Abstract
Conformal symmetry underlies many massless quantum field theories, but little is known about the consequences of this powerful symmetry for on-shell scattering amplitudes. Working in a dimensionally-regularised ϕ3 model at the conformal fixed point, we show that the on-shell renormalised amplitudes satisfy anomalous conformal Ward identities. Each external on-shell state contributes two terms to the anomaly. The first term is proportional to the elementary field anomalous dimension, and thus involves only lower-loop information. We show that the second term can be given as the convolution of a universal collinear function and lower-order amplitudes. The computation of the conformal anomaly is therefore simpler than that of the amplitude at the same perturbative order, which gives our anomalous conformal Ward identities a strong predictive power in perturbation theory. Finally, we show that our result is also of practical importance for dimensionally-regularised amplitudes away from the conformal fixed point.
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References
I.T. Todorov, M.C. Mintchev and V.B. Petkova, Conformal Invariance in Quantum Field Theory, Edizioni della Normale Pisa (1978) [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York, U.S.A. (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
E.S. Fradkin and M.Y. Palchik, New developments in D-dimensional conformal quantum field theory, Phys. Rept. 300 (1998) 1 [INSPIRE].
V.M. Braun, G.P. Korchemsky and D. Müller, The Uses of conformal symmetry in QCD, Prog. Part. Nucl. Phys. 51 (2003) 311 [hep-ph/0306057] [INSPIRE].
C. Coriano, L. Delle Rose, E. Mottola and M. Serino, Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman’s Master Integrals, JHEP 07 (2013) 011 [arXiv:1304.6944] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, JHEP 03 (2014) 111 [arXiv:1304.7760] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Conformal n-point functions in momentum space, Phys. Rev. Lett. 124 (2020) 131602 [arXiv:1910.10162] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Conformal correlators as simplex integrals in momentum space, JHEP 01 (2021) 192 [arXiv:2008.07543] [INSPIRE].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
G.P. Korchemsky and E. Sokatchev, Symmetries and analytic properties of scattering amplitudes in N = 4 SYM theory, Nucl. Phys. B 832 (2010) 1 [arXiv:0906.1737] [INSPIRE].
T. Bargheer et al., Exacting N = 4 Superconformal Symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [INSPIRE].
N. Beisert, J. Henn, T. McLoughlin and J. Plefka, One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N = 4 Super Yang-Mills, JHEP 04 (2010) 085 [arXiv:1002.1733] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [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].
S. Badger et al., Analytic form of the full two-loop five-gluon all-plus helicity amplitude, Phys. Rev. Lett. 123 (2019) 071601 [arXiv:1905.03733] [INSPIRE].
J. Henn, B. Power and S. Zoia, Conformal Invariance of the One-Loop All-Plus Helicity Scattering Amplitudes, JHEP 02 (2020) 019 [arXiv:1911.12142] [INSPIRE].
D. Chicherin and J.M. Henn, Symmetry properties of Wilson loops with a Lagrangian insertion, JHEP 07 (2022) 057 [arXiv:2202.05596] [INSPIRE].
D. Chicherin and E. Sokatchev, Conformal anomaly of generalized form factors and finite loop integrals, JHEP 04 (2018) 082 [arXiv:1709.03511] [INSPIRE].
In collaboration, Conformal Symmetry and Feynman Integrals, PoS LL2018 (2018) 037 [arXiv:1807.06020] [INSPIRE].
D. Chicherin, J.M. Henn and E. Sokatchev, Scattering Amplitudes from Superconformal Ward Identities, Phys. Rev. Lett. 121 (2018) 021602 [arXiv:1804.03571] [INSPIRE].
D. Chicherin, J.M. Henn and E. Sokatchev, Amplitudes from anomalous superconformal symmetry, JHEP 01 (2019) 179 [arXiv:1811.02560] [INSPIRE].
K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240 [INSPIRE].
J.A. Gracey, T.A. Ryttov and R. Shrock, Renormalization-Group Behavior of ϕ3 Theories in d = 6 Dimensions, Phys. Rev. D 102 (2020) 045016 [arXiv:2007.12234] [INSPIRE].
V.M. Braun and A.N. Manashov, Evolution equations beyond one loop from conformal symmetry, Eur. Phys. J. C 73 (2013) 2544 [arXiv:1306.5644] [INSPIRE].
