arXiv:0905.2160v1 [hep-lat] 13 May 2009
JLAB-THY-09-985
A novel quark-field creation operator construction for hadronic physics in lattice QCD
Michael Peardon
∗
School of Mathematics, Trinity College, Dublin 2, Ireland
John Bulava, Justin Foley, and Colin Morningstar
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Jozef Dudek, Robert G. Edwards, B´alint Jo´o, Huey-Wen Lin, and David G. Richards
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
Keisuke Jimmy Juge
Department of Physics, University of the Pacific Stockton, CA 95211, USA
for the Hadron Spectrum Collaboration
(Dated: May 13, 2009)
A new quark-field smearing algorithm is defined which enables efficient calculations of a broad
range of hadron correlation functions. The technique applies a low-rank operator to define smooth
fields that are t o be used in hadron creation operators. The resulting space of smooth fields is small
enough that all elements of the reduced quark propagator can be computed exactly at reasonable
computational cost. Correlations between arbitrary sources, including multi-hadron operators can
be computed a posteriori without requiring new lattice Dirac operator inversions. The method is
tested on realistic lattice sizes with light dynamical quarks.
PACS numbers: 11.15.Ha,12.38.Gc,12.38.Lg
I. INTRODUCTION
One of the goals of lattice calculations is to predict the low-energy hadron spectr um of confined quarks and gluons,
starting solely from the QCD lagrangian. This approach to spectro scopy necessitates methods for measuring the
two-point correlation functions of field operators with the selected quantum numbers under investigation. A complete
understanding of the QCD spectrum must also included excitations of the mesons and baryons as well as exotic
states. Describing the resonance s seen in scattering experiments as states in QCD has long presented a challenge to
lattice calculations as direct ac c e ss to the matrix elements related to decay widths is usually missing in Euclidean
formulations of field theory. The way to circumvent this is well known in principle: decay properties can be inferred
from a detailed study of the dependence of the spectrum in a finite box on the volume of the box [1, 2]. The best means
of making solid deter mina tions of energy levels is to employ a large basis of operators and then use the variational
method to find good approximations to energy eigenstates. As soon as states above threshold are to be explored, the
basis should include operators that resemble multi-hadron systems where the constituents have well-defined momenta.
Access to all elements of the quark propagator [3] relevant to long-range correlation enables these meas urements and
also enables more detailed investigations into physical observables whose accur ate measurement has often eluded
lattice practitioners. One example is the isoscalar meson sector, where mass determinations require the evaluation of
disconnected diagrams as part of the two-point correlation function. As well as this ambitious list of requirements,
precision data will be crucial for these calculations [4].
In a Monte Carlo calcula tion on the lattice, the physically relevant signal in a correlation function falls exponentially
and is rapidly domina ted by statistical fluctuations. Operators that create low-lying energy eigenstates quickly are
invaluable and improve the quality of data extracted exponentially. The most useful tool at the lattice practitioner’s
disposal in building good creation op e rators is smea ring. The best means of understanding the low-energy degrees
of freedom of confinement must thus come from a smearing method that can address all these features; it must be
numerically accessible, do a good job of projecting onto the lowest ener gy states and it must facilitate easy evaluation
of correlation functions involving a broad set of creation operators, including those exciting multi-hadron states.
∗
Electronic address: mjp@maths.tcd.ie
2
A new genera tio n of experiments devoted to hadron spectroscopy, including GlueX at Jefferson Lab, PANDA at
GSI/FAIR and BES III intend to make measurements w ith unprecedented precision and in previously unexplored
mass ranges and quantum-number sectors. The main aim o f the Hadron Spectrum Collaboration is to extract from
realistic lattice simulations predictions for masses, decay properties and releva nt matrix elements in these domains.
Spectr oscopy investigations are almost as old as non-perturbative Monte Carlo c alculations on the la ttice [5, 6].
