Lattice baryon spectroscopy with multi-particle interpolators

2.2 MB·English

by  Adrian L. Kiratidis

Lattice baryon spectroscopy with multi-particle interpolators
Adrian L. Kiratidis,∗ Waseem Kamleh, Derek B. Leinweber, and Benjamin J. Owen
Special Research Centre for the Subatomic Structure of Matter,
Department of Physics, University of Adelaide, South Australia 5005, Australia.
In2+1flavourlatticeQCDthespectrumofthenucleonispresentedforbothparitiesusinglocal
meson-baryon type interpolating fields in addition to the standard three-quark nucleon interpola-
tors. The role of local five-quark operators in extracting the nucleon excited state spectrum via
correlation matrix techniques is explored on dynamical gauge fields with mπ = 293 MeV, leading
to the observation of a state in the region of the non-interacting S-wave Nπ scattering threshold
in the negative-parity sector. Furthermore, the robustness of the variational technique is exam-
ined by studying the spectrum on a variety of operator bases. Fitting a single-state ansatz to
the eigenstate-projected correlators provides robust energies for the low-lying spectrum that are
essentially invariant despite being extracted from qualitatively different bases.
5
1
PACSnumbers: 11.15.Ha,12.38.-t,12.38.Gc
0
2

l I. INTRODUCTION of Gaussian-smeared fermion sources with a variety of
u
widths [13], providing an operator basis that is highly
J
LatticeQCDiscurrentlytheonlyknownab-initionon- suited to accessing radial excitations. The CSSM lat-
8
perturbative approach to study the fundamental quan- tice collaboration has used this technique to study the

] tum field theory governing hadron properties, Quantum nucleon excited state spectrum [14, 15]. In particular,
at Chromodynamics (QCD). While the ability to obtain the CSSM studies were the first to demonstrate that the
l ground state masses is well-understood, an accurate ex- inclusion of very wide quark fields (formed with large
-
p traction of excited states and multi-particle thresholds amountsofGaussiansmearing)iscriticaltoisolatingthe
e remains a challenge. first positive-parity nucleon excited state [8, 16]. This
h The use of variational techniques [1, 2] to study the state was shown to have a quark probability distribution
[ nucleon excited state spectrum has seen remarkable suc- consistent with an N = 2 radial excitation in Ref. [17].
2 cess in recent years. The key feature of these techniques This work also examined the quark probability distri-
v istobeginwithabasisofdifferentoperatorsthatcouple butions for higher positive-parity nucleon excited states,
7 to the quantum numbers of a given state, and then con- revealingthatthecombinationofGaussiansourcesofdif-
6 struct different linear combinations of these operators in ferentwidthsallowsfortheformationofthenodalstruc-
6 order to isolate the ground and higher excited states in tures that characterise the different radial excitations.
7
0 that channel. The negative-parity nucleon channel with its two low-
. Thepositive-paritynucleonchannelhasbeenofsignifi- lying resonances, the N∗(1535) and N∗(1650), has also
1
0 cantinteresttothelatticecommunity[3–8]. Inparticular been of significant interest [6, 15, 18–20]. These S11
5 the first positive-parity JP = 1+ excitation of the nu- states are in agreement with SU(6) based quark model
2
1 cleon, known as the Roper resonance N∗(1440), remains predictions, making an ab-initio study of the low-lying
: a puzzle. In constituent quark models the Roper reso- negative-parityspectrumapotentiallyrewardingendeav-
v
i nanceliesabovethelowest-lyingnegative-paritystate[9– our. Importantly,atnearphysicalquarkmassesthenon-
X 11], the N∗(1535), whereas in Nature it lies 95 MeV interactingπN scatteringthresholdliesbelowthelowest
r below the resonant state. This has led to speculation lying negative-parity state, making it a natural place to
a
about the true nature of this state, with suggestions it lookforthepresenceofmulti-particleenergylevelsinthe
is a baryon with explicitly excited gluon fields, or that extracted spectrum.
it can be understood with meson-baryon dynamics via a
Until recently, the majority of the work in these chan-
meson-exchange model [12].
nels has been performed with three-quark interpolating
In simple quark models, the Roper is identified with
fields,andinthefullquantumfieldtheorytheseinterpo-
an N = 2 radial excitation of the nucleon. Within the
lators couple to more exotic meson-baryon components
variational technique, the choice of an appropriate oper-
such as the aforementioned πN via sea-quark loop in-
ator basis is critical to obtaining the complete spectrum
teractions. However, baryon studies have found that
of low-lying excited states. Recall that we can expand
the couplings of single hadron type operators to hadron-
any radial function using a basis of Gaussians of differ-
hadron type components, suppressed by the lattice vol-
ent widths f(|(cid:126)r|) = (cid:80)icie−εir2. This leads to the use umeas1/√V,aresufficientlylowsoastomakeitdifficult
to observe states associated with scattering thresholds
[6, 20]. Moreover, there is a question as to what extent
∗ [email protected] the presence of multi-particle states might interfere with
2
the extraction of nearby resonances. respectively. We then search for a linear combination of
One solution is to explicitly include hadron-hadron operators
type interpolators [19, 21] by combining single-hadron
operators with the relevant momentum. This creates an φ¯α =χ¯juαj and φα =χiviα (4)
operator that necessarily has a high overlap with the
scattering state of interest thereby enabling its extrac- such that φ and φ¯ couple to a single energy eigenstate.
tion. Instead, in this work we aim to construct meson- That is, we require
baryon type interpolators without explicitly projecting
single-hadronmomenta,andinvestigatetherolethatthe (cid:10)Ω(cid:12)(cid:12)φα(cid:12)(cid:12)β(cid:11) δαβ. (5)
∝
resulting operator plays in the calculation of the nucleon
One can then see from Eq. (3) that
spectrum. Using these operators we construct a basis
containingboththree-andfive-quarkoperators,andper- (t +dt)uα =e−mαdt (t )uα, (6)
form spectroscopic calculations utilising a variety of dif- Gij 0 j Gij 0 j
ferent sub-bases. Examining the resulting spectra then and hence the required values for uα and vα for a given
j i
provides an excellent opportunity to both study the role
choice of variational parameters (t ,dt) can be obtained
0
ofourmulti-particleoperatorsandtesttherobustnessof
by solving the eigenvalue equations
the variational techniques employed.
