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arXiv:nucl-th/0611092 · Physical Review C

New class of quantum bound states: Diprotons in extreme magnetic fields

Danielle Allor, Paulo F. Bedaque, Thomas D. Cohen, Charles T. Sebens

Published Mar 2, 2007• 7 citations
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Abstract

This paper considers the possibility that two charged particles with an attractive short-ranged potential between them which is not strong enough to form bound states in free space, may bind in uniform magnetic fields. It is shown that in the formal limit where Coulomb repulsion is negligible ($q\ensuremath{\rightarrow}0\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{B}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}\phantom{\rule{0.3em}{0ex}}\text{with}\phantom{\rule{0.3em}{0ex}}q{B}_{0}\phantom{\rule{0.3em}{0ex}}\text{fixed}$, where $q$ is the charge and ${B}_{0}$ the field strength) there always exists a bound state for a system of two identical charged particles in a constant magnetic field, provided that there is a short-range uniformly attractive potential between them. Moreover, it is shown that in this limit any potential with an attractive s-wave scattering length will possess bound states provided that the range of the potential is much smaller than the characteristic magnetic length, ${r}_{0}\ensuremath{\equiv}(\frac{q{B}_{0}}{4}){}^{\ensuremath{-}1/2}$. For this case, the binding is computed numerically. We estimate the size of the magnetic field needed to approximately reach a regime where the formal limit considered becomes a good approximation to the dynamics. These numerical estimates indicate that two protons in an extremely strong magnetic field such as might be found in a magnetar will bind to form a diproton.

