Abstract
S-matrix elements are invariant under field redefinitions of the Lagrangian. They are determined by geometric quantities such as the curvature of the field-space manifold of scalar and gauge fields. We present a formalism where scalar and gauge fields are treated together, with a metric on the combined space of both types of fields. Scalar and gauge scattering amplitudes are given by the Riemann curvature Rijkl of this combined space, with indices i, j, k, l chosen to be scalar or gauge indices depending on the type of external particle. One-loop divergences can also be computed in terms of geometric invariants of the combined space, which greatly simplifies the computation of renormalization group equations. We apply our formalism to the Standard Model Effective Field Theory (SMEFT), and compute the renormalization group equations for even-parity bosonic operators to mass dimension eight.
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References
J.S.R. Chisholm, Change of variables in quantum field theories, Nucl. Phys. 26 (1961) 469 [INSPIRE].
S. Kamefuchi, L. O’Raifeartaigh and A. Salam, Change of variables and equivalence theorems in quantum field theories, Nucl. Phys. 28 (1961) 529 [INSPIRE].
H.D. Politzer, Power Corrections at Short Distances, Nucl. Phys. B 172 (1980) 349 [INSPIRE].
C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342 (1995) 189 [hep-ph/9304230] [INSPIRE].
A.V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863 [INSPIRE].
R. Alonso, E.E. Jenkins and A.V. Manohar, A Geometric Formulation of Higgs Effective Field Theory: Measuring the Curvature of Scalar Field Space, Phys. Lett. B 754 (2016) 335 [arXiv:1511.00724] [INSPIRE].
R. Alonso, E.E. Jenkins and A.V. Manohar, Geometry of the Scalar Sector, JHEP 08 (2016) 101 [arXiv:1605.03602] [INSPIRE].
T. Cohen, N. Craig, X. Lu and D. Sutherland, Unitarity violation and the geometry of Higgs EFTs, JHEP 12 (2021) 003 [arXiv:2108.03240] [INSPIRE].
R. Alonso and M. West, Roads to the Standard Model, Phys. Rev. D 105 (2022) 096028 [arXiv:2109.13290] [INSPIRE].
C. Cheung, A. Helset and J. Parra-Martinez, Geometry-kinematics duality, Phys. Rev. D 106 (2022) 045016 [arXiv:2202.06972] [INSPIRE].
T. Cohen, N. Craig, X. Lu and D. Sutherland, On-Shell Covariance of Quantum Field Theory Amplitudes, arXiv:2202.06965 [INSPIRE].
C. Cheung, A. Helset and J. Parra-Martinez, Geometric soft theorems, JHEP 04 (2022) 011 [arXiv:2111.03045] [INSPIRE].
R. Alonso and M. West, On the effective action for scalars in a general manifold to any loop order, IPPP/22/44 (2022), arXiv:2207.02050 [INSPIRE].
A. Helset, E.E. Jenkins and A.V. Manohar, Geometry in scattering amplitudes, Phys. Rev. D 106 (2022) 116018 [arXiv:2210.08000] [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence, JHEP 10 (2013) 087 [arXiv:1308.2627] [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators II: Yukawa Dependence, JHEP 01 (2014) 035 [arXiv:1310.4838] [INSPIRE].
R. Alonso, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology, JHEP 04 (2014) 159 [arXiv:1312.2014] [INSPIRE].
M. Chala, G. Guedes, M. Ramos and J. Santiago, Towards the renormalisation of the Standard Model effective field theory to dimension eight: Bosonic interactions I, SciPost Phys. 11 (2021) 065 [arXiv:2106.05291] [INSPIRE].
M. Accettulli Huber and S. De Angelis, Standard Model EFTs via on-shell methods, JHEP 11 (2021) 221 [arXiv:2108.03669] [INSPIRE].
