JHEP09(2008)065 tion May 6, 2008 August 14, 2008 August 26, 2008 September 11, 2008 phase-space inte- Received: Revised: Accepted: ical calculation of one-loop Published: variant form, compensates for relates one-loop and tree-level lation is realized by a modifica- sit`adi Firenze, ross-sections at next-to-leading e Feynman Tree Theorem. The ies in any relativistic, local and ersity, `encia, Published by Institute of Physics Publishing for SISSA 0 prescription of the Feynman propagators. The new prescrip i NLO Computations, QCD. We derive a duality relation between one-loop integrals and [email protected] [email protected] [email protected] [email protected] [email protected] INFN, Sezione di FirenzeI-50019 and Sesto Dipartimento Fiorentino, di Florence, Fisica, Italy E-mail: Univer Stanford Linear Accelerator Center, StanfordStanford, University, CA 94309, U.S.A. E-mail: Institute for Particle PhysicsDurham Phenomenology, DH1 Durham 3LE, Univ U.K. E-mail: Instituto de Corpuscular, F´ısica CSIC-Universitat deApartado Val de Correos 22085, E-46071E-mail: Valencia, Spain Fermi National Accelerator Laboratory, Batavia, IL 60510, U.S.A. E-mail: SISSA 2008 c Stefano Catani Tanju Gleisberg Frank Krauss Rodrigo Germ´an Jan-Christopher Winter Abstract: grals emerging from them through singletion cuts. of The the duality re customary +

regularizing the propagators, which wethe write absence in of a multiple-cut Lorentz contributionsduality co that relation appear can in be th appliedunitary to field generic theories. one-loop WeGreen’s quantit discuss in functions. detail We the commentscattering duality on amplitudes, that applications and to to theorder. the numerical analyt evaluation of c Keywords: From loops to trees by-passing Feynman’s theorem JHEP09(2008)065 2 3 5 8 19 18 13 15 30 33 12 35 36 41 42 21 26 s 24 – 1 – -point function N 6.2 General 6.1 Two-point function 5.1 General form5.2 of single-cut integrals Duality relation5.3 for the two-point FTT function for the two-point function 14 10.1 Green’s functions 10.2 Scattering amplitudes 2. Notation 3. The Feynman theorem 4. A duality theorem 5. Example: the scalar two-point function 6. Relating Feynman’s theorem and the duality theorem 18 Contents 1. Introduction 11. Final remarks A. Derivation of the duality relation B. An algebraic relation C. Tadpoles and off-forward regularization 7. Dual bases and generalized duality 8. Massive integrals, complex masses and9. unstable particle Gauge theories and gauge poles 10. Loop-tree duality at the amplitude level 29 JHEP09(2008)065 duality he original 1 d unitary) quantum lustrated in section 5 uce our notation. In ute multi-leg one-loop ce integrals of tree-level ey difference is that the of space-time dimensions. lts of this publication: we ue theorem. d uires to properly regularize in section 6. In section 7, we etails about the derivation of agrams are then obtained by . Since the FTT has recently ge theories. In section 10, we eralized duality relation. The grals with multiple-cut phase- elated to tadpole singularities sed in appendix A. The proof een’s functions and scattering le application. The correspon- rals. Using this basic relation nstable particles) when cutting ynman integrals and single-cut which is different from the cus- functions at the loop level with lyze the effect of the gauge poles point of this method is a rals. Although the analogy with p integrals and single-cut phase- nspired methods [7, 8] to evaluate ws from a basic and more elemen- respondence (including similarities heorem has been considered many times – 2 – cuts of the one-loop Feynman diagrams. Both the single cuts (single cuts, double cuts, triple cuts and so forth) of t multiple 0 prescription of the Feynman propagators. The duality is il i The outline of the paper is as follows. In section 2, we introd In this paper, we illustrate and derive the duality relation We have recently proposed a method [3 – 5] to numerically comp Within the context of loop integrals, the use of the residue t 1 section 3, we briefly recallspace how the integrals. FTT relates In one-loopderive section inte and 4, illustrate we the presentspace duality one relation integrals. between of one-loo the Wepropagators main also by resu prove a complex thattomary Lorentz-covariant the prescription, + duality relation req attracted a renewed interest [6] inone-loop the scattering context amplitudes of [9], twistor-i we alsoand discuss differences) its cor with the duality relation. by considering the two-point functiondence as between the the simplest FTT examp andexplore the the duality relation one-to-one is correspondence formalized betweenintegrals one-loop on Fe more mathematical grounds,treatment and of establish particle a masses (including gen loop complex masses integrals is of discussed u in sectionintroduced 8. by the In propagators section of 9,discuss the we the ana gauge fields extension in of localamplitudes. the gau Some duality final relation remarks to arethe presented one-loop duality in Gr section relation 11. by using D of the an residue algebraic theorem relation areare is discus discussed presented in in appendix appendix C. B. Issues r cross sections in perturbative fieldrelation theories. between one-loop The integrals starting andthe phase-space FTT integ is quiteduality close, relation involves there only areFTT and important the differences. duality relation can The be k derived by using the resid loop Feynman diagram. in textbooks and in the literature. It relates perturbative scatteringanalogous amplitudes quantities and at Green’s the treetary level. relation This between relation loop follo loop integrals Feynman and diagrams phase-space canFeynman integ diagrams. be The rewritten corresponding in tree-levelconsidering Feynman terms di of phase-spa 1. Introduction The Feynman Tree Theorem (FTT) [1, 2] applies to any (local an field theories in Minkowsky space with an arbitrary number JHEP09(2008)065 is 1). 0 − k (2.2) (2.1) (2.3) . i p ≡ ,..., i 1 + − ), where , N p k , ta of the internal 0 k and are clockwise , i.e. µ N is not fixed and does = ( , N d µ = diag(+1 0 k i µν 1 ,...,p + continuation in the number g ynman diagrams, namely a µ 2 2 i ss of generality by considering -point scalar integral. less internal lines only. Thus, q notation and the presentation, ,p N ltraviolet or infrared divergent, µ 1 (the convention for the Lorentz- herefore, p N =1 2 d i Y p d 3 ) q , p d 2 π 2) external legs. q d are denoted as k . (2 p ≥ µ k Z i 2. Throughout the paper we consider loop =1 N 1 k X = 0 ( i q √ i / − + p ) – 3 – N 1 q q 1 − 1) with metric tensor N =1 p )= i d X = k − N i with q ± ) N 0 N . It is also convenient to introduce light-cone coordinates ( q k µ N L k ,...,p ,...,d p 2 1 = ( , ,p 1 ± can in general be expressed as: p k = 0 ( ) ) of the external momenta is defined modulo N µ ; they are given by ( N i ( µ i L q L Momentum configuration of the one-loop ), where − , k ⊥ is the loop momentum (which flows anti-clockwise). The momen , k Figure 1: + µ k q The momenta of the external legs are denoted by The number of space-time dimensions is denoted by = ( µ 2. Notation The FTT and the duality relation cantheir be application illustrated with to no lo generic the one-loop basic scalar ingredient integral of any one-loop Fe ordered (figure 1). All arewe taken as also outgoing. limit To ourselves simplify the inthe one-loop the integral beginning to considering mass and momentum conservation results in the constraint lines are denoted by where The space-time coordinates of any momentum The value of the label integrals and phase-space integrals.we always If assume the that integrals theyof are are space-time regularized u dimensions by (dimensional usingnot regularization). analytic necessarily T have integer value. indices adopted here is the energy (time component) of k JHEP09(2008)065 0 + q q , we (2.5) (2.7) (2.4) (2.8) (2.6) (2.9) 2 ⊥ (2.12) (2.11) (2.13) (2.10) q − − 0) reads q , ≥ + . q , 0 q 0 i · · · 0 i = 2 + ) ∓ 2 − ⊥ q = 0, ∓ , q 2 ⊥ q 2 , q 2 2 ⊥ ), the pole with positive q , er in the position of the q 2 − q ± − q ( q , 2 . 0 · · · ≃ Z in eq. (2.1) can be cast into = q 2 he momentum coordinate · · · ) ) ) ± i ± q placed below (above) the real + 2 q N ical massless particle with mo- q q = and to only one pole in the plane gy) of the advanced propagator Z ) q ( . ( ( dq q 0 L 2 0 ( q G + or q or e δ , Z , . dq q · · · N =1 Z i . Y 0 0 0 π i δ Z i i q q q Z 0 0 + ), i − ···≡ Z i q 2 1 2 ( 1 ···≡ ) = 2 + q − q ···≡ A 2 ⊥ 2 d (or, equivalently, 2 )= q ) q p ) q ···≡ G p ( q 2 0 2 N π d δ q q q – 4 – ± − ) 1 ≡ ( d ) ≡ d (2 π δ ) − 0 q ≃± ) d = ) d q d π ) q (2 q 0 ( d 0 ( d 0 ( q q (2 q Z ,...,p G A ( 2 Z − θ G πiθ Z ,p 0 2 1 dq 1 i ⇒ ⇒ − p dq q − ≡ d ( ∞ ) ) d + ) ∞ d π N −∞ q ( + ( Z −∞ (2 e δ L Z + =0 = =0 = i Z 1 1 dq − − − ∞ )] )] q + q ( −∞ ( A Z G [ i G [ − (i.e. an on-shell particle with positive-definite energy: ) denotes the customary Feynman propagator, q q ( G ). , we write We introduce the following shorthand notation: Using this shorthand notation, the one-loop integral − + To be precise, each propagator leads to two poles in the plane q q 2 therefore have and and We recall that theparticle Feynman and poles advanced in propagators the only complex diff plane (figure 2). Using (or or where we have defined (negative) energy of theaxis, Feynman while propagator both is poles slightly (independentlyare of dis slightly the displaced sign of above the the ener real axis. When we factorize off in a loop integral the integration over t Thus, in the complex plane of the variable We also introduce the advanced propagator respectively. The customary phase-spacementum integral of a phys where JHEP09(2008)065 r 0 q are (3.1) (3.2) (3.3) (3.4) 0 q . . Since 0 q = 0 ) plane ± # q ) ( i 0 q e variable ( q A × , which is obtained ) y using the Cauchy G N ( A nced (right) propagators, N =1 L i Y ) with the corresponding l i " . q ( } nergy component ) 0 G i . We have < q L 0 × ( q C A . . Im G ± , { q ) integration can be performed along ) q N =1 ( q or i Y ( 0 Res A ) = 0 0 e q δ q q G N Z X )+ ) q i ), which are located in the upper half-plane. ( i , the q )= q ( q ,...,p Z G N ( – 5 – A 2 A G ,p πi in the lower half-plane of the complex variable G )= 1 → ∞ 2 q p N =1 ( ) plane ( i Y 0 − ,...,p ∞ ) A q ± 2 q N G ( A ( 0 ,p 0 )= 1 L i q dq p q ( ( ) × ): A Z i N ( A q G ( L q A Z N =1 is then equal to the sum of the residues at the poles in the lowe i Y G L )= 0 C N dq × L , which is closed at C L ), in the complex plane of the variable Z ). The only singularities of the integrand with respect to th q C ( ) ,...,p Location of the particle poles of the Feynman (left) and adva q q 2 A in eq. (2.9) by replacing the Feynman propagators ( Z left ) G ,p G 1 N = ( p ( L ) To this end, we first introduce the advanced one-loop integra The proof of eq. (3.2) can be carried out in an elementary way b The advanced and Feynman propagators are related by ) and N ( A q ( L Figure 2: G 3. The Feynman theorem In this section we briefly recall the FTT [1, 2]. from advanced propagators Then, we note that the contour residue theorem and choosing a suitable integration path (figure 3– The loop integral is evaluated by integrating first over the e the poles of the advanced propagators the integrand is convergent when half-plane and therefore it vanishes. The integral along JHEP09(2008)065 (3.6) (3.5) (3.7) (3.8) (3.9) 0 delta ) and . q i , . q m ) ( that give N i G o L . ) C N N × ,...,p perms 2 . ,...,q : 1 × ,p ) ,...,p q 1 2 N . p ( -plane for the advanced ( 0 ,p L ) q 1 j p cut ) q × ( − ( ) + uneq N ons of × ( N G cut N ) L tary identity q i × − ( N 6= =1 ( N , N + j j Y ual number of factors ing eq. (3.4) into the right-hand L ) ) × i . x · · · + ) ...G ) ( q ( ) N N ( ( e δ in the complex · · · )+ L +1 L iπ δ L N m N C =1 ∓ q i )+ X ( and N  G ) q 1 x ) i Z N ) are the contributions with precisely ,...,p ( A )  i 2 m L – 6 – N q q 0 and the one-loop integral ( ( ,p q )= ,...,p e δ e δ 1 ≤ 2 ) N p N ( ,p ( A = PV m . . . L 1 )+ ( L ) i p ) cut C 0 1 q ( i − ( N q ( 1 cut ( ,...,p 1 ) ± G × ) cut 2 e δ L − h N − N x n ( m ,p ( 1 1 L L N =1 )+ q p i Y ( h Z N q vanishes, cf. eq. (3.2), eq. (3.6) results in: − ) cut Z ) − N )= ( 1 N ( A N )= L ,...,p )= L 2 N N × -cut terms ,p 1 m p ,...,p ( ): 2 × ×× × ) i ,...,p ) q ,...,p 2 ,p N Location of poles and integration contour ( ( 2 N 1 e δ ( A ,p p L ,p ( 1 L 1 p = p ( cut ) ( ) ) − Recalling that ), we obtain a relation between N N ( j N ( m ( A q L L ( L e δ (left) and Feynman (right) one-loop integrals, Figure 3: Here, the single-cut contribution is given by which can straightforwardly be obtained by using the elemen where PV denotes theside principal-value of prescription. eq. Insert (3.1) and collecting the contributions with an eq In general, the functions where the sum in the curly bracket includes all the permutati unequal terms in the integrand. JHEP09(2008)065 ) k ly i p q 0 . ( i e δ =1 -point (3.11) (3.10) i k + ell. An N 2 . P ) 1 contribu- i ) − p 2 . q − + aightforward, 0 N +1 q → i i i ( + p p i q + p . The FTT relates 2 j ) he one-loop + ) + cyclic perms k 1 N 1 ( − + to one-loop scattering · · · L by cutting the internal ) j N q ) ) p + ( Feynman propagators by qk N N ˜ . We can repeat the same δ ( . Each delta function ( 2 + q j +1 n L L ful to consider the single-cut i tegral cut p n 6= =1 ) · · · h mary one-particle phase-space Y − j vel diagrams. i N 1 + ( m + ) − i tree diagrams: in this sense, the i L q he loop-integration measure under 2 ( p p e δ vanish, since the corresponding number of m q + N =1 integration variables. Z i m X basic single-cut phase-space integrals. 