You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'The problem of computing the effective nonrelativistic potential $U_{D}$ for the interaction of charged scalar bosons within the context of D-dimensional electromagnetism with a cutoff, is reduced to quadratures. It is shown that $U_3$ cannot bind a pair of identical charged scalar bosons; nevertheless, numerical calculations indicate that boson-boson bound states do exist in the framework of three-dimensional higher-derivative electromagnetism augmented by a topological Chern-Simons term.'
author:
- Antonio Accioly
- Marco Dias
title: |
    Boson-boson effective nonrelativistic potential for higher-derivative

    electromagnetic theories in D dimensions
---

introduction
============

We consider in this Brief Report the problem of determining the effective charged-scalar-boson—charged-scalar-boson low energy potential $U_D$ arising from D-dimensional electromagnetism with a cutoff $a$. The Lagrangian concerning this theory can be written as

$$\begin{aligned}
{\cal L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{a^2}{2}
\partial_\nu F^{\mu\nu} \partial^\lambda F_{\mu\lambda},\end{aligned}$$

where $F_{\mu\nu} \equiv \partial_\nu A_\mu -
\partial_\mu A_\nu$ is the usual electromagnetic tensor field. Lagrangian (1) is gauge and Lorentz invariant; in addition, it leads to local field equations which are linear in the field quantities. At distances much larger than the cutoff, the fields described by Eq. (1) become essentially equivalent to the Maxwell fields.

It worth mentioning that Lagrangian (1) was proposed a long time ago, in 3+1 dimensions, by Podolsky and Schwed [@1], in a rather different context. The main reason for investigating this theory is that recently it was shown that in its framework the electromagnetic mass of a point charge occurs in the equation of motion in a form consistent with special relativity; moreover, the exact equation of motion does not exhibit runaway solutions or non-causal behavior, when the cutoff is larger than half of the classical radius of the electron [@2; @3]. Massive fermions, in turn, have their helicity flipped on account of their interaction with an electromagnetic field described by Podolsky’s generalized electrodynamics, while massless ones seem to be unaffected by the electromagnetic field as far as their helicity is concerned [@4]. Now, as is well known, if one starts from a renormalizable theory and calculates the effective action one obtains typically several operators of dimension 6 (or higher) and not just Podolsky’s term [@5]. This raises the question: Would these latter results change significantly if one takes all these terms into account? Since the amount of spin-flip is determined by the vector nature of the electromagnetic field but not by its spatial distribution, the aforementioned results will not change at all. What about the finiteness of the electromagnetic mass? Since the evaluation of the classical self-force acting on the point charge depends on the cutoff $a$ [@2; @3], one is led to conjecture that the finiteness of the electromagnetic mass is a very specific property of Podolsky’s electrodynamics and not a general feature of higher-derivative theories. It is interesting that the same route that leads to Maxwell’s electrodynamics leads also to Podolsky’s electrodynamics provided we start from Podolsky’s electrostatic force instead of the usual Coulomb’s law [@6; @7]. Unlike Maxwell’s electrodynamics, Podolsky’s generalized electrodynamics implies a finite value for the energy of a point charge in the whole space [@8; @9].

We are motivated by two quite similar developments: In the first, we investigate whether $U_3$ can form “Cooper pairs".

Our second topic is related to three-dimensional Podolsky-Chern-Simons theory. Based on the interesting discussions from Jackiw [@10] about the consistency of the nonrelativistic limit of certain relativistically invariant quantum field theories, it can be shown that the Chern-Simons term alone is unable to form boson-boson bound states [@11]. Nonetheless, numerical calculations indicate that the Podolsky term provides an stabilizing mechanism allowing for the existence of “Cooper pairs".

We use natural units throughout; our signature is $(+,-,-,\cdot\cdot\cdot,-)$.

effective charged-scalar-boson—charged-scalar-boson low energy potential
========================================================================

We begin by describing a method for computing the effective nonrelativistic potential for the interaction of two charged spinless bosons of equal masses via a “Podolskian photon" exchange. The prescription is based on the marriage of nonrelativistic quantum mechanics and quantum field theory in the nonrelativistic limit. An algorithm for calculating the propagator is then presented. The recipe is used afterward to get the propagator for higher-derivative electromagnetism in the Lorentz gauge. Finally, we reduce the problem of computing the effective nonrelativistic potential to quadratures.

