PHYSICAL REVIEW CVOLUME 50, NUMBER 1JULY 1994
Consis en ba yon mapping o qua k sys ems
S. Pi el
Ba ol Resea ch Ins i u e, Uni e si y o Delawa e, Newa k, Delawa e 19716
J.M..A ias
Depa amen o de Fisica A omica, Molecula yNuclea , Uni e sidad de Se illa, Apdo 10.
65, $1080 Se illa, Spain
J.Dukelsky
G upo de Fasica Nuclea -Facul ad de Ciencias, Uniee sidad de Salamanca, 5'7008 Salamanca, Spain
A. F ank
Ins i u o de Ciencias Nuclea es and Iabo a o io de Cue naeaca, Ins i u o de Ecsica,
U u 'e sidad Nacional Au ono na de Mexico, Apdo Pos. al 70-5)8, 04510 Megico, Dis i o Fede al, Megico
(Recei ed 7Decembe 1993)
We p esen anew and consis en mapping o colo less h ee-qua k clus e s on o colo less iple
e mions (ba yons) and es i in he con ex o a h ee-colo ex ension o he Lipkin model. Fo
sys ems wi h wo iple s ( o which he p oblem can be sol ed wi hou app oxima ion bo h be o e
and a e he mapping), we exac ly ep oduce he dynamics o he model o he a ie y o co ela ion
s uc u es conside ed.
PACS numbe (s): 21.60.Gx, 21.60.Fw, 21.30.+y, 12.39.—
x
I. INTRODUCTION
Es ablishing aconnec ion be ween nuclea physics and
/CD has been an a ea o in ense esea ch in he las ew
yea s. Cen al o his e o is he goal o isola ing qua k
e ec s in nuclei.
In ecen yea s, cons i uen qua k models [1]ha e been
applied wi h conside able success o nuclea sys ems wi h
ew pa icles. While many ques ions emain conce ning
he alidi y o such an app oach (e.g., he lack o acon-
nec ion o /CD and i s appa en iola ion o he unde -
lying physics o chi al symme y and spon aneous chi al
symme y b eaking p ocesses), in iew o hese successes
i seems wo hwhile o de elop hese models u he , in
o de o see whe he hey can p o ide apa ial b idge
be ween he physics o /CD and ha o ini e nuclei.
A p esen , such models ha e been di ec ly applied o
one- and wo-ba yon sys ems only. To ea sys ems wi h
la ge numbe s o nucleons, i has p o en necessa y o in-
oduce app oxima ions. One possibili y ha has been
explo ed is o use he esona ing g oup me hod in he six-
qua k p oblem o ex ac an e ec i e nucleon-nucleon in-
e ac ion, which is subsequen ly diagonalized in he space
o se e al nucleons [2]. Un o una ely, many-qua k e -
ec s ha may a ise when mo e han wo nucleons a e
p esen will be missed in such an app oach. To inco -
po a e hem, we would like o bypass he wo-nucleon
p oblem and wo k di ec ly in he space o many qua ks.
Since asys em o 3A qua ks will clus e in o A iple s
(nucleons) a no mal nuclea densi ies, anecessa y in-
g edien in any such app oach is ame hod o handling
s ong h ee-body co ela ions in amany-body en i on-
men .
Recen ly, i has been sugges ed [3—
5] ha mapping
me hods [6] migh p o ide ap ac ical means o accom-
plishing his. The basic idea is o map colo less h ee-
qua k clus e s, which do no sa is y exac e mion an i-
commu a ion ules, on o iple e mions (ba yons) ha
do. Such amapping leads &om he o iginal mul i-
qua k Hamil onian o an e ec i e Hamil onian o hese
ba yons, which igo ously inco po a es he physics o he
Pauli p inciple a he qua k le el.
The i ue o his app oach is no in a educ ion o
he deg ees o eedom —
he e a e in ac mo e s a es
in he mapped space han in he o iginal space —
bu
a he in he ep esen a ion o he dynamics as asys-
em o in e ac ing ba yons. These ba yons con ain as a
subspace he physical nucleons (as well as exci ed nu-
cleonic s a es) and in e ac wi h one ano he in ways
ha should be amenable o he usual e mion many-body
echniques [7], e.g.,Ha ee-Fock, Tamm-Danco , and
andom phase app oxima ions, B ueckne heo y, e c.
Se e al di e en mappings o qua ks o iple e mions
ha e been ecen ly discussed. Pi el, Engel, Dukelsky,
and Ring [3] (he ea e e e ed o as PEDR) p oposed
a wo-s ep p ocedu e, whe eby pai s o qua ks a e 6 s
mapped on o diqua k bosons and hen boson- e mion
pai s a e mapped on o iple e mions. Despi e he suc-
cess o his mapping in ep oducing he dynamics o he
es model o which i was applied, i ne e heless has
some d awbacks. On he one hand, i does no lead di-
ec ly o iple e mions ha a e an isymme ic in hei
h ee indices. In addi ion, as o mula ed, i can only be
applied o qua k Hamil onians domina ed by wo-body
in e ac ions. A oughly he same ime, Nadjako [4]
sugges ed an al e na i e mapping ha leads di ec ly o
an isymme ic iple e mions and, u he mo e, is ap-
plicable o h ee-body in e ac ions. His mapping, how-
0556-2813/94/50(1}/423(12}/$06.00 50 423 1994 The Ame ican Physical Socie y
S.PITTEL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK
e e , is no app op ia e o sys ems ha a e domina ed
by wo-body in e ac ions. Clea ly, wha is needed is a
mapping ha consis en ly ea s bo h wo- and h ee-
qua k in e ac ions.
