{'fraction_non_alphanumeric': 0.04243609240350496, 'fraction_numerical': 0.005576073575204577, 'mean_word_length': 4.443437130469058, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Pinch Technique to All Orders

The generalization of the pinch technique to all orders in perturbation theory is presented. The effective Green's functions constructed with this procedure are singled out in a unique way through the full exploitation of the underlying Becchi-Rouet-Stora-Tyutin symmetry. A simple all-order correspondence between the pinch technique and the background field method in the Feynman gauge is established.

There are considerable conceptual and phenomenological advantages in reformulating the perturbative expansion in terms of off-shell Green's functions which display manifestly the same properties as the physical amplitudes. To begin with, the sharp difference between observables and Green's functions suggests a great deal of redundancy in the conventional diagrammatic formulation of gauge theories, in the sense that extensive underlying cancellations beg to be made manifest and be explicitly exploited as early within a calculation as possible. Implementing these cancellations at an early stage not only renders the book-keeping aspects more tractable [1], but allows for theoretically safe reorganizations or resummations of the perturbative series. For example, identifying and Dyson-resuming the correct sub-set of propagator-like corrections gives rise to physically meaningful Born-improved amplitudes [2]. In addition, the generalization into a non-Abelian context of the characteristic properties of the QED effective charge, has a wide range of phenomenological applications [3].
Finally, n-point functions free of unphysical artifacts could serve, at least in principle, as the new building blocks of manifestly gauge-invariant Schwinger-Dyson equations [4].
It would clearly be preferable to enforce the relevant cancellations already at the level of the functional path-integral defining the theory, and obtain directly from it the desired Green's functions; this is however beyond our powers at the moment. On the other hand, there exists a diagrammatic method, called the pinch technique (PT) [4,5], which systematically exploits the symmetries built into physical observables, such as S-matrix elements, in order to construct off-shell sub-amplitudes that are kinematically akin to conventional Green's functions, but, unlike the latter, are also endowed with desirable properties. The basic observation, which essentially defines the PT, is that there exists a fundamental cancellation between sets of diagrams with different kinematic properties, such as self-energies, vertices, and boxes. This cancellation is driven by the underlying Becchi-Rouet-Stora-Tyutin symmetry [6], and is triggered when longitudinal momenta circulating inside vertex and box diagrams generate (by "pinching" out internal fermion lines) propagator-like terms. The latter are reassigned to conventional self-energy graphs in order to give rise to the aforementioned gauge-invariant effective Green's functions. In its original one- [4,5] and two-loop [7] application, the PT boils down to the study of the kinematic rearrangements produced into individual Feynman diagrams when elementary tree-level Ward identities (WIs) are triggered.
One of the most pressing questions in this context is whether one can extend the PT algorithm to all orders in perturbation theory, thus achieving the systematic construction of effective n-point functions displaying the aforementioned characteristic features. To accomplish this it is clear that one needs to abandon algebraic operations inside individual Feynman graphs, and resort to a more formal procedure. In this Letter we will show that the PT algorithm can be successfully generalized to all orders in perturbation theory, through the collective treatment of entire sets of diagrams. This is accomplished through the judicious use of the Slavnov-Taylor identity (STI) [8] satisfied by a special Green's function, which serves as a common kernel to all higher order self-energy and vertex diagrams.
We will consider for concreteness the S-matrix element for the quark-anti-quark elastic scattering process q(r 1 )q(r 2 ) → q(p 1 )q(p 2 ) in QCD. We set q = r 2 − r 1 = p 2 − p 1 , with s = q 2 the square of the momentum transfer. The longitudinal momenta responsible for the aforementioned kinematical rearrangements stem either from the bare gluon propagators or from the pinching part Γ P αµν (q, k 1 , k 2 ) appearing in the characteristic decomposition of the elementary tree-level three-gluon vertex Γ The above decomposition is to be carried out to "external" three-gluon vertices only, i.e., the vertices where the physical momentum transfer q is entering [7]. In what follows we work in the renormalizable Feynman gauge (RFG); this choice eliminates the longitudinal momenta The subset of the graphs of the quark-anti-quark elastic scattering process which will receive the action of the longitudinal momenta stemming from Γ P . Here ∆ represents the full gluon propagator.