1 ), with ,...,p j + N +1 i − )= i integrals p j p k 3 ,...,p + + 2 + i p – 7 – = ) in the sum on the right-hand side of eq. (3.7), basic phase-space integrals (figure 4): q · · · ( ,p + i , the right-hand side of eq. (3.9) receives contributions on + G N d p 1 ( ,p n =1 1) +2 , we can perform the momentum shift 1 Y i j q cut − ,...,N cut p p ( ) 2 − − N 1 ( , q 1 2 + 1) ( i I ) on the right-hand side of eq. (3.7). Thus, cf. eq. (2.2), we p i − e δ cut q +1 q 3 − = 1 N ( i ( 1 p N =1 Z e δ i p i X I as a sum of to the multiple-cut ) = + ( )= N ) cut ; the terms with larger values of . This is synonymous to setting the respective particle on sh ) = n ( q d i − N L q ( 1 N (or Green’s functions) in perturbative field theories is str ≤ on the right-hand side of eq. (3.9). In the case of single-cut → L j m q q ) cut 1 ,...,k loop) 2 p − N ,...,p − ( 1 2 , k (1 L and 1 The single-cut contribution of the Feynman Tree Theorem to t ,p A k 1 q ( replaces the corresponding Feynman propagator in p ( N cut → ) cut p ) − n i ) cut . They are defined as follows: ( − 1 q − N N I In view of the discussion in the following sections, it is use The extension of the FTT from the one-loop integrals ( m h ( 1 If the number of space-time dimensions is cut ) L 3 L -particle cut decomposes the one-loop diagram in − n ( 1 from the terms with delta functions in the integrand is larger than the number of Figure 4: in the term proportional to shift for each of the terms ( have This equation is the FTT in the specific case of the one-loop in line with momentum tions, the FTT replaces theintegral, one-loop see eqs. integral (2.7) with and the (3.7).translations custo Using of the the invariance loop of momentum t contribution FTT allows us to calculate loop-level diagrams from tree-le the one-loop integral scalar integral. Graphical representation as a sum of in m We denote the basicI one-particle phase-space integrals wit and we can rewrite amplitudes JHEP09(2008)065 and th no (3.12) (3.13) loop) in a loop i − ) to prove (1 M A . in the complex ) 2 i M larities at any finite ) of its loop internal i − q is a linear combination , gators has an effect on ( the convergence of the 2 i q G ( i + g numerator factors in the loop) In the complex plane of the − . . . e various integrands is always unitary, these singularities at es not interfere with the proof local, these numerator factors of a particle mass . This modification then leads (1 e loop. However, as long as the + 4 π i δ . The generalization of eq. (3.9) . A , these differences have harmless ories and, in particular, in the case of q n between one-loop integrals and loop) cut n general result of the present work. , i.e. by starting from loop) − local − ) = 2 cut − 2 m (1 i − cut 2 (1 ) A − M and N A ( m nal issues that arise in gauge theories is postponed − L + 2 i q only by the inclusion of interaction vertices ( δ ) Feynman propagators loop) – 8 – cut ) unitary N − 0 ( − i m 1 (1 q L ( A h ). i πiθ q − ( simply amounts to modifying the corresponding on-shell e δ = i q ) = 2 i loop) M − ; i is affected only by a translation parallel to the real axis, wi (1 q ( 0 A e δ q → ) when this line is cut to obtain i ) q i is obtained in the same way as ( q , as in the case of unitary theories, the position of the poles e δ ( e δ . The only danger, when using the Cauchy theorem as in eq. (3.3 loop) real cut 0 , this polynomial behavior does not lead to additional singu − − q 0 q m (1 A Including interaction vertices has the effect of introducin The proof of eq. (3.12) directly follows from eq. (3.9): Including particle masses in the advanced and Feynman propa This statement is not completely true in the case of gauge the 4 -integration at infinity. Nonetheless, if the field theory is 0 the FTT, stems fromq polynomials of high degree that can spoil 4. A duality theorem In this section we derive and illustrate the duality relatio infinity never occur since thesufficiently degree limited of by the the polynomials unitarity in constraint. th single-cut phase-space integrals. This relation is the mai values of where to arbitrary scattering amplitudes is [1, 2]: delta function effect on the imaginary partof of the the FTT poles. as This given in translation eqs. do (3.1)–(3.9). Therefore, the effect internal line with momentum considering all possible replacements of variable lines with the ‘cut propagators’ to the obvious replacement: integrand of the one-loop integrals.are As at long worst as polynomials the theory of is the integration momentum gauge-dependent quantities. The discussion ofto the section additio 9. provided the corresponding field theory is of one-loop integrals that differ from plane of the variable and, eventually, particle masses. As briefly recalled below consequences on the derivation of the FTT. the location of the polesmasses produced are by the internal lines in th JHEP09(2008)065 N (4.2) (4.4) (4.3) (4.1) (4.5) ), with i q nding on- ( . , epresenta- , G } # ) . i ) 0 prescription i i q ) and therefore q ) ( 2 i − q th pole G j ( q − right + i ( , are not singular at 1 { δ terms, namely the N =1 i η i Y     0 0 ) " N i 6= q. (4.1). j dq , q ( 4.2) is more subtle (see ) has single poles in both } j − ( i 0 lves several subtleties. The q 0 Z om pagators. In contrast, here, 2 j G the propagators < ( n propagator of the internal rivation of the result of this q 0 i q -th pole we write ≥ = G i i 6= =1 N j j Y µ  6= single-cut phase-space integrals. Im (see figure 3– j Y η 0 { ), with     i 0 µ j points, is presented in appendix A. N q η = q  1 + ( ) Res } i = 2 i G q q 2 ( } G X , η th pole } ) 0 refers to the on-shell mode with positive def- − i q i q { Z ≥ + ( th pole   δ 0 − – 9 – G i 0 th pole ) is straightforward and gives πi i i { − 2 q i N =1 ( 1 { + i Y − Res G 2 j , η q  , we directly apply the residue theorem to the compu- Res 0 ) ) i  )= = i N 6= dq ( A , η q j Y  = ( 0 ) L i Z η   G   q ) ( ). Therefore, they can be directly evaluated at this value. j = i = ( q q G N =1 q i Y ( } Z ( µ } G η G vector, ), each of the Feynman propagators is equivalent to cutting that line by including the correspo 0 i ). The insertion of eq. (4.3) in eq. (4.1) directly leads to a r i ). The subscript + of )= : the positive-energy mode is selected by the Feynman q i =1 N q | 2 i ( dq th pole Y j q i N th pole q ( A − L ( q i −   | i { C G G + { Z δ }   = ) j q 0 . We proceed exactly as in eq. (3.3), and obtain ,...,p i q Res ) Z 2 future-like q (  , see eq. (2.12). Considering the residue at the N th pole ( G ,p = 1 − i L i is a p { 6= ( j Y ) poles in the lower half-plane contribute to the residues in e η N   ,...,N ( Res Rather than starting from The sum over residues in eq. (4.1) receives contributions fr The calculation of these residues is elementary, but it invo The calculation of the residue of The calculation of the residue prefactor on the r.h.s. of eq. N L = 1 tation of where we have used the fact that the propagators the integral does not vanishthe as in the case of the advanced pro detailed calculation, including a discussionIn of the its subtle present sectioncomputation. we limit ourselves to sketching the de where shell propagator residues at the poles with negative imaginary part of each of the upper and lower half-planes of the complex variable This result shows thatline considering with the momentum residue of the Feynma the value of the pole of inite energy, At variance with of the propagator tion of the one-loop integral as a linear combination of appendix A) and yields i JHEP09(2008)065 i 0 ⊥ i 0) ηp q 0 i (4.7) (4.6) > )), we − 2 0 1 ), which , η 2 i ) i q p − = (1 + j 0 prescription +1 i i µ i q q ( p η p ( , η ) 0 0) is evaluated at i i i q ), the selected system − + 1 − , 2 j ) 2 − j j q q ) d endence as given by the q q ( ( ( ˜ N δ 1 1 coordinates. Introducing , η / q η is (figure 5) 0 prescription: the original = 0) or time-like ( ⊥ pplying the residue theorem 0 i − ) i ant quantities. The Lorentz- 0 2 ctly obtain the duality relation ing over all the residues. d , N ) values of the coordinates η ( − 1 0 ,...,p η e − L 2 j 2 i one coordinates) and select one of q p real ,p = ( i 1 is a consequence of using the residue 6= =1 N p µ prescription. The dual j j Y dimensional loop integral, we have to ( η µ ) η ) η d i N ( q N =1 i ( X -point scalar integral. Graphical representation e L e δ N , which is determined by the location of the − -th internal line still has the customary form − q j N =1 i – 10 – X at fixed (and 0 prescription is thus a consequence of the fact )= i q 0 N q Z changes the propagators of the other lines in the loop = i )= q 2 ,...,p N . Note that the calculation of the residue at the pole of 2 p 0 η (to be precise, in eq. (4.1) we actually used 3 ,p 1 0 is modified in the new prescription p 0 dependence from the pole has to be combined with the 0 i i = 0 is regularized by a different p ( ,...,p /η ) 2 + j 2 1 q N 2 j − ( with space-time coordinates ,p q d 1 L η µ p 0 prescription or, briefly, the 0 η ( i q ) N q − 1 ( p e 1 L 0 = 0. The − basic dual integrals. i value of the loop momentum d q dependence of the ensuing The duality relation for the one-loop N + = η 2 i N q 1 -dimensional vector that can be either light-like ( p − d 0 prescription. The presence of the vector ′ d i q , its singularity at Inserting the results of eq. (4.2)–(4.4) in eq. (4.1) we dire The complex 2 j /q the auxiliary vector theorem. To apply it to the calculation of the with positive definite energy dependence in the Feynmandual propagator to obtain the total dep Figure 5: specify a system of coordinates (e.g. space-time or light-c them to be integrated over at fixed values of the remaining that the residues atinvariance each of of the the loop poles integral is are recovered not only Lorentz-invari afterbetween summ one-loop integrals and phase-space integrals: as a sum of i.e. a obtain the result in eq. (4.4). of coordinates can bein denoted the in a complex Lorentz-invariant plane form. of A the variable and the internal line with momentum arises from the fact that the original Feynman propagator 1 integral. Although the propagator of the where the explicit expression of the phase-space integral pole at 1 the Feynman prescription we name the ‘dual’ JHEP09(2008)065 , ) n ( ads . ) as . I 2, of . (4.9) (4.8) sheet j N in the ≥ cut ηk ) η − . 0 N m i ( m ) . The basic L ,...,p phase-space 2 − η 2 − 2 j 1 N ,p k (3.11)) and the 1 + , with pole and branch- i p + ( p single j ) cut ) + N ) + cyclic perms − 1 propagators in the qk ( N 1 ion 3. In analogy to ry vector ( m the various single-cut 2 L − originates from the use − L · · · 0 prescription fixes the N i p (see eq. (2.2)). η lations of the momentum n p =1 + N Y j ch only involves single-cut has additional (unphysical) + ) tor ual +1 ) . However, when computing i q defines omment on the role of η ( p N ) is a function of the Lorentz- · · · e-cut contributions ( e δ is actually a N e + L q correlates + ) i ties. In contrast, in the FTT, this tributions Z 2 N idimensional space of the complex η ( p e L in all its contributing dual integrals . dual propagators are denoted by )= + ,...,p η n as a sum of basic dual integrals is just 1 basic phase-space integrals (figure 5): 2 . This compensates for the absence of n ,...,p ) ) ,p N N +1 N 1 ( i ( same p p e L e ( L is independent of the integration momentum ,...,k ) ,...,p 2 + 2 j N i ( p – 11 – q , k L . However, the Feynman propagators present in 1 ,p + − i k i p 1 ; ) cut ( q q − N ( 0 prescription and, thus, it is named ‘dual’ propagator. ,p 1) ( 1 i ) 1 n − L p ( ( N I ( has to be the 1) does not depend on I ) − ) η q N ( N ( N =1 ( e δ i X I e q L Z = 0 prescription of the Feynman propagators selects a Riemann ) = i , so that they are evaluated on the same Riemann sheet: this le ) N )= are well defined for arbitrary values of n N ( ) e L n ( ). This function has a complicated analytic structure, with I ). The j ,...,p j 2 p ,...,k i p 2 i p ,p p 1 , k 0 prescription of the dual propagator depends on the auxilia p i 1 ( k are replaced by dual propagators in ) is the auxiliary vector defined in eq. (4.5). Each of the ( ) N ( n , the future-like vector η Using the invariance of the integration measure under trans The We now comment on the comparison between the FTT (eqs. (3.7)– In our derivation of the duality relation, the auxiliary vec We remark that the expression (4.8) of ( ) ) cut e L I − N N ( ( 1 , we can perform the same momentum shifts as described in sect : it only depends on the momenta of the external legs of the loo e eq. (3.10), we can rewrite eq. (4.7) as a sum of q q L dual integrals L the FTT are completelycontributions absent similar from to the those duality in relation, whi multiple-cut contributions in the duality relation. integrand is regularized by the dual duality relation (eqs. (4.6)–(4.9)). The multiple-cut con Note that the momentum difference and and are defined as follows: The basic one-particle phase-space integrals with (propagators): only then contributions in invariants ( to the cancellation of thecancellation unphysical is single-cut produced singulari by the introduction of the multipl in this multidimensional space and, thus, it unambiguously cut singularities (scattering singularities),variables ( in the mult singularities in the multidimensional complex space. The d of the residue theorem. Independentlyduality of relation. its origin, The we can one-loop c integral position of these singularities. The auxiliary vector a single-valued function. Each single-cut contribution to a matter of notation: for massless internal particles JHEP09(2008)065 , ) 2 (5.1) , the (4.10) − N + i p + . · · · er of particles. 0] ). i + 2 + e. For the FTT, the ,p +1 2 1 i ) p ntext of the duality re- p ( 1 ion Bub, is the simplest p + (2) utting each of the internal i + L 1 f the duality relation to the q the term ‘duality’ precisely to reen’s functions and scattering oint functions will be presented 0] [( ,...,p i 2 ore involved, since the multiple-cut egral as the phase-space integral of s: +1 + p i one) correspondence between dual integrals 2 p q [ + i d . A detailed discussion (including detailed ) q ,p 1 d π i (2) p d p (2 L q ; -dimensional integral over the loop momentum + q d ( q – 12 – Z 1) i − − N ( I 1 )= 2 p 2) contain integrals of expressions that correspond to the N =1 i ,p X ≥ 1 between the ) p ( q 5 m ( The one-loop two-point scalar integral e δ (2) q is the integrand of the dual integral in eq. (4.9). Therefore L Z ) n ≡ (with ( ) I )= 2 1 cut p N ) − N Figure 6: ( m tree-level diagrams over the phase-space for different numb L 1)-dimensional integral over the one-particle phase-spac Bub( ,...,p m − 2 d ,p 1 p ( ) The two-point function (figure 6), also known as bubble funct The simpler correspondence between loops and trees in the co The word duality also suggests a stronger (possibly one-to- N 5 ( e L and loop integrals, which is further discussed in section 7. integral whose integrand is thelines of sum the of loop. the terms In obtained explicit by form, c we can write: where the function duality relation (4.6) directly expressesa the tree-level one-loop quantity. int To namepoint eq. out (4.6), this we direct have relation introduced elsewhere [5] (see also refs. [3, 4]). non-trivial one-loop integral with massless internal line results in analytic form and numerical results) of higher-p relation between loop-level and tree-level quantitiescontributions is m and the ( product of lation is further exploited inamplitudes. section 10, where we discuss G 5. Example: the scalar two-point function In this section weevaluation illustrate of the the one-loop application two-point of function the FTT and o JHEP09(2008)065 ) d 2 on p (5.2) (5.6) (5.7) (5.5) (5.4) (5.8) (5.3) . rescrip- . Note, ǫ . ) 2 0 prescrip- k i ( ǫ − c − 0 prescription, 0 , cf. eq. (3.11),  i i ) and the volume k cut = 4 + ( . Γ 1 − c 2 ec (1) 1 d = 0) and exploited 2 k I  their k 2 0 prescription defines i he single-cut integrals p + 0 iπǫ i e + qk : − ) , 1 2 2 . 0 nd the result is p ied out in four-dimensional p ) k ǫ . The ( 2  ) − θ q
ǫ ( πǫ 0: or, equivalently, for any value 0 − 1 . In particular, it gives Bub( + i . . ions), we set )+ 2 δ  ǫ > 2 ) p , which is the sole available invari- − ) ) cos( 1 p 2 ) k 2 1 ǫ ǫ 2 ( ǫ p − p − , of the single-cut integral. We define: c p q 2 d = ( − ) = +1 we recover ) d 2 θ − k ) d (1) − π reg k ǫ (  ǫ ) (1 I -dimensional phase-space integral is c 0 − ǫ 2 ǫ 2 (2 only depends on the sign of the i ) 2 d − ǫ ) Γ(1 − −