The method
----------

Nonrelativistic quantum mechanics tells us that in the first Born approximation the cross section for the scattering of two indistinguishable massive particles, in the center-of-mass frame (CoM), is given by $\frac{d\sigma}{d\Omega} = \left |
\frac{m}{4\pi} \int e^{-i {\bf p'} \cdot \; {\bf r}} V( r)
e^{i{\bf p} \cdot\; {\bf r}} d^{D-1}\;{\bf{r}} \right |^2,$ where ${\bf p}\; ({\bf p'})$ is the initial (final) momentum of one of the particles in the CoM. In terms of the transfer momentum, ${\bf
k \equiv p' - p}$, it reads

$$\begin{aligned}
\frac{d\sigma}{d\Omega} = \left | \frac{m}{4\pi} \int V( r)
e^{i{\bf k} \cdot\; {\bf r}} d^{D-1}\;  {\bf{r}} \right |^2.\end{aligned}$$

On the other hand, from quantum field theory we know that the cross section, in the CoM, for the scattering of two identical massive scalars bosons by an electromagnetic field, can be written a $\frac{d\sigma}{d\Omega} = \left | \frac{1}{16\pi E}\; {\cal
M}\right |^2,$ where $E$ is the initial energy of one of the bosons and ${\cal M}$ is the Feynman amplitude for the process at hand, which in the nonrelativistic limit $(N.R.)$ reduces to

$$\begin{aligned}
\frac{d\sigma}{d\Omega} = \left | \frac{1}{16\pi m}\; {\cal
M}_{N.R.} \right |^2.\end{aligned}$$

From Eqs. (2) and (3) we come to the conclusion that the expression that enables us to compute the D-dimensional effective nonrelativistic potential has the form

$$\begin{aligned}
V( r) = \frac{1}{4m^2} \frac{1}{(2\pi)^{D-1}} \int d^{D-1}\; {\bf
k}\; {\cal M}_{N.R.}\; e^{-i {\bf k} \cdot \; {\bf r}},\end{aligned}$$

which clearly shows how the potential from quantum mechanics and the Feynman amplitude obtained via quantum field theory are related to each other.

Now, in the Lorentz gauge Podolsky’s scalar QED is described by the Lagrangian

$$\begin{aligned}
{\cal L} &=& -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{a^2}{2}
\partial_\nu F^{\mu\nu} \partial^\alpha
F_{\mu\alpha}-\frac{1}{2\lambda} (\partial_\nu A^\nu)^2
\nonumber\\ &&+ (D_\mu \phi)^* D^\mu \phi - m^2 \phi^* \phi,\end{aligned}$$

where $D_\mu \equiv \partial_\mu + iQA_\mu$. Therefore, the interaction Lagrangian to order $Q$ for the process $S + S
\longrightarrow S + S$, where $S$ denotes a spinless boson of mass $m$ and charge $Q$, is ${\cal L}_{int} = iqQ A^\mu \left( \phi
\partial_\mu \phi^* - \phi^*
\partial_\mu \phi \right).$ The Feynman rule for the elementary vertex is shown in Fig. 1. Accordingly, the Feynman amplitude for the interaction of two charged spinless bosons of equal mass via a “Podolskian photon" exchange (see Fig. 2) is

[cc]{}

![The relevant vertex for boson-boson interaction.](fig1.eps "fig:")&

[l]{}

$V^\mu(p,p')=-Q(p+p')^\mu$\
\
\
\
\
\

![One-Podolskian photon-exchange contibution to the scattering of two identical massive charged bosons.](fig2.eps)

$$\begin{aligned}
{\cal M} =V^{\mu} (p,p') D_{\mu\nu} (k) V^\nu (q,q'),\end{aligned}$$

where $D_{\mu\nu} (k)$ designates the “Podolskian photon" propagator.

