Abstract
We perform a high precision measurement of the static \( q\overline{q} \) potential in three-dimensional SU(N) gauge theory with N = 2, 3 and compare the results to the potential obtained from the effective string theory. In particular, we show that the exponent of the leading order correction in 1/R is 4, as predicted, and obtain accurate results for the continuum limits of the string tension and the non-universal boundary coefficient \( {\overline{b}}_2 \), including an extensive analysis of all types of systematic uncertainties. We find that the magnitude of \( {\overline{b}}_2 \) decreases with increasing N, leading to the possibility of a vanishing \( {\overline{b}}_2 \) in the large N limit. In the standard form of the effective string theory possible massive modes and the presence of a rigidity term are usually not considered, even though they might give a contribution to the energy levels. To investigate the effect of these terms, we perform a second analysis, including these contributions. We find that the associated expression for the potential also provides a good description of the data. The resulting continuum values for \( {\overline{b}}_2 \) are about a factor of 2 smaller than in the standard analysis, due to contaminations from an additional 1/R 4 term. However, \( {\overline{b}}_2 \) shows a similar decrease in magnitude with increasing N. In the course of this extended analysis we also obtain continuum results for the masses appearing in the additional terms and we find that they are around twice as large as the square root of the string tension in the continuum and compatible between SU(2) and SU(3) gauge theory. In the follow up papers we will extend our investigations to the large N limit and excited states of the open flux tube.
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G.S. Bali, K. Schilling and C. Schlichter, Observing long color flux tubes in SU(2) lattice gauge theory, Phys. Rev. D 51 (1995) 5165 [hep-lat/9409005] [INSPIRE].
T. Goto, Relativistic quantum mechanics of one-dimensional mechanical continuum and subsidiary condition of dual resonance model, Prog. Theor. Phys. 46 (1971) 1560 [INSPIRE].
P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Quantum dynamics of a massless relativistic string, Nucl. Phys. B 56 (1973) 109 [INSPIRE].
Y. Nambu, QCD and the string model, Phys. Lett. 80B (1979) 372 [INSPIRE].
M. Lüscher, K. Symanzik and P. Weisz, Anomalies of the free loop wave equation in the WKB approximation, Nucl. Phys. B 173 (1980) 365 [INSPIRE].
A.M. Polyakov, Gauge fields as rings of glue, Nucl. Phys. B 164 (1980) 171 [INSPIRE].
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] [INSPIRE].
O. Aharony and E. Karzbrun, On the effective action of confining strings, JHEP 06 (2009) 012 [arXiv:0903.1927] [INSPIRE].
O. Aharony and N. Klinghoffer, Corrections to Nambu-Goto energy levels from the effective string action, JHEP 12 (2010) 058 [arXiv:1008.2648] [INSPIRE].
O. Aharony and M. Field, On the effective theory of long open strings, JHEP 01 (2011) 065 [arXiv:1008.2636] [INSPIRE].
O. Aharony, M. Field and N. Klinghoffer, The effective string spectrum in the orthogonal gauge, JHEP 04 (2012) 048 [arXiv:1111.5757] [INSPIRE].
M. Billó, M. Caselle, F. Gliozzi, M. Meineri and R. Pellegrini, The Lorentz-invariant boundary action of the confining string and its universal contribution to the inter-quark potential, JHEP 05 (2012) 130 [arXiv:1202.1984] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Effective string theory revisited, JHEP 09 (2012) 044 [arXiv:1203.1054] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity, JHEP 09 (2012) 133 [arXiv:1205.6805] [INSPIRE].
O. Aharony and Z. Komargodski, The effective theory of long strings, JHEP 05 (2013) 118 [arXiv:1302.6257] [INSPIRE].
M. Caselle, D. Fioravanti, F. Gliozzi and R. Tateo, Quantisation of the effective string with TBA, JHEP 07 (2013) 071 [arXiv:1305.1278] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Flux tube spectra from approximate integrability at low energies, J. Exp. Theor. Phys. 120 (2015) 399 [arXiv:1404.0037] [INSPIRE].
M. Caselle, M. Panero, R. Pellegrini and D. Vadacchino, A different kind of string, JHEP 01 (2015) 105 [arXiv:1406.5127] [INSPIRE].
J.F. Arvis, The exact \( q\overline{q} \) potential in Nambu string theory, Phys. Lett. 127B (1983) 106 [INSPIRE].
B.B. Brandt and M. Meineri, Effective string description of confining flux tubes, Int. J. Mod. Phys. A 31 (2016) 1643001 [arXiv:1603.06969] [INSPIRE].
B.B. Brandt, Probing boundary-corrections to Nambu-Goto open string energy levels in 3d SU(2) gauge theory, JHEP 02 (2011) 040 [arXiv:1010.3625] [INSPIRE].
B.B. Brandt, Spectrum of the open QCD flux tube in d = 2 + 1 and its effective string description, PoS (EPS-HEP 2013) 540 [arXiv:1308.4993] [INSPIRE].
