On the scattering over the GKP vacuum
aa r X i v : . [ h e p - t h ] D ec On the scattering over the GKP vacuum
Davide Fioravanti a , Simone Piscaglia a,ba Sezione INFN di Bologna, Dipartimento di Fisica e Astronomia, Universit`a di BolognaVia Irnerio 46, Bologna, Italy b Centro de F´ısica do Porto and Departamento de F´ısica e Astronomia, Universidade do PortoRua do Campo Alegre 687, Porto, Portugal
Marco Rossi c ∗ c Dipartimento di Fisica dell’Universit`a della Calabria and INFN, Gruppo collegato di CosenzaArcavacata di Rende, Cosenza, Italy
Abstract
By converting the Asymptotic Bethe Ansatz (ABA) of N = 4 SYM into non-linear integralequations, we find 2D scattering amplitudes of excitations on top of the GKP vacuum. Weprove that this is a suitable and powerful set-up for the understanding and computation of thewhole S-matrix. We show that all the amplitudes depend on the fundamental scalar-scalarone. ∗ E-mail: fi[email protected], [email protected], [email protected] Introduction
In integrable system and condensed matter theories the study of the scattering of excitations overthe antiferromagnetic vacuum is at least as much important as that over the ferromagnetic one ( cf. one of the pioneering papers [1] and its references). Often the excitations over the ferromagneticvacuum are called magnons, as well as those over the antiferromagnetic one kinks or solitonsor spinons. Also, two dimensional (lattice) field theories (like, for instance, Sine-Gordon) areoften examples with an antiferromagnetic vacuum [2]. If we wish to parallel this reasoning in theframework of the Beisert-Staudacher asymptotic ( i.e. large s ) Bethe Ansatz (ABA) for N = 4SYM [3], we are tempted to choose, as antiferromagnetic vacuum, the GKP long ( i.e. fast spinning) AdS string solution [4]. According to the AdS/CFT correspondence [5], the quantum GKP stringstate is dual to a single trace twist two operator of N = 4 (at high spin); thus let us consider twocomplex scalars, Z , at the two ends of a long series of s (light-cone) covariant derivatives, D + .Then, excitations of the GKP string correspond to insertions of other operators over this Fermisea, thus generating higher twist operators. More precisely, operators associated to one-particle ϕ states are built as O ′ = T r ZD s − s ′ + ϕD s ′ + Z + . . . . (1.1)The set of lower twist (twist three) excitations includes ϕ = Z , one of the three complex scalars; ϕ = F + ⊥ , ¯ F + ⊥ , the two components of the gluon field; ϕ = Ψ + , ¯Ψ + , the 4+4 fermions, respectively.All these fields are the highest weight of a precise representation of the residual so (6) ≃ su (4)symmetry of the GKP vacuum: the scalar of the vector , the fermions of the and ¯4 , respectively,and the gluons of the representation. All these features and the exact dispersion relations forthese excitations in the different regimes have been studied recently in deep and interesting detailby Basso [6]. Now, as for the scattering, we shall consider at least two particles states (twist-4),namely O ′′ = T r ZD s − s − s + ϕ D s + ϕ D s + Z + . . . . (1.2)This situation was already analysed in partial generality in the case when both the excitations areidentical ϕ = ϕ = Z [7]. Importantly, an impressive recent paper [8] proposed a non-perturbativeapproach to 4D scattering amplitudes in N = 4 SYM by using as building blocks the 2D scatteringamplitudes we wish to compute here: we will see some non-trivial checks of their conjectures.Computing the scattering matrix has a long history (see, for instance, [9] and references therein).From this wide literature we can argue that an efficient method of computation rests on the non-linear integral equation (for excited states) [10]. In fact, the same idea of counting function givesa quantisation condition which can be interpreted as (asymptotic) Bethe Ansatz, defining thescattering matrix (elements). In this note we will use this strategy to provide general formulæ forthe scattering amplitudes between the aforementioned excitations. This should give the non-trivialpart (in front) of the scattering matrices, the so-called scalar factor, being the matrix structurefixed by the aforementioned residual symmetry representations. Remarkably, all the scatteringphases (eigenvalues) are expressible in terms of the scalar-scalar one. Moreover, we evaluate oneloop and strong coupling limits of (scalar-scalar and) gluon-gluon scattering amplitudes and findconfirmation of the conjecture of [8]. 