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UMTG–262 Two-loop test of the N = 6 Chern-Simons theory S-matrix 9 0 0 2 n a Changrim Ahn 1 and Rafael I. Nepomechie 2 J 8 2 ] h t - p Abstract e h [ Startingfromtheintegrabletwo-loopspin-chainHamiltoniandescribingtheanoma- 2 lous dimensions of scalar operators in the planar N = 6 superconformal Chern-Simons v theory of ABJM, we perform a direct coordinate Bethe ansatz computation of the cor- 4 3 responding two-loop S-matrix. The result matches with the weak-coupling limit of the 3 3 scalar sector of the all-loop S-matrix which we have recently proposed. In particular, . 1 we confirm that the scattering of A and B particles is reflectionless. As a warm up, 0 we first review the analogous computation of the one-loop S-matrix from the one-loop 9 0 dilatation operator for the scalar sector of planar N = 4 superconformal Yang-Mills : v theory, and compare the result with the all-loop SU(2|2)2 S-matrix. i X r a 1 Department of Physics, Ewha Womans University, Seoul 120-750, South Korea 2 Physics Department, P.O. Box 248046, University of Miami, Coral Gables, FL 33124 USA 1 Introduction Exact factorized S-matrices [1] play a key role in the understanding of integrable models. Planar four-dimensional N = 4 superconformal Yang-Mills (YM) theory (and therefore, according to the AdS /CFT correspondence [2], a certain type IIB superstring theory on 5 4 AdS × S5) is believed to be integrable (see [3]-[6] and references therein). A correspond- 5 ing exact factorized S-matrix with SU(2|2)2 symmetry has been proposed (see [7]-[14] and references therein), which leads [8, 15, 16] to the all-loop Bethe ansatz equations (BAEs) [17]. Aharony, Bergman, Jafferis and Maldacena (ABJM) [18] recently proposed an analogous AdS /CFT correspondencerelatingplanarthree-dimensionalN = 6superconformalChern- 4 3 Simons (CS) theory to type IIA superstring theory on AdS ×CP3. Minahan and Zarembo 4 [19] subsequently found that the scalar sector of N = 6 CS is integrable at the leading two-loop order, and proposed two-loop BAEs for the full theory (see also [20]). Moreover, evidence for classical integrability of the dual string sigma model (large-coupling limit) was discovered in [21, 22, 23]. On the basis of these results, and assuming integrability to all orders, Gromov and Vieira then conjectured all-loop BAEs [24]. Based on the symmetries and the spectrum of elementary excitations [19, 25, 26], we proposed an exact factorized AdS /CFT S-matrix [28]. As a check, we verified that this 4 3 S-matrix leads to the all-loop BAEs in [24]. An unusual feature of this S-matrix is that the scatteringofAandB particlesisreflectionless. (AsimilarS-matrixwhichisnotreflectionless is not consistent with the known two-loop BAEs [29].) For further related developments of the AdS /CFT correspondence, see [30] and references therein. 