Light Meson Form Factors at near Physical Masses

0.31 MB·English

by  Benjamin Owen

Light Meson Form Factors at near Physical Masses
Benjamin J. Owen,1 Waseem Kamleh,1 Derek B. Leinweber,1 M. Selim Mahbub,1,2 and Benjamin J. Menadue1,3
1Special Research Centre for the Subatomic Structure of Matter (CSSM),
∗
School of Chemistry and Physics, University of Adelaide, South Australia 5005, Australia
2Digital Productivity Flagship,
CSIRO, 15 College Road, Sandy Bay, TAS 7005, Australia
3National Computational Infrastructure (NCI),
Australian National University, Australian Capital Territory 0200, Australia
(Dated: January 13, 2015)
Theabilityfor mosthadronstodecayviastronginteractions preventsthedirectmeasurement of
their electromagnetic properties. However, a detailed understanding of how these resonant states
featureinscatteringprocessescanallowonetodisentanglesuchinformationfromphotoproduction
5
processes. In particular, there has been increasing interest in thedetermination of magnetic dipole
1
0 moments using such methods. In a recent study [1], Gudin˜o et al. provide the first experimental
2 determinationofthemagneticdipolemomentoftherhomeson. Tofacilitateacomparisonwiththis
experimental determination, we present a calculation of the rho meson and pion electromagnetic
n
form factors calculated in the framework of Lattice QCD. Using the PACS-CS 2+1 flavour full
a QCDgaugefieldconfigurations, weareabletoaccess lowQ2 valuesat near-physicalquarkmasses.
J
Throughtheuseofvariationaltechniques,wecontrolexcitedstatesystematicsinthematrixelements
2
of the lowest-lying states and gain access to the matrix elements of the first excited state. Our
1
determinationoftherhomesong-factorgρ =2.21(8)isinexcellentagreementwiththisexperimental

determination, but with a significantly smaller uncertainty.
]
t
a
l
- I. INTRODUCTION quarks was a preliminary calculation[6] by the QCD-SF
p collaboration using moderate to heavy pion masses and
e
h Understanding how QCD gives rise to the rich anddi- thus insensitive to the chiral dynamics of full QCD. It is
[ verse structure of the hadrons remains an ongoing effort also limited to a large value of nontrivial 4-momentum
both theoretically and experimentally. Probing the elec- transfer with Q2min = 0.44 GeV2. Thus, there is an ur-
1
tromagnetic structure remains one of the most effective gentneedforadeterminationofthe rhomesonformfac-
v
1 methods. Bymeasuringtheelectromagneticformfactors tors at low Q2 and near-physical quark masses in full
6 ofthesestates,wemapoutthedistributionofchargeand QCD to allow for a direct comparison with experiment.
5 magnetism within and gain valuable insight into the un- The progress of the past few years had lead to lattice
2 derlying dynamics. studiesnearoratphysicalmassesandonlatticevolumes
0
Unlikethenucleonandpion,thevastmajorityofQCD significantly larger than the states that occupy them.
.
1 eigenstates are unstable with respect to the strong in- Working in this new regime has reduced the need for
0 teraction. This makes experimental determinations of chiral extrapolation, but there are new systematics that
5 their properties via conventional means impractical due must be considered to ensure accurate results. As quark
1
to their short lifetime. Instead one is forced to disen- masses decrease, signal weakens and so one is forced to
:
v tangle resonantstate properties fromradiative processes workin earlyEuclidean-timeregionswhereexcitedstate
i in scattering experiments. Recently, there has been in- contamination can be significant. Several studies have
X
terest in extracting magnetic dipole moments [1–3]. In explorednewapproachestoproperlyhandleexcitedstate
r
a particular, Gudin˜o et al. [1] have been able to provide contaminationswiththe mostpopularofthesebeing the
a determination of the magnetic dipole moment of the summation method [8, 9] and the variational approach
rhomeson,the firstsuchexperimentalresultforavector [10–16]. Here we make use of the variational approach.
meson.
In our recent work [12], we demonstrated how this
It is thus timely to consider what results lattice QCD
method provides improvedplateau quality and early on-
can provide as a comparison. The overwhelming focus
set of ground state dominance without the need for the
within the lattice community has understandably been
fine-tuning of source and sink operators. Furthermore,
on the nucleon, and to a lesser extent the pion, where
this improvement allows us to probe the system at ear-
there exists a large body of experimental data [4]. In
lier Euclidean times giving rise to significant improve-
comparison, there have been only a handful of lattice
mentinthe statisticaluncertainties. Preliminarystudies
studies of rho meson structure [5–7] with the majority
ofelectromagneticproperties[15,16]providesimilarcon-
ofthese being performedinthe quenchedapproximation
clusions.
