Structure of next-to-leading order corrections in 1/N C

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

2. Up Bottom • Just general QCD properties: • 4D-QFT description of hadrons Very bottom

3. Just general QCD properties: • 4D-QFT description with hadronic d.o.f. • Chiral symmetry invariance (nf light flavours) • 1/NC expansion around the ‘t Hooft large-NC limit: • NC∞, NCas fixed • Pole structure of amplitudes at large NC (tree-level) • Analiticity + matching QCD short-distance behaviour • (parton logs + as logs + OPE) [ Callam et al.’69] [Colleman et al.’69] [Bando et al.’85] [Ecker et al.’89] [ ‘t Hooft 74 ] p V,V’,… p …, [ Peris et al.’98] [Catà et al.’05] [SC’05]

4. Why going up to NLO in 1/NC? • To validatethe large-NC limit: NLO under control • To show the phenomenological stability of the 1/NC series • To increase the accuracy of the predictions • To make real QFT in 1/NC, not just narrow-width ansate • To understand sub-leading effects (widths, exotica,…) • Because we already have it there (even we don’t know it)

5. Large-NC QCD, NLO in 1/NC and NC=3 QCD

6. QFT description of amplitudes at large NC • Infinite number of hadronic states • + • Goldstones from the ScSB (special) • Infinite set of hadronic operators in L=SliOi • (but don’t panic yet; this already happens in large-NC QCD) • Chiral symmetry invariance • Tree-level description of the amplitudes: • strengths Zk (residues) and masses Mk (pole positions)

7. LO in 1/NC (tree-level) [ ‘t Hooft 74 ] [ Witten 79] p r(770) a1(1260) … Im{q2} Re{q2}

8. LO + NLO in 1/NC (tree-level + one-loop) p r(770) a1(1260) … Im{q2} Re{q2}

9. LO + SLO in 1/NC + Dyson-Schwinger summation (tree-level + one-loop  widths) p r(770) a1(1260) Im{q2} Re{q2} Unphysical Riemann-sheets

10. Truncation of the large-NC spectrum

11. Minimal Hadronical Approximation [Knecht & de Rafael’98] • Lack of precise knowledge on the high-lying spectrum • Relative good knowledge of low-lying states Large-NC(infinite # of d.o.f.)

12. Minimal Hadronical Approximation [Knecht & de Rafael’98] • Lack of precise knowledge on the high-lying spectrum • Relative good knowledge of low-lying states Large-NC(infinite # of d.o.f.) Approximate large-NC(finite # of d.o.f. –lightest ones-)

13. Ingredients of a Resonance Chiral Theory (RcT) [ Ecker et al.’89] • Large NC U(nf) multiplets • Goldstones from ScSB • MHA: First resonance multiplets (R=V,A,S,P) • Chiral symmetry invariance

14. [ Weinberg’79] [ Gasser & Leutwyler’84] [ Gasser & Leutwyler’85] [ Ecker et al.’89] … couplings liRR, liRRR … [Moussallam’95], [Knecht & Nyffeler’01] [ Cirigliano et al.’06] [Pich,Rosell & SC, forthcoming]

15. We must build the RcT that best mimics QCD at large-NC [ Weinberg’79] [ Gasser & Leutwyler’84,’85] • Chiral symmetry invariance: • Ensures the right low-energy QCD structure (cPT), • even at the loop level! • At short-distances: • Demand to the theory the high-energy power behaviour prescribed by QCD (OPE) [ Catà & Peris’02] [Harada & Yamawaki’03] [ Rosell, Pich & SC’04, forthcoming’06] [Shifman et al ’79]

16. s  -∞ • Constraints among the couplings li and masses MR at NC∞ • e.g., Weinberg sum-rules [Weinberg’67]

17. One Loop Diagrams NLO Contributions It is possible to develop the RcT up to NLO in 1/NC RcT at LO [Ecker et al.’89], … [Catà & Peris’02] However, Loops=UV Divergences!! New NLO pieces (NLO couplings)? [Rosell, Pich & SC’04] [Rosell et al.’05] Removablethrough EoM if proper short-distance [Rosell, Pich & SC, forthcoming’06]

18. Again, one must build the RcT that best mimics QCD, but now up to NLO in 1/NC : - Natural recovering of one-loop cPT at low energies - Demanding QCD short-distance power behaviour s  -∞ • Constraints among li and masses MR : LO + NLO contribution • e.g., WSR, [Rosell, Pich & SC, forthcoming’06]

19. However,… plenty of problems [Rosell et al.’05] • The # of different operators is ~102(NOW YOU CAN PANIC!!!) • Even with just the lightest resonances one needs ~30 form-factors Fk(s) to describe all the possible intermediate two-meson states in PLR(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-NC spectrum [Rosell, Pich & SC, forthcoming’06] [SC’05] [Bijnens et al.’03]

20. General properties at NLO in 1/NC

21. Interesting set of QCD matrix elements [SC, forthcoming’06] • QCD amplitudes depending on a single kinematic variable q2 • Paradigm: two-point Green-functions, • e.g., left-right correllator PLR(q2), scalar correllatorPSS(q2), … also two-meson form factors <M1 M2|O|0> ~F(q2) • We consider amplitudes determined by their physical right-hand cut. • For instance, partial-wave projections into TIJ(s) • transform poles in t and u variables into continuous left-hand cut in s variable.

