TOWARDS THE P-WAVE N π SCATTERING AMPLITUDE IN THE ∆ (1232) Interpolating fields and spectra July 27, 2018 Giorgio Silvi Forschungszentrum J¨ ulich
COLLABORATORS Constantia Alexandrou (University of Cyprus / The Cyprus Institute) Giannis Koutsou (The Cyprus Institute) Stefan Krieg (Forschungszentrum J¨ ulich / University of Wuppertal) Luka Leskovec (University of Arizona) Stefan Meinel (University of Arizona / RIKEN BNL Research Center) John Negele (MIT) Srijit Paul (The Cyprus Institute / University of Wuppertal) Marcus Petschlies (University of Bonn / Bethe Center for Theoretical Physics) Andrew Pochinsky (MIT) Gumaro Rendon (University of Arizona) G. S. (Forschungszentrum J¨ ulich / University of Wuppertal) Sergey Syritsyn (Stony Brook University / RIKEN BNL Research Center) July 27, 2018 Slide 1
THE DELTA(1232) the first baryon resonance In nature: ∆ − ∆ 0 ∆ + ∆ ++ (u,d quarks) - mass ∼ 1232 MeV On the lattice: isospin symmetry The unstable ∆( 1232 ) decay predominantly to stable N π Study: Pion-Nucleon scattering J = 3 / 2 , I = 3 / 2, I 3 = + 3 / 2 Orbital angular momentum: L = 1 July 27, 2018 Slide 2
EXPERIMENTAL INFO N π ( → ∆( 1232 )) → N π completely elastic... [Shrestha,Manley (2012)] July 27, 2018 Slide 3
EXPERIMENTAL INFO N π ( → ∆( 1232 )) → N π .. but there are resonances completely elastic... nearby. J P Particle Γ N π [ MeV ] 3 / 2 + ∆( 1232 ) 112 . 4 ( 5 ) 3 / 2 + ∆( 1600 ) 18 ( 4 ) 1 / 2 − ∆( 1620 ) 37 ( 2 ) 3 / 2 − ∆( 1700 ) 36 ( 2 ) . . . . . . [Shrestha,Manley (2012)] July 27, 2018 Slide 3
L ¨ USCHER METHOD L¨ uscher quantization condition for baryons det [ M Jlm , J ′ l ′ m ′ − δ JJ ′ δ ll ′ δ mm ′ cot δ Jl ] = 0 [Gockeler et al. (2012)] This relation connect the energy E from a lattice simulation in a finite volume to the unknown phases δ Jl in the infinite volume via the calculable non-diagonal matrix M Jlm , J ′ l ′ m ′ (depends on symmetry) July 27, 2018 Slide 4
L ¨ USCHER METHOD L¨ uscher quantization condition for baryons det [ M Jlm , J ′ l ′ m ′ − δ JJ ′ δ ll ′ δ mm ′ cot δ Jl ] = 0 [Gockeler et al. (2012)] This relation connect the energy E from a lattice simulation in a finite volume to the unknown phases δ Jl in the infinite volume via the calculable non-diagonal matrix M Jlm , J ′ l ′ m ′ (depends on symmetry) Simplify! With a proper transformation, the matrix M Jlm , J ′ l ′ m ′ can be block diagonalized in the basis of the irreps Λ of the lattice. July 27, 2018 Slide 4
|0,0,1| |0,1,1| |1,1,1| momentum directions MOVING FRAMES Problem Due to quantized momenta p = 2 π n / L we have a energy levels spaced from each other. Chances of hitting the energy region of interest are low. July 27, 2018 Slide 5
MOVING FRAMES Problem Due to quantized momenta p = 2 π n / L we have a energy levels spaced from each other. Chances of hitting the energy region of interest are low. Solution: Moving frames! The Lorentz boost contracts the box giving a different effective value of the size L. Allow access to phase shift at different energies! |0,0,1| |0,1,1| |1,1,1| momentum directions July 27, 2018 Slide 5
