Large-N C resonance relations from partial wave analyses

1. EuroFlavours 07, 14-16 November 2007 Univ. Paris-Sud XI - Orsay Large-NCresonance relations from partial wave analyses J.J. Sanz-Cillero (IFAE - UAB) Z.H. Guo, J.J. Sanz-Cillero and H.Q. Zheng [ JHEP 0706 (2007) 030 ]; arXiv:0710.2163 [hep-ph]

2. Organization of the talk: • Dispersive calculation for pp-scattering at large NC • Matching cPT at low energies • Resonance coupling relations and LEC predictions • Testing phenomenological lagrangians and LEC resonance estimates • Conclusions

3. Motivation • Former large-NC resonance analysis have looked at • a) Form factors, • b) 2-point Green-functions • c) 3-point Green-functions • Scattering amplitudes are the next in the line: • There have been studies on forward scattering • We propose the analysis of the PW scattering amplitudes • In forward scattering s, t and u-channels have similar asymptotics • In PW amplitudes Each has a clearly distinguishable structure

4. Moreover… • In general, the description in terms of couplings of a lagrangian • usually does not provide an intuitive picture of the 1/NC expansion: • Expansion parameter in the hadronic 1/NC theory ? • Model dependence of a lagrangian realization ? • … • However, maybe we can reach a better understanding/agreement • if we express resonance couplings • in terms of physical parameters (like masses and widths)

5. Dispersive calculation of pp-scatering at large-NC

6. T-matrix dispersive relation for pp-pp:[Guo, Zheng & SC’07] • Resonance inputs: • Right-hand cut (s-channel) • Left-hand cut (t- and u-channels)

7. Right-hand cut • At large-NCs-channel narrow-width resonance exchanges: • For the right-hand cut: TJI partial wave  ONLY IJ resonance with

8. R • We substitute this ImT(t) • in the right-hand side dispersive integral and obtain MR , GR which can be identified with the exchange of a tree-level resonance R in the s-channel

9. Left-hand cut • Crossing symmetry relations for right and left-hand cut: (true for any s<0 for large-NC tree-level amplitudes)

10. TJI partial wave  almost EVERY narrow-width state RJ’I’ contributes in the t and u-channels

11. S • By placing ImT(s) in the left-hand cut disperive integral: • Explicit analytical expression TJI(s)tR • for the contribution from the exchange • of a resonance R in the t (and u) channels. • For instance, for R=S and the partial wave T11: MR , GR which can be identified with the exchange of a tree-level scalar resonance in the crossed-channel

12. Final dispersive expression • Putting the different contributions together gives • In our analysis, only the first V and S resonances have been included. • Problems when higher-spin resonances were included.

13. Matching cPT at large-NC

14. We perform a chiral expansion • of the resonance contributions TsR and TtR MR , GR • in powers of s and mp2 • For T(s) and T(0) , we use the values provided by cPT up to O(p6) • (amplitudes expressed in terms of s, mp2 • and mp-independent constants) LECs

15. This produces a matching equation of the form, MR , GR LECs where we match left and right-hand side order by order in (mp2)m sn ,

16. We have taken the matching up to O(p6): • At O(p2) we match the terms • At O(p4) we match the terms • At O(p6) we match the terms O(mp2)  NOT PRESENT O(s) mod-KSRF relation O(mp4)  NOT PRESENT O(s mp2)  Reson. relation O(s2)  L2, L3 O(mp6)  NOT PRESENT O(s mp4)  r2 - 2rf O(s2mp2)  r3, r4 O(s3)  r5, r6

17. Simultaneous analysis of the IJ=11,00,20 channels • Compatible system of 18 equations: rank 9 • number of unknowns = 9 • One must take into consideration that MR and GR • are the physical large-NC masses and they alsodepend on mp: …

18. Matching at O(p2): the O(s1 mp0) term  Modified-KSRF relation (constraint) • To exemplify the matching, we explicitly show this case: IJ=11 : IJ=00 : IJ=20 :

19. The three channels provide exactly the same constraint • which is a modification of the KSRF relation • that takes into account S resonances and crossed exchanges: to be compared to the original result, [Kawarabayashi & Suzuki’66] [Riazuddin & Fayazuddin’66] with The original KSRF relation is recovered in our analysis of the IJ=11 channel if we neglect the impact from S resonances and crossed V exchanges

20. Matching at O(p4): the O(s1 mp2) term  Novel resonance constraint • The three channels provide exactly the same constraint: This new relation provides a constraint between the mp2 corrections to masses and widths.

