Excited meson radiative transitions from lattice QCD using variationally optimized operators

3.5 MB·English

by  Christian J. Shultz

JLAB-THY-15-2004
Excited meson radiative transitions from lattice QCD
using variationally optimized operators
Christian J. Shultz,1,∗ Jozef J. Dudek,1,2,† and Robert G. Edwards2,‡
(for the Hadron Spectrum Collaboration)
1Department of Physics, Old Dominion University, Norfolk, VA 23529, USA
2Theory Center, Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA 23606, USA
(Dated: January 30, 2015)
Weexploretheuseof‘optimized’operators,designedtointerpolateonlyasinglemesoneigenstate,
inthree-pointcorrelationfunctionswithavector-currentinsertion. Theseoperatorsareconstructed
as linear combinations in a large basis of meson interpolating fields using a variational analysis of
matrices of two-point correlation functions. After performing such a determination at both zero
and non-zero momentum, we compute three-point functions and are able to study radiative transi-
tionmatrixelementsfeaturingexcitedstatemesons. Therequiredtwo-andthree-pointcorrelation
5 functions are efficiently computed using the distillation framework in which there is a factorization
1
between quark propagation and operator construction, allowing for a large number of meson oper-
0
ators of definite momentum to be considered. We illustrate the method with a calculation using
2
anisotopic lattices having three flavors of dynamical quark all tuned to the physical strange quark

n mass, considering form-factors and transitions of pseudoscalar and vector meson excitations. The
a dependenceonphotonvirtualityforanumberofform-factorsandtransitionsisextractedandsome
J discussion of excited-state phenomenology is presented.

9
2 PACSnumbers: 12.38.Gc,13.40.Gp,13.40.Hq,14.40.Be

]
at I. INTRODUCTION Production of excited mesons at very forward angles
l using pion beams is also driven by the vector-current
-
p The coupling of mesons and baryons to photons transitionform-factor. InthePrimakoffprocessthepion
e at leading order in α is given by matrix ele- absorbs a nearly on-shell photon from a nucleon or nu-
h ments of the quark-fieldevmector current (cid:10)f(cid:12)(cid:12)jµ(cid:12)(cid:12)i(cid:11) where clear target, with an amplitude to transition to another
[ jµ(x)=ψ¯(x)γµψ(x). In the case that the initial and fi- mesonspeciesgivenbytransitionform-factorsatQ2 ≈0.

1 nal state hadron are the same we express the matrix el- Recentsuchmeasurementsofthecouplingsa2 →πγ and
v π →πγ have been made at COMPASS [4].
ement in terms of Lorentz-invariant form-factors, whose 2
7 dependence on the virtuality of the photon, Q2, can be Incharmoniumandbottomonium,therelativelysmall
5
total widths of the low-lying states means that radia-
4 related to quark charge and current distributions within
tive transition rates between them constitute significant
7 thehadron. Thecurrentcanalsoinduceatransitionfrom
(cid:12) (cid:11) (cid:12) (cid:11) branchingfractions,andcanbemeasureddirectly,ascan
0 onehadroneigenstate(cid:12)i toanother(cid:12)f ,inwhichcasewe
. speak of transition form-factors. For hadrons with non- ratesofdecaytoaphotonpluslight-quarkmesonswhere
1
the heavy quark-antiquark pair annihilates [5, 6].
0 zero spin, there are multiple possible amplitudes which
5 can be labeled by the helicity of the hadrons, or we may In the baryon sector, the Q2 dependence of transition
1 expand the current in terms of multipoles to provide an- form-factors can be measured quite directly in electro-
v: other convenient physically motivated basis. production of excited nucleons off proton and neutron
i In the meson sector, these matrix elements appear in targets. The relative magnitudes of the various multi-
X photo– and electro–production of mesons from nucleon pole or helicity amplitudes and the variation with pho-
r and nuclear targets, where the coupling of the photon ton virtuality have been discussed as a means to study
a to a t-channel meson exchange is described by transi- the internal quark-gluon structure of excited N(cid:63) and ∆(cid:63)
tion form-factors. Measurements of these processes with states [7].
unprecedented statistics will be made in the GlueX and The properties of hadrons constructed from strongly
CLAS12 detectors at the 12 GeV upgraded CEBAF [1]. interactingquarksandgluonsshouldbecalculablewithin
In particular, photoproduction has been proposed as a therelevantgaugefieldtheory,QuantumChromodynam-
means to produce large numbers of exotic JPC hybrids, ics(QCD).Attheenergyscaleofhadrons,QCDdoesnot
those mesons which contain an excitation of the gluonic haveasmallcouplingconstantandmustbetreatednon-
field as well as the usual quark-antiquark pair [2], [3]. perturbatively. The tool we will use to achieve this is
lattice QCD, in which the field theory is discretised on a
finite grid of Euclidean space-time points, and where we
can compute correlation functions as an average over a
∗ [email protected] finite but large number of possible gauge-field configura-
† [email protected] tions.
‡ [email protected] The vector current matrix elements that we are in-
2
terested in, (cid:10)f(cid:12)(cid:12)jµ(cid:12)(cid:12)i(cid:11), appear in three-point correlation the meson sink proceeded using the ‘sequential source’
functionsofgenericform,(cid:10)0(cid:12)(cid:12)Of(tf)jµ(t)Oi†(ti)(cid:12)(cid:12)0(cid:11). Here method, which proved to be a significant limitation on
O†, O† are operators constructed from quark and gluon what could be achieved. In this study we will make use
i f of the distillation framework for correlator construction
fields, capable of interpolating mesons from the vacuum.
