Chiral symmetry patterns of excited mesons with the Coulomb-like linear confinement

0.42 MB·English

by  R. F. Wagenbrunn

Chiral symmetry patterns of excited mesons with the Coulomb-like linear
confinement.
R. F. Wagenbrunn and L. Ya. Glozman
Institute for Physics, Theoretical Physics branch,
University of Graz, Universit¨atsplatz 5, A-8010 Graz, Austria
Thespectrumofq¯qmesonsinamodelwheretheonlyinteractionisalinearCoulomb-likeinstan-
taneous confining potential is studied. The single-quark Green function as well as the dynamical
chiralsymmetrybreakingareobtainedfromtheSchwinger-Dyson(gap)equation. Giventhedressed
quarkpropagator,acompletespectrumof”usual”mesonsisobtainedfromtheBethe-Salpeterequa-
tion. ThespectrumexhibitsrestorationofchiralandU(1)A symmetriesatlargespinsand/orradial
excitations. Thispropertyisdemonstratedbothanalyticallyandnumerically. Atlargespinsand/or
radialexcitationshigherdegreeofdegeneracy isobserved,namelyallstateswith thegivenspinfall
7 into reducible representations [(0,1/2) (1/2,0)] [(0,1/2) (1/2,0)] that combine all possible
⊕ × ⊕
0 chiral multiplets with the given J and n. The structure of the meson wave functions as well as the
0 form of theangular and radial Regge trajectories are investigated.
2
PACSnumbers: 11.30.Rd,12.38.Aw,14.40.-n
n
a
J
INTRODUCTION chiralsymmetryrestorationofthesecondkind. Itshould
4
notbe mixedupwiththechiralsymmetryrestorationat
2
In spite of numerous efforts for 30 years the physics hightemperaturesand/ordensities. Inthelattercasethe

2 systemundergoesaphasetransitionandthe chiralorder
which drives the structure of hadrons in the light fla-
v parameter vanishes - the quark vacuum becomes trivial.
vor sector is not yet completely understood. It is clear,
9
In contrast, nothing happens of course with the quark
3 however,thatherethe mostimportantelements ofQCD
condensate of the vacuum once we are interested in the
0 are spontaneous breaking of chiral symmetry and con-
1 finement. The hadron spectrum is a key to understand given isolated hadron at zero temperature. If a hadron
0 is highly excited, this quark condensate of the vacuum
QCD in the nonperturbative regime, the role and inter-
7 becomessimplyunimportantforphysics,becausetheva-
relations of confinement and chiral symmetry breaking.
0
lence quarks decouple from the condensate.
/ The spontaneous breaking of chiral symmetry is cru-
h
cial for a proper description of the physics of the low- A fundamental origin of this phenomenon is that in
p
lying hadrons. It is obvious, for example, in the case of highly excited hadrons the semiclassical regime is mani-
-
p a pion, which has a dual nature. From the chiral sym- fest and semiclassically the quantum fluctuations of the
e metrypointofviewitisa(pseudo)Goldstoneexcitation quarkfieldsaresuppressedrelativetotheclassicalcontri-
h
: associated with the spontaneously broken axial part of butions that preserve both chiraland U(1)A symmetries
v chiral symmetry [1]. From the microscopical point of [8]. Thisgeneralclaimhasbeenillustrated[9]withinthe
i
X view, it is a highly collective quark-antiquark mode [2]. exactly solvable chirally symmetric confining model [10],
r The latter feature persists in any known microscopical which can be considered as a generalization of the large
a approach to chiral symmetry breaking that is consistent Nc ’tHooftmodelin1+1dimensions[11]to 3+1dimen-
withtheconservationoftheaxialvectorcurrent. Bynow sions. The key issue is that the chiral symmetry break-
it is well understood that both these views are comple- ing Lorentz-scalar dynamical mass of quarks arises via
mentary and consistentwith each other. That the chiral loopdressingandvanishesatlargemomenta,whereonly
symmetry breaking is crucially important for a proper classical contributions survive. When one increases the
physics of the nucleon follows e.g. from the large pion- excitationenergyofahadron,onealsoincreasesthe typ-
nucleon coupling constant or from the Ioffe formula [3] icalmomentumofthe valencequarks. Consequently,the
that relates, though not rigorously, the nucleon mass to chiral symmetry violating dynamical mass of quarks be-
the quark condensate of the vacuum. comes smalland chiralsymmetry gets approximatelyre-
It has been realizedin recent yearsthat the physics in stored in the given hadron. This theoretical expectation
the upper part of the light hadron spectra is quite dif- has been proven by the direct calculation of the heavy-
ferent. In this case the chiral symmetry breaking in the light excited meson spectra with the quadratic confining
vacuum is almost irrelevant, as evidenced by the persis- potential [12] and of the light-light meson spectra with
tence of the approximate multiplets of chiral and U(1) the linear potential [13].
A
groups both in baryon [4, 5, 6] and mesons [7] spectra, The effective chiralrestorationhas transparentlybeen
for a short overview see [8]. This phenomenon, if con- illustratedwithin a simple toy model atthe hadronlevel
firmedexperimentallybydiscoveryofstillmissingstates, thatgeneralizesthesigmamodelandcontainsaninfinite
is referred to as effective chiral symmetry restoration or amount of excited pions and sigmas [14].