One such me hod was in ac p oposed soon he e-
a e by Meye [5]. Howe e , his mapping does no
seem o p ope ly ea wo-body in e ac ions ei he , a
leas when a unca ion o colo less iple e mions is
imposed. Building on he ideas o Meye , we ha e now
succeeded in o mula ing anew ba yon mapping o (col-
o less) h ee-qua k clus e s ha seems o sa is y all he
desi ed equi emen s. In Sec. II, we b ieHy e iew some
gene al ea u es o ba yon Inappings and p esen ou col-
o less e sion.
Bo h PEDR and Meye es ed hei mappings in he
con ex o an exac ly sol able model o qua ks o en e-
e ed o as he Bonn qua k shell model (BQSM) [8].
While his model has some a ac i e ea u es, i also has
some se ious limi a ions. Pe haps he mos signi ican
is ha i does no p oduce spa ially localized colo less
h ee-qua k clus e s (i.e.,nucleons) [9]. Thus asecond
goal o his wo k has been o de elop an al e na i e qua k
model on which o es ou mapping. The model ha we
ha e chosen is a h ee-colo ex ension o he well-known
Lipkin model [10], which in i s adi ional e sion has
been used ex ensi ely o es a ious nuclea many-body
echniques. We desc ibe his model in Sec. III and dis-
cuss i s algeb aic solu ion o small numbe s o pa icles.
Ac ucial componen o his model is ha i admi s, o
di Fe en alues o i s pa ame e s, dynamical one-, wo-,
and h ee-body co ela ions.
In Sec. IV, we apply ou mapping o he h ee-colo
Iipkin model o wo iple s and p esen he esul s. The
bo om line is ha he mapping seems o wo k pe ec ly,
when all colo less ba yon s a es a e included. Fo sys-
ems in ol ing ala ge numbe o iple s, his is clea ly
no possible. Thus, in Sec. V, whe e we summa ize he
p incipal conclusions o ou wo k, we also desc ibe some
u u e ex ensions needed o u he es he applicabili y
o ou me hods o many- iple sys ems.
II. BARYON MAPPINGS OF QUARK SYSTEMS
A. P elimina ies
aS —gq1aq1S )
1
(2.2)
+abed =g~123 &145 q2 q3gq5d q4c
12345 (2.3)
5d —g6123 6456 'VloQ2513 e I"~4d .(
123456
He e, and in he subsequen analysis, we deno e he colo
indices by numbe s and he es by oman le e s.
The idea o aba yon mapping is o eplace he sys em
o in e ac ing qua ks by an equi alen one o in e ac ing
iple e mions. We deno e he c ea ion and annihila ion
ope a o s o he esul ing (mapped) space by Al
253, and
A12/3 espec i ely. They sa is y he mul i-index an i-
commu a ion ela ion
(Al 253 A4de e )—
b(la2b3c, 4d5e6 )
whe e
(2.5)
Ou s a ing poin is anon ela i is ic model o con-
s i uen qua ks. We deno e he qua k c ea ion and an-
nihila ion ope a o s o he model by q,-and q;, espec-
i ely. The i s subsc ip deno es he colo quan um
numbe and he second deno es all he es . These ope -
a o s sa is y he usual e mion an icommu a ion ela ion
(gci Ipc&i& )=bci,c'i' =bcc' bii' (2.l)
QCD conside a ions sugges ha he qua k Hamil o-
nian may include up o h ee-body in e ac ions, all o
which a e colo scala s. Such aHamil onian can always
be exp essed in e ms o he ollowing colo less ope a o s:
b(la2b3c, 4d5e6 ) =bl 4db25, 5ebsc, e +bla, ca&45,e &3c,4d +&la,e b25, 4db3c, 5e
ala, 4db2b, e b3c,ee ala, 5eb2b, 4db3c, e ala, e b2b, eeb3c,4d (2.6)
The ope a o A1 2&3,,by de ini ion, c ea es aba yon
co esponding o h ee qua ks in he s a es la, 26, and
3c. This co espondence is exp essed h ough he e-
qui emen ha bo h A1 2g3 and A1 2p3, a e an isym-
me ic unde in e changes o hei qua k indices, e.g.,
A=—
A~ e c.
la263c 2bla3c~
The space gene a ed by hese ba yon ope a o s is in
ac la ge han ha o he o iginal qua ks. This can be
seen by conside ing he s a e o wo ba yons,
~la2b3c, 4d5e6 )~=Al 253,A4de, e ~0)~, (2.7)
whe e he subsc ip B e e s o s a es in he ba yon space.
The s a e (2.7) is an isymme ic unde he in e change o
he indices co esponding o any wo qua ks wi hin one o
he wo iple s (e.g.,la wi h 2b o 4d wi h 5e) and also
unde he in e change o one iple wi h he o he . How-
e e , i is no an isymme ic unde he in e change o he
quan um numbe s o aqua k in one iple (e.g.,la) wi h
hose o aqua k in he o he (e.g.,4d). A ully an isym-
me ic wo- iple s a e may be eco e ed by aking an
app op ia e linea combina ion o he s a es (2.7). As a
consequence, he e is indeed asubse o wo- iple (and
likewise many- iple ) s a es ha a e ully an isymme ic
and, u he mo e, a e in one- o-one co espondence wi h
he s a es o he o iginal qua k space. This is e e ed
o as he physical subspace. The e is, howe e , ano he
class o s a es ha a e no ully an isymme ic unde in-
e change o qua k indices and which he e o e ha e no
coun e pa s in he o iginal space; his is e e ed o as
he unphysical subspace.