DOI
10.1103/physrevc.75.034001

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This paper considers the possibility that two charged particles with an attractive short-ranged potential between them which is not strong enough to form bound states in free space, may bind in uniform magnetic fields. It is shown that in the formal limit where Coulomb repulsion is negligible (q\ensuremath→0andB0\ensuremath→\ensuremath∞withqB0fixedq\ensuremath{\rightarrow}0\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{B}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}\phantom{\rule{0.3em}{0ex}}\text{with}\phantom{\rule{0.3em}{0ex}}q{B}_{0}\phantom{\rule{0.3em}{0ex}}\text{fixed}q\ensuremath→0andB0​\ensuremath→\ensuremath∞withqB0​fixed, where qqq is the charge and B0{B}_{0}B0​ the field strength) there always exists a bound state for a system of two identical charged particles in a constant magnetic field, provided that there is a short-range uniformly attractive potential between them. Moreover, it is shown that in this limit any potential with an attractive s-wave scattering length will possess bound states provided that the range of the potential is much smaller than the characteristic magnetic length, r0\ensuremath≡(qB04)\ensuremath−1/2{r}_{0}\ensuremath{\equiv}(\frac{q{B}_{0}}{4}){}^{\ensuremath{-}1/2}r0​\ensuremath≡(4qB0​​)\ensuremath−1/2. For this case, the binding is computed numerically. We estimate the size of the magnetic field needed to approximately reach a regime where the formal limit considered becomes a good approximation to the dynamics. These numerical estimates indicate that two protons in an extremely strong magnetic field such as might be found in a magnetar will bind to form a diproton.
Energy levels are investigated for two charged particles possessing an attractive, momentum-independent, zero-range interaction in a uniform magnetic field. A transcendental equation governs the spectrum, which is characterized by a collective Landau-level quantum number incorporating both center-of-mass and relative degrees of freedom. Results are obtained for a system of one charged and one neutral particle, with the interaction chosen to produce a bound state in vanishing magnetic field. Beyond deriving the weak-field expansion of the energy levels, we focus on non-perturbative aspects. In the strong-field limit, or equivalently for a system in the unitary limit, a single bound level with universal binding energy exists. By contrast, excited states are resonances that disappear into the continuum as the magnetic field is raised beyond critical values. A hyperbola is derived that approximates the number of bound levels as a function of the field strength remarkably well.
We study few-body bound states of charged particles subject to attractive zero-range/short-range plus repulsive Coulomb interparticle forces. The characteristic length scales of the system at zero energy are set by the Coulomb length scale D and the Coulomb-modified effective range reff. We study shallow bound states of charged particles with D≫reff and show that these systems obey universal scaling laws different from neutral particles. An accurate description of these states requires both the Coulomb-modified scattering length and the effective range unless the Coulomb interaction is very weak (D→∞). Our findings are relevant for bound states whose spatial extent is significantly larger than the range of the attractive potential. These states enjoy universality – their character is independent of the shape of the short-range potential.
We study charged particles in three dimensions interacting via a short-range potential in addition to the Coulomb potential. When the Bohr radius and the scattering length are much larger than the potential range, low-energy physics of the system becomes independent from details of the short-range potential. We develop the zero-range theory to describe such universal physics in terms of the Bohr radius and the scattering length by generalizing the Bethe-Peierls boundary condition, which is then applied to two charged particles to reveal their bound states and resonances. Infinite resonances are found for a repulsive Coulomb potential, one of which turns into a bound state with increasing inverse scattering length, whereas infinite bound states exist for an attractive Coulomb potential with no resonances at any scattering length. The zero-range theory is also applied to three equally charged particles at infinite scattering length under the variational Born-Oppenheimer approximation. We find that the effective potential between two heavy particles induced by a light particle is an inverse-square attraction at distances shorter than the Bohr radius, leading to infinite deep bound states, whereas shallow ones successively turn into resonances with increasing Coulomb repulsion.
This paper considers the binding of an electron or a positron to a ground-state hydrogen atom in a uniform magnetic field BBB, concentrating on states with zero orbital angular momentum JJJ. It is shown that the affinity of a positron to a hydrogen atom is 2.23b2e2∕a2.23{b}^{2}{e}^{2}∕a2.23b2e2∕a, where aaa is the Bohr radius, $b\ensuremath{\equiv}B∕{B}_{0}$, ${B}_{0}=2.3505\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, b⪡1b⪡1b⪡1 is assumed, and the proton is assumed to have infinite mass. With these assumptions it is also shown that no J=0J=0J=0 singlet or triplet excited state of ${\mathrm{H}}^{\ensuremath{-}}$ exists. These results are compared to a previous estimate of the electron affinity of J=0J=0J=0 magnetically bound ions [V. G. Bezchastnov, P. Schmelcher, and L. S. Cederbaum, Phys. Rev. A 61, 052152 (2000)].
Observing constituent particles with fractional quantum numbers in confined and deconfined states is an interesting and challenging problem in quantum many-body physics. Here, we further explore a computational scheme [Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011)] based on valence-bond quantum Monte Carlo simulations of quantum spin systems. Using several different one-dimensional models, we characterize S=12S=\frac{1}{2}S=21​ spinon excitations using the intrinsic spinon size $\ensuremath{\lambda}$ and confinement length $\mathrm{\ensuremath{\Lambda}}$ (the size of a bound state). The spinons have finite size in valence-bond-solid states, infinite size in the critical region (with overlaps characterized by power laws), and become ill defined (completely unlocalizable) in the N\'eel state (which we stabilize in one dimension by introducing long-range interactions). We also verify that pairs of spinons are deconfined in uniform spin chains but become confined upon introducing a pattern of alternating coupling strengths (dimerization) or coupling two chains (forming a ladder). In the dimerized system, an individual spinon can be small when the confinement length is large; this is the case when the imposed dimerization is weak but the ground state of the corresponding uniform chain is a spontaneously formed valence-bond solid (where the spinons are deconfined). Based on our numerical results, we argue that a system with $\ensuremath{\lambda}\ensuremath{\ll}\mathrm{\ensuremath{\Lambda}}$ is associated with weak repulsive short-range spinon-spinon interactions. In principle, both the length scales $\ensuremath{\lambda}$ and $\mathrm{\ensuremath{\Lambda}}$ can still be individually tuned from small to infinite (with $\ensuremath{\lambda}\ensuremath{\le}\mathrm{\ensuremath{\Lambda}}$) by varying model parameters. In contrast, in the ladder system the two lengths are always similar, and this is the case also in the weakly dimerized systems when the corresponding uniform chain is in the critical phase. In these systems, the effective spinon-spinon interactions are purely attractive and there is only a single large length scale close to criticality, which is reflected in the standard spin correlations as well as in the spinon characteristics.
The existence of bound states of the system ($\ensuremath{\alpha}$, eee, eee, eee) in a magnetic field BBB is studied using the variational method. It is shown that for $B\ensuremath{\gtrsim}0.13\text{ }\text{ }\mathrm{a}.\mathrm{u}.$ this system gets bound with total energy below the one of the ($\ensuremath{\alpha}$, eee, eee) system. It manifests the existence of the stable ${\mathrm{He}}^{\ensuremath{-}}$ atomic ion. Its ground state is a spin doublet $^{2}(\ensuremath{-}1{)}^{+}$ at $0.74\text{ }\text{ }\mathrm{a}.\mathrm{u}.\ensuremath{\gtrsim}B\ensuremath{\gtrsim}0.13\text{ }\text{ }\mathrm{a}.\mathrm{u}.$ and it becomes a spin quartet $^{4}(\ensuremath{-}3{)}^{+}$ for larger magnetic fields. For $0.8\text{ }\text{ }\mathrm{a}.\mathrm{u}.\ensuremath{\gtrsim}B\ensuremath{\gtrsim}0.7\text{ }\text{ }\mathrm{a}.\mathrm{u}.$ the ${\mathrm{He}}^{\ensuremath{-}}$ ion has two (stable) bound states $^{2}(\ensuremath{-}1{)}^{+}$ and $^{4}(\ensuremath{-}3{)}^{+}$.