Geodesic
Functional Integral Approaches to the Bosonization of Effective Multi-Quark Interactions with $U_{\mathrm{A}}(1)$ Breaking
Symmetry, integrability and geometry: methods and applications, Tome 2 (2006) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Low energy hadron phenomenology involving the (u,d,s) quarks is often approached through effective multi-quark Lagrangians with the symmetries of QCD. A very successful approach consists in taking the four-quark Nambu–Jona-Lasinio Lagrangian with the chiral $U_L(3)\times U_R(3)$ symmetry in the massless limit, combined with the $U_{\mathrm{A}}(1)$ breaking six-quark flavour determinant interaction of 't Hooft. We review the present status and some very recent developments related to the functional integration over the cubic term in auxiliary mesonic variables that one introduces to bosonize the system. Various approaches for handling this functional, which cannot be integrated exactly, are discussed: the stationary phase approximation, the perturbative expansion, the loop expansion, their interrelation and importance for the evaluation of the effective action. The intricate group structure rules out the method of Airy's integral. The problem of the instability of the vacuum is stated and a solution given by including eight-quark interactions.
Keywords: field theory; functional integral methods; stationary phase method; 't Hooft interactions; semiclassical corrections; effective action. 
@article{SIGMA_2006_2_a25,
     author = {Brigitte Hiller and Alexander A. Osipov and V\'eronique Bernard and Alex H. Blin},
     title = {Functional {Integral} {Approaches} to the {Bosonization} of {Effective} {Multi-Quark} {Interactions} with $U_{\mathrm{A}}(1)$ {Breaking}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2006},
     volume = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a25/}
}
TY  - JOUR
AU  - Brigitte Hiller
AU  - Alexander A. Osipov
AU  - Véronique Bernard
AU  - Alex H. Blin
TI  - Functional Integral Approaches to the Bosonization of Effective Multi-Quark Interactions with $U_{\mathrm{A}}(1)$ Breaking
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2006
VL  - 2
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a25/
LA  - en
ID  - SIGMA_2006_2_a25
ER  - 
%0 Journal Article
%A Brigitte Hiller
%A Alexander A. Osipov
%A Véronique Bernard
%A Alex H. Blin
%T Functional Integral Approaches to the Bosonization of Effective Multi-Quark Interactions with $U_{\mathrm{A}}(1)$ Breaking
%J Symmetry, integrability and geometry: methods and applications
%D 2006
%V 2
%U http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a25/
%G en
%F SIGMA_2006_2_a25
Brigitte Hiller; Alexander A. Osipov; Véronique Bernard; Alex H. Blin. Functional Integral Approaches to the Bosonization of Effective Multi-Quark Interactions with $U_{\mathrm{A}}(1)$ Breaking. Symmetry, integrability and geometry: methods and applications, Tome 2 (2006). http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a25/

[1] Shifman M. A., Vainshtein A. I., Zakharov V. I., “QCD and resonance physics. Theoretical foundations”, Nucl. Phys. B, 147 (1979), 385–447 | DOI

[2] Weinberg S., “Phenomenological Lagrangians”, Phys. A, 96 (1979), 327–340 | DOI

[3] Gasser J., Leutwyler H., “Chiral perturbation theory to one loop”, Ann. of Phys. (N.Y.), 158 (1984), 142–210 ; Gasser J., Leutwyler H., “Chiral perturbation theory: expansions in the mass of the strange quark”, Nucl. Phys. B, 250 (1985), 465–516 | DOI | MR | DOI

[4] 't Hooft G., “A planar diagram theory for strong interactions”, Nucl. Phys. B, 72 (1974), 461–473 | DOI

[5] Witten E., “Baryons in the $1/N$ expansion”, Nucl. Phys. B, 160 (1979), 57–115 | DOI | MR

[6] Manohar A. V., “Large $N_c$ QCD”, Probing the Standard Model of Particle Interactions, Vol. II (Les Houches LXVIII, 1997), eds. R. Gupta et al., Elsevier, Amsterdam, 1999, 1091–1169; arXiv:hep-ph/9802419

[7] Pich A., “Colourless mesons in a polychromatic world”, The Phenomenology of Large-$N_c$ QCD (January 9–11, 2002, Tempe, Arizona), eds. R. Lebed R, World Scientific, Singapore, 2002, 239–258; arXiv:hep-ph/0205030

[8] Gasiorowicz S., Geffen D. A., “Effective lagrangians and field algebras with chiral symmetry”, Rev. Mod. Phys., 41 (1969), 531–573 ; Volkov M. K., “Meson Lagrangians in a superconductor quark model”, Ann. of Phys. (N.Y.), 157 (1984), 282–303 ; Meißner U.-G., “Low energy hadron physics from effective chiral lagrangians with vector mesons”, Phys. Rep., 161 (1988), 213–362 ; Espriu D., de Rafael E., Taron, J., “The QCD effectve action at long distances”, Nucl. Phys. B, 345 (1990), 22–56 ; Nambu Y., Jona-Lasinio G., “Dynamical model of elementary particles based on an analogy with superconductivity, II”, Phys. Rev., 124 (1961), 246–254 ; Vaks V. G., Larkin A. I., “On the application of the methods of superconductivity theory to the problem of the masses of elementary particles”, Zh. Éksp. Teor. Fiz., 40 (1961), 282–285 | DOI | MR | DOI | DOI | DOI | DOI | Zbl