from the bare propagators, and allows us to focus our attention on the all-order study of the longitudinal momenta originating from Γ P αµν . We will denote by A the subset of the graphs which will receive the action of the longitudinal momenta stemming from Γ P αµν (q, k 1 , k 2 ) (see Fig.1). We have that where λ e are the Gell-Mann matrices, and T ab µν is the sub-amplitude g a µ (k 1 )g b ν (k 2 ) → q(p 1 )q(p 2 ), with the gluons off-shell and the fermions on-shell; for the latter In terms of Green's functions we have Clearly, there is an equal contribution from the Γ P situated on the right hand-side of T .
Let us focus on the STI satisfied by the amplitude T ab µν ; it reads where the Green's function appearing in it are defined in Fig.2. The terms X 1ν andX 1ν die on-shell, since they are missing one fermion propagator. Thus, we arrive at the on-shell STI with Here D and S represent the full ghost and fermion propagators respectively.
. In perturbation theory both T ab µν and S ab 1ν are given by Feynman diagrams, which can be separated into distinct classes, depending on their kinematic dependence and their geometrical properties. Graphs which do not contain information about the kinematical details of the incoming test-quarks are self-energy graphs, whereas those which display a dependence on the test quarks are vertex graphs. The former depend only on the variable s, whereas the latter on both s and the mass m of the test quarks; equivalently, we will refer to them as s-channel or t-channel graphs, respectively. In addition to the s-t decomposition, Feynman diagrams can be separated into one-particle irreducible (1PI) and one-particle reducible (1PR) ones. The crucial point is that the action of the momenta k µ 1 or k ν 2 on T ab µν does not respect, in general, the original s-t and 1PI-1PR separation furnished by the Feynman diagrams (see third paper of [2]). In other words, even though Eq.(5) holds for the entire amplitude, it is not true for the individual sub-amplitudes, i.e., where I (respectively R) indicates the one-particle irreducible (respectively reducible) parts of the amplitude involved. Evidently, whereas the characterization of graphs as propagatorand vertex-like is unambiguous in the absence of longitudinal momenta (e.g., in a scalar theory), their presence tends to mix propagator-and vertex-like graphs. Similarly, 1PR graphs are effectively converted into 1PI ones (the opposite cannot happen). The reason for the inequality of Eq. (7) are precisely the propagator-like terms, such as those encountered in the one-and two-loop calculations; they have the characteristic feature that, when depicted by means of Feynman diagrams contain unphysical vertices, i.e., vertices which do not exist in the original Lagrangian (Fig.3). All such diagrams cancel diagrammatically against each other. Thus, after the aforementioned rearrangements have taken place, for the t-channel irreducible part of the amplitude we will have the equality Eq.(8) merits particular attention, because it is of central importance for the generalization of the PT to all orders. The superscript "PT" on the left hand-side denotes that the corresponding amplitude must be rearranged following the well-defined PT algorithm, as it has been explained in the literature [7]. In particular, one tracks down the rearrangments induced when the action of (virtual) longitudinal momenta (k) on the bare vertices of diagrams trigger elementary WIs. Eventually a WI of the form k µ γ µ = S −1 (k / + p /) − S −1 (p /) will give rise to propagator-like parts, by removing (pinching out) the internal bare fermion propagator S(k / + p /). Depending on the topology of the diagram under consideration this last WI may be activated immediately, or as the final outcome of a sequential triggering of intermediate WIs. We emphasize that, in order to preserve the special unitarity and analyticity properties of the PT Green's functions, "internal" three-gluon vertices should not pinch, nor should one carry out sub-integrations [7].
