Ultraviolet divergences in supersymmetric theories

Kellog Stelle

doi:10.1017/CBO9780511920998.005

5 - Ultraviolet divergences in supersymmetric theories

Published online by Cambridge University Press: 05 August 2012

Edited by

Amanda Weltman and

Kellog Stelle: Affiliation:
Technology and Medicine
Jeff Murugan: Affiliation:
University of Cape Town
Amanda Weltman: Affiliation:
University of Cape Town
George F. R. Ellis: Affiliation:
University of Cape Town

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Summary

Recent advances in calculational techniques permit the study of ultraviolet structure in maximal super Yang–Mills and maximal supergravity theories at heretofore unattainable loop orders. Hints from string theory suggest that maximal supergravity might have a similar ultraviolet behavior in D =4 spacetime dimensions as maximal super Yang–Mills theory and so be ultraviolet convergent. However, what is known of field theoretic non-renormalization theorems suggests only that ½-BPS counterterms are excluded. A key test of the relative finiteness properties of the two theories will be the ultraviolet divergences in D=5 maximal supergravity at the four-loop level. This chapter constitutes a review of the arguments that lead to these remarkable results.

Introduction

Obtaining an acceptable quantum theory of gravity is the key remaining problem in fundamental theoretical physics. A basic difficulty in formulating such a theory was already recognized in the earliest approaches to this problem in the 1930s: the dimensional character of Newton's constant gives rise to ultraviolet divergent quantum correction integrals. Naïve power counting of the degree of divergence Δ of an L-loop diagram in D-dimensional gravity theories yields the result

Δ = (D - 2)L + 2

which grows linearly with loop order, implying the requirement for higher and higher-dimensional counterterms to renormalize the divergences. In the 1970s, this was confirmed explicitly in the first Feynman diagram calculations of the radiative corrections to systems containing gravity plus matter [51]. The time lag between the general perception of the UV divergence problem and its first concrete demonstration was due to the complexity of Feynman diagram calculations involving gravity.

Type: Chapter
Information: Foundations of Space and Time
Reflections on Quantum Gravity
, pp. 85 - 105

DOI: https://doi.org/10.1017/CBO9780511920998.005 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2012

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References

[1] L., Baulieu et al., Finiteness properties of the N = 4 super-Yang–Mills theory in supersymmetric gauge, Nucl. Phys. B753, 252–72 (2006).Google Scholar

[2] L., Baulieu et al., Shadow fields and local supersymmetric gauges, Nucl. Phys. B753, 273–94 (2006).Google Scholar

[3] L., Baulieu et al., Ten-dimensional super-Yang–Mills with nine off-shell supersymmetries, Phys. Lett. B658, 249–52 (2008).Google Scholar

[4] Z., Bern et al., One-loop n-point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B425, 217–60 (1994).Google Scholar

[5] Z., Bern et al., On the relationship between Yang–Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B530, 401–56 (1998).Google Scholar

[6] Z., Bern et al., The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang–Mills theory, Phys. Rev. D75, 085010 (2007).Google Scholar

[7] Z., Bern et al., Three-loop superfiniteness of N=8 supergravity, Phys. Rev. Lett. 98, 161303 (2007).Google Scholar

[8] Z., Bern et al., Manifest ultraviolet behavior for the three-loop four-point amplitude of N =8 supergravity, Phys. Rev. D78, 105019 (2008).Google Scholar

[9] Z., Bern et al., Multi-loop amplitudes with maximal supersymmetry, Talk by L. J. Dixon at the International Workshop on Gauge and String Amplitudes, IPPP Durham, 30th March to 3rd April 2009. (Available on-line at http://conference.ippp.dur.ac.uk) (2009).

[10] Z., Bern et al., The ultraviolet behavior of N =8 supergravity at four loops, Phys. Rev. Lett. 103, 081301 (2009).Google Scholar

[11] N., Berkovits, ICTP lectures on covariant quantization of the superstring, hep-th/0209059 (2002).

[12] N., Berkovits, New higher-derivative R4 theorems, Phys. Rev. Lett. 98, 211601 (2007).Google Scholar

[13] N., Berkovits and P. S., Howe, The cohomology of superspace, pure spinors and invariant integrals, arxiv:0803.3024 (2008).

[14] N., Berkovits et al., Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory, JHEP 11, 063 (2009).Google Scholar

[15] L., Bonara, P., Pasti and M., Tonin, Superspace formulation of 10-D SUGRA+SYM theory a la Green–Schwarz, Phys. Lett. B188, 335 (1987).Google Scholar

[16] L., Bonara, P., Pasti and M., Tonin, Chiral anomalies in higher dimensional supersymmetric theories, Nucl. Phys. B286, 150 (1987).Google Scholar

[17] G., Bossard, P. S., Howe and K. S., Stelle, The ultra-violet question in maximally supersymmetric field theories, Gen. Rel. Grav. 41, 919–81 (2009).Google Scholar

[18] G., Bossard, P. S., Howe and K. S., Stelle, A note on the UV behaviour of maximally supersymmetric Yang–Mills theories, Phys. Lett. B682, 137–42 (2009).Google Scholar

[19] L., Brink, O., Lindgren and B. E.W., Nilsson, The ultraviolet finiteness of the N =4 Yang–Mills theory, Phys. Lett. B123, 323 (1983).Google Scholar

