1. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero Anomalous AVV* amplitude in soft-wall AdS/QCD J.J. Sanz-Cillero ( Bari - INFN) P. Colangelo, F. De Fazio, F. Giannuzzi, S. Nicotri, J.J. SC [PRD 85 (2012) 035013] Ongoing work with F. Zuo QNP’12, April 19 th 2012

2. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero VVA vertex in QCD Holographic model and Chern-Simons term Longitudinal and transverse GF: LR and VVA correlators: Son-Yamamoto relation [ arXiv:1010.0718 [hep-ph] ] Outline:

3. VVA Green's function in AdS/QCD J. J. Sanz Cillero VVA Green’s function in QCD

4. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero This work is focused on the GF, In the soft photon limit k  0, provided by the relation in terms of the VVA correlator The GF is decomposed in T and L Lorentz structures with, JAJA JVJV JAJA JVJV  J EM k0k0 qq

5. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero High-energy OPE for m q =0 High-energy OPE for m q ≠0 with the magnetic susceptibility  : [ Vainshtein ‘03 ]

6. VVA Green's function in AdS/QCD J. J. Sanz Cillero AdS/QCD: Yang-Mills + Chern-Simons

7. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero Setup:  gauge chiral symmetry  Dilaton  AdS Metric  The YM action provides the propagator and 2-point GFs: -  SB through the v.e.v. v(z) - Phase-shift  induced by the axial source A 0 || (x)  Dual operators [ Karch et al. ‘06 ]

8. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero Equations of Motion: Vector EoM  Analytically solvable A 5 =V 5 =0

9. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero Scalar v.e.v. - Explicit breaking: m q - Spontaneous breaking:  [ UV behaviour / short-distance (y  0) ]

10. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero Contribution to A   V (soft k   0) with group factor Chern-Simons action  Chiral anomaly - Chern-Simons term with - Invariant under Vector transf. up to a boundary term (which is removed) (relevant part for AVV)

11. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero This produces the AdS prediction with fixed by for m q =0 All that remains Extract the B-to-b propagators V, A , A ||

12. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero VVA in AdS/QCD

13. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero

14. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero All EoM can be analytically solved ( v(y)=0 ) : In agreemente with exact QCD with m q =0 and no S  SB [ just massless pQCD ] We used this to fix k CS

15. VVA Green's function in AdS/QCD J. J. Sanz Cillero

16. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero At Q 2  ∞ one has the OPE The OPE requires the presence of a logarithmc ln(Q 2 /m q 2 ) at O(1/Q 4 )  Impossible if the UV-b.c. for  is just a constant?

17. VVA Green's function in AdS/QCD J. J. Sanz Cillero

18. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero The Parallel component can be still analytically solved: The perp. component [expansion in 1/Q 2 ] PROBLEM: OPE at high-energies  Our model produces  =0?

19. VVA Green's function in AdS/QCD J. J. Sanz Cillero

20. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero The Parallel component  exp. in 1/Q 2 The perp. Component [expansion in 1/Q 2 ]? ? ISSUES with the OPE:  m q  term: no susceptibility,  =0 !!  m q 2 term: w L : If  (Q 2,0 ) Impossible to recover simply a constant the lnQ 2 terms w T : Impossible to recover the lnQ 2 terms

21. VVA Green's function in AdS/QCD J. J. Sanz Cillero LR-correlator and w T,L (m q =0) : Son-Yamamoto relation

22. VVA Green's function in AdS/QCD J. J. Sanz Cillero Son-Yamamoto proposed the relation [ 2010 ]  SB through IR BC’s [Hirn,Sanz ‘05]  SB through v(y) [Sakai,Sugimoto ’04, ‘05] [Son,Stephanov ‘04] [Karch et al. ‘06] [Colangelo et al. ‘08] ? MHA with  + a 1 [Knecht,De Rafael ‘98]

23. VVA Green's function in AdS/QCD J. J. Sanz Cillero Summary andconclusions

24. VVA Green's function in AdS/QCD J. J. Sanz Cillero For m q =0 one has  = A || = 1 [ topological quantity ] Not determined by EoMs but by b.c. Problems for m q =0 in w T :  =0 !! More ingredients needed? Problems for m q ≠0 : SY relation (at large N C ) :  No 5D-field dual to q   q  No transition q   q    Need for the dual field B  ? [ Gorsky et al. ‘12 ]   (Q,0) ?   =0 again from m q  !!  Are m q corrections understood? Study of  AA ||  Issues in AdS for Q 2  ∞ BUT  BUT it seems to work at Q 2  0  Maybe ‘cause the MHA already does well [ Knecht et al. ‘11 ] [Kampf ‘11 ] [ Cappielo et al. ‘10 ]

25. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero

26. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero

27. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero BACKUP

28. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero Scalar v.e.v.  chiral symmetry breaking -Explicit breaking: -Spontaneous breaking: However, in the simplest model [ Colangelo et al. ’08 ]  C 1 and C 2 related (unlike QCD)  Supossedly solvable by adding a potential V(|X|) We will assume the v.e.v. profile (regardless of its origin)

29. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero Scalar v.e.v.  chiral symmetry breaking -Explicit breaking: -Spontaneous breaking: We will assume the v.e.v. profile (regardless of its origin)

30. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero For our scalar v.e.v. v(y)= m q y/c Notice the relevance of the UV value of the  field !! At Q 2  ∞ one has the OPE The OPE requires the presence of a logarithmc ln(Q 2 /m q 2 ) at O(1/Q 4 )  Impossible if the UV-b.c. for  is just a constant?

31. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero Phenomenology (m q =0)

32. Anomalous AV*V amplitude in soft-wall models J. J. Sanz Cillero For Q 2  0 the EoM can be analytically solved for v(y) =  y 3 /c 3  Experiment [ PDG ’10 ] This work [ Colangelo et al. ‘11] 92.2 8.3 ±1.3 6.3 86.5 For Q 2  ∞ perturbatively solved for g 5 v(y) =  y 3 + O(y 4 ) Experiment [ Prades et al. ’10 ] This work [ Colangelo et al. ‘11] -2.2 ±0.4 - 4.0 [ Prades et al. ’10 ] -3.9 ± 1.0 [ Friot et al. ’04 ] INPUTS: NOT a fit !!!