to QCD. The only existing calculation with dynamical
Inthis workwe presenta calculationofthe rho-meson
and pion form factors utilising variational methods al-
lowing for an accurate determination of these quantities
∗ [email protected] at near-physical masses. Furthermore the variational
2
approach gives us access to the form factors of excited Thevariationalmethod[20,21]hasbecomeastandard
states. The use of moderately large lattice volumes al- approach for the study of the excited state spectrum of
lows us to extract our results at low Q2. QCD.Thegoalofthisapproachistoproduceasetofop-
For the extraction of the form factors we follow the eratorsφα thatcoupledirectlytoindividualQCDenergy
electromagnetic form factor formalism of Hedditch et al. eigenstates
[5]withthenecessarychangesrequiredforusewithinthe
variational approach. Ω φα β,p δαβ. (3)
h | | i∝
Theremainderofthispaperisorganizedasfollows. In
Section 2 we discuss the use of correlation matrix tech- The way in which this is achieved is to take an existing
niques in the evaluation of hadron matrix elements. In basis of operators χi(x) i = 1, ...,n and construct
{ | }
Section 3 we outline the framework used to extract the the optimised operators φα as linear combinations
pion (pseudoscalar)and rho-meson(vector) form factors
on the lattice, while Section 4 summarises the details of φα(x)= viαχi(x), φα†(x)= χ†j(x)uαj . (4)
ourlatticesimulation. InSection5wepresentourresults i j
X X
and finally in Section 6 we provide concluding remarks.
Beginningwith the matrixof two-pointcorrelationfunc-
tions
II. THEORETICAL FORMALISM
G (p~,t)= e−ip~·~x Ω χ (x)χ†(0) Ω ,
ij h | i j | i
x~
On the lattice, the determination of hadronic observ- X
ables begins with the evaluation of the Euclidean-time andmakinguseofEqs.(3)and(4),bothvαG (p~,t)and
i ij
two and three-point correlation functions. Given an in- G (p~,t)uα have a recurrence relation governed by the
ij j
terpolator χ(x) for the state in question, the two-point energy of state α. It follows that the necessary vectors
function is defined as vα anduα are the solutionsof the generalisedeigenvalue
i j
equations
G(p~,t)= e−ip~·~x Ω χ(x)χ†(0) Ω , (1)
h | | i
Xx~ viαGij(p~,t0+δt) =e−Eα(p~)δtviαGij(p~,t0), (5a)
and the three-point function Gij(p~,t0+δt)uαj =e−Eα(p~)δt Gij(p~,t0)uαj . (5b)
G (p~′,~p,t ,t )= e−ip~′·(~x2−~x1)e−ip~·~x1 ItisimportanttonotethatEqs.(5a)and(5b)areevalu-
O 2 1
atedforagiven3-momentump~andsothecorresponding
~xX2,~x1 operators satisfy Eq. (3) for this momentum only. The
Ω χ(x ) (x )χ†(0) Ω ,
×h | 2 O 1 | i correlator for the state α,p can be obtained from the
(2) | i
correlationmatrix by projecting with the relevanteigen-
vectors
where is the current used to probe the state. Here we
areinteOrestedin the electromagneticcurrent =Jµ. In
general, this interpolator χ(x) will have overlOap with all G(p~,t;α)=viα(p~)Gij(p~,t)uαj(p~).
eigenstates consistent with the given quantum numbers.
The extension to three-point correlators follows sim-
Inserting a complete set of states between our interpola-
ply. Beginningwiththematrixofthree-pointcorrelation
tors in Eq. (1)
functions
G(p~,t)= e−Eα(p~)t Ω χ(0) α,p α,p χ†(0) Ω , (G ) (p~′,p~,t ,t )= e−ip~′·(~x2−~x1)e−ip~·~x1
h | | ih | | i O ij 2 1
Xα ~xX2,~x1
we can see that the contribution to the correlationfunc- Ω χ (x ) (x )χ†(0) Ω
×h | i 2 O 1 j | i
tion from each state, α, is exponentially suppressed by
its energy. The standard approach for examining the andhavingevaluatedtheeigenvectorswiththetwo-point
groundstateistoworkatlargeEuclideantimewhereex- correlation matrix, it is a simple matter of projecting
citedstatesaresuppressed. Inthepastthisapproachhas thenecessaryeigenvectorsontothematrixofthree-point
been sufficient for most quantities considered, but with functions, where care is taken to to ensure that the pro-
ensembles at near physical masses and studies seeking jection is done with the correct momenta for source and
precision determinations of hadronic observables, there sink,
hasbeenincreasedconcernastowhetherEuclideantime
G (p~′,p~,t ,t ;α) vα(p~′)(G ) (p~′,p~,t ,t )uα(p~).
evolution is a sufficient measure for eliminating excited O 2 1 ≡ i O ij 2 1 j
state effects [17–19]. In this study we select the varia-
O
tional approach to isolate the individual contribution of This applies equally well to hadron transitions (L R)
→
a particular eigenstate to the two and three-point corre- where one simply projects with relevant eigenvectors for
lators. the differing left (L) and right (R) eigenstates.
3
Theapplicationofthisapproachtostateswithexplicit construct eigenvalue equations for each Lorentz compo-
spin-degrees-of-freedom is straight forward. For a state nent and define the corresponding optimised operators
withspin-1,relevanttothiswork,thematrixoftwo-point as
correlation functions is
φα(x)= vα χ (x), φα†(x)= χ† (x)uα , (6)
σ i,σ i,σ τ j,τ j,τ
G (p~,t)= e−ip~·~x Ω χ (x)χ† (0) Ω , Xi Xj
ij,στ h | i,σ j,τ | i
~x i.e. we determine the eigenvectorsfor each Lorentz com-
X
ponent separately such that the operator maximally iso-
wheretheRomanindicesdenotetheoperatorinthevari- lates the spectrum observed in each Lorentz component
ational basis and the Greek indices denote the Lorentz [22]. Thus our projected two and three-point correlation
indices. Following the arguments set out above, we can functions are
G (p~,t;α) vα (p~)G (p~,t)uα (p~)
στ ≡ i,σ ij,στ j,τ
(G ) (p~′,~p,t ,t ;α) vα (p~′)(G ) (p~′,~p,t ,t )uα (p~).
