BBounds on Crossing Symmetry
Sebastian Mizera [email protected]
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
Proposed in 1954 by Gell-Mann, Goldberger, and Thirring, crossing symmetrypostulates that particles are indistinguishable from anti-particles traveling back intime. Its elusive proof amounts to demonstrating that scattering matrices in differentcrossing channels are boundary values of the same analytic function, as a consequenceof physical axioms such as causality, locality, or unitarity. In this work we reporton the progress in proving crossing symmetry on-shell within the framework ofperturbative quantum field theory. We derive bounds on internal masses above whichscattering amplitudes are crossing-symmetric to all loop orders. They are valid forfour- and five-point processes, or to all multiplicity if one allows deformations ofmomenta into higher dimensions at intermediate steps.
I. INTRODUCTION
Ever since its introduction in 1954 by Gell-Mann, Goldberger, and Thirring, crossingsymmetry has been widely believed to be a fundamental property of Nature [1–3]. It postulatesthat particles are equivalent to anti-particles with opposite energies and momenta, or—moreprecisely—that their scattering amplitudes can be analytically continued between differentcrossing channels. It is routinely taken as an assumption in various bootstrap approachesto the scattering matrix theory, see, e.g., [4–13]. Yet, crossing symmetry does not directly follow from any physical principle and there is only a limited amount of theoretical evidencethat it holds in the Standard Model or even a generic quantum field theory. For instance,crossing between 2 → → → → can be defined in a certain region of the complexified momentumspace [18–22]. We briefly review this point in App. A. At this stage one is tasked witha purely geometric problem of showing that the envelope of holomorphy of this domainintersects physical regions in all crossing channels on the correct sheet, e.g., using versions ofthe edge-of-the-wedge theorem. The connection between scattering amplitudes in differentchannels is achieved via a complex kinematic region of large center-of-mass energy. Forreviews see [23–25] and [26]. Such proofs are prohibitively long and technical [14–17], and a r X i v : . [ h e p - t h ] F e b hile in principle there is no obstruction to attempting generalizations to higher-point cases,they would certainly not improve our physical understanding of crossing symmetry.In view of these difficulties, Witten proposed to prove crossing symmetry entirely on-shell in perturbation theory, where one might reasonably hope for a simpler and more physicalderivation that could potentially extend to higher multiplicity. While work on this problemis ongoing and will be published elsewhere, the purpose of this letter is to demonstrate thateven using simple arguments one can put O (1) bounds on the ratios of masses above whichcrossing symmetry is satisfied to all loop orders.Since for a CPT-invariant theory crossing is already apparent on the level of Feynmandiagrams, the challenge lies in showing that Feynman integrals cannot develop singularitieswhen continued between any pair of crossed processes. To make the problem well-defined weassume that any overall divergences (such as infrared or ultraviolet), if present, have beenregularized or renormalized. As a consequence, one has to consider scalar diagrams of allgraph topologies with an arbitrary number of loops and external legs n . To each of them wecan assign the function V = (cid:88) e α e ( q e − m e ) , (1)which can be thought of as the localized worldline action. Here α e are the Schwinger propertimes associated to the internal edges e , while q µe and m e are the momenta and masses flowingthrough them, sourced by the external momenta p µi . Due to homogeneity in α e ’s, extremizingthe action requires V = 0, which is a necessary condition for a