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PHYSICAL REVIEW D 66, 111901~R! ~2002!

Pinch technique to all orders

Daniele Binosi* and Joannis Papavassiliou†

Departamento de Fı´sica Teo´rica and IFIC, Centro Mixto, Universidad de Valencia–CSIC, E-46100, Burjassot, Valencia, Spain~Received 21 August 2002; published 30 December 2002!

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

DOI: 10.1103/PhysRevD.66.111901 PACS number~s!: 11.15.Bt, 12.38.Bx, 14.70.Dj

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It is well known that, to any finite order, the conventionperturbative expansion gives rise to expressions for physamplitudes which are endowed with crucial propertiS-matrix elements, for example, are independent ofgauge-fixing scheme and parameters chosen to quantiztheory; they are gauge-invariant~current conservation!; theyare unitary~conservation of probability! and well behaved ahigh energies. However, the above properties are in gennot reflected by the individual off-shell Green’s functionwhich are the building blocks of the aforementioned pertbative expansion. The latter depend on the gauge-fixingrameters in a complicated way, grow much faster than phcal amplitudes at high energies, and display unphysthresholds. Evidently, when combining unphysical Greefunctions to form a physical amplitude, subtle fieltheoretical mechanisms are at work, which enforce nontricancellations among them at any given order.

There are considerable conceptual and phenomenoloadvantages in reformulating the perturbative expansionterms of off-shell Green’s functions which display manifesthe same properties as the physical amplitudes. To bwith, the sharp difference between observables and Grefunctions suggests a great deal of redundancy in the contional diagrammatic formulation of gauge theories, in tsense that extensive underlying cancellations beg to be mmanifest and be explicitly exploited as early within a calclation as possible. Implementing these cancellations aearly stage not only renders the bookkeeping aspects mtractable@1#, but allows for theoretically safe reorganizatioor resummations of the perturbative series. For examidentifying and Dyson-resumming the correct subsetpropagatorlike corrections gives rise to physically meaniful Born-improved amplitudes@2#. In addition, the generali-zation into a non-Abelian context of the characteristic proerties of the QED effective charge has a wide rangephenomenological applications@3#. Finally, n-point functionsfree of unphysical artifacts could serve, at least in principas the new building blocks of manifestly gauge-invariaSchwinger-Dyson equations@4#.

It would clearly be preferable to enforce the relevant ccellations already at the level of the functional path integdefining the theory, and obtain directly from it the desir

*Electronic address: Daniele.Binosi@uv.es†Electronic address: Joannis.Papavassiliou@uv.es

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Green’s functions; this is however beyond our powers atmoment. On the other hand, there exists adiagrammaticmethod, called the pinch technique~PT! @4,5#, which system-atically exploits the symmetries built into physical obserables, such asS-matrix elements, in order to construct ofshell subamplitudes that are kinematically akinconventional Green’s functions, but, unlike the latter, are aendowed with desirable properties. The basic observatwhich essentially defines the PT, is that there exists a funmental cancellation between sets of diagrams with differkinematic properties, such as self-energies, vertices,boxes. This cancellation is driven by the underlying BeccRouet-Stora-Tyutin symmetry@6#, and is triggered when longitudinal momenta circulating inside vertex and box dgrams generate~by ‘‘pinching’’ out internal fermion lines!propagatorlike terms. The latter are reassigned to convtional self-energy graphs in order to give rise to the afomentioned gauge-invariant effective Green’s functions. Inoriginal one-@4,5# and two-loop@7# application, the PT boilsdown to the study of the kinematic rearrangements produinto individual Feynman diagrams when elementary trelevel Ward identities~WIs! are triggered.

One of the most pressing questions in this contextwhether one can extend the PT algorithm to all ordersperturbation theory, thus achieving the systematic consttion of effectiven-point functions displaying the aforementioned characteristic features. To accomplish this it is clthat one needs to abandon algebraic operations insidevidual Feynman graphs, and resort to a more formal produre. In this paper we will show that the PT algorithm cansuccessfully generalized toall orders in perturbation theory,through the collective treatment of entire sets of diagramThis is accomplished through the judicious use of tSlavnov-Taylor identity ~STI! @8# satisfied by a speciaGreen’s function, which serves as a common kernel tohigher order self-energy and vertex diagrams.

We will consider for concreteness theS-matrix elementfor the quark–anti-quark elastic scattering proceq(r 1)q(r 2)→q(p1)q(p2) in QCD. We setq5r 12r 25p12p2, with s5q2 the square of the momentum transfer. Tlongitudinal momenta responsible for the aforementionednematical rearrangements stem either from the bare gpropagators or from the pinching partGamn

P (q,k1 ,k2) ap-pearing in the characteristic decomposition of the elementree-level three-gluon vertexGamn

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Gamn[0] ~q,k1 ,k2!5Gamn

F ~q,k1 ,k2!1GamnP ~q,k1 ,k2!,

GamnF ~q,k1 ,k2!5~k12k2!agmn12qngam22qmgan ,

GamnP ~q,k1 ,k2!5k2ngam2k1mgan . ~1!