O στ 2 1 ≡ i,σ O ij,στ 2 1 j,τ
Having obtained the projected two and three-point correlationfunctions, determination of matrix elements follows
in the standard way with the construction of a suitable ratio to isolate the desired terms. We use the ratio defined
by Hedditch, et al. [5]
Gµ (p~′,p~,t ,t ;α) Gµ (p~,p~′,t ,t ;α)
R µ (p~′,~p;α)= h στ 2 1 ih τσ 2 1 i, (7)
σ τ s Gσσ(p~′,t2;α) Gττ(p~,t2;α)
h ih i
where repeated indices are not summed over. We note erator overlapcan be decomposed simply
that the use of the square root requires the sign of the
result to be recovered from the individual three-point Ω φα,p~(0) π (p~) = δαβ α(p~),
functions. Moreover the covariant/contravariant place- h | | β i 2E (p~)Z
α
ment of indices is for clarity only. The advantage of this
construction is that we have exact cancellation of the where α(p~)enumeratesthecopuplingstrengthoftheop-
momentum-dependent α(p~) factors. In the case of the eratorZφα to the state π (p~) . It follows that the two-
β
Z | i
pseudoscalar pion interpolating field, the στ indices are point function can be expressed as
not required and can be ignored.
conAfisgiunraRtieof.ns[5i,n2c3r,e2a4t]inwgeacnonimsipdreorvbedotuhn{bUia}seadnedst{iUm∗a}- G(p~,t2;α)= e−2EEα((p~p~))t2 Zα(p~′)Zα†(p~) (8)
α
tor [25]. This requires the evaluation of both +~q and
~q SST-propagators, thus incorporating parity symme- and the three-point correlation function
−
try. A consequence of this is that our correlators are
perfectly real or imaginary, depending on the matrix el- Gµ(p~′,~p,t ,t ;α)= e−Eα(p~′)(t2−t1)e−Eα(p~)t1
2 1
ement under consideration. As established in Ref. [25], 2 E (p~′)E (p~)
α α
this approach provides improved plateaus and a reduc- α(p~′) αp†(p~) πα(p~′) Jµ(0) πα(p~) ,
tion of statistical uncertainties beyond a naive doubling ×Z Z h | | i
(9)
of the number of configurations considered.
with the matrix element π (p~′)Jµ(0)π (p~) encoding
α α
h | | i
the interaction vertex of the pion with the electromag-
III. MESON FORM FACTORS netic current. For a pseudoscalar meson, this vertex is
parametrized by a single form factor G (Q2)
C
A. Pseudoscalar Mesons
π (p~′) Jµ(0) π (p~) =
α α
h | | i
Havingdefinedsuitable operatorsforcreatingandiso- 1
[p′µ+pµ]Gα(Q2), (10)
lating a particular state α, it now stands to understand 2 E (p~′)E (p~) C
α α
howtheformfactorsareextractedfromtheratiodefined
in Eq. (7). We begin this discussion with π meson. wherepandp′apretheincomingandoutgoing4-momenta
As pseudoscalar mesons are spinless particles, the op- andQ2 = q2isthespace-likemomentumtransfer. Sub-
−
4
stituting Eqs. (8), (9) and (10) into Eq.(7), we obtain which satisfies the spin sum relation
1
Rµ(p~′,~p;α)= [pµ+p′µ]Gα(Q2).
2 E (p~′)E (p~) C
α α
Togetherwithourcphoiceofkinematics,p′ =(E ,p ,0,0)
α x
and p = (m ,0,0,0) with E = m2 +p2, we access
α α α x
Gα(Q2) through
C p
p p
GαC(Q2)= 2E√αm+αmEαα R0(p~x,0;α). (11) Xs ǫασ(p,s)ǫατ∗(p,s)=−(cid:18)gστ − mσ2ατ(cid:19) . (12)
B. Vector Mesons
Fora vectormeson,the overlaptermnow includes the
spin polarization vector ǫ(p,s)
Usingtheseexpressions,thetwo-pointfunctiontakesthe
δαβ
Ω φα,p~(0) ρ (p~,s) = α(p~)ǫα(p,s), form
h | σ | β i 2E (p~)Z σ
α
p
G (p~,t ;α)= e−Eα(p~)t2 α(p~) α†(p~) g pσpτ , (13)
στ 2 − 2E (p~) Z Z στ − m2
α (cid:18) α (cid:19)
while the three-point correlation function is
e−Eα(p~′)(t2−t1)e−Eα(p~)t1
Gµ (p~′,~p,t ,t ;α)= α(p~) α†(p~)ǫ′α(p′,s′) ρ (p~′,s′) Jµ(0) ρ (p~,s) ǫα∗(p,s), (14)
στ 2 1 2 E (p~′)E (p~) Z Z σ h α | | α i τ
α α
where again the matrix elempent ρ (p~′,s′)Jµ(0)ρ (p~,s) contains all of the information governing the interaction
α α
h | | i
with the electromagnetic current. For a spin-1 system, the interaction vertex can be described by three independent
vertex functions [26–28]
1
ρ (p~′,s′) Jµ(0) ρ (p~,s) = ǫ′α∗(p′,s′)Γδµγ(p′,p)ǫα(p,s) (15)
h α | | α i 2 E (p~)E (p~′) δ γ
α α
where p
pµ+p′µ
Γδµγ(p′,p)= gδγ[pµ+p′µ]G (Q2) + [gγµqδ gδµqγ]G (Q2) qδqγ G (Q2) .
