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Form factors in theories with gravity duals
Nuclear Physics B (Proc. Suppl.) 216 (2011) 276–277 www.elsevier.com/locate/npbps
Form factors in theories with gravity duals Alexander V. Zhiboedova a Department of Physics, Princeton University Princeton, New Jersey 08544, USA
The prescription for calculating form factors at strong coupling in the theories with gravity duals was found in [1]. The use of integrability for similar objects culminated in the paper [2] where the Y-system for scattering amplitudes was found. In the paper [3] authors extended this scenario to the case of form factors in AdS3 kinematics. Some exact solutions were present. Here we briefly review the results of [3] and later developments of the problem.
1. Form factors at strong coupling At strong coupling form factors are given by Euclidean classical solutions of string equations of motion in AdS5 . At the boundary the solution reaches a periodic sequence of light-like lines kiμ . The period of the sequence is fixed by the momentum of the operator while light-like segments correspond to the momenta of particles √ λ 1 kiin |O(q)|kjout = e− 2π (Area)T F1 (1 + √ F2 ) λ
where the leading term (Area)T is the area of one period and it was computed in [3]. F1 is the one loop correction and would contain information about the polarizations and the particular operator we are considering. F2 is the two loop √ correction in the 1/ λ expansion.
θ → ±∞ which are obtained through WKB analysis of the flat section problem. Then one can restate these equations in integral form which happened to be of TBA form. The non-trivial part of the area then was given by the free energy of the system which depends on the set of mass parameters in which the kinematics is encoded. However it is much more convenient to have the expressions that depend only on kinematic information namely cross-ratios. It happened that after eliminating the dependence of area on mass parameters the area is given by the critical value of YangYang functional. The fact that we are dealing with periodic solutions – with the period being encoded in the monodromy Ω – can be used to truncate Y-system. For the case of AdS3 and 2n-gluon form factor it takes the form
2. Y-system At strong coupling the algorithm of solving the problem was through the use of the integrability of classical strings on AdS space background. The key steps were: first, introducing the family of flat connections with an arbitrary spectral parameter A(ζ = eθ ). With this flat connection one can consider the flat section problem which allows one to introduce a set of of spectral parameter dependent cross-ratios Yk (θ). Then one finds the set of functional equations which constraints the θ dependence of Y -functions. The system is specified with boundary conditions at 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.05.013
Ys+ Ys− + − Yn−2 Yn−2 Y¯ + Y¯ −
= =
(1 + Ys+1 )(1 + Ys−1 ), (1 + Yn−3 )(1 + Tr[Ω]Y¯ + Y¯ 2 ),
=
1 + Yn−2 .
This Y-system can also be used to study scattering amplitudes in periodic kinematics regime. 3. Exact solutions In several simple cases one can solve the Ysystem written above exactly. First case corresponds to the high temperature limit when all Y ’s
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A.V. Zhiboedov / Nuclear Physics B (Proc. Suppl.) 216 (2011) 276–277
do not depend on the spectral parameter. It corresponds to the regular form factor or zig-zag solution. The Y -functions and the non-trivial part of the area in this case are Ys Y¯
=
s(s + 2)
n−1 π Af ree (n − 1). 6 Another solution corresponds to 4-gluon form factor. In this case the only non-trivial Y -function ¯ takes the form Y¯ = eZ/ζ+Zζ and the non-trivial part of the area is given by the integral φ ∈ (0, π2 ) ∞ |m| sinh t I= dt log(1 + e−2|m| cosh t ). 2π tanh(2t + 2iφ) −∞ = =
Here |m| and φ are related to the cross-ratio in the problem in the known way. 4. Truncation in terms of momentum twistors. AdS5 case. Since the truncation of the Y-system in the case of form factors is purely kinematical it is convenient to think about it in terms of target space. The boundary of AdS can be thought as submanifold Y 2 = 0 in the projective space. Then the i’th cusp location for the case of AdS3 takes the form ˜ i (ζ). Yaia˙ (ζ) = λia (ζ)λ a˙ The periodicity of the contour manifests itself through the fact that λi+n ∝ Ωλi . The truncation condition used in [3] is just the identity Tr[Ω]λ0 , λ1 = λ0 , Ωλ1 − Ωλ0 , λ1 notice that it is manifestly conformal and gauge invariant. Written in this form it can be easily generalized to the case of AdS5 . In this case the position of the cusps are given in terms of momentum twistors i i i−1 Yab (ζ) = λia (ζ)λi−1 b (ζ) − λb (ζ)λa (ζ).
