Hybrid Baryons in QCD

1.4 MB·English

by  Jozef J. Dudek

JLAB-THY-12-1479
Hybrid Baryons in QCD
Jozef J. Dudek1,2,∗ and Robert G. Edwards1,†
(for the Hadron Spectrum Collaboration)
1Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA
2Department of Physics, Old Dominion University, Norfolk, VA 23529, USA
(Dated: January 10, 2012)
WepresentthefirstcomprehensivestudyofhybridbaryonsusinglatticeQCDmethods. Usinga
largebasisofcompositeQCDinterpolatingfieldsweextractanextensivespectrumofbaryonstates
andisolatethoseofhybridcharacterusingtheirrelativelylargeoverlapontooperatorswhichsample
gluonic excitations.
We consider the spectrum of Nucleon and Delta states at several quark masses finding a set of
positive parity hybrid baryons with quantum numbers N , N , N , N , N , and
1/2+ 1/2+ 3/2+ 3/2+ 5/2+
∆ , ∆ at an energy scale above the first band of ‘conventional’ excited positive parity
1/2+ 3/2+
2 baryons. This pattern of states is compatible with a color octet gluonic excitation having JP =1+
1 as previously reported in the hybrid meson sector and with a comparable energy scale for the
0 excitation, suggesting a common bound-state construction for hybrid mesons and baryons.
2
n PACSnumbers: 12.38.Gc,14.20.Gk,12.39.Mk
a
J
I. INTRODUCTION thespectrumofmesonsextractedfromlatticeQCDcom-
1
putations was interpreted in terms of ‘conventional’ qq¯
1
While QCD is the accepted underlying theory of states supplemented with a spectrum of hybrid mesons,

] hadronic physics, with hadrons being viewed as bound some with exotic JPC and some with conventional JPC.
h
p states of strongly coupled quarks and gluons, the role The degeneracy pattern of these states and the form of
- of excited gluonic fields in determining the spectrum the composite QCD operators used to interpolate them
p of mesons and baryons remains unclear. QCD in the from the vacuum strongly suggested that the gluonic ex-
e strongly coupled low-energy regime gives rise to a num- citation present in hybrid mesons is of chromomagnetic
h
[ ber of interesting phenomena that are exhibited in the character.
spectrum of mesons and baryons, such as the sponta- Hybrid baryons have not attracted the same attention
1 neousbreakingofchiralsymmetryyieldingeffective‘con- as hybrid mesons principally because they lack manifest
v stituent’quarkdegrees-of-freedom. Thesequasi-particles ‘exotic’ character. All JP values can be populated by
9
havethequantumnumbersofquarks,andthereareclear states constructed from three quarks having excitation
4
3 signs of a spectrum of excitations corresponding to their in orbital angular momentum, so that hybrids can only
2 relative motion. It seems odd then, considering that the appear in terms of over-population with respect to some
. gluonic field is strongly coupled to itself and to quarks, model of qqq excitations. While a dynamical model may
1
0 thattherearenounambiguoussignalsforstatesfeaturing suggest peculiar decay characteristics for hybrid baryon
2 a gluonic excitation. states that might help to identify them, there is not the
1 The simplest place to look for gluonic excitations ‘smokinggun’quantumnumbersignalpresentintheme-
: wouldseemtobeintheisoscalarmesonspectrum,where son sector. Furthermore the current experimental situa-
v
i states built purely out of glue, the ‘glueballs’, could be tion, in which far fewer baryon resonances are observed
X present. This has proven to be very difficult in prac- than are expected in qqq bound-state models, does not
r tice [1] with suggestions that qq¯excitations mix strongly encourage adding additional gluonic degrees-of-freedom
a
withglueballbasisstates. Anothertargetiselsewherein to the bound-state system.
the spectrum of mesons, where states built from qq¯aug- Despite there being, as yet, no simple way to extract
mented by a gluonic excitation can have JPC quantum them experimentally, there is significant theoretical in-
numbers not available to a pure qq¯state. These ‘exotic sight to be gained from studying hybrid baryons, par-
hybrid mesons’ remain our best hope of a ‘smoking gun’ ticularly in a framework that can simultaneously calcu-
experimental signature, but in practice the current ex- late the properties of hybrid mesons. We might imagine,
perimental situation is at best confused (see the review giventhedifferentnumberofquarks,andthecorrespond-
in[2]). Theoreticalunderstandingofhybridmesonstates ing differing distributions of color sources, that the spec-
in QCD recently took a significant step forward with the trumofgluonicexcitationsinabaryoncouldbedifferent
application of lattice methods to the problem. In [3–6] to that in a meson. We will explore this possibility in
this paper.
The spectrum of hybrid baryons has previously been
considered in a number of QCD-motivated models. In
∗ [email protected] the bag model [7], in which quark and gluon fields are
† [email protected] confined within a cavity with the fields satisfying ap-
2
propriate boundary conditions at the wall of the cavity, andquarklagrangianthatallowsustocomputethespec-
a TE-mode gluonic field (transforming as JPC = 1+−) trum directly from a controlled approximation to QCD.
combines with three quarks in an overall color octet to Inordertoexplorethespectrumofstateswithagivenset
produce a lightest set of hybrid baryons : of quantum numbers it has proved effective [3–5, 12–14]
to construct a large basis of hadron interpolating fields,
N , N , N , N , N featuring combinations of quark and gluon fields having
1/2+ 1/2+ 3/2+ 3/2+ 5/2+
thedesiredexternalquantumnumbers. Amatrixoftwo-
∆ , ∆ .
