Two-loop test of the N=6 Chern-Simons theory S-matrix

0.23 MB·English

by  Changrim Ahn

UMTG–262
Two-loop test of the N = 6 Chern-Simons theory
S-matrix
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n
a Changrim Ahn 1 and Rafael I. Nepomechie 2
J

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h
t
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p Abstract
e
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[ 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-
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3 responding two-loop S-matrix. The result matches with the weak-coupling limit of the
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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,
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we first review the analogous computation of the one-loop S-matrix from the one-loop
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0 dilatation operator for the scalar sector of planar N = 4 superconformal Yang-Mills
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v theory, and compare the result with the all-loop SU(2|2)2 S-matrix.
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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