On the Spectrum of Scalar-Scalar Bound States

22 Pages·0.2 MB·English

by  W. Moedritsch

TUW-96-05
On the Spectrum of Scalar-Scalar Bound States
W. M¨odritsch∗
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1 Institut fu¨r Theoretische Physik, Techn. Univ. Wien

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A-1040 Wien, Wiedner Hauptstraße 8-10
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J

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0 Abstract
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4 A new, exactly solvable, Barbieri-Remiddi like equation for bound states of two scalar
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constituents interacting with massless vector particles is presented, both for stable and
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/
h unstable particles. With the help of this equation the bound state spectrum is calculated
p
- to O(α4) for a SU(N) nonabelian gauge theory. The result for the abelian case reproduces
p
e
h the known result fromthe Foldy-Wouthuysen calculation. It is shown how different graphs
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asinthefermionictheorycontributetothespectrumtothisorder.Furthermorethebound
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state correction to the decay width for a weakly decaying system is calculated. This result
r
a
is equal to its fermionic counterpart. Thus the theorem on bound state corrections for
weakly decaying particles, formulated previously for fermions only, has been extended to
the scalar theory.
revised version, Wien, Dec. 1996
∗ e-mail: [email protected]
1 Introduction
While the discussion of fermionic bound states has a long history [1], much less attention
has been paid to the similar problem with scalar constituents. Only the ladder approxi-
mation with scalar interaction is a well known example and has already been discussed in
the 50-s and 60-s [2] in the framework of the Bethe-Salpeter equation. Indeed, to this day
the only known fundamental matter fields are fermionic. But in supersymmetric theories
for each fermion two scalar partners are required. Since some of them, probably stop or
sbottom, could have masses within the reach of the next generation of e+e− accelerators,
even the observation of bound states of those particles seems possible. These objects and
systems built of scalar composite particles in atomic physics underline the need of an
equally clear and transparent approach as the one developed for the fermionic case [3].
A recent attempt in this direction [4] splits the boson propagator in a particle and anti-
particle propagator in order to be able to treat them like fermions. The spectrum is then
obtained by constructing the Hamiltonian via a Foldy - Wouthuysen transformation and
a perturbation theory `a la Salpeter. This approach does not show significat advantage
over the pure Foldy-Wouthuysen approach [5] and suffers also from the drawback that it
will break down in higher orders due to the appearance of higher powers in the spatial
momentum ~p. The Coulomb field appears in this formalism as an external field which
makes this formalism not very reliable-looking. All these drawbacks can be circumvented
by developing anexactly solvable zero order equation andsubsequently using a systematic
perturbation theory.
To the best of our knowledge there exists no attempt in the literature to construct a
solvablezeroorderequationfortheBSequationcontainingtwochargedscalarsinteracting
via a vector field. This goal will be achieved in section 2.
In section 3 we will review briefly the BS perturbation theory and use it to calculate
the spectrum of bound states forscalar particles with equal mass, bothforthe abelianand
1
nonabelian case to O(α4). This will be of importance if the stop has a narrow width. If the
width becomes comparable to the level splittings this considerations can be understood
as a determination of the scalar-antiscalar potential.
The decay width is also subject of the second application we present in section 4. We
calculate the bound state correction to the decay width Γ of system of scalar constituents
to O(α2Γ).
Finally section 5 is devoted to the conclusions and to the discussion of our results.
2 A bound state equation for scalar particles
2.1 Stable particles
As starting point we present here an exactly solvable equation for stable scalar particles
which interact via a vector field.
We start from the BS equation for a bound state wave function χ
P P d4p′
χBijS(p;P) = −iSii′(2 +p)Sj′j(−2 +p) (2π)4Ki′j′,i′′j′′(P,p,p′)χBi′′Sj′′(p′;P), (1)
Z
where S is the exact scalar propagator, and K is the sum of all two scalar irreducible
graphs. Botharenormalized to beFeynman amplitudes. Furthermore, we have introduced
relative momenta p and p′, a total momentum P = p p , and we choose the center of
1 2
−
mass (CM) frame where P = (P ,~0) = (2m+E,~0).
