The all-loop conjecture for integrands of reggeon amplitudes in \( \mathcal{N}=4 \) SYM

Journal of High Energy Physics, Jun 2018

Abstract In this paper we present the all-loop conjecture for integrands of Wilson line form factors, also known as reggeon amplitudes, in \( \mathcal{N}=4 \) SYM. In particular we present a new gluing operation in momentum twistor space used to obtain reggeon tree-level amplitudes and loop integrands starting from corresponding expressions for on-shell amplitudes. The introduced gluing procedure is used to derive the BCFW recursions both for tree-level reggeon amplitudes and their loop integrands. In addition we provide predictions for the reggeon loop integrands written in the basis of local integrals. As a check of the correctness of the gluing operation at loop level we derive the expression for LO BFKL kernel in \( \mathcal{N}=4 \) SYM.

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The all-loop conjecture for integrands of reggeon amplitudes in \( \mathcal{N}=4 \) SYM

JHE The all-loop conjecture for integrands of reggeon amplitudes in 0 Skobeltsyn Institute of Nuclear Physics 1 Institutskiy Pereulok str. , Dolgoprudny , Russia 2 Joliot-Curie str. , Dubna , Russia 3 The Center for Fundamental and Applied Research 4 A.E. Bolshov 5 National Research Nuclear University , MEPhI 6 Moscow Institute of Physics and Technology, State University 7 Joint Institute for Nuclear Research 8 Leninskiye Gory str. , Moscow , Russia 9 Kashira Highway , Moscow , Russia 10 Sushchevskaya str. , Moscow , Russia 11 Bolshaya Cheremushkinskaya str. , Moscow , Russia In this paper we present the all-loop conjecture for integrands of Wilson line form factors, also known as reggeon amplitudes, in N = 4 SYM. In particular we present a new gluing operation in momentum twistor space used to obtain reggeon tree-level amplitudes and loop integrands starting from corresponding expressions for on-shell amplitudes. The introduced gluing procedure is used to derive the BCFW recursions both for tree-level reggeon amplitudes and their loop integrands. In addition we provide predictions for the reggeon loop integrands written in the basis of local integrals. As a check of the correctness of the gluing operation at loop level we derive the expression for LO BFKL kernel in Scattering Amplitudes; Extended Supersymmetry; Perturbative QCD - 4 SYM HJEP06(218)9 N = 4 SYM. 1 Introduction 2 Form factors of Wilson lines and gluing operation Form factors of Wilson lines operators Gluing operator: transforming on-shell amplitudes into Wilson line form factors HJEP06(218)9 Gluing operation in momentum twistor space Loop integrands Gluing operation and local integrands LO BFKL and gluing operation 5.3.1 5.3.2 Virtual part of LO BFKL Real part of LO BFKL 6 Conclusion A Gluing operation and Grassmannians BCFW for integrands of Wilson lines form factors and correlation functions 1 Introduction In the last two decades the tremendous progress in understanding of the structure of Smatrix (amplitudes) in gauge theories in various dimensions has been achieved. The most prominent examples of such progress are various results for the amplitudes in N = 4 SYM theory. See for review [1, 2]. These results were near to impossible to obtain without plethora of new ideas and approaches to the perturbative computations in gauge theories. These new ideas and approaches mostly exploit analytical properties of amplitudes rather then rely on standard textbook Feynman diagram technique. It is important to note that these analyticity based approaches appear to be effective not only for computations of the amplitudes but for form factors and correlation functions of local and non-local operators in N = 4 SYM and other gauge theories as well [3–12]. So for many important results for the amplitudes in N = 4 SYM their analogs for the form factors and correlation functions were found [3, 4, 6, 8–20]. First of all, new variables such as helicity spinors and momentum twistors appear also useful for the description of the – 1 – form factors and correlation functions [3, 4, 10]. At tree level various recurrence relations (BCFW, CSW e t.c.) were constructed for the form factors of some local [3, 4, 6, 7, 21] as well as non-local [22] operators and various closed solutions for such recurrence relations were obtained [3, 4, 6, 7, 10–12, 19, 20, 23]. Ultimately for the form factors of operators from stress tensor supermultiplet [8] as well as for Wilson line operators [19] the representation in terms of integral over Grassmannian was discovered.1 Also in the case of the Wilson line operators such representation was generalized to the form factors with arbitrary number of Wilson line operator insertions as well as correlation functions [20]. Dual description for such objects in terms of twistor string theories was investigated and in this context different CHY like representations for form factors were obtained [14, 15, 24]. In of external particles [3, 13, 16, 31–41] and connection between form factors and integrable systems [8, 42] was discussed. Interesting results [43, 44] also should be mentioned. The ultimate goal for such investigations, similar to the amplitude case, would be the evaluation, in some closed form, of all factors and correlation functions off all possible operators in N = 4 SYM at arbitrary value of coupling constant. In this note we are going to continue to work in this direction and consider the possibility of constructing recurrence relations for the loop integrands of the Wilson line form factors in N = 4 SYM theory. Wilson lines are non-local gauge invariant operators and are interesting objects not only from pure theoretical but also from phenomenological point of view. They appear, for example, in the description of reggeon amplitudes in the framework of Lipatov’s effective QCD lagrangian2 [22, 46–56], within the context of kT or high-energy factorization [57–60] as well as in the study of processes at multi-Regge kinematics. The Wilson line operators play the role of sources for the reggeized gluons, while their form factors are directly related to amplitudes with reggeized gluons in such framework. The results in this field to a large extent originate from long lasting efforts of St.Peterburg and Novosibirsk groups in the investigation of asymptotic behavior of QFT scattering amplitudes at high energies (Regge limit), which can be tracked back in time to early works [61] of Gribov. These results, in particular, include resummation of leading high energy logarithms (αs ln s)n to all orders in strong coupling constant (LLA resummation) in QCD, which eventually resulted in the discovery of Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [62–66] governing the LLA high energy asymptotic behavior of QCD scattering amplitudes. Today BFKL equation is known at next-to-leading-logarithmic-approximation (NLLA) [67–69]. The current article can be also considered as an effort in this direction, namely towards NNLLA BFKL in the context of N = 4 SYM. More accurately, the results of this article can be considered as a solution to the problem of the reduction of individual Feynman diagrams to a set of 1So in this sense these objects at tree level are known for arbitrary number of external legs. 2We also want to mention recent work [45], where Wilson lines arise in the process of off-shell analytic continuation of light-front quantized Yang-Mills action. – 2 – master integrals in BFKL computations. As in general amplitudes multiloop calculations, in BFKL calculations there are basically two steps in getting the final result. The first one is the reduction of contributing individual Feynman diagrams to a finite set of so called master integrals and the second one is the evaluation of these master integrals themselves. In this paper we discuss only the first part of this problem, which is the easiest one. This article is organized as follows. In section 2 we remind the reader the definition of the Wilson line form factors and correlation functions as well as give the definition of so called gluing operator Aˆi−1i considered in our previous papers [24]. This operator allows one to convert the on-shell amplitudes into the Wilson line form factors and will be heavily used throughout the paper. In section 3 we discuss how the BCFW recurrence relations for the Wilson line form factors are constructed with the use of helicity spinor variables used to describe kinematical data. After that we show how the mentioned BCFW recursion can be derived from the BCFW recursion for on-shell amplitudes by means of application of the gluing operators. Section 4 contains the derivation of the gluing operator in the case when kinematical data are encoded by momentum twistor variables. In section 5 we remind the reader the necessary facts about BCFW recursion for integrands of the on-shell amplitudes in momentum twistor space. After that we show how applying the gluing operator one can formulate similar recurrence relation for the Wilson line form factor integrands as well. We also show how one can directly transform the integrands of on-shell amplitudes into the integrands of the Wilson line form factors on the examples of local on-shell integrands. After that we perform simple but interesting self consistency check of our considerations. Namely starting from our results for tree and one loop level Wilson line form factors we correctly reproduce LO BFKL kernel. Appendix A contains the derivation of the Grassmannian integral representation for the reggeon amplitudes starting from corresponding representation for the on-shell amplitudes. 