Table Of ContentLattice simulations of G -QCD at finite density
2
5
1
0
2
n
a
J
Bjoern H.Wellegehausen∗
7
2 Justus-Liebig-UniversityGiessen
E-mail: [email protected]
]
t
a LorenzvonSmekal†
l
- Justus-Liebig-UniversityGiessenandTUDarmstadt
p
E-mail: [email protected]
e
h
[
1 G2-QCD,inwhichtheexceptionalLiegroupG2 replacestheSU(3)gaugegroupofQCD,does
v notsufferfromafermionsignproblem. Itcanthereforebesimulatedalsoatcomparativelylow
6
temperaturesandhighdensitiesonthelattice,whichhenceallowstomapoutthephasediagram
0
7 ofthisQCD-liketheory. Webrieflyreviewsomeofourpreviousresultsfromsuchfinitedensity
6
simulations to then present further evidence for a first-order transition to what might be called
0
. G2-nuclear matter. In order to isolate diquark condensation effects, we introduce simulations
1
0 with Majoranafermionsand diquarksources. Thisallows to disentanglestates in the spectrum
5 thatareconnectedbychargeconjugation.Wediscusschiralsymmetryinthepresenceofdiquark
1
: sourcesandpresentfirstresultsfromourongoinglarge-scalesimulations.
v
i
X
r
a
32ndInternationalSymposiumonLatticeFieldTheoryLATTICE2014
Juny23-Juni28,2014
ColumbiaUniversity,NewYork,USA
∗Speaker.
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
G -QCDatfinitedensity LorenzvonSmekal
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1. Introduction
G -QCDisaQCD-liketheoryinwhichthegaugegroupSU(3)ofstronginteractionsisreplacedby
2
theexceptionalLiegroupG . Thetheoryisagaugetheorywithfermionicbaryonsandfundamental
2
quarks[1,2]anditcanbesimulatedwithoutsignproblematfinitedensityandtemperature. Unlike
other QCD-like theories such as adjoint QCD or two color QCD, for example, its properties in
the quenched case are very similar to those of QCD [3–8]. Although the center of G is trivial, it
2
showsafirstorderdeconfinementtransitionwhichhasquiteinterestingimplicationsfortheroleof
thecentersymmetryinQCDasreviewedin[9]forLattice2012.
Inthiscontribution webrieflysummarizeourpreviousresults[10]forthehadronspectrumin
the vacuum and the phase structure at zero temperature as seen in the quark density from lattice
simulations on rather small 83×16 lattices in Sections 2 and 3. The observed structures in the
density over the chemical potential can thereby be related to a corresponding hierarchy of mass
scales in the baryon spectrum. In particular, one observes thresholds in the baryon density at
values of the chemical potential that correspond to the pseudo-Goldstone scalar diquark scale,
an intermediate pseudo-scalar and vector diquark scale, and roughly the fermionic baryon scale
set by the G -nucleons and delta baryons. In Section 4 we furthermore present results from our
2
latestsimulations inwhichwehaveaccumulated evidence forazero-temperature first-orderphase
transition towhatmightbecalledG -nuclearmatter.
2
Themost important difference to QCDisthe existence of diquarks inthe hadronic spectrum.
Thelightest twodiquarkstatesarethepseudo-Goldstone bosonsofchiralsymmetrybreaking, and
their quantum numbers differ only bycharge conjugation. Inorder toinvestigate chiral symmetry
breaking in detail we add diquark source terms to disentangle the charge conjugation symmetry.
ThereforewehavetouseMajoranafermionsinthesimulations. InSection5wereviewtheformu-
lation ofG -QCDinterms ofMajorana fermions and discuss the chiral properties in thepresence
2
ofdiquark sources inmoredetail. Firstpreliminary results ofoursimulations atfinitetemperature
anddensity areshowninSections6and7.
2. Chiral symmetry andbaryonnumber inG -QCD
2
TheEuclideanactionofN =1flavourG -QCDwithquarkchemicalpotential m forbaryonsreads
f 2
1
S= d4x − trFmn Fmn +Y¯ D[A,m,m ]Y with
4 (2.1)
Z (cid:26) (cid:27)
D[A,m,m ]=g m (¶ m −gAm )−m+g0m ,
wherethegaugegroupistheexceptional Liegroup G2,andgm =gm† aretheEuclidean g -matrices.
