Flavour blindness in QCD: Sigma - Lambda mixing Paul Rakow for - PowerPoint PPT Presentation

Flavour blindness in QCD: Sigma - Lambda mixing Paul Rakow for QCDSF Flavour blindness in QCD:Sigma - Lambda mixing – p. 1/4

QCDSF R Horsley, J Najjar, Y Nakamura, H Perlt, D Pleiter, PR, G Schierholz, A Schiller, H Stüben, and JM Zanotti PhysRevD.84.054509 (2011) arXiv:1412.0970 [hep-lat] (Lat14) arXiv:1411:7665 [hep-lat] Flavour blindness in QCD:Sigma - Lambda mixing – p. 2/4

Introduction The QCD interaction is flavour-blind. Neglecting electromagnetic and weak interactions, the only difference between flavours comes from the mass matrix. We investigate how flavour-blindness constrains hadron masses after flavour SU(3) is broken by the mass difference between the strange and light quarks, to help us extrapolate 2+1 flavour lattice data to the physical point. We have our best theoretical understanding when all 3 quark flavours have the same masses (because we can use the full power of flavour SU (3) ); nature presents us with just one instance of the theory, with m s /m l ≈ 25 . We are interested in interpolating between these two cases. Flavour blindness in QCD:Sigma - Lambda mixing – p. 3/4

Introduction Standard Theorist’s Approach: Action = Large Piece + Small Piece Treat the Small Piece as a perturbation. Apply this to QCD. Flavour blindness in QCD:Sigma - Lambda mixing – p. 4/4

Introduction This Talk Large Piece = Kinetic Terms + Gluon-Gluon Vertices + Quark-Gluon Vertices + Singlet Quark Mass Term Small Piece = Non-Singlet Quark Mass Terms Perturb about SU (3) symmetric QCD. Flavour blindness in QCD:Sigma - Lambda mixing – p. 5/4

Introduction This Talk Large Piece = Kinetic Terms + Gluon-Gluon Vertices + Quark-Gluon Vertices + Singlet Quark Mass Term Small Piece = Non-Singlet Quark Mass Terms Long history: M. Gell Man, Phys Rev 125 (1962) 1067. S. Okubo, Prog Theor Phys 27 (1962) 949. S. R. Beane, K. Orginos and M. J. Savage, Phys. Lett. B654 (2007) 20 [ arXiv:hep-lat/0604013 ]. Flavour blindness in QCD:Sigma - Lambda mixing – p. 6/4

Introduction This Talk Large Piece = Kinetic Terms + Gluon-Gluon Vertices + Quark-Gluon Vertices + Singlet Quark Mass Term Small Piece = Non-Singlet Quark Mass Terms Not as familiar as chiral perturbation theory, but useful for organising and analysing the data. Flavour blindness in QCD:Sigma - Lambda mixing – p. 7/4

Quark Masses Notation 1 m 3( m u + m d + m s ) ≡ δm u ≡ m u − m δm d ≡ m d − m δm s m s − m ≡ δm u + δm d + δm s = 0 1 m l ≡ 2( m u + m d ) δm l m l − m ≡ Flavour blindness in QCD:Sigma - Lambda mixing – p. 8/4

Quark Masses The quark mass matrix is 0 1 m u 0 0 B C = M 0 m d 0 B C @ A 0 0 m s 0 1 0 1 0 1 1 0 0 1 0 0 − 1 0 0 B C A + 1 B A + 1 C B C = m 2 ( δm u − δm d ) 2 δm s 0 1 0 0 − 1 0 0 − 1 0 B C B C B C @ @ @ A 0 0 1 0 0 0 0 0 2 M has a flavour singlet part (proportional to I ) and a flavour octet part, proportional to λ 3 , λ 8 . In clover case, the singlet and non-singlet parts of the mass matrix renormalise differently. Flavour blindness in QCD:Sigma - Lambda mixing – p. 9/4