As the first studies progressed, it was quickly realis e d that operators creating hadrons which are constructed directly
from the fields in the lattice lagrangian have significant overlap with a large towe r of states and that extracting the
lightest elements of the spectrum was difficult. The solution was to build operators from “smeare d” fields, where some
linear operator was applied first to the qua rk degrees o f freedom on the appropriate time-slice before the creation
operator was formed. The aim of constructing a smeared field was to reduce the component of the fields close to the
cut-off substantially, s ince these modes do not contribute significantly to long-range correlation functions. Initially,
the techniques considered smeared fields that were not gauge covariant. Gauge-covariant schemes were introduced
soon after [7, 8]. In particular, Ref. [8] introduced the Jacobi smearing algorithm, in which a la ttice approximation to
the three-dimensional gauge-covar iant laplacian is constructed and applied iteratively to the field. This acts naturally
as a long-wavelength filter on the modes constituting the quark field.
In this paper, a new method for smearing quark fields is described and tested in large-scale realistic simulations.
In Sec. II, the formulation of the new algorithm is presented, including detail on its application to measurements
of hadron two-point correlation functions and three-point functions where an arbitrary current is inserted. Sec. III
presents the results of initial tests of the effectiveness of the method, and Sec. IV discusses practical issues that arise
with the method and outlines some future research directions.
II. OPERATOR CONSTRUCTION
The energy of an eigenstate of the hamiltonian of a quantum field theory can be determined by computing the
correla tio n function between creation and annihilation operator s χ at Euclidean times t and t
′
;
C(t
′
, t) =
D
χ(t
′
) χ
†
(t)
E
. (1)
Inserting a co mplete set of eigenstates o f the hamiltonian, such that
ˆ
H|ki = E
k
|ki, this correlation function decom-
poses into a sum of contributions from all states in the spectrum with the same quantum numbers as the source
operators,
C(t
′
, t) =
X
k
hχ|ki
2
e
−E
k
(t
′
−t)
. (2)
In order to measure energies of low-lying states, it it crucial to construct operators that overlap predominantly with
these light modes. This exposes the asymptotic behaviour of the correlator at earlier time-separations and enables more
statistically accurate determina tions of energies. There is a good deal of freedom in the construction of appropriate
operators. Once the constraints of symmetries and temporal localisation are impos e d, any function of the fields in
the path integral can be used in principle.
Smearing is a well-established means o f defining an initial step in the construction of creation operators. Rather
than applying a creation operator to the fields dir e c tly, a smoothing function is applied first. The function should
preserve as many symmetries as possible while effectively removing the presence of short-range modes, which make
an insignificant contribution to the low-energy correlation function. In contemporary simulations, a popular form of
this operation in theories where fermion fields couple to gauge bosons is the Jacobi smearing method [8].
Gauge-covariant qua rk smearings based on the lattice Laplacian start with the simplest representation of the
second-order three-dimensional differential operator
−∇
2
xy
(t) = 6δ
xy
−
3
X
j=1
˜
U
j
(x, t)δ
x+ˆ,y
+
˜
U
†
j
(x − ˆ, t)δ
x−ˆ,y
, (3)
where the gauge fields,
˜
U may be cons tructed from an appropriate covariant gauge-field-smea ring algorithm [9]. After
defining the Laplace operator, a simple smearing can be wr itten
J
σ,n
σ
(t) =
1 +
σ∇
2
(t)
n
σ
n
σ
, (4)
3
where σ and n
σ
are tunable parameters that can be used to optimise projection onto the states under investigation.
For large n
σ
, this approximates the exponential of σ∇
2
, i.e.
lim
n
σ
→∞
J
σ,n
σ
(t) = exp
σ∇
2
(t)
. (5)
The re sulting exponential s uppr e ssion of hig her eigenmodes of the lattice Laplace operator means only a small number
of the lowest modes contribute substantially to J.
