A new Perspective on the Scalar meson Puzzle, from Spontaneous Chiral Symmetry Breaking Beyond BCS

17 Pages·0.43 MB·English

by  Pedro Bicudo

A New Perspective on the Scalar Meson Puzzle, from Spontaneous Chiral Symmetry
Breaking Beyond BCS
Pedro J. de A. Bicudo
Departamento de F´ısica and Centro de F´ısica das Interacc¸˜oes Fundamentais, Edif´ıcio Ciˆencia, Instituto Superior T´ecnico,
Av. Rovisco Pais, 1096 Lisboa, Portugal
the breaking of SU(3) due to the m m << m
f u d s
We introduce coupled channels of Bethe-Salpeter mesons ≃
mass difference is nearly the double than expected when
both in the boundstate equation for mesons and in the mass
compared with the vector meson family and with most
gap equation for chiral symmetry. Consistency is insured
baryons. This is only comparable with the splittings in
by the Ward Identities for axial currents, which preserve the
the pseudoscalar family.
Goldstonebosonnatureofthepionandpreventsasystematic
9 shift of thehadron spectrum. Westudythedecayof ascalar
9 meson coupled toa pairof pseudoscalars. Wealso show that [GeV] ?
9 coupledchannelsreducethebreakingofchiralsymmetry,with ?
1 2:
thesameFeynmandiagramsthatappearinthecouplingofa

n scalar meson to a pair of pseudoscalar mesons. Exact calcu-
?
a lations are performed in a particular confining quark model,
J where we find that the groundstate I = 0, 3P0 qq¯meson is

0 the f0(980) with a partial decay width of 40MeV. We also 1:
2 finda 30% reduction of thechiral condensate duetocoupled
channels.
2
v 12.39.Kc, 11.30.Rd, 14.40.Cs, 13.25.Jx FIG0:. 1. We rfe0p;Ir=es0ent, with recat0a;In=gl1es of heightKΓ0(cid:3);,I=th1e=2
8
experimentally observed scalar resonances according to the
5
ReviewofParticleProperties,andmarktheunconfirmedones
0
2 with a ”?”. In a naive quark model, the a0 and K0∗ channels
I. INTRODUCTION
0 should show half of theresonances of thef0 channel.
8
9 The Scalar puzzle The scalar family is also the most interesting place to
/ searchfor the lightest (S-wave) non qq¯states. There are
h
t The scalar mesons form perhaps the most puzzling several theoretical candidates to extra states which may
-
l family in hadronic physics. The first puzzling fact con- be found. The lightest glueball, which is expected from
c
cernstheexperimentalerrorsinthepartialdecaywidths, QCD, should be a scalar [2]. The model of Isgur and
u
the decay widths and even the masses. The lightest Weinstein et al [3] suggests that the narrowest scalars
n
: scalar, f0(400 1200), has a poorly determined mass. are meson-meson molecules. The One Meson Exchange
v −
The confidence on the decay widths of the f (980) and Potential models for the NN interaction usually postu-
i 0
X a (980) has also decreased [1] strongly since 1994. lates a scalar meson σ with a light M 0.5GeV. In
0 ≃
The other puzzling argument concerns the matching stronglycoupledeffectivemesonmodels[4,5],extrapoles
r
a of the the nine observed states with simple SU(3) qq¯ appear in the S matrix when couplings are large. These
f
states. One would expect four different towers of states mesonmodels turn out to be the most successfulmodels
corresponding to the two I = 0 f (which are not de- onessofar,explainingwithcomplexnonlineareffectsnot
0
generate for instance because the quark s has a clearly onlythenarrowa (980)andf (980)whichareduetothe
0 0
larger mass than the quarks u, d) and to the I = 1 a , vicinityoftheKK threshold,butalsothe verywideand
0
andtheI =1/2K . AshortglanceatFig. 1issufficient light f (400 1200).
0∗ 0 −
todiscardthesinglelightandextremelybroadstate,the
f (400 1200) as a simple member of this family. Then
0
−
forthegroundstateswecouldascribethenarroweststates The relevance of chiral symmetry breaking
which are respectively the a (980), K (1430), f (980)
0 0∗ 0
and f0(1500), and for the radial excited states we could Chiral symmetry breaking is important for the study
respectively ascribe the a0(1450), K0∗(1950), f0(1370) ofscalarmesonsandtheirdecaysforseveralreasons. Un-
andf0(2200). HoweverthedecaywidthsΓoftheground- like the vector, axial and tensor mesons, the scalar and
statescalarsarenarrowerthanexpectedwhencompared pseudoscalar mesons are mixed by the chiral rotations,
withotherresonancesdecayinginthe samepseudoscalar
pairsbut with higherangularmomentum, exceptfor the ψ¯ψ cos(θ)ψ¯ψ+isin(θ)ψ¯γ ψ
5
→
only precisely measured one, the K0∗(1430). Moreover ψ¯γ5ψ isin(θ)ψ¯ψ+cos(θ)ψ¯γ5ψ, (1)
→−
1
thus scalars and pseudoscalars are particularly sensitive the nucleon interactions and had good results [12]in the
to the chiral symmetry. The very small mass of the K s-wave scattering, the F and F derivative
N NπN Nπ∆
pseudoscalars is usually explained with chiral symme- couplings and the NN short range interaction.
try breaking. Moreover, since scalars decay essentially However there is a recent trend in the literature to
in pseudoscalars,the pseudoscalar mass is important for reevaluatecoupledchanneleffectsinmanyhadronicphe-
the scalardecay. Thusweexpectthatnotonlythe pseu- nomena. Some years ago they were not supposed to ac-
doscalar mesons but also the scalars must contain the countformore than10%ofahadronmassbut presently
signature of the breaking of chiral symmetry. Inversely, they are supposed to contribute with a negative mass
the breaking of chiral symmetry is generated from the shift of the order of 50% of the bare mass [14–16].
trivial vacuum by scalar condensation, and we also ex- Moreover it is possible to prove that the vacuum so-
pectthatthescalarpropertiesshouldaffectthebreaking lution of the BCS mass gap equation is not exact when
of chiral symmetry. coupled channels are included. Suppose that the mass
These effect have been studied in phenomenological gapequationforchiralsymmetrybreakingwassolvedat
mesonsigmamodels,seeforinstance[5]. Atthemoremi- the BCS level, i.e. without including the coupled chan-
croscopiclevelofquarks,DynamicalSpontaneousChiral nels, then a bare pion with vanishing bare mass would
Symmetry Breaking (χSB) has been worked out in the be found. If the coupled channels were then included,
past with several different quark-quark effective interac- at posteriori, in the bound state equation then the pion
tions, at the same level as Bardeen Cooper and Schrief- mass would be the sum of the small bare mass plus a
fer(BCS)did[6]forSuperconductivity. Since[7]Nambu mass shift and thus would have a resulting nonvanishing
andJona-Lasinio(NJL)anduntilrecently[8,9]the mass mass, which implies that the pion lost its Goldstone bo-
gapequationinchiralphysicshasbeensofaroftheBCS sonnature. Thisresultisunavoidable,andcanbeproved
type, [7] including only the first order contribution from variationally. Thusthemassgapshouldbesolvedbeyond
the quark-quark interaction. In this case the quark con- BCS, especially when scalar mesons are studied.
densate consists [10]of scalar3P quarkantiquarkpairs. Solving this problemis a corner stone of our program.