4 3 Considerable guesswork has entered into the above-mentioned all-loop results. While thereissubstantialevidencefortheall-loopBAEsandS-matrixinthewell-studiedAdS /CFT 5 4 case, the same cannot be said for the rapidly-evolving AdS /CFT case. 4 3 In an effort to further check our proposed S-matrix, we perform here a direct coordinate Bethe ansatz computation of the two-loop S-matrix, starting from the integrable two-loop spin-chain Hamiltonian describing the anomalous dimensions of scalar operators in planar N = 6 CS [19]. The result matches with the weak-coupling limit of the scalar sector of our all-loop S-matrix [28]. In particular, we confirm that the scattering of A and B particles is reflectionless. As a warm up, we first review the analogous computation by Berenstein and Va´zquez [5] of the one-loop S-matrix from the one-loop dilatation operator for the scalar sector of planar N = 4 YM [3], and compare the result with the all-loop SU(2|2)2 S-matrix. The outline of this paper is as follows. In Sec. 2 we review the simpler case of N = 4 YM. In Sec. 3 we analyze the N = 6 CS case, relegating most of the details of A−B scattering 1 to an appendix. We briefly discuss our results in Sec. 4. 2 One-loop S-matrix in the scalar sector of N = 4 YM As is well known, N = 4 YM has six scalar fields Φ (x) (i = 1,...,6) in the adjoint i representation of SU(N). It is convenient to associate single-trace gauge-invariant scalar operators with states of an SO(6) quantum spin chain with L sites, trΦ (x)···Φ (x) ⇔ |Φ ···Φ (cid:105), (2.1) i1 iL i1 iL where Φ on the RHS are 6-dimensional elementary vectors with components (Φ ) = δ . i i j i,j The one-loop anomalous dimensions of these operators are described by the integrable SO(6) quantum spin-chain Hamiltonian [3] L (cid:18) (cid:19) λ (cid:88) 1 Γ = H, H = 1−P + K , (2.2) 8π2 l,l+1 2 l,l+1 l=1 where λ = g2 N is the ’t Hooft coupling, P is the permutation operator, YM P Φ ⊗Φ = Φ ⊗Φ , (2.3) i j j i and the projector K acts as (cid:32) (cid:33) 6 (cid:88) K Φ ⊗Φ = δ Φ ⊗Φ . (2.4) i j ij k k k=1 It is convenient to define the complex combinations X = Φ +iΦ , Y = Φ +iΦ , Z = Φ +iΦ , (2.5) 1 2 3 4 5 6 ¯ and to denote the corresponding complex conjugates with a bar, X = Φ − iΦ , etc. For 1 2 ¯ ¯ ¯ φ ,φ ∈ {X,X,Y ,Y ,Z,Z}, 1 2 P φ ⊗φ = φ ⊗φ , (2.6) 1 2 2 1 and (cid:40) ¯ 0 if φ (cid:54)= φ 1 2 K φ ⊗φ = . (2.7) 1 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ X ⊗X +X ⊗X +Y ⊗Y +Y ⊗Y +Z ⊗Z +Z ⊗Z if φ = φ 1 2 2 2.1 Coordinate Bethe ansatz We take |ZL(cid:105) as the vacuum state, which evidently is an eigenstate of H with zero energy. One-particle excited states (“magnons”) with momentum p are given by L (cid:88) |ψ(p)(cid:105) = eipx|x(cid:105) , (2.8) φ φ x=1 where 1 x L ↓ ↓ ↓ |x(cid:105) = | Z ···Z φ Z··· Z(cid:105) (2.9) φ is the state obtained from the vacuum by replacing a single Z at site x with an “impurity” ¯ ¯ ¯ φ, which can be either X,X,Y,Y (but not Z, which can be regarded as a two-particle bound state). Indeed, one can easily check that (2.8) is an eigenstate of H with eigenvalue E = (cid:15)(p), where (cid:15)(p) = 4sin2(p/2). (2.10) Inordertocomputethetwo-particleS-matrix, wemustconstructallpossibletwo-particle eigenstates. Let 1 x1 x2 L ↓ ↓ ↓ ↓ |x ,x (cid:105) = | Z ··· φ ··· φ ··· Z(cid:105) (2.11) 1 2 φ1φ2 1 2 denote the state obtained from the vacuum by replacing the Z’s at sites x and x with 1 2 impurities φ and φ , respectively, where x < x . Following Berenstein and Va´zquez [5], we 1 2 1 2 distinguish the following three cases: φ = φ : 1 2 ¯ ¯ Thecaseoftwoparticlesofthesametype(i.e., φ = φ ≡ φ ∈ {X,X,Y,Y})isequiva- 1 2 lent to the well-known case originally considered by Bethe in his seminal investigation of the Heisenberg model. (See, e.g., the review by Plefka in [6].) The two-particle eigenstates are given by (cid:88) |ψ(cid:105) = f(x ,x )|x ,x (cid:105) (2.12) 1 2 1 2 φφ x1<x2 where f(x ,x ) = ei(p1x1+p2x2) +S(p ,p )ei(p2x1+p1x2). (2.13) 1 2 2 1 3 Indeed, these states satisfy H|ψ(cid:105) = E|ψ(cid:105) (2.14) with E = (cid:15)(p )+(cid:15)(p ), (2.15) 1 2 where (cid:15)(p) is given by (2.10). It also follows from (2.14) that the S-matrix for φ−φ scattering is given by u −u +i 2 1 S(p ,p ) = , (2.16) 2 1 u −u −i 2 1 where u = u(p ) and j j 1 u(p) = cot(p/2). (2.17) 2 ¯ φ (cid:54)= φ : 1 2 ¯ If the two particles are not of the same type, but φ (cid:54)= φ , then the two-particle 1 2 eigenstates are of the form (cid:88) |ψ(cid:105) = {f (x ,x )|x ,x (cid:105) +f (x ,x )|x ,x (cid:105) } , (2.18) φ1φ2 1 2 1 2 φ1φ2 φ2φ1 1 2 1 2 φ2φ1 x1<x2 where f (x ,x ) = A (12)ei(p1x1+p2x2) +A (21)ei(p2x1+p1x2). (2.19) φiφj 1 2 φiφj φiφj One finds [5] (cid:32) (cid:33) (cid:32) (cid:33)(cid:32) (cid:33) A (21) R(p ,p ) T(p ,p ) A (12) φ1φ2 = 2 1 2 1 φ1φ2 , (2.20) A (21) T(p ,p ) R(p ,p ) A (12) φ2φ1 2 1 2 1 φ2φ1 where the transmission and reflection amplitudes are given by u −u i 2 1 T(p ,p ) = , R(p ,p ) = , (2.21) 2 1 2 1 u −u −i u −u −i 2 1 2 1 respectively. ¯ φ = φ : 1 2 ¯ ¯ ¯ In the case φ = φ ∈ {X,X,Y,Y}, the two-particle eigenstates are given by 1 2 (cid:88) (cid:88) (cid:8) (cid:9) |ψ(cid:105) = f (x ,x )|x ,x (cid:105) +f (x ,x )|x ,x (cid:105) φφ¯ 1 2 1 2 φφ¯ φ¯φ 1 2 1 2 φ¯φ x1<x2φ=X,Y (cid:88) + f (x )|x (cid:105) , (2.22) Z¯ 1 1 Z¯ x1 4 where f (x ,x ) are again given by (2.19), and φiφj 1 2 fZ¯(x1) = AZ¯ei(p1+p2)x1. (2.23) One finds [5]      A (21) R(p ,p ) T(p ,p ) S(p ,p ) S(p ,p ) A (12) XX¯ 2 1 2 1 2 1 2 1 XX¯  A (21)   T(p ,p ) R(p ,p ) S(p ,p ) S(p ,p )  A (12)   X¯X  =  2 1 2 1 2 1 2 1  X¯X  ,(2.24)      A (21) S(p ,p ) S(p ,p ) R(p ,p ) T(p ,p ) A (12)  YY¯   2 1 2 1 2 1 2 1  YY¯  A (21) S(p ,p ) S(p ,p ) T(p ,p ) R(p ,p ) A (12) Y¯Y 2 1 2 1 2 1 2 1 Y¯Y where (u −u )2 2 1 T(p ,p ) = , 2 1 (u −u −i)(u −u +i) 2 1 2 1 −1 R(p ,p ) = , 2 1 (u −u −i)(u −u +i) 2 1 2 1 −i(u −u ) 2 1 S(p ,p ) = . (2.25) 2 1 (u −u −i)(u −u +i) 2 1 2 1 2.2 Comparison with the all-loop S-matrix We now wish to compare the above scattering amplitudes with the weak-coupling limit of the all-loop SU(2|2)⊗SU(2|2) S-matrix [8]-[14]. This check has not (to our knowledge) been presented elsewhere, and will serve as a useful guide for the N = 6 CS case. It is convenient to express the latter in terms of two mutually commuting sets of Zamolodchikov-Faddeev operators A†(p),A˜†(p) (i = 1,...,4), i i A†(p )A†(p ) = (cid:88)S (p ,p )S(cid:98)i(cid:48)j(cid:48)(p ,p )A† (p )A†(p ), i 1 j 2 0 1 2 ij 1 2 j(cid:48) 2 i(cid:48) 1 i(cid:48),j(cid:48) A˜†(p )A˜†(p ) = (cid:88)S (p ,p )S(cid:98)i(cid:48)j(cid:48)(p ,p )A˜† (p )A˜†(p ), i 1 j 2 0 1 2 ij 1 2 j(cid:48) 2 i(cid:48) 1 i(cid:48),j(cid:48) A†(p )A˜†(p ) = A˜†(p )A†(p ). (2.26) i 1 j 2 j 2 i 1 We identify the scalar one-particle states as follows, X(p) = A†(p)A˜†(p), X¯(p) = A†(p)A˜†(p), 