Investigating efficient methods for computing four-quark correlation functions

0.82 MB·

by  Abdou Abdel-Rehim

Investigating efficient methods for computing four-quark
correlation functions
Abdou Abdel-Rehim1,2, Constantia Alexandrou1,3, Joshua Berlin4,
Mattia Dalla Brida5, Jacob Finkenrath1, Marc Wagner4
1Computation-based Science and Technology Research Center, The Cyprus Institute,
20 Kavafi Street, 2121 Nicosia, Cyprus
2Department of Engineering, Science and Mathematics SUNY Polytechnic Institute, Utica,
7
New York 13502, USA
1
0 3Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
2
n 4Goethe-Universit¨at Frankfurt am Main, Institut fu¨r Theoretische Physik,
a Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
J

5 5 Dipartimento di Fisica, Universit`a di Milano-Bicocca & INFN, sezione di Milano-Bicocca,
2
Piazza della Scienza 3, I-20126 Milano, Italy

]
t January 25, 2017
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[ Abstract

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We discuss and compare the efficiency of various methods, combinations of point-to-all propa-
v
gators, stochastic timeslice-to-all propagators, the one-end trick and sequential propagators, to
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2 compute two-point correlation functions of two-quark and four-quark interpolating operators of
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different structure including quark-antiquark type, mesonic molecule type, diquark-antidiquark
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0 type and two-meson type. Although we illustrate our methods in the context of the a0(980),
. they can be applied for other multi-quark systems, where similar diagrams appear. Thus our
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0 results could provide helpful guidelines on the choice of methods for correlation function compu-
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tationforfuturelatticeQCDstudiesofmeson-mesonscatteringandpossiblyexistingtetraquark
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states.
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1 Introduction
In recent years the study of four-quark systems has revived a lot of interest within the field of
lattice QCD. These systems allow to study meson-meson scattering, as well as to investigate
the existence and structure of possibly existing four-quark states, e.g. of mesonic molecule type
or of diquark-antidiquark type. In particular the latter is motivated by experimental results
providing strong indications for the existence of tetraquarks (e.g. the recently observed charged
Z and Z states [1, 2, 3] or the mass ordering of the nonet of light scalar mesons, which is
c b
inverted compared to the expectation from a standard quark-antiquark picture [4]).
In order to perform this study within lattice QCD we need to compute two-point correlation
functions involving both two-quark and four-quark interpolating operators. Depending on the
details of these operators diagrams involving up to four quark propagators with a non-trivial
spacetime structure are present. In contrast to simple quark-antiquark correlation functions,
whichcanbecomputedinastraightforwardway,e.g.byusingstandardpoint-to-allpropagators,
these four-quark correlation functions require in many cases all-to-all propagators (propagators
from any point in space on a timeslice to any other point in space on another timeslice, what
makes them considerably expensive) and, hence, more advanced techniques.
Onepossibilitytocomputethemisthedistillationmethod[5], whichprovidesall-to-allpropaga-
tors between specifically smeared quark field operators (Laplacian Heaviside smearing). Distil-
lation has recently been applied to such four-quark correlation functions (cf. e.g. [6, 7, 8]). The
distillation method however comes with an expensive overhead and only pays if a much larger
number of interpolating fields is used. Moreover, for large volume lattices it is impractical and
one has to apply its stochastic variant, which introduces additional stochastic noise terms for
each quark propagator.
In this work we explore a different strategy, namely combining several traditional techniques to
compute quark propagators and four-quark correlation functions: (A) point-to-all propagators
(cf. e.g. [9, 10]); (B) stochastic timeslice-to-all propagators (cf. e.g. [11, 12]); (C) the one-end
trick (cf. e.g. [13, 14]); (D) sequential propagators (cf. e.g. [15]). There are many different
types of diagrams and for each type we discuss several methods (combinations of the above
mentioned techniques (A) to (D)) and determine numerically the most efficient method. We
study the a (980) channel (quantum numbers I(JP) = 1(0+)) at a lattice spacing of a 0.09fm
0
≈
and spacetime volume of (32a)3 64a with a variety of interpolating operators including quark-
×
antiquarktype,mesonicmoleculetype,diquark-antidiquarktypeandtwo-mesontype(thelatter
describes two independent mesons with total zero momentum). In other words our work is
intended to provide guidelines for future lattice QCD work concerned with arbitrary four-quark
correlation functions, in particular guidelines for a quick and hence time-saving decision, which
diagrams of a given correlation matrix to compute with which combinations of the above listed
techniques.
