Low-lying Lambda Baryons from the Lattice

0.17 MB·English

by  Georg P. Engel

Low-lying Λ Baryons from the Lattice
Georg P. Engel1, C. B. Lang1, and Andreas Scha¨fer2
(BGR [Bern-Graz-Regensburg] Collaboration)
1Institut fu¨r Physik, FB Theoretische Physik, Universita¨t Graz, A–8010 Graz, Austria
2Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D–93040 Regensburg, Germany
(Dated: January 15, 2013)
InalatticeQCDcalculationwithtwolightdynamicalChirallyImproved(CI)quarkswedetermine
groundstateandsomeexcitedstatemassesinallfourΛbaryonchannels1/2±and3/2±. Weperform
an infinite volume extrapolation and confirm the widely discussed Λ(1405). We also analyze the
amount of octet-singlet mixing, which is helpful in comparing states with thequark model.
PACSnumbers: 11.15.Ha,12.38.Gc
One of the central aims of hadron spectroscopy is to oursimulation,aswellasresultsformesons,canbefound
understand the spin-flavor-parity structure of the exci- in [7, 8]. The strange quark is introduced as a valence
3
tation spectra for different quantum numbers. It seems quark and its mass fixed by the Ω-mass.
1
0 inparticularthatthe nucleonandΛspectra showsignif- The Dirac and flavor structure of the interpolating
2 icant differences which, if properly understood, should fields used in our study is motivated by the quarkmodel
n illuminate the influence of quark mass and flavor on [9, 10], see also [11]. Within the relativistic quark model
a hadron structure. The lowest Λ(1−) mass lies below the therehavebeenmanydeterminationsofthehadronspec-
2
4 J RNo∗p(1er5-3li5k)e;uΛn(l1ik6e00th,e12+nu)calenodntsheectnoergitatdioveesphaarviteystnauncdleaordn ttriounms,obnatsheedhoynpceornfifinneiningteproatcetniotinal(sseaen,de.dgi.ff,e[1re2n–t14a]s)s.uTmhpe-
1
singlet, octet and decuplet attribution [11] of the states
level ordering lying between the positive parity ground

has been evaluated based on such model calculations,
] state and the first positive parity excitation. Λ baryons
t e.g., in [15] (see also the summary in [16]).
a canbeflavorsingletsoroctetsor,duetothedifferencein
l light and strange quark masses, mixtures of both. Vari- FortheΛbaryonsweusesetsofupto24interpolating
-
p ouscontinuummodelstudiesdiscussthatmixing. Thisis fields in each quantum channel. The singlet and octet
e thefirstlatticeQCDanalysisoftheΛbaryonsfordynam- combinations of Table I are combined with three possi-
2 [h iacnadl q23u±a-rckhsa,nthnaelts.incWluedeosbtaallinfogurro,unnadmsetlaytethsecoJmPp=ati12b±le- afbonlredcJh(io=1ic,eC1sγaotnfγd5D)ieri(agdchetnmocatotemrdibceibnsya(tΓχio11n,,Γsχ2o2)f=aGn(ad1u,sχCsi3aγn5fo)s,rm(sγeh5ao,rCretd))
v with experiment in three of those. We also study the quarks[127,18]withtwosmearingwidths(n,w). Theop-
2 infinite volume limit and give for the first time lattice
erator numbering is given in Table II. All interpolators
3 results on singlet-octet composition for all four sectors,
are projected to definite parity and all Rarita-Schwinger
0
2 obtainingthemassofthe Λ(1405)andconfirmingitsfla- fields (spin 3 interpolators in Table II) are projected to
. vor singlet nature. definite spin2 3 using the continuum formulation of the
2 2
1 Lattice studies in the quenched case generally had Rarita-Schwinger projector [19]. C denotes the charge
2 problems to find the low lying Λ(1405). Even a study conjugation matrix, γt and γi the time and the spatial
1 with two dynamical light quarks [1, 2] found too large direction Dirac matrices.
v: massvalues. Onlyrecentresults[3]obtainedwithPACS- For point like quark fields, Fierz identities reduce the
i CS (2 + 1)-flavor dynamical quarks configurations [4] actual number of independent operators (see, e.g., [20]).