The e a e a a ie y o possible ways o ensu e ha he
50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS
physics o he o iginal qua k p oblem is p ese ed un-
de he mapping o ba yons. We ollow he app oach
o Belyae and Zele insky, whe e ope a o s in he o ig-
inal space a e mapped on o ope a o s in he new space
so as o p ese e hei (an i)commu a ion ela ions. Im-
plemen a ion o his p esc ip ion gua an ees ha all he
physics o he o iginal sys em is exac ly p ese ed by he
mapping wi hin he physical subspace.
Fo amapping o be o p ac ical use, he unphysical
s a es mus lie high in ene gy ela i e o physical s a es.
O he wise, i will be di icul o disen angle he physical
s a es o in e es &om hose ha a e unphysical, pa icu-
la ly in he p esence o a ia ional app oxima ions. This
places s ingen limi a ions on he kinds o mappings ha
should be conside ed o p ac ical applica ions.
Fo example, i is s aigh o wa d o w i e down amap-
ping o colo less pa icle-hole (p-h) o one-body qua k
ope a o s ha p ese es hei commu a ion ela ions:
1
).q&.nb ~2)A] 23/Alb2 3d (28)
1123cd
Since any o he colo less ope a o s o he ype (2.2)—
(2.4)
can be ew i en in e ms o colo less pa icle-hole qua k
ope a o s, i would seem ha we could simply apply (2.8)
and achie e ou goal. This is un o una ely no he case.
Since apa icle-hole ope a o does no in ol e mo e han
one c ea ion and one annihila ion ope a o , i canno (by
i sel ) inco po a e any in o ma ion on he qua k Pauli
p inciple. As aconsequence, he spec um ha would
esul om apu e p-h mapping ( o e mion sys ems)
would in a iably ha e unphysical s a es lying below he
physical s a es o in e es . To inco po a e qua k Pauli
ei ec s (in aphysically use ul way), we mus map di ec ly
he mul iqua k c ea ion and annihila ion ope a o s ha
appea in he Hamil onian.
In wha ollows we adop he Dyson app oach, which
leads o aba yon Hamil onian ha is non-He mi ian bu
ini e. The non-He mi ici y is adi ec e8ec ion o qua k
Pauli e ec s. Ano el ea u e o ou analysis is ha ,
in con as o ea lie wo k, we ocus on he mapping o
colo less ope a o s. This emo es some o he ambigu-
i ies ha a ise when he mapping is ca ied ou mo e
gene ally. I also leads o amapping ha is ailo ed o
physical applica ions in which a unca ion o colo less
iple s mus be implemen ed.
B.Mapping o colo less one-body ope a o s
We begin wi h adiscussion o he colo less one-body
ope a o Abo (2.2). To map his ope a o , we can make
di ec use o ea lie esul s. Namely, using (2.8), we can
exp ess i s ba yon image as
A his poin , he image o Agis exp essed in e ms o
ba yons wi h colo . We know, howe e , ha i is possi-
ble o desc ibe he ele an physics in abasis o colo less
ba yons only. Towa ds his end, i is use ul o ca y ou
a unca ion o colo less ba yons; his can be done using
he p esc ip ion spelled ou by PEDR. The basic idea is
o ca y ou acolo -SU(3) coupling and o isola e he
piece ha is ully an isymme ic in colo and ully sym-
me ic in he o he noncolo indices. Asimple way o
implemen his is h ough he eplacemen s
A
ca.li2j3~ +6123Paij
~1i2j3& M6123~zjk (2.10)
The ope a o s A -& and A;jg in oduced he e a e ully
symme ic unde he in e change o hei indices. Fu -
he mo e, hey sa is y he an icommu a ion ela ion
(A;,b, A .„j=—S(ijk,lmn),
1
U~lmn (2.ii)
whe e
S(ijk, lmn) =b;)b, bb„+b; b,.„bb) +b;„b,(bb
+4b, bb +b' b,ibb~+b; b,
(2.12)
Inse ing (2.10) in o (2.9) and hen ca ying ou an
explici sum o e he colo indices (gz2z e&2s —
—6), we
a i e a he ollowing esul :
Abm3 )A ,qAb, g.
cd
(2.13)
As an example, conside amapping o he qua k num-
be ope a o Nq =Pz qz qq .Applying (2.13), we
ob ain he expec ed esul
Nq m3) A ~A b, =3N~, (2.14)
whe e N~ is he numbe ope a o o colo less ba yons.
C. Mapping o colo less h ee-body ope a o s
Nex we conside he colo less h ee-body ope a o
Cb,g,yappea ing in (2.4). He e, oo, we can di ec ly
use he mapping gi en by Nadjako and Meye , since
he se o one-body ope a o s plus he se o h ee-qua k
c ea ion and annihila ion ope a o s close unde commu-
a ion. The non-He mi ian mappings o h ee-qua k c e-
a ion and annihila ion ope a o s equi ed o p ese e his
commu a ion algeb a a e [4,5]
~ab =)qj~~qlb ~))A]~2~MAlb2csd ~(2.9)
1123 Cd
ql'q2 q3A: ~~3I2j1; (2.i5)
q.q-q mA . .+—p p p
1
2j sb 12gsb 2)(4js $'2jsbs +4gs 2~ sby s+A4~5 sbAy. 2~s )A4gb s
456,lmn
1
+- )
456789,lmnopq 4l5mli 6n7o2j"8p9q3&~4l6n8p~5m7o9q (2.16)
426 S.PIII'EL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK 50
Applying (2.15)—
(2.16) o he colo less h ee-body ope a o Cbd and ca ying ou asubsequen unca ion o
colo less iple s, we ob ain he ollowing esul :
Cab,d, mCb",d,y—
—36 Ab,Adey+36) (A „Ab, +A„bA, ,+A „,A~)Agh;Adey .
ghi (2.17)
Asupe sc ip nh has been included o indica e ha his is anon-He mi ian ba yon image.
