nLab theta angle

Idea

In Yang-Mills theory and specifically in its application to QCD, the theta angle refers to the prefactor $\theta$ in the expression of the action functional of the theory in front of the piece of topological Yang-Mills theory

$$\nabla \mapsto \frac{1}{g^2 }\int_X F_\nabla \wedge \star F_\nabla \;+\; i \theta \int_X F_\nabla \wedge F_\nabla$$

(see also at S-duality for more).

Phenomenology

In phenomenology the theta angle has to be very close to an integer multiple of $\pi$, see at CP problem. That it is indeed $\theta_{QCD}\sim 0$ instead of $\theta_{QCD} \sim \pi$ that matches experiment is argued at the end of Crewther-DiVecchia-Veneziano-Witten 79, PO discussion.

In string theory and AdS/QCD

Within string theory, with Yang-Mills theory realized on intersecting D-brane models, as on D4-branes for holographic QCD, the theta angle corresponds to the graviphoton RR-field 1-form potential in the higher WZW term of the D4-brane, which is (CGNSW 96 (7.4) APPS97b (51), CAIB 99, 6.1)

$$\mathbf{L}_{D4}^{WZ} \;\propto\; C_1 \wedge \langle F \wedge F\rangle \,.$$

Related concepts

• theta vacuum

References

General

• Rodney Crewther, Paolo Di Vecchia, Gabriele Veneziano, Edward Witten, Chiral estimate of the electric dipole moment of the neutron in quantum chromodynamics, Phys. Lett. B 88 (1979) 123-127 (cds:133382).

• Davide Gaiotto, Anton Kapustin, Zohar Komargodski, Nathan Seiberg, Theta, Time Reversal, and Temperature, JHEP05(2017)091 (arXiv:1703.00501)

Discussion of a similar $\theta$-angle in a 3d field theory, via extended TQFT and stable homotopy theory is in

• Daniel Freed, Zohar Komargodski, Nathan Seiberg, The Sum Over Topological Sectors and $\theta$ in the 2+1-Dimensional $\mathbb{C}\mathbb{P}^1$ $\sigma$-Model (arXiv:1707.05448)

In relation to confinement:

• Massimo D’Elia, Francesco Negro, Theta dependence of the deconfinement temperature in Yang-Mills theories, Phys. Rev. Lett. 109, 2012 (arXiv:1205.0538)

In string theory

The $\theta$-angle as the graviphoton RR-field-potential $C_1$ in the higher WZW term of the D4-brane:

• Martin Cederwall, Alexander von Gussich, Bengt Nilsson, Per Sundell, Anders Westerberg, around (4.6) of The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity, Nucl.Phys. B490 (1997) 179-201 (arXiv:hep-th/9611159)

• Mina Aganagic, Jaemo Park, Costin Popescu, John Schwarz, Section 6 of Dual D-Brane Actions, Nucl. Phys. B496 (1997) 215-230 (arXiv:hep-th/9702133)

• C. Chryssomalakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, Sec. 6.1 of The geometry of branes and extended superspaces, Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 (arXiv:hep-th/9904137)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Theorem 7.4 of Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5

For more see at Green-Schwarz sigma model – References – For D-branes.

Discussion explicitly in view of the Witten-Sakai-Sugimoto model for QCD on D4-branes:

• Si-wen Li, around (3.1) of The theta-dependent Yang-Mills theory at finite temperature in a holographic description (arXiv:1907.10277)