Two-Point Functions on Deformed Spacetime

Josip Trampetić ; Jiangyang You

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

We present a review of the one-loop photon $(\Pi)$ and neutrino $(\Sigma)$ two-point functions in a covariant and deformed $\rm U(1)$ gauge-theory on the 4-dimensional noncommutative spaces, determined by a constant antisymmetric tensor $\theta^{\mu\nu}$, and by a parameter-space $(\kappa_f,\kappa_g)$, respectively. For the general fermion-photon $S_f(\kappa_f)$ and photon self-interaction $S_g(\kappa_g)$ the closed form results reveal two-point functions with all kind of pathological terms: the UV divergence, the quadratic UV/IR mixing terms as well as a logarithmic IR divergent term of the type $\ln(\mu^2(\theta p)^2)$. In addition, the photon-loop produces new tensor structures satisfying transversality condition by themselves. We show that the photon two-point function in the 4-dimensional Euclidean spacetime can be reduced to two finite terms by imposing a specific full rank of $\theta^{\mu\nu}$ and setting deformation parameters $(\kappa_f,\kappa_g)=(0,3)$. In this case the neutrino two-point function vanishes. Thus for a specific point $(0,3)$ in the parameter-space $(\kappa_f,\kappa_g)$, a covariant $\theta$-exact approach is able to produce a divergence-free result for the one-loop quantum corrections, having also both well-defined commutative limit and point-like limit of an extended object.

Détail

Keywords: non-commutative geometry; photon and neutrino physics; non-perturbative effects.

Bibliographie

[1] Abel S., Chu C.-S., Goodsell M., “Noncommutativity from the string perspective: modification of gravity at a mm without mm sized extra dimensions”, J. High Energy Phys., 2006:11 (2006), 058, 27 pp., arXiv: hep-th/0606248 | DOI | MR

[2] Abel S. A., Jaeckel J., Khoze V. V., Ringwald A., “Vacuum birefringence as a probe of {P}lanck scale noncommutativity”, J. High Energy Phys., 2006:9 (2006), 074, 18 pp., arXiv: hep-ph/0607188 | DOI | MR

[3] Alboteanu A., Ohl T., Rückl R., “Probing the noncommutative standard model at hadron colliders”, Phys. Rev. D, 74 (2006), 096004, 9 pp., arXiv: hep-ph/0608155 | DOI

[4] Alboteanu A., Ohl T., Rückl R., “Noncommutative standard model at $\mathcal{O}(\theta^2)$”, Phys. Rev. D, 76 (2007), 105018, 10 pp., arXiv: 0707.3595 | DOI

[5] Alboteanu A., Ohl T., Rückl R., “The noncommutative standard model at colliders”, Acta Phys. Polon. B, 38 (2007), 3647–3651, arXiv: 0709.2359 | MR

[6] Álvarez-Gaumé L., Vázquez-Mozo M. A., “General properties of non-commutative field theories”, Nuclear Phys. B, 668 (2003), 293–321, arXiv: hep-th/0305093 | DOI | MR

[7] Amelino-Camelia G., Ellis J., Mavromatos N. E., Nanopoulos D. V., Sarkar S., “Tests of quantum gravity from observations of gamma-ray bursts”, Nature, 393 (1998), 763–765, arXiv: astro-ph/9712103 | DOI

[8] Arkani-Hamed N., Dimopoulos S., Dvali G., “The hierarchy problem and new dimensions at a millimeter”, Phys. Lett. B, 429 (1998), 263–272, arXiv: hep-ph/9803315 | DOI | MR

[9] Aschieri P., Jurčo B., Schupp P., Wess J., “Noncommutative {GUT}s, {S}tandard {M}odel and $C$, $P$, $T$”, Nuclear Phys. B, 651 (2003), 45–70, arXiv: hep-th/0205214 | DOI | MR | Zbl

[10] Banerjee R., Ghosh S., “Seiberg–{W}itten map and the axial anomaly in non-commutative field theory”, Phys. Lett. B, 533 (2002), 162–167, arXiv: hep-th/0110177 | DOI | MR | Zbl