No e ha acolo less h ee-qua k in e ac ion maps on o colo less one-body plus wo-body in e ac ions only. The
h ee-body piece cancels exac ly in he unca ion o colo less ba yons, because o p ope ies o he 6y23 ac o s.
D. Mapping o colo less wo-body ope a o s
Finally we u n o he wo-body ope a o , o which we canno di ec ly use he esul s o ea lie wo k. Meye
p oposed apossible non-He mi ian ba yon image o wo-body ope a o s, based on he use o he Usui ope a o .
Howe e , as no ed ea lie , his mapping does no seem o wo k when unca ed o colo less iple s and applied o
he es model we p esen la e . Ne e heless, he e is asugges ion in he wo k as o how o build ap ope mapping
o colo less wo-body ope a o s, which we now exploi . A he end, we make some ema ks as o why he esul s a e
no applicable.
As we saw in he p e ious subsec ion, i is possible o map acolo less h ee-body ope a o in non-He mi ian o m.
Since he mapping (2.17) ollowed &om an exac p ese a ion o commu a ion ela ions, i is ce ainly legi ima e. Bu
he e is ano he way o map acolo less h ee-body ope a o ha is equally legi ima e and which leads o aHe mi ian
o m.
The h ee-body ope a o Cbd can be ew i en in e ms o he colo less one-body ope a o s A;~ o (2.2) as
ollows:
Cabcde =AadAbeAc +AadAb Ace +AaeAbdAc +AaeAb Acd +Aa AbdAce +Aa AbeAcd
—
bbd(AaeAc +Aa Ace) —
bbe(AadAc +Aa Acd) —'4 (AadAce+ AaeAcd)
26 d(cA Aaeb +Aa Abe) —
2bce(AadAb +AayAbd) —
2bcj(AadAb, +Aa, Abd)
+2(~bc,e Aad +~bc,d Aae +~bc,deAa )(2.18)
Applying he mapping o colo less one-body ope a o s (2.13) and ocusing on he one-and wo-body pieces, we
ob ain, o he He mi ian image,
Cabcde ~Cabcde 3AabcAde 36 )(AghaAbci +AghbAcai +AghcAabi) (AdghAe i +AeghAd i +A ghAdei)
.
ghi (2.19)
No e ha he se e al 8- unc ion e ms in (2.18) do no su i e a e he mapping. They a e exac ly canceled by o he
e ms ha'. a ise when he ba yon image is pu in no mal o de .
The one-body piece o (2.19) is iden ical o ha gi en in (2.17). The wo-body pa , howe e , is no . One is
He mi ian and he o he non-He mi ian. Howe e , bo h a e o mally jus i ied and hus mus be equi alen in he
physical subspace. I is easy o show ha ei he o he o ms o he wo-ba yon image o Cb,d, can be ans o med
in o he o he by pe o ming he ollowing eplacemen on he wo annihila ion ope a o s:
1
AabcAde j~(AabdAce +AabeAdc +Aab Adec +AadcAbe +AaecAdb +Aa cAdeb
3
+AdbcAae +AebcAda +A bcAdea) (2.20)
These obse a ions sugges ap ocedu e o mapping acolo less wo-body in e ac ion. Namely, we i s ans o m
i o colo less p-h o m, hen map i using he well-known (and o mally jus i ied) colo less p-h mapping (2.13) and
inally ans o m i s wo-body pa o anon-He mi ian o m by ca ying ou he eplacemen (2.20).
We now apply his p esc ip ion o he colo less wo-qua k ope a o Bbdo (2.3). T ans o ming i o p-h o m
leads o he esul
+abed AacAbd +AadAbc ~bdAac ~bcAad (2.21)
Mapping his ope a o in colo less p-h o m and w i ing he esul in no mal o de gi es
h
Babcd ~Babcd =12 )Aaby Acd 9)Aae Abgh(Ace Adgh +Ade Acgh)
e gh (2.22)
Finally, when we impose he eplacemen (2.20) on he wo-ba yon piece, we a i e a
Babcd ~Babcd =12 )AabeAcde +9)Aaey Abgh(AcdeA gh +Ae gAcdh)
ee gh
(2.23)
50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 427
This esul di e s &om he one ha would a ise &om
Meye 's mapping supplemen ed by a unca ion o colo -
less ba yons, eQec ing ou consis en ea znen o colo .
In ac , he non-He mi ian mapping o wo-body op-
e a o s p oposed by Meye can be de i ed in much he
same way by conside ing agene al h ee-body in e ac ion
a he han acolo less one. Such ap ocedu e, howe e ,
is no unique, since agene al h ee-body ope a o can
be ecas as ap oduc o p-h ope a o s in di e en ways.
Explici conside a ion o colo less ope a o s emo es his
ambigui y. The end esul is ap esc ip ion o ans-
o ming &om He mi ian o non-He mi ian images ha
p ope ly inco po a es colo and is hus meaning ul when
iznplemen ing a unca ion o colo single s.
The essen ial esul s o his sec ion, p ope non-
He mi ian ba yon images o colo less one-, wo-, and
h ee-qua k ope a o s, a e con ained in Eqs. (2.13),
(2.23), and (2.17), espec i ely. Al hough he mapping o
wo-body ope a o s was no de i ed by explici conside -
a ion o commu a ion ela ions, we ha e con i med ha
his se o ba yon images does indeed p ese e he com-
nu a o [B~s~a, E, g], whe e E, g =+~2s eq2sq~, qz qsg
We should also emphasize he e ha his se o map-
ping equa ions can be applied o any colo less cons i uen
qua k Hamil onian w i en in uncoupled o m.
