From loops to trees by-passing Feynman's theorem
Stefano Catani, Tanju Gleisberg, Frank Krauss, German Rodrigo, Jan-Christopher Winter
aa r X i v : . [ h e p - ph ] S e p IFIC/08-21, IPPP/08/22, FERMILAB-PUB-08-092-T, SLAC-PUB-13218arXiv:0804.3170 [hep-ph]
From loops to treesby-passing Feynman’s theorem
Stefano Catani ( a ) ∗ , Tanju Gleisberg ( b ) † , Frank Krauss ( c ) ‡ ,Germ´an Rodrigo ( d ) § and Jan-Christopher Winter ( e ) ¶ ( a ) INFN, Sezione di Firenze and Dipartimento di Fisica, Universit`a di Firenze,I-50019 Sesto Fiorentino, Florence, Italy ( b ) Stanford Linear Accelerator Center, Stanford UniversityStanford, CA 94309, USA ( c ) Institute for Particle Physics Phenomenology, Durham University,Durham DH1 3LE, UK ( d ) Instituto de F´ısica Corpuscular, CSIC-Universitat de Val`encia,Apartado de Correos 22085, E-46071 Valencia, Spain ( e ) Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Abstract
We derive a duality relation between one-loop integrals and phase-spaceintegrals emerging from them through single cuts. The duality relation isrealized by a modification of the customary + i ∗ E-mail: stefano.catani@fi.infn.it † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: german.rodrigo@ific.uv.es ¶ E-mail: [email protected]
Introduction
The Feynman Tree Theorem (FTT) [1, 2] applies to any (local and unitary) quantum fieldtheories in Minkowsky space with an arbitrary number d of space-time dimensions. It relatesperturbative scattering amplitudes and Green’s functions at the loop level with analogousquantities at the tree level. This relation follows from a basic and more elementary relationbetween loop integrals and phase-space integrals. Using this basic relation loop Feynmandiagrams can be rewritten in terms of phase-space integrals of tree-level Feynman diagrams.The corresponding tree-level Feynman diagrams are then obtained by considering multiple cuts (single cuts, double cuts, triple cuts and so forth) of the original loop Feynman diagram.We have recently proposed a method [3, 4, 5] to numerically compute multi-leg one-loopcross sections in perturbative field theories. The starting point of this method is a duality relation between one-loop integrals and phase-space integrals. Although the analogy withthe FTT is quite close, there are important differences. The key difference is that theduality relation involves only single cuts of the one-loop Feynman diagrams. Both theFTT and the duality relation can be derived by using the residue theorem ∗ .In this paper, we illustrate and derive the duality relation. Since the FTT has recentlyattracted a renewed interest [6] in the context of twistor-inspired methods [7, 8] to evaluateone-loop scattering amplitudes [9], we also discuss its correspondence (including similaritiesand differences) with the duality relation.The outline of the paper is as follows. In Section 2, we introduce our notation. InSection 3, we briefly recall how the FTT relates one-loop integrals with multiple-cut phase-space integrals. In Section 4, we present one of the main results of this publication: wederive and illustrate the duality relation between one-loop integrals and single-cut phase-space integrals. We also prove that the duality relation requires to properly regularizepropagators by a complex Lorentz-covariant prescription, which is different from the cus-tomary + i ∗ Within the context of loop integrals, the use of the residue theorem has been considered many timesin textbooks and in the literature. Notation
The FTT and the duality relation can be illustrated with no loss of generality by consideringtheir application to the basic ingredient of any one-loop Feynman diagrams, namely ageneric one-loop scalar integral L ( N ) with N ( N ≥
2) external legs. qp q p q q N p N p Figure 1:
Momentum configuration of the one-loop N -point scalar integral. The momenta of the external legs are denoted by p µ , p µ , . . . , p µN and are clockwise or-dered (Fig. 1). All are taken as outgoing. To simplify the notation and the presentation,we also limit ourselves in the beginning to considering massless internal lines only. Thus,the one-loop integral L ( N ) can in general be expressed as: L ( N ) ( p , p , . . . , p N ) = − i Z d d q (2 π ) d N Y i =1 q i + i , (1)where q µ is the loop momentum (which flows anti-clockwise). The momenta of the internallines are denoted by q µi ; they are given by q i = q + i X k =1 p k , (2)and momentum conservation results in the constraint N X i =1 p i = 0 . (3)The value of the label i of the external momenta is defined modulo N , i.e. p N + i ≡ p i .The number of space-time dimensions is denoted by d (the convention for the Lorentz-indices adopted here is µ = 0 , , . . . , d −
1) with metric tensor g µν = diag(+1 , − , . . . , − k µ are denoted as k µ = ( k , k ), where k isthe energy (time component) of k µ . It is also convenient to introduce light-cone coordinates k µ = ( k + , k ⊥ , k − ), where k ± = ( k ± k d − ) / √
2. Throughout the paper we consider loopintegrals and phase-space integrals. If the integrals are ultraviolet or infrared divergent, wealways assume that they are regularized by using analytic continuation in the number ofspace-time dimensions (dimensional regularization). Therefore, d is not fixed and does notnecessarily have integer value. 2e introduce the following shorthand notation: − i Z d d q (2 π ) d · · · ≡ Z q · · · . (4)When we factorize off in a loop integral the integration over the momentum coordinate q or q + , we write − i Z + ∞−∞ dq Z d d − q (2 π ) d · · · ≡ Z dq Z q · · · , (5)and − i Z + ∞−∞ dq + Z + ∞−∞ dq − Z d d − q ⊥ (2 π ) d · · · ≡ Z dq + Z ( q − , q ⊥ ) · · · , (6)respectively. The customary phase-space integral of a physical massless particle with mo-mentum q (i.e. an on-shell particle with positive-definite energy: q = 0, q ≥
0) reads Z d d q (2 π ) d − θ ( q ) δ ( q ) · · · ≡ Z q e δ ( q ) · · · , (7)where we have defined e δ ( q ) ≡ π i θ ( q ) δ ( q ) = 2 π i δ + ( q ) . (8)Using this shorthand notation, the one-loop integral L ( N ) in Eq. (1) can be cast into L ( N ) ( p , p , . . . , p N ) = Z q N Y i =1 G ( q i ) , (9)where G ( q ) denotes the customary Feynman propagator, G ( q ) ≡ q + i . (10)We also introduce the advanced propagator G A ( q ), G A ( q ) ≡ q − i q . (11)We recall that the Feynman and advanced propagators only differ in the position of theparticle poles in the complex plane (Fig. 2). Using q = q − q = 2 q + q − − q ⊥ , we thereforehave [ G ( q )] − = 0 = ⇒ q = ± p q − i , or q ± = q ⊥ − i q ∓ , (12)and [ G A ( q )] − = 0 = ⇒ q ≃ ± p q + i , or q ± ≃ q ⊥ q ∓ + i . (13)Thus, in the complex plane of the variable q (or, equivalently † , q ± ), the pole with positive(negative) energy of the Feynman propagator is slightly displaced below (above) the realaxis, while both poles (independently of the sign of the energy) of the advanced propagatorare slightly displaced above the real axis. † To be precise, each propagator leads to two poles in the plane q and to only one pole in the plane q + (or q − ). ( q ) G A ( q ) q ( q ± ) plane q ( q ± ) plane ×× ×× Figure 2:
Location of the particle poles of the Feynman (left) and advanced (right) propa-gators, G ( q ) and G A ( q ) , in the complex plane of the variable q or q ± . In this Section we briefly recall the FTT [1, 2].To this end, we first introduce the advanced one-loop integral L ( N ) A , which is obtainedfrom L ( N ) in Eq. (9) by replacing the Feynman propagators G ( q i ) with the correspondingadvanced propagators G A ( q i ): L ( N ) A ( p , p , . . . , p N ) = Z q N Y i =1 G A ( q i ) . (14)Then, we note that L ( N ) A ( p , p , . . . , p N ) = 0 . (15)The proof of Eq. (15) can be carried out in an elementary way by using the Cauchyresidue theorem and choosing a suitable integration path C L . We have L ( N ) A ( p , p , . . . , p N ) = Z q Z dq N Y i =1 G A ( q i )= Z q Z C L dq N Y i =1 G A ( q i ) = − πi Z q X Res { Im q < } " N Y i =1 G A ( q i ) = 0 . (16)The loop integral is evaluated by integrating first over the energy component q . Sincethe integrand is convergent when q → ∞ , the q integration can be performed along thecontour C L , which is closed at ∞ in the lower half-plane of the complex variable q (Fig. 3– left ). The only singularities of the integrand with respect to the variable q are the poles ofthe advanced propagators G A ( q i ), which are located in the upper half-plane. The integralalong C L is then equal to the sum of the residues at the poles in the lower half-plane andtherefore it vanishes.The advanced and Feynman propagators are related by G A ( q ) = G ( q ) + e δ ( q ) , (17)4 ( N ) A q C L ××× × × × L ( N ) q C L × ××× × × Figure 3:
Location of poles and integration contour C L in the complex q -plane for theadvanced (left) and Feynman (right) one-loop integrals, L ( N ) A and L ( N ) . which can straightforwardly be obtained by using the elementary identity1 x ± i (cid:18) x (cid:19) ∓ iπ δ ( x ) , (18)where PV denotes the principal-value prescription. Inserting Eq. (17) into the right-handside of Eq. (14) and collecting the contributions with an equal number of factors G ( q i ) and e δ ( q j ), we obtain a relation between L ( N ) A and the one-loop integral L ( N ) : L ( N ) A ( p , p , . . . , p N ) = Z q N Y i =1 h G ( q i ) + e δ ( q i ) i = L ( N ) ( p , p , . . . , p N ) + L ( N )1 − cut ( p , p , . . . , p N ) + · · · + L ( N )N − cut ( p , p , . . . , p N ) . (19)Here, the single-cut contribution is given by L ( N )1 − cut ( p , p , . . . , p N ) = Z q N X i =1 e δ ( q i ) N Y j =1 j = i G ( q j ) . (20)In general, the m -cut terms L ( N )m − cut ( m ≤ N ) are the contributions with precisely m deltafunctions e δ ( q i ): L ( N )m − cut ( p , p , . . . , p N ) = Z q ne δ ( q ) . . . e δ ( q m ) G ( q m +1 ) . . . G ( q N ) + uneq . perms . o , (21)where the sum in the curly bracket includes all the permutations of q , . . . , q N that giveunequal terms in the integrand.Recalling that L ( N ) A vanishes, cf. Eq. (15), Eq. (19) results in: L ( N ) ( p , p , . . . , p N ) = − h L ( N )1 − cut ( p , p , . . . , p N ) + · · · + L ( N )N − cut ( p , p , . . . , p N ) i . (22)This equation is the FTT in the specific case of the one-loop integral L ( N ) . The FTT relatesthe one-loop integral L ( N ) to the multiple-cut ‡ integrals L ( N )m − cut . Each delta function e δ ( q i ) ‡ If the number of space-time dimensions is d , the right-hand side of Eq. (22) receives contributions onlyfrom the terms with m ≤ d ; the terms with larger values of m vanish, since the corresponding number ofdelta functions in the integrand is larger than the number of integration variables. L ( N )m − cut replaces the corresponding Feynman propagator in L ( N ) by cutting the internalline with momentum q i . This is synonymous to setting the respective particle on shell.An m -particle cut decomposes the one-loop diagram in m tree diagrams: in this sense, theFTT allows us to calculate loop-level diagrams from tree-level diagrams. p p p N p q h i − cut = − N X i =1 p i − p i p i +1 q ˜ δ ( q ) 1( q + p i ) + i The single-cut contribution of the Feynman Tree Theorem to the one-loop N -point scalar integral. Graphical representation as a sum of N basic single-cut phase-spaceintegrals. In view of the discussion in the following sections, it is useful to consider the single-cutcontribution L ( N )1 − cut on the right-hand side of Eq. (22). In the case of single-cut contribu-tions, the FTT replaces the one-loop integral with the customary one-particle phase-spaceintegral, see Eqs. (7) and (20). Using the invariance of the loop-integration measure undertranslations of the loop momentum q , we can perform the momentum shift q → q − P ik =1 p k in the term proportional to e δ ( q i ) on the right-hand side of Eq. (20). Thus, cf. Eq. (2), wehave q i → q and q j → q + ( p i +1 + p i +2 + · · · + p i + j ), with i = j . We can repeat the sameshift for each of the terms ( i = 1 , , . . . , N ) in the sum on the right-hand side of Eq. (20),and we can rewrite L ( N )1 − cut as a sum of N basic phase-space integrals (Fig. 4): L ( N )1 − cut ( p , p , . . . , p N ) = I ( N − − cut ( p , p + p , . . . , p + p + · · · + p N − ) + cyclic perms . = N X i =1 I ( N − − cut ( p i , p i + p i +1 , . . . , p i + p i +1 + · · · + p i + N − ) . (23)We denote the basic one-particle phase-space integrals with n Feynman propagators by I ( n )1 − cut . They are defined as follows: I ( n )1 − cut ( k , k , . . . , k n ) = Z q e δ ( q ) n Y j =1 G ( q + k j ) = Z q e δ ( q ) n Y j =1 qk j + k j + i . (24)The extension of the FTT from the one-loop integrals L ( N ) to one-loop scattering am-plitudes A (1 − loop) (or Green’s functions) in perturbative field theories is straightforward,provided the corresponding field theory is unitary and local . The generalization of Eq. (22)to arbitrary scattering amplitudes is [1, 2]: A (1 − loop) = − h A (1 − loop)1 − cut + A (1 − loop)2 − cut + . . . i , (25)6here A (1 − loop)m − cut is obtained in the same way as L ( N )m − cut , i.e. by starting from A (1 − loop) andconsidering all possible replacements of m Feynman propagators G ( q i ) of its loop internallines with the ‘cut propagators’ e δ ( q i ).The proof of Eq. (25) directly follows from Eq. (22): A (1 − loop) is a linear combinationof one-loop integrals that differ from L ( N ) only by the inclusion of interaction verticesand, eventually, particle masses. As briefly recalled below, these differences have harmlessconsequences on the derivation of the FTT.Including particle masses in the advanced and Feynman propagators has an effect onthe location of the poles produced by the internal lines in the loop. However, as long as themasses are real , as in the case of unitary theories, the position of the poles in the complexplane of the variable q is affected only by a translation parallel to the real axis, with noeffect on the imaginary part of the poles. This translation does not interfere with the