This observation suggests the smearing operator can be approximated by forming an eigenvector representation,
truncated to the lowest modes. Since the smea ring operators us e d in lattice calculations are approximated well by
low-rank constructions, a definition of smearing can be chosen to enforce this more absolutely. Let V
M
be the vector
space of scalar fields charge d under the fundamental representation of the gauge group on a particular time-slice.
V
M
has rank M = N
c
× N
x
× N
y
× N
z
where N
c
is the number of colours, and N
x
, N
y
and N
z
are the extents of
the lattice in the three spatial dir e c tions. Now define smearing to be a well-chosen operator of rank N ≪ M. This
class of operator s will be called “distillation” oper ators. There is a substantial benefit to doing this: if the rank of
the operator is sufficiently small, all elements of the propagation matrix fr om this space can be constructed at an
affordable computational cost (which is pr oportional to the rank of the space, N). Conseq uently, correlation functions
involving arbitra rily intricate hadron creation operators c an be measured with a fixed inversion overhead.
Define the distillation operator on time-slice t as a product of an M × N matrix and its hermitian conjugate:
2(t) = V (t)V
†
(t) =⇒ 2
xy
(t) =
N
X
k=1
v
(k)
x
(t)v
(k)†
y
(t). (6)
The k
th
column of V (t) contains the k
th
eigenvector of ∇
2
evaluated on the background of the spatial gauge fields
of time-slice t, once the eigenvectors have been sorted by eigenvalue. This is the projection operator into V
N
, the
subspace spanned by these eigenmodes, so 2
2
= 2. When the number of eigenvectors included is the same as the
dimension of V
M
, i.e. N = M , the distillation op e rator becomes the identity, and fields acted upon are unsmeared.
The Laplace operator inherits many symmetries of the vacuum. It tr ansforms like a scalar under rotations, is
covariant under gauge transforma tion and is parity and charge conjugation invariant. If the action of one of these
symmetries on the Laplace operator ma ps ∇
2
onto
˜
∇
2
, then there is a unitary transformation, R on V
M
such that
R
˜
∇
2
R
†
= ∇
2
. (7)
This implies that if v is an eigenvector of ∇
2
then the eigenvectors of
˜
∇
2
are Rv. Considering the definition of the
distillation op e rator given in Eq. 6, the transformed operator must then obey
R
∼
2
R
†
= 2, (8)
and so correlation functions constructed using distilled fields have the same symmetry properties on the lattice as
those constructed us ing L aplacian smearing methods.
A. Meson two-point correlation functions
Consider the momentum-projected creation and annihilation operators of an isovector meson, ¯uΓ
A
d and
¯
dΓ
B
u,
where Γ acts in spin and color as well as coordinate space. Applying the distillation operator 2 onto ea ch quark field,
the creation operator at three-momentum ~p is written as
χ
†
M
(~p, t) = ¯u
x
(t)2
xy
(t) · e
−ip·y
Γ
A
yz
(t) · 2
zw
(t)d
w
(t), (9)
where there is an implied summation over repeated spatial indices. In a shorthand notation the correlation function
can be written as
C
(2)
M
(t
′
, t) =
D
¯
d(t
′
)2(t
′
)Γ
B
(t
′
)2(t
′
)u(t
′
) · ¯u(t)2(t)Γ
A
(t)2(t)d(t)
E
. (10)
After evaluating the quark field path-integral and inserting the outer-product definition of the distillation operator 2
from Eq. 6, the correlator can be w ritten
C
(2)
M
(t
′
, t) = Tr
h
Φ
B
(t
′
) τ(t
′
, t) Φ
A
(t) τ(t, t
′
)
i
, (11)
4
where
Φ
A
αβ
(t) = V
†
(t)
Γ
A
(t)
αβ
V (t) ≡ V
†
(t)D
A
(t)V (t)S
A
αβ
, (12)
and
τ
αβ
(t
′
, t) = V
†
(t
′
)M
−1
αβ
(t
′
, t)V (t), (13)
with M the lattice representation of the Dirac operator and where the quark spin indices, α, β of Φ and τ have been
explicitly written. Φ has a well-defined momentum, while there is no explicit momentum projection in the definition
of τ. Often, Φ can be decomposed into terms that act only within coordinate and color space D
A
and only within
spin space S
A
. Note that Φ and τ are square matrices of dimension N × N
σ
where N
σ
is the number of components
in a lattice Dirac spinor. Therefore it requires just N × N
σ
operations of the inverse of the fermion matrix on a vector
in order to compute all elements of τ , the “perambulator”. Notice also that the choice of source and sink operators
is entirely independent of the computation of τ; any source and sink operators can be correlated a posteriori once all
elements of the τ matrix have been computed and stored. The method straightforwardly extends to the determination
of correlation functions for mesons composed of different, non-degenerate qua rk flavors.
