Covariant QCD Modeling of Light Meson Physics

6 Pages·0.21 MB·English

by  Peter C. Tandy

Covariant QCD Modeling of Light Meson Physics∗
Peter C. Tandy†
Center for Nuclear Reseach, Department of Physics,
Kent State University, Kent, Ohio 44242 U.S.A.
(Dated: January 7, 2003)
Wesummarizerecentprogressinsoft QCDmodeling basedonthesetofDyson–Schwingerequa-
tions truncated to ladder-rainbow level. This covariant approach to hadron physics accommodates
quarkconfinementandimplementstheQCDone-looprenormalizationgroupbehavior. Wecompare
the dressed quark propagator, pseudoscalar and vector meson masses as a function of quark mass,
and the ρ→ππ coupling to recent lattice-QCD data. The error in the Gell-Mann–Oakes–Renner
relation with increasing quark mass is quantified by comparison to the exact pseudoscalar mass
relation as evaluated within theladder-rainbow Dyson-Schwingermodel.
3
0 I. INTRODUCTION absence of a real mass pole for dressed quark and gluon
0 propagatorshasbeenstudiedandfoundtobeasufficient,
2 but not necessary, condition for confinement [1, 7, 8, 9].
The study of light-quark pseudoscalar and vector

n mesonsisanimportanttoolforunderstandinghowQCD Themodelprovidesanefficientdescriptionofthemasses
a and decay constants of the light-quark pseudoscalar and
worksinthenon-perturbativeregime. Thepseudoscalars
J vector mesons [2, 4], the elastic charge form factors
are important because they are the lightest observed
0 F (Q2) and F (Q2) [10] and the electroweak transition
hadrons and are the Goldstone bosons associated with π K
1 form factors of the pseudoscalars and vectors [11, 12].
dynamical chiral symmetry breaking. The ground state

1 vector mesons are important because, as the lowest spin
v excitations of the pseudoscalars, they relate closely to
0 hadronic q¯q modes that are electromagnetically excited. II. DYSON-SCHWINGER EQUATIONS
4
We use a Poincar´e covariantmodel defined within the
0
framework of the Dyson–Schwinger equations [DSEs] of The dressed quark propagatorS(p) is the solution to
1
QCD;theseformanexcellenttooltostudynonperturba-
0
3 tiveaspectsofhadronproperties[1]. Itisstraightforward S(p)−1 = Z2i/p+Z4m(µ)
0 toimplementthecorrectone-looprenormalizationgroup Λ λi
h/ behavior of QCD [2], and obtain agreementwith pertur- +Z1Z g2Dµν(p−q) 2 γµS(q)Γiν(q,p) ,(1)
q
bation theory in the perturbative region. Provided that
t
- the relevant Ward–Takahashi identities are preserved in
l where D (k) is the renormalized dressed-gluon prop-
c the truncation of the DSEs, the corresponding currents agator, Γµiν(q,p) is the renormalized dressed quark-gluon
u ν
n are conserved. Axial current conservation induces the vertex. Thenotation Λ = Λd4q/(2π)4denotesatrans-
Goldstone nature ofthe pions andkaons [3]; electromag- q
:
v netic current conservation produces the correct electric lationally invariant rRegularRization of the integral with
i mass-scale Λ. The solution is renormalized according to
X chargeof the mesons without fine-tuning. These proper-
S(p)−1 =iγ p+m(µ)atasufficientlylargespacelikeµ2,
r ties are implemented herewithin the rainbowtruncation with m(µ) th·e renormalized quark mass at the scale µ.
a of the DSE for the dressed quark propagators together
The renormalizationconstants Z andZ depend on the
with the ladder approximation for the Bethe–Salpeter 2 4
renormalization mass-scale µ and on the regularization
equation [BSE] for meson bound states.
mass-scaleΛ. ThelimitΛ istobetakenattheend
The model [4] we use has two infrared parameters →∞
of all calculations.