− 1 − 2 − 2m2 3
(cid:26) α (cid:27)
Taking the appropriate linear combination of these vertex functions, we arrive at the Sachs decomposition [27, 28]
G (Q2)=G (Q2) G (Q2)+(1+η)G (Q2),
Q 1 2 3
−
G (Q2)=G (Q2),
M 2
2
G (Q2)=G (Q2)+ ηG (Q2),
C 1 Q
3
where η = Q2 .
4m2
α
Substituting the vertex into Eq. (14) and applying Eq. (12), the three-point function becomes
e−Eα(p~′)(t2−t1)e−Eα(p~)t1
Gµ (p~′,p~,t ,t ;α)= α(p~) α†(p~) µ (p~′,p~),
στ 2 1 2 E (p~′)E (p~) Z Z Aσ τ
α α
where we have grouped all covariantindices intopthe term
p′ p′ p p
µ (p~′,p~)= g σ δ Γδµγ(p′,p) g γ τ .
Aσ τ σδ− m2 γτ − m2
(cid:18) α (cid:19) (cid:18) α (cid:19)
5
Using the symmetry property µ (p~′,p~)= µ (p~,p~′) and our choice of kinematics, the ratio, defined by Eq. (7),
σ τ τ σ
A A
becomes
1 p′p′ −1/2
R µ (p~ ,0)= σ σ g µ (p~ ,0).
σ τ x 2√m E m2 − σσ Aσ τ x
α α (cid:18) α (cid:19)
In particular we consider
E 2 p2
0 (p~ ,0)= α (E +m )G (Q2)+ x G (Q2) ,
1 1 x α α C Q
A m 3m
α (cid:18) α (cid:19)
1 p2
0 (p~ ,0)= 0 (p~ ,0)= (E +m )G (Q2) x G (Q2) ,
2 2 x 3 3 x α α C Q
A A − 3m
(cid:18) α (cid:19)
E
3 (p~ ,0)= αp G (Q2),
1 3 x x M
A −m
α
3 (p~ ,0)=+p G (Q2).
3 1 x x M
A
As such, the form factors for the rho meson are isolated from the following combination of ratio terms
2 √E m
G (Q2)= α α R 0 +R 0 +R 0 , (16)
C 1 1 2 2 3 3
3E +m
α α
√E m (cid:0) (cid:1)
G (Q2)= α α R 3 R 3 , (17)
M 3 1 1 3
p −
x
m √E m(cid:0) (cid:1)
G (Q2)= α α α 2R 0 R 0 R 0 . (18)
Q p2 1 1− 2 2− 3 3
x
(cid:0) (cid:1)
C. Extracting Static Quantities Motivatedbythe observedscalingbehaviourforG and
E
G ofthenucleonatlowQ2,weshallassumethemeson
M
The Sachs form factors G , G and G describe the sector displays similar scaling for each quark sector and
C M Q
distribution of charge, magnetism and charge asymme- use this to extract a value for GM(Q2 =0)
try within the hadron. In particular, the value of these
functions at Q2 = 0 define the hadron’s total charge, q, G (Q2 =0)= Gi(Q2) , (22)
magnetic moment, µ, and quadrupole moment, Q, i GC(Q2)
q = e GC(0), (19a) to facilitate a comparison with the experimental pre-
e diction of [1] and model expectations. Drawing on the
µ= G (0), (19b)
2m M monopole ansatz, we shall also use this for GQ.
e For the mean squared charge radius, we use the stan-
Q= G (0), (19c)
m2 Q dard definition from the small Q2 expansion of the
Fourier transform of the charge distribution which gives
where m is the mass of the hadron. In this calculation,
weworkwithfixed-currentSST-propagators. Thisallows
d
us to explore the form factors of many different hadrons r2 = 6 G (Q2) . (23)
h i − dQ2 C Q2=0
without the need for additional inversions, but limits us
toasingle3-momentumtransferandthusasingleQ2 for Using our monopole form we have (cid:12)(cid:12)
each quark mass.
We make use of a monopole ansatz for the Q2 depen-
6 1
dence r2 = 1 . (24)
h i Q2 G (Q2) −
Λ2 (cid:18) C (cid:19)
G (Q2)= G (Q2 =0), (20)
i Λ2+Q2 i
(cid:18) (cid:19)
IV. SIMULATION DETAILS
as suggested by a Vector Dominance Model hypothesis.
Here we use a conserved current, G (Q2 = 0) = 1, and
C
so we extract the monopole squared-mass Λ2 via Two and three-point correlation functions are evalu-
atedfollowingtheprescriptionoutlinedin[23,24,29,30].