0
At the BCS level, which is very consistent, it is possible It amounts to join the BCS mechanism with the mean
to derive the mass gap equation in several different but field expansion of effective mesons.The logical path of
equivalent methods. The pseudoscalar meson properties the method we will follow is illustrated in Fig. 3. Self
havebeenstudiedingreatdetailandthefullmesonspec- consistencyisinsuredbyWardIdentities,andthescalar-
trumhas been alsobeen calculatedin the literature[11]. pseudoscalarcoupling will turn out to be crucial for this
HowevertheBCSapproachisnotexactinthecasewhere development. The reward of solving chiral symmetry
coupled channels of mesons are included. breaking with coupled channels is a π with a vanishing
positive mass in the chiral limit complying with all the
theoremsofPCAC,anda towerofresonances(including
Coupled Channels of Mesonic qq¯pairs the scalar meson resonances) above the π with higher
masses due to radial, angular, or spin excitations. In
In the case of weak coupled channel effects, it would particulartheresonancesalsohaveanimaginarycompo-
be acceptable to start from boundstates obtained at the nentofthe mass −2iΓ thatdescribesthedecaywidthinto
BCS level and couple them with the help of the anni- the open channels.
hilation diagrams of Fig. 2, without changing the mass
gap equation. In this sense we started some years ago WI
to develop a program [12] to study the coupled channel BCS: Mass Gap ! Bound State
effects in quark models with chiral symmetry breaking.
(cid:16)
(cid:16)
(cid:16)
Q Coupled(cid:16)(cid:16)
QkQ (cid:16)
(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)(cid:27) ; -(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)-(cid:27) ; (cid:24)(cid:24)(cid:24)QsQ(cid:24)9(cid:24)Q(cid:24)(cid:24):(cid:1)(cid:1)QQ(cid:1)(cid:1)(cid:1)(cid:1) (cid:27) (cid:16))(cid:16)(cid:16)(cid:16)(cid:16)WIChannels
FIG. 2(.a) We show (tbh)e quartic d(cia)grams which may con- BbeCySo:nd Mass Gap ! Bound State
FIG.3. Weillustratetheprinciplewhichisfollowedinthis
tribute to the boundstate equation. In the strong coupling
paper to include the coupled channels in the mass gap equa-
BCS, (a) is included in the self energy and (b) is included
tion
in the interaction kernel. Diagram (c) which creates or an-
nihilates quark-antiquark pairs is only used in what we call
beyond BCS.
The paper
The first result of our program was to reproduce [12]
at the BCS level the strong decay of the vector meson The aim of this paper is to study at the quark level
ρ (and of the φ). This has been studied by other au- some meson decays which were studied in the literature
thors recently [13]. Later we extended our program to without including directly the full quark contributions
2
[4,5]. We specialize in the groundstate f π π decay. Global Color Model of ref. [18], and a sophisticated in-
0
→
We also study the effect of the meson coupled channels teractionwithageneraltensorstructure,analmostlinear
on the quark condensed vacuum. long range and pertubative short range is found in ref.
Theremainingofthispaperisorganizedasfollows. In [19].
section II we review the scalar masses and decays at the The other class of models [10,11,20] has the single
BCS level. This includes the choice of an effective inter- drawbackofusinganinstantaneouspotential(exceptfor
action for quarks, the mass gap equation and the Bethe Lorentzinvariantextensions[21]ofthisclass). Butithas
Salpeter equation at the BCS level, the scalar coupling the advantage of being confining which allows to study
to pseudoscalars, and the scalar decays width. In Sec- the whole [11,12,17,22–24] hadron spectrum. This ap-
tion III we produce a finite extension beyond BCS for a proximation also has the advantage to allow a straight-
class of confining effective interactions, derive the mass forward application to low energy nuclear physics. For
gapequationwithcoupledchannels,linkittothescalar- the theoretical foundations of these models, including
pseudoscalarscoupling,andsolvethemassgapequation. the connection to both pertubative and nonpertubative
Results are shown in Section IV together with their dis- QCD, see [25].
cussion. We also include 4 appendices. Thuswechoosetocalculatethe quantitativeresultsof
this paper within the second class of Nambu and Jona-
Lasinio potentials. We use a simple model which is in
II. THE BCS LEVEL FOR A PARTICULAR very good agreement with the experiments in what con-
FORMALISM cerns the hadronic spectroscopy, the decays of the vec-
tor mesons ρ and Φ, the coupling of a π to a N or ∆
A. The choice of an effective interaction and the N N short range interaction [12], moreover it
supports that chiralsymmetry breaking is very stable in
the presence of Nuclear Matter [24]. While confinement
The quantitative results of this paper will be obtained
is an essential physical aspect of the model, the instan-
with a particular chiral invariant strong potential which
taneous approximation simplifies drastically the energy
is an extended version of the NJL potential [7]. For the
dependence of the interaction, and allows to work in a
study of dynamical χSB it is crucial to have a closed
framework which is familiar to Schr¨odinger’ s equations.
model where calculations can be carried until the end,
The 2-body potential for Dirac quarks is,
because precise cancellations occur. At this point we
abandontheexplicitSU(3)gaugeinvariance. Westartby
3~λ ~λ
introducing a class [11,17] of Dirac quark Hamiltonians − γ γ k3r2 U +a~γ ~γk3r2 δ(t) (3)
whichare,inthelimitofmasslessquarks,explicitlychiral 4 2 ·⊗2 0⊗ 0 0 − ·⊗ 0
invariant, and the Four(cid:2)ier trans(cid:0)form of th(cid:1)e potential is, (cid:3)
3~λ ~λ
H = d3x"ψ†(x)(m0β−iα~ ·∇~)ψ(x) + ΩlVl(k)⊗Ωl = −4 2 ·⊗2 γ0⊗γ0 −K03∆k−U
Z l
X +a~γ ~γ (cid:2)K3∆ (cid:0)(2π)3δ3(k) ,(cid:1) (4)
1ψ(x)Ω ψ(x) d4y V (x y)ψ(y)Ω 2ψ(y) (2) e ·⊗ − 0 k
l l l
2 − where we dropped the sum(cid:0) in color(cid:1)(cid:3)and Dirac indices.