21. Matching at O(p4): the O(s2 mp0) term  Prediction for L2 and L3 • The three channels provide two compatible constraints for the LECs: where similar results in terms of widths and masses were also found in previous works [ Bolokhov et al.’93]

22. Matching at O(p6): O(s mp4), O(s2 mp2), O(s3 mp0)  Prediction for r3,4,5,6 and r2 -2rf • The three channels provide compatible constraints for the LECs: [Guo, Zheng & SC’07] with bR and gR given by the chiral corrections,

23. Origin of the relations Once subtracted dispersion relations • Good high-energy behaviour • Good low-energy behaviour cPT matching

24. On the consistency of phenomenological lagrangians

25. We analysed a series of different phenomenological lagrangians: • Linear Sigma Model [’60,’70,’80,’90…] • Gauged Chiral Model [Donoghue et al.’89] • Resonance Chiral Theory (RcT) [Ecker et al.’89] • and extended versions of RcT [Cirigliano et al.’06]

26. For sake of lack of time I will not explain • the first two cases in detail • (although they are exhaustively analysed in [Guo, Zheng & SC’07]) • Nevertheless, the conclusion was that: • - Our dispersive predictions of the LECs • exactly agreed those obtained through the standard procedure • (integrating out the heavy resonances) • - We extracted constraints between resonance couplings • that were intimately related to the asymptotic high-energy behaviour

27. …Hence, I will focus on the last type of lagrangian. First we will analyse the original version of [Ecker et al’89], the Minimal Resonance Chiral Theory

28. Non-linear realization for the Goldstones • No assumptions on the vector and scalar nature • Originally, only linear operators in the resonance fields were considered in the lagrangian: with the linear terms including only O(p2 ) tensors, [Ecker et al.’89] • Procedure: 1) First, we compute MR, GR • 2) Second, we check our relations

29. 1.) We get the widths at LO in 1/NC:

30. 2.) We compare the standard results and our LECs predictions: • Integrating out the resonances in the generating functional, • one gets the LECs corresponding to this action: [Ecker et al.’89] • And using the dispersive predictions one gets a complete agreement: (SIMILAR AGREEMENT WAS FOUND IN THE ANALYSIS OF THE OTHER LAGRANGIANS) In complete agreement with the original lagrangian calculation [Ecker et al.’89]

31. …and study the resonance relations: • From the modified-KSRF constraint we get, • And the aS,V constraint yields, But notice that for both constraints are incompatible

32. What is the problem in this case? • If we introduce the operator cm <S c+> , • it must come together with other operators • (if it is introduced alone, wrong results) • What is special in the cm operator? • It is an operator that couples the scalar to the vacuum • proportionally to mq • This makes fp and the S-pp, V-pp vertices mp dependent • even at large-NC • However, we will see that this mp dependence • may be produced by other operators not considered S p S V p

33. Extensions to Resonance Chiral Theory

34. For a clearer understanding we will focus first on the scalar sector: • Allowing a more general structure in the resonance lagrangian, • the scalar mass and width gain additional O(mp2) corrections • from the extra resonance operators [Cirigliano et al.’06], <RO(p4)> <RRO(p2)> <RRRO(p2)>

35. In order to compute the amplitudes free of scalar tadpoles we perform the mq-dependent shift, [SC’04] • This provides the contribution to the mass and S-pp vertex, • now free of S tadpoles. • The S-pp interaction, in the isospin limit shows the structure, With the mp dependent parameters and MSeff = MS + O(mp2), cmeff = cm + O(mp2),

36. Likewise, the pion decay constant gets mp corrections at large NC, • Hence, the ratio G/M3 for the scalar becomes,

37. Following a similar procedure for the vector we would have an effective coupling, • leading to the ratio,

38. Finally, putting everything together one gets the KSRF and aS,V constraints: which can be easily combined in the single form But what is the meaning of this?

39. At high energies the amplitude behaves like [ SIMILAR RESULT FOR IJ=00,20 ] It is then clear now that the KSRF and aS,V constraints are equivalent to demanding a good behaviour at high (and low) energies Chiral lagrangians

40. Conclusions

41. New dispersive method for the the study • of LECs and resonance constraints at large-NC • Easy implementation of high & low-energy constraints • independent of the realization of the resonance lagrangian • Successfully checked for a wide set of different • phenomenological lagrangians • Useful tool for future studies of other scattering amplitudes

44. 1.) Linear Sigma Model

45. Only Scalar + Goldstones (no Vectors) • For our first check we use the LsM, where the scalar and the Goldstones are introduced in a linear realization: • Simple model with useful properties that give a first insight of the meaning of these constraints. • Procedure: 1) First, we compute MS, GS • 2) Second, we check our relations

46. Renormalizability • Chiral symmetry Good high-energy behaviour Good-low-energy behaviour [ T(s) ~ O(s0) when s∞ ] No place for further constraints KSRF The KSRF and aS,V constraints are trivially fulfilled for any value of l and m aS,V However, renormalizability is not the keypoint, as we will see in the next example.

47. 2.) Gauged Chiral Model

48. Only Vector + Goldstones (no Scalars) • The r and a1 are introduced as gauge bosons • in the O(p2) cPT lagrangian : [Donoghue et al.’89] However, due to the p-a1 mixing, one finds a highly non-trivial interaction, which makes the calculation of the pp-scattering rather involved

49. O(p4) LECs : • Integrating out the resonances in the lagrangian, • one gets the corresponding LECs at large-NC : • If we now use the dispersive predictions we get exactly the same: with

50. Resonance constraints : • The aS,V constraint is trivially obeyed since we find • This is not so for the KSRF constraint, which gives Origin of these constraints?Observe the pp-scattering amplitude at s∞: [ SIMILAR RESULT FOR IJ=00,20 ] = O(mp0) + 0 x O(mp2) KSRF relation TRIVIAL aS,V relation