Abstract
We revisit from a modern bootstrap perspective the longstanding problem of solving QCD in the large N limit. We derive universal bounds on the effective field theory of massless pions by imposing the full set of positivity constraints that follow from 2 → 2 scattering. Some features of our exclusion plots have intriguing connections with hadronic phenomenology. The exclusion boundary exhibits a sharp kink, raising the tantalizing scenario that large N QCD may sit at this kink. We critically examine this possibility, developing in the process a partial analytic understanding of the geometry of the bounds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [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].
A.M. Polyakov, String theory and quark confinement, Nucl. Phys. B Proc. Suppl. 68 (1998) 1 [hep-th/9711002] [INSPIRE].
A.M. Polyakov, The Wall of the cave, Int. J. Mod. Phys. A 14 (1999) 645 [hep-th/9809057] [INSPIRE].
E. Witten, Baryons in the 1/n Expansion, Nucl. Phys. B 160 (1979) 57 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
D. Poland and D. Simmons-Duffin, Snowmass White Paper: The Numerical Conformal Bootstrap, in 2022 Snowmass Summer Study, (2022) [arXiv:2203.08117] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
G. Veneziano, Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories, Nuovo Cim. A 57 (1968) 190 [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP 11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap II: two dimensional amplitudes, JHEP 11 (2017) 143 [arXiv:1607.06110] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part III: higher dimensional amplitudes, JHEP 12 (2019) 040 [arXiv:1708.06765] [INSPIRE].
Y. He, A. Irrgang and M. Kruczenski, A note on the S-matrix bootstrap for the 2d O(N) bosonic model, JHEP 11 (2018) 093 [arXiv:1805.02812] [INSPIRE].
L. Córdova and P. Vieira, Adding flavour to the S-matrix bootstrap, JHEP 12 (2018) 063 [arXiv:1805.11143] [INSPIRE].
A. Homrich, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix Bootstrap IV: Multiple Amplitudes, JHEP 11 (2019) 076 [arXiv:1905.06905] [INSPIRE].
C. Bercini, M. Fabri, A. Homrich and P. Vieira, S-matrix bootstrap: Supersymmetry, Z2, and Z4 symmetry, Phys. Rev. D 101 (2020) 045022 [arXiv:1909.06453] [INSPIRE].
L. Córdova, Y. He, M. Kruczenski and P. Vieira, The O(N) S-matrix Monolith, JHEP 04 (2020) 142 [arXiv:1909.06495] [INSPIRE].
A. Hebbar, D. Karateev and J. Penedones, Spinning S-matrix bootstrap in 4d, JHEP 01 (2022) 060 [arXiv:2011.11708] [INSPIRE].
A.L. Guerrieri, A. Homrich and P. Vieira, Dual S-matrix bootstrap. Part I. 2D theory, JHEP 11 (2020) 084 [arXiv:2008.02770] [INSPIRE].
Y. He and M. Kruczenski, S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem, JHEP 08 (2021) 125 [arXiv:2103.11484] [INSPIRE].
M. Kruczenski, J. Penedones and B.C. van Rees, Snowmass White Paper: S-matrix Bootstrap, arXiv:2203.02421 [INSPIRE].
A.L. Guerrieri, J. Penedones and P. Vieira, Bootstrapping QCD Using Pion Scattering Amplitudes, Phys. Rev. Lett. 122 (2019) 241604 [arXiv:1810.12849] [INSPIRE].
A.L. Guerrieri, J. Penedones and P. Vieira, S-matrix bootstrap for effective field theories: massless pions, JHEP 06 (2021) 088 [arXiv:2011.02802] [INSPIRE].
G. Veneziano, S. Yankielowicz and E. Onofri, A model for pion-pion scattering in large-N QCD, JHEP 04 (2017) 151 [arXiv:1701.06315] [INSPIRE].
B. Lucini and M. Panero, SU(N) gauge theories at large N, Phys. Rept. 526 (2013) 93 [arXiv:1210.4997] [INSPIRE].
T. DeGrand and Y. Liu, Lattice study of large Nc QCD, Phys. Rev. D 94 (2016) 034506 [Erratum ibid. 95 (2017) 019902] [arXiv:1606.01277] [INSPIRE].