Following the outline of standard variational analyses (cid:2) −1(t ) (t +dt)(cid:3) uα =cαuα (7)
in Section II, we construct these hadron-hadron type in- G 0 G 0 ij j i
vα(cid:2) (t +dt) −1(t )(cid:3) =cαvα, (8)
terpolators in the form of five-quark operators in Sec- i G 0 G 0 ij j
tion III. We then develop a method for smearing ele-
mentsofthestochasticallyestimatedlooppropagatorsat where the eigenvalue is cα = e−mαdt. In the ensemble
x, S(x,x) in Section IV. These necessarily arise with the average ij is a symmetric matrix. We work with the
introductionofourfive-quarkinterpolatingfields,dueto improvedGestimator 21(Gij+Gji)ensuringtheeigenvalues
the presence of creation quark fields in our annihilation of Eqs. (7) and (8) are equal. As our correlation matrix
operator and vice versa. Having covered the technology is diagonalised at t0 and t0 +dt by the eigenvectors uαj
required for a spectroscopic calculation, we then outline andviα wecanobtaintheeigenstate-projectedcorrelator
our simulation details in Section V and present nucleon as a function of Euclidean time
spectra for both parities in Section VI.
α(t)=vα (t)uα, (9)
G i Gij j
whichcanthenbeusedtoextractmasses. Moreover,the
II. CORRELATION MATRIX TECHNIQUES
analysis can be performed on a symmetric matrix with
orthogonal eigenvectors. More details can be found in
Correlation matrix techniques [1, 2] are now well- Ref. [15].
established as a method for studying the excited state Atthispointwenotethatiftheoperatorbasisdoesnot
hadron spectrum. The underlying principle is to begin appropriately span the low-lying spectrum, α(t) may
G
with a sufficiently large basis of N operators (so as to contain a mixture of two or more energy eigenstates.
span the space of the states of interest within the spec- There are a number of scenarios in which this might oc-
trum)andconstructanN N matrixofcrosscorrelation cur:
×
functions,
At early Euclidean times the number of states
•
Gij(p(cid:126),t)=(cid:88)e−ip(cid:126)·(cid:126)x(cid:10)Ω(cid:12)(cid:12)χi((cid:126)x,t)χj((cid:126)0,tsrc)(cid:12)(cid:12)Ω(cid:11). (1) sbtero(nmgulychc)onlatrrigbeurttinhgantotthheencuomrrbeleartioofnompeartaritxormsainy
(cid:126)x
the basis.
After selecting p(cid:126) =(cid:126)0 and projecting to a specific parity There may be energy eigenstates present that do
with the operator • not couple or only couple weakly to the opera-
tors used. In particular, it is well known that lo-
1
Γ = (γ I), (2) calthree-quarkinterpolatingfieldscouplepoorlyto
± 0
2 ± multi-hadron scattering states.
we can write the correlator as a sum of exponentials, The nature of the operators selected may be such
•
that it is not possible to construct a linear combi-
(cid:88)
Gij(t)= λαi λ¯αj e−mαt, (3) nation with the appropriate structure to isolate a
α particular state.
where α enumerates the energy eigenstates of mass m It is important to have a strategy to ensure that one can
α
and λ¯α and λα are the couplings of our creation and accuratelyobtaineigenstateenergiesfromthecorrelation
j i
annihilation operators χ and χ at the source and sink matrix. The method we use is to analyse the effective
j i
3
energies of different states from the eigenstate-projected
correlators, 3.5
Eα(t)= 1 log Gα(t) (10) 3.0
n α(t+n)
G 2.5
which is constant in regions where the correlator is dom-
inated by a single state. Neighbouring time slices in the V)2.0
correlation functions are highly correlated in Euclidean Ge
time, and require a covariance-matrix based χ2 analysis. M(1.5
Thebestunbiasedestimatecorrespondstoaχ2/dof 1.
≈ 1.0
We therefore endeavour to obtain a plateau fit of the ef-
fective mass with the χ2/dof close to one. In considering
0.5
an upper limit for the fit, points with errors bars larger
than the central value are discarded. Fits with χ2/dof 0.0
17 18 19 20 21 22 23
>1.20arerejected,asthesefitshavesignificantcontam-
EuclideanTime
ination from nearby states not yet isolated in the corre-
lation matrix analysis [20]. We do not enforce a lower
3.5
bound on acceptable χ2/dof as small values typically re-
flect large uncertainties rather than an incorrect result
3.0
associated with a systematic error. Typically, plateaus
commence three or four time slices after the source, near 2.5
the regime where the generalised eigenvector analysis of
thecorrelationmatrixisdone. Figure1illustratestypical V)2.0
effective mass fits for positive and negative parity states. Ge
Further details of this method can found in Ref. [22]. M(1.5
As we will demonstrate, a careful covariance-matrix
based χ2 analysis to fit the single-state ansatz ensures a 1.0
robust extraction of the eigenstate energies. The physics
0.5
underpinning this robustness is elucidated in detail in
Section VI.