2 Scalar excitations A sl (2) state of L (=twist) scalars and s (=spin) derivatives is described by the counting function Z ( u ) = L Φ( u ) − s X k =1 φ ( u, u k ) , (2.1)where Φ( u ) = Φ ( u ) + Φ H ( u ) , φ ( u, v ) = φ ( u − v ) + φ H ( u, v ) , withΦ ( u ) = − u , Φ H ( u ) = − i ln g x − ( u ) g x + ( u ) , (2.2) φ ( u − v ) = 2 arctan( u − v ) , φ H ( u, v ) = − i " ln − g x + ( u ) x − ( v ) − g x − ( u ) x + ( v ) ! + iθ ( u, v ) , (2.3) θ ( u, v ) being the dressing phase [11] and x ( u ) = u (cid:20) q − g u (cid:21) , x ± ( u ) = x ( u ± i ). One loop or learning the art.–
Let us start by reviewing the one loop formulation, in order tooutline the main steps of our procedure and to enucleate several conceptual features that will beencountered again at all-loops. In the one loop case, the counting function for the twist sector is Z ( u ) = Φ ( u ) − s X k =1 φ ( u, u k ). We remark that by its definition Z ( u ) is a monotonously decreasingfunction. L holes, in positions x h , h = 1 , ..., L are present. Two of them, in positions x , x L areexternal to Bethe roots (i.e. x < u k < x L ); internal or ’small’ holes occupy positions we denotewith x , ...x L − . At one-loop, the counting function satisfies the non-linear integral equation [12] Z ( u ) = Li ln Γ (cid:0) + iu (cid:1) Γ (cid:0) − iu (cid:1) + i L X h =1 ln Γ(1 + iu − ix h )Γ(1 − iu + ix h ) + Z + ∞−∞ dvπ [ ψ (1+ iu − iv )+ ψ (1 − iu + iv )] L ( v ) , (2.4)where L ( u ) = Im ln[1 + ( − L e iZ ( u − i + ) ]. We want to study excitations on top of the GKPstring. The GKP string is dual to the large spin limit of the twist two operator with only thetwo external holes, corresponding to the two Z scalars. Then, excitations arise when L > L − L − Z . In thehigh spin limit and within accuracy O ((ln s ) ), equation (2.4) linearises, because of the expansion R + ∞−∞ dvπ [ ψ (1 + iu − iv ) + ψ (1 − iu + iv )] L ( v ) = − u ln 2 + O (cid:0) s (cid:1) [12], and we can extract scatteringdata in the following way. By definition of counting function( − − L = e − iZ ( x h ) = e iRP ( x h ) L − Y { h ′ =2 , h ′ = h } ( − S ( x h , x h ′ )) ⇒ e iRP ( x h ) L − Y { h ′ =2 , h ′ = h } S ( x h , x h ′ ) = 1 , (2.5)for h = 2 , ..., L −
1, where the last equalities represent the Bethe quantisation conditions: R is giventhe interpretation of the effective length of the closed chain and P ( x h ) that of the momentumof the h -th excitation, so that RP ( x h ) represents the ’free propagation’ phase of this excitation3round the chain (the minus sign in front of iZ ( x h ) is due to our definition of Z ( u ) as a decreasingfunction). On the contrary, S ( x h , x h ′ ) allows for the interaction and is given the interpretation ofphase change due to the scattering between the h -th and the h ′ -th excitation. To ensure unitarityand proper asymptotic behaviour to the scattering, we add and subtract one term in (2.4): Z ( u ) = " − i ( L −
