Hyeon-Dong Son Inha University In collaboration with Prof. H.-Ch. Kim

Transcription

1 Energy-momentum Tensor Form Factors and Transverse Charge Densities of the Pion and the Kaon from the Instanton Vacuum Hyeon-Dong Son Inha University In collaboration with Prof. H.-Ch. Kim APFB 2014 April 7-11, 2014 in Hahndorf, SA, Australia

2 Outline Motivation Energy momentum tensor form factors & Transverse Charge Densities Nonlocal chiral quark model from the Instanton Vacuum Numerical Result Summary

3 Motivation The internal structure of hadrons Generalized Parton Distributions( GPDs) Generalized Form Factors( GFFs) A10, A20, B10, B20... Energy-momentum tensor form factors Transverse Charge Densities( TChDs) : Spatial distribution at the transverse momentum plane SU( 3) symmetry breaking in the mesonic sector: pi, K Reliability of a model: Low-energy theorem

4 Generalized Form Factors [R.L. Jaffe, arxiv:hep-ph/ ] The Hadronic Matrix elements of each operator define the generalized form factors.

5 Generalized Form Factors n=1 n=2

6 n=2: Energy-momentum Tensor Form Factors H.Pagels: Energy-Momentum Structure Form Factors of Particles ( Phys.Rev.144,1250.,1966) Momentum distribution of the partons inside of hadrons Accessible from the vector GPDs by using the polynomiality Lattice calculation: POS( LAT2005) 360 D. Brommel et al( 2005) PhD Thesis, D. Brommel( 2007) PRL.101, D. Brommel et al( 2008) ( QCDSF-UKQCD Collaborations)

7 n=2: Energy-momentum Tensor Form Factors [H.Pagels,Phys.Rev.144,1250.,1966] Θ µν [J.F. Donoghue and H. Leutwyler, Z.Phys.C( 1991) 52, 343] φ(p i ) φ(p f )

8 Transverse Charge Densities P-pole parametrization Transverse charge density Polarized quark spin structure

9 Transverse Charge Densities P-pole parametrization Transverse charge density Logarithmically divergent at b=0 for p=1. Singular when p<1.5 Polarized quark spin structure

10 Transverse Charge Densities P-pole parametrization Transverse charge density Logarithmically divergent at b=0 for p=1. Singular when p<1.5 p=1 for Ano, p=1.6 for Bno Polarized quark spin structure

11 Transverse Charge Densities P-pole parametrization Transverse charge density Polarized quark spin structure Spatial distribution of the quark polarized in the transverse plane inside the pion and the kaon.

12 Nonlocal Chiral Quark Model From the Instanton Vacuum The chiral effective action derived from the instanton vacuum No free parameter -Average Instanton size & separation Nonlocality -Momentum-dependent dynamical quark mass Nicely reproduces pion properties: Fpi, EMFF SU( 3) symmetry breaking [D. Diakonov, Instantons at work, arxiv:hep-ph/ v4]

13 Nonlocal Chiral Quark Model & Gauge Invariance The gauge invariance is broken by the nonlocal interaction of the model The ordinary Noether currents are not conserved due to the broken gauge invariance The gauge invariance is restored: The conserved vector current [M.M. Musakhanov and H.-Ch. Kim, Phys. Lett. B572( 2003) ]

14 Energy-momentum Tensor Operator for the Nonlocal Chiral Quark Model

15 Calculation of the matrix element F a F b k b k c k b k c k a p i p f p i p f F c k c F d k b k a p i p f pi p f k a

16 Pion Form Factors m [MeV] f [MeV] p hr 2 i[fm] p M [GeV]

17 Pion Form Factors A10 ( EM) [S.-i.Nam & H.-Ch.Kim Phys.Rev.D77( 2008),094014] Local Nonlocal Total Exp. [61] hr 2 i 1=2 [fm] :672 0:008 þ hr 2 i 1=2 Kþ [fm] :560 0:031 h 2 i 2 A20 ( present work) m [MeV] f [MeV] p hr 2 i[fm] p M [GeV] p

18 Kaon Form Factors p m K [MeV] f K [MeV] p hr 2 i[fm] p M K [GeV]

19 Kaon Form Factors

20 Transverse Charge Densities of the Pion

21 Transverse Charge Densities of the Pion [G.A. Miller, Phys.Rev.C 79, ( 2009) ] Singular at b = 0

22 Transverse Charge Densities of the Kaon

23 Quark Spin Structure p p n=1 n=2 n=1 n=2 n=1 n=2 s=( 1,0) A,u [S.-i.Nam & H.-Ch.Kim Phys.Lett.B707( 2012),546] 10 (0) M A,u [GeV] B,u (0) M B,u [GeV] hb,u 10 y i[fm] A,u 20 (0) M A,u [GeV] B,u (0) M B,u [GeV] hb,u 20 y i[fm] A K,u 10 (0) M A K,u [GeV] B K,u 10 (0) M 10 B K,u [GeV] hb K,u y i[fm] (0) M A K,u [GeV] B K,u 20 (0) M 20 B K,u [GeV] hb K,u y i[fm] A K,u A K,s 10 (0) M A K,s [GeV] B K,s 10 (0) M 10 B K,s [GeV] hb K,s y i[fm] (0) M A K,s [GeV] B K,s 20 (0) M 20 B K,s [GeV] hb K,s y i[fm] A K,s

24 Quark Spin Structure of the Pion n=1 n=2

25 Quark Spin Structure of the Kaon n=1 n=2

26 Quark Spin Structure of the Kaon n=1 n=2

27 Summary and outlook Generalized FFs A20 of the pion and the kaon have been studied. Transverse charge density : Singular behavior at the center n=2 distribution is narrower than n=1. A22 is under way. Generalized form factors with higher n ( n=3, n=4) are under investigation.

28 Thank you very much!

29 Calculation of the matrix element

30 Nonlocal Chiral Quark Model From the Instanton Vacuum

31 Nonlocal Chiral Quark Model From the Instanton Vacuum

32 Calculation of the matrix element The Quark-Pseudoscalar meson interactions

33 Quark Spin Structure of the Pion

34 Quark Spin Structure of the Kaon

35 Quark Spin Structure of the Kaon