ThefundamentalrepresentationsofG are7-dimensionaland14-dimensional, thelattercoinciding
2
with theadjoint representation. Since G isasubgroup ofSO(7), allrepresentations arereal. The
2
Diracoperator satisfies
D(m )†g =g D(−m ∗) and D(m )∗T =TD(m ∗) (2.2)
5 5
with T =Cg , T∗T =− , T† =T−1 and charge conjugation matrixC. If such a unitary opera-
5
1
tor T exists then the eigenvalues of the Dirac operator come in complex conjugate pairs, all real
2
G -QCDatfinitedensity LorenzvonSmekal
2
U(2)
anomaly
m, YY¯
SU(2)⊗Z(2)B SO(2)V⊗Z(2)B
(cid:10) (cid:11)
m m
m, YY¯
U(1)B U(1)B
(cid:10) (cid:11)
Figure1: PatternofchiralsymmetrybreakinginG -QCD.
2
eigenvalues aredoublydegenerate [11,12]andthus
detD[A,m,m ]≥0 for m ∈ . (2.3)
R
G -QCD with a single Dirac flavour possesses an extended chiral symmetry [11] compared to
2
QCD. The action is invariant under the SO(2)V vector transformations and the usual axial trans-
formations leading toaU(2)symmetry group, inagreement withthe results in[1]. Following the
same arguments as in QCDit is expected that the axialU(1)is broken by the axial anomaly such
that only a SU(2)×Z(2)B chiral symmetry remains. In the presence of a non-vanishing Dirac
mass term (or anon-vanishing chiral condensate) the theory is no longer invariant under the axial
transformations. Thereforethenon-anomalous chiralsymmetryisexpectedtobebrokenexplicitly
(orspontaneously) toitsmaximalvectorsubgroup,
SU(2)⊗Z(2)B7→SO(2)V⊗Z(2)B, (2.4)
Theremainingchiralsymmetryatfinitebaryonchemicalpotential isthesameasinQCD,
SU(2)⊗ (2)B7→U(1)B (2.5)
Z
ThefinalpatternofchiralsymmetrybreakingofG -QCDisshowninFigure1. Ifchiralsymmetry
2
is spontaneously broken, the axial chiral multiplet becomes massless, according to the Goldstone
theorem. Thefollowingoperators generate thetwoGoldstonebosons:
d(0++)=Y¯Cg Y −Y¯g Y C and d(0+−)=Y¯Cg Y +Y¯g Y C. (2.6)
5 5 5 5
They have quark number n = 2 and hence also carry a baryon number of n = 2/3, if baryon
q B
numbercountsthedifferenceofquarksandanti-quarks perG -nucleonasinQCD.TheGoldstone
2
bosons in G -QCDare scalar diquarks instead of pseudoscalar mesons as in QCD.Aslong as we
2
donotintroduce diquark sources, d(0++)andd(0+−)havethesamemass. InSection5belowwe
alsointroducethecorresponding diquarksourceterms,however,inordertodisentangle stateswith
opposite charge quantum numberandinvestigate chiralsymmetrybreaking moreclosely.
3
G -QCDatfinitedensity LorenzvonSmekal
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3. Spectroscopy andthe phasediagram atzero temperature
Thepossiblequarkandgluoncontent of(colorless) boundstatesisdetermined bythetensorprod-
ucts of the appropiate representations of the gauge group G . Quarks in G transform under the
2 2
7-dimensional fundamental representation, gluons under the 14-dimensional fundamental (and at
thesametimeadjoint)representation. Anoverviewoverpossibleboundstatescanbefoundin[10].
We expect to find bound states for every integer quark number n . Mesons have n =0, diquarks
q q
n =2,andnucleons n =3. Inaddition, therearemoreexoticboundstatesofgluons andquarks,
q q
for example a hybrid with n = 1. In the following we give an overview over the bound states
q
consideredhere,whereuandd denoteflavoursofDiracfermions. ForthenucleonsN andthepion
p we make use of the partially quenched approximation in our one-flavour simulations. Table 1
shows bound states that are also present in QCD while Table 2 shows the diquarks. In all tables
Name O T J P C Name O T J P C
p u¯g d SASS 0 - + N Tabc(u¯Cg d )u SAAA 1/2 ± ±
5 a 5 b c
h u¯g5u SASS 0 - + D Tabc(u¯Cagm ub)uc SSAS 3/2 ± ±
Table 1: Bound states of G -QCD with 2 flavours for baryon number n =0 (left) and baryon number
2 B
n =1,i.e.quarknumbern =3(right).Fordetailsseetext.
