JHEP03(2009)054 March 9, 2009 orrections to : February 10, 2009 November 24, 2008 : : Published rdinate-momentum space Accepted Received show that if one represents n field theory follows imme- ution for their corresponding 10.1088/1126-6708/2009/03/054 non-abelian case, we consider [email protected] ulable fluctuations. Combining , ext-to-eikonal corrections. c doi: lines, and obtain a new viewpoint gauge field is represented as a Wil- Published by IOP Publishing for SISSA and Chris D. White [email protected] , d 0811.2067 Gerben Stavenga QCD, Nonperturbative Effects a,b,c We approach the issue of exponentiation of soft gauge boson c [email protected] SISSA 2009 ITFA, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam,ITF, The Utrecht Netherlands University, Leuvenlaan 4, 3584 CE Utrecht,Nikhef The Theory Netherlands Group, Kruislaan 409, 1098 SJ Amsterdam,Institute The for Netherlands Theoretical Physics, UtrechtLeuvenlaan University, 4, 3584 CE Utrecht, The Netherlands E-mail: b c d a c

Abstract: Eric Laenen, ArXiv ePrint: scattering amplitudes from athe path amplitude integral point as of arepresentation, view. first a charged quantized We particle path interacting with integralson a line in soft for a a mixedsuch semi-infinite coo line line segment, segments, together we withdiately show calc that from exponentiation standard in path-integralcolor an combinatorics. singlet abelia hard interactions In with two the for outgoing exponentiation external in terms ofcolor “webs”, factors. with We a investigate closed and form clarify sol the structure ofKeywords: n Path integral approach toexponentiation eikonal and next-to-eikonal JHEP03(2009)054 4 4 6 7 ]. 1 8 11 12 13 14 27 30 31 33 34 40 18 13 – 10 Central to 1 ]. 8 is related to the – 1 ξ atrix element and the 0 (in which the eikonal hese terms to all orders ethods [ → t this indeed occurs [ erturbative gauge theory, be ξ tions. Such radiation generi- e boson corrections and it has re. The statement for the matrix rential cross-sections of the form – 1 – is the coupling constant of the gauge theory, and α , where ]. 9 /ξ ) 1.1.1 Free scalar particle ξ ( m To form an exponential series for a cross-section, both the m See also [ 3.1 Non-abelian3.2 exponentiation at NE order Internal emissions of non-abelian gauge fields 2.2 Next-to-eikonal exponentiation 2.3 Exponentiation for2.4 spinor particles Low’s theorem 1.2 Scalar particle1.3 in an abelian Spinor background particle gauge field 2.1 Eikonal exponentiation 1.1 Propagators as first quantized path integrals log 1 n energy carried away by the soft particles. In the soft limit approximation may be taken),in it perturbation becomes theory, as necessary has to been resum achieved t by a variety of m 1 Introduction Higher order corrections arisingthey from real soft or gauge virtual, bosonscally have leads been in to the p series subjectα of of perturbative many contributions investiga to diffe B Next-to-eikonal Feynman rules C Matrix elements with internal emissions 4 Discussion A Exponentiation of disconnected diagrams 3 Non-abelian gauge theory 2 Soft emissions in scattering processes Contents 1 Introduction resummation is the exponentiationbeen of shown eikonalized for soft both gaug abelian and non-abelian gauge theory tha phase space must exhibit an appropriate factorized structu JHEP03(2009)054 r s, , we (1.1) (1.2) between c G subdiagram of hard outgoing hard lines (including case of hard production n non-abelian perturbation nentiation of disconnected both external lines exactly ssive orders in perturbation ture of higher order diagrams. associated with the web graph. . ore complicated. W exponentiation still holds provided i , C 2 i factored out. By a W c G 0 W by an iterative relation to all orders in ¯ C A . Each web has an associated color factor X W 2 h X C of soft emissions between hard outgoing particle h c – 2 – exp . The non-abelian case is complicated by the G 0 1 exp 0 A A = = A A in abelian gauge theory is simply expressed in the eikonal A ) in terms of disconnected diagrams is reminiscent of anothe , and are diagrams which are two-eikonal irreducible. That i 1.1 ) to are given in terms of webs 1.1 W ¯ C Examples are shown in figure , and the sum in the exponent is over connected subdiagrams 3 ]. ] for a pedagogical exposition. is the amplitude without soft radiation containing a number 13 . Examples of connected diagrams . Examples of webs of soft emissions between external lines i are so-called – 14 A 11 W which is not the same as the normal color factor The nature of eq. ( See [ In the case of more than two external lines, the structure is m 2 3 W ¯ The color factors Figure 1 mean that part ofthe off-shell the photon) Feynman are diagramof removed. which a remains Examples particle-antiparticle after are pair the shown in for figure the perturbation theory. the external lines, with the Born contribution C nontrivial color structure oftheory. the Nevertheless, in Feynman the diagrams case at of two succe external lines, external lines legs in abelian perturbation theory. elements rests upon a thorough analysis of the general struc theory. The result for an amplitude approximation as Figure 2 one cannot partition a webonce into [ webs of lower order by cutting where Here one generalizes eq. ( well-known property of quantum field theory, namely the expo JHEP03(2009)054 ] – 28 15 he collinear ]). A first-quantized 19 how that this is indeed e the problem using the , rmally exponentiate, but ates the field theory path Firstly, it provides a new neralizations of the webs lize the techniques of [ hat of constructing string leading effects with respect he fact that vertices for the 18 e theoretical uncertainty of [ isconnected diagrams. Then perty most naturally emerges e done in practical implemen- orldlines of particles in quan- ikonal corrections. This can interactions of the gauge field t-quantized path integral with ation expansion is sufficient to n of soft radiative corrections appealing for Drell-Yan, Higgs ]. Furthermore, we provide an ]. We will see explicitly that in traightforwardly explore which diating particles moving classi- ion in differential cross-sections also 12 20 , given in terms of normal (rather ns around the classical path. The rom usual combinatoric properties t ourselves here to gluon emissions. , imit and beyond (i.e. corresponding 11 pace, and is deferred for further study. for a brief proof), and thus the suggestion ]. More recently a study was performed [ ]), such that a subset of diagrams arises which A 27 – 3 – 21 – 22 Although at next-to-eikonal order fermion emissions also 4 ]. 29 above). ξ ]. Typically, this involved including subleading terms in t 22 of statistical physics (see e.g. [ Our formalism allows one to straightforwardly consider sub Earlier attempts to include certain sub-eikonal effects wer The essential idea of our approach is as follows. One first rel In the non-abelian case, exponentiation is complicated by t We only consider matrix elements in this paper. Exponentiat 4 ] which have originally been applied in a different context (t depends upon factorization of the multiple particle phase s 17 Feynman diagrams in terms of connected ones. This latter pro using path integral methods (see appendix tations of Sudakov resummation, mostlythe in resummation view [ of gauging th based on a proposal in ref. [ exponentiate. These are then precisely the webs of [ arises of whether itto is the possible gauge theory tothe path relate case, integral. the and the exponentiatio Theperspective result on aim is exponentiation. of important this Secondly,properties for it paper of a allows exponentiation is number survive one at to of to next-to-eikonalto s s l reasons. subleading terms in theoretical analogues of fixedapproach to order Sudakov field resummation has theory also amplitudes appeared in [ explicit closed form solution forthan the modified modified) color color factors weights. to the eikonal limit,be and divided into wediscussed classify a above), subset the and possible which a exponentiatehave next-to-e set an (involving iterative of structure NE remainder in ge generate terms that the which each next order do order. of not the fo perturb evolution kernel in theproduction resummation, which and is related particularly cross sections [ contribute (leading to flavor-changing effects), we restric integral for a particle interactingrespect with to a the gauge particle. fieldtum to That mechanics a is, (rather firs the than external quantum lines field become theory). w Here we uti this representation the eikonalcally, limit and corresponds next-to-eikonal terms to originate thesoft from ra radiation fluctuatio emission verticeswith can a then source, be such interpretedexponentiation that as of individual eikonal corrections emission followsof vertices naturally the form f path d integral. emissions of gluons doreplica trick not commute. However, one can rephras JHEP03(2009)054 (1.4) (1.3) . The propagator T , of square integrable ) 2 is the Green’s function H m , namely hard external prescription. Note that y + per as emerging from a hard iε x will be used in section 2 to he path-integral representa- ical details are presented in We begin with the simplest , and and has a momentum and at final time i  nd spinor particles and in the f x f 3 we consider the extension to x xternal line is created from a p − r for the external line. In the d theory propagators in terms o beyond the eikonal limit, and p nentiate and a remainder term and again examining corrections ns ons in the presence of an abelian = ( , 1 external lines − )] S iε , S ) − ). Soft radiation corrections enter the two- S 3 x,y ( i ( – 4 – x i ) d ( ], which will later on be used in the derivation of δ = [ 17 F – )= ∆ = 0 at space-time point H 15 +). Schematically the propagator may be written as t = 0, and having momentum , (see figure x,y t ( + , F + , )∆ → ∞ − iε at time i T − x S ( i are Hermitian operators working on the Hilbert space F ∆ . Anatomy of an external line, considered throughout this pa S, In the rest of this introduction we review the derivation of t at some final time f interaction at 4-position Figure 3 functions of space-time, andwe we are adopt using the the metric standard ( Feynman particles susceptible to softhard interaction radiation process emission. at time A given e tion of propagators in quantumpresence field and theory, for absence both of scalardemonstrate an a the abelian exponentiation gauge of field.gauge soft field. These radiative results correcti We also considerclassify generalization the of resulting the corrections resultswhich t into mixes a with subset these at whichnon-abelian next-to-eikonal gauge expo order. fields, recovering In the section propertiesto of the webs eikonal limit.the We conclude appendices. in section 4, and some techn 1.1 Propagators as firstThroughout quantized this path paper integrals we will be concerned with for the external line contains the effects of soft radiation. following subsections, we reviewof the first representation quantized path of integrals fiel [ case, that of a free scalar particle. 1.1.1 Free scalar particle The propagator for a free scalar particle between 4-positio point function for the emitting particle i.e. the propagato matrix element exponentiation at eikonal order and beyond. p for the Klein-Gordon equation where JHEP03(2009)054 , 3 and (1.6) (1.5) (1.7) (1.8) (1.9) (1.10) (1.12) (1.11) H steps of . ) is i genstates k . N p ontains the | ates at each  1.10 +1 )) ight-hand-side k x ). To do this we h p,x 1 ( T between a state of − =0 of eq. ( H Y able into k N = U i t − k , ˙ en finds, for small time x x S. n | p , ν 2 1 k ( i . ˆ x rator p T x ) h | dt = er transform from momentum iε iε p T . . . =0 h ) N ˆ − ] 0 Y ral, let us first write the inverse in Weyl ordered form, with all k H − 1 x 2 S ν Z ) ( − 2 # t ˆ i x 2 1 ˆ y t H ( i m · n n [(∆ I, − ip )∆ O + )+ e k ...µ , ...ν + 1 2 , and write the Hamiltonian as T 1 t d µ ( dT e ν p ,x ) H k x )∆ d π H ixp (0) = ) ∞ p ) p e 0 ( n d T π (2 p,x µ Z ( ( d H ˆ p (2 1 2 = ip 1 iH , U – 5 – ) − i − =0 − . . . Z = p e k X  N T | i 1 ( 1 i x µ = − − h ˆ p − ) i exp ˆ " HU x iε =0 | ∞ x )= t X n − D ∆ exp as the Hamiltonian operator of a quantum system, with )= p S 1 x,y iH ( )= ( T D − i ˆ ˆ ( − p H F in the Hilbert space f = 0) and definite momentum (at time e N , which is a unitary operator acting on the Hilbert space | p U . Given we are considering external lines as shown in figure − i i ∆ x, t p x (ˆ p T h )= ST | d ˆ dT 1 2 T H ( i i ...dp (0)= p − x 0 e and Z i dp = x | )= respectively. Note we have expressed are the position and momentum operators whose continuous ei i Z i T i now ensures the convergence of the integral. The integrand c ( t p x N | p ) is the c-number obtained by replacing the operators on the r and inserting a complete set of both position and momentum st ) becomes | ) U t iε T ) with their corresponding variables. Slicing the time vari 1.9 ( p,x and ˆ ( is the number of space-time dimensions, the continuum limit ...dx U and 1.8 | 1 x H d f i p x dx h | Z exponential where the step, eq. ( duration ∆ space. The usual representation then reads where the inverse operator can be defined via an inverse Fouri we must calculate the expectation value of the evolution ope To derive this expression usingKlein-Gordon a operator first-quantized path using integ the Schwinger representation Using the normalization of the basis states satisfying the equation Schr¨odinger definite position (at time where We can therefore interpret introduce states internal time coordinate where ˆ are momentum operators toseparations the ∆ left of position operators. One th of eq. ( where JHEP03(2009)054 d (1.14) (1.16) (1.15) (1.17) (1.13) (1.18) (1.19) ), and 1.6 , i ) gives . .