1/2+ 3/2+
pointcorrelationfunctionscanthenbeevaluatedandby
The overall mass scale of these states is somewhat plas- diagonalisation, a variational best estimate of the spec-
tic, depending upon how one chooses to set parameters trum of states determined. The eigenvectors obtained
withinthemodel-in[7],thefirstexcitedpionresonance, in this procedure indicate the optimum linear combina-
the π(1300), was assumed to be a hybrid meson which tionofinterpolatingfieldsforeachstateinthespectrum.
then sets the scale for Nucleon hybrids starting as light Quark-gluon bound-state structure interpretational in-
as 1.6 GeV. A more plausible modern candidate for a formation follows if a particular state has a large over-
hybrid pion would be the second excited pion resonance, lap with certain characteristic interpolating fields, and
the π(1800), which would place the hybrid Nucleons sig- large overlap onto multiple interpolating fields of differ-
nificantly heavier, somewhere above 2 GeV. entcharacteristicstructuremightsuggestlargemixingof
TheQCDsum-rulestudyin[8]consideredthepossibil- basis states within QCD.
ityofahybridbaryonasthefirst-excitedstateabovethe Our approach will mirror that used to determine a
nucleon with N quantum numbers concluding that spectrum of hybrid mesons in [3, 4, 6], where we found
1/2+
such a state would lie close to 1.5 GeV. that the lightest supermultiplet of hybrid mesons con-
An implementation of the flux-tube model[9, 10], in sists of an exotic JPC =1−+ state and three non-exotic
which quarks sit on the ends of strings, providing a lin- states with JPC = 0−+, 2−+, 1−− which are embedded
ear confining potential, suggests a set of lightest hybrid inaspectrumofqq¯excitations. Thecharacterofgluonic
baryons [11]: excitation that appears to give rise to this spectrum is
chromomagnetic, having JPC = 1+−. Heavier positive
N , N , N , N parity hybrid mesons were also identified that appear to
1/2+ 1/2+ 3/2+ 3/2+
correspondtoquarkorbitalexcitationontopofthesame
∆ , ∆ , ∆ .
1/2+ 3/2+ 5/2+ chromomagnetic gluonic excitation with states featuring
any other type of gluonic excitation apparently signifi-
This differs from the bag model spectrum by having the
JP = 5/2+ state with isospin-3/2 rather than isospin- cantly heavier.
1/2. Themassscaleofthesestatesisfoundtobearound The corresponding procedure for baryons will be fol-
1.9 GeV. The flux-tube model is an example of a frame- lowed in this paper: using a large basis of interpolating
work where we would not necessarily expect there to be fields, including a number which rely upon the presence
a common energy scale for the gluonic excitation in a ofanon-trivialgluonicfield,wewillextractaspectrumof
baryon and in a meson. A meson contains a single flux- baryon states. By considering the relative size of overlap
tubebetweenthequarkandanti-quarkwhichinahybrid of the extracted states onto interpolators of characteris-
is excited in transverse oscillation. In a baryon however tically qqq and hybrid form we will identify those which
there are three tubes which either meet at a junction or we believe to be hybrid baryons.
form a triangle. There would seem to be no particular AswithalllatticeQCDcalculations,wearesomewhat
reason the excitations of this more complicated system limitedinscopebycomputationalrestrictionswhichpre-
need be the same as a single tube. vent us from calculating with quarks whose masses are
As mentioned earlier, a major challenge for the ex- as light as those in the real world. In place of this we
perimental isolation of hybrid baryons is that they have will compute with a range of heavier quark masses and
non-exotic quantum numbers and hence appear embed- study the trend as the quark mass is reduced. Since the
dedinaspectrumofconventionalbaryonstates. Infact, computational cost increases with the volume of space-
worse than this there is, a priori, nothing within QCD timeconsideredwelimitourselvestoasmallregionafew
to prevent hybrids mixing arbitrarily strongly with con- fermi across discretised with grid points a fraction of a
ventional qqq states of the same JP to produced mass fermi apart.
eigenstates which are neither one nor the other. This Inthisworkwewillreportonthefirstsystematicstud-
complicationwillalsoappearinanysufficientlysophisti- ies of hybrid baryons using lattice QCD methods. We
cated theoretical approach. Whether QCD in fact mani- will work with three flavours of dynamical quarks - the
festsstrongmixingwillneedtobedeterminedinexplicit strange quark being fixed at its physical mass and the
calculation. degenerate up and down quarks being varied down from
InthispaperwewillextractaspectrumofQCDeigen- the strange quark mass (where we would have an exact
states from baryon correlators computed using discreti- SU(3) symmetry) to a value where the pion mass is
F
sation of QCD on a finite lattice. This technique is an 400 MeV. This is an extension of the work presented in
ab-initio non-perturbative method starting from a gluon [15] which computed the baryon spectrum on the same
3
lattices, but excluded baryon interpolators featuring a dition to the octet and decuplet states, we’ll extract a
gluonic excitation and which made no observations re- spectrum of B = Λ states, the only member of the
ΣF A
garding hybrid baryons. 1 representation.
F
The Dirac spin factor, (cid:0)SPS(cid:1)n , represents the possi-
ΣS
blecombinationsofupperandlowercomponentsofthree
II. BARYON INTERPOLATORS & THEIR Dirac spinors transforming with angular momentum S
INTERPRETATION and parity PS and having permutational symmetry, ΣS.
For some (cid:0)SPS(cid:1) there are multiple possible construc-
ΣS
The construction of a basis of baryon operators re- tions and they are labelled by the superscript n.
specting the required permutational symmetry of three The derivative factor, D[d] , indicates the number of
L,ΣD
quark fields and featuring up to two gauge-covariant gauge-covariant derivatives applied to the quark fields
derivatives was described in [15]. This leads to a rather (d = 0,1,2) and that they are combined to give objects
large basis of operators that covers all spins and both transformingunderrotationswithangularmomentumL
parities up to JP = 7±. For lower spins there are suffi- and with permutational symmetry ΣD.