[11] Barnich G., Brandt F., Grigoriev M., “Local {BRST} cohomology and {S}eiberg–{W}itten maps in noncommutative {Y}ang–{M}ills theory”, Nuclear Phys. B, 677 (2004), 503–534, arXiv: hep-th/0308092 | DOI | MR | Zbl

[12] Behr W., Deshpande N., Duplančić G., Schupp P., Trampetić J., Wess J., “The $Z \to \gamma \gamma, g g$ decays in the noncommutative standard model”, Eur. Phys. J. C Part. Fields, 29 (2003), 441–446, arXiv: hep-ph/0202121 | DOI

[13] Bichl A., Grimstrup J., Popp L., Schweda M., Grosse H., Wulkenhaar R., “Renormalization of the noncommutative photon self-energy to all orders via {S}eiberg–{W}itten map”, J. High Energy Phys., 2001:6 (2001), 013, 15 pp., arXiv: hep-th/0104097 | DOI | MR

[14] Bichl A. A., Grimstrup J. M., Popp L., Schweda M., Wulkenhaar R., “Perturbative analysis of the {S}eiberg–{W}itten map”, Internat. J. Modern Phys. A, 17 (2002), 2219–2231, arXiv: hep-th/0102044 | DOI | MR | Zbl

[15] Bigatti D., Susskind L., “Magnetic fields, branes, and noncommutative geometry”, Phys. Rev. D, 62 (2000), 066004, 6 pp., arXiv: hep-th/9908056 | DOI | MR

[16] Blaschke D. N., “A new approach to non-commutative ${\rm U}_\star(N)$ gauge fields”, Europhys. Lett., 91 (2010), 11001, 6 pp., arXiv: 1005.1578 | DOI

[17] Blaschke D. N., Grosse H., Kronberger E., Schweda M., Wohlgenannt M., “Loop calculations for the non-commutative ${\rm U}_\star(1)$ gauge field model with oscillator term”, Eur. Phys. J. C Part. Fields, 67 (2010), 575–582, arXiv: 0912.3642 | DOI

[18] Brandt F. T., Das A., Frenkel J., “General structure of the photon self-energy in noncommutative QED”, Phys. Rev. D, 65 (2002), 085017, 13 pp., arXiv: hep-th/0112127 | DOI | MR

[19] Burić M., Latas D., Nikolić B., Radovanović V., “The role of the Seiberg–Witten field redefinition in renormalization of noncommutative chiral electrodynamics”, Eur. Phys. J. C Part. Fields, 73 (2013), 2542, 10 pp., arXiv: 1304.4451 | DOI

[20] Burić M., Latas D., Radovanović V., Trampetić J., “Nonzero $Z \to\gamma\gamma$ decays in the renormalizable gauge sector of the noncommutative standard model”, Phys. Rev. D, 75 (2007), 097701, 4 pp., arXiv: hep-ph/0611299 | DOI

[21] Burić M., Latas D., Radovanović V., Trampetić J., “The absence of the $4\psi$ divergence in noncommutative chiral models”, Phys. Rev. D, 77 (2008), 045031, 7 pp., arXiv: 0711.0887 | DOI

[22] Burić M., Latas D., Radovanović V., Trampetić J., “Chiral fermions in noncommutative electrodynamics: renormalisability and dispersion”, Phys. Rev. D, 83 (2011), 045023, 10 pp., arXiv: 1009.4603 | DOI

[23] Burić M., Radovanović V., Trampetić J., “The one-loop renormalization of the gauge sector in the {$\theta$}-expanded noncommutative standard model”, J. High Energy Phys., 2007:3 (2007), 030, 17 pp., arXiv: hep-th/0609073 | DOI | MR

[24] Calmet X., Jurčo B., Schupp P., Wess J., Wohlgenannt M., “The standard model on non-commutative space-time”, Eur. Phys. J. C Part. Fields, 23 (2002), 363–376, arXiv: hep-ph/0111115 | DOI | MR | Zbl