) 2  J p
. By completing ∗ 1.12 2 1 µ φ J − 1 ther than position. ) into eq. ( ˙ , x iA + − p ( ) iε φ − dt ∗ iε 1.16 f a product of Gaussian − µ T . J 0 ) n function is given by − R ∂ 2 t i i + ( S ′ m ) into eq. ( = . ( φ x x e ∗ les. The measure is such that nd gauge field. Such a system T ) + µ ) J D + n to a scalar particle interacting 2 iε 2 f 1.14 l term in the exponent involving D p t p m − he evolution operator sandwiched f + D + − p . φ ) by expanding around the classical 2 f 2 ) t p
( ( + = m i 2 )=0 iε x i 1 2 T i m ( − x i (0) = 0, and without confusion we can − − ′ p (0)=0 i x µ x x | x + i S ) i Z f ( 2 D )= x ∗ ) and T ip p T t µ | t ( ) ( φ − f ( 2 D ) we can write this as e p U p − 2 ( 1 2 – 6 – | m h ∗ f + = m φ p x 2 f , x i h ) p d i )= ( t + is found by substituting eq. ( d x 1 2 ( p x ) = 0 and ′ | d i µ ( dT F ) p T d − Z ( D i ˜ H into the measure, and made the boundary conditions T ′ ∆ i ∞ x µ + ( p 0  Z f π f Z i D U ip | p h 2 1 − f − exp e p h ). Having reviewed the relationship between propagators an φ )= exp = ( = t )= ( D i 2 f φ S 1.5 ∗ i p p D φ x ( | ∗ ) F D φ ˜ T ∆ ( D Z U | Z f ]= p h are sources for the complex scalar field, and ]= , J ∗ ∗ , J J [ ∗ ) arising from considering a final state of given momentum, ra J Z T J, J [ ( Z x For the present case of a free massive scalar, the Hamiltonia ) T ( The boundary conditions imply The momentum space propagator drop the primed notation from now on. Substituting eq. ( One can now performintegrals the in path the integral intermediate as positionthis and the gives momentum unity, continuum and variab limit one o therefore finds between initial and final positionp states, with an additiona explicit. This is the well-known path-integral result for t in agreement with eq. ( We have absorbed factors of 2 We can perform the path integrations over solution of the equations of motion, given by one finds path integrals in a simple case,with we now a consider gauge the extensio field. 1.2 Scalar particle inWe consider an a abelian charged background scalaris particle gauge described in field an by abelian the backgrou generating functional where the square and defining JHEP03(2009)054 ) A 1.4 · (1.21) (1.22) (1.25) (1.23) (1.26) (1.24) (1.20) p . ) 2 . x )+ m 2  + + ) m t 2 x f + p , A + One then finds 2  t + p
f 5 ( i )+ in normal form (i.e. p 2 1 x ). ¯ A ψη ( , which gives eq. ( S · + ∗ . − A + ] i ∂ φ · ˙ 2 ( 1.14 x e of a background gauge x ical path of the emitting , i ) ( , the gauge field in the ex- p m e have dropped the primes φ ηψ ) p 2 −  T A + + ¯ + in ( A dt 2 , · f f − 1 ψ ) ] T of eq. ( p e position and momentum states ) T p ) ν . d path integral representation of 0 / ) D 2 x 2 Z i m , γ i + ( m −  − µ + − + 2 2 2 γ t [( 2 f p [ p i / f A p )+ D 2 1 2 ( 1 ) p ( 1 2 1 2 T = i The momentum space propagator is then ¯ ( m as before, one may carry out the manipu- ψ − + + 2 − − x i ˙ i 6 } ) x + S m x x x ν A p 2 1 T f ( / d · ( D +  when we discuss next-to-eikonal exponentiation. d ip , γ A = – 7 – 2 ∂ ip µ − · dt i 2 e A γ Z = ( , we can write the operator 2.2 ∂ H T − { i µ ) + i 0 2 + =  2 1  Z i∂ p i m ). + i i · − =  x − exp | exp A ν 1 ) = 1.14 +) throughout. / x ψ γ D , by analogy with the scalar case. Using the standard trick , − T µ exp µ ( 2 ( D D i + p γ A ¯ p , x ψ m U · − | + D D f D p , + p f p = 2 − p h i − ) Z D x F 2 / )= D p i ∆ T ( )=0 ]= (0)= p − = ¯ T η x ( 2 p (0)=0 Z η, x = ( [ m Z = Z S + i i 2 ) x ) = are Grassmann-valued source fields. | A ) T η ( T − , ( f p η U | f We will formalize this statement in section Recall we use the metric ( = ( p 5 6 h S This differs from theponent. free When particle the caseparticle strength due is of to well the approximated the gauge by presence field the of free is particle weak, solution the class the evolution operator sandwiched between the external lin The propagator is given by the inverse of operator quadratic lations of the previous section to obtain the first-quantize where Now we define as before. Using the fact that 1.3 Spinor particle We now consider thefield. case The of system an is emitting described fermion by in the the generating functional presenc given by Again the boundary conditions haveon been the made quantities explicit, defined and in w eq. ( with all momenta operators on the left-hand-side) as where ¯ Now defining the Hamiltonian operator JHEP03(2009)054 ] , 30 µν , (2.1) F (1.27) (1.28) 15 rt from µν hard in- σ 2 1 is the field )+ 2 A µν of the spinning 1 2 ), the contribu- F − 4 , A · . This is a sum of µν ∂ } i 2 i F the path integral over x + µν { A · σ p ond indeed holds, as has ffect of soft gauge boson ed momentum. Each ex- f particle propagators, in- at gives rise to scalar-like soft modes − )+ l lines, considering Green’s n to the issue of corrections 2 2 egral manipulations as in the se. ow the above representations m wn in figure rrections. onentiates. Secondly, that all e field, as first-quantized path ring processes. The proof that m , and consisting of a g this factorized structure. In ing, as it is well known [ ted as an eikonal line. In the + 4 2 howing two things. Firstly, that + p ( . 2 . 2 1 lving couplings to the field strength µ h A − A ˙ 2.4 x p + D ( µ s ], which characterizes the regions of in- dt A A · T 0 33 R D correspond to the position and momentum i i∂ p )+ Z − T – 8 – ( ≡ A x ) · µ and T ( p A x 2 ip eikonally factorized D − − Z 2 p xP e = D p 2 D m f ) is almost identical to the case of a scalar eikonal line, apa p ] are the generators of the Lorentz group, and + i ν x 2 ) with external lines emerging at positions )= ) n T , γ / ( 1.28 D µ (0)= p i x γ [ − Z i 4 ( ,...,x ]). The latter correspond physically to the magnetic moment = − 1 i x 32 i ( = , x | H ) 31 µν T ( σ U | In this introduction, we have reviewed the representation o To make these statements more direct, we begin by separating This is based upon the general analysis of [ f p h of the emitting particle,following section and we the formalize classicalcan these path be statements, used is and to interpre show derive h the exponentiation of soft2 radiative co Soft emissions inWe scattering now processes turn tofunctions the description having of the soft form radiation shown from schematically externa in figure cluding the possible presence ofintegrals. an abelian The background phase gaug space variables emitting particle, and thus have no analogue in the scalar ca we can rewrite subdiagrams containing gauge bosonternal modes line of has asemission, a yet and propagator we unspecifi associated call such with diagrams it summing the e where teraction tions from soft radiationleading on soft the radiation outgoing terms externalthis originate lines paper, from exp we diagrams prove havin been the shown first to be property the andto case the assume elsewhere above that at factorized the eikonal form level. sec (at We NE retur level) inthe section gauge boson field into a product of integrals over hard and the coupling to the fieldthat strength one tensor. can This is castvertices not supplemented fermion by surpris additional actions seagull vertices into invo (see a e.g. [ second-order form th frared sensitivity in Feynman diagrams insoft a radiation number contributions of exponentiate scatte now amountsfor to the s eikonally factorized diagrams defined above (and sho The representation ( strength tensor for the gaugescalar field. case Carrying yields out the path int JHEP03(2009)054 . e 2.2 i (2.2) (2.4) (2.3) n x | iǫ − respectively. S | µ h . n  A y
h ) is a hard interaction x ( ily seen as follows. and . . . ∗ e transformed gauge H i φ µ s 1 ) x A x | ( rm iǫ J xcept for the fact that the ) may well have momentum omentum space analogue of ead, both the soft and hard diation by the fact that one − x , generates connections between ), where ]. Its precise definition does orresponding to soft and hard ( )+ µ s S ξ | . x 33 A 2.5 1 ) ( , y ∗ k ) ), given that the path integral is a h er ( φ ′ x ) ) ( ξ xt-to-eikonal exponentiation in section n x 2.1 ξ µ y ( µ δ k ( However, the fact that such a surface J ∂ S S δJ x 7 )+ d )+ k has yet to be performed. Also, the factors d ( . . . x ( S µ s ) Z µ s µ s,h 1 A – 9 – i y A A δ ( ]+ → → δJ amounts to specifying a surface in the multi-boson µ ) ) H n 1 , whose momentum modes live only in the soft region i x µ h k µ s ( , A ( ∗ is a propagator for the eikonal particle in the presence of th A ∗ µ s A φ S µ s,h A is in the soft and hard regions for D A φ, φ φ k [ and D µ s iS µ h  A A D exp ). Transforming to momentum space, Z x × ( ξ )= n ) is non-zero only if . The factorized form of the Green’s functions of eq. ( ,...,x k 1 ) ( outgoing external lines, and ′ x ( ξ n 2.2 The above separation is not gauge invariant, which can be eas We now formally define the hard interaction as We will consider corrections to this idea when discussing ne H 7 modes. Such a surface is, in general, very complicated [ momentum space that separates it into two distinct regions c with The precise definition of the external lines. background soft gauge field which, after the path integral ov Figure 4 for some function Consider a given soft gaugeof field momentum space. A general gauge transformation has the fo product of integrals over gauge fields of definite momentum. modes defined in thefield will hard have region in of generalgauge both momentum fields soft obey space, and a and hardeq. restricted thus components. ( gauge th invariance Inst given by the m not concern us inmay what neglect follows, where the we recoilexists characterize allows of soft us ra the to introduce eikonal the particles. factorization of eq. ( This is analogous topath the integral over expression soft for gauge a field Green’s modes function, e Here JHEP03(2009)054 , ) ), t- A n + (2.7) (2.6) (2.8) (2.5) p 1.22 ) as no T → ( ,...,p , f p i 1 T p n ) in eq. ( ( . approach its x iε | )  G f ) 1 − f ) x 2 − p T + ) m t ( one multiplies by ), and replaced in f + iε f d of the soft gauge i p 2 f . p + − ( 2.4 εT  i generates all possible external lines emerg- tial position and final 2 1  . ) 1 2 x i S ) ), one may treat each ) ( n T − ( − A | T ( · e i ∞ ( n ∂ x 2.5 f ( the function i p f 2 f d f n eq. ( h dT εT i ip rnal line, and the path inte- ernal leg tor for the emitting particle xternal lines. This is shown d 2 1 x −  ))+ dT f t . . . − de by the residue of the scalar ( icle, and also the fact that the T T ip e i x ) ) d. However, this residue is unity 1 − 2 2  + dT e e t x m m | f  ), one can rewrite each external line ∞ 1 p + + = T 0 2 f 2 f − ) + p p i i Z ) 2 ( ( i x is a full Green’s function of the theory, 1 2 2 1 ( 1.21 m 2 1 iε x i i | ) A + · G 1 − − 2 2 f − ) p − x ( m ) S 0. At this point one can let +˙ 1 2 ( i f dT e | dT e iε + p 1 − → p 2 f ∞ ∞ e − (i.e. such that there are – 10 – . After shifting the integration variable +( h 0 0 p ε 2 ) p ( ˙ S Z d Z x i n ( H dT 1 2 i − − (  = dt ) allows us to simplify the expression for i only |− i (0) (0) ∞ dT f 0 x x ,...,x R 2.7 | f f p i 1 1 h ∞ − − ) x 0 − 2 ( )  Z  x e i i i m iε H D x x x f f f µ s + − , where real and soft radiation is included in the soft blobs a ip ip ip ) in eq. ( A 2 f 4 S ), containing both soft and hard gauge boson modes. Secondly − − − T p D ( (0)=0 e e e } ( ( i i x i f − − − x Z Z { to take account of the fact that the line is external and thus h |− of = = = f 2 )= p )= h n m ) ∞ (i.e. inverse propagators for the particle in the backgroun 2 ( inserted. This latter integral does two things. Firstly, it →∞ i + f i m µ s 2 i x T | A p + ,...,p iǫ 1 ) with the propagator sandwiched between states of given ini 2 f p p ( − ( 2.5 i To obtain the contribution to the scattering amplitude from We now define the further quantity G S | i y written in eikonally factorizedin form. the presence Note of the that background the soft propaga gauge field is removed i The limit mass shell and obtain the simple result a factor (free) propagator attached. Inpropagator, principle arising one from must renormalization also ofdue the divi to scalar the fiel absencegauge of field self-interactions for is the treated eikonal as part a background. As is clear from eq. ( one must truncate each external propagator. That is, for ext In the last step we have taken the limit h the result is a path integral over by performing the Gaussian integral over external line separately. Using thecontribution representation as ( tached to each external line. One thus sees that eq. ( momentum. field) truncate the external legs of the Green’s function. where a propagator factorgral has over been associated with each exte ing at 4-positions subgraphs within the hard interaction it produces soft radiation,schematically both in real figure and virtual, from the e JHEP03(2009)054 ld (2.9) ) and (2.11) (2.13) (2.12) (2.10) t ). One ( x ) will now . 