Quark Structure of the Pion

Transcription

1 Quark Structure of the Pion Hyun-Chul Kim RCNP, Osaka University & Department of Physics, Inha University Collaborators: H.D. Son, S.i. Nam Progress of J-PARC Hadron Physics, Nov. 30-Dec. 01, 2014

2 Interpretation of the Form factors Non-Relativistic picture of the EM form factors Electromagnetic form factors 3-D Fourier Transform Charge & Magnetisation Densities Schroedinger Eq. & Wave functions 2

3 Interpretation of the EMFFs Traditional interpretation of the nucleon form factors Z F 1 (Q 2 )= d 3 xe iq x (r) (r) = X (r) (r) However, the initial and final momenta are different in a relativistic case. Thus, the initial and final wave functions are different. Probability interpretation is wrong in a relativistic case! We need a correct interpretation of the form factors Belitsky & Radyushkin, Phys.Rept. 418, 1 (2005) G.A. Miller, PRL 99, (2007) 3

4 Interpretation of the EMFFs Modern understanding of the form factors Transverse Charge densities (b) Lorentz invariant: independent of any observer. Infinite momentum framework 1 (b) = X q e 2 q Z dxf q q (x, b) GPDs Form factors F (q 2 )= Z d 2 be iq b (b) 4

5 Hadron Tomography b y b y b y 1 Q 2 1 Q 2 b xp b xp x b x x b x x b x Momentum fraction ρ(b ) R dx 0 q(x, b ) q(x) 0 b x Transverse densities of Form factors GPDs b 0 x Structure functions D. Brömmel, Dissertation (Regensburg U.) 5

6 Generalised Parton Distributions Probes are unknown for Tensor form factors and the Energy-Momentum Tensor form factors! Pion Pion 6

7 Generalised Parton Distributions Probes are unknown for Tensor form factors and the Energy-Momentum Tensor form factors! Pion Form factors as Mellin moments of the GPDs Pion 6

8 What we know about the Pion Experimentally, we know about the pion Pion Mass = MeV Pion Spin = 0 Theoretically pseudo-goldstone boson The lowest-lying meson (1 q + 1anti-q + sea quarks + gluons + ) The structure of the pion is still not trivial!

9 The spin structure of the Pion Vector & Tensor Form factors of the pion x Pion: Spin S=0 u b xp z Longitudinal spin structure is trivial. h (p 0 ) 3 5 (p)i = 0 y d What about the transversely polarized quarks inside a pion? Internal spin structure of the pion 8

10 The spin distribution of the quark Spin probability densities in the transverse plane A n0 : Vector densities of the pion, B n0 : Tensor densities of the pion 9

11 The spin distribution of the quark Spin probability densities in the transverse plane A n0 : Vector densities of the pion, B n0 : Tensor densities of the pion 9

12 The spin distribution of the quark Spin probability densities in the transverse plane A n0 : Vector densities of the pion, B n0 : Tensor densities of the pion Vector and Tensor form factors of the pion h (p f ) µ ˆQ (p i )i =(p i + p f )A 10 (q 2 ) 9

13 Nonlocal chiral quark model Gauged Effective Nonlocal Chiral Action S e = N c Tr ln hi /D + im + i p p i M(iD, m)u 5 M(iD, m) 0.3fm,R 1fm µ 600 MeV D µ µ i µ V µ The nonlocal chiral quark model from the instanton vacuum Fully relativistically field theoretic model. Derived from QCD via the Instanton vacuum. Renormalization scale is naturally given. No free parameter Dilute instanton liquid ensemble D. Diakonov & V. Petrov Nucl.Phys. B272 (1986) 457 H.-Ch.K, M. Musakhanov, M. Siddikov Phys. Lett. B 608, 95 (2005). Musakhanov & H.-Ch. K, Phys. Lett. B 572, (2003)

14 Nonlocal chiral quark model Gauged Effective Nonlocal Chiral Action S e = N c Tr ln hi /D + im + i p p i M(iD, m)u 5 M(iD, m) U 5 = U(x) U (x) M(iD, m) =M 0 F 2 (id)f(m) =M 0 F 2 (id) F (k) =2t apple I 0 (t)k 1 (t) I 1 (t)k 0 (t) =1+ i f 5( a a ) "r 1+ m2 d 2 1 t I 1(t)K 1 (t) # m d 1 f 2 t = k 2 M GeV : It is determined by the gap equation. ( a a ) 2 + d GeV D. Diakonov & V. Petrov Nucl.Phys. B272 (1986) 457 H.-Ch.K, M. Musakhanov, M. Siddikov Phys. Lett. B 608, 95 (2005). Musakhanov & H.-Ch. K, Phys. Lett. B 572, (2003)