P. Hernández, C. Pena and F. Romero-López, Large Nc scaling of meson masses and decay constants, Eur. Phys. J. C 79 (2019) 865 [arXiv:1907.11511] [INSPIRE].
M.G. Pérez, A. González-Arroyo and M. Okawa, Meson spectrum in the large N limit, JHEP 04 (2021) 230 [arXiv:2011.13061] [INSPIRE].
J. Baeza-Ballesteros, P. Hernández and F. Romero-López, A lattice study of ππ scattering at large Nc, JHEP 06 (2022) 049 [arXiv:2202.02291] [INSPIRE].
S.R. Coleman and E. Witten, Chiral Symmetry Breakdown in Large N Chromodynamics, Phys. Rev. Lett. 45 (1980) 100 [INSPIRE].
G. Veneziano, Goldstone Mechanism From Gluon Dynamics, Phys. Lett. B 95 (1980) 90 [INSPIRE].
A. Martin, Scattering Theory: Unitarity, Analyticity and Crossing, Springer Berlin Heidelberg, Berlin, Heidelberg, Germany (1969), pp. 1–117 [DOI].
T.N. Pham and T.N. Truong, Evaluation of the Derivative Quartic Terms of the Meson Chiral Lagrangian From Forward Dispersion Relation, Phys. Rev. D 31 (1985) 3027 [INSPIRE].
B. Ananthanarayan, D. Toublan and G. Wanders, Consistency of the chiral pion pion scattering amplitudes with axiomatic constraints, Phys. Rev. D 51 (1995) 1093 [hep-ph/9410302] [INSPIRE].
M.R. Pennington and J. Portoles, The Chiral Lagrangian parameters, l1, l2, are determined by the rho resonance, Phys. Lett. B 344 (1995) 399 [hep-ph/9409426] [INSPIRE].
J. Comellas, J.I. Latorre and J. Taron, Constraints on chiral perturbation theory parameters from QCD inequalities, Phys. Lett. B 360 (1995) 109 [hep-ph/9507258] [INSPIRE].
P. Dita, Positivity constraints on chiral perturbation theory pion pion scattering amplitudes, Phys. Rev. D 59 (1999) 094007 [hep-ph/9809568] [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
A.V. Manohar and V. Mateu, Dispersion Relation Bounds for pi pi Scattering, Phys. Rev. D 77 (2008) 094019 [arXiv:0801.3222] [INSPIRE].
V. Mateu, Universal Bounds for SU(3) Low Energy Constants, Phys. Rev. D 77 (2008) 094020 [arXiv:0801.3627] [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, Energy’s and amplitudes’ positivity, JHEP 05 (2010) 095 [Erratum ibid. 11 (2011) 128] [arXiv:0912.4258] [INSPIRE].
D. Baumann, D. Green, H. Lee and R.A. Porto, Signs of Analyticity in Single-Field Inflation, Phys. Rev. D 93 (2016) 023523 [arXiv:1502.07304] [INSPIRE].
B. Bellazzini, C. Cheung and G.N. Remmen, Quantum Gravity Constraints from Unitarity and Analyticity, Phys. Rev. D 93 (2016) 064076 [arXiv:1509.00851] [INSPIRE].
B. Bellazzini, Softness and amplitudes’ positivity for spinning particles, JHEP 02 (2017) 034 [arXiv:1605.06111] [INSPIRE].
C. Cheung and G.N. Remmen, Positive Signs in Massive Gravity, JHEP 04 (2016) 002 [arXiv:1601.04068] [INSPIRE].
J. Bonifacio, K. Hinterbichler and R.A. Rosen, Positivity constraints for pseudolinear massive spin-2 and vector Galileons, Phys. Rev. D 94 (2016) 104001 [arXiv:1607.06084] [INSPIRE].
C. Cheung and G.N. Remmen, Positivity of Curvature-Squared Corrections in Gravity, Phys. Rev. Lett. 118 (2017) 051601 [arXiv:1608.02942] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Positivity bounds for scalar field theories, Phys. Rev. D 96 (2017) 081702 [arXiv:1702.06134] [INSPIRE].
