Abstract
Recently, infinite families of massive Feynman integrals were found to feature an unexpected Yangian symmetry. In the massless case, similar integrability properties are understood via the interpretation of individual Feynman integrals as correlators in the massless fishnet theory introduced by Gürdoğan and Kazakov. Here we seek for an analogous interpretation of the integrability of massive Feynman integrals. We contrast two approaches to define simple massive quantum field theories in four dimensions. First, we discuss spontaneous symmetry breaking in the massless bi-scalar fishnet theory. We then propose an alternative route to a massive fishnet theory by taking a double-scaling limit of \( \mathcal{N} \) = 4 SYM theory on the Coulomb branch. Both approaches lead to a massive extension of the massless fishnet theory, differing in how masses enter into the propagators. In the latter theory, planar off-shell amplitudes are in one-to-one correspondence with precisely those massive Feynman integrals that were shown to be invariant under the Yangian. This suggests a re-investigation of Coulomb branch \( \mathcal{N} \) = 4 SYM theory with regard to integrability. Finally, we demonstrate that in the case of spontaneous symmetry breaking, the original conformal symmetry leads to soft theorems for scattering amplitudes in the broken phase.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
D. Bombardelli et al., An integrability primer for the gauge-gravity correspondence: An introduction, J. Phys. A 49 (2016) 320301 [arXiv:1606.02945] [INSPIRE].
L. Dolan, C.R. Nappi and E. Witten, A relation between approaches to integrability in superconformal Yang-Mills theory, JHEP 10 (2003) 017 [hep-th/0308089] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
D. Bernard, An introduction to Yangian symmetries, Int. J. Mod. Phys. B 7 (1993) 3517 [hep-th/9211133] [INSPIRE].
N.J. MacKay, Introduction to Yangian symmetry in integrable field theory, Int. J. Mod. Phys. A 20 (2005) 7189 [hep-th/0409183] [INSPIRE].
A. Torrielli, Yangians, S-matrices and AdS/CFT, J. Phys. A 44 (2011) 263001 [arXiv:1104.2474] [INSPIRE].
F. Loebbert, Lectures on Yangian symmetry, J. Phys. A 49 (2016) 323002 [arXiv:1606.02947] [INSPIRE].
O. Gürdoğan and V. Kazakov, New integrable 4D quantum field theories from strongly deformed planar \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 117 (2016) 201602 [Addendum ibid. 117 (2016) 259903] [arXiv:1512.06704] [INSPIRE].
J. Caetano, O. Gürdoğan and V. Kazakov, Chiral limit of \( \mathcal{N} \) = 4 SYM and ABJM and integrable Feynman graphs, JHEP 03 (2018) 077 [arXiv:1612.05895] [INSPIRE].
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller and D.-l. Zhong, Yangian symmetry for bi-scalar loop amplitudes, JHEP 05 (2018) 003 [arXiv:1704.01967] [INSPIRE].
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller and D.-l. Zhong, Yangian symmetry for fishnet Feynman graphs, Phys. Rev. D 96 (2017) 121901 [arXiv:1708.00007] [INSPIRE].
C. Sieg and M. Wilhelm, On a CFT limit of planar γi-deformed \( \mathcal{N} \) = 4 SYM theory, Phys. Lett. B 756 (2016) 118 [arXiv:1602.05817] [INSPIRE].
O. Mamroud and G. Torrents, RG stability of integrable fishnet models, JHEP 06 (2017) 012 [arXiv:1703.04152] [INSPIRE].
N. Gromov, V. Kazakov, G. Korchemsky, S. Negro and G. Sizov, Integrability of conformal fishnet theory, JHEP 01 (2018) 095 [arXiv:1706.04167] [INSPIRE].
D. Grabner, N. Gromov, V. Kazakov and G. Korchemsky, Strongly γ-deformed \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory as an integrable conformal field theory, Phys. Rev. Lett. 120 (2018) 111601 [arXiv:1711.04786] [INSPIRE].
V. Kazakov and E. Olivucci, Biscalar integrable conformal field theories in any dimension, Phys. Rev. Lett. 121 (2018) 131601 [arXiv:1801.09844] [INSPIRE].
B. Basso and D.-l. Zhong, Continuum limit of fishnet graphs and AdS σ-model, JHEP 01 (2019) 002 [arXiv:1806.04105] [INSPIRE].
N. Gromov, V. Kazakov and G. Korchemsky, Exact correlation functions in conformal fishnet theory, JHEP 08 (2019) 123 [arXiv:1808.02688] [INSPIRE].
S. Derkachov, V. Kazakov and E. Olivucci, Basso-dixon correlators in two-dimensional fishnet CFT, JHEP 04 (2019) 032 [arXiv:1811.10623] [INSPIRE].