J.A. Gracey, Four loop renormalization of ϕ3 theory in six dimensions, Phys. Rev. D 92 (2015) 025012 [arXiv:1506.03357] [INSPIRE].
G. Parisi, Conformal invariance in perturbation theory, Phys. Lett. B 39 (1972) 643 [INSPIRE].
S. Giombi, I.R. Klebanov and G. Tarnopolsky, Conformal QEDd, F-Theorem and the ϵ Expansion, J. Phys. A 49 (2016) 135403 [arXiv:1508.06354] [INSPIRE].
A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [INSPIRE].
E. Barnes, D. Vaman, C. Wu and P. Arnold, Real-time finite-temperature correlators from AdS/CFT, Phys. Rev. D 82 (2010) 025019 [arXiv:1004.1179] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Evaluation of conformal integrals, JHEP 02 (2016) 068 [arXiv:1511.02357] [INSPIRE].
H. Isono, T. Noumi and T. Takeuchi, Momentum space conformal three-point functions of conserved currents and a general spinning operator, JHEP 05 (2019) 057 [arXiv:1903.01110] [INSPIRE].
T. Bautista and H. Godazgar, Lorentzian CFT 3-point functions in momentum space, JHEP 01 (2020) 142 [arXiv:1908.04733] [INSPIRE].
M. Gillioz, Conformal 3-point functions and the Lorentzian OPE in momentum space, Commun. Math. Phys. 379 (2020) 227 [arXiv:1909.00878] [INSPIRE].
A. Bzowski, TripleK: A Mathematica package for evaluating triple-K integrals and conformal correlation functions, Comput. Phys. Commun. 258 (2021) 107538 [arXiv:2005.10841] [INSPIRE].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [INSPIRE].
H. Lehmann, K. Symanzik and W. Zimmermann, On the formulation of quantized field theories, Nuovo Cim. 1 (1955) 205 [INSPIRE].
M. Gillioz, M. Meineri and J. Penedones, A scattering amplitude in Conformal Field Theory, JHEP 11 (2020) 139 [arXiv:2003.07361] [INSPIRE].
S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products, Lett. Nuovo Cim. 4S2 (1972) 115 [INSPIRE].
A.W. Knapp and E.M. Stein, Interwining Operators for Semisimple Groups, Annals Math. 93 (1971) 489.
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
V.A. Smirnov, Problems of the strategy of regions, Phys. Lett. B 465 (1999) 226 [hep-ph/9907471] [INSPIRE].
B. Jantzen, Foundation and generalization of the expansion by regions, JHEP 12 (2011) 076 [arXiv:1111.2589] [INSPIRE].
D. Chicherin and G.P. Korchemsky, The SAGEX review on scattering amplitudes Chapter 9: Integrability of amplitudes in fishnet theories, J. Phys. A 55 (2022) 443010 [arXiv:2203.13020] [INSPIRE].
T. Huber and D. Maitre, HypExp: A Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun. 175 (2006) 122 [hep-ph/0507094] [INSPIRE].
T. Huber and D. Maitre, HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters, Comput. Phys. Commun. 178 (2008) 755 [arXiv:0708.2443] [INSPIRE].
B. Power, Conformal symmetry predictions for on-shell scattering amplitudes, MSc thesis, Ludwig Maximilians Universität München, Munich, Germany (2020).
D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun. 180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].
S.E. Derkachov, N.A. Kivel, A.S. Stepanenko and A.N. Vasiliev, On calculation in 1/n expansions of critical exponents in the Gross-Neveu model with the conformal technique, hep-th/9302034 [INSPIRE].
A.N. Vasilev, The field theoretic renormalization group in critical behavior theory and stochastic dynamics, Chapman and Hall/CRC (2004) [https://doi.org/10.1201/9780203483565] [INSPIRE].
F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
F. Chavez and C. Duhr, Three-mass triangle integrals and single-valued polylogarithms, JHEP 11 (2012) 114 [arXiv:1209.2722] [INSPIRE].
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
E.E. Boos and A.I. Davydychev, A Method of the Evaluation of the Vertex Type Feynman Integrals, Moscow Univ. Phys. Bull. 42N3 (1987) 6 [INSPIRE].