B. Bellazzini, F. Riva, J. Serra and F. Sgarlata, Beyond Positivity Bounds and the Fate of Massive Gravity, Phys. Rev. Lett. 120 (2018) 161101 [arXiv:1710.02539] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, UV complete me: Positivity Bounds for Particles with Spin, JHEP 03 (2018) 011 [arXiv:1706.02712] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Massive Galileon Positivity Bounds, JHEP 09 (2017) 072 [arXiv:1702.08577] [INSPIRE].
K. Hinterbichler, A. Joyce and R.A. Rosen, Massive Spin-2 Scattering and Asymptotic Superluminality, JHEP 03 (2018) 051 [arXiv:1708.05716] [INSPIRE].
J. Bonifacio, K. Hinterbichler, A. Joyce and R.A. Rosen, Massive and Massless Spin-2 Scattering and Asymptotic Superluminality, JHEP 06 (2018) 075 [arXiv:1712.10020] [INSPIRE].
B. Bellazzini, F. Riva, J. Serra and F. Sgarlata, The other effective fermion compositeness, JHEP 11 (2017) 020 [arXiv:1706.03070] [INSPIRE].
J. Bonifacio and K. Hinterbichler, Bounds on Amplitudes in Effective Theories with Massive Spinning Particles, Phys. Rev. D 98 (2018) 045003 [arXiv:1804.08686] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Positivity Bounds for Massive Spin-1 and Spin-2 Fields, JHEP 03 (2019) 182 [arXiv:1804.10624] [INSPIRE].
C. Zhang and S.-Y. Zhou, Positivity bounds on vector boson scattering at the LHC, Phys. Rev. D 100 (2019) 095003 [arXiv:1808.00010] [INSPIRE].
B. Bellazzini and F. Riva, New phenomenological and theoretical perspective on anomalous ZZ and Zγ processes, Phys. Rev. D 98 (2018) 095021 [arXiv:1806.09640] [INSPIRE].
B. Bellazzini, M. Lewandowski and J. Serra, Positivity of Amplitudes, Weak Gravity Conjecture, and Modified Gravity, Phys. Rev. Lett. 123 (2019) 251103 [arXiv:1902.03250] [INSPIRE].
S. Melville and J. Noller, Positivity in the Sky: Constraining dark energy and modified gravity from the UV, Phys. Rev. D 101 (2020) 021502 [Erratum ibid. 102 (2020) 049902] [arXiv:1904.05874] [INSPIRE].
C. de Rham and A.J. Tolley, Speed of gravity, Phys. Rev. D 101 (2020) 063518 [arXiv:1909.00881] [INSPIRE].
L. Alberte, C. de Rham, A. Momeni, J. Rumbutis and A.J. Tolley, Positivity Constraints on Interacting Spin-2 Fields, JHEP 03 (2020) 097 [arXiv:1910.11799] [INSPIRE].
L. Alberte, C. de Rham, A. Momeni, J. Rumbutis and A.J. Tolley, Positivity Constraints on Interacting Pseudo-Linear Spin-2 Fields, JHEP 07 (2020) 121 [arXiv:1912.10018] [INSPIRE].
Q. Bi, C. Zhang and S.-Y. Zhou, Positivity constraints on aQGC: carving out the physical parameter space, JHEP 06 (2019) 137 [arXiv:1902.08977] [INSPIRE].
G.N. Remmen and N.L. Rodd, Consistency of the Standard Model Effective Field Theory, JHEP 12 (2019) 032 [arXiv:1908.09845] [INSPIRE].
G. Ye and Y.-S. Piao, Positivity in the effective field theory of cosmological perturbations, Eur. Phys. J. C 80 (2020) 421 [arXiv:1908.08644] [INSPIRE].
M. Herrero-Valea, I. Timiryasov and A. Tokareva, To Positivity and Beyond, where Higgs-Dilaton Inflation has never gone before, JCAP 11 (2019) 042 [arXiv:1905.08816] [INSPIRE].