Z (cid:21)
Thefactor 3/4simplifiesthecolorcontributionforcolor
The quark-quarkinteraction, is an effective color depen- −
singlets.
dent 2-body interaction. In eq. (2) the operators Ω in-
l 0
clude both the color Gell-Mann matrices and the Dirac
V(k)
matrices. The sum in the Dirac matrices must be chiral
invariant. Because the Gell-Mann matrices are traceless -20 -U+k3r2
0
there will be no tadpoles in this scheme. In Hadronic
Physics the effective interaction should be simultane- -40 -Uexp(-k3r2/U)
0
ouslycolorconfining,inthe Minkowskyspace,local,and
Lorentz invariant. However no interaction which com-
-60
plieswithalltheseconstraintshasyetbeenusedtostudy
chiral symmetry breaking.
The models used in the literature divide essentially in -80
two classes, in particular the more popular one springs
directly fromNJL [7], is Euclidean anddue to the struc- -100
ture of the interaction has usually no analytic continua- 0 4 8 12 16 k 20
tiontotheMinkowskyspaceandlacksconfinement. This
classhasbeenextendedinmanydifferentdirections. For FIG. 4. −Ue−kU03r2 is an example of a potential which
instance finite size boundstates were included with the tends to −U +k03r2 in the limit of infinite U. We illustrate
this in thecase where K0 =1, U =100.
3
The γ γ term inthe potential is the limit ofa series d4k
ofattrac0ti·ve0potentials,seeFig. 4,andU isanarbitrarily Γ(p,q)=Γ0+ (2π)4V(k)ΩS(p+k)
Z
large infrared constant. The infinite U reappears in the Γ(p+k,q+k)S(q+k)Ω (6)
self energy and in color singlet channels this cancels the
infinitelyattractivepotential. Anycoloredstatewillhave where the full vertex Γµ is denoted by s . When eq.(6)
a mass proportional to U and will thus be confined, see is iterated, we find that it includes, the Bethe-Salpeter
appendix A. ladder, which will be represented diagrammatically by a
The choice of an harmonic potential is not crucial, a box with 4 emerging lines = =,
⊓⊔
linear [20] or funnel potential has also been used, but a
quadratic form is simpler. In the case of light quarks
the current quark mass m0 is almost vanishing and it -(cid:27) -(cid:27) = -(cid:27) + -(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)-(cid:27) + -(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)-(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)-(cid:27) +::: (7)
essentially affects the family of the π which is a quasi
Goldstone boson. where the ladder represents the mesons, see Appendice
The ~γ ~γ term is introduced in order to have Lorentz C.Inthiswaythequarkpropagator,theverticesandthe
·
invariant pions which are relativistic in the scalar decay. mesons are intertwined.
Clearly the wrong result of a simple γ0 γ0 instanta- Inthiscaseofaninstantaneousinteraction,itisconve-
⊗
neous interaction is the constant fπ which is quite small nient to substitute the Dirac fermions in terms of Weyl
and is not Lorentz invariant. The hope to cure fπ with fermions, in order to find the hadron spectrum. The
covariantextensionsofthe modelturnedouttofailsince Dirac propagator can be decomposed in a quark propa-
they merely [21] increased fπ in 30%. The value for the gatorandanantiquarkpropagator,movingbothforward
parameter a which renders the pion Lorentz invariant is in time.
a= .18,andformostcalculations(exceptfor f which
π
is in−creased by 300%, see subsection D) this a yields a (k ,~k)= i
Dirac 0
resultcomparabletotheonewehadfora=0. Thesmall S k m+iǫ
6 −
a suggests that the cure of fπ may be related with the i 1+ mβ+ kα~ kˆ
pertubative short range [25] quark-quark interaction . = E E · β
k E+iǫ 2
Onceaisfixed,thismodelhasthesinglescaleoftheos- 0−
cillator parameter K . The simplest adimensional units i 1 mβ kα~ kˆ
0 − E − E · β (8)
of K0 = 1 will be used from now on in computations. − k0 E+iǫ 2
− −
When comparingwith experiments we will rescaleK to
0
the value of K = 330MeV which gives the best over- Itisconvenienttouse[11,10]thequarkenergyprojectors,
0
all fit of the meson spectrum. We will also work in the
1
momentum representation and drop the tilde from the Λ+ = 1+Sβ+Ck α~ = u u ,
2 · s †s
potential.
(cid:16) (cid:17) Xs
e 1 b
Λ− = 2 1−Sβ−Ck·α~ = vsvs†, (9)
B. Chiral Symmetry Breaking at the BCS level with (cid:16) (cid:17) Xs
quarks and antiquarks b
where S = sin(ϕ) = m , C = cos(ϕ) = k and ϕ is
E E
a chiral angle which in the non condensed case is equal
At the BCS level and with a color dependent inter- to arctanm0, (m is the current mass of the quark) but
action the Schwinger Dyson equation for the quark self k 0
is not determined from the onset when chiral symmetry
energy (which is also the mass gap equation) is,
breaking occurs. In this case the physical quark mass
is a variational function m = m(k) which is determined
(cid:27)−1 = S0−1− ·····(cid:27)···· by the mass gap equation. This is equivalent to use the
−1 = i p Σ (5) chiral angle ϕ = ϕ(k) as the variational function. In
S − 6 −
Fig. 5 we show examples of non trivial solutions for the
wherethe full(upto the approximationwhichischosen)
function ϕ(k).