[16] within which we will be able to use optimized oper-
In general, such hadron operators with definite quantum
ators of definite momentum at both source and sink as
numbers interpolate not just one QCD eigenstate, but
wellastoinsertavectorcurrentoperatorofdefinitemo-
ratheralinearsuperpositionofallstateswiththosequan-
mentum. This is the first use of distillation to compute
tum numbers. Each state propagates through Euclidean
three-point functions.
timewithafactore−Et suchthatatlargetimesonlythe
In order to demonstrate the technology we perform a
stateoflowestenergysurvives,suggestingthatifwesep-
calculation on dynamical lattices having three flavors of
arate the three operators in the correlation function by
quark all tuned to approximately the physical strange
largetimeintervalswewillobtainthetransitionbetween
quark mass - the spectrum of isovector mesons on these
the lightest states with the quantum numbers of O , O .
i f lattices was previously presented in [17, 18]. We con-
However in many cases we actually wish to study states
siderpseudoscalarandvectormesonsintheSU(3)flavor
which are not the lightest with a given set of quantum
octet,extractingform-factorsandtransitionform-factors
numbers. Ifweuseagenericoperatortointerpolatethem
for both ground and excited states in these channels.
from the vacuum their contribution will be as sublead-
This should be expected to be a challenging undertak-
ing exponential dependences in the correlation function,
ing – transitions between excited-state and ground-state
which become dominated by lighter states as we prop-
vector and pseudoscalar mesons are ‘hindered’ and have
agate in Euclidean time. Determining the amplitude of
relatively small magnitude for small photon virtualities
subleading exponential contributions through fitting the
– as such we will be extracting small signals from corre-
time-dependence proves to be unreliable.
lation functions built using ‘optimal’ excited state oper-
Our solution to this problem is to form ‘optimal’ op-
ators.
erators for interpolation of each state that we wish to
In section II we introduce the decomposition of vector
study, that is operators which have a dominant ampli-
current matrix elements in terms of Lorentz covariant
tude to produce a particular state, and significantly re-
kinematic factors multiplying the unknown form-factors
ducedamplitudetoproduceallotherstates, particularly
we seek to extract. We proceed to introduce the gauge
those lighter than the state in question. In this case the
configurations, operator basis, and background informa-
three-point correlation functions are dominated by the
tionrelevanttotwo-pointcalculationsinSectionsIIIand
desired initial and final states, even if they are not the
IV.SectionVconcernsthree-pointfunctions;wedescribe
lightest in a given quantum number sector. Optimal op-
their calculation within the distillation framework and
erators can be constructed as a linear superposition of
their relation to the matrix elements of interest while
operatorsinalargebasis, wherethereisaparticularlin-
also demonstrating the efficacy of optimized operators.
ear superposition for each state in the tower of excited
We present the results of our calculation in Section VI,
states. The different superpositions are orthogonal in a
comparing with relevant previous calculations, and then
suitable sense. We are able to find appropriate super-
concludewithasummaryandoutlookinSectionVII.Ap-
positions through a variational analysis of the matrix of
pendices describing helicity operators, momentum con-
two-point correlation functions, (cid:10)0(cid:12)(cid:12)Oi(t)Oj†(0)(cid:12)(cid:12)0(cid:11), for a servation in a finite-volume, the improvement of the vec-
set of operators {Oi}. torcurrentandtheflavorstructureofthecurrentfollow.
The technology we will utilize, constructing ‘opti-
mized’ operators as linear superpositions of a large ba-
sis of interpolators, is useful not only for extracting II. FORM-FACTORS AND TRANSITIONS
transitions featuring excited states, but also to improve
ground-state signals by significantly reducing the un-
The photon couples to the electric charges
wantedcontributionsofexcitedstatestocorrelationfunc-
of u,d,s quarks via the vector current
tions. Thepresenceofsuchcontributionsremainsaprob- jµ =+2u¯γµu− 1d¯γµd− 1s¯γµs, up to a factor of
lem for calculations attempting precision extraction of 3 3 3
the magnitude of the electron charge, e. In general a
ground-state matrix elements [7–14]. Increase in statis-
transition induced by this current between a hadron, h,
tical noise precludes separating operators by large time of spin-J and a hadron h(cid:48) of spin-J(cid:48) is described by the
separations, so instead the use of ‘optimized’ operators
matrix element,
seeks to deal with the problem directly by suppressing
creation of the unwanted excited states. (cid:10)h(cid:48)J(cid:48)(λ(cid:48),p(cid:126)(cid:48))(cid:12)(cid:12)jµ(cid:12)(cid:12)hJ(λ,p(cid:126))(cid:11),
The use of optimized operators as a tool to extract
excited-stateradiativetransitionswaspreviouslyconsid- where the spin-state of h is specified in terms of its he-
ered in [15], where the meson source operators, O† were licity, λ, the projection of J(cid:126) along the direction of mo-
i
optimized, while the meson sink operators, O , were mentum p(cid:126). These matrix elements are simply related
f
simple local fermion bilinears. Quark propagation from to the helicity amplitude for the transition γh → h(cid:48) by
3
including the initial-state photon’s polarization vector, having the right parity transformation properties once
current conservation is demanded [19],
M(cid:0)γ(λ ,(cid:126)q)h (λ,p(cid:126))→h(cid:48) (λ,p(cid:126)(cid:48))(cid:1)
γ J =(cid:15)µ(λγ,(cid:126)qJ)(cid:48)(cid:10)h(cid:48)J(cid:48)(λ(cid:48),p(cid:126)(cid:48))(cid:12)(cid:12)jµ(cid:12)(cid:12)hJ(λ,p(cid:126))(cid:11), (cid:10)=ρ+−(λ(cid:2)(cid:48)(,pp(cid:126)+(cid:48))(cid:12)(cid:12)pj(cid:48)µ)µ(cid:12)(cid:12)ρ(cid:15)+∗((λλ,(cid:48),p(cid:126)p(cid:126))(cid:11)(cid:48))·(cid:15)(λ,p(cid:126))(cid:3)G (Q2)