0.0
The CSSM lattice collaboration has used this tech- 17 18 19 20 21 22 23
EuclideanTime
niqueinthecalculationofthenucleonspectrainboththe
positive [8] and negative-parity channels [15] with stan-
FIG. 1. (Colour online). Typical effective mass fits for
dard three-quark interpolators. While largely successful
positive-parity (top) and negative-parity (bottom) nucleon
atidentifyingtowersofexcitedstatesthatwouldbeasso- excitations. The top plot shows a fit to the first positive-
ciatedwithresonancesinNature,ithasbeenshownthat parityexcitationofthe4 4correlationmatrixobtainedfrom
×
with three-quark operators alone it is difficult to detect basis 1 described in Table I of Section V. The fitted mass of
states near multi-particle scattering energy levels [20]. 2.11(4) GeV is shown as a green circle for basis 1 in Fig. 4,
Theconcernisthattheoperatorbasisdoesn’thavesuffi- and provides χ2/dof = 0.17. The bottom plot shows a fit
cientoverlapwithmeson-baryontypecomponents, high- to the lowest-lying state in the negative-parity sector. It is
sourcedfromthe6 6correlationmatrixobtainedfrombasis
lightingtheneedforstudieswithmulti-hadronoperators.
×
3describedinTableIofSectionV.Thefittedmassof1.58(3)
GeVisshownasabluesquareforbasis3inFig. 8,andcorre-
spondstoχ2/dof=0.87. Note,anearlierfitincludingt=20
III. MULTI-PARTICLE STATE
provides χ2/dof =1.22, reflecting the systematic drift in the
CONTRIBUTIONS
effective mass at early times.
Inordertofurtherelucidatethesituation, weconsider
asimpletwo-componenttoymodelwhichconsistsoftwo only has substantial overlap with 1 . That is,
| (cid:105)
QCD energy eigenstates, a and b . We then suppose
that a and b are given|by(cid:105) | (cid:105) (cid:10)Ω(cid:12)(cid:12)χ3(cid:12)(cid:12)1 C and (cid:10)Ω(cid:12)(cid:12)χ3(cid:12)(cid:12)2 C, (13)
| (cid:105) | (cid:105) (cid:105)∝ (cid:105)(cid:28)
(cid:12)(cid:12)a(cid:11)= cosθ(cid:12)(cid:12)1(cid:11)+sinθ(cid:12)(cid:12)2(cid:11), (11) for some constant C. When χ3 acts on the vacuum we
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) therefore create a state that is superposition of the true
(cid:12)b = sinθ(cid:12)1 +cosθ(cid:12)2 , (12) energy eigenstates given by
−
where 1 and 2 denote a single-hadron and meson- (cid:12)(cid:12)1(cid:11)=cosθ(cid:12)(cid:12)a(cid:11) sinθ(cid:12)(cid:12)b(cid:11). (14)
| (cid:105) | (cid:105)
baryon type component respectively, while θ is some ar- −
bitrary mixing angle. Now imagine performing a spec- In the absence of an operator that has substantial over-
troscopic calculation with an interpolating field χ that lap with 2 , it becomes impossible to separate out the
3
| (cid:105)
4
trueQCDeigenstatesofinterest. Thisnaturallyleadsto
two points of concern. Firstly, one cannot extract states
with a significant 2 component and secondly there is
| (cid:105)
possibly contamination of the states that are extracted.
When performing baryon spectroscopy it therefore be-
comes desirable to include interpolating fields that we
expect to have substantial overlap with multi-particle
meson-baryon type states [19]. While projecting single-
hadron momenta in a multi-hadron operator allows for
a clean extraction of states associated with scattering
thresholds, the influence of local five-quark operators
(without explicit momenta assigned to each hadron) on FIG. 2. The Feynman diagrams considered following the in-
the spectrum is less intuitive. It is the purpose of this troductionoffive-quarkinterpolatingfieldstostandardthree-
studyto examinethe role localfive-quark operators play quark operators.
in the spectrum, and to thereby test the robustness of
our variational method.
combined in a correlation matrix. This also ensures the
Starting with standard N and π interpolators we use
standard parity projector of Eq. (2) applies to our five-
the Clebsch-Gordan coefficients to project isospin I =
quark interpolators.
1/2,I = +1/2 and write down the general form of our
3
The presence of creation quark fields in our annihila-
meson-baryon interpolating fields [23, 24],
tion interpolating field and vice versa then leads to the
1 requirement of calculating the more computationally in-
χ (x)= (cid:15)abcγ
Nπ 5 tenselooppropagators,inordertocomputethediagrams
√6 ×
(cid:26) inFig.2. Theliteraturecontainsdifferentwaysofdealing
2(cid:2)uTa(x)Γ db(x)(cid:3)Γ dc(x)(cid:2)d¯e(x)γ ue(x)(cid:3) with such diagrams such as distillation [25], and various
1 2 5
schemes such as the Laplacian Heaviside (LapH) smear-
(cid:2)uTa(x)Γ db(x)(cid:3)Γ uc(x)(cid:2)d¯e(x)γ de(x)(cid:3) ing method [26]. Here we will stochastically estimate
1 2 5
−
(cid:27) inverse matrix elements fully diluting in spin, colour and
+(cid:2)uTa(x)Γ db(x)(cid:3)Γ uc(x)(cid:2)u¯(x)eγ ue(x)(cid:3) , time as outlined below.