2) ln Γ (cid:0) + iu (cid:1) Γ (cid:0) − iu (cid:1) + i L − X h =2 ln Γ (cid:0) + ix h (cid:1) Γ (cid:0) − ix h (cid:1) + i L − X h =2 ln Γ(1 + iu − ix h )Γ(1 − iu + ix h ) + (2.6) " − i ln Γ (cid:0) + iu (cid:1) Γ (cid:0) − iu (cid:1) − i L − X h =2 ln Γ (cid:0) + ix h (cid:1) Γ (cid:0) − ix h (cid:1) + i ln Γ(1 + iu − ix L )Γ(1 + iu − ix )Γ(1 − iu + ix L )Γ(1 − iu + ix ) − u ln 2 . In the large spin limit we have [13] x L = − x = s √ + O ( s ), while x h ∼ s , 2 ≤ h ≤ L − − u ln s − i ln Γ ( + iu ) Γ ( − iu ) + O (cid:16) s ) (cid:17) . When nointernal hole is present at all, the first bracket in (2.6) is absent and we identify the effective lengthof the string and the excitation momentum as R = 2 ln s and P ( x h ) = 2 x h , respectively. On theother hand, to find the scattering matrix involving two scalar excitations, it is convenient to stickto the L = 4 case (two internal holes). Now, the extra phase shift, due to the first bracket in (2.6)is interpreted, via (2.5), as the scattering factor S ( x h , x h ′ ) = − Γ (cid:0) − ix h (cid:1) Γ (cid:0) + ix h ′ (cid:1) Γ(1 + ix h − ix h ′ )Γ (cid:0) + ix h (cid:1) Γ (cid:0) − ix h ′ (cid:1) Γ(1 − ix h + ix h ′ ) , (2.7)between two internal holes with rapidities x h and x h ′ . This expression enjoys unitarity and doesagree with result (3.8) of Basso-Belitsky [15], but seems to be the inverse of (2.13) in [16]. All loops.–
Using results contained in Section 2 of [17], we write for the counting function thefollowing non-linear integral equation Z ( u ) = F ( u ) + 2 Z + ∞−∞ dvG ( u, v ) L ( v ) , (2.8)where the functions F ( u ) and G ( u, v ) are obtained after solving the linear integral equations F ( u ) = f ( u ) − Z + ∞−∞ dvϕ ( u, v ) F ( v ) , G ( u, v ) = ϕ ( u, v ) − Z + ∞−∞ dwϕ ( u, w ) G ( w, v ) , (2.9)with f ( u ) = L Φ( u ) + L X h =1 φ ( u, x h ) , ϕ ( u, v ) = 12 π ddv φ ( u, v ) . (2.10)Equations (2.10) and (2.9) entail the sum F ( u ) = L ˜ P ( u )+ L P h =1 R ( u, x h ) of the two functions R ( u, v ),such that π ddv R ( u, v ) = G ( u, v ), and ˜ P ( u ), solutions respectively of the two linear equations R ( u, v ) = φ ( u, v ) − Z + ∞−∞ dwϕ ( u, w ) R ( w, v ) , ˜ P ( u ) = Φ( u ) − Z + ∞−∞ dwϕ ( u, w ) ˜ P ( w ) . (2.11)4ow, we consider the high spin limit, work out the nonlinear term and - following what we didin the one loop case - identify the momentum of a hole and the scattering phase between twoholes. In fact, the nonlinear term N L ( u ) = 2 R dvG ( u, v ) L ( v ) depends on the function G ( u, v ). Bymanipulating the second of (2.9) in Fourier space and using formula (3.2) of [7], we arrive at this(approximated) integral equation ˆ N L ( k ) = − π ln 2 ik δ ( k ) − − e −| k | R + ∞−∞ dp π ˆ ϕ H ( k, p ) ˆ N L ( − p )+ O (1 /s ),which proves that N L ( u ) start contributing at order O ( s ). Again, as in the one loop case, we usethe quantisation conditions( − − L = exp( − iZ ( x h )) = e iRP ( x h ) L − Y { h ′ =2 , h ′ = h } ( − S ( x h , x h ′ )) (2.12)to define the momentum of a hole/scalar excitation of rapidity u as the function P ( u ) such that − P ( u ) − R ( u, x L ) − R ( u, x ) − L − X h =2 ˜ P ( x h ) − N L ( u ) ≃ R · P ( u ) , (2.13)with R ≃ s (since x L = − x ≃ s √ ) the effective length of the chain. On the other hand, thescattering factor between two holes of rapidities u , v is the function S ( u, v ) defined by i ln ( − S ( u, v )) = R ( u, v ) + ˜ P ( u ) − ˜ P ( v ) = Θ( u, v ) . (2.14)We remark that the following properties hold˜ P ( u ) = − ˜ P ( − u ) , R ( u, v ) = − R ( − u, − v ) , R ( u, v ) = − R ( v, u ) . (2.15)In particular, the last property implies unitarity, i.e. Θ( u, v ) = − Θ( v, u ). To compute this phase,we need first its ’reduced’ version Θ ′ ( u, v ) = R ( u, v ) + ˜ P ( u ), which satisfies, in (double) Fourierspace, ˆΘ ′ ( k, t ) = ˆ φ ( k, t ) + ˆΦ( k )2 πδ ( t ) − Z + ∞−∞ dp π ip ˆ φ ( k, p ) ˆΘ ′ ( − p, t ) , (2.16)upon manipulating and adding equations (2.11). In fact, this function enters the even part (in thesecond variable) of the scattering phase M ( u, v ) = Θ( u,v )+Θ( u, − v )2 = Θ ′ ( u,v )+Θ ′ ( u, − v )2 , whose (double)Fourier transform, by virtue of (2.16), satisfies the equationˆ M ( k, t ) = ˆ φ H ( k, t ) + ˆ φ H ( k, − t )2(1 − e −| k | ) − π δ ( t ) J ( √ gk ) ik sinh | k | ++ 2 π ik e −| k | − e −| k | [ δ ( k + t ) + δ ( k − t )] − Z + ∞−∞ dp π ip ˆ φ H ( k, p )1 − e −| k | ˆ M ( − p, t ) . (2.17)We observe that ˆ M ( k, t ) enjoys the parity propertiesˆ M ( k, t ) = ˆ M ( k, − t ) , ˆ M ( k, t ) = − ˆ M ( − k, t ) ; (2.18)the first property is true by construction, the second one holds since the function ˆ φ H ( k, t ) +ˆ φ H ( k, − t ) is an odd function of k . Now, the key point is that we can relate ˆ M ( k, t ) to functions5e found in the study of high spin twist sector. Let us consider the density corresponding to thefirst generalised scaling function (i.e. the part of the density proportional to ln s L − s , see [18] fordetails ):ˆ σ (1) ( k ) = π sinh | k | [ e − | k | − J ( √ gk )] + ik − e −| k | Z + ∞−∞ dt π ˆ φ H ( k, t ) h π + ˆ σ (1) ( t ) i . (2.19)Then, consider the density ’all internal holes’, which satisfies equation (3.8) of [7]: we formallyput L = 3 and highlight the dependence of the solution of (3.8) of [7] on the position x of the(fictitious) single internal holeˆ σ ( k ; x ) = 2 πe −| k | − e −| k | (cos kx −
1) + ik − e −| k | Z + ∞−∞ dt π ˆ φ H ( k, t ) h π (cos tx −
1) + ˆ σ ( t ; x ) i . (2.20)Fourier transforming with respect to x , it is easy to see that ik ˆ M ( k, t ) = Z + ∞−∞ dxe − itx [ˆ σ (1) ( k ) + ˆ σ ( k ; x )] ⇒ ddu M ( u, v ) = σ (1) ( u ) + σ ( u ; v ) . (2.21)In order to fix M from (2.21), we use properties (2.18): we obtain that M ( u, v ) = Z (1) ( u ) + Z ( u ; v ),where Z (1) ( u ) and Z ( u ; v ) are univocally defined by the conditions ddu Z (1) ( u ) = σ (1) ( u ) , ddu Z ( u ; v ) = σ ( u ; v ) , Z (1) ( u ) = − Z (1) ( − u ) , Z ( u ; v ) = − Z ( − u ; v ) . (2.22)Now, we easily analyse the odd part of the scattering phase N ( u, v ) = Θ( u,v ) − Θ( u, − v )2 = R ( u,v ) − R ( u, − v )2 − ˜ P ( v ), for which properties (2.15) bring about M ( v, u ) = − N ( u, v ). As a con-sequence, the scattering phase Θ( u, v ) can be expressed in terms of only the function M asΘ( u, v ) = M ( u, v ) − M ( v, u ) . (2.23) Two strong coupling limits
Using (2.21, 2.23), we want to study the strong coupling limit of the holes scattering phase. Weanalyse two limits: the so-called non-perturbative regime [19], in which g → + ∞ , with u, v ∼ u = √ g ¯ u , v = √ g ¯ v , and thensend g → + ∞ , with ¯ u , ¯ v fixed. In the latter case, two regimes appear: if | ¯ u | < | ¯ v | <
1, we arein the perturbative regime, if | ¯ u | > | ¯ v | >
1, we are in the giant-hole [20] regime.In the non-perturbative regime we use results (3.21,3.22) of [7], i.e.:ˆ σ (1) ( k ) + ˆ σ ( k ; x ) → ˆ σ (1) lim ( k ) cos kx , ˆ σ (1) lim ( k ) = 2 π " e −| k | − e −| k | − e | k | | k | cosh k . (2.24) In previous literature integral equations are often written by using the ’magic kernel’ ˆ K [11], related to ˆ φ H byˆ φ H ( k, t ) + ˆ φ H ( k, − t ) = 8 iπ g e − t + k ˆ K ( √ gk, √ gt ) , t, k > . g → + ∞ ⇒ ddu M ( u, v ) →
12 [ σ (1) lim ( u − v ) + σ (1) lim ( u + v )] ,σ (1) lim ( u ) = − h ψ (cid:16) − i u (cid:17) + ψ (cid:16) i u (cid:17) − ψ (cid:18) − i u (cid:19) − ψ (cid:18)
12 + i u (cid:19) + 2 π cosh π u i ⇒ M ( u, v ) = − i (cid:0) − i u − v (cid:1) Γ (cid:0) + i u − v (cid:1) Γ (cid:0) − i u + v (cid:1) Γ (cid:0) + i u + v (cid:1) Γ (cid:0) i u − v (cid:1) Γ (cid:0) − i u − v (cid:1) Γ (cid:0) i u + v (cid:1) Γ (cid:0) − i u + v (cid:1) −−
12 gd (cid:18) π ( u − v )2 (cid:19) −
