Lattice QCD study of
Radiative Transitions in
Charmonium
(with a little help from the quark model)
Jo Dudek, Jefferson Lab
with Robert Edwards & David Richards
Charmonium spectrum & radiative transitions
most charmonium states below
threshold have measured
radiative transitions to a lighter charmonium state
transitions are typically E1 or M1
multipoles, although transitions
involving
also admit M2 & E3,
whose amplitude is measured through
angular distributions
Eichten, Lane & Quigg
PRL.89:162002,2002
these transitions are described
reasonably well in quark potential
models
Jo Dudek, Jefferson Lab
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Realization of QCD on a lattice
•
Continuous space-time  4-dim Euclidean lattice
•
•
Derivatives  finite differences
Quarks on sites, gluons on links. Gluons
elements of group SU(3)

O U , ,

Um (x) = exp(iga Am(x))


1
 SG U   M U 
dU
d

d

O
U
,

,

e

Z
1
S U
 '  dU  O U , M 1 U  det  M U   e G  
Z



•
•
Integrate anti-commuting fermion
fields  ! det(M(U)) and M-1(U)
factors
Gauge action ~ continuum:
 
1 4
SG   d x Fa Fa   a 2
4
Jo Dudek, Jefferson Lab
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Monte Carlo Methods
• Numerical (but not direct) integration
• Stochastic Monte Carlo method:
– Generate configurations Um(i)(x) distributed with probability
exp(-SG(U))det(M(U))/Z
– Compute expectation values as averages:

O U , ,
•
•
•
•

1

N
 O U   , M
N
i
i 1
1
U   
i
Statistical errors » 1/sqrt(N), for N configurations
The det(M(U)) is expensive since matrix M is O(VxV), though sparse
Full QCD: include det – improved algorithms, cost ~ O(V5/4)
Quenched approximation: set det(M) = 1, e.g., neglect internal quark
loops.
Jo Dudek, Jefferson Lab
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Lattice QCD – two-point functions and masses
a simple object in QCD – a meson two-point function:
choose a quark bilinear of appropriate quantum numbers
Jo Dudek, Jefferson Lab
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Lattice QCD – two-point functions and masses
e.g.
Jo Dudek, Jefferson Lab
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Lattice QCD – two-point functions and masses
e.g.
excited state
contribution
fermion
formalism
“nasties”
ground state
“plateau”
Jo Dudek, Jefferson Lab
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Smearing
honesty -
did not give us that clear plateau
- this operator has overlap with all psuedoscalar states
we attempt to maximise the overlap of our operator with the ground
state by “smearing it out”
“gauge-invariant Gaussian”
mocking-up the quark-antiquark wavefunction
width of the Gaussian is the smearing parameter
in practice we can compute two-point functions with several smearings and fit
them simultaneously to a sum of exponentials
Jo Dudek, Jefferson Lab
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Smearing & two-point fits
[smeared – local] two-point
Jo Dudek, Jefferson Lab
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Scale setting & the spectrum
this mass is in lattice units – we set the lattice spacing, , using the static
quark-antiquark potential on our lattice
we didn’t “tune” our charm quark mass very accurately – our whole
charmonium spectrum will be too light
Jo Dudek, Jefferson Lab
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Spectrum
appropriate time to own up to some approximations in our simulations
- what have we actually been computing?
and not
QUENCHED
nor
NO DISCONNECTED
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Approximations in the spectrum?
do these approximations show up in the spectrum?
our hyperfine splitting is too small
in quark potential picture hyperfine is a short-distance effect – quenching
could bite here because:
* set scale with long-distance quantity (a mid-point of the potential)
* neglect light quark loops which cause the coupling to run
* coupling doesn’t run properly to short-distances
“Fermilab” Clover
Jo Dudek, Jefferson Lab
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Approximations in the spectrum?
what about the disconnected diagrams
- in the perturbative picture
disconnected diagrams might contribute to the hyperfine splitting
- studies (QCD-TARO, Michael & McNeile) suggest an effect of order 10 MeV in
the right direction
likely to be sensitive also to finite lattice spacing effects
Jo Dudek, Jefferson Lab
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Three-point functions
radiative transitions involve the insertion of a vector current to a quark-line
simple example is the
“transition”
* analogue of the pion form-factor
* couple the vector current only to the quark to avoid charge-conjugation
Jo Dudek, Jefferson Lab
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Three-point functions
need to insert the current far enough away from
have died away (exponentially)
that the excited states
form-factor is defined by the Lorentz-covariant decomposition
compute these three-point functions for a range of lattice momenta
- obtain
by factoring out the
we got from fitting the
two-point functions
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form-factor
we fix the source field at
and the sink field at
- plot three-point as a function of vector current insertion time
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form-factor
plot as a function of
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transition
transitions between different states are no more difficult
has a polarisation
decomposition in terms of one form-factor
smeared-smeared
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transition
radiative width:
Note that this is just one Crystal Ball
measurement
I cooked up an alternative from
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transition
how do we extrapolate back to
?
- take advice from the non-rel quark-model:
this is an M1 transition which proceeds by quark spin-flip
convoluting with Gaussian wavefunctions one obtains
we can fit our lattice points with this form to obtain
and
causes problems for quark
potential models
Jo Dudek, Jefferson Lab
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transition
Jo Dudek, Jefferson Labsystematic
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problems dominate
transition
we have a clue about the systematic difference: scaling of
Jo Dudek, Jefferson Lab
by 1.11
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transition
at
, there are only transverse photons and this transition has only one
multipole – E1
with non-zero
, longitudinal photons provide access to a second: C1
multipole decomposition:
Jo Dudek, Jefferson Lab
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transition
three-point function
several different
combinations have the same
value
are known functions
do the inversion to obtain the multipole form-factors
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transition
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transition
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transition
good plateaux were suggested by the two-point functions
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transition
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transition
we’ll again constrain our
extrapolation using a quark model form
this is an E1 transition which proceeds by the electric-dipole moment
convoluting with Gaussian wavefunctions one obtains
where the photon 3-momentum at virtuality
is given by
in the rest frame of a decaying
Jo Dudek, Jefferson Lab
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transition
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transition
also obtain the physically unobtainable* C1 multipole
*unless someone can measure
quark model form
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transition
two physical multipoles contribute: E1, M2
also one longitudinal multipole: C1
multipole decomposition:
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transition
experimental studies of angular dependence give the M2/E1 ratio
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transition
we might have some trouble with the three-point functions
- go back to the
at rest two-point function
plateau is borderline – our smearing isn’t ideal for this state
non-zero momentum states are even worse
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transition
we anticipate borderline plateaux:
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transition
we anticipate borderline plateaux:
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transition
we anticipate borderline plateaux:
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transition
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transition
quark model structure:
has an additional spin-flip beyond the E1 multipole
- cost of this is
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transition
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transition
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how well do we do?
wavefunction extent:
quark model charmonium wavefunctions (from
Coulomb + Linear potential) have typically
transition widths & multipole ratios:
lat*
PDG
CLEO
lat
PDG
Grotch et al
Jo Dudek, Jefferson Lab
*using lattice simul. masses
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how well do we do?
pretty successful, although still systematics to content with
no obvious sign of our approximations: quenching & connected
- these transitions are long-distance effects
once current simulation run is complete we’ll make a prediction :
excited state transitions are rather tricky – limits the ease with which we can get
quark model extrapolation forms are very powerful – need to verify them with
lattice data points at smaller
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nearer to
our current simulations have points very near to
:
very small, negative
no plateaux
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future
running right now – isotropic lattice (equal spatial & temporal lattice size)
- might cure some of our systematics
larger volume – gets us closer to
but with a simulation-time cost
ultimate aim of this project is to do JLab physics
e.g.
GlueX:
hybrid
meson
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extras
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What were those wiggles?
back to the wiggles at small times in the two-point functions
they come about because of our choice of formalism for fermions on the lattice
- we used Domain Wall Fermions
lattice theorist on learning
that we’re using
anisotropic DWF for
charmonium 3pt functions
Jo Dudek, Jefferson Lab
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Domain Wall Fermions
usually thought of as being good for
chiral quarks
charm quarks at our lattice spacings
are feasible
advantage over Wilson is automatic
O(a) improvement
more computer time, less human
time than tuning a Wilson Clover
action
Jo Dudek, Jefferson Lab
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Domain Wall Wiggles
ghosts – don’t have a quadratic dispersion relation
Jo Dudek, Jefferson Lab
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Anisotropic Lattices
Lattice QCD is a cutoff theory, with
. So to model charmonium
we will need
, i.e. a fine lattice spacing.
we also require a lattice volume large enough to hold the state, so
appear to need a lot of sites,
, and hence a lot of computing time
there’s a handy trick to get us out of this
- the only large scale is
, which is conjugate to time
- typical charmonium momenta are
only need a fine spacing in the time direction – space directions can be
coarse – “Anisotropic Lattice”
fine spacing in the time direction allowed us to
see all the structure in the two-point functions
Jo Dudek, Jefferson Lab
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Anisotropic Lattices
the gluon (Yang-Mills) piece of the action gains a parameter,
this is chosen to get the desired anisotropy
the quark-gluon piece of the action features both
and a second parameter,
sometimes called the “bare speed-of-light” (ratio of spat. to temp. derivatives)
is tuned to ensure physical particles have the correct dispersion relation
i.e.
(up to lattice artifacts)
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,
Dispersion relation tests
we extract the spectrum of states with non-zero momentum using two-point
functions
because our lattice is of finite volume we don’t have continuous momenta
– rather a discrete set
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Dispersion relation tests
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Dispersion relation tests
display the dispersion relation via the quantity
perfect tuning would be
we’ve not tuned perfectly – a hazard of using anisotropy!
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Lattice QCD – two-point functions and masses
e.g.
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ZV
for the vector current we use
, which isn’t conserved on a
lattice. Hence it gets renormalised (multiplicatively)
obtain it from the “pion” form-factor at zero Qsq
NB we see an 11% discrepancy between the value at pf=000 and at pf=100
same 11% doesn’t seen to be present using the rho form-factor?
Jo Dudek, Jefferson Lab
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