1 2 5
(15)
IV. LOOP PROPAGATOR TECHNIQUES
providing us with two five-quark operators, denoted χ
5
and χ(cid:48) which correspond to (Γ ,Γ ) = (Cγ ,I) and
5 1 2 5
(Γ ,Γ ) = (C,γ ) respectively. The square brackets Asobservedintheprecedingsection,spectroscopiccal-
1 2 5
around the diquark contraction denote a Dirac scalar. culationsthatinvolvethefive-quarkoperatorsχ5 andχ(cid:48)5
Under a parity transformation necessarilyinvolvethedeterminationoflooppropagators
atx,denotedS(x,x). AsS(x,x)requiresasourceateach
x x˜=(x , (cid:126)x), (16) lattice point, a different recipe to that of conventional
0
→ −
point-to-all propagators is utilised. For this purpose we
and the quark fields ψ(x) and ψ¯(x) transform as use stochastic estimation of the matrix inverse [27, 28].
Givenasetofrandomnoisevectors η withelements
ψ(x) ψ(x) † =γ0ψ(x˜), drawn from Z such that the average {ove}r noise vectors
→P P 4
ψ¯(x) ψ¯(x) † =ψ¯(x˜)γ . (17) gives
0
→P P
Applying a parity transformation to the standard pion η (x)η† (y) =δ δ δ , (19)
interpolator χ (x)=ψ¯(x)γ ψ(x), and the nucleon inter- (cid:104) aα bβ (cid:105) xy ab αβ
π 5
polators of type χ (x) = (cid:2)ψT(x)(Cγ )ψ(x)(cid:3)ψ(x) of Eq. with colour indices a,b, spin indices α,β and space-time
N 5
(27) we find indicesx,y. Wedefineforeachnoisevectoracorrespond-
ing solution vector
χ (x) ψ¯(x˜)γ ψ(x˜)= χ (x˜),
π 5 π
χ (x)→(cid:2)−ψT(x˜)(Cγ )ψ(x˜−)(cid:3)γ ψ(x˜)=γ χ (x˜). (18) χ=M−1η, (20)
N 5 0 0 N
→
where in this case M is the fermion matrix. Then the
Thus the pion interpolator transforms negatively under
stochasticestimateofapropagatormatrixelementiscal-
parity. To ensure our five-quark baryon interpolator
culated as
formed from the product of pion and nucleon interpo-
lators, transforms in the appropriate manner, the pref- S (x,y) χ (x)η† (y) . (21)
actor of γ is included in Eq. (15). That is, both our ab;αβ (cid:39)(cid:104) aα bβ (cid:105)
5
three-quark and five-quark nucleon operators have the We perform full dilution in time, spin and colour indices
same parity transformation properties and hence can be as a means of variance reduction [29]. That is, given a
5
setoffullnoisevectors η ,wecandefineasetofdiluted 0
noise vectors η[a(cid:48)α(cid:48)t(cid:48)] {by}
5
{ } −
ηa[aα(cid:48)α(cid:48)t(cid:48)]((cid:126)x,t)=δaa(cid:48)δαα(cid:48)δtt(cid:48)ηaα((cid:126)x,t), (22) −10
15
−
where the intrinsic quark field indices are specified by
colour a, spin α, space (cid:126)x and time t respectively and Gπ−20
g
thecolour-spin-timedilutednoisevectorsareenumerated o 25
l−
by the corresponding [a(cid:48)α(cid:48)t(cid:48)] labels. We can similarly
30
enumerate the solution vectors −
χ[a(cid:48)α(cid:48)t(cid:48)] =M−1η[a(cid:48)α(cid:48)t(cid:48)], (23) −4305 nnss==3355,,ssttoacnhdaasrtdic nnss==120000,,ssttaocnhdaasrtdic
− ns=100,stochastic ns=200,standard
which makes it clear that by diluting we increase the 45
− 10 12 14 16 18 20
number of inversions required by a factor of n EuclideanTime
colour
×
nspin ×ntime. The stochastic estimate of the matrix in- −10
verse with dilution is given by
20
S(x,y) (cid:88) χ[a(cid:48)α(cid:48)t(cid:48)](x)η[a(cid:48)α(cid:48)t(cid:48)]†(y) , (24) −
(cid:39)(cid:104) (cid:105) 30
a(cid:48),α(cid:48),t(cid:48) −
where colour and spin indices are taken to be implicit GN−40
for clarity. At this point we remark that, while it is g
computationallyinfeasibletoalsofullydiluteinthespace lo−50
index (cid:126)x, in this extreme limit each diluted noise vector
60
wouldconsistofonlyasinglenon-zeroelement, meaning −
thatweareexactlycalculatingthefullmatrixS(x,y)and ns=35,stochastic ns=100,standard
the above relation becomes an equality rather than an −70 ns=35,standard ns=200,stochastic
ns=100,stochastic ns=200,standard
estimate. Thismakesitclearthatusingdilutionprovides
80
− 10 12 14 16 18 20
an improved stochastic estimate to the matrix inverse. EuclideanTime
AsshowninFig.2,ourconstructionofnucleoncorrela-
tors with five-quark operators combines standard point- FIG. 3. (Colour online). A comparison of correlators calcu-
to-allpropagatorsS(x,0)andstochasticestimatesofthe lated with one stochastically estimated propagator (denoted
loop propagators S(x,x). In order to access the radial “stochastic”)tothosecalculatedwithnostochasticpropaga-
excitations of the nucleon, we make use of multiple lev- tors(denoted“standard”). Resultsarepresentedforthepion
els of Gaussian smearing in our quark fields. Hence, to (top) and the ground state nucleon (bottom).
constructacorrelationmatrixweneedtocalculateprop-
agators with differing levels of source and sink smearing.