1 2 2 1 Y(p) = A†(p)A˜†(p), Y¯(p) = A†(p)A˜†(p). (2.27) 2 2 1 1 The only non-vanishing amplitudes in the scalar sector are 1 1 S(cid:98)aa(p ,p ) = A, S(cid:98)ab(p ,p ) = (A−B), S(cid:98)ba(p ,p ) = (A+B), (2.28) aa 1 2 ab 1 2 2 ab 1 2 2 5 where a,b ∈ {1,2} with a (cid:54)= b. Here x− −x+ A = 2 1 , x+ −x− 2 1 (cid:20)x− −x+ (x− −x+)(x− −x+)(x− +x+)(cid:21) B = − 2 1 +2 1 1 2 2 2 1 , (2.29) x+ −x− (x− −x+)(x−x− −x+x+) 2 1 1 2 1 2 1 2 where x± = x(p )± with i i x+ 1 1 i = eip, x+ + −x− − = , (2.30) x− x+ x− g √ and g = λ/(4π). Moreover, the scalar factor is given by S (p ,p )2 = x−1 −x+2 1− x+11x−2 σ(p ,p )2, (2.31) 0 1 2 x+ −x−1− 1 1 2 1 2 x−x+ 1 2 where σ(p ,p ) is the BES dressing factor [12, 14]. In the weak-coupling (g → 0) limit, 1 2 (cid:18) (cid:19) 1 i x± → u± . (2.32) g 2 Therefore u −u +i 1 2 A → , B → −1, (2.33) u −u −i 1 2 and u −u −i S2 → 1 2 , (2.34) 0 u −u +i 1 2 since σ(p ,p ) → 1. 1 2 For two particles of the same type, the scattering amplitude is evidently given by (cid:16) (cid:17)2 u −u +i S(p ,p ) ≡ S (p ,p )S(cid:98)aa(p ,p ) = S2A2 → 1 2 , (2.35) 1 2 0 1 2 aa 1 2 0 u −u −i 1 2 in agreement with (2.16). ¯ We now consider the case φ (cid:54)= φ , e.g., 1 2 X(p )Y(p ) = T(p ,p )Y(p )X(p )+R(p ,p )X(p )Y(p ). (2.36) 1 2 1 2 2 1 1 2 2 1 It follows from (2.26)-(2.28) and (2.33), (2.34) that 1 u −u T(p ,p ) = S2A(A−B) → 1 2 , 1 2 2 0 u −u −i 1 2 1 i R(p ,p ) = S2A(A+B) → , (2.37) 1 2 2 0 u −u −i 1 2 6 in agreement with (2.21). ¯ Finally, we consider the case φ = φ , e.g., 1 2 ¯ ¯ ¯ X(p )X(p ) = T(p ,p )X(p )X(p )+R(p ,p )X(p )X(p ) 1 2 1 2 2 1 1 2 2 1 ¯ ¯ + S(p ,p )Y(p )Y(p )+S(p ,p )Y(p )Y(p ). (2.38) 1 2 2 1 1 2 2 1 It follows from (2.26)-(2.28) and (2.33), (2.34) that 1 (u −u )2 T(p ,p ) = S2(A−B)2 → 1 2 , 1 2 4 0 (u −u −i)(u −u +i) 1 2 1 2 1 −1 R(p ,p ) = S2(A+B)2 → , 1 2 4 0 (u −u −i)(u −u +i) 1 2 1 2 1 i(u −u ) S(p ,p ) = S2(A−B)(A+B) → 1 2 , (2.39) 1 2 4 0 (u −u −i)(u −u +i) 1 2 1 2 in agreement with (2.25).1 In short, the all-loop AdS /CFT S-matrix correctly reproduces the N = 4 YM one-loop 5 4 scalar-sector scattering amplitudes, as expected. In the next section, we perform a similar check of the AdS /CFT S-matrix. 4 3 3 Two-loop S-matrix in the scalar sector of N = 6 CS The N = 6 CS theory [18] has a pair of scalar fields A (x) (i = 1,2) in the bifundamental i representation (N,N¯) of the SU(N)×SU(N) gauge group, and another pair of scalar fields B (x) (i = 1,2) in the conjugate representation (N¯,N). These fields can be grouped into i SU(4) multiplets YA(x), YA = (A ,A ,B†,B†), Y† = (A†,A†,B ,B ). (3.1) 1 2 1 2 A 1 2 1 2 Following [19], we associate single-trace gauge-invariant scalar operators with states of an alternating SU(4) quantum spin chain with 2L sites, trYA1(x)Y† (x)···YAL(x)Y† (x) ⇔ |YA1Y† ···YALY† (cid:105), (3.2) B1 BL B1 BL where YA on the RHS are 4-dimensional elementary vectors with components (YA) = j δ . The two-loop anomalous dimensions of these operators are described by the integrable A,j 1There is a sign discrepancy in S(p ,p ) which perhaps can be reconciled by a gauge transformation in 1 2 (2.38), e.g., Y →−Y while leaving others unchanged. 