Tr[Ωv ]λ−2 , λ−1 , λ0 , λ1 =
4 k=1
λ−2 , λ−1 , λ0 , λ1 Ω k
λ−2 , λ−1 , λ0 , λ1 Ω k,j
k,j=1;k=j
1 Tr[Ωv ] = (Tr[Ωs ]2 − Tr[Ω2s ]) 2
where, for example, λ−2 , λ−1 , λ0 , λ1 Ω = 1,2 Ωs λ−2 , Ωs λ−1 , λ0 , λ1 etc. The third truncation condition is the same as (1) with the change ¯ Ωs → Ω ¯ s . See [2] for the notations. λ → λ, One can use these identities to truncate Hirota equations [2] for AdS5 . It would be nice to rewrite them in terms of Y-system as it was done for AdS3 in [3]. 5. Weak coupling analysis From the weak coupling side form factors were studied in [4–6]. In [5] the simplicity and similarity with scattering amplitudes of the tree-level form factors of half-BPS operators was revealed. They happen to share many attractive features with the amplitudes, namely BCFW recursion relations and apparently duality with Wilson lines. Acknowledgements I am very grateful to Juan Maldacena for many illuminating and inspiring discussions. REFERENCES 1. L. F. Alday, J. Maldacena, JHEP 0711, 068 (2007). [arXiv:0710.1060 [hep-th]]. 2. L. F. Alday, J. Maldacena, A. Sever, P. Vieira, J. Phys. A A43, 485401 (2010). [arXiv:1002.2459 [hep-th]]. 3. J. Maldacena, A. Zhiboedov, JHEP 1011, 104 (2010). [arXiv:1009.1139 [hep-th]]. 4. W. L. van Neerven, Z. Phys. C 30, 595 (1986). 5. A. Brandhuber, B. Spence, G. Travaglini et al., JHEP 1101, 134 (2011). [arXiv:1011.1899 [hep-th]]. 6. L. V. Bork, D. I. Kazakov, G. S. Vartanov, [arXiv:1011.2440 [hep-th]].
The periodicity is the fact that λi+n ∝ Ωs λi . The truncation conditions take the form of manifestly conformal and gauge invariant identities Tr[Ωs ]λ−2 , λ−1 , λ0 , λ1 =
4
(1)
Form factors in theories with gravity duals Alexander V. Zhiboedova a Department of Physics, Princeton University Princeton, New Jersey 08544, USA
The prescription for calculating form factors at strong coupling in the theories with gravity duals was found in [1]. The use of integrability for similar objects culminated in the paper [2] where the Y-system for scattering amplitudes was found. In the paper [3] authors extended this scenario to the case of form factors in AdS3 kinematics. Some exact solutions were present. Here we briefly review the results of [3] and later developments of the problem.
1. Form factors at strong coupling At strong coupling form factors are given by Euclidean classical solutions of string equations of motion in AdS5 . At the boundary the solution reaches a periodic sequence of light-like lines kiμ . The period of the sequence is fixed by the momentum of the operator while light-like segments correspond to the momenta of particles √ λ 1 kiin |O(q)|kjout = e− 2π (Area)T F1 (1 + √ F2 ) λ
where the leading term (Area)T is the area of one period and it was computed in [3]. F1 is the one loop correction and would contain information about the polarizations and the particular operator we are considering. F2 is the two loop √ correction in the 1/ λ expansion.
θ → ±∞ which are obtained through WKB analysis of the flat section problem. Then one can restate these equations in integral form which happened to be of TBA form. The non-trivial part of the area then was given by the free energy of the system which depends on the set of mass parameters in which the kinematics is encoded. However it is much more convenient to have the expressions that depend only on kinematic information namely cross-ratios. It happened that after eliminating the dependence of area on mass parameters the area is given by the critical value of YangYang functional. The fact that we are dealing with periodic solutions – with the period being encoded in the monodromy Ω – can be used to truncate Y-system. For the case of AdS3 and 2n-gluon form factor it takes the form
2. Y-system At strong coupling the algorithm of solving the problem was through the use of the integrability of classical strings on AdS space background. The key steps were: first, introducing the family of flat connections with an arbitrary spectral parameter A(ζ = eθ ). With this flat connection one can consider the flat section problem which allows one to introduce a set of of spectral parameter dependent cross-ratios Yk (θ). Then one finds the set of functional equations which constraints the θ dependence of Y -functions. The system is specified with boundary conditions at 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.05.013
Ys+ Ys− + − Yn−2 Yn−2 Y¯ + Y¯ −
= =
(1 + Ys+1 )(1 + Ys−1 ), (1 + Yn−3 )(1 + Tr[Ω]Y¯ + Y¯ 2 ),
=
1 + Yn−2 .