1.21 ] s 2.9 is performed. A [ µ s iS A e ) . As seen above, this ∞ ( µ = 0 and in direction , act as a collection of n bation theory. Let us n i µ s f x pace Feynman rule n terms in eq. ( A n diagrams, and one sets . manipulated using simple x ace, as gral over ) 2 n on the soft radiation is de- k diagrams, which in this case A ip ( ht line trajectory k ds. This form ( ct the fluctuations µ − field · ce vertices. By the usual rules soft gauge field, which remains ˜ . A ted at n omentum µ and . ) indicates the particular external ing amplitude for a charged scalar ...e , n  ) A d ) k ∞ . ) µ · k ( · ) ∞ d π nt n x ( f ∂ ( ( n d 1 (2 µ A ) where we integrate over soft gauge field f · − 1 A x dx Z µ 2.9 1 R ip − i e − dtn e p – 11 – ∝ ) ∞ )= 0 n ) Z nt ( i ∞ k ( µ  -field, distributed along the classical trajectory. f A A µ ,...,x 1 exp x ( dt n H ∞ ), and the label of each µ s 0 A Z 2.8 i ) becomes D flows into the vertex. The path integral over the soft gauge fie Z 2.10 k )= = 0 and as well as neglecting the n , p ,...,p ) given by eq. ( = 0 1 ˙ . Then eq. ( p x ∞ ( µ f ( , p S f As a simple one-dimensional path integral, it can be further Inserting this result into the path integral ( ≡ ). This is equivalent to the eikonal approximation in Feynma = 0 t µ ( line. Also we have explicitlyafter factored the out path the action integrals for over the the particle and hard gauge fiel generates all possible diagramsof connecting quantum numbers field of theory, one sour finds connected and disconnected n 2.1 Eikonal exponentiation Now that we havescribed established that by in a the Wilsonconsider, eikonal line without approximati we loss can of analyze generality, an what external happens line in crea pertur classical source terms for the soft in an abelian background field takes the form Thus, the eikonally factorized contribution to the scatter enable us to find all-order expressions for these amplitudes classical methods. The strictest approximation is to negle then finds an Aharanov-Bohm-like phase factor for the straig fluctuations the Wilson lines, being linear in the soft gauge This can be written, after a Fourier transform to momentum sp where momentum x p with Note that this is invariant under rescalings of the eikonal m It can be represented as a 1-photon vertex with the momentum s acts as a source term for the soft gauge field when the path inte JHEP03(2009)054 .  (2.14) (2.15) ) x ), where + al particles ) becomes ) describing 1.14 2.8 ∞ λnt ( ( . This allowed f x A · external lines, to be ∂ i 2 soft radiative corrections )+ action with two outgoing given by eq. ( o the equations of motion x isconnected diagrams. We erpretation of gauge boson he factors tegrals in + m the straight-line path in a s. lso finds disconnected subdia- = 0. Then eq. ( is soft but with non-negligible s are considered. e still expects such corrections 2 j the limit in which the emitting . There one sees a number of e along the latter. The collection n 1 λnt ( . ification also allows one to go easily A i c · G ) x , where . , while the sum on the right is over all j 1 + ˙ X G h λn λn = – 12 – j + ( p = exp 2 ˙ x G 2 1 . However, given the usual property of exponentiation 5  X dt ∞ 0 Z i .  c G exp x D = 0 without loss of generality. Given that the external eikon i x (0)=0 x Z . Example of a disconnected subdiagram between two outgoing ) = ∞ We have thus succeeded in showing that the exponentiation of To illustrate this further, consider the case of a hard inter To formalize this argument, we reconsider the external line ( f compared with the connected subdiagrams of figure Figure 5 The sum on the left is over all subdiagrams connected subdiagrams in the eikonal limitnow can consider what be happens related when to next-to-eikonal the correction exponentiation2.2 of d Next-to-eikonal exponentiation The analysis of thesoft previous radiation section from relied the on external the lines fact are that written t as path in again we take connect the external lines givenof that all the diagrams source exponentiates vertices li in terms of connected diagram eikonal lines, anconnected example subdiagrams of connecting the which externalgrams, is such lines. as shown One that a shown in inof figure figure disconnected diagrams in quantum field theory, one has: the straightforward interpretation ofparticle the follows a eikonal classical limit free path. as beyond However, the this eikonal ident approximation. Ifmomentum, the the emitted classical radiation pathfor is the eikonal still particle, asystematic and good expansion. one approximation can Given t examineemission that vertices deviations this in fro does terms notto of exponentiate. affect disconnected subdiagrams, the on int have light-like momenta, one may write JHEP03(2009)054 . x ), m in λ t , ˙ ( , so i x . At x n 1 (2.16) (2.17) (2.18) x − t/λ | ) which λ t . iǫ ( → x − t  ) in ) 0 proximation, rized Green’s x S | n + , 2.16 y i ). The remaining h
t nt ) ( ( itting particle x x . . . the previous section. A ( · i 1 ¯ ∂ ψη x time variable | λ i 2 he advantage of the above ection. To obtain the NE iǫ ticle. For emitting fermions )+ antifermions etc.), we write sion of eq. ( x − essed by inverse powers of ( hus, following the reasoning in )+ 0 ion in the NE limit, which one onentiate as before, i.e. one has: ψ x S ) orrections is manifest. | + x 1 , ( y ¯ h η nt i ) ( x c n x A d y G ), which can be described as follows. δ · ( d 1 ) ¯ η − ). Truncating the external lines of the full x δ , one may neglect all terms involving X Z λ )), where the propagator in the presence of ( h i + ˙ . . . ), but gives rise to a propagator for 1.28 n 2.5 ) λ ]+ ( – 13 – 1 ∼ O → ∞ µ + ( y δ ( λ 2 ¯ η ˙ ) = exp x ∼ O propagators) diagrams along external lines, and lo- δ ¯ ψ, A 2 λ ∞ n 1 x ( i  ψ, [ f ¯ ψ dt iS D , and show that they agree with the results one obtains in ∞ h ). We examine in detail the NE Feynman rules that result ψ 0 B t Z ( D i x exp µ h  A × ) as remarked above. There is also a NE vertex originating fro 1 D exp − Z λ x ( , and a given NE graph containing such a vertex must then conta , one keeps the first subleading corrections to the eikonal ap D O λ A )= is the free fermion inverse propagator. The eikonally facto · n ) in appendix ) becomes ∂ (0)=0 iǫ x are connected (through Z 2.16 − , leaving precisely the eikonal approximation discussed in 2.15 ) by virtue of being the inverse of the quadratic operator in x c ,...,x 0 A 1 1 S G )= · − x ( λ ∂ ∞ For subsequent purposes it is more convenient to rescale the Firstly, NE graphs must have at most one propagator for the em ( ( H O f By expanding in function has the same form as before (eq. ( That is, fluctuations about the classical free path are suppr is terms generate effective vertices forcan again soft interpret gauge as boson source termsthe emiss for previous the section, soft one finds gauge that field. these T NE corrections exp The first term in the exponent is now One now clearly sees that in the limit the background gauge field is given by eqs. ( standard perturbation theory after expanding to NE order. T representation, however, is that exponentiation of these c 2.3 Exponentiation for spinorWe have particles so far only(with considered a the case similar of expressionthe a for definition scalar of combinations eikonal par the of hard fermions interaction and as and LO, one recovers theapproximation one eikonal must approximation gather of all the terms previous s where cated on the latter by vertices derived by a systematic expan the term in due to its being i.e. corresponding to the next-to-eikonal (NE) limit. where that eq. ( no propagator factors for from eq. ( JHEP03(2009)054 ). ) is 1.25 , and (2.19) (2.20) 2.19 , . In what  6 es are given µν F nally factorized µν σ λ ential of eq. ( 1 2 internal emissions wn in figure )+ not yet a proof that such ed Green’s functions, soft x e magnetic moment vertex s nothing to invalidate the for the additional magnetic not eikonally factorized. At . n for next-to-eikonal photon ) + ) ch a soft emission connects an fixed number of scalar external x ion as r of the following section. given matrix element then has hich include Green’s functions holds for scattering amplitudes s insensitive to the spin of the ikonal level (as is well known), sions which land on an external nt + ( M A nt · ( ∂ A (1 + tly with the inverse free propagator to give a λ · i  as before. The proof of exponentiation ) as involving only the denominator of eq. ( 2 ) . Diagrams with internal emissions have T x NE ( + . However, at NE order there are also cor- U 4 + ˙ t/λ M n + → , the additional vertex is indeed of NE order, as – 14 – E + ( t external λ is a remainder term which does not exponentiate, 2 ). ˙ M r x  2.6 2 λ M  dt ), but where the external line factor is now: = exp ∞ 0 2.9 Z M i  exp x one finds that the eikonally factorized scattering amplitud D 8 collect the eikonal and next-to-eikonal diagrams from eiko as in the scalar case of eq. ( (0)=0 x 2 Z NE m , E + )= 2 f M p ∞ One may worry about ordering of Dirac matrices when the expon The preceding analysis has shown that in eikonally factoriz In the spinor case we defined the evolution operator ( 8 f emitting particle. expected given that in the strict eikonal limit, radiation i moment vertex which, althoughproof. absent in Note that, the due scalar to case, suppression doe by up to NE order proceeds directly as in the scalar case, except and we have rescaled the time variable been studied before in the literature. It has been shown for a Green’s function, factor those originating from external lines as The leftover factor in the numerator indeed combines correc and contains NE contributions from diagrams such as that sho gauge boson corrections exponentiatecorrections up exponentiate to in NE matrixnot order. elements having themselves, This an w is eikonallyone factorized may structure. in fact At ignoreNE strictly contributions order, e from however, diagrams contributions arise whichexternal from are diagrams line in with whi the hard interaction. This is2.4 the subject Low’s theorem In the previous sectionemissions we from have external demonstrated lines.having exponentiatio That the is, eikonally the factorized exponentiation formrections to of the figure exponentiation arisingline, from having soft originated gluon from emis insidethe the schematic form hard (up interaction. to A next-to-eikonal level): follows, we refer to emissions from within the hard interact expanded. However, this isis not NE an and issue thus due occurs to in the each fact diagram that only th once at this order. by the same expression eq. ( Here Green’s functions respectively, and JHEP03(2009)054 , ], ). )) ] s 35 and A [ , the 2.20 s imply 2 (2.21) (2.22) 2.16 iS A e ) . s A  of eq. ( ; ) ) 1 r x x ,p ]. This result is M + 1 + x 34 ( pt nt f + n + actor x . i i i n x ft emission (internal or x oft photon field x ( ip ( − der term does not have a A xternal lines are produced A al gauge invariance, given · ion. This also allows for a · ernal leg factors (eq. ( emissions [ ) ∂ ral formalism adopted in this ...e i x q ) l orders in perturbation theory. . As discussed in section s ositions + ˙ λ e on the eikonal particle’s electric i A 2 A n . Generalization to higher orders ; es from inside the hard interaction. ( 1 ized Green’s function, and adding a soft i + q ,p 1 -particle scattering amplitude (see also + on the gauge field in what follows, and S x S n 2 ( s ˙ x f 1 ]. 