FR02 · How much detail can low energy see?

Heavy Mediator & Effective Field Theory

When can a heavy particle be replaced by a few low-energy interactions? Far below the mediator mass, its propagator can be expanded into local interactions. Each added order suppresses the error by another power of energy squared over mass squared. Data near the pole can force a very precise fit toward the wrong coupling if the omitted terms are ignored.

Raise the collision energy and watch a local interaction lose the shape of a heavy-particle exchange. Compare successive approximations, then fit the same simulated observations with two different assumptions about the missing terms.

Full amplitude $0.650$

Retained expansion $0.554$

Relative remainder $14.8\%$

This is a tree-level real-scalar model in natural units, below the mediator pole. The observations are synthetic real-amplitude measurements with Gaussian noise. The mass is known. The correlated error model keeps only one omitted coefficient; it has no general coverage guarantee. Loop corrections, widths and renormalization-group running are omitted.

The scene is an amplitude schematic: teal shows full exchange, gold the selected expansion. Wave heights share a scale at each energy; motion guides the eye.

One record, two error assumptions

Where does the approximation lose the data?

Full model referenceSelected EFT orderCoral: fitted observations · gray: excluded

Statistical errors only

\widehat\alpha=0.400\pm0.003

One common omitted coefficient

\widehat\alpha=0.400\pm0.005

The smooth curves are model references. Eighteen synthetic real-amplitude observations are loaded on entry; both fits use these exact saved values.

Model, assumptions and sources

\mathcal A=\frac{\alpha}{1-x},\quad\mathcal A_N=\alpha\sum_{n=0}^Nx^n,\quad x=(E/M)^2

The two massless scalar species exchange one heavy real scalar in the s channel. Natural units are used. The matched dimensionless coupling is 0.4. The mass changes the physical energy scale while ratios and coupling stay fixed. The pole is excluded; no artificial width is supplied.

Sampling uses eighteen independent Gaussian amplitude errors and a fixed pseudorandom seed. The second fit integrates out a shared first-omitted coefficient with a zero-mean Gaussian prior of standard deviation 0.4. It is a conditional teaching model; near the pole higher omitted terms remain important. The mass is not fitted.

Matching is performed at the heavy mass. This tree calculation contains no loop logarithms or renormalization-scale running. Scale variation in a loop calculation and variation of the retained energy order answer different questions.

Manohar · EFT lectures, sections 3–6 Furnstahl et al. · Truncation-error assumptions

Read the course: gauge fields and effective theories

Start with the essentials

Focus question: When can a heavy particle be replaced by a few low-energy interactions?
One-sentence intuition: Every extra pair of energy powers buys accuracy only while the energy stays small compared with the heavy mass.

Core mathematical model

A definite scalar exchange

\mathcal L_{\mathrm{int}}=-\frac12 S(g_1\phi^2+g_2\chi^2),\qquad \mathcal A(s)=\frac{g_1g_2}{M^2-s}=\frac{\alpha}{1-x},\quad x=\frac{s}{M^2}

Two light scalars of the first species scatter into two of the second. The light masses vanish, the heavy real scalar has mass M, each coupling has energy dimension one, and alpha is dimensionless. The displayed amplitude is real below the pole.

Matching and the exact remainder

\mathcal A_N=\alpha\sum_{n=0}^N x^n,\qquad R_N=\frac{\alpha x^{N+1}}{1-x},\qquad \frac{R_N}{\mathcal A}=x^{N+1}

LO, NLO and NNLO keep zero, two and four powers of energy beyond the constant term. Here the leading scalar operator has dimension four; adding two derivatives raises the dimension by two. The geometric remainder gives an independent accuracy check.

One correlated model for missing terms

y_i=\alpha f_N(x_i)+b x_i^{N+1}+\epsilon_i,\quad b\sim\mathcal N(0,0.4^2),\quad K_{ij}=\sigma_i^2\delta_{ij}+0.4^2x_i^{N+1}x_j^{N+1}

One common omitted coefficient moves observations at different energies together. Marginalizing it gives the covariance used by the second fit. This first-omitted-term assumption is a teaching choice, not the full geometric remainder or a guaranteed confidence interval.

Common difficulties

Matching scale and expansion order

Typical misconceptionChanging a scale slider should always make a tree-level EFT prediction run.

Better mental modelThis model matches at the heavy mass and has no loop logarithms. Renormalization-scale running requires loop corrections and a stated scheme. Raising the retained energy power is a separate operation.

What was observed?

Typical misconceptionThe points are event counts from a collider.

Better mental modelThis is a tree-level real-scalar model in natural units, below the mediator pole. The observations are synthetic real-amplitude measurements with Gaussian noise. The mass is known. The correlated error model keeps only one omitted coefficient; it has no general coverage guarantee. Loop corrections, widths and renormalization-group running are omitted.

Run the experiment

01
Start below the threshold
Choose “Low-energy agreement”. Predict which approximation changes most when you drag the energy handle.
What to observe: The constant contact response stays flat; the derivative terms restore the slope and curvature.
02
Bring in higher-energy data
Choose “Precision can mislead”. Open “Fit range, mass and sampling” and raise the fit-energy limit while keeping the saved record fixed.
What to observe: A statistical-only fit can be narrow and displaced. The correlated model admits a common missing energy dependence.
03
Test the expansion
Keep the records fixed and change the expansion order. Export the records before changing the highest recorded energy.
What to observe: Higher order reduces the remainder, but convergence slows near the pole. No width is added to rescue that region.

Heavy Mediator & Effective Field Theory

Where does the approximation lose the data?