Fit an effective interaction without outrunning its expansion

A heavy interaction can affect collisions even when its mediator is too heavy to produce. At energies well below the heavy scale, its effect can be expanded in powers of energy divided by that scale. A fit to an energetic tail is useful only while that expansion remains controlled and while ordinary changes in the proton structure, QCD prediction and detector response are included.

Replace a heavy propagator by a low-energy series

A simple massive exchange contains a denominator involving the heavy mass $\Lambda$ and a momentum-transfer scale $Q$. When the magnitude of the ratio is small, the geometric series gives

Each additional term changes the energy dependence. Effective field theory, or EFT, organizes such low-energy effects into local interactions, called operators, with coefficients. The coefficient depends on the operator’s normalization and on the assumed set of interactions. A number multiplying an arbitrary plotted tail has no Wilson-coefficient interpretation until those definitions and the corresponding event prediction are supplied.

The example denominator is a teaching motivation for the expansion. A realistic EFT has several operators, interference patterns, running and matching conditions. Being below a chosen mass scale is necessary in this simple example; a precision calculation must also estimate the omitted terms.

Square the amplitude in a consistent order

A scattering amplitude is a complex number whose absolute square contributes to a rate. Let $\mathcal A_0$ be the standard-model amplitude, $\mathcal A_6$ the coefficient-stripped dimension-six contribution and $c$ a dimensionless coefficient. Keeping a term in the amplitude generates both interference and a square when we calculate the rate:

The interference can be positive or negative, and it can vanish because the amplitudes populate different helicities. A negative interference term lowers a rate; a physical total prediction must still be nonnegative. If a truncated prediction goes negative, that is evidence to reconsider the range or truncation, not a reason to clip bins silently.

At inverse-heavy-scale order four, interference with dimension-eight operators can contribute alongside the dimension-six square. Keeping only that square makes an explicit partial-order assumption. Assessing the omitted dimension-eight contribution requires an additional power-counting or model assumption.

A worked energy-expansion example

Take a deliberately invented real amplitude ratio $\mathcal A/\mathcal A_0=1+cQ^2/\Lambda^2$, with $c=0.5$ and $\Lambda=2$ TeV. At $Q=0.5$ TeV, the amplitude correction is 0.03125. The rate ratio is about 1.06348; the linear-only ratio is 1.0625. Their difference is less than one part in a thousand of the original rate.

At $Q=1.5$ TeV, the amplitude correction is 0.28125. The full square of this chosen amplitude gives about 1.6416, whereas the linear-only result gives 1.5625. The larger disagreement reveals the growing sensitivity to the truncation. An unknown dimension-eight coefficient remains a separate input to the theory prediction.

Make the energy selection part of the fit

High-energy bins often separate a growing EFT contribution from a falling background. They can also lie closest to the expansion’s failure region. Repeat a fit with several upper energy selections, preserve their definitions and show how the result changes. A reconstructed mass cut is a proxy for the relevant hard scale, not a proof that every partonic invariant satisfies a validity criterion.

The collider-likelihood Lab uses a declared illustrative tail template and a signal-strength multiplier. Its upper-mass selection includes complete bins only. Changing $c$ in the amplitude example changes interference and square terms differently; the Lab instead scales their fixed template combination with one strength.

Ordinary physics can imitate a growing tail

A high-momentum spectrum also depends on the parton distributions and QCD radiation. An energy calibration shift moves events across a rapidly falling spectrum. Each can create a shape change that overlaps with an EFT template. A fit that fixes all of them may give a strong-looking coefficient bound with little protection against those alternatives.

The next chapter uses an actual published CMS jet spectrum to test how far a simple shape extrapolates. Both fitted shapes fail at the quoted precision. The useful lesson is how to expose a poor background description before interpreting its residual as a new interaction.

Task and exercises

Task. In the likelihood Lab choose the rising-tail preset. Record the fitted strength and interval at several allowed upper-mass selections. Keep the saved sample and other model settings fixed. Separate two reasons for a changing interval: fewer selected events and a different lever arm for the tail shape.

Exercise 1. In the invented amplitude, double the energy while keeping the heavy scale fixed. How do the amplitude correction, interference and square change?

Answer. The amplitude correction and interference become four times larger; its square becomes sixteen times larger. This different growth is why a tail can become sensitive to the truncation quickly.

Exercise 2. Interference is zero in a chosen channel. Is retaining the dimension-six square automatically a complete calculation at that order?

Answer. No. Dimension-eight interference can appear at the same inverse-scale order. One needs a power-counting assumption, a specified ultraviolet model or an uncertainty treatment to assess it.

Exercise 3. A fitted tail coefficient shrinks when a correlated energy-scale nuisance is added. What should be checked?

Answer. Check whether the nuisance is independently supported, whether its variation is propagated through the event selection and whether it overlaps with the tail’s shape. A shared explanation between two parameters is an identifiability issue that should appear in the reported interval and correlations.