propagator is denoted as usual by (cid:27) . The subindex
S Theenergyprojectorscanbedecomposedinthequark
is reserved for the free functions. The effective quark-
0 spinor u(k) and in the antiquark spinor v(k),
quark 2-body interaction of eq. (2), a chiral invariant
and color dependent interaction, is represented with a
Λ 1+S 1 S
dotted line ....... The Bethe Salpeter equation for the u (k)= − u (0)= + − k α~ u (0)
s s s
vertex is related by the Ward Identities with eq.(5), see 1+S "r 2 r 2 · #
2
Appendice C, q b
Λ 1+S 1 S
v (k)= − v (0)= − k α~ v (0)
s s s
u = Γ0+H(cid:8)(cid:8)···········(cid:8)*HYHH(cid:8)u =−qiσ1+22γS5u∗s(k) . "r 2 −r 2 b· # (10)
4
And finally the Dirac quark propagator is decomposed In that Hamiltonian formalism the mass gap equation
in, is also obtained when the quark antiquark pair creation
operators are postulated to vanish in the Hamiltonian,
(w,~k)=u(k) (p ,~k)u (k)β
SDirac Sq 0 † in order to ensure the vacuum stability against sponta-
v (k) ( p , ~k)v(k)β , (11) neousgenerationofscalars. Withthepresentmethodwe
† q¯ 0
− S − −
project eq(14) with the spinors u u and u v, and
† †
where the quark and antiquark Weyl propagatorsare. ··· ···
we get directly the quark and antiquark energy and the
i mass gap equation,
(w,~k)= (w,~k)= . (12)
q q¯
S S w E(k)+iǫ
− dw′ d3k′
E(k)=u (k) kk α~ +m β iV(k k )
The quark and antiquark formalismis convenientto cal- †s · 0 − 2π (2π)3 l − ′
(cid:26) Z
culate the hadron spectroscopy. With Weyl propagators
the BS equation simplifies into the Salpeter equation, in ΩlΛ+(k′)Ωl b ΩlΛ−(k′)Ωl u (k),
s
w E(k )+iǫ − w E(k )+iǫ
a form which is as close as possible to the more intuitive (cid:20) ′− ′ − ′− ′ (cid:21)(cid:27)
Schr¨odinger equation. In the Feynman rules with Weyl dw′ d3k′
0=u (k) kk α~ +m β iV(k k )
propagators,we choose to redefine the vertices of the ef- †s · 0 − 2π (2π)3 − ′
(cid:26) Z
fectivepotentialwhichnowincludethespinorsu†, u, v† ΩlΛ+(k′)Ωbl ΩlΛ−(k′)Ωl
andv. The” ”signwhichaffectstheantiquarkpropaga- vs′′(k). (15)
− w E(k )+iǫ − w E(k )+iǫ
tor in eq.(11) could also be included in the vertices with (cid:20) ′− ′ − ′− ′ (cid:21)(cid:27)
v , but we prefer to recover the equivalent rules which Inthecaseofaninstantaneousinteraction,theloopinte-
†
are common to nonrelativistic field theory. This ” ” gralin the energy w removesthe pole in the propagator,
−
signtogetherwiththeonefromthefermionloopswillbe
dw i 1
included in the antiquark vertex and in diagrams with = (16)
2π w E(k)+iǫ 2
quark exchange or with antiquark exchange. The Dirac Z −
vertexγ0 isnowreplacedbyu†u,u†v,v†uor v†v when and in the case of a quadratic interaction, the loop inte-
−
thevertexisrespectivelyconnectedtoaquark,apaircre- gralin the momentumis transformedin aLaplacian,see
ation, a pair annihilation or an antiquark;and the Dirac eq.(4). Some useful properties are,
vertex~γ is respectively replacedby u α~u,u α~v,v α~u or
† † †
p−rvo†pα~avg.atWores ochfoqousaerktsheangdraapnhtiicqaulanrkotsa,tion for the Weyl u†sus′ =1δss′ , u†svs′ =0 ~σ·kˆiσ2 ss′
h i
(w,~k)= (cid:27)w,~kD , u†sβus′ =Sδss′ , u†sβvs′ =−C ~σ·kˆiσ2 ss′ (17)
Dirac h i
(wS,~k)= (cid:27)w,~k , ( w, ~k)=-w,~k (13) u†sα~ ·kˆus′ =Cδss′ , u†sα~ ·kˆvs′ =S ~σ·kˆiσ2 ss′
whereSthqeDiagramsusingtShqe¯F−eyn−manrulescorrespond- u†sβα~ ·kˆus′ =0δss′ , u†sβα~ ·kˆvs′ =1h ~σ·kˆiσi2 ss′ .
ingtotheDiracfermionpropagatorswillhaveasubindex h i
We get finally for the quark energy,
in the remaining of the paper. In the case of the Weyl
D
propagators(whichwillbeusedmoreoftenthetheDirac E(k)=kC+m S+ 1 +S∆(S)+Ckˆ ∆(kˆC) + U
0
propagators)thequarkwillberepresentedwithanarrow 2 · 2
pointing to the left while the arrowpointing to the right +a1 3S∆(S)h Ckˆ ∆(kˆC) i
represents an antiquark (both move forward in the time 2 − − ·
direction). U h ϕ˙2 C2 i
= +kC+m S
2 0 − 2 − k2
C2 1
C. The BCS mass gap equation and the quark a SC∆ϕ S2+ ϕ˙2 , (18)
− − k2 − 2
energy (cid:20) (cid:18) (cid:19) (cid:21)
where in color singlets the U/2 term will cancel the U
−
Here we derive the mass gap equation, and the quark termfromthe 2-body quarkpotential. Forthe mass gap
dispersionrelation,replacingthe propagatorof eq. 11 in equation we get,
the Schwinger Dyson equation for the quark self energy
1
(5), 0= kS m C+ C∆(S)+Skˆ ∆(kˆC)
0
− 2 − ·
k m n 1 h i
u (cid:27)−1u† v - −1v† =β6 − 0 β ····(cid:27)··· (14) +a +3C∆(S) Skˆ ∆(kˆC) ~σ kˆiσ2
w,~k − w,~k i − · ·D 2 − · · ss′
h 2SC ioh i
AnotherequivalentmethodistousetheHamiltonianfor-
= ∆ϕ+2kS 2m C
malismfor the quark andantiquarkcreatorsand annihi- − − 0 − k2
lators [10], and find the Bogoliubov Valatin transforma- 1
a 2C2+1 ∆ϕ+2SC ϕ˙2 . (19)
tion which would minimize the vacuum energy density. − − − k2
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:0) (cid:1)
5
The mass gap equation is in general a nonlinear integral energycomponentφ+. TheSchr¨odingerlimit,whereonly
equation,but in this caseof a harmonicpotential it sim- the positiveenergy componentis considered,is therefore
plifies to a differential equation. We solve it numerically acceptable. Ageneralformforthe 3P wave-functionfor
0
with the Runge-Kutta and shooting method, see Fig. 5 the scalar is
for the solution.
[~σ kˆiσ ]
2.0 φ+(k)s1s2 =kφs(k) · √22 s1s2 (21)
the truncated BS amplitude is then,
1.5
2kφ (k) [~σ kˆiσ ]
1.0 χs =−Ωl∆ 1+sS Λ+us1(0) · √22 s1s2vs†2(0)Λ−Ωl
Λ+βα~ ~kφ (k)
s
=Ω ∆ · Ω , (22)
l l
√2
0.5
we get for left hand side of eq. (20) ,
0.0
0 1 2 3 1+a 1 a
k u†(k)χ(k,0)v(k)= ∆ φ+ + − S∆ Sφ+
2 2
FIG.5. WeshowtheBCSchiralangleinunitsofK0 =1. 1 3a (cid:0) (cid:1) 1 3a (cid:0) (cid:1)
We also represent with a doted line the chiral angle that we + − C∆ Cφ+ + − C2φ+ (23)
2 k2
obtain going beyondBCS.