Parts of this work have already been presented at recent conferences [16, 17, 18, 19, 20, 21].
1
The paper is structured as follows. In section 2 we introduce the six interpolating operators we
investigate and the resulting 6 6 correlation matrix. In section 3 we recapitulate the above
×
mentionedfourtechniquesforpropagatorandcorrelationfunctioncomputation. Section4isthe
main section of this work, where various methods are discussed for each element and diagram of
the correlation matrix, with a numerical comparison of their efficiency. In section 5 we conclude,
inparticularwesummarizeafewgeneralrulesregardingthechoiceofanefficientmethod, which
seem to hold for most correlation matrix elements investigated. Section A outlines briefly the
usedlatticesetupandgivesdetailedexamplesforthecalculationofcertaincorrelationfunctions.
2
2 Interpolating operators and correlation matrix
To study the a (980) meson, we consider several interpolating operators j, which create states
0
O
with quantum numbers I(JP) = 1(0+), when applied to the vaccuum:
1 (cid:88)(cid:16) (cid:17)
1 = qq¯ = d¯(x)u(x) (1)
O O √V
s x
2 = KK¯,point = 1 (cid:88)(cid:16)s¯(x)γ u(x)(cid:17)(cid:16)d¯(x)γ s(x)(cid:17) (2)
5 5
O O √V
s x
1 (cid:88)(cid:16) (cid:17)(cid:16) (cid:17)
3 = ηsπ,point = s¯(x)γ s(x) d¯(x)γ u(x) (3)
5 5
O O √V
s x
4 = QQ¯ = 1 (cid:88)(cid:15) (cid:16)s¯ (x)(Cγ )d¯T(x)(cid:17)(cid:15) (cid:16)uT(x)(Cγ )s (x)(cid:17) (4)
O O √V abc b 5 c ade d 5 e
s x
5 = KK¯,2part = 1 (cid:88)(cid:16)s¯(x)γ u(x)(cid:17)(cid:16)d¯(y)γ s(y)(cid:17) (5)
5 5
O O V
s
x,y
1 (cid:88)(cid:16) (cid:17)(cid:16) (cid:17)
6 = ηsπ,2part = s¯(x)γ s(x) d¯(y)γ u(y) (6)
5 5
O O V
s
x,y
(V is the spatial volume, C is the charge conjugation matrix). These interpolating operators
s
are of different structure. qq¯generates a quark-antiquark pair, while the other operators (2) to
(6) generate two quarks anOd two antiquarks. KK¯,point and ηsπ,point are of mesonic molecule
type, i.e. resemble a KK¯ pair or η π pair1 cOentered aroundOthe same spatial point x. QQ¯
s
O
corresponds to a diquark-antidiquark pair2. These three operators are candidates to model the
structure of a possibly existing bound four-quark state, i.e. of a tetraquark. The remaining
two operators KK¯,2part and ηsπ,2part also generate meson pairs (KK¯ and ηsπ), this time,
O O
however, at independent spatial points x and y. They should be suited to resolve low-lying
two-meson states within the I(JP) = 1(0+) sector.
The interpolating operators (1) to (6) enter a 6 6 correlation matrix,
×
(cid:68) (cid:69)
C (t) = j(t ) k (t ) , t = t t > 0, (7)
jk 2 † 1 2 1
O O −
where ... denotes the lattice QCD path integral expectation value. Computing such expecta-
(cid:104) (cid:105)
tions values at affordable numerical costs and with small statistical errors is a highly non-trivial
task, in particular, when four-quark operators like (2) to (6) are involved. One possible strategy,
whichwefollowthroughoutthiswork, istosuitablycombinestandardtechniquesforpropagator
and correlator computation (point-to-all propagators, stochastic propagators, the one-end trick,
1s¯ γ s excitesameson-likestructurecomposedofanss¯pair,whichisexpectedtohavesignificantoverlapto
x 5 x
both η and η(cid:48) and, hence, is denoted by η .
s
2We consider only the lightest (anti-)diquarks, which have spin structure Cγ [22, 23, 24].
5
3
sequential propagators). One of the main goals of this work is to discuss, which combinations of
techniques are possible in principle for each of the matrix elements C and, in a second step, to
jk
determine numerically which combination is most efficient, i.e. results in the smallest statistical
error at comparable computational cost.
