When a reduced width stops being measurable

Take a single-level R-matrix description of a reaction, fit it to data, and obtain an excellent $\chi^2$. Now try to put uncertainties on the resonance parameters by sampling the posterior. The sampler does not converge: the walkers drift away without bound, the transformation between parameter sets starts throwing warnings, and the marginal distributions come out in shapes no Gaussian would recognise.

Nothing is wrong with the sampler, and nothing is wrong with the data. What has happened is that one of the parameters has stopped being a physical quantity and become a redundant coordinate — and the diagnosis, the cure, and the reason both are governed by the channel radius, are what follows.

What R-matrix theory actually does#

R-matrix theory is, before anything else, a division of space. One draws a sphere of radius $a$ — the channel radius — around the pair of fragments and treats the two sides by completely different means:

region	what is there	how it is described
$r < a$	all nucleons interacting, many-body	a few poles $(E_\lambda, \gamma_{\lambda c})$
$r > a$	Coulomb and centrifugal forces only	known functions $(F_\ell, G_\ell)$

The essential point, and the one most often lost, is that this surface is a choice of bookkeeping, not a physical boundary. Nothing happens at $r = a$. A larger $a$ does not mean the fragments are more tightly bound, or more likely to fuse; it means one has drawn a larger box and declared more of the wave function to be "interior". If the model were complete, every observable would be independent of where the surface is put. What does depend on $a$ is how the description divides its labour between the two sides.

The reduced-width amplitude $\gamma_{\lambda c}$ is essentially the amplitude of the level's wave function at the surface. It is therefore not an observable: move the surface and it changes. What can be measured is the observed partial width $\Gamma_c$, the decay rate into channel $c$.

From reduced width to observed width#

The textbook relation between the two is $\Gamma_c = 2\gamma_c^2 P_c$, with $P_c$ the penetrability — the probability of getting out through the Coulomb and centrifugal barriers. That formula is incomplete. The correct one carries a normalisation,

$$\Gamma_c \;=\; \frac{2\,\gamma_c^{2}\,P_c(E_\lambda)}{N}, \qquad N \;=\; 1 + \sum_{c'} \gamma_{c'}^{2}\,\frac{\mathrm{d}S_{c'}}{\mathrm{d}E}\bigg|_{E_\lambda},$$

where $S_c$ is the shift function of channel $c$. Everything in this note follows from reading $N$ correctly, so it is worth doing slowly.

$N$ is a normalisation. A physical state must be normalised over all space, but the pole describes only the inside. Splitting the normalisation integral at $r = a$,

$$\underbrace{1}_{\int_0^a |u|^2\,\mathrm{d}r} \;+\; \underbrace{\sum_c \gamma_c^{2}\,\frac{\mathrm{d}S_c}{\mathrm{d}E}}_{\sum_c \int_a^\infty |u_c|^2\,\mathrm{d}r} \;=\; N .$$

The 1 is the piece of the state inside the channel radius; each $\gamma_c^2\,\mathrm{d}S_c/\mathrm{d}E$ is the piece that has leaked out into channel $c$. Hence the single most useful number in this note,

$$\boxed{\;f_{\mathrm{ext}} \;\equiv\; 1 - \frac{1}{N} \;=\; \text{fraction of the level lying outside the channel radius.}\;}$$

$N = 1$ means the state is entirely inside the box and $\Gamma = 2\gamma^2 P$ is adequate. $N = 2$ means half in, half out. $N = 34$ means 97 % of the state is outside the box one has drawn.

This identification is exact for a channel that is closed at $E_\lambda$, where $\mathrm{d}S/\mathrm{d}E$ is literally the normalisation integral of the exterior Whittaker tail. Above threshold it is the analytic continuation of the same object, and the "fraction outside" reading is the standard physical gloss rather than a theorem. None of the algebra below depends on the interpretation — only the intuition does.