proofof the FTT as given in Eqs. (14)–(22). Therefore, the effect of a particle mass M i in a loopinternal line with momentum q i simply amounts to modifying the corresponding on-shelldelta function e δ ( q i ) when this line is cut to obtain A (1 − loop)m − cut . This modification then leadsto the obvious replacement: e δ ( q i ) → e δ ( q i ; M i ) = 2 π i θ ( q i ) δ ( q i − M i ) = 2 π i δ + ( q i − M i ) . (26)Including interaction vertices has the effect of introducing numerator factors in theintegrand of the one-loop integrals. As long as the theory is local, these numerator factorsare at worst polynomials of the integration momentum q § . In the complex plane of thevariable q , this polynomial behavior does not lead to additional singularities at any finitevalues of q . The only danger, when using the Cauchy theorem as in Eq. (16) to provethe FTT, stems from polynomials of high degree that can spoil the convergence of the q -integration at infinity. Nonetheless, if the field theory is unitary, these singularities atinfinity never occur since the degree of the polynomials in the various integrands is alwayssufficiently limited by the unitarity constraint. In this Section we derive and illustrate the duality relation between one-loop integrals andsingle-cut phase-space integrals. This relation is the main general result of the presentwork.Rather than starting from L ( N ) A , we directly apply the residue theorem to the compu- § This statement is not completely true in the case of gauge theories and, in particular, in the case ofgauge-dependent quantities. The discussion of the additional issues that arise in gauge theories is postponedto Sect. 9. L ( N ) . We proceed exactly as in Eq. (16), and obtain L ( N ) ( p , p , . . . , p N ) = Z q Z dq N Y i =1 G ( q i )= Z q Z C L dq N Y i =1 G ( q i ) = − πi Z q X Res { Im q < } " N Y i =1 G ( q i ) . (27)At variance with G A ( q i ), each of the Feynman propagators G ( q i ) has single poles in boththe upper and lower half-planes of the complex variable q (see Fig. 3– right ) and thereforethe integral does not vanish as in the case of the advanced propagators. In contrast, here,the N poles in the lower half-plane contribute to the residues in Eq. (27).The calculation of these residues is elementary, but it involves several subtleties. Thedetailed calculation, including a discussion of its subtle points, is presented in Appendix A.In the present Section we limit ourselves to sketching the derivation of the result of thiscomputation.The sum over residues in Eq. (27) receives contributions from N terms, namely the N residues at the poles with negative imaginary part of each of the propagators G ( q i ), with i = 1 , . . . , N , see Eq. (12). Considering the residue at the i -th pole we writeRes { i − th pole } " N Y j =1 G ( q j ) = (cid:2) Res { i − th pole } G ( q i ) (cid:3) N Y j =1 j = i G ( q j ) { i − th pole } , (28)where we have used the fact that the propagators G ( q j ), with j = i , are not singular at thevalue of the pole of G ( q i ). Therefore, they can be directly evaluated at this value.The calculation of the residue of G ( q i ) is straightforward and gives (cid:2) Res { i − th pole } G ( q i ) (cid:3) = (cid:20) Res { i − th pole } q i + i (cid:21) = Z dq δ + ( q i ) . (29)This result shows that considering the residue of the Feynman propagator of the internal linewith momentum q i is equivalent to cutting that line by including the corresponding on-shellpropagator δ + ( q i ). The subscript + of δ + refers to the on-shell mode with positive definiteenergy, q i = | q i | : the positive-energy mode is selected by the Feynman i G ( q i ). The insertion of Eq. (29) in Eq. (27) directly leads to a representationof the one-loop integral as a linear combination of N single-cut phase-space integrals.The calculation of the residue prefactor on the r.h.s. of Eq. (28) is more subtle (seeAppendix A) and yields " Y j = i G ( q j ) { i − th pole } = " Y j = i q j + i { i − th pole } = Y j = i q j − i η ( q j − q i ) , (30)where η is a future-like vector, η µ = ( η , η ) , η ≥ , η = η µ η µ ≥ , (31)8.e. a d -dimensional vector that can be either light-like ( η = 0) or time-like ( η > η . Note that the calculation of the residue at the pole ofthe internal line with momentum q i changes the propagators of the other lines in the loopintegral. Although the propagator of the j -th internal line still has the customary form1 /q j , its singularity at q j = 0 is regularized by a different i q j + i q j − i η ( q j − q i ), whichwe name the ‘dual’ i η prescription. The dual i / ( q j + i
0) is evaluated atthe complex value of the loop momentum q , which is determined by the location of thepole at q i + i i i i η µ is a consequence of using the residuetheorem. To apply it to the calculation of the d dimensional loop integral, we have tospecify a system of coordinates (e.g. space-time or light-cone coordinates) and select one ofthem to be integrated over at fixed values of the remaining d − η µ with space-time coordinates η µ = ( η , ⊥ , η d − ), the selected systemof coordinates can be denoted in a Lorentz-invariant form. Applying the residue theoremin the complex plane of the variable q at fixed (and real ) values of the coordinates q ⊥ and q ′ d − = q d − − q η d − /η (to be precise, in Eq. (27) we actually used η µ = (1 , )), we obtainthe result in Eq. (30).The η dependence of the ensuing i p p p N p q = − N X i =1 p i − p i p i +1 q ˜ δ ( q ) 1( q + p i ) − i ηp i Figure 5:
The duality relation for the one-loop N -point scalar integral. Graphical represen-tation as a sum of N basic dual integrals. Inserting the results of Eq. (28)–(30) in Eq. (27) we directly obtain the duality relationbetween one-loop integrals and phase-space integrals: L ( N ) ( p , p , . . . , p N ) = − e L ( N ) ( p , p , . . . , p N ) , (32)where the explicit expression of the phase-space integral e L ( N ) is (Fig. 5) e L ( N ) ( p , p , . . . , p N ) = Z q N X i =1 e δ ( q i ) N Y j =1 j = i q j − i η ( q j − q i ) , (33)9nd η is the auxiliary vector defined in Eq. (31). Each of the N − i q i − q j is independent of the integration momentum q :it only depends on the momenta of the external legs of the loop (see Eq. (2)).Using the invariance of the integration measure under translations of the momentum q ,we can perform the same momentum shifts as described in Sect. 3. In analogy to Eq. (23),we can rewrite Eq. (33) as a sum of N basic phase-space integrals (Fig. 5): e L ( N ) ( p , p , . . . , p N ) = I ( N − ( p , p + p , . . . , p + p + · · · + p N − ) + cyclic perms . = N X i =1 I ( N − ( p i , p i + p i +1 , . . . , p i + p i +1 + · · · + p i + N − ) . (34)The basic one-particle phase-space integrals with n dual propagators are denoted by I ( n ) ,and are defined as follows: I ( n ) ( k , k , . . . , k n ) = Z q e δ ( q ) I ( n ) ( q ; k , k , . . . , k n ) = Z q e δ ( q ) n Y j =1 qk j + k j − i ηk j . (35)We now comment on the comparison between the FTT (Eqs. (20)–(24)) and the dualityrelation (Eqs. (32)–(35)). The multiple-cut contributions L ( N )m − cut , with m ≥
2, of the FTTare completely absent from the duality relation, which only involves single-cut contributionssimilar to those in L ( N )1 − cut . However, the Feynman propagators present in L ( N )1 − cut are replacedby dual propagators in e L ( N ) . This compensates for the absence of multiple-cut contributionsin the duality relation.The i η . The basicdual integrals I ( n ) are well defined for arbitrary values of η . However, when computing e L ( N ) , the future-like vector η has to be the same in all its contributing dual integrals(propagators): only then e L ( N ) does not depend on η .In our derivation of the duality relation, the auxiliary vector η originates from the useof the residue theorem. Independently of its origin, we can comment on the role of η inthe duality relation. The one-loop integral L ( N ) ( p , p , . . . , p N ) is a function of the Lorentz-invariants ( p i p j ). This function has a complicated analytic structure, with pole and branch-cut singularities (scattering singularities), in the multidimensional space of the complexvariables ( p i p j ). The i L ( N ) ( p , p , . . . , p N ) asa single-valued function. Each single-cut contribution to e L ( N ) has additional (unphysical)singularities in the multidimensional complex space. The dual i η correlates the various single-cutcontributions in e L ( N ) , so that they are evaluated on the same Riemann sheet: this leadsto the cancellation of the unphysical single-cut singularities. In contrast, in the FTT, thiscancellation is produced by the introduction of the multiple-cut contributions L ( N )m − cut .We remark that the expression (34) of e L ( N ) as a sum of basic dual integrals is justa matter of notation: for massless internal particles e L ( N ) is actually a single phase-space10ntegral whose integrand is the sum of the terms obtained by cutting each of the internallines of the loop. In explicit form, we can write: e L ( N ) ( p , p , . . . , p N ) = Z q e δ ( q ) N X i =1 I ( N − ( q ; p i , p i + p i +1 , . . . , p i + p i +1 + · · · + p i + N − ) , (36)where the function I ( n ) is the integrand of the dual integral in Eq. (35). Therefore, theduality relation (32) directly expresses the one-loop integral as the phase-space integral ofa tree-level quantity. To name Eq. (32), we have introduced the term ‘duality’ precisely topoint out this direct relation ∗ between the d -dimensional integral over the loop momentumand the ( d − L ( N )m − cut (with m ≥
2) contain integrals of expressions that correspond to theproduct of m tree-level diagrams over the phase-space for different number of particles.The simpler correspondence between loops and trees in the context of the duality re-lation is further exploited in Sect. 10, where we discuss Green’s functions and scatteringamplitudes. In this Section we illustrate the application of the FTT and of the duality relation to theevaluation of the one-loop two-point function L (2) . A detailed discussion (including detailedresults in analytic form and numerical results) of higher-point functions will be presentedelsewhere [5] (see also Refs. [3, 4]). p p q + p q Figure 6:
The one-loop two-point scalar integral L (2) ( p , p ) . The two-point function (Fig. 6), also known as bubble function Bub, is the simplestnon-trivial one-loop integral with massless internal lines:Bub( p ) ≡ L (2) ( p , p ) = − i Z d d q (2 π ) d q + i
0] [( q + p ) + i . (37)Here, we have visibly implemented momentum conservation ( p + p = 0) and exploitedLorentz invariance ( L (2) ( p , p ) can only depend on p , which is the sole available invariant). ∗ The word duality also suggests a stronger (possibly one-to-one) correspondence between dual integralsand loop integrals, which is further discussed in Sect. 7. d = 4 − ǫ . Note, however,that we present results for arbitrary values of ǫ or, equivalently, for any value d of space-timedimensions.The result of the one-loop integral in Eq. (37) is well known:Bub( p ) = c Γ ǫ (1 − ǫ ) (cid:0) − p − i (cid:1) − ǫ , (38)where c Γ is the customary d -dimensional volume factor that appears from the calculationof one-loop integrals: c Γ ≡ Γ(1 + ǫ ) Γ (1 − ǫ )(4 π ) − ǫ Γ(1 − ǫ ) . (39)We recall that the i i p ) as a single-value function of the real variable p . In particular, it gives Bub( p ) animaginary part with an unambiguous value when p > p ) = c Γ ǫ (1 − ǫ ) (cid:0) | p | (cid:1) − ǫ (cid:2) θ ( − p ) + θ ( p ) e iπǫ (cid:3) . (40) To apply the FTT and the duality relation, we have to compute the single-cut integrals I (1)1 − cut and I (1) , respectively. Since these integrals only differ because of their i I (1)reg , of the single-cut integral. We define: I (1)reg ( k ; c ( k )) = Z q e δ ( q ) 12 qk + k + i c ( k ) = Z d d q (2 π ) d − δ + ( q ) 12 qk + k + i c ( k ) . (41)Although c ( k ) is an arbitrary function of k , I (1)reg only depends on the sign of the i c ( k ): setting c ( k ) = +1 we recover I (1)1 − cut , cf. Eq. (24),while setting c ( k ) = − ηk we recover I (1) (see Eq. (35)).The calculation of the integral in Eq. (41) is elementary, and the result is I (1)reg ( k ; c ( k )) = − c Γ πǫ ) 1 ǫ (1 − ǫ ) (cid:20) k k − i k c ( k ) (cid:21) − ǫ (cid:2) k − i k c ( k ) (cid:3) − ǫ . (42)Note that the typical volume factor, e c Γ , of the d -dimensional phase-space integral is e c Γ = Γ(1 − ǫ ) Γ(1 + 2 ǫ )(4 π ) − ǫ . (43)The factor cos( πǫ ) in Eq. (42) originates from the difference between e c Γ and the volumefactor c Γ of the loop integral: e c Γ c Γ = Γ(1 + 2 ǫ ) Γ(1 − ǫ )Γ(1 + ǫ ) Γ(1 − ǫ ) = 1cos( πǫ ) . (44)12e also note that the result in Eq. (42) depends on the sign of the energy k . This followsfrom the fact that the integration measure in Eq. (41) has support on the future light-cone,which is selected by the positive-energy requirement of the on-shell constraint δ + ( q ).The denominator contribution (2 qk + k ) in the integrand of Eq. (41) is positive definitein the kinematical region where k > k >
0. In this region the i I (1)reg has no imaginary part. Outside this kinematical region, (2 qk + k )can vanish, leading to a singularity of the integrand. The singularity is regularized by the i I (1)reg ( k ; c ( k )) = − c Γ πǫ ) ( | k | ) − ǫ ǫ (1 − ǫ ) (cid:8) θ ( − k ) [cos( πǫ ) − i sin( πǫ ) sign( c ( k ))]+ θ ( k ) [ θ ( k ) + θ ( − k ) (cos(2 πǫ ) + i sin(2 πǫ ) sign( c ( k )))] (cid:9) . (45)We note that the functions Bub( k ) and I (1)reg ( k ; c ( k )) have different analyticity properties inthe complex k plane. The bubble function has a branch-cut singularity along the positivereal axis, k >
0. The phase-space integral I (1)reg ( k ; c ( k )) has a branch-cut singularity alongthe entire real axis if k <
0, while the branch-cut singularity is placed along the negativereal axis if k > p p q = 1( q + p ) − i ηp ˜ δ ( q ) − q − p ) + i ηp ˜ δ ( q ) − Figure 7:
One-loop two-point function: the duality relation.
