Does more precise data make a low-energy theory more reliable?

One omitted coefficient affects many energies

Open Furnstahl et al., arXiv:1506.01343v1. Equations (1)–(2) define an observable expansion and its omitted tail. Section II and Table I then specify priors for unknown coefficients. A probability interpretation of an error interval comes from these assumptions.

For the mediator model, write $f_N(x)=\sum_{n=0}^{N}x^n$ and $d_i=\alpha f_N(x_i)+b x_i^{N+1}+\epsilon_i$ . A single omitted coefficient $b$ shared by all energies produces correlated theory errors when marginalized under a Gaussian prior.

K_{ij}=\tau^2x_i^{N+1}x_j^{N+1},\qquad C_{ij}=\sigma_i^2\delta_{ij}+K_{ij}

Use

b\sim\mathcal N(0,\tau^2)

. One coefficient raises or lowers many points together, giving an off-diagonal covariance.

Source: R. J. Furnstahl

What a fit can check

The default teaching record uses $M=2\,\mathrm{TeV}$ , $\alpha=0.4$ and $18$ real-amplitude observations from $E/M=0.08$ to $0.72$ . Gaussian noise has standard deviation $0.012$ and seed $3202$ . Only the coupling ratio is fitted; the mass is known. Refitting reads the same saved observations.

With statistical weights alone, the full positive-coupling amplitude lies above its truncation, so the fit tends to raise the coupling. Higher-energy points intensify this compensation. Compare it with the first-omitted-term model using $\tau=0.4$ .

This tests the specified error model. The exact remainder also contains $1/(1-x)$ , so a single omitted term can still be inadequate near the pole. General coverage requires a separate repeated-sampling study with a stated parameter ensemble.

\widehat\alpha=\frac{f^TC^{-1}d}{f^TC^{-1}f},\qquad \sigma_{\widehat\alpha}^2=(f^TC^{-1}f)^{-1}

These are the linear Gaussian estimates for fixed covariance. A parameter-dependent theory covariance requires a correspondingly defined likelihood.

Genuine running mixes operators

Manohar’s PDF section 5.10.1, equations (5.56)–(5.60), provides an independent one-loop QCD benchmark for two weak four-quark operators. The HTML rendering numbers the ratios and results (68)–(70). For $n_f=5$ , $b_0=23/3$ . A matrix with diagonal entries $-2$ and off-diagonal entries $6$ has eigenvalues $4$ and $-8$ in the sum and difference directions.

Start at $(c_1,c_2)=(1,0)$ . Insert strong couplings $0.118$ at the higher scale and $0.22$ at the lower scale into the powers below. Running generates a second coefficient even though it initially vanishes.

c_\pm=c_1\pm c_2,\qquad \frac{c_+(\mu_1)}{c_+(\mu_2)}=\left[\frac{\alpha_s(\mu_1)}{\alpha_s(\mu_2)}\right]^{-6/23},\qquad \frac{c_-(\mu_1)}{c_-(\mu_2)}=\left[\frac{\alpha_s(\mu_1)}{\alpha_s(\mu_2)}\right]^{12/23}

This leading-log result assumes a fixed five-flavor interval. Crossing quark thresholds requires a new effective theory and threshold matching.

Our recalculation of published equations for strong-coupling ratio 0.22/0.118. These are not fitted experimental values.
Coefficient	Initial	After running
$c_+$	$1$	$0.850010$
$c_-$	$1$	$1.384050$
$c_1$	$1$	$1.117030$
$c_2$	$0$	$-0.267020$

Source: Aneesh V. Manohar

Move the matching point consistently

Write running between scales as $U(\mu_1,\mu_2)$ . Running through an intermediate scale must agree with one direct step. This composition law checks a program and explains why a matching point cannot be moved on only one side. Match both theories at one scale, then run coefficients and operator matrix elements consistently to preserve the physical amplitude to the stated accuracy.

Residual scale dependence at finite order can diagnose some higher-order terms, especially those connected to existing logarithms. A tree prediction with no scale dependence can still have a large energy-expansion remainder. Chapter 4 supplies an exact counterexample.

For a finite-order check, hold endpoint strong couplings at $0.118$ and $0.22$ . Expand the known boundary-running factor through first or second order in $y=0.118/\alpha_s(\mu_m)-1$ at an intermediate scale. Then use exact leading-log running to the endpoint. Moving the intermediate scale leaves the residuals below; another term reduces this known truncation error.

This tests a Taylor expansion of a known logarithmic factor. It contains no unknown finite hard matching between different theories. The residual measures truncation of the known leading-log solution. Probability intervals for unknown higher orders require an additional statistical model.

U(\mu_1,\mu_3)=U(\mu_1,\mu_2)U(\mu_2,\mu_3)

This holds within the same effective theory and approximation. Thresholds require the appropriate matching between running factors.

(1+y)^a=1+ay+\frac{a(a-1)}2y^2+\cdots,\qquad a_+=\frac6{23},\quad a_-=-\frac{12}{23}

Use these exponents for the intermediate boundary factors, return to the original coefficient basis, and compare with the preceding endpoint table.