The worked example#

The $^3\mathrm{H}(d,n)^4\mathrm{He}$ cross section at astrophysical energies is carried by a single level: the $J^\pi = 3/2^+$ state of $^5\mathrm{He}$, 80 keV above the $d+t$ threshold, with an $\ell = 0$ deuteron channel ($a_d = 3.5\ \mathrm{fm}$) and an $\ell = 2$ neutron channel ($a_n = 4.0\ \mathrm{fm}$). It is a large, loosely bound $d+t$ cluster state: a state that barely exists as a bound object has a wave function that extends far.

channel	$\ell$	$a$ [fm]	$P_c$	$\mathrm{d}S_c/\mathrm{d}E$ [MeV−1]	$D_c \equiv (\mathrm{d}S_c/\mathrm{d}E)/P_c$
$d+t$	0	3.5	$5.1485 \times 10^{-2}$	1.1778	22.88
$n+\alpha$	2	4.0	2.4434	$1.8579 \times 10^{-2}$	0.0076

The deuteron channel is barely open and sits under a Coulomb barrier, so its penetrability is small. The neutron channel is 17.7 MeV above its threshold and is wide open.

A fit to 121 data points gives $\gamma_d = 5.3032\ \mathrm{MeV}^{1/2}$, $\gamma_n = 0.78059\ \mathrm{MeV}^{1/2}$ and $\chi^2 \approx 100$. Then

$$N = 34.1, \qquad f_{\mathrm{ext}} = 97.1\ \%, \qquad \theta_d^{2} \equiv \frac{\gamma_d^{2}\,\mu a^{2}}{\hbar^{2}} = 9.95 .$$

The dimensionless reduced width is ten times the single-particle limit, and 97 % of the level is outside the box. Both are symptoms of the same thing.

Why the width saturates#

Now ask the question a fitter asks: what happens to the observable if I turn up the coupling? Let $\gamma$ grow. The numerator of $\Gamma_c$ grows like $\gamma^2$ — but so does $N$, because pushing more amplitude through the surface puts more of the state outside. A ratio of two quantities both growing like $\gamma^2$ tends to a constant:

$$\Gamma_c \;\xrightarrow[\;\gamma \to \infty\;]{}\; \frac{2P_c}{\mathrm{d}S_c/\mathrm{d}E} .$$

The reduced width has disappeared from the answer. Physically: a level cannot decay faster than its channel kinematics allow. Once the state is essentially all outside, increasing the internal coupling adds nothing — there is no more "inside" left to couple to.

The crossover sits at $\gamma_{\mathrm{knee}} = (\mathrm{d}S/\mathrm{d}E)^{-1/2}$, which for the deuteron channel of the example is $0.92\ \mathrm{MeV}^{1/2}$. Below the knee, $\Gamma \propto \gamma^2$ as expected; above it, $\Gamma$ is flat and $\gamma$ is unidentifiable — not merely poorly determined, but absent from the observables. The fitted $\gamma_d = 5.30$ sits 5.8 times past the knee, at 97 % of its ceiling.

For a $\chi^2$ minimisation this is a curiosity: the minimiser finds some point on the plateau and reports it. For an uncertainty analysis it is fatal, because the likelihood possesses an exactly flat direction of infinite extent. The signature is visible in the fit itself: the error matrix of the example returns $\gamma_d = 5.90 \pm 12.41$ (a 210 % error) and $\gamma_n = 0.867 \pm 1.780$, errors whose ratio, 6.97, reproduces the ratio of the values themselves, 6.81. Almost all the uncertainty lies along the direction that scales both couplings together, and almost none across it: the data fix the branching ratio and let the overall scale float.

The same statement as a budget with a hard ceiling#

The saturation argument was about $\gamma$. It is more useful read backwards, in terms of the widths themselves. Multiply the width relation by $D_c \equiv (\mathrm{d}S_c/\mathrm{d}E)/P_c$ and sum over channels:

$$\sum_c \Gamma_c D_c \;=\; \frac{2}{N}\sum_c \gamma_c^{2}\,\frac{\mathrm{d}S_c}{\mathrm{d}E} \;=\; 2\,\frac{N-1}{N} \;=\; 2 f_{\mathrm{ext}} .$$

Since one cannot have more than all of a state outside a sphere, $f_{\mathrm{ext}} \le 1$, and therefore

$$\boxed{\;\sum_c \Gamma_c D_c \;\le\; 2, \qquad D_c = \frac{\mathrm{d}S_c/\mathrm{d}E}{P_c}\;}$$

This is a hard ceiling on the observed widths themselves, and it is the most practically useful form of everything above. Read $D_c$ as the "cost per unit width" of channel $c$, and 2 as the total budget. A set of widths violating the ceiling does not correspond to any reduced width whatsoever: it is not improbable, it is unrepresentable.

