Nonperturbative Iteration, Shallow Scales, and Power-Counting Consistency

A shallow pole introduces a low scale that can make every repetition of a nominally short-range interaction equally important. The correct response is not to iterate the whole EFT indiscriminately: promote the interaction responsible for the pole, resum that leading subset, insert higher-order operators perturbatively, and demand regulator independence at each claimed order. This page derives the leading contact amplitude and its running in pionless two-body EFT.

Required background. Power Counting and Predictive Order supplies the ordinary order lattice. Loops, Counterterms, and Closure of an EFT Expansion supplies local subtraction and the retained-order regulator test. Helpful background. Partial-Wave Unitarity relates the resummed $S$-wave amplitude to phase shifts and pole locations.

A shallow scale defeats naive contact perturbation

For two nonrelativistic particles of equal mass $m$ interacting over a range $R\sim1/\Lambda_b$, the $S$-wave effective-range expansion is

$$k\cot\delta_0(k) = -\frac1a +\frac12r_0k^2 +v_2k^4 +\cdots.$$

Natural short-range dynamics has $a,r_0\sim1/\Lambda_b$. If instead

$$|a|\gg\frac1{\Lambda_b}, \qquad \gamma\equiv\frac1{|a|}\ll\Lambda_b,$$

then $\gamma$ is a new low scale. For momenta $k\sim\gamma$, the product $a k$ is order one. Expanding the amplitude in powers of $a k$ is impossible even though $k/\Lambda_b$ remains small.

Partial-wave unitarity fixes the amplitude up to normalization. With the convention used here,

$$\mathcal A_0(k) = \frac{4\pi}{m} \frac1{k\cot\delta_0(k)-ik}.$$

At leading order in the range expansion,

$$\mathcal A_{\mathrm{LO}}(k) = \frac{4\pi}{m} \frac1{-1/a-ik}.$$

For $a>0$, this has a bound-state pole at

$$k=i\gamma, \qquad \gamma=\frac1a, \qquad E_B=-\frac{\gamma^2}{m}.$$

The pole is shallow because $\gamma\ll\Lambda_b$. Its nonanalytic denominator must be generated at leading order; no finite polynomial in $k$ can reproduce it over $k\sim\gamma$.

Contact bubbles become leading

Use a short-range nonrelativistic EFT with a momentum-independent $S$-wave contact interaction $C_0$. A two-particle bubble scales as

$$I_0(k) \sim \frac{mk}{4\pi}.$$

With natural scattering length, $C_0\sim4\pi/(m\Lambda_b)$ and

$$C_0I_0\sim\frac{k}{\Lambda_b}\ll1,$$

so bubbles are perturbative. A shallow pole requires instead

$$C_0\sim\frac{4\pi}{mQ}, \qquad Q\sim k\sim\gamma,$$

which gives $C_0I_0\sim1$. The contact interaction has been promoted by $\Lambda_b/Q$ relative to its natural estimate. Suppressing convention-dependent overall signs, every bubble in

$$T = C_0 +C_0I_0C_0 +\cdots$$

is therefore the same order. Summing the geometric series gives

$$T_{\mathrm{LO}} = \frac1{C_0^{-1}-I_0},$$

and the Feynman-rule convention below supplies the corresponding overall sign. Renormalizing $C_0$ to the scattering length turns the physical amplitude into the universal form $4\pi/[m(-1/a-ik)]$.

Kaplan, Savage, and Wise introduced a subtraction and RG organization in which this promotion is manifest and all $C_0$ bubbles are leading, while derivative interactions are inserted perturbatively Kaplan, Savage, and Wise 1998, Eqs. (4)–(13), pp. 392–394.

Power-divergence subtraction makes the counting explicit

In dimensional regularization with power-divergence subtraction (PDS), the renormalized bubble is

$$I_0^{\mathrm{PDS}}(k,\mu) = -\frac{m}{4\pi}(\mu+ik).$$

The summed amplitude is

$$\mathcal A_{\mathrm{LO}}(k,\mu) = \frac{-C_0(\mu)} {1+\dfrac{mC_0(\mu)}{4\pi}(\mu+ik)}.$$

Choose

$$C_0(\mu) = \frac{4\pi}{m} \frac1{1/a-\mu}.$$

Substitution gives

$$\mathcal A_{\mathrm{LO}}(k) = \frac{4\pi}{m} \frac1{-1/a-ik},$$

with no $\mu$ dependence. Differentiating $C_0$ yields the exact leading-sector beta function

$$\mu\frac{dC_0}{d\mu} = \frac{m\mu}{4\pi}C_0^2.$$

Thus the counterterm that renormalizes the leading contact loop must itself be leading. Treating it as a higher-order correction would leave the resummed amplitude regulator dependent at leading order.

