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 $a$. For an elastic, three-dimensional two-component gas, matching the vacuum two-body amplitude removes the cutoff divergence and predicts a shallow dimer when $a$ is positive and much larger than the interaction range. This page derives the residual cutoff dependence as well as the zero-range limit.

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 fermion components of equal constituent mass $m$, with a real contact coupling and no open inelastic channel. In vacuum their action is

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

The relative kinetic energy is $E=k^2/m$, because the reduced mass is $m/2$. Restrict relative loop momenta by a sharp spherical cutoff $p<\Lambda$. Define the elastic S-wave amplitude by the large-distance wave $e^{ikz}+f(k)e^{ikr}/r$, and set

$$f(k)=\frac{1}{k\cot\delta(k)-ik}, \qquad \mathcal A(k)=\frac{4\pi}{m}f(k).$$

Here $f(0)=-a$. The low-energy effective-range expansion and this scattering-length sign are those of Braaten and Hammer 2006, § 2.1, pp. 11–13, PDF. At leading zero-range order,

$$k\cot\delta(k)=-a^{-1}, \qquad \mathcal A_{\mathrm{LO}}(k) =\frac{4\pi/m}{-a^{-1}-ik}.$$

This is a definition of the leading approximation. Finite-range corrections belong in $k\cot\delta=-a^{-1}+r_e k^2/2+\cdots$ before expanding the amplitude; a dimensionless additive remainder cannot be added to $\mathcal A$, whose mass dimension is $-2$.

Bubble resummation

For $0<k<\Lambda$, angular integration reduces the center-of-mass loop to

$$I_0(E,\Lambda) =m\int_{p<\Lambda}\!\frac{\mathrm d^3p}{(2\pi)^3} \frac{1}{k^2-p^2+i0^+} =\frac{m}{2\pi^2}\int_0^\Lambda \frac{p^2\,\mathrm dp}{k^2-p^2+i0^+}.$$

Use $1/(x+i0)=\operatorname{PV}(1/x)-i\pi\delta(x)$. The radial delta function gives $-imk/(4\pi)$, while $p^2/(k^2-p^2)=-1+k^2/(k^2-p^2)$ gives the real principal value:

$$\begin{aligned} I_0(E,\Lambda) &=-\frac{m\Lambda}{2\pi^2} +\frac{mk}{4\pi^2}\log\frac{\Lambda+k}{\Lambda-k} -\frac{imk}{4\pi},\\ &=-\frac{m\Lambda}{2\pi^2}-\frac{imk}{4\pi} +\frac{mk^2}{2\pi^2\Lambda} +O\!\left(\frac{mk^4}{\Lambda^3}\right), \qquad \frac{k}{\Lambda}\to0. \end{aligned}$$

For the Lippmann–Schwinger convention $T_{\mathrm{LS}}=V+VG_0^+T_{\mathrm{LS}}$, the regulated contact potential gives

$$T_{\mathrm{LS}}=\frac{1}{C_0^{-1}-I_0}, \qquad f=-\frac{m}{4\pi}T_{\mathrm{LS}}, \qquad \mathcal A=-T_{\mathrm{LS}}.$$

The sign follows already at Born order from the outgoing free resolvent. Requiring $f(0)=-a$ fixes

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

Substitution cancels the linear divergence and yields the exact finite-cutoff result for this contact model:

$$\mathcal A_\Lambda(k) =\frac{4\pi/m}{ -a^{-1}-ik+\dfrac{k}{\pi}\log\dfrac{\Lambda+k}{\Lambda-k}}.$$

Thus $\mathcal A_\Lambda\to\mathcal A_{\mathrm{LO}}$ at fixed positive $k$ as $\Lambda\to\infty$; matching one constant does not remove every finite-$\Lambda$ effect. The residual real denominator is $2k^2/(\pi\Lambda)+O(k^4/\Lambda^3)$, corresponding to an artificial effective range $r_e=4/(\pi\Lambda)$. Other regulators change this residual and the bare-coupling relation, while reproducing the same matched zero-range limit. The imaginary part obeys $\operatorname{Im} f^{-1}=-k$, so $|1+2ikf|=1$: elastic unitarity holds even before removing the cutoff.

To expand an amplitude in a denominator correction $\Delta D$, require $|\Delta D|\ll|-a^{-1}-ik|$. A small coefficient alone is insufficient near a pole after complex continuation. Physical range corrections, beginning with $r_e k^2/2$, obey the same test and are distinct from regulator artifacts.

In the contact-scattering course calculation, $m$ denotes the mass in a one-particle kinetic operator $p^2/(2m)$. For this equal-mass two-body problem, replace that kinetic mass by the reduced mass $m/2$, then use $\mathcal A=-T_{\mathrm{LS}}$.

Shallow pole and the unitary limit

In the zero-range 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 $a>0$ in the universal regime $a\gg R$, where $R$ denotes the interaction range. For $a<0$ the zero-range pole is on the virtual-state sheet rather than a normalizable bound state. Range corrections shift both the pole and residue. The positive binding energy in Braaten and Hammer 2006, § 4.2, p. 48, PDF is $-E_B$ in the energy convention used here.

At finite cutoff the negative-energy radial integral instead gives

$$\begin{aligned} I_0(-\kappa^2/m,\Lambda) &=-\frac{m}{2\pi^2} \left[\Lambda-\kappa\arctan\frac{\Lambda}{\kappa}\right],\\ \kappa\arctan\frac{\Lambda}{\kappa}&=\frac{\pi}{2a} \quad\text{at a bound-state pole}. \end{aligned}$$

The last equation tends to $\kappa=1/a$ when $\kappa/\Lambda\to0$. It must be used if finite-cutoff pole positions are compared; inserting the zero-range value as an exact finite-cutoff pole would miss the same range artifact seen in scattering.

At unitarity, $a^{-1}=0$, the zero-range two-body amplitude has no scattering-length scale. Density, temperature, effective range, three-body data in the sectors that require them, confinement, and external fields can still supply scales to the many-body problem.

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 derivative of the loop’s linear divergence. At nonzero $k$, the logarithmic term still has cutoff dependence; it vanishes in the zero-range limit. Keeping $C_0$ fixed would retain even the linear divergence.

Find the shallow dimer

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

Solution

Set $k=i\kappa$ with $\kappa>0$. The leading denominator becomes $-a^{-1}+\kappa$, so $\kappa=a^{-1}$. A positive $\kappa$ requires $a>0$, and $E=-\kappa^2/m=-1/(ma^2)$. The decaying relative wave is proportional to $e^{-\kappa r}/r$, which is normalizable. The universal interpretation also requires $a\gg R$; a finite-cutoff calculation must instead solve the arctangent equation above.

Continue

Effective Range, Shallow Poles, and Universality Windows adds the first derivative correction. Two-Channel Resonance Models represents an energy-dependent resonance explicitly. Universal Relations and Tan Contact turns the matched short-distance interaction into exact many-body identities.

References

• Braaten, Eric, and Hans-Werner Hammer. “Universality in Few-Body Systems with Large Scattering Length.” Physics Reports 428 (2006): 259–390. DOI. Open PDF, version 3, 18 August 2006; locators above use its printed preprint pages.

Further reading

• Kaplan, David B., Martin J. Savage, and Mark B. Wise. “A New Expansion for Nucleon–Nucleon Interactions.” Physics Letters B 424 (1998): 390–396. DOI.
• Lee, T. D., Kerson Huang, and C. N. Yang. “Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties.” Physical Review 106 (1957): 1135–1145. DOI.