1
where (cid:126)q = p(cid:126)(cid:48)−p(cid:126) and where the photon has a virtuality +(cid:2)(cid:15)µ(λ,p(cid:126))(cid:15)∗(λ(cid:48),p(cid:126)(cid:48))·p+(cid:15)µ∗(λ(cid:48),p(cid:126)(cid:48))(cid:15)(λ,p(cid:126))·p(cid:48)(cid:3)G (Q2)
Q2 =−q2 =(cid:12)(cid:12)p(cid:126)(cid:48)−p(cid:126)(cid:12)(cid:12)2−(cid:0)Eh(cid:48)(p(cid:126)(cid:48))−Eh(p(cid:126))(cid:1)2. −(cid:2)(p+p(cid:48))µ (cid:15)∗(λ(cid:48),p(cid:126)(cid:48))·p(cid:15)(λ,p(cid:126))·p(cid:48) 1 (cid:3)G (Q2),2 (5)
There are relations between these amplitudes which 2m2 3
follow from the constraints of Lorentz invariance, cur-
withacorrespondingsetofthreeindependentdimension-
rent conservation and invariance under parity transfor-
lessform-factorsG ,G ,G . Aconvenientbasishavinga
1 2 3
mations. Thesecanbeaccountedforifwewriteamatrix-
clearer physical motivation is provided by the expansion
element decomposition in terms of a number of Lorentz
of the vector current in terms of multipoles [20], which
invariant form-factors, F (Q2),
i in this case leads to a set of form-factors,
(cid:10)h(cid:48)J(cid:48)(λ(cid:48),p(cid:126)=(cid:48))(cid:12)(cid:12)(cid:88)jµ(cid:12)(cid:12)hJK(λµ(cid:2),hp(cid:126))(cid:48)(cid:11)(cid:0)λ(cid:48),p(cid:126)(cid:48));h (λ,p(cid:126))(cid:3)F (Q2). (1) GC =(cid:16)1+ 6Qm22(cid:17)G1− 6Qm22 G2+ 6Qm22 (cid:16)1+ 4Qm22(cid:17)G3
i J(cid:48) J i G =G
i M 2
(cid:16) (cid:17)
The Lorentz covariant ‘kinematic factors’, Kiµ, are con- GQ =G1−G2+ 1+ 4Qm22 G3, (6)
structed from the meson four-momenta, p , p(cid:48), and ini-
ν ν
tial and final state polarization tensors relevant to the which are proportional to the charge (C ), magnetic
0
spin of the mesons, (cid:15) (λ,p(cid:126)), (cid:15)∗ (λ(cid:48),p(cid:126)(cid:48)). For any dipole (M ), and quadrupole (C ) multipoles respec-
ρσ... ρσ... 1 2
given pair of mesons h,h(cid:48), of definite spin and parity, tively1. At Q2 = 0 they are related to the charge, mag-
there are only a limited number of possible construc- netic moment and quadrupole moment of the vector me-
tions consistent with parity invariance and with the ad- son: G (0)=1, G (0)=2m·µ , G (0)=m2·Q .
C M ρ Q ρ
ditional constraint of current conservation, we can write The other form-factors we considered above may also
explicit decompositions in terms of a few independent be identified with a particular multipolarity – in the
form-factors. ρ→πγ transition case the single form-factor is of mag-
For example, a pseudoscalar particle like the pion has neticdipole(M )type,whilefortheπcasesitisacharge
1
only a single form-factor appearing in its decomposition, form-factor (C ). For stable meson states, time-reversal
0
invariance indicates that the form-factors are real func-
(cid:10)π+(p(cid:126)(cid:48))(cid:12)(cid:12)jµ(cid:12)(cid:12)π+(p(cid:126))(cid:11)=(p+p(cid:48))µFπ(Q2). (2) tions of Q2.
AtQ2 =0,thevectorcurrentmeasuresthechargeofthe
pion in units of e, so Fπ(0)=1 exactly. III. CALCULATION DETAILS
Inatransitionbetweentwodifferentpseudoscalarpar-
ticles, there is again only one form-factor, but the kine-
In this first investigation of the extraction of excited
maticfactordiffersowingtothedifferingmasses(m,m(cid:48))
stateform-factorsusingdistillation, werestrictourselves
of the pseudoscalar particles (π,π(cid:48)),
toasingleensembleofgauge-fieldconfigurations, having
(cid:10)π(cid:48)+(p(cid:126)(cid:48))(cid:12)(cid:12)jµ(cid:12)(cid:12)π+(p(cid:126))(cid:11) three degenerate flavors of dynamical quarks tuned to
approximately the physical strange quark mass. This
(cid:104) (cid:105)
= (p+p(cid:48))µ+m(cid:48)2−m2(p(cid:48)−p)µ F (Q2). (3) set of anisotropic Clover lattices [22, 23] has been used
Q2 π(cid:48)π
previously in studies of the meson spectrum [17, 18, 24–
26],mesondecayconstants[27],baryonspectrum[28–31]
The transition matrix-element between a vector parti-
andmeson-mesonscattering[32–35]. Forthecalculations
cle and a pseudoscalar can be expressed as
reported on in this paper, we used 535 configurations
(cid:10)π+(p(cid:126)(cid:48))(cid:12)(cid:12)jµ(cid:12)(cid:12)ρ+(λ,p(cid:126))(cid:11) of lattice volume (L/as)3 × (T/at) = 163 × 128, with
a spatial grid spacing of a ∼ 0.12fm and a temporal
=(cid:15)µνρσp(cid:48) p (cid:15) (λ,p(cid:126)) 2 F (Q2), (4) s
ν ρ σ mπ+mρ ρπ spacing roughly 3.5 times smaller.
InthiscalculationwehaveanexactSU(3)flavorsym-
and for a vector meson stable under the strong
metry such that all the octet mesons (π, K, η) are de-
interactions, the transition form-factor at Q2 =0
generate with a mass close to 700 MeV. Where results
can be related to the radiative decay width
are expressed in dimensionful units, they are determined
Γ(ρ+ →π+γ)= 4α |q(cid:126)|3 |F (0)|2, where (cid:126)q is the
3 (mρ+mπ)2 ρπ
momentum of the final-state photon in the rest-frame of
the decaying ρ meson.
The vector current matrix element for a stable vector 1 Note that the relationship between G2 and GM was presented
hadronofmassmhasthreepossiblecovariantstructures withatypographicerrorin[21]whichiscorrectedhere.
4
from the dimensionless quantities a E using the scale- A. Variational Analysis
t
setting procedure,
a E Weproposethatwithinanybasisofoperatorsthereisa
E = t ·mphys.. particularlinearcombinationthatismostsuitedtointer-
a m Ω
t Ω polate the lightest state of the spectrum, another linear
where atmΩ is the Ω baryon mass calculated on this lat- combination that optimally interpolates the first excited
tice and mphys. is the experimental value [5]. state, a third combination for the second excited state
Ω
OuruseofaClover-improvedanisotropicquarkaction and so on. Thus optimized interpolators take the form
introduces an improvement term into the vector current Ω† =(cid:80) w(n)O†, where one can show that the best esti-
n i i i
which appears at tree-level. Discussion of the effect of mate for the weights w(n), in a variational sense, comes
improvement and renormalisation of the vector current i
from solving the generalized eigenvalue problem [36–39],
will appear in Section VE.
