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U(\(N\)) Yang-Mills in non-commutative space time. (English) Zbl 1415.83031
Summary: We present an approach to \( \mathrm{U}_{\ast}(N) \) Yang-Mills theory in non-commutative space based upon a novel phase-space analysis of the dynamical fields with additional auxiliary variables that generate Lorentz structure and colour degrees of freedom. To illustrate this formalism we compute the quadratic terms in the effective action focusing on the planar divergences so as to extract the \({\beta}\)-function for the Yang-Mills coupling constant. Nonetheless the method presented is general and can be applied to calculate the effective action at arbitrary order of expansion in the coupling constant and is well suited to the computation of low energy one-loop scattering amplitudes.
MSC:
83C65 Methods of noncommutative geometry in general relativity
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
81T10 Model quantum field theories
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References:
[1] Douglas, MR; Nekrasov, NA, Noncommutative field theory, Rev. Mod. Phys., 73, 977, (2001) · Zbl 1205.81126
[2] Szabo, RJ, Quantum field theory on noncommutative spaces, Phys. Rept., 378, 207, (2003) · Zbl 1042.81581
[3] M.M. Faruk, M. Al Alvi, W. Ahmed, M.M. Rahman and A.B. Apu, Noncommutative scalar fields in compact spaces: quantization and implications, PTEP2017 (2017) 093B02 [arXiv:1707.01643] [INSPIRE].
[4] Connes, A.; Douglas, MR; Schwarz, AS, Noncommutative geometry and matrix theory: Compactification on tori, JHEP, 02, 003, (1998) · Zbl 1018.81052
[5] A.Y. Alekseev, A. Recknagel and V. Schomerus, Noncommutative world volume geometries: Branes on SU(2) and fuzzy spheres, JHEP09 (1999) 023 [hep-th/9908040] [INSPIRE].
[6] Seiberg, N.; Witten, E., String theory and noncommutative geometry, JHEP, 09, 032, (1999) · Zbl 0957.81085
[7] Blumenhagen, R., A Course on Noncommutative Geometry in String Theory, Fortsch. Phys., 62, 709, (2014) · Zbl 1338.81314
[8] Doplicher, S.; Fredenhagen, K.; Roberts, JE, The Quantum structure of space-time at the Planck scale and quantum fields, Commun. Math. Phys., 172, 187, (1995) · Zbl 0847.53051
[9] Minwalla, S.; Raamsdonk, M.; Seiberg, N., Noncommutative perturbative dynamics, JHEP, 02, 020, (2000) · Zbl 0959.81108
[10] R. Bonezzi, O. Corradini, S.A. Franchino Vinas and P.A.G. Pisani, Worldline approach to noncommutative field theory, J. Phys.A 45 (2012) 405401 [arXiv:1204.1013] [INSPIRE]. · Zbl 1255.81219
[11] Y. Kiem, S.-J. Rey, H.-T. Sato and J.-T. Yee, Anatomy of one loop effective action in noncommutative scalar field theories, Eur. Phys. J.C 22 (2002) 757 [hep-th/0107106] [INSPIRE].
[12] N. Ahmadiniaz, O. Corradini, D. D’Ascanio, S. Estrada-Jiménez and P. Pisani, Noncommutative U(1) gauge theory from a worldline perspective, JHEP11 (2015) 069 [arXiv:1507.07033] [INSPIRE].
[13] Y.-j. Kiem, Y.-j. Kim, C. Ryou and H.-T. Sato, One loop noncommutative U(1) gauge theory from bosonic worldline approach, Nucl. Phys.B 630 (2002) 55 [hep-th/0112176] [INSPIRE]. · Zbl 0994.81067
[14] Bern, Z.; Kosower, DA, Efficient calculation of one loop QCD amplitudes, Phys. Rev. Lett., 66, 1669, (1991)
[15] M.J. Strassler, Field theory without Feynman diagrams: One loop effective actions, Nucl. Phys.B 385 (1992) 145 [hep-ph/9205205] [INSPIRE].
[16] Schubert, C., Perturbative quantum field theory in the string inspired formalism, Phys. Rept., 355, 73, (2001) · Zbl 0988.81108
[17] M.B. Halpern and W. Siegel, The Particle Limit of Field Theory: A New Strong Coupling Expansion, Phys. Rev.D 16 (1977) 2486 [INSPIRE].
[18] M.B. Halpern, A. Jevicki and P. Senjanovic, Field Theories in Terms of Particle-String Variables: Spin, Internal Symmetries and Arbitrary Dimension, Phys. Rev.D 16 (1977) 2476 [INSPIRE].