which specify the momentum distribution and strength
The BSE for a a¯b meson is
of the ladder-rainbow kernel at a low scale necessary to
generate an empirically acceptable amount of dynamical Λ
chiral symmetry breaking [5, 6] as measured by the chi- Γa¯b(p+,p−)= K(p,q;P)Sa(q+)Γa¯b(q+,q−)Sb(q−),
Z
ralcondensate. As a corollary,the strong dressingofthe q
(2)
quark propagatorshifts the mass pole significantly away
where K is the renormalized qq¯scattering kernel that is
fromtherealtimelikep2 axis. Theproducedboundstate
irreducible with respect to a pair of qq¯lines. The quark
mesons do not have a q¯q decay width and, in this sense,
momenta are q ; the meson momentum is P =q q
the present model implements quark confinement. The and satisfies P±2 = m2. The relative momentu+m−q −is
−
introducedbyq =q+ηP andq =q (1 η)P where
+ −
− −
η isthemomentumpartitioningparameter. Physicalob-
servables should not depend on η and this provides a
∗Presented at International School on Nuclear Physics, Erice,
September 2002;toappearinProg. Part. Nucl. Phys. convenient check on numerical methods. We employ the
†[email protected] model that has been developed recently for an efficient
2
lattice M(p), ma = 0.036
600 1.1 TABLE I: The pseudoscalar observables that define the
DSE soln, m = 0.6 m
q strange present ladder-rainbow DSE-BSE model, adapted from
lattice Z(p), ma = 0.036, rescaled
Refs. [2, 4].
500 DSE soln, m = 0.6 m 1.0
q strange
experiment calculated
V) 400 0.9 (estimates) († fitted)
e p)
M(p) (M 300 0.8Z( mmµusµ===11dGGeeVV 1005--31000 MMeeVV 152.55 MMeeVV
- hq¯qi0 (0.236 GeV)3 (0.241†)3
200 0.7 µ
mπ 0.1385 GeV 0.138†
100 0.6 fπ 0.131 GeV 0.131†
mK 0.496 GeV 0.497†
0 fK 0.160 GeV 0.155
0 1 2 3 4
p (GeV)
work [2, 4]
FIG.1: DSEsolution[4,13]forquarkpropagatoramplitudes
compared to recent lattice data [14, 15].
G(k2) = 4π2Dk2 e−k2/ω2+ 4π2γm F(k2) ,
k2 ω6 2
1ln τ + 1+k2/Λ2
2 (cid:20) (cid:16) QCD(cid:17) (cid:21)
descriptionofthemassesanddecayconstantsofthelight
(5)
pseudoscalar and vector mesons [2, 4]. This consists of with γ = 12 and (s)= (1 exp( −s ))/s. The
therainbowtruncationoftheDSEforthequarkpropaga- m 33−2Nf F − 4m2t
first term implements the strong infrared enhancement
torandtheladdertruncationoftheBSEforthepionand in the region 0<k2 <1GeV2 required for sufficient dy-
kaon amplitudes. The requiredeffective q¯q interaction is
namical chiral symmetry breaking. The second term
constrained by perturbative QCD in the ultraviolet and
serves to preserve the one-loop renormalization group
has a phenomenologicalinfraredbehavior. In particular,
behavior of QCD. We use m =0.5GeV, τ =e2 1,
the rainbow truncation of the quark DSE, Eq. (1), and t −
N =4, and we take Λ =0.234GeV. The renormal-
the ladder truncation of the BSE, Eq. (2), are f QCD
izationscaleischosentobeµ=19GeVwhichiswellinto
the domain where one-loop perturbative behavior is ap-
Z g2D (k)Γi(q,p) 4πα (k2)Dfree(k)γ λi , (3) propriate[2,4]. Theremainingparameters,ω =0.4GeV
1 µν ν → eff µν ν 2 andD =0.93GeV2 alongwiththequarkmasses,arefit-
ted to give a good description of q¯q , m and f as
π/K π
and h i
shown in Table I. The subsequent values for f and the
K
massesanddecayconstantsofthevectormesonsρ,φ,K⋆
K(p,q;P) 4πα (k2)Dfree(k)λiγ λiγ , (4) are found to be within 10% of the experimental data [4].