E. Physical con en o he eplacemen p ocedu e
Some unde s anding o he eplacemen p ocedu e p o-
posed o gene a e anon-He mi ian wo-body image &om
aHe mi ian one can be ob ained by s udying he map-
ping o acolo less six-qua k s a e,
~abc, de )Q —
——)~»s ~4&s qi~q2sqscq4dqseqs l0)q
123456
(2.24)
whe e he subsc ip qindica es as a e in he o iginal
qua k space. Mapping his s a e wi h (2.16) and impos-
ing a unca ion o colo less ba yons leads o he esul
~abc, de ) p~A&,A&, ~0)a+(A &,Ab, +—
A As,p+AQ As„
3
+A A +A, A +A,„A...+A A, +A..A.,+A.d A...}10) (2.25)
The physical s a e no only in ol es he di ec wo-
ba yon componen bu also asum o e all nine possible
in e changes o he indices o one ba yon wi h hose o
he o he (wi h an o e all ac o o s). I is no di icul
o con i m ha when we ac ei he wi h he ope a o
A~Ap, yon his physical s a e o wi h he eplacemen
o m gi en in (2.20) we a i e a exac ly he same esul .
Thus he p oposed eplacemen indeed sa is ies he de-
si ed c i e ion ha i p oduces he same esul s wi hin
he physical subspace. I is also in e es ing o no e he
co espondence be ween he di ec and exchange pieces
o (2.25) wi h he le and igh hand sides o (2.20).
III. THREE-COLOR LIPKIN MODEL
I
body in e ac ion ha sca e s pai s o pa icles among
he wo le els wi hou changing he p alues.
This znodel can be sol ed exac ly by using g oup he-
o e ical echniques. I is well known ha he se o all
possible bilinea p oduc s o med &om a ini e se o c e-
a ion and annihila ion ope a o s cons i u es aLie alge-
b a. In he case o he Lipkin model, he e a e (20)2such
bilinea p oduc s o c ea ion and annihila ion ope a o s
(gene a o s), and so he ele an Lie algeb a is U(20).
These gene a o s will be deno ed by K,"„, =q „q
Since we a e dealing wi h e mions, all he s a es o he
sys em belong o he i ep [ln] o he U(20) dynamical
g oup. The s uc u e o he p oblem sugges s he decom-
posi ion
U(20) oU(O) gU(2), (3 1)
The h ee-colo Lipkin model is based on he well-
known Lipkin model [10], which can be sol ed analy i-
cally and has been used ex ensi ely in nuclea physics o
es ing many-body app oxima ion me hods. Since many
o he cha ac e is ics o he h ee-colo Lipkin model a e
al eady in he o iginal one, we e iew i b ieBy he e.
The Lipkin model has wo le els, each one 0- old de-
gene a e, sepa a ed by an ene gy A. I is assumed ha
in he unpe u bed g ound s a e N=0pa icles oc-
cupy all he single-pa icle s a es in he lowe le el. The
e mion c ea ion and annihila ion ope a o s o he model
a e w i en as q „and qz, espec i ely. He e, ois a
quan um label which cha ac e izes whe he he pa icle
is in he lowe le el, o. =—,o in he uppe one, o=+,
and pdis inguishes which o he 0degene a e s a es o
ha le el he pa icle occupies. The Hamil onian o he
model includes, in addi ion o he one-body e m, a wo-
whe e he 0ope a o s K„", =PKP~ gene a e he
U(A) algeb a, while he 22 objec s K, =g„K,"„a e
he gene a o s o he U(2) algeb a, and commu e wi h he
K„,.One can easily e i y ha he Lipkin Hamil onian
can be w i en en i ely in e ms o he U(2) gene a o s
and hus ha all he s a es belong o ade ini e i educible
ep esen a ion o U(2) [o SU(2)]. This in u n implies
ha he Hamil onian zna ix can be analy ically e alu-
a ed using he well-known angula moxnen um algeb a
SU(2).
The h ee-colo Lipkin model has amuch iche al-
geb aic s uc u e and analy ic solu ions a e co espond-
ingly mo e di Bcul o de i e. The model in ol es h ee
se s (one o each colo ) o s anda d wo-le el Lipkin
models. Again he lowe le els a e assumed o be com-
ple ely illed in he unpe u bed g ound s a e, which in

428 S. PITTEL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK 50
his case con ains N=30 pa icles. The c ea ion and
annihila ion ope a o s now include alabel i ha ep e-
sen s he colo quan um numbe and a e hus w i en
as q,.„and qi „, espec i ely. The model Hamil onian
now includes one-body, wo-body, and h ee-body in e -
ac ions, which sca e pa icles cohe en ly among he le -
els, wi hou changing he p alues and main aining all
s a es colo less:
H=H1+H2+H3, (3.2)
(V&+„V&+p && &&
—
~)
ip
(3.3)
and
H2 =Og&1236145(g2+ 13+ 15 P14 P+1. „1.-„~+. ~+.