2
Σ From the underlying physics point of view, this model
S0 canbe directly relatedto the Gribov-Zwanzigerscenario
ofconfinementinCoulombgauge[31]. InCoulombgauge
Σ
S S0 S0 S0 aHamiltoniansetuptoQCDcanbeused[32,33]andnu-
A merous studies in different approaches suggest that the
Coulombinteractionintheinfraredturnsouttobeacon-
fining linear potential, see e.g. [34, 35, 36, 37]. Actually
the infinitely increasing Coulomb potential is a neces-
B sarybutnota sufficientconditionforconfinementmech-
anisminQCDandthestrengthoftheCoulombpotential
FIG. 1: Graphical representation of the Schwinger-Dyson
mustbelargerorequaltothatextractedfromtheWilson
(A) and Bethe-Salpeter equations (B) in the ladder approxi-
mation. loop[34]. Then there is no contradictionwith the recent
observation on the lattice that in the deconfining phase
aboveT astrongCoulombconfinementstillpersists[38].
c
Restoration of chiral symmetry requires that hadrons This shows, however, that the actual confining scenario
should decouple from the Goldstone bosons [6, 14, 15, in QCD is richer than a simple linear Coulomb-like po-
16, 17, 18]. A hint for such a decoupling is well seen: tential. Hence some elements of QCD are still missing
The coupling constant for the decay process h∗ h+ in this picture and one needs them in order to obtain a
→
π decreases very fast once one increases the excitation completely realistic picture of hadrons.
energy of a hadron. However,ourpurposeisdifferent. Weusethismodelto
The effective restoration of chiral symmetry has been answera principalquestion whether effective restoration
discussed recently within different approaches in refs. of chiral symmetry in excited hadrons occurs or not and
[19, 20, 21, 22, 23, 24]. if it does - to use the model as a laboratory to get an
In this paper we present a formalism that leads to the insight. This model satisfies all the criteria (i)-(vi) and
meson spectrum reported in our recent Letter [13] as hence is completely adequate for this purpose.
wellas some additionalnumericalresults. We alsostudy
meson wave functions and clarify origins of a very fast
restoration with increasing J and of a slow restoration THE HAMILTONIAN AND THE INFRARED
with increasing the radial quantum number n. REGULARIZATION
Our choice of a model is motivated by the following
constraints. The model must be (i) 3+1 dimensional, The GNJLmodelis describedbythe Hamiltonian[10]
because we know that in 1+1 dimensions the effective
chiral restoration does not occur due to absence of the Hˆ = d3xψ¯(~x,t) i~γ ~ +m ψ(~x,t)
rotational motion and spin; (ii) chirally symmetric; (iii) − ·▽
Z (cid:16) (cid:17)
field-theoretical in nature in order to be able to exhibit + 1 d3xd3y Ja(~x,t)Kab(~x ~y)Jb(~y,t), (1)
spontaneousbreakingof chiralsymmetry; (iv) confining; 2 µ µν − ν
Z
(v) preserving the conservation of the axial vector cur-
with the quark current-current interaction
rent; and (vi) solvable. Then, upon solving a spectrum
within such a model one can judge whether the chiral λa
symmetry patterns appearor not andwhether the phys- Jµa(~x,t)=ψ¯(~x,t)γµ 2 ψ(~x,t) (2)
ical arguments presented above are correct.
The only known model that satisfies all these con- parametrized by an instantaneous confining kernel
straints is the Generalized Nambu and Jona-Lasinio Kab(~x ~y) of a generic form. We use the linear con-
µν −
model of ref. [10]. In this model the only interaction fining potential,
between quarks is the instantaneous confining Lorentz-
vectorpotentialbetweencolorquarkcharges. Thismodel Kµaνb(~x−~y)=gµ0gν0δabV(|~x−~y|), (3)
belongs intrinsically to the class of large N models, be-
c
and absorb the color Casimir factor into the Coulomb
cause only ladders are taken into account upon solving
string tension,
the Schwinger-Dyson and Bethe-Salpeter equations, see
Fig. 1, and there are no vacuum fermion loops. Differ-
ent aspects of this model, in particular chiral symmetry λaλa
breaking,havebeenstudiedinmanyworks,seeforexam- V(r)=σr. (4)
4
ple [25, 26, 27, 28] and the spectrum of the lowest exci-
tationsofthelight-lightmesonswiththelinearpotential The Fourier transform as well as the loop integrals cal-
has been calculated in ref. [29]. It has been recently ap- culated with the linear potential are infrared divergent.