2
cientlymanyoperatorsthatvariationalanalysiscanlead With a single derivative (d=1), excluding a symmet-
to extraction of a significant number of excited states. ric total derivative, only mixed symmetry1 possibilities
The baryon interpolators used can be expressed in a transforming as L=1 arise:
compactnotationwhichexposesthepermutationalsym-
metries, ΣF,S,D =S,M,A, D[1] : (cid:40)DM[1S],m = √16(cid:0)2Dm(3)−Dm(1)−Dm(2)(cid:1)
(cid:16)B ⊗(cid:0)SPS(cid:1)n ⊗D[d] (cid:17)J, (1) L=1,M DM[1A],m = √12(cid:0)Dm(1)−Dm(2)(cid:1)
ΣF ΣS L,ΣD where e.g. D(1) = (cid:126)(cid:15)(m) · D(cid:126)(1) is the gauge-covariant
m
derivative, in a circular polarisation basis, acting on the
wherethethreefactorsdescribetheflavor,Diracspinand
first quark field.
derivativestructureoftheinterpolator. Inthisreportwe
At the two derivative level (d = 2) definite permuta-
will consider only baryons of flavor B = N , ∆ , the
ΣF M S tion symmetry is implemented by the appropriate lin-
strangeness-zero members of 8 , 10 representations,
F F
overarangeofquarkmasses. Theonlyexceptiontothis ear combination of products of D[1] , while the rota-
MS,MA
is the special case of three degenerate flavors of quark, tionaltransformationalpropertiesaresetusinganSO(3)
i.e. having exact SU(3) flavor symmetry, where in ad- Clebsch-Gordan coupling for 1⊗1→L=0⊕1⊕2:
1
D[2] =(cid:104)1m;1m(cid:48)|LM(cid:105)√ (D[1] D[1] +D[1] D[1] ), (2)
L,S 2 MS,m MS,m(cid:48) MA,m MA,m(cid:48)
(cid:40)D[2] =(cid:104)1m;1m(cid:48)|LM(cid:105)√1 (−D[1] D[1] +D[1] D[1] ),
D[2] : L,MS 2 MS,m MS,m(cid:48) MA,m MA,m(cid:48) (3)
L,M D[2] =(cid:104)1m;1m(cid:48)|LM(cid:105)√1 (D[1] D[1] +D[1] D[1] )
L,MA 2 MS,m MA,m(cid:48) MA,m MS,m(cid:48)
1
D[2] =(cid:104)1m;1m(cid:48)|LM(cid:105)√ (D[1] D[1] −D[1] D[1] ). (4)
L,A 2 MS,m MA,m(cid:48) MA,m MS,m(cid:48)
In practice, the interpolators (1) correspond to sums in color, producing overall color-singlet operators, is im-
of terms of generic structure plemented using the Levi-Civita symbol. Given this it is
clear that the flavor, spin and spatial symmetry repre-
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(cid:15)abc Dn112(1±γ0)ψ Dn212(1±γ0)ψ Dn312(1±γ0)ψ sentations, ΣF,ΣS,ΣD must be combined to give overall
a b c symmetric(S)combinationsinorderfortheinterpolator
to obey Fermi antisymmetry.
where each quark field is projected into either upper or
lowerDiraccomponentsandisacteduponbyeitherzero,
oneortwogauge-covariantderivatives. Thespinandfla-
vorcouplingsaresuppressedhere(detailsoftheconstruc- A. “Conventional” qqq interpretation of operators
tion can be found in [15]). The required antisymmetry
A notable subset of the interpolators constructed are
those featuring only upper components of Dirac spinors,
whichwerefertoas“non-relativistic”. AsshowninTable
1 MS,MAindicatessymmetricorantisymmetricbehaviorin1↔2 IVof[15],theseoperatorscanbesimplyclassifiedaccord-
4
ingtoanSU(3) ⊗SU(2) ⊗O(3)symmetryanalogousto observation that leads us to identify these operators as
F S
what one might have in a (non-relativistic) three-quark being ‘hybrid’ - unless an eigenstate contains a gluonic
bound-state model. In this assignment, the derivatives fieldinanon-trivialconfigurationitshouldhavenoover-
play the role of orbital angular momentum between the lap onto an operator of this form.
quarks(theO(3)symmetryinmostbound-statemodels). In the derivative constructions described above
As a concrete example consider the symmetric con- such entries appear via terms of the form
structionoftwoderivativesdefinedinequation(2),D[2] , (cid:104)1m;1m(cid:48)|1M(cid:105)D(n)D(n) ∝ (cid:15)(M) (cid:15) [D ,D ] ∝ B ,
L,S m m(cid:48) i ijk j k M
which can be explicitly written where the chromomagnetic field in a circular polarisa-
tion basis, B ≡(cid:126)(cid:15)(M)·B(cid:126) is related to the commutator
DL[2,]S =3√12(cid:10)1m;1m(cid:48)(cid:12)(cid:12)LM(cid:11) of two gaugeM-covariant derivatives, Bk = 12(cid:15)ijk[Di,Dj].
