t On leave of absence of Istituto di Fisica dell' Universit2 di Parma, Parma, ... in a convincing way and this leaves the question open whether the underlying two-.
I
SLAC-PUB-1395 (T)
ASYMPTOTIC
BEHAVIOR OF FORM FACTORS FOR
TWO- AND THREE-BODY
BOUND STATES
C. Alabiso and G Schierholz Errata Page 1, line 6 (of the text)
.
(iT2)-e (0 < I9 2 1)
Page I, line 9 (of the text)
Corrigenda ( IEl)-l-e
F2E2)
Page 3, line 22
(0 < e)
w(lql)-2-2e
F2(;i2) r(lTl)-2-e F2E2) = (L$l)-2*5-e
Page IO, formula
(0 < 8 5 $),
(20)
Page 1, line 10 (of the text)
i
(4 5 e),
SLAC-PUB-1395 (T) March 1974 ASYMPTOTIC -
BEHAVIOR OF FORM FACTORS FOR
TWO- AND THREE-BODY
-c.
C. Alabisot
BOUND STATES*
and G. Schierholzj-7
Stanford Linear Accelerator Center Stanford. University, Stanford, Calif. 94305 ABSTRACT The asymptotic power behavior of the electromagnetic examined for two- and three-body nonrelativistic.
s-w&e
In the nonrelativistic
form factors are
bound states both relativisitc
and
case we consider local and separable
two-body potentials and we make use of the Faddeev equations in order to define the three-body
bound states.
For local potentials which behave as
G2,- e (0 < 0 5 1) for large momentum transfer, power behavior of the form factors
of the two- and three-body
F2(y2) = ( ly*r) -3-e and F3(q2) = ( 13) -6-2e, potentials
we obtain for the asymptotic
respectively.
V = g( IFI) g( Iktl) and g( IFI) = (l~l)-‘-e
and F3G2) = ( 13) -5-e, respectively. the two- and three-body
For separable
we find F,(2)
For the relativistic
Bethe-Salpeter
bound states
= ( lTl)-2-e
case, we consider
equation in the ladder approximation.
We treat the spin zero case only but we believe that our final conclusions will not be affected by the introduction
of spin l/2 particle.
which behaves as (k 2) -’ at large momentum transfer,
With an interaction we obtain F2(q2) = (q2)-l-’
and F3(q2) = (q2) -2-2e D
(Submitted to Phys. Rev. )
* Work supported in part by the U. S. Atomic Energy Commission. t On leave of absence of Istituto di Fisica dell’ Universit2 di Parma, Parma, Italy. Consiglio Nazionale della Ricerche fellow. tt On leave of absence and present address: II. Institut fiir Theoretische Physik der Universi&t, Hamburg, Germany.
I
I. -
INTRODUCTION,
RESULTS AND CONCLUSIONS
The evaluation of the electromagnetic stant t;sk for the last five years. mentum transfer
l-6
hadron form factors has been a con-
It soon became clear that the large mo-
behavior of the form factors provides a powerful mean of
studying the constituents’of
the hadrons and their dynamics.
It is by now well
accepted that the behaviors Fn(q2) = + and F,p(q2) = +2 are compatible q (4 ) 7 with the experiments. This fact suggests that the pion and the nucleon certainly are of a different
nature as far as the electromagnetic
interaction
are concerned.
It seems also to suggest that the pion is less composite than the nucleon because of the faster decrease of the proton form factor. haviors have been derived from the minimal proton; 899 so far, however,
Recently,
the previous be-
quark structure
the three-particle
of the pion and the
bound state has not been treated
in a convincing way and this leaves the question open whether the underlying and three-particle
structure
can explain the different
two-
behavior of the two form
factors. It is the aim of this paper to investigate the large
q2 behavior of the form
factors of the two- and three-particle
s-wave bound states in a systematic way,
both in relativistic
theories.
and nonrelativistic
sider power behaviors
only, neglecting possible logarithmic
a first approach, we restrict
Here, in
We do not
makes a real difference
on our
This case will be discussed elsewhere. 10
We shall consider the potential scattering First,
factors.
ourselves to spinless constituents.
believe that the case of spin I/2 constituents final conclusions.