plied to study confinement in the diquark systems [30]. Hencewehavetoperformtheinfraredregularizationand
3
suppress the smallmomentum behavior of the linear po- and Σ(p ,p~) is the quark self energy. For an instanta-
0
tentialby introducinga cutoffparameterinto the poten- neous modelthe latter becomes energy(p ) independent
0
tial. Theninthe finalanswerthis cutoffparametermust and is given by
be sentto 0,i.e. the infraredlimit mustbe taken. While
d4q 1
quantities, such as the single quark Green function, can iΣ(p~)= V(p~ ~q )γ γ . (9)
bedivergentintheinfraredlimit,whichmeansthatasin- Z (2π)4 | − | 0S0−1(q0,~q)−Σ(~q) 0
gle quarkcannot be observed,we haveto checkthat any
Inserting (9) into (7) yields an integral equation for
color-singletobservable,likemesonmass,isfiniteandbe
S(p ,~p). With the ansatz
not dependent on the infrared regulator in the infrared 0
limit. Σ(p~)=A(p) m+~γ pˆ(B(p) p), (10)
There are slightly different and physically equivalent − · −
ways to performthe infraredregularizationin the litera- wherehereandinthe followingp means p~ andpˆ=p~/p,
| |
ture. We define the potential in momentum space as in the equation (7) can be cast into a coupled system of
ref [26] integral equations
8πσ 1 d3q M(q)
V(p)= . (5) A(p) = m+ V(k) , (11)
(p2+µ2 )2 2 (2π)3 ω¯(q)
IR Z
1 d3q q
Then, upon the transformation back into a configura- B(p) = p+ V(k)pˆ qˆ . (12)
2 (2π)3 · ω¯(q)
tional space Z
wheretheLorentz-scalarA(p)andLorentz-spatialvector
1 σexp( µ r)
d3peip~~rV(p) = − IR B(p) parts of self-energy contain classical and quantum
− (2π)3 Z − µIR loop contributions [9]. Here
σ
= +σr+ (µ )(6)
−µ O IR A(p)
IR M(p)=p (13)
B(p)
one recoversthat the potential containsthe requiredlin-
ear potential plus a term that diverges in the infrared is called the quark mass function, ω¯(p) = M2(p)+p2
limit µIR → 0. Note that the divergent part is canceled and ~k = ~p −~q. The appearance of a dypnamical mass
exactlyinallphysicalobservablesandthereis nodepen- signalsthespontaneouschiralsymmetrybreaking. Inthe
denceontheparticularwayoftheinfraredregularization. chiral limit m = 0 it has exclusively a quantum (loop)
Given this prescription all the integrals are well de- origin.
finedandwecanproceedbysolvingtheSchwinger-Dyson Noticethatin(11,12)theintegralsaredivergentatk =
(gap)and Bethe-Salpeter equations. Note that there are 0unlessaninfrared(IR)regulatorµ >0isintroduced.
IR
no ultraviolet divergences if only a linear potential is By defining the chiral angle ϕ as
p
used. They will persist, however, once a Coulomb po-
tential is added. Then the ultraviolet regularization and A(p) M(p)
sinϕ = = , (14)
p
renormalization would be required. Since our purpose is ω(p) ω¯(p)
to study the chiral symmetry restoration in the highly B(p) p
cosϕ = = , (15)
excited light-light states, where a role of the Coulomb p ω(p) ω¯(p)
interaction is negligible, such a complication is not re-
quired. where
ω2(p)=A2(p)+B2(p), (16)
THE GAP EQUATION AND THE CHIRAL
(11,12) can be transformed into one integral equation
SYMMETRY BREAKING
psinϕ mcosϕ =
p p
−
Due to interaction with the gluon field the dressed 1 d3q
V(k)(sinϕ cosϕ pˆ qˆcosϕ sinϕ )
Dirac operator for the quark becomes 2 (2π)3 q p− · q p
Z
(17)
D(p0,~p)=iS−1(p0,p~)=D0(p0,p~) Σ(p0,~p), (7) forthe chiralangle. The integralin (17)is convergentat
−
k = 0 since the infrared divergences arising by integrat-
where the bare Dirac operatorwith the bare quark mass ing over the two terms of the integrandmutually cancel.
m is Hence the chiral angle and the dynamical mass
D (p ,p~)=iS−1(p ,p~)=p γ ~p ~γ m, (8) M(p)=ptanϕ , (18)
0 0 0 0 0 0− · − p
4
This yields
σ
A(p) = sinϕ +A (p), (22)
p f
2µ
µ = 10-1 σIR
IR B(p) = cosϕ +B (p), (23)
µ = 10-2 2µIR p f
IR σ
M(p)0.1 µIR = 10-3 ω(p) = 2µIR +ωf(p), (24)
µ = 10-4
IR where Af(p), Bf(p), and ωf(p) are infrared-finite func-
tions.
Numerically the integration in (17) with µ = 0
IR
is feasible but demanding. Alternatively one can also
solve (11,12) with sufficiently small but finite values for
00 0.5 1 1.5 2 µIR =0. Still, in particular for very small values of µIR,
p specialcarehastobetakenforthenumericalintegration
in the vicinity of ~q = p~. The convergence of M(p) for
1.5 µ 0 in the chiral limit (m = 0) has been demon-
IR
→
strated in Refs. [26, 30]. The mass function and the
µ = 10-1 chiral angle, which are in agreement with the previous
IR
µ = 10-2 studies, are shownin the upper and lower plot of Fig. 1,
1 IR respectively. The result for µ = 10−4√σ is already a
p µ = 10-3 IR
ϕ IR very good approximationof the infrared limit which can
µ = 10-4 be determined numerically by extrapolating the results
IR
to µ =0.
0.5 IR
An important issue is a momentum dependence of the
chiral angle and dynamical mass. Their nonzero values
are uniquely related in the chiral limit with the sponta-
0 neoussymmetrybreaking. Theeffectofsymmetrybreak-
0 0.5 1 1.5 2
p ing is large at low momenta and the chiral angle goes to
π/2 at zero momenta. At large momenta the mass func-
FIG. 2: Mass function and chiral angle in thechiral limit for tion and the chiral angle go to zero. This property is
different values of the infrared regulator µIR. All quantities crucial for a proper understanding of chiral symmetry
are given in appropriate unitsof √σ. restorationin excited hadrons.
are infrared finite. Then the quark condensate is given CHIRAL AND U(1)A SYMMETRY PROPERTIES
as OF THE NONINTERACTING TWO-QUARK
AMPLITUDE
N ∞
q¯q = C dpp2sinϕ . (19) Given a dressed quark Green function, one can obtain
h i − π2 p
Z0 aspectrumofthequark-antiquarkboundstatesfromthe
homogeneous Bethe-Salpeter equation in the rest frame.
