Large-NC resonance relations from partial wave analyses EuroFlavours 07, 14-16 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:0710.2163 [hep-ph]

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

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

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)

Dispersive calculation of pp-scatering at large-NC

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)

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

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

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)

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

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

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.

Matching cPT at large-NC

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

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 ,

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

… 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 : …

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 :

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

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.

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]

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,

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

On the consistency of phenomenological lagrangians

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]

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

…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

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

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

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]

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

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

Extensions to Resonance Chiral Theory

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)>

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),

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

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

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?

[ 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

Conclusions

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

1.) Linear Sigma Model

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

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.

2.) Gauged Chiral Model

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

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

[ 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) + 0 x O(mp2) KSRF relation TRIVIAL aS,V relation

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,

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]