Table Of ContentCharmonia in a Contact Interaction
Marco A. Bedolla,1 J.J. Cobos-Mart´ınez,1 and Adnan Bashir1
1 Instituto de F´ısica y Matema´ticas, Universidad Michoacana de San Nicol´as Hidalgo,
Apartados Postals 2-82, Morelia, Michoaca´n 58040, Mexico
(Dated: June 14, 2016)
For the flavour-singlet heavy quark system of charmonia, we compute the masses of the ground
state mesons in four different channels: pseudo-scalar (η (1S)), vector (J/Ψ(1S)), scalar (χ (1P))
c c0
and axial vector (χ (1P)), as well as the weak decay constants of the η (1S) and J/Ψ(1S) and
c1 c
the charge radius of η (1S). The framework for this analysis is provided by a symmetry-preserving
c
Schwinger-Dysonequation(SDEs)treatmentofavector×vectorcontactinteraction(CI).Theresults
found for the meson masses and the weak decay constants, for the spin-spin combinations studied,
are in fairly good agreement with experimental data and earlier model calculations based upon
Schwinger-Dyson and Bethe-Salpeter equations (BSEs) involving sophisticated interaction kernels.
6 The charge radius of η (1S) is consistent with the results from refined SDE studies and lattice
c
1 Quantum Chromodynamics (QCD).
0
2 PACSnumbers: 12.38.-t,11.10.St,11.15.Tk,14.40.Lb
n
a
J I. INTRODUCTION reported.
1 The extension of this program to the complicated ex-
2 Quantum Chromodynamics (QCD) and the resulting otic and baryonic states, decay rates and form factors
hadron bound states form a challenging sector of the is, numerically, not straightforward at all. A few years
h] StandardModelofparticlephysics. Inthenonperturba- ago, an alternative was explored to initially study pion
p tive regime of these interactions, the emergent phenom- properties assuming that quarks interact not via mass-
- enaofchiralsymmetrybreakingandconfinementgovern lessvector-bosonexchangebutinsteadthroughasymme-
p
their spectrum and properties. Within the framework trypreservingvector-vectorcontactinteraction(CI)[19–
e
h of Schwinger-Dyson equations (SDEs), we can study the 23]. One then proceeds by embedding this interaction
[ structure of strongly interacting bound states through in a rainbow-ladder truncation of the SDEs. Confine-
first principles in the continuum. SDEs for QCD have ment is implemented by employing a proper time regu-
1
beenextensivelyappliedtothestudyoflightquark[1–3] larization scheme. This scheme systematically removes
v
9 andgluonpropagators[4–6],theirinteractions[7–9],me- quadratic and logarithmic divergences ensuring that the
3 son spectra below the masses of 1 GeV as well as their axial-vector Ward-Takahashi identity (axWTI) is sat-
6 static and dynamic properties. isfied. One can also explicitly verify the low energy
5 First explorations for heavy mesons, both charmonia Goldberger-Treiman relations. A fully consistent treat-
0 ment of the CI model is simple to implement and can
and bottomonia, with a consistent use of the rainbow-
1. ladder truncation in the kernels of the gap and Bethe- helpusprovideusefulresultswhichcanbecomparedand
0 Salpeter equations (BSEs), were undertaken by Jain contrasted with full QCD calculation and experiment.
6 and Munczek in Ref. [1]. They found the mass spec- This interaction is capable of providing a good de-
1 trum and the decay constants of pseudoscalar mesons in scription of the masses of meson and baryon ground and
:
v good agreement with experiments. This work was re- excited-states for light quarks [19–22]. The results ob-
i peated with the Maris-Tandy model for c¯c bound states tained from the CI model are quantitatively comparable
X
in Refs. [10, 11] with extrapolations to the¯bb systems in to those obtained using sophisticated QCD model inter-
ar Refs.[12,13]. Afullnumericalsolutionforflavour-singlet actions, [14, 24–26]. Interestingly and importantly, this
pseudoscalar mesons again yielded charmonia and bot- simpleCIproducesaparity-partnerforeachground-state
tomonia masses and decay constants consistent with ex- thatisalwaysmoremassivethanitsfirstradialexcitation
perimental data; predictions for states with exotic quan- so that, in the nucleon channel, e.g., the first JP =1/2−
tum numbers were also made in Refs. [14, 15]. The state lies above the second JP =1/2+ state [21].
effect of the quark-gluon interaction in the gap equa- We take this as a sufficient justification to employ this
tion and the vertex-consistent Bethe-Salpeter kernel was interaction for the analysis of the quark model heavy
investigated in Ref. [16]. More recently, employing a mesons for spins J = 0,1 and study the mass spec-
parametrization of the quark propagator to analytically trum and weak decay constants for charmonia. Without
continue it into the complex plane, heavy quark systems parameter readjustment, we find good agreement with
were studied in detail in Ref. [17]. A more direct ap- charmonia masses. However, we need to modify the set
proach through the numerical computation of the quark ofparameterstosimultaneouslyaccountfortheweakde-
propagator in the required region of the complex plane cayconstantsoftheηc(1S)andJ/Ψ(1S),andthecharge
wasemployedinRef.[18]. Therethemassspectrumand radius of ηc(1S).
decay constants for flavor singlet JP = 0− mesons were The paper is organized as follows: in Section II we
2
present the necessary details of the SDE-BSE approach m M E F E m m f f
π π ρ π ρ π ρ
to mesons; while in Section III and Section IV we intro- 0 0.358 3.568 0.459 1.520 0 0.919 0.100 0.130
duce the interaction used and the consequences of the 0.007 0.368 3.639 0.481 1.531 0.140 0.928 0.101 0.129
axWTI; Section V outlines the general forms of the
Bethe-Salpeteramplitude(BSA)forthemesonsstudied; TABLE I: Results for light pseudoscalar and vector mesons
Section VI contains numerical analysis of the results ob- obtained with mg =0.8GeV, αIR =0.93π, ΛIR =0.24GeV,
tained without readjusting the parameters of the light ΛUV =0.905GeV. Thesemodelparametersweredetermined
in Refs. [21, 22]
quark sector; while in Section VII we minimally modify
the CI model parameters and re-evaluate the charmonia
masses, decay constants of η (1S) and J/Ψ(1S), and the
c
charge radius of η (1S); finally in Section VIII, we state
c III. RAINBOW-LADDER TRUNCATION AND
our findings and present our conclusions and discussion.
