Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Structure of next-to-leading order corrections in 1/N C J.J. Sanz Cillero, IPN-Orsay Hadrons & Strings, Trento, July 21 st 2006

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Bottom Up Very bottom Just general QCD properties: 4D-QFT description of hadrons

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Just general QCD properties: 4D-QFT description with hadronic d.o.f. Chiral symmetry invariance (n f light flavours) 1/N C expansion around the ‘t Hooft large-N C limit: N C  ∞, N C  s fixed Pole structure of amplitudes at large N C (tree-level) Analiticity + matching QCD short-distance behaviour (parton logs +  s logs + OPE) V,V’,  …   [ ‘t Hooft 74 ] [ Callam et al.’69] [Colleman et al.’69] [Bando et al.’85] [Ecker et al.’89] …, [ Peris et al.’98] [Catà et al. ’05] [SC’05]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Why going up to NLO in 1/N C ? To validate the large-N C limit: NLO under control To show the phenomenological stability of the 1/N C series To increase the accuracy of the predictions To make real QFT in 1/N C, not just narrow-width ansate To understand sub-leading effects (widths, exotica,…) Because we already have it there (even we don’t know it)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large-N C QCD, NLO in 1/N C and N C =3 QCD

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 QFT description of amplitudes at large N C Infinite number of hadronic states + Goldstones from the S  SB (special) Infinite set of hadronic operators in L =  i O i (but don’t panic yet; this already happens in large-N C QCD) Chiral symmetry invariance Tree-level description of the amplitudes: strengths Z k (residues) and masses M k (pole positions)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 …  a 1 (1260)  (770) LO in 1/N C (tree-level) Re{q 2 } Im{q 2 } [ ‘t Hooft 74 ] [ Witten 79]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 …  a 1 (1260)  (770) LO + NLO in 1/N C (tree-level + one-loop) Re{q 2 } Im{q 2 }

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006  a 1 (1260)  (770) LO + SLO in 1/N C + Dyson-Schwinger summation (tree-level + one-loop  widths) Re{q 2 } Im{q 2 } Unphysical Riemann-sheets

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Truncation of the large-N C spectrum

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Minimal Hadronical Approximation [Knecht & de Rafael’98] Large-N C (infinite # of d.o.f.) Lack of precise knowledge on the high-lying spectrum Relative good knowledge of low-lying states

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Minimal Hadronical Approximation [Knecht & de Rafael’98] Large-N C (infinite # of d.o.f.) Approximate large-N C (finite # of d.o.f. –lightest ones-) Lack of precise knowledge on the high-lying spectrum Relative good knowledge of low-lying states

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Ingredients of a Resonance Chiral Theory (R  T) Large N C  U(n f ) multiplets Goldstones from S  SB MHA: First resonance multiplets (R=V,A,S,P) Chiral symmetry invariance [ Ecker et al.’89]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 … … [Moussallam’95], [Knecht & Nyffeler’01] [ Cirigliano et al.’06] [Pich,Rosell & SC, forthcoming] [ Ecker et al.’89] [ Weinberg’79] couplings i RR, i RRR [ Gasser & Leutwyler’84] [ Gasser & Leutwyler’85]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 We must build the R  T that best mimics QCD at large-N C Chiral symmetry invariance: Ensures the right low-energy QCD structure (  PT), even at the loop level! At short-distances: Demand to the theory the high-energy power behaviour prescribed by QCD (OPE) [Shifman et al ’79] [ Weinberg’79] [ Gasser & Leutwyler’84,’85] [ Catà & Peris’02] [Harada & Yamawaki’03] [ Rosell, Pich & SC’04, forthcoming’06]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 s  -∞ Constraints among the couplings i and masses M R at N C  ∞ e.g., Weinberg sum-rules [Weinberg’67]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 It is possible to develop the R  T up to NLO in 1/N C One Loop DiagramsNLO Contributions However, Loops=UV Divergences!! New NLO pieces (NLO couplings) ? Removable through EoM if proper short-distance R  T at LO [Rosell, Pich & SC’04] [Ecker et al.’89], … [Catà & Peris’02] [Rosell, Pich & SC, forthcoming’06] [Rosell et al.’05]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 s  -∞ Constraints among i and masses M R : LO + NLO contribution e.g., WSR, Again, one must build the R  T that best mimics QCD, but now up to NLO in 1/N C : - Natural recovering of one-loop  PT at low energies - Demanding QCD short-distance power behaviour [Rosell, Pich & SC, forthcoming’06]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 However,… plenty of problems The # of different operators is ~10 2 (NOW YOU CAN PANIC!!!) Even with just the lightest resonances one needs ~30 form-factors F k (s) to describe all the possible intermediate two-meson states in  LR (s) Systematic uncertainty due to the MHA Eventually, inconsistences between constraints when more and more amplitudes under analysis Need for higher resonance multiplets Even knowing the high-lying states, serious problems to manage the whole large-N C spectrum [Rosell et al.’05] [Rosell, Pich & SC, forthcoming’06] [SC’05] [Bijnens et al. ’03]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 General properties at NLO in 1/N C