15 EM Form factor of the pion EM form factor (A 10 ) h (p f ) µ ˆQ (p i )i =(p i + p f )A 10 (q 2 ) S.i. Nam & HChK, Phys. Rev. D77 (2008)

16 EM Form factor of the pion EM form factor (A 10 ) h (p f ) µ ˆQ (p i )i =(p i + p f )A 10 (q 2 ) : Local vertices S.i. Nam & HChK, Phys. Rev. D77 (2008)

17 EM Form factor of the pion EM form factor (A 10 ) h (p f ) µ ˆQ (p i )i =(p i + p f )A 10 (q 2 ) : Local vertices : Nonlocal vertices S.i. Nam & HChK, Phys. Rev. D77 (2008)

18 EM Form factor of the pion r 2 =0.675 fm r 2 =0.672 ± fm (Exp) M(Phen.): GeV M(Lattice): GeV M(XQM): GeV S.i. Nam & HChK, Phys. Rev. D77 (2008)

19 Tensor Form factor of the pion + (p f ) (0) µ (0) + (p i ) = pµ q q µ p B 10 (Q 2 ) m p =(p f + p i )/2, q= p f p i S.i. Nam & H.-Ch.K, Phys. Lett. B 700, 305 (2011). 14

20 Tensor Form factor of the pion RG equation for the tensor form factor Brommel et al. PRL101 B 10 (Q 2,µ)=B 10 (Q 2,µ 0 ) apple (µ) (µ 0 ) N f p-pole parametrization for the form factor B 10 (Q 2 ) = B 10 (0) apple1+ Q2 pm 2 p p S.i. Nam & H.-Ch.K, Phys. Lett. B 700, 305 (2011). 15

21 Tensor Form factor of the pion S.i. Nam & H.-Ch.K, Phys. Lett. B 700, 305 (2011). 16

22 Tensor Form factor of the pion S.i. Nam & H.-Ch.K, Phys. Lett. B 700, 305 (2011). 17

23 Spin density of the quark Unpolarized Polarized spin Polarization 18

24 Spin density of the quark Significant distortion appears for the polarized quark! Results are in a good agreement with the lattice calculation! 19

25 Spin density of the quark 20

26 Stability of the pion Isoscalar vector GPDs of the pion 2 ab H I=0 (x,, t) = 1 2 Z d 2 eix (P n) h a (p 0 ) ( n/2)/n[ n/2, n/2] ( n/2) b (p)i The second moment of the GPD Z dx xh I=0 (x,, t) =A 20 (t)+4 2 A 22 (t) 21

27 Stability of the pion Isoscalar vector GPDs of the pion 2 ab H I=0 (x,, t) = 1 2 Z d 2 eix (P n) h a (p 0 ) ( n/2)/n[ n/2, n/2] ( n/2) b (p)i The second moment of the GPD Z dx xh I=0 (x,, t) =A 20 (t)+4 2 A 22 (t) : Generalized form factors of the pion 21

28 Stability of the pion Isoscalar vector GPDs of the pion 2 ab H I=0 (x,, t) = 1 2 Z d 2 eix (P n) h a (p 0 ) ( n/2)/n[ n/2, n/2] ( n/2) b (p)i The second moment of the GPD Z dx xh I=0 (x,, t) =A 20 (t)+4 2 A 22 (t) : Generalized form factors of the pion Energy-momentum Tensor Form factors (Pagels, 1966) h a (p 0 ) T µ (0) b (p)i = ab 2 [(tg µ q µ q n u) 1 (t)+2p µ P 2 (t)] T µ (x) = 1 2 (x) { } (x) : QCD EMT operator 21