On the other hand, A(p), B(p) and consequently also
Before solving the Bethe-Salpeter equation we would
ω(p)divergelikeµ−1. Thelattercanbedeterminedfrom
IR like to discuss a limiting case when the self-interaction
A(p) and B(p) with Eq. (16) or directly by combining
is switched off, M(p) = 0. In such a case there is no
(11,12) to
self-energy and the quark condensate is identically zero.
The chiral and U(1) symmetries are not broken. Then
ω(p)=msinϕ +pcosϕ + A
p p
1 d3q the Bethe-Salpeter amplitude is reduced to a system of
2 (2π)3 V(k)(sinϕqsinϕp+pˆ·qˆcosϕqcosϕp). massless quarks with definite chirality. In such a case
Z properties of the system are unambiguously determined
(20)
by the Poincar´e invariance as well as chiral and U(1)
The infrared divergences can be explicitly projected out A
symmetries [7]. The Poincar´e invariance prescribes the
by using
standard quantum numbers JPC. The chiral and U(1)
A
µ 1 propertiesofthesystemthenareuniquelydeterminedby
lim IR d3q f(~q)=
µIR→0 π2 Z (k2+µ2IR)2 (21) itshecoimndpeaxtibofleawrietphrethseengtaivteionnJoPfCSaUn(d2)iLso×spSinUI(.2)TRhtehnaat
d3q δ(p~ ~q)f(~q)=f(p~).
− complete list of states is
Z
5
J = 0 BETHE-SALPETER EQUATION FOR MESONS
AND CANCELLATION OF THE INFRARED
DIVERGENCES
(1/2,1/2) : 1,0−+ 0,0++
a
←→
(1/2,1/2) : 1,0++ 0,0−+, (25) The homogeneous Bethe–Salpeter equation (BSE) for
b
←→
a quark-antiquark bound state with total and relative
J = 2k, k=1,2,... four momenta P and p, respectively, can generally be
written in covariant form as
(0,0) : 0,J−− 0,J++ d4q
←→ χ(P,p) = i K(P,p,q)S(q+P/2)
(1/2,1/2) : 1,J−+ 0,J++ − (2π)4
a ←→ Z
(1/2,1/2) : 1,J++ 0,J−+ χ(P,q)S(q P/2), (29)
b ←→ × −
(0,1) (1,0) : 1,J++ 1,J−− (26)
⊕ ←→ where K(P,p,q) is called the Bethe-Salpeter kernel and
χ(P,p) is the meson vertex function. In the rest frame,
i.e. with the four momentum Pν =(µ,P~ =0) the latter
J = 2k-1, k=1,2,...
dependsonthemesonmassµandallfourcomponentsof
p. Finally,fortheinstantaneousinteractionofourmodel
(0,0) : 0,J++ 0,J−− it is energy independent and BSE becomes
←→
(1/2,1/2) : 1,J+− 0,J−−
(1/2,1/2)ab : 1,J−− ←→0,J+− χ(µ,~p) = −i (2dπ4q)4V(|p~−~q|)γ0S(q0+µ/2,~k)
←→
(0,1)⊕(1,0) : 1,J−− ←→1,J++ (27) χ(µZ,~q)S(q0 µ/2,~k)γ0. (30)
× −
The sign indicates that both given states belong to
←→ In order to solve the Bethe-Salpeter equation for
the same representation and must be degenerate. Note,
mesons with the set of quantum numbers JPC with the
that the structure of the chiral multiplets is different for
given total spin J, parity P, and C-parity C, we expand
mesonswithdifferentspin. Inparticular,forsomeofthe
themesonvertexfunctionχPC(µ,p~)intoasetofallpos-
JPC the chiral symmetry requires a doubling of states. JM
sible independent Poincar´e-invariant amplitudes consis-
The U(1) multiplets combine the opposite spatial
A tent with JPC. This is done in the Appendix A. Then
paritystatesfromthedistinct(1/2,1/2) and(1/2,1/2)
a b the Bethe-Salpeter equation transforms into a system of
multiplets of SU(2) SU(2) .
L× R coupledintegralequations. Suchasystemofequationsis
The energy of the rotating massless quark is the same
derived in the Appendix B. The expansion of the meson
irrespective whether it is right or left, because there is
vertex functions as well as the system of coupled equa-
no spin-orbit force in this case [6]. Then one expects
tionslookslightlydifferentlyforthreepossiblecategories
a degeneracy of all chiral multiplets with the same J.
of q¯q mesons.
This means that the states with the same J fall into a
Category 1. To this category mesons with J−+ for
reducible representation
J = 2n and J+− for J = 2n+1 belong. In the non-
relativistic limit these mesons have wave functions with
orbital angular momentum L = J and the sum of the
[(0,1/2) (1/2,0)] [(0,1/2) (1/2,0)], (28)
⊕ × ⊕ two quark spins S =0.
that combines all possible chiral representations of the Category 2. Mesons with J++ for J = 2n and J−−
quark-antiquark system with the same spin. for J = 2n + 1 are in this category. In this case the
Certainly, once the self-energy is switched on, the chi- nonrelativisticwavefunctionshaveL=J 1andS =1.
