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On the Spectrum of Scalar-Scalar Bound States

A new, exactly solvable, Barbieri-Remiddi like equation for bound states of two scalar constituents interacting with massless vector particles is presented, both for stable and unstable particles. With the help of this equation the bound state spectrum is calculated to O(\alpha^4) for a SU(N) nonabelian gauge theory. The result for the abelian case reproduces the known result from the Foldy-Wouthuysen calculation. It is shown how different graphs as in the fermionic theory contribute to the spectrum to this order. Furthermore the bound state correction to the decay width for a weakly decaying system is calculated. This result 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.


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 approximation 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 antiparticle 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 theoryà 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 an exactly solvable zero order equation and subsequently using a systematic perturbation theory.
To the best of our knowledge there exists no attempt in the literature to construct a solvable zero order equation for the BS equation containing two charged scalars interacting 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 for scalar particles with equal mass, both for the abelian and nonabelian case to O(α 4 ). 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(α 2 Γ).
Finally section 5 is devoted to the conclusions and to the discussion of our results.

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 where S is the exact scalar propagator, and K is the sum of all two scalar irreducible graphs. Both are normalized to be Feynman amplitudes. Furthermore, we have introduced relative momenta p and p ′ , a total momentum P = p 1 − p 2 , and we choose the center of mass (CM) frame where P = (P 0 , 0) = (2m + E, 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 For a nonablelian theory with gauge group SU(N) we use In this case χ has to be a singlett in order that K C represents an attractive force. The kernel (2) has the drawback that it is p 0 dependent and the exact solution of eq. (1) with (2) is not known. However, in the nonrelativistic regime by the scaling argument [6] we can start with an instantaneous approximation to the kernel since p 0 is of O(α 2 m) in this region and may be included in the corrections afterwards. Doing this, we can perform the zero component integration on the propagator ( and it is quite easy to show that gives a solvable equation with the normalized solutions to the eigenvalues Eq. (8) is dictated by the requirement thatχ should acquire the same analytic properties as the underlying field correlators 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 with instead of that for the BS wave function and using again (6) we find with and G C denotes the well known Coulomb Green function in momentum space. These solutions can be used for a systematic BS perturbation theory for scalar constituents, as will be demonstrated in the next section.

Unstable Particles
As has been shown recently by the author [7] While (17) leads to expressions for the BS wave functions which contain unpleasant expressions 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 The eigenvalues for P 0 are 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 0 − iΓ instead of P 0 + iΓ were projected away by the choice of an appropriate kernel K. This 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.

Perturbation Theory
Perturbation theory for the BS equation starts from the BR equation for the Green function G 0 (eq. (12)) of the scattering of two fermions [9] which is exactly solvable. D 0 is the product of two zero order propagators, K 0 the corresponding kernel. The exact Green function G may be represented as where the corrections are contained in the insertions H and G reg is the part of G regular at P 0 = P n . It is easy to show that H can be expressed by the full kernel K and the full propagators D: 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 0 − P n , the mass shift is obtained [6,10]: Here the BS-expectation values are defined as e.g.
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 expectation value and ... for the usual nonrelativistic expectation value Of course, (23) reduces to an ordinary "expectation value" involving d 3 p and Φ( p), whenever h does not depend on p 0 and p ′ 0 . In (22) h i and g i represent the expansion coefficients of H and G 0 near the pole at P n , respectively, i.e.

