1. Mixing properties of a 1 (1260) axial vector meson ~ Composite and elementary natures ~ Hideko NAGAHIRO (Nara Women’s University) H. Nagahiro, K. Nawa, S. Ozaki, D. Jido and A. Hosaka, arXiv:1101.3623 [hep-ph] Collaborate with : K. Nawa, S. Ozaki, D. Jido, A. Hosaka “Hadron Nuclear Physics 2011” “Quarks in hadrons, nuclei, and hadronic matter” Feb. 21(Mon), 2011, POSCO International Center, Pohang, Republic of Korea

2. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 2 Introduction Exotic hadrons : not simple qq (or qqq) state › , f 0 (980), a 0 (980), …, a 1 (1260), K 1 (1270), …, N*(1535),  (1405), … etc…  molecule, qqqq state, or mixed states among them ?  molecule, qqqq state, or mixed states among them ?   as a dynamically generated resonance [as  composite particle] [coupled-channel BS] Lutz-Kolomeitsev, NPA730(04)392, …  a1a1a1a1 as an elementary field (or qq) : candidate for the chiral partner of   [Hidden local sym.] Bando-Kugo-Yamawaki; PR164(88)217; Kaiser-Meissner, NPA519(90)671, … [qq-NJL] M.Wakamatsu et al,. ZPA311(88)173, A.Hosaka, PLB244(90)363-367, …  [Lattice QCD] M. Wingate et al., PRL74(95)4596,... [Holographic QCD] T. Sakai, S. Sugimoto, PTP113 (05) 843; ibid.114(05)1083, … [Chiral Unitary model] Roca-Oset-Singh, PRD72(05)014002, … in coupled-channel approach based on the chiral effective theory Nature of axial vector meson a 1 (1260) : m = 1230  40 MeV,  =260 to 600 MeV [PDG]

3. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 3 a 1 (1260) axial vector meson … a good example we have some models that include the elementary a 1 meson (dominated by qq bar ) as well as the  and  mesons we have some models that include the elementary a 1 meson (dominated by qq bar ) as well as the  and  mesons the same model gives an attractive potential between the  and  mesons, that would generate the composite a 1 meson the same model gives an attractive potential between the  and  mesons, that would generate the composite a 1 meson  We can treat these two pictures in a unified way  We can treat these two pictures in a unified way a good example to study the mixing nature of elementary and composite comp. » We focus on the nature of hadrons consisting of multiple components, and study how to disentangle their mixture appearing in the physical states. » We focus on the nature of hadrons consisting of multiple components, and study how to disentangle their mixture appearing in the physical states. 1. large N C classification 2. wave functions by residues 1. large N C classification 2. wave functions by residues a1a1a1a1 phys = C1C1C1C1+ C2C2C2C2 + …  a1a1a1a1

4. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 4 Large-N C, large-  (=  YM 2 N C ) QCD can be analyzed by the classical supergravity Lagrangians : holographic QCD (HQCD)  Important concept to avoid the double-counting of molecule and bare comp. [Sakai, Sugimoto, PTP113(05)843; PTP114(05)1083, Nawa, Suganuma, Kojo, PRD75(07)086003 etc.] a 1 meson in HQCD [E. Witten, Nucl. Phys. B 160, 57 (1979)] elementary a 1 meson does not have molecular component [large Nc limit] elementary a 1 meson does not have molecular component [large Nc limit] » essentially the same as those of the hidden-local symmetry       a1a1a1a1 g 4 = 1, g a 1   = 0.26 g 4 = 1, g a 1   = 0.26 f  = 92.4MeV, m   = 776MeV f  = 92.4MeV, m   = 776MeV m a1 = 1189 MeV and couplings are determined automatically m a1 = 1189 MeV and couplings are determined automatically » a 1 meson appears through the mode expansion of A  (x, z) [5D gauge field] i where

5. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 5 Dynamically generated resonances : composite a 1 meson a 1 meson as a  -composite : chiral unitary model   generated in s-wave  scattering trough non-perturbative dynamics t WT ( s ) = v WT ( s ) + v WT (s) G (s) t WT (s) Bethe-Salpeter equation for unitarized s-wave amplitude = v WT + v WT G v WT + v WT G v WT G v WT + … L.Roca, E.Oset and J.Singh, PRD72(05)014002 ++++++ ……=    tt tt = + v G v WT t WT ( s ) = 1− v WT G formal solution If v WT is attractive, a pole appears in the amp.

6. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 6 Dynamically generated resonances : composite a 1 meson WT potential v WT obtained from s-wave projection of WT int. a 1 meson as a  -composite : chiral unitary model L.Roca, E.Oset and J.Singh, PRD72(05)014002  K*K*K*K*  − + coupled-channel calculation  a 1 pole as mainly  composite at 1011-84i MeV Loop function G with on-shell  energies (low energy amplitude ~ threshold) =  = … a pheno. = −1.85 (  =900MeV) a parameter : subtraction constant … arbitrary. Suitably chosen to reproduce data : a pheno v WT

7. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 7 Dynamically generated resonances : composite a 1 meson Loop function G = = = … a parameter : subtraction constant … arbitrary. Suitably chosen to reproduce data : a pheno Choice of regularization constant : natural renormalization A choice of the subtraction constant a pheno ≠ a natural is equivalent to the introduction of the CDD (or elementary) pole that is not included in the model space of the scattering problem. [T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203] A choice of the subtraction constant a pheno ≠ a natural is equivalent to the introduction of the CDD (or elementary) pole that is not included in the model space of the scattering problem. [T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203]  a(  ) =  0.2 [natural] a(  )=  1.85 (  =900MeV) [Roca05] to ensure that the resulting state is a purely hadronic composite without quark-core

8. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 8 Formalism : elementary a 1 field through additional interaction (coupled-channel) Bethe-Salpeter eq. ++ ++++ ++++……++ potential WT interaction    a 1 pole term where     bare a 1 ++++++……= ×x×x

9. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 9 Numerical result 1 : pole-flow of T-matrix 0  100  200  300  400  500  600 900100011001200130014001500160017001800 [Roca et al., PRD72] a=   channel only a=   s p = 1011  84i MeV  s p = 1012  98i MeV  channel only a=  [natural] s p = 1012  221i MeV single channel natural       +  

10. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 10 Numerical result 1 : pole-flow of T-matrix 0  100  200  300  400  500  600 900100011001200130014001500160017001800 elementary a 1 pole m a = 1189 MeV (HQCD)  s p = 1012  221i MeV where V = + bare a 1 ×x×x    channel only a=  [natural] a1a1

11. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 11 Numerical result 1 : pole-flow of T-matrix 0  100  200  300  400  500  600 900100011001200130014001500160017001800  s p = 1012  221i MeV x  1x  1x  1x  1 x  1x  1x  1x  1 x = 1 x = 0  channel only a=  [natural]   elementary a 1 pole m a = 1189 MeV (HQCD) a1a1 where V = + ×x×x bare a 1

12. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 12 Numerical result 1 : pole-flow of T-matrix 0  100  200  300  400  500  600 100011001200130014001500160017001800 x = 1 x = 0 Pole from “  -composite” is closer to the real axis than that of “elementary” pole pole-a pole-b   bare a 1 x   a1a1 ? ? Q. What is the nature of these physical ( x =1) poles ?

13. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 13 Resonant structure tested by behaviors in large N C limit » To see the nature of the physical resonant states ( x = 1 ), we first check the N C (color number) dependence of the pole positions. Natures of mesons in large Nc limit [’t Hooft, NPB72(74)461] qq bar (elementary) meson becomes stable as N C  ∞ M  M 1 M 2  O(N C −1/2 ) m qq -meson  O(1) −  qq -meson  O(1/ N C ) − T MM  MM  O(1/N C ) hadronic composite meson is vanished as N C  ∞  scaling law of the pion decay constant f   O(N C 1/2 )

14. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 14 Numerical result 1 : large N C flow 0  100  200  300  400  500  600 900100011001200130014001500160017001800  bare a 1 x x = 0 x = 1 x = 0 N C  large N C =5 N C =10 x = 0.7 x = 0.65 N C  large To see the nature of poles : and N C  large pole-a L.Geng et al., EPJA39(09)81 pole-b ? ? flip!   a1a1

15. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 15 Alternative expression for the full  scattering amplitude T = ( g R, g ) sspsspsspssp s  m a1 2  gGg R gGg g R Gg 1111 gRgRgRgR g ++ ++ ++…… = a 1 pole term    bare a 1  -composite a 1 pole, = ()  1111 1111

16. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 16 Alternative expression for the full  scattering amplitude T, = ()  1111, = () + + … = + + + + + 1111 = ( g R, g ) sspsspsspssp s  m a1 2  gGg R gGg g R Gg 1111 gRgRgRgR g = + ++ gRgRgRgR gRgRgRgR g g g g gRgRgRgR gRgRgRgR D 11 D 21 D 12 D 22 In this form, we can analyze the mixing nature of the physical a 1 in terms of the original two bases: and composite a 1 elementary a 1

17. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 17 Features of the full propagators The propagators have two poles exactly at the same position as T full The propagators have two poles exactly at the same position as T full the residues z a ii have the information on the mixing rate the residues z a ii have the information on the mixing rate + regular term  Probability of finding the original composite a 1 in pole-a D 11 In this form, we can analyze the mixing nature of the physical a 1 in terms of the original two bases: and composite a 1 elementary a 1

18. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 18 -0.1 -0.2 -0.3 -0.4 0 11.11.21.31.41.51.61.71.80.9 x=0 x=1 pole-a pole-b mixing parameter x 010.80.60.40.2 2 3 4 1 0 |z| zbzbzbzb11 zazazaza22 pole flow residue x=1 zbzbzbzb22 zazazaza11 Large residue  due to the ene-dep. + (regular) Residues : probabilities of finding two a 1 ’s in pole-a and -b interchange

19. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 19 Residues : probabilities of finding two a 1 ’s in pole-a and -b -0.1 -0.2 -0.3 -0.4 0 11.11.21.31.41.51.61.71.80.9 x=0 x=1 pole-a pole-b mixing parameter x 010.80.60.40.2 2 3 4 1 0 |z| zbzbzbzb11 zazazaza22 non-zero comp. of pole flow residue x=1 zbzbzbzb22 zazazaza11 at physical point (x=1) pole-a has a component of the elementary a 1 meson comparable to that of composite a 1. (pole-a at x=1 is possibly observed one) pole-a has a component of the elementary a 1 meson comparable to that of composite a 1. (pole-a at x=1 is possibly observed one) + (regular) ~ ~

20. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 20 Last question : large N C procedure vs. nature of poles -0.1 -0.2 -0.3 -0.4 0 11.11.21.31.41.51.61.71.80.9 x=0 x=1 pole-a pole-b pole flow for pole-b, large N C flip point ~ residue-interchange for pole-b, large N C flip point ~ residue-interchange for pole-a, for pole-a, large N C flip point  residue-interchange 2 3 4 1 0 |z| NCNCNCNC 3 4 567 8 9 10 zbzbzbzb11 zazazaza 22 zazazaza11 zbzbzbzb22 Residues interchange by large N C procedure. x=0.8 x=0  1/ N C Large N C itself changes the nature of resonances  1/N C  

21. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 21 Last question : large N C procedure vs. nature of poles 2 3 4 1 0 |z| NCNCNCNC 3 4 567 8 9 10 zbzbzbzb11 zazazaza 22 zazazaza11 zbzbzbzb22 Residues interchange by large N C procedure. x=0.8  1/ N C Large N C limit doesn’t always reflect the world at N C =3.  1/N C   two sources of N C dependence: a 1  vertex controlling the mixing strength a 1  vertex controlling the mixing strength WT interaction determining the pole position s p of the basis state for composite a 1 WT interaction determining the pole position s p of the basis state for composite a 1 ssp(NC)ssp(NC)ssp(NC)ssp(NC) s  m a1 2  g (N C ) Gg R (N C ) 1111 g R (N C ) Gg (N C ) g (N C ) Gg (N C ) D =D =D =D = ^

22. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 22 Conclusions » We discussed the mixing properties of a 1 (1260) meson as the superposition of the hadronic  composite and elementary a 1 based on the holographic QCD Lagrangian. ›bare a 1 … doesn’t have molecule nature ›  molecule … “natural” regularization » We analyzed the pole nature by residues Important to avoid the double-counting the pole expected to be observed is pole-a: having finite comp. the pole expected to be observed is pole-a: having finite comp. Non-trivial N C dependence pole-nature  ?  large N C Non-trivial N C dependence pole-nature  ?  large N C Future works phenomenological interests »  -decay spectrum with our model parameter [Wagner and Leupold, PRD78(08)053001, …] [Wagner and Leupold, PRD78(08)053001, …] » radiative decay width [H. Nagahiro, L. Roca, A. Hosaka, E. Oset, PRD79(09)014015, …] [H. Nagahiro, L. Roca, A. Hosaka, E. Oset, PRD79(09)014015, …] » etc… to see how the nature of poles affects observables

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24. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 24 L.Roca, et al., PRD72(05)014002 WT interaction    (coupled-channel) Bethe-Salpeter eq.  a(  ) =  0.2 [natural] L. Roca et al., PRD72(05)014002 [T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203]  a 1 pole as  composite at 1011-84i MeV a(  )=  1.85 (  =900MeV) [Roca et al.] regularization constant to avoid the double-counting. = ++++++……= =  dynamics in a 1 channel : chiral unitary approach

25. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 25 effective a 1 field that is equivalent to composite a 1 ++ ++ ++ …… = ++ ++ = …… Or considering the quantum effects as.. CRITERION = Ref. D. Lurie, “Particles and fields”, Interscience, D. Lurie and A.J. Macfarlane, PR136(64)B816. D. Lurie and A.J. Macfarlane, PR136(64)B816.

26. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 26 G function : 700 80090010001100120013001400 ImG  0.018  0.016  0.014  0.012  0.010  0.006  0.008  0.004  0.002 0 0.002 ReG(a(  )=  0.2 : natural) ReG(a(  )=  1.85) mmmm “natural” renormalization [T. Hyodo, D.Jido, A.Hosaka, PRC78(08)025203] G(E=m T )=0 so that T(E=m T )=V WT

27. Loop function G L.Roca et al, PRD72 T. Hyodo, D. Jido, A. Hosaka, Phys.Rev.C78:025203,2008;

28. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 280.80.91.01.11.21.31.41.5 GeV a(  GeV) =  0.2 & energy-indep. 3 point vertex. 0 -100 -200 -300 -50 -150 -250 x=0 x=1 x=0 x=0.8 at physical point (x=1) there is a single pole (pole-b) there is a single pole (pole-b)1.21.0 0.8 0.6 0.4 0.2 0 |z| zazazaza 11 zazazaza 22 zbzbzbzb 11 zbzbzbzb 22 0 0.2 0.40.60.81 mixing parameter x

29. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 29 |T| 2 spectrum : a=  1.85 w/o bare a 1 vs. a=  0.2 w/ bare a 1 |T| 2 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 ×10 4 11.11.2 1.31.4 1.51.61.7 1.8 a=  1.85 a=  0.2 + GeV

30. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 30 Numerical result 1 : pole flow 0  100  200  300  400  500  600 900100011001200130014001500160017001800 x = 1 x = 0 To see the nature of poles : and N C  large Pole from “WT-molecule” is closer to the real axis than that of “bare” pole pole-a pole-b   bare a 1 x |T| 2  s p = 1012  98i MeV WT-  molecule (a=  1.85) only WT-  molecule (a=  1.85) only pole-b 1728  314i MeV 0.51.01.522.53.0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 ×10 4 The present case pole-a 1033  107 i MeV

31. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 31 nature of two poles analyzed by residue » To see the nature of two poles, one can see the residues of T-matrix at poles. c.f.  (1405) with double-pole structure in chiral unitary approach [c.f. Jido, Oller, Oset, Ramos, Meißner, NPA725(03)181] KN N    » This case : K  To know the mixing strength of “elementary a 1 ” and “composite a 1 ”, we define the appropriate base as elementary a 1 and composite a 1 we define the appropriate base as elementary a 1 and composite a 1 bare a 1

32. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 32 2D-matrix propagator expand full T-matrix element    bare a 1 1 st term ++ ++ ++…… 2 nd term ++ = ++ ++

33. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 33,, 2D-matrix propagator = ++ ++ ++ ++ ++ ++ …… == (()) ++ ++ …… ~ 2D-matrix propagator ~ Residues of G full = comp. of wave function Residues of G full = comp. of wave function These residues are normalized to be 1 at x=0. These residues are normalized to be 1 at x=0. expand full T-matrix element

34. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 34 Numerical result 1 : pole flow 0  100  200  300  400  500  600 900100011001200130014001500160017001800

35. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 35 C. Amsler et al. (Particle Data Group), PL B667, 1 (2008) a 1 (1260) meson

36. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 36 mixing properties analyzed by Residue -0.1 -0.2 -0.3 -0.4 0 11.11.21.31.41.51.61.71.80.9 x=0 x=1 pole-a pole-b 16 14 12 10 8 6 4 2 0 mixing parameter x 010.80.60.40.2 zbzbzbzb 11 zbzbzbzb 22 zazazaza 22 zazazaza 11 |z| pole flow residue x=1 x=0 Large residue  due to the ene-dep.

37. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 37 -0.1 -0.2 -0.3 -0.4 0 11.11.21.31.41.51.61.71.80.9 x=0 x=1 pole-a pole-b mixing parameter x 010.80.60.40.2 2 3 4 1 0 |z| zbzbzbzb11 zazazaza22 pole flow residue x=1 zbzbzbzb22 zazazaza11 Large residue  due to the ene-dep. + (regular) Residues : probabilities of finding two a 1 ’s in pole-a and -b

38. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 38 Probabilities of finding two a 1 ’s in pole-a & -b : residues -0.1 -0.2 -0.3 -0.4 0 11.11.21.31.41.51.61.71.80.9 x=0 x=1 pole-a pole-b mixing parameter x 010.80.60.40.2 2 3 4 1 0 |z|interchange zbzbzbzb11 zazazaza22 non-zero comp. of pole flow residue x=1 zbzbzbzb22 zazazaza11 at physical point (x=1) pole-a remains as a “composite-a 1 ” (bare comp.  molecule comp.) pole-a remains as a “composite-a 1 ” (bare comp.  molecule comp.)  both poles have molecule comp. pole-b changes into a “composite-a 1 ” pole-b changes into a “composite-a 1 ” Difference of transition points  energy-dependence (mainly) of  energy-dependence (mainly) of + (regular) ~ >

39. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 39 Last question : large N C procedure vs. nature of poles -0.1 -0.2 -0.3 -0.4 0 11.11.21.31.41.51.61.71.80.9 x=0 x=1 pole-a pole-b pole flow for pole-b, large N C flip point ~ residue-exchange for pole-b, large N C flip point ~ residue-exchange for pole-a, for pole-a, large N C flip point  residue-exchange 2 3 4 1 0 |z| NCNCNCNC 3 4 567 8 9 10 zbzbzbzb11 zazazaza 22 zazazaza11 zbzbzbzb22 Residues interchange by large N C procedure. x=0.8 x=0  1/ N C Large N C procedure itself changes strength of the mixing int.  1/N C  

40. HNP2011 @ POSCO International Center, Pohang, Republic of Korea 400  100  200  300  400  500  600 100011001200130014001500160017001800 x = 1 x = 0 pole-a pole-b   bare a 1 x Numerical result 1 : pole-flow of T-matrix Pole from “  -composite” is closer to the real axis than that of “elementary” pole 900 |T| 2 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 11.11.2 1.31.4 1.51.61.7 1.8 GeV pole-a coming from the composite a 1 pole-b [Roca05] amplitude on the real axis