X
show a level ordering which is compatible with experi- In particular, there are no non-vanishing point-like in-
ar ment. terpolators for ∆(12) and singlet Λ(32). We use different
Westudythebaryonsforasetofsevenensembleswith quark smearing widths such that the Fierz identities do
two dynamical Chiral Improved (CI) quarks [5, 6]. The not apply. Hence χ1, χ2 and χ3 are all independent, for
CI fermion action consists of several hundred terms and J = 3 all interpolators are non-vanishing.
2
obeys the Ginsparg-Wilson relation in a truncated ap- From the cross-correlation matrix Cik(t) =
proximation. The pion mass ranges from 255 to 596 hOi(t)Ok(0)†i we obtain the energy levels with help of
MeV, the lattice spacing lies between 0.1324 and 0.1398 the variational method [21, 22]. One solves the gener-
fm. The bulk of our results were obtained for lattices of alized eigenvalue problem C(t)~un(t) = λn(t)C(t0)~un(t)
size 163×32. For two ensembles with light pion masses in order to approximately recoverthe energy eigenstates
(255 and 330 MeV) we also used 24×48 lattices. Thus |ni. The eigenvalues λn(t) ∼ exp(−Ent) allow us to
mπLwhichcontrolsthefinitesizeeffectsis4.08and5.61 get the energy values and the eigenvectors serve as
inthesetwosituations. Furtherdetailsontheactionand fingerprints of the states, indicating their content in
2
Spin Flavor Name Interpolator Quark Numberingof associated interpolators
1 Singlet Λ(1,i) ǫ Γ(i)u (dTΓ(i)s −sTΓ(i)d ) smearing types Λ(1,i) Λ(8,i) Λ(1,i) Λ(8,i)
2 1/2 abc 1 a b 2 c b 2 c 3/2 3/2 1/2 1/2
+cyclic permutations ofu,d,s (nn)n 1 9 1,9,17 25,33,41
1 Octet Λ(8,i) ǫ Γ(i)s (uTΓ(i)d −dTΓ(i)u ) (nn)w 2 10 2,10,18 26,34,42
2 1/2 abch 1 a b 2 c b 2 c
+Γ(i)u (sTΓ(i)d )−Γ(i)d (sTΓ(i)u ) (nw)n 3 11 3,11,19 27,35,43
1 a b 2 c 1 a b 2 c i
(nw)w 4 12 4,12,20 28,36,44
3 Singlet Λ(1,i) ǫ γ u (dTCγ γ s −sTCγ γ d )
2 3/2 abc 5 a b 5 i c b 5 i c (wn)n 5 13 5,13,21 29,37,45
+cyclic permutations ofu,d,s
(wn)w 6 14 6,14,22 30,38,46
3 Octet Λ(8,i) ǫ γ s (uTCγ γ d −dTCγ γ u )
2 3/2 abch 5 a b 5 i c b 5 i c (ww)n 7 15 7,15,23 31,39,47
+γ u (sTCγ γ d )−γ d (sTCγ γ u (ww)w 8 16 8,16,24 32,40,48
5 a b 5 i c 5 a b 5 i ci
TABLEI. Baryoninterpolators: Thepossiblechoicesforthe TABLEII. Baryoninterpolators: Quarksmearingtypes(n/w
DiracmatricesΓ(i) inthespin 1channelsarediscussedinthe fornarrow/wide)andnamingconventionfortheinterpolators
1,2 2 in the different channels. The three columns for the J = 1
text. Summation convention applies; for spin 3 observables, 2
2 interpolators refer to χ –χ .
theopenLorentzindex(afterspinprojection)issummedafter 1 3
taking theexpectation valueof correlation functions.