ANGULAR MOMENTUM ON THE LATTICE In the continuum, states are classified according to angular momentum J and parity P label of the irreps of the symmetry group SU ( 2 ) July 27, 2018 Slide 6
ANGULAR MOMENTUM ON THE LATTICE In the continuum, states are classified according to angular momentum J and parity P label of the irreps of the symmetry group SU ( 2 ) On the lattice the rotational symmetry is broken [R.C. Johnson (1982) ] : The symmetry left is the O h group of 48 elements (13 axis of symmetry) July 27, 2018 Slide 6
ANGULAR MOMENTUM ON THE LATTICE In the continuum, states are classified according to angular momentum J and parity P label of the irreps of the symmetry group SU ( 2 ) On the lattice the rotational symmetry is broken [R.C. Johnson (1982) ] : The symmetry left is the O h group of 48 elements (13 axis of symmetry) For half-integer J we need the double cover O D h (96 elements) which include the negative identity (2 π rotation) July 27, 2018 Slide 6
ANGULAR MOMENTUM ON THE LATTICE In the continuum, states are classified according to angular momentum J and parity P label of the irreps of the symmetry group SU ( 2 ) On the lattice the rotational symmetry is broken [R.C. Johnson (1982) ] : The symmetry left is the O h group of 48 elements (13 axis of symmetry) For half-integer J we need the double cover O D h (96 elements) which include the negative identity (2 π rotation) Each of the infinite irreps J P in the continuum get mapped to one of the finite irreps Λ of the group O D h on the lattice. July 27, 2018 Slide 6
GROUND PLAN Frames, Groups & Irreps Λ (with ang. mom. content) + ) + ) Λ( J ) : π ( 0 − ) Λ( J ) : N ( 1 Λ( J ) : ∆( 3 P ref [ N dir ] Group N elem 2 2 O D G 1 g ( 1 2 , 7 2 , ... ) ⊕ G 1 u ( 1 2 , 7 H g ( 3 2 , 5 2 , ... ) ⊕ H u ( 3 2 , 5 ( 0 , 0 , 0 ) [ 1 ] 96 A 1 u ( 0 , 4 , ... ) 2 , ... ) 2 , ... ) h C D G 1 ( 1 2 , 3 G 1 ( 1 2 , 3 2 , ... ) ⊕ G 2 ( 3 2 , 5 ( 0 , 0 , 1 ) [ 6 ] 16 A 2 ( 0 , 1 , ... ) 2 , ... ) 2 , ... ) 4 v C D G ( 1 2 , 3 G ( 1 2 , 3 ( 0 , 1 , 1 ) [ 12 ] 8 A 2 ( 0 , 1 , ... ) 2 , ... ) 2 , ... ) 2 v C D G ( 1 2 , 3 G ( 1 2 , 3 2 , ... ) ⊕ F 1 ( 3 2 , 5 2 , ... ) ⊕ F 2 ( 3 2 , 5 ( 1 , 1 , 1 ) [ 8 ] 12 A 2 ( 0 , 1 , ... ) 2 , ... ) 2 , ... ) 3 v C 4v D C 2v D C 3v D axis of symmetry July 27, 2018 Slide 7
SINGLE HADRON OPERATORS Delta interpolators: ∆ ( 1 ) i µ = ǫ abc u a µ ( u bT C γ i u c ) (1) ∆ ( 2 ) i µ = ǫ abc u a µ ( u bT C γ i γ 0 u c ) (2) Nucleon interpolators: N ( 1 ) = ǫ abc u a µ ( u bT C γ 5 d c ) (3) µ N ( 2 ) = ǫ abc u a µ ( u bT C γ 0 γ 5 d c ) (4) µ Pion interpolator: π = ¯ d γ 5 u (5) July 27, 2018 Slide 8
PROJECTION METHOD how it works... O G D , Λ , r , m ( p )= d Λ r , r (˜ R φ ( p ) U − 1 R ∈ G D Γ Λ � R ) U ˜ ˜ [C. Morningstar et al. (2013)] ˜ g GD R to get an operators O G D , Λ , r , m ( p ) for a specific: July 27, 2018 Slide 9
PROJECTION METHOD how it works... O G D , Λ , r , m ( p )= d Λ r , r (˜ R φ ( p ) U − 1 R ∈ G D Γ Λ � R ) U ˜ ˜ [C. Morningstar et al. (2013)] ˜ g GD R to get an operators O G D , Λ , r , m ( p ) for a specific: momentum p July 27, 2018 Slide 9