THE CONTACT INTERACTION
It has been shown [19–22] that a momentum-
independent vector×vector CI is capable of providing a
II. THE BETHE-SALPETER AND THE GAP
descriptionoflightpseudoscalarandvectormesonsstatic
EQUATIONS
properties, which is comparable to that obtained using
more sophisticated QCD model interactions [14, 24–26];
Meson bound states appear as poles in a four-point
see for example Table I. We employ this interaction for
function. Theconditionfortheappearanceofthesepoles
theanalysisofthequarkmodelcharmoniaspectrum. We
inaparticularJPC channelisgivenbythehomogeneous
therefore use
BSE [27–29]
4πα 1
g2D (k)= IRδ ≡ δ , (4)
(cid:90) d4q µν m2 µν m2 µν
[Γ (k;P)] = χ(q;P) Krs(q,k;P), (1) g G
H tu (2π)4 sr tu
in Eq. (3), where m = 800 MeV is a gluon mass scale
g
which is in fact generated dynamically in QCD, see for
where χ(q;P) = S (q )Γ (q;P)S (q ); q = q +ηP,
f + H g − +
example [34], and α = 0.93π is a parameter that de-
q =q−(1−η)P;k(P)istherelative(total)momentum IR
−
termines the interaction strength. For the quark-gluon
of the quark-antiquark system; S is the f-flavour quark
f
vertex, the rainbow truncation will be used:
propagator; Γ (q;P) is the meson BSA, where H = fg
H
specifies the flavour content of the meson; r,s,t,u rep-
λa
resent colour, flavour, and spinor indices; and K is the Γa(p,q)= γ . (5)
µ 2 µ
quark-antiquark scattering kernel. For a comprehensive
recent review of the BSE and its applications, see [24].
Once the elements of the kernel in the quark SDE have
The f-flavour dressed-quark propagator, Sf, that en- been specified, we can proceed to obtain and analyse its
tersEq.(1)isobtainedasthesolutionofthequarkSDE, solution. The general form of the f-flavoured dressed
the so called gap equation [30–33]: quark propagator, obtained as the solution of Eq. (2), is
given in terms of two Lorentz-scalar dressing functions,
S−1(p)=iγ·p+m +Σ (p), (2) written in two different but equivalent forms as:
f f f
(cid:90) d4q λa
Σ (p)= g2D (p−q) γ S (q)Γa(p,q),(3) S−1(p) = iγ·pA (p2)+B (p2) (6)
f (2π)4 µν 2 µ f ν f f f
= Z−1(p2)(cid:0)iγ·p+M (p2)(cid:1) . (7)
f f
where g is the strong coupling constant, D is the
µν
dressed-gluon propagator, Γa is the dressed-quark-gluon In the latter expression, Z(p2) is known as the wave-
ν
vertex, andmf isthebaref-flavourcurrent-quarkmass. function renormalization, and Mf(p2) is the dressed,
Since the CI, to be defined later in Section III, is non- momentum-dependent quark mass function, which con-
renormalizable,itisnotnecessarytointroduceanyrenor- nects current and constituent quark masses [30–32].
malziation constant, and the chiral limit is obtained by UsingEq.(4)andEq.(5),thequarkequation,Eq.(2),
setting m =0 [30–32]. takes the following simple form
f
BothD andΓa satisfytheirownSDE,whichinturn
are coupleµdνto highνer n-point functions and so on ad in- S−1(p)=iγ·p+m + 4 1 (cid:90) d4q γ S (q)γ . (8)
finitum. Therefore, the quark SDE, Eq. (2), is only one f f 3m2 (2π)4 µ f µ
G
oftheinfinitesetofcouplednonlinearintegralequations.
A tractable problem is defined once a truncation scheme The solution to this equation is of the form
has been specified, i.e., once the gluon propagator and
the quark-gluon vertex are defined. S−1(p)=iγ·p+M , (9)
f f
3
sincethelasttermontheright-handsideofEq.(8)isin- where the appropriate indices are contracted and
dependent of the external momentum. The momentum- K(k,q;P) is the Bethe-Salpeter kernel that appears in
independent mass, M , is determined as the solution of the bound state Eq. (1). A similar equation is satisfied
f
by the pseudoscalar vertex.
M (cid:90) ∞ 1
M =m + f dss . (10) CombiningtheSDEssatisfiedbythepseudovectorand
f f 3π2m2G 0 s+Mf2 pseudoscalar vertices with the axWTI one arrives at [2]
Since Eq. (10) is divergent, we have to specify a regular- (cid:90) d4q
ization procedure. We employ the proper time regular- (2π)4 Ktu;rs(k,q;P)[γ5S(q−)+S(q+)γ5]sr
ization scheme, [35], and write:
=[Σ(k )γ +γ Σ(k )] , (16)
1 = (cid:90) ∞dτe−τ(s+M2) →(cid:90) τIR2dτe−τ(s+M2), thus constraining the conten+t of5 the5qua−rk-tauntiquark
s+M2
0 τ2 scattering kernel K(p,q;P) if an essential symmetry of
UV
e−τU2V(s+M2)−e−τI2R(s+M2) thestronginteractions,anditsbreakingpattern,istobe
= , (11) faithfully reproduced.