2 − π )Γ Z | ) Γ(1 + 2 ǫ (1) reg ǫ ǫ 2 2 − 0 I 1 (4 − p = 2 | k k , – 13 – − ) (see eq. (4.9)). k )  ) Γ(1 (1 ) Γ(1 π ǫ ǫ k ǫ , of the ) ( ) ): setting (4 (1) Γ ǫ Γ(1 + Γ(1 c ǫ Γ k ec 2 I 2 c ( 0 c i ≡ = 1 − 1 − Γ Γ Γ(1 + + )= c 1 c e Γ(1 + 2 (1 (1 2 2 ǫ ǫ ) can only depend on p k = 2 Γ ) -dimensional volume factor that appears from the calculati c + Γ Γ ,p d ec c πǫ 1 we recover Bub( qk p Γ ( )= c 2 0 prescription in eq. (5.2) follows from the corresponding p 2 i ηk ) p (2) q 2 cos( − L ( e δ − ) in eq. (5.6) originates from the difference between q Bub( )= Z πǫ k ( , respectively. Since these integrals only differ because of )) = c ) is an arbitrary function of k k ( (1) )) = ( c I k c ; is the customary of the loop integral: ( k c Γ ( Γ ; ) as a single-value function of the real variable c c and 2 k ( (1) reg p The calculation of the integral in eq. (5.5) is elementary, a We recall that the The result of the one-loop integral in eq. (5.1) is well known I cut (1) reg − I (1) 1 tion, i.e. on the sign of the function while setting Bub( of one-loop integrals: where factor Note that the typical volume factor, The factor cos( Here, we have visibly implemented momentum conservation ( Although 5.1 General form of single-cutTo integrals apply the FTTI and the duality relation, we have to compute t Lorentz invariance ( tion of the Feynman propagators in the integrand of eq. (5.1) we introduce a more general regularized version, an imaginary part with an unambiguous value when of space-time dimensions. ant). Since mostfield of theories the (or one-loop in calculations their have dimensionally-regularized been carr vers however, that we present results for arbitrary values of JHEP09(2008)065 1 ) 2 ηp ηk k 0 tive − i (5.9) + ). (5.12) (5.10) (5.11) + 2 . qk q 1 2 ) = ( ) k 1 + ))] (

p δ c k ( − c . This follows ) )))] 0 ng the positive q q k ( k ( ( 0 prescription is . ˜ δ c i  ) sign( ) πǫ ηk ) sign( 2 k πǫ sin( , of this example. The dual the energy i − .  − 1 sign( pport on the future light-cone, ngularity is regularized by the on-shell constraint sin(2 0 p ary part. The result in eq. (5.6) ) by written as ) ) i ≥ πǫ πǫ πǫ 2 ↔− )+ k ) is obtained by setting 1 )) has a branch-cut singularity along k 1 sin( k cos( ( πǫ p if ( 0. In this region the ηp ) [cos( i c  0 )) have different analyticity properties in 2 (1) i ; k k > I − , k ( ) has the following important property: ( − c − 1 0 k )+ ) ) ( ; ( k 1  1 2 q 0 θ ) (cos(2 (1) c k ) reg p ( ǫ k 0 ( 1 ) in the integrand of eq. (5.5) is positive definite I (  ˜ δ − 2 k p – 14 – ) )
ǫ (1) k reg (1) ǫ ǫ − 0 − I + I ( 2 2 i +
0 and θ q | ( − 2 − − qk > k )= 2 | )+ (1 ) = sign( 2 (1 k 2 ) and ǫ 0 ǫ 2 k − k ,p ηk k − ) ( 1 θ p ( πǫ Γ 2 ) [ Γ c 2 sign( = One-loop two-point function: the duality relation. c (2) 0, while the branch-cut singularity is placed along the nega k e L − ( has no imaginary part. Outside this kinematical region, (2 θ < 2 cos( 2 0 − + (1) p reg is a future-like vector, )= k I k µ ( η (1) )) = 0. Figure 7: I plane. The bubble function has a branch-cut singularity alo k 0. The phase-space integral ( 2 > c k ; > 0 k k 2 q ( k (1) reg I The denominator contribution (2 1 p 0 prescription, which also produces a non-vanishing imagin the entire real axis if We also note that the result in eq. (5.6) depends on the sign of real axis if in the kinematical regioninconsequential, where and real axis, 5.2 Duality relation for theWe two-point now function consider the duality relation (figure 7) in the context representation of the one-loop two-point function is given from the fact that thewhich integration measure is in selected eq. by (5.5) the has positive-energy su requirement of the can vanish, leading to ai singularity of the integrand. The si explicitly shows these expected features, since it can be re the complex cf. eqs. (4.8) and (4.9). The basic dual integral We note that the functions Bub( Using this property, the result in eq. (5.6) can be written as in eq. (5.6). Since JHEP09(2008)065 , and, ) 2 (5.16) (5.13) (5.14) (5.15) k ) 1 p + → − ) ) 1 q q k ( p (( ˜ δ , δ , + ) ǫ q i ( 10 − ˜ δ ) ) p
ǫ 2 0 2 i + ). two-point function. To ,p 2 1 , 0 1 − − p q p − ( 2 1 ( ginary contribution in the  (1 θ p -point function. However, 1 ǫ ) p cut − 2 Bub( . For the FTT, the two-point 0 − q i (2) 2 ( − Γ 0). Taken together, δ L c + ↔− ) 2 0 1 − ) )= q p 1 )+ 1 ( 2 2  p = θ ,p ,p  2 − 1 1 1 p )+ − q p ) p ( q d 1 ( ( ) q d p ( ( d π (2) ˜ δ cut e L (2 − ↔− cut (2) 1 – 15 – − 1 L (1) 1 Z p I h  i − )= )+ )= 2 1 1 − p )= p ,p ( 1 N + p (1) ( q I = ), this contribution is odd under the exchange ( cut e δ ηk − ) ,...,p )= (2) 1 q 2 2 One-loop two-point function: the Feynman Tree Theorem ( L 2 e δ ,p p ,p 0 1 sign( 1 i p q p − ( ( Z + ) (2) q 2 (2) ( ) L 1 e ) = 1 ˜ L δ )= p 2 Figure 8: ηk ,p + q − 1 q p ( ( cut − − (2) 2 1 L p Comparing this expression with eq. (5.2), we see that the ima square bracket issince responsible sign( for the difference with the two which fully agrees with the duality relation therefore, it cancels when eq. (5.12) is inserted in eq. (5.1 5.3 FTT for the two-point function We now would like tothis discuss end, the FTT we (figure wantfunction 8) to in is check the cast the case into relations of the of the form eqs. (3.8)–(3.11) where the single-cut and double-cut contributions are and JHEP09(2008)065 . . ) cut 2 − ) (2) 2 (5.20) (5.19) (5.18) (5.17) p L − . q  ) (( . 2 1 δ p )  0 −  q ( ) 0 θ − k ) ) 0 p πǫ πǫ . ( ) cancels (this part is ) ) 0 θ the imaginary part of q ) ) across its branch-cut q k double-cut contribution ( ) 2 1 2 sin( ) sign( cos( ˜ δ 2 p p 2 i − ginary term with support putation of the two-point q le-cut contribution k ( − f eqs. (5.13) and (5.18) are p ) results in − ( ( δ ( 1 θ 0 prescription with the Feyn- θ ˜ ) can be obtained by applying δ i  0 ) ) to sign( q ǫ erting eq. (5.17) into eq. (5.15), f Bub( ( )+ the FTT expression of eq. (5.14), − nd (5.17), we see that the imagi- πǫ πǫ θ 2 )
ǫ k 0 2 2 i − − sin( cos( q ( d − − ) of eq. (5.15) is obtained by setting ) = ) d θ 2 1 ) 0 k ǫ  d π (1 p ǫ ( p ǫ − ) 2 ) ( (2 −
θ | cut πǫ − πǫ 2 1 − Γ Z p (1) i 1 c | (1 I i ǫ − sin( cos( – 16 – i Γ = )=  − q i c 1 1 p − −  p ǫ ( e δ − ↔− )= ), while the remaining part does not: )
) 1 2 ǫ k 0 q p 2 i q ( 0, whereas the two-point function is purely real in the same ,p  e δ 1 − − One-loop two-point function: the imaginary part. p < →− 2 ( )+ q (1 2 1 k 1 Z k ǫ p p cut − ( − (2) 2 )= p cut Γ L 0 2 − c p (1) 1 ( I θ h − Figure 9:  ) )= 2 2 Im )= p i k ,p ( 1 2 p ( cut is different from the unitarity-cut contribution that gives Bub( −  0 prescription. In particular, the difference is a purely ima (1) 1 cut i I The calculation of the double-cut contribution in eq. (5.16 To conclude this illustration of the FTT, we add a remark. The − cut (2) 1 ) = +1 in eq. (5.6); we then have Im − k L (2) 2 i ( man on the space-like region region. In the FTT, this difference is compensated by the doub Inserting eqs. (5.18) and (5.19) intowe the find right-hand agreement side of withfunction. the result from the direct one-loop com L We see that also thedifferent: sum the difference of is the due two to single-cut the replacement contributions of o the dual respectively. The basic single-cut integral the bubble function (or, equivalently, the discontinuity o singularity). The imaginarythe part Cutkosky of rules the (figure two-point 9): function 2 Comparing the individual single-cut results,nary contributions eqs. in (5.12) the a squarethe brackets part are of different. the Ins imaginary contribution that is proportional c odd under the exchange JHEP09(2008)065 has )= k ηq (5.21) (5.22) (5.24) (5.23) (5.26) (5.25) , + o q ) i 0 ) q 2 ( ) θ . p − , ) − i 0 ) q 0 imply sign( ) p . p 0 (( . − > p δ ) ( 0 . 0 q ) ed in a direct way (i.e., y performing the shift q θ q − 0 ( ↔− ) ) q − θ ) of the two-point function, 2 − 2 p p n p ) ηp 0 − ( ) ( .20) are different due to the p 2 e 0). The computation of the δ 0 θ ) nes. Once the energy of the , p p ator, we have ) ) + p ting eq. (5.21) in eq. (5.20), we ) + ( ( 0 ) = +1. Therefore eq. (5.26) q duality relation. The derivation p θ q ( − πǫ ( e > δ ηp ) q (( ) sign( 0 0 δ 2 (1) q (( q q ) ( I h δ p Z ) ) θ 2 sin( q 2 ) ( + Im q − ) ǫ e δ ( ǫ h q ηp − δ 2 = )
(( q q 0 | δ − Z 2 o Z p = 0 and ( p ) ) ) | 2 ) 0 0 q (1 ) 0 ( q ǫ p iθ – 17 – q p ( e δ ( ( 2 θ Γ θ ) = sign( − θ ) q − 0 . The energy constraints in eq. (5.24) result in p − ) i c p Z q ηp ), in complete agreement with the result (5.4) of the ( ) 2 π 0 ηp θ = ( )= p p 0  2 ) p − q sign( 2 ( )=+ )] = 2 sign( p 0 θ q ) p q (  ( − πi ) θ π i 2 (1) p 2 (2 p ( n Bub( I  e δ − − ) is fixed to be positive, the on-shell line with momentum 0 ) p = q Im [ Im q ( ( Bub( ) = i e θ δ  0 2 p q ( Im Z θ  i ) 2 2 p )) = +1 (see eq. (5.11)) and, hence, sign( q of the integration variable − p Bub( p  − ( η q We also note that the Cutkosky rules in eq. (5.20) can be deriv Im i 2 → determination of the positive-energyline with flow momentum in the internal li We see that the double-cut contributions in eq. (5.16) and (5 positive energy in eq.double-cut (5.16) integral in and eq. negative (5.20) energy yields in eq. (5.2 which indeed differs from theobtain expression the in imaginary eq. (5.19). part of Inser Bub( without the explicit computation of anyis integrals) as from the follows. Applying the identity (3.5) to the dual propag one-loop integral. We now use the dualitywhich relation is to given compute by the imaginary part where the firstq term in the square bracket has been rewritten b Inserting eq. (5.22) in eq. (5.23), we obtain We observe that the constraints This can be inserted in eq. (5.24) to obtain sign( becomes identical to eq. (5.20). JHEP09(2008)065 ), 0. on. ηk > (6.4) (6.3) (6.1) (6.2) (6.5) = 2. ( θ 0 . . q N ) ) 2 2 . ,p ,p ) 1 1 N p p i ( ( ) ), which finally 2 2 = 0 and ) ) cut cut 2 k k . − − q (2) 2 (2) 2 ,...,p + + i 2 L L an propagators. The ) q q i ider the case of is direct and purely ) k ,p ) 1 (( (( 1 2 . + p + ηp ( δ ηp q ( ( 0 and the constraint heorem to the evaluation of πi δ ( ted in a single-cut integral. θ = 0, from eq. (5.11) we thus θ cut ng momentum conservation, e δ ) → 2 − ), we have 1) and (5.16), we obtain ) 2 ) N ) ) = 1 ) TT and the duality relation. q ( N )+ tiple-cut Feynman integrals. It )+ 1 2 k ( 1 om eq. (3.5). The equality on the 1 ηk ηk L ) e ηq > δ p ( ( k + ηp ( θ θ ηp + q ( − + cut ( θ q θ h · · · − )+ )+ (1) 1 k k (( I δ )+ )+ + + )+ 2 1 0. Using : q q N 0 with ( )= ( ( ,p 1 ηp (2) 1 ( G G p > e p L θ h h ( – 18 – ) + ηq > ) ) k ,...,p q q q cut ( ( ( 2 )= + − e δ e e δ δ 2 (2) 1 ,p q ) ( 1 L ηp q = = p η ( ( ( e δ θ q )= ηk ) cut 2 Z 0 − p )+ N ) i ( 1 ( 1 1 L − can be expressed by using either the FTT or the duality relati (1) ηp ηp 2 ) 1 ( I ( k θ θ N )= ( + )+ L N 1 )+ 0. Combining p 1 qk ( p 2 > ( (1) ) ) 0. This enables the replacement ,...,p I q cut k 2 ( − > e δ (1) 1 + ,p I 0 )= 1 q 2 k p ( ( η ) ,p + )= = 0, we find 1 N 1 0 ( 2 p p q ( p e ( L Our starting point is a basic identity between dual and Feynm In this section we present another proof of eq. (6.1). The pro (2) + (1) e L 1 I 6. Relating Feynman’s theorem and the dualityThe theorem one-loop integral we have Comparing eq. (3.9) with eq. (4.6), we thus derive have This relation is equivalentp to eq. (6.1), since by merely usi We can now use this equation to compute yields eq. (6.2). 6.1 Two-point function The relation (6.2) can be used to prove eq. (6.1). We first cons Inserting eq. (6.2) in eq. (4.9) and comparing with eqs. (3.1 identity applies to theThen dual propagators when they are inser algebraic. It further clarifies the connection between the F This expression relates single-cuthas dual been integrals derived with inone-loop an mul indirect integrals. way, by applying the residue t The equality on the first linesecond of line eq. is (6.2) directly obtained follows as fr follows. Using the constraint Therefore, from eq. (5.11) we thus have JHEP09(2008)065 ) n k (6.6) (6.7) (6.9) (6.8) ) (6.12) (6.11) (6.10) . + . n ) q k )  ( n . N k + , in eq. (6.7) . q ) + ) , ( n ...G n q ( m,η ) ) ( perms I n . = 1 ,...,p . ) +1 with multiple-cut 2 . ...G n o m ) ) atorics, and it does . ,p = 0. The derivation ...G ,...,k k n 1 ( i 2 ) +1 p p I ,...,k + ( m , k 2 +1 ), in eq. (6.7). q k + uneq 1 i ,...,k =1 ( m cut k N i , k st rewrite perms 2 ) k ( +  1 − G . ηk N ) k ( , k P q ( m ) ( n + 1 ( θ ( m,η L ) tegrals m k + 1 q 1 I G g eq. (3.11), we obtain n ( k ( 6.1). ( m,η ) = ) , m n G =1 . n + ( η δI m ) X I m )) + cyclic perms q k o − . ( )+ uneq 1 m ) e δ k + − )+ m m )+ )+ n q N n n + ( ηk . . . p ηk e ( δ q as in eq. (4.8), and using eqs. (6.6), (3.10) ) ( perms ( 1 + . e δ k 1) . . . ,...,k − ...θ ) · · · -cut integral. Eventually, the main ingredient + ...θ ,...,k ,...,k 2 . . . 1 ) N and the multiple-cut contributions of the FTT 2 2 ) ( 1 q k ) m – 19 – + 1 , k ( I 1 )+ cyclic perms ) , k , k 1 e 2 δ k 1 + ηk n 1 1 + uneq k ηk p ( ( m,η n − k k ( ( q I θ + ( ( ( θ N ) + ) e δ -point function, we employ the following relation: q  q p n × 1 ( ( cut cut ( m,F n ) ) p N e δ + I e δ − − n n ( ) × ( ( 1 1 n η q q I I ( · · · ( ) Z )= e δ q + = n ( 2 q e δ ) = p Z ... θ n + 1 1 q + Z 1 )) m 2 )= ,...,k p n 2 )= ,...,k + , k n )= 2 1 1 n ,...,p k p 2 , k -point function ( ( p 1 ,...,k ) η k 2 N ( n + ( ( m,η ,...,k θ ) 1 I , k 2 n ,...,k ) ( 1 2 ,p 1 I k , k 1 ( 1 , k p ηp ) k ( 1 ( ( n k θ ( ,η 1) m,η ) ( 1 I − n To simplify the combinatorics in the proof of eq. (6.8), we fir We note that the proof of eq. (6.8) is mainly a matter of combin For a proof in the case of the ) ( − m,η N n ( m ( m,F δI I I More generally, the identity (6.2) relates the basic dual in 6.2 General and Feynman integrals. Inserting eq. (6.2) in eq. (4.9) and usin where of eq. (6.9) is presented in appendix B. Note that the key difference between and (6.8), we straightforwardly obtain the relation in eq. ( not require the explicitof evaluation the of proof any is the following algebraic identity It is a direct consequence of momentum conservation, namely as Summing over the cyclic permutations of (see eq. (3.8)) is the presence of the momentum constraints, where JHEP09(2008)065 i is , N P
-cut ) 1 1 N (6.17) (6.16) (6.13) (6.15) (6.14) m . The − − 1 N m − i is a sum of Q , so that ,...,p 1 2 fixed internal cut − of the internal ) i . − ,p and i N 1 m Q ( m q i p L ( Q − = 0 i has cut . ) , the factor Q − ide is a sum of N puted according to the 1) ,F ( m e Feynman propagators enta 1 = − L . (3.8), ee, the numbers given by − N , i N momenta ( m = P wing to their symmetry, the n give weight unity to each I )! resent discussion. The other . ssarily proportional, and the ide of eq. (6.13) is also a fully below, this coefficient vanishes m in eq. (6.13). − , . (see eq. (6.11)), and the factor 1)! -cut contributions on both sides of N ) )! ,F 1) m − m ) + cyclic perms 1 − 1 m Q − N − 1)! ( N ( ( ( ! m − e δ N I − p N cut lines, and we denote the momenta of N . . . m + ) + cyclic perms ( ! ( m ) 1 equals the total number of permutations in the 2 − m · · · – 20 – 1 m Q N ( cut + = p ) e δ . We can organize these contributions in a sum of (figure 10). We define = − 2 ) N ) +  p ( m N 1 m N ( L m N + -cut contributions: the symmetrization follows from the Q L · · · (  1  e δ m 1 1 + 2 − − ,...,Q p 2 m N ,...,p + 2 ,Q 1  p 1 is the weight of each contribution to Q 1 + m 1 ,...,p ,p /m 2 1 p p ( + ,η 1) 1 1 − − N ,p ( m 1 p δI ( ,F 1) 1 diagrams as on the right-hand side of eq. (3.8): each diagram − We consider one of the diagram with The relation (6.14) can be proven as follows. The left-hand s The relation (6.13) can be proven as follows. According to eq −