±
ralsymmetry getsbrokenandthere cannotbe exactchi- Category 3. This category contains mesons with J−−
ralandU(1) multiplets. Theeffectiverestorationofchi- for J =2(n+1) and J++ for J =2n+1. Their nonrel-
A
ralsymmetryinexcitedhadronsisdefinedtooccurifthe ativistic wave functions have L=J and S =1.
following conditions are satisfied: (i) the states fall into The relativistic quark-antiquark states with JPC =
approximate multiplets of SU(2) SU(2) and U(1) 0−− do not exist in the BSE framework with an instan-
L R A
×
andthesplittingswithinthemultiplets(∆µ=µ µ ) taneous interaction. In the nonrelativistic limit L = 0
+ −
−
vanish at n and/or J ; (ii) the splitting and S = 1 would have to couple to J = 0, which is im-
→ ∞ → ∞
within the multiplet is much smaller than between the possible. The quark-antiquark states of category 4, i.e.,
two subsequent multiplets. with J+− for J = 2n and J−+ for J = 2n+1, do not
To verify this prediction we have to solve the Bethe- exist in the BSE framework with the instantaneous in-
Salpeter equation, which will be done in the following teraction, too, as is also shown in Appendix B. There
sections. are also no corresponding nonrelativistic states. Mesons
6
with such quantum numbers are called exotic (mesons a matter of dicsussion. We do not consider them in the
with exotic quantum numbers) and can be realized in present paper, however.
form of hybrids, i.e., as qq¯states with explicit excitation Now we want to study the infraredlimit of the Bethe-
of the gluonic field, or as four-quark states. If they can Salpeter equation and cancellation of the infrared sin-
also be described as quark-antiquark states in the the gularities. To this end let us take a closer look at the
BSE framework with a non-instantaneous interaction is integral equations for mesons of category 1, which are
1 d3q µ2
ω(p)h(p) = V(k)P (pˆ qˆ) h(q)+ g(q) , (31a)
2 (2π)3 J · 4ω(q)
Z (cid:18) (cid:19)
µ2
ω(p) g(p) = h(p)
− 4ω(p)
(cid:18) (cid:19)
1 d3q M(p)M(q)PJ(pˆ·qˆ)+pq 2JJ++11PJ+1(pˆ·qˆ)+ 2JJ+1PJ−1(pˆ·qˆ)
+ V(k) g(q)(.31b)
2 (2π)3 (cid:16)ω¯(p)ω¯(q) (cid:17)
Z
Here the functions h(p),g(p) are as defined in the Ap- The term with g(q) in the integrandof (31a) comes with
pendix B whichalong with the mesonmass µ haveto be afactor1/ω(q)= (µ )andthe integraloverthis term
IR
found from the given equations and~k =~p ~q. becomes the IR finOite expression
−
According to (24) and (21) there are IR divergent
terms µ2
g(p). (34)
σ 4
h(p) (32)
2µ
IR
Similarly there are IR divergent terms
and
σ
2µσ d3qδ(~k)PJ(pˆ·qˆ)h(q)= 2µσ h(p) (33) 2µIRg(p) (35)
IR Z IR
on the left and right hand sides of (31a), respectively. and
M(p)M(q)P (pˆ qˆ)+pq J+1 P (pˆ qˆ)+ J P (pˆ qˆ)
σ d3qδ(~k) J · 2J+1 J+1 · 2J+1 J−1 · g(q)= σ g(p) (36)
2µ (cid:16)ω¯(p)ω¯(q) (cid:17) 2µ
IR Z IR
on the left and right hand sides of (31b), respectively. The second term in the bracket on the left hand side has a
factor ω(p)−1 = (µ ) which vanishes in the IR limit. On the right hand side there is an IR finite term h(p). One
IR
O
sees that the IR divergent terms on both sides of both equations cancel and we end up with the coupled system of
integral equations
µ2 1 d3q
ω (p)h(p) = g(p)+ V (k)P (pˆ qˆ)h(q), (37a)
f 4 2 (2π)3 f J ·
Z
ω (p)g(p) = h(p)
f
1 d3q M(p)M(q)PJ(pˆ·qˆ)+pq 2JJ++11PJ+1(pˆ·qˆ)+ 2JJ+1PJ−1(pˆ·qˆ)
+ V (k) g(q), (37b)
2 (2π)3 f (cid:16)ω¯(p)ω¯(q) (cid:17)
Z
with
σ
V (k)=V(k) (2π)3δ(~k), (38)
f
− µ
IR
which has a finite IR limit.