As an example of how such an object enters into our
(cid:34)
(cid:16) (cid:17) operator basis, consider the projection of equations (3)
2 D(1)D(1)+D(2)D(2)+D(3)D(3)
m m(cid:48) m m(cid:48) m m(cid:48) into L=1:
−(cid:16)D(1)D(2)+D(2)D(1)(cid:17) D[2] = √1 (cid:10)1m;1m(cid:48)(cid:12)(cid:12)1M(cid:11)
m m(cid:48) m m(cid:48) L=1,MS 3 2
−(cid:16)D(1)D(3)+D(3)D(1)(cid:17) (cid:34) (cid:16) (cid:17)
m m(cid:48) m m(cid:48) 2 D(1)D(1)+D(2)D(2)−2D(3)D(3)
(cid:35) m m(cid:48) m m(cid:48) m m(cid:48)
(cid:16) (cid:17)
− D(3)D(2)+D(2)D(3) , (5) (cid:16) (cid:17)
m m(cid:48) m m(cid:48) − D(1)D(3)+D(3)D(1)
m m(cid:48) m m(cid:48)
(cid:16) (cid:17)
where expressed more fully a term D(3)D(2) would have − D(2)D(3)+D(3)D(2)
m m(cid:48) m m(cid:48) m m(cid:48)
the color-spatial structure (cid:15)abcψa(Dm(cid:48)ψ)b(Dmψ)c. A (cid:35)
(cid:16) (cid:17)
simple model interpretation of the action of D[2] fol- −2 D(1)D(2)+D(2)D(1) .
L,S m m(cid:48) m m(cid:48)
lows if we treat the gauge-covariant derivatives as or-
dinary, commuting, derivative operators (neglecting the
presence of the gauge-field) and use Jacobi co-ordinates
(ρ(cid:126) = √12((cid:126)r1−(cid:126)r2),(cid:126)λ = √16((cid:126)r1+(cid:126)r2−2(cid:126)r3)) for the quark DL[2=]1,MA =√16(cid:10)1m;1m(cid:48)(cid:12)(cid:12)1M(cid:11)
positions. In that case we find that
(cid:34)
(cid:16) (cid:17)
DL[2,]S ∼(cid:10)1m;1m(cid:48)(cid:12)(cid:12)LM(cid:11)(cid:104)(p(cid:126)ρ)m(p(cid:126)ρ)m(cid:48) +(p(cid:126)λ)m(p(cid:126)λ)m(cid:48)(cid:105) −Dm(1)Dm(1(cid:48))+Dm(2)Dm(2(cid:48))
∼(cid:2)p2YM(pˆ )Y0(pˆ )+p2YM(pˆ )Y0(pˆ )(cid:105)δ , +(cid:16)D(1)D(3)+D(3)D(1)(cid:17)
ρ L ρ 0 λ λ L λ 0 ρ L,even m m(cid:48) m m(cid:48)
(cid:35)
sothatifL=2,wecouldinterpretthisasthesymmetric (cid:16) (cid:17)
− D(2)D(3)+D(3)D(2) .
combinationofaD-waveontheρ-“oscillator”coupledto m m(cid:48) m m(cid:48)
an S-wave on the λ-“oscillator” and a D-wave on the λ-
“oscillator” coupled to an S-wave on the ρ-“oscillator”. In each case there are terms which are symmetric in
Thiskindofoperator,coupledto“non-relativistic”Dirac m ↔ m(cid:48) while (cid:10)1m;1m(cid:48)(cid:12)(cid:12)1M(cid:11) is antisymmetric - since
spinorscaninterpolatefromthevacuumexactlythekind derivativeoperatorsactingondifferent quarkscommute,
of structures used in qqq “quark models” [16–19]. For thesetermsarezero. Theremainingterms,inwhichboth
example, the interpolator (cid:16)N ⊗ (cid:0)1+(cid:1)1 ⊗ D[2] (cid:17)32+ derivativesactonthesamequarkfield,givecolor-spatial
M 2 M L=2,S structures proportional to
wouldbeidentifiedwiththemodelstateN2D 3+. Table
S 2 MS∼(cid:15) ((B ψ) ψ ψ +ψ (B ψ) ψ −2ψ ψ (B ψ) ),
IV in [15] presents qqq model interpretations for “non- abc M a b c a M b c a b M c
relativistic” interpolators. MA∼(cid:15)abc (−(BMψ)aψbψc+ψa(BMψ)bψc).
The form of the three-quark color coupling can be ex-
B. “Hybrid” qqqG interpretation of operators posed by expressing the chromomagnetic field in an ad-
joint basis, B =BAtA, so that
M M
Thesubsetofoperatorswhichweidentifyasessentially
(ψψψ)A ∼(cid:15) (cid:0)(tAψ) ψ ψ +ψ (tAψ) ψ −2ψ ψ (tAψ) (cid:1),
hybrid in nature appears at two derivatives and follows MS abc a b c a b c a b c
from combinations which correspond to commutation of (ψψψ)A ∼(cid:15) (cid:0)−(tAψ) ψ ψ +ψ (tAψ) ψ (cid:1),
MA abc a b c a b c
twogauge-covariantderivativesactingonthesamequark
field. Suchoperatorsarezeroinatheorywithoutgauge- which indicates that the three quarks are coupled to a
fieldsandcorrespondtothechromomagneticcomponents color-octet as either (3 ⊗ 3 → 3) ⊗ 3 → 8 ∼ MS or
of the gluonic field-strength tensor. This is the central (3⊗3→6)⊗3→8∼MA. Thechromomageticfield(a
5
L=0,2 L=1 We mention in passing that the antisymmetric com-
D[2] (qqq) 0 bination of two derivatives defined in equation (4), has
S 1c
D[2] (qqq) (cid:2)(qqq) G (cid:3) structure
M 1c 8c 8c 1c
D[2] 0 (qqq) (cid:16) (cid:17)
A 1c D[2] = √1 [D(1),D(2)]+[D(2),D(3)]+[D(3),D(1)] ,
A 6
TABLE I. Summary of simplest bound-state interpretations
of two-derivative structures. In practice the derivatives are which does not feature the commutator of two gauge-
actually implemented as finite differences through gauge- covariantderivativesactingonthesamequarkfield. The
covariant parallel transport (links) and this gives rise to cor- antisymmetricnatureissuchthatonlywithL=1dowe
rectionstotheaboveassignmentsthataresuppressedbypow-
get non-zero operators but these are not of hybrid na-
ers of the lattice spacing.