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Interesting set of QCD matrix elements QCD amplitudes depending on a single kinematic variable q 2 Paradigm: two-point Green-functions, e.g., left-right correllator  LR (q 2 ), scalar correllator  SS (q 2 ), … also two-meson form factors ~ F (q 2 ) We consider amplitudes determined by their physical right-hand cut. For instance, partial-wave projections into T I J (s) transform poles in t and u variables into continuous left-hand cut in s variable. [SC, forthcoming’06]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Essentially, we consider amplitude with an absorptive part of the form This information determines the QCD content of the two-point Green-functions

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Exhaustive analysis of the different cases: 1.Unsubtracted dispersive relations Infinite resonance large-N C spectrum 2.m-subtracted dispersive relations Straight-forward generalization

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Unsubtracted dispersion relations This is the case when  (s)  0 for |s|  ∞ In this case one may use the analyticity of  (s) and consider the complex integral Providing at LO in 1/N C the correlator expresion R 1, R 2,…

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Up to NLO in 1/N C one has tree-level + one-loop topologies The finite (renormalized) amplitudes contain up to doble poles …so the dispersive relation must be performed a bit more carefully…

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 ZOOM

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 s M k,r 2  ZOOM

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 s M k,r 2  ZOOM

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 s M k,r 2  ZOOM with the finite contribution

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 …where, in addition to the spectral function (finite), one needs to specify the value of: Each residue Each double-pole coefficient Each renormalized mass

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 What’s the meaning of all this is in QFT language? Consider separately the one-loop contributions  (s) 1-loop Absorptive behaviour of  (s) 1-loop =  (s) OPE at |s|  ∞ Possible non-absorptive in  (s) 1-loop ≠  (s) OPE at |s|  ∞ (but no physical effect at the end of the day) Counterterms in  (s) tree = behaviour as  (s) OPE at |s|  ∞

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 If one drops appart the any “nasty” non-absorptive contribution in  (s) 1-loop  (s) 1-loop fulfills the same dispersion relations as  (s) LO+NLO Same finite function UV divergences + …

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 But, the LO operators are precisely those needed for the renormalization of these UV-divergences Renormalization of the Z k and M k 2 up to NLO in 1/N C : Finite renormalized couplings Counter-terms NNLO in 1/N C

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 …leading to the renormalization conditions, with  c k (1) and  c k (2) setting the renormalization scheme (for instance,  c k (1) =  c k (2) =0 for on-shell scheme ) Hence, the amplitude becomes finally finite:

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 …leading to the renormalization conditions, with  c k (1) and  c k (2) setting the renormalization scheme (for instance,  c k (1) =  c k (2) =0 for on-shell scheme ) Hence, the amplitude becomes finally finite: On-shell scheme On-shell scheme