[25] Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994 | MR | Zbl

[26] Douglas M. R., Nekrasov N. A., “Noncommutative field theory”, Rev. Modern Phys., 73 (2001), 977–1029, arXiv: hep-th/0106048 | DOI | MR | Zbl

[27] Ettefaghi M. M., Haghighat M., “Massive neutrino in noncommutative space-time”, Phys. Rev. D, 77 (2008), 056009, 8 pp., arXiv: 0712.4034 | DOI

[28] Filk T., “Divergencies in a field theory on quantum space”, Phys. Lett. B, 376 (1996), 53–58 | DOI | MR | Zbl

[29] Gomis J., Mehen T., “Space-time noncommutative field theories and unitarity”, Nuclear Phys. B, 591 (2000), 265–276, arXiv: hep-th/0005129 | DOI | MR | Zbl

[30] Grimstrup J. M., Wulkenhaar R., “Quantisation of {$\theta$}-expanded non-commutative {QED}”, Eur. Phys. J. C Part. Fields, 26 (2002), 139–151, arXiv: hep-th/0205153 | DOI | MR | Zbl

[31] Grosse H., Lechner G., “Noncommutative deformations of {W}ightman quantum field theories”, J. High Energy Phys., 2008:9 (2008), 131, 29 pp., arXiv: 0808.3459 | DOI | MR

[32] Grosse H., Wulkenhaar R., “Renormalisation of {$\phi^4$}-theory on noncommutative {${\mathbb R}^4$} in the matrix base”, Comm. Math. Phys., 256 (2005), 305–374, arXiv: hep-th/0401128 | DOI | MR | Zbl

[33] Grozin A. G., Lectures on perturbative HQET, I, arXiv: hep-ph/0008300

[34] Hayakawa M., “Perturbative analysis on infrared aspects of noncommutative {QED} on {${\mathbb R}^4$}”, Phys. Lett. B, 478 (2000), 394–400, arXiv: hep-th/9912094 | DOI | MR | Zbl

[35] Hayakawa M., Perturbative analysis on infrared and ultraviolet aspects of noncommutative QED on ${\mathbb R}^4$, arXiv: hep-th/9912167 | MR

[36] Hinchliffe I., Kersting N., Ma Y. L., “Review of the phenomenology of noncommutative geometry”, Internat. J. Modern Phys. A, 19 (2004), 179–204, arXiv: hep-ph/0205040 | DOI | MR | Zbl

[37] Horvat R., Ilakovac A., Kekez D., Trampetić J., You J., “Forbidden and invisible $Z$ boson decays in a covariant $\theta$-exact noncommutative standard model”, J. Phys. G: Nuclear Part. Phys., 41 (2014), 055007, 13 pp., arXiv: 1204.6201 | DOI

[38] Horvat R., Ilakovac A., Schupp P., Trampetić J., You J., “Neutrino propagation in noncommutative spacetimes”, J. High Energy Phys., 2012:4 (2012), 108, 28 pp., arXiv: 1111.4951 | DOI

[39] Horvat R., Ilakovac A., Schupp P., Trampetić J., You J., “Yukawa couplings and seesaw neutrino masses in noncommutative gauge theory”, Phys. Lett. B, 715 (2012), 340–347, arXiv: 1109.3085 | DOI

[40] Horvat R., Ilakovac A., Trampetić J., You J., “On {UV}/{IR} mixing in noncommutative gauge field theories”, J. High Energy Phys., 2011:12 (2011), 081, 11 pp. | DOI | MR

[41] Horvat R., Ilakovac A., Trampetić J., You J., “Self-energies on deformed spacetimes”, J. High Energy Phys., 2013:11 (2013), 071, 27 pp., arXiv: 1306.1239 | DOI

[42] Horvat R., Kekez D., Schupp P., Trampetić J., You J., “Photon-neutrino interaction in $\theta$-exact covariant noncommutative field theory”, Phys. Rev. D, 84 (2011), 045004, 10 pp., arXiv: 1103.3383 | DOI