F. Gliozzi, M. Pepe and U.J. Wiese, The width of the confining string in Yang-Mills theory, Phys. Rev. Lett. 104 (2010) 232001 [arXiv:1002.4888] [INSPIRE].
N. Cardoso, M. Cardoso and P. Bicudo, Inside the SU(3) quark-antiquark QCD flux tube: screening versus quantum widening, Phys. Rev. D 88 (2013) 054504 [arXiv:1302.3633] [INSPIRE].
M. Caselle, M. Panero and D. Vadacchino, Width of the flux tube in compact U(1) gauge theory in three dimensions, JHEP 02 (2016) 180 [arXiv:1601.07455] [INSPIRE].
P. Cea, L. Cosmai and A. Papa, Chromoelectric flux tubes and coherence length in QCD, Phys. Rev. D 86 (2012) 054501 [arXiv:1208.1362] [INSPIRE].
P. Cea, L. Cosmai, F. Cuteri and A. Papa, Flux tubes in the SU(3) vacuum: London penetration depth and coherence length, Phys. Rev. D 89 (2014) 094505 [arXiv:1404.1172] [INSPIRE].
M. Lüscher, G. Munster and P. Weisz, How thick are chromoelectric flux tubes?, Nucl. Phys. B 180 (1981) 1 [INSPIRE].
M. Caselle, F. Gliozzi, U. Magnea and S. Vinti, Width of long color flux tubes in lattice gauge systems, Nucl. Phys. B 460 (1996) 397 [hep-lat/9510019] [INSPIRE].
D. Förster, Dynamics of relativistic vortex lines and their relation to dual theory, Nucl. Phys. B 81 (1974) 84 [INSPIRE].
J.-L. Gervais and B. Sakita, Quantized relativistic string as a strong coupling limit of Higgs model, Nucl. Phys. B 91 (1975) 301 [INSPIRE].
K.-M. Lee, The Dual formulation of cosmic strings and vortices, Phys. Rev. D 48 (1993) 2493 [hep-th/9301102] [INSPIRE].
P. Orland, Extrinsic curvature dependence of Nielsen-Olesen strings, Nucl. Phys. B 428 (1994) 221 [hep-th/9404140] [INSPIRE].
M. Sato and S. Yahikozawa, ‘Topological’ formulation of effective vortex strings, Nucl. Phys. B 436 (1995) 100 [hep-th/9406208] [INSPIRE].
E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov and M.A. Zubkov, Quantum theory of strings in Abelian Higgs model, Phys. Rev. D 53 (1996) 2087 [hep-th/9505070] [INSPIRE].
C.J. Morningstar, K.J. Juge and J. Kuti, Where is the string limit in QCD?, Nucl. Phys. Proc. Suppl. 73 (1999) 590 [hep-lat/9809098] [INSPIRE].
K.J. Juge, J. Kuti and C. Morningstar, Fine structure of the QCD string spectrum, Phys. Rev. Lett. 90 (2003) 161601 [hep-lat/0207004] [INSPIRE].
K.J. Juge, J. Kuti and C. Morningstar, QCD string formation and the Casimir energy, in the proceedings of the Color confinement and hadrons in quantum chromodynamics international conference (Confinement 2003), July 21-14, Wako, Japan (2003), hep-lat/0401032 [INSPIRE].
A. Athenodorou, B. Bringoltz and M. Teper, Closed flux tubes and their string description in D = 3+1 SU(N) gauge theories, JHEP 02 (2011) 030 [arXiv:1007.4720] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Evidence from lattice data for a new particle on the worldsheet of the QCD flux tube, Phys. Rev. Lett. 111 (2013) 062006 [arXiv:1301.2325] [INSPIRE].
A. Athenodorou and M. Teper, On the mass of the world-sheet ‘axion’ in SU(N ) gauge theories in 3 + 1 dimensions, Phys. Lett. B 771 (2017) 408 [arXiv:1702.03717] [INSPIRE].
A.M. Polyakov, Fine structure of strings, Nucl. Phys. B 268 (1986) 406 [INSPIRE].
G. German and H. Kleinert, Perturbative two loop quark potential of stiff strings in any dimension, Phys. Rev. D 40 (1989) 1108.
J. Ambjørn, Y. Makeenko and A. Sedrakyan, Effective QCD string beyond the Nambu-Goto action, Phys. Rev. D 89 (2014) 106010 [arXiv:1403.0893] [INSPIRE].
B.B. Brandt and P. Majumdar, Spectrum of the QCD flux tube in 3d SU(2) lattice gauge theory, Phys. Lett. B 682 (2009) 253 [arXiv:0905.4195] [INSPIRE].
A.M. Polyakov, Confining strings, Nucl. Phys. B 486 (1997) 23 [hep-th/9607049] [INSPIRE].
E. Braaten, R.D. Pisarski and S.-M. Tse, The static potential for smooth strings, Phys. Rev. Lett. 58 (1987) 93 [Erratum ibid. 59 (1987) 1870] [INSPIRE].
G. German and H. Kleinert, Comment on ‘effective string tension in the finite temperature smooth string model, Phys. Rev. D 40 (1989) 4199.