Q2 Forthis calculationwemakeuse ofthe PACS-CS(2+1)-
Λ2 = G (Q2). (21)
1 G (Q2) C flavour dynamical-QCD gauge field configurations [31]
C
−
6
sets ofconfigurationsin the evaluationofourcorrelation
TABLE I.Ensemble parameters used in this calculation.
functions. Rather than performing further matrix inver-
κud mπ (MeV) Ncfgs Nsrcs sions on the {U∗} configurations, we make use of the
fermion matrix property
0.13700 702 350 350
0.13727 570 350 350 ∗
M( U∗ )= C˜M( U )C˜−1
0.13754 411 350 700 { } { }
0.13700 296 350 700 (cid:16) (cid:17)
with C˜ = Cγ to give us the propagators for the U∗
0.13781 156 197 ∼1600 5
{ }
from the existing U propagators[23]
{ }
S(x,0;U∗)=(C˜S(x,0;U)C˜−1)∗.
made available through the ILDG [32]. These configu-
rations are generated using a non-perturbatively (a)- Forthefixed-currentSST-propagatorsthisidentitytakes
O
improved Wilson fermion action and Iwasaki gauge ac- the form
tion on a 323 64 lattice with periodic spatialboundary
conditions. T×he scale is determined via the static quark S(x,0;t,~q,µ;U∗)=(C˜S(x,0;t, ~q,µ;U)C˜−1)∗.
−
potential. The value β = 1.90 gives a lattice spacing
a=0.0907(13)fm resulting in a physical volume of spa- Thus, both ~q and ~q momentum insertions are required
−
cial extent L = 2.9 fm. We have access to 5 light-quark as discussed at the end of Section 2. Our error analysis
masses, with the strange quark mass held fixed. The is performed using a second-order jackknife where the
resulting pion masses range from 702 MeV to 156 MeV. χ2/dof for our fits is obtained through the covariance
A fixed boundary condition is applied to the temporal matrix.
directionandourquarksourcesareinsertedatt =16. As the primary goal of this work is to cleanly extract
src
Wehaveverifiedthatreflectionsassociatedwiththefixed matrix elements of the ground state, we choose to work
boundary are negligible at Euclidean times greater than with a small variational basis. Our operators are local
16 time slices away from the boundary. Due to the lim- meson operators of varying widths. This is achieved by
ited number of configurations at the lightest mass we applying increasing levels of gauge invariant Gaussian
make use of multiple quark sources on each configura- smearing[34]toourquarksourcesandsinks. Thesmear-
tion. This is done by using two maximally separated ing procedure is:
spacial sources with a relative temporal boundary shift
of 8 time slices. The temporal boundary is then shifted ψi(x,t)= F(x,x′)ψi−1(x′,t), (25)
by multiples of 16 time slices for each of these spacial x′
X
sources. Multiple sources are also used for the next two
where
lightest quark masses with a single spacial position, sep-
arated temporally by 32 time slices. Table I provides a
summary of the simulation details for each κ-value and F(x,x′)=(1−α)δx,x′ (26)
the corresponding value for m2. α 3
Thethree-pointcorrelationfπunctionsareevaluatedus- + 6 Uµ(x)δx′,x+µˆ+Uµ†(x−µˆ)δx′,x−µˆ ,
ing a sequential source technique with the current held µX=1(cid:2) (cid:3)
fixed as outlined in [23, 24, 29, 30]. For the current we
where the parameter α = 0.7 is used in our calculation.
use an (a)-improvedconservedvectorcurrentobtained
O We use the four different levels of smearing examined in
usingthe Noetherprocedurewiththeimprovementterm
Ref. [35] to provide an optimal basis for these ensem-
constructedintheformofatotalfour-divergence[23,33].
bles. Table II lists the corresponding RMS radii for the
The current is inserted at t = 21 relative to the quark
c quark sources. We focus on a single spin-flavour con-
insertion time at t =16. The resulting source-current
src struction for the meson interpolators and draw on the
separationwouldin generalbe too smallandso give rise
various source/sink smearing widths to enable the effi-
to excited state contamination, but as was highlighted
cient and accurate isolation of states in the variational
in [12] our use of the variational approach gives rise to
approach. The use of a variety of widths enables the re-
rapid ground state dominance after the source allow-
sulting operators to form nodal structures in the radial
ing for considerably early current insertion and smaller
wave functions of the excited states [36] and is central
source-current-sink separations.
to our ability to rapidly isolate states in our two- and
Given our choice of correlation-function ratio, we re-
three-point correlation functions.
quire SST-propagators for both +~q and ~q where we
choose ~q = 2πxˆ. The current is polarized−with µ = 3,4 For the spin-flavour form of our local interpolator we
L choose to use,
as required by our evaluations of the form factors. As
outlined in Ref. [25], the invariance of the lattice action
χ (x)=d(x)γ u(x)
under U U∗ implies the link variables U and U∗ π 5
→ { } { } χ (x)=d(x)γ u(x)
areof equalweightandso we chooseto accountfor both ρ i
7
for the π and ρ meson respectively. When coupled with 3.0
our four smearing widths this allows for the construc-
tion of up to a 4 4 correlation matrix. The use of the 2.5
×
alternate bi-linear forms γ γ and γ γ was considered,
0 5 0 i
but these were found to notprovide any additionalbasis 2.0
spanwhenusedwithmorethantwosmearinglevels. We LV
e
G 1.5
considered all combinations of variational parameters t
0 H
andδtinrange17-20and1-4respectivelywhereasuper- M
1.0
position of states can be used to constrain the analysis.