→− eff µν 2 µ⊗ 2 ν
In Fig. 1 we compare the DSE model propagator am-
where Dµfrνee(k = p−q) is the free gluon propagator in plitudes defined by S(p)= Z(p2)[i/p+ M(p2)]−1 with
Landau gauge. These two truncations are consistent in
the most recent results in lattice QCD using staggered
thesensethatthecombinationproducesvectorandaxial-
fermions in Landau gauge [14, 15]. These simulations
vectorverticessatisfyingtherespectiveWTIs. Intheax-
weredonewiththeAsqtadimprovedstaggeredquarkac-
ial case, this ensures that in the chiral limit the ground tion,whichhaslatticeerrorsoforder (a4)and (a2g2).
state pseudoscalar mesons are the massless Goldstone O O
Fig. 1 shows both M(p) and Z(p) obtained with a bare
bosons associated with chiral symmetry breaking [2, 3].
lattice mass of ma = 0.036 in lattice units, which corre-
In the vector case, this ensures electromagnetic current
sponds to a bare mass of 57 MeV in physical units. The
conservation. The “effective coupling” α (k2) defines
eff DSE calculations use a current mass value of 75 MeV
the model. The ultravioletbehavior is chosen to be that
atµ = 1 GeV to match the lattice mass function around
of the QCD running coupling α(k2); the ladder-rainbow
3GeV;thiscurrentmassisabout0.6m . Thereisagree-
s
truncation then generates the correctperturbative QCD
ment in the qualitative infrared structure of M(p) and
structureoftheDSE-BSEsystemofequations. Thephe-
Z(p). Sincethelatticesimulationproducestheregulated
nomenologicalinfrared form of α (k2) is chosenso that
eff butun-renormalizedpropagator,thescaleofZ(p)isarbi-
the DSE kernel contains sufficient infrared enhancement
traryand we have rescaledthe lattice Z(p) to match the
to produce anempirically acceptableamountofdynami-
DSEsolutionat3GeV.ForZ(p)theladder-rainbowDSE
calchiralsymmetrybreakingasrepresentedbythechiral
model saturates much slower than does the lattice; this
condensate [18].
maysignaladeficiencyofthebaregluon-quarkvertex. A
WeemploytheAnsatzfoundtobesuccessfulinearlier recent study of the coupled ghost-gluon-quarkDSEs has
3
2.2
3.2 2
2.8 1.8
1.6
2.4
V) 1.4
e
2 G
GeV)1.6 M (V1.21
M (V1.2 DJ/ψSE− ηcca l(cE xpt) M , P0.8 ρP,S K m*a, s sD r*e l a(Etixopnt ()DSE fit)
fit (DSE calc + J/ψ−η) M DSE calc
c 0.6 V
0.8 Lattice [CP-PACS] M DSE calc
Lattice [UK QCD] 0.4 π, PK, D (Expt)
0.4 0.2 GMOR mass relation
00 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 00 0.2 0.4 0.6 0.8 1 1.2 1.4
m (µ=1 GeV) (GeV)
M (GeV) Q
P
FIG. 3: MV(mQ) and MP(mQ) for unequal flavor u − Q
mesons. The DSEmodel-exact mass relation reveals thesize
FIG. 2: DSE calculation of equal flavor vector meson mass
of the correction to theGMOR relation.
variationwithpseudoscalarmeson massas mq isvariedcom-
pared to lattice data from CP-PACS [16] and UK-QCD [17].
densate at scale µ, the current masses are determined at
the same scale,andf0 isthe chirallimit electroweakde-
P
found that the quark-gluon vertex dressing can produce cayconstant(inthef =92.4MeVconvention). Foru/d
π
achangeofthischaracterintheinfraredstructureofthe quarks the GMOR relation is satisfied to high accuracy
quark amplitudes [19, 20]. (within 0.2% in the present DSE model). For current
In Fig. 2 we compare the rainbow-ladder DSE model masses of the order of ms 120 MeV and above, the
∼
with unquenched lattice data for the variation of vector question of the size of the error in the GMOR relation
andpseudoscalarmesonmasseswithquarkcurrentmass is not settled. An exact mass relation for pseudoscalar
in the case of equal flavor quarks. The DSE calculation,
shownbythediscretecircles,islimitedtothemassrange
where it is reliable. The solidline is a fit to those results
0.28
plus the experimental J/ψ η point. The curvature at
c
lTohwismcaosmspiasrcisoonnsiisstecnotnswisitteh−ntMwVit∝h tmheq kannodwMnPpr∝op√ermtieqs. 0.26 Dπ,S KE (cEaxlcpt) + D, B (Lattice)
0.24
of the DSE results: M is 5% too low for the ρ and 5% fit to DSE calc + D, B (Lattice)
V
too high for the φ, while the pseudoscalar masses in the 0.22
u-quark and s-quark regions are fit to experiment. (If V) 0.2
the evident fit is continued to the Υ vector mass, the Ge
predicted ηb mass would be 10.0 GeV.) f (P0.18
0.16
0.14
III. PSEUDOSCALAR MESON MASS
RELATION 0.12
0.1
As the current quark mass is raised from zero, the ex- 0 1 2 3 4 5
m (µ = 1 GeV) (GeV)
plicitbreakingofchiralsymmetryaddsmasstotheGold- Q
stone boson modes. The way in which the pseudoscalar
FIG. 4: Current quark mass dependence of fP, the elec-
meson mass grows with quark mass is described, at low troweak decay constant for u−Q mesons. [fπ = 131 MeV
mass, by the GMOR relation. This is convention.]