[19] H.-T. Sato and M.G. Schmidt, Worldline approach to the Bern-Kosower formalism in two loop Yang-Mills theory, Nucl. Phys.B 560 (1999) 551 [hep-th/9812229] [INSPIRE].
[20] P. Dai, Y.-t. Huang and W. Siegel, Worldgraph Approach to Yang-Mills Amplitudes from N = 2 Spinning Particle, JHEP10 (2008) 027 [arXiv:0807.0391] [INSPIRE].
[21] M. Reuter, M.G. Schmidt and C. Schubert, Constant external fields in gauge theory and the spin 0\(,\) 1\(/\)2\(,\) 1 path integrals, Annals Phys.259 (1997) 313 [hep-th/9610191] [INSPIRE].
[22] N. Ahmadiniaz, F. Bastianelli and O. Corradini, Dressed scalar propagator in a non-Abelian background from the worldline formalism, Phys. Rev.D 93 (2016) 025035 [arXiv:1508.05144] [INSPIRE].
[23] Corradini, O.; Edwards, JP, Mixed symmetry tensors in the worldline formalism, JHEP, 05, 056, (2016) · Zbl 1388.83210
[24] Edwards, JP; Corradini, O., Mixed symmetry Wilson-loop interactions in the worldline formalism, JHEP, 09, 081, (2016) · Zbl 1390.81589
[25] P. Mansfield, The fermion content of the Standard Model from a simple world-line theory, Phys. Lett.B 743 (2015) 353 [arXiv:1410.7298] [INSPIRE].
[26] J.P. Edwards, Unified theory in the worldline approach, Phys. Lett.B 750 (2015) 312 [arXiv:1411.6540] [INSPIRE].
[27] J.P. Edwards and O. Corradini, Worldline colour fields and non-Abelian quantum field theory, EPJ Web Conf.182 (2018) 02038 [arXiv:1711.09314] [INSPIRE].
[28] T. Krajewski and R. Wulkenhaar, Perturbative quantum gauge fields on the noncommutative torus, Int. J. Mod. Phys.A 15 (2000) 1011 [hep-th/9903187] [INSPIRE]. · Zbl 0963.81055
[29] C.P. Martin and D. Sánchez-Ruiz, The one loop UV divergent structure of U(1) Yang-Mills theory on noncommutative R4, Phys. Rev. Lett.83 (1999) 476 [hep-th/9903077] [INSPIRE].
[30] Moyal, JE, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc., 45, 99, (1949) · Zbl 0031.33601
[31] Groenewold, HJ, On the Principles of elementary quantum mechanics, Physica, 12, 405, (1946) · Zbl 0060.45002
[32] L. Álvarez-Gaumé and S.R. Wadia, Gauge theory on a quantum phase space, Phys. Lett.B 501 (2001) 319 [hep-th/0006219] [INSPIRE].
[33] A. Iskauskas, A Remark on Polar Noncommutativity, Phys. Lett.B 746 (2015) 25 [arXiv:1503.03684] [INSPIRE].
[34] J.P. Edwards, Non-commutativity in polar coordinates, Eur. Phys. J.C 77 (2017) 320 [arXiv:1607.04491] [INSPIRE].
[35] E. Chang-Young, D. Lee and Y. Lee, Noncommutative BTZ Black Hole in Polar Coordinates, Class. Quant. Grav.26 (2009) 185001 [arXiv:0808.2330] [INSPIRE].
[36] A.F. Ferrari et al., Towards a consistent noncommutative supersymmetric Yang-Mills theory: Superfield covariant analysis, Phys. Rev.D 70 (2004) 085012 [hep-th/0407040] [INSPIRE].
[37] B.S. DeWitt, Quantum Theory of Gravity. 2. The Manifestly Covariant Theory, Phys. Rev.162 (1967) 1195 [INSPIRE]. · Zbl 0161.46501
[38] L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon.B 13 (1982) 33 [INSPIRE].
[39] L.F. Abbott, M.T. Grisaru and R.K. Schaefer, The Background Field Method and the S Matrix, Nucl. Phys.B 229 (1983) 372 [INSPIRE].
