Phase transitions and theta vacuum energy

Miguel Aguado ; Manuel Asorey

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of modern theoretical and mathematical physics: Gauge theories and superstrings, Tome 272 (2011), pp. 9-19.

Résumé

The analytic behaviour of $\theta$-vacuum energy is related to the existence of phase transitions in QCD and $\mathbb C\mathrm P^N$ sigma models. The absence of singularities different from Lee–Yang zeros only permits $\wedge$ cusp singularities in the vacuum energy density and never $\vee$ cusps. This fact, together with the Vafa–Witten diamagnetic inequality, provides a key missing link in the Vafa–Witten proof of parity symmetry conservation in vector-like gauge theories and $\mathbb C\mathrm P^N$ sigma models. However, this property does not exclude the existence of a first phase transition at $\theta=\pi$ or a second order phase transition at $\theta=0$, which might be very relevant for interpretation of the anomalous behaviour of the topological susceptibility in the $\mathbb C\mathrm P^1$ sigma model.

Bibliographie

[1] Vafa C., Witten E., “Parity conservation in quantum chromodynamics”, Phys. Rev. Lett., 53 (1984), 535 | DOI

[2] Lüscher M., “Does the topological susceptibility in lattice $\sigma $ models scale according to the perturbative renormalization group?”, Nucl. Phys. B, 200 (1982), 61 | DOI | MR

[3] Asorey M., Mitter P.K., “Regularized, continuum Yang–Mills process and Feynman–Kac functional integral”, Commun. Math. Phys., 80 (1981), 43 | DOI | MR | Zbl

[4] Aguado M., Asorey M., “Vafa–Witten theorem and Lee–Yang singularities”, Phys. Rev. D, 80 (2009), 127702 | DOI | MR

[5] Azcoiti V., Galante A., “Parity and CT realization in QCD”, Phys. Rev. Lett., 83 (1999), 1518 | DOI

[6] Cohen T.D., “Spontaneous parity violation in QCD at finite temperature: On the inapplicability of the Vafa–Witten theorem”, Phys. Rev. D, 64 (2001), 047704 | DOI

[7] Einhorn M.B., Wudka J., “Vafa–Witten theorem on spontaneous breaking of parity”, Phys. Rev. D, 67 (2003), 045004 | DOI | MR

[8] Ji X., “On absence of spontaneous parity breaking in QCD”, Phys. Lett. B, 554 (2003), 33 | DOI

[9] Seiler E., Gauge theories as a problem of constructive quantum field theory and statistical mechanics, Lect. Notes Phys., 159, Springer, Berlin, 1982 | MR

[10] Osterwalder K., Seiler E., “Gauge field theories on a lattice”, Ann. Phys., 110 (1978), 440 | DOI | MR

[11] Aguado M., Seiler E., “Clash of positivities in topological density correlators”, Phys. Rev. D., 72 (2005), 094502 | DOI

[12] Osterwalder K., Schrader R., “Axioms for Euclidean Green's functions. I, II”, Commun. Math. Phys., 31 (1973), 83 ; 42 (1975), 281–305 | DOI | MR | Zbl | DOI | MR | Zbl

[13] Yang C.N., Lee T.D., “Statistical theory of equations of state and phase transitions. I: Theory of condensation”, Phys. Rev., 87 (1952), 404 | DOI | MR | Zbl

[14] Lee T.D., Yang C.N., “Statistical theory of equations of state and phase transitions. II: Lattice gas and Ising model”, Phys. Rev., 87 (1952), 410 | DOI | MR | Zbl

[15] Asorey M., Falceto F., “Geometric regularization of gauge theories”, Nucl. Phys. B, 327 (1989), 427 | DOI | MR

[16] Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Univ. Press, Cambridge, 1986 | MR

[17] Aguado M., Asorey M., “CP symmetry and phase transitions”, Gribov–80 memorial volume: Quantum chromodynamics and beyond, eds. Yu. Dokshitzer, P. Lévai, J. Nyíri, World Sci., Singapore, 2011

