Light-Front Hadronic Models

Transcription

1 Light-Front Hadronic Models Siraj Kunhammed Wayne Polyzou Department of Physics and Astronomy The University of Iowa May 17, 2018 Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

2 Introduction Table of Contents 1 Introduction 2 3 Meson Model Cont.. 4 String Breaking Vertex Matrix elements of the vertex Vertex strength 5 Dynamics 6 Summary Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

3 Introduction Introduction A class of phenominological models of hadronic systems that have dual represenations as models of mesons and nucleons or constituent quarks. Quarks, antiquarks and gluons in confined color singlets interact by string breaking interaction. The degrees of freedom are gauge invariant sub systems of quarks, antiquarks and gluons Locally and globally color invarant, hence no color indices Model using mass operator with kinetic and potential term Interaction between confined color singelts is by string breaking Fully relativistic model with light front kinematic symmetry Assymptotically linear confinement Vertices can be calculated analytically The role of sea quarks in spectral calculations, lifetime calculations, scattering and electromagnetic calculations Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

4 Table of Contents 1 Introduction 2 3 Meson Model Cont.. 4 String Breaking Vertex Matrix elements of the vertex Vertex strength 5 Dynamics 6 Summary Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

5 Vectors in the irreducible representation space H mj are square integrable functions of the light-front momentum components p = (p +, p ) and the eigenvalues µ of the ˆn component of the light-front spin,j f ˆn, Poincaré transformation on the basis states can be expressed as U(Λ, a) (m, j), p, µ = j µ = j For a non-interacting quark-antiquark pair (m, j) p, µ ψ. (1) (m, j), p, µ e ia p p + p + Dj µ µ [B 1 P = p q + p q M 0 = P 2 Q = P/M 0. f (p /m)λb f (p/m)] (2) The momentum of the quark in the rest frame of the non-interacting quark-anti-quark pair is defined by transforming to the rest frame with a light-front preserving boost The light-front momentum fractions are defined by k = k f = k q = B 1 f (Q)p q k f q = B 1 f (Q)p q. ξ = k+ M 0 = p + q /P+ 1 ξ = p + q /P+ (3) which can be used to express k = p q ξp k + = M 0ξ (4) Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

6 The tensor product and two particle Poincaré irreducible light-front bases are related by νq, ν q,µq,µ q,m,µs With the interaction Poincaré transformation U(Λ, a) ( M, j) P, µ(l, s) = ( M 0, j) P, µ(l, s) = (m q, j q) p q, ν q (m q, j q) p q, ν q D jq νq µq [R fc (k q/m q)]d j q ν q µ q [R fc (k q/m q)] j q, µ q, j q, µ q s, µ s Y m l (ˆk) s, µ s, l, m j, µ p+ q p + q (ωq(kq) + ω q(k q)) ω q(k q)ω q(k q)(p q + + p + q ) (5) j µ = j M = M 0 + V (6) (M, j), P, µ e ia P P + P + Dj µ µ [B 1 f (P /m)λb f (P/m)] (7) Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

7 Meson Model Table of Contents 1 Introduction 2 3 Meson Model Cont.. 4 String Breaking Vertex Matrix elements of the vertex Vertex strength 5 Dynamics 6 Summary Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

8 Meson Model The confined singlets contain quark-antiquark pairs. The Mass operator with the quadratic confining interaction is, M c = k 2 + V c + mq 2 + k 2 + V c + m2 q. (8) The resulting bare meson mass eigenvalues are M nl V c = λ2 4 2 k + V0 (9) m 2q + λ(2n + l + 32 ) + m 2 q + λ(2n + l ) (10) Adding a spin term M nls m 2q + λ(2n + l + 32 ) + m 2 q + λ(2n + l ) + δ l0(a + b (s(s + 1) 3/2))) (11) 2 We can solve for a and b to get the correct π and ρ masses a = 1 4 (3mρ + mπ) m 2 q + λ 3 2 m 2 q + λ 3 2 ) (12) b = m ρ m π (13) Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

9 Meson Model Cont.. The harmonic oscillator wave functions in the coordinate and momentum-space have the form r n, l, m = R nl (r)y lm (ˆr). (14) k n, l, m = R nl (k)y lm (ˆk). (15) The RMS relative momentum and displacement of the pair is 2 r 2 1/2 = λ (2n + l + 3 λ 2 ) k2 1/2 = 2 (2n + l ) (16) In the limit 2n + l gets large, M 2λ r 2 1/2 Mass 2 vs L Mass vs r 14 n=0 n=1 n=2 n=3 12 n=4 4 s=0 s= Mass 2 (Gev) 8 6 Mass(Gev) L <r>' Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

10 String Breaking Table of Contents 1 Introduction 2 3 Meson Model Cont.. 4 String Breaking Vertex Matrix elements of the vertex Vertex strength 5 Dynamics 6 Summary Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