7 alternating SU(4) quantum spin-chain Hamiltonian [19] 2L (cid:18) (cid:19) (cid:88) 1 Γ = λ2H, H = 1−P + {K ,P } , (3.3) l,l+2 l,l+1 l,l+2 2 l=1 where λ = N/k is the ’t Hooft coupling2, P is the permutation operator, and the projector K acts as 4 4 (cid:88) (cid:88) K YA ⊗Y† = δA YC ⊗Y†, K Y† ⊗YA = δA Y† ⊗YC . (3.4) B B C B B C C=1 C=1 That is, 2 (cid:88)(cid:16) (cid:17) KA ⊗A† = KB† ⊗B = δ A ⊗A† +B† ⊗B , i j i j ij k k k k k=1 2 (cid:88)(cid:16) (cid:17) KA† ⊗A = KB ⊗B† = δ A† ⊗A +B ⊗B† , i j i j ij k k k k k=1 KA ⊗B = KB ⊗A = KA† ⊗B† = KB† ⊗A† = 0. (3.5) i j i j i j i j 3.1 Coordinate Bethe ansatz Following [25, 26], we take the state with L pairs of (A B ), i.e., 1 1 |(A B )L(cid:105) (3.6) 1 1 as the vacuum state, which evidently is an eigenstate of H with zero energy. It is convenient to label the (A B ) pairs by x ∈ {1,...,L}. There are two types of one-particle excited 1 1 states with momentum p, called “A-particles” and “B-particles.” The former are given by L (cid:88) |ψ(p)(cid:105)A = eipx|x(cid:105)A, (3.7) φ φ x=1 where 1 x L ↓ ↓ ↓ |x(cid:105)A = | (A B ) ··· (φB ) ··· (A B )(cid:105) (3.8) φ 1 1 1 1 1 is the state obtained from the vacuum by replacing the A from pair x with an “impurity” 1 φ, which can be either A or B† (but not B†, which can be regarded as a two-particle bound 2 2 1 state). Similarly, the “B-particles” are given by L (cid:88) |ψ(p)(cid:105)B = eipx|x(cid:105)B, (3.9) φ φ x=1 2The action has two SU(N) Chern-Simons terms with integer levels k and −k, respectively. 8 where 1 x L ↓ ↓ ↓ |x(cid:105)B = | (A B ) ··· (A φ) ··· (A B )(cid:105) (3.10) φ 1 1 1 1 1 is the state obtained from the vacuum by replacing the B from pair x with an “impurity” 1 φ, which can be either A† or B (but not A†, which can be regarded as a two-particle bound 2 2 1 state). Indeed, both (3.7) and (3.9) are eigenstates of H with eigenvalue E = (cid:15)(p), where (cid:15)(p) is given by (2.10). Inordertocomputethetwo-particleS-matrix, wemustconstructallpossibletwo-particle eigenstates. 3.1.1 A−A scattering Let 1 x1 x2 L ↓ ↓ ↓ ↓ |x ,x (cid:105)AA = | (A B ) ··· (φ B ) ··· (φ B ) ··· (A B )(cid:105) (3.11) 1 2 φ1φ2 1 1 1 1 2 1 1 1 denote the state obtained from the vacuum by replacing the A ’s from pairs x and x with 1 1 2 impurities φ and φ , respectively, where x < x and φ ∈ {A ,B†}. We distinguish two 1 2 1 2 i 2 2 cases: φ = φ : 1 2 The case of two A-particles of the same type (i.e., φ = φ ≡ φ ∈ {A ,B†}) is again 1 2 2 2 the same as in the Heisenberg model. The two-particle eigenstates are given by (cid:88) |ψ(cid:105) = f(x ,x )|x ,x (cid:105)AA (3.12) 1 2 1 2 φφ x1<x2 where f(x ,x ) is given by (2.13). These states have energy (2.15), and the S-matrix 1 2 is again given by (2.16), u −u +i 2 1 S(p ,p ) = . (3.13) 2 1 u −u −i 2 1 φ (cid:54)= φ : 1 2 If the two A-particles are not of the same type (e.g., φ = A ,φ = B†), then the 1 2 2 2 two-particle eigenstates are of the form (cid:88) (cid:8) (cid:9) |ψ(cid:105) = f (x ,x )|x ,x (cid:105)AA +f (x ,x )|x ,x (cid:105)AA , (3.14) φ1φ2 1 2 1 2 φ1φ2 φ2φ1 1 2 1 2 φ2φ1 x1<x2 where f (x ,x ) is again given by (2.19). Since K on these states is zero, the S- φiφj 1 2 matrix is again given by (2.21), u −u i 2 1 T(p ,p ) = , R(p ,p ) = . (3.15) 2 1 2 1 u −u −i u −u −i 2 1 2 1 9