The propagator
--------------

We propose now an algorithm for computing the propagator for electromagnetic theories with higher-derivatives, based on the usual transverse and longitudinal vector projector operators, namely $\theta_{\mu\nu} = \eta_{\mu\nu} - \frac{\partial_\mu
\partial_\nu}{\Box},\; \omega_{\mu\nu} = \frac{\partial_\mu
\partial_\nu}{\Box},$ which satisfy the relations $\theta_{\mu\rho} {\theta^\rho}_\nu =
\theta_{\mu\nu},\;\omega_{\mu\rho} {\omega^\rho}_\nu =
\omega_{\mu\nu},\;\theta_{\mu\rho} {\omega^\rho}_\nu = 0,$ where $\eta_{\mu\nu}$ is the Minkowski metric. The set of operators $\{
\theta,\omega\}$ is a complete set of projector operators for rank-one tensors. Indeed, they are idempotent, mutually orthogonal and satisfy the completness relation $[\theta + \omega]_{\mu\nu} =
\eta_{\mu\nu} \equiv I_{\mu\nu}.$

Let ${\bar{\cal L}}$ be the Lagrangian for electromagnetism with higher-derivatives. Since ${\bar{\cal L}}$ is a gauge -invariant Lagrangian, we add to it a gauge-fixing Lagrangian ${\cal
 L}_{gf}$, which implies that ${\cal L} \equiv {\bar{\cal L}} +
 {\cal L}_{gf}$ can be written as ${\cal L} = \frac{1}{2} A^{\mu} O_{\mu\nu} A^\nu.$ Expanding $O$ in the basis $\{ \theta,\omega\}$, yields $O = x_1 \theta + x_2 \omega.$ Accordingly, $O^{-1} = y_1 \theta + y_2 \omega,$ where $O^{-1}$ is the propagator and $y_1$ and $y_2$ are parameters to be determined. Now, taking into account that $O O^{-1} = I,$ we promptly obtain

$$O^{-1} = \frac{1}{x_1} \theta + \frac{1}{x_2} \omega,$$

where we are supposing that both $x_1$ and $x_2$ are nonvanishing. Note that the procedure we have just outlined is quite straightforward: on the one hand it reduces the work of calculating the propagator to a trivial algebraic exercise; on the other hand it great simplifies calculations involving the contraction of conserved currents $(\partial_\nu J^\nu = 0)$ with the propagator since in this case the alluded contraction simply gives

$$\begin{aligned}
{O^{-1}}_{\mu\nu} J^\mu = \frac{J_\nu}{x_1}.\end{aligned}$$

From the above we find that the propagator for Podolsky’s electrodynamics in the Lorentz gauge assumes the form

$$\begin{aligned}
D_{\mu\nu} (k) = \frac{M^2}{k^2 (k^2 - M^2)}\; \theta_{\mu\nu}
-\frac{\lambda}{k^2}\; \omega_{\mu\nu},\end{aligned}$$

where $M^2 \equiv \frac{1}{a^2}.$

The potential
-------------

From Eqs. (6), (7) and (8), we get immediately ${\cal M} =
\frac{M^2 Q^2 (2p -k)\cdot(2q + k)}{k^2 (k^2-M^2)},$ which implies

$$\begin{aligned}
{\cal M}_{N.R.} = \frac{4m^2M^2Q^2}{{\bf k}^2 ({\bf k}^2 + M^2)}.\end{aligned}$$

Inserting Eq. (9) into Eq. (4), we obtain

$$\begin{aligned}
V(r) &=& {\int_0}^{\;\infty} f(|{\bf k}|) |{\bf k}|^{n-1}\; d|{\bf
k}|\; {\int_0}^{\; 2\pi} d\theta_1 \nonumber \\ &&\times
{\int_0}^{\; \pi}\sin \theta_2 \; d\theta_2 \;{\int_0}^{\; \pi}
{\sin ^2} \theta_3 \; d\theta_3 \nonumber  \\ &&... {\int_0}^{\;
\pi} e^{-i |{\bf k}| r \cos \theta_{n-1}} \sin^{n-2}
\theta_{n-1}\; d\theta_{n-1}, \nonumber\end{aligned}$$

where $2 <n \equiv D-1$ and $ f(|{\bf k}|) \equiv
 \frac{Q^2}{(2 \pi)^n} \left( \frac{1}{\;{\bf k}^2} - \frac{1}{\;{\bf
 k}^2 - M^2} \right).$ Now, taking into account that

$$\begin{aligned}
{\int_0}^{\; \pi} \sin^{m} \theta \; d\theta = \frac{\sqrt{\pi}\;
\Gamma \left( \frac {m - 1}{2} \right)}{\Gamma \left( \frac{ m +
2}{2} \right)}, \nonumber\end{aligned}$$