LetS(m,n)(x,y)denoteapropagatorwithmiterations Eqs.(21)and(25)thatanestimateofthesmearedprop-
of smearing applied at the sink and n iterations applied agator is given by
at the source. In the case of point-to-all propagators
S(m,n)(x,y)= χ(m)(x)η(n)†(y) , (26)
S(m,n)(x,0) the source point is fixed, y =0, and starting (cid:104) (cid:105)
with a point source ψ(0), we apply n iterations of Gaus- whereχ(m) =Hmχistheresultof miterationsofGaus-
siansmearingpre-inversiontoobtainthesmearedsource
sian smearing applied to the (diluted) solution vectors,
ψ(n) =Hnψ(0), where and η(n) = Hnη is similarly constructed from the (di-
luted) noise vectors. Note that the smearing is applied
α (cid:88)3 after (any dilution and) the solution vectors have been
Hψ(x)=(1 α)ψ(x)+ U (x)ψ(x+aµˆ)
− 6 { µ calculated. The construction of a smeared loop prop-
µ=1 agatorS(m,m)(x,x)issimplyanapplicationoftheabove
+U†(x aµˆ)ψ(x aµˆ)(cid:9), (25) formulae in the case y =x.
µ − −
Inordertodeterminehowmanynoisevectorspercon-
and α specifies the smearing fraction. Sink smearing figuration are sufficient to provide similar statistical er-
is applied to the propagator post-inversion to obtain rors for our point-to-all and stochastic propagators, cor-
S(m,n)(x,0). relators are calculated with a stochastic estimate of the
The application of smearing to construct a stochastic point-to-all propagator, and compared to those obtained
estimate for the quark propagator S(m,n)(x,y) is some- usingpoint-to-allpropagatorscalculatedinthestandard
what different. The set of (diluted) noise and solution way. Aseachindependentquarklineinahadroncorrela-
vectors η,χ isfirstconstructed,wherebyitfollowsfrom tor requires an independent noise source to ensure unbi-
{ }
6
ased estimation [26] we insert one stochastic propagator operators
into the aforementioned correlators. Furthermore, as a
testofoursmearingtechniqueforstochasticpropagators, χ1 =(cid:15)abc[uaT (Cγ5)db]uc
we perform this comparison using a variety of smearing χ =(cid:15)abc[uaT (C)db]γ uc (27)
2 5
levels. Note that, as smearing of both the source and so-
lution vectors is performed post-inversion, the stochastic in order to form the seven bases we study that are out-
method effectively provides different smearing levels for lined in Table I.
free.
The comparison is performed on 75 203 40 lattice
× TABLE I. Table of the various operators used in each basis.
gauge configurations, with the FLIC fermion action [30].
The lattice spacing is 0.126 fm in both the temporal and Basis Number Operators Used
spatial direction providing a physical lattice volume of
(2.52 fm)3. Four full noise vectors are used per stochas- 1 χ1, χ2
tic propagator, which are then colour-spin-time diluted. 2 χ1, χ2, χ5
Asthesourcetimesliceisfixedinthiscase,eachstochas- 3 χ1, χ2, χ(cid:48)5
ticpropagatorrequiresncolour nspin inversionspernoise 4 χ1, χ2, χ5, χ(cid:48)5
vector. Recall standard point×-to-all propagators require 5 χ1, χ5, χ(cid:48)5
ncolour × nspin inversions, although source smearing is 6 χ2, χ5, χ(cid:48)5
applied pre-inversion unlike the stochastic case. Three 7 χ5, χ(cid:48)5
different levels of smearing are used, n = 35,100,200
s
sweeps with α = 0.7. Fig. 3 shows good agreement
across all smearing levels between those correlators con- Throughout this work we employ Gauge-invariant
taining a stochastically estimated propagator and those Gaussiansmearing[36]atthesourceandsinktoincrease
that do not, demonstrating that using four noise vectors the basis size via altering the overlap of our operators
per quark line provides a comparable statistical uncer- with the states of interest. We choose n = 35 and
s
tainty to that of a standard propagator. We note here n = 200 sweeps of smearing providing bases of sizes 4,
s
thatultimatelyweutilisethismethodtocalculateS(x,x) 6 and 8. Stochastic quark lines are calculated using four
not S(x,xsource), meaning we get the added benefit of random Z4 noise vectors that are fully diluted in colour,
spatial averaging for our loop propagators. spin and time.
VI. RESULTS
V. SIMULATION DETAILS
A. Positive-parity Spectrum
For the baryon spectroscopy results presented herein
weusethePACS-CS2+1flavourdynamical-fermioncon- The results for the nucleon spectrum in the positive-
figurations [31] made available through the ILDG [32]. parity sector are shown in Fig. 4. Solid horizontal lines
These configurations use the non-perturbatively (a)- are added to guide the eye, with their values set by the
O
improved Wilson fermion action and the Iwasaki gauge states in basis number 4, since this basis contains all the
action [33]. The lattice size is 323 64 with a lat- operators studied and has the largest span.