q¯q 0
M2 m (µ),m (µ) =[m (µ)+m (µ)] |h iµ|+ (m2) , mesons in QCD, applicable for all values of the quark
P(cid:0) 1 2 (cid:1) 1 2 (fP0)2 O masses,has been established [3]. With allowancefor dif-
(6) ferent quark flavors,it takes the form
in the general case where the two quark flavors are dif-
ferent. Here hq¯qi0µ = −Z4Nctr qΛS0(q) is the chiralcon- MP2 fP(m1,m2)=(m1+m2)RP(m1,m2) , (7)
R
4
where R is the projection of the meson wave function P2 =0 to be cancelled and thus M =0. The analysis
P P
onto γ at the origin of q¯q separation and is given by [3] alsoprovidesrelationshipsbetweentheBSamplitudeand
5
the quark propagator, e.g., f0 E0(q;0)=B (q2), where
P P 0
Λ Γ (q;P)= iγ E + . Since the quark mass function
P 5 P
RP =−iZ4NcZ tr γ5Sf1(q+)ΓP(q;P)Sf2(q−) , (8) is proportionalto B0(·q·2·), this latter relationmeans that
q (cid:2) (cid:3)
dynamical chiral symmetry breaking is necessarily ac-
with all renormalized quantities taken at the same scale companiedbyamasslesspseudoscalarboundstate. This
µ. The chiral limit of this quantity can be shown to is the familar Goldstone’s theorem; notice that the com-
beR (m =m =0)= q¯q 0/f0,andthustheGMOR posite, distributed nature of the pion amplitude requires
P 1 2 −h iµ P
relation follows as a collorary of the exact relation, a running quark mass function.
Eq. (7), at low mass. The origin of this exact mass rela- At small m, both fP and RP are constant leading to
tion is the axial vector Ward-Takahashi relation the GMOR behavior MP √m. The error in this has
∼
to increase with mass since the heavy quark limiting be-
iP Γ (q;P)= S−1(q )γ +γ S−1(q ) havior is [21] f 1/√m and R √m which leads to
− µ 5µ f1 + 5 5 f2 − P ∼ P ∼
the linear behavior M m.
(m1+m2)Γ5(q;P) . (9) P ∼
−
The dressedvertices Γ (q;P) and Γ (q;P) satisfy inho-
5µ 5
mogeneousintegralequations that havethe same kernel, 0
10
the irreducible q¯q scattering amplitude; the inhomoge- DSE results
VMD dipole
neous terms are Z γ andZ γ respectively. Thus both
2 µ 4 5
vertices have poles corresponding to the pseudoscalar bare vertices
meson bound states. [The axial vector poles in Γ 10-1 (4/3) π2 f2 / Q2
5µ π
have transverse residues and do not contribute.] The
exact mass relation, Eq. (7), arises from the equality 2)
Q
of the pseudoscalar pole residues from both sides of 2, 10-2
Q
Eq. (9). The residue of Γ5(q;P) is −iRP ΓP(q;P), and F(
the residue of Γ (q;P) is P f Γ (q;P), where Γ is
5µ µ P P P
the pseudoscalar bound state BS amplitude. The ex-
-3
10
pression for f P is the same as for R in Eq. (8) ex-
P µ P
cept that γ is replaced by γ γ and iZ is replaced
5 5 µ 4
−
by Z /√2 (in the convention where the physical f is
2 π
-4
92.4 MeV). In the chiral limit, the last term of Eq. (9) 10
2 10-2 10-1 100 101 102 103
2 2
Q [GeV ]
1.9
FIG. 6: Our DSE results for the symmetric γ∗π → γ∗ form
1.8 factor, compared to the pQCD asymptotic 1/Q2 behavior.
