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Scattering and Unitarity Methods in Two Dimensions

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1st Karl Schwarzschild Meeting on Gravitational Physics

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 170))

Abstract

The standard unitarity-cut method is applied to several massive two-dimensional models, including the world-sheet AdS\(_5 \times S^5\) superstring, to compute \(2\rightarrow 2\) scattering S-matrices at one loop from tree level amplitudes. Evidence is found for the cut-constructibility of supersymmetric models, while for models without supersymmetry (but integrable) the missing rational terms can be interpreted as a shift in the coupling.

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Notes

  1. 1.

    For the three-dimensional case see [3–5].

  2. 2.

    This is nothing but the application of the optical theorem. The case where the loop amplitude is subdivided into more than two pieces is referred to as generalized unitarity.

  3. 3.

    It would be interesting to analyze models which are just supersymmetric and not integrable.

  4. 4.

    Without loss of generality, one can consider in (20.1) the amplitudes associated to the first product of \(\delta \)-functions \(\delta (\text {p}_1 - \text {p}_3)\delta (\text {p}_2 - \text {p}_4)\). The denominator in (20.3) is required to make contact with the standard definition of the S-matrix in two dimensions.

  5. 5.

    At two loops, to constrain completely the four components of the two momenta circulating in the loops one needs four cuts, each one giving an on-shell \(\delta \)-function. Two-particle cuts at two loops would result in a manifold of conditions for the loop momenta.

  6. 6.

    This is like using \(f(x)\delta (x-x_0)=f(x_0)\delta (x-x_0)\) where f(x) are the tree-level amplitudes in the integrals.

  7. 7.

    This corresponds to the choice \(p_3=p_1\), \(p_4=p_2\).

  8. 8.

    In (20.8), \(\tilde{S}^{(0)}(p_1,p_2)=4 (\varepsilon _2\,\mathrm{p}_1-\varepsilon _1\,\mathrm{p}_2) S^{(0)}(p_1,p_2)\) and the denominator on the right-hand side comes from the Jacobian \(J(p_1,p_2)\) assuming a standard relativistic dispersion relation (for the theories we consider, at one-loop this is indeed the case).

  9. 9.

    Because its bubble integral \(I_0\) can only contribute to rational terms, the t-channel contribution has been neglected in  [7], where all rational terms were determined from symmetry considerations.

  10. 10.

    In some sense this is natural as, in general dimensions, quantum field theory amplitudes have the form (20.1), while the \(\delta \)-function identity (20.2) is specific to two dimensions.

  11. 11.

    See [6] for the generalization to the case which includes fermions.

  12. 12.

    In the sine-Gordon case the agreement is exact. For \(n\ge 2\) the shift in the coupling is by the dual Coxeter number of the group \(G=\text {SO}(n)\), a structure appears regularly in the quantization of WZW and gauged WZW models, where k is the quantized level (see for example [13]).

  13. 13.

    The reduced AdS\(_2 \times S^2\) theory is in fact given by the \(\mathscr {N} = 2\) supersymmetric sine-Gordon model and hence is supersymmetric. The reduced AdS\(_3 \times S^3\) and AdS\(_5 \times S^5\) theories have a non-local \(\mathscr {N}=4\) and \(\mathscr {N}=8\) supersymmetry respectively, which manifests as a q-deformation of the S-matrix symmetry algebra. Furthermore, we have also checked that the unitarity-cutting procedure matches the perturbative result at one-loop in the \(\mathscr {N}=1\) supersymmetric sine-Gordon model [18].

  14. 14.

    Notice that this is a non-relativistic model, as seen quantizing it perturbatively and noticing that the choice of a flat Minkowski worldsheet metric is incompatible with Virasoro constraints (see for example [24]).

  15. 15.

    Notice that the non-relativistic dispersion relation \(\varepsilon (\mathrm{p})=\sqrt{1+\frac{\lambda }{\pi ^2}\sin ^2\frac{\mathrm{p}}{2}}\) [25, 37], when expanded in the near-BMN limit \(\mathrm{p}\rightarrow \zeta \mathrm{p}\), corresponding to the perturbative regime, leads to a relativistic energy \(\varepsilon _i=\sqrt{1+\mathrm{p}_i^2}\).

  16. 16.

    In the comparison with the world-sheet calculation all dimensional quantities (such as the spin-chain length and the momenta) should be rescaled via a factor of \(\sqrt{\lambda }/(2\pi )\) [31], for us \(\mathrm{p}\rightarrow \zeta \,\mathrm{p}\).

  17. 17.

    This is done in the so-called constant-J gauge \(a = 0\).

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Authors and Affiliations

  1. Institut für Physik und Integrative Research Institute for the Sciences IRIS Adlershof, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489, Berlin, Germany

    Valentina Forini, Lorenzo Bianchi & Ben Hoare

Authors
  1. Valentina Forini
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  2. Lorenzo Bianchi
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  3. Ben Hoare
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Correspondence to Valentina Forini .

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Editors and Affiliations

  1. Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany

    Piero Nicolini

  2. Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama, USA

    Matthias Kaminski

  3. Department of Physics, Loyola Marymount University, Los Angeles, California, USA

    Jonas Mureika

  4. Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany

    Marcus Bleicher

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Forini, V., Bianchi, L., Hoare, B. (2016). Scattering and Unitarity Methods in Two Dimensions. In: Nicolini, P., Kaminski, M., Mureika, J., Bleicher, M. (eds) 1st Karl Schwarzschild Meeting on Gravitational Physics. Springer Proceedings in Physics, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-319-20046-0_20

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