×
tice spacing of 0.0907 fm providing a physical volume Of particular interest is the robustness of the varia-
of (2.90 fm)3. β = 1.90, the light quark mass is set tional techniques employed. While changing bases may
≈
by the hopping parameter κud = 0.13770 which gives a effectwhetherornotaparticularstateisseen,theenergy
pion mass of mπ = 293 MeV, while the strange quark of the extracted states is consistent across the different
mass is set by κs =0.13640. Fixed boundary conditions bases, even though they contain qualitatively different
are employed in the time direction removing backward operators.
propagating states [34, 35], and the source is inserted at Despitetheuseof5-quarkoperators,nostatenearthe
tsrc =nt/4=16, well away from the boundary. System- non-interacting P-wave Nπ scattering threshold is ob-
atic effects associated with this boundary condition are served. This is understood by noting that none of our
negligible for t>16 slices from the boundary. The main operators have a source of the back-to-back relative mo-
resultsofourvariationalanalysisisperformedatt0 =17 mentum between the nucleon and pion necessary to ob-
anddt=3,providingagoodbalancebetweensystematic serveanenergylevelintheregionofthisscatteringstate.
and statistical uncertainties. Uncertainties are obtained The corresponding eigenvector components for the
via single elimination jackknife while a full covariance positive-parity states are shown in Fig. 5 as a function
matrix analysis provides the χ2/dof which is utilised to ofbasisandvariationalparameterdt,witht =17fixed.
0
select fit regions for the eigenstate-projected correlators. Thevaluesofdtrangefrom1through4. Theupperlimit
In addition to the five quark operators χ and χ(cid:48) pre- of dt = 4 was chosen as the largest value for which the
5 5
sentedinSectionIIIweusetheconventionalthree-quark variationalanalysisconvergedforeachofthesevenbases.
7
ns =35+200 State 1
45 P-waveN+π 1234567→→→→→→→χχχχχχχ1111125+++++++χχχχχχχ22225505+++++χχχχχ50550505+χ05 onent 01..50 uuuχ3χ2χ35051120
p
m uχ2
V) Co 0.0 200
M(Ge3 vector uuχ3χ555
n 200
e 0.5
2 Eig− uχ3550
1.0 uχ20500
1 −
1 2 3 4 5 6 7
BasisNumber/VariationalParameters
1 2 3 4 5 6 7
BasisNumber State 2
1.0 uχ1
FIG. 4. (Colour online). The positive-parity nucleon spec- 35
trum with various operator bases constructed with 35 and uχ1
200 sweeps of smearing. Horizontal solid lines are present to nt 200
guide the eye and are drawn from the central value of the ne 0.5 uχ2
o 35
states in basis 4, while the dashed line marks the position of mp uχ2
the non-interacting P-wave Nπ scattering threshold o 200
C
0.0
. tor uχ355
c
ve uχ5
n 200
e 0.5
regTahrdelegsrsouonfdth-setaatbesennucceleoornpirsesoebnsceervoefdaipnaretviecruylarbaospis- Eig− uχ3550
erator. If χ1 is present then this provides the dominant 1.0 uχ20500
contribution, with χ(cid:48) coupling strongly to the ground −
5
state in bases where χ1 is absent. An interesting inter- 1 2 3 4 5 6 7
play between 35 and 200 sweep smeared χ is observed BasisNumber/VariationalParameters
1
with the smaller source diminishing in importance as dt
State 3
is increased. This may be associated with the Euclidean
time evolution of highly excited states which are sup- 1.0 uχ1
35
pressed with increasing dt.
uχ1
Turning our attention to state 2, we see that χ1 plays nt 200
a critical role in the extraction of the first excited state, ne 0.5 uχ2
o 35
whichisassociatedwitharadialexcitationoftheground p
m uχ2
state [17]. Here the 35 and 200 sweep χ interpolators o 200
1 C
0.0
ethnetenrowdiethstsrimucitlaurrestorfenagtrhadbiaulteoxpcpitoastitioens.igχns1,dseotmtiinngatueps vector uuχ3χ555
the construction of the optimised operator for this state n 200
e 0.5
for bases 1 through 5, whereas basis 6 and 7 which lack Eig− uχ3550
χ do not observe this state.
1The eigenvectors for state 3, the second excited state, 1.0 uχ20500
are dominated by χ components with the same sign −
2
when this operator is present (bases 1-4,6). This state 1 2 3 4 5 6 7
BasisNumber/VariationalParameters
is not observed in basis 5 (where χ is absent). Inter-
2
estingly, in basis 7 which only contains five-quark opera-
tors it appears that it is possible to form this state using FIG.5. (Colouronline). Eigenvectorcomponentscorrespond-
χ(cid:48) components at two different smearings with opposite ing to the low-lying positive-parity nucleon states. State 1
5
corresponds to the ground state, with states 2 and 3 corre-
sign.
sponding to the first and second excited states respectively.
We observe that the overall structure of the eigenvec-
Thecolumnnumbersdenotebasisnumberwhiletheminorx
torsforeachofthethreestatesishighlyconsistentacross
axistickscorrespondtothevaluesofthevariationalparame-
different bases and different values of the variational pa- terdtwhichrunsfrom1throughto4. t0 =17hasbeenused
rameterdt.Thestructureoftheeigenvectorscanbecon- throughout. The subscripts 35 and200 in thelegend referto
the number of smearing sweeps applied.
8
sidered to be a signature or fingerprint of the extracted State 1
2.0
state, and this consistency across bases confirms that it
Massfromprojectedcorrelatorfit
is the same state being identified.