0
As a first approximation to eq. (1) we would like to use beside the free relativistic
scalar propagators the kernel due to the Coulomb interaction
(P +p +p′)(P p p′)
K (p,p′) = 4πα 0 0 0 0 − 0 − 0 . (2)
C
(p~ ~p′)2
−
For a nonablelian theory with gauge group SU(N) we use
N2 1 g2
α = − . (3)
2N 4π
2
In this case χ has to be a singlett in order that K represents an attractive force. The
C
kernel (2) has the drawback that it is p dependent and the exact solution of eq. (1) with
0
(2) is not known. However, in the nonrelativistic regime by the scaling argument [6]
p O(mα2), p~ O(mα), (4)
0
≈ | | ≈
P 2m O(mα2)
0
≈ −
we can start with an instantaneous approximation to the kernel since p is of O(α2m) in
0
this region and may be included in the corrections afterwards. Doing this, we can perform
the zero component integration on the propagator ( E = √m2 +p~2)
p
dp 1
0
i =
− 2π [(P0 +p )2 E2 +iǫ][( P0 +p )2 E2 +iǫ]
Z 2 0 − p − 2 0 − p
1 1 1 1
= = (5)
2EpP0 "2Ep −P0 − 2Ep +P0# Ep(4Ep2 −P02)
and it is quite easy to show that
4m EpEp′
K (p,p′) = 4πα (6)
0 q~q2
gives a solvable equation with the normalized solutions
E (P2 4E2)
χ(p) = i p 0 − p φ(p~) (7)
√2P [(P0 +p )2 qE2 +iǫ][( P0 +p )2 E2 +iǫ]
0 2 0 − p − 2 0 − p
χ¯(p,ǫ) = χ∗(p, ǫ) (8)
− −
to the eigenvalues
α
P = M(0) = 2m 1 σ2, σ = . (9)
0 n − n n 2n
q
Eq. (8) is dictated by the requirement that χ¯ should acquire the same analytic properties
as the underlying field correlators
x x
χ(p) = eipx 0 TΦ†( )Φ( ) P , (10)
n
h | 2 −2 | i
Z
x x
χ¯(p) = e−ipx P TΦ( )Φ†( ) 0 > . (11)
n
h | 2 −2 |
Z
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Using the integral representation for the step function which is included in the time
ordered product, one derives eq. (8).
Taking the equation for the Green function
iG = D +D K G , (12)
0 0 0 0 0
−
with
(2π)4δ4(p p′)
D = − , (13)
0 [(P0 +p )2 E2 +iǫ][( P0 +p )2 E2 +iǫ]
2 0 − p − 2 0 − p
instead of that for the BS wave function and using again (6) we find
G (E,~p,~p′)
G = F(p) C F(p′) (14)
0
− 4m
b
with
P2 4m2
E = 0 − (15)
4m
and b
E (P2 4E2)
F(p) = p 0 − p . (16)
[(P0 +p )2 Eq2 +iǫ][( P0 +p )2 E2 +iǫ]
2 0 − p − 2 0 − p
G denotes the well known Coulomb Green function in momentum space. These solutions
C
can be used for a systematic BS perturbation theory for scalar constituents, as will be
demonstrated in the next section.
2.2 Unstable Particles
As has been shown recently by the author [7] for the fermionic case, an important sim-
plification can be achieved in some calculations if the width of the bound state is already
included in the zero order equation. Furthermore, if the width becomes comparable to
the level shifts, this approach even becomes indispensible. For the scalar case this can be
done by the replacement
E E2 iΓm. (17)
p → p −
q
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While (17) leads to expressions for the BS wave functions which contain unpleasant ex-
pressions for the particle poles it has the advantage that the propagator has the form as
expected from the phase space of an unstable particle. Furthermore the above calculation
remains essentially unchanged if we define the square root in (17) to be that with the
negative imaginary part (clearly we demand Γ > 0 and m > 0). Only the energy in the
resulting equation for the Green function and thus in (14) changes to
P2 4m2
E = 0 − +iΓ. (18)
4m
b
The eigenvalues for P are
0
Γ mσ4 Γ2 σ2Γ
P = 2m 1 σ2 i 2m mσ2 n + iΓ i n (19)
0,n s − n − m ≈ − − 4 4m − − 2
In the case of the fermions we managed to construct wave functions independent of
Γ. This was possible because the small components of the propagator containing P iΓ
0
−
instead of P + iΓ were projected away by the choice of an appropriate kernel K. This
0
cannot be achieved in the scalar case and thus, surprisingly enough, the scalar wave
functions look more complicated than the fermionic ones. A version for a zero order
equation for decaying particles where the propagator is chosen in close analogy to the
fermionic case has been developed in [8]. In our present work we, instead, proceed in the
spirit of our generalized approach.
3 Perturbation Theory
Perturbation theory for the BS equation starts from the BR equation for the Green
function G (eq. (12)) of the scattering of two fermions [9] which is exactly solvable. D is
0 0
the product of two zero order propagators, K the corresponding kernel. The exact Green
0
function G may be represented as
1 ∞
G = χBS χ¯BS +G = G (HG )ν, (20)
nl P P nl reg 0 0
Xl 0 − n νX=0
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where the corrections are contained in the insertions H and G is the part of G regular
reg
at P = P . It is easy to show that H can be expressed by the full kernel K and the full
0 n
propagators D:
H = K +K +iD−1 iD−1. (21)
− 0 − 0
Thus the perturbation kernel is essentially the negative difference of the exact BS-kernel