[20] M., Cederwall, B.E.W., Nilsson and D., Tsimpis, Spinorial cohomology and maximally supersymmetric theories, JHEP 02, 009 (2002).Google Scholar

[21] E., Cremmer et al., Spontaneous symmetry breaking and Higgs effect in supergravity without cosmological constant, Nucl. Phys. B147, 105 (1979).Google Scholar

[22] F., Delduc and J., McCabe, The quantization of N =3 super Yang–Mills off-shell in harmonic superspace, Class. Quant. Grav. 06, 233 (1989).Google Scholar

[23] S., Deser and B., Zumino, Consistent supergravity, Phys. Lett. B62, 335 (1976).Google Scholar

[24] S., Deser, Renormalizability properties of supergravity, Phys. Rev. Lett. 38, 527 (1977).Google Scholar

[25] J.A., Dixon, Supersymmetry is full of holes, Class. Quant. Grav. 07, 1511–21 (1990).Google Scholar

[26] J.M., Drummond et al., Integral invariants in N =4 SYM and the effective action for coincident D-branes, JHEP 08, 016 (2003).Google Scholar

[27] M. J., Duff and D. J., Toms, Kaluza–Klein kounterterms. Presented at the 2nd Europhysics Study Conference on Unification of Fundamental Interactions, Erice, Sicily, Oct 6–14, 1981.

[28] B., Eden et al., Partial non-renormalisation of the stress-tensor four-point function in N =4 SYM and AdS/CFT, Nucl. Phys. B607, 191–212 (2001).Google Scholar

[29] D. Z., Freedman, P., van Nieuwenhuizen and S., Ferrara, Progress toward a theory of supergravity, Phys. Rev., D13, 3214–18 (1976).Google Scholar

[30] A., Galperin et al., Unconstrained off-shell N =3 supersymmetric Yang–Mills theory, Class. Quant. Grav. 02, 155 (1985).Google Scholar

[31] A., Galperin et al., N = 3 supersymmetric gauge theory, Phys. Lett. B151, 215–18 (1985).Google Scholar

[32] S. J., Gates Jr. et al., Component actions from curved superspace: normal coordinates and ectoplasm, Phys. Lett. B421, 203–10 (1998).Google Scholar

[33] M. B., Green, J.G., Russo and P., Vanhove, Ultraviolet properties of maximal supergravity, Phys. Rev. Lett. 98, 131602 (2007).Google Scholar

[34] P. J., Heslop and P. S., Howe, Anote on composite operators in N =4 SYM, Phys. Lett. B516, 367–75 (2001).Google Scholar

[35] P. S., Howe and U., Lindstrom, Higher order invariants in extended supergravity, Nucl. Phys. B181, 487 (1981).Google Scholar

[36] P. S., Howe, K. S., Stelle and P.K., Townsend, Superactions, Nucl. Phys. B191, 445 (1981).Google Scholar

[37] P. S., Howe, K. S., Stelle and P.K., Townsend, The relaxed hypermultiplet: an unconstrained N =2 superfield theory, Nucl. Phys. B214, 519 (1983).Google Scholar

[38] P. S., Howe, K. S., Stelle and P.K., Townsend, Miraculous ultraviolet cancellations in supersymmetry made manifest, Nucl. Phys. B236, 125 (1984).Google Scholar

[39] P. S., Howe, G., Papadopoulos and K.S., Stelle, The background field method and the nonlinear sigma model, Nucl. Phys. B296, 26 (1988).Google Scholar

[40] P. S., Howe and K. S., Stelle, The ultraviolet properties of supersymmetric field theories, Int. J. Mod. Phys. A4, 1871 (1989).Google Scholar

[41] P. S., Howe, U., Lindstrom and P., White, Anomalies and renormalization in the BRST-BV framework, Phys. Lett. B246, 430–34 (1990).Google Scholar

[42] P. S., Howe, Pure spinors, function superspaces and supergravity theories in ten-dimensions and eleven-dimensions, Phys. Lett. B273, 90–94 (1991).Google Scholar

[43] P. S., Howe, Pure spinors lines in superspace and ten-dimensional supersymmetric theories, Phys. Lett. B258, 141–4 (1991).Google Scholar

[44] P. S., Howe, O., Raetzel and E., Sezgin, On brane actions and superembeddings, JHEP 08, 011 (1998).Google Scholar

[45] P. S., Howe and K. S., Stelle, Supersymmetry counterterms revisited, Phys. Lett. B554, 190–96 (2003).Google Scholar

[46] P. S., Howe and D., Tsimpis, On higher-order corrections in M theory, JHEP 09, 038 (2003).Google Scholar

[47] R. E., Kallosh, Counterterms in extended supergravities, Phys. Lett. B99, 122–7 (1981).Google Scholar

[48] R., Kallosh, On a possibility of a UV finite N =8 supergravity, arxiv:0808.2310 (2008).

[49] S., Mandelstam, Light cone superspace and the ultraviolet finiteness of the N = 4 model, Nucl. Phys. B213, 149–68 (1983).Google Scholar

[50] N., Marcus and A., Sagnotti, The ultraviolet behavior of N =4 Yang–Mills and the power counting of extended superspace, Nucl. Phys. B256, 77 (1985).Google Scholar

[51] G., 't Hooft and M.J.G., Veltman, One loop divergencies in the theory of gravitation, Ann. Poincare Phys. Theor. A20, 69–94 (1974).Google Scholar

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5 - Ultraviolet divergences in supersymmetric theories

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