Can a resonance chiral theory be a renormalizable theory ?

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

ANSWER: We cannot say about the whole theory But, we can confirm this for some sectors

Organization of the talk: • Motivation • Meson field redefinitions: • Simplifications in the hadronic action • Analysis of the Spp 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.

Motivation

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(p2) 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 !!

A chiral theory for resonances • Here, we denote as resonance chiral theory (RcT) • to the most general chiral invariant theory including: • The Goldstones from the spontaneous c symmetry breaking • + • The mesonic resonances (See the last two speakers)

Building blocks [Ecker et al., NPB 321 (1989) 311] Goldstone fields ( xL , xR ) ( gLxL ht , gRxR h t ) with xR=xLt=u = exp{ip/√2 F} Covariant transformations, X h X h t with X=um ,c± ,f±mn qq resonance multiplets X h X h t with X=S, V… g  G g  G g  G

And their covariant derivativesa … m X with X=R, um ,c± ,f±mn • Putting these elements together and taking flavour traces • one gets the different chiral-invariant operators for the lagrangian. • For instance, • < aX1amX2 ··· > • < X1abX2 ··· > • < X1> <aX2 ··· > • … [Ecker et al., NPB 321 (1989) 311] [Cirigliano et al., NPB 753 (2006) 139]

The aim of this talk (work) is to show that, indeed, it is possible to build a RcT that provides a model independent description of QCD J=s, p, vm, am , tmn From this we will be able to extract some deeper implications about the structure of the hadronic QFT Renormalizable sectors

Challenges in the construction of hadronic lagrangians • What is needed? • Formal pertubation theory: 1/NC expansion  loop expansion • Short-distance matching: RcT  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]

Goal in the development of a QFT for hadrons The action may contain an infinite number of operators (like e.g. in cPT) … But, for a given amplitude at a given order in the perturbative expansion, only a finite number of operators is required (again, like in cPT)

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)

… and just to remind what is the meson field redefinition invariance, F F + dF W[J] W[J] ( keeping covariance )

Meson field redefinition

The intuitive picture: = DR-1 DR The contribution from some operators may look like a non-local resonance exchange… … but they always appear through local structures e.g., l <… (∂2+MR2) R >  So we would like to remove these redundant operators

A more formal procedure: • Meson field redefinitions in the RcT lagrangian …

We start from a completely general RcT 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 ua ub

Goldstone field transformation: • We perform a shift such that [Xiao & SC’07]

Goldstone field transformation: • We perform a shift such that • Scalar meson field transformation: • By means of the decomposition [Xiao & SC’07]

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]

Analysis of the Spp decay amplitude

Thanks to these transformation we will proof that the Spp decay amplitude is ruled at tree-level by a finite # of operators in the RcT lagrangian • The most general form for operators contributing to Spp is given (in the chiral limit) by • withouth any a priori constraint on the number of derivatives S, um, c±, f±mn P, C, h.c.

The simplest operator of this kind is the cd term With l=cd/2 in Ecker et al. NPB 321 (1989) 321

The terms with covariant derivatives were exhaustively analysed by regarding the possible contractions for the indices mi • mir (or nj s) • mimj (or ni nj) • minj • mis and ni r [Xiao & SC’07]

The only surviving case yields an equivalent operator • with a lower number of derivatives • Iteratively it is then possible to reduce ANY OPERATOR to the cd term • simply using the chiral identities • and the former field transformations

If one also considers multi-trace operators (subleading in 1/NC ) there are another three operators la < S > < um um > lb < S um > < um > lc < S > <um > < um > exhausting the list of independent chiral-invariant operators contributing to Spp

Conclusions and prospects

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pp amplitude is given at large-NC by just the cd term • For instance, the S-meson contribution to pppp • is given at large-NC by just this operator • The remaining information would be in the • local cPT-like operators • and other resonance exchanges • (which must be taken into account both if one makes the simplifications or not) p p S p p p p p p R’ + p p p p

What implications does this have on the renormalizability ? • The only available chiral-invariant structures • for the UV divergences appearing in Spp at the loop level • are these 4 operators • The renormalization of the 4 couplings cd, la, lb, lc • renders this amplitude finite

The existence of a finite basis of independent operators… • Might be just one lucky situation for a particular amplitude (not true; preliminary results) • Valid for a wide set of amplitudes (the most likely) • A general feature of the lagrangian (unlikely but appealing enough to study it)

F4 F4 F3 F3 F2 F2 … … F5 F5 F1 F1 • 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 local UV div.

Summarising • There are only 4 independent Spp operator • at any order in perturbation theory • There are only 4 independent Spp UV-divergent structures • at any order in perturbation theory

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

Can a resonance chiral theory be a renormalizable theory ?