Abstract
We calculate the energy spectrum of ac onfining flux tube that is closed around a spatial torus, as a function of its length l. We do so for various SU(N) gauge theories in 3+ 1 dimensions, and for various values of spin, parity and longitudinal momentum. We are able to present usefully accurate results for about 20 of the lightest such states, for a range of l that begins close to the (finite volume) deconfining phase transition at l√σ ∼1.6, and extends up to l√σ ∼6 (where σ is the string tension). We find that most of these low-lying states are well described by the spectrum of the Nambu-Goto free string theory in flat space-time. Remarkably, this is so not only at the larger values of l, where the gap between the ground state energy and the low-lying excitations becomes small compared to the mass gap, but also down to much shorter lengths where these excitation energies become large compared to √σ, the flux-tube no longer ‘looks’ anything like a thin string, and an expansion of the effective string action in powers of 1/l no longer converges. All this is for flux in the fundamental representation. We also calculate the k = 2 (anti)symmetric ground states and these show larger corrections at small l. So far all this closely resembles our earlier findings in 2+ 1 dimensions. However, and in contrast to the situation in D = 2+ 1, we also find that there are some states, with J P=0−quantum numbers, that show large deviations from the Nambu-Goto spectrum. We investigate the possibility that (some of) these states may encode the massive modes associated with the internal structure of the flux tube, and we discuss how the precocious free string behaviour of most states constrains the effective string action, on which much interesting theoretical progress has recently been made.
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References
P. Goddard, From dual models to string theory, arXiv:0802.3249 [SPIRES].
R.C. Brower, String/gauge duality: (re)discovering the QCD string in AdS space, Acta Phys. Polon. B 34 (2003) 5927 [hep-th/0508036] [SPIRES].
G. ’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B 72 (1974) 461 [SPIRES].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [SPIRES].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [SPIRES].
M. Lüscher and P. Weisz, String excitation energies in SU(N) gauge theories beyond the free-string approximation, JHEP 07 (2004) 014 [hep-th/0406205] [SPIRES].
J.M. Drummond, Universal subleading spectrum of effective string theory, hep-th/0411017 [SPIRES].
J.M. Drummond, Reply to hep-th/0606265, hep-th/0608109 [SPIRES].
N.D. Hari Dass and P. Matlock, Universality of correction to Luescher term in Polchinski-Strominger effective string theories, hep-th/0606265 [SPIRES].
N.D. Hari Dass and P. Matlock, Field definitions, spectrum and universality in effective string theories, hep-th/0612291 [SPIRES].
N.D. Hari Dass and P. Matlock, Covariant calculus for effective string theories, arXiv:0709.1765 [SPIRES].
O. Aharony and E. Karzbrun, On the effective action of confining strings, JHEP 06 (2009) 012 [arXiv:0903.1927] [SPIRES].
N.D. Hari Dass, P. Matlock and Y. Bharadwaj, Spectrum to all orders of Polchinski-Strominger effective string theory of Polyakov-Liouville type, arXiv:0910.5615 [SPIRES].
N.D. Hari Dass and Y. Bharadwaj, Spectrum to all orders of Polchinski-Strominger effective string theories of the Drummond type, arXiv:0910.5620 [SPIRES].
N.D.H. Dass, All conformal effective string theories are isospectral to Nambu-Goto theory, arXiv:0911.3236 [SPIRES].
B. Bringoltz and M. Teper, A precise calculation of the fundamental string tension in SU(N) gauge theories in 2+1 dimensions, Phys. Lett. B 645 (2007) 383 [hep-th/0611286] [SPIRES].
A. Athenodorou, B. Bringoltz and M. Teper, The closed string spectrum of SU(N) gauge theories in 2+1 dimensions, Phys. Lett. B 656 (2007) 132 [arXiv:0709.0693] [SPIRES].
O. Aharony, presented at Confining flux tubes and strings, ECT, Trento Italy (2010).
A. Athenodorou, B. Bringoltz and M. Teper, The spectrum of closed loops of fundamental flux in D =3+1 SU(N) gauge theories, PoS LAT 2009 (2009) 223[arXiv:0912.3238] [SPIRES].
H.B. Meyer, Poincaré invariance in effective string theories, JHEP 05 (2006) 066 [hep-th/0602281] [SPIRES].
P. Olesen, Strings and QCD, Phys. Lett. B 160 (1985) 144 [SPIRES].
J. Polchinski and A. Strominger, Effective string theory, Phys. Rev. Lett. 67 (1991) 1681 [SPIRES].
M. Lüscher, K. Symanzik and P. Weisz, Anomalies of the free loop wave equation in the WKB approximation, Nucl. Phys. B 173 (1980) 365 [SPIRES].
M. Lüscher, Symmetry breaking aspects of the roughening transition in gauge theories, Nucl. Phys. B 180 (1981) 317 [SPIRES].
J.F. Arvis, The exact \( q\overline q \) potential in Nambu string theory, Phys. Lett. B 127 (1983) 106 [SPIRES].
B. Lucini, M. Teper and U. Wenger, The deconfinement transition in SU(N) gauge theories, Phys. Lett. B 545 (2002) 197 [hep-lat/0206029] [SPIRES].
B. Lucini, M. Teper and U. Wenger, The high temperature phase transition in SU(N) gauge theories, JHEP 01 (2004) 061 [hep-lat/0307017] [SPIRES].
