The QCD theta-parameter in canonical quantization
Abstract: The role of the QCD theta-parameter is investigated in pure Yang-Mills theory in the spacetime given by the four-dimensional Euclidean torus. While in this setting the introduction of possibly unphysical boundary conditions is avoided, it must be specified how the sum over the topological sectors is to be carried out. To connect with observables in real time, we perceive the partition function as the trace over the canonical density matrix. The system then corresponds to one of a finite temperature on a spatial three-torus. Carrying out the trace operation requires canonical quantization and gauge fixing. Fixing the gauge and demanding that the Hermiticity of the Hamiltonian is maintained leads to a restriction of the Hilbert space of physical wave functionals that generalizes the constraints derived from imposing Gauss' law. Consequently, we find that the states in the Hilbert space are properly normalizable under an inner product that integrates over each physical configuration represented by the gauge potential one time and one time only. The observables derived from the constrained Hilbert space do not violate charge-parity symmetry. We note that an exact hidden symmetry of the theory that is present for arbitrary values of theta in the Hamiltonian is effectively promoted to parity conservation in this constrained space. These results, derived on a torus in order to avoid the introduction of boundary conditions, also carry over to Minkowski spacetime when taking account of all possible gauge transformations.
- R. Jackiw and C. Rebbi, “Vacuum Periodicity in a Yang-Mills Quantum Theory,” Phys. Rev. Lett. 37 (1976) 172–175.
- C. G. Callan, Jr., R. F. Dashen, and D. J. Gross, “The Structure of the Gauge Theory Vacuum,” Phys. Lett. B 63 (1976) 334–340.
- C. G. Callan, Jr., R. F. Dashen, and D. J. Gross, “Toward a Theory of the Strong Interactions,” Phys. Rev. D 17 (1978) 2717.
- W.-Y. Ai, J. S. Cruz, B. Garbrecht, and C. Tamarit, “Consequences of the order of the limit of infinite spacetime volume and the sum over topological sectors for CP violation in the strong interactions,” Phys. Lett. B 822 (2021) 136616, arXiv:2001.07152 [hep-th].
- W.-Y. Ai, J. S. Cruz, B. Garbrecht, and C. Tamarit, “The limits of the strong CP𝐶𝑃CPitalic_C italic_P problem,” PoS DISCRETE2020-2021 (2022) 084, arXiv:2205.15093 [hep-th].
- F. Yu, “Primer on Axion Physics,” Annalen Phys. 2023 (8, 2023) 2300106, arXiv:2308.08612 [hep-ph].
- Y. Nakamura and G. Schierholz, “The strong CP problem solved by itself due to long-distance vacuum effects,” Nucl. Phys. B 986 (2023) 116063, arXiv:2106.11369 [hep-ph].
- N. Yamanaka, “Unobservability of the topological charge in nonabelian gauge theory: Ward-Takahashi identity and phenomenological aspects,” arXiv:2212.11820 [hep-ph].
- G. Torrieri and H. D. Truran, “The strong CP problem, general covariance, and horizons,” Class. Quant. Grav. 38 no. 21, (2021) 215002, arXiv:2007.13183 [hep-th].
- E. Fradkin, Quantum Field Theory: An Integrated Approach. Princeton University Press, 3, 2021.
- S. Okubo and R. E. Marshak, “Argument for the nonexistence of the ’strong CP problem’ in QCD,” Prog. Theor. Phys. 87 (1992) 1059–1062.
- H. Leutwyler and A. V. Smilga, “Spectrum of Dirac operator and role of winding number in QCD,” Phys. Rev. D 46 (1992) 5607–5632.
- G. ’t Hooft, “A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories,” Nucl. Phys. B 153 (1979) 141–160.
- P. van Baal, “Some Results for SU(N) Gauge Fields on the Hypertorus,” Commun. Math. Phys. 85 (1982) 529.
- M. Luscher, “Some Analytic Results Concerning the Mass Spectrum of Yang-Mills Gauge Theories on a Torus,” Nucl. Phys. B 219 (1983) 233–261.
- R. Jackiw, “Introduction to the Yang-Mills Quantum Theory,” Rev. Mod. Phys. 52 (1980) 661–673.
- L. Di Luzio, M. Giannotti, E. Nardi, and L. Visinelli, “The landscape of QCD axion models,” Phys. Rept. 870 (2020) 1–117, arXiv:2003.01100 [hep-ph].
- J. M. Gracia-Bondía and J. C. Várilly, “Ensuring locality in QFT via string-local fields,” 7, 2022. arXiv:2207.06522 [hep-th].
- W.-Y. Ai, B. Garbrecht, and C. Tamarit, “CP𝐶𝑃CPitalic_C italic_P conservation in the strong interactions.” to appear.
- V. F. Tokarev, “Does the strong CP problem really exist?,” Mod. Phys. Lett. A 8 (1993) 531–541.
- F. K. Guo, R. Horsley, U. G. Meissner, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, and J. M. Zanotti, “The electric dipole moment of the neutron from 2+1 flavor lattice QCD,” Phys. Rev. Lett. 115 no. 6, (2015) 062001, arXiv:1502.02295 [hep-lat].
- A. Shindler, T. Luu, and J. de Vries, “Nucleon electric dipole moment with the gradient flow: The θ𝜃\thetaitalic_θ-term contribution,” Phys. Rev. D 92 no. 9, (2015) 094518, arXiv:1507.02343 [hep-lat].
- J. Dragos, T. Luu, A. Shindler, J. de Vries, and A. Yousif, “Confirming the Existence of the strong CP Problem in Lattice QCD with the Gradient Flow,” Phys. Rev. C 103 no. 1, (2021) 015202, arXiv:1902.03254 [hep-lat].
- L. Giusti and M. Lüscher, “Topological susceptibility at T>Tc𝑇subscript𝑇cT>T_{\rm c}italic_T > italic_T start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT from master-field simulations of the SU(3) gauge theory,” Eur. Phys. J. C 79 no. 3, (2019) 207, arXiv:1812.02062 [hep-lat].
- D. Albandea, G. Catumba, and A. Ramos, “The Strong CP Problem in the Quantum Rotor,” arXiv:2402.17518 [hep-lat].
- M. Shifman, Advanced Topics in Quantum Field Theory. Cambridge University Press, 4, 2022.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.