Two features of the budget are worth noting. First, the cost is dominated by the barely-open channel: in the example the $d+t$ channel spends 97 % of the budget and the wide-open $n+\alpha$ channel spends 0.03 %, because a large $P$ makes $D$ small. Sub-threshold and near-threshold charged channels are where this pathology lives. Second, the budget consumed falls steadily as the channel radius grows:

$a$ [fm]	3.5	4.0	4.5	5.0	6.0	7.0
$\sum_c \Gamma_c D_c$ at the measured widths	1.94 (97 %)	1.80 (90 %)	1.68 (84 %)	1.58 (79 %)	1.38 (69 %)	1.22 (61 %)

What the ceiling does to the inverse transformation#

Analyses that fit observed widths rather than reduced ones must invert the width relation. Solving the budget identity for the normalisation and substituting gives

$$\gamma_c^{2} \;=\; \frac{\Gamma_c}{P_c\,\mathcal{D}}, \qquad \mathcal{D} \;=\; 2 - \sum_{c'} \Gamma_{c'} D_{c'} \;=\; \frac{2}{N} \;=\; 2\,(1-f_{\mathrm{ext}}).$$

The denominator $\mathcal{D}$ is exactly the unspent budget. Two consequences follow immediately, and they are the operational heart of the matter.

The inverse map has a pole at the ceiling. As $\sum_c \Gamma_c D_c \to 2$, $\mathcal{D} \to 0$ and the reduced width required to produce the requested observed width diverges. A model sitting at 97 % of its budget is sitting next to a singularity.

Past the ceiling, $\mathcal{D}$ turns negative. A negative $\mathcal{D}$ means the requested widths would need more than 100 % of the level outside the surface. Implementations that take the absolute value at this point do not fail — they return a valid-looking but different model, and the map folds back on itself: requesting a larger width returns a smaller one. At $a_d = 3.5$ fm in the example the ceiling is 87.4 keV; asking for 120 keV returns a 69 keV model. This is the mechanism behind transformation warnings that nevertheless leave a fit running.

What the channel radius does#

Everything so far was at fixed $a$. Now move the surface, and watch the two quantities that make up the cost $D_c$ compete:

• $P_c$ rises. The penetrability is the probability of tunnelling out starting from $r = a$; start further out and less barrier remains. For the $d+t$ channel of the example, $P_d$ goes from $5.15 \times 10^{-2}$ at 3.5 fm to $1.54 \times 10^{-1}$ at 7 fm, a factor 3.0.
• $\mathrm{d}S_c/\mathrm{d}E$ also rises, but more slowly: 1.18 to 2.22, a factor 1.9.

$P$ wins, so $D_c$ falls (from 22.9 to 14.4), the ceiling $2/D_c$ rises, and $f_{\mathrm{ext}}$ falls. That competition is the entire mechanism. The physical reading is simply that a bigger box holds more of the state, so less of it is "outside", so the reduced width recovers its meaning. Nothing about the nuclear force has changed; one has stopped describing a large object with a small box.

The table below is the same data fitted at a ladder of channel radii, both channels moved together, each row an independent converged refit.