(1,2,11,20,25,26,33,34,43). After extrapolation to the
terms of the lattice interpolators. The quality of the physical point our lowest energy level agrees well with
results depends on the statistics and the provided set of the experimental Λ(1116). The systematic error esti-
lattice operators. The dependence on t0 is used to study mated from different combinations of interpolators and
the systematic error; in the final analysis we use t0 = 1 fit ranges is indicated in the summary Fig. 6. Analyzing
(withtheoriginat0). Thestatisticalerrorisdetermined the eigenvectors, we find that the ground state is domi-
with single-elimination jack-knife. For the fits we use natedbyoctetinterpolatorsofDiracstructureχ1 andχ3
single exponential behavior but check the stability with (inagreementwitharelativisticquarkmodelcalculation
double exponential fits; we take the correlation matrix [15]). Our first excitation is dominated by singlet in-
for the correlated fits from the complete sample [8]. terpolators (first Dirac structure) matching the Λ(1810)
For the extrapolation to the physical pion mass we fit (singlet in the quark model). The Roper-like Λ(1600)
to the leading order chiral behavior, which is linear in (octet in the quark model) seems to be missing. This re-
m2π. Two ensembles (at pion masses 255 MeV and 588 semblesthe situationinthe N(21+)channel[7]. The sec-
MeV)sufferfromaslightmistuningofthe strangequark ond and third excitations are again dominated by octet
mass, which are therefore discarded in the extrapolation interpolators.
tothephysicalpionmass,whenevertheeffectsarefound Infinite volume extrapolation: We performed a vol-
significant. Thisisthecaseforthelowestenergylevelsin ume analysisfor severalsets of interpolatorsandvarious
each channel (three lowest ones in 1−). The quoted sys- fit ranges. The results in finite volume and the infinite
2
tematic errors for these levels include the corresponding volume extrapolations for the ground state for specific
deviationandthe dependence ofthe energylevelsonthe interpolators are shownin Fig. 2. The extrapolation fol-
choice of interpolatorsand fit ranges for the eigenvalues. lows the method of [26]. A stable choice is the set of
Inthepresentstudy wearerestrictedto3-quarkoper- interpolators A and tmin = 5a. The corresponding sys-
ators for the baryon. Note that ideally one should take tematic error estimated from different interpolators and
into account also meson-baryon interpolators (see, e.g., fit ranges is indicated in the summary Fig. 6. Our final
the discussion in [23]). This leads to many more con- resultis1126(17)(11)MeV(statisticalandsystematicer-
tributing graphs and necessitates also the inclusion of ror), which agrees nicely with the experimental Λ(1116)
backtracking quark loops. The related computational mass.
andalgorithmiceffortpreventedsuchlatticecalculations JP = 1−, Finite volume: We use different sets of in-
2
so far, although such studies are in progress [24]. Due terpolators and fit ranges. We stress that our basis is
to sea quarks, in principle, 3-quark operators have over- large compared to that of other studies with three types
lapwithmeson-baryonstatesaswell. Thecorresponding of Dirac structures for both singlet and octet interpola-
coupling was however found to be weak in actual simu- tors. Wecanextractthelowestfourenergylevels,shown
lations [7, 25]. We will argue below that in particular in in Fig. 3, using interpolators (2,3,10,18,26,27,34,42). We
the s-wave channels we find hints of such coupling even find that the ground state energy level agrees well with
for our interpolators. the experimental Λ(1405). The dependence of the lev-
JP = 1+, Finite volume: In Fig. 1 we show re- els on the tuning of the strange quark mass appears to
2
sults for the four lowest eigenstates for interpolator set be sizeable, albeit it an accident of our simulation. This
3
3 glet interpolators of all three Dirac structures. There is,
however, a considerable mixing with octet interpolators
2.5
(secondDiracstructure)of15-20%inensembleA66,i.e.,
V] 2 at mπ ≈ 255 MeV (see Fig. 4). This mixing is expected
Ge to increase towards the physical point, which may com-
s [1.5 plicate the functional dependence of the energy levelson
mas 1 the pion mass. The first and second excitation are both
dominatedbyoctetinterpolators,bythesecondandfirst
0.5 Λ 1/2+ (L~2.2 fm) Dirac structure, respectively.
0 Thefirstandsecondexcitedenergylevelarebothabit
0 0.1 0.2 0.3 0.4
low but compatible with the experimental Λ(1670) and
2 2
mπ [GeV ] Λ(1800). IngeneralintheJP = 1− channelonemayex-
2
pectcouplingtoπΣandKN ins-wave. In[27,28]theex-
FIG. 1. Energy levels for Λ spin 1+ in a finite box of linear
size L ≈ 2.2 fm. Stars denote the2experimental values [16], pectedenergylevelsinfinitevolumesarediscussedbased
on a continuum hadron exchange model. There (with
other symbols denote results from the simulation. The full
lines show the extrapolations linear in the m2, the broken physical hadron masses), the low-lying scattering state
curvesindicate the statistical error bands. π levels in the 12− channelare well separatedfor mπL.3.