[9] Sov. Phys. JETP, 13 (1961), 192–193 ; Arbuzov B. A., Tavkhelidze A. N., Faustov R. N., “On the fermion mass in the $\gamma_5$-invariant model of the quantum field theory”, Docl. Akad. Nauk. SSSR, 139 (1961), 345–347 | DOI | DOI | MR | Zbl | Zbl

[10] Sov. Phys. Dokl., 6 (1962), 598–600 | DOI | DOI | MR | Zbl | Zbl

[11] Eguchi T., “New approach to collective phenomena in superconductivity models”, Phys. Rev. D, 14 (1976), 2755–2763 | DOI | MR

[12] Volkov M. K., Ebert D., “Four-quark interactions as a common dynamical basis of the $\sigma$ model and the vector dominance model”, Sov. J. Nucl. Phys., 36 (1982), 736–742; Ebert D., Volkov M. K., “Composite meson model with vector dominance based on $U(2)$ invariant four quark interactions”, Z. Phys. C, 16 (1983), 205–210 | DOI

[13] Dhar A., Wadia S., “The Nambu–Jona-Lasinio model: an effective Lagrangian for quantum chromodynamics at intermediate length scales”, Phys. Rev. Lett., 52 (1984), 959–962 ; Dhar A., Shankar R., Wadia S., “Nambu–Jona-Lasinio type effective Lagrangian: anomalies and nonlinear Lagrangian of low-energy, large-N QCD”, Phys. Rev. D, 31 (1985), 3256–3267 ; Ebert D., Reinhardt H., “Effective chiral hadron lagrangian with anomalies and Skyrme terms from quark flavour dynamics”, Nucl. Phys. B, 271 (1986), 188–226 ; Schüren C., Arriola E. R., Goeke K., “Explicit chiral symmetry breaking in the Nambu–Jona-Lasinio model”, Nucl. Phys. A, 547 (1992), 612–632 | DOI | MR | DOI | DOI

[14] 't Hooft G., “Computation of the quantum effects due to a four-dimensional pseudoparticle”, Phys. Rev. D, 14 (1976), 3432–3450 ; Erratum Phys. Rev. D, 18 (1978), 2199 | DOI | DOI

[15] Diakonov D., “Instantons at work”, Progr. Part. Nucl. Phys., 51 (2003), 173–222 ; ; Shuryak E. V., “Applying the many-body theory to quarks and gluons”, Phys. Rept., 391 (2004), 381–428 arXiv:hep-ph/0212026 | DOI | DOI

[16] Simonov Yu. A., “Effective quark Lagrangian in the instanton-gas model”, Phys. Lett. B, 412 (1997), 371–376 ; arXiv:hep-th/9703205 | DOI

[17] Bali G. S., “QCD forces and heavy quark bound states”, Phys. Rept., 343 (2001), 1–136 ; arXiv:hep-ph/0001312 | DOI | Zbl

[18] Witten E., “Instantons, the quark model, and $1/N$ expansion”, Nucl. Phys. B, 149 (1979), 285–320 | DOI

[19] Simonov Yu. A., “Chiral Lagrangian with confinement from the QCD Lagrangian”, Phys. Rev. D, 65 (2002), 094018, 10 pp., ages ; arXiv:/hep-ph/0201170 | DOI

[20] Bernard V., Jaffe R. L., Meissner U.-G., “Flavor mixing via dynamical chiral symmetry breaking”, Phys. Lett. B, 198 (1987), 92–98 ; Bernard V., Jaffe R. L., Meissner U.-G., “Strangeness mixing and quenching in the Nambu–Jona-Lasinio model”, Nucl. Phys. B, 308 (1988), 753–790 | DOI | MR | DOI

[21] Reinhardt H., Alkofer R., “Instanton induced flavour mixing in mesons”, Phys. Lett. B, 207 (1988), 482–488 | DOI

[22] Osipov A. A., Hiller B., “Path integral bosonization of the 't Hooft determinant: quasiclassical corrections”, Eur. Phys. J. C, 35 (2004), 223–241 ; arXiv:hep-th/0307035 | DOI

[23] Klimt S., Lutz M., Vogl U., Weise W., “Generalized $SU(3)$ Nambu–Jona-Lasinio model. Part 1. Mesonic modes”, Nucl. Phys A, 516 (1990), 429–468 | DOI

[24] Bernard V., Blin A. H., Hiller B., Meißner U.-G., Ruivo M. C., “Strong and radiative meson decays in a generalized Nambu–Jona-Lasinio model”, Phys. Lett. B, 305 (1993), 163–167 ; ; Dmitrasinovic V., “$U_A(1)$ symmetry breaking, scalar mesons and the nucleon spin problem in an effective chiral field theory”, Nucl. Phys. A, 686 (2001), 379–392 ; arXiv:hep-ph/9302245arXiv:hep-ph/0010047 | DOI | MR | DOI