C(t)v(n) =λ (t)C(t )v(n). (7)
n 0
IV. TWO-POINT FUNCTION ANALYSIS
Here C(t) is the N ×N matrix whose elements are the
correlation functions C (t) constructed from the basis
ij
In order to determine radiative transition amplitudes
of N operators, {O }, and v(n) is a generalized eigen-
i
between meson states within QCD we must first obtain
vector. The generalized eigenvalues, or principal corre-
the spectrum of states and find operators, constructed
lators, λn(t) behave like e−En(t−t0) at large times, and
fromquarkandgluonfields, thatreliablyinterpolatethe
can be used to determine the spectrum of energy eigen-
states of interest from the vacuum. In general a color-
states. The vectors v(n) are orthogonal on a metric,
singlet operator Oi† having definite JPC can produce all v(m)†C(t0)v(n) = δmn, where t0 is a reference timeslice.
QCD eigenstates having those quantum numbers,
Examinationoftheorthogonalityconditionsuggeststhat
(cid:88) t should be chosen to be sufficiently large such that
O†|0(cid:105)= |n(cid:105)(cid:104)n|O†|0(cid:105). 0
i n i thecorrelationfunctionsaredominatedbytheN lowest-
lyingstates,withheavierstateshavingdecayedexponen-
Weseektodetermineoptimizedinterpolators,Ω†,which
n tially to a negligible level. Further considerations on the
when acting on the vacuum strongly interpolate only a
choice of t are presented in [38, 39].
0
single state with much reduced contributions from other
In practice we solve Eqn 7 independently on each
states,
timeslice, t, so that for each state, n, we obtain a time
Ω†|0(cid:105)=|n(cid:105)(cid:104)n|Ω†|0(cid:105)+ (cid:88)|m(cid:105)(cid:104)m|Ω†|0(cid:105) seriesofgeneralizedeigenvectorsv(n)(t,t0),whichweob-
n n n serve to be essentially time-independent with a suitably
m(cid:54)=n
large choice of t . In practice we use the mean values
(cid:88) 0
=|n(cid:105)(cid:104)n|Ω†n|0(cid:105)+ |m(cid:105)εm. (over the ensemble of gauge configurations) of the ele-
m(cid:54)=n ments v(n) chosen on a single timeslice to construct the
i
optimized operators as
Inessenceweseekaprocedurebywhichwecanminimize
the ε (m (cid:54)= n) relative to the strength with which our
operamtor creates the n’th state, (cid:104)n|Ω†n|0(cid:105). Ω†n =(cid:112)2Ene−Ent0/2(cid:88)ivi(n)Oi†, (8)
Wewillproceedbyusingabasis ofinterpolators,{O },
i
to construct two-point correlation functions of the form, where the coefficients multiplying the sum are chosen to
give the normalization (cid:104)n|Ω†|0(cid:105)=2E , which will prove
n n
Cij(t)=(cid:104)0|Oi(t)Oj†(0)|0(cid:105), tobeconvenientwhenconsideringthree-pointfunctions.
The procedure of solving the generalized eigenvalue
where operators O are color-singlet constructions built
i problem in a basis of interpolating fields is carried out
fromthebasicquarkandgluonfieldsofQCD,havingthe
independentlyforeachquantumnumberchannelateach
quantum numbers of the desired hadrons. Such correla-
possible allowed momentum value.
tion functions can be expressed as,
C (t)=(cid:88) 1 (cid:104)0|O (0)|n(cid:105)(cid:104)n|O†(0)|0(cid:105)e−Ent, B. Meson operator construction
ij 2E i j
n
n
A straightforward approach to constructing a basis of
where the spectrum of eigenstates is seen to control the
operators capable of interpolating meson states is to use
Euclidean time dependence2.
fermionbilinearscontainingsomenumberofspatiallydi-
rected gauge-covariant derivatives, that is operators of
generic structure,
2 Wehaveintroducedourparticularchoiceofstatenormalization
←→ ←→
inafinite-volumehere-wediscussthisinAppendixB. O ∼ψ¯ΓD ··· Dψ, (9)
5
←→ ←− →−
where D ≡ D−D. Byexpressingthevector-likederiva- are summed. The object Γ can be non-local in (cid:126)x-space,
tivesandGammamatricesinacircularbasis,wecaneas- and may for example feature gauge-covariant derivatives
ilyconstructoperatorsofdefinitespinusingthestandard as discussed in the previous section.
SO(3)Clebsch-Gordoncoefficients[17,18]. Operatorsof An advantage of the distillation framework is that it
this type with non-zero momentum can be constructed leadstoafactorizationoftwo-pointfunctionsintomatri-
to have definite helicity, as described in [40]. ces describing quark propagation, called perambulators,
In this calculation, QCD is discretised on a grid of andmatricesdescribingoperatorconstruction. Ageneric
pointswhosespatialstructurehasacubicsymmetryand connected two-point function using fermionbilinear con-
as such we have not the complete continuous rotational structions, in which we explicitly show the smearing op-
symmetry of the continuum, but rather a reduced sym- erators, can be decomposed as
mreeletvrayn:tthliettlseymgrmoueptriynoafmthoevicnugbfircagmroe.upAactonressetquaenndcethies (cid:10)0(cid:12)(cid:12)ψ¯(cid:3)Γf(cid:3)ψ(t)ψ¯(cid:3)Γi(cid:3)ψ(0)(cid:12)(cid:12)0(cid:11)
that instead of having an infinite number of irreducible =−τ (0,t)Φf (t)τ (t,0)Φi (0)
nm mp pq qn
representationslabeledbyintegerspinJ (atrest)weonly
haveaccesstothefinitenumberofirreduciblerepresenta- wheretheperambulators,τpq(t,0)=ξ(p)†(t)Mt−,01ξ(q)(0),
tions, or irreps, of the cube, labelled A , T , T , E, and can be obtained by inverting the Dirac matrix M
1 1 2
A2. A corresponding argument applies in-flight, where on sources {ξ(q)}q=1...ND at timeslice 0, and where
thecontinuumhelicitylabelingisbrokendowntoafinite Φ (t)=ξ(m)†(t)Γξ(p)(t)encodestheoperatorconstruc-
mp
number of irreps of the little group, see [40] for details. tion (here we include the momentum projection in Γ).