2 λ x 1 S  36 ip – dt – 15 – − e 34 ∞ ) s 0 Z A H i ;  n exp x ,...,x Low-Burnett-Kroll theorem 1 D x f ( p , and was generalized to the case of spinor external lines in [ i H x )= ’s depend on the soft gauge field s f ]. The fact that graphs with an extra emission can be related s ∞ A ( p (0)= 36 D x Z 6= 0 i.e. Z i x )= and the s )= n Low’s theorem H , A . Furthermore, we drop the subscript . An example of a diagram which contributes to the remainder f f i )) q ,p i ,...,p 2.9 x Our starting point is the expression for the Both 1 ( p f ( S Figure 6 separation between soft and hard gauge modes leaves a residu eq. ( on the soft gauge field.at We 4-positions also now consider the fact that the e Such contributions are formed bygluon taking an emission eikonally which factor lands on an external line, but originat lines that, up toexternal) NE can be order, related theknown to sum as the of scattering amplitude diagramsan with containing no extension a known so aswas considered the in [ to those without anformal emission exponential means structure, it that,In has although an this the iterative section remain form we topaper, discuss al in these properties order in togeneralization the complete of path the our integ ideas discussion presented of in NE [ exponentiat where we have explicitly indicated the dependence of the ext We have also indicated the dependence of the hard part on the s leave implicit the path integrals over the external line 4-p rescaled this field so as tocharge explicitly display the dependenc JHEP03(2009)054 ) x (2.28) (2.25) (2.29) (2.23) (2.27) (2.24) (2.26) . ) . ) n x x ) generates x Λ( Λ( ; n n  iq ) ). Because Λ( + j µ . , x ... , ) ) ) , j n )+ − ) 1 x A ,...,x n x ; 1 x . ). In momentum space ) ( f − x ) Λ( δ x ( amily of gauge transfor- n 1 e to charge conservation j ( x t this in the case in the ,p µ q ( i iq µ 2.25 δ e ft emissions from inside the x k,...,p H j A ) ,...,p ( j n ) ) has only soft modes when q 1 f X + llows. First, one may expand A x x p ) ; ,...,p ) i j ( ; j actors transform as Lorentz index n 1 )) = n x n n X p x H ( Λ( ) µ n iq Λ( j H ∂ − µ ,...,p µ e ∂ ∂p 1 j ,...,x ,...,x ,...,x ∂ 1 p 1 µ 1 ( x k x ∂p x must transform as ,...,x ( j ( )+ ( j H 1 q x q µ j Λ) = H H x ( H q ∂ ( µ j  n j n X ). X A j x xH n + to obtain iH and in Λ, which gives d d X – 16 – )( − d d Λ) = k A 2.7 x A ; )= ∂ ; f )= k Z Z )= n ; + )= x i k ,p n ; k i ; A ; n + n )+ x ; n ( n n f ,...,x 1 ), but where the function Λ( ,...,p → x 1 is an arbitrary soft momentum, one may write ,...,x ( ,...,p p ) 1 ,...,p 1 µ 2.2 ,...,x µ ( 1 ,...,x x p A 1 µ k 1 p ( ( ), the hard function ; x ( µ µ x f ( H xH µ ( µ d 2.2 H H ) to first order in ,p H k H d H i µ µ ∂ x k − ( Z → 2.24 − f − ) A )+ ; n n ,...,x ,...,x 1 = 0. Now, because 1 x j In order for the path integral to remain invariant under the f ( x q ( j H H by the usual form of eq. ( where theP zeroth order term on the right hand side vanishes du When the path integral over the soft gauge field is performed, this becomes mations given by eq. ( transformed to momentum space.following. Under such It a is transformation, implicitly the external assumed line tha f which follows from the definition of eq. ( is arbitrary we infer One can expand this up to first order in We can now use the gaugehard invariance interaction to relate to diagrams similar withboth diagrams so sides with of no eq. emission, ( as fo where we have integrated by parts on the left hand side of eq. ( hard interactions which a single soft photon emission (with JHEP03(2009)054   which (2.33) (2.32) (2.30) (2.31) n ) p i k · . To see . , ( x n r µ ) ) ix A A A M ns − ) ; ; ... n n n teraction with − 1 ,p ,p ), and up to NE p · 1 (0 (0 s. Such corrections ix ,...,p )+ 2.29 1 − k e p ( ...f ...f ( µ  ) ) s. Leading eikonal loga- tion of this result. The ) eq. ( A H ks using more traditional A A structure to all orders in k ; ; d  k ( emainder term corrections, where each NE 1 1 ) roximation, and the second iven by k s into account also leads to · µ A d π j , ,p ,p · ∂ d n garithms do not exponentiate, ) (2 k ∂p (0 (0 e due to the bracketed prefactor n tion with the same hard interac- k ( . tial positions of the external lines ) f f · k Z A − k · · × · n ix  ν j ) j e n ∂ n ) ) ), and do not exponentiate. However, ix k ∂p k k ( − ν · · ( k A d 2.20 · n k ix ) k ,...,p · µ j n d π 1 j n d . p d (2 ( n ) k C – 17 – (1 + d π  H d Z d j in eq. ( µ (2 ) q k j j d q r π ∂ d j Z n ∂p (2 j X M q j X q d Z − )  k ) is suppressed by one power of momentum and is thus a j q ) n d π k X n ) contain exponentiated eikonal and NE terms, as discussed d · − (2 A − x ; i Z   ,p ,...,x A A 1 (0 x f D D ( H Z Z we ignored the fact that the external lines emerge at positio d n 2 )= )= n n ...dx ), which contains a sum over different possible NE correction d 1 ) shows that they can be obtained as derivatives of the hard in dx 2.33 ,...,p ,...,p 1 1 2.33 p p Z . Performing these integrals, the scattering amplitude is g For scattering amplitudes, one may summarize this as follow In section ( ( } i S S x no soft emissions.perturbation Thus, theory. the remainder term has anrithms iterative arising from soft gluon emission exponentiate. NE lo Some comments are inexternal line order factors regarding the form and interpreta where we have explicitly instated the{ integrals over the ini previously. Corrections to the NEin exponentiation eq. then ( aris This can be expanded as This simply relates internal emission from the hard interac tion but with no emission.methods, we For a refer simple the example reader of to how appendix this wor contribute to the remainder term eq. ( where the first bracketedterm term (involving corresponds the to factor theNE eikonal correction. app We now combine these terms with the factors of order the scattering amplitude isfactor given occurs by at a most sum once. over One all then such finds a NE contribution this, we write the eikonal one-photon source term as: are integrated over andcorrections thus of in NE general order (and non-zero. beyond), Taking which thi enter the above r JHEP03(2009)054 (3.1) (3.2) (2.34) ) then ucts of vanish, 2 i . , t the soft p ) 1 2.33 roup, such j . ] 1 A i s ; # A n [  ) ) ) iS ,p . eq. ( x 2 e 2 + n ) (0 t p ) as an extra vertex · f · p ∞ ) can be reexpressed k 1 ( + p k 2 i ...f j − 2.34 · x 2 ) ( ) can be written ntiate, and a remainder i 2.33 2 1 2 A A f · n n ; 2 . ( ∂ onsider the case where the 2.9 long the external line. As · n the case where the gauge 1 x i 2 2 k to the exponent being linear . ( ,p ( l Feynman diagram treatment ip n the abelian case, the source soft gauge field when the path ple to all orders in the coupling µ 2 ))+ hat derivatives w.r.t. − cted diagrams between sources. tiation of soft gauge boson cor- t (0 s (corresponding to Mandelstam ( n e tude for eikonally factorized dia- f x ) # + + is a generator of the gauge group. t ) ∞ f ) ( k p A 1 ( 1 j t + µ n 1 i i 1 · x A ( f ) k 1 A n · k x ) 1 − · x and summation over repeated color indices is ip 2 1 +˙ , where f − } n n p ) have the form e k · A – 18 – ,...,p ) j t 1 +( 2 k { ∞ µ A 2 ( = 2) there is only one scalar p ( ˙ x ( µ 1 k ,x A 1 2 j n 1 n k ( H . Here we consider the simple case of a hard interaction i x = ) 2  ( dt f 2 2 i d µ p ∞ 0 1 ) k · i R A d π ∂ i 1 d e H p (2 ( µ s P ∂ A x Z D × D " i A x Z D is a scalar. Then it can only depend on Lorentz invariant prod are indices in the fundamental representation of the gauge g (0)= Z )= } x is performed. However, it is no longer immediately clear tha H 2 k Z j ), and one may interpret the bracketed factor in eq. ( µ s " ,p { )= C 1 A n p ( )= and S ∞ } ( k 1 ,...,p i j 1 { 1 i p To further clarify the above formulae, it is instructive to c ( f S but can be separated into the sum of a series which does expone sum which does not exponentiate, butconstant. is obtainable in princi hard interaction Here i.e. similar to before,in but the matrix-valued non-abelian in gauge color field space due This is precisely the form one(see expects appendix based on a conventiona implied. The external line factors becomes describing soft emission from within the hard interaction. 3 Non-abelian gauge theory So far we have consideredfield an is abelian non-abelian, background the gauge derivationgrams field. of proceeds similarly the I to scattering section ampli with two outgoing external lines. That is, the analogue of eq in terms of derivatives with respectinvariants in to the products hard of scattering 4-vector process). One may verify t that the outgoing particles have indices Furthermore, there is abefore, path the external ordering line ofintegral factors the act over color as source matrices terms a for the corrections exponentiate. In therections abelian was case, identified the with exponen theCrucial exponentiation to of the disconne combinatorics of this result is the fact that i and in the case of two external lines ( momenta. The derivatives with respect to the 4-momenta in eq JHEP03(2009)054 (3.9) (3.3) (3.5) (3.7) (3.4) (3.8) (3.6) the emitting increases along t n of disconnected ions where e.g. an , one may consider − . ] xponentiation in the of statistical physics s 2.4 = . , A [ ) i s x ) iS ( 2 dering of the exponential n-abelian case, due to the e x A · ( e strict eikonal limit. Fur- ne has the property ) . color singlet structure A ) · dx mplitude in a form such that ction of soft gluon emissions ∞ 2 x combine the two external line R ( ( . , i ction of two hard final charged 2 dx A of diagrams. These can then be 2 0; · e i ij ∞ 2 R 1 ng momentarily the color indices, i replica trick ), where i , P dx f δ e s ) R F ( ) = i P 1 s< ) 2 e ∞ 2 i x ( j 11 ) acts as a collection of sources for ) ) N B y x . Here we consider this up to ( ( i j A ). However, the expression for a given 2 3.16 ...t t 7 A A 1 ) ) for each diagram orders the color matrices B N y x operator t ( ( [ j i R 1 A A t . . . ] ( – 21 – 1 n 1 A 0 t’ )] = y ( ...t j 1 0 A A 1 t ) t ), one must consider contracting gluons emitted from two [ replica ordering x ( i 3.14 and adjoint color index A . [ ′ 0 i R . Given that the replicas do not interact with each other, the >t i ′ 1 = 0. Also, each diagram has a multitude of similar