N.I. Usyukina and A.I. Davydychev, An Approach to the evaluation of three and four point ladder diagrams, Phys. Lett. B 298 (1993) 363 [INSPIRE].
N.I. Usyukina and A.I. Davydychev, Some exact results for two loop diagrams with three and four external lines, Phys. Atom. Nucl. 56 (1993) 1553 [hep-ph/9307327] [INSPIRE].
A.I. Davydychev, Recursive algorithm of evaluating vertex type Feynman integrals, J. Phys. A 25 (1992) 5587 [INSPIRE].
N.I. Usyukina and A.I. Davydychev, New results for two loop off-shell three point diagrams, Phys. Lett. B 332 (1994) 159 [hep-ph/9402223] [INSPIRE].
A.I. Davydychev, Explicit results for all orders of the epsilon expansion of certain massive and massless diagrams, Phys. Rev. D 61 (2000) 087701 [hep-ph/9910224] [INSPIRE].
T.G. Birthwright, E.W.N. Glover and P. Marquard, Master integrals for massless two-loop vertex diagrams with three offshell legs, JHEP 09 (2004) 042 [hep-ph/0407343] [INSPIRE].
P. Wasser, Analytic properties of Feynman integrals for scattering amplitudes, Ph.D. thesis, Institut für Physik (IPH), Johannes Gutenberg-Universität Mainz (JGU), Germany (2018) [INSPIRE].
J. Henn, B. Mistlberger, V.A. Smirnov and P. Wasser, Constructing d-log integrands and computing master integrals for three-loop four-particle scattering, JHEP 04 (2020) 167 [arXiv:2002.09492] [INSPIRE].
C. Dlapa, J. Henn and K. Yan, Deriving canonical differential equations for Feynman integrals from a single uniform weight integral, JHEP 05 (2020) 025 [arXiv:2002.02340] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP 07 (2013) 128 [arXiv:1306.2799] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
S. Borowka et al., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222 (2018) 313 [arXiv:1703.09692] [INSPIRE].
T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP 12 (2016) 030 [arXiv:1608.01902] [INSPIRE].
T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP 07 (2019) 031 [arXiv:1905.08019] [INSPIRE].
O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].
D. Maitre, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
W. Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics 14, Interscience Publishers John Wiley & Sons Inc., New York-London-Sydney (1965).
S. Caron-Huot et al., Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in \( \mathcal{N} \) = 4 Super Yang-Mills and \( \mathcal{N} \) = 8 Supergravity, JHEP 10 (2020) 188 [arXiv:2003.03120] [INSPIRE].
K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831 [INSPIRE].
F. Brown, Iterated integrals in quantum field theory, in the proceedings of the 6th Summer School on Geometric and Topological Methods for Quantum Field Theory, Villa de Leyva Colombia, July 6–23 (2009) [Cambridge University Press (2013), p. 188–240] [https://doi.org/10.1017/CBO9781139208642.006] [INSPIRE].
A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].
B. Jantzen, A.V. Smirnov and V.A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically, Eur. Phys. J. C 72 (2012) 2139 [arXiv:1206.0546] [INSPIRE].
Acknowledgments
We are grateful to Vladimir Braun, Einan Gardi, Grisha Korchemsky, Emery Sokatchev and Alexander Manashov for many insightful discussions. We thank Bláithín Power for collaboration at the early stage of this project [51]. JMH thanks Erik Panzer and Yang Zhang for collaboration on canonical differential equations for Feynman integrals with non-integer powers. SZ wishes to thank Robin Brüser for helping with Qgraf. We have used JaxoDraw [52] to draw Feynman diagrams. This project has received funding from the European Union’s Horizon 2020 research and innovation programmes Novel structures in scattering amplitudes (grant agreement No 725110), and High precision multi-jet dynamics at the LHC (grant agreement No 772099). DC is supported by the French National Research Agency in the framework of the Investissements d’avenir program (ANR-15-IDEX-02). DC and SZ gratefully acknowledge the computing resources provided by the Max Planck Institute for Physics. This research was supported by the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311.
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Chicherin, D., Henn, J. & Zoia, S. Anomalous Ward identities for on-shell amplitudes at the conformal fixed point. J. High Energ. Phys. 2023, 110 (2023). https://doi.org/10.1007/JHEP06(2023)110
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DOI: https://doi.org/10.1007/JHEP06(2023)110