Nucleons from 5D Skyrmions Giuliano Panico Physikalisches Institut der Universität Bonn Planck 2009 26 May 2009 Based on G. P. and A. Wulzer 0811.2211 [hep-ph] and A. Pomarol and A. Wulzer 0807.0316 [hep-ph]
Outline 1 Introduction 2 The Model 3 Properties of the Model The Mesonic Sector The Baryonic Sector 4 Conclusions and Outlook
Introduction Introduction QCD is a very relevant theory for Nature, but is still far from being completely understood. We will consider the low-energy regime of QCD in which the theory is strongly coupled: numerical methods (lattice) limited theoretical tools (large N c expansion) From large-n c considerations, QCD in the IR limit can be described as a weakly-interacting theory of mesons Baryons arise automatically as solitons in the non-linear theory of mesons.
Introduction Introduction This scenario has been implemented in the Skyrme model and its generalizations [Skyrme; Adkins, Nappi, Witten;...] baryon observables related to the meson sector and completely fixed by the meson properties results in reasonable agreement with the experiments But... the energy scale of the skyrmion is comparable with the cut-off 1 ρ m ρ Λ 4πF π m ρ The model has a problem of calculability, an expansion parameter is lacking.
Introduction Introduction This scenario has been implemented in the Skyrme model and its generalizations [Skyrme; Adkins, Nappi, Witten;...] baryon observables related to the meson sector and completely fixed by the meson properties results in reasonable agreement with the experiments But... the energy scale of the skyrmion is comparable with the cut-off 1 ρ m ρ Λ 4πF π m ρ The model has a problem of calculability, an expansion parameter is lacking.
Introduction Introduction A possible solution is given by Holographic QCD models in 5D [Son, Stephanov (et al.); Da Rold, Pomarol] These are gauge theories on a 5D space provide a (partial) UV completion of the non-linear σ-model describing the pion have a natural expansion parameter related to the cut-off can be interpreted as QCD at large N c The calculability issue is under control in the mesonic and in the baryonic sector
The Model A 5D Model for QCD We want to describe QCD with two massless flavors. Model on AdS 5 space ds 2 = ( L/z ) 2 (ηµν dx µ dx ν dz 2 ) with bulk gauge group U(2) L U(2) R Chiral symmetry breaking at the IR boundary (L µ R µ ) z=zir = 0 (L µ5 + R µ5 ) z=zir = 0 Dirichlet condititions at the UV boundary L µ z=zuv = 0, R µ z=zuv = 0 UV AdS 5 sources lµ rµ U(2) L xu(2) R z=0 IR U(2) V U(2) A z=l
The Model A 5D Model for QCD The 5D bulk Lagrangian for the gauge fields is S g = d 5 x a(z) M 5 2 { Tr [L MN L MN] } + α2 2 L MN LMN + {L R} We must also introduce a Chern Simons term N c S CS = i 24π 2 d 5 x { ω 5 (L) ω 5 (R) }, where ω 5 (A) = 3 2ÂTr [F 2] + 1 4Â ( dâ) 2. required to reproduce the Adler Bardeen anomaly. At two derivatives order only 3 parameters: L, M 5, α
The Model A 5D Model for QCD The 5D bulk Lagrangian for the gauge fields is S g = d 5 x a(z) M 5 2 { Tr [L MN L MN] } + α2 2 L MN LMN + {L R} We must also introduce a Chern Simons term N c S CS = i 24π 2 d 5 x { ω 5 (L) ω 5 (R) }, where ω 5 (A) = 3 2ÂTr [F 2] + 1 4Â ( dâ) 2. required to reproduce the Adler Bardeen anomaly. At two derivatives order only 3 parameters: L, M 5, α
Properties of the Model The Mesonic Sector Mesons from Kaluza Klein Modes We can identify the mesons with the KK modes of the 5D gauge fields There is a massless Goldstone boson: the pion described by the usual χpt Lagrangian prediction for the decay constant F π = 2 M5 relations between the O(p 4 ) terms L 2 = 2L 1, L 3 = 6L 1, L 9 = L 10 L
Properties of the Model The Mesonic Sector Mesons from Kaluza Klein Modes The higher KK modes give the massive mesons and provide a (partial) UV completion of the theory. Vector Meson Dominance automatically holds Decay constants and meson couplings scale as F i M 5, g i 1/ M 5 Correct N c scaling recovered with α, L N 0 c and M 5 N c Excellent agreement with the experiments (error 10%) 1/L 340 MeV, M 5 L 0.017, α 0.94
Properties of the Model Baryons as 5D Skyrmions The Baryonic Sector The model admits non-trivial static solutions with conserved topological charge B identified with the baryon number B = 1 32π 2 d 3 [Lˆµˆν x dz εˆµˆν ˆρˆσ Tr Lˆρˆσ R ˆµˆν R ˆρˆσ] where ˆµ, ˆν,... label the four spatial coordinates Single baryon states are described by zero-mode time-dependent fluctuations of the static solution with B = 1. We need to quantize the collective coordinates corresponding to 3D rotations. Quantization analogous to the 4D skyrmion case.