diagrams ob t 2 2 p and 0 >t =0or 1 2 1 . Example of radiative corrections generated by the source t p >t 2 t As before (and by analogy with conventional Feynman perturb matrices for replica i Abelian gauge fields), the single exponent in eq. ( of matrix-valued fields into a sequenceproduct of increasing is replic no longernumber strictly remain time time ordered, ordered. although the mat Figure 7 with obvious generalization to higher numbers of operators actually contribute in eq. ( diagram, as dictated by the source terms arising from eq. ( the soft gauge field.containing multiple replica The emissions path alongnot the integration necessarily eikonal over li ordered the along soft the line gauge (see figure products of operators, eachassociated with of each which diagram is involvesperturbation not a the theory, color same but as matrix rather thatmatrix-valued that which wou associated fields. with the Thecolor gi subset structure, of as is diagrams known which to exponent beor the more case vertices. for webs Givenare that only the [ tangled gauge through color fieldthe structure), replicas same one do replica can not number clearly in o gauge boson replicas arecase) shown. Each set of replica emission where we have introduced the i.e. a product of stringsn of color matrices, with one string f by permuting the replica labels. The operator in the form on the same segmentgauges) ultimately give contributions proporti in the scattering amplitudecontributions from (i.e. vertices on up different to segments of two the com gluon lines). Fir JHEP03(2009)054 = the ) is m F (3.21) (3.20) (3.19) P C N 3.18 (a). There 8 actor of this j, B (b,c). For each of (c) 8 e group, and e , because of the fact j which implements the i, A ame. There are 6= t this diagram is clearly AB i hown in figure δ roportional to the identity. rdered perturbation theory 1 2 the factors in eq. ( color factor in conventional x x th the subdiagrams formed d contribution to the scattering ) has two color indices in the et of diagrams obtained from ted when more than one gluon and and j 3.19 , or singlet structure, for one and two ] i = ussed in the text. n ABC i 1 A , f j, B 2 F , C ...t F 1 = C A 1 t B j [ (b) = t i one has a color factor n A i A is the number of different replica species present j – 22 – t t ...A A A i = m 1 t t i, A i A i acts differently in the two cases. B j t K 1 2 x x R N =1 i ) are contracted independently of the other strings. That Y in the diagram being considered. Each of the strings of color i 3.18 i, A (a) , denoting a color structure that exponentiates. The color f is a combination of factors involving N (b), in the case where 8 i n )! such permutations, where ...A 1 is a generator in the fundamental representation of the gaug m 1 2 . Diagrams that potentially contribute to the exponentiate A x i x A − t K N For figure For one gluon emission, there is only one possible diagram, s For two gluon emission, one has the two diagrams shown in figur ( / ! amplitude in the casegluon of emissions. two external In lines fact, connected only by (a) a and col (c) contribute as disc indices of each string in eq. ( Figure 8 that the replica ordering operator color contractions for replica diagram is matrices corresponding to a given replica number in eq. ( where unimportant, so that thea color given factors diagram associated merely with the by s permuting replica numbers are the s is a sum overproportional the to replica number of the exchanged gluon, so tha these, one must consider separately the cases where is, the color factor for each diagram has the form fundamental representation, thus by Schur’sHence, Lemma the must color be factor p is of a the complete product diagramfrom of in each the the replica replica individual o separately. color factors Furthermore, associated the ordering wi of in the diagram. N relevant Casimir invariant. Noteperturbation that theory, this although is things become theis more same involved. complica as the where JHEP03(2009)054 of 1) (a), − 8 (3.22) (3.23) (3.25) (3.24) N ( N ). 2 = ), and involv- N 2 4 S ( P α ( O N O luon emissions, a , and . gher orders in pertur- j )(b) is are interchanged). The combine the results for  8 = ], but we prove it in the A j i such diagrams in each case, plitude at C 12 2 aracterized by the fact that operator i.e. one has F ssion diagram of figure i, C , N lor factor associated with the C and ctor equal to the ordinary one. nnect each diagram by cutting R 11 i ociated with each replica. When − (b) does not contribute due to . (b) 8  A i, A C N 2 i, B . F 2 F 2 F = C C has a color factor: C 2 F . There are − 2 1 x 9 x = C diagrams where = N 2 ) F (but where are now different, such that the two cases do B j B j C N t N t j (c). Figure − B j 8 = B j t – 23 – 6= t B j i i > j A i A i + ( t t t A i A i  A i i, C t t t ) get reordered by the and A B j t j C 2 F (a) ). There are A i 3.22 = t 2 C i one also has diagrams containing gluon self-interactions, i, A case has the color factor N − ( j S i, B (c) has a color factor which differs from that of conventional 2 O F α 8 = C i 2 1  x x ]. . Thus the term linear in N j 13 – 6= i 11 (c), the . Then one sees that the contribution from figure ( 8 j one has ), and figure 6= 2 j i N , with a similar expression for . Diagrams contributing to the exponentiated scattering am ( 6= i O and , the color matrices in eq. ( We now consider the generalization of the above remarks to hi Note at this order in The above discussion can be summarized as follows. Up to two g For figure j j i < j 6= = where we have explicitly indicated which color matrix is ass ing the 3-gluon vertex. i for so that they also exponentiate (i.e. are webs) with abation color theory. fa The subdiagrams whichthey exponentiate are can two-eikonal be irreducible. ch eikonal That lines is, in one two cannot places. disco This property is well-known [ Figure 9 color factors of thesei diagrams are the same, and thus one may following using the methods outlined above. When and the crossedbeing gluon diagram ofperturbation figure theory, and indeedknown is webs precisely of the [ modified co subset of diagrams exponentiates. Namely, the one gluon emi not combine to give a term Note that the color factors for which there are two possibilities, shown in figure diagrams where JHEP03(2009)054 in i (3.29) (3.27) (3.26) (3.28) . Then for each i i H is the color contains no S . The indices ectively. Now , m i G i (where each of . . G ) ) I, ...B j j S 10 1 ( ( B ∝ , C C n ) i i i i j 6= 6= i ( +1 ) as j j Y Y ...A H H 1 ntribution in color space n H C o disconnected components n A × × i i H A m n figure H A ) contains the results of all 6= i i G A 3.27 then gives a color factor j K en gauge boson replica is j Y 1 ( l to the identity i.e. the only +1 B H . ( ...t × C ...t 10 10 H 1 ...t 1 i m A H 1 Q A H m t ...t G A H t , and B H h A G h j t H has two indices in the fundamental H m H ...t m ih ...t H B m G t n m m A G ...B as a product of factors A G ...B B H 1 t ih i 1 t B B +1 K ih ...t H ih H H n n n n m A G +1 A H G A H – 24 – ...A ...A contains emissions with replica number n 1 i 1 i that do not involve replica number ...t A G A H ...t A H t ...t H h 1 S K H +1 m A H m H t m m A G . According to the Feynman rules of the replica-ordered n h t i A G ...B ...B ...B t m ih G +1 +1 +1 n introduced above. That is, ih ...B H A G H H 1 i 1 m m m B H B as well as , and consider first the case where subdiagram B B B n n ...t n n 10 G ...t ...A +1 ...A 1 ...A ...A H H A i +1 m n +1 +1 H B A G H K H H t t as well as those of n i n n i i h h A G A A G G denote emissions from the lower and upper eikonal lines resp G in figure × × K K K } k H B ) is the color factor associated with replica { . A two-eikonal reducible diagram, in the simplest case of tw j ( is the identity in color space. This follows given that the co C I and Consider the general two-eikonal reducible diagram shown i } k A with respect to the external lines. Figure 10 gauge bosons with replica number perturbation theory described above, the diagram of figure the subdiagrams could itselfsubdiagram be reducible). We focus on a giv We may use the fact that where contraction factor for replica where { consider the case where from a given replicarepresentation. in By a Schur’s disconnected Lemma, subdiagram possible this two-index must invariant be tensor. proportiona We may then rewrite eq contractions in (due to the reducible structure) one can write subdiagram, so that the color factor associated with figure JHEP03(2009)054 , ), g ), are 3.30 (3.32) (3.33) (3.30) g ( C ), such that (in general, ) is a closed ). The color 2 s, two-eikonal 12 me color factor al perturbation N , such that the 3.33 ) of subgraphs 3.26 is thus given by N ∝ P -dependent factors ( N G n ∝ connected pieces (i.e. c . . Eq.( . n ) odified color factor asso- rtition. The only depen- n for the color factors of ) g 2 east g er, i.e. one subgraph with ( ny given diagram, in that as the property, as it must nsidering a different replica N (c). There are two possible ( C 8 ing of O P e is at least ollows that the number of ways . The total color factor of the graph ∈ For each such diagram, we then g Y a color factor given by eq. ( g + , which has the form ) N 1)! P is not a web. As a simple example ) (3.31) ( g n − which is linear in 1)! ( ). Permuting the replica numbers (but G P ) C − G 11 N , P ) P ) ( ∈ g n P Y g ( ( ( ) is then the same as eq. ( ) . i.e. one can absorb all 1 n C P ( − ( ) H P 1 – 25 – n 3.29 is the product of the number of ways of forming ∈ P − Y ( g ) P n P N 10 ( 1) n P − X 1) ( − P . These are sets containing a number X = ( ) )= P operator). The color factors on the right hand side, ( G n ( R intact) corresponds to the same partition (see figure P ), such that eq. ( ¯ C partitions j g N ( of C } Q is the number of two-eikonal irreducible subdiagrams). Thu P distinct diagrams in each partition, each of which has the sa { ) M P than has been considered in , rather than the color factor one would obtain in convention ( n G G P ) is the color factor associated with subgraph g N ( C , where M In fact, one can proceed further and obtain a general solutio N The contribution from a given complete diagram number in This has the same color structure as would arise if one were co contribution from diagrams which are two-eikonal reducibl The bar on theciated left with hand side denotes the fact that this is the m and weighted by the numberdence of on diagrams the replica represented number by resides each in pa the factor partitions. Either the two gluons have the same replica numb keeping the subgraphs there are form solution formodified the color modified factors color onlydo factor appear from associated on the the with above left-hand a considerations, side. of being It zero h if structures of the subdiagrams areof thus independent, forming and the it total f diagramthe in individual figure subdiagrams. For each subdiagram this is at l theory (i.e. without the each of which contains only one replica (see figure into the product (already encountered above), consider the diagram of figure the ordinary color factors associated with each replica sub where ∝ complete diagram is now given by the diagrams to all orders. Consider a given diagram consist reducible diagrams do not exponentiate. gluons connected by self interactionsconsider or the fermion set bubbles). i.e. one sums over all possible partitions, each of which has JHEP03(2009)054 . 