With regard to state isolation and the stability of the
0.5
analysis, the optimal choice is t = 17 and δt = 3 for
0
the three heavier masses and t = 17 and δt = 2 for the
0 0.0
two remaining lighter masses. The use of an earlier t 0.0 0.1 0.2 0.3 0.4
0
value relative to baryon studies [35, 37, 38] on the same m 2HGeV2L
Π
ensembles is unsurprising given the larger energy gaps
displayed between the ground state and first excitation FIG.1. Low-lyingeigenstatesinthepion(0+−)channelfrom
in the meson sector. our4×4correlationmatrixofsmearedsourcesandsinks. The
In performing our variational analysis we choose to left-most points at the physical pion mass are experimental
use the symmetrisation procedure outlined in Ref. [39]. measurements of the spectrum.
Namely we exploit the ensemble average symmetry
G (p~,t)=G (p~,t), The resulting eigenvectors then give us a direct measure
ij ji
of the relative contribution from each interpolator. Ac-
and consider the improved unbiased estimator cordingly, we also normalise the three-point correlators
using the relevant two-point functions
1
[G (p~,t)+G (p~,t)]
2 ij ji 1 1
Gµ(p~′,p~,t ,t ) .
Enforcing this symmetry allows us to re-express the Gii(p~′,tsrc) ij 2 1 Gjj(p~,tsrc)
eigenvalueequations(5a)and(5b)intermsofarealsym-
p p
metric matrix
V. RESULTS
[G−1/2(p~,t )G(p~,t +δt)G−1/2(p~,t )] wα =e−Eα(p~)δtwα.
0 0 0 ij j j
A. Low lying meson spectrum
Theresultingeigenvectorsfromthis formulationforman
orthogonal basis. These orthogonal eigenvectors, wα(p~),
i
can also be obtained from the eigenvectors uα(p~) using In Figs. 1 and 2 we display the resulting spectrum be-
i low 3 GeV in the 0−+ and 1−− channels respectively
wα(p~)=G1/2(p~,t )uα(p~). obtained from the 4 4 correlation matrix of smeared
i ij 0 j ×
sources and sinks. The masses for the states considered
Uponnormalisingthisbasis,weareabletosorttheeigen- in the form factorsanalysis are summarisedin Table III.
vectors consistently across jackknife complement sets, Beginning with the pion channel we find three well
and track eigenstates across momenta using the approx- separated eigenstates, consistent with the spectrum ob-
imate orthogonality condition served in previous works examining the entire isovector
meson sector [40–42]. At the lightest mass we find that
w~(p~′)α w~(p~)β δαβ.
· ≃ our first excited state is consistent with the π(1300). A
Correlators are normalized to be (1) by considering notable feature in our spectrum is a significant shift in
∼O the mass for the first excited state at the second light-
1 1 est mass and a similar jump for the second excited state
G (p~,t) .
ij
G (p~,t ) G (p~,t ) at the middle mass. This feature is observed across the
ii src jj src
range of variational parameters considered and on cor-
p p
responding 3 3 analyses formed from subsets of the
×
variational basis. A similar feature is observed in posi-
TABLE II. The RMS radii for the various levels of smearing
tive parity spectrum of the nucleon [35, 39]. Examina-
considered in thiswork.
tion of the wave functions for these nucleon excitations
Sweepsof smearing RMS radius (fm) [36] show significant finite volume effects for the light-
16 0.216 est two masses which may give rise to an increase in the
35 0.319 eigenstate energies. It is possible that we are observing
100 0.539 a similar effect here.
In the rho meson channel, we again observe three well
200 0.778
separated eigenstates. However, in this channel we ex-
8
TABLE III. Masses for the first two π and ρ meson eigenstates (ground state and first excitation) for various values of the
hoppingparameter, κ, extracted from our 4×4 correlation matrix of smeared sources and sinks.
κ m2π (GeV2) mπ (GeV) mρ (GeV) mπ∗ (GeV) mρ∗ (GeV)
0.13700 0.3876(11) 0.6226(9) 0.981(5) 1.66(7) 1.68(7)
0.13727 0.2647(9) 0.5145(9) 0.917(6) 1.59(4) 1.72(5)
0.13754 0.1509(7) 0.3884(9) 0.867(6) 1.55(8) 1.77(5)
0.13770 0.0811(6) 0.2848(11) 0.832(10) 1.77(4) 1.81(7)
0.13781 0.0260(10) 0.1613(31) 0.793(14) 1.50(23) 1.77(11)
3.0 An important feature in the QCD spectrum is the
possibility of multi-particle intermediate states. In the
2.5 infinite-volumelimitthisrendersthemajorityofhadrons
unstable under the strong interaction. However on the
2.0 finite-volumelattice,the QCDeigenstatesarestableand
LV are composed of admixtures of both single-particle and
e
G 1.5
multi-particle states.
