Physics Department, University of Ljubljana and Institute J' Stefan

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Messages from inclusion of positive parity heavy
mesons in heavy meson chiral
perturbation theory
Svjetlana Fajfer
Physics Department, University of Ljubljana and
Institute J. Stefan Ljubljana, Slovenia
in collaboration with J. Kamenik, D. Becirevic
and J. O. Eeg
Troia 07, 30 Aug. 3. Sept. 2007, Canakkale,
Turkey
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Outline

• Low energy chiral perturbation theory
• Heavy meson chiral perturbation theory
• Inclusion of positive parity heavy mesons
• Chiral loop corrections in strong charm decays
• Chiral loop corrections to B meson oscillations
• Chiral loop corrections to Isgur -Wise functions
• Lattice computation and chiral limit
• Summary

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SU(3)xSU(3) chiral perturbation theory
At energies bellow 1 GeV QCD is in a highly
non-perturbative regime Very difficult to
describe hadronic interactions!
However, the hadronic spectrum is simple at low
energies and the interactions among the
pseudoscalar mesons become weak in the limit of
E -gt 0.
The QCD can be treated perturbatively at low
energies with a suitable choice of degrees of
freedom.
Instead of quark and gluons in perturbative QCD
one uses light pseudoscalar mesons as
interacting objects of Chiral Perturbation
Theory.
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CHPT is an example of Effective Quantum field
theory
The basic principle Only few degrees of freedom
are relevant in a given energy range- the heavy
ones can be integrated out.
Chiral symmetry
Chiral group
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Callan, Coleman and Zumino have analyzed general
construction of non-linear realisations of a
spontaneously broken group in terms of its
Goldstone boson fields. Gasser and Leutwyler have
introduced SU(3)x SU(3) chiral perturbation
theory.
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Simplest choice
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Electroweak interactions are introduced by using
gauge fields as external fields
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Gasser and Leutwyler have obtaied all 10
paramters (the finite part of them) at the 4-th
order in chiral expansion to be related to
vector, axial-vector, scalar and pseudoscalar
resonaces
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Resonances and CHPT
G.Ecker, et al. 1989
Note that masses of all these resonances larger
than masses of Goldstone bosons in SU(3) octet.
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Extension of chiral perturbation theory to heavy
mesons
Mass spectrum of open charm mesons
Question What is their impact on strong and weak
charm meson decays?
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Strong decays of positive and negative parity
charmed mesons

• average of Belle and Focus values
• average of Belle and CLEO values

There are many studies quark models, QCD sum
rules, HMChPT, lattice
S.F. and J. Kamenik, hep-ph/0606278 (Phys. Rev D
74, 074023 (2006))
Strong couplings are investigated including
chiral loop corrections within Heavy meson
chiral perturbation theory (I.W. Stewart, NP
B529, 62 (1998), P. Colangelo et al., Phys. Rev.
D 52, 6422 (1995), T. Mehen and R. Springer, PRD
72, 034006 (2005).
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Framework
The leading order of HMChPT in chiral and heavy
quark expansion
These three couplings are being discussed within
this study
One is free to set
But, in the loop calculations enter
Note, that previous calculations extracted g
coupling without positive parity states and h
only at tree level.
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Extraction of bare couplings
Wave function renormalization
Vertex corrections
residual masses of the final and initial states
Scale dependence of the loops cancelled by the
counterterms. However, many new parameters
appear in counterterms which cannot be fixed by
existing data.
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We constrain the range of the fitted bare
couplings by using existing knowledge of their
dressed values and assuming the first order loop
corrections to be moderate and thus also
maintaining convergence of the perturbation
theory.
We assume
Then Monte -Carlo randomized least square fit
for all three couplings is performed using
experimental values for the decay rates to
compute . It is found that and
g 0.66, h0.47
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Counterterms
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Effects of counterterms
We take counterterm couplings entering our decay
modes to be randomly distributed at
in the interval and
5000 values of
are generated near original fitted solution by
minimizing at each counterterm sample.
For each solution the average absolute value of
the randomized couterterm couplings
is computed. It is assumed that counterterm
contributions do not exceed values of the order
.
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The inclusion of counterterms spreads the fitted
values of the tree couplings.
Extracted bare couplings
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at

• We could not include terms
  due to very large number of new
• parameters, however lattice studies indicate
  that they do not contribute significantly
  

• The result is in a way complementary to the
  study of renormalization scale
• dependence. Both are important since although it
  is possible to trade the
• counterterms contributions for a specific choice
  of the renormalization scale,
• the latter will be different for different
  amplitudes where the combination
• of counterterms will be different.

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Chiral extrapolation
Lattice QCD performs calculations using large
light quark masses and then makes chiral limit.
The inclusion of heavy excited mesons in the
chiral loops introduces large scale dependence
into the renormalization of the coupling
constants. It looks as excited states dominates
loop contributions.
Note that pions in the loops can be real, what
introduces uncontrollable FSI.