Even though we focus on the a (980) meson with quantum numbers I(JP) = 1(0+), our find-
0
ings are of general interest regarding the study of tetraquark systems and unstable mesonic
resonances. The study of such systems usually requires the computation of correlation matrices
of identical or similar structure as (7) with interpolating operators (1) to (6). For example, after
replacing the quark flavors according to (u,d,s) (c,s,u/d), one would obtain a matrix suited
→
to study the JP = 0+ D (2317) meson, and probe whether two- or for-quark structures are
s∗0
dominating. Similarly, (u,d,s) (u,d,c) would allow to study certain charmonium states, for
→
which are tetraquark structure is frequently discussed, e.g. the Z(4430) meson.
Toeasenotation,wewilloftenpicturecorrelationmatrixelementsinadiagrammaticway,where
quark propagators are represented by arrows. These diagrams do not exhibit information about
color and spin, but clearly display the spacetime structure. In particular one can read off, which
combinations of methods is suited to compute a diagram. In our case, each correlation matrix
element (7) corresponds to either a single diagram or a sum of two diagrams. Due to the flavor
structure of the interpolating operators (either ud¯or ud¯ss¯) there are within each diagram either
two or four quark propagators connecting the timeslices at t and t . The diagrams are, hence,
1 2
denoted as “2 connected” and “4 connected”, respectively (for an example cf. Figure 1). The
× ×
diagrammatic representation of the full 6 6 correlation matrix is shown in Figure 2. Note that
×
certain correlation matrix elements have identical diagrammatic representations, since they only
differ in their color and spin structure.
t2 t2
t1 t1
(4 connected) (2 connected)
× ×
Figure 1: Diagrammatic representation of C .
22
4
t x t x t x t x t x t x
t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ y′ t0 x′ y′
t x t x t x t x t x t x t x t x t x t x t x
t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ y′ t0 x′ y′ t0 x′ y′ t0 x′ y′
t x t x t x t x t x t x t x t x t x t x t x
t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ y′ t0 x′ y′ t0 x′ y′ t0 x′ y′
t x t x t x t x t x t x t x t x t x t x t x
t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ y′ t0 x′ y′ t0 x′ y′ t0 x′ y′
t x y t x y t x y t x y t x y t x y t x y t x y t x y t x y t x y
t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ y′ t0 x′ y′ t0 x′ y′ t0 x′ y′
t x y t x y t x y t x y t x y t x y t x y t x y t x y t x y t x y
t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ t0 x′ y′ t0 x′ y′ t0 x′ y′ t0 x′ y′
Figure 2: Diagrammatic representation of the 6 6 correlation matrix C (eq. (7)).
jk
×
5
3 Techniques for propagator computation
The lattice action for quarks is bilinear in the quark fields q and q¯,
S(q) = (cid:88)q¯ (x)D(q) (x;y)q (y), (8)
a,A a,A;b,B b,B
x,y
where D(q) denotes the Dirac operator for quark flavor q and indices a,b,..., A,B,... and
x,y,... label color, spin and spacetime, respectively. The propagator G(q) is the inverse of the
Dirac operator, i.e. the solution of the linear system
(cid:88) (q) (q)
D (x;y)G (y;z) = δ δ δ(x;z). (9)
a,A;b,B b,B;c,C a;c A;C
y
In the following subsections we discuss several standard techniques for propagator computa-
tion which are well-know in the literature, namely: (1) point-to-all propagators, (2) stochastic
propagators, (3) the one-end trick and (4) sequential propagators. We illustrate some of these
techniques in the context of a simple example, the correlation function of the interpolating oper-
ator 1 = qq¯= (1/√V )(cid:80) d¯(x)u(x) (eq. (1)), i.e. correlation matrix element C in (7) and
O O s x 11
Figure2. IntegratingovertheGrassmannvaluedquarkfieldsallowsustoexpressthecorrelation
function in terms of quark propagators,
1 (cid:88)(cid:68) (cid:16) (cid:17)(cid:69)
C (t) = Tr G(d)(x,t ;y,t )G(u)(y,t ;x,t ) =
11 1 2 2 1
−Vs U
x,y
1 (cid:88)(cid:68) (cid:16) (cid:16) (cid:17) (cid:17)(cid:69)
= Tr γ G(d)(y,t ;x,t ) †γ G(u)(y,t ;x,t ) , (10)
5 2 1 5 2 1
−Vs U
x,y
where γ hermiticity has been used, Tr(...) denotes the trace in spin and color space and
5
... U is the average over gauge link configurations distributed proportionally to e−Seff =
(cid:104) (cid:105)
e (Sgauge ln(det(Q))).