s+M2
From a practical point of view, Eq. (16) provides a
whereτ2 andτ2 are,respectively,infraredandultravi- way of obtaining the quark-antiquark scattering kernel
IR UV
olet regulators. A nonzero value for τIR ≡ 1/ΛIR imple- if we can solve this constraint, given an expression for
mentsconfinementbyensuringtheabsenceofquarkpro- the quark self-energy. However, this is not always possi-
ductionthresholds [36]. Furthermore,sinceEq.(4)does ble, see e.g. [40], and we must find an alternative way of
not define a renormalizable theory, τUV ≡1/ΛUV cannot preserving the chiral symmetry properties of the strong
be removed, but instead plays a dynamical role and sets interactions. In principle, one may construct a quark-
the scale for all dimensioned quantities. Note that the antiquark scattering kernel satisfying Eq. (16) from a
role of ultraviolet cut-off in Nambu–Jona-Lasinio type functionalderivativeofthequarkself-energywithrespect
models has also been discussed in Refs. [37, 38]. Thus tothequarkpropagator[41],withintheframeworkofthe
effective action formalism for composite operators devel-
M
M =m + f C (M2;τ ,τ ), (12) oped in [42].
f f 3π2m2 01 f IR UV
G Fortunately, for the CI model under study, Eq. (16)
where can be easily satisfied. The resulting expression for the
scattering kernel is called the rainbow-ladder (RL) trun-
(cid:0)M2(cid:1)ν
C (M2;τ ,τ )= Γ(β−2,τ2 M2,τ2 M2), cation. This kernel is the leading-order term in a non
αβ IR UV Γ(β) UV IR perturbative, symmetry-preserving truncation scheme,
(13) which is known and understood to be accurate for pseu-
with ν = α−(β −2) and Γ(a,z ,z ) is the generalized doscalar and vector mesons. Moreover, it guarantees
1 2
incomplete gamma function. electromagnetic current conservation [36]:
1 (cid:20)λa (cid:21) (cid:20)λa (cid:21)
K(p,q;P) =− δ γ γ . (17)
IV. AXIAL-VECTOR WARD-TAKAHASHI tu;rs m2 µν 2 µ 2 ν
G ts ru
IDENTTIY
Using the interaction we have specified, Eqs. (4,5), the
homogeneous BSE for a meson (η = 1) takes a simpler
Thephenomenologicalfeaturesofchiralsymmetryand form,
its dynamical breaking in QCD can be understood by
4 1 (cid:90) d4q
means of the axWTI. In the chiral limit, it reads
Γ (k;P)=− γ S (q+P)Γ (q;P)S (q)γ .
H 3m2 (2π)4 µ f H g µ
−iPµΓ5µ(k;P)=S−1(k+)γ5+γ5S−1(k−). (14) G (18)
Sincetheinteractiondoesnotdependontherelativemo-
TheaxWTIrelatestheaxial-vectorvertex,Γ ,thepseu-
5µ mentumofthequarks,asymmetry-preservingregulariza-
doscalar vertex, γ and the quark propagator. This in
5 tionofEq.(18)willyieldsolutionswhichareindependent
turn implies a relationship between the kernel in the
ofit. ItfollowsthatiftheinteractionofEq.(4)produces
BSE and that in the quark SDE. It must be preserved
bound states, then the relative momentum between the
by any viable truncation scheme of the SDE-BSE cou-
quarkandtheantiquarkcanassumeanyvaluewithequal
pled system. It is the preservation of this identity which
probability. Thisisthedefiningcharacteristicofapoint-
proves useful in obtaining the defining characteristics of
like particle.
theoctetofpseudoscalarmesons,namelytheirlowmass,
their masslessness in the chiral limit, and the hadron
mass splittings [2, 39]
A. A Corollary of the Axial-Vector WTI
The axial-vector vertex satisfies its own SDE, namely
(cid:90) d4q There are further non trivial consequences of the
Γ (k;P)=γ γ + K(k,q;P)χ (q;P), (15)
5µ 5 µ (2π)4 5µ axWTI and the CI. They define our regularization pro-
4
cedure, which must maintain whereM isamassscale,tobedefinedlater. Resultswill
be independent of its choice. A charge-conjugated BSA
(cid:34) (cid:35)
0=(cid:90) d4q P ·q+ − P ·q . (19) is obtained, in general, via
(2π)4 q2 +M2 q2 +M2
+ f − g Γ (k;P)=C†Γ(−k;P)TC, (26)
H
This ensures that Eq. (14) is satisfied. Now analyzing
the integrands, using a Feynman parametrization, one where T denotes the transposing of all matrix indices
arrives at the following identity for P2 = 0 = m, and and C = γ2γ4 is the charge conjugation matrix, with
Mf =Mg =M: C† =−C, and [C,γ5]=0. We thus have
0=(cid:90) d4q 12q2+M2 . (20) C†γµTC = −γµ, (27)
(2π)4 (q2+M2)2 C†σT C = −σ , (28)
µν µν
Eq. (20) states that the axWTI is satisfied if, and only C†γ5TC = γ5, (29)
if, the model is regularized so as to ensure there are C†γ σT C = −γ σ , (30)
5 µν 5 µν
no quadratic or logarithmic divergences. Unsurprisingly,
C†(γ σ )TC = γ γ , (31)
these are the circumstances under which a shift in inte- 5 µν 5 µ
gration variables is permitted, an operation required in
and therefore
order to prove Eq. (14).