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 And what about those “nasty” non-absorptive terms? This terms are not linked to any ln(-s) dependence  Purely analytical contributions They would require the introduction of local counter-terms Nevertheless, when summing up, they both must vanish (so  (s)  0 for |s|  ∞) UV divergences NLO local couplings

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 m-subtracted dispersion relations Other Green-functions shows a non-vanishing behaviour  (s)  s m-1 when |s|  ∞ In that situations, one need to consider not  (s) but some m-subtracted quantity like the moment of order m: This contains now the physical QCD information, and can be obtained from the spectral function:

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 To recover the whole  (s) one needs to specify m subtraction constants at some reference energy s=s O These subtractions are not fixed by QCD (e.g., in the SM,  VV (s O ) is fixed by the photon wave-function renormalization)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Providing at LO in 1/N C the pole structures R 1, R 2,…

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 …but at the end of the day, at NLO one reaches the same kind of renormalization conditions …and an analogous structure for the renormalized moment: Finite (from the spectral function) Renormalized tree-level

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 …but at the end of the day, up to NLO one reaches exactly the same renormalization conditions …and an analogous structure for the renormalized moment: Finite (from the spectral function) Renormalized tree-level On-shell scheme On-shell scheme

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Renormalizability?

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 R  T descriptions of  (s) inherites the “good renormalizable properties” from QCD, through the matching in the UV (short-distances) Caution on the term “renormalizability”: Infinite # of renormalizations The LO operators cover the whole space of possible UV divergences (for this kind of  (s) matrix elements) Inner structure of the underlying theory: The infinity of renormalizations are all related and given in terms of a few “hidden” parameters (N C and N C  s in our case) (see, for instance, the example of QED 5 [Álvarez & Faedo’06] )

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 General “renormalizable” structures in other matrix elements? Appealing!! Larger complexity  (s 1,s 2,…) Multi-variable dispersion relations, crossing symmetry,… Next step: three-point GF and scattering amplitudes

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Conclusions

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 General QCD properties + 1/N C expansion: Already valuable information Decreasing systematic errors Increasing accuracy Proving that QCD NC=3 has to do with QCD NC  ∞ MHA: Relevance of NLO in 1/N C -Introduces systematic uncertainties -Makes calculation feasible Nevertheless, at some point the 4D-QFT becomes unbearably complex

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 AdS dual representations of QCD are really welcome: They provide nice/compact/alternative description of QCD Extremely powerful technology However, there are several underlying QCD features that must be incorporated: - Chiral Symmetry and Goldstones from S  SB - Short-distance QCD (parton logs +  s logs + OPE) - “Renormalizable” structure for  (s) amplitudes at NLO in 1/N C in terms of a few AdS parameters

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Chiral order parameter: No pQCD contribution Isolates the effective  PT coupling L 8 (quark mass pGoldstone mass ) Less trivial case than the J=1 correlators Two-point Green functions: We focus the attention on the SS-PP with I=1 Interest of this correlator

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Resonance Chiral Theory framework (R  T): Construction of the lagrangian PROGRAM:

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Resonance Chiral Theory framework (R  T): Construction of the lagrangian 2-body form-factors at LO in 1/N C : QCD short-distance constraints on the FF at LO in 1/N C PROGRAM: Tree-level

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Resonance Chiral Theory framework (R  T): Construction of the lagrangian 2-body form-factors at LO in 1/N C : QCD short-distance constraints on the FF at LO in 1/N C Derivation of  S-P (dispersive relations): QCD short-distance constraints on  S-P up to NLO in 1/N C PROGRAM: Tree-level 1-loop

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Resonance Chiral Theory framework (R  T): Construction of the lagrangian 2-body form-factors at LO in 1/N C : QCD short-distance constraints on the FF at LO in 1/N C Derivation of  S-P (dispersive relations): QCD short-distance constraints on  S-P up to NLO in 1/N C Recovering  PT at low energies: Low energy constants up to NLO in 1/N C : L 8 PROGRAM: Tree-level 1-loop