B. Bellazzini, J. Elias Miró, R. Rattazzi, M. Riembau and F. Riva, Positive moments for scattering amplitudes, Phys. Rev. D 104 (2021) 036006 [arXiv:2011.00037] [INSPIRE].
A.J. Tolley, Z.-Y. Wang and S.-Y. Zhou, New positivity bounds from full crossing symmetry, JHEP 05 (2021) 255 [arXiv:2011.02400] [INSPIRE].
S. Caron-Huot and V. Van Duong, Extremal Effective Field Theories, JHEP 05 (2021) 280 [arXiv:2011.02957] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, The EFT-Hedron, JHEP 05 (2021) 259 [arXiv:2012.15849] [INSPIRE].
A. Sinha and A. Zahed, Crossing Symmetric Dispersion Relations in Quantum Field Theories, Phys. Rev. Lett. 126 (2021) 181601 [arXiv:2012.04877] [INSPIRE].
T. Trott, Causality, unitarity and symmetry in effective field theory, JHEP 07 (2021) 143 [arXiv:2011.10058] [INSPIRE].
Y.-J. Wang, F.-K. Guo, C. Zhang and S.-Y. Zhou, Generalized positivity bounds on chiral perturbation theory, JHEP 07 (2020) 214 [arXiv:2004.03992] [INSPIRE].
C. Zhang and S.-Y. Zhou, Convex Geometry Perspective on the (Standard Model) Effective Field Theory Space, Phys. Rev. Lett. 125 (2020) 201601 [arXiv:2005.03047] [INSPIRE].
C. Zhang, SMEFTs living on the edge: determining the UV theories from positivity and extremality, arXiv:2112.11665 [INSPIRE].
Z.-Z. Du, C. Zhang and S.-Y. Zhou, Triple crossing positivity bounds for multi-field theories, JHEP 12 (2021) 115 [arXiv:2111.01169] [INSPIRE].
J. Davighi, S. Melville and T. You, Natural selection rules: new positivity bounds for massive spinning particles, JHEP 02 (2022) 167 [arXiv:2108.06334] [INSPIRE].
S.D. Chowdhury, K. Ghosh, P. Haldar, P. Raman and A. Sinha, Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds, arXiv:2112.11755 [INSPIRE].
J. Henriksson, B. McPeak, F. Russo and A. Vichi, Rigorous bounds on light-by-light scattering, JHEP 06 (2022) 158 [arXiv:2107.13009] [INSPIRE].
Z. Bern, D. Kosmopoulos and A. Zhiboedov, Gravitational effective field theory islands, low-spin dominance, and the four-graviton amplitude, J. Phys. A 54 (2021) 344002 [arXiv:2103.12728] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Sharp boundaries for the swampland, JHEP 07 (2021) 110 [arXiv:2102.08951] [INSPIRE].
X. Li, H. Xu, C. Yang, C. Zhang and S.-Y. Zhou, Positivity in Multifield Effective Field Theories, Phys. Rev. Lett. 127 (2021) 121601 [arXiv:2101.01191] [INSPIRE].
C. de Rham, S. Kundu, M. Reece, A.J. Tolley and S.-Y. Zhou, Snowmass White Paper: UV Constraints on IR Physics, in 2022 Snowmass Summer Study, (2022) [arXiv:2203.06805] [INSPIRE].
S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, Causality constraints on corrections to Einstein gravity, arXiv:2201.06602 [INSPIRE].
J. Henriksson, B. McPeak, F. Russo and A. Vichi, Bounding Violations of the Weak Gravity Conjecture, arXiv:2203.08164 [INSPIRE].
B. Alvarez, J. Bijnens and M. Sjö, NNLO positivity bounds on chiral perturbation theory for a general number of flavours, JHEP 03 (2022) 159 [arXiv:2112.04253] [INSPIRE].
A. Bose, A. Sinha and S.S. Tiwari, Selection rules for the S-matrix bootstrap, SciPost Phys. 10 (2021) 122 [arXiv:2011.07944] [INSPIRE].
A. Bose, P. Haldar, A. Sinha, P. Sinha and S.S. Tiwari, Relative entropy in scattering and the S-matrix bootstrap, SciPost Phys. 9 (2020) 081 [arXiv:2006.12213] [INSPIRE].
