Abstract
Superstring field theory expresses the perturbative S-matrix of superstring theory as a sum of Feynman diagrams each of which is manifestly free from ultraviolet divergences. The interaction vertices fall off exponentially for large space-like external momenta making the ultraviolet finiteness property manifest, but blow up exponentially for large time-like external momenta making it impossible to take the integration contours for loop energies to lie along the real axis. This forces us to carry out the integrals over the loop energies by choosing appropriate contours in the complex plane whose ends go to infinity along the imaginary axis but which take complicated form in the interior navigating around the various poles of the propagators. We consider the general class of quantum field theories with this property and prove Cutkosky rules for the amplitudes to all orders in perturbation theory. Besides having applications to string field theory, these results also give an alternative derivation of Cutkosky rules in ordinary quantum field theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
20 September 2018
"Superstring field theory expresses the perturbative S-matrix of superstring theory as a sum of Feynman diagrams each of which is manifestly free from ultraviolet divergences. The interaction vertices fall off exponentially for large space-like external momenta making the ultraviolet finiteness property manifest, but blow up exponentially for large time-like external momenta making it impossible to take the integration contours for loop energies to lie along the real axis. This forces us to carry out the integrals over the loop energies by choosing appropriate contours in the complex plane whose ends go to infinity along the imaginary axis but which take complicated form in the interior navigating around the various poles of the propagators. We consider the general class of quantum field theories with this property and prove Cutkosky rules for the amplitudes to all orders in perturbation theory. Besides having applications to string field theory, these results also give an alternative derivation of Cutkosky rules in ordinary quantum field theories."
References
A. Sen, BV Master Action for Heterotic and Type II String Field Theories, JHEP 02 (2016) 087 [arXiv:1508.05387] [INSPIRE].
E. Witten, Interacting Field Theory of Open Superstrings, Nucl. Phys. B 276 (1986) 291 [INSPIRE].
R. Saroja and A. Sen, Picture changing operators in closed fermionic string field theory, Phys. Lett. B 286 (1992) 256 [hep-th/9202087] [INSPIRE].
N. Berkovits, SuperPoincaré invariant superstring field theory, Nucl. Phys. B 450 (1995) 90 [Erratum ibid. B 459 (1996) 439] [hep-th/9503099] [INSPIRE].
N. Berkovits, The Ramond sector of open superstring field theory, JHEP 11 (2001) 047 [hep-th/0109100] [INSPIRE].
Y. Okawa and B. Zwiebach, Heterotic string field theory, JHEP 07 (2004) 042 [hep-th/0406212] [INSPIRE].
N. Berkovits, Y. Okawa and B. Zwiebach, WZW-like action for heterotic string field theory, JHEP 11 (2004) 038 [hep-th/0409018] [INSPIRE].
T. Erler, S. Konopka and I. Sachs, Resolving Witten‘s superstring field theory, JHEP 04 (2014) 150 [arXiv:1312.2948] [INSPIRE].
H. Kunitomo, The Ramond Sector of Heterotic String Field Theory, PTEP 2014 (2014) 043B01 [arXiv:1312.7197] [INSPIRE].
T. Erler, S. Konopka and I. Sachs, NS-NS Sector of Closed Superstring Field Theory, JHEP 08 (2014) 158 [arXiv:1403.0940] [INSPIRE].
H. Matsunaga, Nonlinear gauge invariance and WZW-like action for NS-NS superstring field theory, JHEP 09 (2015) 011 [arXiv:1407.8485] [INSPIRE].
H. Kunitomo, Symmetries and Feynman rules for the Ramond sector in open superstring field theory, PTEP 2015 (2015) 033B11 [arXiv:1412.5281] [INSPIRE].
T. Erler, Y. Okawa and T. Takezaki, A ∞ structure from the Berkovits formulation of open superstring field theory, arXiv:1505.01659 [INSPIRE].
T. Erler, S. Konopka and I. Sachs, Ramond Equations of Motion in Superstring Field Theory, JHEP 11 (2015) 199 [arXiv:1506.05774] [INSPIRE].
K. Goto and H. Matsunaga, On-shell equivalence of two formulations for superstring field theory, arXiv:1506.06657 [INSPIRE].
S. Konopka, The S-matrix of superstring field theory, JHEP 11 (2015) 187 [arXiv:1507.08250] [INSPIRE].
H. Kunitomo and Y. Okawa, Complete action for open superstring field theory, PTEP 2016 (2016) 023B01 [arXiv:1508.00366] [INSPIRE].
K. Goto and H. Matsunaga, A ∞ /L ∞ structure and alternative action for WZW-like superstring field theory, arXiv:1512.03379 [INSPIRE].
T. Erler, Y. Okawa and T. Takezaki, Complete Action for Open Superstring Field Theory with Cyclic A ∞ Structure, JHEP 08 (2016) 012 [arXiv:1602.02582] [INSPIRE].
S. Konopka and I. Sachs, Open Superstring Field Theory on the Restricted Hilbert Space, JHEP 04 (2016) 164 [arXiv:1602.02583] [INSPIRE].
B. Jurčo and K. Muenster, Type II Superstring Field Theory: Geometric Approach and Operadic Description, JHEP 04 (2013) 126 [arXiv:1303.2323] [INSPIRE].
R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].
M. Fowler, Introduction to Momentum Space Integration Techniques in Perturbation Theory, J. Math. Phys. 3 (1962) 936.
M.J.G. Veltman, Unitarity and causality in a renormalizable field theory with unstable particles, Physica 29 (1963) 186 [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Diagrammar, NATO Sci. Ser. B 4 (1974) 177 [INSPIRE].
S. Bloch and D. Kreimer, Cutkosky Rules and Outer Space, arXiv:1512.01705 [INSPIRE].
A. Sen, Supersymmetry Restoration in Superstring Perturbation Theory, JHEP 12 (2015) 075 [arXiv:1508.02481] [INSPIRE].
K. Aoki, E. D’Hoker and D.H. Phong, Unitarity of Closed Superstring Perturbation Theory, Nucl. Phys. B 342 (1990) 149 [INSPIRE].
E. Witten, Superstring Perturbation Theory Revisited, arXiv:1209.5461 [INSPIRE].
R. Donagi and E. Witten, Supermoduli Space Is Not Projected, Proc. Symp. Pure Math. 90 (2015) 19 [arXiv:1304.7798] [INSPIRE].
R. Donagi and E. Witten, Super Atiyah classes and obstructions to splitting of supermoduli space, arXiv:1404.6257 [INSPIRE].
A. Sen and E. Witten, Filling the gaps with PCO’s, JHEP 09 (2015) 004 [arXiv:1504.00609] [INSPIRE].
A. Berera, Unitary string amplitudes, Nucl. Phys. B 411 (1994) 157 [INSPIRE].
E. Witten, The Feynman iϵ in String Theory, JHEP 04 (2015) 055 [arXiv:1307.5124] [INSPIRE].
G.F. Sterman, An Introduction to quantum field theory, Cambridge University Press (1993).
B. Zwiebach, Closed string field theory: Quantum action and the B-V master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650.
T.D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. 133 (1964) B1549.
F. Bloch and A. Nordsieck, Note on the Radiation Field of the electron, Phys. Rev. 52 (1937) 54 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1604.01783
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Pius, R., Sen, A. Cutkosky rules for superstring field theory. J. High Energ. Phys. 2016, 24 (2016). https://doi.org/10.1007/JHEP10(2016)024
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2016)024