7
In an analogous manner all IR divergences from the integral equations for the mesons of category 2 and 3 can be
removed. The IR finite integral equations for mesons in category 2 are
µ2 1 d3q J J +1
ω (p)h (p) = g (p)+ V (k) P (pˆ qˆ)+ P (pˆ qˆ) h (q)
f 1 4 1 2 (2π)3 f 2J +1 J+1 · 2J +1 J−1 · 1
Z (cid:26)(cid:18) (cid:19)
M(q) J(J +1)
+ (P (pˆ qˆ) P (pˆ qˆ))h (q) , (39a)
J+1 J−1 2
ω¯(q) p2J +1 · − · )
1 d3q pqPJ(pˆ·qˆ)+M(p)M(q) 2JJ+1PJ+1(pˆ·qˆ)+ 2JJ++11PJ−1(pˆ·qˆ)
ω (p)g (p) = h (p)+ V (k) g (q)
f 1 1 2 (2π)3 f  (cid:16)ω¯(p)ω¯(q) (cid:17) 1
Z 
M(p) J(J +1)

+ (P (pˆ qˆ) P (pˆ qˆ))g (q) (39b)
J+1 J−1 2
ω¯(p) p2J +1 · − · )
µ2 1 d3q pqPJ(pˆ·qˆ)+M(p)M(q) 2JJ++11PJ+1(pˆ·qˆ)+ 2JJ+1PJ−1(pˆ·qˆ)
ω (p)h (p) = g (p)+ V (k) h (q)
f 2 4 2 2 (2π)3 f  (cid:16)ω¯(p)ω¯(q) (cid:17) 2
Z 
M(p) J(J +1)

+ (P (pˆ qˆ) P (pˆ qˆ))h (q) , (39c)
J+1 J−1 1
ω¯(p) p2J +1 · − · )
1 d3q J +1 J
ω (p)g (p) = h (p)+ V (k) P (pˆ qˆ)+ P (pˆ qˆ) g (q)
f 2 2 2 (2π)3 f 2J +1 J+1 · 2J +1 J−1 · 2
Z (cid:26)(cid:18) (cid:19)
M(q) J(J +1)
+ (P (pˆ qˆ) P (pˆ qˆ))g (q) . (39d)
J+1 J−1 1
ω¯(q) p2J +1 · − · )
Note that the number of the auxiliary functions h ,g ,h ,g as well as the number of coupled equations in the
1 1 2 2
present case is twice larger than for mesons of category 1. This is related to the number of independent components
in χPC(µ,p~) for J > 0. However, for the states 0++ within the same category only the last two equations (39c) -
JM
(39d) apply, because in this case there are only two independent functions h ,g .
2 2
For mesons in category 3 we have:
µ2
ω (p)h(p) = g(p)
f
4
1 d3q M(p)M(q)PJ(pˆ·qˆ)+pq 2JJ+1PJ+1(pˆ·qˆ)+ 2JJ++11PJ−1(pˆ·qˆ)
+ V (k) h(q), (40a)
2 (2π)3 f (cid:16)ω¯(p)ω¯(q) (cid:17)
Z
1 d3q
ω (p)g(p) = h(p)+ V (k)P (pˆ qˆ)g(q). (40b)
f 2 (2π)3 f J ·
Z
From the integral equations the normalization of the by
vertex functions cannot be determined. One can derive
a normalizationconditionby demanding that the charge d3p
tr ψPC †(µ,p~)ψPC (µ,~p)
of the isovector mesons with isospin projection +1 must (2π)3 +JM +JM
Z
be one, orequivalently bynormalizingthe wavefunction (cid:0)ψPC †(µ,p~)ψPC (µ,p~) =2µ (41)
− −JM −JM
(cid:1)
(see the Appendix C for the definition of the wave func-
tion), where traces in Dirac space and color space have
to be taken, the latter giving just a factor 3. Then for
8
mesons of categories 1 and 3 this yields =sin2α and =cos2α . Obviously in the limit
1 M 2 M
N N
M(p) 0 the mixing vanishes.
3 d3p →
h(p)g(p)=1. (42)
4π (2π)3
Z
EFFECTIVE RESTORATION OF CHIRAL
Formesonsofcategory2thenormalizationconditionbe-
SYMMETRY FOR MESONS
comes
3 2 d3p In this section we demonstrate that in the limit of the
h (p)g (p)= + =1, (43)
4π (2π)3 i i N1 N2 vanishing dynamical mass M(p) 0 the Bethe-Salpeter
i=1Z →
X equation of the previous section recovers all the antici-
Here and represent the normalization factors of pated chiral multiplets (25)-(27).
1 2
N N
two coupled amplitudes, (h ,g ) and (h ,g ). Since For M(p) = 0 the integral equations (37), which de-
1 1 2 2
these amplitudes are mixed by the dynamical mass of scribe mesons of category 1, i.e. , with J−+ for J = 2n
quarks, M(p), a mixing angle α can be defined by and J+− for J =2n+1, become
M
µ2 1 d3q
ω (p)h(p)= g(p)+ V (k)P (pˆ qˆ)h(q), (44a)
f 4 2 (2π)3 f J ·
Z
1 d3q (J +1)P (pˆ qˆ)+JP (pˆ qˆ)
J+1 J−1
ω (p)g(p)=h(p)+ V (k) · · g(q). (44b)
f 2 (2π)3 f 2J +1
Z
For mesons of category 2, i.e., with J++ for J = 2n (J = 0) and J−− for J = 2n+1, the four coupled integral
6
equations (39) decouple into two independent systems of equations
µ2 1 d3q JP (pˆ qˆ)+(J +1)P (pˆ qˆ)
J+1 J−1
ω (p)h (p)= g (p)+ V (k) · · h (q), (45a)
f 1 4 1 2 (2π)3 f 2J +1 1
Z
1 d3q
ω (p)g (p)=h (p)+ V (k)P (pˆ qˆ)g (q) (45b)
f 1 1 2 (2π)3 f J · 1
Z
and
µ2 1 d3q
ω (p)h (p)= g (p)+ V (k)P (pˆ qˆ)h (q), (45c)
f 2 4 2 2 (2π)3 f J · 2
Z
1 d3q (J +1)P (pˆ qˆ)+JP (pˆ qˆ)
J+1 J−1
ω (p)g (p)=h (p)+ V (k) · · g (q). (45d)
f 2 2 2 (2π)3 f 2J +1 2
Z
For the 0++ states within this category only equations (45c)-(45d) apply.