The operators of definite J (or helicity) constructed
above can be projected into irreps of the relevant sym-
metry group using a procedure called subduction, D. Meson spectra & optimized operators
J
O[J] = (cid:88) SJMOJM (10) We computed correlation matrices for irreps corre-
Λµ Λµ spondingtoJP =0−, 1−atrest,andtomagnitudeofhe-
M=−J
licity|λ|=0,1withnonzeromomentum. Wemadeuseof
where Λ labels the cubic irrep and µ is the ‘row’ of the (cid:126)n = [0,0,0], [0,0,1], [0,1,1], [1,1,1], [0,0,2], where the
p(cid:126)
irrep (µ = 1...dim(Λ)). The subduction coefficients, momentumisexpressedinunitsof2π/L,p(cid:126)=(2π/L)(cid:126)n .
p(cid:126)
SJM aretabulatedin[18,40]. Theseoperatorshavebeen The quark flavor constructions were chosen to give ac-
Λµ
usedextensivelytostudytheexcitedspectrumofmesons cess to the members of the SU(3) octet – in this case
F
[17, 18, 24–26, 34, 35]. the two-point function Wick contraction contains only a
single connected diagram.
In the rest frame we used operator constructions of
C. Correlator construction through distillation the type in Eq. 9 with up to three derivatives while for
nonzero momentum we used up to two-derivative con-
Operatorswhichinterpolatehadronsfromthevacuum structions. Further details about the construction of
have long been known to do so more effectively if the derivative operators and the basis used in this calcula-
quark fields are suitably smeared over space [41, 42]. tion can be found in [17, 18, 26, 40].
An extremely convenient method to do this is provided Theresultingcorrelationmatriceswereanalyzedusing
by distillation [16], where the smearing operator is con- Eq.7,witheachprincipalcorrelator,λn(t),beingfitwith
structed on time-slice t as an outer product of vectors in the form,
color and (cid:126)x−space,
λn(t)=(1−An)e−En(t−t0)+Ane−En(cid:48)(t−t0), (12)
(cid:3)(cid:126)x(cid:126)y(t)= (cid:88)ND ξ(cid:126)x(n)(t)ξ(cid:126)y(n)†(t), (11) wherethefitparametersareEn,En(cid:48),andAn –thesecond
exponential is present to absorb the effect of any states
n=1
other than |n(cid:105) remaining in the principal correlator. In
where the N vectors should be constructed to have
D practice, for suitably large values of t , we find that the
0
strong overlap onto the low-energy quark modes most energy scale of E(cid:48) is typically at or above the largest
relevanttolow-lyinghadronstates. Asuitablechoicefor n
energy extracted, E .
n=dim(C)
the vectors are the eigenvectors of the gauge-covariant
A typical example is presented in Figure 1, where the
three-dimensional Laplacian on a time-slice ordered by
principal correlators for the lightest three states in the
their eigenvalue. (cid:126)n = [0,0,0], ΛPC = T−−(A−+) irreps, which contain
Smearing each quark field, a meson creation operator 1−p(cid:126)−(0−+) mesons, are s1hown.1 We make use of a basis
of fermion bilinear form with momentum p(cid:126) is of 26 operators in the T−− channel and 12 operators for
1
O†(p(cid:126))=ψ¯(cid:126)x(cid:3)(cid:126)x(cid:126)ye−ip(cid:126)·(cid:126)yΓ(cid:126)y(cid:126)z(cid:3)(cid:126)zw(cid:126)ψw(cid:126) A−1+.
Another example is shown in Figure 2(a) for the
where time, color, and spin indices have been sup- case (cid:126)n = [0,0,1], ΛC = A+, which contains the he-
p(cid:126) 2
pressed for brevity and where repeated position indices licity zero component of mesons of ‘unnatural parity’
6
1.2 1.2
1.0 1.0
0.8 0.8
0 5 10 15 20 25 0 5 10 15 20 25
1.2 1.2
1.0 1.0
0.8 0.8
0 5 10 15 20 25 0 5 10 15 20 25
1.2 1.2
1.0 1.0
0.8 0.8
0 5 10 15 20 25 0 5 10 15 20 25
FIG. 1. Left(Right) columns: Principal correlators, plotted as eEn(t−t0)λn(t), for lightest three states in irrep A−1+(T1−−).
Twoexponentialfitsshown,withtheresultingmassspectrumof0−+ and1−− mesonsshowninthecentralcolumn. Radiative
transitionsandform-factorstobepresentedinthispapershownbythelinesjoiningthestates(solidlinesare‘charge’transitions,
dashed lines are ‘magnetic’ transitions).
(JP =0−, 1+, 2−,...), where we present the effective to suppress or amplify the contribution of unwanted ex-
mass of the principal correlator, m = 1 log λ(t) cited states, but since we do not have this information
eff δt λ(t+δt)
in advance of the calculation, the above time separation
with δt = 3a . As will be discussed in the next sub-
t
serves as a reasonable estimation of the minimum.
section, these states appear to correspond (in order of
increasingenergy)totheground-stateπ,thea ,thefirst- The source and sink cannot be separated arbitrarily
1
excited π(cid:48), and the π . farasthestatisticalnoiseontheentirethree-pointcorre-
2
lation function grows exponentially with increasing sep-
We note that in both figures we observe that there is
negligible curvature present for t (cid:38) t , indicating that a aration. We can obtain an estimate of the maximum
0
practicaltimeseparationforthree-pointfunctionsbyex-
single state is dominating the correlation function. This
amining the growth of noise in two-point function prin-
observation only holds for sufficiently large choices of t ,
0
cipal correlators. As an example in Figure 1 we see that
and is one guide in the selection of a suitable t value.