singularity, equivalent toputting propagators on-shell.Were it not for the requirement of causality, scattering amplitudes would be analyticalong complex paths connecting any two real non-singular points in the space of kinematicinvariants p i · p j , because along such a deformation (cid:88) e α e (cid:12)(cid:12) q e − m e (cid:12)(cid:12) > , (2)ergo, it is impossible to simultaneously put all propagators on-shell. However, such analyticcontinuations generically violate causality, which requires that Im V > iε prescription. Its consistent implementation is whatputs bounds on crossing symmetry.One way of ensuring causality is analytic continuation via a region where V < m e cannotbe too light, or more precisely m e > √ n √ (cid:114) max i (cid:16) M i , (cid:80) j M j − M i n − (cid:17) , (3)where M i are the external masses. These are bounds for crossing symmetry to be satisfiedon-shell to all loop orders in perturbation theory.For instance, the above bounds are satisfied for scattering of massless particles with allthe exchanged states having arbitrary non-zero mass. This result implies crossing symmetry2or a range of low-energy effective field theories, which at present does not have a counterparton the non-perturbative level [15, 17].For n = 4 , M i = M , we have respectively m e (cid:38) . M, m e (cid:38) . M, (4)and for scattering of the lightest states, i.e., m e (cid:62) M , crossing symmetry is valid for n < n = 4 with the samebound in the equal-mass case [27], and for n = 5 [27–30] without attempts to put bounds.The above strategy relies on linear deformations of the kinematic invariants p i · p j ratherthan the momenta p µi themselves. The advantage of doing so is that we can continue betweencrossed processes involving a different number of incoming/outgoing particles, such as acontinuation from 2 → → p µi will in general span an ( n − n (cid:62) p µi and outgoing − p µi , such that the momentum conservationreads (cid:80) i p µi = 0. II. REVIEW OF FEYNMAN INTEGRALS AND THEIR SINGULARITIES
We find it most intuitive to interpret Feynman diagrams in the worldline formalism, whereSchwinger proper times α e are the only dynamical variables. A scalar diagram with n (cid:62) (cid:90) ∞ d E α e U D / e i V / (cid:126) , (5)where V is the localized action and U is the determinant of the Laplacian of the diagram. Forreal kinematics, it is then customary to use the rescaling invariance α e → λα e to integrateout the overall scale λ , which gives up to normalization (cid:90) d E α e δ ( (cid:80) e α e − U D / V E − LD / , (6)where E − LD / V → V + iε with infinitesimal ε , or as a contourdeformation, see App. B. Here we will not deform α e ’s, but instead implement causality bydeforming the external kinematics such that Im V > i, j = 1 , , . . . , n and e = 1 , , . . . , E): V := − (cid:88) i Singularities of Feynman integrals are governed by Landau equations [33], which in therepresentation (6) read [34, 35]: α e ∂ V ∂α e = 0 (10)for all edges e . Since a solution involving α e (cid:48) = 0 gives Landau equations for a simplergraph with the edge e (cid:48) contracted, and we already take into account all graph topologies, weonly need to consider leading Landau equations with α e (cid:54) = 0 (in other words, the analyticcontinuation we will employ avoids sub-leading Landau singularities just as well as the leading4nes). One can interpret them as the classical limit of the action V where all propagators goon-shell according to (1). See App. D for more details [36]. Recent work on Landau equationsincludes [37–41].Given the definition in (7), V is a degree-one homogeneous function in α e ’s, which meanson the solution of Landau equations we have V = (cid:88) e α e ∂ V ∂α e = 0 . (11)This is a necessary (but not sufficient) condition for a singularity. Since leading Landauequations require α e > 0, the definitions (8) and (9) give U > G ij > B. Upper Bound on the Graph