rank-four vacuum polarization tensor. But we have just seen that, although the two ways to implement the kinematic limit relevant for a HLxL µ give different results for the pion pole, they are both consistent with the content of Eq. (3.4), and the confrontation between the two options in Eq. (3.25) never shows up.
In order to understand the origin of this apparent paradox, let us come back to the combination w 0 + w 1 that actually appears on the right-hand side of Eq. (3.4). Even in the chiral limit, the structure of this function remains quite different from the simple form taken by w 3 L and given in Eq. (3.24), where ∆w 3 (q 2 3 , q 2 4 , (q 3 + q 4 ) 2 ) represents the part of w 0 + w 1 that is regular at (q 3 + q 4 ) 2 = 0 in the chiral limit. The first equality gives the version of Eq. (3.9) corresponding to the chiral limit. In the second equality I have isolated the contribution to the pole coming from o F πγ * γ * (0, 0) alone, and have identified it, in the third equality, with Eq. (3.24). Taking now the limit where q 4 vanishes or, equivalently, the combined limit q 2 4 → 0 and (q 3 + q 4 ) 2 → q 2 3 , we see that the relation which follows from Eq. (2.35), rests on an exact cancellation between a contribution that comes from a part of the pion pole, namely the one involving the momentum dependence of the pion transition form factor, and the contribution that is regular at (q 3 + q 4 ) 2 = 0, The computation in Appendix B shows that this cancellation indeed happens at one loop in the low-energy expansion. But it is in fact an exact property of QCD in the chiral limit, and a direct consequence of the relation (2.34). Besides its confirmation in the low-energy expansion, it can also be illustrated in a simple resonance model like the one of Ref. [76]. A straightforward calculation yields The manner in which the parameters b, c 1 , c 2 are related to the resonance couplings and to the mass M V of the vector resonance in this model need not concern us here. What matters instead is to observe that the cancellation (3.28) indeed takes place when either one of the limit (3.12) or (3.17) is considered. Whatever one decides to call the pseudo-paradox (3.25) at the origin of the debate in the recent literature, it rests on a wrong identification, in the chiral limit, of the pion-pole contribution on the right-hand side of the short-distance constraint in Eq. (3.4), and which itself arises from the identification of the two functions w L and w 0 + w 1 in the kinematic limit relevant for a HLxL µ . This second identification is correct from the functional point of view, but the quite different physical contents of these two functions have not been given sufficiently close attention. Once this is done, the debate loses its raison d'être.

Pseudoscalar poles
We may now extend the discussion to pseudoscalar poles in general. Strictly speaking, poles appear only for the lightest of these states, the pseudo-Goldstone mesons π 0 , η, η ′ . Heavier pseudoscalar states, like for instance the π(1300) isotriplet J P = 0 − resonance, are often too broad to be described just as poles on the real axis of the complex s-plane. Such a description would require a narrow-width approximation, which finds some justification by considering, for instance, the large-N c limit. Let us adopt the latter framework for the present discussion. In the case of the three-point function W µνρ , these poles are again to be found in the function w 0 (3.30) Here the sum runs over all the J P = 0 − states with masses M P , decay constants F a P , defined by the matrix elements Ω|A a ρ (0)|P (p) = iF a P p ρ , (3.31) and with transitions form factors F P γ * γ * defined in analogy with the case of the pion in Eq. (3.6). At the level of the four-point function, each of these pseudoscalar states produces a contribution analogous to the one of the pion,Ŵ (P ) (3.32) Since the dynamical singularities have to match on both sides of the short-distance constraint (3.4), we need to check that the relationŴ holds for asymtotic Euclidian values of the momentumq. That this is indeed the case follows again from Eq. (2.9). It is thus possible to consider the two limits discussed previously for the pion-pole contribution to a HLxL µ . Without surprise, the outcomes are again different lim q4→0 lim q 2 →−∞Ŵ Finally, in the combined large-N c and three-flavour chiral limit, each one of the flavour-diagonal axial currents defined in Eq. (2.10) is conserved, so that the decay constants vanish as F a P ∼ O(m q ) + O(1/N c ) for P = π 0 , η, η ′ , and the non-Goldstone pseudoscalar poles inŴ 1 contribute only to subleading terms of the short-distance expansion.

Summary and conclusion
This note proposes a critical, albeit only partial, discussion of the implications of the short-distance constraint of Ref. [60] for one of the invariant functions describing the rank-four hadronic vacuum polarization tensor. This study is focused on a very specific issue under debate in the recent literature, with the hope that it may contribute positively to this discussion. To this effect, I have first re-derived the short-distance constraints in the more general case of a generic kinematic configuration, and, more importantly, expressed them in terms of functions that are free of kinematic singularities. This allows to state a certain number of general properties that have to be met and that have been listed after Eq. (3.4).