N ( m -cut contribution and simply count the number of -cut contributions with a fully symmetric dependence on the N m I and two other factors.of the One uncut factor internal stems lines from and the it product is of inconsequential to th the p computation of the diagram involves the factor is the total external momentum between the cut lines with mom these internal lines as where the factor 1 This leaves us with the task to prove the relations lines that have beenexpression cut. on the The left-hand sidealgebraically, coefficient thus of of yielding eq. each the (6.14). diagram result As in is discussed eq. com (6.14). and The number of terms on the left-hand side of eq. (6.13) is lines of the loopsymmetric integral. linear The combination of expression onsum the over left-hand the s permutations inleft-hand eqs. side (6.11) and and the (6.13).proportionality right-hand Hence, coefficient side o is of just eq.m unity. (6.13) are To nece showeq. this, (6.13). we The ca number of terms in the number of permutations that contribute to curly bracket of eq. (3.8), namely is the number ofeqs. cyclic (6.15) permutations and in (6.16) eq. coincide, (6.13). thus yielding As the we equality contributions can of s the loop integral m Obviously, eqs. (6.10), (6.13) and (6.14) imply eq. (6.8). JHEP09(2008)065 1 − . In (7.1) m corre- (6.18) ce gen- . 1 ,...,m . 2 − , o . delta functions, = 1 one-to-one m i , can be expressed as a ) of ) ) 1 i q N − ( ( ) with i i m + cyclic perms L α k Q ect to the argument of tum conservation constraint -hand side of eq. (6.12). We different contributions: each  ( G e general terms the nature of re not in a + e δ 1 containing that of the one-loop m q m als in this section. N ( =1 )) + cyclic perms ly the left-hand side of eq. (6.17) i Y → h Feynman and advanced propa- e δ 2 1 − 6.14) and performing the sum over ) P − q q )) Z m 3 ( . . . 1 2 e δ P P ) − , but the opposite statement is not true. Q 1 m + )= k 1) P 1 N − + Q · · · N + q ( , α ( I + e δ N · · · 2 ) – 21 – P q 1 + ( P 2 e δ + cut lines. Each blob denotes a set of internal lines that P 1 ,...,p 2 m P + ( -cut diagram of the left-hand side of eq. (6.14) has a , α 1 η form a linear basis of the functional space generated by ( 2 P m m ) m ( n ,p P Q ( η 1 ( I ... θ , α 1 )) p ... θ 2 ( ) P as a linear combination of loop integrals, we have to introdu )) N 2 ( + 1) P L 1 − involves the product P + ( N ( 1 η 1) ,η I ( 1 P − θ ( − ) N A one-loop diagram with η ( m 1 ( θ δI ) ηP = 0. Therefore, each 1 ( i θ P ηP n ( To express Using the duality relation, any one-loop Feynman integral θ = =1  m i note that are not cut. factor arises from the term in the square bracket on the right but the term in the square bracket is symmetric only with resp conclusion, the square-bracket term contributesby to a multip factor proportional to the following expression: delta functions. Therefore, inserting eq.the (6.12) permutations, into the eq. term ( in the square bracket leads to contribution corresponds to one of the assignments Figure 10: Therefore, the dual integrals This expressionP vanishes, because of eq. (6.9) and the momen vanishing coefficient. 7. Dual bases and generalized duality One-loop Feynman integrals and single-cut dual integrals a linear combination of the basic dual integrals the loop integrals, but overallFeynman they integrals. generate a larger space eralized one-loop integrals, whosegators. integrands contain We bot define them through spondence. Starting from thisthe observation correspondence we between discuss one-loop in and mor single-cut integr JHEP09(2008)065 1) = . − 2 , A N (7.2) (7.4) α ( ) and )). I i q = 1 ) ( . Using = 1 p A i N G p 1 − ) α G i j , α ) 1 k , = k p we obtain the ↔− , ( +  and i + ) i 1 q · · · 1 ) (2) ) p ( q ) is the Feynman p j k ( G ( i L A = ) (7.3) k q j G ( + + − 2 G ). k -loop integrals. Note + ) j q ) q G α − ( j ( q + racket is odd under the ) ( A ηk A = , A k q ηk A 1 ± G ( G ( 1 spondence between single- ) = p ( t the dual integrals i θ G + alized one-loop integrals of ) ) α G θ nation of q − ) q q ) ( ) consistently reproduces the j ( ( ηk q. (7.3) is not invertible. The j se 1 F ( )+ single-cut integrals by a proper A G p ηk j θ , F, G ηk 7). However, the single-cut inte- ( 1 G k q − θ of both dual and advanced propa- p − (  Z ( ( o-point function ) + )+ + ) θ (1) 1 k q (2) h I ηk ( , and ( ) as a linear combination of L + ηp G θ q ( n ) =1 q ( ) θ Y j 1 )+ ( j e δ 1 F, A G − )+ p  ηp ηk ( ) ) k ) = − q 1 − ( ( ( (1) + p ηk i ) θ θ – 22 – I 1 G α q h − + p h ( ( − q θ G + ( ) n =1 )+ h h Y j q q 1 G ) ) ( ( p and the generalized one-loop integrals in eq. (7.1) is q q ) ) A ( ( G q q − e e δ δ ( ( = 1, eq. (7.3) reads: G , ) 1) e δ 1 A q  − n p ( q = = appears only in the coefficients q G ( N Z ) G ( Z n q I (2) ( ηk q = Z I L )= Z 0 ) ) is the advanced propagator. In particular, when i n ) ) 1 i 1 1 q ( − ηp ηp ηp A 2 1 . Therefore the sum − can take two values, k 1 ( − − G ( ( ,...,k i p θ + 2 θ θ  α , k we recover the one-loop Feynman integral in eq. (2.9), while − − + qk 1 2 ↔ − k F = ( ) 1 ) dependence of ) = q n p = ( ( 1 η e δ I p N ( α ). The right-hand side of eq. (7.3) is a sum of generalized one (1) The relation between In the simplest case, with More generally, the linear relation in eq. (7.3) implies tha Nonetheless, we have not yet established a one-to-one corre q I = ( A where the label which can be inserted in eq. (4.9). We thus obtain propagator, while obtained by rewriting the dual propagators as a linear combi eqs. (3.4) and (6.2) we have: one-loop advanced integral in eqs. (3.1) and (3.2) for the ca that the where we have used eq.exchange (3.2). Note that the term in the square b G · · · where again we have used eq. (3.4) to express duality relation (i.e., equivalently, it reproduces the tw belong to the functionaleq. space (7.1) that is generated by the gener cut and one-loop integrals.generalized In one-loop fact, integrals can the be correspondencegeneralization expressed of in in the e terms duality of grals relation of in this eqs. generalized (4.6) relationgators. and involve (4. the integration JHEP09(2008)065 (7.6) (7.5) (7.7) (7.8)  , 2 i . ) # 2  ηp ) . ) p j 0 2 i q e with each p + ( ). The right- i ) + 1 A . + 1 1 p 2 , A G p 1 ) 3 i p 1 + + ) p ,A q j + k q ( α + ( e q δ ,F,p δ e δ + ( q 2 ) e δ ( ) q 1 + 2 ( ) consists of rewriting the ) p ) ow its application to the e δ 2 2 lgebraic procedure similar i ηp p ,F,p ) + q the integrand of the phase- ( sely, in the case of the dual ηp 1 ee appendix A). p integral are cut; the uncut q θ rals to single-cut phase-space + p − ηk ( − rs. The advanced propagators grals of expressions involving ( 7) and (7.8) coincide. g eq. (3.4) to replace Feynman ( 1 e δ j ight-hand side of eq. (7.5) reads − − θ p q ( ) (3) ( − θ 1 1 η L − + ) p 1 0 ) − q ,F i 2 p ( i + ) p e δ α k q − δ + ( + + 2 j ) q + A i q ( 1 2 ) q q p G A q ( ( ( e  δ ηp A G ,F + ) A j 0 2 ) G i q α G p 2 ( h N =1 δ i – 23 – p X − A " ) q + i q 2 Z + G 1 ( q ) 6= =1 N p  1 e δ 2 j j Z Y − 1 ) p p 1 + − × = p + + q )= ( 1 q + 2 e δ ηk ( )= p p q e δ 0 N ( i + + + e δ h q , α 1 − )+ ( p ) 2 N 2 1 ) q p k ( 1 + p A + q + ( 1 G + p G ,...,p qk q q 2 ( 2 ) Z + e 1 δ ) , α p q −  q 2 ( propagators. The resulting expressions can be shown to agre ( ) + A e δ ,p = q q 1 G ( ( ) A , α G 1 G 1 p ) p q q ( ( + ) Z advanced A q N The generalized duality relation is: To exemplify this algebraic procedure, we can explicitly sh Alternatively, eq. (7.5) can also be derived by applying an a The generalized duality in eq. (7.5) relates one-loop integ ( ( − G e δ L h q hand side of eq. (7.1) yields other. The rewrite of eqs.and (7.1) dual and propagators (7.5) with ispropagators, achieved advanced by eqs. propagators. usin (3.4) More and preci (7.2) give: simple, though non-trivial, case of the one-loop integral This result can be derived by applying the residueto theorem the (s one usedright-hand in side section of 6 eqs.only to (7.1) prove eq. and (6.1). (7.5) as This multiple-cut procedure inte where we have also used eq. (3.2). After using eq. (7.6), the r By simple inspection, we see that the expressions in eqs. (7. integrals. Note that onlyFeynman the propagators Feynman are propagators instead of replaced by theof dual loo the propagato loop integral arespace not integral. cut, and they appear unchanged in × Z JHEP09(2008)065 ) i q ( e δ (8.3) (8.1) (8.2) (7.9) . in the ) i i k M + q ( A -th internal line j s G k =1 i Y i lds are real. If some of amounts to modifying ) j i icle masses and massive s . and single-cut integrals is k q ingle-cut integrals of dual + ator of the (see eqs. (4.7) and (7.5)) of ion of self-energy insertions, alent dual basis of the same Im he duality relation, since this q integrals is thus equivalent to ) ( ose analogy with the derivation nd (7.3), we indeed obtain: > ing on-shell delta function N A hand side of eq. (7.9). The one- ( on of the FTT from the massless 0 , l) treatment of the corresponding e L G ) y recovered (modulo higher-order terms) ) (such as the propagator used in the j i , imaginary contributions in the prop- s> q C s ηk ( − G -th line — the cut line. , the effect of a particle mass θ i − 1 j ) 2 q i finite ( q k real )+ η j 0 prescription of the dual propagators is not 0 + k i i 1 with Re q )= ( – 24 – s + − ; A q 2 j q ( ( G M G s , C k ) =1 G i Y j − Im 2 j   ηk i q j − of the mass in the + ( mass of the unstable particle: i ηk θ s 0 h M i − =1 m Y j [10]) is 2 j 1 = Re . Note also that the k 6 )  s complex ) N + q ( ( j L G qk 2 − ) , the corresponding dual propagator is =1 q m j Y j ( M A   denotes the ) G q s  ( q e δ In any unitary quantum field theory, the masses of the basic fie Moreover, the correspondence in eq. (7.5) between one-loop Z In the complex-mass scheme, unitarity can be perturbativel q 6 Z = has mass affected by the masses. More precisely, if the Feynman propag independently of the value where order by order. (according to the replacement in eq. (3.13)) the correspond when this line is cut to obtain the dual representation these fields describe unstable particles,propagators a in proper perturbation (physica theorywhich requires produces a finite-width Dyson effects summat introducing invertible. Using the same algebraic steps as in eqs. (7.2) a the loop integral Feynman propagator of a loop internal line with momentum The functional space generatedthe by the space generated generalized by one-loop theloop single-cut integrals integrals of on Feynman theand and left- advanced advanced propagators propagators can and thereforefunctional the space. be s regarded as equiv 8. Massive integrals, complex masses andAs unstable discussed particle atpropagators the does end not lead of tocase. difficulties section in The the same 3, discussion generalizati and therelation the can same introduction be conclusions derived of apply by to applying part of t the residue the theorem FTT. in Therefore, cl as long as the mass is complex-mass scheme agators. A typical form of the ensuing propagator JHEP09(2008)065 (8.6) (8.4) (8.5) , otes the i s . + 2 i ) q N √ s of the Feynman = ed far off the real 0 , i q     ,...,p 2 rms of single-cut phase are those with complex- . . . es in the context of the i . . . ,p opagators of the loop are s or, more generally, with 1 i s of the uncut lines. Note p i ane of the loop integration elation that is analogous to ( 6= 6= introduced in both tree-level y complex-mass propagators ) j .7)), while the complex-mass Y ion is positive, as we discuss j Y oop and phase-space integrals of eq. (4.6) has to be modified: N on 4. The only difference is the .6) the ‘on-shell’ delta function ( C   ary part of the complex masses.   e L i   ) s i infinitesimal imaginary displacement has )+ M . . . imaginary part of the complex mass. + ; 2 i N i 1 i q q ( 6= j Y e δ q 2   0 displacement produced by the residue at the pole F i ,...,p to denote these cut lines). Taken together ∈ ) denotes those from the poles of the complex- 2 i X C i ∈ ) C s of the loop with a Feynman propagator (we use i X ,p ; – 25 – q N 1 i i ∈ ( C Z p q 1 e i ( ( L ) q e δ − 1 d N )= ) ( − C d e π N L ∈ i X d (2 q → -point scalar integral (see eq. (2.9)) where one or more of Z Z ) N ,...,p N 2 = 7 is similar to eq. (8.5), but the cut lines . ) ) = ,p 1 N N ( C p ( e L ,...,p ) 2 N ( to denote these cut lines). The term in the square bracket den ,p e L ,...,p 1 2 F p unchanged ( ) ,p ∈ 1 N i ( p ( denotes the terms that correspond to the residues at the pole e L ) , the complex-mass propagators produce poles that are locat ) 0 is thus expressed as N ). To derive a representation of this one-loop integral in te q ( C N ) i ( e L s N e L ; ( i e We consider a one-loop L The expression of q The dual propagators arise from the infinitesimal ( 7 C G the Feynman propagators of the internal lines are replaced b space integrals, we then apply thepresence same of procedure as the in complex-mass secti variable propagators. In theaxis, complex the displacement pl being controlledUsing by the the finite Cauchy imagin theorem aseq. in (4.6). eq. The (4.1), only we difference derive is a that duality the r the right-hand side of the Feynman propagator,no see effect section on 4 the and complex-mass appendix propagators, A. owing to This the finite and one-loop Feynman diagrams.present paper is A whether natural(and the question the duality FTT, that relation as between aris propagators well) one-l of can deal unstable with particles.below. complex-mass The propagator answer to this quest These complex masses, together with complex couplings, are where the term inthat the in the square integral bracket representation contains on the the propagator first line of eq. (8 Here, propagators of the loop integral, while mass propagators. where the sum refers to the internal lines product of the propagatorsreplaced of by the the uncut corresponding lines.propagators dual are The propagators Feynman (as pr in eq. (4 mass propagators (we use the notation the notation JHEP09(2008)065 . cut ent, ) − (8.7) N ( m ) of the L . . . . ) ), that depends 2 N ingle cuts of the q nced propagators expression of the ( s and in the FTT by the loop integration ernal particles. The mass propagators do out the residue at the ,...,p 2 0 prescription in any of i o be used to explain how tors of the internal lines. (and negative) imaginary ,p s from the complex-mass ce the advanced one-loop ays, we conclude that the 1 e rewritten in two different egral (i.e. the integral with