Transcript Document 7817367

QCD@Work, June 19th 2007
Can
a resonance chiral theory
be a renormalizable theory ?
J.J. Sanz-Cillero
(Peking U.)
[email protected]
L.Y.Xiao and J.J.Sanz-Cillero [hep-ph/0705.3899]
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
ANSWER:
We cannot say
about the whole theory
But, we can confirm this
for some sectors
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
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.
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
Motivation
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
•
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 !!
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
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)
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
Building blocks
[Ecker et al.,
NPB 321 (1989) 311]
Goldstone fields
( xL , xR )
gG
( gL xL ht , gR xR h t )
with xR=xLt =u = exp{ip/√2 F}
Covariant transformations,
X
gG
h X ht
with X=um ,c± ,f±mn
h X ht
with X=S, V…
qq resonance multiplets
X
J.J. Sanz-Cillero
gG
Can a RcT be a renormalizable theory ?
And their covariant derivatives
with X=R, um ,c± ,f±mn
a … m X
• 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 ··· >
…
J.J. Sanz-Cillero
[Ecker et al.,
NPB 321 (1989) 311]
[Cirigliano et al.,
NPB 753 (2006) 139]
Can a RcT be a renormalizable theory ?
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
J.J. Sanz-Cillero
Renormalizable
sectors
Can a RcT be a renormalizable theory ?
Challenges in the construction
of hadronic lagrangians
What is needed?
• Formal pertubation theory:
1/NC expansion  loop expansion
[‘t Hooft, NPB 72 (1974) 461]
• Short-distance matching:
RcT  OPE + pQCD
[Ecker et al., PLB 223 (1989) 425]
• Numerical convergence of the perturbative expansion
• (Chiral) Symmetry constrains the lagrangian
BUT, a priori, it still allows an infinite # of operators
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
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)
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
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
Can a RcT be a renormalizable theory ?
… and just to remind what is the meson field
redefinition invariance,
F
W[J]
J.J. Sanz-Cillero
F + dF
( keeping covariance )
W[J]
Can a RcT be a renormalizable theory ?
Meson
field
redefinition
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• The intuitive picture:
DR-1
DR
The contribution
from some operators
may look like
a non-local resonance exchange…
e.g.,
=
… but they
always appear
through local structures
l <… (∂2+MR2) R >
 So we would like to remove these redundant operators
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• A more formal procedure:
Meson field redefinitions in the RcT lagrangian
…
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• We start from a completely general RcT lagrangian:
~ S u a ub
with the remaining part containing any other possible operator,
• In this work, we consider two kinds of transformations
•Goldstone field transformation
•Scalar field transformation
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• Goldstone field transformation:
We perform a shift such that
[Xiao & SC’07]
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• Goldstone field transformation:
We perform a shift such that
[Xiao & SC’07]
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• Goldstone field transformation:
We perform a shift such that
[Xiao & SC’07]
• Scalar meson field transformation:
By means of the decomposition
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• Goldstone field transformation:
We perform a shift such that
[Xiao & SC’07]
• Scalar meson field transformation:
By means of the decomposition
and the transformation
We end up with the simplified lagrangian:
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
Analysis of
the Spp
decay amplitude
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
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
S, um, c±, f±mn
P, C, h.c.
withouth any a priori constraint on the number of derivatives
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• The simplest operator of this kind is the cd term
With l=cd/2 in
J.J. Sanz-Cillero
Ecker et al. NPB 321 (1989) 321
Can a RcT be a renormalizable theory ?
• The terms with covariant derivatives were exhaustively analysed by
regarding the possible contractions for the indices
1. mi r
mi
[Xiao & SC’07]
(or nj  s)
2. mi mj (or ni  nj)
3. mi nj
4. mi s and ni  r
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• 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
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
If one also considers multi-trace operators (subleading in 1/NC )
there are another three operators
la < S > < u m um >
lb < S um > < um >
lc < S > <um > < um >
exhausting the list of independent chiral-invariant operators
contributing to Spp
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
Conclusions
and
prospects
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• 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
p
is given at large-NC by just this operator
The remaining information would be in the
local cPT-like operators
p
p
and other resonance exchanges
p
p
+
S
p
p
p
p
p
R’
p
p
(which must be taken into account both if one makes the simplifications or not)
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
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
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
• 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
Can a RcT be a renormalizable theory ?
• 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
F4
F3
F2
F1
J.J. Sanz-Cillero
…
F5
F3
F2
F1
local
UV
div.
…
F4
F5
Can a RcT be a renormalizable theory ?
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
J.J. Sanz-Cillero
Can a RcT be a renormalizable 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
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?
J.J. Sanz-Cillero
Can a RcT be a renormalizable theory ?