ture;intheJacobibasispresentedintheprevioussection
they transform like (cid:10)1m;1m(cid:48)(cid:12)(cid:12)1M(cid:11)(p(cid:126)ρ)m(p(cid:126)λ)m(cid:48), i.e. they
can be viewed as a P-wave on each “oscillator” in the
language of the constituent quark model, giving states
color octet) is coupled in to give an overall color singlet
labeled 2,4P .
baryon operator, (ψψψ)ABA. These are the operators A
M The interpretations of the D[2] constructions are sum-
which we identify with hybrid baryons.
marisedinTableI.Theseoperators,excluding thoseiden-
AtfirstglanceitwouldappearthatD[2] ,asdefined tifiedaboveasbeingofhybridcharacter,wereusedin[15]
L=1,S
in equation (2) and written out in full in equation (5) to extract a spectrum of baryons. In the next section
should also have hybrid character owing to the presence of this paper we will present spectrum results including
of terms featuring (cid:104)1m;1m(cid:48)|1M(cid:105)D(n)D(n). In fact this thehybridoperators,identifyinganumberofstateswith
m m(cid:48) large overlap onto these gluonic operators that we inter-
operatoriszero-D[2] givesrisetoathree-quarkcolor
L=1,S pret as being hybrid baryons.
structure
We have determined that operators featuring D[2]
L=1,M
can be associated with hybrid structure - now we can
(ψψψ)A ∼(cid:15) (cid:0)(tAψ) ψ ψ +ψ (tAψ) ψ +ψ ψ (tAψ) (cid:1), askwhatflavor-spinconstructionscanbecombinedwith
S abc a b c a b c a b c D[2] to give operators which respect Fermi antisym-
L=1,M
metry? Asisthecasewithqqqinterpretations,theeasiest
which cannot be a color-octet as the antisymmetric hybrid interpolators to interpret phenomenologically are
combination of three 3 representations gives a color- those with “non-relativistic” Dirac spin structure. The
c
singlet. We do not include any operators constructed explicitflavor-spin-colorstructureoftheseoperators,fea-
using D[2] in our basis. turing D[2] , is as follows:
L=1,S L=1,M
21: (cid:18)Λ1,A⊗(cid:16)21+(cid:17)1M⊗DL[2=]1,M(cid:19)JP=21+,32+ ∼φA√12(ψ3¯χMA+ψ6χMS)=φA(ψχ)S =[ψφχ]A
28: (cid:18){N,Σ8,Λ8,Ξ8}M⊗(cid:16)21+(cid:17)1 ⊗DL[2=]1,M(cid:19)JP=12+,32+ ∼ 12[ψ3¯(φMAχMA−φMSχMS)−ψ6(φMAχMS+φMSχMA)]
M
= √12[ψ3¯(φχ)MS−ψ6(φχ)MA]=[ψφχ]A
48:(cid:18){N,Σ8,Λ8,Ξ8}M⊗(cid:16)23+(cid:17)1S ⊗DL[2=]1,M(cid:19)JP=12+,32+,52+∼ √12χS(ψ3¯φMS−ψ6φMA)=χS(ψφ)A =[ψφχ]A
210: (cid:18){∆,Σ10,Ξ10,Ω}S⊗(cid:16)21+(cid:17)1M⊗DL[2=]1,M(cid:19)JP=21+,32+ ∼ √12φS(ψ3¯χMS−ψ6χMA)=φS(ψχ)A =[ψφχ]A (6)
where ψ3¯, ψ6 are the MA, MS color combinations (cor- Diracspinlowercomponentsgivesasomewhatlargerset
responding to the MS, MA derivative combinations) and ofoperators,includingsomewithnegativeparity,asindi-
φ,χaretheflavor,spinstructurespresentedinAppendix cated in Table II, although their model interpretation is
A of [15]. This set of operators corresponds to a [70,1+] somewhat more involved and will not be described here.
intheSU(6)FS assignmentscheme, arepresentationnot In order to account for the reduced (cubic) rotational
present for conventional three-quark baryons. symmetryofthelattice,wesubduce allinterpolatorsinto
Coupling the hybrid derivative structures (D[2] ) to irreducible representations of the cubic group and form
L=1,M
6
positive parity negative parity
N(1+) 32{4}[8](2) ∆(1+) 16{2}[4](1) N(1−) 32{2}[8](0) ∆(1−) 16{1}[4](0)
2 2 2 2
N(3+) 36{5}[8](2) ∆(3+) 19{3}[4](1) N(3−) 36{2}[8](0) ∆(3−) 19{1}[4](0)
2 2 2 2
N(5+) 19{3}[3](1) ∆(5+) 9{2}[1](0) N(5−) 19{1}[3](0) ∆(5−) 9{0}[1](0)
2 2 2 2
N(7+) 4{1}[0](0) ∆(7+) 3{1}[0](0) N(7−) 4{0}[0](0) ∆(7−) 3{0}[0](0)
2 2 2 2
TABLEII.Numberofoperatorsconstructedineachquantumnumberchannel: total{non-relativisticconventional}[hybrid](non-
relativistic hybrid)
correlators according to that symmetry. In [15] we show clover configurations can be found in [21, 22]. The two-
that the spectrum of states extracted from variational point correlators used in the variational analysis were
analysis of these correlators can be spin-identified with computedviadistillation[23]inwhichallquarkfieldsare
confidenceandhereinwewillpresentthespectralabelled smeared over space by an operator (cid:50) = (cid:80)Nξ ξ† con-
n n n
according to the continuum spin. structed using the lowest N = 56 eigenvectors of the
three-dimensional gauge-covariant laplacian (−D(cid:126)2ξ =
n
λ ξ ). Allgauge-linksenteringintheoperatorconstruc-
n n
III. BARYON SPECTRUM
tions are stout-smeared [24].