The non-trivial step for generalizing the PT to all orders is then the following: Instead of going through the arduous task of manipulating the left hand-side of Eq. (8), following the aforementioned rules, in order to determine the pinching parts and explicitly enforce their cancellation, use directly the right-hand side, which already contains the answer! Indeed, the right-hand side involves only conventional (ghost) Green's functions, expressed in terms of normal Feynman rules, with no reference to unphysical vertices. That this must be so follows from the same PT rules mentioned above: due to the absence of external three-gluon vertices the right-hand side cannot be pinched further, i.e. its separation into propagatorand vertex-like graphs is unambiguous, since there is no possibility (without violating the PT rules) to obtain further mixing. Thus, the right-hand side of Eq. short) in order to establish whether the known correspondence persists to all orders; as we will see, this is indeed the case (for an extended list of related references see [9]).
To begin with, it is immediate to recognize that in the RFG box diagrams of arbitrary order n, to be denoted by B We then continue with the construction of the 1PI PT gluon-quark-anti-quark vertex Γ e α . We start from the corresponding vertex in the RFG, to be denoted by Γ e α , and focus only on the class of vertex diagrams containing an external bare three-gluon vertex; we will denote this subset by Γ e A 3 ,α [ Fig.4(a)]. All other types of graphs contributing to Γ e α are inert as far as the PT procedure is concerned, because they do not furnish pinching momenta [7]. The next step is to carry out the vertex decomposition of Eq.(1) to the external three-gluon vertex Γ eab,[0] αµν appearing in Γ e A 3 ,α . This will result in the obvious separation Γ e A 3 ,α = Γ F, e A 3 ,α + Γ P, e A 3 ,α . The part Γ F, e A 3 ,α is also inert, and will be left untouched. Thus, the only quantity to be further manipulated is Γ P, e A 3 ,α ; it reads where ≡ µ 2ε d d k/(2π) d , d = D − 2ε, D is the space-time dimension, and µ is the 't Hooft mass. Following the discussion presented above, the pinching action amounts to the replacement k ν [T ab µν ] t,I → [k ν T ab µν ] t,I = S ab 2µ (−k + q, k) t,I and similarly for the term coming from the momentum (k − q) µ , i.e., [(k − q) µ T ab µν ] t,I = − S ab 1ν (−k + q, k) t,I , or, equivalently, At this point the construction of the effective PT vertex Γ e α has been completed. The next important point is to study the connection between Γ e α and the vertex Γ e α in the BFG. To begin with, all "inert" terms contained in the original Γ e α carry over to the same sub-groups of graphs obtained in the BFG; most notably, the Γ F, e A 3 ,α is precisely the Γ e AA 2 ,α part of Γ e α , where A is the background gluon. The only exception are the ghost-diagrams contributing to Γ e α [ Fig.4(b)]; the latter do not coincide with the corresponding ghost contributions in the BFG.
This concludes the proof that Γ e α ≡ Γ e α . We emphasize that the sole ingredient in the above construction has been the STI of Eq.onshSTI; in particular, at no point have we employed a priori the background formalism. Instead, its special ghost sector has arisen dynamically, once the PT rearrangement has taken place.
The final step is to construct the (all orders) PT gluon self-energy Π ab µν . Notice that at this point one would expect that it too coincides with the BFG gluon self-energy Π ab µν , since both the boxes as well as the vertex do coincide with the corresponding quantities in BFG, and the S-matrix is unique (renormalization may be carried out order-by-order without any complications, see second paper in [7]). We will carry out a proof based on the strong induction principle, which states that a given predicate P (n) on N is true ∀ n ∈ N, if P (k) is true whenever P (j) is true ∀ j ∈ N with j < k. We will use a schematic notation, suppressing Lorentz, color, and momentum indices. At one-and two-loop, we know that the result is true [4,7]. Assuming then that the PT construction has been successfully carried out up to the order n − 1, we will show that the PT gluon self-energy is equal to the BFG gluon self-energy at order n, hence proving that this equality holds true at any given n.
At this point, by means of the explicit construction presented for the vertex, we have that Γ [n] ≡ Γ [n] , so that one immediately gets Π [n] ≡ Π [n] . Hence, by strong induction, the above relation is true for any given perturbative order n, i.e., we have Π ab µν ≡ Π ab µν , q.e.d. In conclusion, we have shown that the use of the STIonshSTI satisfied by the special Green's functionsgf, allows for the generalization of the PT procedure to all orders. It would