[40] Ahmadiniaz, N.; Schubert, C.; Villanueva, VM, String-inspired representations of photon/gluon amplitudes, JHEP, 01, 132, (2013)
[41] N. Ahmadiniaz and C. Schubert, A covariant representation of the Ball-Chiu vertex, Nucl. Phys.B 869 (2013) 417 [arXiv:1210.2331] [INSPIRE]. · Zbl 1262.81185
[42] N. Ahmadiniaz and C. Schubert, Gluon form factor decompositions from the worldline formalism, PoS(LL2016)052 (2016) [INSPIRE].
[43] N. Ahmadiniaz and C. Schubert, QCD gluon vertices from the string-inspired formalism, Int. J. Mod. Phys.E 25 (2016) 1642004 [arXiv:1811.10780] [INSPIRE].
[44] Schwinger, JS, On gauge invariance and vacuum polarization, Phys. Rev., 82, 664, (1951) · Zbl 0043.42201
[45] Bastianelli, F.; Corradini, O.; Zirotti, A., BRST treatment of zero modes for the worldline formalism in curved space, JHEP, 01, 023, (2004) · Zbl 1243.81211
[46] O. Corradini and M. Muratori, String-inspired Methods and the Worldline Formalism in Curved Space, Eur. Phys. J. Plus133 (2018) 457 [arXiv:1808.05401] [INSPIRE].
[47] F. Bastianelli and P. van Nieuwenhuizen, Path integrals and anomalies in curved space, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2006) [INSPIRE]. · Zbl 1120.81057
[48] Bastianelli, F.; Bonezzi, R.; Corradini, O.; Latini, E., Particles with non abelian charges, JHEP, 10, 098, (2013) · Zbl 1342.81660
[49] F. Bastianelli, R. Bonezzi, O. Corradini, E. Latini and K.H. Ould-Lahoucine, A worldline approach to colored particles, 2015, arXiv:1504.03617 [INSPIRE].
[50] Bastianelli, F.; Corradini, O.; Latini, E., Higher spin fields from a worldline perspective, JHEP, 02, 072, (2007)
[51] Bastianelli, F.; Bonezzi, R.; Corradini, O.; Latini, E., Massive and massless higher spinning particles in odd dimensions, JHEP, 09, 158, (2014)
[52] Howe, PS; Penati, S.; Pernici, M.; Townsend, PK, A Particle Mechanics Description of Antisymmetric Tensor Fields, Class. Quant. Grav., 6, 1125, (1989)
[53] Bastianelli, F.; Benincasa, P.; Giombi, S., Worldline approach to vector and antisymmetric tensor fields, JHEP, 04, 010, (2005)
[54] F. Bastianelli, P. Benincasa and S. Giombi, Worldline approach to vector and antisymmetric tensor fields. II., JHEP10 (2005) 114 [hep-th/0510010] [INSPIRE].
[55] A. Barducci, R. Casalbuoni and L. Lusanna, Anticommuting Variables, Internal Degrees of Freedom and the Wilson Loop, Nucl. Phys.B 180 (1981) 141 [INSPIRE].
[56] Bastianelli, F.; Corradini, O.; Latini, E., Spinning particles and higher spin fields on (A)dS backgrounds, JHEP, 11, 054, (2008)
[57] Corradini, O., Half-integer Higher Spin Fields in (A)dS from Spinning Particle Models, JHEP, 09, 113, (2010) · Zbl 1291.81238
[58] Bastianelli, F.; Bonezzi, R.; Corradini, O.; Latini, E., Effective action for higher spin fields on (A)dS backgrounds, JHEP, 12, 113, (2012) · Zbl 1397.81189
[59] R. Bonezzi, Induced Action for Conformal Higher Spins from Worldline Path Integrals, Universe3 (2017) 64 [arXiv:1709.00850] [INSPIRE].
[60] F. Bastianelli and R. Bonezzi, U(\(N\)) spinning particles and higher spin equations on complex manifolds, JHEP03 (2009) 063 [arXiv:0901.2311] [INSPIRE].
[61] F. Bastianelli and R. Bonezzi, U(\(N\) |\(M\)) Quantum Mechanics on Kähler Manifolds, JHEP05 (2010) 020 [arXiv:1003.1046] [INSPIRE].
[62] F. Bastianelli and R. Bonezzi, Quantum theory of massless (p, 0)-forms, JHEP09 (2011) 018 [arXiv:1107.3661] [INSPIRE]. · Zbl 1301.81101
[63] F. Bastianelli, R. Bonezzi and C. Iazeolla, Quantum theories of (p, q)-forms, JHEP08 (2012) 045 [arXiv:1204.5954] [INSPIRE]. · Zbl 1397.81132
[64] M. Reuter, Metaplectic spinor fields and global anomalies, Int. J. Mod. Phys.A 10 (1995) 65 [INSPIRE].
