Abstract
We present the superspace formulation of the local RG equation, a framework for the study of supersymmetric RG flows in which the constraints of holomorphy and R-symmetry are manifest. We derive the consistency conditions associated with super-Weyl symmetry off-criticality and initiate the study of their implications. As examples, we derive an expression for the a-function, and present an analog of the a-maximization equation, which is valid off-criticality. We also apply this machinery to the study of conformal manifolds and give a simple proof that the metric on such manifolds is Kähler.
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References
H. Osborn, Derivation of a Four-dimensional c Theorem, Phys. Lett. B 222 (1989) 97 [INSPIRE].
I. Jack and H. Osborn, Analogs for the c Theorem for Four-dimensional Renormalizable Field Theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rept. 569 (2015) 1 [arXiv:1302.0884] [INSPIRE].
F. Baume, B. Keren-Zur, R. Rattazzi and L. Vitale, The local Callan-Symanzik equation: structure and applications, JHEP 08 (2014) 152 [arXiv:1401.5983] [INSPIRE].
I. Jack and H. Osborn, Constraints on RG Flow for Four Dimensional Quantum Field Theories, Nucl. Phys. B 883 (2014) 425 [arXiv:1312.0428] [INSPIRE].
N. Seiberg, Naturalness versus supersymmetric nonrenormalization theorems, Phys. Lett. B 318 (1993) 469 [hep-ph/9309335] [INSPIRE].
V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Exact Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories from Instanton Calculus, Nucl. Phys. B 229 (1983) 381 [INSPIRE].
M.A. Shifman and A.I. Vainshtein, Solution of the Anomaly Puzzle in SUSY Gauge Theories and the Wilson Operator Expansion, Nucl. Phys. B 277 (1986) 456 [Sov. Phys. JETP 64 (1986) 428] [Zh. Eksp. Teor. Fiz. 91 (1986) 723] [INSPIRE].
D.Z. Freedman and H. Osborn, Constructing a c function for SUSY gauge theories, Phys. Lett. B 432 (1998) 353 [hep-th/9804101] [INSPIRE].
I. Jack and C. Poole, The a-function for gauge theories, JHEP 01 (2015) 138 [arXiv:1411.1301] [INSPIRE].
N. Arkani-Hamed, G.F. Giudice, M.A. Luty and R. Rattazzi, Supersymmetry breaking loops from analytic continuation into superspace, Phys. Rev. D 58 (1998) 115005 [hep-ph/9803290] [INSPIRE].
K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
D. Kutasov and A. Schwimmer, Lagrange multipliers and couplings in supersymmetric field theory, Nucl. Phys. B 702 (2004) 369 [hep-th/0409029] [INSPIRE].
E. Barnes, K.A. Intriligator, B. Wecht and J. Wright, Evidence for the strongest version of the 4d a-theorem, via a-maximization along RG flows, Nucl. Phys. B 702 (2004) 131 [hep-th/0408156] [INSPIRE].
D. Kutasov, New results on the ’a theorem’ in four-dimensional supersymmetric field theory, hep-th/0312098 [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].
V. Asnin, On metric geometry of conformal moduli spaces of four-dimensional superconformal theories, JHEP 09 (2010) 012 [arXiv:0912.2529] [INSPIRE].
B. Kol, On conformal deformations, JHEP 09 (2002) 046 [hep-th/0205141] [INSPIRE].
D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, Exactly marginal deformations and global symmetries, JHEP 06 (2010) 106 [arXiv:1005.3546] [INSPIRE].
B. Keren-Zur, The local RG equation and chiral anomalies, JHEP 09 (2014) 011 [arXiv:1406.0869] [INSPIRE].
I.L. Buchbinder and S.M. Kuzenko, Ideas and methods of supersymmetry and supergravity: A Walk through superspace, IOP Publishing, Bristol U.K. (1995).
J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].
L. Bonora, P. Cotta-Ramusino and C. Reina, Conformal Anomaly and Cohomology, Phys. Lett. B 126 (1983) 305 [INSPIRE].
S. Ferrara and B. Zumino, Transformation Properties of the Supercurrent, Nucl. Phys. B 87 (1975) 207 [INSPIRE].
Z. Komargodski and N. Seiberg, Comments on Supercurrent Multiplets, Supersymmetric Field Theories and Supergravity, JHEP 07 (2010) 017 [arXiv:1002.2228] [INSPIRE].
Y. Nakayama, Supercurrent, Supervirial and Superimprovement, Phys. Rev. D 87 (2013) 085005 [arXiv:1208.4726] [INSPIRE].
H. Osborn, N = 1 superconformal symmetry in four-dimensional quantum field theory, Annals Phys. 272 (1999) 243 [hep-th/9808041] [INSPIRE].
J. Erdmenger, C. Rupp and K. Sibold, Conformal transformation properties of the supercurrent in four-dimensional supersymmetric theories, Nucl. Phys. B 530 (1998) 501 [hep-th/9804053] [INSPIRE].
J.-F. Fortin, B. Grinstein, C.W. Murphy and A. Stergiou, On Limit Cycles in Supersymmetric Theories, Phys. Lett. B 719 (2013) 170 [arXiv:1210.2718] [INSPIRE].
L. Bonora, P. Pasti and M. Tonin, Cohomologies and Anomalies in Supersymmetric Theories, Nucl. Phys. B 252 (1985) 458 [INSPIRE].
L. Bonora and S. Giaccari, Weyl transformations and trace anomalies in N = 1, D = 4 supergravities, JHEP 08 (2013) 116 [arXiv:1305.7116] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [INSPIRE].