2 ) ( (3.34) (3.35) (3.36) C . From − ) ( h total color C ]. )= 12 , ( 3 present distinct replica ¯ three loop examples. 11 ) C mine the subset of dia- ( ) three possible partitions C and notation by ) and en in [ ( lica gluons all have the same grams which exponentiate are ) + 2 . This is related to the original C . ) ( , N , ) ot interact with each other. Thus C 3 ) = 0, as expected for a non-web. ( ) ) )= ( (b) replica numbers. Permuting colors gives C ( ( ( C ) ( ) ¯ ¯ C C C ¯ ( C ( − − C C ) ) − − ) ( ( ) ¯ C C ) ) ( – 26 – ( ( ( C C ¯ C C ), and thus 2 2 )= )= ( − − C ( ) ) ) ( ¯ C ) gives a total modified color factor ( ¯ C ( ( , for which one has C C C 3.33 = )= )= ( as required. ( C ¯ A C C F ), or they have different color factors i.e. two subgraphs wit (a) C ). Then eq. ( 2 1 ( ( C − C . Examples of diagrams in the same partition, where colors re ) . Examples of (a) a diagram containing 3 connected pieces; (b ), one has schematically ( C 3.33 To summarize, the above replica trick has allowed us to deter To further clarify the discussion, we consider here a pair of Our second example is Figure 11 gauge theory (with noself-interactions replicas) and as scalar-gluon interactions, follows. but Firstly, do the n rep generated by this diagram, wherethe colors same represent partition. distinct eq. ( grams which exponentiate in non-abelianthose which theory have i.e. a the term dia linear in the number of replicas We consider first the diagram represented in the above shorth Figure 12 color factor numbers. which gives where in the second lineThe we second have used line the then fact agrees that with the modified color factor giv factor Reducibility implies JHEP03(2009)054 ), B (3.37) (3.38) -scaling , λ  ) ′ µν F nt ( µν ν σ A ) λ ′ β 1 2 ∂ ) nt ) ′ ( onic emitting particles, der, using the nal lines emerging from − ν nt ) the scattering amplitude ) nt ( n space in the exponent. A ( x µ ) iagrams are precisely those nt α ν ( x n the next section. A ∂ . luons to yield the same kine- + A ) α A  . The structure of two-eikonal ) + on in terms of the path integral ) ly that of web exponentiation, ∂ · belian NE terms exponentiates. ons in the non-abelian case ex- ish scalar from fermion emitters nt nt ν nt ( ( nt ( nt n µ ( A ( µ µ µ · dt ∂ A n A A ν A i i · ∂ ∞ ) ) n α ′ ′ ) 0 λ i i t ∂ t ) x Z 2 ( ( ′ ν t β ν ∂ λ ( + ˙ 1 + i x x 2 α ) ) ) ˙ n t t t x ( ( ( ) − t α α µ ) + ( ( x x x – 27 – (as detailed for the abelian case in appendix µ ˙ ) h h 2 ′ nt ˙ t ′ ˙ x x ( x ( h dt ν 2 dt ′ λ A x · ∞  h dt ∞ 0 µ 0 dt Z ∞ Z . We have already shown that a subset of NE corrections 0 dt n ∞ Z dt dt n dt 0 ∞ 2.2 Z 0 dt ∞ ∞ ∞ i 0 0 0 Z  ∞ i Z Z Z 0  i 1 2 2 2 1 Z ), but where the external line factor is given by exp + + − − exp P 3.10 x P ]. D 14 )= , ∞ 12 ( (0)=0 f x Z )= ∞ The above discussion proceeds similarly in the case of fermi In this section we again consider the simple case of two exter ( f technique discussed in section One next performs the path integral in given that the gauge boson emissiononly vertices appear which distingu at NE order. We discuss subleading corrections i 3.1 Non-abelian exponentiation at NEThe order significance of the above derivationmethod of is web exponentiati that one may then easily extend the analysis to NE or in any diagram one can replace replica gluons with original g matic result. Thenfound the above color i.e. weights modified of withirreducible respect the diagrams to exponentiating with the d original modifieddescribed theory color in factors [ is precise which gives exponentiate in the abelianponentiate. case, Thus, and it that is eikonal not correcti surprising that a subset of non-a a hard interactionfactorizes which as has in a eq. ( color singlet structure. Then For reasons that will become clear, we have stayed in positio JHEP03(2009)054 ) 3.39 (3.39) A t ) nt ( ). There are A µ A ν B.18 ∂ ν ∂ µ . represents a vertex of NE  in the third and fourth ) becomes: dt n • A B t t ) ∞ bs that appear at two gluon (d) (b) two gluon vertex. For these 0 ors of eq. ( A 3.38 nt B ared in some of the terms in Z t t ( on up to NE webs, where each non-abelian gauge fields. Note ) al eikonal emissions may occur ′ λ A i the replica trick) carry forward A 2 t A nt tices in the third line of eq. ( ) · ( ′ shown about the horizontal axis. + B ν )), eq. ( } nt ( A B dt ∂ B ν β 2 1 2 1 ,t ∂ x , which involve correlated emissions from x x x A ∞ B.18 A ) 0 t α 14 Z { ∂ nt ) ( 1 2 λ 1 ) A µ 2 nt ( A nt ) — those that depend on a single time (in the – 28 – − ( A µ α ∂ B ν A A ν t ν A 3.39 ) n n ) µ nt n µα nt ( ′ ( fields (see eqs. ( η t A ′ ′ A µ x A i, A dt dt A · t t . At higher orders, the NE webs again have the property of 0 0 µν (c) (a) Z Z 13 dt n dt η dt dt ∞ 0 ∞ ∞ ∞ 0 0 0 Z Z Z Z i  1 1 1 λ λ λ 1 2 1 2 − + − exp x x x x P . Webs involving local vertices occurring at NE order, where )= ∞ ( f Figure 13 Here we have explicitly factored outthat color path matrices ordering from the (i.e.the time exponent, ordering due in to this the case) Θ has function appe occurring in the correlat order. Additional diagrams arise by reflecting each diagram Inserting the correlators of the being two-external-line irreducible. The timegenerate ordered diagrams ver such as those showndifferent in positions figure on the external line, but where addition first two lines), and those that depend on two different times ( two types of vertex occurring in eq. ( lines). Of the formervertices, type, the there arguments are of onewith the gluon minimal previous vertices modification, section and and a (involving oneNE has vertex eikonal must exponentiati occur only onceorder per diagram. are The shown additional in we figure JHEP03(2009)054 ). 3.14 (3.42) (3.40) q. ( ). This gives B.30 o applies in the NE ) has no abelian ana- ), which correlates gluons rules. ] vanishes for an abelian ndled similarly to those iated with the NE webs.  their eikonal equivalents. B actors in the eikonal case sition space expressions 3.42 tex ,t between gluons in separate (pairs of gluons). There are belian gauge boson emission ame web. Secondly, one has A t involves a subset of diagrams, el (i.e. are a product of webs), re is such that they form a web l-line irreducible. . ) ase). The time ordered vertices t ]( wn here, further eikonal emissions may B ,t ; (3.41) A µν t A [ t F b ν µν µν A σ σ ) t ) ( t ∞ ( 0 A µ – 29 – A ν Z A A λ i µ µν 2 ∂ σ ) i.e. the non-abelian analogue of eq. ( ∞ ∞ 0 0 ) associated with each subgraph involving replica number Z Z j 3.37 ( i 1 λ 2 C ). Note that the two-gluon vertex eq. ( ). The sum over all such diagrams then enters the exponent of e 3.39 14 . Example of a web involving the NE vertex (denoted here by )) is independent of the eikonal approximation. Thus, it als 3.33 Some comments are in order regarding the color factors assoc In the case of a spinor emitting particle, the additional ver are the normal color factors one obtains using the NE Feynman j Both of these verticesoccurring depend in on eq. a ( single time, and thus are ha Figure 14 appears in the exponent of eq. ( in between. One must then sum over all possible correlations emitted from different positions onoccur the between external the line. correlated As NE sho emissions. two types of such diagrams.at Firstly, diagrams eikonal whose level structu (i.e.then are implement two-eikonal correlations irreducible between indiagrams pairs that that of would c gluons be in two-eikonalbut reducible the which at s become eikonal irreducible lev webs at (e.g. figure NE level due to correlations We note that the(eq. derivation ( of the formula for modified color f gauge field. rise to both a one- and a two-gluon vertex, derived from the po logue, which can be seen from the fact that the commutator [ case, where the color factors To summarize, the set ofoutside NE of corrections the resulting hard from interactionNE non-a exponentiate. webs, some The of exponent whichHowever, are they more still complicated share in the structure property than of being two-externa JHEP03(2009)054 ) of 2.4 g (3.45) (3.46) . ) 1 f section ) (3.44) x i = 0. However, ) (3.43) ( x µ 1 ( x − 1 A ; − U , f ) ransformation law of U A 2 t ) ,p ) i µ j ,x x tion considered above, a 1 A x ( ; x f e hard interaction with no ( f − ard interaction (which give ) e but could be obtained it- , the amplitude for internal we briefly discuss how this ge field. As in the previous hich exponentiate (i.e. those i inate from e associated with soft gauge H nteraction). There were then ,p x x = 0. The diagrams containing i an case, one still has to worry ) ( ( 2 δ x t utgoing eikonal lines. ( j U x an internal emission is related to sections) then transforms as g ( ), and the coupling constant f A = 0 i.e. where the eikonal segments U j t )= t . The hard interaction (in the two ) X 1 )= x ) A 1 − ( )= 2 − 1 A U − ,x µ U iθ , as indeed it must be from group theory 1 for an antiparticle. This has the same 1 µ U A x − µ t ( iU∂ iU∂ ), and using the fact that the latter is arbitrary iH − – 30 – iU∂ x − 1 ( = a 1 − − ) = exp( θ )). This is the subject of the next section. − 1 x A U t ( − ) can be interpreted as an extra vertex for soft gluon µ U ) U µ x U 2.20 and ; µ 2 has one adjoint index and two fundamental indices). 3.46 , UA A µ f , UA ,x ) 1 f , UA H ,p x 2 i ( ,p i x µ 2.26 ,x ( x 1 ( f H x f µ ( ∂ → H → ) − µ ) → µ A ) ; A f ; f , A ,p 2 i ,p meet. There are terms (also by analogy with the abelian case o i x ,x = +1 for an eikonal particle, and ( x 2 1 ), which is ( f j x x f ( g H 2.24 We begin with the non-abelian analogue of the abelian gauge t The above discussion shows that, for the simple hard interac As in the abelian case, eq. ( and 1 eikonal line, color singlet case considered in the previous these also take the form of an additional vertex localized at Expanding this to first order in the non-abelian gauge field has been absorbed in which correct for the fact that the external lines do not orig arising from soft gauge bosonremainder emissions terms, outside which of dideratively the not hard to have i all an orders exponentialstructure structur can in be the generalized perturbationsection, to we expansion. the consider case the Here ofinternal case emissions a where has non-abelian a the gau color matrix singlet element structure, for with th twoeq. o ( emission. This vertex is located on the Wilson line at 3.2 Internal emissions ofIn the non-abelian abelian gauge case, fields we identified a subset of NE corrections w subset of corrections exponentiatesboson up emission to outside the NE hardabout order interaction. corrections i.e. As due thos in torise the eikonal to abeli the emissions remainder from term within in the eq. h ( gives the analogue of eq. ( for an external particle, and where for an external antiparticle, where the amplitude with noemission such is emission. proportional In to the the non-abelian color case matrix interpretation as in the abelian case i.e. the amplitude for considerations (the quantity x JHEP03(2009)054 n e abelian nentiates, and the ctions which do not can be analyzed using e of two external lines, er is conceptually equiv- otal matrix element (up agrams with no internal ral can be performed by ective Feynman rules for hich exponentiate (i.e. NE which the outgoing eikonal ntegral point of view. This ich mixes with the exponen- ily have a simple structure, n corrections for matrix ele- , a subset of next-to-eikonal e. Then the diagrams which hich differ from those of the esentation, where the integral olving an internal emission. gauge fields. Diagrams which tter, which corresponds to sys- cture in perturbation theory, in hich act as source terms for the ams are then precisely the webs red perturbation theory arising tween eikonal photons or gluons tering amplitude then factorizes rom the usual exponentiation of N The propagator for each external (as before), and form a remainder nteractions with a given number of ) have the property of being two-external ). These terms are associated with Low’s N ( O 2.20 – 31 – ). 