[65] J.P. Edwards and C. Schubert, Quantum Mechanical Path Integrals in the First Quantised Approach to Quantum Field Theory, in preparation (2018).
[66] F. Bastianelli, O. Corradini and A. Zirotti, dimensional regularization for N = 1 supersymmetric σ-models and the worldline formalism, Phys. Rev.D 67 (2003) 104009 [hep-th/0211134] [INSPIRE].
[67] W. Bietenholz, J. Nishimura, Y. Susaki and J. Volkholz, A Non-perturbative study of 4-D U(1) non-commutative gauge theory: The Fate of one-loop instability, JHEP10 (2006) 042 [hep-th/0608072] [INSPIRE].
[68] A. Armoni, Comments on perturbative dynamics of noncommutative Yang-Mills theory, Nucl. Phys.B 593 (2001) 229 [hep-th/0005208] [INSPIRE].
[69] R. Fresneda, D.M. Gitman and A.E. Shabad, Photon propagation in noncommutative QED with constant external field, Phys. Rev.D 91 (2015) 085005 [arXiv:1501.04987] [INSPIRE].
[70] D. D’Ascanio, P. Pisani and D.V. Vassilevich, Renormalization on noncommutative torus, Eur. Phys. J.C 76 (2016) 180 [arXiv:1602.01479] [INSPIRE].
[71] Sheikh-Jabbari, MM, Renormalizability of the supersymmetric Yang-Mills theories on the noncommutative torus, JHEP, 06, 015, (1999) · Zbl 0961.81123
[72] H. Grosse and R. Wulkenhaar, Renormalization of phi**4 theory on noncommutative R2in the matrix base, JHEP12 (2003) 019 [hep-th/0307017] [INSPIRE].
[73] E. Langmann, R.J. Szabo and K. Zarembo, Exact solution of noncommutative field theory in background magnetic fields, Phys. Lett.B 569 (2003) 95 [hep-th/0303082] [INSPIRE]. · Zbl 1059.81608
[74] Langmann, E.; Szabo, RJ; Zarembo, K., Exact solution of quantum field theory on noncommutative phase spaces, JHEP, 01, 017, (2004) · Zbl 1243.81205
[75] A. de Goursac, J.-C. Wallet and R. Wulkenhaar, Noncommutative Induced Gauge Theory, Eur. Phys. J.C 51 (2007) 977 [hep-th/0703075] [INSPIRE].
[76] H. Grosse and M. Wohlgenannt, Induced gauge theory on a noncommutative space, Eur. Phys. J.C 52 (2007) 435 [hep-th/0703169] [INSPIRE]. · Zbl 1189.81217
[77] J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J.C 16 (2000) 161 [hep-th/0001203] [INSPIRE].
[78] Goursac, A.; Masson, T.; Wallet, J-C, Noncommutative epsilon-graded connections, J. Noncommut. Geom., 6, 343, (2012) · Zbl 1275.58003
[79] D.N. Blaschke, H. Grosse, E. Kronberger, M. Schweda and M. Wohlgenannt, Loop Calculations for the Non-Commutative U_{∗}(1) Gauge Field Model with Oscillator Term, Eur. Phys. J.C 67 (2010) 575 [arXiv:0912.3642] [INSPIRE].
[80] F. Bastianelli and A. Zirotti, Worldline formalism in a gravitational background, Nucl. Phys.B 642 (2002) 372 [hep-th/0205182] [INSPIRE]. · Zbl 0998.81064
[81] F. Bastianelli, The Path integral for a particle in curved spaces and Weyl anomalies, Nucl. Phys.B 376 (1992) 113 [hep-th/9112035] [INSPIRE].
[82] M.G. Schmidt and C. Schubert, Worldline Green functions for multiloop diagrams, Phys. Lett.B 331 (1994) 69 [hep-th/9403158] [INSPIRE].
[83] N. Chair and M.M. Sheikh-Jabbari, Pair production by a constant external field in noncommutative QED, Phys. Lett.B 504 (2001) 141 [hep-th/0009037] [INSPIRE].
[84] Riad, I.; Sheikh-Jabbari, MM, Noncommutative QED and anomalous dipole moments, JHEP, 08, 045, (2000) · Zbl 0989.81126
[85] Ilderton, A.; Lundin, J.; Marklund, M., Strong Field, Noncommutative QED, SIGMA, 6, 041, (2010) · Zbl 1217.81138
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