It is fascinating to see that for state 1 in bases 6 and 1.8 Eigenmass
7, where χ(cid:48) takes the role of the absent χ operator, Extractedmassused
5 1
the values of the two dominant eigenvector components 1.6
(whichindicatethemixtureofthetwodifferentsmearing
)
levels used) are extremely similar to the χ components V
1 Ge1.4
in bases 1-5. Interestingly, at dt = 2 the error bars for (
M
the dominant components of states 2 and 3 blow up. As
we shall explain below, this is due to an accidental de- 1.2
generacy in the eigenmasses for this choice of variational
parameters. 1.0
In order to further test the robustness of our varia-
tional method we conduct a comparison of the masses
0.8
obtainedfromfittingtheeigenstate-projectedcorrelators 1 2 3 4 5 6 7
BasisNumber/VariationalParameters
as a function of the variational parameters for each ba-
sis. TheseresultsarepresentedinFig.6. Alsoshownfor State 2
3.2
comparison are the eigenmasses, m , that result from
α Massfromprojectedcorrelatorfit
solving the generalised eigenvalue equation of Eqs. (7)
or (8) with cα =e−mαdt. 3.0 Eigenmass
Studyingstate1,thenucleongroundstate,weobserve Extractedmassused
2.8
that the masses obtained from projected correlator fits
are approximately invariant across different bases and
)2.6
V
choices of the variational parameter. In contrast, the
e
G
eigenmass lies well above the fitted mass, dropping in M(2.4
value as the variational parameter dt is varied from 1 to
4. While the eigenmass is directly related to the princi-
2.2
palcorrelatorandthusshouldapproachthegroundstate
mass in the large time limit, it is clear that the values
2.0
of dt we examine here are insufficient for this to occur.
It is worth noting that, in bases 6 and 7 where χ is ab-
1 1.8
sent we see that the eigenmass value rises significantly. 1 2 3 4 5 6 7
BasisNumber/VariationalParameters
Nevertheless, the fitted mass remains remarkably consis-
tentwiththevaluesobtainedinbases1-5. Weemphasize State 3
3.2
how strong the variational parameter dependence of the
Massfromprojectedcorrelatorfit
eigenmass contrasts the more consistent structure of the
eigenvectors. Insensitivityoftheeigenvectorstothevari- 3.0 Eigenmass
ational parameters is a key component of the invariance Extractedmassused
2.8
of the masses obtained from the projected correlator.
Turning to state 2, we see that the eigenmass shows
)2.6
similar behaviour to state 1, lying above the extracted V
e
mass and dropping with dt. Interestingly, for state 3 in G
M(2.4
bases 1-4 and 6 the eigenmass shows constant behaviour
fordt=2 4butsystematicallyliesbelowtheextracted
− 2.2
mass. Inbasis7,thestate3eigenmassisverydifferentto
the previous bases, lying above the extracted mass and
2.0
showing a similar downward trend to states 1 and 2 as
dt varies.
1.8
As for state 1, the fitted masses for states 2 and 3 1 2 3 4 5 6 7
BasisNumber/VariationalParameters
provide highly consistent values and uncertainties across
the different bases and values of dt, with the notable ex-
ception of dt = 2. As observed previously in Fig. 5, we FIG. 6. (Colour online). Comparisons of eigenmasses to
see in Fig. 6 considerably larger error bars at the varia- masses obtained from a projected correlator fit for low-lying
tional parameter set (t ,dt) = (17,2) in both the eigen- states in the positive-parity nucleon channel. The column
0
numbersdenotebasisnumberwhiletheminorxaxistickscor-
vector components and projected mass fits for the first
respondtothevaluesofthevariationalparameterdt=1...4.
and second excited states. To understand this, we turn
to Fig. 7, where the eigenmasses for states 2 and 3 are t0 =17 has been used throughout. The line denoting the ex-
tracted mass is set using basis 4 with dt=3.
9
Eigenmass comparison olds discussed in this section and the next, refer to the
3.2
non-interacting threshold. In contrast to the positive-
State2Eigenmass
parityresults, wedoobserveastate nearthe S-wave Nπ
3.0 State3Eigenmass
scatteringthresholdinthenegative-paritychannel(bases
5,6,7), alsonotingthattheP-waveNππ thresholdsliein
2.8
theregionofstate3seeninbases3,4and5. Itisimpor-
tanttonotethatevenaftertheintroductionofoperators
)2.6
V thatpermitaccesstoastatenearthelow-lyingscattering
e
G
state, the energies of the higher states in the spectrum
M(2.4
are consistent, demonstrating the robustness of the vari-
ational techniques employed.
2.2
n =35+200
s
2.0
1.8 1 BasisNumb2er/Variational3Parameters 4 56 S-waveN+π 1234567→→→→→→→χχχχχχχ1111125+++++++χχχχχχχ22225505+++++χχχχχ50550505+χ05
FIG. 7. (Colour online). A plot showing the eigenmasses for
bothstates 2 and 3, illustrating the accidental degeneracy at V)4
dt=2. Ge
(
M
3
plottedagainstthevariationalparameterdtineachbasis.
Note that at (t ,dt)=(17,2) there is an approximate
0
2
degeneracyintheeigenmassforstates2and3. Asacon-
sequence,thecorrespondingeigenvectorscanthereforebe
arbitrarily rotated within the state 2/state 3 subspace
1
while remaining a solution to the eigenvalue problem. 1 2 3 4 5 6 7
Whenconstructingthejackknifesub-ensemblestocalcu- BasisNumber
latetheerrorinthefittedenergy,weneedtosolveforthe
eigenvectors on each sub-ensemble. Due to the approx- FIG. 8. (Colour online). The negative-parity nucleon spec-
imate degeneracy, the particular linear combination of trum with various operator bases using 35 and 200 sweeps of
state 2 and state 3 that we obtain for each sub-ensemble smearing. Solid horizontal lines are present to guide the eye
can vary. Indeed, we observe that the dot-product be- andaredrawnfromthecentralvalueofthestatesinbasis4,
tween the ensemble average and sub-ensemble can drop since this basis is the largest. The dashed line marks the po-
significantly for dt = 2 in comparison to other values sitionofthenon-interactingS-waveNπ scatteringthreshold.
of dt. This causes a large variation in the sub-ensemble The variational parameters used herein are (t0,dt)=(17,3).