12345,pg p2
(3.4)
X3 )
123456)pi p2p3
&123&456 g1y+ 'V2+ 13+ 'V6 —95—g4—+14—95—16—13+P 12+ Ql+ )(3.5)
In his case he e a e (6O)2 gene a o s K& ", ,—
—q „qs „,leading o he Lie algeb a U(6O), and he s uc u e o
he model sugges s ha we ca y ou he classi ica ion o s a es in e ms o he chain
U(6O) zU(O) U(6) zU(O) 13 U(3) U(2) .(3.6)
The O2 ope a o s K", =P,.K,'. ~, gene a e U(O), while he 62 objec s K&, —
—gK& ~a e he U(6) gene a o s, in
e ms o which we shall ew i e he Hamil onian (3.2)—
(3.5) below. We can u he decompose U(6) by con ac ing
again o he 22 ope a o s K, =g,.K,',,which gene a e U(2), o o he 32 gene a o s Kl', —
—PK& co esponding
o U(3). The la e g oup is indeed necessa y in he classi ica ion, since all physically admissible s a es should be
colo less; i.e., hey should belong o he (O, O, O) U(3) ep esen a ion [which co esponds o he (A, p) =(0, 0)
scala ep esen a ion in Ellio sSU(3) no a ion]. The si ua ion is mo e complex han in he s anda d Lipkin model,
howe e , since di e en U(6) ep esen a ions can con ain hese s a es and, mo eo e , o each o hem se e al U(2)
ep esen a ions a e connec ed by he Hamil onian.
F om (3.2)—
(3.5), we see ha he model in ol es h ee pa ame e s, one each o he one-, wo-, and h ee-body
in e ac ions. As men ioned be o e, i may be ew i en in e ms o he U(6) gene a o s as
H1 ——
AJ, ,(3.7)
ik
(3 8)
and
2(J++J)+2J+) K„'+K,
"+ +J)K„'+K,
"+
ik ik )(K +Kq+ K,". ++K,*+K„'+K,
"+ ), (3.9)
ilk
whe e J„J~,and J,de ined by J, =2(K++ —
K), J+ =K+, and J=K+, a e he SU(2) subg oup gene a o s,
which oge he wi h he numbe ope a o N=P,.K,'comp ise he U(2) g oup in (3.6). The o he ope a o s in
(3.8) and (3.9), namely, g,.&K&+ K,". +and g1& Kl*+K&+K,"+, oge he wi h hei He mi ian conjuga es, clea ly lie
ou side SU(2). Using hei commu a ion ela ions wi h he SU(2) gene a o s, we eadily conclude ha hey beha e
as ank 2and ank 3 enso s T=T2 and T3 in SU(2), espec i ely.
(e) (2) (3) .
Using his no a ion, he model Hamil onian acqui es he simple o m
H1 —
—AJ, ,(3.10)
X2 (J2 J2 )+X2 (T(2) +T(2)) (3.11)
and
50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 429
o which we shall e e hence o h.
Since we a e dealing wi h asys em o e mions, he
s a es o he model belong o he [ls+] ep esen a ion o
U(6O), while he U(O) and U(6) ep esen a ions a e com-
pleinen a y; i.e.,once he U(6) ep esen a ion is de e -
mined, he U(O) ep esen a ion is ixed. Since he U(3)
ep esen a ions a e he colo less ones men ioned be o e,
i should be clea ha he basis s a es can be unambigu-
ously deno ed by
~[&i h2 li3 h4 h5 hs], ~jm), (3.13)
The s a e ~G) has h ee pa icles wi h p=1, h ee wi h
p=2,...,and h ee wi h p=O. De ining P=P,.p;, we
see ha he s a e ~G) has aunique P alue o 3O(O+1)/2.
In gene al, di Fe en U(6) i educible ep esen a ions
appea which con ain he U(3) colo less i eps in he e-
duc ion U(6) DU(3) U(2) o (3.6). Since he Hamil o-
nian (3.10)—
(3.12) is buil solely in e ms o U(6) gene -
a o s, he ene gy ma ix will sepa a e in o blocks, each
co esponding o ade ini e U(6) i ep.
Fo 0=1, he g oup analysis is pa icula ly simple.
The e is only asingle U(6) i ep, (1,1,1,0,0,0), which con-
ains he (1,1,1)(j =3/2) and he (2, 1,0) (j =1/2)
U(3) SU(2) i eps. Howe e , only he i s one belongs
o ou (colo less) space. Fu he mo e, all s a es ha e
P=3.
Fo 0=2 he si ua ion is mo e complex in se e al e-
spec s. Fi s , as we shall enume a e sho ly, he e a e
se e al possible U(6) i eps ha con ain colo less s a es.
All con ain s a es wi h P=9, while se e al also con ain
s a es wi h o he P alues. These o he s a es, howe e ,
can be gene a ed om he co esponding P=9s a es by
he U(O) aising and lowe ing ope a o s Ki', wi h pgp',
and a e hus degene a e in ene gy wi h hem. Fo his
eason, we need only conside he s a es wi h P=9 o
ully exhaus he spec um. In Table I, we display he
ou possible O=2U(6) i eps ha con ain colo less
P=9s a es and also indica e he associa ed angula
momen a. Asimila analysis can also be ca ied ou o
highe 0 alues.
TABLE I. Colo less s a es and g oup labels o 0=2.
U(6) labels
[2,2,2,o,o,o]
[2,2,1,1,0,0]
[2,1,1,1,1,0]
[1,1,1,1,1,1]
SU(2) label
j=1,3
j=0,2
j=1
j=0
Degene acy
10
6
3
1
whe e [hi, ...,hs] labels he U(6) ep esen a ions, jm a e
he SU(2) DSO(2) quan um numbe s, and ais an ex a
label ha may be needed o dis inguish ei he epea ed
U(6) i educible ep esen a ions (i eps) wi hin he same
U(6O) ep esen a ion, epea ed j's wi hin he same U(6)
ep esen a ion, o any o he necessa y quan um numbe s.
Fo example, he unpe u bed g ound s a e o he model,
de ined as he s a e o which all o=—le els a e illed,
co esponds o
[G) =[O, O, O, 0,0, 0], j=,m=—.(3.14)
30 30
In he ollowing sec ion we es ou mapping p ocedu e
o he case o O=2. He e, we illus a e he algeb aic
e alua ion o ma ix elemen s o he [2,2,2,0,0,0] O=2
subma ix, o which we will need o conside s a es wi h
j=3and j=1(see Table I). Analogous calcula ions
ha e also been done o he o he s a es and a e included
in he esul s ha we p esen .