2 2.1 Form factors of Wilson lines and gluing operation Form factors of Wilson lines operators To describe the form factors of Wilson line operators we will use the definition in [22]: Wpc(k) = Z d4xeix·kTr 1 c πg t P exp √ ig Z ∞ 2 −∞ ds p · Ab(x + sp)tb (2.1) where tc is SU(Nc) generator,3 k (k2 6= 0) is the off-shell reggeized gluon momentum and p is its direction or polarization vector, such that p2 = 0 and p · k = 0. The polarization vector and momentum of the reggeized gluon are related to each other through the so called kT -decomposition of the latter: μ kμ = xpμ + kT , x ∈ [0, 1] . (2.2) 3The color generators are normalized as Tr(tatb) = δab. – 3 – HJEP06(218)9 It is convenient to parametrize such decomposition by an auxiliary light-cone four-vector qμ, so that kTμ (q) = kμ − x(q)pμ with x(q) = and q2 = 0. q · k q · p write down the latter in the basis of two “polarization” vectors4 [46]: Noting that the transverse momentum kTμ is orthogonal to both pμ and qμ vectors, we may kTμ (q) = − 2 [pq] κ hp|γμ|q] − 2 κ∗ hq|γμ|p] hqpi with κ = hq|k/|p] , κ∗ = hp|k/|q] . hqpi [pq] It is easy to check, that k2 = −κκ∗ and both κ and κ∗ variables are independent of auxiliary four-vector qμ [46]. Also, it turns out convenient to use spinor helicity decomposition of the light-cone four-vector q as q = |ξi[ξ|. Wpc(k) is non-local gauge invariant operator and plays the role of source for the reggeized gluon [22, 70], so the form factors of such operators, or off-shell gauge invariant scattering amplitudes in our terminology, are closely related to the reggeon scattering amplitudes, and we will use words off-shell gauge invariant scattering amplitudes, reggeon amplitudes and Wilson line form factors hereafter as synonyms. Both usual and color ordered reggeon amplitudes with n reggeized and m usual onshell gluons could be then written in terms of the form factors with multiple Wilson line insertions as [22]: n Y j=1 n Y j=1 (2.3) (2.4) Here asterisk denotes an off-shell gluon and p, k, c are its direction, momentum and color index. Next h{ki, ǫi, ci}im=1| = Nm i=1hki, εi, ci| and hki, εi, ci| denotes an on-shell gluon state with momentum ki, polarization vector εi− or εi+ and color index ci, pj is the direction of the j’th (j = 1, . . . , n) off-shell gluon and kj is its off-shell momentum. To simplify things, here we are dealing with color ordered amplitudes only. The usual amplitudes are then obtained using their color decomposition, see [19, 71]. For example, the color ordered amplitude with one reggeon and two on-shell gluons with opposite helicity at tree level is given by the following expression: A2+1(1−, 2+, g3∗) = ∗ δ4(λ1 λ˜1 + λ2λ˜2 + k3) κ ∗ 3 hp3 1i4 . When dealing with N = 4 SYM we may also consider other on-shell states from N = 4 supermultiplet. The easiest way to do it is to consider color ordered superamplitudes defined on N = 4 on-shell momentum superspace [72, 73]: Am+n Ω1, . . . , Ωm, gm∗+1, . . . , gn∗+m ∗ = hΩ1 . . . Ωm| Wpm+j (km+j )|0i, (2.7) Am+n 1±, . . . , m±, gm∗+1, . . . , gn∗+m ∗ = h{ki, ǫi, ci}im=1| Wpcmm++jj (km+j )|0i . (2.5) sometime abuse spinor helicity formalism notations and write hq|γμ|p]/2 ≡ |p]hq|, λq ≡ hq| and λ˜q ≡ [p|. – 4 – where hΩ1Ω2 . . . Ωm| ≡ Nim=1h0|Ωi and Ωi (i = 1, . . . , m) denotes an N = 4 on-shell chiral superfield [73]: Ω = g + + η˜AψA + 1 2! η˜Aη˜BφAB + 1 1 3! η˜Aη˜Bη˜C ǫABCDψ¯D + 4! η˜Aη˜Bη˜C η˜DǫABCDg−. (2.8) ¯ Here, g+, g− are creation/annihilation operators of gluons with +1 and −1 helicities, ψA, ψA stand for creation/annihilation operators of four Weyl spinors with negative helicity −1/2 and four Weyl spinors with positive helicity correspondingly, while φAB denote creation/annihilation operators for six scalars (anti-symmetric in the SU(4)R R-symmetry indices AB). The A∗m+n Ω1, . . . , gn∗+m superamplitude is then the function of the following kinematic5 and Grassmann variables Ak,m+n Ω1, . . . , gm∗+n = Ak,m+n {λi, λ˜i, η˜i}im=1; {ki, λp,i, λ˜p,i}i=m+1 . ∗ ∗ m+n (2.9) and encodes in addition to the amplitudes with gluons also amplitudes with other on-shell states similar to the case of usual on-shell superamplitudes [1]. Here, additional index6 k in A∗k,m+n denotes the total degree of A∗k,m+n in Grassmann variables ηi, which is given by 4k − 4n. For example the supersymmetrised (in on-shell states) version of (2.6) is given by: δ4(λ1λ˜1 + λ2λ˜2 + k3) δ4 (η˜1hp3 1i + η˜2hp3 2i) . κ ∗ 3 Then it was conjectured that the following relation holds at least at tree level: Ak,m+n Ω1, . . . , gm∗+n = hV1, . . . VmVmge+n.1, . . . , Vmge+n.niworldsheet fields. ∗ Here h. . .i means average with respect to string worldsheet fields. This conjecture was successfully verified at the level of Grassmannian integral representations for the whole 5We used helicity spinor decomposition of on-shell particles momenta. 6We hope there will be no confusion with momentum labels. – 5 – Here we have k = 2, m = 2 and n = 1. We also for simplicity will often drop ∂4/∂η˜4pi projectors in further considerations. 2.2 Gluing operator: transforming on-shell amplitudes into Wilson line form In [24, 74] it was conjectured that one can compute the form factors of Wilson line operators by means of the four dimensional ambitwistor string theory [75]. In an addition to the standard vertex operators V and Ve, which describe Ωi on-shell states in N = 4 SYM field theory, one can introduce, so called, generalised vertex operators V Vj gen. ∼ Z A2,2+1(Ωj , Ωj+1, g∗) ∗ Y i=j,j+1 d2λid2λ˜i Vi Vol[GL( 1 )] gen. [24]: d4η˜i. A2,2+1(Ω1, Ω2, g3∗) = ∗ factors (2.10) (2.11) (2.12) tree level S-matrix [24, 74] and on several particular examples [24] with fixed number of external states. Effectively the evaluation of the string theory correlation function in (2.12) can be reduced to the action of some integral operator Aˆ on the on-shell amplitudes. In the case of one Wilson line operator insertion the relation between on-shell amplitude and the Wilson line form factor looks like: An+1 = Aˆn+1,n+2[An+2] , ∗ (2.13) where An+2 is the usual on-shell superamplitude with n + 2 on-shell external states and the gluing integral operator Aˆn+1,n+2 acts on the kinematical variables associated with the The action of Aˆn+1,n+2 on any function f of variables {λi, λ˜i, η˜i}in=+12 is formally given by This expression can be simplified. Performing integration over λ˜n+1, λ˜n+2, η˜n+1 and η˜n+2 variables [24] in (2.14) we get Aˆn+1,n+2[f ] = hpn+1ξn+1i Z dβ1 ∗ κn+1 dβ2 1 where ∗ denotes substitutions {λi, λ˜i, ηi}i=n+1 7→ {λi(β), λ˜i(β), η˜i(β)}in=+n2+1 with n+2 λn+1(β) = λn+1 +β2λn+2 , λ˜n+1(β) = β1λ˜n+1 + β2 (1+β1) λ˜n+2 , η˜n+1(β) = −β1η˜n+1 , β1β2 (1+β1) λn+1 , λ˜n+2(β) = −β1λ˜n+2 −β1β2λ˜n+1 , η˜n+2(β) = β1β2η˜n+1 , A3,3+1(1−, 2+, 3−, g4∗) = δ4 ∗ 3 i=1 X λiλ˜i + k4 ! 1 [2p4]4 κ4 [12][23][3p4][p41] – 6 – λn+2(β) = λn+2 + and where [19, 46] λn+1 = λp, λ˜n+1 = hξ|k , η˜n = η˜p; λn+2 = λξ, λ˜n+2 = hp|k , η˜n+2 = 0. hξpi hξpi All other variables left unshifted. The integration with respect to β1,2 will be understood as a residue form [76] and will be evaluated by means of the composite residue in points resβ2=0 ◦ resβ1=−1. For example, one can obtain [24] the Wilson line form factor A∗3,3+1(1−, 2+, 3−, g4∗) from 5 point NMHV on-shell amplitude A∗3,5(1−, 2+, 3−, 4−, 5+): A3,3+1(1−, 2+, 3−, g4∗) = Aˆ45[A3∗,5(1−, 2+, 3−, 4−, 5+)], ∗ (2.18) (2.16) (2.17) = Z dα α where There is another way of representing the action of gluing operator. One can note that (2.14) is in fact equivalent to the action of a pair of consecutive BCFW bridge operators, in terminology of [77], on the f function weighted with an inverse soft factor. Namely, function f of the arguments {λi, λ˜i, η˜i}in=+12, 1 ≤ i, j ≤ n + 2 according to: if one [77] defines [i, ji BCFW shift operator as Br(i, j) (see figure 1) which acts on the Br(i, i + 1) hf . . . , λi, λ˜iη˜i, . . . , λj , λ˜j , η˜j , . . . i = ˆ f . . . , λi, λ˜i, η˜ˆi, . . . , λˆj , λ˜j , η˜j , . . . Z dα α f . . . , λi, λ˜i − αλ˜j , η˜i − αη˜j , . . . , λj + αλi, λ˜j , η˜j , . . . , then one can see that the following relation holds: Aˆn+1,n+2[ f ] = Br(n + 1, n + 2) ◦ Br(n + 2, n + 1) S−1(1, n + 2, n + 1) f , Several Wilson line operator insertions correspond to the consecutive action of several gluing operators. For example A∗3,0+3(g1∗, g2∗, g3∗) can be obtained [24] from 6 point NMHV amplitude A3,6(1−2+3−4+5−6+): A3,0+3(g1∗, g2∗, g3∗) = (Aˆ12 ◦ Aˆ34 ◦ Aˆ56)[A3,6(1−2+3−4+5−6+)], ∗ where A∗3,0+3 is given by (P′ is the permutation operator which shifts all spinor and momenta labels by +1 mod 3.): A3,0+3(g1∗, g2∗, g3∗) = δ4(k1 + k2 + k3) 1 + P′ + P′2 f, ∗ and function the f depends on {λi, λ˜i, η˜i}in=+12 arguments. Note also that since Br(i, j) operators act naturally on on-shell diagrams [77] one can easily consider the action of Aˆn+1,n+2 operator on the top-cell diagram corresponding to the Ak,n+2 tree level on-shell amplitude. The top-cell for Ak,n+2 in its turn can be represented as the integral over Grassmannian Lkn+2 [77] (here let’s ignore integration contour for a moment): dk×n+2C δk×2(C · λ˜)δk×4(C · η˜)δ(n+2−k)×2(C⊥ · λ) Vol[GL(k)] (1 · · · k)(2 · · · k + 1) · · · (n + 2 · · · k − 1) . Then one can see that the following relation also holds: h k i Aˆn+1,n+2 Ln+2 = Ωn+1 , k – 7 – i where Ωkn+2 is the Grassmannian integral representation for the off-shell amplitude A∗k,n+1, with the Wilson line insertion positioned after the on-shell state with number n [19], if the appropriate integration contour is chosen for Ωkn+2: 7 HJEP06(218)9 with and k Ωn+2 = Z dk×(n+2)C′ Vol[GL(k)] δk×2 (C′ · λ˜) δk×4 (C′ · η˜) δ(n+2−k)×2 C′⊥ · λ (1 · · · k) · · · (n + 1 · · · k − 2)(n + 2 1 · · · k − 1) , Reg. = hξn+1pn+1i (n + 2 1 · · · k − 1) , ∗ κn+1 (n + 1 1 · · · k − 1) λi = λi, λ˜i = λ˜i, η˜i = η˜i, i = 1, . . . n, i = 1, . . . n, i = 1, . . . n, λn+1 = λpn+1 , λ˜n+1 = hξn+1|kn+1 , hξn+1 pn+1i η˜n+1 = η˜pn+1 , λn+2 = ξn+1 λ˜n+2 = − hξn+1 pn+1i hpn+1|kn+1 , η˜n+2 = 0. The action of several Aˆi,i+1 operators can be considered among the same lines and the result reproduces Grassmannian representation of the form factors with multiple Wilson line operator insertion obtained in [20]. At the end of this section let us make the following comment. Both on-shell and offshell amplitudes (Wilson line form factors) can be represented by means of the BCFW recursion relations. But due to different analytical properties (Wilson line form factors will have additional type of poles corresponding to the Wilson line propagators [46]) the recursion for on-shell and off-shell amplitudes looks rather different. However, from the examples similar to ones considered above (namely (2.18) and (2.18)) one can note that 7One can think of this as alternative derivation of the results of appendix A of [24]. See also appendix of the current article for notation explanation. (2.27) (2.28) (2.29) – 8 – not only gluing operator maps on-shell amplitudes to off-shell ones but one can choose representation for the on-shell amplitude in terms of the BCFW recursion in such a way that each BCFW term from on-shell amplitude will be mapped one-to-one to the terms from the BCFW recursion for the off-shell amplitudes. So a natural question to ask is whether it is possible to derive the BCFW recursion for the Wilson line form factors from the BCFW recursion for the on-shell amplitudes. We will address this question in the next section. BCFW recursion for Wilson line form factors Off-shell BCFW from analyticity First let us remind the reader the main results of [46] and comment on supersymmetric extension of the off-shell BCFW recursion. The off-shell BCFW recursion for the reggeon amplitudes with an arbitrary number of off-shell reggeized gluons was worked out in [46]. Similar to the BCFW recursion [78, 79] for the on-shell amplitudes it is based on the observation, that a contour integral of an analytical function f vanishing at infinity equals where the sum is over all poles of f and resif (z) is a residue of f at pole zi. Using this, one can relate the off-shell amplitude to the sum over contributions of its factorisation channels, which in turn can be represented as the off-shell amplitudes with smaller number of external states. In the original on-shell BCFW recursion the z-dependence of scattering amplitude is obtained by a z-dependent shift of particle’s momenta. Similarly, the off-shell gluon BCFW recursion of [46] is formulated using a shift of momenta for two external gluons i and j with a vector so that 1 2 eμ = hpi|γμ|pj ], pi · e = pj · e = e · e = 0, kˆiμ(z) ≡ kiμ + zeμ = xi(pj )piμ − kˆjμ(z) ≡ kj − zeμ = xj (pi)pjμ − 2 μ κj hpj |γμ|pi] [pj pi] − κi − [pipj ]z hpi|γμ|pj ] 2 This shift does not violate momentum conservation and we still have pi · kˆi(z) = 0 and pj · kˆj (z) = 0. We would like to note, that the overall effect of shifting momenta is that the values of κi and κj∗ shift, while κi∗ and κj stay unshifted. In the on-shell limit the above to zero, that is and the integration contour expands to infinity. Taking the above integral by residues I dz f (z) 2πi z iμ + ki+1 + · · · + kjμ and h is an internal on-shell gluon helicity or a summation μ index over all on-shell states in the Nair on-shell supermultiplet in the supersymmetric case discussed later. Here and below we use the convention that double lines may stand both for off-shell and on-shell gluons. The coil crossed with a line correspond to the off-shell gluons (Wilson line operator insertion). The thick solid lines stand for on-shell particles. The off-shell coil lines can be bent apart to form a single eikonal quark lines [22, 46]. According to this kjμ in kiμ,j can be either off-shell or on-shell depending on the context. Let’s now discuss each type of the terms encountered in (3.6) in more details. The Ai,h terms are usual on-shell BCFW terms, which correspond to the z-poles at which denominator of internal gluon (and also fermion or scalar) propagator kˆ12,i(z) vanishes: shift corresponds to the usual [i, ji BCFW shift. Note also, that we could have chosen another shift vector eμ = 12 hpj |γμ|pi] and shift κi∗ and κj instead. The off-shell amplitudes we consider in this paper do also have a correct large z (z → ∞) behavior [46], so that we should not worry about boundary terms at infinity. The sum over the poles (3.2) for z-dependent off-shell gluon scattering amplitude is given by the following graphical representation8 [46]: (3.6) This is standard BCFW on-shell condition for physical states of N = 4 SYM supermultiplet. The Bi term is a new one and is unique to the BCFW recursion for the off-shell amplitudes. It originates from the situation when the denominators of eikonal propagators coming from Wilson line expansion vanish, that is 8We are considering the color ordered scattering amplitudes and without loss of generality may use shift of two adjacent legs 1 and n. kˆ12,i(z) = 0. pi · kˆi,n(z) = 0 and piμ is the direction of the Wilson line associated with the off-shell gluon. It is important to understand that condition pi · kˆi,n(z) = 0 fixes only the direction of momentum flowing through the Wilson line kˆi,n. The off-shell momenta kˆL = ki−1 + . . . + kˆ1 and kˆR = ki + . . . + kˆn, which belongs to the off-shell amplitudes in eq. (3.7), term Bi, are different (kˆL = −kˆR), but satisfy the same condition pi · kˆL/R = 0. We also want to stress that this term is present only if i labels an off-shell external gluon. In addition, let us note that κi∗ factors in the pair of off-shell amplitudes, which contribute to this term, are given explicitly by the following expressions: [piqi] [piqi] κi,L = hpi|kˆ1 + . . . + ki−1|qi] ∗ , κi∗,R = hpi|kˆn + . . . + ki+1|qi] , (3.10) where κi∗,L and κi∗,R belong to the off-shell amplitudes positioned to the left and to the right The C term is only present if the gluon number 1 is off-shell. It is also unique to the BCFW recursion for the off-shell amplitudes. It appears due to vanishing of the external momentum square Similarly, the D term is due to vanishing of the external momentum square kˆn2(z). It turns out that both these contributions could be calculated in terms of the same BCFW term with the off-shell gluons 1 or n exchanged for the on-shell ones. The helicity of the on-shell 12 hpj |γμ|pi]) used. We refer the reader to [46] for further details and examples. gluons depends on the type of the term (C or D) and the shift vector eμ ( 12 hpi|γμ|pj ] or The use of shifts involving only on-shell legs also allows one to perform the supersymmetrization of the off-shell BCFW recursion introduced in [46]. Indeed, it is easy to see, that the supersymmetric shifts of momenta and corresponding Grassmann variables are given by the on-shell BCFW [i, ji super-shifts:9 |ˆi] = |1] + z|j], |ˆji = |ji − z|ii, ηˆAi = ηAi + zηAj. (3.12) No other spinors or Grassmann variables shift. 3.2 Off-shell BCFW from gluing operation The aim of this section is to derive the off-shell recursion relations described above from the BCFW recursion for the on-shell amplitudes by means of the gluing operator. Before proceeding with general derivation let us consider a simple example first: we will take the BCFW recursion for the on-shell 6-point NMHV amplitude A3,6(1−, 2+, 3−, 4+, 5−, 6+) and transform it into the three point off-shell amplitude A∗0+3(g1∗, g2∗, g3∗) considered in [46]. This off-shell amplitude in its turn also can be obtained from the off-shell BCFW recursion, when external momenta 1 and 3 are shifted. Contributions corresponding to this shift are given in figure 2, and the sum of these three terms is given by (2.21). 9These shifts respect both momentum and supermomentum conservation. external momenta. The first and the third terms labeled A) and C) are type C and D contributions, while the second, labeled B), term is type B contribution of the off-shell BCFW recursion (3.6). representation of A3,6(1−, 2+, 3−, 4+, 5−, 6+) amplitude. So let’s consider A3,6(1−, 2+, 3−, 4+, 5−, 6+) amplitude represented via the standard BCFW shift [1−, 6+i. The amplitude is then given, once again, by three terms (see figure 3) A3,6(1−, 2+, 3−, 4+, 5−, 6+) = A + B + C, where A = A2,5(3−, 4+, 5−, ˆ6+, Pˆ+) 2 A2,3(1ˆ−, 2+, −Pˆ−), B = A2,4(4+, 5−, ˆ6+, Pˆ−) 2 A2,4(1ˆ−, 2+, 3−, −Pˆ+), C = A1,3(5−, ˆ6+, Pˆ+) 2 A3,5(1ˆ−, 2+, 3−, 4+, −Pˆ−). 