Can a resonance chiral theory be a renormalizable theory ?

Can a RT be a renormalizable theory ?

J.J. Sanz-Cillero

Can

a resonance chiral theory

be a renormalizable

theory ?

J.J. Sanz-Cillero (Peking U.)[email protected]

QCD@Work, June 19th 2007

L.Y.Xiao and J.J.Sanz-Cillero [hep-ph/0705.3899]


J.J. Sanz-Cillero

ANSWER:ANSWER:

We cannot say

about the whole theory

But, we can confirm this

for some sectors


J.J. Sanz-Cillero

Organization of the talk: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.


J.J. Sanz-Cillero

MotivationMotivation


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 (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 !!


J.J. Sanz-Cillero

A chiral theory for resonances

Here, we denote as resonance chiral theory (RT)

to the most general chiral invariant theory including:

•The Goldstones from the spontaneous symmetry breaking

+•The mesonic resonances

(See the last two speakers)


J.J. Sanz-Cillero

Building blocks

Goldstone fieldsGoldstone fields

( L , R ) ( gL L ht , gR R h t )

with R=Lt =u =

exp{i/√2 F}

Covariant transformations,

X h X h t with X=u ,± ,f±

qq qq resonance multipletsresonance multiplets

X h X h t with X=S, V…

g G

g G

g G

[Ecker et al., NPB 321 (1989)

311]


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,

< X1 X2 ··· >

< X1 X2 ··· >

< X1 > < X2 ··· >

…

[Ecker et al., NPB 321 (1989)

311][Cirigliano et al.,

NPB 753 (2006) 139]


J.J. Sanz-Cillero

The aim of this talk (work)

is to show that, indeed,

it is possible to build a RRT T

that provides

a model independent description of QCDa model independent description of QCD

From this we will be able to extract From this we will be able to extract

some deeper implications some deeper implications

about the structure of the hadronic about the structure of the hadronic QFT QFT

Renormalizable

sectors

J=s, p, v, a, t


J.J. Sanz-Cillero

Challenges in the construction

of hadronic lagrangians

What is needed?What is needed?

• Formal pertubation theory: 1/NC expansion loop expansion

• Short-distance matching: RT 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]


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)


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)


J.J. Sanz-Cillero

W[J] W[J]

( keeping covariance )

… and just to remind what is the meson field

redefinition invariance,


J.J. Sanz-Cillero

Meson Meson

field field

redefinitionredefinition


J.J. Sanz-Cillero

• The intuitive picture:

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.,

<… (∂2+MR2) R >

So we would like to remove these redundant operators


J.J. Sanz-Cillero

• A more formal procedure:

Meson field redefinitions in the RT lagrangian

…


J.J. Sanz-Cillero

• We start from a completely general RT 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


J.J. Sanz-Cillero

• Goldstone field transformation:

We perform a shift such that

[Xiao & SC’07]


J.J. Sanz-Cillero



[Xiao & SC’07]


J.J. Sanz-Cillero



• Scalar meson field transformation:

By means of the decomposition

[Xiao & SC’07]


J.J. Sanz-Cillero



• Scalar meson field transformation:

By means of the decomposition

and the transformation

We end up with the simplified lagrangian:

[Xiao & SC’07]


J.J. Sanz-Cillero

Analysis of Analysis of

the Sthe S

decay amplitudedecay amplitude


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 RT 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.


J.J. Sanz-Cillero

• The simplest operator of this kind is the cd term

With =cd/2 in Ecker et al. NPB 321 (1989) 321


J.J. Sanz-Cillero

• The terms with covariant derivatives were exhaustively analysed by regarding the possible contractions for the

indices i

1. i (or j )

2. i j (or i j)

3. i j

4. i and i

[Xiao & SC’07]


J.J. Sanz-Cillero

• The only surviving case yields an equivalent operator

with a lower number of derivatives

• Iteratively it is then possible to reduce ANY OPERATOR to reduce ANY OPERATOR to the cthe cdd term term

simply using the chiral identities

and the former field transformations


J.J. Sanz-Cillero

If one also considers multi-trace operators (subleading in 1/NC )

there are another three operators

a < S > < u u >

b < S u > < u >

c < S > <u > < u >

exhausting the list of independent chiral-invariant operators

contributing to S


J.J. Sanz-Cillero

ConclusionsConclusions

andand

prospectsprospects


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-NC by just the

cd term

• For instance, the S-meson contribution to

is given at large-NC 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’

+


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 cd, a, b, c

renders this amplitude finite

What implications does this have

on the renormalizability ?


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)


J.J. Sanz-Cillero

• What if this is a general feature ?

If any amplitude 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

1

2

4

5 …

3

1

2

4

5

localUVdiv.


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


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


J.J. Sanz-Cillero

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Can a resonance chiral theory be a renormalizable theory ?

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