For $\mu\sim Q$ and $1/a\sim Q$,

$$C_0(\mu)\sim\frac{4\pi}{mQ}.$$

The subtraction scale exposes the intended counting but is not physical. Other regulators are allowed if their running coefficients reproduce the same amplitude and residual errors.

Range corrections remain perturbative

The next contact interaction carries two derivatives, with coefficient $C_2$. Matching the effective range gives in the PDS convention

$$C_2(\mu) = \frac{2\pi}{m} \frac{r_0}{(1/a-\mu)^2}.$$

For $r_0\sim1/\Lambda_b$ and $\mu\sim Q$,

$$C_2k^2 \sim \frac{4\pi}{mQ} \frac{Q}{\Lambda_b}.$$

A single $C_2k^2$ insertion dressed on both sides by the leading bubble sum is suppressed by $Q/\Lambda_b$. Expanding the effective-range amplitude gives

$$\mathcal A_0(k) = \frac{4\pi}{m} \frac1{-1/a-ik} \left[ 1- \frac{r_0k^2/2}{-1/a-ik} +O\!\left(\frac{Q^2}{\Lambda_b^2}\right) \right].$$

The leading denominator is kept exact because $1/a$ and $k$ are both $O(Q)$. The range correction is expanded because $r_0k\sim Q/\Lambda_b$. Iterating $C_2$ without a separate counting would generate selected terms of all orders and new ultraviolet sensitivity not represented by a nominal next-to-leading-order counterterm set.

Epelbaum, Hammer, and Meißner summarize pionless EFT as an expansion around the large-scattering-length limit, with the leading contact resummed and effective-range effects perturbative, in Epelbaum, Hammer, and Meißner 2009, §§ I.D and II.A.

The promoted order lattice

The right panel of the shared diagram now has a precise meaning: $I$ is the two-particle bubble, $C_0I\sim1$, and the infinite $C_0$ chain defines the leading amplitude. The derivative operator appears once at its assigned correction order; the dashed box is the first omitted structure.

Predictive order requires closure. Panel (a) shows generic orders $q^\nu,q^{\nu+\Delta},\ldots$; each retained column must include every tree or insertion, loop, and local counterterm assigned to it, while the dashed column is the first omitted order. Panel (b) shows the distinct case $C_0I\sim1$, where a shallow scale promotes the entire $C_0$ iteration to leading order and higher-derivative structures remain perturbative. The diagram is schematic: $\Delta$ and the relative order of $C_2$ are theory dependent.

The promotion is regime specific. It applies to a short-range channel with $|a|\Lambda_b\gg1$ and momenta below the range scale. It does not establish a universal rule for pion exchange, singular tensor forces, relativistic bound states, many-body resummations, or strong coupling in other partial waves.

Cutoff independence as an order-by-order test

With a sharp momentum cutoff $\Lambda_{\mathrm{reg}}$, the leading bubble contains

$$I_0(k;\Lambda_{\mathrm{reg}}) = -\frac{m\Lambda_{\mathrm{reg}}}{2\pi^2} -i\frac{mk}{4\pi} +O\!\left(\frac{mk^2}{\Lambda_{\mathrm{reg}}}\right).$$

The linear term must be canceled by the cutoff dependence of $C_0(\Lambda_{\mathrm{reg}})$ at leading order. After fitting $a$, the remaining $k^2/\Lambda_{\mathrm{reg}}$ dependence has the form of an effective-range correction and is removed or demoted consistently when $C_2$ enters.

A useful cutoff test is:

1. choose several regulator values above the low momenta but within a range where the EFT implementation is meaningful;
2. refit every coefficient present at the tested order for each regulator;
3. compare predictions for withheld energies or observables; and
4. verify that the spread scales with the first omitted power of $Q/\Lambda_b$.

It is not enough that one fitted datum is cutoff independent; the running coefficient guarantees that by construction. Nor should one take $\Lambda_{\mathrm{reg}}\to\infty$ blindly when an iterated truncated kernel samples momenta far beyond the EFT domain. Epelbaum, Hammer, and Meißner explain that iterating a truncated potential generates ultraviolet divergences in its Neumann series and can require counterterms beyond the truncation; regulator choice and promoted counterterms must therefore be tied to a consistent counting Epelbaum, Hammer, and Meißner 2009, § II.C.3.

What may be resummed

A justified nonperturbative subset satisfies all of the following:

• a diagnosed small denominator or infrared enhancement makes every repetition the same order;
• the resummation preserves the symmetries and analytic structure required in the domain;
• all counterterms needed by the resummed ultraviolet behavior are included at the promoted order;
• subleading interactions have a declared perturbative insertion rule;
• regulator variation leaves observables stable through the retained order; and
• spurious deep poles or cutoff-scale states remain outside the claimed domain and do not contaminate low-energy predictions.