B q
Name O T J P C
d(0++) u¯Cg u+c.c. SASS 0 + +
5
d(0+−) u¯Cg u−c.c. SASS 0 + -
5
d(0−+) u¯Cu+c.c. SASS 0 - +
d(0−−) u¯Cu−c.c. SASS 0 - -
d(1++) u¯Cgm d−d¯Cgm u+c.c. SSSA 1 + +
d(1+−) u¯Cgm d−d¯Cgm u−c.c. SSSA 1 + -
d(1−+) u¯Cg5gm d−d¯Cg5gm u+c.c. SSSA 1 - +
d(1−−) u¯Cg5gm d−d¯Cg5gm u−c.c. SSSA 1 - -
Table2: Boundstateswithbaryonnumbern =2/3,i.e.quarknumbern =2.
B q
O is the interpolating operator used to extract the mass in simulations, the string T represents the
behaviour of the wave function under change of position, spin, colour and flavour (S stands for
symmetric,Aforanti-symmetric), andJ,P,Carethespin,parityandchargeconjugation quantum
numbers. The difference between the h and the diquark correlation function is only the discon-
nected contribution. Therefore, thediquark withpositive parity hasthesamemassathepion with
negative parity, md(0+) =mp (0−). In [10] it is shown that for every diquark there is a flavour non-
singlet meson with the same mass but opposite parity. In the following we discuss two different
ensembles with lattice parameters as listed in Table 3. A physical scale is set by the proton mass,
m =938 MeV. The mass spectrum for both ensembles is shown in Figure 2. In the heavy en-
N
semble the diquark masses and all parity even and odd states are almost degenerate. In the light
ensemble the diquark masses are no longer degenerate. We observe a significant mass splitting
4
G -QCDatfinitedensity LorenzvonSmekal
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Ensemble b k m a m a m [MeV] a[fm] a−1 [MeV] MC
d(0+) N d(0+)
Heavy 1.05 0.147 0.59(2) 1.70(9) 326 0.357(33) 552(50) 7K
Light 0.96 0.159 0.43(2) 1.63(13) 247 0.343(45) 575(75) 5K
Table3: Parametersforthetwodifferentensembles.Allresultsarefrom83×16lattices.
between parity even and odd states as well as between scalar and vector diquarks. Especially the
Goldstone boson becomes the lightest state, with the h also being somewhat heavier. For the nu-
cleons we also observe different masses for parity even and odd states and the spin 1/2 and spin
3/2 representations. In particuclar, we find three clearly different scales in the light spectrum: a
pseudo-Goldstone scale, an intermediate boson scale set by the remaining diquarks, and the nu-
cleon scale set by the N and D masses. Thismass hierarchy ofthe spectrum seems tobe reflected
in various structures of the quark density at zero temperature which one might thus attribute to
different bosonic and fermionic phases at finite density, see Figure 3. With increasing chemical
potential, the quark number density first remains consistent with zero until it very quickly rises
to a very small but nonzero value. When we compare the critical chemical potential m for this
c
onset transition tothemassofthe lightest baryon m ,thepseudo-Goldstone 0+ diquark inour
d(0+)
case, we find that numerically very good agreement with the expectation from the Silver Blaze
property, i.e. m =m /2. Thegroundstatechanges fromthevacuum toafinite-density ground
c d(0+)
stateonlywhenthequarkchemicalpotential reachesthemassofthelightestbaryon dividedbyits
quark number so that the corresponding excitation energy vanishes. For bosonic excitations one
might expect Bose-Einstein-condensation in a continuous second-order quantum phase transition
at m =m /2,without binding energy, andourdataiscertainly consistent withthat. Forlarger
c d(0+)
values ofthechemical potential plateaus develop wherethequark number density remains almost
constant. Especially in the light ensemble, the step towards the second plateau conicides with the
mass of the heavier bosonic diquark states divided by their quark number. It appears that the two
bosonic baryon mass scales are not sufficiently separated from each other to resolve these two
distinct transitions intheheavierensemble.
At around am =0.6 for the heavy ensemble and am =0.55 for the light ensemble the quark
2000
3.5 N∗ N∗ D ∗ D ∗ 3.5 N∗ N∗ D ∗ D ∗ 2000
3.0 3.0
1500
1500
2.5 2.5 d∗
a2m.0 d∗ d∗ h ∗ N N d∗ d∗ D D 938 a2m.0 d∗ hd∗∗ N N d∗ D D 938
1.5 1.5
minMeV minMeV
d
1.0 1.0
500 500
0.5 d d h d d 326 0.5 d hd d 247
0.0 0 0.0 0
0+ 0− 1+ 1− 1+ 1− 3+ 3− 0+ 0− 1+ 1− 1+ 1− 3+ 3−
2 2 2 2 2 2 2 2
Figure2: Massspectrumfortheheavy(left)andlight(right)ensembleinG -QCD.