ItisnotablethatEq.(14)isalsovalidforarbitraryP. Γ0++(P) = Γ0++(P), (32)
Using a Feynman parametrization of the integrand, and
making an appropriate change of variables (q →q−xP) Γ1−−(P) = −Γ1−−(P), (33)
µ µ
to diagonalize the denominator, we find, for non zero P2
Γ1++(P) = Γ1++(P). (34)
(cid:90) 1 (cid:90) d4q 1q2+M2 µ µ
0= dx 2 , (21)
(2π)4 (q2+M2)2 It is a well known feature of the rainbow-ladder trun-
0
cationoftheSDE-BSEsystemthatitdescribesthepseu-
where M2 = M2x + M2(1 − x) + x(1 − x)P2. This doscalar and vector mesons well, but not their parity
f g
constraint will be implemented in all our calculations so partners, namely, the scalar and axial-vector mesons.
that Eq. (14) is preserved. In more realistic kernels, see for example [7], when the
quark-gluon vertex is fully dressed, it was found that
dynamical chiral symmetry breaking (DCSB) generates
V. CLASSIFICATION OF BSA IN A CONTACT a large dressed-quark anomalous chromo-magnetic mo-
INTERACTION
ment in the infrared. Consequently, the associated cor-
rections cancel in the pseudoscalar and vector channels
We are interested in the static properties of several butaddinthescalarandaxialvectorchannels,resulting
mesons. We thus begin with their classification and the in a magnified splitting between parity partners. This
generalformoftheirBSAintheCIweareworkingwith. effect is specially important for mesons made up of light
Table II lists the spin quantum numbers of the quark quarks. With this in mind, following Ref. [22], we have
model mesons we will study. introduced a spin-orbit repulsion into the scalar- and
axial-vector-mesonchannelsthroughtheartificeofaphe-
L JPC Type L JPC Type nomenological coupling g2 ≤ 1, introduced as a factor
SO
0 0−+ Pseudoscalars 1 0++ Scalars multiplyingthescalarandaxial-vectorkernels. Thevalue
0 1−− Vectors 1 1++, 1+− Axial Vectors g = 0.24 was chosen in Ref. [22] so as to obtain the
SO
experimental value for the a -ρ mass splitting which is
1
TABLE II: Quark model mesons knowntobeachievedbythecorrectionsdescribedabove
(without the spin-orbit coupling g , the mass differ-
SO
ence between the a and the ρ is 0.15 GeV, a factor of 3
1
With the dependence on the relative momentum for-
smaller than the experimental value, It implies that the
bidden by the CI, the general form of the BSAs for the
spin-orbit coupling has increased the mass of the a by
1
mesons listed in Table II are [43]:
28%, which is a large effect).
(cid:20) (cid:21)
1
Γ (P) = γ iE + γ·PF , (22)
0−+ 5 0−+ 2M 0−+
VI. NUMERICAL RESULTS
Γ (P) = 1E , (23)
0++ 0++
1
Γ (P) = γTE + σ P F , (24) ThemassandBSAofamesondependonitsquantum
1−−µ µ 1−− 2M µν ν 1−− numbersandcanbefoundbysolvingEq.(1). Inorderto
(cid:20) (cid:21)
1 dothis,wewillintroduceafictitiouseigenvalueλ tothe
Γ (P) = γ γTE + σ P F , (25) H
1++µ 5 µ 1++ 2M µν ν 1++ boundstateequation. Thus,themassoftheboundstate
5
in a particular channel, m , will be such that λ (P2 =
H H
−m2 ) = 1, where P is the meson’s momentum. In any
H
channel, the form of the homogeneous BSE for the CI
will be
Experimental data
3.6 Contact Interaction
K (m )·Γ (m )=λ (m )Γ (m ), (35)
H H H H H H H H
]
where KH is a 2×2 matrix, and the subscript H indi- eV3.4
cates the dependence of the explicit expressions on the G
[
quantum numbers of the meson under consideration, see s
Eqs.(22-25). Equation(35)isaneigenvalueequationfor Mas3.2
the vector Γ (m ) = (E (m ),F (m ))T with solu-
H H H H H H
tions for discrete values of P2 = −m2 . Explicit expres-
H 3
sions for every channel given in Table II are presented in
Appendix A.
2.8 0-+ 1-- 0++ 1++
masses PC
J
m m m m
ηc(1S) J/Ψ(1S) χc0(1P) χc1(1P)
Experiment [44] 2.983 3.096 3.414 3.510
Contact Interaction 2.983∗ 2.979 3.412 3.442 FIG. 1: Contact interaction results for the cc mass spectrum
- - 3.293 3.344 usingmodelparametersfittedinthelightsector,seeTableIII
JM [1] 2.821 3.1 3.605 - PDG-labeled data is from [44].
BK [13] 2.928 3.111 3.321 3.437
S1rp [17] 3.035 3.192 - -
RB1 [18] 3.065 - - -
obtainedfrommoresophisticatedSDE-BSEmodelcalcu-
RB2 [18] 3.210 - - -
lations[13,18],latticeQCDforthecharmsector[45,46]
as well as experimental values [44].
TABLE III: Ground state charmonia masses obtained with
the light sector parameter set: m = 0.8GeV, α = 0.93π, ThataRLtruncationwithaCIdescribeswellthemass
g IR
Λ = 0.24GeV, Λ = 0.905GeV. The current-quark spectrum of ground state charmonia can be understood
IR UV
mass is m = 1.578∗GeV, and the dynamically generated inasimpleway. Sincethewavefunctionrenormalization
c
constituent-like mass is Mc = 1.601GeV. The value imme- andquarkmassfunctionaremomentum-independentthe
diately below the CI results is obtained without a spin-orbit heavyquark-gluonvertexcanthereforereasonablybeap-
couplingg =0.24. Foradirectcomparison,wequotevalues
so proximated by a bare vertex, ensuring the vector and
fromotherSDEapproachestocalculatethemassesoflowly-
axWTI.
ingcharmonia. DimensionedquantitiesareinGeV.(∗ =The
currentquarkmasswasfittedtoobtainthemassofthepseu-
doscalar meson).
