Nonrelativistic Power Counting and Universality

Nonrelativistic power counting ranks operators only after a scaling regime has been named. About the vacuum fixed point, time scales as momentum squared and a contact interaction has dimension $2-d$ . At finite density, modes are instead organized by distance from a Fermi surface or by collective hydrodynamic scales. Universality means insensitivity to omitted short-distance data within a quantified window, not independence from every physical parameter.

Required background. Galilean Fields, Scales, and Low-Energy Degrees of Freedom supplies $z=2$ kinematics. Power Counting and Predictive Order and Universality Classes and Scaling Functions supply the general expansion and fixed-point concepts.

Helpful background. Momentum-Shell Integration and Rescaling derives the Wilsonian flow behind the dimensional analysis.

Vacuum scaling with dynamical exponent two

Use momentum dimension $[\mathbf x]=-1$ , $[t]=-2$ , and require the action to be dimensionless. From

S_0=\int\mathrm dt\,\mathrm d^d x\, \psi^\dagger\left(i\partial_t+\frac{\nabla^2}{2m}\right)\psi,

one obtains

[\psi]=\frac d2, \qquad [m]=0.

For the normal-ordered contact operator $\mathcal O_0=\psi^\dagger\psi^\dagger\psi\psi$ ,

[C_0]=d+2-2d=2-d.

It is relevant in $d=1$ , classically marginal in $d=2$ , and irrelevant by Gaussian counting in $d=3$ . This classification is about the free vacuum fixed point. A large scattering length in three dimensions tunes the theory near a non-Gaussian fixed point and promotes repeated $C_0$ insertions to leading order.

A two-derivative four-field operator has $[C_2]=-d$ . Relative to $C_0$ , it is suppressed by two powers of momentum if its coefficient is natural. More generally,

\mathcal L=\mathcal L_{\rm LO} +\sum_{n\ge1}C_{2n}\mathcal O_{2n}, \qquad \frac{\Delta\mathcal A_{2n}}{\mathcal A_{\rm LO}} \sim \left(\frac Q{\Lambda_b}\right)^{2n},

unless a shallow pole, symmetry, or fine tuning changes the assignment.

Natural and promoted operators

Naive dimensional analysis assumes dimensionless coefficients at the breakdown scale are order one. It is a prior, not a theorem. If $|a|\gg R$ , where $a$ is a scattering length and $R\sim\Lambda_b^{-1}$ is the range, then the ratio $a/R$ is unnatural and the contact interaction must be resummed.

For two particles of mass $m$ in three dimensions, the leading amplitude can be written

\mathcal A_0(k)=\frac{4\pi/m}{-a^{-1}-ik}.

Every term $(ika)^n$ is equally important when $ka\sim1$ , even though $kR\ll1$ . Resumming the promoted interaction preserves an expansion in $kR$ around the large- $a$ fixed point Kaplan, Savage, and Wise 1998, §§ II–III.

The correct lesson is not that power counting failed. The expansion point was changed, and the fine-tuned parameter was treated nonperturbatively.

Vacuum and finite-density counting are different

At a Fermi surface, write $\mathbf p=\mathbf k_F+\boldsymbol\ell$ with $\ell_\perp$ normal to the surface. Low energy scales $\omega$ and $\ell_\perp$ , while tangential momentum labels a patch and is not scaled in the same way. Four-fermion interactions are then classified by kinematic channels; forward scattering and the Cooper channel can remain marginal even though a local four-fermion operator was irrelevant by vacuum $d=3$ counting Polchinski 1992, §§ 3–4.

For a Bose condensate, the infrared fields are phase and density fluctuations, and their derivative expansion differs from vacuum particle counting. At nonzero temperature, Matsubara zero modes can lead to a classical long-distance fixed point. Therefore the statement “this operator is irrelevant” is incomplete without the state and scaling transformation.

A universality window

Suppose a zero-range prediction $X_0$ is corrected by effective range $r_e$ , a three-body scale $\kappa_*$ , and thermal or density momenta. A useful error model is

X=X_0\left[ 1+c_rQr_e+c_2(QR)^2+\cdots \right],

with additional dependence on $Q/\kappa_*$ when the observable is three-body sensitive. The universality window requires all omitted ratios to be small and stability under regulator variation after refitting the retained data.

One should report:

the fixed point and scaling variables;
the promoted interactions and why they are promoted;
the physical breakdown scale, distinct from the regulator;
the expected first omitted order; and
an observable or cutoff-variation test of that order.

A result can be universal while depending on a small set of relevant parameters such as $a$ , density, temperature, or $\kappa_*$ . Universality means no additional short-distance detail enters at the claimed accuracy.

A controlled example

For a dilute natural three-dimensional Bose gas, $Q\sim n^{1/3}$ and $a\sim R$ . The dimensionless gas parameter is $na^3$ . Mean-field energy density scales as

\mathcal E_{\rm MF}\sim\frac{a}{m}n^2,

and the leading quantum correction is relatively $O(\sqrt{na^3})$ . If $na^3\ll1$ , the hierarchy is useful. Taking $a\to\infty$ at fixed $n$ destroys this expansion even though the interaction range may remain short; one must reorganize around the unitary regime and include three-body and loss scales for bosons.

Common pitfalls

Quoting canonical dimensions without the fixed point. Vacuum, Fermi-surface, condensate, and thermal scaling classify the same operator differently.

Calling an unnatural coefficient inconsistent. A large scattering length is evidence of tuning and operator promotion. The error lies in keeping the tuned term perturbative.

Equating regulator independence with universality. Renormalization removes regulator artifacts. Universality additionally requires that omitted physical scales remain suppressed.

Exercises

Classify an $N$ -body contact operator

At the free vacuum fixed point, find the dimension of the coupling multiplying $(\psi^\dagger\psi)^N$ .

Solution

The operator has dimension $Nd$ , while the Lagrangian density has dimension $d+2$ . Therefore

[C_N]=d+2-Nd=2-(N-1)d.

This is only Gaussian vacuum counting; anomalous dimensions or a different fixed point can change it.

Build an error estimate

A two-body observable is calculated at momenta $k$ with $ka=O(1)$ , $kr_e=0.08$ , and the next shape correction estimated as $(kR)^2=0.01$ . Which effect is resummed and what is the nominal truncation error?

Solution

Because $ka$ is not small, scattering-length dependence must be resummed. If the effective-range term is included explicitly, the next stated correction is about one percent. If it is omitted, the nominal error is instead eight percent. Coefficients and additional scales can enlarge either estimate and should be tested by variation.

Continue

Short-Range Scattering Data as Many-Body Inputs implements the promoted contact interaction. Effective Range, Shallow Poles, and Universality Windows identifies the first range correction. The Fermi Gas and Fermi-Surface Kinematics supplies the finite-density scaling geometry.

References

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. Open PDF.
Polchinski, Joseph. “Effective Field Theory and the Fermi Surface.” In Recent Directions in Particle Theory, 235–274. Singapore: World Scientific, 1992. Abstract. Open PDF.