For the pion masses of our study, the non-interacting
1.3 meson-baryon thresholds lie close but (except for one
point) above the lowest energy level observed. E.g., for
1.2
the lowest pion mass, the values are mΣ +mπ ≈ 1.52
]1.1 GeV, mN + mK ≈ 1.62 GeV, above the lowest level.
GeV 1 Λ 1/2+: A66 lTiehriswroerskemabrgleusetdhethsaittutahteioncoiunptlhinegNo(f21s−in)gclehabnanreylo.nEsatro-
[
ss baryon-meson channels may be too weak to lead to ob-
a1.3
m servable effects [7, 25]. However, in our case, in s-wave
1.2 scattering,wecannotexcludethatoneoreventwoofthe
Λ 1/2+: C77 observedthreelowestenergylevelscorrespondtoscatter-
1.1 ing states. Note that the measured ground state energy
A,4 A,5 A,6 A,7 B,4 B,5 B,6 B,7
levelisalways(exceptforonepoint)belows-wavethresh-
(set of interpolators, start of fit range)
old, thus supporting the Λ(1405) identification.
FIG.2. SystematicerroroftheΛ 1+groundstateenergylev- It has been conjectured from Chiral Unitary Theory
2
els. Thelevelsareshownfordifferentchoicesofinterpolators
that the lowest state may have a double-pole [23, 29]
andfitranges,labelled onthehorizontalaxis. E.g.,“A4”de-
andaidentificationstrategyforlatticesimulationsissug-
notesthesetofinterpolator“A”andafitrangefortheeigen-
gested in [30]. This would require asymmetric boxes or
values from t = 4a to the onset of noise. “A” denotes inter-
polators (2,3,10,18,26,27,34,42), “B” denotes (3,11,18,27,34). moving frames.
Foreachsetofinterpolatorsandfitrange,resultsforsmallto
Infinite volume extrapolation: We study the volume
largelattices(spatialsize16,24forensembleA66,and12,16,
dependence ofthe three loweststates fordifferentsets of
24 for C77, for notation see [8]) are shown from left to right.
The corresponding infinite volume limits are the rightmost interpolators and various fit ranges. We emphasize that
points. the stability of the signals of the excitations is compa-
rable to the ones of the ground state. The volume de-
pendence of all three low states appears fairly flat in our
is one reason of the large systematic error depicted in simulation, in a few cases showing even negative finite
Fig. 6. volume corrections. These features are compatible with
The chosen set of interpolators is particularly suitable significant contributions of an attractive s-wave scatter-
for an analysis of the content of the states. The spatial ing state. For interpolators (2,3,10,18,26,27,34,42) and
support of the quark fields is equivalent in all interpola- tmin = 5a, after infinite volume extrapolation, we show
tors and hence does not require additional renormaliza- the result of the final extrapolation of the ground state
tion when comparing the contribution of different inter- energy level to the physical pion mass in Fig. 6. The fi-
polators to the eigenstate. In addition, we use several nal result for the ground state agrees very well with the
interpolators for each given combination of flavor and experimental Λ(1405). Both the first and the second ex-
Dirac structures, which allows us to identify a possibly citation appear to be a bit low but are compatible with
higher number of states with similar structure. Within the experimental Λ(1670) and Λ(1800) (see Figs. 3 and
the basis used, the ground state is dominated by sin- 6)andmightalsopossiblybes-waveπΣandKN signals.