Green’s Functions In the following steps we will need an upper bound on G ij that does not depend on thenumber of loops, edges, or external states. As a proxy for its derivation, let us briefly considerthe case n = 2 off-shell, where − p · p = p is allowed to vary, and anomalous thresholds areabsent. We have V = p G − (cid:88) e m e α e (cid:54) p G − m , (12)where m is the lightest of m e > (cid:80) e α e = 1. Since V < p = 0, the actionhas to stay negative before encountering the first physical threshold at p = (cid:16) (cid:88) e ∈ R m e (cid:17) (cid:62) | R | m , (13)where R is the set of | R | intermediate particles, as in Fig. 1. This implies G (cid:54) / | R | . Sincethe labeling of the momenta was arbitrary, we have G ij (cid:54) , (14)because | R | (cid:62) p p R Figure 1. Normal threshold for n = 2, where a subset R of propagators goes on-shell. We remind the reader that analyticity of higher-point on-shell amplitudes is not well-understood because of the presence of anomalous thresholds. The above trick circumventsthis issue by deriving bounds on the individual G ij which are the building blocks entering (7)for arbitrary n . 5 II. BOUNDS ON CROSSING SYMMETRY We will show how to analytically continue between two non-singular points in the realkinematic space, denoted by p (0) i · p (0) j and p (1) i · p (1) j , i.e., where Landau equations are not satisfied.The deformation takes place in the n ( n − / p i · p j , whose pre-image in the momentum vectors p µi can be only realized in an( n − A. Naive Approach We introduce a complex variable z and linearly deform the kinematic invariants accordingto p i · p j = p (0) i · p (0) j + z (cid:0) p (1) i · p (1) j − p (0) i · p (0) j (cid:1) , (15)as well as consider a path in the upper-half plane approaching the two kinematic points at z = 0 and z = 1, see Fig. 2. This deformation preserves momentum conservation and on-shellconditions, p i = M i . z Figure 2. Path of deformation in the upper-half plane. Since V responds linearly to changes in kinematics, we have V = V + z ( V − V ) , (16)where V α := V ( p ( α ) i · p ( α ) j ). Remaining on the original integration contour with α e real, the realand imaginary parts of the leading Landau equations are ∂ V ∂α e + Re z (cid:18) ∂ V ∂α e − ∂ V ∂α e (cid:19) = 0 , (17)Im z (cid:18) ∂ V ∂α e − ∂ V ∂α e (cid:19) = 0 , (18)for all edges e . These cannot be simultaneously satisfied along the deformation path withIm z > 0: vanishing of the imaginary part implies ∂ V /∂α e = 0 for all e , which is acontradiction. Hence there are no singularities in the upper-half plane of z .Nevertheless, this deformation cannot be used because by utilizing the imaginary part of V for the deformation we lost a reliable way of imposing the iε prescription near both of two6hysical points. Put differently, the path of analytic continuation will in general veer awayfrom the physical sheet.An exception to this point are planar diagrams, which have vastly simpler analyticityproperties and crossing for n (cid:54) D+1 can be proven without any constraints on masses, seeApp. E. B. Fixing the iε In order to guarantee the correct iε prescription we will add an intermediate step in thedeformation, which passes through an open set { p ( ∗ ) i · p ( ∗ ) j } for which V ∗ < p i · p j = p ( ∗ ) i · p ( ∗ ) j + z (cid:0) p (1) i · p (1) j − p ( ∗ ) i · p ( ∗ ) j (cid:1) (20)followed by an analogous continuation connecting p ( ∗ ) i · p ( ∗ ) j to p (0) i · p (0) j . We further restrict toRe z (cid:62) 0. There are two cases depending on the sign of V − V ∗ . When V > V ∗ we haveIm V = Im z ( V − V ∗ ) > , (21)which is the correct causal prescription. Otherwise, when V (cid:54) V ∗ , we haveRe V = V ∗ + Re z ( V − V ∗ ) < , (22)since V ∗ < iε is not needed.It remains to prove that a set of { p ( ∗ ) i · p ( ∗ ) j } with V ∗ < C. Bounds on Masses We consider kinematics with − p ( ∗ ) i · p ( ∗ ) j < c for