For the mesons of category3, i.e., with J++ for J =2n+1 and J−− for J =2(n+1), the Bethe-Salpeter equation
takes the form
µ2 1 d3q JP (pˆ qˆ)+(J +1)P (pˆ qˆ)
J+1 J−1
ω (p)h(p)= g(p)+ V (k) · · h(q), (46a)
f 4 2 (2π)3 f 2J +1
Z
1 d3q
ω (p)g(p)=h(p)+ V(k)P (pˆ qˆ)g(q). (46b)
f 2 (2π)3 J ·
Z
Weseethattheequations(44a)-(44b)becomeidenti- caltotheequations(45c)-(45d)aswellastheequations
9
(45a) - (45b) are identical to equations (46a) - (46b).
TABLE I: Masses of isovector mesons in unitsof √σ.
Note, that there is no isospin dependence of the interac-
tion,i.e. eachsystemofcoupledequationsdescribesboth chiral JPC radial excitation n
I=0 and I=1 states which are exactly degenerate. Then multiplet 0 1 2 3 4 5 6
we recover that the equations above fall into chiral mul- (1/2,1/2)a 0−+ 0.00 2.93 4.35 5.49 6.46 7.31 8.09
tiplets (25)-(27) and both SU(2)L×SU(2)R and U(1)A (1/2,1/2)b 0++ 1.49 3.38 4.72 5.80 6.74 7.57 8.33
are manifest. Hence if the typical momentum of quarks (1/2,1/2)a 1+− 2.68 4.03 5.16 6.14 7.01 7.80 8.53
is large,asitis anticipatedin the highly-excitedmesons, −−
(1/2,1/2)b 1 2.78 4.18 5.32 6.30 7.17 7.96 8.68
the spectrum should be organized into chiral and U(1)A (0,1) (1,0) 1−− 1.55 3.28 4.56 5.64 6.57 7.40 8.16
multiplets , because at large momenta dynamical mass ⊕
(0,1) (1,0) 1++ 2.20 3.73 4.95 5.98 6.88 7.69 8.43
of quarks M(p) vanishes. ⊕
(1/2,1/2)a 2−+ 3.89 4.98 5.94 6.80 7.59 8.31 8.99
(1/2,1/2)b 2++ 3.91 5.02 6.00 6.88 7.67 8.40 9.09
NUMERICAL RESULTS FOR SPECTRUM (0,1) (1,0) 2++ 3.60 4.67 5.64 6.51 7.32 8.06 8.75
⊕
−−
(0,1) (1,0) 2 3.67 4.80 5.80 6.68 7.49 8.23 8.91
⊕
The method of solution for the BSE used in this work (1/2,1/2)a 3+− 4.82 5.77 6.62 7.41 8.13 8.81 9.45
−−
is described in Appendix D. In Table 1 we present our (1/2,1/2)b 3 4.82 5.78 6.65 7.44 8.17 8.86 9.50
results for the spectrum for the two-flavor (u and d) (0,1) (1,0) 3−− 4.68 5.63 6.48 7.26 7.99 8.67 9.30
⊕
mesons in the chiral limit m = 0. Due to chiral sym- (0,1) (1,0) 3++ 4.69 5.66 6.53 7.32 8.06 8.74 9.39
⊕
metry breaking there is a mixing between (h1,g1) and (1/2,1/2)a 4−+ 5.59 6.45 7.23 7.96 8.64 9.28 9.89
(h2,g2) amplitudes for mesons of category2 (but not for (1/2,1/2)b 4++ 5.59 6.45 7.24 7.97 8.66 9.30 9.91
the0++ states). Inthelanguageofchiralrepresentations (0,1) (1,0) 4++ 5.51 6.36 7.15 7.88 8.56 9.19 9.80
this mixing means that for mesons of category 2 with ⊕ −−
(0,1) (1,0) 4 5.51 6.37 7.16 7.90 8.58 9.23 9.84
J =2n+1 the representations (0,0) and (1/2,1/2) are ⊕
mixedinthemesons0,J−−aswellastherepresentaations (1/2,1/2)a 5+− 6.27 7.05 7.78 8.47 9.11 9.72 10.3
−−
(0,1) (1,0) and (1/2,1/2) are mixed in the mesons (1/2,1/2)b 5 6.27 7.06 7.79 8.47 9.12 9.73 10.3
1,J−−⊕. Formesonsofcategobry2withJ =2n,J >0the (0,1)⊕(1,0) 5−− 6.21 7.00 7.73 8.41 9.06 9.67 10.2
(0,1) (1,0) 5++ 6.21 7.00 7.73 8.42 9.07 9.68 10.3
representations (0,0) and (1/2,1/2) are mixed in the
a ⊕
0,J++ states and the representations (0,1) (1,0) and (1/2,1/2)a 6−+ 6.88 7.61 8.29 8.94 9.55 10.1 10.7
(1/2,1/2)b are mixed in the mesons 1,J++. ⊕The mesons (1/2,1/2)b 6++ 6.88 7.61 8.29 8.94 9.56 10.1 10.7
of the category 2 in the Table are assigned to a defi- (0,1) (1,0) 6++ 6.83 7.57 8.25 8.90 9.51 10.1 10.7
⊕
nite chiral representation according to the chiral Bethe- (0,1) (1,0) 6−− 6.83 7.57 8.26 8.90 9.52 10.1 10.7
⊕
Salpeter amplitude which dominates in the meson wave
function, which is determined by the relative magnitude
of the normalization factors N1 and N2 in (43). (12,21)a (12,21)b
Note that within the present model there are no vac- 10 10 10 10
uum fermion loops. Then since the interaction be- 9 9 9 9
tween quarks is isospin-independent the states with the 8 8 8 8
7 7 7 7
same JPC but different isospins from the distinct mul-
6 6 6 6
tiplets (1/2,1/2)a and (1/2,1/2)b as well as the states 5 5 5 5
with the same JPC but different isospins from (0,0) 4 4 4 4
and (0,1) (1,0) representations are exactly degener- 3 3 3 3
⊕ 2 2 2 2
ate. Hence it is enough to show a complete set of the
1 1 1 1
isovector (or isoscalar) states.