Flavour Hierarchy Large Piece Kinetic Terms = + Gluon-Gluon Vertices + Quark-Gluon Vertices + Singlet Quark Mass Term Small Piece Non-Singlet Quark Mass Terms = All terms in Large Piece are flavour singlets, leave SU (3) unbroken. Small Piece is pure flavour octet. Higher SU (3) representations completely absent from QCD action. Flavour blindness in QCD:Sigma - Lambda mixing – p. 10/4

Flavour Hierarchy Higher representations of SU (3) are absent from the QCD action, but they appear at higher orders in the perturbation. Square an octet — generates 27-plet. δm 0 1 1 q δm 1 8 8 q δm 2 1 1 8 27 2! 8 × 9 = 36 q 1 δm 3 1 8 10 27 64 10 3! 8 × 9 × 10 = 120 q Flavour blindness in QCD:Sigma - Lambda mixing – p. 11/4

Flavour Hierarchy Decuplet mass matrix 10 ⊗ 10 = 1 ⊕ 8 ⊕ 27 ⊕ 64 ∆ 0 ∆ + ∆ ++ Σ ∗ 0 Σ ∗ + Ξ ∗ 0 ∆ − Σ ∗− Ξ ∗− Ω − SU (3) 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 2 8 − 1 − 1 − 1 − 1 3 9 27 3 3 3 − 5 − 5 − 5 − 3 − 3 4 64 − 1 − 1 − 1 − 1 4 4 4 − 6 − 6 Flavour blindness in QCD:Sigma - Lambda mixing – p. 12/4

Flavour Hierarchy 4 M ∆ + 3 M Σ ∗ + 2 M Ξ ∗ + M Ω = 13 . 82 GeV singlet + M Ξ ∗ + M Ω − 2 M ∆ = 0 . 742 GeV octet 4 M ∆ − 5 M Σ ∗ − 2 M Ξ ∗ + 3 M Ω = − 0 . 044 GeV 27 − plet − M ∆ + 3 M Σ ∗ − 3 M Ξ ∗ + M Ω = − 0 . 006 GeV 64 − plet , [PDG masses] Strong Hierarchy: 1 8 27 64 ( m s − m l ) 0 ( m s − m l ) 1 ( m s − m l ) 2 ( m s − m l ) 3 Flavour blindness in QCD:Sigma - Lambda mixing – p. 13/4

Strategy Keep Large Piece constant, Vary Small Piece until we reach the physical point. Flavour blindness in QCD:Sigma - Lambda mixing – p. 14/4

Strategy Start from a point with all 3 sea quark masses equal, m u = m d = m s ≡ m 0 and extrapolate towards the physical point, keeping the average sea quark mass m ≡ 1 3( m u + m d + m s ) constant. Starting point has m 0 ≈ 1 3 m phys s As we approach the physical point, the u and d become lighter, but the s becomes heavier. Pions are decreasing in mass, but K and η increase in mass as we approach the physical point. Flavour blindness in QCD:Sigma - Lambda mixing – p. 15/4

Singlet Quantities Consider a flavour singlet quantity (eg plaquette P ) at the symmetric point ( m 0 , m 0 , m 0 ) . ∂P = ∂P = ∂P . ∂m u ∂m d ∂m s If we keep m u + m d + m s constant, dm s = − dm u − dm d so ∂P ∂P ∂P dP = dm u + dm d + dm s = 0 ∂m u ∂m d ∂m s The effect of making the strange quark heavier exactly cancels the effect of making the light quarks lighter, so we know that P must have a stationary point at the symmetrical point. Flavour blindness in QCD:Sigma - Lambda mixing – p. 16/4

Singlet Quantities Any permutation of the quarks, eg u ↔ s, u → d → s → u doesn’t really change physics, it just renames the quarks. Group S 3 , permutations of three objects, symmetry group of the equilateral triangle. Any quantity unchanged by all permutations will also be flat at the symmetric point. Flavour blindness in QCD:Sigma - Lambda mixing – p. 17/4

Singlet Quantities 2 M ∆ + M Ω 2( M ∆ + M Σ ∗ + M Ξ ∗ ) 2( M N + M Σ + M Ξ ) M Σ + M Λ M Σ ∗ Flavour blindness in QCD:Sigma - Lambda mixing – p. 18/4