A. Zahed, Positivity and geometric function theory constraints on pion scattering, JHEP 12 (2021) 036 [arXiv:2108.10355] [INSPIRE].
K. Kawarabayashi and M. Suzuki, Partially conserved axial vector current and the decays of vector mesons, Phys. Rev. Lett. 16 (1966) 255 [INSPIRE].
Riazuddin and Fayyazuddin, Algebra of current components and decay widths of rho and K* mesons, Phys. Rev. 147 (1966) 1071 [INSPIRE].
E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156 (1979) 269 [INSPIRE].
G. Veneziano, U(1) Without Instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].
H. Leutwyler, On the 1/N expansion in chiral perturbation theory, Nucl. Phys. B Proc. Suppl. 64 (1998) 223 [hep-ph/9709408] [INSPIRE].
R. Kaiser and H. Leutwyler, Large Nc in chiral perturbation theory, Eur. Phys. J. C 17 (2000) 623 [hep-ph/0007101] [INSPIRE].
M. Correia, A. Sever and A. Zhiboedov, An analytical toolkit for the S-matrix bootstrap, JHEP 03 (2021) 013 [arXiv:2006.08221] [INSPIRE].
R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, The Pomeron and gauge/string duality, JHEP 12 (2007) 005 [hep-th/0603115] [INSPIRE].
J.R. Pelaez and F.J. Yndurain, The Pion-pion scattering amplitude, Phys. Rev. D 71 (2005) 074016 [hep-ph/0411334] [INSPIRE].
S.J. Brodsky and G.R. Farrar, Scaling Laws at Large Transverse Momentum, Phys. Rev. Lett. 31 (1973) 1153 [INSPIRE].
G.P. Lepage and S.J. Brodsky, Exclusive Processes in Perturbative Quantum Chromodynamics, Phys. Rev. D 22 (1980) 2157 [INSPIRE].
S. Caron-Huot, Z. Komargodski, A. Sever and A. Zhiboedov, Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude, JHEP 10 (2017) 026 [arXiv:1607.04253] [INSPIRE].
C. Lovelace, A novel application of Regge trajectories, Phys. Lett. B 28 (1968) 264 [INSPIRE].
J.A. Shapiro, Narrow-resonance model with Regge behavior for pi pi scattering, Phys. Rev. 179 (1969) 1345 [INSPIRE].
M. Bianchi, D. Consoli and P. Di Vecchia, On the N-pion extension of the Lovelace-Shapiro model, JHEP 03 (2021) 119 [arXiv:2002.05419] [INSPIRE].
N. Arkani-Hamed, L. Eberhardt, Y.-t. Huang and S. Mizera, On unitarity of tree-level string amplitudes, JHEP 02 (2022) 197 [arXiv:2201.11575] [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
L.-Y. Chiang, Y.-t. Huang, W. Li, L. Rodina and H.-C. Weng, Into the EFThedron and UV constraints from IR consistency, JHEP 03 (2022) 063 [arXiv:2105.02862] [INSPIRE].
Y.-t. Huang, J.-Y. Liu, L. Rodina and Y. Wang, Carving out the Space of Open-String S-matrix, JHEP 04 (2021) 195 [arXiv:2008.02293] [INSPIRE].
F. Caracciolo and V.S. Rychkov, Rigorous Limits on the Interaction Strength in Quantum Field Theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].
D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
I. Zahed and G.E. Brown, The Skyrme Model, Phys. Rept. 142 (1986) 1 [INSPIRE].
E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
G.S. Adkins, C.R. Nappi and E. Witten, Static Properties of Nucleons in the Skyrme Model, Nucl. Phys. B 228 (1983) 552 [INSPIRE].
J. Bijnens, G. Colangelo and J. Gasser, K(l4) decays beyond one loop, Nucl. Phys. B 427 (1994) 427 [hep-ph/9403390] [INSPIRE].
L. Girlanda, M. Knecht, B. Moussallam and J. Stern, Comment on the prediction of two loop standard chiral perturbation theory for low-energy pi pi scattering, Phys. Lett. B 409 (1997) 461 [hep-ph/9703448] [INSPIRE].