(cid:0) (cid:1)
and the radial Salpeter equation for the scalar in the
center of mass is,
D. The pseudoscalar and scalar solutions to the
Salpeter equation d2 2 ϕ˙2 C2 d2
2E(k) M + a C2
− − dk2 − k2 − 2 k2 − − dk2
(cid:20) (cid:18) (cid:19) (cid:18)
The homogeneous Salpeter equation for a meson (a d C2 1+2C2
color singlet quark antiquark boundstate) is, according +2SCϕ˙dk − k2 +SCϕ¨+ 2 ϕ˙2 k2φs =0 (24)
to Appendix C, (cid:19)(cid:21)
Solving the bound state equation we find that the solu-
+M(P)−Ei(k1)−E(k2)φ+(k,P)=−iu†(k1)χ(k,P)v(k2) tion of the equation is very close to a Gaussian,
−M(P)−Ei(k1)−E(k2)φ−t(k,P)=−iv†(k1)χ(k,P)u(k2) −k2
χ(k,P)=R (d23πk)′3 V+l(vk(k−1′)kφ′−)Ωt(lk(cid:0)′u,P(k)1′u)†φ(+k2(′k)′,ΩPl)v†(k2′()20) φs(k)≃ eN2αs2s , Ns−1 = 4√p3απ√5s/2π , αs ≃.476 (25)
and the mass is M = 2.94K = 970MeV which is close
(cid:17) 0
where k = k+ P , k = k P and P is the total mo- tothemostprobableexperimentalmassofthef0ground-
1 2 2 − 2 state.
mentum of the meson. We use the Bethe-Salpeter am-
Wenowstudythepseudoscalargroundstateinthelow
plitude χ as an intermediate step to compute the contri-
P limit, which was already studied extensively in the
butionof interactionV to the boundstate equation. The
wave functions φ+ and φ are equivalent to the Bethe- literature [11,12]. For vanishing P we find that φ+ =
−
φ and both are proportional to sin(ϕ). This is due
Salpeter amplitude χ . For color singlets the contribu- −
−
to the Goldstone boson nature of the π, see the result
tion of the infinite infrared constant U cancels, see Ap-
of Appendix C. However this component of the wave-
pendix A. The equation is also flavor independent, and
functionhaszeronorm,anditisnecessarytoincludethe
we will now concentrate on the momentum spin part
⊗ next orderof the expansionin P to determine the norm.
ofthe wave-functions. We willnow dropthe U termand
The most general low P pseudoscalar wave-function is
the color dependence from the equations. In this section
then,
the matrices Ω will only include the Dirac structure,
l
Ω Ω =γ γ +a~γ ~γ. Withtheaimofstudyingthe
feq0lu⊗daetciaolynienq0au⊗aptaio0irnoffoπr,·t⊗wheewscialllanrofw0sionlvietsthceenbtoeurnodf mstaatses φ+ =Np−1 S+ Mk(P)f1+ig1Pk~ ·kˆ×~σ!i√σ22
frameandthe equationforthe pseudoscalargroundstate
M(P) P~ iσ
π in the limit of small P and in the limit of large P. φ = 1 S+ f ig kˆ ~σ 2 , (26)
Due to the large mass of the scalar meson f0 in this − Np− − k 1− 1k · × !√2
model, it turns out that the negative energy φ compo-
−
nent for the groundstate is less than 10% of the positive where the norm is a function of the π energy,
6
2 = 2f(t) 2M , M2(P)=M2(0)+P2 fπ(s) othWerelnimowit dofislcaursgsetmheompseenutudmoscPal.arIngrtohuisndcastsaettehienntehge-
Np π vufπ(t) ative energy components are suppressed by a factor of
(cid:0) (cid:1) u
2m ψ¯ψ d3k t 1/P. The chiral angle ϕ, depicted in Fig. [5] vanishes
M2(0)= 0h i , ψ¯ψ = 6 S (27) completely, and the spinors are simpler, for instance,
− fπ(t)2 h i − Z (2π)3
1+α~ kˆ 2
and where in the case of an instantaneous interaction us(k1) · 1us(0) , kˆ1 Pˆ+ ~k (31)
there are [11,12] usually 2 different f(t) and f(s), ≃ √2 ≃ P ⊥
π π
where the index denotes the projection~k (~k P)P of
f(t) = 3 ∞dk kf S (28) avector~k inthe⊥planeperpendiculartoP~. −Thev·ertices,
π sπ2 Z0 1 up to first order in 1/P are for instance, b b
qfπ(s)fπ(t) =s2π12 Z0∞dk −k2Sϕ˙ +4kC(g1+1/2). u†s(k1)us′(k1′)≃δss′ − P1i~σss′Pˆ×(~k⊥−~k⊥′ ) (32)
(~k +~k )+i~σ (~k ~k )
Substituting the wave-functions of eq. (26) in eq. (20), u†s(k1)α~us′(k1′)≃Pˆδss′ + ⊥ ⊥′ P × ⊥− ⊥′
andexpandingtheresultingequationuptothefirstorder
inP,wegettheequationsforthef andg components, Up to first order in 2/P the equation for positive and
1 1
negative energy boundstate functions φ (k)iσ / is,
1 ± 2 √2
f¨ = kS+(2Ck)f
1 1+a(2S2+1) − 1 2k2
+4a SC∆ϕ hC2/k2+S2ϕ˙2+ϕ˙2/2 f 0≃ P + P ∓M −(1−a)∆k φ±(k)
− − 1 (cid:18) (cid:19)
4aS(cid:0)Cϕ˙(f˙1 f1/k) , (cid:1) (29) 2i(1 a)P ~ ~σ,φ (k)
− − k ±
− − P ×∇ ·
1 i
g¨ = kC+(2kC+2S2/k2)g b 2 (cid:8) (cid:9)
1 1+a(2S2−1)h 1 −a∆kφ∓(k)− P2~σ⊥·φ∓(k)~σ⊥ (33)
+2a(S2/k2 2SC∆ϕ 2S2ϕ˙2)g
1
− − We find that the wave-function has a component with
4aSCϕ˙(g˙1 g1/k) . (30) structure i~σ P ~k. However this component is smaller
− − · ×
i than the s-wavecomponent by a factor of less than 1/P.