This Y-system can also be used to study scattering amplitudes in periodic kinematics regime. 3. Exact solutions In several simple cases one can solve the Ysystem written above exactly. First case corresponds to the high temperature limit when all Y ’s
277
A.V. Zhiboedov / Nuclear Physics B (Proc. Suppl.) 216 (2011) 276–277
do not depend on the spectral parameter. It corresponds to the regular form factor or zig-zag solution. The Y -functions and the non-trivial part of the area in this case are Ys Y¯
=
s(s + 2)
n−1 π Af ree (n − 1). 6 Another solution corresponds to 4-gluon form factor. In this case the only non-trivial Y -function ¯ takes the form Y¯ = eZ/ζ+Zζ and the non-trivial part of the area is given by the integral φ ∈ (0, π2 ) ∞ |m| sinh t I= dt log(1 + e−2|m| cosh t ). 2π tanh(2t + 2iφ) −∞ = =
Here |m| and φ are related to the cross-ratio in the problem in the known way. 4. Truncation in terms of momentum twistors. AdS5 case. Since the truncation of the Y-system in the case of form factors is purely kinematical it is convenient to think about it in terms of target space. The boundary of AdS can be thought as submanifold Y 2 = 0 in the projective space. Then the i’th cusp location for the case of AdS3 takes the form ˜ i (ζ). Yaia˙ (ζ) = λia (ζ)λ a˙ The periodicity of the contour manifests itself through the fact that λi+n ∝ Ωλi . The truncation condition used in [3] is just the identity Tr[Ω]λ0 , λ1 = λ0 , Ωλ1 − Ωλ0 , λ1 notice that it is manifestly conformal and gauge invariant. Written in this form it can be easily generalized to the case of AdS5 . In this case the position of the cusps are given in terms of momentum twistors i i i−1 Yab (ζ) = λia (ζ)λi−1 b (ζ) − λb (ζ)λa (ζ).
Tr[Ωv ]λ−2 , λ−1 , λ0 , λ1 =
4 k=1
λ−2 , λ−1 , λ0 , λ1 Ω k
λ−2 , λ−1 , λ0 , λ1 Ω k,j
k,j=1;k=j
1 Tr[Ωv ] = (Tr[Ωs ]2 − Tr[Ω2s ]) 2
where, for example, λ−2 , λ−1 , λ0 , λ1 Ω = 1,2 Ωs λ−2 , Ωs λ−1 , λ0 , λ1 etc. The third truncation condition is the same as (1) with the change ¯ Ωs → Ω ¯ s . See [2] for the notations. λ → λ, One can use these identities to truncate Hirota equations [2] for AdS5 . It would be nice to rewrite them in terms of Y-system as it was done for AdS3 in [3]. 5. Weak coupling analysis From the weak coupling side form factors were studied in [4–6]. In [5] the simplicity and similarity with scattering amplitudes of the tree-level form factors of half-BPS operators was revealed. They happen to share many attractive features with the amplitudes, namely BCFW recursion relations and apparently duality with Wilson lines. Acknowledgements I am very grateful to Juan Maldacena for many illuminating and inspiring discussions. REFERENCES 1. L. F. Alday, J. Maldacena, JHEP 0711, 068 (2007). [arXiv:0710.1060 [hep-th]]. 2. L. F. Alday, J. Maldacena, A. Sever, P. Vieira, J. Phys. A A43, 485401 (2010). [arXiv:1002.2459 [hep-th]]. 3. J. Maldacena, A. Zhiboedov, JHEP 1011, 104 (2010). [arXiv:1009.1139 [hep-th]]. 4. W. L. van Neerven, Z. Phys. C 30, 595 (1986). 5. A. Brandhuber, B. Spence, G. Travaglini et al., JHEP 1101, 134 (2011). [arXiv:1011.1899 [hep-th]]. 6. L. V. Bork, D. I. Kazakov, G. S. Vartanov, [arXiv:1011.2440 [hep-th]].
The periodicity is the fact that λi+n ∝ Ωs λi . The truncation conditions take the form of manifestly conformal and gauge invariant identities Tr[Ωs ]λ−2 , λ−1 , λ0 , λ1 =
4
(1)