12 gd (cid:18) π ( u + v )2 (cid:19) . (2.25)Therefore, for the scattering phase Θ( u, v ) we have the expression [21] g → + ∞ ⇒ Θ( u, v ) → − i ln Γ (cid:0) − i u − v (cid:1) Γ (cid:0) + i u − v (cid:1) Γ (cid:0) i u − v (cid:1) Γ (cid:0) − i u − v (cid:1) − gd (cid:18) π ( u − v )2 (cid:19) , (2.26)which depends only on the difference of the rapidities.We now rescale the rapidity u = √ g ¯ u and then send g → + ∞ , with ¯ u fixed. It is easierto compute the double derivative of the scattering factor Θ( u, v ), since it depends on the density σ ( u ; v ) only: ddu ddv Θ( u, v ) = ddv σ ( u ; v ) − ddu σ ( v ; u ) . (2.27)On the other hand, the function dd ¯ x σ ( u ; x ) is written (at the leading order g ) as dd ¯ x σ ( u ; x ) ∼ = Z + ∞ d ¯ t √ g cos ¯ t ¯ u " dd ¯ x Γ − (¯ t ; ¯ x ) − dd ¯ x Γ + (¯ t ; ¯ x ) − e − ¯ t √ g − e − ¯ t √ g ¯ t sin ¯ t ¯ x , (2.28)where the functions dd ¯ x Γ(¯ t ; ¯ x ) satisfy the integral equation, valid for | ¯ u | ≤ Z + ∞ d ¯ t (cid:20) e i ¯ t ¯ u dd ¯ x Γ − (¯ t ; ¯ x ) − e − i ¯ t ¯ u dd ¯ x Γ + (¯ t ; ¯ x ) (cid:21) = Z + ∞ d ¯ te i ¯ t ¯ u ¯ t sin ¯ t ¯ x sinh ¯ t √ g ∼ = 2 √ g ¯ x ¯ x − ¯ u , (2.29)We set Γ + (¯ t ; ¯ x ) = R dk cos k ¯ t ˜Γ( k ; ¯ x ), Γ − (¯ t ; ¯ x ) = − R dk sin k ¯ t ˜Γ( k ; ¯ x ) and solve (2.29): dd ¯ x ˜Γ( k ; ¯ x ) = −√ g [ δ ( k − ¯ x ) − δ ( k + ¯ x )] + O (1 /g ) , | ¯ x | < dd ¯ x ˜Γ( k ; ¯ x ) = − gπ " (cid:0) k − k (cid:1) / (cid:0) ¯ x − x +1 (cid:1) / ¯ x − k + (cid:0) k − k (cid:1) / (cid:0) ¯ x +1¯ x − (cid:1) / ¯ x + k + O (1 /g ) , | k | < , | ¯ x | > | ¯ x | , | k | > k ; ¯ x ) is exponentially small. Plugging (2.30, 2.31) into (2.28), we find thefollowing behaviour at the leading order g : dd ¯ x σ ( √ g ¯ u ; √ g ¯ x ) = − H (¯ u − H (¯ x − h (cid:0) ¯ x − x +1 (cid:1) / (cid:0) ¯ u − u +1 (cid:1) / + (cid:0) ¯ x +1¯ x − (cid:1) / (cid:0) ¯ u +1¯ u − (cid:1) / ¯ u + ¯ x ++ (cid:0) ¯ x − x +1 (cid:1) / (cid:0) ¯ u +1¯ u − (cid:1) / + (cid:0) ¯ x +1¯ x − (cid:1) / (cid:0) ¯ u − u +1 (cid:1) / ¯ x − ¯ u i , (2.32)7ith H ( x ) the Heaviside function. Inserting (2.32) into (2.27), we obtain the leading order dd ¯ v dd ¯ w Θ( √ g ¯ v, √ g ¯ w ) = √ gH (¯ v − H ( ¯ w − (cid:0) ¯ v +1¯ v − (cid:1) / (cid:0) ¯ w − w +1 (cid:1) / + (cid:0) ¯ v − v +1 (cid:1) / (cid:0) ¯ w +1¯ w − (cid:1) / ¯ v − ¯ w . (2.33)Result (2.33) agrees with corresponding formula coming from the scattering phase (2.34) of [16]. The excitations of gauge fields on GKP string correspond to insertions of partons of the type D l − ⊥ F + ⊥ or ¯ D l − ⊥ ¯ F + ⊥ . As for the ABA, the field D l − ⊥ F + ⊥ is represented as a stack of l + 1 roots u , l u and l − u roots [6] of the Beisert-Staudacher equations [3] (similarly, ¯ D l − ⊥ ¯ F + ⊥ comesout from the replacement ( u , u , u ) → ( u , u , u )). Then, these large s equations (an operatorwith some fields D l − ⊥ F + ⊥ on a sea of covariant derivatives D + ( u roots)) can take on the form • e − ip k L ′ s Y j =1 S (44) ( u k , u j ) Y m N m Y i =1 S (4 g ) m ( u k , u mi ) (3.1) • Y m N m Y i =1 S ( gg ) lm ( u lk , u mi ) s Y j =1 S ( g l ( u lk , u j ) , (3.2)where S (44) ( u k , u j ) describes the scattering involving two type-4 Bethe roots, while S (4 g ) m ( u k , u mi )that of a type-4 root colliding with a gluonic stack (of length m and real centre u mi ) and finally S ( gg ) lm ( u lk , u mi ) represents the matrix for the scattering of gluonic bound states, in terms of real cen-tres. We are going to take into account a system composed of Q gluonic bound states, representedby stacks of length m k and real centre ˜ u k , with k = 1 , . . . , Q , together with L − L ′ appearing in (3.1) equals L + Q . As amatter of fact, in order to accommodate a gluonic excitation, a type-4 root has to be ’pulled away’from the sea, and a gluon substitute it, then. The state we are thus considering is characterisedby L + Q missing main roots in the sea, the vacuum corresponding to L = 2 , Q = 0.The counting function for a gluonic stack, with real centre u and length l , which collides onlywith scalars and other gluonic excitations (centre rapidity ˜ u k , length m k ) is Z g ( u | l ) = Q X k =1 ˜ χ ( u, ˜ u k | l, m k ) − Z + ∞−∞ dv π χ ( v, u | l ) ddv [ Z ( Qg ) ( v ) − L ( Qg ) ( v )] − L X h =1 χ ( x h , u | l ) , (3.3)whereas the counting function (2.1) adapted to the case at hand, including the Q gluonic stacks,takes on the form: Z ( Qg ) ( v ) = ( L + Q )Φ( v ) + Z + ∞−∞ dw π φ ( v, w ) ddw [ Z ( Qg ) ( w ) − L ( Qg ) ( w )] ++ L X h =1 φ ( v, x h ) + Q X k =1 χ ( v, ˜ u k | m k ) . (3.4)8n addition to definitions (2.2, 2.3), we introduced χ ( v, u | l ) ≡ χ ( v − u | l + 1) + χ H ( v, u − il χ H ( v, u + il , (3.5)˜ χ ( u, v | l, m ) ≡ χ ( u − v | l + m ) − χ ( u − v | l − m ) + 2 l − X γ =1 χ ( u − v | l + m − γ ) , (3.6)where we split the function χ into its one-loop and higher than one loop parts, respectively • χ ( u | l ) ≡ ul = i ln il + 2 uil − u • χ H ( u, v ) ≡ i ln − g x − ( u ) x ( v ) − g x + ( u ) x ( v ) ! . (3.7)Equation (3.4) suggests that Z ( Qg ) ( v ) satisfies the nonlinear integral equation Z ( Qg ) ( v ) = F ( Qg ) ( v ) + 2 Z + ∞−∞ dw G ( v, w ) L ( Qg ) ( w ) , (3.8)where the function F ( Qg ) ( v ) is written as F ( Qg ) ( v ) = ( L + Q ) ˜ P ( v ) + L X h =1 R ( v, x h ) + Q X k =1 T ( v, ˜ u k | m k ) , (3.9)with R ( v, u ) and ˜ P ( v ) solutions of (2.11) respectively, and T ( v, ˜ u | m ) equals T ( v, ˜ u | m ) = χ ( v, ˜ u | m ) − Z + ∞−∞ dw G ( v, w ) χ ( w, ˜ u | m ) . (3.10)We are interested in the scattering factors involving gluons, which correspond to l = 1 stacks.Therefore, we restrict to this case and, for clarity’s sake, denote the scalar-scalar factor as S ( ss ) ( x h , x h ′ ) = − exp[ − i Θ( x h , x h ′ )]. The quantisation conditions for holes and gluons are • ( − L − = e − iZ ( Qg ) ( x h ) = e iRP ( x h ) L − Y h ′ =2 ,h ′ = h (cid:0) − S ( ss ) ( x h , x h ′ ) (cid:1) Q Y j =1 S ( sg ) ( x h , ˜ u j ) (3.11) • ( − Q − = e − iZ g (˜ u k | = e iRP g (˜ u k ) Q Y j =1 ,j = k (cid:0) − S ( gg ) (˜ u k , ˜ u j ) (cid:1) L − Y h ′ =2 S ( gs ) (˜ u k , x h ′ ) . (3.12)In order to gain S ( gg ) we consider (3.3) when the system is composed just of two gluons ( Q = 2and l = 1) with rapidities ˜ u and ˜ u and no internal holes are present ( L = 2): Z g ( u |
1) = − Z ∞−∞ dv π χ ( v, u | ddv ˜ P ( v ) + Z ∞−∞ dvπ dL (2 g ) dv ( v ) T ( v, u | −− X h =1 T ( x h , u |
1) + X k =1 (cid:20) ˜ χ ( u, ˜ u k | , − Z ∞−∞ dv π χ ( v, u | ddv T ( v, ˜ u k | (cid:21) . (3.13)9ia quantisation condition (3.12), from (3.13) we can identify the momentum P ( g ) ( u ) of a gluonwith rapidity u T ( x , u |
1) + T ( x L , u | − Z ∞−∞ dvπ dL ( Qg ) dv ( v ) T ( v, u | − L − X h =2 ˜ P ( x h ) + (3.14)+2 Z + ∞−∞ dv π χ ( v, u | ddv ˜ P ( v ) + Q X k =1 Z + ∞−∞ dv π χ ( v, ˜ u k | ddv ˜ P ( v ) ≃ R · P ( g ) ( u ) , with effective length R ≃ s and the scattering phase between gluons with rapidities u and ˜ u : i ln (cid:0) − S ( gg ) ( u, ˜ u ) (cid:1) = ˜ χ ( u, ˜ u | , − Z ∞−∞ dv π χ ( v, u | ddv T ( v, ˜ u | − Z ∞−∞ dv π χ ( v, u | ddv ˜ P ( v ) ++ Z ∞−∞ dv π χ ( v, ˜ u | ddv ˜ P ( v ) = ˜ χ ( u, ˜ u | , − Z + ∞−∞ dv π [ χ ( v, u |
1) + Φ( v )] ddv [ χ ( v, ˜ u |
1) + Φ( v )] ++ Z + ∞−∞ dv π Z + ∞−∞ dw π [ χ ( v, u |
1) + Φ( v )] (cid:20) ddv ddw Θ( v, w ) (cid:21) [ χ ( w, ˜ u |
1) + Φ( w )] , (3.15)where Θ (2.23) enters the hole-hole scattering phase. This expression at one loop reduces to S ( gg ) ( u, ˜ u ) = − Γ (1 + i ( u − ˜ u ))Γ (1 − i ( u − ˜ u )) Γ (cid:0) − iu (cid:1) Γ (cid:0) + iu (cid:1) Γ (cid:0) + i ˜ u (cid:1) Γ (cid:0) − i ˜ u (cid:1) , (3.16)which agrees with relations (7) and (11) of [8]. Moreover, the gluonic counting function (3.3) allowsus to retrieve the scattering matrix between a gluon ( l = 1 and rapidity ˜ u ) and a scalar excitation(internal hole with rapidity x h ), once properly rewritten after fixing Q = 1 and L = 3: Z g ( u |
1) = − Z ∞−∞ dv π χ ( v, u | ddv ˜ P ( v ) + Z ∞−∞ dvπ dL (1 g ) dv ( v ) T ( v, u |
1) + − X h =1 T ( x h , u |
1) + (cid:20) ˜ χ ( u, ˜ u | , − Z ∞−∞ dv π χ ( v, u | ddv T ( v, ˜ u | (cid:21) . (3.17)Therefore, the gluon-scalar scattering phase reads i ln (cid:0) S ( gs ) (˜ u, x h ) (cid:1) = − χ ( x h , ˜ u | − Φ( x h ) + Z + ∞−∞ dw π (cid:20) ddw Θ( x h , w ) (cid:21) ( χ ( w, ˜ u |
1) + Φ( w )) == Z + ∞−∞ dw π (cid:20) ddw Θ( x h , w ) − πδ ( x h − w ) (cid:21) ( χ ( w, ˜ u |