$a$ [fm]	$\chi^2$	$\gamma_d$ [MeV$^{1/2}$]	$\sigma(\gamma_d)/\gamma_d$	$\theta_d^2$	$f_{\mathrm{ext}}$	headroom in $\Gamma_d$
3.00	97.9	42.58	45 %	471	99.95 %	0.1 %
3.25	97.8	11.58	42 %	40.9	99.33 %	0.7 %
3.50	97.7	6.654	40 %	15.7	98.11 %	1.9 %
4.00	97.6	2.241	25 %	2.32	87.23 %	14.7 %
4.50	97.3	1.379	14 %	1.11	74.82 %	33.7 %
5.00	96.8	1.072	12 %	0.83	66.78 %	49.8 %
5.50	96.4	0.903	10 %	0.71	60.99 %	64.0 %
6.00	96.4	0.794	9 %	0.66	56.58 %	76.8 %
7.00	97.9	0.667	8 %	0.63	50.68 %	97.3 %

Three columns deserve attention.

$\chi^2$ is flat. It varies between 96.4 and 97.9 across a factor of more than two in radius. The data are genuinely indifferent to where the surface is placed, exactly as the theory says they should be. Goodness of fit cannot be used to choose $a$.

The reduced width becomes identified. Its fractional error falls monotonically from 45 % to 8 %, and $\theta_d^2$ falls from 471 to below unity. The same data, the same physics, a different coordinate — and a parameter that was invisible becomes measurable.

The headroom opens up. From 0.1 % at 3 fm to 97 % at 7 fm. This column predicts, quantitatively, how badly a width-space fit or sampler will misbehave: at 3.5 fm a proposal only 2 % away is already unrepresentable, while at 6 fm it takes a 77 % excursion.

None of this makes $a$ a free parameter to be pushed as far as convenient. Too small, and a genuinely extended cluster state is mostly external — the pathology of this note. Too large, and the assumption that only Coulomb and centrifugal forces act beyond $a$ fails, while the interior grows too big to be described by a few poles. There is a window, and the diagnostics above tell one which end of it a given model is sitting at.

Consequences for uncertainty quantification#

The two parameterisations fail in two different ways, and it is worth seeing both because the symptoms look nothing alike.

Sampling reduced widths. The likelihood is exactly flat along the direction that scales all $\gamma_c$ together at fixed ratio. With an unbounded prior the posterior is improper: walkers drift in $\log\gamma$ without limit while the calculated cross section stays put, because the observables stopped depending on the scale beyond the knee. The chain looks broken; it is in fact faithfully reporting that the model contains a redundant coordinate.

Sampling observed widths. The same degeneracy is compressed into a finite interval that ends at the ceiling, and it is the singular map, not the flat direction, that does the damage. At 3.5 fm the profile likelihood is flat to $\Delta\chi^2 < 0.01$ across the whole saturated range, its minimum sits on the ceiling itself, and the $\Delta\chi^2 = 1$ interval runs from about 58 keV to 89 keV — a flat-topped slab with a hard wall on one side. That is not a shape any sampler can render as a Gaussian. Past the wall the folded branch re-enters the model space backwards and contributes a second, spurious lobe. At 6 fm the profile near the mode is an ordinary parabola and the width is genuinely measured.

One caution. Enlarging the radius removes the failures and restores a Gaussian core, but it does not by itself remove the unbounded direction: the folded branch survives, merely displaced far from the mode, and there it returns to $\Delta\chi^2 \approx 1$. The reason is transparent in the algebra — on the folded branch $\gamma_c^2 = \Gamma_c/(P_c|\mathcal{D}|)$ and, as $\Gamma_d \to \infty$, $\mathcal{D} \to -\Gamma_d D_d$, so $\gamma_d \to (\mathrm{d}S_d/\mathrm{d}E)^{-1/2}$, which is precisely the knee. When the knee happens to lie near the best-fit value, as it does at 6 fm, the far branch is a genuinely decent model. A bounded prior is required regardless of the radius.

An alternative: an attractive potential outside the surface#

There is a second way to attack the same problem. Instead of moving the surface outward, one admits that the nuclear force does not stop abruptly at $r = a$: a Woods–Saxon (or Gaussian) well is added to the Coulomb and centrifugal potential in the external region, and the external wave functions are obtained by numerical integration inward from a large matching radius — where the pure Coulomb solution is imposed — instead of analytically. $P_c$ and $\mathrm{d}S_c/\mathrm{d}E$ are then evaluated from those solutions.