We now consider the duality relation (Fig. 7) in the context of this example. The dualrepresentation of the one-loop two-point function is given by e L (2) ( p , p ) = I (1) ( p ) + (cid:16) p ↔ − p (cid:17) , (46)cf. Eqs. (34) and (35). The basic dual integral I (1) ( k ) is obtained by setting c ( k ) = − ηk inEq. (42). Since η µ is a future-like vector, c ( k ) has the following important property:sign( ηk ) = sign( k ) , if k ≥ . (47)Using this property, the result in Eq. (42) can be written as I (1) ( k ) = − c Γ − k − i − ǫ ǫ (1 − ǫ ) (cid:20) − i sin( πǫ )cos( πǫ ) sign( k ηk ) (cid:21) . (48)13omparing this expression with Eq. (38), we see that the imaginary contribution in thesquare bracket is responsible for the difference with the two-point function. However, sincesign( − ηk ) = − sign( ηk ), this contribution is odd under the exchange k → − k and, therefore,it cancels when Eq. (48) is inserted in Eq. (46). Taken together, e L (2) ( p , p ) = I (1) ( p ) + (cid:16) p ↔ − p (cid:17) = − c Γ ( − p − i − ǫ ǫ (1 − ǫ ) , (49)which fully agrees with the duality relation e L (2) ( p , p ) = − Bub( p ). p p q =1( q + p ) + i δ ( q ) − q − p ) + i δ ( q ) − ˜ δ ( q )˜ δ ( q + p ) − Figure 8:
One-loop two-point function: the Feynman Tree Theorem
We now would like to discuss the FTT (Fig. 8) in the case of the two-point function.To this end, we want to check the relations of Eqs. (21)–(24). For the FTT, the two-pointfunction is cast into the form L (2) ( p , p , . . . , p N ) = − h L (2)1 − cut ( p , p ) + L (2)2 − cut ( p , p ) i , (50)where the single-cut and double-cut contributions are L (2)1 − cut ( p , p ) = I (1)1 − cut ( p ) + (cid:16) p ↔ − p (cid:17) , (51)and L (2)2 − cut ( p , p ) = Z q e δ ( q ) e δ ( q + p ) = i Z d d q (2 π ) d − θ ( q ) δ ( q ) θ ( q + p ) δ (( q + p ) ) , (52)respectively. The basic single-cut integral I (1)1 − cut ( k ) of Eq. (51) is obtained by setting c ( k ) = +1 in Eq. (42); we then have I (1)1 − cut ( k ) = − c Γ − k − i − ǫ ǫ (1 − ǫ ) (cid:20) − i sin( πǫ )cos( πǫ ) (cid:2) θ ( − k ) + θ ( k ) sign( k ) (cid:3)(cid:21) . (53)14omparing the individual single-cut results, Eqs. (48) and (53), we see that the imaginarycontributions in the square brackets are different. Inserting Eq. (53) into Eq. (51), thepart of the imaginary contribution that is proportional to sign( k ) cancels (this part is oddunder the exchange k → − k ), while the remaining part does not: L (2)1 − cut ( p , p ) = I (1)1 − cut ( p ) + (cid:16) p ↔ − p (cid:17) = − c Γ ( − p − i − ǫ ǫ (1 − ǫ ) (cid:20) − i sin( πǫ )cos( πǫ ) θ ( − p ) (cid:21) . (54)We see that also the sum of the two single-cut contributions of Eqs. (49) and (54) aredifferent: the difference is due to the replacement of the dual i i p <
0, whereas the two-point function is purely real inthe same region. In the FTT, this difference is compensated by the double-cut contribution L (2)2 − cut .The calculation of the double-cut contribution in Eq. (52) results in L (2)2 − cut ( p , p ) = − i c Γ ( | p | ) − ǫ ǫ (1 − ǫ ) sin( πǫ )cos( πǫ ) θ ( − p ) . (55)Inserting Eqs. (54) and (55) into the right-hand side of the FTT expression of Eq. (50),we find agreement with the result from the direct one-loop computation of the two-pointfunction. 2 i Im h p q i θ ( p ) = ˜ δ ( q )˜ δ ( p − q )Figure 9: One-loop two-point function: the imaginary part.
To conclude this illustration of the FTT, we add a remark. The double-cut contribution L (2)2 − cut is different from the unitarity-cut contribution that gives the imaginary part ofthe bubble function (or, equivalently, the discontinuity of Bub( p ) across its branch-cutsingularity). The imaginary part of the two-point function can be obtained by applyingthe Cutkosky rules (Fig. 9):2 i Im (cid:2) Bub( p ) (cid:3) θ ( p ) = Z q e δ ( q ) e δ ( p − q ) = i Z d d q (2 π ) d − θ ( q ) δ ( q ) θ ( p − q ) δ (( q − p ) ) . (56)We see that the double-cut contributions in Eq. (52) and (56) are different due to thedetermination of the positive-energy flow in the internal lines. Once the energy of the linewith momentum q is fixed to be positive, the on-shell line with momentum q + k has positiveenergy in Eq. (52) and negative energy in Eq. (56). The computation of the double-cut15ntegral in Eq. (56) yields Z q e δ ( q ) e δ ( p − q ) = + i c Γ ( | p | ) − ǫ ǫ (1 − ǫ ) 2 sin( πǫ ) θ ( p ) θ ( p ) , (57)which indeed differs from the expression in Eq. (55). Inserting Eq. (57) in Eq. (56), weobtain the imaginary part of Bub( p ), in complete agreement with the result (40) of theone-loop integral.We also note that the Cutkosky rules in Eq. (56) can be derived in a direct way (i.e.,without the explicit computation of any integrals) from the duality relation. The derivationis as follows. Applying the identity (18) to the dual propagator, we haveIm [ I (1) ( p )] = π Z q e δ ( q ) δ (( q + p ) ) sign( ηp ) . (58)We now use the duality relation to compute the imaginary part of the two-point function,which is given by2 i Im (cid:2) Bub( p ) (cid:3) θ ( p ) = − i θ ( p ) (cid:2) Im I (1) ( p ) + ( p ↔ − p ) (cid:3) . (59)Inserting Eq. (58) in Eq. (59), we obtain2 i Im (cid:2) Bub( p ) (cid:3) θ ( p ) = − π i sign( ηp ) θ ( p ) Z q e δ ( q ) h δ (( q + p ) ) − δ (( q − p ) ) i = − (2 π i ) sign( ηp ) θ ( p ) Z q δ ( q ) δ (( q − p ) ) n θ ( q − p ) − θ ( q ) o , (60)where the first term in the square bracket has been rewritten by performing the shift q → q − p of the integration variable q . The energy constraints in Eq. (60) result in θ ( p ) n θ ( q − p ) − θ ( q ) o = − θ ( q ) θ ( p − q ) . (61)This can be inserted in Eq. (60) to obtain2 i Im (cid:2) Bub( p ) (cid:3) θ ( p ) = sign( ηp ) Z q e δ ( q ) e δ ( p − q ) . (62)We observe that the constraints q = ( p − q ) = 0 and q > , p − q > ηq ) =sign( η ( p − q )) = +1 (see Eq. (47)) and, hence, sign( ηp ) = +1. Therefore Eq. (62) becomesidentical to Eq. (56). The one-loop integral L ( N ) can be expressed by using either the FTT or the duality relation.Comparing Eq. (22) with Eq. (32), we thus derive e L ( N ) ( p , p , . . . , p N ) = L ( N )1 − cut ( p , p , . . . , p N ) + · · · + L ( N )N − cut ( p , p , . . . , p N ) . (63)16his expression relates single-cut dual integrals with multiple-cut Feynman integrals. Ithas been derived in an indirect way, by applying the residue theorem to the evaluation ofone-loop integrals.In this Section we present another proof of Eq. (63). The proof is direct and purelyalgebraic. It further clarifies the connection between the FTT and the duality relation.Our starting point is a basic identity between dual and Feynman propagators. Theidentity applies to the dual propagators when they are inserted in a single-cut integral.Then e δ ( q ) 12 qk + k − i ηk = e δ ( q ) h G ( q + k ) + θ ( ηk ) 2 πi δ (( q + k ) ) i = e δ ( q ) h G ( q + k ) + θ ( ηk ) e δ ( q + k ) i . (64)The equality on the first line of Eq. (64) directly follows from Eq. (18). The equality on thesecond line is obtained as follows. Using the constraint e δ ( q ), we have q = 0 and q > ηq >
0. Using ηq > θ ( ηk ), wehave η ( q + k ) >
0. Combining η ( q + k ) > q + k ) = 0, from Eq. (47) we thus have q + k >
0. This enables the replacement δ (( q + k ) ) → δ + (( q + k ) ), which finally yieldsEq. (64). The relation (64) can be used to prove Eq. (63). We first consider the case N = 2. InsertingEq. (64) in Eq. (35) and comparing with Eqs. (24) and (52), we obtain I (1) ( p ) = I (1)1 − cut ( p ) + θ ( ηp ) Z q e δ ( q ) e δ ( q + p ) = I (1)1 − cut ( p ) + θ ( ηp ) L (2)2 − cut ( p , p ) . (65)We can now use this equation to compute e L (2) : e L (2) ( p , p ) = I (1) ( p ) + I (1) ( p ) = L (2)1 − cut ( p , p ) + h θ ( ηp ) + θ ( ηp ) i L (2)2 − cut ( p , p ) . (66)This relation is equivalent to Eq. (63), since by merely using momentum conservation, p + p = 0, we find θ ( ηp ) + θ ( ηp ) = θ ( ηp ) + θ ( − ηp ) = 1 . (67) N -point function More generally, the identity (64) relates the basic dual integrals I ( n ) with multiple-cutFeynman integrals. Inserting Eq. (64) in Eq. (35) and using Eq. (24), we obtain I ( n ) ( k , k , . . . , k n ) = I ( n )1 − cut ( k , k , . . . , k n ) + I ( n ) η ( k , k , . . . , k n )= I ( n )1 − cut ( k , k , . . . , k n ) + n X m =1 I ( n ) m,η ( k , k , . . . , k n ) , (68)17here I ( n ) m,η ( k , k , . . . , k n ) = Z q e δ ( q ) ne δ ( q + k ) . . . e δ ( q + k m ) G ( q + k m +1 ) . . . G ( q + k n ) × θ ( ηk ) . . . θ ( ηk m ) + uneq . perms . o . (69)Note that the key difference between I ( n ) m,η and the multiple-cut contributions of the FTT(see Eq. (21)) is the presence of the momentum constraints, θ ( ηk i ), in Eq. (69).For a proof in the case of the N -point function, we employ the following relation: I ( N − m − ,η ( p , p + p , . . . , p + p + · · · + p N − ) + cyclic perms . = L ( N )m − cut ( p , p , . . . , p N ) . (70)Summing over the cyclic permutations of I ( N − as in Eq. (34), and using Eqs. (68), (23)and (70), we straightforwardly obtain the relation in Eq. (63).We note that the proof of Eq. (70) is mainly a matter of combinatorics, and it does notrequire the explicit evaluation of any m -cut integral. Eventually, the main ingredient ofthe proof is the following algebraic identity θ ( ηp ) θ ( η ( p + p )) . . . θ ( η ( p + p + · · · + p N − )) + cyclic perms . = 1 . (71)It is a direct consequence of momentum conservation, namely P Ni =1 p i = 0. The derivationof Eq. (71) is presented in Appendix B.To simplify the combinatorics in the proof of Eq. (70), we first rewrite I ( n ) m,η in Eq. (69)as I ( n ) m,η ( k , k , . . . , k n ) = I ( n ) m,F ( k , k , . . . , k n ) + δI ( n ) m,η ( k , k , . . . , k n ) , (72)where I ( n ) m,F ( k , k , . . . , k n ) = 1 m + 1 Z q e δ ( q ) ne δ ( q + k ) . . . e δ ( q + k m ) G ( q + k m +1 ) . . . G ( q + k n )+ uneq . perms . o , (73)and δI ( n ) m,η ( k , k , . . . , k n ) = Z q e δ ( q ) ne δ ( q + k ) . . . e δ ( q + k m ) G ( q + k m +1 ) . . . G ( q + k n ) × (cid:20) θ ( ηk ) . . . θ ( ηk m ) − m + 1 (cid:21) + uneq . perms . (cid:27) . (74)This leaves us with the task to prove the relations I ( N − m − ,F ( p , p + p , . . . , p + p + · · · + p N − ) + cyclic perms . = L ( N )m − cut ( p , p , . . . , p N ) , (75)and δI ( N − m − ,η ( p , p + p , . . . , p + p + · · · + p N − ) + cyclic perms . = 0 . (76)Obviously, Eqs. (72), (75) and (76) imply Eq. (70).18he relation (75) can be proven as follows. According to Eq. (21), L ( N )m − cut is a sum of m -cut contributions with a fully symmetric dependence on the momenta q i of the internallines of the loop integral. The expression on the left-hand side of Eq. (75) is also a fullysymmetric linear combination of m -cut contributions: the symmetrization follows from thesum over the permutations in Eqs. (73) and (75). Hence, owing to their symmetry, theleft-hand side and the right-hand side of Eq. (75) are necessarily proportional, and theproportionality coefficient is just unity. To show this, we can give weight unity to each m -cut contribution and simply count the number of m -cut contributions on both sides ofEq. (75). The number of terms in L ( N )m − cut equals the total number of permutations in thecurly bracket of Eq. (21), namely (cid:18) Nm (cid:19) = N ! m ! ( N − m )! . (77)The number of terms on the left-hand side of Eq. (75) is1 m (cid:18) N − m − (cid:19) N = 1 m ( N − m − N − m )! N , (78)where the factor 1 /m is the weight of each contribution to I ( N − m − ,F , the factor (cid:0) N − m − (cid:1) is thenumber of permutations that contribute to I ( N − m − ,F (see Eq. (73)), and the factor N is thenumber of cyclic permutations in Eq. (75). As we can see, the numbers given by Eqs. (77)and (78) coincide, thus yielding the equality in Eq. (75).The relation (76) can be proven as follows. The left-hand side is a sum of m -cutcontributions of the loop integral L ( N ) . We can organize these contributions in a sum of (cid:0) Nm (cid:1) diagrams as on the right-hand side of Eq. (21): each diagram has m fixed internallines that have been cut. The coefficient of each diagram is computed according to theexpression on the left-hand side of Eq. (76). As discussed below, this coefficient vanishesalgebraically, thus yielding the result in Eq. (76). P Q P Q Q m P m P Figure 10:
A one-loop diagram with m cut lines. Each blob denotes a set of internal linesthat are not cut. We consider one of the diagram with m cut lines, and we denote the momenta of theseinternal lines as Q , Q , . . . , Q m (Fig. 10). We define P i = Q i − Q i − , so that P i is the totalexternal momentum between the cut lines with momenta Q i and Q i − . The computationof the diagram involves the factor e δ ( Q ) e δ ( Q ) . . . e δ ( Q m ) , (79)19nd two other factors. One factor stems from the product of the Feynman propagatorsof the uncut internal lines and it is inconsequential to the present discussion. The otherfactor arises from the term in the square bracket on the right-hand side of Eq. (74). Wenote that δI ( N − m − ,η involves the product e δ ( q ) e δ ( q + k ) . . . e δ ( q + k m − ) of m delta functions,but the term in the square bracket is symmetric only with respect to the argument of m − m different contributions: eachcontribution corresponds to one of the assignments e δ ( q ) → e δ ( Q i ) with i = 1 , , . . . , m . Inconclusion, the square-bracket term contributes to multiply the left-hand side of Eq. (79)by a factor proportional to the following expression: (cid:20) θ ( ηP ) θ ( η ( P + P )) . . . θ ( η ( P + P + · · · + P m − )) − m (cid:21) + cyclic perms . = n θ ( ηP ) θ ( η ( P + P )) . . . θ ( η ( P + P + · · · + P m − )) + cyclic perms . o − . (80)This expression vanishes, because of Eq. (71) and the momentum conservation constraint P mi =1 P i = 0. Therefore, each m -cut diagram of the left-hand side of Eq. (76) has a vanishingcoefficient. One-loop Feynman integrals and single-cut dual integrals are not in a one-to-one corre-spondence. Starting from this observation we discuss in more general terms the nature ofthe correspondence between one-loop and single-cut integrals in this section.Using the duality relation, any one-loop Feynman integral L ( N ) can be expressed as alinear combination of the basic dual integrals I ( N − , but the opposite statement is not true.Therefore, the dual integrals I ( n ) form a linear basis of the functional space generated bythe loop integrals, but overall they generate a larger space containing that of the one-loopFeynman integrals.To express I ( N − as a linear combination of loop integrals, we have to introduce general-ized one-loop integrals, whose integrands contain both Feynman and advanced propagators.We define them through L ( N ) ( p , α , p , α , . . . , p N , α N ) = Z q N Y i =1 G α i ( q i ) , (81)where the label α i can take two values, α i = F, A , and G F ( q i ) = G ( q i ) is the Feynmanpropagator, while G A ( q i ) is the advanced propagator. In particular, when α = α = · · · = α N = F we recover the one-loop Feynman integral in Eq. (9), while we obtain the one-loopadvanced integral in Eqs. (14) and (15) for the case α = α = · · · = α N = A .The relation between I ( N − and the generalized one-loop integrals in Eq. (81) is ob-tained by rewriting the dual propagators as a linear combination of G and G A . Using20qs. (17) and (64) we have: e δ ( q ) 12 qk + k − i ηk = e δ ( q ) h G ( q + k ) + θ ( ηk ) (cid:16) G A ( q + k ) − G ( q + k ) (cid:17) i = e δ ( q ) h θ ( − ηk ) G ( q + k ) + θ ( ηk ) G A ( q + k ) i , (82)which can be inserted in Eq. (35). We thus obtain I ( n ) ( k , k , . . . , k n ) = Z q e δ ( q ) n Y j =1 h θ ( − ηk j ) G ( q + k j ) + θ ( ηk j ) G A ( q + k j ) i = Z q (cid:16) G A ( q ) − G ( q ) (cid:17) n Y j =1 h θ ( − ηk j ) G ( q + k j ) + θ ( ηk j ) G A ( q + k j ) i , (83)where again we have used Eq. (17) to express e δ ( q ) as a linear combination of G ( q ) and G A ( q ). The right-hand side of Eq. (83) is a sum of generalized one-loop integrals. Notethat the η dependence of I ( n ) appears only in the coefficients θ ( ± ηk j ).In the simplest case, with n = 1, Eq. (83) reads: I (1) ( p ) = − θ ( − ηp ) Z q G ( q ) G ( q + p )+ (cid:20) θ ( − ηp ) Z q G A ( q ) G ( q + p ) − θ ( ηp ) Z q G ( q ) G A ( q + p ) (cid:21) (84)= − θ ( − ηp ) L (2) ( p , − p ) + h θ ( − ηp ) L (2) ( p , F, − p , A ) − ( p ↔ − p ) i , where we have used Eq. (15). Note that the term in the square bracket is odd under theexchange p ↔ − p . Therefore the sum I (1) ( p ) + I (1) ( − p ) consistently reproduces theduality relation (i.e., equivalently, it reproduces the two-point function L (2) ( p , − p )).More generally, the linear relation in Eq. (83) implies that the dual integrals I ( N − belong to the functional space that is generated by the generalized one-loop integrals ofEq. (81)Nonetheless, we have not yet established a one-to-one correspondence between single-cut and one-loop integrals. In fact, the correspondence in Eq. (83) is not invertible. Thegeneralized one-loop integrals can be expressed in terms of single-cut integrals by a propergeneralization of the duality relation in Eqs. (32) and (33). However, the single-cut integralsof this generalized relation involve the integration of both dual and advanced propagators.The generalized duality relation is: L ( N ) ( p , α , p , α , . . . , p N , α N ) = − Z q N X i =1 e δ ( q i ) δ α i ,F × N Y j =1 j = i (cid:20) δ α j ,F q j − i η ( q j − q i ) + δ α j ,A G A ( q j ) (cid:21) . (85)21his result can be derived by applying the residue theorem (see Appendix A).Alternatively, Eq. (85) can also be derived by applying an algebraic procedure similarto the one used in Sect. 6 to prove Eq. (63). This procedure consists of rewriting theright-hand side of Eqs. (81) and (85) as multiple-cut integrals of expressions involvingonly advanced propagators. The resulting expressions can be shown to agree with eachother. The rewrite of Eqs. (81) and (85) is achieved by using Eq. (17) to replace Feynmanand dual propagators with advanced propagators. More precisely, in the case of the dualpropagators, Eqs. (17) and (82) give: e δ ( q ) 12 qk + k − i ηk = e δ ( q ) h G A ( q + k ) − θ ( − ηk ) e δ ( q + k ) i . (86)To exemplify this algebraic procedure, we can explicitly show its application to thesimple, though non-trivial, case of the one-loop integral L (3) ( p , F, p , F, p , A ). The right-hand side of Eq. (81) yields Z q G A ( q ) G ( q + p ) G ( q + p + p ) = − Z q G A ( q ) × he δ ( q + p ) G A ( q + p + p ) + e δ ( q + p + p ) G A ( q + p ) − e δ ( q + p ) e δ ( q + p + p ) i , (87)where we have also used Eq. (15). After using Eq. (86), the right-hand side of Eq. (85)reads − Z q G A ( q ) (cid:20) e δ ( q + p ) 1( q + p + p ) − i ηp + e δ ( q + p + p ) 1( q + p ) + i ηp (cid:21) = − Z q G A ( q ) he δ ( q + p ) (cid:16) G A ( q + p + p ) − θ ( − ηp ) e δ ( q + p + p ) (cid:17) + e δ ( q + p + p ) (cid:16) G A ( q + p ) − θ ( ηp ) e δ ( q + p ) (cid:17)i . (88)By simple inspection, we see that the expressions in Eqs. (87) and (88) coincide.The generalized duality in Eq. (85) relates one-loop integrals to single-cut phase-spaceintegrals. Note that only the Feynman propagators of the loop integral are cut; the uncutFeynman propagators are instead replaced by dual propagators. The advanced propagatorsof the loop integral are not cut, and they appear unchanged in the integrand of the phase-space integral.Moreover, the correspondence in Eq. (85) between one-loop and single-cut integrals isinvertible. Using the same algebraic steps as in Eqs. (82) and (83), we indeed obtain: Z q e δ ( q ) m Y j =1 qk j + k j − i ηk j ! k Y i =1 G A ( q + k i )= Z q (cid:16) G A ( q ) − G ( q ) (cid:17) m Y j =1 h θ ( − ηk j ) G ( q + k j ) + θ ( ηk j ) G A ( q + k j ) i k Y i =1 G A ( q + k i ) . (89)The functional space generated by the generalized one-loop integrals is thus equivalent tothe space generated by the single-cut integrals on the left-hand side of Eq. (89). The one-loop integrals of Feynman and advanced propagators and the single-cut integrals of dual22nd advanced propagators can therefore be regarded as equivalent dual basis of the samefunctional space. As discussed at the end of Sect. 3, the introduction of particle masses and massive propa-gators does not lead to difficulties in the generalization of the FTT from the massless case.The same discussion and the same conclusions apply to the duality relation, since this re-lation can be derived by applying the residue theorem in close analogy with the derivationof the FTT. Therefore, as long as the mass is real , the effect of a particle mass M i inthe Feynman propagator of a loop internal line with momentum q i amounts to modifying(according to the replacement in Eq. (26)) the corresponding on-shell delta function e δ ( q i )when this line is cut to obtain the dual representation e L ( N ) (see Eqs. (33) and (85)) ofthe loop integral L ( N ) . Note also that the i j -th internal linehas mass M j , the corresponding dual propagator is1 q j − M j − i η ( q j − q i ) , (90)independently of the value M i of the mass in the i -th line – the cut line.In any unitary quantum field theory, the masses of the basic fields are real. If some ofthese fields describe unstable particles, a proper (physical) treatment of the correspondingpropagators in perturbation theory requires a Dyson summation of self-energy insertions,which produces finite-width effects introducing finite imaginary contributions in the prop-agators. A typical form of the ensuing propagator G C (such as the propagator used in thecomplex-mass scheme † [10]) is G C ( q ; s ) = 1 q − s , (91)where s denotes the complex mass of the unstable particle: s = Re s + i Im s , with Re s > > Im s . (92)These complex masses, together with complex couplings, are introduced in both tree-leveland one-loop Feynman diagrams. A natural question that arises in the context of thepresent paper is whether the duality relation between one-loop and phase-space integrals(and the FTT, as well) can deal with complex-mass propagators or, more generally, withpropagators of unstable particles. The answer to this question is positive, as we discussbelow.We consider a one-loop N -point scalar integral (see Eq. (9)) where one or more ofthe Feynman propagators of the internal lines are replaced by complex-mass propagators † In the complex-mass scheme, unitarity can be perturbatively recovered (modulo higher-order terms)order by order. C ( q i ; s i ). To derive a representation of this one-loop integral in terms of single-cut phasespace integrals, we then apply the same procedure as in Sect. 4. The only difference is thepresence of the complex-mass propagators. In the complex plane of the loop integrationvariable q , the complex-mass propagators produce poles that are located far off the realaxis, the displacement being controlled by the finite imaginary part of the complex masses.Using the Cauchy theorem as in Eq. (27), we derive a duality relation that is analogous toEq. (32). The only difference is that the the right-hand side of Eq. (32) has to be modified: e L ( N ) ( p , p , . . . , p N ) → e L ( N ) ( p , p , . . . , p N ) + e L ( N ) C ( p , p , . . . , p N ) . (93)Here, e L ( N ) denotes the terms that correspond to the residues at the poles of the Feynmanpropagators of the loop integral, while e L ( N ) C denotes those from the poles of the complex-mass propagators. e L ( N ) is thus expressed as e L ( N ) ( p , p , . . . , p N ) = Z q X i ∈ F e δ ( q i ; M i ) " Y j = i . . . , (94)where the sum refers to the internal lines i of the loop with a Feynman propagator (we usethe notation i ∈ F to denote these cut lines). The term in the square bracket denotes theproduct