[18] Blythe R.A., Evans M.R., “Lee–Yang zeros and phase transitions in nonequilibrium steady states”, Phys. Rev. Lett., 89 (2002), 080601 | DOI | MR | Zbl

[19] Evans M.R., Blythe R.A., “Nonequilibrium dynamics in low-dimensional systems”, Physica A, 313 (2002), 110 | DOI | Zbl

[20] Blythe R.A., Evans M.R., “The Lee–Yang theory of equilibrium and nonequilibrium phase transitions”, Braz. J. Phys., 33 (2003), 464 | DOI

[21] Asorey M., García Álvarez D., Muñoz-Castañeda J.M., “Casimir effect and global theory of boundary conditions”, J. Phys. A: Math. and Gen., 39 (2006), 6127 | DOI | MR | Zbl

[22] Olejník Š., Schierholz G., “On the existence of a first order phase transition at small vacuum angle $\theta $ in the $\mathrm {CP}^3$ model”, Nucl. Phys. B. Proc. Suppl., 34 (1994), 709 | DOI

[23] Azcoiti V., Galante A., Laliena V., “$\theta $-Vacuum: Phase transitions and/or symmetry breaking at $\theta =\pi $”, Prog. Theor. Phys., 109 (2003), 843 | DOI | Zbl

[24] Allés B., D'Elia M., Di Giacomo A., “Topological susceptibility at zero and finite $T$ in $SU(3)$ Yang–Mills theory”, Nucl. Phys. B, 494 (1997), 281 | DOI

[25] Allés B., D'Elia M., Di Giacomo A., “Analyticity in $\theta $ on the lattice and the large volume limit of the topological susceptibility”, Phys. Rev. D, 71 (2005), 034503 | DOI

[26] Vicari E., Panagopoulos H., “$\theta $ dependence of $SU(N)$ gauge theories in the presence of a topological term”, Phys. Rep., 470 (2009), 93 | DOI

[27] Zamolodchikov A.B., Zamolodchikov Al.B., “Factorized $S$-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models”, Ann. Phys., 120 (1979), 253 | DOI | MR

[28] Polyakov A., Wiegmann P.B., “Theory of nonabelian goldstone bosons in two dimensions”, Phys. Lett. B, 131 (1983), 121 | DOI | MR

[29] Campostrini M., Rossi P., Vicari E., “Monte Carlo simulation of $\mathrm {CP}^{N-1}$ models”, Phys. Rev. D, 46 (1992), 2647 | DOI | MR

[30] D'Adda A., Di Vecchia P., Lüscher M., “Confinement and chiral symmetry breaking in $\mathrm {CP}^{n-1}$ models with quarks”, Nucl. Phys. B, 152 (1979), 125 | DOI

[31] Ahmad S., Lenaghan J.T., Thacker H.B., “Coherent topological charge structure in $\mathrm {CP}^{N-1}$ models and QCD”, Phys. Rev. D, 72 (2005), 114511 | DOI

[32] Burkhalter R., Imachi M., Shinno Y., Yoneyama H., “$\mathrm {CP}^{N-1}$ models with a $\theta $ term and fixed point action”, Prog. Theor. Phys., 106 (2001), 613 | DOI

[33] Beard B.B., Pepe M., Riederer S., Wiese U.-J., “Study of $\mathrm {CP}(N-1)$ $\theta $-vacua by cluster simulation of $\mathrm {SU}(N)$ quantum spin ladders”, Phys. Rev. Lett., 94 (2005), 010603 | DOI

[34] Imachi M., Kanou S., Yoneyama H., “Two-dimensional $\mathrm {CP}^2$ model with $\theta $-term and topological charge distributions”, Prog. Theor. Phys., 102 (1999), 653 | DOI

[35] Plefka J.C., Samuel S., “Monte Carlo studies of two-dimensional systems with a $\theta $ term”, Phys. Rev. D, 56 (1997), 44 | DOI

[36] Blatter M., Burkhalter R., Hasenfratz P., Niedermayer F., “Instantons and the fixed point topological charge in the two-dimensional $\mathrm O(3)$ $\sigma $ model”, Phys. Rev. D, 53 (1996), 923 | DOI