22. Essentially, we consider amplitude with an absorptive part of the form This information determines the QCD content of the two-point Green-functions

23. Exhaustive analysis of the different cases: • Unsubtracted dispersive relations • Infinite resonance large-NC spectrum • m-subtracted dispersive relations • Straight-forward generalization

24. Unsubtracted dispersion relations • This is the case when P(s)0 for |s|∞ • In this case one may use the analyticity of P(s) and consider the complex integral • Providing at LO in 1/NC the correlator expresion R1, R2,…

25. Up to NLO in 1/NC 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…

27. ZOOM

28. e s Mk,r2 ZOOM

29. e s Mk,r2 ZOOM

30. e s Mk,r2 ZOOM with the finite contribution

31. …where, in addition to the spectral function (finite), one needs to specify the value of: • Each residue • Each double-pole coefficient • Each renormalized mass

32. What’s the meaning of all this is in QFT language? • Consider separately the one-loop contributions P(s)1-loop • Absorptive behaviour of P(s)1-loop=P(s)OPE at |s|∞ • Possible non-absorptive in P(s)1-loop≠P(s)OPE at |s|∞ • (but no physical effect at the end of the day) • Counterterms in P(s)tree= behaviour as P(s)OPE at |s|∞

33. If one drops appart the any “nasty” non-absorptive contribution in P(s)1-loop P(s)1-loop fulfills the same dispersion relations as P(s)LO+NLO Same finite function UV divergences + …

34. But, the LO operators are precisely those needed • for the renormalization of these UV-divergences • Renormalization of the Zk and Mk2 up to NLO in 1/NC: Finite renormalized couplings Counter-terms NNLO in 1/NC

35. …leading to the renormalization conditions, with Dck(1) and Dck(2) setting the renormalization scheme (for instance, Dck(1)=Dck(2)=0 for on-shell scheme ) • Hence, the amplitude becomes finally finite:

36. …leading to the renormalization conditions, with Dck(1) and Dck(2) setting the renormalization scheme (for instance, Dck(1)=Dck(2)=0 for on-shell scheme ) • Hence, the amplitude becomes finally finite: On-shell scheme

37. 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 P(s)0 for |s|∞) UV divergences NLO local couplings

38. m-subtracted dispersion relations • Other Green-functions shows a non-vanishing behaviour • P(s)sm-1 when |s|∞ • In that situations, one need to consider not P(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:

39. To recover the whole P(s) one needs to specify m subtraction constants at some reference energy s=sO • These subtractions are not fixed by QCD • (e.g., in the SM, PVV(sO) is fixed by the photon wave-function renormalization)

40. Providing at LO in 1/NC the pole structures R1, R2,…

41. …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

42. …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: On-shell scheme Finite (from the spectral function) Renormalized tree-level

43. Renormalizability?

44. RcT descriptions of P(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 P(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 (NC and NCas in our case) • (see, for instance, the example of QED5[Álvarez & Faedo’06])

45. General “renormalizable” structures in other matrix elements? • Appealing!! • Larger complexity P(s1,s2,…) • Multi-variable dispersion relations, crossing symmetry,… • Next step: three-point GF and scattering amplitudes

46. Conclusions

47. General QCD properties + 1/NC expansion: • Already valuable information • Decreasing systematic errors • Increasing accuracy • Proving that QCDNC=3 has to do with QCDNC∞ • MHA: Relevance of NLO in 1/NC -Introduces systematic uncertainties -Makes calculation feasible Nevertheless, at some point the 4D-QFT becomes unbearably complex

48. 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 ScSB • - Short-distance QCD (parton logs + as logs + OPE) • - “Renormalizable” structure for P(s) amplitudes at NLO in 1/NC • in terms of a few AdS parameters

50. Two-point Green functions: • We focus the attention on the SS-PP with I=1 Interest of this correlator • Chiral order parameter: No pQCD contribution • Isolates the effective cPT coupling L8(quark mass <-> pGoldstone mass ) • Less trivial case than the J=1 correlators