[43] Horvat R., Kekez D., Trampetić J., “Spacetime noncommutativity and ultra-high energy cosmic ray experiments”, Phys. Rev. D, 83 (2011), 065013, 5 pp., arXiv: 1005.3209 | DOI | MR

[44] Horvat R., Trampetić J., “Constraining noncommutative field theories with holography”, J. High Energy Phys., 2011:1 (2011), 112, 8 pp., arXiv: 1009.2933 | DOI | MR | Zbl

[45] Jackiw R., Pi S.-Y., “Covariant coordinate transformations on noncommutative space”, Phys. Rev. Lett., 88 (2002), 111603, 4 pp., arXiv: hep-th/0111122 | DOI | MR

[46] Jaeckel J., Khoze V. V., Ringwald A., “Telltale traces of {${\rm U}(1)$} fields in noncommutative standard model extensions”, J. High Energy Phys., 2006:2 (2006), 028, 21 pp., arXiv: hep-ph/0508075 | DOI | MR

[47] Jurčo B., Schupp P., Wess J., “Nonabelian noncommutative gauge theory via noncommutative extra dimensions”, Nuclear Phys. B, 604 (2001), 148–180, arXiv: hep-th/0102129 | DOI | MR | Zbl

[48] Latas D., Radovanović V., Trampetić J., “Noncommutative {${\rm SU}(N)$} gauge theory and asymptotic freedom”, Phys. Rev. D, 76 (2007), 085006, 7 pp., arXiv: hep-th/0703018 | DOI | MR

[49] Madore J., An introduction to noncommutative differential geometry and its physical applications, London Mathematical Society Lecture Note Series, 257, 2nd ed., Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl

[50] Magnen J., Rivasseau V., Tanasa A., “Commutative limit of a renormalizable noncommutative model”, Europhys. Lett., 86 (2009), 11001, 6 pp., arXiv: 0807.4093 | DOI

[51] Martín C. P., “The gauge anomaly and the {S}eiberg–{W}itten map”, Nuclear Phys. B, 652 (2003), 72–92, arXiv: hep-th/0211164 | DOI | MR | Zbl

[52] Martín C. P., “The minimal and the new minimal supersymmetric grand unified theories on noncommutative space-time”, Classical Quantum Gravity, 30 (2013), 155019, 15 pp., arXiv: 1302.3732 | DOI | MR | Zbl

[53] Martín C. P., Sánchez-Ruiz D., “One-loop {UV} divergent structure of {${\rm U}(1)$} {Y}ang–{M}ills theory on noncommutative {${\mathbb R}^4$}”, Phys. Rev. Lett., 83 (1999), 476–479, arXiv: hep-th/9903077 | DOI | MR | Zbl

[54] Martín C. P., Tamarit C., “Noncommutative GUT inspired theories and the UV finiteness of the fermionic four point functions”, Phys. Rev. D, 80 (2009), 065023, 6 pp., arXiv: 0907.2464 | DOI

[55] Martín C. P., Tamarit C., “Renormalisability of noncommutative GUT inspired field theories with anomaly safe groups”, J. High Energy Phys., 2009:12 (2009), 042, 18 pp., arXiv: 0910.2677 | DOI

[56] Matusis A., Susskind L., Toumbas N., “The {IR}/{UV} connection in non-commutative gauge theories”, J. High Energy Phys., 2000:12 (2000), 002, 18 pp., arXiv: hep-th/0002075 | DOI | MR | Zbl

[57] Mehen T., Wise M. B., “Generalized {$\star$}-products, {W}ilson lines and the solution of the {S}eiberg–{W}itten equations”, J. High Energy Phys., 2000:12 (2000), 008, 10 pp., arXiv: hep-th/0010204 | DOI | MR | Zbl

[58] Meljanac S., Samsarov A., Trampetić J., Wohlgenannt M., “Scalar field propagation in the $\phi^4$ $\kappa$-Minkowski model”, J. High Energy Phys., 2011:12 (2011), 010, 23 pp., arXiv: 1111.5553 | DOI | MR