H
M
Some insight into the composition of states can be
1.0
taken from the physical spectrum and scattering thresh-
olds. However, the position of these thresholds change
0.5
on the finite volume. Multi-particle states are forced to
overlap in the finite volume, giving rise to a volume-
0.0
0.0 0.1 0.2 0.3 0.4 dependent interaction energy. Mixing with single-
m 2HGeV2L particledominatedstatesfurtherdistortsthespectrumto
Π
the pointwhere intuitionfrominfinite-volumescattering
FIG.2. Low-lyingeigenstatesintherhomeson(1−−)channel thresholdsandthephysicalspectrumbecomesirrelevant,
from our 4 × 4 correlation matrix of smeared sources and particularly in volumes with lattice length L 3 fm.
sinks. The left-most points at the physical pion mass are Below the finite-volume modified two-parti∼cle scatter-
experimental measurements of theρ-meson spectrum.
ing threshold, states are generally single-particle dom-
inated but still contain important contributions from
nearby scattering channels. The position of states in the
pect to see two eigenstates near 1600 MeV separated by spectrum can be changed by varying the quark mass or
about 250 MeV. This suggests that a basis of local op- the volumeofthe lattice andthe eigenstatescanbecome
erators is not able to isolate both of these eigenstates. maximally mixed making their traditional identification
A similar conclusion was found in Ref. [41]. In Ref. [42] as scattering states or resonantstates impossible. In the
they found that the ρ(1450) required a basis that con- casewherealow-lyingfinite-volumescatteringthreshold
tained displaced, derivative operators. Quark model ex- sits well below the resonant state, then the lowest-lying
pectationspredictthatforthesetwostates,oneisS-wave statemayberegardedasatwo-ormulti-particlescatter-
dominant while the other D-wave dominant [4]. Given ing state and the single-particle dominated state is now
the radial symmetry of our operators, it is not possible the higher eigenstate.
to formD-wavestates andsoitis notsurprisingthatwe In the case of the vector meson under study here, we
are unable to isolate both states. must be careful to ensure that the state we are excit-
The close agreementof our first excited state with the ing on the lattice is in fact the single-particle dominated
ρ(1700) and the absence of the ρ(1450) from our spec- resonant state. Though there is strong evidence to sug-
trum suggeststhat the ρ(1700)is the S-wavestate while gestthat localmesonoperatorscouple poorlyto scatter-
the ρ(1450) is the D-wave state. ing states, especially on larger volumes such as that un-
Dudek et al. [40]wereable to successfullyisolateboth der investigation here, we perform a check to determine
of these states at larger quark masses and found a trend whether the eigenstate isolatedin ourcorrelation-matrix
consistent with our identification of states. They found analysis is single particle in nature.
the S-wave dominated state to be the lighter state at Forallourensembles,thegroundstateρmesonatrest
their heavier quark masses, and observed that with de- iswellbelowtheππ threshold,andwillbesingle-particle
creasingquarkmassthemasssplittingbetweenthesetwo dominated. However, upon applying the boost to mo-
states became smaller, with the states nearly degenerate mentum~q,theextractedenergyeigenstatenowsitsabove
at their lightest mass of m 400 MeV. A continua- the lowest-lying bare ππ energy allowed by momentum
π
≃
tion of this trend to lighter quark masses places D-wave conservation. Inordertodeterminewhetherthestatewe
dominated state lower in mass. have isolated in the boosted case is the finite-volume ρ-
9
2.0 to the resonant ρ meson.
1.5 B. Form Factor Determination
LV
Akeyfindingfromourcalculationofg usingthevari-
Ge 1.0 A
H ational approach [12], was that the optimised operators
E
obtained from the correlation matrix analysis composed
of various smeared source/sink operators gave rise to
0.5
significant improvement in the quality and duration of
plateausfromwhichthe matrixelementswereextracted.
Comparison with a modest historically-typical choice of
0.0
0 1 2 3 4 5 smearinghighlightedthat excitedstate effects are signif-
n icant and suppress the value of g .
p A
We are able to draw similar conclusions here. In
FIG. 3. An example of our comparison between the disper- Fig. 4 we present a comparison of the correlation func-
sion relation E = pm2+np|p~min|2 (blue line), and the en- tionratiosprovidingtherho-mesoncharge,magneticand
ergies extracted from the finite-momentum correlators (blue quadrupole form factors using both the standard single-
circles) for the rho meson over a range of momenta. Here sourceapproachwithamodestlevelofsmearingandour
m is taken from thezero-momentum correlator, |p~min| is the variational approach.
magnitudeofthelowestnon-trivialmomentumonthelattice.
The magnetic form factor, shown in Fig. 4(b), is the
Thered diamonds providethecorresponding non-interacting
most striking example. Here we see a clear difference in
ππ-energies allowed by momentum conservation. Results at
the quality of the plateau. For the correlation matrix
theother values of mπ are similar.
approach, single-state dominance follows the current at
t =21 immediately, allowing fits as early as t =22.
c S
For the standard approach, the excited states act to
mesonorthelower-lyingππscatteringstate,wecompare
suppressthevalueofG atearliertimeslicesforcingone
the extracted eigenstate energy against the expected en- M
to wait until at least t =24 before an adequate χ2 is
ergygivenbythe dispersionrelationforasingleparticle. S dof
obtained. However,the centralvalues show a systematic
In Fig. 3 corresponding to the second-lightest quark
upward trend following this time slice and consequently
mass considered, we overlay the dispersion relation ex-
there is no indication that a plateau has been obtained.
pectation (blue band) with the energies extracted from
Similar conclusions can be drawn for the quadrupole
the finite-momentum correlators(blue data points) for a
formfactorinFig.4(c). Herewefindthatboththecorre-
range of momenta. Here we can see that the energies
lation matrix and the standard approachgive consistent
extracted from the boosted correlator are in excellent
values immediately following the current, but diverge as
agreement with dispersion. This is observed across all
we move out to later time-slices. Again there is a clear
the masses considered.