B. Lucini, M. Teper and U. Wenger, Properties of the deconfining phase transition in SU(N) gauge theories, JHEP 02 (2005) 033 [hep-lat/0502003] [SPIRES].
H.B. Meyer and M.J. Teper, GlueballRegge trajectories and the pomeron: a lattice study, Phys. Lett. B 605 (2005) 344 [hep-ph/0409183] [SPIRES].
H.B. Meyer and M.J. Teper, GlueballRegge trajectories in (2+1) dimensional gauge theories, Nucl. Phys. B 668 (2003) 111 [hep-lat/0306019] [SPIRES].
Ape collaboration, M. Albanese et al., Glueball masses and string tension in lattice QCD, Phys. Lett. B 192 (1987) 163 [SPIRES].
Ape collaboration, M. Albanese et al., Glueball masses and the loop loop correlation functions, Phys. Lett. B 197 (1987) 400 [SPIRES].
M. Teper, An improved method for lattice glueball calculations, Phys. Lett. B 183 (1987) 345 [SPIRES].
M. Teper, The scalar and tensor glueball masses in lattice gauge theory, Phys. Lett. B 185 (1987) 121 [SPIRES].
H.B. Meyer, GlueballRegge trajectories, hep-lat/0508002 [SPIRES].
S. Perantonis and C. Michael, Static potentials and hybrid mesons from pure SU(3) lattice gauge theory, Nucl. Phys. B 347 (1990) 854 [SPIRES].
C. Michael, Adiabatic surfaces from the lattice: Excited gluonic potentials, hep-ph/9809211 [SPIRES].
M. Caselle, R. Fiore, F. Gliozzi, M. Hasenbusch and P. Provero, String effects in the Wilson loop: A high precision numerical test, Nucl. Phys. B 486 (1997) 245 [hep-lat/9609041] [SPIRES].
M. Caselle, F. Gliozzi, U. Magnea and S. Vinti, Width of long colour flux tubes in lattice gauge systems, Nucl. Phys. B 460 (1996) 397 [hep-lat/9510019] [SPIRES].
K.J. Juge, J. Kuti, F. Maresca, C. Morningstar and M.J. Peardon, Excitations of torelon, Nucl. Phys. Proc. Suppl. 129 (2004) 703 [hep-lat/0309180] [SPIRES].
K.J. Juge, J. Kuti and C. Morningstar, Fine structure of the QCD string spectrum, Phys. Rev. Lett. 90 (2003) 161601 [hep-lat/0207004] [SPIRES].
K.J. Juge, J. Kuti and C. Morningstar, QCD string formation and the Casimir energy, hep-lat/0401032 [SPIRES].
J. Kuti, Lattice QCD and string theory, PoS LAT 2005 (2006) 001 [hep-lat/0511023] [SPIRES].
H. Meyer and M. Teper, Confinement and the effective string theory in SU(N→∞): A lattice study, JHEP 12 (2004) 031 [hep-lat/0411039] [SPIRES].
M. Lüscher and P. Weisz, Quark confinement and the bosonic string, JHEP 07 (2002) 049 [hep-lat/0207003] [SPIRES].
B. Lucini and M. Teper, Confining strings in SU(N) gauge theories, Phys. Rev. D 64 (2001) 105019 [hep-lat/0107007] [SPIRES].
B. Lucini, M. Teper and U. Wenger, Glueballs and k-strings in SU(N) gauge theories: Calculations with improved operators, JHEP 06 (2004) 012 [hep-lat/0404008] [SPIRES].
B. Lucini and M. Teper, SU(N) gauge theories in four dimensions: Exploring the approach to N = ∞, JHEP 06 (2001) 050 [hep-lat/0103027] [SPIRES].
A. Athenodorou, B. Bringoltz and M. Teper, On the spectrum of closed k=2 flux tubes in D =2+1 SU(N) gauge theories, JHEP 05 (2009) 019 [arXiv:0812.0334] [SPIRES].
B. Lucini and M. Teper, The k = 2 string tension in four dimensional SU(N) gauge theories, Phys. Lett. B 501 (2001) 128 [hep-lat/0012025] [SPIRES].
L. Del Debbio, H. Panagopoulos, P. Rossi and E. Vicari, k-string tensions in SU(N) gauge theories, Phys. Rev. D 65 (2002) 021501 [hep-th/0106185] [SPIRES].
O. Aharony and N. Klinghoffer, Corrections to Nambu-Goto energy levels from the effective string action, JHEP 12 (2010) 058 [arXiv:1008.2648] [SPIRES].
O. Aharony and M. Field, On the effective theory of long open strings, JHEP 01 (2011) 065 [arXiv:1008.2636] [SPIRES].
O. Aharony, The effective action on long strings, presented at Confining flux tubes and strings, ECT, Trento Italy (2010) http://www.ect.it/Meetings/ConfsWksAndCollMeetings/ConfWksDocument/2010/talks/Workshop05072010/talks.htm
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Athenodorou, A., Bringoltz, B. & Teper, M. Closed flux tubes and their string description in D =3+1 SU(N) gauge theories. J. High Energ. Phys. 2011, 30 (2011). https://doi.org/10.1007/JHEP02(2011)030
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DOI: https://doi.org/10.1007/JHEP02(2011)030