Short-Range Scattering Data as Many-Body Inputs

A zero-range coupling is not an observable: it changes with the regulator. The physical many-body input is the on-shell scattering amplitude, parameterized at leading order by the scattering length $a$ . Matching the two-body $T$ matrix replaces the bare coupling by $a$ and simultaneously fixes the shallow bound-state pole.

Required background. Coherent-State Path Integrals supplies the nonrelativistic action, Tree-Level Matching and Classical Elimination supplies matching logic, and Power Counting and Predictive Order supplies the truncation criterion.

Helpful background. Spectra, Resolvents, and Functional Calculus clarifies the pole–bound-state correspondence.

Regulated contact theory

Consider two distinguishable fermions of equal mass $m$ in three spatial dimensions,

\mathcal L =\sum_{\sigma=\uparrow,\downarrow} \psi_\sigma^\dagger \left(i\partial_t+\frac{\nabla^2}{2m}\right)\psi_\sigma -C_0(\Lambda) \psi_\uparrow^\dagger\psi_\downarrow^\dagger \psi_\downarrow\psi_\uparrow.

Loop momenta are restricted by a sharp spherical cutoff $p<\Lambda$ . With the amplitude convention

\mathcal A(k)=\frac{4\pi/m}{-a^{-1}-ik}+O(k^2R^2),

the bubble sum is cutoff independent at leading order when

\frac{1}{C_0(\Lambda)} =\frac{m}{4\pi a}-\frac{m\Lambda}{2\pi^2}.

The numerical coefficients depend on the regulator, while $\mathcal A(k)$ does not. Dimensional regularization with power-divergence subtraction or a lattice cutoff gives a different relation for $C_0$ and the same matched low-energy amplitude.

Bubble resummation

The center-of-mass two-particle loop at energy $E=k^2/m$ is

I_0(E,\Lambda) =m\int_{p<\Lambda}\!\frac{\mathrm d^3p}{(2\pi)^3} \frac{1}{k^2-p^2+i0^+} =-\frac{m\Lambda}{2\pi^2}-\frac{imk}{4\pi} +O(k^2/\Lambda).

Summing repeated contact interactions gives an inverse amplitude proportional to $C_0^{-1}-I_0$ . Substituting the matching relation cancels the linear divergence and yields the stated $-a^{-1}-ik$ denominator, up to the overall scattering-amplitude convention. This is the central renormalization check: predictions depend on $a$ , not on $C_0$ or $\Lambda$ .

The imaginary term is fixed by two-body unitarity. In the normalization $f(k)=1/(k\cot\delta-ik)$ and $\mathcal A=(4\pi/m)f$ , leading order has $k\cot\delta=-1/a$ .

Shallow pole and the unitary limit

Continue $k$ to $i\kappa$ with $\kappa>0$ . The denominator vanishes for

\kappa=\frac1a, \qquad E_B=-\frac{1}{ma^2},

so a shallow dimer exists for $a>0$ . For $a<0$ the pole is on the virtual-state sheet rather than a normalizable bound state. The result is universal only when $|a|\gg R$ , where $R$ denotes the interaction range; range corrections shift both the pole and residue.

At unitarity, $a^{-1}=0$ , the two-body amplitude has no scattering-length scale. Density, temperature, effective range, three-body data, confinement, and external fields can still supply scales to the many-body problem. The universal zero-range discussion and its domain are reviewed in Braaten and Hammer 2006, §§2–3.

Many-body use and limits

Vacuum matching must be completed before inserting medium occupation factors. At leading order a dilute-gas calculation is organized in $k_Fa$ when $|k_Fa|\ll1$ , or around unitarity with range corrections controlled by $k_FR$ . Deep molecular states and inelastic channels have been integrated out; if they are accessible, their effects require additional operators, complex couplings, or explicit fields.

A cutoff plateau is necessary but not sufficient. It must occur with $k\ll\Lambda$ and $\Lambda$ below scales where the zero-range model’s omitted physics becomes relevant, or within a renormalization scheme whose counterterms cover that range.

Common pitfalls

Treating $C_0$ as measured. Its value changes with regulator and cutoff. Quote $a$ and the matching prescription.

Using the wrong reduced mass. For two equal masses the relative energy is $k^2/m$ because the reduced mass is $m/2$ .

Calling every pole a bound state. The sheet and sign of $a$ distinguish a physical dimer from a virtual-state pole.

Exercises

Verify cutoff running

Differentiate the matching relation with respect to $\Lambda$ at fixed $a$ .

Solution

$\mathrm d(C_0^{-1})/\mathrm d\Lambda=-m/(2\pi^2)$ , so $\mathrm dC_0/\mathrm d\Lambda=(m/2\pi^2)C_0^2$ . This running cancels the corresponding cutoff derivative of the loop. Keeping $C_0$ fixed would make the amplitude regulator dependent.

Find the shallow dimer

Continue the leading amplitude to negative energy and locate its normalizable pole.