Itturnsoutthattheparameterahaslittleeffectonmost Thus we findtbhat the momentum spin solutionupto
⊗
functions, except for f . The homogeneous equation for highestorderin,isessentiallyapositiveenergyGaussian
1
f hasthesolutiona = .195anthusf α1/a a . This function φ (k)iσ2 ,
1 0 − 1 − 0 p √2
will essentially affect f(t), f(s) and the pion velocity c.
π π
W69eMfienVd.foTrhais≃sh−o.w18s athcalteacr=im1p,rfoπ(vt)em=efnπt(s)of=th0e.2m1Kod0e≃l, φp(k)≃ e−2αk2p2 , Np−1 = 2α√π 32 , α2p = 1−2 aP (34)
with a correct relativistic pion and a better fπ. Np (cid:18) p (cid:19) r
3 For large momentum P we find that φ+ is quite flat in
k, while φ is almost negligible,
−
f /k
1
a
2 φ− ∆φ+ . (35)
≃ 2p
This result is consistent with the relativistic space con-
1
S traction. We checked that the components that we ne-
glected here would yield a small contribution to the f
0
0 decay.
g /k
1
-1 E. The coupling of a scalar to a pair of pseudoscalars
0 1 2 3
k
FIG. 6. We represent the π wave-functions functions S, TheformfactorF(P)forthecouplingofascalarf to
0
f1/k and g1/k respectively with solid, dashed and dotted apairofπcanbedecomposedindiagramswhereaquark
lines, in the adimensional units K0 =1. (antiquark) line either emits (absorbs)a pseudoscalaror
ascalar. WeusethetruncatedBethe-Salpeteramplitude
χ, as an intermediate step to compute the coupling of a
7
meson to a quark line u χu. F(P) is represented with We now consider the opposite limit of large pion mo-
†
a large triangle and the boundstate amplitudes χ and φ mentum P. In this case the negative energy component
are represented with small triangles, of the pion is quite small. We expect that the dominant
diagrams are the ones of the first line of eq. (34), which
include only the positive energy φ+. This is present in
(cid:17) uy either the coupling of a pion to the quark line, or the
(cid:17) (cid:17)(cid:16)(cid:16))Q(cid:31)Q(cid:25) (cid:16))(cid:16)(cid:16)Q+Q(cid:25) coupling of a π to an antiquark line,
Q(cid:17)f0Q&(cid:25)(cid:25) Q(cid:17)Qf0PuPqP(cid:17)PP(cid:17)PiQ+(cid:17)Q(cid:25) Q(cid:17)Q(cid:17)f0(cid:16)PqPvyQ(cid:16)(cid:31)(cid:16)1(cid:17)Q(cid:16)(cid:25) (cid:17)(cid:17) d3k
Q = (cid:17) + v(cid:17) F(P)= φ (k ,0)( ∆ ) φ (k ,P)φ (k, P)t
(2π)3 s 1 − ′k p ′ p − q
(cid:27)(cid:23) Z h
u
+ Q(cid:17)(cid:0)Q(cid:17)(cid:25)PQ(cid:17)Pq(cid:31)Q(cid:17)PfPiP0(cid:16)1vvP(cid:16)y (cid:17)Q+(cid:17)Q(cid:25) + (cid:26)Q(cid:17)(cid:22)(cid:0)Q(cid:17)(cid:25)(cid:16)Q(cid:17)(cid:16)(cid:16))(cid:31)Q(cid:17)f(cid:16)1(cid:16)0PuPi(cid:16)y (cid:17)Q+(cid:17)Q(cid:25) (36) where t +anφdp(tk,Par)eφpr(eks′p,e−cPtiv)teq¯lyi(cid:12)(cid:12)(cid:12)(cid:12)kt′h=ekt,races in Dirac(4in1)-
q q¯ (cid:12)
This includes the qq¯pair creation or annihilation of Fig. dices,
2. The same irreducible interaction for quarks which is
uthseedaninnihthileatbioonu.ndInsteaqte. (e3q6u)attihoensloiospaelsnoer[g2i6e]s uarseedtrfiovr- tq =tr(−√iσ22~σ·~k1u†(k1)Ωlu(k1′)i√σ22v†(k2′)Ωlu(k2)i√σ22)
ial, see for instance eq. (A3), and we now compute the
moWmeefinrtsutmco⊗nssipdienrtchoentlrimibiuttoiofna.couplingtopionsoflow =tr(~σ√·~2k1(1+2 β)(1+√α~2·kˆ1)Ωl(1+√α~2·kˆ1′)
momentum P. The coupling of the pseudoscalar to the
quark is derivative and thus is suppressed. For instance 1 (1+β)γ (1−α~ ·kˆ2′)Ω (1+α~ ·kˆ2) 1 (42)
5 l
in the case of a massless π in the center of mass, using √2 2 √2 √2 √2)
the wave-function of eq. (26), we find that
and t , of the second diagram , with the coupling of the
q¯
χ= (1 3a)√2 1∆(S) βγ +o(P) pion to the antiquark yields the same result. The total
− − Np− 5
coupling is a functional of the scalar and pseudoscalar
h i
u†(k)χ(k,0)u(k)=o(P) , (37) wavefunctions, which are described in eqs. (25,34) by
⇒
Gaussians. We nowapplythe Laplacianto the functions
whichisconsistentwiththederivativecouplingofapion
ofk . Thedominanttermoftheexpansionin2/P comes
toaquark. The dominantcontributionincludes thecou- ′
from the derivatives of φ . A derivative of φ (k ,P)
pling of the 3P scalar meson to the quark (antiquark) P P ′
0 will be proportional to P/2 when the Gaussian integral
line. The coupling F(P) to a pair of pions with low mo-
is performed , while a derivative of t or t which are
mentum P is, q q¯
functions of respectively kˆ or kˆ is proportional to 1/P
1′ 2′
d3k and will not produce a dominant term. The traces then
F =tr (2π)3φ−† u†χsuφ++φ+v†χsv (38) simplify to,
Z
(cid:0) (cid:1)
where for instance we get for the scalar coupling t =t =a k1 1 (kˆ kˆ )2 (43)
u†(k)χs(k)u(k) to the quark line, q q¯ √2 − 1· 2
h i
u χ (k)u = δs1s2 (1 a)Ckˆ ∆(kˆSkφ ) WenowapplytheLaplaciantoφ ,andexpandthekˆ kˆ
†s1 s s2 2√2 − · s in a series of 1/P. It is convenienPt to define α2 =2α12·+2
h T s
−(1−3a)S∆(Ckφs) (39) α2p, and we get finally find for the momentum ⊗ spin
contribution,
i
except for the i factor which goes with any potential
iisntimsienprlttiihfioiensdasewcccitotihrodn−ti.hnegInhttoeelgtphraeotfFintehgyenbmmyaapnsasrrgtusaleptsh.eeqWeuqea.t(wi3oi8lnl)dacinasdcnawbrdee F(P)=a321−6α42pπ+43 αα2Ts12α22pα3T P2 +o(cid:18)P2(cid:19)e−(αP22T)2 (44)
get,  
(cid:0) (cid:1)
where the dominantterm is the a term which is of the
F =Z0.0(12d1π3Kk)32k√φ2s0(cid:20).0−5k2SM+2(ϕP˙d)dk +aSC∆(cid:21)(cid:16)φ−†φ+(cid:17) oTrhdTeehrefloafcvo0ol.ro4r3faKfact0c−ot1or/r2foffoorrrftP0heaonfcdothuπeploicnrodgloerrofosfian2gKslec0ta.slairs i1s/o√sin3-.