G.P. Korchemsky, Exact scattering amplitudes in conformal fishnet theory, JHEP 08 (2019) 028 [arXiv:1812.06997] [INSPIRE].
A.C. Ipsen, M. Staudacher and L. Zippelius, The one-loop spectral problem of strongly twisted \( \mathcal{N} \) = 4 Super Yang-Mills theory, JHEP 04 (2019) 044 [arXiv:1812.08794] [INSPIRE].
B. Basso, J. Caetano and T. Fleury, Hexagons and correlators in the fishnet theory, JHEP 11 (2019) 172 [arXiv:1812.09794] [INSPIRE].
V. Kazakov, E. Olivucci and M. Preti, Generalized fishnets and exact four-point correlators in chiral CFT4 , JHEP 06 (2019) 078 [arXiv:1901.00011] [INSPIRE].
R. de Mello Koch, W. LiMing, H.J.R. Van Zyl and J.P. Rodrigues, Chaos in the fishnet, Phys. Lett. B 793 (2019) 169 [arXiv:1902.06409] [INSPIRE].
N. Gromov and A. Sever, Derivation of the holographic dual of a planar conformal field theory in 4D, Phys. Rev. Lett. 123 (2019) 081602 [arXiv:1903.10508] [INSPIRE].
A. Pittelli and M. Preti, Integrable fishnet from γ-deformed \( \mathcal{N} \) = 2 quivers, Phys. Lett. B 798 (2019) 134971 [arXiv:1906.03680].
N. Gromov and A. Sever, Quantum fishchain in AdS5, JHEP 10 (2019) 085 [arXiv:1907.01001] [INSPIRE].
S. Dutta Chowdhury, P. Haldar and K. Sen, On the Regge limit of fishnet correlators, JHEP 10 (2019) 249 [arXiv:1908.01123] [INSPIRE].
G.K. Karananas, V. Kazakov and M. Shaposhnikov, Spontaneous conformal symmetry breaking in fishnet CFT, Phys. Lett. B 811 (2020) 135922 [arXiv:1908.04302] [INSPIRE].
T. Adamo and S. Jaitly, Twistor fishnets, J. Phys. A 53 (2020) 055401 [arXiv:1908.11220] [INSPIRE].
N. Gromov and A. Sever, The holographic dual of strongly γ-deformed \( \mathcal{N} \) = 4 SYM theory: derivation, generalization, integrability and discrete reparametrization symmetry, JHEP 02 (2020) 035 [arXiv:1908.10379] [INSPIRE].
B. Basso, G. Ferrando, V. Kazakov and D.-l. Zhong, Thermodynamic Bethe ansatz for biscalar conformal field theories in any dimension, Phys. Rev. Lett. 125 (2020) 091601 [arXiv:1911.10213] [INSPIRE].
S. Derkachov and E. Olivucci, Exactly solvable magnet of conformal spins in four dimensions, Phys. Rev. Lett. 125 (2020) 031603 [arXiv:1912.07588] [INSPIRE].
F. Levkovich-Maslyuk and M. Preti, Exploring the ground state spectrum of γ-deformed N = 4 SYM, arXiv:2003.05811 [INSPIRE].
G.K. Karananas, Aspects of spontaneous breaking of conformal invariance in the fishnet CFT, PoS CORFU2019 (2020) 116 [arXiv:2003.13716] [INSPIRE].
J.-b. Wu, J. Tian and B. Chen, Loop operators in three-dimensional \( \mathcal{N} \) = 2 fishnet theories, JHEP 07 (2020) 215 [arXiv:2004.07592] [INSPIRE].
S. Derkachov and E. Olivucci, Exactly solvable single-trace four point correlators in χCFT4, arXiv:2007.15049 [INSPIRE].
S.D. Chowdhury, P. Haldar and K. Sen, Regge amplitudes in generalized fishnet and chiral fishnet theories, JHEP 12 (2020) 117 [arXiv:2008.10201] [INSPIRE].
F. Loebbert, D. Müller and H. Münkler, Yangian bootstrap for conformal Feynman integrals, Phys. Rev. D 101 (2020) 066006 [arXiv:1912.05561] [INSPIRE].
L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N = 4 super Yang-Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [INSPIRE].
F. Loebbert, J. Miczajka, D. Müller and H. Münkler, Massive conformal symmetry and integrability for Feynman integrals, Phys. Rev. Lett. 125 (2020) 091602 [arXiv:2005.01735] [INSPIRE].
O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 95 [hep-th/9503121] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, A piece of cake: the ground-state energies in γi-deformed \( \mathcal{N} \) = 4 SYM theory at leading wrapping order, JHEP 09 (2014) 078 [arXiv:1405.6712] [INSPIRE].
M. Sogaard, Bilocal phase operators in beta-deformed super Yang-Mills, Phys. Rev. D 86 (2012) 085016 [arXiv:1112.1906] [INSPIRE].