− −
3.1 Point-to-all propagators
The exact computation of a propagator G(q)(x;y) from any point in spacetime x to any other
point y is numerically not feasible, because for typical lattices with e.g. 324 lattice sites both
D(q) and G(q) are matrices with (107) entries. However, using translation invariance it is often
O
possible to simply compute propagators from a single spacetime point x to any other point y.
(cid:80)
For example in (10), can be replaced by the spatial volume V ,
x s
(cid:28) (cid:29)
(cid:88) (cid:16) (cid:16) (cid:17) (cid:17)
C (t) = Tr γ G(d)(y,t ;x,t ) †γ G(u)(y,t ;x,t ) , (11)
11 5 2 1 5 2 1
−
y U
6
where x denotes an arbitrary but fixed point in space. The spacetime column index of both
propagators is now (x,t ), i.e. it is not anymore necessary to compute the full matrix G(u/d),
1
but only 12 columns of this matrix (3 color 4 spin), which is numerically feasible.
×
Explicitly, one has to solve 12 linear systems,
(cid:88) (q) (q)
D (x;y)φ (y)[c,C,z] = ξ (x)[c,C,z] , ξ (x)[c,C,z] = δ δ δ(x;z), (12)
a,A;b,B b,B a,A a,A a;c A;C
y
where c = 1,2,3 and C = 1,2,3,4 label 12 different point sources. (Each solution φ corresponds
to a single column of the inverse of the Dirac matrix D; φ are, therefore, also commonly called
“inversions”). The propagator ending at spacetime point x, a so-called point-to-all propagator
(cf. e.g. [9, 10]), is then
(q) (q)
G (y;x) = φ (y)[a,A,x], (13)
b,B;a,A b,B
The example correlation function (10) expressed in terms of such point-to-all propagators is
(cid:28) (cid:18) (cid:19)(cid:29)
(cid:88)
C (t) = (γ ) φ(d)(y,t )[a,B,x,t ] γ φ(u)(y,t )[a,A,x,t ] . (14)
11 5 A;B 2 1 † 5 2 1
−
y U
Of course, for each diagram, translation invariance allows one to replace only a single spatial
(cid:80)
sum by an arbitrary fixed x. Diagrams, where all propagators either start or end at the
x
same spacetime point can, hence, be expressed exclusively in terms of point-to-all propagators
(e.g. C , C , left diagram of C ). However, since this is not the case for the majority of
11 12 22
diagrams, additional methods to compute propagators are necessary.
(cid:80)
Afterreplacingaspatialsum byafixedx, spatialaveragingtoreducestatisticalerrorsisnot
x
implemented anymore. For each set of 12 inversions (12) and each diagram only a single sample
is computed. Of course, it is possible to compute additional samples by choosing different x
values. This however requires additional sets of 12 inversions for each different x, and quickly
becomes expensive.
3.2 Stochastic timeslice-to-all propagators
While it is not possible in practice to compute the propagator from any spacetime point x to
any other spacetime point y exactly, one can at least estimate it stochastically. Quite common
are so-called stochastic timeslice-to-all propagators (cf. e.g. [11, 12]): stochastically estimated
propagators from any space point x in a given time-slice t , to any other spacetime point y.
0
Again linear systems have to be solved labeled now by n = 1,...,N,
(cid:88) (q) (q)
D (x;y)φ (y)[t ,n] = ξ (x)[t ,n] , ξ (x)[t ,n] = δ(x ,t )Ξ (x)[n], (15)
a,A;b,B b,B 0 a,A 0 a,A 0 0 0 a,A
y
7
where Ξ (x)[n] are random numbers satisfying
a,A
N
1 (cid:88)
Ξ (x)[n]Ξ (y)[n] = δ δ δ(x;y)+unbiased noise. (16)
∗a,A b,B a;b A;B
N
n=1
A convenient choice is Ξ (x)[n] Z(2) Z(2) which results in an unbiased noise proportional
a,A
(cid:16) (cid:17) ∈ ×
to O 1 . As usual since the noise average and the average over the gauge field commute, in
√N
practice one can take a fairly small number N of noise sources per gauge configuration, but not
smaller than the number of propagators in the diagram.