p ot contribute, since they are s pole. Independently of its ( phase-space integrals ) ors is realized by the following N -mass propagators are not cut), spectively. In general, the term ives exactly the contribution in ues at the poles of the complex- on of a gauge-fixing procedure, Cauchy theorem as in eq. (3.3). finite ( C mplex mass, ces singularities that are located e (poles, branch cuts, n the right-hand side of eq. (3.6), e L )+ N 2) to the FTT in addition to the real-mass ,...,p ≥ , which has a 2 , we can remove the i s m ,p – 26 – +) 1 + C, 0 p ( i ( 2 i is thus given in theq second line of eq. (8.6). Owing q ) ) cut q N − N ( C ( 1 e L = L +) → C, 0 ) ( i q N = -cut contributions ( 0 of the propagator. We can also include a non-resonant compon i m q ,...,p q 2 ,p 1 p in eq. (3.8). of eq. (3.1) with a one-loop integral that contains both adva ( is the usual contribution (see eq. (3.7)) emerging from the s ) is given by eq. (8.6). Note, in particular, that the complex- cut ) cut . Such contributions can be included in the duality relation N ) imaginary distance from the real axis in the complex plane of ) cut ( A ) 0 − − N N q − 1 ( N L N m ( 1 ( ( C L ) of the cut propagator has a formal meaning, since it singles L L e L i has a form that differs from eq. (8.6) and depends on the actual s finite The outcome of our discussion of the duality relation can als We add a final comment on one-loop integrals with unstable int produce further ) ; i N q ( C ( e e the Feynman propagators inside the square bracket. the FTT can beFollowing generalized the to derivation deal of withintegral the complex-mass FTT propaga in section 3, we can repla and complex-mass propagators.ways. This First one-loop (exploiting integral eq. (3.4)), canin it b terms can be of expressed, a asFeynman and i linear complex-mass propagators) combination and of of multiple-cut the required one-loop int to the finite imaginary component of part. The explicit expression of complex-mass pole, δ Alternatively, it can be evaluatedThis directly direct by evaluation applying leads the to themass computation propagators of (the the poles resid ofplaced the outside advanced the propagators integration do contour): n eq. the (8.6). computation g Comparinggeneralization the of the expressions FTT obtained toreplacement in include in complex-mass these propagat the two right-hand w side of eq. (3.9): sole Feynman propagators of the internalwhile lines (the complex on the momentum Here, not in addition to thespecific resonant form, the contribution propagator of of theat the unstable a complex-mas particle produ variable propagator of an unstablepropagator in particle eq. (8.2). can We can have introduce, a for instance, form a co that differ terms performing the replacements in eq. (8.4) and in eq. (8.7), re L propagator and, in particular, on the singularity structur propagator in the complex plane. 9. Gauge theories and gauge poles The quantization of gauge theories requires the introducti JHEP09(2008)065 = ζ uchy (9.1) (9.2) gerous, ) is q ( G . On the other the only relevant : the polarization q G elicate, since they ) does not interfere ), which arises from q q is obtained by multi- ( ies. Here, the gauge ysical particles. The ( q µν ve to take into account produced by the gauge µν ℓ tion. This is done in a , which have to be taken integrals. Therefore, the d ) propagates longitudinal . harmless q Hooft-Feynman gauge ( deal with one-loop integrals ( loop. Therefore, to derive arization tensor is ns in spontaneously broken ) e-cut phase-space integral of sons and the ensuing content q µν ( ℓ . On the one hand, the specific . G resence of the poles of the gauge- ν p integral correspond to fictitious G addeev-Popov ghosts in unbroken q e cuts of the Feynman propagators 2 ) µ q q M ( the effect is − µν ℓ , or ν 0) 1 i q 1) ) with the tensor q ( − + 2 G ζ , or q µ – 27 – ( q ζ + ( dependence on the momentum µν g )= q = 0), the gauge-mode propagator does not depend on − ( and, in particular, it produces poles with respect to the G ζ q G )= q ( polynomial ’t Hooft-Feynman gauge µν , and the form of the gauge-mode propagator d M ) (the presence of polynomial terms from and factorizes off the loop integration. When applying the Ca ) (we call it ‘gauge-mode’ propagator) has a potentially dan q . ( q q ( G µν ) has a G g q G ( G − µν ℓ ) is not relevant in the context of the following discussion; q ( µν ℓ When considering one-loop quantities in gauge theories, we We first discuss the case of spontaneously broken gauge theor The fictitious particles have their own Feynman propagators The impact of the propagators ofThe the propagator gauge of the particles (spin is 1) more gauge boson d with momentum and factorizes off the loop integration in any of the one-loop The second term on the right-hand side is absent only in the ’t form of tensor is simply hand, the factor point is that the sum of the spin polarizations. The general form of the pol 1). In any other gauge, this term is present and the tensor residue theorem as inthe sections possible 3 additional contributions and that 4mode arise from in propagator the any p other gauge, we ha non-polynomial dependence on containing gauge bosonthe propagators FTT as or internal thepolarization lines tensors. duality of In relation, the the we have to consider the effect polarizations, which are proportional to momentum variable with the residue theorem). boson has a finite mass plying the customary Feynman propagator of (possible) compensating fictitiousnon-Abelian particles (e.g. gauge the theories, F orgauge theories). the would-be Goldstone boso into account when applyingstraightforward manner: either if some the internal FTT linesparticles, in or a they one-loo the have dualitymultiple-cut to phase-space rela be integrals cut ofthe exactly the duality relation FTT in will include and the theof contributions the same from these singl th fictitious way particles. as for ph introduce ‘gauge poles’. This point is discussed below. which specifies the spin polarization vectors of the gauge bo q Considering the unitary gauge ( JHEP09(2008)065 (9.3) (9.4) untouched nd the du- is the space cording to a q = 0) or planar A · tional single-cut and n -loop computations in ) is absent. ), where d contains gauge-mode gauge boson is massless. q 0. The extension of the ator remains gator is of the duality relation is q ( i , G µ fixed while performing the a single-cut dual integral. − , G η ) leads to a (first- or second- e, since its specific form has = 0. Applying the residue duality relation by using the q d 4 to one-loop computations q = ( We will not pursue this issue = (0 2 t ges and physical gauges. e gauge-mode propagators. be specified by some imaginary 2 utions from this type of second- G q µ q where G n (as in section 4), we have to take . If we instead consider a generic ). In axial ( 0 q q . , 0 = 1 or 2 q ), i.e. aligned along the time direction. 0 i 0 , 1 0 = ( + η pole when 2 µ q , k q 6= 0, we see that the gauge-mode propagator = ( (see section 4 and appendix A), the gauge pole ) is located at ζ k – 28 – µ q q ) )= ) is equal to the Feynman propagator, the two η ( 0. This is an additional pole with respect to the q q q i G 1 · ( ( G G G n − ( G G 0) from the associated Feynman propagator. For the i /ζ second-order 2 )= − q M ( 2 gauge) with G ζ = M G R 2 q = = 0. In Coulomb gauge we have 2 q q · . Therefore, the auxiliary future-like vector n plane at fixed values of 0 (see appendix A) to be q 0 q has only inconsequential implications on the use of the FTT a This issue is not pursued any further in the present paper. is a fixed external vector and the pole has to be regularized ac 8 fixed gauge, one has to properly consider the introduction of addi denotes an auxiliary gauge vector. We see that µ n ζ µ n R We now discuss the case of unbroken gauge theories, where the In covariant gauges, we have In physical gauges, the typical form of the gauge-mode propa We now consider a generic one-loop integral, whose integran In Coulomb gauge, the pole of Of course, this does not apply to the ’t Hooft-Feynman gauge, 8 introduces a pole when where multiple-cut contributions from gauge-modeany further propagators. in the present paper. We separately consider two classes of gauges: covariant gau order) pole when Since the gauge-mode propagator in the extension of the FTT and the duality relation of sections 3 an unitary gauge physical pole (when necessarily FTT and the dualitycovariant gauges relation requires a of proper sectionsorder treatment of 3 poles. the and contrib 4 to hold for one component of the gauge boson momentum ality relation for one-looprenormalizable calculations gauge in (or gauge theories propagators together generate a integration over gauges, propagators in addition to Feynmanresidue propagators. theorem To derive in a the complex plane of the variable proper prescription (the precise positiondisplacement of from the the pole real has axis),no to which effect we on do the not discussion that specify follows. her into account the possible contributions from the poles of th theorem in the in going fromNote, the however, that one-loop this integral conclusion follows to from its having kep representation as does not contribute. We conclude that the gauge-mode propag JHEP09(2008)065 ); in i and = 0. M ; (10.1) 1 i ⊥ on the q − is time- q ( d . Hence, e δ µ q 1 specifying 1 n − loop) − ′ µ d d − q η untouched 1 n (1 − e A d − n 0 . To be precise, in q − µ 0 be extended to eval- η n = in eqs. (4.6) and (4.7). = ) y the corresponding dual nq FTT. s unitary and local. Some N equires to properly include ion in physical gauges can ( re left unchanged by going nq ) and apply (see section 4) : f section 9) about its appli- 1 e L easoning (see also sections 8 amplitude level. l vector ingle-cut dual integral. Note, − d neously broken gauge theories, zation tensors can play a role, T is valid in its customary form, e validity of the FTT. The only ing from the gauge-polarization is valid only if the gauge vector ,n , (these contributions depend on the ⊥ n is beyond the scope of this paper. 0 , we have . We conclude that the gauge-mode , 1 0 , and the pole of the gauge-mode propagator 0 0 q − loop) q n d − 0 and consider all possible replacements of n (1 /n ) is located at e = ( 0 at fixed values of the coordinates = q A . Therefore, since the vector n ( 1 µ but also complete one-loop quantities (such 0 loop) G − − nq n q ) − d = G N – 29 – = (1 0) and, moreover, the dual vector is fixed to be 0 ( /n A 0 L . ≤ /η n 0 1 q 2 loop) − = 0. These requirements are not fulfilled if n d = − ( η η (1 0 · = 0, we have A /η 0 n ) of its loop internal lines with the cut propagator 1 A i − q d ( η . G light-like . Setting 0 the regularization prescription of its gauge pole, is or loop) /η − 1 (1 − e d A η generically denotes a one-loop quantity. The expression 0 to q including − space-like loop) to the gauge vector, 1 loop) − − d − (1 q (1 A The derivation of the duality relation in time-like gauges r A ) does not depend on the integration variable = In axial or planar gauges, the pole of Our discussion and conclusions regarding the duality relat The analogue of eq. (3.12) is the following duality relation The duality relation (10.1) is valid in any field theory that i q 9 For example, in the axial gauge is either 1 ( 9 − G µ ′ d does not decouple from the integration over Without loosing generality, we can assume propagator, going from the one-loophowever, integral that to we its have representation set as a s the residue theorem in the complex plane q G the dual prescription isn future-like, the above conclusion orthogonal like. contributions from cuts of the gauge-polarization tensors specific regularization of the gauge poles): this derivatio straightforwardly be used to drawdifference similar conclusions is on that th in the latter case there is no auxiliary dua right-hand side of eq. (10.1) is obtained in the same way as Coulomb gauge and in space-like or light-likewithout gauges, introducing the FT any multiple-cut contributionstensors. stemm In time-likeand gauges, their the effect poles has of to be the taken gauge-polari into account10. when Loop-tree applying the duality at the amplitudeIn level the final partuate of not section only 3, basic we one-loop have integrals discussed how the FTT can We start from any Feynman diagram in where as Green’s functions andand scattering 9) amplitudes). applies The to same the extension r of the duality relation to the the uncut Feynman propagatorspropagators. in the All loop the are otherfrom then factors replaced in b the Feynman diagrams a words of caution are,cability however, to needed theories (see with the local gauge conclusions symmetries. o In sponta each Feynman propagator JHEP09(2008)065 , t 2 i +2 ex- M n be (tree) (tree) N (10.2) (10.3) N A A = 2 i , respec- , p teractions ) l ( loop) N − is the one-loop (1 oop internal lines A . g explicit form: ,...,p is time-like). q 1 loop) ν − − and n (1 , q,p A − eynman diagrams on the is a Green’s function. We and q, If (tree) q loop Feynman diagrams and A ( nd in the unitary gauge. In t consider the case with only (tree) P A y with a single real scalar field sical process is obtained from ). To be precise, we consider ta 10 A ty he right-hand side of eq. (10.1) +2 i , provided the auxiliary duality e ’t Hooft-Feynman gauge; it is e one-loop scattering amplitude p ν ) (tree) N are e n P A ( . f ) A . an [2] in the context of the FTT. of the free propagators of the exter- is given by the sum of the tree-level X 2 o . d M ) related to the tree-level expression − (tree) P P -th line is 2 A i M q ; ( (tree) P q + ( δ amputated A e – 30 – δ 1 0 for an incoming particle) and including the appro- − P q d and X ) d ≥ d π q 0 = 0 (this excludes gauges where i on the left-hand side. The tree-level counterpart Z , how is (2 p on the right-hand side is exactly the tree-level counterpar η ) denotes a generic off-shell Green’s function with A ∼ − · Z N n loop) . The integrand denotes the sum over the degrees of freedom (such as spin, 1 2 − (tree) P ) connected loop) by specifying the renormalization procedure. N (1 A P − ( A . (1 ,...,p f . 