The attraction pulls amplitude inward, so both quantities increase sharply. For the example, with a well of depth 150 MeV, $R = 3.6$ fm, $a_{\mathrm{ws}} = 0.6$ fm, at the original $a_d = 3.5$ fm:

	pure Coulomb	with the potential	ratio
$P_d$	$5.148 \times 10^{-2}$	$4.323 \times 10^{-1}$	× 8.40
$\mathrm{d}S_d/\mathrm{d}E$	1.1778	8.4328	× 7.16
$D_d$	22.88	19.51	× 0.85

Both move by nearly an order of magnitude, but they move together, so the ceiling — which depends only on their ratio — shifts by just 15 %. The real gain appears on refitting: because $P_d$ is eight times larger, a far smaller $\gamma_d$ reproduces the same cross section, and the model leaves the saturated regime altogether:

configuration	$\chi^2$	$\gamma_d$	$\theta_d^2$	$f_{\mathrm{ext}}$	headroom
$a = (3.5, 4.0)$, pure Coulomb	97.7	5.303	9.95	97.07 %	3.0 %
$a = (3.5, 4.0)$, with the potential	96.5	0.478	0.08	66.79 %	49.7 %
$a = 5.0$ fm, pure Coulomb	96.8	1.072	0.83	66.78 %	49.8 %
$a = 6.0$ fm, pure Coulomb	96.4	0.794	0.66	56.58 %	76.8 %

The second and third rows are numerically indistinguishable: $N = 3.011$ against 3.010, $f_{\mathrm{ext}} = 66.79$ % against 66.78 %. Adding an attractive tail at 3.5 fm does the same job as moving the surface to 5 fm, and for the same reason — both get the description to where the state actually is.

Two cautions. First, the effect is not monotonic in the well depth. The external potential has single-particle states of its own, and when one of them falls near the level energy, $P$ and $\mathrm{d}S/\mathrm{d}E$ diverge and $\mathrm{d}S/\mathrm{d}E$ can change sign. In the example, a depth of 75 MeV drives $\mathrm{d}S/\mathrm{d}E$ to $-110$ and $N$ below unity, while 100 MeV gives $N = 54.6$ — worse than using no potential at all. A negative $\mathrm{d}S/\mathrm{d}E$ means the channel adds normalisation budget instead of spending it, and the interpretation of the split integral breaks down. Second, one has traded a single arbitrary choice, $a$, for four: $a$ and the three potential parameters, none of which the data meaningfully constrain. The diagnostics must be checked afterwards either way.

Practical summary#

1. Compute $f_{\mathrm{ext}} = 1 - 1/N$ before doing anything statistical. It costs nothing — $N$ is already formed inside the width transformation — and it is the single number that predicts whether a fit is well conditioned. Above roughly 90 %, the reduced width of that level is not a measurement. Below roughly 70 %, the model is in good shape.
2. Check $\theta_c^2$ as well. A dimensionless reduced width far above unity is the same warning in different units.
3. Do not use $\chi^2$ to choose the channel radius. It is flat against $a$ by construction. Use $f_{\mathrm{ext}}$ and $\theta^2$.
4. Widths above the budget are not merely unlikely — they do not exist. If a negative denominator is reported and the calculation continues, treat every subsequent number from that evaluation as unreliable, and check whether the proposal was rejected or silently replaced.
5. Bound the prior. Even a well-conditioned radius leaves the folded branch somewhere. Either $\theta_c^2 \le 1$ or, most directly, $\sum_c \Gamma_c D_c < 2$, which is the exact statement of "physically representable".

The broader lesson generalises beyond R-matrix theory. A model parameter is only a measurement if the observables depend on it. Here the dependence is switched off smoothly, by a normalisation that grows in step with the coupling, and the switch is thrown by a quantity the analyst chooses rather than measures. Any level whose normalisation $N$ departs appreciably from unity should be treated as a candidate for this behaviour before, not after, an uncertainty analysis is attempted.