1 q5,6 1 q1,3 1 Here qa,b = Pib=a qi and qi denote on-shell particle momenta. Now we are going to consider the action of our gluing operators on A3,6 on-shell amplitude, which will convert all pairs of i−, (i + 1)+ gluons into Wilson line operator insertions (reggeized gluons). As we have discussed in previous sections to do this we have to take into account the action of the following combination of the gluing operations on A3,6: A0+3(g1∗, g2∗, g3∗) = Aˆ12 ◦ Aˆ34 ◦ Aˆ56[A3,6(1−, 2+, 3−, 4+, 5−, 6+)]. (3.17) Let’s consider each contribution in details. We will start with A contribution first: Aˆ12 ◦ Aˆ34 ◦ Aˆ56[A] = Aˆ34 ◦ Aˆ56[A2,5(6ˆ+, 5−, 4+, 3−, Pˆ−)]Aˆ12 2 A2,3(1ˆ−, 2+, −Pˆ+) . It turns out, that both the value of the BCFW shift parameter z as well as shifted spinors are regular after we made ∗ substitutions (see (2.15) and (2.2)) corresponding to gluing operations and took limits β2 → 0, β1 → −1. Let us introduce the following notation for the spinors entering kT -decomposition of momenta of three reggeized gluons, each spinor will be labeled by corresponding gluing operator: The original value of the on-shell z parameter (z = [[6221]] ) transformed after the action of Aˆ12 into and helicity spinor decomposition of momentum Pˆ is given now by where we used kT -decomposition of the first reggeized gluon g1∗ momentum k1: z = [21] Then, it is easy to see that Aˆ34 ◦ Aˆ56[A2,5(3−, 4+, 5−, ˆ6+, Pˆ+)] = A1+2(g2∗, gˆ3∗, Pˆ+), ∗ where gˆ3∗ denotes reggeized gluon g3∗ with momentum shifted as For the term we have kˆ3 = k3 + κ1 [p1 p3] |p1i[p3|. Aˆ12 2 A2,3(1ˆ−, 2+, −Pˆ−) = resβ1=−1 ◦ resβ2=0[ωA], # (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) (3.26) ωA = − = 1 hp1 ξ1i κ ∗ 1 1 1 k2 1 h12ih2Pˆi h1Pˆi3 ! dβ1 ∧ dβ2 + less singular terms. where z = where helicity decomposition of momentum Pˆ is given by Aˆ12 ◦ Aˆ34 ◦ Aˆ56[C] = A1+2(gˆ1∗, g2∗, Pˆ−) , ∗ This is precisely the D term from the off-shell BCFW recursion [46] for A∗0+3(g1∗, g2∗, g3∗) amplitude. Now let us turn to B contribution to A3,6 (see figure 2). The value of z parameter in this case transforms under the action of the gluing operator as Taking residues and combining everything together we finally get for A term which is precisely the C term from the off-shell BCFW recursion [46] for A∗0+3(g1∗, g2∗, g3∗) reggeon amplitude (see (3.6)). The C term can be analysed similarly. In this case we get HJEP06(218)9 where ωB = hp2 ξ2i ∗ κ 2 Evaluating corresponding residues we get Aˆ34[B] = 1 h5p2i4 1 κ1 1 κ3 1 1 q1,3 hˆ1 3i4 (3.27) (3.28) (3.29) (3.30) (3.31) (3.32) (3.33) Here it is convenient to consider first the action of Aˆ34. In this case the value of Pˆ momentum is given by β2 2 hξ2 p2i Pˆ = qˆ1 + q2 + (1 + β1) κ∗ |p2i[p2| + less singular terms, before residues evaluation. So for the whole B term after Aˆ34 action we have Aˆ34[B] = Aˆ34 A2,4(4+, 5−, ˆ6+, Pˆ−) 2 A2,4(1ˆ−, 2+, 3−, −Pˆ+) " # = resβ1=−1 ◦ resβ2=0[ωB] h5 Pˆi4 1 2 h4 5ih5 6ˆih6ˆ PˆihPˆ 4i q1,3 hˆ1 2ih2 3ih3 PˆihPˆ 1i ! ∗ β12β2 β1β2 1 dβ1 ∧ dβ2 . (3.34) κˆ2,L h6ˆ5ih5p2ihp26ˆi × hp2|k1 + k2|p2] × κˆ∗2,R h12ih2p2ihp21i ∗ 1 h1p2i4 (3.35) The action of Aˆ12 ◦ Aˆ56 can be evaluated in similar fashion and finally we arrive at Aˆ12 ◦ Aˆ56 ◦ Aˆ34[B] = A0+2(g2∗, gˆ3∗) ∗ 1 which is exactly B term in the off-shell BCFW recursion [46] for A∗0+3(g1∗, g2∗, g3∗) amplitude. So we see the pattern here: the contributions C and D from the off-shell BCFW recursion for A∗0+3(g1∗, g2∗, g3∗) amplitude are reproduced from the on-shell BCFW recursion for A3,6(1−, 2+, 3−, 4+, 5−, 6+) when gluing operator is acting on three point MHV3 or MHV3 sub-amplitudes (with degenerate kinematics), while B contribution is reproduced by the action of gluing operator on both sides of the BCFW bridge, see figure 3. These observations can be immediately generalized to the situation with arbitrary on-shell amplitude. One can obtain the off-shell BCFW recursion for A∗0+n(g1∗, . . . , gn∗) with the shift of the off-shell momenta k1 and kn from the BCFW recursion for An,2n(1−, 2+, . . . , (2n)+) on-shell amplitude represented by [1−, (2n)+i shift. Indeed, the terms C and D are reproduced when gluing operator acts on MHV3 or MHV3 on-shell amplitudes. Repeating the steps identical to the previous discussion (see (3.28)) we get: κˆ2,L = hp2|q5 + qˆ6|ξ] , κˆ2,R = hp2|q2 + qˆ1|ξ] . ∗ ∗ [p2ξ] [p2ξ] " |Pˆi = " where where with (3.36) (3.37) (3.38) (3.39) # (3.40) (3.41) (3.42) Aˆ2n−1 2n ◦ . . . ◦ Aˆ12 An−1,2n−1 3−, . . . , (d2n)+, Pˆ+ 1 q1,2 # = 1 κ1 A1+(n−1)(g2∗, . . . , gˆn∗, Pˆ+), ∗ Similarly for MHV3 we have: 1 x(pn)|p1i − hpnp1i κ ∗ |pni , |Pˆ] = |p1]. Aˆ2n−1 2n ◦ . . . ◦ Aˆ12 A1,3 (2n − 1)−, (d2n) , −Pˆ+ + 1 2 q2n−1,2n An,2n−1(1ˆ−, . . . , Pˆ−) 1 = κ∗ A1+(n−1)(gˆ1∗, . . . , gn∗−1, Pˆ−), ∗ n x(p1)|pn] − [pnp1] |p1] , κ1 |Pˆi = |pni. When gluing operator Aˆii+1 acts on legs separated by the BCFW bridge the B type contribution is reproduced. In this case the on-shell BCFW shift z is replaced by z = hpi|k1 + . . . + ki|pi] , hp1 pii[pi pn] shell [1−, n+i recurcion in general case. A) diagrams will give A type terms of the off-shell BCFW recurcion, B) type diagrams will give B type terms, while C) diagrams will give C and D type terms of the off-shell BCFW recurcion. and the factor 1/q12,i is replaced by (3.43) HJEP06(218)9 1 hpi|k1 + . . . + ki|pi] times βi factors. The explicit proof that such contribution in general case gives us the B term can be done by induction and can be sketched as follows: one can decompose each Aki,ni on-shell amplitude in individual BCFW terms via on-shell diagram representation [77] into combination of MHV3 and MHV3 vertexes (on-shell diagrams). The action of the gluing operator on such on-shell diagrams was considered in the previous section. After that one have to reassemble A∗2,2+1, MHV3 and MHV3 amplitudes together. As the result one can obtain that: " . . . ◦ Aˆii+1 ◦ . . . Ak1,n1 (i + 1)+, . . . , (d2n)+, Pˆ− 2 Ak2,n2 (1ˆ−, . . . , i−, −Pˆ+) 1 q1,i = A0∗+i(gˆ1∗, . . . , gi∗) 1 hpi|k1 + . . . + ki|pi] A0+(n−i)(gi∗, . . . , gˆn∗). ∗ # (3.44) Also, presented above relation is implicitly guaranteed by the Grassmannian integral representation of the on-shell and off-shell amplitudes. It was shown [24, 74] that the latter could be easily related with each other by means of the same gluing operations (see (2.26) and appendix A) which imply similar relation for the individual residues of the top-forms. We see that the gluing operator transforms ordinary 1/P 2 propagator type poles of the on-shell amplitudes into eikonal ones, when external legs, on which the gluing operator acts, are separated by the BCFW bridge (see figure 4 B). The value of the BCFW shift parameter z is adjusted accordingly to match the off-shell BCFW recursion term B. All other contributions (see figure 4) reproduce A type terms from [46], which also can be shown by induction. Indeed it is easy to see that the pole factor remains of the same 1/P 2 type in this case, which corresponds to the propogators of the on-shell states of N = 4 SYM, and the value of z is adjusted to match A term of the off-shell BCFW. The fact that one can transform each individual term in the BCFW recursion for the on-shell amplitudes into terms of the BCFW recursion for the Wilson line form factors using gluing operators Aˆi,i+1 in fact is not (very)surprising and in some sense trivial. Indeed, as was mentioned before, if the relation (2.26) holds on the level of the Grassmannian integrals (top-cell diagrams) then it likely will hold for the individual residues (boundaries of topcells) as well.10 And it is also natural, that in the case of Ωkn+2, and its generalisations to multiple Wilson line insertions, the residues of Ωkn+2 can be identified with the individual BCFW terms of recursion for the off-shell amplitudes in full analogy with the on-shell case. However observation that one can transform the BCFW recursion for the on-shell amplitudes into the BCFW recursion for the Wilson line form factors at tree level opens up exiting possibility. It is known that one can formulate the BCFW recursion not only for the tree level on-shell amplitudes in N = 4 SYM theory but for the loop integrands as well [77, 80]. If we can transform on-shell BCFW recursion into off-shell one at tree level, then what about recursion for the loop integrands ? The next sections will be dedicated to discussion of this question. Namely we will investigate what will happen if we apply the analog of the gluing operators to the BCFW recursion for N = 4 SYM loop integrands of the on-shell amplitudes. Our ultimate goal is to present arguments that using the gluing operator one can transform the integrands of the on-shell amplitudes into the integrands of the Wilson line form factors at any given number of loops and external states. 4 Gluing operation in momentum twistor space The integrands of the on-shell amplitudes in the planar limit are naturally formulated using momentum super twistors variables, in particular this is the case for theis BCFW recursion representation [80]. So, to proceed with our main goal we need to fprmulate our gluing operation in momentum twistor variables as well. To do this, let us recall how the momentum twistor variables are introduced. We start with so called zone super variables or dual super coordinates yi and ϑi which are related with on-shell momenta and its supersymmetric counterpart as [1, 81]: qi = λiλ˜i = yi+1 − yi, λiηi = ϑi+1 − ϑi. (4.1) The introduction of dual super coordinates helps to trivialize the conservation of super momentum [1, 81] and figure 5 shows the momentum conservation geometrically for the case of n = 4 on-shell and one off-shell momenta as an example. There we have a contour in the dual space formed by on-shell particles momenta together with two auxiliary on-shell momenta 5 and 6 used to describe off-shell momentum. The momentum super twistor variables Zi = (λi, µ i, ηi) [81] are then defined through the following incidence relations µ i = λiyi = λiyi+1, η˜i = λiϑi = λiϑi+1. (4.2) 10This can be explicitly seen for some particular case considering integration contours for tree level 3 3 amplitudes Ln+2 and Ωn+2, and probably can be easily generalised for the case of arbitrary number of Wilson line insertions and arbitrary value of k [24]. Here however we avoided considerations of integration contours completely. on-shell legs. In contrast to the case of the on-shell amplitudes, the n on-shell momenta do not add up to zero but to the off-shell gluon momentum k: q1 + . . . + q4 = k, which in its turn can be decomposed as a pair of auxiliary on-shell momenta k = q5 + q6. The bosonic part of Zi will be labeled as Zi = (λi, µ i). Inverting presented above relations δi−1 j hi i + 1i + δij hi + 1 i − 1i + δi+1 j hi − 1 ii hi − 1 iihi i + 1i . (4.3) Here λ˜ ≡ (λ˜1 · · · λ˜n), η˜ ≡ (η˜1 · · · η˜n) and it is assumed that Pin=1 qi = 0. The transition from momentum twistors to helicity spinors could be performed with a formula like:11 µ = Q˜ · λ˜ , η = Q˜ · η˜ , ˜ Qij = (hj ii if 1 < j < i 0 otherwise (4.4) Note, that momentum super twistors trivialize both on-shell condition qi2 = 0 and mentioned above conservation of super momentum. To construct the gluing operator acting in momentum twistor space let us recall that the initial gluing operator in helicity spinor variables can be represented as an action of two consecutive BCFW bridges times some regulator12 factor (2.23). The BCFW bridge operators can also be defined in momentum twistor space using special version of the onshell diagrams [84]. The action of [i, i + 1i the BCFW shift bridge operator br(ˆi, i + 1) in momentum twistor representation on the function Y of {Zi}in=1 variables is given by [80, 84]: Y ′(Z1, . . . , Zn) = br(ˆi, i + 1) [Y (Z1, . . . , Zn)] ≡ Y (Z1, . . . , Zˆi, . . . Zn), (4.5) Z dc c cZi+1. We also do not require Y and Y ′ to be Yangian invariants. where Y , Y ′ are both functions of n momentum super twistors variables and Zˆi = Zi + 11The matrix Q˜ij is a formal inverse of singular map Qij, see [82, 83] for details. 