Selective iteration merely because it improves a fit is not a power counting. It can mix incomplete higher orders into the result, hide missing counterterms, and make the apparent error smaller without improving predictivity.

The three-body sector supplies an important warning. For identical particles with large scattering length, repeated two-body interactions can generate ultraviolet dependence that no two-body coefficient absorbs. A three-body counterterm is then promoted, with limit-cycle running in the Efimov regime. That is a new closure result for a new sector, not evidence that every many-body force is leading.

A common uncertainty and validation checklist

Resummation changes the leading reference amplitude but does not collapse the error budget.

Component	Record explicitly	Diagnostic or failure trigger
Domain and expansion parameters	Observable, kinematic window, $q_i(Q)$, hard scales, thresholds, and correlations among small parameters	A threshold enters, some $q_i\not\ll1$, or the assumed relation among parameters fails
Retained order and inventory	Highest order $k$, every tree, loop, insertion, counterterm, and parameter correction included	An omitted contribution has the same assigned order as a retained one
Coefficient assumptions	Operator normalization, scheme and scale, expected coefficient sizes, symmetry suppressions, and any priors	Coefficients drift with fit window or require unexplained enhancement
EFT truncation	First omitted powers, reference size, correlation model across energies and observables, and interval interpretation	Residuals do not scale with the predicted powers or coverage fails on withheld data
Input and fit uncertainty	Experimental or synthetic inputs, covariance, fitted combinations, and propagation method	Results are unstable under admissible input or fit-window changes
Numerical uncertainty	Solver, discretization, integration, rounding, convergence tolerance, and reproducibility data	Numerical changes are not parametrically below the claimed EFT error
Matching and running	Matching order and scale, anomalous dimensions, threshold sequence, and residual $\mu$ dependence	Scale cancellation fails through the retained order or a threshold is double counted
Regulator, basis, and scheme checks	Regulator range, required counterterms, field/basis map, and scheme transformation	Predictions depend on an auxiliary choice at or below the claimed order
Model discrepancy and breakdown	Effects not represented by the EFT, validation observables, stopping rule, and alternative field content	Persistent structured residuals, new nonanalyticity, or failure across observables

For the shallow contact EFT, the record must add the fitted scattering length, effective range when included, pole convention, regulator window, and the list of iterated versus perturbative operators. Numerical solution error must be smaller than the first omitted range correction.

Common pitfalls

A large coupling must be iterated. The relevant criterion is the dimensionless product of the interaction and loop kernel. A numerically large coefficient can remain perturbative, while a small coefficient near a pole can be enhanced.

Exact two-body unitarity proves EFT consistency. A geometric resummation can satisfy elastic unitarity and still have uncontrolled regulator dependence or omit same-order operators.

All derivative interactions should be put into the potential and iterated. Iterating a nominal correction generates infinitely many higher-order terms and can demand new counterterms. Insert it according to the derived counting unless a separate enhancement promotes it.

Cutoff independence means taking the cutoff to infinity. The test is independence through the claimed order within a regulator window compatible with the EFT. Sending an incomplete kernel arbitrarily above $\Lambda_b$ can probe physics the EFT does not contain.

Exercises

Show directly that the PDS amplitude is independent of $\mu$ when $C_0(\mu)=4\pi/[m(1/a-\mu)]$.

Solution

Substitute into

$$\mathcal A = \frac{-C_0}{1+mC_0(\mu+ik)/(4\pi)}.$$

Multiplying numerator and denominator by $1/a-\mu$ gives

$$\mathcal A = -\frac{4\pi/m}{1/a-\mu+\mu+ik} = \frac{4\pi/m}{-1/a-ik}.$$

The subtraction scale cancels exactly in the resummed leading sector.

For $a>0$, compute the pole energy and estimate the expansion parameter of the range correction.

Solution

The pole occurs at $k=i/a$, so with nonrelativistic energy $E=k^2/m$,

$$E_B=-\frac1{ma^2}.$$

If $r_0\sim1/\Lambda_b$, the range correction near the pole is controlled parametrically by $r_0/a\sim Q/\Lambda_b$ with $Q=1/a$. It is small only when the bound state is shallow relative to the range scale.

References

• Epelbaum, Evgeny, Hans-Werner Hammer, and Ulf-G. Meißner. “Modern Theory of Nuclear Forces.” Reviews of Modern Physics 81 (2009): 1773–1825. DOI · Open PDF
• 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