1 ? In the next subsections, we show how the duality relation ca o . The particles are self-interacting through polynomial in A )=+ . p loop) A ( d − N M N (1 P A ) by setting the external momenta on their physical mass shel A N ,...,p ). In this case, the duality relation (10.2) has the followin 1 4 p ( denotes the sum over the particles that can propagate in the l φ is chosen such that ) of mass ,...,p P φ 1 or µ p loop) η P 3 ( = 0 for an outgoing particle, − φ ∗ The structure of eq. (10.2) implies a natural question. Equation (10.1) establishes a correspondence between one- To simplify the illustration of the duality relation, we firs N (1 Issues related to similar questions were discussed by Feynm ≥ φ A ( 10 (tree) 0 i (e.g. Feynman diagrams that areleft-hand obtained side. by cutting the one-loop F formulated directly at the amplitude level, when the quanti expression of a specific quantity of the same quantity where that are cut, and colors, . . . ) of the particle the phase-space integral of tree-levelcan Feynman be diagrams. written T in the following sketchy form: involves two additional external lines with outgoing momen vector also discuss the case of on-shell scattering10.1 amplitudes. Green’s functions In the following, ternal lines (the outgoing momentum of the one type of massive scalarφ particles. We thus refer to a theor where the integrand factor of the one-loop quantity priate wave-function factors ofis the obtained external from particles. Th the duality relation is valid in the ’t Hooft-Feynman gauge a unbroken gauge theories, the dualityalso relation valid in is physical valid gauges in specified th by a gauge vector tively. The tree-level scatteringA amplitude for a given phy p nal lines. The tree-level and one-loop expressions of Green’s functions that are JHEP09(2008)065 ) ) e , of a ) in ,... ernal ) (10.4) (10.5) ) q,... ) from tor) to q denotes ( − particles , q,... ,... P q, i ) ,... − ( q ) q,... ( ← q, q both +2 ( − ( P ) color (tree) N , q P q, ( +2 | ( A ← P ) (tree) N spin ) -cut recipe, namely ( q A ( on the physical mass- P (tree) ,... e P q A ) . We find: ( (tree) q tors. The tree-level ex- es, the generalization of is a scattering amplitude plicit examples: P A tion − in +2 ve-function factors of the ( self-interacting sources (the (tree) N +2 P e pes A , (tree) N generically denote the (spin- N forward-scattering process ) ) A q | e ( More precisely, , ) P ) in the Feynman propagators of q ( P ( ( σ ηq P includes the sum over ) h (2 +2 2 / P P (tree) N µ ) on the right-hand side of eq. (10.4) is a M η P P and |A 0 is a linear combination of the external mo- − ( i ) ) is obtained from i is a bosonic particle (e.g. spin 0 Higgs boson, σ 2 q j is a fermionic particle (e.g. spin 1/2 fermion, ( − q k ( ) ( P P µ – 31 – ,... q + P h q j ) ( δ ) by setting the momentum q k P ( ,... . We can write: | denotes the replacement of some of the Feynman → + q P external legs. This expression is obtained from the P 1 if ,... X q µ − ) − ) is the amplitude for the forward-scattering process color q q ← N 1 X , )=+1 if − − ) (tree) q d ( q P e and ) = ) ,... d A spin ( ( of the two additional external lines of ) P d π q P external legs are denoted by ‘dots’, since they play no activ σ . The quantum numbers (spin, color, . . . ) of the incoming , q P ( q ) ( ( (2 ). The coefficient in the external field produced by P q σ )= − N P ( P q Z P of 0) and including the proper wave-function factors of the ext (tree) ( 6= ← ,... e 2 1 i ) and A p ≥ P q +2 ) ( q q 0 (tree) N ( P A P ). ), we assign a dual propagator (rather than a Feynman propaga , q )=+ ( ← 2 P ) ) in the field of the +2 . . . q M q ( ( q,... q,... (tree) N ( P = − − A P ). We note that this step can also be performed by using a short ( i loop) 2 q, q, p q ( ( − +2 → N (1 We illustrate the general notation in eq. (10.5) with a few ex The tilde superscript in The momenta As in eq. (10.3), In a theory with different types of particles and antiparticl (tree) N ) external legs). A q A (tree) (tree) ( and antiparticles (if role on both sides of the equation. Note that where the (‘ket’ and ‘bra’) vectors dependent, color-dependent, . .forward-scattered . ) particle incoming and outgoing wa the coherent sum over them. propagators with dual propagators. More precisely, to obta and outgoing wave functions are fixed to be equal, and the nota menta eq. (10.3) are on their physical mass-shell: in this respect Bose-Fermi statistics factor: where the momenta by applying the momentum shift legs with outgoing momenta (there are no wave-function factors for scalar particles). A shell ( A Green’s function spin 1 gauge boson), and is the tree-level physical amplitude that corresponds to th particle with momentum each internal line with momentum Faddeev-Popov ghost). P by the replacement ofpression Feynman propagators with dual propaga eq. (10.3) is obtained by including a sum over the particle ty N JHEP09(2008)065 ) ) ). ) q ← q q ( ( ( ) ) ) ) . . . s q s ( λ β ( (10.7) (10.8) (10.6) ( β ( ( ν v ) is the ε u P q ( ( , loop) ij αβ ) µν ab , ij αβ  λ −   ( ν +2 ) i ) ) ε N (1 i (tree) N ij -dimensional, ∗  e , ij A d A )  )) ,... ) ,... ,... q ) is are color indices) ) ) ( are color indices) q µν ab ) q q µ (  ,... λ q are Lorentz indices; ) ( µ − ) on of − ,... i, j Q ( ε ( ) les and antiparticles. q i, j , ( ( q ) Q λ , g q Q , ) − ,... µ,ν ) , ( ) q − P ) q ( q ( finition of dimensional reg- Q ( q man diagrams. Therefore, g , − Q ( l and external particles. As Q ) the Feynman propagators of − ( ( )= ( ( ve definition, the antiparticle q q ( +2 , g ( Q -level integrand +2 ) ( +2 (tree) N Q y of the momenta of the external q (tree) N ( µν p momentum ( (tree) N A d g +2 A  ( A  can equivalently be defined to just +2 (tree) N  ∗ ) (tree) N ) are Dirac indices;  q A +2 P q ) (  A ( ) (tree) N are Dirac indices; q  ) s P ( ( α s A ) ( α as sum over both particle and antiparticle i,j α, β v  λ X u ( µ i,j X ε P α, β )  a,b ) i,j X X P i,j M X – 32 – M ) a,b − X q α,β + X ( q / α,β X ( ) is absent, and the corresponding particle contribu- q / 2 µν h ( , 2 µ,ν , h d X =1 X Tr ,... s =1 X ) s λ labels the spin; µ,ν q − X X − ( s = Tr ) must be multiplied by a factor of 2. In view of the issue P labels the spin; = = )= )= )= s ← ,... ) ,... ) q ) ,... q ,... ( ) q ( ) ( q q P ( P ( Q labels the spin-polarization or helicity states; ( g Q ) is the customary Dirac spinor for spin 1/2 fermions; ) is the customary Dirac spinor for spin 1/2 anti-fermions. ) is the gluon-polarization vector and ← λ q q q ← +2 ( ← ( ( ) ← ) ) ) ) (tree) N q s ) s ) λ q e ( ( β ( β ( ν q A ( q u v ( ε ( P Q g ( ( Q ( are color indices) yields ( ) of eq. (10.4) are exactly the same as specified in the definiti +2 = massive anti-quark ( = gluon ( = massive quark ( +2 +2 (tree) N +2 (tree) N (tree) N P where yields where where a, b P corresponding polarization tensor; P yields e (tree) N A ,... Note that, as stated below eq. (10.4), we sum over both partic We recall that, at the level of one-loop computations, the de A A ) • • • A q ( However, on the right-hand side of eq. (10.4), refer to the sumcontribution over particles. According to this alternati there is still freedom inparticles the and definition of of the the dimensionalit remarked number below of eq. (10.1), polarizationsthe the of loop, duality both relation leaving interna acts unchangedthe only all dimensional-regularization on rules the to other be factors used in in the the tree Feyn tion discussed in appendix C, the definition of ularization involves some arbitrariness. Although the loo contributions has to be preferred on general grounds. P JHEP09(2008)065 ) . he ) (10.9) (10.10) N loop; ex ). − N (1 -dimensional ,... ) d A q -dimensional ( ,...,p d external lines, internal lines). 1 P p ( has always to be N ← ) µ q q ( (tree) N P 2 particle. ( A / amplitude, um 0 i +2 , (tree) N ) + e j ) -th line refers to a spin 1 particle, . A p j 2 zed (mass and wave-function j ’s functions (the introduction ber of space-time dimensions. ( -dimensional one-loop integrals; the j , we introduce the following d M tudes, the only relevant point to htforward). − cuts of a finite loop integral, each single- i D loop; in dentify the different cut momenta of the 2 j loop) − in the integrand on the right-hand ators (the propagators with momen- p − N (1 ) N (1 12 j oop calculations [9]: the customary reduction larization procedure. The same intermediate A is required also if the one-loop Green’s sible cancellations of the singularities from the re applied to the separate single-cut terms. In can be carried out by exploiting p A in terms of µ + − of the particle in the internal line with ot globally) spoiled by the loop-momentum shifts q ) , . j p -th line refers to a spin 1 13 ( j in eq. (10.4) plays simply the role of an intermediate µ – 33 – q loop; ex loop) − = 4) of the space-time. The use of a − of the tree-level expression N (1 d 2 (1 q A A arbitrary in the sense of dimensional regularization. In cannot directly be evaluated on-shell owing to the kine- = d ) . ) (cfr. eqs. (9.1) and (10.6)) if the =1 N p j X ( loop) µν (i.e. if it has no infrared and ultraviolet divergences) in t d − )= loop; ex N (1 11 N -dimensionality of (cfr. eq. (10.7)) if the − d A N (1 M A + ) denotes p / is the contribution from one-loop insertions on the ,...,p p is finite 1 is the remaining contribution (i.e. one-loop insertions on ( ) . p j ) ( . D ) . -dimensional, with -dimensional on-shell momentum . is finite, the loop) d j d ) is the spin-polarization factor − p j loop; ex ) denotes N (1 p loop; in p − ( ( loop) loop; ex − A j j N (1 − − N (1 in eq. (10.10)). Thus, to calculate the one-loop scattering D N (1 D A N (1 j A A p A As is well known, We remark that in eq. (10.4) the on-shell integration moment Considering the off-shell Green’s function To be explicit, Even if some of these terms correspond to the sum of the single If is necessary since, in general, the various terms 13 12 11 µ eq. (10.4) the momentumloop shifts with are the implemented common external to momentum be able to i (compare eqs. (3.7) or (3.10) with eqs. (4.7) or (4.8)) that a cut contribution may not bevarious separately single-cut finite. contributions can Moreover, pos be locally (though, n whereas has to be firstrenormalization), evaluated off-shell, before then considering it its has on-shell to limit. be renormali matical singularity arising from its external-linetum propag unitarity techniques. computational tool is usedof in tensor other integrals methods to to scalarcomputation perform integrals of one-l has finite to rational be terms carried in out one-loop amplitudes computational tool, rather than the role of a necessary regu where momentum q where while side of eq. (10.4) are not separately integrable in a10.2 fixed Scattering num amplitudes To extend the discussion of sectionbe 10.1 examined to is scattering ampli the on-shellof limit the of wave-function factors the of corresponding the Green external lines is straig In explicit form, we have decomposition: considered as particular, a function original and fixed dimensionality (e.g. JHEP09(2008)065 ) ) . . as ion ) and . ors) to (10.12) (10.11) loop; ex loop; ex . loop) − − ntroduce loop; in − o N (1 N (1 ) − N (1 A A N N (1 A A ,...,p 1 ) p ( P ( n the right-hand side of ). The integrand factor σ j p ) (tree) N ted in the duality relation. -hand side contains a sum 2 − P A -shell limit. , he algebraic computation of .9) to express gration of complete tree-level on of the (on-shell) tree-level j 0 on-shell limit. The singularity . M he on-shell limit. However, the i , i.e. by removing before setting the external lines e-particle phase-space to obtain amplitude is not included. This ,p ) ) ) . − ) . ) + on in the form of eq. (10.1): s of the lines with momenta equal j . can directly be computed in the ies, so as to directly evaluate the q 2 p 2 j ( ) ( q . j ( P M + loop; in δ loop; in loop; in − i D ← − − external lines are on-shell, and the two − 2 j N (1 ) loop; in N (1 p P . N (1 q e − X ) A N ( A A j N (1 p P 1 − ( loop) ), and (the duality-propagator version of) A + 2 external legs. This is because a subset of − ) − q − d N , = – 34 – ) N d j N (1 N d ) π . (tree) 4 . Then we use eq. (10.10) to rewrite A ,p ) (2 . ) A q ( ,...,p ,...,p Z 1 P . 1 loop; in p ) . ( 2 1 − loop; ex ,p ← ) and ) N (1 − q ) N A ( N (1 q (tree) N +2 external legs. Having performed the tree-level computat ( P A loop; in A )=+ P − N ( N ← N (1 ,...,p ) and A 1 q ( (tree) 4 . Finally, we express the full one-loop Green’s functions ,p of the external lines. Various procedures can be devised to i e ) P A ,...,p ( equal (modulo the replacement of Feynman with dual propagat q j loop) 1 ( p p − loop) P =1 ( +2 N N (1 − j ) X . not (tree) N 2 (1 A e ← − A tree-level Feynman diagrams (the A ) n q in terms of the duality relation (10.4). ( loop; in × − P ( N (1 loop) In contrast, the one-loop contribution The duality relation (10.12) involves the phase-space inte We consider the internal-line contribution This ‘missing’ subset of tree-level diagrams can be reinser We point out that the integrand of the phase-space integral o A +2 − on-shell 2 (1 (tree) N difference of The derivation of this equation is simple. We first use eq. (10 in terms of Green’s functions, namely A A the tree-level scattering amplitude with subset has been removed by considering only the diagrams that enter the complete tree-level scattering on-shell limit. In particular, we can write a duality relati in the curly bracket on thetwo right-hand terms side in is well the defined curly inis bracket t purely are kinematical; separately it singular simply in arisesto the from the the propagator momenta from the complete one-loop expression an intermediate regularization oftwo the terms separate close singularit to on-shell kinematical configurations. Here, the integrandof of the phase-space integral on the right However, as discussed below, this makes more delicate the on on-shell. We can write the following duality relation: the integrand is thus completelyscattering amplitude analogous with to the computati additional lines from cutting the loop are also on-shell). T eq. (10.11) is of the integrand, the resultthe can full be one-loop integrated over term the singl JHEP09(2008)065 butions, leading h the FTT and the ed when propagators e computation of tree- lation between one-loop integrals to Green’s func- . This simple modification in the complex plane by a η for applications to analytic (NLO) requires the separate ere the duality relation can ve propagators and unstable integrals, have an attractive plane of any of the space-time istance from the real axis, and that appear in the FTT. The riant prescription, named dual he dual prescription. Unstable auge poles. We have discussed lex plane of the pole of the cut le-cut terms from the Feynman itudes can be obtained starting ality discussed in this paper, as ering amplitude. single-cut phase-space integrals, rm. Using the duality relation, ), where (some of) the internal be integrated over the multipar- r propagators. The poles of the , from Feynman diagrams that vel counterpart is then integrated ctions in a form that closely par- (virtual) radiative corrections are ngle-cut terms from the absorptive vector N lation of the residues is elementary, This close correspondence can help – 35 – 0 Feynman prescription of the uncut propagators. One- i cancels, as expected, in the sum of all the single-cut contri η 0 prescription of the usual Feynman propagators can be remov i We have generalized the duality relation for internal massi The computation of cross sections at next-to-leading order In recent years much progress [11 – 13] has been achieved on th Finally, we have extended the duality relation from Feynman Particular care has to be taken with gauge propagators in bot -independent results. η to particles. Real massestranslation just parallel to modify the the realparticles position axis, introduce and of