N. Beisert and A. Garus, Yangian algebra and correlation functions in planar gauge theories, SciPost Phys. 5 (2018) 018 [arXiv:1804.09110] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γi-deformed N = 4 SYM theory, J. Phys. A 47 (2014) 455401 [arXiv:1308.4420] [INSPIRE].
A. Dymarsky, I.R. Klebanov and R. Roiban, Perturbative search for fixed lines in large N gauge theories, JHEP 08 (2005) 011 [hep-th/0505099] [INSPIRE].
E. Pomoni and L. Rastelli, Large N field theory and AdS tachyons, JHEP 04 (2009) 020 [arXiv:0805.2261] [INSPIRE].
G.C. Wick, Properties of Bethe-Salpeter wave functions, Phys. Rev. 96 (1954) 1124 [INSPIRE].
R.E. Cutkosky, Solutions of a Bethe-Salpeter equations, Phys. Rev. 96 (1954) 1135 [INSPIRE].
M. Gillioz, M. Meineri and J. Penedones, A scattering amplitude in conformal field theory, JHEP 11 (2020) 139 [arXiv:2003.07361] [INSPIRE].
T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N = 4 superconformal symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [INSPIRE].
A. Sever and P. Vieira, Symmetries of the N = 4 SYM S-matrix, arXiv:0908.2437 [INSPIRE].
T. Bargheer, N. Beisert and F. Loebbert, Exact superconformal and yangian symmetry of scattering amplitudes, J. Phys. A 44 (2011) 454012 [arXiv:1104.0700] [INSPIRE].
T. Bargheer, N. Beisert, F. Loebbert and T. McLoughlin, Conformal anomaly for amplitudes in \( \mathcal{N} \) = 6 superconformal Chern-Simons theory, J. Phys. A 45 (2012) 475402 [arXiv:1204.4406] [INSPIRE].
M.S. Bianchi, M. Leoni, A. Mauri, S. Penati and A. Santambrogio, One loop amplitudes in ABJM, JHEP 07 (2012) 029 [arXiv:1204.4407] [INSPIRE].
D. Chicherin and E. Sokatchev, Conformal anomaly of generalized form factors and finite loop integrals, JHEP 04 (2018) 082 [arXiv:1709.03511] [INSPIRE].
T. Dennen and Y.-t. Huang, Dual conformal properties of six-dimensional maximal super Yang-Mills amplitudes, JHEP 01 (2011) 140 [arXiv:1010.5874] [INSPIRE].
R.H. Boels and W. Wormsbecher, Spontaneously broken conformal invariance in observables, arXiv:1507.08162 [INSPIRE].
P. Di Vecchia, R. Marotta, M. Mojaza and J. Nohle, New soft theorems for the gravity dilaton and the Nambu-Goldstone dilaton at subsubleading order, Phys. Rev. D 93 (2016) 085015 [arXiv:1512.03316] [INSPIRE].
I. Low and A.V. Manohar, Spontaneously broken space-time symmetries and Goldstone’s theorem, Phys. Rev. Lett. 88 (2002) 101602 [hep-th/0110285] [INSPIRE].
Y.-t. Huang and C. Wen, Soft theorems from anomalous symmetries, JHEP 12 (2015) 143 [arXiv:1509.07840] [INSPIRE].
H. Lüo and C. Wen, Recursion relations from soft theorems, JHEP 03 (2016) 088 [arXiv:1512.06801] [INSPIRE].
M. Bianchi, A.L. Guerrieri, Y.-t. Huang, C.-J. Lee and C. Wen, Exploring soft constraints on effective actions, JHEP 10 (2016) 036 [arXiv:1605.08697] [INSPIRE].
L. Rodina, Scattering amplitudes from soft theorems and infrared behavior, Phys. Rev. Lett. 122 (2019) 071601 [arXiv:1807.09738] [INSPIRE].
F. Loebbert, M. Mojaza and J. Plefka, Hidden conformal symmetry in tree-level graviton scattering, JHEP 05 (2018) 208 [arXiv:1802.05999] [INSPIRE].
N. Beisert, A. Garus and M. Rosso, Yangian symmetry and integrability of planar N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 118 (2017) 141603 [arXiv:1701.09162] [INSPIRE].
N. Beisert, A. Garus and M. Rosso, Yangian symmetry for the action of planar \( \mathcal{N} \) = 4 super Yang-Mills and \( \mathcal{N} \) = 6 super Chern-Simons theories, Phys. Rev. D 98 (2018) 046006 [arXiv:1803.06310] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2008.11739
Rights and permissions
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.
About this article
Cite this article
Loebbert, F., Miczajka, J. Massive fishnets. J. High Energ. Phys. 2020, 197 (2020). https://doi.org/10.1007/JHEP12(2020)197
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)197