J. Grosse, Quantum Field Theories Coupled to Supergravity: AdS/CFT and Local Couplings, Fortsch. Phys. 56 (2008) 183 [arXiv:0711.0444] [INSPIRE].
E. Guadagnini, K. Konishi and M. Mintchev, Nonabelian Chiral Anomalies in Supersymmetric Gauge Theories, Phys. Lett. B 157 (1985) 37 [INSPIRE].
I.N. McArthur and H. Osborn, Supersymmetric Chiral Effective Action and Nonabelian Anomalies, Nucl. Phys. B 268 (1986) 573 [INSPIRE].
M. Marinkovic, Wess-Zumino Effective Action for Supersymmetric Yang-Mills Theories, Nucl. Phys. B 366 (1991) 74 [INSPIRE].
Y. Ohshima, K. Okuyama, H. Suzuki and H. Yasuta, Remark on the consistent gauge anomaly in supersymmetric theories, Phys. Lett. B 457 (1999) 291 [hep-th/9904096] [INSPIRE].
S.J. Gates Jr., M.T. Grisaru and S. Penati, Holomorphy, minimal homotopy and the 4-D, N = 1 supersymmetric Bardeen-Gross-Jackiw anomaly, Phys. Lett. B 481 (2000) 397 [hep-th/0002045] [INSPIRE].
S.J. Gates Jr., M.T. Grisaru, M.E. Knutt, S. Penati and H. Suzuki, Supersymmetric gauge anomaly with general homotopic paths, Nucl. Phys. B 596 (2001) 315 [hep-th/0009192] [INSPIRE].
S.L. Adler and W.A. Bardeen, Absence of higher order corrections in the anomalous axial vector divergence equation, Phys. Rev. 182 (1969) 1517 [INSPIRE].
W.A. Bardeen and B. Zumino, Consistent and Covariant Anomalies in Gauge and Gravitational Theories, Nucl. Phys. B 244 (1984) 421 [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].
D. Anselmi, J. Erlich, D.Z. Freedman and A.A. Johansen, Positivity constraints on anomalies in supersymmetric gauge theories, Phys. Rev. D 57 (1998) 7570 [hep-th/9711035] [INSPIRE].
J. Babington and J. Erdmenger, Space-time dependent couplings in N = 1 SUSY gauge theories: Anomalies and central functions, JHEP 06 (2005) 004 [hep-th/0502214] [INSPIRE].
J.L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
Z. Komargodski, The Constraints of Conformal Symmetry on RG Flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
J.-F. Fortin, B. Grinstein and A. Stergiou, Limit Cycles and Conformal Invariance, JHEP 01 (2013) 184 [arXiv:1208.3674] [INSPIRE].
M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the Asymptotics of 4D Quantum Field Theory, JHEP 01 (2013) 152 [arXiv:1204.5221] [INSPIRE].
D. Erkal and D. Kutasov, a-Maximization, Global Symmetries and RG Flows, arXiv:1007.2176 [INSPIRE].
O. Aharony, B. Kol and S. Yankielowicz, On exactly marginal deformations of N = 4 SYM and type IIB supergravity on AdS 5 × S 5, JHEP 06 (2002) 039 [hep-th/0205090] [INSPIRE].
B. Kol, On Conformal Deformations II, arXiv:1005.4408 [INSPIRE].
S.J. Gates, M.T. Grisaru, M. Roček and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200] [INSPIRE].
S. Abel, M. Buican and Z. Komargodski, Mapping Anomalous Currents in Supersymmetric Dualities, Phys. Rev. D 84 (2011) 045005 [arXiv:1105.2885] [INSPIRE].
M. Buican, A Conjectured Bound on Accidental Symmetries, Phys. Rev. D 85 (2012) 025020 [arXiv:1109.3279] [INSPIRE].
H. Elvang et al., On renormalization group flows and the a-theorem in 6d, JHEP 10 (2012) 011 [arXiv:1205.3994] [INSPIRE].
B. Grinstein, A. Stergiou and D. Stone, Consequences of Weyl Consistency Conditions, JHEP 11 (2013) 195 [arXiv:1308.1096] [INSPIRE].
B. Grinstein, D. Stone, A. Stergiou and M. Zhong, Challenge to the a Theorem in Six Dimensions, Phys. Rev. Lett. 113 (2014) 231602 [arXiv:1406.3626] [INSPIRE].
H. Osborn and A. Stergiou, Structures on the Conformal Manifold in Six Dimensional Theories, JHEP 04 (2015) 157 [arXiv:1501.01308] [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
P.S. Howe and R.W. Tucker, Scale Invariance in Superspace, Phys. Lett. B 80 (1978) 138 [INSPIRE].
V. Kaplunovsky and J. Louis, Field dependent gauge couplings in locally supersymmetric effective quantum field theories, Nucl. Phys. B 422 (1994) 57 [hep-th/9402005] [INSPIRE].
A. Schwimmer and S. Theisen, Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching, Nucl. Phys. B 847 (2011) 590 [arXiv:1011.0696] [INSPIRE].
T.E. Clark, O. Piguet and K. Sibold, The Absence of Radiative Corrections to the Axial Current Anomaly in Supersymmetric QED, Nucl. Phys. B 159 (1979) 1 [INSPIRE].
K. Konishi, Anomalous Supersymmetry Transformation of Some Composite Operators in SQCD, Phys. Lett. B 135 (1984) 439 [INSPIRE].
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Auzzi, R., Keren-Zur, B. Superspace formulation of the local RG equation. J. High Energ. Phys. 2015, 150 (2015). https://doi.org/10.1007/JHEP05(2015)150
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DOI: https://doi.org/10.1007/JHEP05(2015)150