K p, K p l l on the Lattice

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K ? p??, K ? p ll- on the Lattice

• A feasibility study

Paolo Turchetti LNF Spring School Frascati, 19th
May
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Rare decays the present procedure
Rare decays K?p ??, K/-?p/- ll, KS?p0 ll
Top, W, beauty, charm integrated out
Perturbation Theory renormalization group
Experiments ?PT
Wilson coefficients
Operator matrix elements
Phenomenological Predictions
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Problems with this procedure
The problems affecting this procedure are
connected with the importance of long distance
physics in the description of this decays. For
example in the K ? p?? family there is only the
weak contribution. In this case a hard GIM
mechanism is active
By courtesy of G. Isidori
The errors affecting the theoretical estimates of
charm contribution (the long distance one) are
relevant O(20-30).
Error sources

• Operators with dimension greater than 6
• Truncated perturbative expression for Wilson
  Coefficients (NNLO terms are not included)
• Is perturbation theory valid at a such low
  energy scale (lt 1 GeV)?

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Lattice QCD and Rare decays
Now charm is a dynamical degree of freedom
Rare decays K?p ??, K/-?p/- ll, KS?p0 ll
Top, W, beauty, integrated out
Perturbation Theory renormalization group
Experiments LQCD
Wilson coefficients
Operator matrix elements
Phenomenological Predictions
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Impact of LQCD computations
By means of LQCD we could improve our
phenomenological predictions lowering the
theoretical errors affecting our estimates of
total decay amplitude by an order of magnitude
for both K ? p ?? and K? p l l- decays
Owing to the great sensibility of these decays to
Vtd and to physics beyond the Standard Model, an
improved theoretical comprehension of these
transitions through LQCD computation could have
an important impact on our understanding of the
dynamics of quark-flavour mixing.
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LQCD approach
If we consider the charm as a dynamical degree of
freedom (dof), we can restrict ourselves to
compute, through LQCD numerical simulations at a
scale greater than the charm mass, the physical
amplitude in which are involved the operators

• In this way
• we can take into account all the long distance
  contributions exactly
• we dont need to evolve Wilson coefficients down
  to a too low scale, in such a way that NNLO
  contributions are small
• LQCD computations are based only on first
  principles

What do we need to compute?
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What do we really compute?
The physical quantities of our interest are
encoded in
Where J is the electro-magnetic or weak current.
The main issue one has to face is the possible
presence of power divergences in the expression
of the T-product.
These divergences introduce ambiguities that make
the extraction of physical informations through
numerical simulations impossible.
The task of our work is to make an analysis as
complete as possible of these power divergences.
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Classifications of power divergences
The power divergences potentially present in our
T-product originate from two diverse singularity
sources.
We can face the first one renormalizing the
operators involved in the T-product. In this way
one can get rid of the divergences associated
with the diagrams