1.7 [Here fπ =131 MeV.] The naive VMD model suggests a
01/3>] 1.6 dipole behavior which is correct only in the infrared.
q
q The evolution of both vector and pseudoscalar meson
< 1.5
H / masseswithincreasingcurrentquarkmasshasbeenstud-
>
q 1.4 iedwithinthe DSEmodelinboththe equalandunequal
q
< flavor cases. An example is shown in Fig. 3 for u Q
[ 1.3 −
mesons as a function of m (µ=1 GeV) up to the limit
Q
1.2 ofaccuracyofthe calculations. The fitshownis adapted
from Ref. [22] by evolving the m (µ = 19 GeV) used
1.1 Q
there to µ=1 GeV for ease of comparison with conven-
10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 tionally quoted values. Thus here we have MP =α′+
m (µ =1 GeV) (GeV) β′√mQ+ γ′mQ; with both masses in GeV, the parame-
Q tersare(α′,β′,γ′)=(0.083,0.842,0.880). WiththeDSE
model-exact values of f0 and q¯q 0, the GMOR relation
P h iµ
FIG. 5: A measure of the current quark mass dependence of is explicitly M2 = 0.00955+ 1.724m (µ=1 GeV) and
P Q
thein-hadron condensate for qQ pseudoscalars. thisiscomparedtotheexactmassrelationinFig.3. For
the K meson the GMOR error is 4%, at m = 0.4 GeV
Q
is not present and the right hand side is not singular. A the error is 14%, while at the D meson the error is 30%.
systematic expansion in powers of P reveals [3] that in In the D meson region, only 50% of the mass comes
µ
∼
leadingorderthe poleinthe lefthandside mustmoveto fromthelinearterm;theheavyquarkdomainisathigher
5
is lighter. As an independent check, coupling constants
TABLE II: The coupling constants gv→pp calculated in the arealsoextractedfromthetimelikeelectroweakformfac-
DSEmodelcomparedtoresultsfromafittothetimelikeform
torsnearthevectormesonpoles[24,25]. Theagreement
factor pole [24, 25] and lattice-QCD [26].
is encouraging considering that with eight independent
gv→pp Expt this work pole fit lattice-QCD covariantsfor the vector BS amplitude and four each for
gρ→ππ 6.02 5.14 5.2 6.08+−21..0040 the pseudoscalars, there are 128 distinct quark loop in-
tegrals for each physical decay. Also shown in Table II
g 4.64 4.25 4.3 -
φ→KK
is a recent lattice-QCD result [26] for ρ ππ from the
gK⋆+→K0π+ 4.60 4.81 4.1 - UKQCD Collaboration. Although the la→ttice data is at
m /m =0.578,whichcorrespondsto thes-quarkmass,
π ρ
and thus no physical decay of the ρ can take place, the
mass.
amplitude ρππ isaccessiblethroughstudyofstatemix-
TheexactmassrelationM (m ),Eq.(7),differsfrom h | i
P Q ing on the lattice.
the GMOR behavior due to the mass dependence of f
P Since the width of the ρ is almost 20% of its mass
and R . Instead of the latter quantity, one can de-
P while the widths of the φ and K⋆ are significantly less
fine q¯q H = f R asaneffective“in-hadron”conden-
h iµ − P P important, we expect the ladder approximation for the
sate[21]whichallowstheexactmassrelationtotakethe
BSE kernel (which omits the strong channels ππ, KK
GMOR-like form M2 f2 = (m +m ) q¯q H . From
P P 1 2 µ|h iµ| and Kπ respectively) to be less accurate for the ρ than
this relation we extract the quark mass dependence of
for the φ and K⋆. Accordingly we speculate that this is
q¯q H at µ= 1 GeV. The low mass DSE results for M
ahndiµf are fitted to forms that respect the heavy quarPk largely the reason why the result for gρ→ππ in Table II
P deviatesfromexperimenttwice asmuch(15%)as do the
limits, are consistent with D and B meson masses, and
other decay constants.
are consistent with lattice results [23] for f of the D
P
and B mesons. We adapt the f (M ) fit from Ref. [21]
P P
to produce f (m ) and to accommodate the DSE re-
P Q
V. PQCD LIMIT OF FORM FACTORS
sults at low mass. The result is displayed in Fig. 4.