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 R  T lagrangian

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Ingredients of R  T Large N C  U(n f ) multiplets Goldstones from S  SB ( ,K,  8,  0 ) MHA: First resonance multiplets (V,A,S,P) Chiral symmetry invariance Just O (p 2 ) operators Chiral limit

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 … … [Moussallam’95], [Knecht & Nyffeler’01] [ Cirigliano et al.’06] [Pich,Rosell & SC, forthcoming] [ Ecker et al.’89] [ Weinberg’79] couplings i RR, i RRR [ Gasser & Leutwyler’84] [ Gasser & Leutwyler’85]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st body form-factors

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Optical theorem and the 1/N C expansion At LO in 1/N C,  t  is given by tree-level (1-particle intermediate states) 2 1-P cuts: asymptotic behaviour

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st P cuts: asymptotic behaviour?? At NLO in 1/N C, 2-particle intermediate states: 2

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 [Brodsky & Lepage’79] ARGUMENTS:

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 General FF analysis: V , V, A, … , V, ,  … A V, A, S, … , V, ,  … S , V, A, … , V, ,  … P V, A, S, … , V, ,  …

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 SS-PP correlator at one loop

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 The example of L 8 : SS-PP correlator At LO in 1/N C one has the resonance exchange

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 The example of L 8 : SS-PP correlator At LO in 1/N C one has the resonance exchange which at low energies becomes,

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Matching OPE for  S-P : [ Golterman & peris’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Matching OPE for  S-P : one gets at low energies, [ Golterman & peris’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Matching OPE for  S-P : one gets at low energies,  ??? [ Golterman & peris’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Up to NLO in 1/N C  S-P shows the general structure with the 2-P contributions from dispersion relations depending on the correponding couplings i, fixed before at LO in 1/N C in the FF analysis

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Exact definition of the integral: t MR2MR2

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Example:  contribution

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Example:  contribution Tree-level SFF

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Example:  contribution Tree-level SFF Short-distance SFF (correlator)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Example:  contribution Tree-level SFF Short-distance SFF (correlator) Optical theorem

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Example:  contribution Tree-level SFF Short-distance SFF (correlator) Optical theorem Dispersion relations

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st particle channels: Goldstone-Goldstone (  ) Resonance-Goldstone (R  ) Resonance-Resonance Suppressed  Neglected

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Full recovering of  PT at one loop

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Result in  PT within U(n f ): Low energy expansion at one loop TO NOTICE: Exact cancellation of  dependence Presence of the massless ln(-q 2 ) from  loop Analytical part (L 8 coupling constant)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Tree level:  Analytical LO +NLO

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Tree level: Intermediate state    Chiral log NLO  Analytical LO +NLO

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Tree level: Intermediate state   Intermediate state  R  constant  Chiral log NLO  Analytical NLO  Analytical LO +NLO

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Tree level: Intermediate state   Intermediate state  R   NEGLECTEDIntermediate state  RR  NEGLECTED constant  Chiral log NLO  Analytical NLO  Analytical LO +NLO

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Matching OPE for  S-P (q 2 ) ~ 1/q 4 up to NLO in 1/N C with ( not considered, competition vs. NLO)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Matching OPE for  S-P (q 2 ) ~ 1/q 4 up to NLO in 1/N C with ( not considered, competition vs. NLO)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Matching OPE for  S-P (q 2 ) ~ 1/q 4 up to NLO in 1/N C with ( not considered, competition vs. NLO)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Inputs Parameters needed at LO in 1/N C Parameters needed at LO in 1/N C (appearing only NLO in  S-P ) U(3)  SU(3) [ Kaiser & Leutwyler’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Inputs Parameters needed at LO in 1/N C Parameters needed at LO in 1/N C (appearing only NLO in  S-P ) (appearing at LO+NLO in  S-P ) U(3)  SU(3) SD matching up to NLO Short-distance matching at LO [ Kaiser & Leutwyler’00 ] Parameters needed up to NLO in 1/N C Parameters needed up to NLO in 1/N C