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 α 4 s . Since in the zero order equation we have replaced the exact one Coulomb exchange (2) by K 0 as given in (6) we have now to calculate the contribution of −K C + K 0 to the energy levels. This is shown in fig. (1)a . With we obtain The fact that the p-integrations are well behaved and the result is of O(α 4 ) proves the usefulness of our zero order kernel.
The transverse gluon of fig. 1b gives rise to a kernel Performing the zero component integrations exactly and expanding in terms of the spatial momenta one obtains to leading order (c.f. [11]) .
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(α 5 ln α) 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. - 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.
where I 0 is decomposed according to fig. 3 for a generic SU(N) theory: Using the abbreviations we have written the result for Pauli Villars regularization to make the cancellation of the liner divergent parts obvious.
For the double Coulomb exchange graph from fig. 2 we obtain for the time component Collecting everything from above we have which leads with eq. (35),(40) and (3) immediately to the result The net result for the spectrum of two scalars bound by an abelian gauge field is equal to that of ref [4]. However, we showed which graphs contribute in a pure BS approach which can be used as a basis for any higher order calculation.
We have also checked the derivative ∂K 0 /∂P 0 contributing to h 1 and the X- in the fermionic case [12]. Possible large contributions of lighter particles to the vacuum polarization as depicted in (4.c) can be treated as in the fermionic case [13].
In supersymmetric theories a |Φ| 4 term is part of the lagrangian. Clearly it can be put in by hand into the Langragian of an ordinary qantum field theory. The contribution from an interaction term of the form −λ/2(Φ † T a Φ)(Φ † T a Φ) is easily calculated: and gives a contribution of the same form as the Darwin term (usually interpreted as a zitterbewegung contribution) which is suppressed by two orders in α in the scalar theory.
There may exist a small chance that this term may be helpful for the determination of the supersymmety parameters of the theory contained in λ.
For an ordinary quantum field theory without a direct interaction on the tree level it was shown first by Rohrlich [14] that a counter term of this form is needed for the scattering of two scalars (e.g the graphs of fig. 3  The spectrum calculated so far is common for the abelian and the nonabelian theory.
Collecting all pieces a we have where ∆M j F,nl originates in the contribution of j light fermions to the vacuum polarization and can be found in [13].
It has been pointed out first in [15]  (4.d) has been calculated in [15] for the fermionic case.
Here we will give a calculation of the same contribution for scalar constituents. After performing the color trace the perturbation kernel for the second graph in fig. (4.d) reads Performing the k 0 integration and using the scaling to extract the leading contribution in α we find that Adding the similar contribution from the first graph in fig. (4.d) gives This result differs by a factor 4m 2 from the fermionic result which is compensated by a corresponding difference in the wave functions to give eventually precisely the same result as in the fermionic case .  fig. (4.h) gives the same result as in the fermionc case, which was calculated recently [16].
The box graph with two Coulomb lines crossed, vanishes due to the color trace. Another box graph with the Coulomb vertices on one scalar line replaced by a seagull vertex can be shown to contribute to O(α 5 ). However, they are in principle needed to cancel the Coulomb singularities.
Thus the difference in the spectrum of the scalar bound state to O(α 4 ) compared to the fermionic case is entirely due to the graphs also present in the abelian theory discussed above. 4 Bound state corrections to the decay width Assuming that the scalar particle under consideration decays into two other particles, the decay width is the imaginary part of the self energy function Σ at the mass shell. Focusing on the stop quark a possible scenario could bet R → b +χ i [17]. We shall be interested in terms of the order O(α 2 Γ) where Γ is the tree level decay width. The first part of the perturbation kernel due to the exact inverse propagator p 2 − m 2 − Σ(p 2 ) for the bound state corrections to the decay width reads In the derivation of eq. (51) we expanded the self energy function around the mass shell and we assumed that the decay width used in the zero order equation (e.g. in D 0 ) is given by As has been first shown in [18], the gauge dependence contained in the off shell contribution Σ ′ is cancelled by parts of the vertex correction depicted in fig. 4.f. It gives rise to a perturbation kernel with Λ 0 representing the vertex correction. The color trace is already included in α. As in the fermionic case [7] it is possible to derive a Ward identity which guaranties the cancellation of the gauge dependent terms (the T a 's are the SU(N) generators) Λ a µ (p, q = 0) = −2gT a p µ ∂ ∂p 2 Σ(p 2 ), ImΛ 0 (p = (m, 0), q = 0) = −2mImΣ ′ (m 2 ). (55) But the detailed calculation shows differences to the fermionic case: The sum of the contributions from H 1 and H 2 vanishes to the desired order with the help of the zero order equation On the other hand, we observed above that the wave functions for decaying fermions and scalars were very different. While it was possible to obtain the same wave functions It is easy to see that the only difference comes from the fact that the perturbation has to be taken at the position of the pole (19) which leads to the replacement in eq. (28). We thus get a relativistic correction to the decay width of the bound state This has to be added to the O(α 2 Γ) term of (19) to yield the final result for the boundstate correction to the decay width: This result generalizes the result of [7,18] to the bosonic case. We can thus say that the effect of the bound state corrections to the decay width can be interpreted entirely as a time dilatation effect as was first conjectured for the fermionic theory [19].

Conclusion
We have presented a consistent formalism for the calculation of bound state properties for scalar particles interaction with an abelian or nonabelian spin one vector field. This is done by deriving a solvable relativistic zero order equation similar to that of Barbieri and Remiddi both for stabel and unstable scalars. Based on this equation a systematic perturbation theoy can be built which allows especially the calculation of the position of the bound state poles to higher orders.
Using this approach the bound state spectrum was calculated to O(α 4 ). We found that we had to take into account the abelian box graphs to this order. This is not the case in the fermionic theory. All the relativistic Coulomb corrections only reproduce the p 4 term from the expansion of √ m 2 + p 2 indicating that a fully relativistic formulation is not really economic for the lowest orders in perturbation theory. However, the advantage of the presented formalism is that it is straigtformward applicable to any higher order calculation. We calculated also the nonabelian contributions to O(α 4 ). It would be very interesting to observe a particle where the above mentioned predictions could be tested. Today it seems that the stop-antistop system could be a candidate.
It will be heavy enough to allow a perturbative treatment even for the nonabelian case.
Whether the decay width will be small enough to allow a detailed study of the spectrum remains open to speculation at present. But even for a quite large decay width the scalarscalar potential will provide the basis for interesting threshold calculations for this case [20], analogous to the ones for the top-antitop system [19,21].
Thus we have to the desired accuracy It should be noted that dimensional regularization does not show up linear divergencies.
Instead the use of the regularization (39) leads to a visible liner divergent term (of higher order in α) iΛ/(8| t − k|) which has to be cancelled by similar contributions from graphs where the two Coulomb gloun vertices on one or both scalar lines are double Coulomb vertices.