1) + Φ( w )) = − i ln (cid:0) S ( sg ) ( x h , ˜ u ) (cid:1) ; (3.18)at one loop, it becomes S ( gs ) (˜ u, x h ) = Γ (1 + i (˜ u − x h ))Γ (1 − i (˜ u − x h )) Γ (cid:0) + ix h (cid:1) Γ (cid:0) − ix h (cid:1) Γ (cid:0) − i ˜ u (cid:1) Γ (cid:0) + i ˜ u (cid:1) = [ S ( sg ) ( x h , ˜ u )] − . (3.19)Finally, we consider scattering between F + ⊥ , with rapidity ˜ u , and ¯ F + ⊥ , with rapidity ˜¯ u . We stickto L = 2 and consider Z g , counting function of F + ⊥ and Z ( g ¯ g ) , counting function of scalars in the10resence of F + ⊥ and ¯ F + ⊥ : Z g ( u |
1) = ˜ χ ( u, ˜ u | , − Z + ∞−∞ dv π χ ( v, u | ddv [ Z ( g ¯ g ) ( v ) − L ( g ¯ g ) ( v )] − X h =1 χ ( x h , u | , (3.20) Z ( g ¯ g ) ( v ) = 4 ˜ P ( v ) + X h =1 R ( v, x h ) + T ( v, ˜ u |
1) + T ( v, ˜¯ u |
1) +
N L ( v ) . (3.21)Plugging (3.21) into (3.20), computing Z g in u = ˜ u and neglecting non linear terms, we obtain Z g (˜ u |
1) = − Z + ∞−∞ dv π χ ( v, ˜ u | (cid:20) ddv ˜ P ( v ) + ddv T ( v, ˜¯ u | (cid:21) − T ( x , ˜ u | − T ( x L , ˜ u | . (3.22)The quantisation condition reads 1 = e − iZ g (˜ u | = e iRP ( g ) (˜ u ) S ( g ¯ g ) (˜ u , ˜¯ u ), from which, using (3.14,3.15) we obtain i ln S ( g ¯ g ) (˜ u , ˜¯ u ) = i ln (cid:0) − S ( gg ) (˜ u , ˜¯ u ) (cid:1) − χ (˜ u , ˜¯ u | , ⇒ S ( g ¯ g ) (˜ u , ˜¯ u ) = S ( gg ) (˜ u , ˜¯ u ) ˜ u − ˜¯ u − i ˜ u − ˜¯ u + i . (3.23)Analogously, considering the counting function of ¯ F + ⊥ together with Z ( g ¯ g ) , we find S (¯ gg ) (˜¯ u , ˜ u ) =[ S ( g ¯ g ) (˜ u , ˜¯ u )] − . Eventually, we also obtain S (¯ gs ) ( u, v ) = S ( gs ) ( u, v ). Perturbative strong coupling gluon-gluon scattering
We want to study the gluonic scattering matrix (3.15) in the limit g → + ∞ , with u = ¯ u √ g ,˜ u = ¯˜ u √ g , ¯ u , ¯˜ u fixed and ¯ u <
1, ¯˜ u <
1. We have i ln (cid:0) − S ( gg ) ( u, ˜ u ) (cid:1) = I + I + I , (3.24)where I = ˜ χ ( u, ˜ u | ,
1) = − √ g (¯˜ u − ¯ u ) = − π sgn(¯˜ u − ¯ u ) + √ g (¯˜ u − ¯ u ) + O (1 /g ) , (3.25) I = − (cid:20) g (¯˜ u − ¯ u ) √ (cid:21) − g (¯˜ u − ¯ u ) √
2] + 4 arctan " √ g (¯˜ u − ¯ u )3 = O (1 /g ) . (3.26)For what concerns the last term I in the right hand side of (3.15), we find convenient to performthe change of variables v = √ g ¯ v , w = √ g ¯ w I ∼ = Z + ∞−∞ d ¯ v π Z + ∞−∞ d ¯ w π √ g ¯ v − √ g ¯ u √ g ¯ w − √ g ¯˜ u dd ¯ v dd ¯ w Θ( √ g ¯ v, √ g ¯ w ) . (3.27)Plugging formula (2.33) into (3.27) and performing the integrations we arrive at I = 12 √ g (¯ u − ¯˜ u ) " − (cid:18) u − ¯ u (cid:19) (cid:18) − ¯˜ u u (cid:19) − (cid:18) − ¯ u u (cid:19) (cid:18) u − ¯˜ u (cid:19) + O (1 /g ) . (3.28)11ow, summing up (3.25, 3.26, 3.28) we obtain the final result for the gluon-gluon scattering matrixat the order O (1 /g ) S ( gg ) ( u, ˜ u ) = exp " i √ g (¯ u − ¯˜ u ) (cid:18) u − ¯ u (cid:19) (cid:18) − ¯˜ u u (cid:19) + 12 (cid:18) − ¯ u u (cid:19) (cid:18) u − ¯˜ u (cid:19) + O (1 /g ) ! , (3.29)which agrees with the result coming from (7), (15) and (16) of [8]. To parametrise the dynamics of a fermionic excitation, the rapidity to look at is actually x ,which is related to the Bethe rapidity u via the Zhukovski map u ( x ) = x + g x . To properlyinvert the Zhukovski map for the complete range of values of x , we need to glue two u -planestogether, each corresponding to a Riemann sheet. The two sheets are related to two distinctregimes of the fermionic excitations [6]: large fermions, embedded in Beisert-Staudacher equations[3] as u roots, which do carry energy and momentum even at one-loop; small fermions, corre-sponding to u roots, which couple to main root equations just at higher loops. The function x ( u ) can be analytically continued from the u Riemann sheet to the u -sheet by means of themap x ( u ) −→ ( g ) / (2 x ( u )), and Beisert-Staudacher equations are invariant under this exchange u ↔ u , provided we modify the spin-chain length (see [3] for details). Exactly the same reason-ing applies for anti-fermions by