2.20 then exponentiate, and crucially only a subset of diagrams i N , where the sources are placed along the external lines. In th µ s A ). Those which contribute at 3.16 ]. As in the abelian case, a subset of NE corrections also expo 14 , The case of a non-abelian gauge field is more complicated, but In both the abelian and non-abelian cases, there are NE corre Thus, the all-order structure of matrix elements up to NE ord 12 line irreducible, andcorresponding also diagrams have in the (in originalof general) theory. [ color These diagr factors w the replica trick, in which one considers an ensemble of term analogously to that of eq. ( case, exponentiation of eikonaldisconnected corrections diagrams then in follows quantum f corrections field can theory. also Furthermore beradiation in shown the to NE exponentiate, limit and is obtained. a set of eff these additional vertices do not necessarily exponentiate exponentiate, and which formto a NE remainder order) term hastheorem, such and the that the form relevant the diagrams shown involve t on contractions in the be eq. external lines, ( andlines a meet. NE Furthermore, vertex these localized terms have at an the iterative cusp stru at alent to the abelian case.webs), and One a has set of a corrections subsettiated which of NE form a NE corrections. remainder corrections term The w but wh remainder at term least doesemission not has are necessar an sufficient to iterative generate structure higher due order diagrams to inv the fact4 that di Discussion In this paper, we havements considered in the abelian issue and ofinvolves non-abelian considering soft gauge factorized gauge diagrams theories boso involving from hardexternal i a lines, path each i of whichline emits can further then soft radiation. be castis into a a first sum quantized overexpanding path about integral paths repr the for classical the straighttematic line corrections emitting to path particle. the for eikonal the approximation.into emi This a The hard scat path interaction integ plussoft factors gauge for field each external line w exponent contains a sum of webs up to NE level. have a term linear in the theory have suchconnected a by property. a hardexponentiate We interaction contain considered sources with the arising colorfrom from simple singlet eq. the cas structur ( replica orde JHEP03(2009)054 NE (4.1) (4.2) M . pectively, and (NNE) plements next- O +  2 on. Up to NE order, of one-particle phase | vide a new viewpoint (when exponentiated) any exponentiation of do not. However, it is ntributions which would (E) NE r d here, clearly generalizes related to the total energy lly factorized diagrams as ximation, the phase space s NE corrections resulting due to the color structures fact that the terms in |M M ξ M eikonal expansion. Whether ll phase space. one Na¨ıvely,
ne with a cusp. The general grams at a lower order in the der terms. The exponentiated ) are contained in this paper. ian case we considered only the e phase space of the final state .e. conservation of energy is a ture publication. Nevertheless, ternal lines with a color singlet er may provide a useful starting NE nt implies exponentiation of soft (NE) rix elements, in both abelian and iable 4.2 s-sections. This is not necessarily M PS d + r Z M + ). The precise nature of this latter term is 2 ξ | 1+ (NE)  – 32 – E |M M  (E) PS ) is then of eikonal order, and the bracketed term is NE, d = exp 4.2 Z  M + 2 | (E) |M denote the eikonal and next-to-eikonal matrix elements res ) is particularly useful if the contribution from represents that part of the multi-gluon phase space which im (E) 4.1 PS d (NE) ) can be expanded to give (E, NE) Z PS the eikonal phase space, consisting of a factorized product d = M 2.20 The proof of exponentiation in the abelian case, as presente We have only considered matrix elements in this paper. Thus, A comment is in order regarding the nature of NE exponentiati To conclude, the path integral methods used in this paper pro (E) dξ dσ PS gives the dominant contribution toor higher not order this terms is in the the case is presumably processto higher dependent. numbers of eikonal lines.simplest possible However, in hard the interaction, non-abel namelystructure. that with This two ex couldcase be of easily higher related numbers toinvolved. of a Nevertheless, the external single methods lines Wilson introducedpoint is li in in this more addressing pap complicated NE corrections in these situations. NE corrections pertains only beforegauge any bosons integration has over th beenfactorizes performed. into a In productsubleading the effect), of strict thus single exponentiation eikonal in particle appro thelogarithms matrix phase in eleme spaces differential (i (butthe partially case integrated) cros beyond thefrom eikonal the approximation, eikonal matrix whereexpects element one a with given expect integration differential cross-section over (e.g. the in fu some var all of the ingredients for the first bracketed term in eq. ( unclear, and an investigation of its effect is deferred to a fu fraction carried by soft gluons) to have the form i.e. one may eitherbeing consider in the the exponent, NE or terms kept to arising linear from order. eikona This masks the to-eikonal corrections (i.e. subleading terms in eq. ( that the extra vertices that contribute can be related to dia perturbation expansion. Here for the exponentiation of soft radiative corrections to mat where d spaces. The first term in eq. ( genuinely do exponentiate, whereas the remainder terms in true that the exponentiated NEthen terms mix lead with to higher NNE, order NNNE (inform etc. the of co eikonal eq. expansion) ( remain JHEP03(2009)054 N (A.2) (A.3) (A.6) (A.5) (A.4) (A.1) interacting , N diagrams contribute oundation for Funda- ribed by the generat- Jφ ed Feynman diagram, ea for valuable discus- R nteractions between the i on for Scientific Research ]+ N , φ ) [ t-to-eikonal corrections. This 2 grams in quantum field theory, en. Furthermore, the approach iS N of webs is rephrased such that a n of soft radiative corrections to ( , ). This has generating functional O ...e Jφ copies. It follows that } . 1 . R ]) n i N Jφ J ]) + [ N ]+ ]) R φ J i Z [ [ N, J [ ,...,n ]+ iS Z 1 1 ∝ Z φ [ c φe copies. By the same reasoning, disconnected ∈ { log( iS G D = log( is the classical action. Now consider defining – 33 – i e ] = ( c ( N N ) one has N J Z S i [ G X φ φ N D A.3 ]= Z X J [ , and . . . ]=1+ φ 1 Z J ) gives φ [ D N A.5 Z Z 2 constituent parts have ), ( . From eq. ( ≥ ]= of statistical physics. Although we consider a single self- N J A.4 n [ N Z , the proof generalizes easily to other systems. φ replica trick denotes a connected diagram. Furthermore, no disconnected is a source for the field c J G The Green’s functions of a given quantum field theory are desc which clearly satisfies replicas of the theory, involving fields The Feynman rules forfields. each field Thus, are thereand similar, can connected and be diagrams there no therefore are more no have than i one field in each connect closed form solution for the modifiednaturally color encompasses factors can the be exponentiation giv approach of should prove classes fruitful ofall in nex orders the in further perturbation investigatio theory. Acknowledgments We would like to thanksions. Jan-Willem The van work Holten of andmental EL Lorenzo Research and Magn of CDW Matter is(NWO). supported (FOM) by and the the Netherlands National F Organizati A Exponentiation of disconnectedHere diagrams we briefly prove the exponentiationusing of the disconnected dia non-abelian gauge theories. In particular, the discussion diagrams containing and comparing eqs. ( where terms proportional to scalar field ing functional where JHEP03(2009)054 (B.3) (B.1) (B.2) (A.7) . , and show   2.2 , and consider φ ∗ λn J ntiate, as required. = + p ∗ onsider the case of a n rules for the eikonal -scaling as discussed in -photon emission. The Jφ λ of the symmetry factors that the symmetry factor + s for the eikonal and next- , his paper demonstrate that µ φ e theory and its Green func- ) les one can describe emissions , ∗ rting from exact perturbation k ) φ ion of soft radiative corrections 2 scussed in section iε − m , p − µ − 2 ) i (2 . c i 2 1 φ = 0 and use m p 2 µν G µ ) η + k + m D 2 2 ∗ 1 i 1 X − ) p p h ( ( φ − p i i ( µ i D , this corresponds to the eikonal approximation, ( – 34 – representing the NE approximation. 2 − 2 2 λ p p  ] = exp x J [ d d Z p Z i k p  1 1 p p exp p−k ∗ φ . From this functional one derives the Feynman rules D µ φ D iA . As discussed in section − Z µ ∂ ]= →∞ = 1. This is the statement that disconnected diagrams expone ∗ = λ . That is, we consider an external particle of 4-momentum µ N 2 J, J D [ Z The above proof shortcuts the nontrivial combinatoric fact We first consider the propagator-vertex combination for one We begin with the conventional perturbative approach, and c associated with aof disconnected each Feynman of graph the isthe constituent same the connected combinatoric product reasoning graphs. appliesfrom to The fast-moving the results external exponentiat of particles. t B Next-to-eikonal Feynman rules In this appendix, we showto-eikonal how approximations to in derive the path effective integral Feynman rule approach di and sets diagram and corresponding expression is Finally, one writes this as: with the first subleading corrections in charged scalar interacting with ations background are gauge defined field. via the Th generating functional explicitly that these agreetheory with before the taking rules the one NE obtains limit. by sta where Note the symmetry factor in thefrom seagull vertex. a With charged these scalar ru and line, NE and approximation. from To them do we so, can we shall derive take Feynma section the limit JHEP03(2009)054 (B.4) (B.8) (B.7) (B.9) (B.5) (B.6) ) yields /λ , (1  O issions, as shown in ) l ( µ · n ) to . n the remainder as the NE ssible two-photon sources. ) ese sources graphically as . so that the Feynman rules − . rtex and propagator factors diation is insensitive to the B.3 ionic emitting particle, and, 2 µν )  η ator-vertex combination: k 2 2 µ  ne expects this given that there · i ) p/λ ) p k l p − µ · ( , , → 2( n + 2 . k n ν ) k 2 µ · l k n n ·  k µ p 2( ( l p k · p 2 − µ . · − 2 k ) n n k 1 − p l − − − − +  k µν p k · ( µ η ( i ) yields +  k · and expanding eq. ( n – 35 – k 2  (1) (2a) (2b) ν /λ n · n k  1 λn λ (1 p µ · p 1 n λ O − = l n ) is to be contracted with a background gauge field, so p p p  p + − k B.4 µ ·  n k k k n ) k l − + ν k p−k−l n ( · n −  and gathering terms up to . At eikonal level these diagrams give a contribution p 15 We next examine the contribution of two individual photon em We recognize the firstcontribution. term Each as term the in eq. eikonal ( approximation, and which one may rearrange to give: (to the left of the vertex). Setting where we have taken this diagram to represent the combined ve that we can treat each as a one-photon source. We represent th figure Evidently, at the eikonal level theis seagull no term such is seagull absent. vertexas in O is the exact well Feynman known, rulesparticle’s in for spin. the a ferm eikonal approximation the emitted ra Note that here and in following graphs, we replace Scaling are given in terms ofStarting the with physical momenta. the seagull We now term, consider we po take the following propag JHEP03(2009)054 p ) can (B.10) (B.12) (B.13) (B.11) . , B.13 .   