Multigluon Exchange Forces in Heavy Mesons
M. Dillig * ** , F. C. Pilotto and C. A. Z. Vasconcellos *
* Instituto di Fisica, Universidade Federal do Rio Grande do Sul
Porto Alegre, RS, Brazil
** Institute for Theoretical Physics III, University Erlangen – Nürnberg
Erlangen, Germany
PACS:
Keywords:
Abstract
Starting from the covariant three-dimensional reduction of the Bethe-Salpeter equation we
derive the leading irreducible multi – gluon exchange contributions in the static one-body
limit for qQ mesons with a light u,d quark and a heavy s,c and b antiquark. The resulting
divergent expansion reflects the nonlocal propagation of the light quark and is dominated by
rapidly increasing colour factors. In a local appromximation we sum the series by
nondiagonal Pade´ approximants. The resulting qQ potential interpolates the single gluon
exchange and the asymptotic linear confinment.
Email: mdillig@theorie3.physik.uni-erlangen.de
Email:
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Supported in part by the Kernforschungszentrum KFZ Jülich, Germany
One genuine feature of Quantum Chromodynamics (QCD) as the theory of strong interactions
is the nonlinear structure of the quark – quark (or antiquark) interaction (1). Thus mesonic and
baryonic systems are under intensive investigation in a variety of theoretical attempts, among
them lattice calculations as the most promising approach (2). Another line for the
investigation of relativistic bound systems in a covariant manner is the manifestly covariant
fourdimensional Bethe – Salpeter equation (BSE; 3). Coventionally, due due technical and
conceptional difficulties, the BSE is reduced to still covariant 3-dimensional quasipotential
equations (QPE, 4). This reduction is not unique (5): the projection on the relative energy
variable is guided by intuitive physical constraints. one constraint for qQ mesons with a very
light quark q (mass m ) and a very heavy (anti)quark Q (mass m ) is the correct one body –
limit for m -( the Gross limit; 6). This makes the investigation of mesons with a very
heavy (anti)quark attractive; in addition the heavy mass limit is highly interesting with respect
to recent developments in Heavy Quark Effective Theory (HQET, 7).
These different steps for the formulation of the QPE are easily summarized. For the qQ
system the vertex function (in the center-of-mass system with the total and relative momenta
P and p, respectively)
(1)
involves the irreducible qQ kernel K(p,k,P). The two-body 4-dimensional Greensfunction
G(k,P) for the qQ system (we relate Q to Q by charge conjugation)
(2a)
is reduced in the spectator limit m
shell to
/ m --
by putting the heavy quark Q on it s mass
(2b)
where is the full energy of the light quark. Implementing the same one-body limes for the
BS amplitude
(3)
results in the static single particle Dirac QPE
(4)
which is easily reduced to a corresponding two-comonent Pauli - Schrödinger equation for
the relative binding energy of the qQ system (note, that above the mass of the heavy quark
disappears in the heavy mass limit, the spin of the heavy quark decouples and the theory
becomes static)..
A quantitative understanding of qQ mesons requires a detailed derivation of the kernel of the
BSE equation as the sum of all irreducible single and multigluon exchange contributions (a
similar expansion can be find for example for modern nucleon – nucleon effective models;
(8)). In principle, with the quark masses, the running coupling constant and the gluon
propagator as given input (we do not attempt to solve a coupled Dyson-Schwinger and BSE
equation for the quark-gluon vertex and the gluon propagator (9)), the derivation of the
irreducible kernel is in principle straightforward. In practice, however, the complexity of the
evaluation of the corresponding Feynman diagrams, particularly due to the nonabelian nature
of QCD with the self couplings of the gluons (1) defies a systematic perturbative expansion of
the kernel order by order. Again qQ systems here offer an attractive opportunity: all Feynman
diagrams in an expansion in the running coupling constant up to the 3. order and the full
class of irreducible diagrams for noninteracting gluons are reduced to the static limit, which
allows (as sketched below) an analytic representation in coordinate space.
A schematical representation of the irreducible qQ kernel is given in Fig. 1.
Fig.1 Irrreducible contributions in the running qQ coupling constant up the 3. order (a - c) and
for noninteracting gluons in arbitrary order (d) (for all diagrams only one specific time
ordered contribution is shown).
The linear and nonlinear quark and gluon vertices are given in the standard notation by ( 10)
(5a)
(5b)
(5c)
with
and g = (
); (here the
Mann matrices and structure constants, respectively).
and f
are the SU(3) Gell-
We examplify the further steps of the calculation for the two-gluon exchange diagram (Fig.
1(b)) in the qQ CMS (i. e. p = - p = p in the inital and correrspondingly in the final state).
