Mesonic Light – Cone Wavefunctions for
Covariant Harmonic Confinment.
M. Dillig * **, F. Pilotto * and C.A.Z. Vasconcellos *
* Instituto di Fisica, Universidade Federal do Rio Grande do Sul
Porto Alegre, Brazil
** Institute for Theoretical Physics, University of Erlangen – Nuernberg
Erlangen, Germany
PACS:
Keywords:
Abstract
Starting from a covariant quasipotential equation in the instant form we derive for qQ mesons
analytic wavefunctions in the front form (on the light cone) for a covariant harmonic
confinement. Compared are constraints of the four dimensional Bethe-Salpeter for quarks
with equal masses and in the limit of a very heavy and a very light (anti) quark. For the
invariant distribution amplitude and the corresponding momelnta we compare our findings
with current parametrizations in the literature.
Email: mdillig@theorie3.physik.uni-erlangen.de
Email:
Email:
Supported in part by the Kernforschungszentrum KFZ, Jülich, Germany
It is well understood that even for consituent quark models an appropriate formulation of
the dynamics of bound systems (mesons and baryons) with u, d or s quarks requires a
covariant, relativisic formulation. One approach, which garantees a covariant description
of bound systems for covariant kernels is the manifestly covariant four – dimensional
Bethe – Salpeter equation (BSE) (1,2). Though in principle this approach is rigorous – in a
diagrammatic expansion the exact kernel is an infinite sum of well defined irreducible
diagrams – presently a quantiative application to even two – body quark – antiquark
systems is too abitious: on the one side, the BS kernel cannot be rigorously derived from
Quantumchromodynamics (QCD), on the other side the technical solution of the BSE is
even for simple kernel extremely complex. With further serious problems for truncated
kernels for tlhe BSE (such as states of negative norm, the interpretation of unequal particle
times or the appropriate one – body limit (3)), for phenomenological applications the BSE
is commonly reduced to still covariant threedimensional Quasipotential Equations (QPE).
However, such a reduction is not unique (4): it has to be motivated by pysical intuition on
the (off-shell) propagation of the interacting particles (5).
Solutions for the BSE or the QPEs have been generally obtained in the instant form (i. e.
in equal time coordinates). More recently – initiated by the pionieering work of Dirac (6) it has been realized that the description of relativistic bound states in the front form
(equivalently on the light cone (LC) (for recent reviews compare refs 7) or in the infinite
momentum frame (8)) has various advantages compared to the standard description such
as the natural transition to the LC in deep inelastic scattering, the reduction of dynamical
generators of the Poincare group and the kinematical nature of Lorentz boosts, or the
decoupling of particle – antiparticle mixing, resulting in a trivial LC vacuum. As as
consequence for bound systems it allows a systematic expansion in the components of the
Fock space (7,9).
Motivated by numerous studies of mesonic qQ systems in the recent literature, we
investigate in this note the structure of mesons on the LC in the framework of QP
equations. As we are looking for analytical solutions for the corresponding LC
wavefunctions, we incorporate the genuine feature of QCD by a covariant scalar harmonic
confinment (attempts in the instant form in this direction have been followed for example
by Mitra and coworkers (10)). In view of recent activities at charm ( C ) and beauty (B)
factories (11) we focus on momentum distriubtions both for light qq mesons (with u and d
quarks) and light – heavy qQ mesons (with an s, c or b antiquark).
Our starting point for a quark q(mass m and momentum p ) and and antiquark Q (mass m
and momentum p ) is the BSE
(1)
with the two-particle propagator
(2)
We eliminate the dependence of eq. (1) on the relative enrgy variable and convert the BSE
into a QPE by resricting the propagation of the qQ system on the mass shell (12), i. e.
(3)
with the free parameter 0
1 and the total and relative moment p, P of the qQ
system, respectively (above in eq. (3) we keep only the positive energy pole + in the
function).
The covariant nature of the BSE allows an immediate transition to LC variables (7,8)
(4)
The resulting equations is in it s full structure a coupled system of 16 differential
equations due to the spin ½ nature of the interacting fermions. As we are focussing mainly
on the x – dependence of the LC wave function we reduce this complex system to a single
radial equation, by an appropriate simplification of the spin structure of the system
(5)
where in an helicity representation R(
) is determined in the Mock representation
(assuming no interaction in the spin sector (13) or a free Melosh rotation from the qQ rest
system to the IMF (14). The resulting radial QPE for
(x,p ,M ) on the LC is then
obtained with the appropriate LC projection for the two – particle Greens function (15)
(6)
which leads to
(7)
with M being the invariant squared qQ mass and m
= m + p (we absorb all
irrelevant factors from the projection of the Greens function in the kernel K).