Be o e p oceeding, we w i e down he commu a ion
ela ion o he U(6) gene a o s, which will be used ex-
ensi ely in ou analysis:
(Kq+~ +Kq )~j,m) =bi,;A ~j,m) .
To 6nd he alue o A, we no e ha
)(K&++K& )~j,m) =N ~j,m) =6~ j,m),
o 0=2. Since all colo s should be equally ep esen ed
in acolo less s a e, we a i e a he use ul ela ion
(K„'++K„' )~j,m) =2bi,;]j,m) .(3.16)
The basic idea o ou analysis is o conside he ac ion
o he ope a o s Tz and Tz on he s a es ~j,m)
(2) (~)
~3, —
3) and ]1,—
1), namely,
T, ~3, -3) =ass ]3, -1)+us 2] 1, -1),
(2) (3.17)
T( )
i3, —
3) =bs,o]3,0) +bs, 2i1,0), (3.18)
~2"' ll-1) =aiol1+1) +oi,2]3,+1), (3»)
and
&s" ]1-1) =bi,2]3+2) .(3.20)
We use ano a ion whe eby he 6 s index in he ex-
pansion coe icien s deno es he SU(2) label o he ini-
ial s a e and he second gi es he inc emen needed o
ob ain he SU(2) label o he inal s a e.
I we can de e mine all o he independen aand bex-
pansion coe icien s in (3.17)—
(3.20), we will ha e e ec-
i ely sol ed he p oblem. We can hen use he Wigne -
Ecka heo em o de e mine all he ele an educed ma-
ix elemen s o T& ~and T& ~and om hem de e mine
all o he ma ix elemen s o he Hamil onian o any
choice o he model pa ame e s. We now ou line asim-
ple algeb aic p ocedu e o e alua ing hese expansion
coeKcien s.
We 6 s conside he calcula ion o he coe icien s ap-
pea ing in (3.17) and (3.18). By using he U(6) commu-
a ion ela ion (3.15) epea edly, we ind ha
[K„' ', ,K,"*]=Si,„b, ,K,''—
bah, ~,K„"'
,.(3.15)
We simpli y he no a ion and w i e he [2, 2, 2, 0, 0, 0] se
o s a es as ~j,m). We also no e ha he U(3) ope a o s
K& —
—gK& ac as colo aising o lowe ing ope a o s
i kgi. Thus, ac ing on acolo less s a e, hey gi e ze o
unless k =i, i.e.,
430 S.PITTEL, J.M. ARIAS, J.DUKELSKY, AND A. FRANK 50
[J,T2 ]=(4J, +2)T2 +J+ )[K'+ (K" —
K"+)+(K„' —
Ki,+)K +), (3.21)
which leads o
J'T' 'IG) =2T2 '
IG) —2J+ IG) .(3.22)
Using he Wigne -Ecka heo em o ela e
(1,1[T,
'"~3, —
1) o (1,—
1~T,
''~3, —
3)
Doing he same o T3, we ob ain
(3)
J'Ts~'l ~G) =-3J+T,'" ~G) —3J+' [G) (3.23)
(3, —
1~JT2 ~3, —
3) =2asQ —4~15 (3.24)
To calcula e as Q, we mul iply (3.22) by (3, —
1~and
use (3.18). Remembe ing also ha ~G) =~3, —
3), we
ind ha
and he ea lie esul s o a3 pa3 2, and b3 2, we ind
ha
42
'l2' xp5(3.33)
F om he aand bcoe Bcien s e alua ed abo e, we can
de e mine all he emaining coe Ficien s as well as all pos-
sible educed ma ix elemen s o in e es by using he
Wigne -Ecka heo em; he esul s a e
Applying J o he le and no ing ha
asQ —
—(3, —
1iT~~ l
i3, —
3),
we ob ain
(3.2s) 0=3iT"i~ =3) =-6 14
5(3.34)
G3 p=—
2(3.26) 5
To calcula e as 2, we need he o e lap JV =(G
(T2 l) T2 ~G) .Asimple calcula ion using he com-
mu a o ela ion (3.15) gi es
(3.3s)
(3.36)
which leads o
N' =60
G3 2=12
(3.27)
(3.28) 0=3iiT&'& ii& =»=»5(3.37)
b3 P12
~s (3.29)
Likewise, o ob ain bs 2, ™1~~ply
(3.23) by (1,0~;
he esul is
36
b3 5(3.30)
Finally, we u n o he las independen coe%cien aqp,
which appea s in (3.19). We mus i s in oduce ano he
enso ope a o T4 —
—(T2 ),whose ac ion on he un-
(4) (2) 2
pe u bed g ound s a e ~G) can be eadily shown o be
JT~~ ~
~G) =4J+T2 ~G) —
8J—
+Ts ~G) .(3.31)
To de e mine ai,Q, we mul iply (3.31) by (1,1~and apply
J2 o he le . This leads o
2a3Q(1, 1~T2 ~3, —
1) +2 as 2aiQ
8as,—
2—
8V2 b3, —
2.(3.32)
The e emains an unde e mined o e all sign o his co-
e Ecien , which can be chosen a bi a ily wi h no change
in he inal esul s .
Asimila p ocedu e can be used o e alua e b3 pand
b3 —
2~In pa icula , o ob ain b3 Qwe mul iply (3.23) by
(3,0~, which leads o
15
(3.39)
These esul s coupled wi h u he use o he Wigne -
Ecka heo em pe mi us o de e mine all ma ix ele-
men s o he h ee-colo Lipkin Hamil onian, which be-
cause o he small numbe o basis s a es can be easily
diagonalized.