eigenvector components and a correspondingly large er-
Plots of the corresponding eigenvectors for the low-
ror bar. The simplest way to avoid the problem of this
lying negative-parity states as a function of basis and
accidental degeneracy is to select a different value of the
variational parameter dt = 1...3 are presented in Fig-
variational parameter.
ures 9 and 10. The upper limit of dt=3 was chosen as
the largest value for which the variational analysis con-
verged for all seven bases. The eigenvector components
B. Negative-parity Spectrum
for state 0 (when it is present) are dominated by the
multi-particle operators χ and χ(cid:48), suggesting that this
5 5
The negative-parity nucleon spectrum is presented in state should be identified as a scattering state. The ex-
Fig. 8. Solid horizontal lines have been added to guide tracted energy for this state is in the region of the non-
the eye, with their values set by the states in the largest interacting S-wave Nπ scattering threshold (which lies
basis (number 4). Once again, while changing bases ef- below the first negative-parity resonant state). The un-
fects whether or not we observe a given state, the ex- certainty in bases 6 and 7 are relatively large compared
tracted states display an impressive level of consistency tobasis5,indicatingthatthepresenceofχ mayalsobe
1
across the different bases. required to cleanly isolate this scattering state. Indeed,
The dashed line indicating the energy of the non- we note that in basis 5 there is a significant contribution
interacting(infinite-volume)scattering-statethresholdis to state 0 from the χ (n =200) operator.
1 s
also indicated with the caution that mixing with nearby It is also important to note that either χ or χ(cid:48) can
5 5
states in the finite volume can alter the threshold posi- be the dominant interpolator exciting this lowest-lying
tion [37, 38]. We note here that all scattering thresh- state. Giventhatχ ispredominantlyassociatedwiththe
2
10
State 0 State 1
1.0 uχ1 1.0 uχ1
35 35
uχ1 uχ1
t 200 t 200
n n
ne 0.5 uχ2 ne 0.5 uχ2
o 35 o 35
p p
m uχ2 m uχ2
o 200 o 200
C C
0.0 0.0
tor uχ355 tor uχ355
c c
ve uχ5 ve uχ5
n 200 n 200
e 0.5 e 0.5
Eig− uχ3550 Eig− uχ3550
1.0 uχ20500 1.0 uχ20500
− −
1 2 3 4 5 6 7 1 2 3 4 5 6 7
BasisNumber/VariationalParameters BasisNumber/VariationalParameters
State 2
FIG.9. (Colouronline). Eigenvectorcomponentscorrespond-
ing to State 0 which is in the region of the non-interacting 1.0 uχ1
35
S-waveN+π scatteringthreshold. Thecolumnnumbersde-
notebasisnumberwhiletheminorxaxistickscorrespondto t uχ2010
n
the values of the variational parameter dt which runs from ne 0.5 uχ2
1 through to 3. t0 = 17 has been used throughout. The po 35
subscripts 35 and 200 in the legend refer to the number of m uχ2
o 200
smearing sweeps applied. C 0.0
tor uχ355
c
ve uχ5
n 200
tmhiigrhdtsntaaitveeliynetxhpeepctosχit(cid:48)ivweo-upladribtyeasesscotocirataetd2w.4itGheSV-woanvee Eige−0.5 uχ3550
5
scattering states near 2.7 GeV. Remarkably it creates 1.0 uχ20500
a scattering state near 1.35 GeV. Thus one should use −
caution in predicting the spectral overlap of five-quark
1 2 3 4 5 6 7
operators by examining the spectral overlap of the pion BasisNumber/VariationalParameters
and nucleon components of the five-quark operators sep-
State 3
arately. In light of the quark field operator contractions
required in calculating the full two-point function this 1.0 uχ1
35
result is not surprising.
uχ1
In accord with previous studies [15, 39], we find that t 200
n
the χ1 interpolating field is crucial for extracting state one 0.5 uχ352
1, associated with the lowest-lying negative-parity reso- p
m uχ2
nance, as we do not observe this state when χ1 is absent Co 200
0.0
ainsainntbcaosnetsri6buatniodn7.toAsstaetxepe1c,tewdh,iχch1 pisroavsisdoecsiattheeddwoimth- ctor uχ355
ve uχ5
the S11(1535) in Nature. Similarly, we see that χ2 has a en 0.5 200
high overlap with state 2, the next resonant state. Basis Eig− uχ3550
5doesnotseestate2duetotheabsenceofχ . However,
unlikestate1,thereisanimportantmixingo2fχ1 andχ2 1.0 uχ20500
in isolating the eigenstate. It is interesting to note that −
in basis 7 we are able to form this state by combining χ 1 2 3 4 5 6 7
5
and χ(cid:48). BasisNumber/VariationalParameters
5
The consistency of the eigenvector structure for the
low-lying states 1 and 2 is strong. Despite the appear- FIG. 10. (Colour online). Eigenvector components corre-
ance of a state near the S-wave Nπ threshold, state 0 in sponding to low-lying negative-parity nucleon states. States
1 and 2 correspond to the two lowest-lying resonant states,
basis 5, the eigenvector components for state 1 are re-
whilestate3interestinglyliesintheregionoftheP-wavescat-
markably consistent with those in other bases where this
tering thresholds. The column numbers denote basis num-
lower-lying state is absent. If we look at basis 6, where
ber while the minor x axis ticks correspond to the values of
state 0 is present but state 1 is absent, the eigenvec- thevariationalparameterdtwhichrunsfrom1throughto3.
tor components for state 2 are in good agreement with t0 =17hasbeenusedthroughout. Thesubscripts35and200
inthelegendrefertothenumberofsmearingsweepsapplied.