of the propagators of the uncut lines. The Feynman propagators of the loop arereplaced by the corresponding dual propagators (as in Eq. (33)), while the complex-masspropagators are unchanged ‡ .The expression of e L ( N ) C is similar to Eq. (94), but the cut lines i are those with complex-mass propagators (we use the notation i ∈ C to denote these cut lines). Taken together e L ( N ) C ( p , p , . . . , p N ) = Z q X i ∈ C e δ ( q i ; s i ) " Y j = i . . . = Z d d − q (2 π ) d − X i ∈ C p q i + s i " Y j = i . . . q i = √ q i + s i , (95)where the term in the square bracket contains the propagators of the uncut lines. Notethat in the integral representation on the first line of Eq. (95) the ‘on-shell’ delta function e δ ( q i ; s i ) of the cut propagator has a formal meaning, since it singles out the residue at thecomplex-mass pole, q i = q ( C, +) i = p q i + s i , which has a finite (and negative) imaginarypart. The explicit expression of e L ( N ) C is thus given in the second line of Eq. (95). Owing tothe finite imaginary component of q ( C, +) i , we can remove the i ‡ The dual propagators arise from the infinitesimal i L ( N ) A of Eq. (14) with a one-loop integral that contains both advanced propagatorsand complex-mass propagators. This one-loop integral can be rewritten in two differentways. First (exploiting Eq. (17)), it can be expressed, as in the right-hand side of Eq. (19),in terms of a linear combination of the required one-loop integral (i.e. the integral withFeynman and complex-mass propagators) and of multiple-cut phase-space integrals L ( N )m − cut .Alternatively, it can be evaluated directly by applying the Cauchy theorem as in Eq. (16).This direct evaluation leads to the computation of the residues at the poles of the complex-mass propagators (the poles of the advanced propagators do not contribute, since they areplaced outside the integration contour): the computation gives exactly the contribution inEq. (95). Comparing the expressions obtained in these two ways, we conclude that thegeneralization of the FTT to include complex-mass propagators is realized by the followingreplacement in the right-hand side of Eq. (22): L ( N )1 − cut ( p , p , . . . , p N ) → L ( N )1 − cut ( p , p , . . . , p N ) + e L ( N ) C ( p , p , . . . , p N ) . (96)Here, L ( N )1 − cut is the usual contribution (see Eq. (20)) emerging from the single cuts of thesole Feynman propagators of the internal lines (the complex-mass propagators are not cut),while e L ( N ) C is given by Eq. (95). Note, in particular, that the complex-mass propagators do not produce further m -cut contributions ( m ≥
2) to the FTT in addition to the real-massterms L ( N )m − cut in Eq. (21).We add a final comment on one-loop integrals with unstable internal particles. Thepropagator of an unstable particle can have a form that differs from the complex-masspropagator in Eq. (91). We can introduce, for instance, a complex mass, s ( q ), that dependson the momentum q of the propagator. We can also include a non-resonant component,in addition to the resonant contribution of the complex-mass pole. Independently of itsspecific form, the propagator of the unstable particle produces singularities that are locatedat a finite imaginary distance from the real axis in the complex plane of the loop integrationvariable q . Such contributions can be included in the duality relation and in the FTT byperforming the replacements in Eq. (93) and in Eq. (96), respectively. In general, the term e L ( N ) C has a form that differs from Eq. (95) and depends on the actual expression of thepropagator and, in particular, on the singularity structure (poles, branch cuts, . . . ) of thepropagator in the complex plane. The quantization of gauge theories requires the introduction of a gauge-fixing procedure,which specifies the spin polarization vectors of the gauge bosons and the ensuing contentof (possible) compensating fictitious particles (e.g. the Faddeev–Popov ghosts in unbrokennon-Abelian gauge theories, or the would-be Goldstone bosons in spontaneously brokengauge theories).The fictitious particles have their own Feynman propagators, which have to be takeninto account when applying either the FTT or the duality relation. This is done in astraightforward manner: if some internal lines in a one-loop integral correspond to fictitious25articles, they have to be cut exactly in the same way as for physical particles. The multiple-cut phase-space integrals of the FTT and the single-cut phase-space integral of the dualityrelation will include the contributions from the cuts of the Feynman propagators of thesefictitious particles.The impact of the propagators of the gauge particles is more delicate, since they intro-duce ‘gauge poles’. This point is discussed below.The propagator of the (spin 1) gauge boson with momentum q is obtained by multiplyingthe customary Feynman propagator G ( q ) with the tensor d µν ( q ), which arises from the sumof the spin polarizations. The general form of the polarization tensor is d µν ( q ) = − g µν + ( ζ − ℓ µν ( q ) G G ( q ) . (97)The second term on the right-hand side is absent only in the ’t Hooft–Feynman gauge ( ζ =1). In any other gauge, this term is present and the tensor ℓ µν ( q ) propagates longitudinalpolarizations, which are proportional to q µ , or q ν , or q µ q ν . On the one hand, the specificform of ℓ µν ( q ) is not relevant in the context of the following discussion; the only relevantpoint is that ℓ µν ( q ) has a polynomial dependence on the momentum q . On the otherhand, the factor G G ( q ) (we call it ‘gauge-mode’ propagator) has a potentially dangerous,non-polynomial dependence on q and, in particular, it produces poles with respect to themomentum variable q .When considering one-loop quantities in gauge theories, we deal with one-loop integralscontaining gauge boson propagators as internal lines of the loop. Therefore, to derivethe FTT or the duality relation, we have to consider the effect produced by the gaugepolarization tensors. In the ’t Hooft–Feynman gauge the effect is harmless : the polarizationtensor is simply − g µν and factorizes off the loop integration. When applying the Cauchyresidue theorem as in Sects. 3 and 4 in any other gauge, we have to take into account thepossible additional contributions that arise from the presence of the poles of the gauge-mode propagator G G ( q ) (the presence of polynomial terms from ℓ µν ( q ) does not interferewith the residue theorem).We first discuss the case of spontaneously broken gauge theories. Here, the gauge bosonhas a finite mass M , and the form of the gauge-mode propagator G G ( q ) is G G ( q ) = 1 ζ ( q + i − M . (98)Considering the unitary gauge ( ζ = 0), the gauge-mode propagator does not depend on q and factorizes off the loop integration in any of the one-loop integrals. Therefore, the unitary gauge has only inconsequential implications on the use of the FTT and the du-ality relation for one-loop calculations in gauge theories. If we instead consider a genericrenormalizable gauge (or R ζ gauge) with ζ = 0, we see that the gauge-mode propagatorintroduces a pole when q = M /ζ − i
0. This is an additional pole with respect to thephysical pole (when q = M − i
0) from the associated Feynman propagator. For theextension of the FTT and the duality relation of Sects. 3 and 4 to one-loop computationsin the R ζ gauge, one has to properly consider the introduction of additional single-cut andmultiple-cut contributions from gauge-mode propagators. We will not pursue this issue anyfurther in the present paper. 26e now discuss the case of unbroken gauge theories, where the gauge boson is massless.We separately consider two classes of gauges: covariant gauges and physical gauges.In covariant gauges, we have G G ( q ) = 1 q + i . (99)Since the gauge-mode propagator G G ( q ) is equal to the Feynman propagator, the twopropagators together generate a second-order pole when q = − i
0. The extension ofthe FTT and the duality relation of Sects. 3 and 4 to hold for one-loop computations incovariant gauges requires a proper treatment of the contributions from this type of second-order poles § . This issue is not pursued any further in the present paper.In physical gauges, the typical form of the gauge-mode propagator is G G ( q ) = 1( n · q ) k , k = 1 or 2 , (100)where n µ denotes an auxiliary gauge vector. We see that G G ( q ) leads to a (first- or second-order) pole when n · q = 0. In Coulomb gauge we have n µ = (0 , q ), where q is the spacecomponent of the gauge boson momentum q µ = ( q , q ). In axial ( n · A = 0) or planargauges, n µ is a fixed external vector and the pole has to be regularized according to aproper prescription (the precise position of the pole has to be specified by some imaginarydisplacement from the real axis), which we do not specify here, since its specific form hasno effect on the discussion that follows.We now consider a generic one-loop integral, whose integrand contains gauge-modepropagators in addition to Feynman propagators. To derive a duality relation by using theresidue theorem in the complex plane of the variable q (as in Sect. 4), we have to take intoaccount the possible contributions from the poles of the gauge-mode propagators.In Coulomb gauge, the pole of G G ( q ) is located at q = 0. Applying the residue theoremin the q plane at fixed values of q (see Sect. 4 and Appendix A), the gauge pole does notcontribute. We conclude that the gauge-mode propagator remains untouched in going fromthe one-loop integral to its representation as a single-cut dual integral. Note, however, thatthis conclusion follows from having kept q fixed while performing the integration over q .Therefore, the auxiliary future-like vector η µ of the duality relation is necessarily fixed (seeAppendix A) to be η µ = ( η , ), i.e. aligned along the time direction.In axial or planar gauges, the pole of G G ( q ) is located at nq = n q − n d − q d − = 0.Without loosing generality, we can assume n µ = ( n , ⊥ , n d − ) and apply (see Sect. 4)the residue theorem in the complex plane q at fixed values of the coordinates q ⊥ and q ′ d − = q d − − q η d − /η . Setting η d − /η = n /n d − , we have nq = − n d − q ′ d − . Hence, G G ( q ) does not depend on the integration variable q . We conclude that the gauge-modepropagator, including the regularization prescription of its gauge pole, is untouched ingoing from the one-loop integral to its representation as a single-cut dual integral. Note,however, that we have set η d − /η = n /n d − . Therefore, since the vector η µ specifyingthe dual prescription is future-like, the above conclusion is valid only if the gauge vector § Of course, this does not apply to the ’t Hooft–Feynman gauge, where G G ( q ) is absent. µ is either space-like or light-like ( n ≤
0) and, moreover, the dual vector is fixed to be orthogonal to the gauge vector, n · η = 0. These requirements are not fulfilled if n µ is time-like ¶ . The derivation of the duality relation in time-like gauges requires to properly includecontributions from cuts of the gauge-polarization tensors (these contributions depend onthe specific regularization of the gauge poles): this derivation is beyond the scope of thispaper.Our discussion and conclusions regarding the duality relation in physical gauges canstraightforwardly be used to draw similar conclusions on the validity of the FTT. The onlydifference is that in the latter case there is no auxiliary dual vector η µ . To be precise, inCoulomb gauge and in space-like or light-like gauges, the FTT is valid in its customary form,without introducing any multiple-cut contributions stemming from the gauge-polarizationtensors. In time-like gauges, the poles of the gauge-polarization tensors can play a role,and their effect has to be taken into account when applying the FTT.
