Abstract
The geometry of field space governs on-shell scattering amplitudes. We formulate a geometric description of effective field theories which extends previous results for scalars and gauge fields to fermions. The field-space geometry reorganizes and simplifies the computation of quantum loop corrections. Using this geometric framework, we calculate the fermion loop contributions to the renormalization group equations for bosonic operators in the Standard Model Effective Field Theory up to mass dimension eight.
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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].
K. Meetz, Realization of chiral symmetry in a curved isospin space, J. Math. Phys. 10 (1969) 589 [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].
D.V. Volkov, Phenomenological lagrangians, Sov. J. Part. Nucl. 4 (1973) 1.
C. Cheung, A. Helset and J. Parra-Martinez, Geometric soft theorems, JHEP 04 (2022) 011 [arXiv:2111.03045] [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].
R. Alonso, K. Kanshin and S. Saa, Renormalization group evolution of Higgs effective field theory, Phys. Rev. D 97 (2018) 035010 [arXiv:1710.06848] [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].
A. Helset, A. Martin and M. Trott, The geometric standard model effective field theory, JHEP 03 (2020) 163 [arXiv:2001.01453] [INSPIRE].
C. Hays, A. Helset, A. Martin and M. Trott, Exact SMEFT formulation and expansion to \( \mathcal{O} \)(v4/Λ4), JHEP 11 (2020) 087 [arXiv:2007.00565] [INSPIRE].
T. Cohen, N. Craig, X. Lu and D. Sutherland, Is SMEFT enough?, JHEP 03 (2021) 237 [arXiv:2008.08597] [INSPIRE].
T. Corbett, A. Helset, A. Martin and M. Trott, EWPD in the SMEFT to dimension eight, JHEP 06 (2021) 076 [arXiv:2102.02819] [INSPIRE].
T. Corbett, A. Martin and M. Trott, Consistent higher order σ(\( \mathcal{GG} \) → h), Γ(h → \( \mathcal{GG} \)) and Γ(h → γγ) in geoSMEFT, JHEP 12 (2021) 147 [arXiv:2107.07470] [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].
A. Martin and M. Trott, More accurate σ(\( \mathcal{GG} \) → h), Γ(h → \( \mathcal{GG} \), \( \mathcal{AA} \), \( \overline{\Psi}\Psi \)) and Higgs width results via the geoSMEFT, arXiv:2305.05879 [INSPIRE].
V. Gattus and A. Pilaftsis, Minimal supergeometric quantum field theories, Phys. Lett. B 846 (2023) 138234 [arXiv:2307.01126] [INSPIRE].
K. Finn, S. Karamitsos and A. Pilaftsis, Frame covariant formalism for fermionic theories, Eur. Phys. J. C 81 (2021) 572 [arXiv:2006.05831] [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, Phys. Rev. Lett. 130 (2023) 041603 [arXiv:2202.06965] [INSPIRE].
A. Helset, E.E. Jenkins and A.V. Manohar, Geometry in scattering amplitudes, Phys. Rev. D 106 (2022) 116018 [arXiv:2210.08000] [INSPIRE].
N. Craig, Y.-T. Lee, X. Lu and D. Sutherland, Effective field theories as Lagrange spaces, arXiv:2305.09722 [INSPIRE].
A. Helset, E.E. Jenkins and A.V. Manohar, Renormalization of the standard model effective field theory from geometry, JHEP 02 (2023) 063 [arXiv:2212.03253] [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization group evolution of the standard model dimension six operators. Part 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. Part 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. Part 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].
S. Das Bakshi, M. Chala, Á. 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].
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 and Á. Díaz-Carmona, Renormalisation of SMEFT bosonic interactions up to dimension eight by LNV operators, JHEP 06 (2023) 123 [arXiv:2301.07151] [INSPIRE].
E.E. Jenkins, A.V. Manohar and P. Stoffer, Low-energy effective field theory below the electroweak scale: operators and matching, JHEP 03 (2018) 016 [arXiv:1709.04486] [INSPIRE].
E.E. Jenkins, A.V. Manohar and P. Stoffer, Low-energy effective field theory below the electroweak scale: anomalous dimensions, JHEP 01 (2018) 084 [arXiv:1711.05270] [INSPIRE].
L. Alvarez-Gaume and D.Z. Freedman, Geometrical structure and ultraviolet finiteness in the supersymmetric sigma model, Commun. Math. Phys. 80 (1981) 443 [INSPIRE].
R. Nagai, M. Tanabashi, K. Tsumura and Y. Uchida, Scalar and fermion on-shell amplitudes in generalized Higgs effective field theory, Phys. Rev. D 104 (2021) 015001 [arXiv:2102.08519] [INSPIRE].
B.S. DeWitt, Supermanifolds, Cambridge University Press, Cambridge, U.K. (2012) [https://doi.org/10.1017/CBO9780511564000] [INSPIRE].
A. Rogers, Supermanifolds: theory and applications, World Scientific, Singapore (2007).
R.L. Arnowitt and P. Nath, Riemannian geometry in spaces with Grassmann coordinates, Gen. Rel. Grav. 7 (1976) 89 [INSPIRE].
G. ’t Hooft, An algorithm for the poles at dimension four in the dimensional regularization procedure, Nucl. Phys. B 62 (1973) 444 [INSPIRE].
H. Neufeld, J. Gasser and G. Ecker, The one loop functional as a Berezinian, Phys. Lett. B 438 (1998) 106 [hep-ph/9806436] [INSPIRE].
B. Henning, X. Lu and H. Murayama, One-loop matching and running with covariant derivative expansion, JHEP 01 (2018) 123 [arXiv:1604.01019] [INSPIRE].
G. Buchalla et al., Complete one-loop renormalization of the Higgs-electroweak chiral Lagrangian, Nucl. Phys. B 928 (2018) 93 [arXiv:1710.06412] [INSPIRE].
G. Buchalla, A. Celis, C. Krause and J.-N. Toelstede, Master formula for one-loop renormalization of bosonic SMEFT operators, arXiv:1904.07840 [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].
C.W. Murphy, Dimension-8 operators in the standard model effective field theory, JHEP 10 (2020) 174 [arXiv:2005.00059] [INSPIRE].
H.-L. Li et al., Complete set of dimension-eight operators in the standard model effective field theory, Phys. Rev. D 104 (2021) 015026 [arXiv:2005.00008] [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].
E. Fermi, Trends to a theory of beta radiation (in Italian), Nuovo Cim. 11 (1934) 1 [INSPIRE].
W. Dekens and P. Stoffer, Low-energy effective field theory below the electroweak scale: matching at one loop, JHEP 10 (2019) 197 [Erratum ibid. 11 (2022) 148] [arXiv:1908.05295] [INSPIRE].
Acknowledgments
We thank Xiaochuan Lu for helpful discussions. This work is supported in part by the U.S. Department of Energy (DOE) under award numbers DE-SC0009919 and DE-SC0011632 and by the Walter Burke Institute for Theoretical Physics. This work is also supported in part by the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
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Assi, B., Helset, A., Manohar, A.V. et al. Fermion geometry and the renormalization of the Standard Model Effective Field Theory. J. High Energ. Phys. 2023, 201 (2023). https://doi.org/10.1007/JHEP11(2023)201
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DOI: https://doi.org/10.1007/JHEP11(2023)201