B. Decay constants
A. Charmonia mass spectrum
amplitudes
η (1S) J/Ψ(1S) χ (1P) χ (1P)
c c0 c1
It has been shown [19–22] that a momentum- E 6.028 3.024 0.437 0.298
H
independent vector×vector interaction is capable of pro- - - 1.905 1.153
viding a description of light pseudoscalar and vector F 1.711 - - -
H
mesonsstaticpropertieswhichiscomparabletothatob- decay constants
tainedusingmoresophisticatedQCDmodelinteractions Experiment [44] 0.361 0.416 - -
[14, 24–26]. Here we assess the capability of this model Contact Interaction 0.084 0.080 - -
to provide a description of static properties of charmo-
nia. The parameter set used in this calculation is the TABLE IV: Ground state charmonia amplitudes and de-
cay constants obtained with the light sector parameter set
same as that obtained used in the light sector. Only the
adopted in Table III:
current-quark mass for the charm quark is an input pa-
rameter, and it is fixed such that the experimental mass
of the pseudoscalar is reproduced. The rest of the me-
son masses are predictions of the model. As can be seen Since the BSE is a homogeneous equation, the BSA
from Table III, the predictions for the masses of the re- has to be normalized by a separate condition. In the
maining mesons are in good agreement with the results Rainbow-Ladder truncation of the BSE, that condition
6
takes a simple form (we choose η =1): It is not difficult to understand why the decay con-
stants come out to be much smaller than what one ex-
∂ (cid:90) d4q (cid:2) (cid:3) pectsfromtheQCDbasedSDEwithrunningquarkmass
P =N Tr Γ (−Q)S(q )Γ (Q)S(q) ,
µ c∂P (2π)4 H + H function. As noticed in Refs. [3, 16, 49], the decay con-
µ
(36) stant is influenced by the high momentum tails of the
at Q = P, with P2 = −m2 , which ensures that the dressed-quark propagator and the BSAs. This high mo-
H
residue at the mass pole is unity. Here, Γ is the nor- mentum region probes the wave-function of quarkonia
H
malized BSA and Γ its charge conjugated version. at origin. The CI, on the other hand, yields constant
H
For every channel, we will re-scale Γ such that masswithnoperturbativetailforlargemomenta. There-
H
Eq. (36) is satisfied. Thus, we replace Γ with Γ /N , fore, this artefact of quarkonia has to be built into the
H H H
where N is the normalization constant and now Γ is model in an alternative manner. We know that with in-
H H
the non-normalized BSA, the amplitude that is obtained creasing mass of the heavy quarks, they become increas-
by solving the homogeneous BSE. Thus the normaliza- ingly point-like in the configuration space. The closer
tion constant is obtained from thequarksget,thefurtherthecouplingstrengthbetween
them decreases. Therefore, we cannot expect the decay
N2P2 =N P ∂ (cid:90) d4q Tr(cid:2)Γ (−Q)S(q )Γ (Q)S(q)(cid:3) , constants to be correctly reproduced with the parame-
H c µ∂Pµ (2π)4 H + H ters of the light quark sector. In the next section, we
(37)
considerthepossibilityofextendingthesimpleCImodel
at Q = P, with P2 = −m2 . For the vector and axial
H to the heavy sector by reducing the effective coupling.
vector channels there is an additional factor of 1/3 on
However, the reduction in the strength of the kernel has
therighthandsidesincewehavetotakeintoaccountall
to be compensated by increasing the ultraviolet cut-off.
three meson polarizations.
ThismakessensebyobservingthattheΛ (highesten-
UV
OncetheBSAhasbeennormalizedcanonically,wecan
ergy scale associated with the system) used in the light
calculate observables from it. The pseudoscalar leptonic
quarksectoris,infact,lessthanthecurrentcharmquark
decay constant, f , is defined by (in a more realistic
0−+ mass. Therefore, it needs to be modified. We look for a
interaction,thereisafactorofZ ontheright-handside,
2 balancebetweentheeffectivecouplingandtheultraviolet
and of course the BSA depends on the relative momen-
cut-off to describe the static properties of charmonia.
tum)
(cid:90) d4q
P f =N Tr[γ γ S(q )Γ (P)S(q )] . VII. CONTACT INTERACTION MODEL FOR
µ 0−+ c (2π)4 5 µ + 0−+ − CHARMONIA
(38)
Similarly, the vector decay constant, f , is defined by
1−− As we mentioned in the last section, we set out to
m f = Nc (cid:90) d4q Tr(cid:2)γ S(q )Γ S(q )(cid:3) , rweedaekfindeectahyecpoanrsatamnettsearsndofththeechCaIrgteorsatduidiyoftchhearmmaosnseias,.
1−− 1−− 3 (2π)4 µ + µ1−− −
(39)
masses
wherem isthemassofthe1−− boundstate, andthe
1−− m m m m
factorof3inthedenominatorcomesfromsummingover ηc(1S) J/Ψ(1S) χc0(1P) χc1(1P)
Experiment [44] 2.983 3.096 3.414 3.510
the three polarizations of the spin-1 meson. Explicit ex- Contact Interaction 2.950∗ 3.129 3.407 3.433
pressions for the normalization condition in every chan- 3.194 3.254
nel and the decay constants f0−+ and f1−− are given in JM [1] 2.821 3.1 3.605 -
Appendix B. BK [13] 2.928 3.111 3.321 3.437
As can be seem from Table IV, the pseudoscalar and RB1 [18] 3.065 - - -
vector decay constants, for the model parameters used, RB2 [18] 3.210 - - -
are strongly underestimated, in disagreement both with
model calculations and experimental data. Numerically, TABLE V: Ground state charmonia masses obtained with
this is because the corresponding BSAs are too small. the best-fit parameter set: m = 0.8GeV, α = 0.93π/20,
g IR
Changing ΛUV, e.g., to 1.843GeV, keeping the other ΛIR = 0.24GeV, ΛUV = 2.788GeV. The current-quark
parameters fixed, except for the current-quark masses, mass is mc = 0.956∗GeV, and the dynamically generated
constituent-like mass is M = 1.497GeV. The value imme-
which are taken from Ref. [44], improves the situation c
diately below the CI results is obtained without a spin-orbit
by about a factor of 2. However, there is still a signifi-
coupling g = 0.24/3. Dimensioned quantities are in GeV.
cant mismatch between our results, model calculations, so
(∗ = This parameter set was obtained from a best-fit to the
and experiment. We observe that this disagreement per-
massanddecayconstantofthepseudoscalarandvectorchan-
sists despite the fact that our calculation for the pseu-
nels).
doscalars are in perfect agreement with the Gell-Mann–
Oakes–Renner relation, which is valid for every 0− me-
son, irrespective of the magnitude of the current-quark We retain the parameters m and Λ of the light sec-
g IR
mass, [13, 47, 48]. tor. Modern studies of the gluon propagator indicate
7
sults. The spin-dependent interactions are proportional
to the chromomagnetic moment of the quark (simulated
by a spin orbit coupling). Predictably, these effects are
substantially suppressed for charmonia as compared to
Experiment
3.5 light mesons.