In the determination of an isoscalar meson correlation function, evaluation of disconnected terms is required. The
disconnected diagram can be similarly repres e nted once distilled fields are used in creation and annihilation operators.
Such a term would comprise two separate traces over the distillation space:
C
(2,disc)
M
(t
′
, t) = Tr
h
Φ
A
(t)τ(t, t)
i
Tr
h
Φ
B
(t
′
)τ
†
(t
′
, t
′
)
i
. (14)
An exact determination of this expressio n requires computing τ (t, t) for all time-slices t.
Since distilled single-particle operato rs can be projected onto definite momentum at both source and sink, multi-
meson correlators can a lso be constructed using creation and annihilation operators of the form
χ
MM
(~q = ~p
1
− ~p
2
; t) = χ
M
(~p
1
, t)χ
M
(−~p
2
, t), (15)
where p
1
and p
2
are the three-momenta of the single particle operators in Eq. 9. After integration over the quark
fields, the resulting diagrams can again be computed by taking trac es over products of constr uction operators in the
distillation space with the perambulators. The precise structure of these functions depends on the quark flavor content
of the source and sink operators, so details are not presented here. As before , these matrices ar e small compared to
the dimension of the space of quark fields, and once the quark propag ation is encoded in the perambulator, these
multi-hadro n corr e lation function can be computed.
B. Baryon two-point correlation functions
The factorization technique can also be applied to baryons. To illustrate the concepts involved, consider just the
isospin-1/2 sector although the technique generalizes naturally to other baryon multiplets. An annihilation operator
involving displace ments as well as coefficients in spin space is written:
χ
B
(t) = ǫ
abc
S
α
1
α
2
α
3
(D
1
2d)
a
α
1
(D
2
2u)
b
α
2
(D
3
2u)
c
α
3
(t), (16)
where the color indices of the quark fields acted upon by the displacement operators D
i
are contracted with the
antisymmetric tensor, and repeated spin indices are summed. After integration over quark fields, the correlation
function
C
(2)
B
(t
′
, t) = −
D
χ
B
(t
′
) ¯χ
B
(t)
E
(17)
can be facto red into perambulator terms , E q. 13 as well a s creatio n and annihilation operators, where the latter can
be wr itten as
Φ
(i,j,k)
α
1
α
2
α
3
(t) = ǫ
abc
D
1
v
(i)
a
D
2
v
(j)
b
D
3
v
(k)
c
(t) S
α
1
α
2
α
3
. (18)
The spin terms factor from the antisymmetric contraction of the vectors v.
5
There are two terms in the resulting cor relation function that involve tensor contractions of the creation and
annihilation operators with the perambulators
C
(2)
B
[τ
d
, τ
u
, τ
u
](t
′
, t) = Φ
(i,j,k)
(t
′
)τ
(i,
¯
i)
d
(t
′
, t)τ
(j,
¯
j)
u
(t
′
, t)τ
(k,
¯
k)
u
(t
′
, t)Φ
(
¯
i,
¯
j,
¯
k)∗
(t)
− Φ
(i,j,k)
(t
′
)τ
(i,
¯
i)
d
(t
′
, t)τ
(j,
¯
k)
u
(t
′
, t)τ
(k,
¯
j)
u
(t
′
, t)Φ
(
¯
i,
¯
j,
¯
k)∗
(t), (19)
where there is an implicit tensor contraction over the internal s pin indices, and where the quark labels d and u are
used to denote the corresponding flavors of the quark perambulator terms in Eq. 13.