B. Lucini and M. Panero, SU(N) gauge theories at large-N , Phys. Rept. 526 (2013) 93 [arXiv:1210.4997] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, The QCD potential at O(1/m), Phys. Rev. D 63 (2001) 014023 [hep-ph/0002250] [INSPIRE].
A. Pineda and A. Vairo, The QCD potential at O(1/m 2): complete spin dependent and spin independent result, Phys. Rev. D 63 (2001) 054007 [Erratum ibid. D 64 (2001) 039902] [hep-ph/0009145] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, The (mΛQCD)1/2 scale in heavy quarkonium, Phys. Lett. B 580 (2004) 60 [hep-ph/0307159] [INSPIRE].
N. Brambilla, A. Pineda, J. Soto and A. Vairo, Effective field theories for heavy quarkonium, Rev. Mod. Phys. 77 (2005) 1423 [hep-ph/0410047] [INSPIRE].
N. Brambilla, M. Groher, H.E. Martinez and A. Vairo, Effective string theory and the long-range relativistic corrections to the quark-antiquark potential, Phys. Rev. D 90 (2014) 114032 [arXiv:1407.7761] [INSPIRE].
O. Andreev, Exotic hybrid quark potentials, Phys. Rev. D 86 (2012) 065013 [arXiv:1207.1892] [INSPIRE].
O. Andreev, Exotic hybrid pseudopotentials and gauge/string duality, Phys. Rev. D 87 (2013) 065006 [arXiv:1211.0930] [INSPIRE].
T.R. Klassen and E. Melzer, The thermodynamics of purely elastic scattering theories and conformal perturbation theory, Nucl. Phys. B 350 (1991) 635 [INSPIRE].
V.V. Nesterenko and I.G. Pirozhenko, Justification of the zeta function renormalization in rigid string model, J. Math. Phys. 38 (1997) 6265 [hep-th/9703097] [INSPIRE].
M. Lüscher, Symmetry breaking aspects of the roughening transition in gauge theories, Nucl. Phys. B 180 (1981) 317 [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
U. Kol and J. Sonnenschein, Can holography reproduce the QCD Wilson line?, JHEP 05 (2011) 111 [arXiv:1012.5974] [INSPIRE].
V. Vyas, Intrinsic thickness of QCD flux-tubes, arXiv:1004.2679 [INSPIRE].
V. Vyas, Heavy quark potential from gauge/gravity duality: a large D analysis, Phys. Rev. D 87 (2013) 045026 [arXiv:1209.0883] [INSPIRE].
D. Giataganas and N. Irges, On the holographic width of flux tubes, JHEP 05 (2015) 105 [arXiv:1502.05083] [INSPIRE].
M. Lüscher and P. Weisz, Locality and exponential error reduction in numerical lattice gauge theory, JHEP 09 (2001) 010 [hep-lat/0108014] [INSPIRE].
R. Sommer, A new way to set the energy scale in lattice gauge theories and its applications to the static force and α s in SU(2) Yang-Mills theory, Nucl. Phys. B 411 (1994) 839 [hep-lat/9310022] [INSPIRE].
B. Lucini and M. Teper, SU(N) gauge theories in (2 + 1)-dimensions: Further results, Phys. Rev. D 66 (2002) 097502 [hep-lat/0206027] [INSPIRE].
B. Bringoltz and M. Teper, String tensions of SU(N) gauge theories in 2 + 1 dimensions, PoS (LAT2006) 041 [hep-lat/0610035] [INSPIRE].
D. Karabali, C.-j. Kim and V.P. Nair, On the vacuum wave function and string tension of Yang-Mills theories in (2+1)-dimensions, Phys. Lett. B 434 (1998) 103 [hep-th/9804132] [INSPIRE].
M.J. Teper, SU(N) gauge theories in (2 + 1)-dimensions, Phys. Rev. D 59 (1999) 014512 [hep-lat/9804008] [INSPIRE].
N.D. Hari Dass and P. Majumdar, Continuum limit of string formation in 3 − D SU(2) LGT, Phys. Lett. B 658 (2008) 273 [hep-lat/0702019] [INSPIRE].
N. Cabibbo and E. Marinari, A new method for updating SU(N) matrices in computer simulations of gauge theories, Phys. Lett. 119B (1982) 387 [INSPIRE].
A.D. Kennedy and B.J. Pendleton, Improved heat bath method for monte carlo calculations in lattice gauge theories, Phys. Lett. 156B (1985) 393 [INSPIRE].
M. Creutz, Overrelaxation and Monte Carlo simulation, Phys. Rev. D 36 (1987) 515.
M. Lüscher and P. Weisz, Quark confinement and the bosonic string, JHEP 07 (2002) 049 [hep-lat/0207003] [INSPIRE].
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Brandt, B.B. Spectrum of the open QCD flux tube and its effective string description I: 3d static potential in SU(N = 2, 3). J. High Energ. Phys. 2017, 8 (2017). https://doi.org/10.1007/JHEP07(2017)008
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DOI: https://doi.org/10.1007/JHEP07(2017)008