Contact Interaction
Thedecayconstantsfortheη (1S)andJ/Ψ(1S)chan-
c
3.4 nels are reported in Table VI. For the pseudoscalar me-
]
V son, the result aligns nicely with the experimental value.
e3.3
G However, thisisnotexactlythecaseforthevectorchan-
[
ss 3.2 nel. Furthermore, we note that the decay constant for
a J/Ψ(1S) is smaller than that for η (1S). The correct
M c
3.1 ordering can be recovered by reducing the interaction
strength by a large factor. However, this is something
3 we consider contrived and, therefore, not pursue further.
Notice that one of the SDE results yields the J/Ψ(1S)
2.9 0-+ 1-- 0++ 1++ decay constant even smaller than our value, [17].
PC
J
decay constants
f f
ηc J/Ψ
FIG.2: Contactinteractionresultsfortheccmassspectrum, Experiment [44] 0.361 0.416
see Table V. PDG-labelled data is taken from Ref. [44]. S1rp [17] 0.239 0.198
S3ccp [17] 0.326 0.330
BK [13] 0.399 0.448
Contact Interaction 0.305 0.220
that in the infrared, the dynamically generated gluon
mass scale virtually remains unaffected by the introduc-
TABLE VI: The decay constants for the states η (1S) and
tion of heavy dynamical quark masses, see for example c
J/Ψ(1S) obtained with m = 0.8GeV, α = 0.93π/20,
Refs. [50, 51]. The rest of the parameters are obtained g IR
Λ =0.24GeV, Λ =2.788GeV. The current-quark mass
from a best-fit to the mass and decay constant of the IR UV
is m =0.956GeV. Dimensioned quantities are in GeV.
pseudoscalar (η ) and vector (J/Ψ) channels. c
c
One can now readily calculate the masses of the
ground state pseudo-scalar, vector, scalar, and axial vec-
tormesons. TheresultsareshowninTableVandFig.2.
η (1S) charge radius
c
They are in very good agreement with experimental val-
SDE [11] Lattice [53] CI
ues and comparable to the best SDE results with refined
0.219 fm 0.25 fm 0.21 fm
truncations. Itistruethatthemassesofcharmoniafrom
quenched quark models can be shifted by large amounts
TABLEVII: Thechargeradiusforthestateη (1S)obtained
whenconsideringhadronloops, seee.g. [52]. Thisobser- c
with the new parameter set discussed in the text.
vation appears to invalidate the quenched quark model
results. However, as noted in Ref. [52], since this scale
of mass shift is common to all low-lying states, it can As another test for the new parameter set for the CI
therefore be absorbed in a change of model parameters. Model for charmonia, we compute the charge radius of
Thus, instead of consistently adding hadron loops into the η (1S), r2 =−6∂F (Q2)/∂Q2| where F (Q2)
our Contact Interaction calculation, we have mimicked is itscelectromηcagnetic foηrcm factor wQh2i=ch0 we willηrceport
the effect by fitting the model parameters (coupling con- elsewhere and compare it with the results presented in
stant and ultraviolet cutoff). It ensures, for example, a Ref. [11] from previous SDE studies and Refs. [11, 53]
constituent-like charm quark mass of the order of 1 GeV from Lattice QCD. The results are presented in Ta-
and a correct value for the experimental mass of the ηc. ble VII. As can be seen, our calculated charge radius is
For the case of the χ , we find a mass of 3.254 GeV very close to the one obtained by employing the Maris-
c1
withoutaspin-orbitcoupling,whichis7%lowerthanthe Tandy Model and the one reported in the lattice stud-
experimentalvalue; ourcalculatedvalueisevencloserto ies [11, 53].
that of the experimental J/ψ mass. A similar pattern
is observed with the pseudoscalar and scalar channels.
Therefore, to achieve an acceptable mass difference be-
tween parity partners, we have introduced a spin-orbit
coupling of g =0.08. This gives the values in the sec-
SO
ondrowofTableV.Ascanbeseenfromthethesevalues,
themassoftheχ hasincreasedonlyby5%. Thissmall
c1
effect is in line with the heavy-quark spin symmetry re-
8
VIII. CONCLUSIONS Weonlyhavetoensurethatthereductioninthestrength
of the kernel is appropriately compensated by increasing
We compute the quark model ground state spin-0 the ultraviolet cut-off, a natural requirement for study-
and spin-1 charmonia masses and decay constants us- ingheavyquarks. Wefindthatwithamodifiedchoiceof
ing a rainbow-ladder truncation of the simultaneous set two parameters, not only the masses of the ground state
of SDE and BSE with a CI model of QCD, developed mesons, i.e., pseudo-scalar (ηc(1S)), vector (J/Ψ(1S)),
and tested for the light quark sector [19–23]. As the scalar (χc0(1P)), and axial vector (χc1(1P)), but also
modelisnon-renormalizable,weemploypropertimereg- theirweakdecayconstants,areinmuchbetteragreement
ularization scheme which ensures confinement is imple- with the experiments [44] as well as earlier SDE calcu-
mentedthroughtheabsenceofquarkproductionthresh- lations with QCD based refined truncations [13]. As a
old. Moreover, the relevant Ward identities and the low further test of the model, we evaluate the charge radius
energy theorems such as Goldberger-Triemann relations of ηc(1S) and found it reasonably close to the results
are satisfied. Without parameter readjustment, we find computed in Refs. [11, 53]. This is an encouraging first
that the masses of the studied mesons are in reason- step towards a comprehensive study of heavy mesons in
ably good agreement with experimental data and other thisapproach. Furtherstepswillinvolveflavoredmesons
model calculations. Moreover, the Gell-Mann–Oakes– andbaryons. Ourgoalistoprovideaunifieddescription
Renner relation, valid for every current-quark mass in of light and heavy hadrons within the CI model.
