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arXiv:nucl-th/0605074

Dimer scattering in the epsilon expansion

Gautam Rupak

Published May 29, 2006• 1 citation
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Abstract

The atom-dimer and dimer-dimer scattering lengths are analytically calculated in an expansion around four spatial dimensions for fermions with a large 2-body scattering length 'a'. We find the atom-dimer scattering length a_ad/a = 4/3-2/9 epsilon and the dimer-dimer scattering length a_dd/a = epsilon -0.344 epsilon^2, where epsilon = 4-d and d is the number of spatial dimensions. These ratios a_ad/a = 1.11 and a_dd/a = 0.656 at epsilon = 1 are to be compared with the non-perturbatively calculated numerical results 1.18 and 0.6 respectively. The neutron-deuteron scattering length in the quartet channel a_nD = 4.78 fm is in reasonable agreement with the experimental value 6.35 fm, considering 2-body effective range corrections r_0/a = 0.4 were excluded. The deuteron-deuteron scattering length a_DD = 3.15 fm in the spin-2 channel.

DOI
10.48550/arxiv.nucl-th/0605074

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The atom-dimer and dimer-dimer scattering lengths are analytically calculated in an expansion around four spatial dimensions for fermions with a large 2-body scattering length 'a'. We find the atom-dimer scattering length a_ad/a = 4/3-2/9 epsilon and the dimer-dimer scattering length a_dd/a = epsilon -0.344 epsilon^2, where epsilon = 4-d and d is the number of spatial dimensions. These ratios a_ad/a = 1.11 and a_dd/a = 0.656 at epsilon = 1 are to be compared with the non-perturbatively calculated numerical results 1.18 and 0.6 respectively. The neutron-deuteron scattering length in the quartet channel a_nD = 4.78 fm is in reasonable agreement with the experimental value 6.35 fm, considering 2-body effective range corrections r_0/a = 0.4 were excluded. The deuteron-deuteron scattering length a_DD = 3.15 fm in the spin-2 channel.
We investigate universal behavior in elastic atom-dimer scattering below the dimer breakup threshold calculating the atom-dimer effective-range function $ak\mathrm{cot}\ensuremath{\delta}$. Using the He-He system as a reference, we solve the Schr\"odinger equation for a family of potentials having different values of the two-body scattering length aaa and we compare our results to the universal zero-range form deduced by Efimov, $ak\mathrm{cot}\ensuremath{\delta}={c}_{1}(ka)+{c}_{2}(ka)\mathrm{cot}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\phi}(ka)]$, for selected values of the three-body parameter ${\ensuremath{\kappa}}_{*}$. Using the parametrization of the universal functions ${c}_{1},{c}_{2},\ensuremath{\phi}$ given in the literature, a good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections. Furthermore, we show that the same parametrization describes a very different system: nucleon-deuteron scattering below the deuteron breakup threshold. Our analysis confirms the universal character of the process, and relates the pole energy in the effective-range function of nucleon-deuteron scattering to the three-body parameter ${\ensuremath{\kappa}}_{*}$.
We present continuum and lattice calculations for elastic scattering between a fermion and a bound dimer in the shallow binding limit. For the lattice calculation we use the finite-volume method of L\"uscher. We take into account topological finite-volume corrections to the dimer binding energy which depend on the momentum of the dimer. After subtracting these effects, we find from the lattice calculation $\ensuremath{\kappa}{a}_{fd}=1.174(9)$ and $\ensuremath{\kappa}{r}_{fd}=\ensuremath{-}0.029(13)$. These results agree well with the continuum values $\ensuremath{\kappa}{a}_{fd}=1.17907(1)$ and $\ensuremath{\kappa}{r}_{fd}=\ensuremath{-}0.0383(3)$ obtained from the STM equation. We discuss applications to cold atomic Fermi gases, deuteron-neutron scattering in the spin-quartet channel, and lattice calculations of scattering for nuclei and hadronic molecules at finite volume.
Collisions of two fermionic dimers near the unitary limit are studied using exact four-particle equations for transition operators in momentum space. Universal properties of dimer-dimer phase shifts and effective range expansion (ERE) parameters are determined. The inclusion of the fourth-order momentum term in the ERE significantly extends its validity to higher collision energies. The dimer-dimer scattering length and effective range are determined in the unitary limit as well as their corrections arising due to the finite range of the two-fermion interaction. These results are of considerably higher accuracy as compared to previous works, but confirm most of the previous results except for the lattice effective field theory calculations.
We present a diagrammatic approach for the dimer-dimer scattering problem in two or three spatial dimensions, within the resonance approximation where these dimers are in a weakly bound resonant state. This approach is first applied to the calculation of the dimer-dimer scattering length aB{a}_{B}aB​ in three spatial dimensions, for dimers made of two fermions in a spin-singlet state, with corresponding scattering length aF{a}_{F}aF​, and the already known result aB=0.60{a}_{B}=0.60aB​=0.60 aF{a}_{F}aF​ is recovered exactly. Then we make use of our approach to obtain results in two spatial dimensions for fermions as well as for bosons. Specifically, we calculate bound-state energies for three bbbbbbbbb and four bbbbbbbbbbbb resonantly interacting bosons in two dimensions. We consider also the case of a resonant interaction between fermions and bosons, and we obtain the exact bound-state energies of two bosons plus one fermion bbfbbfbbf, two bosons plus two fermions $b{f}_{\ensuremath{\uparrow}}b{f}_{\ensuremath{\downarrow}}$, and three bosons plus one fermion bbbfbbbfbbbf.
We consider two-component fermions with short-range interactions and large scattering length. This system has universal properties that are realized in several different fields of physics. In the limit of large fermion–fermion scattering length aff and zero-range interaction, all properties of the system scale proportionally with aff. For the case with shallow bound dimers, we calculate the dimer–dimer scattering phase shifts using lattice effective field theory. We extract the universal dimer–dimer scattering length add/aff=0.618(30) and effective range rdd/aff=−0.431(48). This result for the effective range is the first calculation with quantified and controlled systematic errors. We also benchmark our methods by computing the fermion–dimer scattering parameters and testing some predictions of conformal scaling of irrelevant operators near the unitarity limit.
We derive the scattering length of composite bosons (cobosons) within the framework of the composite-boson many-body formalism that uses correlated-pair states as a basis instead of free-fermion states. The integral equation constructed from this physically relevant basis makes transparent the role of fermion exchange in the coboson-coboson effective scattering. Three potentials used for Cooper pairs, fermionic-atom dimers, and semiconductor excitons are considered. While the sss-wave scattering length for the BCS-like potential is just equal to its Born value, the other two are substantially smaller. For fermionic-atom dimers and semiconductor excitons, our results, calculated within a restricted correlated-pair basis, are in good agreement with those obtained from procedures numerically more demanding. We also propose model coboson-coboson scatterings that are separable and thus easily workable and that produce scattering lengths which match quantitatively well with the numerically obtained values for all fermion mass ratios. These separable model scatterings can facilitate future works on many-body effects in coboson gases.