4
3.5 1
02 [Singlet]
s
3 nt 09 [Octet]
ne 0 10 [Octet]
2.5 o 16 [Octet]
] p
GeV 2 om Λ (3/2+): 1st excit. (A66)
c-1
ass [1.5 ctor 1
m e
1 v
n 0
e
g
0.5 Λ 1/2- (L~2.2 fm) Ei +
Λ (3/2 ): ground state (A66)
0 -1
0 0.1 0.2 0.3 0.4 2 3 4 5 6 7
m 2 [GeV2] t
π
FIG. 5. Eigenvectors for Λ spin 3/2+ ground state and
FIG. 3. Energy levels for Λ spin 1− in a finite box of linear first excitations for ensemble A66 (m ≈255 MeV). We em-
size L ≈ 2.2 fm, for legend see c2aption of Fig. 1. Fits are phasize the domination of singlet intπerpolators for the first
omitted for clarity. excitation. Suchinterpolatorsarenon-vanishingonlyforbro-
ken Fierz identities, which is realized by the use of different
quark smearing widths.
1
02 [Singlet χ ]
1
ts 03 [Singlet χ ]
n 1
ne 0 10 [Singlet χ2] in this channel, even though those interpolators are van-
po 18 [Singlet χ3] ishing exactly for symmetric point-like quark fields.
r com-11 Λ (1/2−): 1st excit. (A66) 223674 [[[OOOcccttteeettt χχχ11]]] noItnofibnsietervveolaumcleeaerxtvroalpuomlaetiodne:pendWenicthe.inTerhreorfisnwalerdeo-
o 2 sultagreeswiththeexperimentalΛ(1890)mass,butwith
ct 42 [Octet χ ]
e 3 large uncertainty.
v
gen 0 JP = 23−, Finite volume: We choose interpolators
i (2,7,9,10,15) and find a clear gap between ground state
E −
Λ (1/2 ): ground state (A66) and excitations. The extrapolation of the ground state
-1
energy level lies clearly above the experimental ground
2 3 4 5 6 7 8 9
t state Λ(1520) and is compatible with the Λ(1690). The
excitations extrapolate close to the Λ(2325).
FIG. 4. Eigenvectors for Λ spin 1/2− ground state and first
From the eigenvectors we find that the two lowest
excitation for ensemble A66 (m ≈ 255 MeV). The ground
π states are dominated by octet, the second excitation by
state is dominated by singlet interpolators of all three Dirac
structures. The first (and also the second excitation, not singletinterpolators. ThequarkmodelshowsforΛ(1520)
shown)isdominatedbyoctetinterpolators. Notethatacon- singlet dominance (like for its companion Λ(1405)). We
siderable mixing of singlet and octet is observed (15-20% for concludethatwemightmissasignalforthegroundstate
theground state). altogether, or, alternatively, there is a strong deviation
from the leading chiral behavior towards the physical
point. The latter case is intriguing as it might be re-
JP = 3+,Finitevolume: Inspin 3 channels,forsym- lated to strong coupling to an s-wave πΣ(1385) state,
2 2
metric quark fields, singlet interpolators vanish exactly which is discussed, e.g., in [31, 32].
due to Fierz identities. We use different quark smear- Infinite volume extrapolation: The volume depen-
ing widths and thus circumvent the Fierz identities con- denceturnsouttobefairlyflat,inafewcasesevencom-
structingsingletinterpolatorsnevertheless. Wederivere- patiblewithnegativefinitevolumecorrections. Thefinal
sults for the lowest three energy levels of the variational result in the infinite volume limit at the physical point
analysis of interpolators (2,9,10,16). Only the ground again misses the experimental Λ(1520) and agrees with
statecanbeclearlyidentifiedanditsextrapolationagrees the Λ(1690) mass.
with the experimental Λ(1890). Within the finite basis Summary: We presentacomprehensivestudyofspin
used, this state is dominated by octet interpolators. 1 and 3 Λbaryongroundstatesandexcitations,utilizing
2 2
The first excitation is dominated by singlet interpola- a large basis of interpolators in the variational analysis
tors with non-negligible octet contributions at our light- includingdifferentlysmearedquarksources(whichallows
estpionmass(seeFig.5). Wewanttoemphasizetheim- to sidestep the Fierz identities), three Dirac structures,
portance of singlet interpolators for the low lying states and singlet and octet forms. Fig. 6 shows the results for
5
3.0 L ~ 2.2 fm L →∞ 3.0 the LRZ Munich and on local clusters at the University
of Graz. G.P.E. and A.S. acknowledge support by the
2.5 2.5 DFG project SFB/TRR-55.
]
V
e2.0 2.0
G
[

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