10 Loop-tree duality at the amplitude level
In the final part of Sect. 3, we have discussed how the FTT can be extended to evaluate notonly basic one-loop integrals L ( N ) but also complete one-loop quantities (such as Green’sfunctions and scattering amplitudes). The same reasoning (see also Sects. 8 and 9) appliesto the extension of the duality relation to the amplitude level.The analogue of Eq. (25) is the following duality relation: A (1 − loop) = − e A (1 − loop) , (101)where A (1 − loop) generically denotes a one-loop quantity. The expression e A (1 − loop) on theright-hand side of Eq. (101) is obtained in the same way as e L ( N ) in Eqs. (32) and (33).We start from any Feynman diagram in A (1 − loop) and consider all possible replacements ofeach Feynman propagator G ( q i ) of its loop internal lines with the cut propagator e δ ( q i ; M i );the uncut Feynman propagators in the loop are then replaced by the corresponding dualpropagators. All the other factors in the Feynman diagrams are left unchanged by goingfrom A (1 − loop) to e A (1 − loop) .The duality relation (101) is valid in any field theory that is unitary and local. Somewords of caution are, however, needed (see the conclusions of Sect. 9) about its applicabilityto theories with local gauge symmetries. In spontaneously broken gauge theories, theduality relation is valid in the ’t Hooft–Feynman gauge and in the unitary gauge. Inunbroken gauge theories, the duality relation is valid in the ’t Hooft–Feynman gauge; it isalso valid in physical gauges specified by a gauge vector n ν , provided the auxiliary dualityvector η µ is chosen such that n · η = 0 (this excludes gauges where n ν is time-like).Equation (101) establishes a correspondence between one-loop Feynman diagrams andthe phase-space integral of tree-level Feynman diagrams. The right-hand side of Eq. (101) ¶ For example, in the axial gauge A = 0, we have nq = n q , and the pole of the gauge-mode propagatordoes not decouple from the integration over q . A (1 − loop) ∼ Z q X P e δ ( q ; M P ) X d . o . f . ( P ) A (tree) P , (102)where P P denotes the sum over the particles that can propagate in the loop internal linesthat are cut, and P d . o . f . ( P ) denotes the sum over the degrees of freedom (such as spin,colors, ..) of the particle P . The integrand A (tree) P is given by the sum of the tree-levelFeynman diagrams that are obtained by cutting the one-loop Feynman diagrams on theleft-hand side.The structure of Eq. (102) implies a natural question k . If A (1 − loop) is the one-loopexpression of a specific quantity A , how is A (tree) P related to the tree-level expression A (tree) of the same quantity A ? In the next subsections, we show how the duality relation can beformulated directly at the amplitude level, when the quantity A is a Green’s function. Wealso discuss the case of on-shell scattering amplitudes. In the following, A N ( p , . . . , p N ) denotes a generic off-shell Green’s function with N externallines (the outgoing momentum of the i -th line is p i ). To be precise, we consider Green’sfunctions that are connected and amputated of the free propagators of the external lines.The tree-level and one-loop expressions of A are A (tree) and A (1 − loop) , respectively. The tree-level scattering amplitude for a given physical process is obtained from A (tree) ( p , . . . , p N )by setting the external momenta on their physical mass shell ( p i = M i , p i ≥ − p i ≥ A (1 − loop) by specifying the renormalization procedure.To simplify the illustration of the duality relation, we first consider the case with onlyone type of massive scalar particles. We thus refer to a theory with a single real scalar field φ ( φ ∗ = φ ) of mass M . The particles are self-interacting through polynomial interactions(e.g. φ or φ ). In this case, the duality relation (102) has the following explicit form: A (1 − loop) N ( p , . . . , p N ) = + 12 Z d d q (2 π ) d − δ + ( q − M ) e A (tree) N +2 ( q, − q, p , . . . , p N ) , (103)where the integrand factor A (tree) on the right-hand side is exactly the tree-level counterpartof the one-loop quantity A (1 − loop) N on the left-hand side. The tree-level counterpart A (tree) N +2 involves two additional external lines with outgoing momenta q and − q .The tilde superscript in e A (tree) denotes the replacement of some of the Feynman propaga-tors with dual propagators. More precisely, to obtain e A (tree) ( q, − q, . . . ) from A (tree) ( q, − q, . . . ),we assign a dual propagator (rather than a Feynman propagator) to each internal line withmomentum q + k j ( k j is a linear combination of the external momenta p i ). We note that this k Issues related to similar questions were discussed by Feynman [2] in the context of the FTT. q µ → q µ − i η µ / (2 ηq ) in the Feynman propagators of A (tree) ( q, − q, . . . ).The momenta q and − q of the two additional external lines of A (tree) N +2 ( q, − q, . . . ) inEq. (103) are on their physical mass-shell: in this respect, A (tree) N +2 is a scattering amplitude(there are no wave-function factors for scalar particles). More precisely, A (tree) N +2 ( q, − q, . . . )is the tree-level physical amplitude that corresponds to the forward-scattering process of aparticle with momentum q in the external field produced by N self-interacting sources (the N external legs).In a theory with different types of particles and antiparticles, the generalization ofEq. (103) is obtained by including a sum over the particle types P . We find: A (1 − loop) N ( . . . ) = + 12 Z d d q (2 π ) d − X P δ + ( q − M P ) σ ( P ) e A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) , (104)where the momenta p i of N external legs are denoted by ‘dots’, since they play no activerole on both sides of the equation. Note that P P includes the sum over both particlesand antiparticles (if P = P ). The coefficient σ ( P ) on the right-hand side of Eq. (104) is aBose–Fermi statistics factor: σ ( P ) = +1 if P is a bosonic particle (e.g. spin 0 Higgs boson,spin 1 gauge boson), and σ ( P ) = − P is a fermionic particle (e.g. spin 1/2 fermion,Faddeev–Popov ghost).As in Eq. (103), e A (tree) ( P ( q ) ← P ( q ) , . . . ) is obtained from A (tree) ( P ( q ) ← P ( q ) , . . . )by the replacement of Feynman propagators with dual propagators. The tree-level ex-pression A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) is the amplitude for the forward-scattering process P ( q ) → P ( q ) in the field of the N external legs. This expression is obtained from theGreen’s function A (tree) N +2 ( P ( q ) , P ( − q ) , . . . ) by setting the momentum q on the physical mass-shell ( q = M P , q ≥
0) and including the proper wave-function factors of the external legswith outgoing momenta q and − q . We can write: A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) = X spin , color , .. h P ( q ) | A (tree) N +2 ( P ( q ) , P ( − q ) , . . . ) | P ( q ) i , (105)where the (‘ket’ and ‘bra’) vectors | P ( q ) i and h P ( q ) | generically denote the (spin-dependent,color-dependent, ...) incoming and outgoing wave-function factors of the forward-scatteredparticle P . The quantum numbers (spin, color, ...) of the incoming and outgoing wavefunctions are fixed to be equal, and the notation P spin , color , .. denotes the coherent sumover them.We illustrate the general notation in Eq. (105) with a few explicit examples: • P = gluon ( λ labels the spin-polarization or helicity states; µ, ν are Lorentz indices; a, b are color indices) yields A (tree) N +2 ( g ( q ) ← g ( q ) , . . . ) = X λ X µ,ν X a,b (cid:0) ε ( λ ) µ ( q ) (cid:1) ∗ (cid:2) A (tree) N +2 ( g ( q ) , g ( − q ) , . . . ) (cid:3) µ νab ε ( λ ) ν ( q )= X µ,ν d µν ( q ) X a,b (cid:2) A (tree) N +2 ( g ( q ) , g ( − q ) , . . . ) (cid:3) µ νab , (106)30here ε ( λ ) ν ( q ) is the gluon-polarization vector and d µν ( q ) = P λ ( ε ( λ ) µ ( q )) ∗ ε ( λ ) ν ( q ) is thecorresponding polarization tensor; • P = massive quark ( s labels the spin; α, β are Dirac indices; i, j are color indices)yields A (tree) N +2 ( Q ( q ) ← Q ( q ) , . . . ) = X s =1 , X α,β X i,j u ( s ) α ( q ) (cid:2) A (tree) N +2 ( Q ( q ) , Q ( − q ) , . . . ) (cid:3) ijα β u ( s ) β ( q )= Tr h ( /q + M ) X i,j (cid:2) A (tree) N +2 ( Q ( q ) , Q ( − q ) , . . . ) (cid:3) ij i , (107)where u ( s ) β ( q ) is the customary Dirac spinor for spin 1/2 fermions; • P = massive anti-quark ( s labels the spin; α, β are Dirac indices; i, j are color indices)yields A (tree) N +2 ( Q ( q ) ← Q ( q ) , . . . ) = − X s =1 , X α,β X i,j v ( s ) α ( q ) (cid:2) A (tree) N +2 ( Q ( − q ) , Q ( q ) , . . . ) (cid:3) ijα β v ( s ) β ( q )= − Tr h ( /q − M ) X i,j (cid:2) A (tree) N +2 ( Q ( − q ) , Q ( q ) , . . . ) (cid:3) ij i , (108)where v ( s ) β ( q ) is the customary Dirac spinor for spin 1/2 anti-fermions.Note that, as stated below Eq. (104), we sum over both particles and antiparticles.However, on the right-hand side of Eq. (104), P P can equivalently be defined to just referto the sum over particles. According to this alternative definition, the antiparticle con-tribution e A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) is absent, and the corresponding particle contribution e A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) must be multiplied by a factor of 2. In view of the issue dis-cussed in Appendix C, the definition of P P as sum over both particle and antiparticlecontributions has to be preferred on general grounds.We recall that, at the level of one-loop computations, the definition of dimensional reg-ularization involves some arbitrariness. Although the loop momentum q µ is d -dimensional,there is still freedom in the definition of the dimensionality of the momenta of the ex-ternal particles and of the number of polarizations of both internal and external par-ticles. As remarked below Eq. (101), the duality relation acts only on the Feynmanpropagators of the loop, leaving unchanged all the other factors in the Feynman dia-grams. Therefore, the dimensional-regularization rules to be used in the tree-level integrand e A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) of Eq. (104) are exactly the same as specified in the definition of A (1 − loop) N ( . . . ).We remark that in Eq. (104) the on-shell integration momentum q µ has always to beconsidered as d -dimensional, with d arbitrary in the sense of dimensional regularization. Inparticular, a d -dimensional on-shell momentum q µ is required also if the one-loop Green’sfunction A (1 − loop) N is finite ∗∗ (i.e. if it has no infrared and ultraviolet divergences) in the ∗∗ If A (1 − loop) N is finite, the d -dimensionality of q µ in Eq. (104) plays simply the role of an intermediatecomputational tool, rather than the role of a necessary regularization procedure. The same intermediate d = 4) of the space-time. The use of a d -dimensional q µ is necessary since, in general, the various terms †† in the integrand on the right-hand sideof Eq. (104) are not separately integrable in a fixed number of space-time dimensions. To extend the discussion of Sect. 10.1 to scattering amplitudes, the only relevant point tobe examined is the on-shell limit of the corresponding Green’s functions (the introductionof the wave-function factors of the external lines is straightforward).Considering the off-shell Green’s function A (1 − loop) N , we introduce the following decom-position: A (1 − loop) N = A (1 − loop; ex . ) N + A (1 − loop; in . ) N , (109)where A (1 − loop; ex . ) N is the contribution from one-loop insertions on the N external lines,while A (1 − loop; in . ) N is the remaining contribution (i.e. one-loop insertions on internal lines).In explicit form, we have A (1 − loop; ex . ) N ( p , . . . , p N ) = N X j =1 A (1 − loop)2 ( p j , − p j ) i D j ( p j ) p j − M j + i A (tree) N ( p , . . . , p N ) (110)where D j ( p j ) is the spin-polarization factor ‡‡ of the particle in the internal line with mo-mentum p j .As is well known, A (1 − loop; ex . ) N cannot directly be evaluated on-shell owing to the kine-matical singularity arising from its external-line propagators (the propagators with momen-tum p j in Eq. (110)). Thus, to calculate the one-loop scattering amplitude, A (1 − loop; ex . ) N has to be first evaluated off-shell, then it has to be renormalized (mass and wave-functionrenormalization), before considering its on-shell limit.In contrast, the one-loop contribution A (1 − loop; in . ) N can directly be computed in the on-shell limit. In particular, we can write a duality relation in the form of Eq. (101): A (1 − loop; in . ) N = − e A (1 − loop; in . ) N . (111)Here, the integrand of the phase-space integral on the right-hand side contains a sumof on-shell tree-level Feynman diagrams (the N external lines are on-shell, and the two computational tool is used in other methods to perform one-loop calculations [9]: the customary reductionof tensor integrals to scalar integrals has to be carried out in terms of d -dimensional one-loop integrals; thecomputation of finite rational terms in one-loop amplitudes can be carried out by exploiting d -dimensionalunitarity techniques. †† Even if some of these terms correspond to the sum of the single cuts of a finite loop integral, each single-cut contribution may not be separately finite. Moreover, possible cancellations of the singularities fromthe various single-cut contributions can be locally (though, not globally) spoiled by the loop-momentumshifts (compare Eqs. (20) or (23) with Eqs. (33) or (34)) that are applied to the separate single-cut terms.In Eq. (104) the momentum shifts are implemented to be able to identify the different cut momenta of theloop with the common external momentum q of the tree-level expression e A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ). ‡‡ To be explicit, D j ( p ) denotes d µν ( p ) (cfr. Eqs. (97) and (106)) if the j -th line refers to a spin 1 particle,whereas D j ( p ) denotes /p + M (cfr. Eq. (107)) if the j -th line refers to a spin 1 / N +2 external legs. Having performed the tree-level computationof the integrand, the result can be integrated over the single-particle phase-space to obtainthe full one-loop term A (1 − loop; in . ) N .We point out that the integrand of the phase-space integral on the right-hand side ofEq. (111) is not equal (modulo the replacement of Feynman with dual propagators) tothe tree-level scattering amplitude with N + 2 external legs. This is because a subset ofthe diagrams that enter the complete tree-level scattering amplitude is not included. Thissubset has been removed by considering only A (1 − loop; in . ) N , i.e. by removing A (1 − loop; ex . ) N from the complete one-loop expression A (1 − loop) N .This ‘missing’ subset of tree-level diagrams can be reinserted in the duality relation.However, as discussed below, this makes more delicate the on-shell limit.We consider the internal-line contribution A (1 − loop; in . ) N before setting the external lineson-shell. We can write the following duality relation: A (1 − loop; in . ) N ( p , . . . , p N ) = + 12 Z d d q (2 π ) d − X P δ + ( q − M P ) σ ( P ) × n e A (tree) N +2 ( P ( q ) ← P ( q ) , p , . . . , p N ) (112) − N X j =1 e A (tree)4 ( P ( q ) ← P ( q ) , p j , − p j ) i D j ( p j ) p j − M j + i A (tree) N ( p , . . . , p N ) o . The derivation of this equation is simple. We first use Eq. (109) to express A (1 − loop; in . ) N as difference of A (1 − loop) N and A (1 − loop; ex . ) N . Then we use Eq. (110) to rewrite A (1 − loop; ex . ) N in terms of A (1 − loop)2 . Finally, we express the full one-loop Green’s functions A (1 − loop) N and A (1 − loop)2 in terms of the duality relation (104).The duality relation (112) involves the phase-space integration of complete tree-levelGreen’s functions, namely A (tree) N ( p , . . . , p N ), and (the duality-propagator version of) A (tree) N +2 ( P ( q ) ← P ( q ) , p , . . . , p N ) and A (tree)4 ( P ( q ) ← P ( q ) , p j , − p j ). The integrand factor inthe curly bracket on the right-hand side is well defined in the on-shell limit. However, thetwo terms in the curly bracket are separately singular in the on-shell limit. The singularityis purely kinematical; it simply arises from the propagators of the lines with momenta equalto the momenta p j of the external lines. Various procedures can be devised to introducean intermediate regularization of the separate singularities, so as to directly evaluate thetwo terms close to on-shell kinematical configurations.