S. Das Bakshi, M. Chala, A. Díaz-Carmona and G. Guedes, Towards the renormalisation of the Standard Model effective field theory to dimension eight: bosonic interactions II, Eur. Phys. J. Plus 137 (2022) 973 [arXiv:2205.03301] [INSPIRE].
K. Meetz, Realization of chiral symmetry in a curved isospin space, J. Math. Phys. 10 (1969) 589 [INSPIRE].
G. Buchalla, A. Celis, C. Krause and J.-N. Toelstede, Master Formula for One-Loop Renormalization of Bosonic SMEFT Operators, LMU-ASC~15/19 (2019), arXiv:1904.07840 [INSPIRE].
J. Honerkamp and K. Meetz, Chiral-invariant perturbation theory, Phys. Rev. D 3 (1971) 1996 [INSPIRE].
J. Honerkamp, Chiral multiloops, Nucl. Phys. B 36 (1972) 130 [INSPIRE].
A. Helset, M. Paraskevas and M. Trott, Gauge fixing the Standard Model Effective Field Theory, Phys. Rev. Lett. 120 (2018) 251801 [arXiv:1803.08001] [INSPIRE].
G. ’t Hooft, An algorithm for the poles at dimension four in the dimensional regularization procedure, Nucl. Phys. B 62 (1973) 444 [INSPIRE].
C.W. Murphy, Dimension-8 operators in the Standard Model Eective Field Theory, JHEP 10 (2020) 174 [arXiv:2005.00059] [INSPIRE].
H.-L. Li, Z. Ren, J. Shu, M.-L. Xiao, J.-H. Yu and Y.-H. Zheng, Complete set of dimension-eight operators in the standard model effective field theory, Phys. Rev. D 104 (2021) 015026 [arXiv:2005.00008] [INSPIRE].
B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10 (2010) 085 [arXiv:1008.4884] [INSPIRE].
A. Helset, A. Martin and M. Trott, The Geometric Standard Model Effective Field Theory, JHEP 03 (2020) 163 [arXiv:2001.01453] [INSPIRE].
A.V. Manohar, The HQET / NRQCD Lagrangian to order αs/m3, Phys. Rev. D 56 (1997) 230 [hep-ph/9701294] [INSPIRE].
A. Manohar and H. Georgi, Chiral Quarks and the Nonrelativistic Quark Model, Nucl. Phys. B 234 (1984) 189 [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Naive Dimensional Analysis Counting of Gauge Theory Amplitudes and Anomalous Dimensions, Phys. Lett. B 726 (2013) 697 [arXiv:1309.0819] [INSPIRE].
B.M. Gavela, E.E. Jenkins, A.V. Manohar and L. Merlo, Analysis of General Power Counting Rules in Effective Field Theory, Eur. Phys. J. C 76 (2016) 485 [arXiv:1601.07551] [INSPIRE].
R. Alonso, E.E. Jenkins and A.V. Manohar, Holomorphy without Supersymmetry in the Standard Model Effective Field Theory, Phys. Lett. B 739 (2014) 95 [arXiv:1409.0868] [INSPIRE].
C. Cheung and C.-H. Shen, Nonrenormalization Theorems without Supersymmetry, Phys. Rev. Lett. 115 (2015) 071601 [arXiv:1505.01844] [INSPIRE].
Z. Bern, J. Parra-Martinez and E. Sawyer, Structure of two-loop SMEFT anomalous dimensions via on-shell methods, JHEP 10 (2020) 211 [arXiv:2005.12917] [INSPIRE].
P. Baratella, D. Haslehner, M. Ruhdorfer, J. Serra and A. Weiler, RG of GR from on-shell amplitudes, JHEP 03 (2022) 156 [arXiv:2109.06191] [INSPIRE].
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Helset, A., Jenkins, E.E. & Manohar, A.V. Renormalization of the Standard Model Effective Field Theory from geometry. J. High Energ. Phys. 2023, 63 (2023). https://doi.org/10.1007/JHEP02(2023)063
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DOI: https://doi.org/10.1007/JHEP02(2023)063