G. Amoros, J. Bijnens and P. Talavera, Kℓ4 form-factors and π − π scattering, Nucl. Phys. B 585 (2000) 293 [Erratum ibid. 598 (2001) 665] [hep-ph/0003258] [INSPIRE].
P. Haldar, A. Sinha and A. Zahed, Quantum field theory and the Bieberbach conjecture, SciPost Phys. 11 (2021) 002 [arXiv:2103.12108] [INSPIRE].
P. Raman and A. Sinha, QFT, EFT and GFT, JHEP 12 (2021) 203 [arXiv:2107.06559] [INSPIRE].
G. Ecker, J. Gasser, A. Pich and E. de Rafael, The Role of Resonances in Chiral Perturbation Theory, Nucl. Phys. B 321 (1989) 311 [INSPIRE].
J.F. Donoghue, C. Ramirez and G. Valencia, The Spectrum of QCD and Chiral Lagrangians of the Strong and Weak Interactions, Phys. Rev. D 39 (1989) 1947 [INSPIRE].
M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, Is rho Meson a Dynamical Gauge Boson of Hidden Local Symmetry?, Phys. Rev. Lett. 54 (1985) 1215 [INSPIRE].
M. Bando, T. Kugo and K. Yamawaki, Nonlinear Realization and Hidden Local Symmetries, Phys. Rept. 164 (1988) 217 [INSPIRE].
Z. Komargodski, Vector Mesons and an Interpretation of Seiberg Duality, JHEP 02 (2011) 019 [arXiv:1010.4105] [INSPIRE].
J.J. Sakurai, Currents and mesons, The University of Chicago Press, Chicago, IL, U.S.A. (1969).
J.J. Sakurai, Eight Ways of Determining the rho-Meson Coupling Constant, Phys. Rev. Lett. 17 (1966) 1021 [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, PTEP 2020 (2020) 083C01 [INSPIRE].
Z. Komargodski and D. Simmons-Duffin, The Random-Bond Ising Model in 2.01 and 3 Dimensions, J. Phys. A 50 (2017) 154001 [arXiv:1603.04444] [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
O. Aharony and Z. Komargodski, The Effective Theory of Long Strings, JHEP 05 (2013) 118 [arXiv:1302.6257] [INSPIRE].
S. Dubovsky and V. Gorbenko, Towards a Theory of the QCD String, JHEP 02 (2016) 022 [arXiv:1511.01908] [INSPIRE].
S. Dubovsky and G. Hernandez-Chifflet, Yang-Mills Glueballs as Closed Bosonic Strings, JHEP 02 (2017) 022 [arXiv:1611.09796] [INSPIRE].
J. Elias Miró, A.L. Guerrieri, A. Hebbar, J. Penedones and P. Vieira, Flux Tube S-matrix Bootstrap, Phys. Rev. Lett. 123 (2019) 221602 [arXiv:1906.08098] [INSPIRE].
J. Elias Miró and A. Guerrieri, Dual EFT bootstrap: QCD flux tubes, JHEP 10 (2021) 126 [arXiv:2106.07957] [INSPIRE].
C.A. Meyer, Light and exotic mesons, in Hadronic Structure — 14’th Annual HUGS at CEBAF, World Scientific, Singapore (2001) [DOI].
J.R. Pelaez, From controversy to precision on the sigma meson: a review on the status of the non-ordinary f0(500) resonance, Phys. Rept. 658 (2016) 1 [arXiv:1510.00653] [INSPIRE].
V.M. Banda Guzmán, R. Flores-Mendieta, J. Hernández and F.d.J. Rosales-Aldape, Spin and flavor projection operators in the SU(2Nf) spin-flavor group, Phys. Rev. D 102 (2020) 036010 [arXiv:2006.14169] [INSPIRE].
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: 2203.11950
Rights and permissions
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.
About this article
Cite this article
Albert, J., Rastelli, L. Bootstrapping pions at large N. J. High Energ. Phys. 2022, 151 (2022). https://doi.org/10.1007/JHEP08(2022)151
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2022)151