Abstract
We study the form factors of the Konishi operator, the prime example of non-protected operators in \( \mathcal{N}=4 \) SYM theory, via the on-shell unitarity method. Since the Konishi operator is not protected by supersymmetry, its form factors share many features with amplitudes in QCD, such as the occurrence of rational terms and of UV divergences that require renormalization. A subtle point is that this operator depends on the spacetime dimension. This requires a modification when calculating its form factors via the on-shell unitarity method. We derive a rigorous prescription that implements this modification to all loop orders and obtain the two-point form factor up to two-loop order and the three-point form factor to one-loop order. From these form factors, we construct an IR-finite cross-section-type quantity, namely the inclusive decay rate of the (off-shell) Konishi operator to any final (on-shell) state. Via the optical theorem, it is connected to the imaginary part of the two-point correlation function. We extract the Konishi anomalous dimension up to two-loop order from it.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, On-Shell Methods in Perturbative QCD, Annals Phys. 322 (2007) 1587 [arXiv:0704.2798] [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE].
J.M. Henn and J.C. Plefka, Scattering Amplitudes in Gauge Theories, Lect. Notes Phys. 883 (2014) 1.
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
W.L. van Neerven, Infrared Behavior of On-shell Form-factors in a \( \mathcal{N}=4 \) Supersymmetric Yang-Mills Field Theory, Z. Phys. C 30 (1986) 595 [INSPIRE].
T. Gehrmann, J.M. Henn and T. Huber, The three-loop form factor in \( \mathcal{N}=4 \) super Yang-Mills, JHEP 03 (2012) 101 [arXiv:1112.4524] [INSPIRE].
R.H. Boels, B.A. Kniehl, O.V. Tarasov and G. Yang, Color-kinematic Duality for Form Factors, JHEP 02 (2013) 063 [arXiv:1211.7028] [INSPIRE].
A.H. Mueller, On the Asymptotic Behavior of the Sudakov Form-factor, Phys. Rev. D 20 (1979) 2037 [INSPIRE].
J.C. Collins, Algorithm to Compute Corrections to the Sudakov Form-factor, Phys. Rev. D 22 (1980) 1478 [INSPIRE].
A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys. Rev. D 24 (1981) 3281 [INSPIRE].
Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].
A. Brandhuber, B. Spence, G. Travaglini and G. Yang, Form Factors in \( \mathcal{N}=4 \) Super Yang-Mills and Periodic Wilson Loops, JHEP 01 (2011) 134 [arXiv:1011.1899] [INSPIRE].
L.V. Bork, D.I. Kazakov and G.S. Vartanov, On form factors in \( \mathcal{N}=4 \) SYM, JHEP 02 (2011) 063 [arXiv:1011.2440] [INSPIRE].
A. Brandhuber, O. Gurdogan, R. Mooney, G. Travaglini and G. Yang, Harmony of Super Form Factors, JHEP 10 (2011) 046 [arXiv:1107.5067] [INSPIRE].
L.V. Bork, D.I. Kazakov and G.S. Vartanov, On MHV Form Factors in Superspace for \( \mathcal{N}=4 \) SYM Theory, JHEP 10 (2011) 133 [arXiv:1107.5551] [INSPIRE].
L.V. Bork, On NMHV form factors in \( \mathcal{N}=4 \) SYM theory from generalized unitarity, JHEP 01 (2013) 049 [arXiv:1203.2596] [INSPIRE].
O.T. Engelund and R. Roiban, Correlation functions of local composite operators from generalized unitarity, JHEP 03 (2013) 172 [arXiv:1209.0227] [INSPIRE].
L.V. Bork, On form factors in \( \mathcal{N}=4 \) SYM theory and polytopes, JHEP 12 (2014) 111 [arXiv:1407.5568] [INSPIRE].
A. Brandhuber, G. Travaglini and G. Yang, Analytic two-loop form factors in \( \mathcal{N}=4 \) SYM, JHEP 05 (2012) 082 [arXiv:1201.4170] [INSPIRE].