• The loop integral depends on two scales
• the mass of pseudogoldstone boson (its value can
  be as large
• as 1 GeV in the lattice calculation, small in
  CHPT)
• the scale contains the splitting and it
  is not protected either by heavy
• quark or chiral symmetry to be small

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These contributions dominate the chiral limit
If we instead use the loop integral expansion we
effectively replace by
. Furthermore, these terms
then become of the order
and formally contribute only to
next-to-leading chiral log running.
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Taming resonance contribution decoupling limit
The diverging analytic and logarithmic parts
cancel out exactly washing out any leading order
contributions to the chiral running from such
loops
The message is that below
the presence of the nearby opposite parity
states does not affect the leading order pionic
logarithmic behavior of
1/
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We are interested in the low momentum part of the
integral and assume that
the counterterms account high momentum
contributions.
We effectively throw out all contributions coming
from positive powers of x since they originate
from hard pseudo-Goldstone exchange and shifting
them into counterterms.
The relevant energy scales of the effective
theory are
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Chiral extrapolation of the effective meson
couplings
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• Without better experimental data and/or lattice
  QCD inputs, the phenomenology of strong decays of
  charmed mesons cannot be considered reliable at
  this stage.

• Due to large number of parameters
  (MehenSpringer) in 1/M corrections and chiral
  loop corrections it is not possible to determine
• them all

• We found that for the chiral extrapolation of
  the coupling g full loop
• contributions give sizable effects in modifying
  slope and curvature in the case
• when

• If we instead use
  expansion the effect is reduced
• h coupling contributions are reduced to be of the
  order 5.

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with D. Becirevic and J. Kamenik, hep-ph/0612224,
JHEP 0706, 003 (2007)
indication that top quark is heavy
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(d or s quark)
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the SUSY basis
VSA vacuum saturation approximation
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(tilde means that operators are considered in the
static limit of HQET) The HQET states are
normalized as
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Scope of work
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Bosonisation of the operators leads to
The bag parameters are
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Chiral logarithmic corrections to bag parameters
in SUSY basis
Nonfactorisable contributions

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Impact of heavy mesons with positive parity
Corrections to decay constants
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Bag parameters
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Conclusions
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Chiral loop corrections to weak decays of B
mesons to positive and
negative parity charmed mesons
With J. Kamenik and J.O. Eeg, arXiv0705.4567 ,
JHEP 0707,078 (2007)
impact of the lowest-lying positive parity
states on determinations of the Isgur-Wise
functions
and

corrections appear at the second order
important in the extraction of the CKM parameter
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Isgur-Wise function
Heavy-quark symmetry reduces six form factors to
only one!


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Chiral limit of our results necessary for the
lattice studies
We use the same framework as in two previous
cases!
One has to bosonise weak current. We use only
leading order in chiral and heavy quark expansion.
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where
heavy quark symmetry dictates
We consider leading chiral loop corrections to
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Chiral extrapolation
In order to tame the chiral behavior of the
amplitudes containing the mass gap between the
ground state and the excited heavy meson states
we use expansion of the chiral
loop integrals
(au,d)
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Gell-Mann Okubo mass formula
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Conclusions

• Motivated by previous result that pionic chiral
  logarithms are not changed by
• the inclusion of even parity heavy meson states
  we consider chiral extrapolation
• of Isgur-Wise functions

• Leading pionic contribution is not changed by
  the inclusion of positive parity
• heavy meson states of the Isgur Wise function
  .

• Our results on the chiral corrections are
  crucial in assuring validity
• of the form factor extraction and error
  estimation coming from the lattice QCD
• extraction

• Our estimation of the leading
  also constrain the accuracy of such
• extrapolations.

• The decays are not
  approached by experiment yet. Our result
• contains chiral corrections to this decay
  amplitude. Due to the strange quark
• there is no leading pion logarithmic corrections.
  

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Final messages
In chiral corrections
can be calculated in many processes. However,
due to huge number of counterterms and 1/M
corrections it is impossible to build whole
picture as it is done for low energy physics.

Inclusion of positive parity heavy meson usually
contributes in the chiral corrections as much
as strange pseudo-Goldstone bosons. (Do not
forget new counterterms and 1/M corrections!)
The pionic corrections are decoupled from
contributions of higher lying states. Therefore,
the chiral extrapolation, necessary for the
lattice QCD studies is possible when pion mass
is smaller than the heavy mesons mass splitting.
The effects of excited heavy mesons are then
included into higher order corrections to a
theory without dynamical excited heavy mesons.
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Feynman rules
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Chiral loop integrals
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Wave function renormalisation
Vertex corrections
we consider
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