$$\begin{aligned}
 {\int_0}^{\; \pi} e^{-i |{\bf k}| r
\cos \theta_{n-1}} \sin^{n-2} \theta_{n-1}\; d\theta_{n-1}&=&
\frac{ 2^{\frac {n-2}{2}}\; \Gamma \left(\frac{1}{2}
\right)}{\left( |{\bf k}| r
 \right)^{\frac{n-2}{2}}} \nonumber \\ && \times
 \Gamma \left( \frac{n-1}{2} \right) \nonumber \\ && \times J_{\frac{n-2}{2}} \left(|{\bf k}| r
 \right), \nonumber\end{aligned}$$

where $J$ denotes the Bessel function, we arrive at the following expression for the potential

$$U_{D}(r) = \frac{Q}{(2\pi)^{\frac{D-1}{2}}\; r^{\frac{D-3}{2}}}
{\int_0}^{\;\infty} \left( \frac{1}{\;{\bf k}^2} - \frac{1}{\;{\bf
k}^2 -M^2} \right)|{\bf k}|^{\frac{D-1}{2}}\; J_{\frac{D-3}{2}}
\left(|{\bf k}| r \right) \; d|{\bf k}|,$$

where $U_{D} (r) \equiv  \frac{V(r)}{Q}$ and $D>3.$

On the other hand , it is trivial to show that $U_3$ can be evaluated from the expression

$$\begin{aligned}
U_3(r) = \frac{Q}{2\pi} {\int_0}^{\;\infty} \left( \frac{1}{\;{\bf
k}^2} - \frac{1}{\;{\bf k}^2 -M^2} \right)|{\bf k}| J_{0}\left(
|{\bf k}| r \right) \; d|{\bf k}|,\nonumber\end{aligned}$$

which allows us to conclude that Eq. (10) can also be applied to the case $D=3$. Hence, the problem of computing the effective nonrelativistic potential for $D>2$ was reduced to quadratures.

If Eq. (10) is correct , it must reproduce the Podolskian potential in 3+1 dimensions. Performing the computation for $D=4$, we get

$$\begin{aligned}
U_4 (r) = \frac{Q}{4 \pi} \frac{1 - e^{ - \frac{r}{a}}}{r},
\nonumber\end{aligned}$$

which is just the same result as that obtained in Podolsky’s electromagnetic theory \[1\].

Planar quadratic electromagnetism
=================================

For $D=3$, Eq. (10) yields

$$\begin{aligned}
U_3(r) = -\frac{Q}{2 \pi} \left[ \ln {\frac{r}{r_0}} + K_0(Mr)
\right],\end{aligned}$$

where $r_0$ is an infrared regulator and $K$ is the modified Bessel function.

We discuss now the existence of boson-boson bound states in the context of planar quadratic electromagnetism. The corresponding time-independent Schrödinger equation can be written as

$$\begin{aligned}
{\cal{H}}_l {\cal{R}}_{nl} &=& -\frac{1}{m} \left( \frac{d^2}{d
r^2} {\cal{R}}_{nl} + \frac{1}{r} \frac{d}{d r} {\cal{R}}_{nl}
\right) + V^{eff}_l {\cal{R}}_{nl} \nonumber \\ &=& E_{nl} {\cal
R}_{nl}, \nonumber\end{aligned}$$

$$\begin{aligned}
V^{eff}_l &\equiv& \frac{l^2}{m r^2} + Q U_3(r) \nonumber \\ &=&
\frac{l^2}{m r^2} - \frac{Q^2}{2\pi} \left[\ln{\frac{r}{r_0}} +
K_0(Mr) \right], \nonumber\end{aligned}$$

where ${\cal R}_{nl}$ is the $n$th normalizable eigenfunction of the radial Hamiltonian ${\cal H}_l$ whose corresponding eigenvalue is $E_{nl}$ and $V^{eff}_l$ is the $l$th partial wave effective potential. On the other hand,

$$\begin{aligned}
\frac{d}{d r} V^{eff}_l = -\frac{2l^2}{m} \frac{1}{r^3} -
\frac{Q^2}{2 \pi} \frac{1}{r} + \frac{Q^2 M}{2
\pi}K_1(Mr),\nonumber\end{aligned}$$

which allows us to conclude that $\frac{d}{d r}
V^{eff}_l < 0$ in the interval $ 0 < r < \infty$, implying that $V^{eff}_l$ is strictly decreasing in this interval. Consequently, in the framework of planar quadratic electromagnetism, no bound state concerning the two charged scalar bosons system exists.