12We call inverse soft factor (2.24) regulator because it makes soft holomorphic limit with respect to one of the auxiliary on-shell momenta, which encodes off-shell one, regular [19, 46]. require that sumed): As new on-shell diagrams in momentum twistor space are no longer built from ordinary MHV3 and MHV3 vertexes (amplitudes), then in the definition of the gluing operator we will have, in principle, to change the form of regulator factor. So, to construct the gluing operator in momentum twistors we will consider the following ansatz: Aˆim−.t1w,iistor [. . .] = N br(ˆi, i + 1) ◦ br(i [+1, i) [M . . .] with two unknown rational functions of helicity spinors λi (first components of momentum twistors) — measure M and normalization coefficient N . To fix N , M functions we Aˆnm+.t1w,nis+to2r hLkn+2i = ωnk+2 , where (k is N(k−2)MHV degree and the use of an appropriate integration contour is ask ωn+2 = Reg. = 1 , , is the momentum twistor Grassmannian integral representation for the ratio of amplitudes with one Wilson line operator insertion A∗k,n+1/A∗2,n+1 and Ln+2 is the Grassmannian k representation for Ak,n+2/Ak=2,n+2 on-shell amplitude ratio: That is our gluing operation should transform the Grassmannian integral representation of on-the shell amplitudes into corresponding Grassmannian integral representation for the off-shell amplitudes. From this requirement we get (4.6) (4.7) (4.8) (4.10) (4.11) HJEP06(218)9 where S is the usual soft factor M = N −1 = S(i + 1, i, i − 1) , S(i + 1, i, i − 1) = ∗ κi−1hi − 1 i + 1i hi i + 1ihi − 1 ii . Computation details can be found in appendix A. So, finally, we have the following expression for the gluing operation in momentum twistor space Aˆim−.t1w,iistor [. . .] = S(i + 1, i, i − 1)−1 br(ˆi, i + 1) ◦ br(i [+1, i) [S(i + 1, i, i − 1) . . .] . (4.12) It may be, at first glance, surprising that here in momentum twistor space we used [i, i + 1i BCFW shift and not [i − 1, ii as for the gluing operation in the helicity spinors representation. In fact, mentioned before the two BCFW shifts are equivalent, see, for example, Aˆ5,6 P64 = Aˆ5,6 P64 = 1 1 1 + hhpp55ξ15ii hh13344556ii [13456] + 1 + hhpp55ξ15ii hh12233556ii [12356] + [12345], A∗2,4+1 A∗3,4+1 (Ω1, . . . , Ω4, g5∗) , Aˆ3,4 ◦ Aˆ5,6 P64 = c35[12345] + c36[12356] + c46[13456], Aˆ3,4 ◦ Aˆ5,6 P64 = A∗2,2+2 A∗3,2+2 (Ω1, Ω2, g3∗, g4∗) , 1 1 1 1 1 + hhpp33pξ34iihh11223354ii , c36 = 1 + hhpp44ξ14ii hh12233556ii , c46 = 1 + hhpp33pξ34ii hh11334566ii 1 + hhpp44ξ14ii hh13344556ii . These results are in complete agreement with previously obtained results from the off-shell BCFW [46] and Grassmannian integral representation [19, 20]. In general the on-shell ratio function Pk,n+2 = Pn+2 4(k−2)(Z1, . . . , Zn+2) can be found for fixed n and k via the solution of the on-shell BCFW recursion in momentum twistor and14 with c35 = Pk,n(Z1, . . . , Zn) = Pk,n−1(Z1, . . . , Zn−1) + X[j − 1, j, n − 1, n, 1]Pk1,n+2−j (ZIj , Zj , Zj+1, . . . , Zˆnj )Pk2,j (ZIj , Z1, Z2, . . . , Zj−1) , where15 Znj = (n − 1, n) ∩ (1, j − 1, j), ZˆIj = (j − 1, j) ∩ (1, n − 1, n), k1 + k2 + 1 = k. We will make more comments about the structure of this recursion relation in the next section. From practical point of view the easiest way to compute Wilson line form factor with f on-shell states and m Wilson line operator insertions is to solve (4.27) for n = f + 2m and then apply m gluing operators via (4.16) rule. Now, when we have the definition of the gluing operator Aˆii+1 in momentum twistor space and some practice with the tree level answers we are ready to consider loop integrands. 5 Loop integrands The natural way to define planar loop integrands unambiguously is to use momentum twistors or dual coordinates. The loop integrand IkL,n for on-shell L-loop amplitude AkL,n 13It is assumed that the momentum super twistors Z5 and Z6 are sent to corresponding off-shell kinematics related to off-shell momenta of g5 reggeized gluon. ∗ reggeized gluons g3 and g4 . ∗ ∗ 15(i, j) ∩ (k, p, m) ≡ Zihjkpmi + Zjhikpmi. 14We again assume corresponding off-shell kinematics for momentum super twistors Z3-Z6 describing in this language is defined as16 Z L reg m=1 A(kL,n)/A(20,n) = Y d4lmIk(L,n)(Z1, . . . , Zn; l1, . . . , lL) , functions n−2 j=2 where momentum super twistors Z1, . . . , Zn describe kinematics of external particles and reg stands for regularization needed by loop integrals. Here Ik,n is a rational function of both loop integration and external kinematical variables. Moreover, Ik,n is cyclic in external momentum super twistors. It is also assumed that loop integrand is completely symmetrized in loop variables l1, . . . , lL. Rewriting the latter in terms of bi-twistors (lm ≡ (AmBm) ≡ (AB)m) the loop integration measure takes the form [80]: where we dropped out factors hλA λBi = hZAZBI∞i as the integrands in N = 4 SYM are always dual conformal invariant. Here I∞ denotes infinity bi-twistor [81]. The integral over the line (AB) is given by the integrals over the points ZA, ZB modulo GL( 2 ) transformations leaving them on the same line. BCFW for integrands of Wilson lines form factors and correlation Now let us see what modifications occur to the on-shell integrand BCFW recursion in the off-shell case. The loop-level BCFW for on-shell amplitudes in N = 4 SYM was worked out in detail in [80] (see also [85, 86] for situation with less SUSY) and the result for Zˆn = Zn + wZn−1 shift reads Ik(L,n) = Ik(L,n)−1(Z1, . . . , Zn−1) +X[j −1, j, n−1, n, 1]Ik(L1,1n)+2−j (ZIj , Zj , Zj+1, . . . , Zˆnj )Ik(L2,2j)(ZIj , Z1, Z2, . . . , Zj−1) + Z d4|4ZAd4|4ZB Z Vol[GL( 2 )] GL( 2 ) [A, B, n−1, n, 1]Ik(L+−1,1n)+2(Z1, Z2, . . . , ZˆnAB , ZA, ZB) , where Zˆnj = (n−1, n)∩(1, j−1, j), ZIj = (j−1, j)∩(1, n−1, n), ZˆnAB = (n−1, n)∩(A, B, 1) and k1+k2+1 = k. The RGL( 2 ) integral is defined as follows. First we set ZA → ZA+αZB ≡ ZA′ and ZB → ZB + βZA ≡ ZB′, which is equivalent to moving points ZA and ZB without changing the line they span. Then we calculate composite residue in α, β such that hA′, 1, n − 1, ni → 0 and hB′, 1, n − 1, ni → 0, what is equivalent to taking points A′, B′ to lie on the plane h1, n − 1, ni: Z GL( 2 ) Z dα Z hA′,1,n−1,ni→0 hB′,1,n−1,ni→0 dβ (1 − αβ)2 . the Jacobian factor (1 − αβ)2 makes poles in α, β simple. Taking the residue as above is equivalent to setting ZA′, ZB′ to (A, B) ∩ (1, n − 1, n) and 16Here by dividing on MHV amplitude we mean that we are factoring out h12i . . . hn1i product and dropping momentum conservation delta function. (5.1) (5.2) (5.3) (5.4) scalar integrals for two loop n = 4 example. Red arrows indicate propagators which we are cutting when evaluating residues. Term C) is actually absent in N = 4 SYM case as well as A). HJEP06(218)9 Next, let us make some comments about the origin of different terms in (5.3). The first two terms, namely n−2 j=2 Ik(L,n)−1(Z1, . . . , Zn−1) + X[j − 1, j, n − 1, n, 1]Ik(L1,1n)+2−j (ZIj , . . . , Zˆnj )Ik(L2,2j)(ZIj , . . . , Zj−1) originate from the poles in the BCFW shift parameter w coming from propagators which does not contain loop momentum dependence: hi − 1in − 1nˆ(w)i = 0. that is from propagators connecting loop integrals, see figure 7 A. These contributions are identical both at tree and loop level. The term containing GL( 2 ) integration Z GL( 2 ) [A, B, n − 1, n, 1]Ik(L+−1,1n)+2(Z1, Z2, . . . , ZˆnAB , ZA, ZB) is present only at the loop level. It originates from the poles in the BCFW shift parameter w coming from propagators containing loop momenta [80], see figure 7 B. At L loop level for n point amplitude the residue at such pole corresponds to the so called forward limit of L−1 loop n + 2 point amplitude. Indeed, if we consider L loop integrand17 of some amplitude at the pole 1/lL2 corresponding to L’th loop integration we will get (see figure 8) In(L)({p1, . . . , pn}, l1, . . . , lL), where {p1, . . . , pn} are external momenta and consider residue ReslL2=0 In(L) ∼ In(L+−21)({p1, . . . , pn, −lL, lL}, l1, . . . , lL−1). In momentum twistor space residue can be evaluated as follows. For simplicity let’s consider L = 1 example to make formulas more readable. The generalization for general L is trivial. The n-point amplitude integrand is the function of the following variables 17Here we