1. Large-NC resonance relations from partial wave analyses
EuroFlavours 07, November 2007
Univ. Paris-Sud XI - Orsay
Large-NC resonance 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:  [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. MR , GR 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 R

9. for large-NC tree-level amplitudes)
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. MR , GR 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 S

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. MR , GR LECs 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. MR , GR LECs This produces a matching equation of the form,
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 also depend 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. with bR and gR given by the chiral corrections,
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. with the linear terms including only O(p2 ) tensors,
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. both constraints are incompatible
…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 V S 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. This provides the contribution to the mass and S-pp vertex,
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. which can be easily combined in the single form
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. [ SIMILAR RESULT FOR IJ=00,20 ]
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. No place for further constraints
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. [ SIMILAR RESULT FOR IJ=00,20 ]
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) x O(mp2)
KSRF relation
TRIVIAL aS,V relation

51. Through the explicit integration of the heavy resonances in the generating functional one gets the LECs corresponding to this action:
We also get the widths at LO in 1/NC,

52. with the original lagrangian calculation [Ecker et al.’89]
If we now use the dispersive predictions we get exactly the right results for the LECs:
In complete agreement
with the original lagrangian calculation [Ecker et al.’89]