podolsky-chern-simons planar electromagnetism
=============================================

Since boson-boson bound states do not show up in Podolsky planar electromagnetism, we investigate here whether the effective boson-boson low energy potential related to Podolsky-Chern-Simons (PCS) planar theory can bind a pair of identical charged scalar bosons. The Lagrangian for PCS scalar QED, in the Lorentz gauge, can be written as

$$\begin{aligned}
{\cal L} &=& -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{a^2}{2}
\partial_\nu F^{\mu\nu} \partial^\alpha
F_{\mu\alpha}-\frac{1}{2\lambda} (\partial_\nu A^\nu)^2
\nonumber\\ &&+ (D_\mu \phi)^* D^\mu \phi - m^2 \phi^* \phi
+\frac{s}{2} \varepsilon_{\mu\nu\rho} A^\mu \partial ^\nu A^\rho,\end{aligned}$$

where $s>0$ is the topological mass.

In the basis $\{\theta,\omega,S\}$, where $S_{\mu\nu} \equiv
\varepsilon_{\mu\rho\nu} \partial^\rho$, the propagator assumes the form

$$\begin{aligned}
O^{-1} = \frac{(a^2 k^4-k^2) \theta}{\left( a^2k^4-k^2 \right)^2 -
s^2 k^2} -\frac{\lambda \omega}{k^2} - \frac{s S}{\left(
a^2k^4-k^2 \right)^2- s^2k^2} . \nonumber\end{aligned}$$

Now, in the nonrelativistic limit the Feynman amplitude for the process shown in Fig. 2 reduces to

$$\begin{aligned}
{\cal M}_{NR} =  \left[ \frac{(a^2{\bf k}^4 + {\bf k}^2
)4Q^2m^2}{(a^2{\bf k}^4 + {\bf k}^2 )^2 + s^2{\bf k}^2} +
\frac{8ismQ^2 \; {\bf k} \wedge {\bf P} }{(a^2{\bf k}^4 + {\bf
k}^2 )^2 + s^2{\bf k}^2} \right], \nonumber\end{aligned}$$

where ${\bf P} \equiv \frac{1}{2} ({\bf p} -{\bf q})$ is the relative momentum of the incoming charged scalar bosons in the CoM. On the other hand, is trivial to show that if $a < \frac{2
\sqrt 3}{9s}$ the equation

$$\begin{aligned}
x^3 + \frac{2x^2}{a^2}  + \frac{x}{a^4} + \frac{s^2}{a^4} = 0,\end{aligned}$$

where $ x \equiv {\bf k}^2 $, has three distinct negative real roots. In this case ${\cal M}_{NR}$ can be rewritten as

$$\begin{aligned}
{\cal M}_{NR}&=&  \frac{8ismQ^2\;{\bf k} \wedge {\bf P}}{a^4}
\left[ \sum_{j=1}^{3} \frac{ B_j}{{\bf k}^2 -x_j} + \frac{a^4}{s^2
{\bf k}^2} \right] \nonumber \\ &&+ \frac{4Q^2 m^2}{a^4}
\sum_{j=1}^{3} \frac{A_j}{{\bf k}^2 - x_j}, \nonumber\end{aligned}$$

where $x_1, x_2$ and $x_3$ are the roots of Eq. (13) and $A_1 \equiv \frac{1 +a^2 x_1}{(x_1-x_2)(x_1-x_3)}$,$A_2 \equiv
\frac{1 +a^2 x_2}{(x_2-x_1)(x_2-x_3)}$, $A_3 \equiv \frac{1
+a^2 x_3}{(x_3-x_1)(x_3-x_2)}$,$B_1 \equiv
\frac{-(1+a^2x_1)^2}{s^2(x_1-x_2)(x_ 1-x_3)}$ ,$B_2 \equiv
\frac{-(1+a^2x_2)^2}{s^2(x_2-x_1)(x_2-x_3)}$,$B_3 \equiv
\frac{-(1+a^2x_3)^2}{s^2(x_3-x_1)(x_3-x_2)}$.