The fit (in the f = 131 MeV convention) is f2 =N/D,
π P
where N =a+bm and D =1.0+cm +dm2, with Besides the soft physical characteristics of light
Q Q Q
(a,b,c,d)= (0.017,0.068,0.649,0.391) when m is in mesons, the present DSE model should also reproduce
Q
GeV. The resulting estimate of the mass dependence for perturbative QCD limits. This has been checked for the
q¯q H at µ = 1 GeV is shown in Fig. 5 and indicates uv behavior of the quark mass function M(p2); both the
h 1iµ5% increase over the chiral limit value for an m leading log behavior away from the chiral limit, and the
Q
r∼elevantto K, while for the D mesonthe in-hadroncon- coefficientoftheleading1/p2 behaviorinthechirallimit
densate is about 70% enhanced. reproduce the exact 1-loop results of QCD [2]. A more
difficult task is to test the asymptotic behaviorof meson
form factors against pQCD predictions. This is compli-
IV. VECTOR MESON STRONG DECAYS cated by the fact that covariant ladder-rainbow calcula-
tions that link the dressed quark propagator, the BSE,
andtheimpulseapproximationforformfactorshaveonly
SincethisDSEmodeldescribestheelasticchargeform
beencarriedoutinEuclideanmetricforpracticalreasons.
factors of the pseudoscalars very well [10] in impulse
The mass-shell constraint for mesons then requires an
approximation, the strong decays of the vector mesons
analytic continuation which entails complex quark mo-
should be well-described without parameter adjustment.
mentain loopintegrals. This greatlyhinders the asymp-
Inimpulseapproximation,theamplitudeforthedecayof
totic analysis.
ρ with 4-momentum Q=p +p to ππ with 4-momenta
1 2
p ,p is given by [13] A case that is free of these difficulties is the symmet-
1 2
ric γ∗π γ∗ transition where the photons are taken
Λ to have e→qual virtuality Q2 and there is only one mass-
2PT g = √2N tr S(q)Γ (q,q )S(q )
µ ρ→ππ c Zk P + + shell constraint. Since m2π is negligible compared to all
Γ (q ,q )S(q )Γ¯ (q ,q), (10) other scales in the problem, all involved quark momenta
µ + − − P −
× are essentially real and spacelike. In Fig. 6 we show the
where no distinction is made between the u/d quarks, result [11] of the present DSE model compared to the
P =(p p )/2, q = k+P/2 and q = k P/2 Q/2. pQCDasymptotic behavior [27]obtained fromthe light-
1 2 ±
− − ±
The appropriate generalizations of the flavor structure cone operator product expansion. (In this case, log cor-
appropriate to φ KK and K∗ Kπ are straightfor- rections occur at sub-leading order.) The numerically
→ →
ward [13]. In Eq. (10), the component of P transverse generated asymptotic behavior of the DSE-based model
µ
to Q is indicatedonthe left handside to coverthe case reproducesthepQCDlimitasitmust. Byabout2GeV2
µ
of unequal decay products. the dressing of the photon vertices becomes negligible;
The results shown in Table II are within 5-10% of ex- howeverthe3-pointfunctiondoesnotbecomeaneffective
periment with the error being larger if the vector meson 2-point function (thereby generating the required power
6
off ) untilabout 15-20GeV2 [11]. Sucha highscale for amount of cancellation between repulsive and attractive
π
the onset of pQCD behavior is consistent with an earlier corrections [30, 31]. The preservation of the axial vec-
observation[28]inaDSE-basedmodelstudyofF (Q2). tor WTI is what makes the pseudoscalar meson sector a
π
robust and ideal base for parameter fixing; the rainbow-
ladder truncation may be used as a convenience in that
VI. DISCUSSION sector. It is hoped that future interplay between lattice
simulationsandcontinuummodelingwillincreaseourun-
Recent reviews [1, 29] put this model in a wider per- derstanding of QCD for hadron physics.
spective and compile results for both meson and baryon
physics,ananalysishowquarkconfinementismanifestin
solutions of the DSEs, and both finite temperature and Acknowledgments
finite density extensions. The question of the relevance
and accuracy of the ladder-rainbow truncation has also ThanksaredueA.FaesslerandT.Gutschefororgani-
received some attention; it has been shown to be partic- zationofastimulatingschoolinErice. Veryhelpfulcom-
ularly suitable for the flavor octet pseudoscalar mesons ments from Pieter Maris are acknowledged. This work
sincethenext-ordercontributionstotheBSEkernel,ina was supported in part by NSF grants No. PHY-0071361
quark-gluonskeletongraphexpansion,have a significant and INT-0129236.
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