Abstract
BPS Wilson loops in supersymmetric gauge theories have been the subjects of active research since they are often amenable to exact computation. So far most of the studies have focused on loops that do not intersect. In this paper, we derive exact results for intersecting 1/8 BPS Wilson loops in \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory, using a combination of supersymmetric localization and the loop equation in 2d gauge theory. The result is given by a novel matrix-model-like representation which couples multiple contour integrals and a Gaussian matrix model. We evaluate the integral at large N , and make contact with the string worldsheet description at strong coupling. As an application of our results, we compute exactly a small-angle limit (and more generally near-BPS limits) of the cross anomalous dimension which governs the UV divergence of intersecting Wilson lines. The same quantity describes the soft anomalous dimension of scattering amplitudes of W -bosons in the Coulomb branch.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Yu.M. Makeenko and A.A. Migdal, Exact equation for the loop average in multicolor QCD, Phys. Lett. B 88 (1979) 135 [Erratum ibid. B 89 (1980) 437] [INSPIRE].
Yu. Makeenko and A.A. Migdal, Quantum chromodynamics as dynamics of loops, Nucl. Phys. B 188 (1981) 269 [INSPIRE].
Y. Makeenko, Large N gauge theories, NATO Sci. Ser. C 556 (2000) 285 [hep-th/0001047] [INSPIRE].
A.A. Migdal, Loop equations and 1/N expansion, Phys. Rept. 102 (1983) 199 [INSPIRE].
V.A. Kazakov and I.K. Kostov, Nonlinear strings in two-dimensional U(∞) gauge theory, Nucl. Phys. B 176 (1980) 199 [INSPIRE].
V.A. Kazakov, Wilson loop average for an arbitrary contour in two-dimensional U(N ) gauge theory, Nucl. Phys. B 179 (1981) 283 [INSPIRE].
D.V. Boulatov, Wilson loop on a sphere, Mod. Phys. Lett. A 9 (1994) 365 [hep-th/9310041] [INSPIRE].
J.-M. Daul and V.A. Kazakov, Wilson loop for large N Yang-Mills theory on a two-dimensional sphere, Phys. Lett. B 335 (1994) 371 [hep-th/9310165] [INSPIRE].
B. Rusakov, Wilson loops in large N QCD on a sphere, Phys. Lett. B 329 (1994) 338 [hep-th/9310134] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
V. Pestun, Localization of the four-dimensional N = 4 SYM to a two-sphere and 1/8 BPS Wilson loops, JHEP 12 (2012) 067 [arXiv:0906.0638] [INSPIRE].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Wilson loops: from four-dimensional SYM to two-dimensional YM, Phys. Rev. D 77 (2008) 047901 [arXiv:0707.2699] [INSPIRE].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S3 , JHEP 05 (2008) 017 [arXiv:0711.3226] [INSPIRE].
J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].
N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].
S. Giombi and V. Pestun, Correlators of local operators and 1/8 BPS Wilson loops on S2 from 2d YM and matrix models, JHEP 10 (2010) 033 [arXiv:0906.1572] [INSPIRE].
S. Giombi and V. Pestun, Correlators of Wilson loops and local operators from multi-matrix models and strings in AdS, JHEP 01 (2013) 101 [arXiv:1207.7083] [INSPIRE].
S. Giombi, V. Pestun and R. Ricci, Notes on supersymmetric Wilson loops on a two-sphere, JHEP 07 (2010) 088 [arXiv:0905.0665] [INSPIRE].
S. Giombi and V. Pestun, The 1/2 BPS ’t Hooft loops in N = 4 SYM as instantons in 2d Yang-Mills, J. Phys. A 46 (2013) 095402 [arXiv:0909.4272] [INSPIRE].
A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai and D. Young, Correlators of supersymmetric Wilson-loops, protected operators and matrix models in N = 4 SYM, JHEP 08 (2009) 061 [arXiv:0905.1943] [INSPIRE].
A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai and D. Young, Correlators of supersymmetric Wilson loops at weak and strong coupling, JHEP 03 (2010) 038 [arXiv:0912.5440] [INSPIRE].
M. Bonini, L. Griguolo and M. Preti, Correlators of chiral primaries and 1/8 BPS Wilson loops from perturbation theory, JHEP 09 (2014) 083 [arXiv:1405.2895] [INSPIRE].
M. Bonini, L. Griguolo, M. Preti and D. Seminara, Bremsstrahlung function, leading Lüscher correction at weak coupling and localization, JHEP 02 (2016) 172 [arXiv:1511.05016] [INSPIRE].