Calculability Properties of the Model The Baryonic Sector The presence of a solution is due to the CS term which fixes the size of the skyrmion ρ γ 2/3 N c L γ = 16π 2 M 5 Lα From the meson fit γ 1 1 ρ 1 L 340 MeV The cut-off of the theory is determined by the CS term Λ 5 24π3 M 5 N 2/3 c 2 GeV The theory is perturbative and higher-dimensional operators are under control Λ 5 L Λ 5 ρ 5N 1/3 c
The Large N c Limit Properties of the Model The Baryonic Sector The mass M and the moment of inertia λ scale as in QCD M N c λ N c There are cancellations in the form factors G V E and GS M : J 0 S GS E N c J i,a V GV M N c N c J i S GS M N c N 0 c J 0,a V GV E N0 c J i,a A G A N c GE V (0) = 1/2 because nucleons are in the 1/2 isospin rep. Cancellation in G S M also happens in the naive quark model All the scalings analogous to the Skyrme model [T. D. Cohen al. [arxiv:0903.2662]]
Properties of the Model The Baryonic Sector Static Properties of the Nucleons Experiment AdS 5 Deviation M N 940 MeV 1130 MeV 20% µ S 0.44 0.34 30% µ V 2.35 1.79 31% g A 1.25 0.70 79% re,s 2 0.79 fm 0.88 fm 11% re,v 2 0.93 fm rm,s 2 0.82 fm 0.92 fm 12% rm,v 2 0.87 fm ra 2 0.68 fm 0.76 fm 12% µ p/µ n 1.461 1.459 0.1% RMS error 36% Significantly better agreement than the original Skyrme model Deviations compatible with the expected 1/N c corrections
Properties of the Model The Baryonic Sector Static Properties of the Nucleons Experiment AdS 5 Deviation M N 940 MeV 1130 MeV 20% µ S 0.44 0.34 30% µ V 2.35 1.79 31% g A 1.25 0.70 79% re,s 2 0.79 fm 0.88 fm 11% re,v 2 0.93 fm rm,s 2 0.82 fm 0.92 fm 12% rm,v 2 0.87 fm ra 2 0.68 fm 0.76 fm 12% µ p/µ n 1.461 1.459 0.1% RMS error 36% chiral limit two orders in 1/N c Significantly better agreement than the original Skyrme model Deviations compatible with the expected 1/N c corrections
Conclusions and Outlook Conclusions and Outlook Holographic QCD models can capture some features of QCD in a perturbative framework existence of an expansion parameter (analogous to 1/N c ) mesons are recovered as KK modes of the 5D fields the model is highly predictive (only 3 parameters) Baryons arise as stable soliton solutions calculable and insensitive to the UV cut-off fair agreement with the data (RMSE 36%) Future directions: Introduction of a pion mass (with A. Pomarol and A. Wulzer) Inclusion of the U(1) A anomaly and the η mass (some ideas with A. Wulzer)
Appendix Fit on the Mesonic Observables From a global fit on the mesonic observables we find 1/L 340 MeV M 5 L 0.017 α 0.94 Experiment (MEV) AdS 5 (MEV) Deviation m ρ 775 825 6% m ω 782 825 5% m a1 1230 1347 9% F π 87 88 1% F ρ 153 169 10% F ρ/f ω 0.88 0.94 6% g ρππ 6.0 5.4 10% L 9 6.9 10 3 6.2 10 3 10% L 10 5.5 10 3 6.2 10 3 12% Γ(ω πγ) 0.75 0.81 8% Γ(ω 3π) 7.6 6.7 11% Γ(ρ πγ) 0.068 0.077 13% Γ(ω πµµ) 8.2 10 4 7.3 10 4 10% Γ(ω πee) 6.5 10 3 7.3 10 3 12% RMS error 10%