a thus the finite docomplex-mass not imaginary poles propagators affect contribution are t located in at thei athe finite + imaginary d level amplitudes, including resultsthis in amount compact of analytic informationcalculations fo at at the the tree one-loop level level. can be exploited evaluation of real and virtualgiven radiative by corrections. multileg Real tree-level (one-loop)ticle matrix phase-space elements of to the physicalwell process. as other The methods loop-treefeature du that [3, relates 14 – one-loop 16]: andallels they phase-space the recast contribution the of virtual the radiative real corre radiative corrections. duality relation owing to thethis presence issue, of and unphysical extra havebe g identified applied the in its different originalpropagators. gauge form, This choices which avoids the wh includes introduction the ofcontribution additional sole of si sing unphysical gauge poles. tions and scattering amplitudes.from One-loop scattering tree-level ampl scatteringenter amplitudes the (or, computation more ofpropagators precisely are tree-level replaced scattering by dual amplitudes propagators.over This a tree-le single-particle phase space to get the one-loop scatt of unstable particles are cut. compensates for the absencedependence of on multiple-cut contributions with propagators regularized by a newprescription. complex It Lorentz-cova is defined through a future-like auxiliary coordinates of the loopintegrals and momentum single-cut we phase-space integrals. havebut The derived calcu introduces a several duality subtleties. re propagator The modifies location the in original the + comp 11. Final remarks Applying directly the Cauchy residue theorem in the complex loop integrals are then written as a linear combination of JHEP09(2008)065 ). We i q ( G depends on the poles, according BF is gratefully i , the computation q N 0 e derivation of the q hristian Schwinn for under contracts FLA- 35505), and MCnet S.C., F.K., G.R.) were -th residue in eq. (4.2), Zoltan Nagy and Dave i able oop Feynman diagrams is uted numerically, together the propagator ver, it involves some subtle oss sections. In particular, first introduce the notation ´nunderi´on grants FPA2007- )), where ependence on the subscripts i l/numerical techniques to the al results along these lines are q rared or ultraviolet) divergent x. y evaluated in dimensional reg- y relation in eqs. (4.6) and (4.7) n of the ics in Florence: we wish to thank − j q + ( i q ( -th pole, i.e. the location of the pole with G i – 36 – ) = j q ( is a linear combination of the external momenta (see G ji k Advancing Collider Physics: from Twistors to Monte Carlos )= i reduces to the evaluation of the residues at q ) − N ( j q L to explicitly denote the location of the Applying the residue theorem in the complex plane of the vari The extension of the duality relation from one-loop to two-l Major fractions of the work were completed while three of us ( S.C. would like to thank Antonio Bassetto, Michael Kr¨amer, The evaluation of the residues in eq. (4.2) is a key point in th 0 (+) i of the one-loop integral A. Derivation of the duality relation In section 4 we have illustrated the derivation of the dualit Soper for discussions.pointing ref. We [15] wish out to to us. thank Gudrun Heinrich and C by using the residuepoints. theorem. These The points derivation are is discussed simple. in Howe detail in this appendi to directly combine real and virtual contributions to NLO cr Acknowledgments Financial support by the Ministerio de e Educaci´on Innovac using the duality relation,evaluation we of can apply the [3] one-looppart mixed virtual of analytica the contributions. correspondingularization. dual The integrals (inf The can finite bewith part analitycall the of finite the part dualpresented of in integrals the refs. can real [3, emission be 4] contribution. comp and Parti further work isunder in investigation progress. [5]. 60323 and CPAN (CSD2007-00042),VIAnet by (MRTN-CT-2006-035482), the HEPTOOLS European (MRTN-CT-2006-0 Commission (MRTN-CT-2006-035606), by theacknowledged. INFN-MEC agreement and by BM participating in the Workshop at the Galileo Galilei Institute (GGI)the for GGI Theoretical for Phys its hospitality and the INFN for partial support. to eqs. (4.1) and (4.2). duality relation. To makeq this point as clear as possible, we loop momentum while ( negative imaginary part (see eq. (2.12)) that is produced by further simplify our notationthat with label respect the momenta. to the We explicit write d eq. (2.2)). Therefore, to carry out the explicit computatio JHEP09(2008)065 ). 0, q (+) 0 ( q with → (A.4) (A.5) (A.1) (A.3) (A.2) G 2 q me that the esidue of the : therefore the (+) 0 1 q q 2 . . = ) second square-bracket | , 2 q  | q 0 ( = 0 prescription; under this i 0 ality in the second line of i + t singularity. The second (+) q 0 qs. (4.3) and (4.4)) and we idue prefactor in eq. (A.1), δ q + cted the pole with negative he definition of the on-shell   0 prescription either in 1) and (A.1)) of the residue = 0 i 2 gration over 2 0 1 ) q q dq j   k − ) , becomes singular when 1 Z j 2 0 + k ) 1 q = . 2 q − ) ( + q , and we simply evaluate the term 2 ( 0 1 0) j q q i i (+) + 0 ( k j δ q Y |} − G − + q 0 | − 2 0 prescription of the Feynman propagators   2 q i = j q 0 dq q 0 Y q q → ( ) gives → { p p Z   – 37 – q  j ( = q (+) 0 i = 0 prescription from the previous expression. This q G ) i ) and its prefactor — the associated factor arising = ( 2 q = (+) 0 q ): ( q 0 (+) ( 0 q 1 q 2 and q 1 G ) = Res − G (see eq. (A.3)). However, we observe that the result →   ) p q } q /q ) 0 | ( 2 q j 1 ) are not singular when evaluated at the poles of q k (+) G (+) 0 0 | j → ). q q = } k → j i + = q ) = lim k = 0 ) q (+) 0 q + q q ( q { ( 0 ( + , the prescription eventually singled out the on-shell mode q = q G G ( q G 0 ( q } Res G (+) 0 { j ) ( h G q Y (+) 0 q ). q (   = Res 2 = G q 0 0 ( q q { + ) in eq. (A.1). On the basis of this observation, we might assu δ q ( 0 prescription has no effect on the actual calculation of the r Res i G ) or in | q → | We now consider the evaluation of the residue prefactor (the The computation of the residue of (+) 0 q 0 prescription has no regularization effect on such end-poin 0 prescription also has no effect on the calculation of the res positive definite energy, in eq. (A.3) can be obtained by removing (neglecting) the Hence, the where (see eq. (2.12)) we re-label the momenta by and this corresponds toi an end-point singularity in the inte factor in eq. (A.1)). We first recall that the ( thus leading to theeq. result (A.3) in is eq. obtained (4.3). by removing Note the that the first equ propagator equality in the seconddelta line function of eq. (A.3) simply follows from t has played an importanttheorem role to in the the computationimaginary application of part, (see the eqs. loop (4. integral: having sele is fully justified. The term ( i since the propagators We might thus compute the residue prefactor by removing the In the next paragraphs,separately we compute follow the the steps residue of of section 4 (see e from the propagators assumption we obtain JHEP09(2008)065 , ) s 0 j i 0). k 2 j i . . . mal k + finite (A.6) (A.7) q + ( vanish. . + j 2 G in the ex- | ) k j cut q · | k − 0 as a = (+) 3 0 q i 0 + q 2 q 1 L 0 prescription. q i   − = + 0 . 0 0 j q /q k cut | 0 j at small values of q − | 2 k 0 prescription in both (+) 0 = factors ( i 0 L q 0 i (+) 0 q 2 ned and leads to a well- q he evaluation of the one-   fined, but, when inserted − 1 0). The first equality on the 0 j 2 j i i 0 prescription in the residue Y k cuts, i treatment of the Feynman rem (as in eq. (4.1)) involves − ght-hand side of eq. (A.6) into integral, since the singularities ually not true as shown by the + -defined result: the integration = + from setting ual (see eqs. (3.7) and (3.9)) to A consistent treatment requires 2 = eq. (4.1). The possible singulari- j ) 1 j 2 (+) 0 q qk k q 2 = + 0 q q j (   Y 0   ; then we obtain i | j Y q = end of the computation, thus leading to the + | . Performing the limit of infinitesimal values are regularized by the displacement produced | 0 2 (i.e. from expanding   ) 2 q = | 1 | j ) /q – 38 – q j 0 / k → 0 | very q j 0 k / j 0 prescription in the Feynman propagators k + 0 i k + 0 (+) 0 i q i 0 q q ( i ( = are now regularized by the Feynman |− 0 / , of the FTT. The ensuing contradiction with the FTT − q q j 2 | 2 ) Y j   2 cut j ) k j − 0 prescription as mathematical distribution. Applying thi k   i 1 ≃ k + 1 L = + 0), though possibly small, quantity; the limit of infinitesi j + i k (+) q 0 q ( · q (+) 0 ( 6= / q q G = 0 2 0 i q j has to be dealt with by considering the imaginary part at its on-shell value −   Y ) 0 0 j j q   (+) 0 k k | q q + | 2 q ( computation of the residue prefactor in eq. (A.1): the 0 has to be taken only at the G i j ) and Y j j k is singular at any phase-space points where the denominator = Y q + The discussion of the previous paragraph illustrates that t The result in eq. (A.7) for the residue prefactor is well-defi strict   q ( interpretation of the ensuing G strict procedure, we obtain To recover a well-defined result, we have to reintroduce the prefactor. We might thus maintain the Inserting (through eqs. (A.1) andeq. (4.2)) (4.1), eq. we (A.3) arrive and atfrom the a ri the well-defined result propagators for 1 the one-loop pression on the square-bracket (note, in particular, that The expression on the right-hand side of eq. (A.5) is well-de (through eqs. (A.1) and (4.2))over in eq. (4.1), it leads to an ill (thus, for instance, 2 However, this result for thethe one-loop sole integral 1-cut is contribution, exactly eq values of The last equality on the first line of eq. (A.7) simply follows to the FTT vanishes; thisexplicit is one-loop obviously calculations unlikely, and performed it in is section act 5. loop integrals by thesome direct subtleties. application The of subtleties the mainly residue concern theo the correct and still keeping prescription in the calculationthe of the residue prefactors. can be resolved only if the total contribution from multiple by the associated imaginary amount defined (i.e. non singular) expressionties once from it each is of inserted the in propagators 1 second line follows from 2 JHEP09(2008)065 , , 0 i j i q ηk here- (A.8) (A.9) → 0 in the i (A.12) (A.11) (A.10) q ∞ → , 0. 0  > # /q 0 ) i 0 i − j q q k ( + , 0 2 G i | ⊥ q | q N =1 dependence in the 1 Y i , = 0 − η q " − ordinates (see eq. (2.6))   } q (+) + j 0 q , + gration contour at ify this point, we explicitly < ) (see eqs. (A.7) and (A.8)). e constraint = q ηk putation in eqs. (A.1), (A.3) covered by reintroducing the one-loop integral can then be + 0 g to the replacements + 2 0 q 0 q be introduced by applying the , ) . i ) i 0 tion performed in this appendix   > q Im η ) ) ( − { 2 j (+) + − 1 q 2 G k q q ( ) = ( j Res + + − k N µ =1 δ i Y q η + ( + q + ( X G q with + ( dq  ) 0; we finally obtain j dq ⊥ Y , j Z q (see figures 2 and 3). We can now compute the > Y (+) + ) gives , plane) of the pole with negative imaginary part Z   q – 39 – q 0 = 0 −   + ( i q η + q → (  ). Thus (see eq. (2.12)), we have q ) − G = ) Z − q + − 1 q q ⊥ ( q q 2 ( 2 q 2 , G (+) 0 πi ⊥ ) ) lim G − q 2 q q } − − = ) with ( q q 0 0 − Z q ( ( = (+) + , θ θ q   0 = ) = η j = (+) ) = + + k q q ) = { N 0 prescription (and not its actual magnitude) is relevant. T = ( i q + ( µ q η G Res (  } G ,...,p 2 (+) + j q Y ,p = 1 +   p q ( is positive, in eq. (A.7) we can perform the replacement { ) , see the discussion above eq. (A.1)). 0 N i denotes the location (in the q ( q Res is the vector L − µ (+) + j η q q The computation of the residue of In the following we explain in more detail the origin of the The analogue of the term in eq. (A.1) is 0, only the sign of the → i j fore, since where of where the requirement of negative imaginary part leads to th which is the resultoriginal in labels eq. of (4.4) the (to momenta be of precise, the eq. loop (4.4) integral accordin is re k prescription of the dualleads propagators. to The the explicit introduction calcula of the future-like vector residues in eq. (A.9)and by (A.7). closely following the analogous com that is produced by the propagator lower half-plane of the complex variable where where we have applied the residue theorem by closing the inte As discussed in sectionresidue 4, theorem different in future-like vectors differentshow can systems the of application coordinates.rather of than To in the clar space-timeevaluated coordinates residue as (as theorem follows: in eq. in (4.1)). light-cone The co JHEP09(2008)065 s µ η are 0’ is µ (A.13) η . → at fixed 0 in the − 0 i 0 /q q e vector 2 → ⊥ ding on-shell ), eqs. (A.8) enerates dual q iρ 2 = q + ( q plane +   δ j re as in eqs. (A.7) ηk xiliary vectors 2 j 0 i he left-hand side to the k d advanced propagators. 4.6) and (4.7), while the dix are very general: they 0) in eq. (A.8), whereas it ). − + on (by means of the residue y the residue theorem. The ng the limit of infinitesimal te the advanced propagator 1 nfinitesimal limit ‘ 7.5). The generalized one- 2 j .13)). > ng the integration contour at 0, the Feynman propagators ) ⊥ j 2 0. k k η → · > + 0 ⊥ 2 q i q ( √ 2 , such that the poles of the advanced 0 0 j η − is positive, we have thus implemented q is kept finite). Finally, we perform the Y 1 + = ρ   − j q k + = is time-like ( η − q − η /q – 40 – 2 + 2 ⊥ that depends on the specific system of coordinates q − j = η where, in the present case, we have introduced the k + q j = 0) with   (+) + ηk q − − and, evaluating the one-loop integral, we perform first the 2 0 i , η 1 /q ⊥ j − − j 0 Y → ) ] , k 0 0 = + q 0. We thus obtain eq. (7.5), whereas the advanced propagator i − ) in the Feynman propagators and then the limit η ρ 0 in both equations, 1 /q − (+) + → q − 2 j = ( leads to a residue prefactor with exactly the same light-lik > j sign( = k k µ 0 iρ + ⊥ 0 η q iρ η + ) (see eqs. (4.3), (A.3) and (A.12)); the residue prefactor g i q   2 j − = 0) in eq. (A.13). ) q j ) always replaces the Feynman propagator with the correspon ( 2 2 qk k q q + η and 2 ( δ 0 (at fixed + 0, analogously to eq. (A.8). Since G − i j q q Y ( → ) = [   q G 0 ( i = A The residue prefactor is evaluated by using the same procedu It is important to note that, owing to the on-shell condition We also note that the use of the residue theorem in the complex We conclude this appendix by briefly describing the derivati The main features of the calculation presented in this appen j in the lower half-plane of the complex variable Y G   infinitesimal limit the replacement advanced propagators. We apply the∞ residue theorem by closi future-like vector and (A.8). We obtain We see that the residue produces the same factor as in eq. (A.3 and (A.13) have the same form, although the corresponding au performed in the two differentas types of propagators. We rewri have not been alteredphase-space integral by on going the from right-hand side. the one-loop integral on t limit different: though The last equality invalues this equation of has been found by performi behave exactly asadvanced in propagators the remain case unchanged (since of the duality relation in eqs. ( propagators do not contribute. Performing the limit is light-like ( that has been actually employed (see eqs. (4.4), (A.8) and (A theorem) of theloop generalized integral duality on relationBefore the stated applying left-hand the in residue side eq. theorem, contains we ( can both specify Feynman how the an i propagator as in eq. (A.13). are valid in any systemresidue of of coordinates that can be used to appl propagators with an auxiliary vector values of JHEP09(2008)065 0, of , is ives real 1 > (B.4) (B.5) (B.3) (B.6) (B.1) (B.2) λ 1 n ), and , λ n ) . . . ) = 0. To 1 . . 