0
the ground-state signal remains of high statistical qual-
The time beyond which excited state contributions are
negligible, which we can call t(cid:63), plays an important role ity out to 25 timeslices (and beyond), while the first and
second excited states begin to show significant fluctua-
in the construction of our three-point functions in terms
tions above t/a ∼ 15. Later we will find that while
of selecting the time separation of the source and sink t
optimized ground-state three-point correlation functions
meson operators. We desire our three-point functions to
are still statistically precise for time separations as high
feature a time region in which the correlation function
as 36 timeslices, excited-state correlation functions are
is dominated by the transition of interest, with contri-
not well determined for separations larger than around
butions from other states having decayed away. In order
20 timeslices.
to achieve this we should separate the source and sink
projected operators (Eq. 8) by at least t(cid:63) + t(cid:63). It re- In practice we solve the matrix problem, Eq. 7, inde-
i f
mains possible that the vector current insertion may act pendently on each timeslice (as described in [18]). For
7
patible with zero for t(cid:38)t .
0
0.2 0.4 0.6 0.8
Weconcludethissectionbydemonstratingthatanop-
0.6
timized ground-state operator constructed as described
2.5
0.5 above, does indeed significantly reduce the contribution
0.4 2.0 of excited states to a correlation function, relaxing to
the ground-state more quickly. This can be seen in Fig-
0.3 1.5 ure3wheretheoptimizedground-stateoperatorinA−+
1
0.2 1.0 is compared with the (distillation smeared) ψ¯γ ψ opera-
5
tor.
0 5 10 15 20 25
1.04
1.2
E. Meson dispersion relations
1.00 1.0
0.96 0.8
As previously mentioned, we independently compute
0 5 10 15 20 25 0 5 10 15 20 25
the energy spectrum of states at each allowed value of
1.2 1.2 p(cid:126) for each relevant irrep, but we expect to see the same
mesonsappearingateachmomentumwithanenergyde-
1.0 1.0
termined by their rest mass and the relevant dispersion
0.8 0.8 relation. After identifying the mesons at each momen-
tum(bytheiroverlapwithcharacteristicoperators[40]),
0 5 10 15 20 25 0 5 10 15 20 25
wemayexaminetheirdispersionrelation,E(|p(cid:126)|). Thisis
presented in Figure 4 for mesons, π,π(cid:48),ρ,ρ(cid:48) and mesons
FIG. 2. (a) Effective masses of principal correlators for a ,a ,b ,π (notusedinthisanalysis)wheretheyareall
0 1 1 1
four lightest states in the A+ irrep for momentum direction observedtobecompatiblewiththerelativisticdispersion
2
(cid:126)np(cid:126)=[1,0,0]alongwiththeenergydeterminedfromatwoex- relation, E2 =m2+p2, or in temporal lattice units,
ponential fit. (b) ‘Optimized’ operator correlation functions,
(2En)−1eEnt(cid:10)Ωn(t)Ω†n(0)(cid:11), for the four states shown above. (cid:0)a E(cid:1)2 =(cid:0)a m(cid:1)2+(cid:16) 2π (cid:17)2|(cid:126)n |2,
t t ξ(L/as) p(cid:126)
0.2 0.6 1.0
for a meson of mass m, with momentum
1.3
a p(cid:126)= 1a p(cid:126)= 1 2π [n ,n ,n ], with the anisotropy
t ξ s ξL/as x y z
0 taking the value ξ = 3.44. Making use of optimized
1.2
operators with momenta up to [0,0,2] allows us to
sample many values of Q2 in the form-factor extraction.
1.1
0.2 0.6 1.0
1.0
6.0
0 5 10 15 20 25 30 0.25
5.0
0.20
FIG. 3. The rest-frame pion correlation function us- 4.0
ing the (distillation smeared) ψ¯γ ψ operator (red) ver- 0.15
5 3.0
sus using the ‘optimized’ operator, Ω (blue). Plotted is
2m emπt(cid:104)0|O(t)O†(0)|0(cid:105)/|(cid:104)0|O|π(cid:105)|2. 0 0.10 2.0
π
0.05 1.0
sufficiently large t0 we find that the elements of the 0 1 2 3 4
eigenvectors so obtained, v(n)(t), are essentially flat for
i
t (cid:38) t , and in practice we construct our projected op- FIG.4. Squaredenergiesasafunctionof|(cid:126)n |2 =(L)2|p(cid:126)|2for
0 p(cid:126) 2π
erators, Eq. 8, using ensemble mean values taken from selectedmesonstates. Thepointsrepresentextractedenergies
a single timeslice. In Figure 2(b) we show the corre- for all irreps considered in the variational analysis (e.g. a
lation functions (cid:10)0(cid:12)(cid:12)Ωn(t)Ω†n(0)(cid:12)(cid:12)0(cid:11), observing that they Avec,tBor,mBes)o,nwlhikieleththeeρlianpepsesahroswintthherereeliartrievpisstiwcitdhis[p1e,r1s,io0n]:
behave in the manner we expect for optimized opera- 1 1 2
relation for an anisotropy ξ=3.44.
tors. The corresponding off-diagonal correlation func-
tions (cid:10)0(cid:12)(cid:12)Ωn(t)Ω†n(cid:48)(0)(cid:12)(cid:12)0(cid:11) for n (cid:54)= n(cid:48) are statistically com-
8
V. THREE-POINT FUNCTIONS
We now turn to the three-point correlation functions
usedinthisanalysiswhichcontainthevector-currentma-
trix elements of interest. Their basic form is
Cfµi(∆t,t)=(cid:10)0(cid:12)(cid:12)Of(∆t)jµ(t)Oi†(0)(cid:12)(cid:12)0(cid:11),
wheretheoperatorsO arecapableofinterpolatingme- FIG. 5. Graphical depiction of a connected three-point cor-
i,f
relator. The gray blobs represent the source and sink oper-
son states of definite momentum from the vacuum – a
ators carrying the quantum numbers of the initial and final
suitable basis was discussed in the previous section. The
state mesons. The blue line corresponds to a perambulator
relationbetweenthecorrelationfunctionandthedesired
(τ (∆t,0) = ξ(p)†(∆t)M−1 ξ(q)(0)) while the red line rep-
matrix-element is exposed by inserting complete sets of pq ∆t,0
resents a generalized perambulator as described in the text,
eigenstateswiththequantumnumbersofOi andOf,and GΓ(∆t,t,0) = ξ(p)†(∆t)M−1 Γ M−1ξ(q)(0), which carries
pq ∆t,t t t,0
evolving all operators back to the origin of Euclidean the momentum and quantum numbers of the current inser-
time, tion.