The above decomposition is to be carried out to ‘‘externthree-gluon vertices only, i.e., the vertices where the physmomentum transferq is entering@7#. In what follows wework in the renormalizable Feynman gauge~RFG!; thischoice eliminates the longitudinal momenta from the bpropagators, and allows us to focus our attention on theorder study of the longitudinal momenta originating froGamn

P . We will denote byA the subset of the graphs whicwill receive the action of the longitudinal momenta steming from Gamn

P (q,k1 ,k2) ~see Fig. 1!. We have that

A5 ig2u~r 1!le

2gav~r 2! f eabGP,amn~q,k1 ,k2!

3T mnab~k1 ,k2 ,p1 ,p2!, ~2!

wherele are the Gell-Mann matrices, andT mnab is the subam-

plitude gma (k1)gn

b(k2)→q(p1)q(p2), with the gluonsoff-shell and the fermions on-shell; for the latte

v(p2)S21(p2)up” 25m5S21(p1)u(p1)up” 15m50, where S(p)is the ~full ! quark propagator. In terms of Green’s functiowe have

T mnab5 v~p2!@C rs

ab~k1 ,k2 ,p1 ,p2!Dmr ~k1!Dn

s~k2!#u~p1!.~3!

Clearly, there is an equal contribution from theGP situatedon the right-hand side ofT.

FIG. 1. The subset of the graphs of the quark–anti-quark elascattering process which will receive the action of the longitudimomenta stemming fromGP. Here D represents the full gluonpropagator.

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Let us focus on the STI satisfied by the amplitudeT mnab ; it

reads

k1mCmn

ab1k2nG1ab2 ig f bcdQ1n

acd2gX1nab1gX1n

ab50, ~4!

where the Green’s function appearing in it is defined in F2. The termsX1n andX1n die on-shell, since they are missinone fermion propagator. Thus, we arrive at the on-shell Sfor T mn

ab

k1mT mn

ab5S 1nab, ~5!

with

S 1nab5 v~p2!@ ig f bcdQ 1n

acd~k1 ,k2 ,p1 ,p2!D~k1!

2k2nG 1ab~k1 ,k2 ,p1 ,p2!D~k1!D~k2!#u~p1!, ~6!

where we have defined

Q1nacd~k1 ,k2 ,p1 ,p2!

5Q 1nacd~k1 ,k2 ,p1 ,p2!D~k1!S~p1!S~p2!.

In perturbation theory bothT mnab and S 1n

ab are given byFeynman diagrams, which can be separated into distclasses, depending on their kinematic dependence andgeometrical properties. Graphs which do not contain infmation about the kinematical details of the incoming tequarks are self-energy graphs, whereas those which dispdependence on the test quarks are vertex graphs. The fodepend only on the variables, whereas the latter depend oboth s and the massm of the test quarks; equivalently, wwill refer to them ass-channel ort-channel graphs, respectively. In addition to thes-t decomposition, Feynman diagrams can be separated into one-particle irreducible~1PI!and one-particle reducible~1PR! ones. The crucial point isthat the action of the momentak1

m or k2n on T mn

ab doesnotrespect, in general, the originals-t and 1PI-1PR separatiofurnished by the Feynman diagrams~see the third paper o@2#!. In other words, even though Eq.~5! holds for the entireamplitude, it is not true for the individual subamplitudes, i.

k1m@T mn

ab #x,YÞ@S 1nab#x,Y , x5s,t; Y5I,R, ~7!

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FIG. 2. Diagrammatic representation of thGreen’s function appearing in the STI of Eq.~4!.HereD andSrepresent the full ghost and fermiopropagators respectively.

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PINCH TECHNIQUE TO ALL ORDERS PHYSICAL REVIEW D66, 111901~R! ~2002!

where I~R! indicates the one-particleirreducible ~reducible!parts of the amplitude involved. Evidently, whereas the chacterization of graphs as propagatorlike and vertexlike isambiguous in the absence of longitudinal momenta~e.g., in ascalar theory!, their presence tends to mix-propagatorliand vertexlike graphs. Similarly, 1PR graphs are effectivconverted into 1PI ones~the opposite cannot happen!. Thereason for the inequality of Eq.~7! is precisely the propagatorlike terms, such as those encountered in the one- andloop calculations; they have the characteristic feature twhen depicted by means of Feynman diagrams they conunphysical vertices, i.e., vertices which do not exist inoriginal Lagrangian~Fig. 3!. All such diagrams canceldia-grammaticallyagainst each other. Thus, after the aforemtioned rearrangements have taken place, for thet channelirreducible part of the amplitude we will have the equality

@k1mT mn

ab # t,IPT[@S 1n

ab# t,I . ~8!