Using (15) and (16) it is straightforward to show
N
1 (cid:88)
G(q)(y;x,t ) = φ(q)(y)[t ,n]ξ(x,t )[t ,n] +unbiased noise. (17)
0 0 0 0 †
N
n=1
The example correlation function (10) expressed in terms of stochastic timeslice-to-all propaga-
tors is
1 (cid:88) 1
C (t) =
11
−N(N 1) V
s
− n=n˜
(cid:54)
(cid:33)
(cid:28)(cid:18) (cid:18) (cid:19)(cid:29)
(cid:88) (cid:88)
φ(d)(y,t )[t ,n˜] γ φ(u)(y,t )[t ,n] ξ(x,t )[t ,n] γ ξ(x,t )[t ,n˜] .
2 1 † 5 2 1 1 1 † 5 1 1
y x U
(18)
Note that each propagator needs to be estimated by a different pair of stochastic sources ξ[n]
(cid:80)
and corresponding inversions φ[n] (guaranteed here by ).
n=n˜
(cid:54)
(cid:80)
Stochastic timeslice-to-all propagators are most flexible, i.e. replacing a spatial sum by a
x
fixedxasinthecaseofpoint-to-allpropagatorsisnotnecessary. Inprinciplealldiagramsofthe
correlation matrix (7) can be computed using exclusively stochastic timeslice-to-all propagators.
A severe drawback of these propagators is, however, that they introduce additional stochastic
noise. Thenumberofstochasticnoisetermsis VM (number of signal terms),whereM isthe
s
≈ ×
number of stochastic timeslice-to-all propagators in a diagram3. While using a single stochastic
propagator, i.e. M = 1, typically leads to acceptable signal-to-noise ratios, the noise grows quite
rapidly with the number of stochastic propagators. Already for M > 2, the signal can easily be
lostinthenoiseifthesetechniquesareappliednaively. Therefore, apromisingstrategymightbe
to combine a single stochastic timeslice-to-all propagator with several point-to-all propagators,
as we shall see later on.
3For example the number of signal terms in (18) is ∝V2 (all (x,t ) connected by a pair of propagators with
s 1
all (y,t )), while the number of stochastic noise terms is ∝V4.
2 s
8
3.3 The one-end trick
The one-end trick is an efficient technique to estimate the product of two propagators stochas-
tically (cf. e.g. [13, 14]). The product must be of the form
(cid:88)
G(q1)(x;y,t)ΓG(q2)(y,t;z), (19)
y
i.e. the propagators are connected at spacetime point (y,t) with a sum over y, but no further
propagatorsstartingorendingat(y,t). Theone-endtrickisthusparticularlysuitedtocompute
correlation matrix elements, where at least one of the two interpolating operators is either a qq¯
or a two-meson operator, i.e. qq¯, KK¯,2part or ηsπ,2part defined in section 2.
O O O
One has to solve 2N linear systems labeled by n = 1,...,N,
(cid:88)D(q1) (x;y)φ(q1)(y)[t ,n] = ξ (x)[t ,n] (20)
a,A;b,B b,B 0 a,A 0
y
(cid:88)D(q2) (x;y)φ˜(q2)(y)[t ,Γ,n] = (γ Γ ξ) (x)[t ,n], (21)
a,A;b,B b,B 0 5 † a,A 0
y
where ξ is a stochastic timeslice source defined in (15). With the resulting φ and φ˜ the product
of propagators can be estimated as,
N
(cid:88) 1 (cid:88)
G(q1)(x;y,t)ΓG(q2)(y,t;z) = φ(q1)(x)[t,n]φ˜(q2)(z)[t,Γ,n] γ +unbiased noise. (22)
† 5
N
y n=1
Applying the one-end trick to the example correlation function (10) results in
N (cid:28) (cid:29)
1 (cid:88) 1 (cid:88)
C (t) = φ˜(d)(y,t )[t ,1,n] γ φ(u)(y,t )[t ,n] . (23)
11 2 1 † 5 2 1
−N V
n=1 s y U
Note that there is a big advantage in efficiency when using the one-end trick, compared to using
two ordinary stochastic timeslice-to-all propagators discussed in the previous subsection. The
one-end trick introduces V (number of signal terms) stochastic noise terms, i.e. for the
s
≈ ×
example correlation function (23) V3 stochastic noise terms. In contrast to that using two
s
∝
stochastic timeslice-to-all propagators (18) will generate V4 stochastic noise terms.
s
∝
3.4 Sequential propagators
Quite similar to the one-end trick the technique of sequential propagators (cf. e.g. [15]) is appli-
cable when two propagators are connected at spacetime point (y,t) with a sum over y, but no
9