Singlet Quantities X 2 ( M 2 π + 2 M 2 = K ) / 3 π X ρ = ( M ρ + 2 M K ∗ ) / 3 X N = ( M N + M Σ + M Ξ ) / 3 X ∆ = (2 M ∆ + M Ω ) Multiplet Centre-of-Mass Use octet baryons ( X N ) to set scale for the other three multiplets. Flavour blindness in QCD:Sigma - Lambda mixing – p. 19/4

Singlet Quantities 1.50 S = ∆ κ 0 =0.12090 S = ρ 1.25 S = π 1.00 X S /X N 0.75 0.50 0.25 0.00 0.00 0.25 0.50 0.75 1.00 1.25 2 /X π 2 M π X S so flat because we keep m u + m d + m s constant. Choose initial m 0 to make X S /X N equal to physical value. Flavour blindness in QCD:Sigma - Lambda mixing – p. 20/4

SU(3) classification Classify physical quantities by SU (3) and permutation group S 3 (which is a subgroup of SU(3)). Classify quark mass polynomials in same way. Quantity of Known Symmetry = Polynomials of Matching Symmetry Taylor expansion about ( m 0 , m 0 , m 0 ) strongly constrained by symmetry. Flavour blindness in QCD:Sigma - Lambda mixing – p. 21/4

SU(3) classification Polynomial S 3 SU (3) 1 A 1 1 � ( m − m 0 ) A 1 1 E + δm s � 8 ( δm u − δm d ) E − 8 � ( m − m 0 ) 2 A 1 1 E + ( m − m 0 ) δm s 8 ( m − m 0 )( δm u − δm d ) E − 8 δm 2 u + δm 2 d + δm 2 A 1 1 27 � s 3 δm 2 s − ( δm u − δm d ) 2 E + 8 27 � δm s ( δm d − δm u ) � E − 8 27 Flavour blindness in QCD:Sigma - Lambda mixing – p. 22/4

SU(3) classification Polynomial S 3 SU (3) ( m − m 0 ) 3 A 1 1 ( m − m 0 ) 2 δm s E + 8 ( m − m 0 ) 2 ( δm u − δm d ) E − 8 ( m − m 0 )( δm 2 u + δm 2 d + δm 2 s ) A 1 1 27 ˆ 3 δm 2 s − ( δm u − δm d ) 2 ˜ E + ( m − m 0 ) 8 27 ( m − m 0 ) δm s ( δm d − δm u ) E − 8 27 δm u δm d δm s � A 1 1 27 64 δm s ( δm 2 u + δm 2 d + δm 2 E + s ) 8 27 64 � ( δm u − δm d )( δm 2 u + δm 2 d + δm 2 s ) E − 8 27 64 � ( δm s − δm u )( δm s − δm d )( δm u − δm d ) A 2 10 10 64 � Flavour blindness in QCD:Sigma - Lambda mixing – p. 23/4

SU(3) classification Flavour blindness in QCD:Sigma - Lambda mixing – p. 24/4

SU(3) classification ( m u + m d + m s ) = const, m q ≥ 0 Flavour blindness in QCD:Sigma - Lambda mixing – p. 25/4

SU(3) classification Flavour blindness in QCD:Sigma - Lambda mixing – p. 26/4

SU(3) classification + � A E E A 1 2 0 ( Æ m ) q 1 ( Æ m ) q 2 ( Æ m ) q 3 ( Æ m ) q Flavour blindness in QCD:Sigma - Lambda mixing – p. 27/4

2 + 1 Simulation Tree-level Symanzik glue, β = 5 . 50 Clover Fermions, non-pert c SW . To to keep the action highly local, the hopping terms use a stout smeared link (‘fat link’) with α = 0 . 1 ‘mild smearing’ for the Dirac kinetic term and Wilson mass term. Symmetric point κ 0 = 0 . 12090 24 3 × 48 lattices and 32 3 × 64 lattices Flavour blindness in QCD:Sigma - Lambda mixing – p. 28/4