To facilitate comparisons of the spectrum at different
Mass spectra and matrix-elements follow from vari- quark masses, the ratio of hadron masses with the Ω
ational analysis of a matrix of correlators, Cij(t) = baryon mass obtained on the same ensemble is used to
(cid:10)0(cid:12)(cid:12)Oi(t)Oj†(0)(cid:12)(cid:12)0(cid:11), which can be shown to correspond to removetheexplicitscaledependence, followingRef.[22].
solution of the generalised eigenvalue problem
C(t)v =λ (t,t )C(t )v . A. Nucleons & Deltas at m =524 MeV
n n 0 0 n π
(cid:12) (cid:11)
Each state (cid:12)n that can be interpolated by some lin- Figure1showsthespectrumofpositiveparityNucleon
ear combination of O† has associated with it a “prin- and Delta states extracted on the 524 MeV lattice along
i
cipal correlator”, λn(t,t0) that at large times behaves with histograms showing the relative sizes of matrix ele-
like e−En(t−t0), yielding the energy of the state, and an ments (cid:10)n(cid:12)(cid:12)O†(0)(cid:12)(cid:12)0(cid:11) for a set of “non-relativistic” interpo-
i
eigenvector vn which indicates the optimum interpolator lators characteristic of qqq and hybrid basis states. The
for the state, (cid:80) viO†. Details of our procedure for effi- histograms are normalised such that the largest overlap
i n i
ciently and accurately implementing such a solution can onto a given operator across the entire extracted spec-
be found in [4, 20]. trum has unit value (the tallest bars in the figure are of
Totheextentthatonehasabound-stateinterpretation unit value).
of the interpolator Oi, built from quark and gluon fields, Focussing first on the JP = 1+ nucleon states we ob-
the relative sizes of the matrix elements (cid:10)n(cid:12)(cid:12)Oi†(0)(cid:12)(cid:12)0(cid:11) can servealightgroundstatenucleo2nhavingdominant2over-
b(cid:12)(cid:12)ne(cid:11).usTedhetsoeimntaetrrpixretelethmeenbtosuanrde-sttraitveiaslltyrurcetluarteedoftosttahtee lapontoa“non-relativistic”interpolator(cid:0)NM⊗(cid:0)12+(cid:1)1M⊗
1+
eigenvectorsofthegeneralisedeigenvalueproblemsolved D[2] (cid:1)2 with qqq structure 2S . The band of four ex-
above. Such a procedure was followed in [4], leading to L=0,S S
cited 1+ states between 2.0 and 2.5 GeV can be char-
identification of the low-lying meson spectrum as being 2
acterised as being dominantly admixtures of the qqq ba-
ratherliketheconstituentquarkmodelqq¯spectrum,sup-
sis 2S , 2S , 4D , 2P , and while this basis is clearly not
plemented with a spectrum of hybrid mesons in which S M M A
manifested diagonally in the spectrum, the mixing is
a chromomagnetic gluonic field was dominant. We will
modest. Notably, the extracted spectrum lacks a light
take the same approach here in the baryon spectrum in
first-excitedstatethatwemightassociatewiththebroad
order to isolate the role of hybrid baryon basis states in
enhancement observed experimentally and known as the
the spectrum.
Weperformedcalculationsonlatticesofsize163×128, “Roper”resonance. Wewillreturntothislaterwhenwe
discuss the role of hadronic decays on the spectrum.
corresponding to a physical spatial volume of approxi-
mately(2.0fm)3, withthreeflavorsofdynamicalquarks. Slightly heavier than the first-excited band of four
One quark has mass corresponding to the strange quark 1+ states we observe two states with dominant over-
2
and the other two correspond to degenerate u and d lap onto non-relativistic interpolators we have identified
quarks. One calculation had all three quarks degenerate
at the strange quark mass, corresponding to 702 MeV
octet pseudoscalar bosons, and the remaining two cal- 2 Theobservedsub-dominantoverlapofthegroundstatenucleon
culations had lighter u,d quarks giving pion masses of ontooperatorsfeaturinganexcitedgluonicfieldisalsorequired
524and396MeV.Detailsofthesedynamicalanisotropic todescribeQCDsum-rulesoftheN1/2+ channel[8]
7
3.0
2.5
2.0
1.5
1.0
3.0
2.5
2.0
1.5
1.0
FIG. 1. Extracted spectrum of Nucleon and Deltas states by JP at a pion mass of 524 MeV. Rectangle height indicates the
statisticaluncertaintyinmassdetermination. Histogramsindicatetherelativesizeofmatrixelements(cid:10)n(cid:12)(cid:12)O†(0)(cid:12)(cid:12)0(cid:11)fora“non-
i
relativistic”subsetoftheoperatorsused-normalisationisasdescribedinthetextwiththelighterareaattheheadofeachbar
beingthestatisticaluncertainty. OperatorlabelingisasinTableIVof[15]withtheadditionof2hyb=(cid:0)N ⊗(cid:0)1+(cid:1)1 ⊗D[2] (cid:1)
M 2 M L=1,M
and 4hyb=(cid:0)N ⊗(cid:0)3+(cid:1)1 ⊗D[2] (cid:1). Asterisks indicate states having dominant overlap onto hybrid operators.
M 2 S L=1,M
as being of hybrid nature, (cid:0)N ⊗(cid:0)1+(cid:1)1 ⊗D[2] (cid:1)12+ in passing that in [15], the spectrum extracted without
M 2 M L=1,M hybrid interpolators showed one badly-determined state
1+
and (cid:0)N ⊗ (cid:0)3+(cid:1)1 ⊗ D[2] (cid:1)2 . They appear to be in this same mass region, indicating the importance of
M 2 S L=1,M including interpolators with good overlap onto all types
largely admixtures of two basis states with three-quark
spinS =1/2, 3/2thatwelabel2hyb, 4hyb. Wemention of state present in the spectrum.