We ha e limi ed ou algeb aic analysis he e o 0=2,
since ha is he case o which we ca y ou mapping
es s in he nex sec ion, bu i is possible o use simila
algeb aic echniques o la ge alues o O. The me hod,
howe e , apidly becomes mo e complica ed o inc eas-
ing O. The eason is ha mo e j alues appea o la ge
0and an i e a i e p ocedu e is equi ed o de e mine he
ac ion o T2( ~and T3 on p og essi ely smalle angula
momen um s a es. As we ha e seen in he 0=2case,
each s ep in he i e a i e p ocedu e equi es he in o-
duc ion o anew enso ope a o , buil ou o he un-
damen al ones. VVe ha e al eady succeeded in ob aining
algeb aic esul s o 0=3using simila me hods. Thei
gene aliza ion o a bi a y 0is cu en ly being in es i-
ga ed .
50 CONSISTENT BARYON MAPPING OF QUARK SYSTEMS 431
IV. TEST OF MAPPING ON THE
THREE-COLOR LIPKIN MODEL
TABLE II. Numbe o dis inc colo less one- iple s a es
.p, spg ~0)S O agiVen O al P=p1 +pg +pg.
In his sec ion we apply he colo less ba yon mapping
de eloped in Sec. II o he h ee-colo Lipkin model. We
ca y ou he analysis and he esul ing compa isons o
0=2only, o which he numbe o ba yons is likewise 2.
Diagonaliza ion o he e Fec i e iple Hamil onian can
be done exac ly o his case, leading o adi ec es o
he mapping.
P
4
5
6
No. o s a es
4
6
6
4
A. Cons uc ion o he colo less ba yon space
The colo less s a es o he model, a e ca ying ou he
mapping, a e cons uc ed in e ms o ba yons wi h quan-
~~ numbe s c ~pq, 0'2@2, and T3p3. As in he p e ious
sec ion, wo noncolo quan um numbe s a e needed o
speci y he s a e o each o he h ee qua ks ep esen ed
by he colo less ba yon. Since all h ee qua ks ha e di e -
en colo s, he e is no Pauli es ic ion on hese quan um
numbe s.
As no ed in he p e ious sec ion, ause ul way o cha -
ac e ize s a es o he model is in e ms o he o al P
alue, which o asingle ba yon is P=p1+p2 +p3, In
Table II, we enume a e he numbe o dis inc colo less
one-ba yon s a es o each possible alue o P, anging
om P=3—
6(= 30).
The wo-ba yon s a es o pa icula in e es a e hose
wi h o al P=Pq +P2 ——9. The e a e wo ways o
achie e P=9, ei he wi h one iple ha ing P=3
and he o he P=6, o wi h one ha ing P=4and
he o he P=5. F om Table II, we see ha he num-
be o dis inc wo- iple s a es wi h P=9is 52; 16
ha e (P1,P2) =(3,6) and 36 ha e (P1,P2) =(4, 5). This
is signi6can ly la ge han he numbe o P=9s a es
in he o iginal qua k model (see Table I), which is 20.
The eason is ha he wo- iple space includes bo h
physical and unphysical s a es. Acen al heme o ou
analysis will be o con6 m ha ou non-He mi ian map-
ping no only ep oduces he spec um o physical s a es
(as ob ained in he algeb aic analysis o Sec. III) bu also
pushes up he unphysical s a es ela i e o he He mi ian
(pu e p-h) mapping.
B.Mapping he Hamil onian
HmH„h —
—T„h +V„h, (4.1)
whe e
The gene al h ee-colo Lipkin Hamil onian, gi en by
(3.2)—
(3.5), can be mapped ei he in non-He mi ian o
He mi ian o m. We will be pa icula ly in e es ed in he
non-He mi ian mapping, since i is expec ed o p o ide a
mo e p ac ical inco po a ion o qua k Pauli e ec s. How-
e e , in wha ollows, we p esen bo h, o see whe he ou
expec a ions a e indeed ealized.
The non-He mi ian (nh) mapping is implemen ed by
using (2.13) o he one-qua k e m, (2.23) o he wo-
qua k in e ac ion, and (2.17) o he h ee-qua k in e ac-
ion. The esul ing e ec i e Hamil onian Hh o colo -
less ba yons is gi en by
36
Tnh =2lA
J1' +pq as peag pg +ps asps asps —
pq e g ps asps px aspeas ps )
A
px ps ps~as
X& A
)+pl+pgaspg pl pgagps —
pl-psagpg +pl+psagpg )
P1PQPS ~S
36y3 A
Q2 l~1+Pl+Ps+Ps Pl ps Pg +pc —
ps —
pg +Pl +Ps+Ps )
P&P&PS
(4.2)
and
Vh —
—— +A A
)+pqaspsaepe' +pgaspsaepe(' Px PsasPs—
'aepe—
aspsaepe +aspsaepeaepe Pl PgasPS)
P1~P 3CTS ~Cog
~cs pqc spsaepe —
pgc spsc epe1 +p—
x+Pgasps aepeaspsaepe aspsaepeaepe +Pc+Psasps))
A+A A
108y3
02 (+Pl c ePec sPs +Pg+PsaePe aspsaspsaepe Pc Pg Ps
A AA
p1~pe~e ~~e
pc e e peas ps pg psaepe aepe—
a—
spsa—
epe +pa+ps+ps )
AA(4 3)
The He mi ian (h) mapping is implexnen ed by using (2.13) o he one-body e m, (2.22) o he wo body e m,
and (2.19) o he h ee-body in e ac ion. The inal esul o his mapping is