11 String Breaking Vertex String Breaking Vertex This allowes the confined singlets to interact. Its a string breaking interaction that assumes a quark-antiquark pair is created with equal probability at any point on the path between the quark and antiquark causing it to break up into a pir of confined singlets. The string breaking interaction in r space have the form 1 v(r 1, r 2, r 12; r) := γδ(r 2r 12) dηδ λ (r 1 ηr)δ λ (r 2 (1 η)r) (17) The expression for kernel of the vertex as an expansion in harmonic oscillator states is v(r 1, r 2, r 12; r) := g λ δ(r 2r12)( λ 4π )3 4π 3 ( 2 1 λ )3/2 dηe λ 2 r2 12 (η2 +(1 η) 2) η l 1 +2n 1 (1 η) l 2 +2n 2 0 λ ( 2 r12)2n 1 +l 1 +2n 2 +l 2 ψ n 1l 1m 1 (r 1)Yl 1 m (ˆr 1 12)ψ n2 l 2 m 2 (r 2)Y l 2 m (ˆr 2 12) (18) 2n 1!Γ(n 1 + l 1 + 2n 32 ) 2!Γ(n 2 + l ) Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

12 String Breaking Matrix elements of the vertex Matrix Elements of the Vertex Integrating the vertex against three oscillator states n 1, l 1, m 1, n 2, l 2, m 2, r 12 v n, l, m = ψ n 1 l 1 m (r 1 1 )ψ n 2 l 2 m (r 2 2 )v(r 1, r 2, r 12 ; r)ψ nlm (x)dr 1 dr 2 dr = g λ R nl (2r 12 )(2λ) 3/2 ( λ2 r 12 ) 2n 1 +l 1 +2n 2 +l 2 2n 1!Γ(n 1 + l ) 2n 2!Γ(n 2 + l ) e λ 4 r2 12 k 1 +k 2 =2r 1 (l 1 + 2n 1 )!(l 2 + 2n 2 )! k 1!k 2!(l 1 + 2n 1 k 1 )!(l 2 + 2n 2 k 2 )! ( )k 2 ( 1 2 )l 1 +2n 1 +l 2 +2n 2 2r + 1 M( r, 3 2 The following is added to include the spin dependence 2 λr12 + r, 4 )Y lm(ˆr 12 )Y l 1 m (ˆr 1 12 )Y l 2 m (ˆr 2 12 ) (19) 0, 0 1, m l, 1, m s 1, m s 1 2, µ 3, 1 2, µ 4 Y 1 m 1 (ˆr 12 ) (20) Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

13 String Breaking Vertex strength Vertex strength v(r) 2 vs r q 2 v 2 vs q n=0 n=1 n=2 n= n=0 n=1 n=2 n=3 n= r 2 v q 2 v r q Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

14 Dynamics Table of Contents 1 Introduction 2 3 Meson Model Cont.. 4 String Breaking Vertex Matrix elements of the vertex Vertex strength 5 Dynamics 6 Summary Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

15 Dynamics Dynamics The model Hilbert space is the orthogonal direct sum of the one-singlet Hilbert space with the tensor product of two copies of the one-singlet Hilbert space H = H Q (H Q H Q ) (21) The mass operator on H is given by M = M c0 + V (22) and the mass eigenvalue problem has the form ( Mc v 1:2 v 2:1 Mc1 2 + q2 12 M + c2 2 + q2 12 ) ( ψ1 ψ 2 ) ( ψ1 = η ψ 2 ) (23) I to label the one and two singlet channels, and 1 1 v 1:2 v 2:1 Ψ 1 = 0 (24) η M c η Mc1 2 + q Mc2 2 + q2 12 M n := M nls M n1,n 2,q 12 := q M2 + q n 1,l 1,s M2 (25) n 2,l 2,s 2 to label the mass eigenvalues of the bare meson systems. In this notation the mass eigenvalue problem has the form δ nm 1 q 2 dq n v 1:2 n 1, n 2, q n 1, n 2, q v 2:1 m m Ψ 1 = 0. (26) m n 1,n 0 η M 2 n η M n1 n 2 q Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

16 Summary Table of Contents 1 Introduction 2 3 Meson Model Cont.. 4 String Breaking Vertex Matrix elements of the vertex Vertex strength 5 Dynamics 6 Summary Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17

17 Summary Summary A simple relativistic quark model with a string breaking enteraction between confined color singlets Study the effect of sea quarks in the spectral calculation, lifetime calculations, scattering and Electromagnetic phenomena. See if it can be extended for baryon studies with diquark approximation THANK YOU Siraj Kunhammed Wayne Polyzou (Department of Physics Light-Front and Astronomy Hadronic Models The University of Iowa) May 17, / 17