systematic drift in the results obtained using the stan-
As a further check, we compare the non-interacting dard approach and it would be difficult to select a fit
ππ energies(reddatapoints)allowedbymomentumand regionwheretheformfactorcanbedeterminedwithcon-
parityconservation. Forthesingleunitoflatticemomen- fidence. Giventhatweseekaregionwherethe extracted
tum relevant to our form factor analysis, we find that formfactorisconstantoversuccessivetimeslices,wecan
the mass separation between the dispersion result and clearly see the improvement offered by the correlation
the non-interacting ππ energy is significant. Moreover, matrix approach.
the attractive finite-volume interactionin the ππ system In the case of the charge form factor, Fig. 4(a), the
wouldact to further increase the separationbetween the two methods are in closer agreement and display a sim-
single and multi-particle dominated states. As a matter ilar quality in plateau. In either case, some Euclidean
of principle, we do expect a ππ scattering state to reveal time evolution is required before a plateau is observed,
itself in the long Euclidean-time tail of our correlation but we find that the correlation matrix approach gives
function. However,ourinterpolatingfieldshaverendered a systematically lower value following the current and
this contribution to be negligible at the finite-Euclidean plateausa couple oftime-slices earlierthan the standard
times considered. approach.
Thisindicatesthatourcorrelationfunctionsareindeed Though the examples presented are selected to high-
dominated by the resonant-like state of interest and not light the improvement using the variational method, we
the lower-lying finite volume scattering state. Similar see significant improvement across all masses and form
separations are found for all other masses with the ex- factorsconsidered. Throughthismethodweconsistently
ception of the middle mass. Through this process we findthatoneisabletoobtainsingle-statedominanceear-
have determined that the state we have isolated in the lier and often with significant improvement in the qual-
boosted system is in fact the state most closely related ity of the plateau. Through the use of the variational
10
approachwe are able to:
0.9
• Isolate an eigenstate at earlier Euclidean times,
• Insert the conserved vector current at earlier Eu-
clidean times,
0.8
GC • Fit the correlation function at earlier Euclidean
times,
• Observe robust plateau behaviour,
0.7 • Identify large Euclidean-time fit windows,
• Determine the form factors with significantly re-
20 22 24 26 28 30 32
duced systematic errors, and
t
(a)Chargeformfactor,GC,formπ =296MeV. • Determine the form factors with significantly re-
ducedstatisticaluncertaintiesduetotheadmission
2.2 of analysis at earlier Euclidean times.
2.1
2.0
C. Form Factors and Static Quantities
1.9
L 1.8
N Inextractingformfactorsacrossawiderangeofquark
Μ
H 1.7 masses, it is important to note that for each mass there
M
G 1.6 will be a slight change in the value of Q2 due to the
1.5 variation in the mass of the hadron. To ensure that
the comparisonbetween quark masses and eigenstates is
1.4
meaningful,we makeuse ofthe monopole ansatzto shift
1.3
the extracted values for our form factors to a common
1.2 20 22 24 26 28 30 Q2. Fig. 5 demonstrates this shift for the pion. For the
pion system, we shift Q2 values to a common value of
t
Q2 = 0.1 GeV2, while for the ρ-meson system we select
(b)Magneticformfactor,GM,formπ =156MeV. Q2 = 0.16 GeV2. These values are selected to minimise
the shift for the extractions at the lightest quark mass.
0.00 We use different values to ensure that we minimise the
shift for each system. As the pion is significantly lighter
thantheρmeson,asmallervalueforQ2 arisesnaturally.
-0.01 Before we examine our form factor results, we note
that all quantities presented are the quark sector contri-
L
2m butions for unit-charge quarks. Here we choose to label
Hf -0.02 these as the quark sector contributions to the positive-
Q
G charge eigenstate of the corresponding iso-triplet. That
is, the quark contribution is labelled as the u-quark sec-
-0.03
tor while the anti-quark contribution is labelled as the
d-quark sector. As we are working with exact isospin
-0.04 symmetry, these quark sector contributions are equiva-
20 22 24 26 28 30
lentandsowe chooseto presentthe u-quarksectoronly.
t As we are working in full QCD, the hadron form factor
(c)Quadrupoleformfactor,GQ,formπ =296MeV. will have contributions from the sea quarks. However
these contributions have been neglected in our calcula-
tion. Thus to make meaningful comparison with exper-
FIG.4. Acomparisonofρ-mesonformfactorsasafunctionof imental data, one should consider the iso-vector quanti-
Euclideansinktimeforasinglelevelofsmearingandourvari-
ties,
ationalapproach. Thedatasetsobtainedfromratiosofthree-
and two-point functions are offset for clarity. The (blue) cir- 1
1 = ( 1 1 ),
cles denote the results from the variational approach while h Vi 2 h +i−h −i
the(red) squaresillustrate traditional results using thestan-
dard single-source method with amoderate levelof smearing where 1± labels the positive and negative members of
h i
(35 sweeps). The vertical dashed line indicates the position the iso-triplet of the pion and ρ meson. However, un-
ofthecurrentinsertion. Thefittedvaluefromthevariational derexactisospinsymmetry one finds thatthis iso-vector
approachhasbeenincluded(shadedband)tohighlightwhere
the single source approach is consistent with our improved
method.