For a given interaction kernel the solution of the equations above directly results in the
light cone distribution of qQ system. As so far a detailed derivation of the LC amplitude
from QCD is till lacking (only results in the large Q
limit or estimates from QCD
sum rules exist (16,17), the main two goals of this short note are very modest: on the one
side we would like to use a kernel which can be derived explicitly from the corresponding
covariant representation in the instant form and we aim for an analytical solution for the
momentum distribution. A QCD inspired kernel, which includes confinment as the unique
property of QCD, is a covariant harmonic confinment kernel
(8a)
,equivalently in momentum space
(8b)
(the strength parameter g is related to the running quark-gluon coupling constant; ist value
is fixed from lthe spectrum of B – mesons). Following the same steps for the reduction to
the QPE we obtain explicitly
(9)
where the projection on the relative energy variable fixes p
from eq.(6). In the
following we focus on the two most interesting cases for the parameter :
- light mesons with u,d quarks m = m
limit (18))
= m with
= ½ (Blankenbecler – Sugar BBS
(10a)
- mesons with a light u,d and a heavy quark m > m with = 0 (Gross limit (19); this
limit bears some relations to the heavy quark effective theory (20))
(10b)
We should mention, that for light mesons with = ½ , but m = m , p is only shifted by the
difference of the squared masses and does not effect the radial solution; only the spin
dependence, which we do not consider here, is modified.
The next steps are now well defined. The radial equations for the two cases above are:
- BBS limit (eq. (10a))
(11a)
- Gross limit (eq. (10b))
(11b)
The solutions for these equations are found in textbooks (21). Explicitly they are given as
BBS
(12a)
Gross
(12b)
Where the functions F(a,b,z) denotes confluent hypergeometrical functions, while H (p )
represent standard Hermite polynomials (22). The eigenvalues for the invariant mass M , i. e.
the mesonic spectrum, is then definied by the eigenvalue conditions
F(a,b x=0) = Fa,b, x=1) =0
(13)
which yields the eigenvalues
BBS:
(14a)
Gross:
(14b)
Together with normalization 1= ½
completely.
dx dp
(x,p )
this fixes the LC wave function
We compare our derivation with current parametrizations in the literature. Various simple
analytical forms can be found (23). In order to compare the influence of unequal quark masses
we compare with three current momentum distributions:
- the Lepage – Brodsky parametrization (24)
(15)
- the BSW model (25) and
(16)
- the parametrization as a simple power law (26)
(17)
(we extended the parametrization to unequal quark masses). The different parametrizations
we compare with two important quantities: the invariant longitudinal distribution amplitude
(IDA) (7, 27)
(18a)
and the corresponding momenta (28)
(18b)
The characteristic results with parameters from the literature are compared in Fig. 1 and Table
1. In Fig. 1 (a) we compare the IDA for two quarks with equal mass for the BBS projection;
Fig. 1(b) shows the same quantitiy for the D(uc) meson; finally, fig. 1(c) shows the shift of
the maximum of the momentum distribution from the rho, K*, D up to the B meson in the
Gross projection. As a general trend in (a), (b) we find, that the various parametrizations
differ mainly around the endpoints x -- 0 and x --- 1 and in the width of the distribution
around the maximum x
= m /m
. Comparison (c) confirms a systematic shift of the
dominant momentum components towards x = 1 for qQ mesons with increasing heavy quark
mass (a similar, however, less pronounced trend is found for the BBS projection for unequal
masses). Similar quantiative differences show up for the various momenta of the different
parametrizations (Table 1): here the differences are most pronounced for B mesons, as the
momenta depend with increasing power depend sensitively on the maximum of the
momentum distribution (here significant differences are found for the different projections:
BBS versus Gross projection).
We summarize. In this note we invstigated the momentum distribution of mesons with
unequal quark masses on the light cone. Without considering spin effects in detail, the radial
dependence was derived in a covariant fashion starting from the BSE and projecting it in a
covariant and physically motivated way onto the light cone. Rigorous analytical solutions
were derived and compared for a covariant confining kernel for two constraints for the
relative energy variable of the qQ system.
We find, that our results reproduce qualitatively the trends of purely phenomenological
parametrizations of LC distribution amplitudes. In detail, however, particularly in the
endpoint behaviour both in the longitudinal and in the transversal components we find serious
quantiative difference due to the implementation of our boundary conditions. Our simple
model shows various advantages. First of all it starts from a covariant interaction kernel,
which can be easily extended to more realistic cases (such as to parametrize the influence of
the one-gluon exchange). Furthermore, the transition from the BSE to QPE allows a direct
transition from the instant form to the front form for systems with two very heavy quarks (i. e.
in the nonrelativistic limit).
Of course, the rather selective results presented yield only little insight in the details of the
model. Presently we are extending our approach (including additional elements of the qQ
interaction) and test it on other quantities, such as decay constants and form factors (29).
Fig. 1: Comparison of the invariant distribution amplitude the parametrizations from eqs. (1517) with the BBS-limit (a; eq(10a)) and Gross-limit (b; eq.(10b)) and for different mesons in
the Gross-limit (c).
Table 1: Momenta n=0 to n=3 for the parametrizations (15-17) and the Gross-limit for the B
meson.
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QCD in the improved ladder approximation – arXiv:hep-ph/0101110