(cid:88) 1 1
Cfµi(∆t,t)=ni,nf 2Enf 2Enie−Enf(∆t−t)e−Enit A. Correlator construction & distillation
×(cid:10)0(cid:12)(cid:12)Of(0)(cid:12)(cid:12)nf(cid:11)(cid:10)nf(cid:12)(cid:12)jµ(0)(cid:12)(cid:12)ni(cid:11)(cid:10)ni(cid:12)(cid:12)Oi†(0)(cid:12)(cid:12)0(cid:11). Three-point correlation functions featuring a current
Thesummationrunsoverallstates,butclearlyifthesep- insertion require a slight extension of the distillation
arations between the operators are large, ∆t (cid:29) t (cid:29) 0, framework presented in Section IVC since the quark
only the lightest states in the i and f channels will con- fields in the current should be those which appear in the
tribute and we can extract the vector-current matrix el- action, which are not smeared. Exposing the smearing
ement between them. However, at modest time sepa- operators, the general form required is,
rations there will remain subleading exponential contri-
butions from excited states, and these ‘polluting’ terms (cid:10)0(cid:12)(cid:12)ψ¯(cid:3)Γf(cid:3)ψ(∆t)·ψ¯Γψ(t)·ψ¯(cid:3)Γi(cid:3)ψ(0)(cid:12)(cid:12)0(cid:11),
can be a source of systematic error in the extraction of
ground state matrix elements [7–14, 43]. Reducing ex- which,consideringfornowonlythecompletelyconnected
cited state pollution by simply separating operators by Wick contraction, can be decomposed as
longerEuclideantimesisnotalwayspractical,duetothe
increase in statistical noise with increasing separation. (cid:10)0(cid:12)(cid:12)ψ¯(cid:3)Γf(cid:3)ψ(∆t)·ψ¯Γψ(t)·ψ¯(cid:3)Γi(cid:3)ψ(0)(cid:12)(cid:12)0(cid:11)
(cid:104) (cid:105)
One of our major aims here is to extract excited-state =−τ (0,∆t)Φf (∆t) ξ(p)†(∆t)M−1 ΓM−1ξ(q)(0) Φi (0)
matrix elements, which we may access using the opti- nm mp ∆t,t t,0 qn
mizedoperatorsdescribedintheprevioussection. Using =−τ (0,∆t)Φf (∆t)GΓ(∆t,t,0)Φi (0),
nm mp pq qn
theoptimizedoperatorforanexcitedstateshouldleadto
athree-pointcorrelationfunctionwhoseleadingbehavior where the object in square brackets, GΓ(∆t,t,0), is a
pq
atlargetimesisgivennotbythegroundstate,butrather generalized perambulator. It can be obtained through in-
by the relevant excited state. If we are interested in the version from sources {ξ(q)} at timeslice 0 to ob-
q=1...ND
ground-state there is also an advantage to using the ap- tainM−1ξ(q)(0),inversionfromsources{ξ(p)} at
propriateoptimizedoperatorinthatitwillhavemuchre- timeslict,e0 ∆t to obtain M−1 ξ(p)(∆t) which cp=an1...bNeDre-
duced overlap onto low-lying excitations (relative to any t,∆t
single operator in the original basis, for example ψ¯γ5ψ, lated to ξ(p)†(∆t)M∆−t1,t using γ5 hermiticity, and con-
tractionwiththeoperatorinsertion,Γ,ateachvalueoft
c.f. Figure 3), leading to a corresponding reduction in
between 0 and ∆t. In this calculation we do not average
the excited state pollution in the three-point correlator.
over multiple time-sources separated by the same value
Three-point correlation functions using optimized op-
of ∆t, although this could be done to increase statistics.
erators take the form,
In our application, exposing the Dirac spin,
Cnfµni(∆t,t)==(cid:10)e−0(cid:12)(cid:12)EΩnnf(f∆(∆t−tt))jeµ−(Et)niΩt†n(cid:10)in(0f(cid:12)(cid:12))j(cid:12)(cid:12)µ0((cid:11)0)(cid:12)(cid:12)ni(cid:11)+... (cid:126)xΓ−(cid:126)αx,βs(cid:126)yp;aa,bce=aγnµαdβec−oiq(cid:126)lo·(cid:126)xrδ(cid:126)xin,(cid:126)ydδiac,ebs.,Inthteheccuarsreenotf ainnsiemrtpioronveids
vector current, which we consider later, we require also
(13)
Γαβ =σαβe−iq(cid:126)·(cid:126)xδ δ ,yieldinganothersetofgen-
(cid:126)x,(cid:126)y;a,b 4k (cid:126)x,(cid:126)y a,b
where the leading time-dependence is that of the states eralized perambulators.
(|n(cid:105),|n (cid:105)), selected by the choice of optimized operators. Within this construction we are able to project each
i f
Theellipsisrepresentstheresidualcontributionsofother operator into definite momentum, and as such we only
states,whichshouldbesignificantlysuppressedwhenus- compute correlation functions in which the momentum
ing optimized operators – we will explore the degree to is conserved, p(cid:126) =p(cid:126) +(cid:126)q. Some discussion of momentum
f i
which this is manifested in explicit calculation. conservation in a finite volume appears in Appendix B.
9
B. Correlation functions using optimized operators lie beyond the reach of our variational basis, and indeed
curvature away from flat behavior as we approach the
source or sink timeslice is observed in Figures 6 and 7.