11 Final remarks
Applying directly the Cauchy residue theorem in the complex plane of any of the space-time coordinates of the loop momentum we have derived a duality relation between one-loop33ntegrals and single-cut phase-space integrals. The calculation of the residues is elementary,but introduces several subtleties. The location in the complex plane of the pole of the cutpropagator modifies the original + i N single-cut phase-space integrals,with propagators regularized by a new complex Lorentz-covariant prescription, named dualprescription. It is defined through a future-like auxiliary vector η . This simple modificationcompensates for the absence of multiple-cut contributions that appear in the FTT. Thedependence on η cancels, as expected, in the sum of all the single-cut contributions, leadingto η -independent results.We have generalized the duality relation for internal massive propagators and unstableparticles. Real masses just modify the position of the poles in the complex plane by atranslation parallel to the real axis, and thus do not affect the dual prescription. Unstableparticles introduce a finite imaginary contribution in their propagators. The poles of thecomplex-mass propagators are located at a finite imaginary distance from the real axis, andthe + i Acknowledgements
Financial support by the Ministerio de Educaci´on y Ciencia (MEC) under grants FPA2007-60323 and CPAN (CSD2007-00042), by the European Commission under contracts FLA-VIAnet (MRTN-CT-2006-035482), HEPTOOLS (MRTN-CT-2006-035505), and MCnet(MRTN-CT-2006-035606), by the INFN-MEC agreement and by BMBF is gratefully ac-knowledged.Major fractions of the work were completed while three of us (S.C., F.K., G.R.) wereparticipating in the Workshop
Advancing Collider Physics: from Twistors to Monte Carlos at the Galileo Galilei Institute (GGI) for Theoretical Physics in Florence: we wish to thankthe GGI for its hospitality and the INFN for partial support.S.C. would like to thank Antonio Bassetto, Michael Kr¨amer, Zoltan Nagy and DaveSoper for discussions. We wish to thank Gudrun Heinrich and Christian Schwinn forpointing Ref. [15] out to us.
A Appendix: Derivation of the duality relation
In Sect. 4 we have illustrated the derivation of the duality relation in Eqs. (32) and (33)by using the residue theorem. The derivation is simple. However, it involves some subtlepoints. These points are discussed in detail in this Appendix.Applying the residue theorem in the complex plane of the variable q , the computationof the one-loop integral L ( N ) reduces to the evaluation of the residues at N poles, accordingto Eqs. (27) and (28).The evaluation of the residues in Eq. (28) is a key point in the derivation of the dualityrelation. To make this point as clear as possible, we first introduce the notation q (+) i toexplicitly denote the location of the i -th pole, i.e. the location of the pole with negativeimaginary part (see Eq. (12)) that is produced by the propagator G ( q i ). We further sim-plify our notation with respect to the explicit dependence on the subscripts that label themomenta. We write G ( q j ) = G ( q i + ( q j − q i )), where q i depends on the loop momentumwhile ( q j − q i ) = k ji is a linear combination of the external momenta (see Eq. (2)). There-fore, to carry out the explicit computation of the i -th residue in Eq. (28), we re-label the35omenta by q i → q and q j → q + k j , and we simply evaluate the term h Res { q = q (+)0 } G ( q ) i " Y j G ( q + k j ) q = q (+)0 , (113)where (see Eq. (12)) q (+)0 = p q − i . (114)In the next paragraphs, we follow the steps of Sect. 4 (see Eqs. (29) and (30)) and weseparately compute the residue of G ( q ) and its prefactor – the associated factor arisingfrom the propagators G ( q + k j ).The computation of the residue of G ( q ) givesRes { q = q (+)0 } G ( q ) = lim q → q (+)0 (cid:26) ( q − q (+)0 ) 1 q − q + i (cid:27) = 12 q (+)0 = 12 p q = Z dq δ + ( q ) , (115)thus leading to the result in Eq. (29). Note that the first equality in the second line ofEq. (115) is obtained by removing the i q (+)0 ) − = ( p q − i − becomes singular when q →
0, andthis corresponds to an end-point singularity in the integration over q : therefore the i δ + ( q ).We now consider the evaluation of the residue prefactor (the second square-bracketfactor in Eq. (113)). We first recall that the i q = q (+)0 , the prescription eventually singled out the on-shell mode withpositive definite energy, q = | q | (see Eq. (115)). However, we observe that the resultin Eq. (115) can be obtained by removing (neglecting) the i q (+)0 ( q (+)0 → | q | ) or in G ( q ) ( G ( q ) → /q ):Res { q = q (+)0 } G ( q ) = Res { q = | q |} q = Z dq δ + ( q ) . (116)Hence, the i G ( q ) in Eq. (113). On the basis of this observation, we might assume that the i G ( q + k j ) are not singular when evaluated at the poles of G ( q ).We might thus compute the residue prefactor by removing the i " Y j G ( q + k j ) q = q (+)0 → " Y j q + k j ) q = | q | . (117)36he expression on the right-hand side of Eq. (117) is well-defined, but, when inserted(through Eqs. (113) and (28)) in Eq. (27), it leads to an ill-defined result: the integrationover q is singular at any phase-space points where the denominator factors ( q + k j ) vanish.To recover a well-defined result, we have to reintroduce the i i G ( q + k j )and still keeping q at its on-shell value q = | q | ; then we obtain " Y j G ( q + k j ) q = q (+)0 → " Y j q + k j ) + i q = | q | . (118)Inserting (through Eqs. (113) and (28)) Eq. (115) and the right-hand side of Eq. (118) intoEq. (27), we arrive at a well-defined result for the one-loop integral, since the singularitiesfrom the propagators 1 / ( q + k j ) are now regularized by the Feynman i L − cut , of the FTT. The ensuing contradiction with the FTTcan be resolved only if the total contribution from multiple cuts, L − cut + L − cut + . . . ,to the FTT vanishes; this is obviously unlikely, and it is actually not true as shown by theexplicit one-loop calculations performed in Sect. 5.The discussion of the previous paragraph illustrates that the evaluation of the one-loop integrals by the direct application of the residue theorem (as in Eq. (27)) involvessome subtleties. The subtleties mainly concern the correct treatment of the Feynman i strict computation of the residue prefactor in Eq. (113): the i G ( q + k j ) and q (+)0 has to be dealt with by considering the imaginary part i finite (thus, for instance, 2 i = i i very end of the computation, thus leading to theinterpretation of the ensuing i " Y j G ( q + k j ) q = q (+)0 = " Y j q + k j ) + i q = q (+)0 = Y j q (+)0 k j − q · k j + k j = Y j | q | k j − q · k j + k j − i k j / | q | = "Y j qk j + k j − i k j /q q = | q | . (119)The last equality on the first line of Eq. (119) simply follows from setting q = q (+)0 in theexpression on the square-bracket (note, in particular, that q = − i q (+)0 ≃ | q | − i / | q | (i.e. from expanding q (+)0 at small valuesof i / ( q + k j ) are regularized by the displacement produced by theassociated imaginary amount i k j /q . Performing the limit of infinitesimal values of i i q is positive, in Eq. (119) we can perform the replacement i k j /q → i ηk j , where37 µ is the vector η µ = ( η , ) with η >
0; we finally obtain " Y j G ( q + k j ) q = q (+)0 = "Y j q + k j ) − i ηk j q = | q | , (120)which is the result in Eq. (30) (to be precise, Eq. (30) is recovered by reintroducing theoriginal labels of the momenta of the loop integral according to the replacements q → q i , k j → q j − q i , see the discussion above Eq. (113)).In the following we explain in more detail the origin of the η dependence in the i η µ = ( η , ) (see Eqs. (119) and (120)). Asdiscussed in Sect. 4, different future-like vectors can be introduced by applying the residuetheorem in different systems of coordinates. To clarify this point, we explicitly show theapplication of the residue theorem in light-cone coordinates (see Eq. (6)) rather than inspace-time coordinates (as in Eq. (27)). The one-loop integral can then be evaluated asfollows: L ( N ) ( p , p , . . . , p N ) = Z ( q − , q ⊥ ) Z dq + N Y i =1 G ( q i )= − πi Z ( q − , q ⊥ ) X Res { Im q + < } " N Y i =1 G ( q i ) , (121)where we have applied the residue theorem by closing the integration contour at ∞ in thelower half-plane of the complex variable q + (see Figs. 2 and 3). We can now compute theresidues in Eq. (121) by closely following the analogous computation in Eqs. (113), (115)and (119).The analogue of the term in Eq. (113) is h Res { q + = q (+)+ } G ( q ) i " Y j G ( q + k j ) q + = q (+)+ , (122)where q (+)+ denotes the location (in the q + plane) of the pole with negative imaginary partthat is produced by the propagator G ( q ). Thus (see Eq. (12)), we have q (+)+ = q ⊥ − i q − , with q − > , (123)where the requirement of negative imaginary part leads to the constraint q − > G ( q ) givesRes { q + = q (+)+ } G ( q ) = θ ( q − ) lim q + → q (+)+ (cid:26) ( q + − q (+)+ ) 12 q + q − − q ⊥ + i (cid:27) = θ ( q − ) 12 q − = Z dq + δ + ( q ) . (124)38e see that the residue produces the same factor as in Eq. (115).The residue prefactor is evaluated by using the same procedure as in Eqs. (119) and(120). We obtain " Y j G ( q + k j ) q + = q (+)+ = Y j q (+)+ k j − + 2 q − k j + − q ⊥ · k ⊥ j + k j = "Y j qk j + k j − i k j − /q − q + = q ⊥ /q − = "Y j q + k j ) − i ηk j q + = q ⊥ /q − . (125)The last equality in this equation has been found by performing the limit of infinitesimalvalues of i
0, analogously to Eq. (120). Since q − is positive, we have thus implementedthe replacement i k j − /q − → i ηk j where, in the present case, we have introduced thefuture-like vector η µ = ( η + , ⊥ , η − = 0) with η + = η √ > δ + ( q ), Eqs. (120) and(125) have the same form, although the corresponding auxiliary vectors η µ are different:though η > η is time-like ( η >
0) in Eq. (120), whereas it is light-like( η = 0) in Eq. (125).We also note that the use of the residue theorem in the complex plane q at fixed valuesof q − and q ⊥ leads to a residue prefactor with exactly the same light-like vector η µ as inEq. (125).The main features of the calculation presented in this Appendix are very general: theyare valid in any system of coordinates that can be used to apply the residue theorem. Theresidue of G ( q ) always replaces the Feynman propagator with the corresponding on-shellpropagator δ + ( q ) (see Eqs. (29), (115) and (124)); the residue prefactor generates dualpropagators with an auxiliary vector η that depends on the specific system of coordinatesthat has been actually employed (see Eqs. (30), (120) and (125)).We conclude this Appendix by briefly describing the derivation (by means of the residuetheorem) of the generalized duality relation stated in Eq. (85). The generalized one-loopintegral on the left-hand side contains both Feynman and advanced propagators. Beforeapplying the residue theorem, we can specify how the infinitesimal limit ‘ i →
0’ is per-formed in the two different types of propagators. We rewrite the advanced propagator as G A ( q ) = [ q − iρ sign( q ) ] − and, evaluating the one-loop integral, we perform first thelimit i → ρ ) in the Feynman propagators and then the limit iρ → ∞ in the lower half-plane of the complex variable q , such that the poles of the advancedpropagators do not contribute. Performing the limit i →