I have then discussed the two kinematic limits that are currently considered in applications to the anomalous magnetic moment of the muon, for both the contribution from the pion pole or from narrow non-Goldstone pseudoscalar states. Working with functions that are free of kinematic singularities warrants that both kinematic limits can be taken without problem or ambiguity, and lead to coherent results if the same limit is taken on both sides of Eq. (3.4). They are, however, definitely different limits, and as such simply give... different results for the contribution from these poles. In itself, this needs not necessarily constitute a problem, since the pion pole is but one contribution to a HLxL µ , although an important one. But an evaluation of a HLxL µ at a level of precision of 10% in relative terms requires also to include other contributions in a controled manner, and a comparison between different approaches or prescriptions is only meaningful once this task has been completed.
If one follows the evolution of the pion pole through the different limits that are taken, no ambiguity in its identification arises, and a cancellation mechanism that necessarily needs to be at work (in QCD) in order to bring the two functions w L (q 2 3 , 0, q 2 3 ) and w 0 (q 2 3 , 0, q 2 3 ) + w 1 (q 2 3 , 0, q 2 3 ) to an identical form is brought out. This mechanism is clearly evidenced in the regime of small momentum transfers, where the low-energy expansion can be used. The function that appears on the right-hand side of the short-distance constraint is w 0 (q 2 3 , 0, q 2 3 ) + w 1 (q 2 3 , 0, q 2 3 ), whose pion-pole contribution in the chiral limit is only partially given by the pion pole of w L (q 2 3 , 0, q 2 3 ). Narrow pseudoscalar states other than π, η, η ′ contribute to both sides of Eq. (3.4) in a perfectly consistent manner. And this consistency persists in the chiral limit, where the non-singlet and non-Goldstone pseudoscalar states disappear altogether from the right-hand side while their contribution to the left-hand side becomes sub-leading in the short-distance expansion. If one takes in addition the large-N c limit, then this situation extends to all non-Goldstone pseudoscalar states.
Finally, let me point out that although I have refered several times to the constraint (3.4) or to its more general version (2.12) as the short-distance condition, the plural form would actually be more appropriate, since it really is a constraint on the fourth rank vacuum polarization tensor for each value of the momentum transfers q 2 3 , q 2 4 and (q 3 + q 4 ) 2 . And even in the kinematic regime relevant for the evaluation of a HLxL µ it still gives a condition for each value of q 2 3 and not only when q 2 3 becomes large in the Euclidian region, as it is most of the time being used. No phenomenological approach or model designed for the evaluation of a HLxL µ I am aware of has, so far, exploited the full content of the condition of Ref. [60] in this broader sense.
for instance, why in the definition ofŴ 1 in Eq. (3.3) there is a relative minus sign between the two terms, whereas one finds a plus sign in Eq. (2.15) of Ref. [23].
Coming back to Eq. (3.1), each function W i has, in the limit under consideration, an expansion of the form The value of n i , which determines the leading power behaviour, can be fixed in the following manner: the tensors T µνρσ i have dimension 4 for i = 1, . . . , 6, dimension 8 for i = 31, . . . , 36, and dimension 6 in all other cases, whereas the tensor W µνρσ is dimensionless. Furthermore, we are looking for relations of the type with some numerical coefficients c ik , and where the functions w k (q 3 , q 4 ) have dimension −2. This means that one has n i = 2 for i = 1, . . . , 6, n i = 6 for i = 31, . . . , 36, and n i = 4 for the remaining values of i. It is then possible to proceed upon going through the following steps: • First, one notices that the highest power inq of each tensor T µνρσ i (q +q,q −q, q 3 , q 4 ), which is given by also varies from case to case. This highest power is simply equal to 1 for i = 1, to 2 for i = 2, . . . , 6,9,12,13,15,17,18,29,30,32,37,48,49, and so on. Since we are looking for a behaviour that does not decrease faster than 1/q whenq becomes large, we are eventually left with only the cases to consider.