Turning first to the case of three-point functions with
In order to make maximal use of the time-series data,
pion-like operators at the source and sink, we plot in
in particular in those regions where there remains some
Figure 6 the form-factor (as defined in Eqn. 2) extracted
unwanted excited-state contribution, we opt to perform
from the three-point function,
(cid:104)0|O (∆t,p(cid:126) )jµ(t,(cid:126)q)O†(0,p(cid:126) )|0(cid:105)
π f π i
where O represents either ψ¯γ ψ (in red) or the opti-
π 5 0.3
mized operator Ω (in blue). The sink operator, located
π
at ∆t = 28a ∼ 0.9 fm, is in the ΛC = A+ irrep of mo-
t 2 0.2
mentum(cid:126)n =[1,0,0],whilethesourceoperator,located
p(cid:126)f
at t = 0, is at rest in the ΛPC = A−+ irrep. We clearly 0.1
1
observethattheoptimizedoperatorsgiverisetoasignal
which is flat over a number of timeslices away from the 0
sourceandsink,correspondingtothecontributionofjust
the ground-state pion, while the simpler ψ¯γ ψ operators -0.1
5
overthistimerangealwaysretainanon-negligiblepollu-
-0.2
tion from excited states. Such behavior is expected from 0 2 4 6 8 10 12 14 16 18 20
our two-point function analysis: for example, at rest we 0.3
(cid:12) (cid:12)
find (cid:12)(cid:104)0|ψ¯γ5ψ|n=1(cid:105)(cid:12)∼0.73, so the distillation smeared op-
(cid:12)(cid:104)0|ψ¯γ5ψ|n=0(cid:105)(cid:12) 0.2
erator (cid:2)ψ¯γ ψ(cid:3)†, acting on the vacuum, creates both the
5
0.1
ground and first excited state with comparable strength.
Ourprincipalmotivationforusingoptimizedoperators
0
is to get access to transitions involving excited states.
In Figure 7 we show matrix elements extracted from -0.1
three-point correlation functions computed using either
theground-stateπ orfirst-excitedstateπ(cid:48) optimizedop- -0.2
0 2 4 6 8 10 12 14 16 18 20
erator at the source (ti = 0, p(cid:126)i = [-1,0,-1]) and either 0.3
the ground-state ρ or first-excited state ρ(cid:48) operator at
the sink (t = 20a , p(cid:126) = [1,0,-1]). We observe that 0.2
f t f
thereareclearstatisticallysignificantsignalsforexcited-
state transitions when using the appropriate optimized 0.1
operators.
0
Ingeneral,evenforoptimizedoperators,theremaystill
be some residual contamination coming from states that
-0.1
-0.2
0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 16 18 20
0.3
0.8 0.2
0.1
0.7
0
0.6
-0.1
0
0.5
-0.2
0 2 4 6 8 10 12 14 16 18 20
0 5 10 15 20 25
FIG.6. Form-factorfromvectorcurrentthree-pointfunction FIG. 7. Form-factors extracted from optimized three-point
with pion operators at source (t = 0, (cid:126)n = [0,0,0]) and sink correlationfunctionswithaπorπ(cid:48)operatorwithp(cid:126)=[-1,0,-1]
(∆t = 28a , (cid:126)n = [1,0,0]). Red points correspond to using att=0andaρorρ(cid:48) operatorwithp(cid:126)=[1,0,-1]att=20a .
t t
the ‘unoptimized’ bilinear ψ¯γ ψ; the ‘optimized’ operator is Thesource-sinkseparationinphysicalunitsisroughly0.7fm.
5
shown in blue.
10
1.0
0.8
0.6
0.4
0 0.01 0.02 0.03 0.04 0.05
0.9
0.7
0.5
0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
FIG.8. Upperpanelshowsthepionground-stateform-factorextractedfromcorrelationfunctionswithsource-sinkseparations
of ∆t = (12,16,28)a using the procedure of fitting with Eqn. 14. The lower panel illustrates this using the example of
t
(cid:126)n =[1,0,0],(cid:126)n =[2,0,0].
p(cid:126)i p(cid:126)f
a correlated fit over a time range with the form, In practice, while we extract a very large number of
form-factordeterminationsatmanyQ2-values,wechoose
F(Q2;t)=F(Q2)+ffe−δEf(∆t−t)+fie−δEit (14) to make use of only those where application of Eq. 14 to
F(Q2;t) shows modest excited-state contributions. Any
where f ,δE ,f ,δE and F(Q2) are real fit parameters.
f f i i cases where a clear trend toward a constant value is not
We make further use only of the constant term, which
visible are discarded.
corresponds to the desired form-factor. Fitting the data
to this form also exposes the energy scale of the pollu-
tion terms, δE and δE . Generically, when present, we
f i C. Extracting multiple form-factors
find that these energies lie at or above the scale of the
largestenergieswereliablyextractinourtwo-pointfunc-
Eqn. 1 presents the general form of the decomposi-
tion variational analysis. In cases where there is a clear
tion of a vector-current matrix element into independent
extendedplateauregion,wemayexcludetheexponential
form-factors, F (Q2), and the corresponding kinematical
terms and perform a fit to a constant value. i
factors, K , which depend upon momenta and helicities.
The dependence upon source-sink separation, ∆t, for i
Moving to a more complete notation including momen-
theground-statepionform-factorcanbeseeninFigure8.
tum and helicity labels, our three-point correlation func-
Thelowerpanelshowstheillustrativecase(cid:126)n =[1,0,0],
p(cid:126)i tions may be written,
(cid:126)n =[2,0,0], where we observe that there is only a vis-
p(cid:126)f
ible plateau region for ∆t = 28at, while for the shorter (cid:10)0(cid:12)(cid:12)Ω (∆t)jµ(t)Ω† (0)(cid:12)(cid:12)0(cid:11)
Eseqpna.ra1t4io.nsT,h∆etr=esu1l2tiantg,1v6aaltu,ewseofmFaπke(Qu2s)e aorfeaofibtseursvinedg =e−nEf,np(cid:126)ff,(λ∆ft−t)e−q(cid:126)Enit(cid:10)nnfi,,p(cid:126)p(cid:126)if,,λλif(cid:12)(cid:12)jµ(0)(cid:12)(cid:12)ni,p(cid:126)i,λi(cid:11)+...
tpolobreedcwomitphastiimbliela–r roetshuelrtst.imTeheseuppapraetriopnasnewlesrheoawlssotheaxt- =e−Enf(∆t−t)e−Enit(cid:88)Kiµ(cid:0)nf,p(cid:126)f,λf;ni,p(cid:126)i,λi(cid:1)Fi(Q2)+...,
this procedure is generally applicable and leads to form- i
factors from each time-separation that are in agreement where as previously the ellipsis represents possible pol-
across a range of Q2. lution from states other than (|n(cid:105),|n (cid:105)), which will have
i f