S. Cordes, G.W. Moore and S. Ramgoolam, Lectures on 2D Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. Proc. Suppl. 41 (1995) 184 [hep-th/9411210] [INSPIRE].
D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in N = 4 super Yang-Mills, JHEP 06 (2012) 048 [arXiv:1202.4455] [INSPIRE].
S. Giombi and S. Komatsu, Exact correlators on the Wilson loop in N = 4 SYM: localization, defect CFT and integrability, JHEP 05 (2018) 109 [Erratum ibid. 11 (2018) 123] [arXiv:1802.05201] [INSPIRE].
H. Münkler, The cross anomalous dimension in maximally supersymmetric Yang-Mills theory, JHEP 10 (2018) 162 [arXiv:1805.06448] [INSPIRE].
G.P. Korchemsky, On near forward high-energy scattering in QCD, Phys. Lett. B 325 (1994) 459 [hep-ph/9311294] [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, High-energy scattering in QCD and cross singularities of Wilson loops, Nucl. Phys. B 437 (1995) 127 [hep-ph/9409446] [INSPIRE].
S. Giombi, J. Jiang and S. Komatsu, Defect CFT correlators on the giant Wilson loops, to appear.
B. Basso, S. Komatsu and P. Vieira, Structure constants and integrable bootstrap in planar N = 4 SYM theory, arXiv:1505.06745 [INSPIRE].
Y. Jiang, S. Komatsu, I. Kostov and D. Serban, Clustering and the three-point function, J. Phys. A 49 (2016) 454003 [arXiv:1604.03575] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of correlation functions, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
B. Eden, Y. Jiang, M. de Leeuw, T. Meier, D. le Plat and A. Sfondrini, Positivity of hexagon perturbation theory, JHEP 11 (2018) 097 [arXiv:1806.06051] [INSPIRE].
F. Coronado, Perturbative four-point functions in planar N = 4 SYM from hexagonalization, JHEP 01 (2019) 056 [arXiv:1811.00467] [INSPIRE].
T. Bargheer, F. Coronado and P. Vieira, Octagons II: strong coupling, arXiv:1909.04077 [INSPIRE].
I. Kostov, V.B. Petkova and D. Serban, Determinant formula for the octagon form factor in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 122 (2019) 231601 [arXiv:1903.05038] [INSPIRE].
I. Kostov, V.B. Petkova and D. Serban, The octagon as a determinant, JHEP 11 (2019) 178 [arXiv:1905.11467] [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Exact null octagon, JHEP 05 (2020) 070 [arXiv:1907.13131] [INSPIRE].
M. De Leeuw, B. Eden, D. Le Plat, T. Meier and A. Sfondrini, Multi-particle finite-volume effects for hexagon tessellations, arXiv:1912.12231 [INSPIRE].
A.V. Belitsky and G.P. Korchemsky, Octagon at finite coupling, arXiv:2003.01121 [INSPIRE].
S. Giombi and S. Komatsu, More exact results in the Wilson loop defect CFT: bulk-defect OPE, nonplanar corrections and quantum spectral curve, J. Phys. A 52 (2019) 125401 [arXiv:1811.02369] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Analytic solution of Bremsstrahlung TBA II: turning on the sphere angle, JHEP 10 (2013) 036 [arXiv:1305.1944] [INSPIRE].
N. Drukker, 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP 09 (2006) 004 [hep-th/0605151] [INSPIRE].
V. Forini, A.A. Tseytlin and E. Vescovi, Perturbative computation of string one-loop corrections to Wilson loop minimal surfaces in AdS5 × S5 , JHEP 03 (2017) 003 [arXiv:1702.02164] [INSPIRE].
D. Medina-Rincon, A.A. Tseytlin and K. Zarembo, Precision matching of circular Wilson loops and strings in AdS5 × S5 , JHEP 05 (2018) 199 [arXiv:1804.08925] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [Erratum ibid. 111 (2013) 199905] [arXiv:0901.0722] [INSPIRE].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].
L.J. Dixon and I. Esterlis, All orders results for self-crossing Wilson loops mimicking double parton scattering, JHEP 07 (2016) 116 [Erratum ibid. 08 (2016) 131] [arXiv:1602.02107] [INSPIRE].
S. Caron-Huot, L.J. Dixon, F. Dulat, M. von Hippel, A.J. McLeod and G. Papathanasiou, Six-gluon amplitudes in planar N = 4 super-Yang-Mills theory at six and seven loops, JHEP 08 (2019) 016 [arXiv:1903.10890] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Spacetime and flux tube S-matrices at finite coupling for N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].
