Chiral and angular momentum content of rho and rho mesons from dynamical lattice calculations L. Ya. Glozman Institut für Physik, FB Theoretische Physik, Universität Graz With Christian Lang and Markus Limmer. PRL 0 (00) 0 Few Body Systems (00) 00. [hep-lat] - submitted to PRL L.Ya. Glozman
Contents of the Talk Chiral symmetry and origin of hadron mass Generalized t Hooft Model Chiral and S+ L J content of mesons on lattice p.0/
What is a hadron mass origin in QCD? Gell-Mann - Levy sigma model, Nambu - Jona-Lasinio mechanism, many "Bag-like" microscopical models,...: Dynamical chiral symmetry breaking is the source of the hadron mass in the light quark sector. Is it true? Or, better to say, is it entirely true?
Low and high lying baryon spectra. 000 [MeV] 00 * * 000 * * * * * * * * * * * * 00 000 Low-lying spectrum: spontaneous breaking of chiral symmetry dominates physics. High-lying spectrum: parity doubling is indicative of EFFECTIVE chiral symmetry restoration. Mass of excited baryons comes mostly from the chirally symmetric dynamics; baryons decouple from the quark condensate of the vacuum. L.Ya.G., 000; T.D.Cohen and L.Ya.G., 00; L.Ya.G., Phys. Rep. (00)
Low and high lying meson spectra. An alternative: N = n+l without chiral symmetry (Afonin, 00; Klempt and Zaitsev, 00; Shifman and Vainshtein, 00) Angular momenta ( S+ L J )- like in the nonrelativistic -body quantum mechanics. Requires L,S to be separately conserved quantum numbers! M............................. π f a ρ a h ω b f a ρ η f π ω ρ a h ω b f a ρ η f π ω ρ a h η 0 0 0 0 ω... The high-lying mesons are from LEAR. Systematic appearance of SU() L SU() R and U() A multiplets (L.Ya.G., 00, 00). Missing chiral partners for highest spins. Large symmetry: N = n+j plus chiral symmetry(l.ya.g. and A.V. Nefediev, 00).
Do missing chiral partners exist? If yes, why didn t Anisovich, Bugg, Sarantsev observe them in 000-00? pp annihilation around GeV. - Strong centrifugal repulsion! Observed states have positive parity, f and a, and are produced in the L = partial wave. All missing states,η,ρ,π,ω, have negative parity and can be produced only in the L = partial wave. Additional centrifugal suppression 0-00 times! -00-0 -00 ln L -0-00 -0-00..... M(pp ) (GeV) L.Ya.G., A. Sarantsev, PRD (00) 00
B ± Nπ decays (L.Ya.G., PRL (00) 0) If a baryon is a member of an approximate chiral multiplet, then its decay into Nπ must be suppressed, (f BNπ /f NNπ ). If, on the contrary, this excited nucleon has no chiral partner and hence its mass is due to chiral symmetry breaking in the vacuum, then it should strongly decay into Nπ, (f BNπ /f NNπ ). Spin Chiral multiplet Representation (f B+ Nπ/f NNπ ) (f B Nπ/f NNπ ) / N + (0) N () (/,0) (0,/) 0. 0.0 / N + (0) N (0) (/,0) (0,/) 0.000 0.0 / N + (0) N (00) (/,0) (0,/) 0.0 0. / N + (0) N () (/,0) (0,/) 0. 0.0 / N + (?) N (0)?? 0.000 / N + (0) N (0)? 0.0000 0.000000 / N + (?) N (00)?? 0.0000000 / N (0) no chiral partner. A 00% correlation of decays with the parity doublet patterns
Meson Spectra (R.F. Wagenbrunn and L.Ya.G., 00) (, ) a J = 0 (, ) b 0 0 0 0, 0 + 0, 0 ++ 0, 0 +, 0 ++ (0, 0) (, ) a J = (, ) b (0, ) (, 0) 0 0 0 0 0 0 0 0 0, ++ 0,, + 0, ++ 0, +, ++, ++,