each of the n ( n − / V ∗ and some positive constant c . Using the upper bound on G ij from (14) andcalling m = min e ( m e ) the lightest internal mass one finds V ∗ < n ( n − c − m . (23)Therefore, requiring that V ∗ < − p ( ∗ ) i · p ( ∗ ) j < c < n ( n − m . (24)7n terms of the external masses M i this translates to two types of constraints. Usingmomentum conservation requires on the one hand M i = − p ( ∗ ) i · (cid:88) j (cid:54) = i p ( ∗ ) j < n m (25)and on the other (cid:88) j M j − M i = − (cid:88) j (cid:54) = i p ( ∗ ) j · (cid:88) k (cid:54) = i,j p ( ∗ ) k < n − n m (26)for all i . They together imply the constraints (3). These are the conditions for crossingsymmetry to be satisfied to all loops and multiplicities. IV. OUTLOOK Let us comment on two natural directions for future work: optimizing the bounds andpreventing the momenta p µi from wandering into higher dimensions.With respect to the former, let us notice that the upper bounds G ij (cid:54) / | R | are saturatedon configurations where α e ≈ / | R | for each of the intermediate lines e ∈ R , and α e ≈ G ij ’s simultaneously,because G ij are not mutually independent (for example, they satisfy G ij + G jk (cid:62) G ik ). It is notunlikely that exploiting such inter-dependencies can improve bounds on crossing symmetry,though probably not significantly so for generic quantum field theories. On the other hand,implementing conservation laws for specific processes might improve the bounds, perhapsalong the lines of previous work on dispersion relations [43, 44].Remaining in four dimensions for Im z > × p i · p j treated as a matrix, which would violate linear dependence on z that our argumentshinged upon. Instead, one should employ a deformation directly on the four-momenta p µi that correspond to linear shifts of p i · p j , such as those used in on-shell recursion relations [45].Nonetheless, in some situations it might be possible to get away without doing so, such as inthe case of four-point scattering in two dimensions with equal external masses, M i = M . Inthis setup we have ( p + p ) = 0 and repeating the steps from previous sections gives V ∗ (cid:54) 12 ( s ∗ + M ) − m (27)with s ∗ = ( p ( ∗ ) + p ( ∗ ) ) . Since we can choose s ∗ to be arbitrarily small, it guarantees crossingsymmetry and maximal analyticity for m e > M/ √ CKNOWLEDGMENTS The author thanks Edward Witten for illuminating discussions. He gratefully acknowledgesthe funding provided by Frank and Peggy Taplin as well as the grant DE-SC0009988 fromthe U.S. Department of Energy. APPENDICESAppendix A: Why Crossing Symmetry is Still a Conjecture Here we give a lightning review of the core arguments of Gell-Mann, Goldberger, andThirring [1] in a more modern formulation which can be found in [24, 46]. We start with a localquantum field theory with a mass gap and consider (scalar for simplicity) charged fields ϕ a ( − x )and ϕ † b ( x ) at spacelike separation, x < 0. Following the Lehmann–Symanzik–Zimmermanprocedure we introduce the currents j a ( − x ) = ( (cid:3) − x − M a ) ϕ a ( − x ) , (A1) j † b ( x ) = ( (cid:3) x − M b ) ϕ † b ( x ) , (A2)but do not take the on-shell limit until the very end. For the purposes of this discussion wewill ignore irrelevant normalization factors. The quantity of our interest is C = (cid:90) d x e i ( p b − p a ) · x (cid:104) out | [ j † b ( x ) , j a ( − x )] | in (cid:105) (A3)for p µa and p µb future and past timelike respectively. The remaining n − (cid:104) out | and | in (cid:105) are arbitrary. We can expand the commutator in two different ways. One of them separates x µ into the future and past lightcone,[ j † b ( x ) , j a ( − x )] = θ ( x )[ j † b ( x ) , j a ( − x )] + θ ( − x )[ j † b ( x ) , j a ( − x )] . (A4)The first term is a retarded commutator, which in the integrand can be replaced by atime-ordered product of the two currents, giving (cid:90) d x e i ( p b − p a ) · x (cid:104) out |T j † b ( x ) j a ( − x ) | in (cid:105) . (A5)It is the Green’s function for the process { in , p a } → { out , − p b } with the overall momentumconservation delta function stripped away. Assuming causality and temperedness (polynomialboundedness) of the bra-ket, the integrand has support only when x µ is a future timelikevector. This implies that the Fourier transform is convergent when Im( p b − p a ) is futuretimelike, since only then the integrand is damped by a factor of e − Im( p b − p a ) · x .On the other hand, after relabeling x → − x , the second term in (A4) is a retarded9ommutator for the crossed process, giving − (cid:90) d x e i ( p a − p b ) · x (cid:104) out |T j a ( x ) j † b ( − x ) | in (cid:105) . (A6)The additional minus sign came from reversing the commutator. This is the Green’s functionfor { in , p b } → { out , − p a } scattering, where the bar denotes an anti-particle. It suggests that C might be the difference between the two Green’s functions. However, repeating previousarguments one finds that the integral (A6) converges only when Im( p b − p a ) is past timelike,which has no overlap with the domain of analyticity of (A5). Therefore in order to relatethe two processes one needs to show that C can be analytically continued between the twokinematic regions.To this end, let us evaluate C using the more obvious way of expressing the commutator,[ j † b ( x ) , j a ( − x )] = j † b ( x ) j a ( − x ) − j a ( − x ) j † b ( x ) . (A7)We then use unitarity to insert a complete basis of states = (cid:82) d p I (cid:80) I | I (cid:105)(cid:104) I | and translationinvariance so that C evaluates to (cid:88) I (cid:104) out | j † b (0) | I (cid:105) (cid:104) I | j a (0) | in (cid:105) δ ( p b − p a − p in + p out − p I ) − (cid:88) I (cid:104) out | j a (0) | I (cid:105) (cid:104) I | j † b (0) | in (cid:105) δ ( p b − p a + p in − p out +2 p I ) , (A8)where p µI are the momenta of the intermediate states, which by assumption of the non-zeromass gap satisfy p I > 0. Therefore C = 0 in a region Φ of the Re( p b − p a ) space that lies belowthe production threshold for the lightest intermediate states in both sums.Since the regions of analyticity of (A5) and (A6) border Φ, within which C vanishes,the edge-of-the-wedge theorem guarantees that the two crossed processes must be analyticcontinuations of each other in the region where Re( p b − p a ) ∈ Φ and Im( p b − p a ) belongs tothe union of future and past lightcones.This argument does not yet imply crossing symmetry for two reasons. Firstly, the abovedomain of analyticity has no intersection with on-shell kinematics. In order to see this wecan use a (complex) Lorentz frame, such that in lightcone coordinates p µa = ( M a , M a , , , p µb = ( p + b , p − b , , , (A9)where M a > 0. The constraint of Im( p b − p a ) being timelike implies(Im p + b )(Im p − b ) > . (A10)However, on-shell we need p + b p − b = M b > 0, whose real and imaginary parts give respectively(Re p + b )(Re p − b ) − (Im p + b )(Im p − b ) > , (A11)(Re p + b )(Im p − b ) + (Im p + b )(Re p − b ) = 0 . (A12)10he constraint (A10) together with (A12) mean that Re p + b and Re p − b have to have oppositesigns, which is in contradiction with (A11).The second problem concerns the assumption of temperedness made before, which wasnot necessarily justified. To check it, one ought to consider the setup where (cid:104) in | and | out (cid:105) are vacuum states and all n particles are represented as fields. Repeating essentially thesame steps as above with more book-keeping, one arrives at the so-called primitive domain of analyticity (with no support on-shell) [18–22], where all crossed Green’s functions agree.It is then a geometric problem to show that the envelope of holomorphy of the primitiveregion intersects the real on-shell regions for all crossed processes [14, 16]. Moreover, for2 → → Appendix B: Causal Contour Deformations Let