2
5
G -QCDatfinitedensity LorenzvonSmekal
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m inMeV m inMeV
0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 450
0.12 0.12
D (3−)
0.10 D (3−) 0.10 2
2 D (3+)
0.08 D (3+) 0.08 2
2
nq 0.06 d(0−) nq 0.06
0.04 d(0+) 0.04 d(0−)
0.02 0.02 d(0+)
0.00 0.00
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
am am
Figure3: Quarknumberdensityfortheheavy(left)andlight(right)ensembleinG -QCDatfinitedensity
2
andzerotemperature.
number density starts increasing again and no further plateau is observed. This transition appears
to coincide with the mass scale of nucleon and D divided by three. In both ensembles the general
pattern thus seems to be that the various transitions in the quark number density are related to
the various baryon masses in units of their quark number. While for bosonic baryons the density
bendstowardsthezero-axiswithplateausformingaftereachtransition,atthescaleofthefermionic
baryons thequarknumberdensityisconvex,asseenmostclearlyinthelightensemblewithbetter
separation of scales, and continues to further increase with increasing chemical potential until
saturation sets in, eventually, when the lattice starts to get filled with the maximum number of
quarks per site, i.e. at a3n = 14 here. This clearly is a lattice artifact beyond the range of any
q
hadronic interpretation ofthedensityanditistherefore notshownhereagain,see[10].
m inMeV
180 200 220 240 260
0.035
0.030
0.025
nq 0.020
0.015
0.010
0.005
0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46
am
Figure4: Quarknumberdensityforthelightensembleinthevicinityofthetransition.
6
G -QCDatfinitedensity LorenzvonSmekal
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4. Evidence ofa first ordernuclear matter transition
Inbothensemblesweobserveastrongtransitionatam ≈0.52(heavyensemble)andam ≈0.38
(light ensemble) that does not appear to correpond to any of our spectroscopic states. In Figure 4
we show the vicinity of this transition in the light ensemble in more detail. The quark number
density rises between am =0.36 and am =0.40 from a lower value n ≈0.010 to a higher value
q
n ≈0.025. InFigure5weshowthequarknumberdensity asafunction ofMonte-Carlo timeand
q
observe tunneling between these two states. This might indicate that there is a first-order phase
transition atam ≈0.38 in the phase diagram at zero temperature. Whether this phase transition is
indeed the analogue of the liquid-gas transition of nuclear matter as expected in QCD remains to
be shown by further simulations. If this is the case, then either the binding energy per nucleon is
comparatively largeorthemassesofnucleon andD change withdensityintheregimeofthefinite
bosonic baryondensity inthegroundstatebeforethistransition whichisnotpossible inQCD.
0.04 0.04 0.04
0.03 0.03 0.03
0.02 0.02 0.02
0.01 0.01 0.01
0.00 0.00 0.00
-0.01 -0.01 -0.01
0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500
MonteCarlotime MonteCarlotime MonteCarlotime
0.04 0.04 0.04
0.03 0.03 0.03
0.02 0.02 0.02
0.01 0.01 0.01
0.00 0.00 0.00
-0.01 -0.01 -0.01
0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 3000 3500
MonteCarlotime MonteCarlotime MonteCarlotime
Figure5:QuarknumberdensityasafunctionofMonte-Carlotimefordifferentvaluesofchemicalpotential.
Fromlefttorightintheupperrowam =0.36,am =0.37andam =0.38andinthelowerrowam =0.39,
am =0.40andam =0.41.
5. Simulations ofG2-QCDwithMajorana fermions
In the present section wediscuss the introduction of diquark sources in G -QCDfor tworeasons:
2
First, on larger lattices and especially for values of the chemical potential in the vicinity of the
firstordertransition thesimulations becomemoreandmoreexpensive. Anobviousreasonforthis
mightbethepresenceofverylightdiquarkexcitationsinthesimulationsinthisregionofthephase
diagram. Withtheintroduction ofdiquark sourcetermssimulations should becomemorefeasible.