PROJECTION METHOD how it works... O G D , Λ , r , m ( p )= d Λ r , r (˜ R φ ( p ) U − 1 R ∈ G D Γ Λ � R ) U ˜ ˜ [C. Morningstar et al. (2013)] ˜ g GD R to get an operators O G D , Λ , r , m ( p ) for a specific: momentum p double group G D and irreducible representation Λ July 27, 2018 Slide 9
PROJECTION METHOD how it works... O G D , Λ , r , m ( p )= d Λ r , r (˜ R φ ( p ) U − 1 R ∈ G D Γ Λ � R ) U ˜ ˜ [C. Morningstar et al. (2013)] ˜ g GD R to get an operators O G D , Λ , r , m ( p ) for a specific: momentum p double group G D and irreducible representation Λ row r and occurence m July 27, 2018 Slide 9
PROJECTION METHOD how it works... O G D , Λ , r , m ( p )= d Λ r , r (˜ R φ ( p ) U − 1 R ∈ G D Γ Λ � R ) U ˜ ˜ [C. Morningstar et al. (2013)] ˜ g GD R to get an operators O G D , Λ , r , m ( p ) for a specific: momentum p double group G D and irreducible representation Λ row r and occurence m is needed : 1 representation matrices Γ Λ July 27, 2018 Slide 9
PROJECTION METHOD how it works... O G D , Λ , r , m ( p )= d Λ r , r (˜ R φ ( p ) U − 1 R ∈ G D Γ Λ � R ) U ˜ ˜ [C. Morningstar et al. (2013)] ˜ g GD R to get an operators O G D , Λ , r , m ( p ) for a specific: momentum p double group G D and irreducible representation Λ row r and occurence m is needed : 1 representation matrices Γ Λ 2 elements ˜ R of the double group – rotations + inversions July 27, 2018 Slide 9
PROJECTION METHOD how it works... O G D , Λ , r , m ( p )= d Λ r , r (˜ R φ ( p ) U − 1 R ∈ G D Γ Λ � R ) U ˜ ˜ [C. Morningstar et al. (2013)] ˜ g GD R to get an operators O G D , Λ , r , m ( p ) for a specific: momentum p double group G D and irreducible representation Λ row r and occurence m is needed : 1 representation matrices Γ Λ 2 elements ˜ R of the double group – rotations + inversions 3 single/multi hadron operator φ ( p ) July 27, 2018 Slide 9
PROJECTION METHOD how it works... O G D , Λ , r , m ( p )= d Λ r , r (˜ R φ ( p ) U − 1 R ∈ G D Γ Λ � R ) U ˜ ˜ [C. Morningstar et al. (2013)] ˜ g GD R to get an operators O G D , Λ , r , m ( p ) for a specific: momentum p double group G D and irreducible representation Λ row r and occurence m is needed : 1 representation matrices Γ Λ 2 elements ˜ R of the double group – rotations + inversions 3 single/multi hadron operator φ ( p ) 4 proper transformation matrices U ˜ R July 27, 2018 Slide 9
OCCURENCES OF IRREPS It is possible to find the occurence ( ∼ multiplicity) m of the irrep Γ Λ in the transformation matrices U ˜ R using the character χ : [Moore,Fleming (2006) ] R ∈ G D χ Γ Λ (˜ R ) χ U (˜ 1 m = � R ) ˜ g GD D D D D O h C 4v C 2v C 3v NUCLEON [0,0,0] [0,0,1] [0,1,1] [1,1,1] G 1g G 1 G G U R̃ [4x4] G 1u G 1 G G July 27, 2018 Slide 10
OCCURENCES OF IRREPS D O h G 1g [0,0,0] G 1u DELTA H g U R̃ [12x12] H u D D D C 4v C 2v C 3v G 1 G G [0,0,1] [0,1,1] [1,1,1] G 1 G G F 1 G 1 G F 2 G 2 G G F 1 G 1 G F 2 G 2 G G July 27, 2018 Slide 11
Recommend
More recommend