and of the zero order approximation.
Expanding both sides of equation (20) in powers of P P , the mass shift is obtained
0 n
−
[6, 10]:
∆Γ
∆M i = h (1+ h )+ h g h +O(h3). (22)
0 1 0 1 0
− 2 h i h i h i
Here the BS-expectation values are defined as e.g.
d4p d4p′
h χ¯ij(p)hii′jj′(p,p′)χi′j′(p′), (23)
hh ii ≡ (2π)4 (2π)4
Z Z
We emphasize the four-dimensional p-integrations which correspond to the generic case,
rather than the usual three dimensional ones in a completely nonrelativistic expansion.
We distinguish these two case by introducing the notation ... for a four-dimensional
hh ii
expectation value and ... for the usual nonrelativistic expectation value
h i
d3p d3p′
V(p~,~p′) = φ∗(p~′)V(p~,~p′)φ(p~) (24)
h i (2π)3 (2π)3
Z Z
Of course, (23) reduces to an ordinary ”expectation value” involving d3p and Φ(p~), when-
ever h does not depend on p and p′.
0 0
In (22) h and g represent the expansion coefficients of H and G near the pole at P ,
i i 0 n
respectively, i.e.
∞
H = h (P P )m (25)
m 0 n
−
m=0
X
∞
G = g (P P )m−1 (26)
0 m 0 n
−
m=0
X
Similar corrections arise for the wave functions [6, 10]:
1
χ(1) = (g h + h )χ(0) (27)
1 0 1
2h i
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3.1 Fine structure
As an application of this perturbation theory as well as of the new zero order equation
for scalar particles developed in the last section, we will present here the calculation of
the fine structure of two stable scalar particles interacting via a vector particle. Existing
calculations [4] rely on a mix of Fouldy-Wouthuysen transformation and the iterated
Salpeter perturbation theory. Our present approach is much more transparent and allows
in principle the inclusion of any higher order effect in a straightforward manner. First we
will calculate the fine structure for two scalars of equal mass interacting by an abelian
vector field. Then we consider also the nonabelian case which could be of interest for the
stop-antistop system. In this case we will calculate the spectrum up to order α4.
s
Since in the zero order equation we have replaced the exact one Coulomb exchange
(2) by K as given in (6) we have now to calculate the contribution of K +K to the
0 C 0
−
energy levels. This is shown in fig. (1)a . With
d4p d4p′ (P +p +p′)(P p p′)
K = 4πα χ¯(p) 0 0 0 0 − 0 − 0 χ(p′) =
C
hh− ii − (2π)4(2π)4 (p~ ~p′)2
Z −
P02 +2Ep2 +2Ep2′ 4πα
= = (28)
−h 4P0 EpEp′ ~q2 i
2m qσ2 4πα
= n
−h P0 − 2 ! ~q2 i
2m 4πα
K = (29)
hh 0ii P h ~q2 i
0
we obtain
σ2 4πα mα4
∆M := K +K = n = . (30)
C hh− C 0ii 2 h ~q2 i 16n4
The fact that the p-integrations are well behaved and the result is of O(α4) proves the
usefulness of our zero order kernel.
The transverse gluon of fig. 1b gives rise to a kernel
4πα (p~2 ~p′2)2
H = (p~+~p′)2 − (31)
T
q2 − ~q2 !
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-
K
BR
Fig. 1
Performing the zero component integrations exactly and expanding in terms of the spatial
momenta one obtains to leading order (c.f. [11])
4πα p~2 (p~~q)2
∆M = H = (32)
T T
hh ii − m2 h~q2 − ~q4 i
1 δ 3
= mα4 + l0 . (33)
8n4 8n3 − 16n3(l+ 1)!
2
Due to the fact that scalars can only form spin zero bound states, the the annihilation
graph into one gauge particle (with spin one) contributes only for p-waves and thus is
supressed by two additional powers in α. Furthermore, as in the fermionic case, it vanishes
for the nonabelian theory due to the color trace since the bound states are color singlets.
As can be seen from the above results the contribution of the transverse gauge field
is equal for fermions and bosons. However, the relativistic correction to the Coulomb
exchange appears to be different. Let us therfore check the contribution of this Coulomb
correction from second order perturbation theory (fig. 2.a). These contributions give only
rise to O(α5lnα) effects in the fermionic theory. Since the leading Coulomb singularity
is cancelled we may hope that we can replace the Green function by the free propagator.
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- - =
K G K
BR 0 BR
=
K K K
BR BR BR
Fig. 2
Indeed it can be shown that the next terms of the Green function give only higher order
contributions.
Due to the presence of the zero component momentum in the scalar-Coulomb gluon
vertex we observer that the contribution from fig.2.b diverges linearly. However, it is an
easy exercise to show that in the sum of graphs (fig.2.b + fig. 3) this linear divergence
cancels. Thus we regularise all the single graphs, sum up and find a finite result. We have
used dimensional regularization as well as a one dimensional Pauli-Villars regularization.
Both give the same result for the finte parts of the intgrals.
(3)
∆M = h + ( K +K )g ( K +K ) (34)
box hh 0 ii hh − C 0 1 − C 0 ii
d3k I
0
h = i (35)
hh 0ii hh− (2π)3~k2(~q ~k)2ii
Z
−
where I is decomposed according to fig. 3 for a generic SU(N) theory:
0
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