= 0 − K1/2 . (40) glet (uu¯ + dd¯)/√2 to a pair of pseudoscalar isovec-
fπ2M(P) 0 tors, say ud¯and du¯, with a flavor independent quark-
−
Thiscouplingisverysensitivetothepiondecayconstant antiquark annihilation is 1/√2. The total coupling is
−
f and to the energy M(P) of the pion. then F(P)/√6.
π
−
8
ThefunctionF(P)isverycumbersometoderiveinthe symbol which is used in other sections for the vector or
caseofintermediatemomenta. Formomentaoftheorder axial vertices. The width Γ is a simple function of the
of K , see Fig. 7, matching the high and low P limits imaginary component of ∆M,
0
with an interpolating function is a possible approxima-
Γ 1
tion. Im(∆M)= i , Γ= P M F(P)2 (47)
0.8 − 2 4π f0| |
|F(P)|
whereP = 1 M2 4M2isthemomentumoftheemit-
2 f0 − π
0.6
ted π. Let uqs consider the case of a scalar mass Mf0 of
the order of 1GeV, where P is larger than the scale of
P the interactionbya factorof1.4. Inthis case,itis sensi-
0.4 ble to use the limit of large momentum P for F(P), see
eq. (44). We finally find a partial decay width of just
40MeV for F ππ which lies within the experimental
0
0.2 limits. We exp→ect that a complete calculation without
the large P approximation would not deviate from this
P 0
by more than a factor of 2.
0.0 This is also compatible with the narrow resonances
0.0 0.5 1.0 1.5 2.0 2.5 3.0
P f (1500) and a (980) which are possible groundstates.
0 0
FIG. 7. We show F(P), in the adimensional units of ConcerningK0∗(1430)whichiswiderandhasdecayprod-
K0 =1. The dotted line and dashed line correspond respec- ucts with a larger momentum P, the function Γ(P) of
tively low P and high P limits. eq.(47)hasthe correctqualitativebehaviorofbeing pro-
portional to P3 for intermediate momenta. However in
thismodeltheexponentialdecreaseistoostrong,andthe
F. The f0(980)→ππ decay modelneedssomeimprovementinordertoreproducethe
correct K (1430).
0∗
Thedecaywidthofaf inapairofπcanbecalculated
0
fromthetheBreit-Wignerpoleinthemesonpropagator.
We call bare the ladder meson, see eqs. (7), (C1) and III. GOING BEYOND BCS WITH FINITE
COUPLED CHANNEL EFFECTS
(C13). The bare mass M is real and is a solution of
0
the Bethe-Salpeter equation. When coupled channels of
mesons are included, the bare meson is dressed. The A. The Mass Gap Equation and the self energy
dressed pole is composed by the bare mass M plus the
0
coupled channel contribution which includes a real mass We find in Appendix D that the minimal extension of
shift, and an imaginary term in the case where the mass the mass gap equation beyond BCS is achieved with a
is above the coupled thresholds. The mass is then, new tadpole term in the self energy,
M =M +∆M , ∆M = iΣ
0 − (cid:7)O (cid:4)
?
Q(cid:17)f0(cid:17)&(cid:17) < Q.Qf0Q(cid:17) Q(cid:17)f0(cid:17)&(cid:17)JJJJ(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)Q.Qf0Q(cid:17) (cid:6) = (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:27)(cid:1)(cid:1)(cid:1)(cid:1) + (cid:6)(cid:1)(cid:1)?(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)6(cid:5)6 ,
(cid:6)f0 = Q(cid:25)Q(cid:25) < (cid:17)(cid:25)(cid:25)(cid:17) + Q(cid:25)Q(cid:25)(cid:10)(cid:10) JJJJ(cid:17)(cid:25)(cid:25)(cid:17) (45) '~k;w0 (cid:27) (cid:27) ~k;w0$
(cid:19)- - (cid:16)
whereweonlyincludedaloopofbareπ,whichisthesim- O(~k;w)= (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ~k0(cid:0)P~;w0(cid:0)W (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) ;
plest contribution to the scalar self energy. The integral ~k(cid:0)~k0(cid:18)~k(cid:0)P~(cid:27);w(cid:0)W ~(cid:17)k(cid:0)~k0 (48)
in the loop energy provides an extra 2πi factor, which D
verifies that in general ∆M includes a real component, where the sub-diagram is defined as an intermediate
O
and we find that, step. ThisamountstoextendtheMGEfortheselfenergy
ofthequarkswiththesimpleonemesonexchange. Using
F(q)∗ F(q)
d3q the Weyl fermions, and expanding the ladder in meson
∆M =6 √6 √6 (46)
poles, we find that the self energy of the quark (anti-
(2π)3M 2 q2+M2+iǫ
Z f0 − π quark) has a diagonal component Σ which contributes
d
where the factor of 6 inclpudes the 3 different flavors of to the dynamical mass of the quark (antiquark),
the isovector π, and the factor 2 from the direct and
exchange diagram of the self energy in eq. (45). In this (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
section we represent as usually the width of a resonance (cid:6)d = (cid:1) (cid:27)(cid:1) + (cid:24)X(cid:24)-(cid:1) X(cid:24)X(cid:1) ; (49)
by a Γ, which should not to be confused with the same
9
and the energy of 1 quark is identical to the BCS one of order U0. The first pair of diagrams are BCS dia-
of eq. (18) except for the expected changes of the chiral grams. It turns out that the new diagrams are the same
angle ϕ. In eq.(49) we only included the nonvanishing diagrams which contribute to the f π π coupling,
0
→ ⊗
diagrams which remain from an expansion in powers of except for the negative energy wave function of the π
1/U. The free Green functions are proportional to U 1. and for the integral in the π momentum P. The neg-
−
The interactions without a pair creation or annihilation ative energy wave-function φ always vanishes for high
−
areproportionaltotheinfiniteinfraredconstantU,while momentum P and in the case of low momentum P it
the remaining interactions are finite. It turns out that is relevantonly andfor the pseudoscalarfamily of the π.
the new coupledchannel diagramsvanish. This happens Wesupposethatthelargenumberofexcitedstatesisnot
because in the limit of an infinite U the box diagrams sufficienttocompensatethesmallnessofφ ,andwewill
−
−
in eq. (48) which contribute to the quark energy vanish. not consider this ultraviolet problem. Thus we will only
This is the case for instance of diagrams (a) (c), include the coupledchannelcontributionoftheπ family.