us review how to implement infinitesimal contour deformation that replaces the iε prescription. We start by deforming Schwinger parameters α e intoˆ α e = α e + iεβ e iε (cid:80) e (cid:48) β e (cid:48) , (B1)which satisfy (cid:80) e ˆ α e = 1. Here β e = β e ( α e (cid:48) ) are yet to be determined functions, which areassumed to be ε -independent and vanish at α e = 0 and α e = 1 in order not to alter theendpoints of integration. Using homogeneity of V , the deformed ˆ V = V ( ˆ α e ) readsˆ V = V ( α e + iεβ e )1 + iε (cid:80) e β e . (B2)Expanding in ε gives ˆ V = (1 − iε (cid:80) e β e ) (cid:32) V + iε (cid:88) e (cid:48) β e (cid:48) ∂ V ∂α e (cid:48) (cid:33) + O ( ε )= V + iε (cid:88) e β e (cid:18) ∂ V ∂α e − V (cid:19) + O ( ε ) . (B3)This suggest a natural choice for β e , β e = α e (1 − α e ) (cid:18) ∂ V ∂α e − V (cid:19) , (B4)which satisfies all the required properties and gives the deformed actionˆ V = V + iε (cid:88) e α e (1 − α e ) (cid:18) ∂ V ∂α e − V (cid:19) + O ( ε ) . (B5)11ence for sufficiently small ε and real kinematics, it implements Im ˆ V > Appendix C: Acnode Diagram Example p α α α α α p p p Figure 3. Acnode diagram. Let us consider a classic example of the acnode diagram, illustrated in Fig. 3. Accordingto the definition (8), U is given as a sum over eight spanning trees, U = α ( α + α + α + α ) + ( α + α )( α + α ) . (C1)The graph Green’s functions G ij between vertices where p µi and p µj enter the diagram aregiven by the expressions (9), which yield explicitly G = U α [( α + α )( α + α ) + α α ] , (C2) G = U α ( α + α )( α + α ) , (C3) G = U α [( α + α )( α + α ) + α α ] , (C4) G = U α [( α + α )( α + α ) + α α ] , (C5) G = U [( α + α )( α α + α α + α α ) + α α ( α + α )] , (C6) G = U α [( α + α )( α + α ) + α α ] . (C7)One can confirm that on the support of the constraint (cid:80) e α e = 1, all the G ij ’s are upper-bounded by 1 / 4, except for G , whose bound can be improved to 1 / M = M = M and all theremaining M i = m e = m . The acnodes and real cusps appear when m (cid:54) M (cid:112) √ (cid:39) . M (C8)and hence below our bound (4). 12 ppendix D: Physical Interpretation of Landau Equations Landau equations can be interpreted as the classical limit of Feynman integrals, in whichon-shell propagators describe particles traveling in space-time [50, 51]. For the purpose ofthe discussion below we specialize to the strictly massive case, m e > q µe in terms of Schwinger parameters α e (cid:48) . Let us assign arbitrary orientations to each internaledge. Momentum conservation at each of the V vertices v reads p µv + (cid:88) e η ve q µe = 0 , (D1)where η ve equals +1 when q µe is incoming towards v , − p µv is the total external momentum flowing into the vertex v . Similarly, we have theconservation law for each of the L oriented loops (cid:96) , (cid:88) e α e η (cid:96)e q µe = 0 , (D2)where η (cid:96)e equals +1 when the orientations of the loop (cid:96) and the edge e agree, − q µe = q µe ( α e (cid:48) ) interms of Schwinger parameters and external kinematics we have V = (cid:88) e α e ( q e − m e ) . (D3)Since q e are degree-zero homogeneous functions of α e (cid:48) ’s, the leading Landau equations areequivalent to putting all propagators on-shell, q e − m e = 0 . (D4)Let us trivialize the constraints (D2) by introducing a Lorentz vector x µv associated toeach vertex v . Calling ∆ x µe the difference between x v ’s at the end and beginning of the edge e we assign α e q µe = ∆ x µe = (cid:88) v η ve x µv , (D5)which automatically satisfies (D2). In the classical limit each x µv has an interpretation ofposition of the vertex v in space-time. Let us confirm this by evaluating a scalar Feynmanintegral in position space. Up to normalization it can be written as (cid:90) d DV x v e − i (cid:80) v p v · x v / (cid:126) (cid:89) e G F (∆ x e , m e ) , (D6)where G F (∆ x e , m e ) denotes the Feynman propagator between points at