assume some specific “appropriate” choice of loop momenta. The corresponding ambiguity in the choice of loop momenta can be removed [80] if one considers dual (or momentum twistor) variables and planar limit, which we are interested in. (5.5) (5.6) (5.7) (5.8) A L−1 = l A L−1 Red arrow indicates which propagator we are cutting. which propagator we are cutting. In( 1 )({Z1 . . . , Zn}, ZA, ZB). The residue at the point (we consider Zˆn = Zn + wZn−1 shift and take residue with respect to w parameter) is given by: where hAB1nˆ(w)i = 0 ReshAB1nˆi=0 In( 1 ) ∼ Atnr+ee2(Z1, . . . , Zˆn, ZˆB, ZˆB), ˆ Zˆn = (n − 1, n) ∩ (A, B, 1), ZB = (A, B) ∩ (n − 1, n, 1). (5.9) (5.10) (5.11) This is analog of (5.8) in momentum twistor space, see also figure 9 and 10. The first expression for Zˆn solves hAB1nˆi = 0. The second expression for ZˆB is the consequence of the first one and the forward limit. See [1] for detailed derivation and discussion. The expression (5.10) in this limit could be obtained from the expression for Atnr+ee2(Z1, . . . , Zˆn, ZA, ZˆB) at general kinematics18 by introducing GL( 2 ) integration with [A, B, n−1, n, 1] weight (5.7). Now let’s see how similar to (5.3) the recurrence relation for Wilson line form factors can be constructed. Let’s consider integrand Ik∗,(nL+) 1 of A∗k(,nL+)1(Ω∗1, . . . , Ωn, gn∗+1) Wilson line 18General in a sense that there are no collinear twistors in contrast to (5.10). form factor. As one will try to reconstruct it via Zˆi = Zi+wZi−1 shift he/she will encounter two types of contributions. The first type will be given by the residues with respect to propagators which does not contain loop momentum dependence. These can be considered along the same lines as in sections 3 and 4. The second type of contribution is the residues with respect to propagator poles with loop momentum dependence. Now in contrast to the case of on-shell amplitudes we have two types of propagator poles. Ordinary 1/l2 poles and eikonal ones 1/hp|l|p]. To simplify discussion let’s consider one-loop case. Generalization to higher loops can be easily done by induction. The residue evaluation with respect to 1/l2 poles is identical to the on-shell amplitudes case and is given by forward limit of tree level Wilson line form factor with n + 2 on-shell legs (k as usual is off-shell momentum with direction p and {q1, . . . , qn} are on-shell momenta): ∼ A(∗n(t+re2e))+1({q1, . . . , qn, −lL, lL}, {p, k}). (5.12) The terms which include eikonal propagator pole residue are a little more complicated. Surprisingly, here similar to the on-shell case we also have forward like limit. For example, consider Wilson line form factor at one loop level In∗( 1 )({q1, . . . , qn}, {k, p}, l). Here once again {q1, . . . , qn} are on-shell momenta and k is off-shell momentum with direction p. Using decomposition (4.13) we can decompose our off-shell momentum into pair of onshell momenta k′ = |pik|ξi/hpξi, k′′ = |ξik|pi/hξpi and formally write this integrand as In∗( 1 )({q1, . . . , qn, k′′, k′}, l). Considering residue for the pole 1/hp|l|p] we enforce on loop momentum l condition hp|l|p] = 0, l2 6= 0 (see also (3.9) and discussion there). This results can set l′ and k′ collinear to each other: where l′ = |pil|ξ′i/hξ′pi, l′′ = |ξ′il|pi/hξ′pi. Now using the freedom in the choice of |ξ′i one (5.13) (5.14) (5.15) (5.16) (l′)μ = −(k′)μ/hξ′pi up to scalar factor hξ′pi. This resembles the on-shell forward limit kinematics of (5.8) for n + 4 point off-shell amplitude. So presumably the residue with respect to Wilson line propagators can, in principle, be evaluated in momentum twistor space along the same lines as (5.10) and (5.11). Consideration of this eikonal residue type, however, can be avoided entirely if one will choose BCFW shift in such a way that w parameter will not appear in eikonal propagators at all. To see this let’s consider once again one-loop case, that is we take the solution of (5.3) for n external particles Ik(L,n=1) and apply gluing operator Aˆn−1n to it Ik∗,((Ln=−12))+1 = Aˆn−1n hIk(L,n=1)i . We will assume that the tree level form factors and on-shell amplitudes are related as Aˆn−1n[Ak,n(Ω1 . . . , Ωn)] = Ak,(n−2)+1(Ω1 . . . , Ωn−2, gn∗−1). ∗ x n + 1 x 1 x n + 2 x 2 Z n Z n Z n−1 Z 1 Z B Z n + 2 gluing xn+1 with x1 while keeping xn fixed in such a way that x21n = 0. In momentum twistor space this equivalent to gluing Zn+1 and Zn+2 with ZˆB. The same is also true for their supersymmetric What we are going to show now is that Ik∗,((Ln=−12))+1 will have appropriate factorization properties19 for one loop Wilson line form factor and that it can be obtained from recurrence relation similar to (5.3), where only poles of (5.12) type will contribute. I.e. there always will be possibility to choose the BCFW shift in such a way that only 1/P 2 type poles will contribute to recursion. To show this let us consider all possible BCFW shifts in Ik∗,((Ln=−12))+1. But first let us note that in the case under consideration the pair of axillary momentum twistor variables Zn and Zn−1 is used to encode information about off-shell momentum k according to (4.13). So, the only possible propagators which contain loop momentum and which will be affected by gluing operator Aˆn−1n are given by hABn − 1ni and hABn1i. More accurately, only hABn − 1ni will be transformed into eikonal propagator hABn − 1n∗i since hABn∗1i = hABn1i. Equivalently one can note that due to the cyclical symmetry the only possible eikonal propagator with loop momentum dependence in A∗k(,(Ln=−12))+1 will depend on ZA, ZB, Zn−1, Zn, Z1 momentum twistors. Now let’s return to the shifts. If we shift Zi as Zˆi = Zi + wZi−1 for i = 1, . . . , n − 2, then the shift parameter w will not affect the eikonal propagator and the corresponding residues with respect to w can be evaluated according to (5.12), so that the result will be given by the forward limit of the tree level Wilson line form factor with n on-shell states. These is precisely the desired factorization property. For this form factor we also know that the relation (5.16) holds. So we see that in such cases the gluing operation indeed transforms solutions of (5.3) into Wilson line form factors similar to tree level. As for the shifts involving Zn−1 and Zn, we can always choose to shift Zˆn = Zn + wZn−1, so that the w parameter drops out of hABn − 1n∗i bracket and will remain only in hABnˆ1i bracket, which is again not affected by the action of Aˆn−1n gluing operator. This gives us ReshABnˆ1i=0 Aˆn−1n[In( 1 )] ∼ Aˆn−1n[Atnr+ee2] = A(n)+1(Z1, . . . , Zn−1, Zˆn∗, ZˆB, ZˆB), ∗ (5.17) 19That is the corresponding residue will be given by forward limit of tree level Wilson line form factor with n + 2 on-shell states. where Zˆn∗ = (n − 1, n∗) ∩ (A, B, 1), Zn∗ = Zn + hpξi hp1i Z1. Here we see that Zˆn∗ solves hABnˆ∗1i = hABnˆ1i = 0 — the same condition as in the case of the on-shell amplitudes. So once again we have appropriate factorization properties and we also see that the gluing operation indeed transforms solutions of (5.3) into the Wilson Equivalently using the same arguments as above one can show that in A∗k(,(Ln=−12))+1 in pair Zn−1, Zn one can always choose to shift Zˆn = Zn + wZn−1 so that w will drop out from eikonal propagator. That is for all Zˆi, which describe both on-shell and offshell momenta, one can choose such shifts that will not affect eikonal propagators with loop momentum dependance and the corresponding recurrence relations will contain only contribution of (5.5) and (5.12) type. This considerations can be easily generalized by induction to arbitrary loop level and ˆ to arbitrary number of gluing operators applied. So we may conclude that application of Ai−1i to (5.3) will likely result in a valid recursion relation for loop integrands of Wilson line form factors (off-shell amplitudes) similar to tree level case. For example if we chose i = n, to match our previous considerations, we will get recurrence relation for the integrand Ik∗,((Ln)−2)+1 of Wilson line form factor when operator is inserted after on-shell state with Ik∗,((Ln)−2)+1 = Ik(L,n)−1(Z1, . . . , Zn−1) + X[j − 1, j, n − 1, n∗, 1]Ik(L1,1n)+2−j (ZIj , Zj , Zj+1, . . . , Zˆn∗j )Ik(L2,2j)(ZIj , Z1, Z2, . . . , Zj−1) (5.18) (5.19) [A, B, n − 1, n∗, 1]Ik(L+−1,1n)+2(Z1, Z2, . . . , Zˆn∗AB , ZA, ZB) , where Zˆnj = (n − 1, n∗) ∩ (1, j − 1, j), ZIj = (j − 1, j) ∩ (1, n − 1, n), ZˆnAB = (n − 1, n∗) ∩ (A, B, 1) and k1 + k2 + 1 = k. Zn∗ is given by (5.18). As before, to encode off-shell momenta we use twistor variables with numbers n − 1 and n. p and ξ are light-cone vectors entering kT -decomposition of this off-shell momentum k. Spinors |pi and |ξi are obtained from corresponding vectors. One can also skip the solution of this new recursion and apply Aˆi−1i directly to the solutions of on-shell recursion relation (5.3), that is to the on-shell integrands, similar to the tree level case (4.16). In the next section we will consider such action using local form of integrands instead of non-local form produced directly by BCFW recursion. At the end of this section we want to make the following note: in general forward limits may not be well defined [80], because on the level of integrands one may encounter contributions from tadpoles and bubble type integrals on external on-shell legs (see figure 7 C as an example). However, such contributions are absent in N = 4 SYM on-shell amplitudes due to the enhanced SUSY cancellations [1, 80]. Their analogs are also absent for the Wilson line form factors (off-shell reggeon amplitudes) — there are no tadpoles diagrams involving closed Wilson line propagators and bubbles on external Wilson