replacing u → u and u → u : turning on u ( u ) roots meansexciting fermionic fields Ψ + , while u ( u ) corresponds to ¯Ψ + . Hence, we can extend (3.1, 3.2)to include N F large fermions u F = u , of physical rapidities x Fj = x ( u Fj ) with the arithmeticsquare root for x ( u ) = ( u/ h p − (2 g ) /u i , and n f small fermions u f = u , of rapidities x fj = ( g ) / (2 x ( u fj )): • e − ip k L ′ s Y j = k S (44) ( u k , u j ) N F Y j =1 S (4 F ) ( u k , u Fj ) n f Y j =1 S (4 f ) ( u k , u fj ) Y m N m Y j =1 S (4 g ) m ( u k , u mj ) (4.1) • s Y j =1 S ( F ( u Fk , u j ) Y m N m Y j =1 S ( F g ) m ( u Fk , u mj ) (4.2) • s Y j =1 S ( f ( u fk , u j ) Y m N m Y j =1 S ( fg ) m ( u fk , u mj ) (4.3) • Y m N m Y i =1 S ( gg ) lm ( u lk , u mi ) s Y j =1 S ( g l ( u lk , u j ) N F Y j =1 S ( gF ) l ( u lk , u Fj ) n f Y j =1 S ( gf ) l ( u lk , u fj ) , (4.4)where L ′ = L + N F + Q and, in addition to previously defined matrices, we introduce the scatteringmatrices describing the collision between a type-4 root and a large fermion S (4 F ) ( u k , u Fj ) or smallfermion S (4 f ) ( u k , u fj ) together with their inverses (respectively S ( F ( u Fk , u j ), S ( f ( u fk , u j )); thematrices S ( gF ) l ( u lk , u Fj ), S ( F g ) m ( u Fk , u mj ) (or S ( gf ) l ( u lk , u fj ) and S ( fg ) m ( u fk , u mj )) describe the scattering12nvolving gluonic stacks and large (small) fermions. Explicitly, these scattering matrices over theGKP vacuum are listed here (the definition χ F ( v, u ) = χ ( v − u |
1) + χ H ( v, u ) is used): • large (anti)fermion-large (anti)fermion: S ( F F ) ( u, u ′ ) = S ( F ¯ F ) ( u, u ′ ) = S ( ¯ F F ) ( u, u ′ ) = S ( ¯ F ¯ F ) ( u, u ′ ), i log S ( F F ) ( u, u ′ ) = − Z dv π dw π [ χ F ( v, u ) + Φ( v )] ddv (cid:18) πδ ( v − w ) − d Θ dw ( v, w ) (cid:19) [ χ F ( w, u ′ ) + Φ( w )](4.5) • large (anti)fermion-small (anti)fermion: S ( F f ) ( u, u ′ ) = S ( F ¯ f ) ( u, u ′ ) = S ( ¯ F f ) ( u, u ′ ) = S ( ¯ F ¯ f ) ( u, u ′ ), i log S ( F f ) ( u, u ′ ) = Z dv π dw π [ χ F ( v, u ) + Φ( v )] ddv (cid:18) πδ ( v − w ) − d Θ dw ( v, w ) (cid:19) χ H ( w, u ′ ) (4.6) • scalar-large (anti)fermion: S ( sF ) ( u, u ′ ) = S ( s ¯ F ) ( u, u ′ ), i log S ( sF ) ( u, u ′ ) = − Z dv π (cid:20) d Θ dv ( u, v ) − πδ ( u − v ) (cid:21) ( χ F ( v, u ′ ) + Φ( v )) (4.7) • gluonic stack-large (anti)fermion: S ( gF ) l ( u, u ′ ) = S (¯ g ¯ F ) l ( u, u ′ ), S ( g ¯ F ) l ( u, u ′ ) = S (¯ gF ) l ( u, u ′ ), i log( − S ( gF ) l ( u, u ′ )) = χ ( u − u ′ | l ) + i log S (¯ gF ) l ( u, u ′ ) = (4.8)= χ ( u − u ′ | l ) − Z dv π dw π [ χ ( v, u | l ) + Φ( v )] ddv (cid:18) πδ ( v − w ) − d Θ dw ( v, w ) (cid:19) [ χ F ( w, u ′ ) + Φ( w )] . We checked that unitarity holds, i.e. that S ( sF ) ( u, u ′ ) = [ S ( F s ) ( u ′ , u )] − , S ( gF ) l ( u, u ′ ) =[ S ( F g ) l ( u ′ , u )] − and so on. By virtue of the map between small and large (anti)fermions, fromthe expressions above we recover the corresponding ones for small (anti)fermions by replacing χ F ( v, u ) + Φ( v ) −→ − χ H ( v, u ). Eventually, we note that all these scattering phases depend onlyon the ’basic’ scalar-scalar one, except known functions. Note Added : When completing to write this work, [22] on scalars appeared (one day in advance).In fact, [22] focuses on scalars giving for them the complete S-matrix (namely the g dependentscalar-scalar (pre)factor, which is also presented in this letter, times a matrix fixed by the O (6)symmetry, as in the O (6) NLSM). In this letter we derive all the (pre)factors concerning the otherfields, i.e. gluons and fermions. Acknowledgements
We enjoyed discussions with B. Basso, D. Bombardelli, N.Dorey and P.Zhao. This project was partially supported by INFN grants IS FI11 and PI14, the Italian MIUR-PRIN contract 2009KHZKRX-007, the ESF Network 09-RNP-092 (PESC) and the MPNS–COSTAction MP1210.
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