Chiral and S+ L J content of mesons on lattice One needs a complete and orthogonal basis for the qq component in mesons. Representations of SU() L SU() R is such a basis. J = 0 (/,/) a :,0 + 0,0 ++ (/,/) b :,0 ++ 0,0 +, Even J > 0 (0,0) : 0,J 0,J ++ (/,/) a :,J + 0,J ++ (/,/) b :,J ++ 0,J + (0,) (,0) :,J ++,J Odd J > 0 (0,0) : 0,J ++ 0,J (/,/) a :,J + 0,J (/,/) b :,J 0,J + (0,) (,0) :,J,J ++ L.Ya.G., Phys. Lett. B (00)
Chiral and S+ L J content of mesons on lattice Example: ρ(i,j PC =, ) (,0)+(0,) : O V = qτγ i q = Rτγ i R+ Lτγ i L Chiral partners: (/,/) b : O T = qτσ 0i q = Rτσ 0i L+ Lτσ 0i R (,0)+(0,) : ρ(, ) a (, ++ ) (/,/) b : ρ(, ) h (0, + ) T. D. Cohen and X. Ji, PRD () 0 L.Ya.G., Phys. Lett. B (00)
Chiral and S+ L J content of mesons on lattice A unitary transformation exists from the chiral basis to the {I; S+ L J } basis (L.Ya.G. and A. V. Nefediev, PRD (00) 000; PRD 0 (00) 00): R;IJ PC = LS χ RPI λ q λ q λ q λ q L+ J + CSΛ λ q CJΛ λ q L0SΛ I; S+ L J. ρ : (0,)+(,0); = ρ : (/,/) b ; = ; S + ; D, ; S ; D. However: a : (0,)+(,0); ++ = ; P h : (/,/) b ;0 + = 0; P
Chiral and S+ L J content of mesons on lattice Some elements of lattice technology. Assume we have a hadron with excitation energies n =,,,... C(t) ij = O i (t)o j (0) = n a (n) i a (n) j e E(n) t () where The generalized eigenvalue problem: a (n) i = 0 O i n. Ĉ(t) ij u (n) j = λ (n) (t,t 0 )Ĉ(t 0) ij u (n) j. () Each eigenvalue and eigenvector corresponds to a given state. If a basis O i is complete enough, one extracts energies and "wave functions" of all states. C(t) ij u (n) j C(t) kj u (n) j = a(n) i a (n) k. ()
Smear the local interpolators in spatial directions over the range R in a gauge-invariant way. Then you will study a hadron "wave function" at the resolution scale fixed by R. Chiral and S+ L J content of mesons on lattice We want to study ρ = ρ(0) and its first excitation ρ = ρ(0).we need energies, chiral as well as the angular momentum decomposition of the states. Then a sufficient basis of interpolators: (,0)+(0,) : O V = q(x)τγ i q(x) = R(x)τγ i R(x)+ L(x)τγ i L(x) (/,/) b : O T = q(x)τσ 0i q(x) = R(x)τσ 0i L(x)+ L(x)τσ 0i R(x) If local interpolators are used, then we extract the "wave functions" at the origin (more exactly, at the scale µ (lattice spacing) fixed by the lattice spacing): a (n) i = 0 O i (µ) n. But we want to know the "wave functions" at the infrared scales, where mass is generated! What to do?
Chiral and S+ L J content of mesons on lattice Gauge-invariant Gaussian smearing and resolution scale R definition. QUARK QUARK GAUGE CONNECTOR ANTIQUARK ANTIQUARK R SPATIAL DIRECTIONS
Chiral and S+ L J content of mesons on lattice Lattice size. fm. Two light sea flavors. Lüscher-Weisz gauge action. Chirally Improved Dirac operator ( C.Gattringer - 00; C. Gattringer, I. Hip, C. Lang - 00). Two smearings sizes: n = 0. fm and w = 0. fm. Four interpolating operators. O n V = u nγ i d n, O w V = u w γ i d w, () O n T = u nγ t γ i d n, O w T = u w γ t γ i d w. ().. m V [GeV].. 0. 0. 0. 0.0 0.00 0.0 0.0 0. 0.0 0. 0.0 0. m π [GeV ] ρ ρ exp.
Chiral and S+ L J content of mesons on lattice..0 a V / a T..0 0. ρ ρ 0.0 0.0 0. 0. 0. 0. 0. 0. 0. 0. R [fm] ρ: at the scale of the ρ size - strong chiral symmetry breaking: (0,)+(,0) (/,/) b. ρ 0. S 0.0 D S ρ : at the scale of the ρ size ( fm) - weak chiral symmetry breaking: (0,)+(,0) (/,/) b 0. ρ(0) in the infrared is mostly (/,/) b. Its chiral partner likely is h (0). ρ / S / D ρ is not a radial excitation of ρ ( S ). The quark model and similar pictures are ruled out for ρ.