the pseudo-scalar channel, is always satisfied. However,
the decay constants of pseudo-scalar as well as vector
mesons are significantly underestimated. Appendix A: BSE Kernels
We realize that the extension of the CI model to the
heavysectorrequiresareductionoftheeffectivecoupling, Herewegiveexplicitexpressionsforthekernelinevery
whichmimicsthehighmomentumtailofthequarkmass channelconsideredinthisarticle. Thegeneralexpression
function obtained in the SDE studies of QCD [3, 16, 49]. for the kernel of matrix elements is
(cid:104) (cid:105)
Kij = −4 1 (cid:90) d4q Tr PHi (P)γµSˆf(q+)DHj (P)Sˆg(q−)γµ (A1)
H 3m2 (2π)4 (q2 +M2)(q2+M2)
G + f g
(cid:104) (cid:105)
4 1 (cid:90) 1 (cid:90) d4q Tr PHi (P)γµSˆf(q+)DHj (P)Sˆg(q−)γµ
= − dx q→q−xP ,
3m2 (2π)4 (q2+M2)2
G 0
whereSˆ (k)=−iγ·k+M ,andDi ,Pi are,respectively, Thus
f f H H
suitable Dirac covariants projectors for a given channel.
In order to write compact expressions below, we define
the following expressions
J(x)=x(1−x), K(x)=M x+M (1−x),
f g
L(x)=M2x+M2(1−x), L±(x)=M M ±L(x).
f g f g
For notational simplicity, we shall omit an overall factor
of1/3π2m2 whichmultipliesthekernelineverychannel. (cid:90) 1
G K11 = dx(cid:2)C (M2)+(cid:0)L−(x)−2J(x)P2(cid:1)C (M2)(cid:3),
0−+ 01 02
0
(A4)
1. Pseudoscalar kernel
P2 (cid:90) 1
K12 = dxK(x)C (M2), (A5)
For the pseudoscalar channel, 0−+ 2M 0 02
(cid:90) 1
1 K21 =M dxK(x)C (M2), (A6)
D1 =iγ , D2 = γ γ·P , (A2) 0−+ 02
0−+ 5 0−+ 2M 5 0
P01−+ =−4iγ5, P02−+ =−2MP2γ5γ·P . (A3) K02−2+ =−21(cid:90)01dxL+(x)C02(M2). (A7)
9
2. Vector kernel the decay constants of pseudoscalar and vector mesons.
The general expression for the normalization condition
For the vector channel, can be written as
(cid:88)
1 N2 = NijFi Fj , (B1)
D1 =γT , D2 = σ P , (A8) H H H H
1−− µ 1−− 2M µν ν i,j=1
1 M
P1 = γT , P2 = σ P . (A9)
1−− 12 µ 1−− 6P2 µν ν wheretheupperlimitsonthesummationsdependonthe
numberofnonzerodressingfunctionsinagivenchannel.
Thus A factor of 1/16π2 multiplies every Nij and recall the
H
1(cid:90) 1 factor of 1/3 for the vector and axial vector channels,
K11 = dx(L−(x)−2J(x)P2)C (M2), (A10) stemming from all three polarizations.
1−− 2 02
0
P2 (cid:90) 1
K12 =− dxK(x)C (M2), (A11)
1−− 4M 02
0 1. Pseudoscalar channel
K21 =0, (A12)
1−−
K22 =0. (A13) In this channel, the normalizations are:
1−−
(cid:90) 1
N11 = 8 dxJ(x)(cid:2)2(cid:0)L−(x)−2J(x)P2(cid:1)C (M2)
3. Scalar kernel 0−+ 03
0
+3C (M2)(cid:3) , (B2)
For the scalar channel, 02
D01++ = 1, (A14) N01−2+ =−M2 (cid:90)01dxK(x)(cid:2)−4J(x)P2C03(M2)
P1 = 11. (A15) +C02(M2)(cid:3) , (B3)
0++ 4
N21 =− 2 (cid:90) 1dxK(x)(cid:2)−4J(x)P2C (M2)
Thus 0−+ M 03
0
K11 =(cid:90) 1dx (cid:2)C(M2)−(L+(x)+2J(x)P2)C (M2)(cid:3) . +C02(M2)(cid:3) , (B4)
0++ 02 4P2 (cid:90) 1
0 N22 =− dxJ(x)L+(x)C (M2). (B5)
(A16) 0−+ M2 03
0
4. Axial vector kernel
a. Pseudoscalar decay constant
For the axial vector channel,
Explicit expression for the decay constant is as follows:
1
D11++ =γ5γµT , D12++ =γ52MσµνPν, (A17) f0− = Nc(cid:2)K01−F01− +K02−F02−(cid:3) , (B6)
1 M 1 (cid:90) 1
P11++ = 12γµTγ5, P12++ = 6P2σµνPνγ5. (A18) K01− = 16π2 dxK(x)C02(M2),
0
Thus 1 2 (cid:90) 1
K1 = − dxL+(x)C (M2).
0− 16π2M 02
1(cid:90) 1 0
K11 = − dx(L+(x)+2J(x)P2)C (M2),
1++ 2 02
0
(A19)
2. Vector channel
K12 = 0, (A20)
1++
K21 = 0, (A21) The normalization condition in this channel is:
1++
K22 = 0. (A22)
1++
(cid:90) 1
N11 =48 dxJ(x)(cid:2)(cid:0)L−(x)−2J(x)P2(cid:1)C (M2)
1−− 03
0
Appendix B: Normalization + C (M2)(cid:3) . (B7)
02
In the appendix, we give explicit expressions for nor-
malization condition in every channel considered and for
10
a. Vector decay constant 4. Axial vector channel
The vector decay constant is given by:
N The normalization condition for the axial vector channel
f = c K1 F1 , (B8)
1−− 3m 1−− 1−− is:
1−−
K1 = 12 (cid:90) 1dx (cid:0)L−(x)−2J(x)P2(cid:1)C (M2).