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Results (for comparisson; exactly scale independent expression) Contributions: U(3)  SU(3) tree  VV SS AA PP [ Kaiser & Leutwyler’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Results (for comparisson; exactly scale independent expression) Contributions: U(3)  SU(3) tree  VV SS AA PP Uncertainties: MSrMSr dmrdmr Fmomo MAMA MVMV truncation MPrMPr [ Kaiser & Leutwyler’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Results (for comparisson; exactly scale independent expression) Contributions: U(3)  SU(3) tree  VV SS AA PP Uncertainties: MSrMSr dmrdmr Fmomo MAMA MVMV truncation MPrMPr to be compared to the  PT result, [ Kaiser & Leutwyler’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Results (for comparisson; exactly scale independent expression) Contributions: U(3)  SU(3) tree  VV SS AA PP Uncertainties: MSrMSr dmrdmr Fmomo MAMA MVMV truncation MPrMPr to be compared to the  PT result, [ Kaiser & Leutwyler’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Results (for comparisson; exactly scale independent expression) Contributions: U(3)  SU(3) tree  VV SS AA PP Uncertainties: MSrMSr dmrdmr Fmomo MAMA MVMV truncation MPrMPr to be compared to the  PT result, [ Kaiser & Leutwyler’00 ]

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Conclusions

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large N C is meaningful: it is possible to control NLO

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large N C is meaningful: it is possible to control NLO Systematic expansion of QCD amplitudes in 1/N C

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large N C is meaningful: it is possible to control NLO Systematic expansion of QCD amplitudes in 1/N C General analysis of the 2-body FF

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large N C is meaningful: it is possible to control NLO Systematic expansion of QCD amplitudes in 1/N C General analysis of the 2-body FF General structure of  (t) (dispersive analysis)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large N C is meaningful: it is possible to control NLO Systematic expansion of QCD amplitudes in 1/N C General analysis of the 2-body FF General structure of  (t) (dispersive analysis) Short-distance matching order by order in 1/N C

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large N C is meaningful: it is possible to control NLO Systematic expansion of QCD amplitudes in 1/N C General analysis of the 2-body FF General structure of  (t) (dispersive analysis) Short-distance matching order by order in 1/N C Full recovering of  PT at low q 2 : -Example of L 8

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large N C is meaningful: it is possible to control NLO Systematic expansion of QCD amplitudes in 1/N C General analysis of the 2-body FF General structure of  (t) (dispersive analysis) Short-distance matching order by order in 1/N C Full recovering of  PT at low q 2 : -Example of L 8 Manifestation of the uncertainty origin and full control of the “saturation” scale

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Large N C is meaningful: it is possible to control NLO Systematic expansion of QCD amplitudes in 1/N C General analysis of the 2-body FF General structure of  (t) (dispersive analysis) Short-distance matching order by order in 1/N C Full recovering of  PT at low q 2 : -Example of L 8 Manifestation of the uncertainty origin and full control of the “saturation” scale Straight-forward extension to O (p 6 ) LECs

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 How is it possible to compute hadronic loops? (Why and how it works? How loops do not blow up at high/low energies? …) How is the transition from high to low energy QCD? (How can the d.o.f. change from Goldstones  Resonances  pQCD Continuum? How do we have this progressive change in the amplitudes? …) How can we relate hadronic and quark-gluon parameters? Energy regimes? Weinberg sum-rules? Narrow-width approximations, do they have some systematic physics behind or they just fix “experimental” numbers? How well do we understand hadronic interactions?

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 QCD expansion in 1/N C ? QCD at any q 2 QCD at any q 2 (MESONS)

Structure of NLO corrections in 1/N C J. J. Sanz Cillero - Hadrons & Strings, Trento, July 21 st 2006 Resonance FF, does it make any sense? (1 st ) VMDWSR (2 nd ) [Weinberg’67] … [Cata & Peris’ 02] [Pich, Rosell & SC’04] (3 rd ) [Pich, Rosell & SC; forthcoming]   a 1     a1a1   R R’ R R