l le insertions l l · k ν ν · · n n n n + . kn l ) and ( l · − −  · · l k n k · µ n ) · ν one must sum over all l ). n µ   B.11 p−k n n ) 2 kn n µ + l ves a contribution µ ν · B.5 − )) n ) ), ( k k l l + n ( l menting correlated photon ) d NE emissions, while the · l k + +  B.7 ( µ + 2 n k k 2 · + n ( · k )) · n s. ( k . is non-factorizable into terms ( + l n ( 2 n ν ·  izes into a product of uncorrelated +  + n ν 2 iven in eq. ( 2( n µ k ) n 2 l l  + ( ) ν · · l k  µ n − · n n l p−k−l n + · n ). There are possible vertices, as given in  2( ν 2( k l l n 2 ( 2 − · l 15 ) 2 ν l − l n − 2   +  ) – 36 – k + l )  ν ( l )  k l  + − p µ · k + 2 µ n k µ  · ) + n ( l k n n ( ν · k · n + − · n n n (2 −   n 2  2 2( ) + −  2 k l k +  µ  · + − 2 n  n )  k k · k 2( µ µ  · · 2 µ k n ) n l p−k n k k 2 n 2 − 2( + µ 2 k n k l  ( l   · − ν · ) ) l l ), corresponding to the NE limit, one must sum over all possib n n n + + −  /λ − ν ν l . Diagrams contributing to two photon emission. At NE level,  k k n n ν · (1  ( ( n · · ). The vertex (2a) yields an expression n O n n − B.5 At − −  p−k−l   of a NE one-photon emission vertex in figure ( possible insertions of a NE one-photon emission vertex, as g which can be rearranged to give which can be rewritten as Figure 15 Again this is the sumemission. of an The uncorrelated various part, correlated and contributions a given term in imple eq The contribution from the soft emissionsemissions, explicitly as factor is well-known. eq. ( Notice that the firstlast two term terms represents correspond adependent to correlated on two two-photon a uncorrelate emission single (i.e photon momentum). The NE vertex (2b) gi JHEP03(2009)054 (B.18) (B.15) (B.17) (B.14) (B.16) ) as ), which . . λ 2.16 B.16  ) x + t f p , + l ) i . l · x l · x p sion. To this end we need mitted next to the on-shell · ( es these vertices using the is given by eq. ( µ . , , + t all orders of perturbation ) ) , which we need below, are A necessarily follow that these p kp ) ′ t pectively by full proof within conventional t · kp ν x · l nt , · k · − p ∂ t,t − ) ) p ism. The path integral represen- + ′ t k ′ λ l + ) t i , ( i t ν l and ˙ 2 ( ) ( . x k x p l + θ δ ( ) · + x µ min( + ′ i λ i k  λ p + A p i ( k λ 2 field, and their scaling with · µν ν 2 ( · t,t k = = p η · ) ∂ ( (thus is of NE order), and satisfies the ∂t p x µ · ) x ) p l ′ ′ )= iλ p ′ ]. Here we will only demonstrate that these ′ + ˙ min( /λ − t,t  t,t 37 ( n ∂t t,t i ( λ ) ( t G ∂t∂t ( – 37 – G 2 ∂G + ( x ∂ )= 2 1 2 ′ = ˙ = x i = i dt 2 ) λ t,t (3) + (4a) (4b) + ) ′ i ( ′ t )  t ∞ ( ′ ( G 0 t x x ( dt Z ) ) t x t p p p ( ) ˙ ∞ − ( l l l t x 0 ˙ x ( h Z h ˙ x i h  k k k exp is given by the inverse of the quadratic operator in eq. ( x x D ) = 0. Other two-point correlators of ′ ,t (0)=0 x (0 kinetic term is given by Z G x )= ∞ The ( f Our task is toto derive determine the the photon propagator source and terms vertices from for this the expres be represented as new two-photon vertices, and are given res condition Note that it is symmetric, proportional to 1 is found to be terms are precisely reproduced intation our of path-integral a formal charged scalar coupled to a background gauge field We have shown that these verticeseikonal apply when line. two photons From are our e vertices analysis apply in this to appendix, emissionstheory. it anywhere That does on this not thepath is integral external indeed methods line, described the in a case thisperturbation paper, is theory and to clear we a defer when forthcoming a one paper [ rederiv The propagator for JHEP03(2009)054 ♠ = . t source (B.21) (B.20) (B.22) (B.19) (B.24) (B.23)  ) ) k · A x . n ♥ ( i  + e t t ) are ) t f ♦ k ( x · p ) is given by κ n t ( ( x . Using the dis- + i t ) i ) X x e i t ) k ) ( · t x t + ♣ ( ν n ( ( ( x i ν A . , ) e x ) t t · , ote that these ) dtx ˙ ) ) ( ) k t t k ∂ ( ( · x k ( = 0 ∞ h ν µ n ( λ µ 0 i ( ) ˜ µ i ˙ 2 labeled them with symbols. x A Z X ˜ e A ) k κ dt x dt t,t t µ · k k ( he derivative ˙ µ )+ k )+ ν · ∞ ∞ µ 2 does not appear in the result- n n x k k 0 0 ˙ / x n ) ( d Z Z min( ) µ k + propagator in the eikonal limit, we k d ν ( ν ) ) ˜ d π k A t dt i − k µ k x d d π ( ) nt ) ) ˜ k (2 ) ǫ d A x ∞ µ k k · (2 µ ( 0 + /λ ( k ǫ,t Z µ − n µ i Z n (1 ˜ Z ˜ ) x ) d ǫ d A A λ + ( ) ǫ 1 ) k k k O µ 2 d − t ( d π d π ) A k n only for a NE analysis. The terms without a + µ d d · d π d (2 ˜ (2 t ) (2a). Terms with one power of A d ) – 38 – ) k x ( )= )= (2 d d π x min( x ) Z Z k d path-integral. This amounts to using the Feynman ǫ nt nt B.5 or ˙ (2 d π 0 + ˙ i Z ( ( 2 λ ↓ x 0 d 1 ǫ ↓ 2 lim (2 x n A A − − ǫ Z lim · · i λ = − Z + ( )+ )= ν = 2 dt∂ k dtn t ˙ x x ( ( i , and thus the factor of 1 ) )= )= ) µ κ ∞ ∞ 2 field can be obtained by Taylor expansion of the other terms λ x are t t ˜ 0 0 x ( ( A nt nt )  x Z ν Z x ( ( x t k ), keeping terms to i ) x µ µ ( µ · λ t ) dt 1 ν ˙ n ( 2 A A n x ν x µ nt ∞ ) − h ˙ ( d ∂ B.23 0 x ) k µ µ Z nt d ˙ π dt ( x i A ), ( d µ  ν (2 dt ∞ ∂ A 0 µ κ ∞ Z Z exp ∂ B.17 0 i is quadratic in ν Z − ). Due to the subleading nature of the dtn x are ∂ i  ♠ µ D ∞ x 0 B.15 Z exp dtn i (0)=0 ∞ The next step is to carry out the x 0 Z Z i 2 in eq. ( where we have also represented the terms in momentum space. N We will also need the properties of the equal time correlator The vertices involving the To distinguish the verticesTerms in with our two powers discussion of below, we have terms correspond to the vertex in eq. ( and thus one has shall need them to second order in The term ing vertex. rules in eqs. ( power of cretization of space-time adopted throughout this paper, t JHEP03(2009)054 ). ). = B.5 B.14 (B.25) (B.27) (B.28) (B.26) (B.29) ) l · . . , ) ′ ) ) ) lt l in l l · n − . = + + + + 2 )( ) k ) µ ′ µν k k kt k ( l · l time correlator ( ( k lt η 2 · · l ν · n · · · ( k n · l . n i then show that NE · n + e in ln − ( λn  kn in 2 kt k combination together · l ) − ) · combination gives · -photon source terms ) l ( · − n k l ( ( ( i = ν ln n λn ρσ µ e + ˜ ) ν · ) ˜ η A ′ ams, where terms beyond th a sum over all possible l A k ) k lt ( µκ ♣−♣ · ( d µ  k µ η ) n λn ♣−♦ ) · k ( e precisely reproduced in our n ′ ˜ d π ν µ + . In the tree graph, there are A t  ) d ˜ n + ) kt A ♠ kn (2 ′ · k − µ t d ( · vertex, however, does contribute n k ate to all orders. 2 1 t ns of section ( ) n k µ ). The · i ( ) d − π ˜ l e ln δ ♠ A ). The − n and d ( t · i d (2 λ µ ( µ µν ) l k δ ˜ d B.22 η ♥ = ν  A d π ) ) i k λn λ ) ) are the vertices (1) and (2a) of ( ′ B.14 , d n ′ d π t kt k (2 ·  d ♦ ( combination gives − d (2 i − µ in , ) ′ k  ˜ t B.21 dt dt A d π ) ♣ ( t e Z l d d ). δ ( ∞ (2 ) i k λ 2 1 i – 39 – 0 ν dt dt d π λ ˜ ♦−♦ Z − d A B.5 Z dt (2 ∞ ′ ) σ l 0 k d ( vertices from ( Z Z ρ ) k µ ) k d π σ x ). dt dt ˜ κ d ν k A (2 k ). The ρ d n ∞ k ) k − 0 ) has in principle two possible choices of vertex from ( µ ) ( d π Z B.14 Z n k ν d ) ) ( 2 1 B.14 (2 l l n µ ( ( ) d B.24 ˜ l ν ν ) A k ( ˜ ˜ d d π ν A A ) ) ) d k ˜ A (2 d π k k ) ( ( d k (2 µ µ ( Z ˜ ˜ d µ A A ) 1 2 k ˜ d d A d π ) ) vertex does not actually contribute, as it involves the equa k k d d d d π π ) (2 k d d ♥ correlator gives d π (2 (2 d Z x d d (2 ) ) i k k d , which was shown above to be zero. The d d π π 2 1 ) i k d d ) d π (2 (2 t − ) are neglected. The first two terms ( d ( (2 x The loop graph in eq. ( We conclude that, to next-to-eikonal order, the one- and two Z Z ) /λ t Z 2 1 2 1 ( (1 ˙ x This precisely reproduces source term (3) of eq. ( O We have written the result as the exponent of connected diagr corrections from eikonally factorized diagrams exponenti However, the The third and fourthinsertions terms of represent the tree vertices and denoted loop above graphs, by wi with a two- found by approximations in standardfirst-quantized path-integral perturbation approach. theory The ar consideratio This is source term (4a) of ( which produces term (4b) of ( which can be rewritten as This finally yields source term (2b) of ( h three different combinations of one- and gives JHEP03(2009)054 , µ int (C.1) , and (B.30) (B.31) ) to be ntaining 16 2 )]+Γ p k · + 1 y contributes 2 p ,p ring amplitude ( 2 2 2 1 p p · ,p 1 2 1 ]. p ,p k 2 38 ) Γ( abelian generalization , k ) ) k ≡ + ( ithin the hard interaction µ he above discussion. How- 2 h containing a given hard H ˜ p ch is part of the content of A e vertex ( B.15 n using the normal Feynman ] , only modification arising due ne Γ µ 2 1 t of e.g. [ ictly eikonal limit is insensitive nal Feynman diagram methods, p . ] calar particles. The Lorentz index , γ µ ν Γ[ 2 1 p p γ µ shown pictorially in figure [ , γ ) ν ν k k ]). To clarify this discussion, we here µ k γ · k as [ + d 36 2 ν ) k µ p 2 k d π 2 p d (2 − (2 H Z ]+ – 40 – 2 1 λ p p · − ] (see also [ ) + + 2 1 = p k p 35 , k k − µν 1 34 F p ( µν , 2 2 σ ,p ∞ corresponding to all possible soft emissions from a graph co 2 0 H ) Z µ k . λ 3 i − 2 1 p = Γ[( µ ) k ), as expected from the fact that a magnetic moment vertex onl , we show from our path integral representation of the scatte k · /λ 1 2.4 − . Amplitude Γ p (1 1 2 p O − (2 The above discussion assumes an abelian gauge field. The non- We have considered the case of scalar emitting particles in t We consider the momentum-space amplitude Γ = µ Figure 16 for particles having non-zeroto spin, the and spin radiation of in the the emitting str particle. is discussed in section Γ ever, things proceed similarlyto for the the presence spinor of an case, additional with term the in the exponent of eq. ( a given hard interaction,corresponds in to the the emitted simple photon. case of two outgoing s in the case of two scalar lines. Our presentation follows tha present how one would obtain a similar result using traditio how NE corrections arisingcan from be soft related photon tothe emissions the Low-Burnett-Kroll from hard theorem w interaction [ with no emissions, whi corresponding to a singleinteraction with gluon two emission external emitted scalar from particles. a We grap first defi C Matrix elements withIn internal section emissions the hard interaction amplitude withrules no for photon scalar emission. electrodynamics, The one may write Γ where the right-hand-side corresponds to the momentum spac This is JHEP03(2009)054 , ) es . 2 C.4 p (C.4) (C.2) (C.5) (C.6) (C.7) (C.3) · Γ , 1 ). Then ∂ ∂p  C.3  k ) · , 2 1 2 the limit where p p · . 2 (corresponding to k in eq. ( 2 ross-sections k p k + µ − · · ) so that one obtains Γ k k 1 2 1 ∂ · p p 1 C.3 · 1 ∂p p ) k les for soft emission up to 2 2 ( . p µ 2 ) to obtain  is the amplitude for emission p µ int . µ + nd diagram on the right-hand- 2 16 k 1 + µ int p p +Γ · ) Γ + ( 1 1  ∂ ·  p 2 2 k · , , ∂p p p µ i · µ i k µ 2 Γ · − Γ Γ ∂ p k 1 ∂ ∂ 1 ∂ 2 1 − ∂p ∂p · , one sees that the first term of eq. ( Γ − 2 ∂p 1 ∂ ∂p ∂p p = i p 2 ) is precisely the contribution contained in k 2.4 k · k X p · j back into eq. ( · · · k p Γ 1 − 2 ( C.4 µ 1 1 · p ∂ – 41 – ]. ]. ]. p µ 1 ∂ ). ) to next-to-leading order in p i = p 2 + ∂p −  = 0, one may simplify eq. 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