• Second, for each of these cases, one extracts from the tensor T µνρσ i (q +q,q −q, q 3 , q 4 ) the part, denoted as T µνρσ i (q, q 3 , q 4 ), that is either linear inq, i.e. With these pieces at hand, one can then construct a set of other useful relations involving the tensors K µνρσ i (q, q 3 , q 4 ) defined in Eq. (3.2): In these identities x 1,2,3 and y 1,2,3 are real parameters belonging to the interval [0, 1], but can otherwise be chosen arbitrarily. To these, one also has to add the two following relations: • Next, one expands the fonctions W i as explained in Eq. (A.2), taking into account the symmetry properties of these functions that are listed in Ref. [22]. It then remains to collect in the four-point function all the terms that do not decrease faster than 1/q and to require that their sum matches the right-hand side of Eq. 5,µ (q 3 , q 4 ) + W [5] 10,µ (q 3 , q 4 ) − W [5] 10,µ (q 4 , q 3 ) + q 3µ W where A 3 = −N c /12π 2 = (3/4)A and the loop functionJ P P , P = π, K, is defined in Ref. [80] and can be conveniently expressed as the integralJ Furthermore, µ denotes the chiral renormalization scale. The low-energy constant C W 7 is µ-independent, while the µdependence of the renormalized constant C W 22 (µ) is compensated by the log µ 2 terms, see Ref. [82]. Notice that despite the suggestive notation, and as the symbol is meant to remind of, F πγ * γ * (q 2 1 , q 2 2 ) is not yet the pion transition form factor F πγ * γ * (q 2 1 , q 2 2 ). The relation between the two is given by In the semi-off-shell case the expression of F πγ * γ * ((p/2 ± q) 2 , 0) one obtains this way reproduces the one that is given in Ref. [85]. From these formulas, one deduces, through the relations given in Eq. (2.34), the one-loop expression of the remaining functions The kinematic singularity, at (q 1 + q 2 ) 2 = 0, of w 3 L is immediately visible in this expression. It also shows how, in the chiral limit, this kinematic singularity transforms into a dynamical singularity due to the massless pion pole, but with a constant residue, fixed by the anomaly, lim The combination that appears in the short-distance condition (3.4) forŴ 1 is completely different already at one loop, since w 3 0 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) + w 3 1 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) = F π F πγ * γ * (q 2 1 , q 2 2 ) (q 1 + q 2 ) 2 − M 2 As stated in the text, it exhibits a pion pole, with residue given by F π F πγ * γ * (q 2 1 , q 2 2 ) that retains a non-trivial momentum dependence even in the chiral limit. The difference between the two expressions can be given a suggestive form, w 3 0 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) + w 3 1 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) − 1 8π 2 w 3 L (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) It clearly exhibits the cancellation that takes place in the limit q 2 → 0. The corresponding expressions in the chiral limit m q → 0 can be easily worked out from the formulas given above, using Whether one then takes the limit where the four-vector q 2 vanishes, or the combined, "dispersive-friendly", limit q 2 2 → 0, (q 1 + q 2 ) 2 → q 2 1 , one obtains the same result, lim q2→0 lim mq→0 w 3 0 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) + w 3 1 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) = lim q2→0 lim mq→0 1 8π 2 w 3 L (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) = A 3 q 2 1 . (B.11) But the manner how this result comes about is totally different in the two cases. To see this in an easy manner, let me consider the combined chiral and large-N c limit, where one finds the simple expressions [in the large-N c limit, L W 22 scales as O(N c ) and becomes independent of the renormalization scale µ, and recall that A 3 is also proportional to N c , whereas F π scales as O( lim mq →0 Nc→∞ w 3 0 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) + w 3 1 (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) = (B.12) As is well known, there are no corrections to the above expression of w 3 L (q 2 1 , q 2 2 , (q 1 + q 2 ) 2 ) in the chiral limit [86,87]. In the case of w 3 L , it is straightforward to understand how the limit in Eq. (B.11) arises. It simply reflects the fact that in the limit under consideration all that survives is the kinematic pole that has actually become a dynamical pion pole, with constant residue fixed by the anomaly, and there is nothing else, even before the limit q 2 → 0 is taken, as shown in the last expression in Eq. (B.12). In the case of the sum w 3 0 + w 3 1 , the situation is somewhat more subtle. There are other contributions besides a pion pole with constant residue in Eq. (B.12) before the limit q 2 → 0 is taken: the momentum-dependent residue of the pole is given by whatever is left over from F π F πγ * γ * (q 2 1 , q 2 2 ) in the combined chiral and large-N c limit, i.e. here a contribution proportional to C W 22 , and there are other, non-pole, contributions, also proportional to C W 22 . When the limit q 2 → 0 is taken, these two different contributions combine such as to leave only a part of the full pion pole, the one with a constant residue F 0 o F πγ * γ * (0, 0) = A 3 , behind. That this will happen that way to higher, and in fact, to all orders in the low-energy expansion, is guaranteed by Eq. (2.34), so that Eq. (B.11) actually constitutes an exact result of QCD. But as far as w 3 0 + w 3 1 is concerned, it only reproduces a truncated part of the full pion pole that was present to start with. In a nutshell, sometimes the two operations of taking the limit q 2 → 0 and of extracting the pion pole do not commute.