Equation~8! merits particular attention, because it iscentral importance for the generalization of the PT toorders. The superscript ‘‘PT’’ on the left-hand side denothat the corresponding amplitude must be rearranged folling the well-defined PT algorithm, as it has been explainethe literature@7#. In particular, one tracks down the rearranments induced when the action of~virtual! longitudinal mo-menta~k! on the bare vertices of diagrams trigger elementWIs. Eventually a WI of the formkmgm5S21(k”1p” )2S21(p” ) will give rise to propagatorlike parts, by removin~pinching out! the internal bare fermion propagatorS(k”1p” ). Depending on the topology of the diagram under cosideration this last WI may be activated immediately, or

FIG. 3. Diagrammatic representation of the tree-level inequties of Eq.~7!. After the PTs-t channel cancellation occurs we aleft with the equality of Eq.~8!.

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the final outcome of a sequential triggering of intermediWIs. We emphasize that, in order to preserve the speunitarity and analyticity properties of the PT Green’s funtions, ‘‘internal’’ three-gluon vertices should not pinch, nshould one carry out subintegrations@7#.

The nontrivial step for generalizing the PT to all ordersthen the following: Instead of going through the arduous taof manipulating the left-hand side of Eq.~8!, following theaforementioned rules, in order to determine the pinchparts and explicitly enforce their cancellation, use directhe right-hand side, which already contains the answer.deed, the right-hand side involves only conventional~ghost!Green’s functions, expressed in terms of normal Feynmrules, with no reference to unphysical vertices. That tmust be so follows from the same PT rules mentioned abodue to the absence of external three-gluon vertices the rihand side cannot be pinched further, i.e. its separationpropagatorlike and vertexlike graphs is unambiguous, sithere is no possibility~without violating the PT rules! toobtain further mixing. Thus, the right-hand side of Eq.~8!serves as a practical definition of the PT to all orders.

After these observations, we proceed to the PT consttion to all orders. Once the effective Green’s functions habeen derived, they will be compared to the correspondGreen’s functions obtained in the Feynman gauge ofbackground field method~BFG for short! in order to estab-lish whether the known correspondence persists to all ordas we will see, this is indeed the case~for an extended list ofrelated references see@9#!.

To begin with, it is immediate to recognize that in thRFG box diagrams of arbitrary ordern, to be denoted byB[n] , coincide with the PT boxesB[n] , since all three-gluonvertices are ‘‘internal,’’ i.e., they do not provide longitudinmomenta. Thus, they coincide with the BFG boxes,B[n] , i.e.,B[n]5B[n]5B[n] for everyn.

We then continue with the construction of the 1PI Pgluon-quark–anti-quark vertexGa

e . We start from the corre-sponding vertex in the RFG, to be denoted byGa

e , and focusonly on the class of vertex diagrams containing anexternalbare three-gluon vertex; we will denote this subset byGA3,a

e

@Fig. 4~a!#. All other types of graphs contributing toGae are

inert as far as the PT procedure is concerned, becausedo not furnish pinching momenta@7#. The next step is tocarry out the vertex decomposition of Eq.~1! to the external

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FIG. 4. The Green’s functions identified in th

construction of the all order PT vertexGae . The

Green’s functions~b! and ~c! receive a contribu-tion from similar terms with the ghost arrows reversed~not shown!.

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three-gluon vertexGamneab,[0] appearing inGA3,a

e . This will re-sult in the obvious separationGA3,a

e5GA3,a

F,e1GA3,a

P,e . The partGA3,a

F,e is also inert, and will be left untouched. Thus, the onquantity to be further manipulated isGA3,a

P,e ; it reads

GA3,aP,e

5g febaE @~k2q!mgan 1knga

m#@T mnab # t,I , ~9!

where*[m2«*ddk/(2p)d, d5D22«, D is the space-timedimension, andm is the ’t Hooft mass. Following the discussion presented above, the pinching action amounts toreplacement kn@T mn

ab # t,I→@knT mnab # t,I5@S 2m

ab(2k1q,k)# t,I

and similarly for the term coming from the momentumk2q)m, i.e., @(k2q)mT mn

ab # t,I52@S 1nab(2k1q,k)# t,I , or,

equivalently,

GA3,aP,e

~q!→g febaE ~@S 2aab# t,I2@S 1a

ab# t,I!. ~10!