29 Stability of the pion Isoscalar vector GPDs of the pion 2 ab H I=0 (x,, t) = 1 2 Z d 2 eix (P n) h a (p 0 ) ( n/2)/n[ n/2, n/2] ( n/2) b (p)i The second moment of the GPD Z dx xh I=0 (x,, t) =A 20 (t)+4 2 A 22 (t) : Generalized form factors of the pion Energy-momentum Tensor Form factors (Pagels, 1966) h a (p 0 ) T µ (0) b (p)i = ab 2 [(tg µ q µ q n u) 1 (t)+2p µ P 2 (t)] T µ (x) = 1 2 (x) { } (x) : QCD EMT operator EMTFFs (Gravitational FFs) 21

30 Stability of the pion Isoscalar vector GPDs of the pion 2 ab H I=0 (x,, t) = 1 2 Z d 2 eix (P n) h a (p 0 ) ( n/2)/n[ n/2, n/2] ( n/2) b (p)i The second moment of the GPD Z dx xh I=0 (x,, t) =A 20 (t)+4 2 A 22 (t) : Generalized form factors of the pion Energy-momentum Tensor Form factors (Pagels, 1966) h a (p 0 ) T µ (0) b (p)i = ab 2 [(tg µ q µ q n u) 1 (t)+2p µ P 2 (t)] T µ (x) = 1 2 (x) { } (x) : QCD EMT operator EMTFFs (Gravitational FFs) 21

31 Stability of the pion Time component of the EMT matrix element gives the pion mass. The sum of the spatial component of the EMT matrix element gives the pressure of the pion, which should vanish! Zero in the chiral limit (Based on the local model) H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 22

32 Stability of the pion Pressure of the pion beyond the chiral limit H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 23

33 Stability of the pion Pressure of the pion beyond the chiral limit Quark condensate H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 23

34 Stability of the pion Pressure of the pion beyond the chiral limit Quark condensate Pion decay constant H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 23

35 Stability of the pion Pressure of the pion beyond the chiral limit Quark condensate Pion decay constant by the Gell-Mann-Oakes-Renner relation to linear m order H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 23

36 Energy-momentum Tensor FFs in the chiral limit H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 24

37 Energy-momentum Tensor FFs in the chiral limit The difference arises from the explicit chiral symmetry breaking. H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 24

38 Transverse charge density of the pion H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 25

39 Transverse charge density of the pion H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 26

40 Transverse charge density of the pion The transverse charge density is divergent at b=0. H.D. Son & H.-Ch.K, to be published in PRD Rapid Comm. 26

41 Effective chiral Lagrangian Low-Energy Constants Derivative expansions: pion momentum as an expansion parameter Weinberg term Gasser-Leutwyler terms HChK et al. Prog. Part. Nucl. Phys. Vol.37, 91 (1996)

42 Effective chiral Lagrangian Weinberg Lagrangian Gasser-Leutwyler Lagrangian H.A. Choi and HChK, PRD 69, (2004)

43 Low-energy constants in flat space Gasser-Leutwyler Lagrangian H.A. Choi and HChK, PRD 69, (2004)

44 Low-energy constants in flat space Gasser-Leutwyler Lagrangian H.A. Choi and HChK, PRD 69, (2004)

45 Effective chiral Lagrangian in curved space Gasser-Leutwyler Lagrangian

46 Effective chiral Lagrangian in curved space Gasser-Leutwyler Lagrangian Low-energy constants in curved space

47 Effective chiral Lagrangian in curved space Gasser-Leutwyler Lagrangian Low-energy constants in curved space They can be derived either by expanding the action by the heat-kernel method or expand the EMTffs with respect to t.

48 Low-energy constants in curved space Present Results XPT Results (Gasser & Leutwyler)

49 Summary & Conclusion

50 Summary We have reviewed recent investigations on the quark structures of the pion, based on the nonlocal chiral quark model from the instanton vacuum. We have derived the EM and tensor form factors of the pion, from which we have obtained the quark transverse spin densities inside a pion. We also have shown the energy-momentum tensor (gravitational or generalised) form factors of the pion. The pressure of the pion nontrivially vanishes because of the Gell-Mann-Oakes-Renner relation. We also have presented the higher-order transverse charge density of the pion. 33

51 Outlook The transverse charge and spin densities for the transition processes can be studied (K-pi transition is done, see the next talk by Hyeon-Dong Son). The excited states for the nucleon and the hyperon can be investigated (Generalisation of the XQSM is under way). Internal structure of Heavy-light quark systems (Derivation of the Partition function is close to the final result.) New perspective on hadron tomography 34

52 Though this be madness, yet there is method in it. Hamlet Act 2, Scene 2 Thank you very much!