Known-log benchmark with five fixed flavors and no thresholds. Entries give the largest absolute endpoint-coefficient error. The boundary parameter vanishes when the intermediate point equals the initial point.
$\alpha_s(\mu_m)$	First-order boundary	Second-order boundary
$0.090$	$0.0274750$	$0.00684656$
$0.118$	$0$	$0$
$0.150$	$0.0157440$	$0.00272729$
$0.180$	$0.0439221$	$0.0123466$

Source: Aneesh V. Manohar

Which amplitude is constrained by positivity?

Open Adams et al., arXiv:hep-th/0602178v2, section 4, equation (20), with the forward poles in figures 5–6. We now use its linear sigma model for identical Goldstone elastic scattering. This differs from the distinguishable-scalar channel of chapter 4; an arbitrary coupling product there need not be positive.

Let the heavy mass be $M_h$ and the stable model’s coupling $\lambda>0$ . Massless external states obey $s+t+u=0$ . At zero momentum transfer $t=0$ , crossing pairs the poles into an even forward amplitude. Differentiate it directly to obtain positive curvature.

\mathcal M(s,t)=\frac{\lambda}{M_h^2}\left[\frac{s^2}{M_h^2-s}+\frac{t^2}{M_h^2-t}+\frac{u^2}{M_h^2-u}\right]

This is an equivalent form of equation (20), retaining all three exchange channels.

A(s)=\mathcal M(s,0)=\frac{2\lambda s^2}{M_h^4-s^2},\qquad A^{\prime\prime}(0)=\frac{4\lambda}{M_h^4}>0

Differentiation is with respect to a variable of energy-squared dimension; the curvature has inverse-energy-fourth units.

Recalculation of the paper’s tree benchmark for coupling 0.2 and heavy mass 2 TeV. Curvature is 0.05 per TeV to the fourth power.
$x=s/M_h^2$	$A(s)$	Leading term $2\lambda x^2$
$0.25$	$0.0266667$	$0.0250000$
$0.50$	$0.1333333$	$0.1000000$
$0.75$	$0.5142857$	$0.2250000$

Source: A. Adams

From a pole benchmark to a research question

The paper deforms the contour onto physical cuts and uses unitarity to relate absorption to total cross sections. That step turns a particular model’s curvature into a condition on completions with suitable analyticity, crossing, high-energy boundedness and treatment of low-energy singularities.

Massless forward exchange poles, especially with gravity, can invalidate the simple forward argument. A negative fitted coefficient need not violate positivity: identify the relevant elastic-amplitude combination and align basis, scale and truncation first.

A concrete research question now follows. On the same saved observations, how stable is a positivity-relevant posterior under changes in order, energy cutoff and omitted-coefficient prior? Change one assumption at a time and report fit quality. The reproduced scope here is the tree-pole and leading-log benchmarks.

Try the reasoning in an experiment

Separate fit bias from statistical noise

Predict the direction of a positive-coupling fit shift when higher-energy observations are included.
Save one seeded record. Raise the fit-energy limit and compare statistical weights with the correlated omitted-term model. Record center, width and order.
Check a basis change on the same prediction, then identify differences caused by omitted physics and by statistical assumptions.

Transform a physical combination

Choose a linear observable and record its coefficient-space direction.
Transform that direction and the covariance together. Check the mean and variance.
Explain why the sign of one new coefficient usually cannot replace a physical-amplitude positivity condition.

Exercises and explanations

How many independent theory-error directions does a single outer-product covariance contain?
Read the explanation
One, when its energy vector is nonzero. A shared omitted coefficient drives all points. Replacing it with independent errors changes the statistical model.
Double $M_h$ in the positivity example at fixed coupling. How does the curvature change?
Read the explanation
It falls by a factor of sixteen, since it scales as inverse mass to the fourth power.
A tree model has no renormalization-scale variation. How do you test the claim that its theory error is zero?
Read the explanation
Calculate the geometric remainder in chapter 4. A finite truncation has a nonzero remainder at nonzero energy. Vanishing tree-scale variation only says there are no quantum logarithms here.

Sources and reproduced scope

Derivations, tables and the plot are authored for this course. The reading locations below separate source results from our teaching extensions. Numerical examples are model references or labeled synthetic observations. No experimental dataset or source figure is copied.

Aneesh V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863v1 (2018)
Sections 3.5 and 4 (integrating out and power counting), 5.5–5.7 (matching), 5.10.1, PDF equations (5.51)–(5.60) (QCD running and operator mixing), and 6.2–6.3 (field redefinitions). The scalar-channel examples are derived for this course. The known-log boundary expansion checks truncation of the published running factors; it does not calculate finite hard matching.
R. J. Furnstahl, N. Klco, D. R. Phillips and S. Wesolowski, Quantifying truncation errors in effective field theory, arXiv:1506.01343v1; Physical Review C 92, 024005 (2015)
Equations (1)–(2) define the expansion and remainder. Section II and Table I specify coefficient priors; equation (29) retains several omitted terms. The present numerical benchmark checks these definitions with a geometric series, not the paper’s neutron–proton calculations.
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, Analyticity and an IR Obstruction to UV Completion, arXiv:hep-th/0602178v2; JHEP 10 (2006) 014
Section 4, equations (19)–(23), printed pages 14–16: a tree-level linear-sigma-model amplitude and its forward poles. We reproduce equation (20) at zero momentum transfer and its small-energy coefficient. Figures 5–6 are reading references; no source artwork is copied.