8
ThenextnucleonstateresolvedwithJP = 1+issome- set to the strange quark mass - here the lightest pseu-
2
what heavier than the hybrids and has no obvious inter- doscalar meson has a mass of 702 MeV. We show the
pretationintermsofoverlapswiththe“non-relativistic” mass spectrum for 8 , 10 , 1 representations.
F F F
operators used (with up to two gauge-covariant deriva- We observe that the 8 , 10 spectra strongly resem-
F F
tives). Stillheavierstatesarepresentwiththesequantum bletheNucleon,Deltaspectraextractedatthe524MeV
numbers3. pionmassshowninfigure1. The1 spectrumconsistsof
F
In the JP = 3+ nucleon channel, we see a lightest the states expected in a qqq model, lying roughly degen-
2
band of states roughly degenerate with the first-excited erate with the first excited band in the octet, decuplet
band in 1+. This band features five states observed to spectrum,andinaddition,twostatesofhybridcharacter
be domin2ated by the qqq basis 4S , 2D , 4D , 2D , 2P , having JP = 1+, 3+. As in the octet and decuplet, the
M S M M A 2 2
withthestatedominatedby2P beingheaviestasitwas hybrid states are slightly heavier than the qqq states.
A
for JP = 1+. As in the nucleon 1+ case we observe
2 2
a pair of states somewhat above this band which have
dominant overlap onto 2hyb, 4hyb. A few hundred MeV C. Quark mass dependence
heavier there are more states of unidentified structure.
In the JP = 5+ nucleon channel there are three states In order to determine if the hybrid spectrum might
2
inthe“first-excited”massregion,apparentlyconstructed be strongly quark mass dependent we also computed the
from superpositions of 4DS, 2DM, 4DM. Somewhat heav- NucleonandDeltaspectrumwithu,dquarkmassessuch
ier is a single state having large overlap onto a hybrid thatthepionweighs396MeV.Theresultsarepresented
operator of character 4hyb. Unidentified 5+ states are in Figure 3, where we see that the gross structure of the
2
present a little heavier than the hybrid. spectrum is as it was at m =524MeV.
π
In the JP = 7+ nucleon channel, there is a single
2
state in the “first-excited” region having dominant over-
lap onto 4D . Heavier states of unidentified structure IV. CONCLUSIONS
M
arepresentabove3GeV,withnoidentifiedhybridstates.
Weremindthereaderthatthereareno“non-relativistic” From the extracted baryon spectra presented in the
hybridinterpolatorshavingJP = 7+ constructedatD[2] previous section we can draw a number of conclusions
2
level. regarding the nature of hybrid baryons and more gener-
In the Delta sector we again see a band of “first- allygluonicexcitationswithinQCD.Thestatesidentified
excited” states whose population and overlaps are in ac- havelargeoverlapontointerpolatingfieldscontainingthe
cordwithexpectationsofaqqq model. Stateswithdom- operator D[2] which transforms like a chromomag-
inantoverlapontoourhybridinterpolatorsappearabove L=1,M
netic field (color octet, JPC = 1+−)4. The form of the
that band, but below other unidentified states. The hy-
operators suggest that within a hybrid baryon the three
brids appear in the JP = 1+, 3+ channels and are of
2 2 quarksarearrangedinacoloroctetwiththechromomag-
2hyb character.
neticgluonicexcitationmakingthestateanoverallcolor
The spectrum of negative parity nucleons and Deltas
singlet. Thelow-lyinghybridstatesoverlapstronglyonto
below2GeVisalmostidenticaltothatpresentedin[15],
the “non-relativistic” subset of our hybrid interpolators,
obtained without inclusion of any hybrid interpolators,
those constructed using only upper components of Dirac
and agrees with qqq model state counting. Negative par-
spinors. We can interpret this as suggesting that the
ity states having significant overlap onto the hybrid op-
quarkswithinthelightesthybridbaryonsaredominantly
erators in the basis first appear above 3 GeV, somewhat
in S-waves.
heavier than the positive parity hybrid states. The particular set of flavor-JP states observed,
N , N , N , N , N , plus ∆ , ∆
1/2+ 1/2+ 3/2+ 3/2+ 5/2+ 1/2+ 3/2+
andΛ , Λ ,suggeststhatallcolorsingletstates
1,1/2+ 1,3/2+
B. 1 , 8 , 10 at an SU(3) symmetric point
F F F F are present that can be constructed from antisymmetric
(qqq) with quarks in an S-wave coupled to a chromo-
8c
The flavor singlet construction in equation (6), Λ1,A⊗ magnetic (8c, 1+) gluonic excitation (see equations (6)).
(cid:0)1+(cid:1)1 ⊗D[2] , is most simply explored at an SU(3) Furthermore the fact that we are easily able to pick the
2 M L=1,M F
point where the states interpolated cannot mix with hybrid states out of a dense spectrum of qqq states indi-
Lambda states in an 8 representation. In Figure 2 we cates that they tend to retain their character despite the
F
showthespectrumextractedwithallthreequarkmasses possibilityofmixingstronglywithqqq statesofthesame
JP.
3 Thereareatotalof32 1+statesextracted,equaltothenumber
2
of interpolators used, but the statistical precision and stability 4 see [6] for a simple model in which this operator interpolates a
ofextractionofthemassesdecreasesasonegoesupinenergy quasi-gluoninaP-wave
9
3.5
3.0
2.5
2.0
1.5
FIG.2. Extractedspectrumofflavoroctet,decupletandsingletstatesbyJP withthreequarkflavorsallatthestrangequark
mass,correspondingtoanoctetpseudoscalarmassof702MeV.Greyboxesareconventionalqqq statesandblueboxesarethe
states identified as hybrid baryons.
this to be meaningful we must take some account of the
differingnumberofquarks-inaconstituentquarkmodel
we would subtract twice the constituent quark mass in
3.0 mesons and three times the mass in baryons. With the
latticedataweopttosubtracttheρmassfromthemeson
2.5 spectrum and the nucleon mass from the baryon spec-
trum. This is presented in Figure 4 where it would ap-
pear that there is a common energy scale, in the region
2.0
of 1.3 GeV, for the lowest gluonic excitation in mesons
and baryons.