1−− 16π2 02
0
(cid:90) 1
3. Scalar channel N11 =48 dxJ(x)(cid:2)(cid:0)L+(x)+2J(x)P2(cid:1)C (M2)
1++ 03
0
The normalization condition in this channel is: − C (M2)(cid:3) . (B10)
02
(cid:90) 1
N11 =8 dxJ(x)(cid:2)−(cid:0)L+(x)+2J(x)P2(cid:1)C (M2)
0++ 02
0
+ 3C (M2)(cid:3) . (B9)
02
[1] P.JainandH.J.Munczek,Phys.Rev.D48,5403(1993), 1009.0067.
hep-ph/9307221. [21] C. Chen, L. Chang, C. D. Roberts, S. Wan, and D. J.
[2] P. Maris, C. D. Roberts, and P. C. Tandy, Phys. Lett. Wilson, Few Body Syst. 53, 293 (2012), 1204.2553.
B420, 267 (1998), nucl-th/9707003. [22] H.L.Roberts,L.Chang,I.C.Cloet,andC.D.Roberts,
[3] P. Maris and P. C. Tandy, Phys. Rev. C60, 055214 Few Body Syst. 51, 1 (2011), 1101.4244.
(1999), nucl-th/9905056. [23] H. Roberts, A. Bashir, L. Gutierrez-Guerrero,
[4] P.Boucaud,J.Leroy,A.LeYaouanc,J.Micheli,O.Pene, C. Roberts, and D. Wilson, Phys.Rev. C83, 065206
et al., JHEP 0806, 099 (2008), 0803.2161. (2011), 1102.4376.
[5] A. Aguilar, D. Binosi, and J. Papavassiliou, Phys.Rev. [24] A. Bashir, L. Chang, I. C. Cloet, B. El-Bennich, Y.-
D78, 025010 (2008), 0802.1870. X. Liu, et al., Commun.Theor.Phys. 58, 79 (2012),
[6] M. Pennington and D. Wilson, Phys.Rev. D84, 119901 1201.3366.
(2011), 1109.2117. [25] G. Eichmann, R. Alkofer, I. Cloet, A. Krassnigg, and
[7] L.ChangandC.D.Roberts,Phys.Rev.Lett.103,081601 C. Roberts, Phys.Rev. C77, 042202 (2008), 0802.1948.
(2009), 0903.5461. [26] I.Cloet,A.Krassnigg,andC.Roberts,eConf C070910,
[8] A.KizilersuandM.Pennington,Phys.Rev.D79,125020 125 (2007), 0710.5746.
(2009), 0904.3483. [27] F.Gross,Relativisticquantummechanicsandfieldtheory
[9] A. Bashir, R. Bermudez, L. Chang, and C. Roberts, (Wiley, New York, 1993), first ed.
Phys.Rev. C85, 045205 (2012), 1112.4847. [28] E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232
[10] A. Krassnigg and P. Maris, J.Phys.Conf.Ser. 9, 153 (1951).
(2005), nucl-th/0412058. [29] M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951).
[11] M. Bhagwat, A. Krassnigg, P. Maris, and C. Roberts, [30] C.D.Roberts,M.S.Bhagwat,A.Holl,andS.V.Wright,
Eur.Phys.J. A31, 630 (2007), nucl-th/0612027. Eur. Phys. J. ST 140, 53 (2007), 0802.0217.
[12] P. Maris and P. Tandy, Nucl.Phys.Proc.Suppl. 161, 136 [31] A. Holl, C. D. Roberts, and S. V. Wright (2006), nucl-
(2006), nucl-th/0511017. th/0601071.
[13] M. Blank and A. Krassnigg, Phys.Rev. D84, 096014 [32] P.MarisandC.D.Roberts,Int.J.Mod.Phys.E12,297
(2011), 1109.6509. (2003), nucl-th/0301049.
[14] P. Maris, AIP Conf.Proc. 892, 65 (2007), nucl- [33] R. Alkofer and L. von Smekal, Phys. Rept. 353, 281
th/0611057. (2001), hep-ph/0007355.
[15] A.Krassnigg,Phys.Rev.D80,114010(2009),0909.4016. [34] P.Boucaud,J.Leroy,A.L.Yaouanc,J.Micheli,O.Pene,
[16] M. Bhagwat, A. Holl, A. Krassnigg, C. Roberts, et al., Few Body Syst. 53, 387 (2012), 1109.1936.
and P. Tandy, Phys.Rev. C70, 035205 (2004), nucl- [35] D. Ebert, T. Feldmann, and H. Reinhardt, Phys.Lett.
th/0403012. B388, 154 (1996), hep-ph/9608223.
[17] N. Souchlas, Phys.Rev. D81, 114019 (2010). [36] C. Roberts, Prog.Part.Nucl.Phys. 61, 50 (2008),
[18] E. Rojas, B. El-Bennich, and J. de Melo, Phys.Rev. 0712.0633.
D90(7), 074025 (2014), 1407.3598. [37] R. Farias, G. Dallabona, G. Krein, and O. Battistel,
[19] L. Gutierrez-Guerrero, A. Bashir, I. Cloet, and Phys.Rev. C73, 018201 (2006), hep-ph/0510145.
C. Roberts, Phys.Rev. C81, 065202 (2010), 1002.1968. [38] R. Farias, G. Dallabona, G. Krein, and O. Battistel,
[20] H. Roberts, C. Roberts, A. Bashir, L. Gutierrez- Phys.Rev. C77, 065201 (2008), hep-ph/0604203.
Guerrero,andP.Tandy,Phys.Rev.C82,065202(2010), [39] W. Weise (2005), nucl-th/0504087.