In the case of low momentum, the π family has a ex-
tremely large φ , however the coupling to a quark u χu
(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)=0; (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)H-(cid:27)H(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) =0; (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)-(cid:27)H(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) = H(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) itshedelarisvtatliinvee oafn−deqv.(a5n2i)s.heFs.inTalhlyiswperocmanptrsemusovteo,nweigt†lhecat
(a) (b) (c) . (50) functionalderivative,thevariationalscalarwavefunction
φ+. The result after integrating in all the loop energies
We find that the quark energyE =E0 Σd remains the f0
− can be represented,
BCS one of eqs. (15) to (18),
1 d3k
E(k)=u†(k)α~ ·~ku(k)− 2 u†(k) (2π)′3V(k−k′) (cid:1)k(cid:0)(cid:1)k(cid:1)0 (cid:1)k(cid:0)(cid:1)k(cid:1)0
ΩlhΛ+(k′)−Λ−(k′n)iΩlu(kZ)o . (51) (cid:6)a = 12 kk(cid:24)s1(cid:24)(cid:1)(cid:1)(cid:1)s(cid:24)2 (cid:24)k0s(cid:24)2(cid:1)(cid:1)s(cid:1)5 + 21 kkXs5X(cid:1)(cid:1)s(cid:1)2XXk0sXs1(cid:1)2(cid:1)(cid:1)
However the last diagram of eq. (50), which contributes
to the mass gap equation is finite. (cid:17)k0 k0Q (cid:17) k0+P Q
The mass gap equation is obtained when we impose (cid:17) Q (cid:17) s3 Q
P(cid:17) S (cid:19) QPP(cid:17) Q P
that the antidiagonal components Σa of the self energy Q(cid:0) SkSS0(cid:0)(cid:19)P (cid:0) (cid:17) Q (cid:0) k0 k0 (cid:0) (cid:17)
must cancel. As in eq.(15) this component in obtained QQ (cid:19)(cid:19)S(cid:19)Ss3(cid:17)(cid:17) QQ@ (cid:0)(cid:17)(cid:17)
wduitchesthaefpurnocjteicotnioΣnaofwtihthe stphienoqrusaun†tuamndnvu.mbTehriss opfroa- + kk(cid:24)s1(cid:24)s2(cid:1)(cid:1)(cid:19)(cid:24)(cid:1)(cid:19)(cid:1)(cid:1)k(cid:1)(cid:1)(cid:24)(cid:0)(cid:1)(cid:1)(cid:1)k(cid:1)(cid:1)(cid:24)0(cid:1)S(cid:1)(cid:1)(cid:1)s(cid:1)s45 + kk s1s(cid:1)(cid:1)(cid:0)2(cid:1)(cid:1)(cid:1)k(cid:1)(cid:1)(cid:0)@@(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)@(cid:0)sk(cid:1)4(cid:1)@0(cid:1)(cid:1)(cid:1)(cid:1)(cid:28)(cid:1)s5 ; (53)
scalar, see eqs. (19) and (21). In order to use the re- wherethelines onlyrepresentthespinorsu, v, u or
†
sults of the preceding section, it is convenient to fold Σa v†,the integralsandthetracesandnolongerincludethe
withagenericscalarwavefunctionφ+f0. Thentheresult- quarkoranti-quarkpropagators. Themassgapequation
ing productmustvanishfor anyφ+f0. Infactthis ensures 0=S0−1a−Σa is now,
vacuumstabilitysincethispreventsthevacuumtodecay
inscalarmodes. Thediagramsthatcontributetothean- 0=+u†s1(k)α~ ·~kvs5(k)
tidiagonal component of the self energy in the mass gap d3k
′
equation are now, − u†s1(k) (2π)3V(k−k′)Ωl
n Z h
δ d3P
trnφ+f0(k)†Σao= Q(cid:17)Q(cid:17)f0(cid:16)(cid:16)(cid:27)(cid:16)1(cid:1)(cid:16)(cid:1)(cid:1)(cid:1)(cid:1)(cid:16)(cid:27)(cid:1)(cid:1)(cid:1)(cid:1)+ Q(cid:17)Q(cid:17)f0-PPiPP-(cid:1)(cid:1)(cid:1)(cid:1)PP(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) +uφs2−s(3ks4′)†(cid:16)(Ps,22ks′4−−PZ/2()2πu)3†s4φ(−sk2′s)3(P,k′−P/2)
δ d3(cid:17)P
(cid:17)(cid:16)(cid:16))uyQ(cid:31)Q(cid:25) (cid:20)(cid:24) (cid:16))(cid:16)(cid:16)Q(cid:0)Q(cid:25) −vs2(k′) s22s4 − (2π)3φ−s2s3†(P,k′−P/2)
+ Q(cid:17)Qf0PuPqPP(cid:17)PPi(cid:17)(cid:17)Q(cid:0)(cid:17)Q(cid:25) + Q(cid:17)Q(cid:17)f0Pq(cid:16)Pvvy(cid:17)Q(cid:16)(cid:31)(cid:16)1(cid:17)Q(cid:16)(cid:25) (cid:17)(cid:17)(cid:21)(cid:25) φ−s3s4((cid:16)P,k′−PZ/2) vs†4(k′) Ωlvs5(k), (54)
(cid:17) i
where the sum over repeated spin indexes s is assumed.
i
(cid:27)(cid:23) < $$
u
+ Q(cid:17)(cid:0)Q(cid:17)(cid:25)Q(cid:17)PqPQ(cid:17)P(cid:31)fPiP0(cid:16)1vvP(cid:16)y (cid:17)Q(cid:0)(cid:17)Q(cid:25) + (cid:26)Q(cid:17)(cid:22)(cid:0)Q(cid:17)(cid:25)(cid:16)Q(cid:17)(cid:16)(cid:16))(cid:31)Q(cid:17)f(cid:16)1(cid:16)0P>uPi(cid:16)y (cid:17)Q(cid:0)(cid:17)Q(cid:25)%%. (52) B. Model indepencdheanntneefflsects of the coupled
In eq.(52) we only show the diagrams which are nonva- Thedominanteffectofcoupledchannelsisto multiply
nishing in orders of 1/U, and in fact they all are finite, the potential term in the mass gap equation by a factor
10