Abstract
A first quantised approach to loop amplitudes based on the pure spinor particle is applied to the systematics of four-particle amplitudes in maximally supersymmetric field theories. Counting of fermionic zero modes allows the identification of momentum factors multiplying \( {\mathcal{R}^4} \) in the case of supergravity (and F 4 in the Yang-Mills case) thereby making manifest their ultraviolet properties as a function of dimension, D. For L = 2, 3, 4 loops the leading supergravity divergence is in D = 4 + 6/L dimensions and proportional to \( {\partial^{2L}}\ {\mathcal{R}^4} \), in line with earlier field theory calculations. However, at five loops there is a radical change in the systematics, suggesting the presence of a contribution with an explicit L = 5 logarithmic ultraviolet divergence when D = 24/5 that is proportional to \( {\partial^8}{\mathcal{R}^4} \). We further argue that \( {\partial^8}{\mathcal{R}^4} \) should receive contributions from all loops, which would imply that \( \mathcal{N} = 8 \) supergravity (with D = 4) is not protected by supersymmetry from a seven-loop divergence.
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Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [SPIRES].
Z. Bern et al., Three-loop superfiniteness of N = 8 supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [SPIRES].
Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The ultraviolet behavior of N = 8 supergravity at four loops, Phys. Rev. Lett. 103 (2009) 081301 [arXiv:0905.2326] [SPIRES].
M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 supergravity as limits of string theories, Nucl. Phys. B 198 (1982) 474 [SPIRES].
M.B. Green, J.G. Russo and P. Vanhove, Non-renormalisation conditions in type-II string theory and maximal supergravity, JHEP 02 (2007) 099 [hep-th/0610299] [SPIRES].
M.B. Green and M. Gutperle, Effects of D-instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [SPIRES].
M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999) 046006 [hep-th/9808061] [SPIRES].
N. Berkovits, Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, JHEP 09 (2004) 047 [hep-th/0406055] [SPIRES].
M.B. Green and P. Vanhove, Duality and higher derivative terms in M-theory, JHEP 01 (2006) 093 [hep-th/0510027] [SPIRES].
N. Berkovits, New higher-derivative R 4 theorems, Phys. Rev. Lett. 98 (2007) 211601 [hep-th/0609006] [SPIRES].
M.B. Green, J.G. Russo and P. Vanhove, String theory dualities and supergravity divergences, JHEP 06 (2010) 075 [arXiv:1002.3805] [SPIRES].
N. Berkovits, Super-Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [SPIRES].
N. Berkovits, Covariant quantization of the superstring, Nucl. Phys. Proc. Suppl. 127 (2004) 23 [Int. J. Mod. Phys. A 16 (2001) 801] [hep-th/0008145] [SPIRES].
N. Berkovits, M.B. Green, J.G. Russo and P. Vanhove, Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory, JHEP 11 (2009) 063 [arXiv:0908.1923] [SPIRES].
M.B. Green, J.G. Russo and P. Vanhove, Ultraviolet properties of maximal supergravity, Phys. Rev. Lett. 98 (2007) 131602 [hep-th/0611273] [SPIRES].
J. Björnsson, Loop amplitudes using pure spinors in theories with maximal supersymmetry, in preparation.
N. Berkovits, Covariant quantization of the superparticle using pure spinors, JHEP 09 (2001) 016 [hep-th/0105050] [SPIRES].
L. Anguelova, P.A. Grassi and P. Vanhove, Covariant one-loop amplitudes in D = 11, Nucl. Phys. B 702 (2004) 269 [hep-th/0408171] [SPIRES].
N. Berkovits, Pure spinor formalism as an N = 2 topological string, JHEP 10 (2005) 089 [hep-th/0509120] [SPIRES].
N. Berkovits and P.S. Howe, Ten-dimensional supergravity constraints from the pure spinor formalism for the superstring, Nucl. Phys. B 635 (2002) 75 [hep-th/0112160] [SPIRES].
P.A. Grassi and L. Tamassia, Vertex operators for closed superstrings, JHEP 07 (2004) 071 [hep-th/0405072] [SPIRES].
P. Dai and W. Siegel, Worldline green functions for arbitrary Feynman diagrams, Nucl. Phys. B 770 (2007) 107 [hep-th/0608062] [SPIRES].
N. Berkovits and N. Nekrasov, Multiloop superstring amplitudes from non-minimal pure spinor formalism, JHEP 12 (2006) 029 [hep-th/0609012] [SPIRES].
Y. Aisaka and N. Berkovits, Pure spinor vertex operators in Siegel gauge and loop amplitude regularization, JHEP 07 (2009) 062 [arXiv:0903.3443] [SPIRES].
P.A. Grassi and P. Vanhove, Higher-loop amplitudes in the non-minimal pure spinor formalism, JHEP 05 (2009) 089 [arXiv:0903.3903] [SPIRES].
C.R. Mafra, Four-point one-loop amplitude computation in the pure spinor formalism, JHEP 01 (2006) 075 [hep-th/0512052] [SPIRES].
N. Berkovits and C.R. Mafra, Equivalence of two-loop superstring amplitudes in the pure spinor and RNS formalisms, Phys. Rev. Lett. 96 (2006) 011602 [hep-th/0509234] [SPIRES].
L. Dixon, Multi-loop amplitudes with maximal supersymmetry, talk given at the meeting International workshop on gauge and string amplitudes, March 30–April 3, Durham, U.K. (2009).
Z. Bern, J.J.M. Carrasco, L. Dixon, R. Roiban and H. Johansson, The complete four-loop four-point amplitude in N = 4 Super-Yang-Mills theory,” arXiv:1008.3327, to appear.
Z. Bern, J.J.M. Carrasco, H. Johansson and D.A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [SPIRES].
P.S. Howe and U. Lindström, Higher order invariants in extended supergravity, Nucl. Phys. B 181 (1981) 487 [SPIRES].
G. Bossard, P.S. Howe and K.S. Stelle, A note on the UV behaviour of maximally supersymmetric Yang-Mills theories, Phys. Lett. B 682 (2009) 137 [arXiv:0908.3883] [SPIRES].
P. Vanhove, The critical ultraviolet behaviour of N = 8 supergravity amplitudes, arXiv:1004.1392 [SPIRES].
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, arXiv:1004.0476 [SPIRES].
N. Berkovits, Superstring multiloop amplitudes and non-renormalization theorems, lectures given at the Spring school on superstring theory and related topics, March 22–30, ITCP, Trieste, Italy (2010).
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ArXiv ePrint: 1004.2692
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Björnsson, J., Green, M.B. 5 loops in 24/5 dimensions. J. High Energ. Phys. 2010, 132 (2010). https://doi.org/10.1007/JHEP08(2010)132
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DOI: https://doi.org/10.1007/JHEP08(2010)132