The presented results are accurate within the quoted 1,0−+ 0,0++ 0,0−+ 1,0++
digits at leastfor states with small J and for states with
higher J but small n. At larger J for larger n numerical
errors accumulate in the second digit after comma. FIG. 3: Spectra of J =0 mesons (masses in unitsof √σ).
The axialanomalyis absentwithinthe presentmodel,
becausetherearenovacuumfermionloopshere. Evenso
therearenoexactU(1) multiplets,becausethissymme- metry restorations are satisfied. We observe a very fast
A
tryisbrokenalsobythequarkcondensateofthevacuum. restorationofbothSU(2)L SU(2)R andU(1)A symme-
×
The mechanismofthe U(1) breakingandrestorationis trieswithincreasingJ andessentiallymoreslowrestora-
A
exactly the same as for SU(2) SU(2) . tionwithincreasingofn. ThespectrumoftheJ =0,1,2
L R
×
The excited mesons fall into approximate chiral and mesons is shown in Figs. 3 - 5.
U(1) multiplets and all conditions of the effective sym- InFig. 6theratesofthesymmetryrestorationagainst
A
10
1.5
(0,0) (12,21)a
10 10 10 10
9 9 9 9 1
8 8 8 8 M
∆
7 7 7 7
6 6 6 6 0.5
5 5 5 5
4 4 4 4
3 3 3 3 00 1 2 3 4 5 6
2 2 2 2 J
1.5
1 1 1 1
0,1−− 0,1++ 1,1+− 0,1−−
1
(12,21)b (0,1)⊕(1,0) ∆M
10 10 10 10 0.5
9 9 9 9
8 8 8 8
7 7 7 7 0
0 1 2 3 4 5 6
6 6 6 6 n
5 5 5 5
4 4 4 4 FIG. 6: Mass splittings in units of √σ for isovector mesons
3 3 3 3 of the chiral multiplets (1/2,1/2)a and (1/2,1/2)b (circles)
2 2 2 2 andwithinthemultiplet(0,1) (1,0)(squares)againstJ for
1 1 1 1 ⊕
n=0 (top) and against n for J =0 and J =1, respectively
0,1+− 1,1−− 1,1−− 1,1++ (bottom). The full line in the bottom plot is 0.7pσ/n.
theradialquantumnumbernandspinJ areshown. Itis
FIG. 4: Spectra of J =1 mesons (masses in unitsof √σ).
seenthatwith the fixedJ the splitting within the multi-
plets∆M decreasesasymptoticallyas1/√n,dictatedby
the asymptotic linearity of the radial Regge trajectories
(0,0) (12,21)a with different intercepts. With the fixed n the J-rate of
10 10 10 10 the symmetry restoration is much faster.
9 9 9 9 Inthelimitn orJ oneobservesacomplete
8 8 8 8 degeneracyofall→m∞ultiplets→,w∞hichmeansthatthestates
7 7 7 7
fall into representation (28) that combines all possible
6 6 6 6
chiral representations for the systems of two massless
5 5 5 5
4 4 4 4 quarks [7]. This means that in this limit the loop effects
3 3 3 3 disappear completely and the system becomes classical
2 2 2 2 [8, 9]. The analytical proof for this larger degeneracy
1 1 1 1
will be given in the next section.
0,2++ 0,2−− 1,2−+ 0,2++ In Fig. 7 the angular and radial Regge trajectories
are shown. Both kinds of trajectories exhibit asymptot-
(12,21)b (0,1)⊕(1,0) ically linear behavior. This has to be expected a-priory,
10 10 10 10 because at large J or n all higher Fock components are
9 9 9 9 suppressedandasymptoticallyvanish(itwillbelaterwell
8 8 8 8 seen from the meson wave functions).Then the semiclas-
7 7 7 7
sical description of the states with large n or J with the
6 6 6 6
5 5 5 5 linearpotentialrequiresthelinearReggetrajectories,see,
4 4 4 4 e.g., ref. [39]. There are deviations from the linear be-
3 3 3 3 havior at finite J and n, however. This fact is obviously
2 2 2 2
related to the chiral symmetry breaking effects for lower
1 1 1 1
mesons. Note, that the chiral symmetry requires a dou-
0,2−+ 1,2++ 1,2++ 1,2−− blingofsomeoftheradialandangularReggetrajectories
for J = 1,2,... This is a highly nontrivial prediction of
chiral symmetry. For example, some of the rho-mesons
FIG. 5: Spectra of J =2 mesons (masses in unitsof √σ). lie on the trajectory that is characterized by the chiral
index (0,1)+(1,0),while the other fit the trajectory with