.5cm

It follows that the effective nonrelativistic potential can be calculated from the expression

$$\begin{aligned}
U_3(r) &=& \frac{i s Q}{\pi m a^4} \left[ \frac{a^4}{s^2}
\lim_{\sigma \rightarrow 0} \int_{0}^{\infty} \frac{({\bf k}
\wedge {\bf P}) J_0({|\bf k|}r) |{\bf k}|\; d|{\bf k}|}{{\bf k}^2
+ \sigma^2} \right. \nonumber \\
 &&+ \left. \sum_{j} \int_{0}^{\infty} \frac{({\bf k} \wedge {\bf P}) B_j J_0(|{\bf k}|r)
|{\bf k}|\; d|{\bf k}|}{{\bf k}^2 - x_j}  \right] \nonumber \\ &&+
\frac{Q}{2 \pi a^4} \sum_{j} \int_{0}^{\infty} \frac{A_j}{{\bf
k}^2 - x_j} J_0(|{\bf k}|r) |{\bf k}| d|{\bf k}|. \nonumber\end{aligned}$$

Performing the computations, we obtain

$$\begin{aligned}
U_3(r) = - \frac{s Q}{\pi m a^4}\left[ \frac{ a^4}{s^2}
\frac{1}{r^2} + \frac{1}{r} \sum_{j} B_j {\sqrt{|x_j|}}
\;K_1({\sqrt{|x_j|}}\; r) \right] {\bf L } + \frac{Q}{2 \pi a^4}
\left[ \sum_{j} A_j K_0({\sqrt{|x_j|}}\; r) \right],\end{aligned}$$

where ${\bf L} \equiv {\bf r} \wedge {\bf P}$ is the orbital angular momentum.

Using Jackiw’s arguments [@10], one can show that the topological term alone is unable to bind the charged scalars bosons [@11].

We return now to the problem of probing whether “Cooper pairs" exist in the framework of PCS scalar QED. In this case the radial Schrödinger equation is

$$\left[\frac{d^2}{dr^2}+\frac{1}{r} \frac{d}{dr} \right]{\cal R}
_{nl}+m\left[E_{nl}-V^{eff}_l\right]{\cal R}_{nl}=0,$$

where

$$\begin{aligned}
V^{eff}_l(r) = - \frac{s Q^2}{\pi m a^4}\left[ \frac{ a^4}{s^2}
\frac{1}{r^2} + \frac{1}{r} \sum_{j} B_j {\sqrt{|x_j|}}
\;K_1({\sqrt{|x_j|}}\; r) \right]l  + \frac{Q^2}{2 \pi a^4} \left[
\sum_{j} A_j K_0({\sqrt{|x_j|}}\; r) \right] + \frac{l^2}{m r^2}.
\nonumber\end{aligned}$$

Employing the dimensionless parameters $ y \equiv sr $ ,$ \alpha \equiv \frac{Q^2}{\pi s}$ , $b_j \equiv
\frac{s^2}{a^4}  B_j$ , $X_j \equiv \frac{|x_j|}{s}$ , $\beta \equiv \frac{m}{s}$ , $a_j \equiv \frac{A_j}{a^4}$ and $ {\tilde E}_{nl} \equiv \frac{m E_{nl}}{s^2}$, we can rewrite Eq. (15) as

$$\begin{aligned}
\left[\frac{d^2}{dy^2}+\frac{1}{y} \frac{d}{d y} \right]{\cal R}
_{nl}+\left[{\tilde E}_{nl}-{\tilde V}^{eff}_l\right]{\cal
R}_{nl}=0,\end{aligned}$$

with

$$\begin{aligned}
{\tilde V}^{eff}_l &\equiv& - \frac{l(\alpha -l)}{y^2} +
\frac{\alpha\beta}{2} \sum_{j} a_j K_0(X_j\; y)  \nonumber \\ &&-
\frac{\alpha l}{y} \sum_{j} b_j X_j K_1(X_j \;y).\nonumber\end{aligned}$$