0, the Feynman propagators be-have exactly as in the case of the duality relation in Eqs. (32) and (33), while the advancedpropagators remain unchanged (since ρ is kept finite). Finally, we perform the infinitesimallimit iρ →
0. We thus obtain Eq. (85), whereas the advanced propagators have not beenaltered by going from the one-loop integral on the left-hand side to the phase-space integralon the right-hand side. 39
Appendix: An algebraic relation
Here, we provide a proof of the relation (71). More generally, we consider a set of n realvariables λ i , with i = 1 , , . . . , n , that fulfill the constraint n X i =1 λ i = 0 . (126)We shall prove the following relation: θ ( λ ) θ ( λ + λ ) . . . θ ( λ + λ + · · · + λ n − ) + cyclic perms . = 1 . (127)Equation (71) simply follows from setting λ i = η p i and is just a consequence of mo-mentum conservation, namely Eq. (126). Note that the future-like nature of the vector η plays no role in Eq. (71).To present the proof of Eq. (127), we first define the following function F n : F n ( λ , · · · , λ n ) = θ ( λ ) θ ( λ + λ ) . . . θ ( λ + λ + · · · + λ n − ) + cyclic perms . . (128)Then, we proceed by induction. Assuming that Eq. (127) is valid for n − F n − = 1), we shall prove that it is valid for n variables (i.e. F n = 1).The proof is simple. We first note two properties: owing to Eq. (126), at least one of thevariables λ i must have a positive value; F n ( λ , · · · , λ n ) has a fully symmetric dependenceon the n variables λ i . If we can show that F n = 1 when one of the variables, say λ , ispositive, from these two properties it follows that F n is always equal to unity.We consider the various terms on the right-hand side of Eq. (128) and, setting λ > θ ( λ ) θ ( λ + λ ) . . . θ ( λ + λ + · · · + λ n − ) = θ ( λ + λ ) . . . θ ( λ + λ + · · · + λ n − ) , (129) θ ( λ ) θ ( λ + λ ) . . . θ ( λ + · · · + λ n ) = 0 , (130) θ ( λ i ) . . . θ ( λ i + · · · + λ n ) θ ( λ i + · · · + λ n + λ ) θ ( λ i + · · · + λ n + λ + λ ) . . . = θ ( λ i ) . . . θ ( λ i + · · · + λ n ) θ ( λ i + · · · + λ n + λ + λ ) . . . , ( i ≥ . (131)The equality in Eq. (129) simply follows from θ ( λ ) = 1. To obtain Eq. (130), we exploitmomentum conservation to get θ ( λ + · · · + λ n ) = θ ( − λ ), and then we use θ ( − λ ) = 0. Toobtain Eq. (131) we simply use θ ( λ i + · · · + λ n + λ ) = 1, which follows from the presenceof θ ( λ i + · · · + λ n ) and from λ > F n ( λ , λ , · · · , λ n ); the sum of the corresponding terms on the right-hand side gives F n − ( λ + λ , · · · , λ n ) (note that the two variables λ and λ are replaced by the singlevariable λ + λ ). Therefore, we obtain † F n ( λ , λ , · · · , λ n ) = F n − ( λ + λ , · · · , λ n ), andhence F n ( λ , λ , · · · , λ n ) = 1 from the induction assumption. This completes the proof ofEq. (127). † Note that, starting from λ i >
0, we would have obtained F n ( · · · , λ i , λ i +1 , · · · ) = F n − ( · · · , λ i + λ i +1 , · · · ). Starting from λ i <
0, we can analogously obtain F n ( · · · , λ i − , λ i , · · · ) = F n − ( · · · , λ i − + λ i , · · · ). Appendix: Tadpoles and off-forward regularization
The one-loop Feynman diagrams that contribute to a generic quantity include diagramswith tadpoles. Among them there are also ‘1-particle tadpoles’, namely tadpoles linkedto a single line of the diagram (Fig. 11– left ). This single line necessarily corresponds tothe zero-momentum propagation of a particle K with no associated antiparticle (i.e. theparticle K is the quantum of a real bosonic field). If the particle K is massless , its zero-momentum propagator is ill-defined (it gives 1 / (+ i p i q K p i q qK Figure 11:
A one-loop Feynman diagram with a 1-particle tadpole (left), and the tree-leveldiagram that is obtained by cutting the tadpole (right). The black disk denotes a generictree diagram.
In any consistent theories, the diagrams with 1-particle tadpoles linked to a masslessline are considered to be vanishing, by definition. Therefore, they are harmless in any directcomputations at one-loop level: they are simply removed from the set of one-loop diagramsto be computed. However, their effect may appear to be ‘dangerous’ in the context ofloop-tree duality at the amplitude level.To illustrate the origin of the possible ‘danger’, we consider the right-hand side of theduality relation in Eq. (104). Here, the integrand is related to the tree-level forward-scattering amplitude A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ). This amplitude is the full tree-level am-plutude and, therefore, it includes also the tree-level diagrams that are obtained by cut-ting 1-particle tadpoles (see Fig. 11– right ). If the 1-particle tadpole is linked to the ill-defined propagator 1 / (+ i
0) of a massless particle K , the corresponding diagram in thetree-level scattering amplitude is also ill-defined. To make Eq. (104) a well-defined rela-tion, e A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) has to be defined starting from a regularized version of the(possibly ill-defined) amplitude A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ). This regularization procedurehas to be consistent: the only effect that it can eventually produce in the right-hand ofEq. (104) is the cancellation of the terms that correspond to vanishing tadpole diagramsat one-loop level.We introduce a very simple regularization procedure of tadpole-induced (forward-scattering)41ingularities: the two momenta of the on-shell particle P are displaced slightly off-forward.We thus consider the following off-forward scattering amplitude (cf. Eq. (105)): A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) = X spin , color , .. h P ( q ) | A (tree) N +2 ( P ( q ) , P ( − q ) , . . . ) | P ( q ) i , (132)where q = q , although both q and q are on-shell. It is important to note that theexpression in Eq. (132) includes the wave-function factors of the on-shell external lines withmomenta q and q ; in particular, it includes the coherent sum over the spins and coloursof the wave functions of the incoming and outgoing particles P . The possibly ill-definedpropagators 1 / (+ i / (( q − q ) + i
0) when considering A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ).As discussed in Sect. 10, the amplitude e A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) is obtained by startingfrom A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) and replacing Feynman propagators with dual propagators.The off-forward regularization is obtained by starting from the corresponding regularizedversion of A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ). The regularized version is defined as follows: • if P has no corresponding antiparticle, we consider the limit q → q of A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ); • if P has a corresponding antiparticle P , we first combine the particle and antiparticlecontributions and then we consider the limit q → q of the sum A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ) + A (tree) N +2 ( P ( q ) ← P ( q ) , . . . ).The key point of the off-forward regularization is simple: rather than considering theforward-scattering limit at fixed values of the spin and colour, we first sum over spins,colours and, possibly, particle and antiparticle, and then we consider the forward-scatteringlimit.Within the Standard Model of strong and electroweak interactions, the massless parti-cles K that can produce tadpole-induced singularities are gluons and photons (Fig. 12). Weconsider these explicit examples to illustrate how the off-forward regularization consistentlyleads to the cancellation of tadpole-induced singularities.The gluon case is very trivial, since the colour sum on the right-hand side of Eq. (132)directly cancels any tadpole-induced singularities. The cancellation is eventually the con-sequence of colour conservation. To be precise, the coupling P ( q ) P ( q ) g ∗ (see Fig. 12) isproportional to the colour matrix T acc , where a is the color index of the gluon, and c and c are the colour indeces of P ( q ) and P ( q ), respectively. The sum over the colours of theparticle P thus gives Tr( T a ) = 0, independently of the specific case (gluon, quark, ghost,..) of particle P .In the photon case, the particle P is charged and thus P = P . In this case, thecancellation of the tadpole-induced singularity is eventually due to charge conservation,and it is achieved by summing the contributions of P (Fig. 12– left ) and P (Fig. 12– right ).To be precise, we can consider explicitly the three cases: P is a charged scalar, P is acharged vector boson and P is a charged fermion.42 ( q ) P ( q ) p i q qµ g, γ P ( q ) P ( q ) p i − q − qµ g, γ Figure 12:
Off-forward regularization of tree-level diagrams with tadpole-induced singulari-ties: contributions from particle (left) and antiparticle (right) scattering. If P is a charged scalar particle, the couplings P ( q ) P ( q ) γ ∗ and P ( q ) P ( q ) γ ∗ lead tothe factors ( q + q ) µ and − ( q + q ) µ , respectively ( µ is the Lorentz index of the photon).These two factors simply differ by the overall sign, and thus they cancel each other.If P is a charged vector boson, the cancellation occurs as in the case of scalar particles.To be precise, the scalar vertex ( q + q ) µ is replaced by the vertex Γ νµν ( q, q − q, − q ) =( q + q ) µ g νν + . . . , where ν and ν are the Lorentz indeces of the vector bosons P ( q ) and P ( q ), respectively. Including the wave-function polarization vectors of the charged vectorbosons, we can define V ( λ ) µ ( q, q ) ≡ X ν,ν ( ε ( λ ) ν ( q )) ∗ Γ νµν ( q, q − q, − q ) ε ( λ ) ν ( q ) . (133)The couplings P ( q ) P ( q ) γ ∗ and P ( q ) P ( q ) γ ∗ lead to the factors V ( λ ) µ ( q, q ) and − V ( λ ) µ ( q, q ),respectively. Therefore, these two contributions cancel each other for any fixed polarizationstate λ of the vector boson.If P is a charged (massive or massless) fermion, the cancellation takes place after sum-ming over the spin states s = 1 , P ( q ) P ( q ) γ ∗ and P ( q ) P ( q ) γ ∗ produces the factor X s =1 , u ( s ) ( q ) γ µ u ( s ) ( q ) − X s =1 , v ( s ) ( q ) γ µ v ( s ) ( q ) , (134)which identically vanishes.To show that the expression in Eq. (134) vanishes, we use the following relations: X s =1 , u ( s ) α ( q ) u ( s ) β ( q ) = " ( /q + M )(1 + γ )( /q + M )2 p ( q + M )( q + M ) αβ , − X s =1 , v ( s ) α ( q ) v ( s ) β ( q ) = " ( − /q + M )(1 − γ )( − /q + M )2 p ( q + M )( q + M ) αβ , (135)43 s =1 , u ( s ) ( q ) γ µ u ( s ) ( q ) = Tr [ γ µ ( /q + M )(1 + γ )( /q + M )]2 p ( q + M )( q + M ) , − X s =1 , v ( s ) ( q ) γ µ v ( s ) ( q ) = Tr [ γ µ ( /q − M )(1 − γ )( /q − M )]2 p ( q + M )( q + M ) , (136)Tr [ γ µ ( /q + M )(1 + γ )( /q + M )] = − Tr [ γ µ ( /q − M )(1 − γ )( /q − M )] (137)= 4 (cid:20) M ( q + q ) µ + ( q q µ + q q µ ) + 12 g µ ( q − q ) (cid:21) . The two relations in Eq. (135) are directly derived by using the explicit expressions of theDirac spinors u ( s ) and v ( s ) from the solutions of the Dirac equation. The two relations inEq. (136) are obtained from Eq. (135), and Eq. (137) is the result of an elementary com-putation of Dirac γ matrices. Using the relations in Eqs. (136) and (137), we immediatelysee that the expression in Eq. (134) is equal to zero. References [1] R. P. Feynman, Acta Phys. Polon. (1963) 697.[2] R. P. Feynman, Closed Loop And Tree Diagrams, in Magic Without Magic , ed.J. R. Klauder, (Freeman, San Francisco, 1972), p. 355, in
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