B. Penante, B. Spence, G. Travaglini and C. Wen, On super form factors of half-BPS operators in \( \mathcal{N}=4 \) super Yang-Mills, JHEP 04 (2014) 083 [arXiv:1402.1300] [INSPIRE].
A. Brandhuber, B. Penante, G. Travaglini and C. Wen, The last of the simple remainders, JHEP 08 (2014) 100 [arXiv:1406.1443] [INSPIRE].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].
J. Maldacena and A. Zhiboedov, Form factors at strong coupling via a Y-system, JHEP 11 (2010) 104 [arXiv:1009.1139] [INSPIRE].
Z. Gao and G. Yang, Y-system for form factors at strong coupling in AdS 5 and with multi-operator insertions in AdS 3, JHEP 06 (2013) 105 [arXiv:1303.2668] [INSPIRE].
M. Wilhelm, Amplitudes, Form Factors and the Dilatation Operator in \( \mathcal{N}=4 \) SYM Theory, JHEP 02 (2015) 149 [arXiv:1410.6309] [INSPIRE].
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].
T. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. 133 (1964) B1549.
L.V. Bork, D.I. Kazakov, G.S. Vartanov and A.V. Zhiboedov, Construction of Infrared Finite Observables in \( \mathcal{N}=4 \) Super Yang-Mills Theory, Phys. Rev. D 81 (2010) 105028 [arXiv:0911.1617] [INSPIRE].
C.R. Schmidt, \( H\to ggg\left(gq\overline{q}\right) \) at two loops in the large M(t) limit, Phys. Lett. B 413 (1997) 391 [hep-ph/9707448] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, From correlation functions to event shapes, Nucl. Phys. B 884 (2014) 305 [arXiv:1309.0769] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, Event shapes in \( \mathcal{N}=4 \) super-Yang-Mills theory, Nucl. Phys. B 884 (2014) 206 [arXiv:1309.1424] [INSPIRE].
L. Bianchi, V. Forini and A.V. Kotikov, On DIS Wilson coefficients in \( \mathcal{N}=4 \) super Yang-Mills theory, Phys. Lett. B 725 (2013) 394 [arXiv:1304.7252] [INSPIRE].
W. Siegel, Supersymmetric Dimensional Regularization via Dimensional Reduction, Phys. Lett. B 84 (1979) 193 [INSPIRE].
D.M. Capper, D.R.T. Jones and P. van Nieuwenhuizen, Regularization by Dimensional Reduction of Supersymmetric and Nonsupersymmetric Gauge Theories, Nucl. Phys. B 167 (1980) 479 [INSPIRE].
W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Deep Inelastic Scattering Beyond the Leading Order in Asymptotically Free Gauge Theories, Phys. Rev. D 18 (1978) 3998 [INSPIRE].
D. Anselmi, M.T. Grisaru and A. Johansen, A Critical behavior of anomalous currents, electric-magnetic universality and CFT 4 in four-dimensions, Nucl. Phys. B 491 (1997) 221 [hep-th/9601023] [INSPIRE].
D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, Universality of the operator product expansions of SCFT in four-dimensions, Phys. Lett. B 394 (1997) 329 [hep-th/9608125] [INSPIRE].
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, On the logarithmic behavior in \( \mathcal{N}=4 \) SYM theory, JHEP 08 (1999) 020 [hep-th/9906188] [INSPIRE].
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Anomalous dimensions in \( \mathcal{N}=4 \) SYM theory at order g 4, Nucl. Phys. B 584 (2000) 216 [hep-th/0003203] [INSPIRE].
B. Eden, C. Schubert and E. Sokatchev, Three loop four point correlator in \( \mathcal{N}=4 \) SYM, Phys. Lett. B 482 (2000) 309 [hep-th/0003096] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in \( \mathcal{N}=4 \) SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [hep-th/0404092] [INSPIRE].