line are also equal to 0 on integrand level (see Feynman rules in [46]). Gluing operation and local integrands Now, following our discussion in the previous subsection we conclude that the integrands for the planar off-shell L-loop amplitudes could be obtained from the corresponding onshell integrands by means of the same gluing procedure as was used by us at tree level. Namely, for reggeon amplitude with n reggeized gluons (Wilson line operator insertions) and no on-shell states Ik∗,(0L+)n(g1∗, . . . , gn∗) we should have: Ik∗,(0L+)n = Aˆ2n−1 2n ◦ . . . ◦ Aˆ12hIk,2n i = Ik(L,2)n h1 2i Z1, Z2 − h1 3i Z3, . . . , Z2n−1, Z2n − h2n − 1 2ni h2n − 11i Z1 . (5.20) Here it is assumed that Ik∗,(0L+)n is normalized by A∗2,(00+)n similar to the definition of onshell integrands (5.1). The loop integrands for reggeon amplitudes (Wilson line form factors) with on-shell states can be obtained from (5.20) by removing necessary number of ˆ Ai−1i operators. The loop integrands produced by the BCFW recursion are non-local in general [80]. However, it is still possible to rewrite the integrands in a manifestly local form.20 Moreover, one may choose as a basis the set of chiral integrals with unit leading singularities [80, 87]. The leading singularities are generally defined as the residues of a complex, multidimensional integrals of integrands in question over C4L, where L is the loop order. The computation of residues for the integrands expressed in momentum twistors is then ultimately related to the classic Schubert problem in the enumerative geometry of CP3 [87]. When the residues of integral associated to at least one of its Schubert problems are not the same then the integral is called chiral. In the case when the integral has at most one non-zero residue for the solutions to each Schubert problem then the integral is called completely chiral. If all non-vanishing residues are the same up to a sign then it is possible to normalize them, so that all residues are ±1 or 0. The integrals with this property are called pure integrals or integrals with unit leading singularities. The application of the gluing operation to the on-shell integrands written in the local form follows the general rule (5.20). Let’s see some particular examples. At one-loop for MHV n-point integrand we have21 [80, 87]: I2(1,n) = X hAB(i − 1 i i + 1) ∩ (j − 1 j j + 1)ihXiji i<j hAB XihAB i − 1 iihAB i i + 1ihAB j − 1 jihAB j j + 1i . (5.21) 20This procedure spoils the Yangian-invariance of each term in the on-shell case however. 21(i − 1 i i + 1) ∩ (j − 1 j j + 1) ≡ Zi−1Zihi + 1j − 1jj + 1i + ZiZi+1hi − 1j − 1jj + 1i + Zi−1Zi+1hij − 1jj + 1i. I2∗,((1n)−2)+1 = X where Zn∗ is given by: responds to eikonal propagators with shifted twistor. Wavy line corresponds to numerator of the form hAB(ij)W i, where (ij)W = (i − 1 i i + 1) ∩ (j − 1 j j + 1). This expressions is cyclic invariant and sum in the above expression is independent from X, but contains spurious poles hAB Xi term by term. If we choose X = (k k + 1) then all poles are manifestly physical but cyclic invariance will be lost. To obtain corresponding expression I∗( 1 ) 2,(n−2)+1 for the amplitude with one off-shell leg in place of two last on-shell legs A∗2,((1n)−2)+1(Ω1, . . . , Ωn−2, gn∗−1) we just shift momentum super twistor Zn. Also it is convenient to choose X = (n − 1n): hAB(i − 1 i i + 1) ∩ (j − 1 j j + 1)ihn − 1n∗iji i<j hAB n − 1 n∗ihAB i − 1 iihAB i i + 1ihAB j − 1 jihAB j j + 1i Zn∗ = Zn − hp1i hpξi Z1. , (5.22) (5.23) See figure 11 A. Legs n−1 and n describe off-shell momentum, so that p and ξ are light-cone vectors entering kT -decomposition of this momentum k. Next, taking the expression for the integrand of 2-loop 4-point MHV on-shell amplitude [80, 87]: I2( 2,4 ) = h2341ih3412ih4123i hAB41ihAB12ihAB23ihCD23ihCD34ihCD41ihABCDi +cyclic, no repeat (5.24) and applying Aˆ3,4 gluing operation we get for the integrand of A∗2,(22+)1(Ω1, Ω2, g3∗) (See h3412ih4123ih1234i (5.25) 1 2 2 1 2 Vertical red line represents cuts of corresponding propagators. Grey blobs are on-shell amplitudes with k = 2, 3. Dark grey blobs are Wilson line form factors with k = 2, 3. HJEP06(218)9 where Z4∗ is given by: Z4∗ = Z4 − hp1i hpξi Z1. As always we assume off-shell kinematics for legs 3 and 4, so that p and ξ are light-cone vectors entering kT -decomposition of the off-shell gluon momentum k. Note also that this result is consistent with two and three particle unitarity cuts. See figure 12. The introduced gluing operation also allows us easily obtained expressions for integrands of off-shell remainder functions starting from their on-shell counterparts. Indeed, starting from integrand for 1-loop on-shell remainder function R(k1,n) = Ik,n − Pn ( 1 ) 4(k−2)I2(1,n) and applying gluing operation Aˆn−1,n we may obtain the expression for off-shell remainder function with one off-shell leg in place of two last on-shell legs Rk,(n−2)+1 = Ik∗,((1n)−2)+1 − Aˆn−1,n Pn ∗( 1 ) That is, for example taking integrand for R3∗,(61) on-shell remainder function written in terms of chiral octagons [87]: R3∗,(61) = 1 2 1 where and (see figure 13) ([1, 2, 3, 4, 5]+[1, 2, 3, 5, 6∗]+[1, 2, 3, 6∗, 4])I8(1, 3, 4, 6∗)+ [1, 2, 3, 4, 6∗]I8odd(1, 3, 4, 6∗) 1 − 6 ([1, 3, 4, 5, 6∗]−[1, 2, 3, 4, 5])I8odd( 1, 3, 4, 5 )+ 6 ([1, 2, 4, 5, 6∗]+[1, 3, 4, 5, 6∗])I8odd(1, 4, 5, 6∗) , 1 6 I8odd(i, j, k, l) ≡ I8(i, j, k, l) − I8(j, k, l, i) I8(i, j, k, l) = hABijihAB(j − 1 j j + 1) ∩ (k − 1 k k + 1)i hABi − 1 iihABi i + 1ihABj − 1 jihABj j + 1i × hABklihAB(l − 1 l l + 1) ∩ (i − 1 i i + 1)i hABk − 1 kihABk k + 1ihABl − 1 lihABll + 1i (5.26) (5.27) (5.28) (5.29) (5.30) (5.31) k j numerator of the form hABiji. As before Z6∗ is defined as Z6∗ = Z6 − hp1i Z1 , hpξi and we again assume off-shell kinematics for legs 5 and 6 with p and ξ denoting light-cone vectors entering kT -decomposition of reggeized gluon momentum. Now we would like to show one simple but interesting test both for our tree and loop level constructions (4.16), (5.20) and obtain the expression for LO BFKL kernel with gluing operation. LO BFKL and gluing operation at large center of mass energy √ s and fixed momentum transfer √ Within BFKL approach [62–66] amplitudes of scattering of some quantum states A + B → A′ + B′, which can be partons in hadron, hadrons themselves, high energy electrons etc., −t, s ≫ |t| can be represented as AAB A′B′ = hΦA′A|eαsN ln(s/s0) KBFKL |ΦB′Bi , where the so called impact factors hΦA′A| and |ΦB′Bi are process dependent functions and describe the transitions A → A′ and B → B′. This scattering, in the mentioned above regime, can be described via interaction with special quasiparticles — so called reggeized gluons. BFKL kernel KBFKL describes the self interaction of these reggeized gluons. s0 is some process related energy scale. See for example [70] for detailed discussion. Let us now calculate the LO kernel of BFKL equation in N = 4 SYM with the use of our gluing operation. At LO order it is given by two contribution so called real and virtual one. Consider virtual contribution first (also see figure 14 A). 5.3.1 Virtual part of LO BFKL To compute virtual contribution to the LO BFKL we need the Regge trajectory. The latter could be conveniently extracted from the one-loop correlation function of two Wilson lines playing the role of sources for reggeized gluons [70]. Namely, we have to compute the (5.32) (5.33) following off-shell amplitude: h0|Wp1 (k)Wp2 (−k)|0i = A2,0+2(g1∗, g2∗) = Aˆ12 ◦ Aˆ34 A2,4(1−, 2+, 3−, 4+) . ∗ where β1,( 1 ), β2,( 1 ) parameters correspond to Aˆ12 gluing operation and those with ( 2 ) subscripts to Aˆ34. Evaluating β2,(i) = 0 we get substitutions and taking composite residues at β1,(i) = −1, HJEP06(218)9 A∗2,(00+)2 = − hpκ1∗pκ2∗i2 . 1 2 Now we should recall that the Wilson lines were used here to describe scattering of two fast moving particles at high energy.23 This restricts further our kinematics, so that p1 ·p2 = s/2 (s is the usual Mandelstam variable) and momentum transfer between two particles is restricted by two orthogonality conditions k · p1 = k · p2 = 0. The latter two conditions allow us to write down transverse momentum transfer as k = c1λp1 λ˜p2 + c2λp2 λ˜p1 , so that t ≡ k2 = c1c2s and24 Then for A∗2,(00+)2 amplitude we have κ∗1κ∗2 = c2hp1 p2i [p1 p2] c1hp2 p1i [p2 p1] = −c1c2hp1 p2i2 = − s hp1 p2i2. t [p1 p2] [p2 p1] Now let us turn to the integrand of the corresponding one-loop amplitude. The latter is given for n = 4 by (5.21): A∗2,(00+)2 = t I2∗,(01+) 2 = h1234i2 h12∗ABih23ABih34∗ABih41ABi Z2∗ = Z2 − hp1p2i hp1ξ1i Z3, Z4∗ = Z4 − hp2p1i hp2ξ2i Z1. , 1 (5.34) (5.35) (5.36) (5.37) (5.38) (5.39) (5.40) (5.41) with This expression can be rewritten in spinor helicity variables as: 22The gluing details are similar to those considered in sections 2 and 3. 23For the introduction to corresponding description see [70]. 