Throughout the paper we con-
case (II) for two main reasons.
because many features of composite particle
means of the nonrelativistic
models can be explained by
quark model; 11 moreover,
equation in the ladder approximation
the Bethe-Salpeter
reduces to a nonrelativistic -2 -
form in the
large momenta limit, -
as it can be recovered from various (equivalent)
three-
12-15 dimensional equations. The second good reason for studying the potential c, theory is the firm mathematical ground on which the nonrelativistic threetheory in the form of the Faddeev equations 16 is based (we do not con-
particle
sider three-particle
forces).
For both two- and three-particle potentials
V( IQ)
Ifl=L
03
0 > 0, and the separable potentials
( IiTpe,
V(g G) = g( lk7) g( Ik7l) with g( Ia) the potentials limiting
is determined
behavior ( lk7)-l
cases, we shall assume the two-body local
(,kf)424
IjqZ
by simple reasons.
is characteristic
, 0 > 0. Our choice of
03
For the local potentials,
the
of the singular potential (-h/r2)
produces the unpleasant feature of a wave function fall-off
which
depending on the
coupling constant. 17,18 On the other hand, an even more singular potential gives rise to the exponential decrease of the wave function and of the form factor both, lg and this does not seem to be the physical case.
As far as the separable
potential is concerned,
the choice f3 > 0 is imposed by the very existence of
scattering
The use of nonlocal potentials
processes,
is suggested both by the
existence of tensor forces in the spin l/2 case, and by the structure tivistic
potential as recovered
in the three-dimensional
of the rela-
version of the Bethe-
Salpe ter equation D10,12-15 Our results are as follows.
For the two-body and three-body
form factors we obtain F2(T2) = ( Ia) -3-e and F,(2) potentials,
whereas we obtain F,(z)
= ( 13) -24
bound states
= ( I~)-s-2e~ with local
and F3G2) = ( I$J-~-~’
with
separable potentials ,, In the framework
of relativistic
theories,
we consider (III) the s-wave bound
states of two and three particles
described by the two-body Bethe-Salpeter
tion in the ladder approximation
(III A) and by the relativistic -3 -
Faddeev
equa-
equations (III B) . 12-15 ’ 2o We shall assume a two-body interaction
of the form
8 > 0. Our interactions correspond to the Aq3 theory for 2 = (k2)-‘, kern I 8 = 1 and to the A(p4 theory for the limiting case 0 = 0. For the latter case it
V(k)
has been proved 2,4,18 that the high momentum transfer
behavior of the two-
body wave functions and form factors depends on the coupling constant,
as in
the singular (- A/r2) potential,, Our results for the asymptotic behavior of the two- and three-body factors are F2(q2) = (q 2 ) -l-‘,
F3(q2)
form
N (q2)-2-2e;
Since the Arp4 theory leads to that strange dependence on the coupling constant, we define the physical form factors as given by our superrenormalizable interaction “nucleon”
in the limit
8 - 0; the asymptotic behavior of our “pion”
form factors turns out to be (q2)-l
The spin $ constituents,
structure
of the three-body
integration
and (q2)-2, respectively.
which are more interesting
for the physical situa-
apart from the complicate
tion, present some technical difficulties: wave function,
and
spin-
there appears a delicate region of
so that one has to be more careful than in the spin zero case.
How-
ever, we do not agree with Ref. 4, note 25, where the author claims that the consistency argument, renormalizable Finally, the predictions
widely applied in our paper, does not work for super-
interactions. it is worthwhile
to remark
that our results are in agreement with
given in Ref 0 8 and in Ref 0 9. Furthermore,
turn out to be integrable
our wave functions
as it was assumed in Ref. 8 as a crucial hypothesis.
-4-
II. In the framework asymp;tic
POTENTIAL
SCATTERING
of a potential scattering
behavior of the two- and three-body
theory we shall discuss the bound states form factors at
large momentum transfer;
we shall consider s-wave bound states only.
more, in order to simplify
things, we shall always assume that only one particle
is charged.
Let us start with the two-body case.
Further-
Here the charge form factor
reads: .-
where the wave function
zj satisfies the homogeneous Schroedinger equation:
GiT= --L
di?V(r-
-i;i z/J(~
(2)
q2-E
If we now consider a central potential which behaves at large I,@
N ---A-Iqi+e
Ikf as
(3)
bo,
2
we get the following behavior for 11, and F2: (4) In the limiting
case 0 = 0 which corresponds to the potential (-h/r’) the form factor - -2-2 ,\/$Y behaves like ( Iql) (0