timelike separation13 x e and mass m e , G F (∆ x e , m e ) = (cid:90) d D q e (2 π ) D e − iq e · ∆ x e / (cid:126) q e − m e + iε . (D7)It can be expressed as a Bessel function, which up to overall normalization reads (cid:90) ∞ d α e α D / e exp (cid:20) − i (cid:126) (cid:18) ∆ x e α e + α e ( m e − iε ) (cid:19)(cid:21) . (D8)Therefore the Feynman integral in (D6) evaluates to (cid:90) d DV x v d E α e ( (cid:81) e α e ) D / e i V / (cid:126) , (D9)where, ignoring the iε factor, V in the exponent is given by V = − (cid:88) v p v · x v − (cid:88) e (cid:18) ∆ x e α e + m e α e (cid:19) . (D10)Extremizing V with respect to x v yields p µv + (cid:88) e η ve ∆ x µe α e = 0 (D11)for each vertex v . This is just the momentum conservation (D1). Similarly, varying α e weget ∆ x e α e − m e = 0 (D12)for all edges e , which are the on-shell conditions (D4). We conclude that the classical limit inthe position space reproduces Landau equations and x µv can be interpreted as positions whereparticles interact in space-time. This is the origin of the term “Schwinger proper time” sinceaccording to (D12) the α e ’s measure the proper time (cid:112) ∆ x e elapsed between two interactions,normalized by the particle mass m e .While the most obvious saddles of (D10) are those lying on the original integration contourwith α e and x µv real, in general solutions of Landau equations are complex. Their physicalmeaning remains nebulous. See [52] for a recent review of observable effects of anomalousthresholds at particle colliders.For completeness let us mention that Landau equations have an interpretation in terms ofelectric circuits [35, 53]. This should not be surprising because both can be described as ascalar field on a graph. In this interpretation, component by component, x µv measures voltageat v , q µe is the current flowing through e , and α e its resistance. Then (D1) and (D2) are theKirchhoff’s circuit laws. When a unit current is applied flowing from the vertex where p µi p µj does so, the graph Green’s function G ij = (cid:88) e α e q e (D13)measures the power dissipated in the circuit. Appendix E: Crossing Symmetry for Planar Diagrams An alternative representation of V which is particularly suitable for planar diagrams reads V = (cid:88) S p S F S − (cid:88) e m e α e , (E1)where the first sum goes over all 2 n − − S , withoutdouble-counting the complements ¯ S = { , , . . . , n }\ S . Here F S = F ¯ S := 1 U (cid:88) F S (cid:89) e/ ∈ F S α e (E2)involve sums over all two-forests F S = T S (cid:116) T ¯ S such that T S and T ¯ S only contain verticeswhere particles from the sets S and ¯ S enter the diagram respectively. For example, for thediagram in Fig. 3 we have F = U α α α , F = U α α ( α + α + α ) , (E3) F = U α α α , F = U α α ( α + α + α ) , (E4) F = U α α α , F = U α α α , F = 0 , (E5)where F vanishes due to planarity of the diagram. Comparing the definitions (9) and (E2)it is easily seen that G ij can be expressed as sums G ij = (cid:88) S (cid:54)(cid:51){ i,j } F S ∪ i , (E6)which range over all 2 n − sets S not including labels i and j .What is special about planar amplitudes is that, not counting p i = M i , only n ( n − / p S appear in (E1), where S have the appropriate planar ordering. It meansthey can be deformed independently of each other in the upper-half planes Im p S > p µi are embedded in at least n − p i · p j . Since F S > S , along such a deformationwe have Im V > , (E7)which at the same time imposes the correct iε prescription and allows one to deform between15ny two points in the real kinematic space, thus proving crossing symmetry without anyconstraints on masses.In the case of the acnode diagram in D (cid:62) s = ( p + p ) , t = ( p + p ) in Im s/ Im t > 0, while all the non-analyticity (beyond a certainmass threshold) are confined to Im s/ Im t < NOTES AND REFERENCES [1] M. Gell-Mann, M. 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