At this point the construction of the effective PT vertexGae

has been completed. The next important point is to studyconnection betweenGa

e and the vertexGae in the BFG. To

begin with, all ‘‘inert’’ terms contained in the originalGae

carry over to the same subgroups of graphs obtained inBFG; most notably, theGA3,a

F, e is precisely theG AA2,ae part of

Gae , whereA is the background gluon. The only exceptio

are the ghost-diagrams contributing toGae @Fig. 4~b!#; the

latter donot coincide with the corresponding ghost contribtions in the BFG.

The important step is to recognize that the BFG ghsector is provided precisely by combining the RFG ghowith the right-hand side of Eq.~8!. Specifically, one arrivesat both thesymmetricvertexG Acc

e , characteristic of the BFG

as well as at the four-particle ghost vertexG AAcce , which is

totally absent in the conventional formalism@Fig. 4~c!#. In-deed we find~omitting the spinors!

E @S 1aab# t,I5E D~2k1q!$2ka@G 1

ab~2k1q,k!# t,ID~k!

1 ig f bcd@Q 1aacd~2k1q,k!# t,I%. ~11!

A similar equation, in which we have to trade theG 1ab and

Q 1aacd Green’s functions for their Bose symmetric onesG 2

ab

and Q 2aacd , holds for theS2a term. It is then easy to show

that

G Acc,ae

~q![GAcc,ae

~q!1g feba

E $ka@G 1ab~2k1q,k!# t,I

1~k2q!a@G 2ab~2k1q,k!# t,I%

3D~2k1q!D~k!,

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~q![ ig2f ebaE $ f acd@Q 2acdb~2k1q,k!# t,ID~k!

2 f bcd@Q 1aacd~2k1q,k!# t,ID~2k1q!%.

~12!

This concludes the proof thatGae[Ga

e . We emphasize thathe sole ingredient in the above construction has beenSTI of Eq.~5!; in particular, at no point have we employedapriori the background formalism. Instead, its special ghsector has arisendynamically, once the PT rearrangemenhas taken place.

The final step is to construct the~all orders! PT gluonself-energyPmn

ab . Notice that at this point one would expe

that it too coincides with the BFG gluon self-energyPmnab ,

since both the boxes as well as the vertex do coincide wthe corresponding quantities in BFG, and theS-matrix isunique ~renormalization may be carried out order-by-ordwithout any complications, see the second paper in@7#!. Wewill carry out a proof based on the strong induction principwhich states that a given predicateP(n) on N is true ; nPN, if P(k) is true wheneverP( j ) is true ; j PN with j,k. We will use a schematic notation, suppressing Lorencolor, and momentum indices. At one- and two-loop, wknow that the result is true@4,7#. Assuming then that the PTconstruction has been successfully carried out up to the on21, we will show that the PT gluon self-energy is equalthe BFG gluon self-energy at ordern, hence proving that thisequality holds true at any givenn. From the inductive hy-pothesis, we know thatP [ ,][P [ ,] , G [ ,][G [ ,] , and B[ ,]

[B[ ,][B[ ,] , with ,51, . . . ,n21. Now, theS-matrix ele-ment of ordern, to be denoted asS[n] , assumes the formS[n]5$GDG% [n]1B[n] . Moreover, since it is unique, regardless of whether it is written in the RFG, in the BFG, as was before and after the PT rearrangement, we have thatS[n]

[S[n][S[n] . Using then the fact thatB[ ,][B[ ,] holds trueeven when ,5n, we find that $GDG% [n][$GDG% [n]

[$GDG% [n] . These amplitudes can then be split into 1Pand 1PI parts; in particular, the 1PR part after the PT rerangement coincides with the 1PR part written in the BFsince$GDG%R

[n]5G [n1]D [n2]G [n3] with n1 ,n2 ,n3,n, andn1

1n21n35n. This implies in turn the equivalence of the 1Pparts, i.e.,

~ G [n]2G [n] !D [0]G [0]1G [0]D [0]~ G [n]2G [n] !

1G [0]D [0]~P [n]2P [n] !D [0]G [0][0. ~13!

At this point, by means of theexplicit construction pre-sented for the vertex, we have thatG [n][G [n] , so that oneimmediately getsP [n][P [n] . Hence, by strong inductionthe above relation is true for any given perturbative orden,i.e., we havePmn

ab[Pmnab , QED.

In conclusion, we have shown that the use of the STI~5!satisfied by the special Green’s function~3!, allows for the

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generalization of the PT procedure to all orders. It wouldinteresting to further explore the physical meaning ofn-point functions obtained@10#, and establish possible connections with related formalisms@11#.

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This work has been supported by the CICYT GranAEN-99/0692 and BFM2001-0262. D.B. thanks the Theorical Physics Department of the University of Trento, whepart of this work has been carried out.

.

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