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In the second class are included all the
divergences due to the singularities appearing in
when
i.e. those divergences associated with contact
terms and with the diagrams
This last kind of divergences cant be removed by
simply renormalizing the relevant operators.
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The Bubble (1)
These topologies are connected by Fierz
transformations so that we are allowed to
restrict our attention only on one of them
At this point is useful to emphasize an important
issue regarding the differences existing on the
lattice between electromagnetic and weak current
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The Bubble (2)
The Z0 boson is too heavy (MZ 80 GeV) and cant
be considered as a dynamical dof on the lattice
because the lattice cutoff at our disposal are
only O(3/4 GeV).
It must be integrated out and the weak
interactions mediated by a Z boson have to be
considered as local ones.
Of course this is not the case for photon. So we
can associate with the photon a non-local
interaction.
This is a fundamental observation because it
brings with it some very important consequences
about the way one handles this two cases on the
lattice.
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Bubbles on a lattice
In fact, on the lattice we can associate with the
photon a gauge-current that is implemented by a
splitted (non local) current.
With weak current we have to associate a local
current that, on the lattice is not a gauge
current.
This difference is clear if we consider the
Feynman rules associated with the two currents
Gauge current
Local current
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Photon case
In this lucky case the constraints imposed by
gauge symmetry cause the algebraic cancellation
of power divergences. The residual divergences
are only logarithmic.
We have worked out explicit computations in
different QCD regularizations on euclidean
lattice (Wilson, clover, twisted mass) and found
that
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Where
are constants and
C
and
L
In all lattice regularizations the power
divergences are absent!
The only difference between different
regularizations concerns the finite term L.
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Z boson case
As outlined earlier in this case we dont have a
gauge current and the gauge symmetry doesnt work
anymore. So we dont have any constraints
implying the algebraic cancellation of power
divergences
As a result we HAVE quadratic divergences left if
we are working with Wilson regularization of
lattice QCD!!!!!
But
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Twisted mass fermions
Appling the recent theory with twisted mass
fermions (Frezzotti, Rossi hep-lat/0306014) we
have explicitly demonstrated, through the first
computation ever done, to our knowledge, in
perturbation theory with this regularization in
the physical basis, that
Twisted mass fermions
Z0 case (logarithmically divergent)
Z0 case (quadratically divergent)
GIM mechanism
In all this work one has to cope with the
perturbative lattice computations. In particular
the twisted mass case is the most involved.
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First conclusion
G. Isidori, G. Martinelli and P. Turchetti to be
published.
We dont have any power divergences associated
with contact terms both in photon case and in Z
boson case!
We can handle the residual logarithmically
divergent terms by usual perturbative methods.
The perturbative procedure is very laborious, but
doesnt imply any conceptual problems.
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The renormalization ambiguity
Lets now face the problem of effective
hamiltonian renormalization.
The Wilson term, necessary to solve the fermion
doubling problem of naive discretization, breaks
chiral symmetry explicitly even if we are working
with massless fermions. This implies that under
the renormalization procedure a generic operator
mixes with other operators of equal or lower
dimension (in energy) having, in general,
different chiral properties. In our case we have
that
Where
are the renormalized operators
are the bare ones
is the field stenght tensor and
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Symmetries
By dimensional counting we see that
But, if we consider the constraints introduced by
GIM mechanism and CPS symmetry with
We can infer that A and B are at most
logarithmically divergent and
So we need non perturbative methods to subtract
these power divergences.
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Renormalization condition
Just limit ourselves to the photon case. For
parity reasons the pseudoscalar density doesnt
appear in this case and the renormalization
condition we need to impose takes the form
Where C is a constant, J is the electromagnetic
current and p and K are the operators
interpolating the pion and the Kaon
The question is
Who can fix the value on r.h.s?
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Perturbation theory would be the ideal tool to
impose the correct renormalization condition, but
the presence of power divergences prevents us to
use it.
The arbitrariness in the renormalization
condition introduces the outlined ambiguities and
thus it makes the extraction of reliable
physical informations through numerical
simulations impossible.
In correspondence of every different value of
finite term we are given a different estimate of
physical quantities this is the ambiguity.
Can this ambiguity really affect the physical
quantities?
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Can this ambiguity really affect the physical
quantities?
The answer to this question we found is very
encouraging.
The physical quantities we are looking for
doesnt need any subtraction. This implies that
it isnt affected by any ambiguities!
By means of Ward identities
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Ward identities
we are given this relation, derivable by means
of usual methods,
Where S(x) is the scalar density. From this
expression we deduce that the insertion of this
quantities in our correlator, needed to subtract
the power divergences present in the insertion of
Q/-, has a singularity structure completely
different from that associated with the physical
quantity we are interested in.
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Pole structure (1)
As can be derived by the previous Ward identity
the power divergences are proportional to this
pole structure
Minkowski
Euclidean
i.e. they are proportional to the sum of two
correlators characterized each by a single double
pole.
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Pole structure (2)
The physical quantities we are interested in are
characterized by the product of two simple poles,
one of which is associated with the mass of the
Kaon and the other one with the mass of the pion
Minkowski
Euclidean
so that any other quantity having another
singularity structure doesnt interfere with it.
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Second Result
G. Isidori, G. Martinelli and P. Turchetti to be
published.
As a matter of fact the power divergences, due to
their pole structure, cant affect the physical
quantities we want to extract.
We can estimate the matrix elements of Q/-
operators without any ambiguities.
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Summary
In conclusion we can summarize our results

• We have demonstrated, through explicit
  computations, the algebraic cancellation of power
  divergences due to the contact terms present in
  the relevant T-product

• We have demonstrated the absence of any
  ambiguities in the extraction of physical
  quantities due to renormalization conditions.

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Fermion doubling problem
Naive discretization
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Physical content
16 solutions
16 lattice degrees of freedom
One of the main consequences is that this theory
is anomaly-free
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Wilson Action
Wilson term. This term breaks the chiral symmetry
explicitly even if we are considering massless
fermions
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Twisted mass action
Where the quarks are organized in mass-degenerate
doublet so that
And where
Twisted Wilson Term
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LSZ reduction formula
LSZ reduction formulas assert that we can extract
S-matrix elements for a particular transition
taking into account the Fourier transforms of
T-product of appropriate operators and than going
on-shell. For example if we consider a four
particles transition well have
Where mi are the masses of the particles involved
in this process and Oi are the operators
interpolating the particles.