[25] Klevansky S. P., “The Nambu–Jona-Lasinio model of quantum chromodynamics”, Rev. Mod. Phys., 64 (1992), 649–708 ; Hatsuda T., Kunihiro T., “QCD Phenomenology based on a chiral effective Lagrangian”, Phys. Rept., 247 (1994), 221–367 ; arXiv:hep-ph/9401310 | DOI | MR | DOI

[26] Osipov A. A., Hiller B., Bernard V., Blin A. H., “Aspects of $U_A(1)$ breaking in the Nambu and Jona–Lasinio model”, Ann. of Phys. N.Y., 321:11 (2006), 2504–2534 ; arXiv:hep-ph/0507226 | DOI | Zbl

[27] Osipov A. A., Hiller B., da Providência J., “Multi-quark interactions with a globally stable vacuum”, Phys. Lett. B, 634 (2006), 48–54 ; arXiv:hep-ph/0508058 | DOI

[28] Coleman S., Weinberg E., “Radiative corrections as the origin of spontaneous symmetry breaking”, Phys. Rev. D, 7 (1973), 1888–1910 | DOI

[29] Okubo S., “Phi meson and unitary symmetry model”, Phys. Lett. B, 5 (1963), 165–168 ; Zweig G., An $SU(3)$ model for strong interaction symmetry and its breaking. 2, CERN Report No. 8419/TH412, 1964 ; Iizuka I., “A systematics and phenomenology of meson family”, Progr. Theor. Phys., 37–38, suppl. (1966), 21–34 | DOI | MR | Zbl | Zbl | DOI

[30] Witten E., “Current algebra theorems for the $U(1)$ “Goldstone boson””, Nucl. Phys. B, 156 (1979), 269–283 | DOI | MR

[31] Diakonov D., Chiral symmetry breaking by instantons, Lectures at the Enrico Fermi school in Physics (June 27 – July 7, 1995, Varenna)

[32] Dorokhov A. E., Zubov Y. A., Kochelev N. I., “Manifestations of QCD vacuum structure in composite quark models”, Sov. J. Part. Nucl., 23 (1992), 522–531; Shuryak E. V., “Correlation fuctions in the QCD vacuum”, Rev. Mod. Phys., 65 (1993), 1–46 | DOI

[33] Diakonov D., Petrov V., “A theory of light quarks in the instanton vacuum”, Nucl. Phys. B, 272 (1986), 457–489 ; Diakonov D., Petrov V., “Spontaneous breaking of chiral symmetry in the instanton vacuum”, Hadron Matter under Extreme Conditions, eds. G. Zinoviev and V. Shelest, Naukova Dumka, Kiev, 1986, 192–225 ; Diakonov D., Petrov V., Pobylitsa P., “A chiral theory of nucleons”, Nucl. Phys. B, 306 (1988), 809–848 ; Diakonov D., Petrov V., “Diquarks in the instanton picture”, Quark Cluster Dynamics, Lecture Notes in Physics, 417, eds. K. Goeke, P. Kroll and H.-R. Petry, Springer-Verlag, 1992, 288–297 | DOI | DOI

[34] Kerbikov B. O., Simonov Yu. A., Path integral bosonization of three flavor quark dynamics, arXiv:hep-th/9503146 | Zbl

[35] Osipov A. A., Hiller B., “Path integral bosonization of the 't Hooft determinant: fluctuations and multiple vacua”, Phys. Lett. B, 539 (2002), 76–84 ; arXiv:hep-ph/0204182 | DOI | Zbl

[36] Osipov A. A., Hiller B., “Effective chiral meson Lagrangian for the extended Nambu–Jona-Lasinio model”, Phys. Rev. D, 62 (2000), 114013, 10 pp., ages ; ; Osipov A. A., Hansen H., Hiller B., “Long distance expansion for the NJL model with $SU(3)$ and $U_A(1)$ breaking”, Nucl. Phys. A, 745 (2004), 81–103 ; arXiv:hep-ph/0007102arXiv:hep-ph/0406112 | DOI | DOI

[37] Weinberg S., “Effective theories in the large $N$ limit”, Phys. Rev. D, 56 (1997), 2303–2316 ; arXiv:hep-th/9706042 | DOI | MR

[38] Bjorken J., Drell S., Relativistic quantum mechanics, McGraw-Hill, New York, 1964, see in Chapter VIII, Section 8.2 | MR

[39] Weinberg S., The quantum theory of fields, Vol. 1, Cambridge University Press, 1995, see in Chapter IX, in the end of § 9.3

[40] Abers E. S., Lee B. W., “Gauge theories”, Phys. Rep., 9 (1973), 1–141, see equation (11.20) there | DOI

[41] Wolfram S., The Mathematica Book, 5th ed., Wolfram Media, 2003 | MR

[42] Osipov A. A., Hiller B., Moreira J., Blin A. H., “Stationary phase corrections in the process of bosonization of multi-quark interactions”, Europ. Phys. J. C, 46:1 (2006); arXiv:hep-ph/0601074