[59] Minkowski P., Schupp P., Trampetić J., “Neutrino dipole moments and charge radii in non-commutative space-time”, Eur. Phys. J. C Part. Fields, 37 (2004), 123–128, arXiv: hep-th/0302175 | DOI

[60] Minwalla S., Van Raamsdonk M., Seiberg N., “Noncommutative perturbative dynamics”, J. High Energy Phys., 2000:2 (2000), 020, 31 pp., arXiv: hep-th/9912072 | DOI | MR

[61] Ohl T., Reuter J., “Testing the noncommutative standard model at a future photon collider”, Phys. Rev. D, 70 (2004), 076007, 10 pp., arXiv: hep-ph/0406098 | DOI | MR

[62] Okawa Y., Ooguri H., “Exact solution to the {S}eiberg–{W}itten equation of noncommutative gauge theory”, Phys. Rev. D, 64 (2001), 046009, 11 pp., arXiv: hep-th/0104036 | DOI | MR

[63] Rovelli C., “Loop quantum gravity: the first 25 years”, Classical Quantum Gravity, 28 (2011), 153002, 35 pp., arXiv: 1012.4707 | DOI | MR | Zbl

[64] Rovelli C., Zakopane lectures on loop gravity, PoS Proc. Sci., QGQGS 2011_003, 2011, 60 pp., arXiv: 1102.3660

[65] Schupp P., Trampetić J., Wess J., Raffelt G., “The photon-neutrino interaction induced by non-commutativity and astrophysical bounds”, Eur. Phys. J. C Part. Fields, 36 (2004), 405–410, arXiv: hep-ph/0212292 | DOI

[66] Schupp P., You J., “U{V}/{IR} mixing in noncommutative {QED} defined by {S}eiberg–{W}itten map”, J. High Energy Phys., 2008:8 (2008), 107, 10 pp., arXiv: 0807.4886 | DOI | MR

[67] Seiberg N., Witten E., “String theory and noncommutative geometry”, J. High Energy Phys., 1999:9 (1999), 032, 93 pp., arXiv: hep-th/9908142 | DOI | MR

[68] Susskind L., Lindesay J., An introduction to black holes, information and the string theory revolution. The holographic universe, World Scientific Publishing Co. Inc., Hackensack, NJ, 2005 | MR | Zbl

[69] Susskind L., Thorlacius L., Uglum J., “The stretched horizon and black hole complementarity”, Phys. Rev. D, 48 (1993), 3743–3761, arXiv: hep-th/9306069 | DOI | MR

[70] Szabo R. J., “Quantum field theory on noncommutative spaces”, Phys. Rep., 378 (2003), 207–299, arXiv: hep-th/0109162 | DOI | MR | Zbl

[71] Szabo R. J., “Quantum gravity, field theory and signatures of noncommutative spacetime”, Gen. Relativity Gravitation, 42 (2010), 1–29, arXiv: 0906.2913 | DOI | MR

[72] 't Hooft G., “On the quantum structure of a black hole”, Nuclear Phys. B, 256 (1985), 727–745 | DOI | MR

[73] Trampetić J., “Signal for space-time noncommutativity: the $Z \to \gamma \gamma$ decay in the renormalizable gauge sector of the $\theta$-expanded NCSM”, Modern Mathematical Physics, Proceedings of 4th Summer School (September 3–14, 2006, Belgrade, Serbia), eds. B. Dragovich, Z. Rakić, Institute of Physics, Belgrade, Serbia, 2007, 379–390, arXiv: 0704.0559

[74] Trampetić J., “Renormalizability and phenomenology of {$\theta$}-expanded noncommutative gauge field theory”, Fortschr. Phys., 56 (2008), 521–531, arXiv: 0802.2030 | DOI | MR | Zbl

[75] Zeiner J., Noncommutative quantumelectrodynamics from Seiberg–Witten maps to all orders in $\theta^{\mu \nu}$, Ph.D. Thesis, Julius-Maximilians-Universität Würzburg, 2007