Secondly,wewouldeventuallyliketoresolvethecompletediquarkspectruminordertoinvestigate
chiral symmetry breaking atfinitedensity. Anydiquark source necessarily consists ofanoperator
7
G -QCDatfinitedensity LorenzvonSmekal
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with quark number n =2 and therefore contains a pair of charge conjugated Dirac spinors Y for
q
the quarks. In order to integrate over the fermion fields in the path integral, however, one needs
a bilinear expression in Y¯ and Y . Unlike two-color QCD, however, for a single flavor of Dirac
fermions it is then necessary to introduce corresponding Majorana fermions in the simulations.
SincethegaugefieldsinG2 satisfy ATm =−Am =−Aam Ta itispossible torewritethematterpartof
theactionin(2.1)asasumover2for m =0atfirstuncoupled Majoranaspinors l =(c ,h ),
S[Y ,A]= d4xY¯ g m (¶ m −gAm )−m Y = d4xl¯ g m (¶ m −gAm )−m l (5.1)
Z Z
Herel obeystheMajorana c(cid:0)ondition l C=Cl¯T(cid:1)=l ,l¯C=−(cid:0)l TC−1=l¯,andit(cid:1)isrelated tothe
Diracspinor asY =c +ih ,Y¯ =c¯ −ih¯,Y C=c −ih andY¯C=c¯ +ih¯. The(baryon) chemical
potential m isanoff-diagonal terminMajoranaflavourspacesuchthat
L =Y¯ D(m,m )Y =l¯ M(m,m )l with
(5.2)
D(m,m )=D/−m+g m and M(m,m )=(D/−m)s −mg s
0 0 0 2
where s = and the Pauli matrix s act on the 2 Majorana flavours c and h which are now
0 2
1
coupled toeach otherbytheoffdiagonal ms -term. Integration inthepath integral overtheMajo-
2
rana fermions leads to the Pfaffian instead of the fermion determinant and we can proove that the
Pfaffianispositiveasexpected.
It is possible to introduce two different diquark sources in G2-QCD, one for the scalar (Lg5)
andoneforthepseudoscalar diquarks (L ),
1
1 1
Lg5(J˜)= 2 J˜Y¯Cg5Y −J˜∗Y¯g5Y C , L1(J)= 2 JY¯CY +J∗YY¯ C . (5.3)
(cid:16) (cid:17) (cid:16) (cid:17)
IntheMajoranadecomposition thesetermsread(J=J +iJ ,J˜=J˜ +iJ˜)
1 2 1 2
c¯ J J c
Lg5 =i h¯ J2 −J1 g5 h =il¯ J˜1s 1+J2s 3 g5l ,
! 1 2! !
(cid:0) (cid:1) (5.4)
c¯ J −J c
L = 1 2 =l¯ (J s −J s )l .
1 h¯! −J2 −J1! h ! 1 3 2 1
TheLagrangedensityforthematterpartofthetheoryisthengivenby
L =l¯ (D/−m−m g )s −mg s +(iJ˜g −J )s +(J +iJ˜g )s l . (5.5)
5 5 0 0 2 1 5 2 1 1 2 5 3
Similartothecasew(cid:2)ithoutdiquarksourcesonecanshowthatthePfaffianisrealifJ(cid:3) =J˜ =0,butit
2 2
isnotnecessarilypositiveanylonger. Nevertheless,weexpectthatforsmallvaluesofthechemical
potential thesign problem isnotpresent andthisexpectation isconfirmedinoursimulations. The
first derivatives of the partition function with respect to J, J˜and m define the chiral and diquark
condensates,
1 ¶ ln(Z(m,J,J˜))
S = = YY¯ =hcc¯ +hh¯ i,
V ¶ m
1 ¶ ln(Z(m,J,J˜)) (cid:10) (cid:11)
S = = Y¯CY +c.c. =hcc¯ −hh¯ i, (5.6)
1 V ¶ J
1 ¶ ln(Z(m,J,J˜)) D E
S = = Y¯Cg Y +c.c. =ihcg¯ h +hg¯ c i,
5 V ¶ J˜ 5 5 5
D E
8
G -QCDatfinitedensity LorenzvonSmekal
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Operator Parameter OA,1 OA,3 OV,2 Goldstonebosons Massivestate
l¯l m x x X d(0++),d(0+−) f(0++)
l¯g s l J˜ x X x d(0++), f(0++) d(0+−)
5 1 1
l¯g s l J˜ X x x d(0+−), f(0++) d(0++)
5 3 2
l¯g l m x x X d(0−+),d(0−−) h (0−+)
5 5
l¯s l J x X x d(0−+),h (0−+) d(0−−)
1 2
l¯s l J X x x d(0−−),h (0−+) d(0−+)
3 1
l¯g s l m x x X - -
0 2
Table 4: The table shows the transformation behaviour of bilinears under the chiral transformations, X
means invariant, x not invariant. In the last two columns the corresponding Goldstone bosons and the
massivestatesareshown.
that weinvestigate inthe following. Forasingle Dirac flavour the chiral symmetry is SU(2)L=R∗.