In the limit m / m -- 0 it is given as (after the integration over k )
(6)
(the colour and the spin factor in the heavy mass limit are included above). With the
Fouriertransform
(7)
and appropriate new variables p + k = q; p + k = q ,the various integrations factorize and the
result in coordinate space can be represented in a compact form as
V (r,r’) =
(8)
where V (r) is full one – gluon exchange contribution, including the complete dependence on
the running coupling constant. As expected that two-gluon exchange contributation is the
product of the one-gluon exchange contributions folded with the nonlocality from the
propagation of the light quark (the mass of the heavy quark as a static source disappears in the
heavy mass limit). For further steps we localize eq. (8) at the expense of an effective mass for
the intermediate light quark
m
=
<
fm
(9)
(for a constitutent u,d quark mass of aroung 300 MeV and an avarage momentum k
which then yields the compact expression
V (r) =
m* x (V (r )/m*) **2 with c =
),
(10)
The contributions of 3. order in
are technically more involved and highly nonlocal(for a
detailed derivation we refer to a forthcoming article), as shown for diagröam (i) in fig. 1c:
; in the localaized limit the structure simplifies cosiderably and we obtain
for the time ordered diagrams (i), (ii) and (iii) in fig. 1c.
The contribution from the diagrams up to
are presented in the local approximation in fig.
2. Here we supplement the one-gluon exchange by the Richardson one-gluon exchange
potential (22) with an infrared running coupling constant (which yields a Coulombic
singularity for small r and a linear divergence for large r)
with the parameters
(10GeV/c) =
and
= 200 MeV; this yields a plausible
extrapolation of the experimental running coupling coupling constants for 3-momenta below
1GeV/c (
).
Fig. 2(a) shows the relative importance of the uncorrelated and correlated 2-gluon exchange
to
(compare fig1 (c)). All contriubtions are qualitatively similar, the main difference
originates from the different colour factors (which all add up coherently). In fig. 2(b) the total
contributions in
,
+
and
are compared. The different contributions differ by their colour factors:
(only the colour factor for the two-gluon exchange contribution changes sign, which holds
also for all higher order contributions investigated). More stricking (but not unexpected) is the
behaviour of the different orders for large and small r: the singularities at small and large r
Increase order by order, indicating already here, that the perturbation series diverging, except
for V(r ) = 0. This is confirmed in fig. 2c for the uncorrelated multi-gluon exchanges up the
8. order: extrapolating to N ---the qQ exchange potential develops a sharp barrier at r
0.7 fm: at the barrier the potentials jumps from to +
, it is completely unphysical. It
is expected, that this behaviour persists even for QQ systems with two heavy quarks (like bb
mesons), as the number of diagrams and correspondingly the total colour factor increases
porportional to the order N of the gluon exchange.
Fig. 2: Comparison of the gluon exchange contributions of various orders in the localaized
limit. (a) comparison of the contributions (i),..,(iii) from fig 1(c) in
; (b) radial
dependence of the one, two and threee – gluon exchange; (c) uncorrelated multigluon
exchange contributions in 2.,5. and 8. order in
. Full line at r
0.7 fm indicates the
extropolation in N ---- .
We attempt to resum the divergent perturbative series
For the uncorrleated multi-gluon exchange (the coefficients are colour factors with ratios c /c
/ ..../c
given as 1/
(27). To construct a convergent series we follow the
resummations as Pade´ approximants (28-32), which consists in the resummation of eq.( ) as
a ratio of two finite order power series
P (M,N) =
where M and N are the order of the power series in the numerator and denominator,
respectively. The convergence properties of Pade´ approximants one ar rigorous mathematical
level are not fully understood (33,34), however, for problems with an exact solution
Pade´approximants have shown excellent convergence properties. It is not fully clear, what
Pade`approximant to formulate. However, from
(M+1,M) Pade´ approimants show the most reliable convergence (35,36).
it is known, that
The qualitative structure of the Pade´summed perturbation theory is easily anticipated: in all
approximants (1,0) (i. e. the one-gluon exchange itself) up to order (4,3), preserve the
asymptotic
behaviour of the one-gluon exchange for very small and large intequark distances. The main
difference comes for intermediate distances r, where higher contributions interpolate the onegluon exchange and lead to characteristic new patterns (fig. 3). However, one has to keep in
mind, that just in regions with a dominant multigluon exchange nonlocalities from the
propagation of the light quark should introduce major addiational modifications (which
cannot be represented in this local approximation). For QQ systems we expect, that the sum of
higher order contributions only renormalize the effective strength and range of the one-gluon
exchange, without changing the radial dependence significantly and thus – not unexpected –
confirm the results of the lattice calculations for two heavy (static) quarks (1-4; 37,..,40)).
Fig. 3: Pade´approximants (M+1,M) for M = 0,1,2 and 3 (full, dashed, dashed-dotted and
dotted lines, respectively)
Our results above suggest different conclusions. Particularly the nonlinear self coupling of the
gluons and the dramatically rising colour factors make a systematic expansion of the qQ
kernel to very high order in
fairly hopeless; beqond that the picture is skrutinized due to
strong nonlocalities from the propagation of the light quark (evidently, this pattern shows up
even more strongly for qq systems with two light quarks, i. e. mesons below 1 GeV).
Furthermore, a realistic prediction of the qq interaction requires a systematic and unified
treatment of both the single gluon – quark interaction and the multigluon qq exchanges in a
coupled Dyson-Schwinger and BS approach (41,..,44). Alternatively, for the asymptotic
behaviour of the qq interaction asymptotic expansions in the interquark distances and in
colour space may be more adequate to shed light on fundamental questions of QCD.
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