N. Drukker, A new type of loop equations, JHEP 11 (1999) 006 [hep-th/9908113] [INSPIRE].
A.M. Polyakov and V.S. Rychkov, Gauge field strings duality and the loop equation, Nucl. Phys. B 581 (2000) 116 [hep-th/0002106] [INSPIRE].
A.M. Polyakov and V.S. Rychkov, Loop dynamics and AdS/CFT correspondence, Nucl. Phys. B 594 (2001) 272 [hep-th/0005173] [INSPIRE].
P.D. Anderson and M. Kruczenski, Loop equations and bootstrap methods in the lattice, Nucl. Phys. B 921 (2017) 702 [arXiv:1612.08140] [INSPIRE].
H.W. Lin, Bootstraps to strings: solving random matrix models with positivity, arXiv:2002.08387 [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
S. Giombi, R. Roiban and A.A. Tseytlin, Half-BPS Wilson loop and AdS2 /CFT1 , Nucl. Phys. B 922 (2017) 499 [arXiv:1706.00756] [INSPIRE].
N. Kiryu and S. Komatsu, Correlation functions on the half-BPS Wilson loop: perturbation and hexagonalization, JHEP 02 (2019) 090 [arXiv:1812.04593] [INSPIRE].
M. Beccaria, S. Giombi and A. Tseytlin, Non-supersymmetric Wilson loop in N = 4 SYM and defect 1d CFT, JHEP 03 (2018) 131 [arXiv:1712.06874] [INSPIRE].
M. Beccaria, S. Giombi and A.A. Tseytlin, Correlators on non-supersymmetric Wilson line in N = 4 SYM and AdS2 /CFT1 , JHEP 05 (2019) 122 [arXiv:1903.04365] [INSPIRE].
D. Correa, M. Leoni and S. Luque, Spin chain integrability in non-supersymmetric Wilson loops, JHEP 12 (2018) 050 [arXiv:1810.04643] [INSPIRE].
A. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multileg scattering, Phys. Rev. Lett. 117 (2016) 172002 [arXiv:1507.00047] [INSPIRE].
A. Almelid, C. Duhr, E. Gardi, A. McLeod and C.D. White, Bootstrapping the QCD soft anomalous dimension, JHEP 09 (2017) 073 [arXiv:1706.10162] [INSPIRE].
B. Fiol, E. Gerchkovitz and Z. Komargodski, Exact Bremsstrahlung function in N = 2 superconformal field theories, Phys. Rev. Lett. 116 (2016) 081601 [arXiv:1510.01332] [INSPIRE].
C. Gomez, A. Mauri and S. Penati, The Bremsstrahlung function of N = 2 SCQCD, JHEP 03 (2019) 122 [arXiv:1811.08437] [INSPIRE].
L. Bianchi, M. Billò, F. Galvagno and A. Lerda, Emitted radiation and geometry, JHEP 01 (2020) 075 [arXiv:1910.06332] [INSPIRE].
D. Correa, J. Henn, J. Maldacena and A. Sever, The cusp anomalous dimension at three loops and beyond, JHEP 05 (2012) 098 [arXiv:1203.1019] [INSPIRE].
M. Kim, N. Kiryu, S. Komatsu and T. Nishimura, Structure constants of defect changing operators on the 1/2 BPS Wilson loop, JHEP 12 (2017) 055 [arXiv:1710.07325] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve and structure constants in N = 4 SYM: cusps in the ladder limit, JHEP 10 (2018) 060 [arXiv:1802.04237] [INSPIRE].
D.H. Correa, P. Pisani and A. Rios Fukelman, Ladder limit for correlators of Wilson loops, JHEP 05 (2018) 168 [arXiv:1803.02153] [INSPIRE].
D. Correa, P. Pisani, A. Rios Fukelman and K. Zarembo, Dyson equations for correlators of Wilson loops, JHEP 12 (2018) 100 [arXiv:1811.03552] [INSPIRE].
J. McGovern, Scalar insertions in cusped Wilson loops in the ladders limit of planar N = 4 SYM, JHEP 05 (2020) 062 [arXiv:1912.00499] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2003.04460
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Giombi, S., Komatsu, S. Loop equation and exact soft anomalous dimension in \( \mathcal{N} \) = 4 super Yang-Mills. J. High Energ. Phys. 2020, 75 (2020). https://doi.org/10.1007/JHEP06(2020)075
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2020)075