B. Eden, C. Jarczak and E. Sokatchev, A three-loop test of the dilatation operator in \( \mathcal{N}=4 \) SYM, Nucl. Phys. B 712 (2005) 157 [hep-th/0409009] [INSPIRE].
C. Sieg, Superspace computation of the three-loop dilatation operator of \( \mathcal{N}=4 \) SYM theory, Phys. Rev. D 84 (2011) 045014 [arXiv:1008.3351] [INSPIRE].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Wrapping at four loops in \( \mathcal{N}=4 \) SYM, Phys. Lett. B 666 (2008) 100 [arXiv:0712.3522] [INSPIRE].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Anomalous dimension with wrapping at four loops in \( \mathcal{N}=4 \) SYM, Nucl. Phys. B 805 (2008) 231 [arXiv:0806.2095] [INSPIRE].
V.N. Velizhanin, The four-loop anomalous dimension of the Konishi operator in \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory, JETP Lett. 89 (2009) 6 [arXiv:0808.3832] [INSPIRE].
Z. Bajnok and R.A. Janik, Four-loop perturbative Konishi from strings and finite size effects for multiparticle states, Nucl. Phys. B 807 (2009) 625 [arXiv:0807.0399] [INSPIRE].
Z. Bajnok, A. Hegedus, R.A. Janik and T. Lukowski, Five loop Konishi from AdS/CFT, Nucl. Phys. B 827 (2010) 426 [arXiv:0906.4062] [INSPIRE].
G. Arutyunov, S. Frolov and R. Suzuki, Five-loop Konishi from the Mirror TBA, JHEP 04 (2010) 069 [arXiv:1002.1711] [INSPIRE].
J. Balog and A. Hegedus, 5-loop Konishi from linearized TBA and the XXX magnet, JHEP 06 (2010) 080 [arXiv:1002.4142] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky, V.A. Smirnov and E. Sokatchev, Five-loop Konishi in \( \mathcal{N}=4 \) SYM, Nucl. Phys. B 862 (2012) 123 [arXiv:1202.5733] [INSPIRE].
S. Leurent, D. Serban and D. Volin, Six-loop Konishi anomalous dimension from the Y-system, Phys. Rev. Lett. 109 (2012) 241601 [arXiv:1209.0749] [INSPIRE].
Z. Bajnok and R.A. Janik, Six- and seven loop Konishi from Lüscher corrections, JHEP 11 (2012) 002 [arXiv:1209.0791] [INSPIRE].
S. Leurent and D. Volin, Multiple zeta functions and double wrapping in planar \( \mathcal{N}=4 \) SYM, Nucl. Phys. B 875 (2013) 757 [arXiv:1302.1135] [INSPIRE].
D. Volin, Quantum spectral curve for AdS 5 /CFT 4 spectral problem, talk given at Integrability in Gauge and String Theory (IGST), Utrecht, The Netherlands (2013).
C. Marboe and D. Volin, Quantum spectral curve as a tool for a perturbative quantum field theory, arXiv:1411.4758 [INSPIRE].
V.P. Nair, A Current Algebra for Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
S. Penati and A. Santambrogio, Superspace approach to anomalous dimensions in \( \mathcal{N}=4 \) SYM, Nucl. Phys. B 614 (2001) 367 [hep-th/0107071] [INSPIRE].
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Progress in one loop QCD computations, Ann. Rev. Nucl. Part. Sci. 46 (1996) 109 [hep-ph/9602280] [INSPIRE].
C. Sieg and A. Torrielli, Wrapping interactions and the genus expansion of the 2-point function of composite operators, Nucl. Phys. B 723 (2005) 3 [hep-th/0505071] [INSPIRE].
C. Boucher-Veronneau, L. Dixon and J. Pennington, Unpublished notes.
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev. D 75 (2007) 125019 [arXiv:0704.1835] [INSPIRE].
J.C. Collins, Renormalization. An introduction to renormalization, the renormalization group, and the operator product expansion, Cambridge University Press, Cambridge U.K. (1984).