24It is convenient here to chose ξ1 = p2 and ξ2 = p1. I2∗,(01+) 2 = t2hp1 p2i2 1 = st 4κ∗1κ∗2 l2(l + k)2 l · p1 l · p2 4 l2(l + k)2 l · p1 l · p2 (5.42) The same result can also be obtained within helicity spinor picture where we arrange loop momenta as: I2∗,(01+) 2 = Aˆ12 ◦ Aˆ34 hI2( 1,4 )(1−, 2+, 3−, 4+)i , I2( 1,4 )(1−, 2+, 3−, 4+) = l2(l + q2)2(l + q1 + q2)2(l − q3)2 . (q1 + q2)2(q2 + q3)2 In the expression above qi, (i = 1, . . . , 4) are momenta of external gluons and l is loop momentum. In LO BFKL regime we are interested in leading logarithmic approximation (LLA) to high-energy scattering amplitude. The latter could be obtained using Sudakov decomposition of loop integration momentum and retaining only logarithmic in Mandelstam invariant s contribution. That is 2 dDl = dα dβ dD−2l⊥, l = αp1 + βp2 + l⊥, pi · l⊥ = 0, k · pi = 0, and we are interested in the following regime (here m is some problem related mass scale): (5.43) (5.44) (5.45) (5.46) (5.47) (5.48) (5.49) (5.50) (5.51) (5.52) (5.53) (5.54) Then Now taking residue in β at 0 and integrating over α from m2/s to 1 we get 1 ≫ α ≫ β ∼ ≪ 1. p2 · l = αs/2 + p1 · l⊥ = αs/2 p1 · l = βs/2 + p2 · l⊥ = βs/2 l2 = αβs/2 − l⊥ 2 (l + k)2 = αβs/2 − (l⊥ + k⊥)2. – 34 – Z 1 (2π)4 l2(l + k)2(p1l)(p2l) Z 2(2π)4 1 4π3s log dα dβ dD−2l⊥ αs/2 βs/2 [αβs/2 − l⊥2][αβs/2 − (l⊥ + k⊥)2] m2 Z A∗2,(00++21) = A∗2,(00+)2 g 2 1 − 16π3 log m2 Z k2 d2l⊥ ⊥ l⊥2(l⊥ + k⊥)2 This expression tells us that in LLA approximation25 with account for color factor (CA = N for SU(N ) gauge group) for reggeized gluon propagator we get h0|Wp1 (k)Wp2 (−k)|0i LLA ∼ k2 1 ⊥ s m2 ω(t) = −αsN Z d2+εl⊥ k 2 N αs 2(k⊥2)ε/2 ε is the famous LO BFKL Regge trajectory, which at LO is the same in QCD and N = 4 SYM. See for example [88–90]. Using this result for virtual part of BFKL kernel we can write [70]: − αsN KBVFKL = − 12 δ( 2 )(k − k′) (ω(k⊥) + ω(k⊥ − r⊥)) . Here r is the momentum transfer for A → A′ scattering r = pA′ − pA. In conclusion we would like also to note the following interesting fact. In N = 4 SYM the four point on-shell amplitude A2,4 has a remarkable property27 of being Regge exact, i.e. the contribution of the gluon Regge trajectory to the amplitude (c(t) is the gluon A2,4(s, t) = c(t)2 + subleading terms in | | , t coincides with the exact expression for A2,4(s, t) as a function of arbitrary s and t. 5.3.2 Real part of LO BFKL Now let’s consider real contribution. This contribution is given by the integrated product of two, so called, Lipatov’s Lμ RRP vertexes [70], see figure 14 B. To compute this contribution we may note, that Lipatov’s RRP Lμ vertex (tree level reggeon-reggeon-particle amplitude) is related to reggeon amplitudes A∗2,2+1(g1∗, g2∗, 3+) and A∗3,2+1(g1∗, g2∗, 3−) as Lμ(k, k′) = (k′ + k)μ + n μ k 2 k′− − k + n k+ − k′− , A2∗,2+1(g1∗, g2∗, 3+) = δ4(k − k′ − q3)A2∗,2+1(k, −k′, −q3) = δ4(k − k′ − q3) 3 A3∗,2+1(g1∗, g2∗, 3−) = δ4(k − k′ − q3)A3∗,2+1(k, −k′, −q3) = δ4(k − k′ − q3) 3 ǫμ,+Lμ(k, k′) ǫμ,−Lμ(k, k′) which in their turns could be obtained with two our gluing operations applied to 5-point onand n± are normalized light like directions for reggeized gluons shell amplitude A2,5. Here k, k′ are reggeized gluons g1∗ and g1∗ momenta with k −k′ −q3 = 0 n − = √ , 2p1 s n + = 2√p2 , (n−n+) = 2, (kn±) ≡ k±, and ǫ3± are polarization vectors of on-shell gluon with momentum −q3. It is assumed that in the definitions of A∗2,2+1(g1∗, g2∗, 3+) and A∗3,2+1(g1∗, g2∗, 3−) amplitudes one has to take in 26Here we introduced dimensional regularization of otherwise divergent integral. 27See the discussion in [91]. (5.55) (5.56) (5.57) (5.58) (5.59) (5.60) B B’ B B’ A) B) N = 4 SYM. At large center of mass energy √ s and fixed momentum transfer √ totical behaviour of an amplitude is given by its imaginary part [50, 70]: AAB Grey squares represents impact factors, wavy lines represents gluon propagators, vertical red line represents cuts of corresponding propagators and impact factors. Diagrams of type A) gives congive contribution to KBRFKL and their total sum is equivalent to evaluation of (5.62). tribution to KBVFKL and their total sum is equivalent to evaluation of (5.53). Diagrams of type B) A B ′ ′ |s≫1 ∼ Im[AAABB ]. kT decomposition of k and k′ momenta direction vectors as p1 = n− and p2 = n+. We have also defined functions A∗2,2+1 and A∗3,2+1 which are given by corresponding Wilson line form factors stripped from momentum conservation delta functions: A∗2,2+1(k1, k2, q3) = A∗3,2+1(k1, k2, q3) = [n−n+]3 κ∗1κ∗2 h3n+ihn−3i κ1κ2 [3n+][n−3] . , Performing Sudakov decomposition28 of reggeized gluon momentum the contribution of real radiation to BFKL kernel takes the form [70]: KBRFKL(k⊥, k⊥′, r) ln m2 = s Z 2 dαkdβk′ Lμ(k, k′)Lμ(r − k, r − k ) ′ δ(αkβk′ s + (k⊥ − k⊥′)2) ′ k⊥2(r⊥ − k⊥′)2 Note that factor Lμ(k, k′)Lμ(r − k, r − k′) = gμν Lμ(k, k′)Lν (r − k, r − k′) can be rewritten purely in terms of Wilson line form factors. Namely using gauge invariance of A∗2,2+1(g1∗, g2∗, 3+) and A∗3,2+1(g1∗, g2∗, 3−) we can replace 28See for example [70] for details. gμν 7→ X ǫ(μi)ǫ(νi), i=± (5.61) . (5.62) (5.63) HJEP06(218)9 so that (r − k ≡ m and r − k′ ≡ m′) Lμ(k, k′)Lμ(m, m′) = A∗2,2+1(k, −k′, −q3)A2∗,2+1(m, −m′, −q3) + A∗3,2+1(k, −k′, −q3)A3∗,2+1(m, −m′, −q3). (5.64) r 2 r 2 αk is performed over the interval [ ms2 , 1]. This way we get Also note that in this case no other particles besides gluons from N = 4 SYM supermultiplet give contribution to real radiation. This happens due to the R-charge conservation. The integral over βk′ is taken with the help of δ-function, while the integration over KBRFKL(k⊥, k⊥′, r) = − k⊥′2(r⊥ − k⊥′)2 + ′ ⊥ ⊥ k⊥2(k⊥ − k⊥′)2 + k 2 (r⊥ − k⊥)2 (r⊥ − k⊥′)2(k⊥ − k⊥′)2 . (5.65) Altogether with account for the Regge trajectories contributions we recover LO expression for BFKL kernel KBFKL = KBRFKL + KBVFKL [70]: KBFKL(k⊥, k⊥′, r) = − k⊥′2(r⊥ −k⊥′)2 ⊥ + ′ k 2 (r⊥ −k⊥)2 (r⊥ −k⊥′)2(k⊥ −k⊥′)2 − 12 δ( 2 )(k −k ) ′ Z d2l⊥ 4π2 k 2 (k⊥ −r⊥)2 (l⊥ −r⊥)2(k⊥ −l⊥)2 . (5.66) 6 In this paper we considered the derivation of the BCFW recurrence relation for the Wilson line form factors and correlation functions (off-shell reggeon amplitudes) both at tree and at integrand level. We have shown that starting from the BCFW recursion for on-shell amplitudes and using so called “gluing operator” one can obtain recursion relations for the Wilson line form factors. The latter is true both at tree and integrand level in helicity spinor and momentum twistor representations. The gluing operation also allows one easily convert known local integrands of the on-shell amplitudes into integrands of the Wilson line form factors. These results are condensed in formulas (4.16), (4.17) and (5.19), (5.20) for tree and loop level correspondingly. We have verified our considerations by reproducing LO BFKL kernel. We also made some predictions regarding the structure of the integrands of Wilson line form factors at higher loops/large number of external states. As far as we can understand our construction is not limited to the Wilson line operators only. Indeed, using [24] similar gluing operator the form factors of stress tensor operator supermultiplet could be constructed. Also, presumably, analogs of gluing operator for all other type of local operators in N = 4 SYM theory should exist. The only real obstacle in this direction is that at the level of integrands for local single trace operators we should account for nonplanar contributions. So the notion of “integrand” in this case is somewhat obscure at first sight. Nevertheless we think that one can still introduce integrands in setup similar to considerations in [92], where nonplanar contributions to the on-shell amplitudes were considered in momentum twistor variables. We hope that the presented results will be interesting and useful for people both from N = 4 SYM “amplitudology” and BFKL/reggeon physics communities. Acknowledgments This work was supported by RFBR grants # 17-02-00872, # 16-02-00943, contract # 02.A03.21.0003 from 27.08.2013 with Russian Ministry of Science and Education and Heisenberg-Landau program. The work of L.V. Bork was supported by the grant # 17-13325-1 of the “Basis” foundation for theoretical physics. A Gluing operation and Grassmannians Let’s see how the use of the gluing operation in momentum twistors could easily reproduce known Grassmannian integral representations for the tree-level off-shell amplitudes [19, 20]. We start with the Grassmannian integral representation for the on-shell amplitudes in momentum twistors: Lk,n+2 = Ak,n+2 = Z d(k−2)×(n+2)D δ4(k−2)|4(k−2)(D · Z) Vol[GL(k − 2)] (1 . . . k − 2) . . . (n + 2 . . . k − 3) , here (also similar notations are used in (2.26)) δ4(k−2)|4(k−2)(D · Z) = k−2 Y δ4|4 a=1 X DaiZi , and (i1, . . . , ik−2) is minor constructed from columns of D matrix with numbers i1, . . . , ik−2. Applying to this expression the gluing operation Aˆn+1,n+2 amounts to the following shifts of momentum super twistors: (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) so that where ′ Z1 → (1 + α1α2)Z1 + α1Zn+2 ≡ Z1 , ′ Zn+2 → Zn+2 + α2Z1 ≡ Zn+2 , D1′ = (1 + α1α2)D1 + α2Dn+2 , ′ Dn+2 = α1D1 + Dn+2 ′ ′ D1 = D1 − α2Dn+2 , Dn+2 = −α1D1′ + (1 + α1α2)Dn′+2 . transformation from D’s to D′’s is then given by All other momentum super twistors are unshifted and we have Di′ = Di. The inverse With these transformations it is easy to write down transformation rules for minors. For example, we have (1 . . . k−2) → (1 . . . k−2)′ −α2(n+2 2 . . . k−2)′ (n−k+5 . . . n+2) → −α1(n−k+5 . . . n+1 1)′ +(1+α1α2)(n−k+5 . . . n+2)′ Finally performing transition in the Grassmannian integral from D’s to D′’s and taking residues at α1 = 0 and α2 = − hn+1 1i hn+1 n+2i = − hpn+1 1i hpn+1 ξn+1i we get A∗k,n+1 = A∗2,n+1 Z d(k−2)×(n+2)D′ 1 (1 ... k−2)′ Γ Vol[GL(k−2)] 1+ hpn+1 ξn+1i (n+2 2 ... k−2)′ (1 . . . k−2)′ . . . (n+2 . . . k−3)′ δ4(k−2)|4(k−2)(D′ ·Z) , (A.12) Similarly applying several gluing operations we recover formula [20]. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Yang-Mills and Periodic Wilson Loops, JHEP 01 (2011) 134 [arXiv:1011.1899] [INSPIRE]. [4] A. Brandhuber, O. Gurdogan, R. Mooney, G. Travaglini and G. 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A. E. Bolshov, L. V. Bork, A. I. Onishchenko. The all-loop conjecture for integrands of reggeon amplitudes in \( \mathcal{N}=4 \) SYM, Journal of High Energy Physics, 2018, 129, DOI: 10.1007/JHEP06(2018)129