1.5
Itisworthmentioningherethatinthemesonsectorwe
used an operator basis featuring up to three derivatives.
1.0
Thelow-lyingspectrumofnegative-parityhybridmesons
was largely insensitive to the presence or absence of the
FIG.3. ExtractedspectrumofNucleonandDeltastatesata three-derivativeoperators. Conversely,accuratedetermi-
pionmassof396MeV.Greyboxesareconventionalqqqstates
nation of the higher-lying positive-parity hybrid mesons
and blue boxes are the states identified as hybrid baryons.
required inclusion of these operators. We expect that a
similarsituationwouldholdtrueinthebaryonspectrum
- where we to include a set of three-derivative opera-
Thepositionofhybridbaryonswithinthespectrumof tors, the low-lying positive parity hybrid baryon spec-
expectedqqqstatesisnowdetermined: thelighteststates trum would be largely unchanged, but we would likely
are of positive parity and lie slightly heavier than the resolve more clearly a spectrum of higher-lying negative
first-excited positive parity qqq states. Negative parity parity hybrid baryons.
hybrid baryons appear to be heavier than this.
There are features of our extracted spectrum that we
In [6] we presented the spectrum of hybrid mesons ex- cannot explain without more detailed modelling, in par-
tracted on the same lattices. The distribution of states ticular the nature of the dynamics giving rise to the fine
across JPC (the supermultiplet structure) is also com- structure in the spectrum is not known to us. It is inter-
patible with a color octet chromomagnetic field coupled estingtonotethattheextractedstructureisratherclose
to quarks in a color octet (in this case (q¯q) ). This to that presented in [7] following from perturbative cor-
8c
leads us to think that the gluonic excitation form may rections (quark-spin, gluon Compton etc ...) to the bag
be common to hybrid mesons and baryons. To deter- spectrum. One possible exercise using the spectrum ob-
minewhethertheenergyscaleofthegluonicexcitationis tained in this paper would be to explore the minimal set
commonwechoosetoplotthespectrumofhybridmesons of model components required to describe the observed
alongside the spectrum of hybrid baryons. Of course for fine structure.
10
2000
1500
1000
500
0
FIG. 4. Spectrum of hybrid mesons and baryons for three light quark masses. Mass scale is m−m for mesons and m−m
ρ N
for baryons to approximately subtract the effect of differing numbers of quarks.
With optimum linear combinations of baryon interpo- In summary we have extracted a spectrum of baryons
latorsdeterminedinthevariationalsolutionofthematrix from QCD computations that features ‘expected’ qqq
of two-point correlators, we can in the near future con- statesbuiltfromsuperpositionsofflavor-spinrepresenta-
sider computations of radiative transition form-factors. tions[56,0+],[70,0+],[70,2+],[56,2+],[20,1+]inposi-
Such calculations, which involve three-point hadron cor- tive parity and [70,1−] in negative parity, supplemented
relators, have been done for charmonium [25], and for with a set of positive parity hybrid baryons that form
spin-1/2 baryons, albeit with very small operator bases a [70,1+] and which lie above the band of first-excited
[26, 27]. Radiative transition form-factors are of phe- positive parity baryons. In light of this it seems unlikely
nomenological interest because they can be measured that the Roper is dominantly of hybrid character as has
in baryon electroproduction experiments[28] where the been speculated in the past. The structure of the opera-
photon Q2 dependence and the relative size of multipole tors that interpolate the hybrid states from the vacuum,
amplitudes might provide some insight into the internal along with the observation that the energy scale of glu-
structure of the excited state [29]. onic excitations appears to be common for mesons and
baryons, provides evidence that the gluonic excitation
Thephysicsofexcitedstatesasresonances,abletode-
sector in QCD may turn out to be relatively simple. We
cay into meson-baryon final states is incomplete in this
suggest that a chromomagnetic excitation (JPC = 1+−)
study. Alongwithlargerthanphysicalquarkmasses,this
is lightest with an energy scale in the region of 1.3 GeV.
may be the cause of the lack of a light Roper candidate
state within the calculation. In order to observe such
decay physics in the extracted finite volume spectrum,
whichwould correspondtoadditionalvolume-dependent ACKNOWLEDGMENTS
energy levels, we must supplement our basis of inter-
polating fields with some resembling multi-hadron con-
We thank Ted Barnes, Simon Capstick, and Frank
structions (e.g. πN, ππN). Such work is underway and
Closefortheirinputonthephysicsofhybridbaryonsand
mayeventuallyleadtophenomenologicaldecayfiltersfor
our colleagues within the Hadron Spectrum Collabora-
hybrid character that would be useful for experimental
tion. Chroma[30]andQUDA[31,32]wereusedtoperform
searches.
this work on clusters at Jefferson Laboratory under the
A more complete understanding of hybrid baryons in USQCDInitiativeandtheLQCDARRAproject. Gauge
QCD will of course require calculations at lighter quark configurations were generated using resources awarded
masses, approaching the true physical values, coupled from the U.S. Department of Energy INCITE program
with larger lattice volumes to accommodate the increas- at Oak Ridge National Lab, the NSF Teragrid at the
ing physical size of the bound states and such work is Texas Advanced Computer Center and the Pittsburgh
now warranted given the success of this first calculation. SupercomputerCenter,aswellasatJeffersonLab. RGE