G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].
Z. Bern, A. De Freitas, L.J. Dixon and H.L. Wong, Supersymmetric regularization, two loop QCD amplitudes and coupling shifts, Phys. Rev. D 66 (2002) 085002 [hep-ph/0202271] [INSPIRE].
A.J. Buras and P.H. Weisz, QCD Nonleading Corrections to Weak Decays in Dimensional Regularization and ’t Hooft-Veltman Schemes, Nucl. Phys. B 333 (1990) 66 [INSPIRE].
W. Siegel, Inconsistency of Supersymmetric Dimensional Regularization, Phys. Lett. B 94 (1980) 37 [INSPIRE].
L.V. Avdeev, G.A. Chochia and A.A. Vladimirov, On the Scope of Supersymmetric Dimensional Regularization, Phys. Lett. B 105 (1981) 272 [INSPIRE].
L.V. Avdeev, Noninvariance of Regularization by Dimensional Reduction: An Explicit Example of Supersymmetry Breaking, Phys. Lett. B 117 (1982) 317 [INSPIRE].
L.V. Avdeev and A.A. Vladimirov, Dimensional Regularization and Supersymmetry, Nucl. Phys. B 219 (1983) 262 [INSPIRE].
S.S. Gubser and I.R. Klebanov, Absorption by branes and Schwinger terms in the world volume theory, Phys. Lett. B 413 (1997) 41 [hep-th/9708005] [INSPIRE].
D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [INSPIRE].
W.L. van Neerven, Dimensional Regularization of Mass and Infrared Singularities in Two Loop On-shell Vertex Functions, Nucl. Phys. B 268 (1986) 453 [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, \( \mathcal{N}=6 \) superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
A. Brandhuber, O. Gürdogan, D. Korres, R. Mooney and G. Travaglini, Two-loop Sudakov Form Factor in ABJM, JHEP 11 (2013) 022 [arXiv:1305.2421] [INSPIRE].
D. Young, Form Factors of Chiral Primary Operators at Two Loops in ABJ(M), JHEP 06 (2013) 049 [arXiv:1305.2422] [INSPIRE].
M.S. Bianchi et al., ABJM amplitudes and WL at finite N , JHEP 09 (2013) 114 [arXiv:1306.3243] [INSPIRE].
L. Bianchi and M.S. Bianchi, Nonplanarity through unitarity in the ABJM theory, Phys. Rev. D 89 (2014) 125002 [arXiv:1311.6464] [INSPIRE].
G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].
Z. Bern and G. Chalmers, Factorization in one loop gauge theory, Nucl. Phys. B 447 (1995) 465 [hep-ph/9503236] [INSPIRE].
C. Anastasiou, E.W.N. Glover and C. Oleari, Application of the negative dimension approach to massless scalar box integrals, Nucl. Phys. B 565 (2000) 445 [hep-ph/9907523] [INSPIRE].
V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1.
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
T. Gehrmann, T. Huber and D. Maître, Two-loop quark and gluon form-factors in dimensional regularisation, Phys. Lett. B 622 (2005) 295 [hep-ph/0507061] [INSPIRE].
G. Passarino and M.J.G. Veltman, One Loop Corrections for e + e − Annihilation Into μ + μ − in the Weinberg Model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].
Z.-G. Xiao, G. Yang and C.-J. Zhu, The rational parts of one-loop QCD amplitudes I: The general formalism, Nucl. Phys. B 758 (2006) 1 [hep-ph/0607015] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γ i -deformed \( \mathcal{N}=4 \) SYM theory, J. Phys. A 47 (2014) 455401 [arXiv:1308.4420] [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: 1410.8485
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
Nandan, D., Sieg, C., Wilhelm, M. et al. Cutting through form factors and cross sections of non-protected operators in \( \mathcal{N}=4 \) SYM. J. High Energ. Phys. 2015, 156 (2015). https://doi.org/10.1007/JHEP06(2015)156
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2015)156