Similar to the meson case, the choice of s ource and sink operators is independent of the co mputation of the
perambulators τ
d
and τ
u
. The computation of these matrices can also be sha red with computation of meson correlators.
A baryon correlation function can be evaluated a posteriori using the contractions of the vectors with displacements
in Eq. 18 which can also be shared among the source and sink operators. These contractions do no t involve spin
components, thus making the storage of Φ manage able.
C. Meson three-point correlation functions
A generic meson three-point function is written
C
(3)
(t
f
, t, t
i
) =
D
¯
d2Γ
B
2u
(t
f
) ·
¯uΓu
(t) ·
¯u2Γ
A
2d
(t
i
)
E
, (20)
where there is no smearing in the operator on timeslice t. The completely connected Wick contraction can be expressed
as
C
(3)
conn
(t
f
, t, t
i
) = Tr
h
Φ
B
(t
f
) J(t
f
, t, t
i
) Φ
A
(t
i
) γ
5
τ
†
(t
f
, t
i
)γ
5
i
(21)
where the “generalized perambulator” is defined by
J
αβ
(t
f
, t, t
i
) = V
†
(t
f
)
M
−1
(t
f
, t) Γ(t) M
−1
(t, t
i
)
αβ
V (t
i
). (22)
In general, a disconnected term may also appear; this can be factorised as the product of a two-point function
(evaluated according to the algorithm of Sec. II A) and a disconnected insertion. Since the disconnected trace does
not involve distilled fields, the evaluation of this insertion is not discussed further at this point. To compute the
connected three-point cor relation function, there are two distinct sets of inversions needed. The first is from the
distillation vectors at the source time-slice t
i
and the second from the sink time-slice t
f
. The generalised perambulator
of Eq. 22 is then constr ucted by contracting the two solutions from these two sets of inversions. The current insertion
is encoded in the choice of operator Γ. Any momentum insertion at the current operator involves a Fourier transform
on each time-slice t.
D. Baryon three-point correlation functions
Baryon three-point correla tion functions can also be e xpressed in terms of the gener alized perambulators of Eq. 22.
Consider a three-point correlation function where a baryon is created on time-slice t
i
, then is acted on by a current
operator on time-slice t and is subsequently annihilated at t
f
:
C
(3)
B
(t
f
, t, t
i
) = −
D
B(t
f
) ·
¯uΓu
(t) ·
¯
B(t
i
)
E
. (23)
After quark-field integration, this can be written as a sum of both a connected three-point contribution and the
product o f a disconnected insertion and a two-point correlator. As b efore, the discussion here is restricted to isospin-
1/2 light baryons although the technique is quite general. Note that the disconnected inse rtion involves unsmea red
fields and must be evaluated through some other technique. The two-point co ntribution follows from Sec. II B. For
an up-quark insertion in the correlation function
C
(3)
conn.
(t
f
, t, t
i
) = − ǫ
abc
ǫ
¯a
¯
b¯c
S
α
1
α
2
α
3
¯
S
¯α
1
¯α
2
¯α
3
D
(D
1
2d)
a
α
1
(D
2
2u)
b
α
2
(D
3
2u)
c
α
3
(t
f
)
·
¯uΓu
(t) ·
(
¯
d2
¯
D
1
)
¯a
¯α
1
(¯u2
¯
D
2
)
¯
b
¯α
2
(¯u2
¯
D
3
)
¯c
¯α
3
(t
i
)
E
,
(24)