Thegenerators forthesymmetrytransformations aregivenbyPaulimatrices
TV= ⊗s 2, TA=g5⊗{s 1,s 3}, (5.7)
1
andthechiraltransformations read
OA,1l =eiag 5s 1l and l¯ →l¯ eiag 5s 1
OA,3l =eiag 5s 3l and l¯ →l¯ eiag 5s 3 (5.8)
OV,2l =eias 2l and l¯ →l¯ e−ias 2.
Possiblebilinear boundstatesforasingleDiracflavourare
d(0+−)=l¯g s l =cg¯ h , d(0++)=l¯g s l =cg¯ c −hg¯ h ,
5 1 5 5 3 5 5
d(0−−)=l¯s l =ch¯ , d(0−+)=l¯s l =cc¯ −hh¯ , (5.9)
1 3
f(0++)=l¯l =cc¯ +hh¯ , h (0−+)=l¯g l =cg¯ c +hg¯ h .
5 5 5
Table 4 shows their behaviour under the chiral transformations together with the corresponding
Goldstone bosons. UnderthechiralSU(2)thesebilinersdecompose as
2¯⊗2=3⊕1, (5.10)
andwecanidentify apositiveandnegativeparitytriplet,
f(0++), d(0++), d(0+−) and h (0−+), d(0−+), d(0−−) . (5.11)
(cid:16) (cid:17) (cid:16) (cid:17)
Since the negative parity multiplet obtains a contribution to its mass from the chiral anomaly we
will set the corresponding sources J =J =m =0. Under a general (infinitesimal) chiral trans-
1 2 5
formation dl =i(ag s +bg s +gs )l theLagrangian densitytransforms as
5 1 5 2 2
d L =2il¯ g s (−a m+g J˜)+g s (−b m−g J˜)+i(a J˜ +b J˜)+img g (bs −as ) l ,
5 1 2 5 3 1 1 2 0 5 1 2
(5.12)
(cid:0) (cid:1)
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G -QCDatfinitedensity LorenzvonSmekal
2
m m J˜ Solution Symmetry Generator
1,2
X X X SU(2) T ={g s ,g s ,s }
5 1 5 3 2
X x X a =b =0 U(1) T =s
2
X X x g =0,a J˜ =−b J˜ U(1) T =g (J˜s −J˜s )
1 2 5 2 1 1 3
X x x a m=g J˜,b m=−g J˜ U(1) T =g (J˜s −J˜ s )+ms
2 1 5 2 1 1 3 2
x X X a =b =0 U(1) T =s
2
x x X a =b =0 U(1) T =s
2
x X x a =b =g =0 - -
x x x a =b =g =0 - -
Table5: ChiralsymmetryofG -QCDinthepresenceofdiquarksourcesandchemicalpotential,X means
2
=0,xmeans6=0.
andweobtainthefollowingsystemofequations fortheinvariance oftheLagrangian
ma =0, mb =0, a m=g J˜ , b m=−g J˜ , a J˜ =−b J˜. (5.13)
2 1 1 2
Possible solutions are shown in Table 5. The Goldstone bosons are then linear combinations in
thecorresponding multiplet. Anillustration ofthechiralsymmetrybreaking isshowninFigure6.
At vanishing chemical potential, any linear combination of mass m and diquark sources J˜ and J˜
1 2
lin(m,J˜,J˜)
1 2
SU(2)L=R∗ U(1)B
m
U(1)B lin(J˜1,J˜2)
m m m
m U(1)B lin(J˜1,J˜2)
U(1)B -
lin(m,J˜,J˜)
1 2
Figure6: ChiralsymmetrybreakinginG -QCDinthepresenceofdiquarksources.
2
breaks thechiralsymmetrydowntoaU(1)subgroup andbaryon numberisconserved. Thedirec-
tion(generator) fortheinvariantsubgroupU(1)oftheSU(2)symmetryisshowninTable5. Since
the massmandthechemical potential m break theSU(2)inthesamedirection, baryon number is
10
