1. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Can a resonance chiral theory be a renormalizable theory ? J.J. Sanz-Cillero (Peking U.) cillero@th.phy.pku.edu.cn QCD@Work, June 19 th 2007 L.Y.Xiao and J.J.Sanz-Cillero [hep-ph/0705.3899]

2. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero ANSWER: We cannot say about the whole theory But, we can confirm this for some sectors

3. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Organization of the talk: Motivation Meson field redefinitions: Simplifications in the hadronic action Analysis of the S   decay amplitude: Minimal basis of operators Conclusions: 1.) Fully model-independent calculation of the amplitude 2.) Finite # of local chiral-invariant structures for UV div.

4. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Motivation

5. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Rosell et al., JHEP 12 (2005) 020, calculated the one-loop generating functional W[J] from a LO lagrangian with only spin-0 mesons and O (p 2 ) operators They computed the UV divergences and found a huge amount of new NLO structures (operators) but not all you were expecting From a later work, Rosell et al., hep-ph/0611375 (PRD at press), they realised that after imposing the proper high energy behaviour there were no new UV divergent structures in the one-loop SS-PP correlator All one needed was a renormalization of the parameters in the LO lagrangian !!

6. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero A chiral theory for resonances Here, we denote as resonance chiral theory (R  T) to the most general chiral invariant theory including: The Goldstones from the spontaneous  symmetry breaking + The mesonic resonances (See the last two speakers)

7. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Building blocks Goldstone fields (  L,  R ) ( g L  L h t, g R  R h t ) with  R =  L t =u = exp{ i  /√2 F } Covariant transformations, X h X h t with X=u ,  ±,f ±  qq resonance multiplets X h X h t with X=S, V… g  G [Ecker et al., NPB 321 (1989) 311]

8. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero And their covariant derivatives   …   X with X=R, u ,  ±,f ±  Putting these elements together and taking flavour traces one gets the different chiral-invariant operators for the lagrangian. For instance, … [Ecker et al., NPB 321 (1989) 311] [Cirigliano et al., NPB 753 (2006) 139]

9. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero The aim of this talk (work) is to show that, indeed, R  T it is possible to build a R  T that provides a model independent description of QCD From this we will be able to extract some deeper implications about the structure of the hadronic QFT Renormalizable sectors J=s, p, v , a , t 

10. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Challenges in the construction of hadronic lagrangians What is needed? Formal pertubation theory: 1/N C expansion  loop expansion Short-distance matching: R  T  OPE + pQCD Numerical convergence of the perturbative expansion (Chiral) Symmetry constrains the lagrangian BUT, a priori, it still allows an infinite # of operators [‘t Hooft, NPB 72 (1974) 461] [Ecker et al., PLB 223 (1989) 425]

11. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Goal in the development of a QFT for hadrons  The action may contain an infinite number of operators (like e.g. in  PT) …  But, for a given amplitude at a given order in the perturbative expansion, only a finite number of operators is required (again, like in  PT)

12. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero How to find this minimal basis of operators ? How can we simplify the structure of the lagrangian ?  By demanding a good low-energy behaviour (chiral symmetry) Just putting meson fields together is not enough  By demanding a good high-energy behaviour A hadronic action is only QCD for a particular value of the couplings  Through meson field redefinitions of the generating functional W[J] Some operators in the action are redundant (unphysical)

13. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero  W[J] ( keeping covariance ) … and just to remind what is the meson field redefinition invariance,

14. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Mesonfieldredefinition

15. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero The intuitive picture: RR  R -1 = The contribution from some operators may look like a non-local resonance exchange… … but they always appear through local structures e.g.,  So we would like to remove these redundant operators

16. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero A more formal procedure: Meson field redefinitions in the R  T lagrangian …

17. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero We start from a completely general R  T lagrangian: with the remaining part containing any other possible operator, In this work, we consider two kinds of transformations Goldstone field transformation Scalar field transformation ~ S u  u 

18. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Goldstone field transformation: We perform a shift such that [Xiao & SC’07]

19. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Goldstone field transformation: We perform a shift such that [Xiao & SC’07]

20. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Goldstone field transformation: We perform a shift such that Scalar meson field transformation: By means of the decomposition [Xiao & SC’07]

21. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Goldstone field transformation: We perform a shift such that Scalar meson field transformation: By means of the decomposition and the transformation We end up with the simplified lagrangian: [Xiao & SC’07]

22. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Analysis of the S   decay amplitude

23. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Thanks to these transformation we will proof that the S   decay amplitude is ruled at tree-level by a finite # of operators in the R  T lagrangian The most general form for operators contributing to S   is given (in the chiral limit) by withouth any a priori constraint on the number of derivatives S, u ,  ±, f ±  P, C, h.c.

24. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero The simplest operator of this kind is the c d term With =c d /2 in Ecker et al. NPB 321 (1989) 321

25. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero The terms with covariant derivatives were exhaustively analysed by regarding the possible contractions for the indices  i  i   (or j   )  i   j (or i  j )  i  j  i   and i   [Xiao & SC’07]

26. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero The only surviving case yields an equivalent operator with a lower number of derivatives reduce ANY OPERATOR to the c d termIteratively it is then possible to reduce ANY OPERATOR to the c d term simply using the chiral identities and the former field transformations

27. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero If one also considers multi-trace operators (subleading in 1/N C ) there are another three operators a b c exhausting the list of independent chiral-invariant operators contributing to S  

28. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Conclusionsandprospects

29. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero This provides a clear example of the possibility of constructing fully model-independent resonance lagrangians The action may contain an infinite # of operators but the S   amplitude is given at large-N C by just the c d term For instance, the S-meson contribution to    is given at large-N C by just this operator The remaining information would be in the local  PT-like operators and other resonance exchanges (which must be taken into account both if one makes the simplifications or not) S        R’     +

30. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero The only available chiral-invariant structures for the UV divergences appearing in S   at the loop level are these 4 operators The renormalization of the 4 couplings c d, a, b, c renders this amplitude finite What implications does this have on the renormalizability ?

31. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero The existence of a finite basis of independent operators… 1.Might be just one lucky situation for a particular amplitude (not true; preliminary results) 2.Valid for a wide set of amplitudes (the most likely) 3.A general feature of the lagrangian (unlikely but appealing enough to study it)

32. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero What if this is a general feature ? If any amplitude M is always given at tree-level by a finite # of chiral-invariant operators, then the local UV divergences in the generating functional would have this same structure … 33 11 22 44 55 … 33 11 22 44 55 local UV div.

33. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Summarising There are only 4 independent S   operator at any order in perturbation theory There are only 4 independent S   UV-divergent structures at any order in perturbation theory

34. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero Outlook To extend this kind of simplifications for a wider set of amplitudes Other S-meson processes Other resonances Heavy meson sector Green-functions Preliminary results on the SFF, PFF and correlators look very promising

35. Can a R  T be a renormalizable theory ? J.J. Sanz-Cillero