Note that ${\tilde V}^{eff}_l$ behaves as $\frac{l^2}{y^2}$ at the origin and as $\frac{l(l-\alpha)}{y^2}$ asymptotically. On the other hand, the derivative of this potential with respect to $y$ is given by

$$\begin{aligned}
\frac{d}{d y}{\tilde V}^{eff}_l &=& \frac{2l(\alpha -l)}{y^3} +
\alpha \sum_{j}\left[ \frac{2l}{y^2} b_j - \frac{\beta a_j}{2}
\right ] X_j K_1(X_j \; y) \nonumber \\ &&+ \frac{\alpha l}{y}
\sum_{j} b_j X^2_j K_0(X_j  \;y). \nonumber\end{aligned}$$

In order to find out whether or not boson-boson bound states could be formed, we shall analyze how $\frac{d}{d y}{\tilde
V}^{eff}_l$ behaves for small values of the cutoff $a$. Indeed, only if $a \lll 1$ will the well recognized properties of $QED_3$ be preserved. In this limit, we get

$$\begin{aligned}
\frac{d}{d y}{\tilde V}^{eff}_l \sim \frac{2l(\alpha -l)}{y^3} -
\left[ \frac{2 \alpha l}{y^2} + \frac{\alpha \beta}{2} \right]
K_1(y) - \frac{\alpha l}{y}K_0(y).\nonumber\end{aligned}$$

We assume from now on $a \lll 1$ and $l> 0$, without any loss of generality. It is trivial to see that if $l > \alpha$, the potential is strictly decreasing, which precludes the existence of bound states. The remaining possibility is $l < \alpha$. In this interval ${\tilde V}^{eff}_l$ approaches $ + \infty$ at the origin and $0^{-}$ for $y \rightarrow  + \infty$, which is indicative of a local minimum. Therefore, the existence of “Cooper pairs" is subordinated to the conditions $a \lll 1$ and $0<l< \alpha$.

Of course, it is impossible to solve Eq. (16) analytically; however, it can be solved numerically. To do that, we rewrite beforehand the radial function as $ {\cal R}_{nl} \equiv
\frac{u_{nl}}{\sqrt{y}}$. As a consequence, Eq. (16) takes the form

$$\begin{aligned}
\left[ \frac{d^2}{d y^2} + \frac{1}{4 y^2} \right] u_{nl} + \left[
{\tilde E}_{nl} - {\tilde V}^{eff}_l \right] u_{nl} = 0.\end{aligned}$$

Using the Numerov algorithm [@12], we solved Eq. (17) numerically for several values of the parameters $\alpha
\;,\; \beta$ and $l$, keeping the cutoff $a$ fixed. The latter was chosen equal to $\frac{2}{3} r_e = 9.52033 \times 10^{-3}
MeV^{-1}$, where $r_e$ is the fourdimensional classical radius of the electron. It is worth mentioning that the anomalous factor of $\frac{4}{3}$ in the inertia related to the Abraham-Lorentz model for the electron does not show up if if $a > \frac{1}{2} r_e$ \[2,3\].

In Fig. 3 we present our numerical results for the potential and the corresponding radial eigenfuntions concerning the first three bound states in the specific case of $l= 4$. The associated energies are

$ E_{14} = - 6.37501 \times 10^{-7} MeV \;,\; E_{24} = -1.2536
\times 10^{-7} MeV \;,\; E_{34} = - 5.22441 \times 10^{-9} MeV.$

The graphics shown in Fig. 3 exhibit, in a sense, the generic features of the potential and of the radial eigenfunctions, although they have been composed using particular values of the parameters $\alpha \;,\; \beta$ and $a$. A detailed study of the modifications of the effective potential induced by radioactive corrections, as well as the corresponding alterations to the eigenvalue structure, will be published elsewhere [@11]

To conclude, we remark that “Cooper pairs" exist in the context of PCS scalar QED if $a \lll 1$ and $0< l < \alpha$.

![$V^{eff}_4$ with the lowest three allowed energies and the corresponding energy eigenfunctions. Here $\left[ V^{eff}_4
\right]=eV$, $[r]=MeV^{-1}$, $\alpha =8$, $\beta =2000$ and $a=0.00952\,MeV^{-1}.$ ](fig3.eps)

A.A. thanks CNPq-Brazil and M.D. thanks CAPES-Brazil for financial support.

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