1 ) − λ 2 : n . . n λ λ − ,...,λ ( F 2 + 1 real variables + θ 3) λ 1 = 1 − ≥ λ + · · · . i n 1 ( + nd (B.6), we obtain + = 1). λ 2 ( n n λ 1 .3) and, setting function λ (B.1), at least one of the F − , + n + id for 0, we can analogously obtain are replaced by the single 1 ) + cyclic perms re-like nature of the vector F λ 1 ... , 2 ( · · · y, we consider a set of < − ) λ and is just a consequence of n ) = 0 2 i )= + i λ n λ λ n i λ ... θ λ + + and ), and then we use ηp ( ) + 1 ) + cyclic perms 1 2 θ ) has a fully symmetric dependence 1 1 is always equal to unity. λ λ · · · n ) λ = λ − 1 n − · · · variables (i.e. ) = 1, which follows from the presence ,...,λ n ) = 1. To obtain eq. (B.5), we exploit i ( + + + λ 1 2 F . 1 λ θ 1 n λ + 2 λ n λ + λ = 1 when one of the variables, say , λ ( 2 λ λ + ( 1 + θ n ,...,λ λ θ n λ )= ( + 1 = 0 + λ n ( F n · · · λ i 1 λ n ( λ + λ λ · · · F )= + n – 41 – ( + ). Starting from 1 + ... θ F 2 n + =1 − · · · ). 14 λ i ) X n i · · · ,... 3 · · · + λ λ ... θ + λ ( ,... +1 i + + i i ) 1 θ + λ i 2 2 λ λ + 0. λ ( ) λ λ ( λ 2 θ n + ( ( + · · · > , that fulfill the constraint λ ) λ θ θ 1 i + ( n 1 + − θ i 1 + λ 0, we would have obtained ,λ 2 λ ... θ ) λ λ ,λ 2 ( ) > + · · · θ · · · 2 λ i + ( ,...,n ( ) λ · · · λ 1 1 2 θ ( 1 + · · · − ) = 1 from the induction assumption. This completes the proof , 1 λ i λ + n ( n − ( + . If we can show that λ ) (note that the two variables F 1 n i θ ( i n F λ = 1 λ λ ( i ( θ ... θ ) and from ); the sum of the corresponding terms on the right-hand side g )= )= ) = ) n ). Therefore, we obtain ) ... θ n ,...,λ n 1 2 λ 2 2 ) λ λ λ ,...,λ i ... θ ,... ,... ( + i 2 must have a positive value; , λ λ θ ) , with + + 1 ( λ +1 i i i ,λ i 1 = 1), we shall prove that it is valid for θ λ 1 1 variables λ λ λ · · · ,...,λ ( ,...,λ λ ( + λ ,λ 1 − 2 1 i i ( n θ = n + 1 − θ λ F n i ,λ ,λ , λ λ ( ) ( 1 λ F 1 n We consider the various terms on the right-hand side of eq. (B The proof is simple. We first note two properties: owing to eq. To present the proof of eq. (B.2), we first define the following Summing the terms on the left-hand side of eqs. (B.4), (B.5) a Equation (6.9) simply follows from setting 1 ( λ Note that, starting from λ F · · · · · · ( θ − ( ( ( plays no role in eq. (6.9). 14 n n n n θ we have: on the The equality in eq. (B.4) simply follows from variables momentum conservation to get (i.e. positive, from these two properties it follows that Then, we proceed by induction. Assuming that eq. (B.2) is val obtain eq. (B.6) we simply use momentum conservation, namely eq. (B.1).η Note that the futu F B. An algebraic relation Here, we provide a proof of the relation (6.9). More generall of variables F F We shall prove the following relation: F hence variable eq. (B.2). JHEP09(2008)065 , its zero- i p massless , and the tree-level is e ‘dangerous’ in the are displaced slightly ole-induced (forward- K P K uantity include diagrams er the right-hand side of dpoles linked to a massless es’, namely tadpoles linked e, they are harmless in any ved from the set of one-loop o vanishing tadpole diagrams q q black disk denotes a generic tree lated to the tree-level forward- e eq. (10.4) a well-defined rela- particle grams that are obtained by cut- ). This regularization procedure ishes. , the corresponding diagram in the ,... K ) q 0)). In this case, the theory is consistent with no associated antiparticle (i.e. the ( i P K (+ ). This amplitude is the full tree-level am- / ). This single line necessarily corresponds to ← ). If the 1-particle tadpole is linked to the ) q ,... – 42 – left ( ) q right P ( ( P +2 ← (tree) N ) A effect that it can eventually produce in the right-hand of q ( i ) has to be defined starting from a regularized version of the P p 0) of a massless particle ( i only ,... +2 (+ ) (tree) N / q ( A propagation of a particle P K ← ) q q ( A one-loop Feynman diagram with a 1-particle tadpole (left) P ( is the quantum of a real bosonic field). If the particle +2 K (tree) N e A We introduce a very simple regularization procedure of tadp To illustrate the origin of the possible ‘danger’, we consid In any consistent theories, the diagrams with 1-particle ta zero-momentum C. Tadpoles and off-forward regularization The one-loop Feynman diagrams that contribute to a generic q diagram that isdiagram. obtained by cutting the tadpole (right). The with tadpoles. Amongto them a there single are line of also the ‘1-particle diagram tadpol (figure 11– Figure 11: the ill-defined propagator 1 particle tree-level scattering amplitude istion, also ill-defined. To mak plutude and, therefore, itting includes 1-particle also tadpoles the tree-level (see dia figure 11– momentum propagator is ill-defined (it gives 1 has to be consistent: the eq. (10.4) is the cancellation of the terms that correspond t at one-loop level. scattering) singularities: the two momenta of the on-shell line are considered todirect computations be at vanishing, one-loop bydiagrams level: definition. to they be are Therefor context computed. simply of remo loop-tree However, duality at their the effect amplitude level. may appearthe to duality b relation inscattering eq. amplitude (10.4). Here, the integrand is re (perturbatively stable) only if the 1-particle tadpole van (possibly ill-defined) amplitude JHEP09(2008)065 (C.1) , i ) nd colours 1 i q ( p ly replaced by ) P of | q (see figure 12) is ( ) q ∗ P g ) q → ) is obtained by start- 1 ,... − 1 q ) g, γ ( q 1 pole-induced singularities: and photons (figure 12). q ht-hand side of eq. (C.1) tions, the massless parti- P ,... . The possibly ill-defined ) 1 ) − ther than considering the µ q q ( q P ( ( ur, we first sum over spins, ). − P P P , -forward regularization consis- ). ) ) g of the sum 1 tering amplitude (cf. eq. (10.5)): q q ( ← ,... of the on-shell external lines with arities. ancellation is eventually the con- q ( ) we consider the forward-scattering ) ,... P 1 P ) q ( q → ( 1 ( q 1 P +2 ( P q ( (tree) N t) scattering. P ← +2 ) |A ← (tree) N q ) e ) ( A q q , we first combine the particle and antiparticle ( ( P ( P P – 43 – ). The regularized version is defined as follows: are on-shell. It is important to note that the P h ( 1 +2 q ,... (tree) N +2 ,... ) is obtained by starting from the corresponding regu- (tree) N A q ( A color and X , ) and replacing Feynman propagators with dual propaga- P q i )+ ); spin ← p ,... ) ) ) q q ,... ,... ( q )= ( ) ) ( P 1 1 P ( P q q ( ( ,... q ← ) +2 P P 1 g, γ ) (tree) N q q ; in particular, it includes the coherent sum over the spins a 0), related to forward-scattering kinematics, are obvious ( ← ← ( A i 1 ) ) P q µ 1 P q q 0) when considering ( q (+ ( i ( , although both / ← 1 Off-forward regularization of tree-level diagrams with tad P P +2 + ) q ) ( and ( 1 q 2 (tree) N off-forward regularization has no corresponding antiparticle, we consider the limit q ( has a corresponding antiparticle q ) ( 6= +2 +2 A 1 P q P P P (tree) N (tree) N ( that can produce tadpole-induced singularities are gluons q A contributions and then we consider the limit A if if − +2 K The gluon case is very trivial, since the colour sum on the rig Within the Standard Model of strong and electroweak interac As discussed in section 10, the amplitude q • • (tree) N (( A / 1 We consider these explicit examplestently to leads illustrate to how the the off cancellation of tadpole-induced singul directly cancels any tadpole-inducedsequence singularities. of The colour c conservation. To be precise, the couplin The key pointforward-scattering of limit the at off-forward fixedcolours regularization and, values possibly, is particle of and simple: the antiparticle,limit. and spin then ra and colo cles Figure 12: where expression in eq. (C.1) includesmomenta the wave-function factors contributions from particle (left) and antiparticle (righ off-forward. We thus consider the following off-forward scat of the wave functions of the incoming and outgoing particles propagators 1 ing from tors. The larized version of JHEP09(2008)065 , P )= and 1 (C.4) (C.3) (C.5) (C.2) ) and q c ) and q lead to − 1 ( ∗ P q, γ q,q (figure 12– ) ( − 1 µ q 1 ) P ( λ ( P , q,q ) V ( , , . q 1 ( ) i i n takes place after . In this case, the αβ 1 , ) and ) ) , l each other for any P q # νµν P ( ase of scalar particles. is a charged scalar, ) ) ) M M ollowing relations: αβ 1 left q λ ( ν 6= ( P ) ) # M − + and ) ε ) ) s 1 q / ∗ ( ) + M M ) P q / 1 M v γ M hey cancel each other. )( 1 q ) + + )( 0 q / M µ 1 + 0 + γ − 0 0 n vectors of the charged vector q γ − γ 0 q q ( + q / ) q ally due to charge conservation, q, )( 0 produces the factor )( )( 1 P − )( 0 (figure 12– )( q ) q 0 − ∗ γ ( q M M γ )( )(1 + ) is the Lorentz index of the photon). ( 1 γ M P s )(1 ) − ( + + P 1 µ M M v + q lead to the factors M q,q ( ( 10 10 2 + + )(1 , 1 10 is the color index of the gluon, and q q )(1 + − P ∗ 1 ( ( q ) 10 ( M γ q q / / =1 a X q M q ) νµν s ( ( p p ( ( 1 µ µ p + Γ 2 2 q + − P 2 γ γ ( q / ), respectively. The sum over the colours of the ∗ p h 1 h – 44 – is replaced by the vertex Γ ) 1 2 is charged and thus P 1 q / − )) q ) µ ( ( q q ( ) ( q ( and Tr Tr P " " , where ) 1 ( ) P s 1 q ∗ λ ( , respectively ( P ( ν a are the Lorentz indeces of the vector bosons cc γ 2 of the fermion and antifermion contributions. Indeed, µ u ε ) + ) , T ( 1 ) = ) = ) = ) = 1 1 µ 1 is a charged fermion. q ν 1 1 q q q q γ ( ( q q ( ) and ) ) ( ( ν,ν X = 1 and ) q P s s ) ) + P ( ( ( β q s s ) s ( β ( ( and ≡ v q u ∗ q P ) ) = 0, independently of the specific case (gluon, quark, ghost ( v u ( s ) ) γ µ ν a ( ) 1 ) 1 of the vector boson. − µ P γ u 1 q T q γ ( ( q ) λ ) 2 ) ( 1 q,q ) , s s ( q ( q α ( α and P ( ( µ v =1 ) u X ) ) ) s s µ q s λ 2 , where ( ( 2 ) ( ( , , . v 1 u P V q =1 P =1 X . . . 2 X 2 s , s , + + − =1 =1 X q X ), respectively. Therefore, these two contributions cance s 1 s 1 thus gives Tr( νν − is a charged (massive or massless) fermion, the cancellatio is a charged scalar particle, the couplings is a charged vector boson, the cancellation occurs as in the c g P q,q µ ( P ) P P µ 1 polarization state ) ). To be precise, we can consider explicitly the three cases: q ), respectively. Including the wave-function polarizatio λ If If To show that the expression in eq. (C.3) vanishes, we use the f In the photon case, the particle If 1 ( are the colour indeces of q + V ( 1 q the sum of the couplings summing over the spin states is a charged vector boson and right which identically vanishes. − proportional to the colour matrix c fixed cancellation of the tadpole-inducedand singularity it is is eventu achieved by summing the contributions of the factors ( . . . ) of particle These two factors simply differ by the overall sign, and thus t particle To be precise, the scalar vertex ( P ( bosons, we can define The couplings JHEP09(2008)065 n 4 . , →  (C.6) 2 − (2006) 200 ) e , 1 + q e 01 Nucl. − , ediately see Ann. Phys. q ( , 7). 0 µ JHEP nt generation for g , i Phys. Rept. 2 1 ) Commun. Math. , , , presented at the e M , J.R. Klauder ed., (1963) 697. )+ , presented at the − µ q 1 24 , presented at the q / 10 , September 2006, Zurich q )( ult of an elementary compu- d GDE Joint Meeting) ]; 0 he explicit expressions of the + γ er, in preparation. µ 1 groups/particle/hp2/; A-GDE2006/; − q 0 q Selected papers of Richard Feynman )(1 rch Predictions for all processes Electroweak corrections to ]. + ( M µ Magic without magic ) − , March 2007 Heidelberg Germany q q / Acta Phys. Polon. ( From trees to loops and back , + hep-ph/9904472 , in µ 1 γ q ]. h ]. ( – 45 – On-shell methods in perturbative QCD ]. M Tr Twistor space structure of one-loop amplitudes in  − hep-th/0406177 (1999) 33 [ = =4 i Multiparton amplitudes in gauge theories ) ]. M B 560 fermion processes: technical details and further results + (DPG Spring Meeting) hep-ph/0505042 from the solutions of the Dirac equation. The two relations i 4 q / (2004) 074 [ arXiv:0704.2798 ) hep-th/0312171 07 s )( → ]. ( 0 10 v − γ e + , Ph.D. thesis, University of Dresden, Dresden Germany (200 e : High Precision for Hard Processes at the LHC Closed loop and tree diagrams Quantum theory of gravitation and Nucl. Phys. 2 matrices. Using the relations in eqs. (C.5) and (C.6), we imm JHEP Semi-numerical calculation of N-point loop integrals Automating methods to improve precision in Monte-Carlo eve (2005) 247 [ ) )(1 + , s A new method to compute multileg one-loop cross sections , γ ( γ hep-th/0509223 Perturbative gauge theory as a string theory in twistor spac (2004) 189 [ M A new method to compute multileg one-loop cross sections (2007) 1587 [ HP u + 1 252 B 724 322 q / hep-th/0510253 ( µ γ h Switzerland http://www-theorie.physik.unizh.ch/resea (1991) 301 [ 142 [ Phys. L.M. Brown ed., World Scientific, Singapore (2000) pg. 867. Freeman, San Francisco U.S.A. (1972), pg. 355 and in Workshop T. Gleisberg, November 2006, Valencia Spain http://ific.uv.es/˜ilc/ECF gauge theory http://www.dpg-tagungen.de/program/heidelberg/. G. Rodrigo, Conference Heidelberg A. Denner, S. Dittmaier, M. Roth and L.H. Wieders, charged-current International Linear Collider (ILC) Workshop (ILC-ECFA an particle colliders Phys. (NY) fermions + Tr [7] E. Witten, [2] R.P. Feynman, [3] S. Catani, [8] F. Cachazo, P. and Svrˇcek E. Witten, [5] S. Catani, T. Gleisberg, F.[6] Krauss, A. G. Brandhuber, Rodrigo B. and Spence J. and Wint G. Travaglini, [1] R.P. Feynman, [4] T. Gleisberg, [9] Z. Bern, L.J. Dixon and D.A. Kosower, that the expression in eq. (C.3) is equal toReferences zero. eq. (C.5) are obtained fromtation eq. of Dirac (C.4), and eq. (C.6) is the res The two relations in eq.Dirac (C.4) spinors are directly derived by using t [11] M.L. Mangano and S.J. Parke, [10] A. Denner, S. Dittmaier, M. Roth and D. Wackeroth, JHEP09(2008)065 , , , D , , auge Nucl. Phys. , Phys. 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Phys. J. (2004) 006 [ , (1988) 759; 09 (2000) 014009 [ hep-ph/9804454 arXiv:0802.4171 Yang-Mills theory R. Britto, F. Cachazo, B. Feng and E. Witten, P. Draggiotis, R.H.P. Kleiss andamplitudes C.G. Papadopoulos, R. Britto, F. Cachazo and B. Feng, Nucl. Phys. M. Kramer and D.E. Soper, collisions Phys. Rev. Ph.D. Thesis, DESY-THESIS-2007-042, University of Hambur B 306 F. Caravaglios and M. Moretti, Feynman graphs 62 [ JHEP [14] D.E. Soper, [15] T. Kleinschmidt, [12] F.A. Berends and W.T. Giele, [16] M. Moretti, F. Piccinini and A.D. Polosa, [13] F. Cachazo, P. and Svrˇcek E. Witten,