Why can a new operator basis appear to tighten a constraint?

A coordinate change leaves the prediction fixed

Write a normalized low-energy amplitude as $y(x)=c_0+c_1x$ , with a dimensionless energy variable $x$ and dimensionless coefficients. The row vector $r=(1,x)$ and column $c=(c_0,c_1)^T$ give $y=rc$ .

Choose an invertible matrix $T$ and define $c^\prime=Tc$ . Substitution of $c=T^{-1}c^\prime$ determines $r^\prime=rT^{-1}$ . The same experiment now has different coordinate labels.

c^\prime=Tc,\qquad r^\prime=rT^{-1},\qquad y=r^\prime c^\prime=rc

For an action written as the inner product of coefficients and operators, the operator column transforms by the inverse transpose.

The error ellipse must follow

Let the coefficient mean be $\bar c$ and its covariance $\Sigma=\langle(c-\bar c)(c-\bar c)^T\rangle$ . Each error vector acquires a factor of $T$ , so its outer product acquires a matrix on both sides.

The inverse matrix in the prediction vector cancels the corresponding matrix in the new covariance. An ellipse can stretch and tilt without changing the prediction’s uncertainty. Assess the information by comparing uncertainty in the same physical prediction.

\bar c^\prime=T\bar c,\qquad \Sigma^\prime=T\Sigma T^T

Transform the mean and covariance together, retaining every correlation term.

\operatorname{Var}(y)=r\Sigma r^T=r^\prime\Sigma^\prime r^{\prime T}

This observable variance provides the comparison. Ellipse area and covariance determinant can change with coordinate units.

Calculate a shear explicitly

Start with mean $(1,2)^T$ and independent standard deviations $0.2$ and $0.3$ . Set $c_0^\prime=c_0+c_1$ and $c_1^\prime=c_1$ . The new mean is $(3,2)^T$ ; its covariance has diagonal entries $0.13,0.09$ and off-diagonal entry $0.09$ .

At $x=0.2$ , the original expression $c_0+0.2c_1$ gives $1.4$ . The new expression $c_0^\prime-0.8c_1^\prime$ gives the same value. Correlation cancels part of the new diagonal contribution, leaving variance $0.0436$ in either basis.

Analytic reference. The last row deliberately omits the covariance transformation.
Calculation	Mean	Variance	Standard deviation
Original basis	$1.4$	$0.0436$	$0.208806$
Complete transformation	$1.4$	$0.0436$	$0.208806$
New prediction vector, old covariance	$1.4$	$0.0976$	$0.312410$

Resetting a prior changes the question

Suppose the original coefficients have independent Gaussian priors of equal width. Assigning independent equal-width Gaussians again after a shear selects a different set of physical functions. The original circular prior becomes an ellipse; drawing another circle changes the assumption.

For data $d=Xc+\epsilon$ , transform the design matrix as $X^\prime=XT^{-1}$ . The Gaussian likelihood’s quadratic form is then unchanged. A full probability-density comparison also includes the Jacobian of the variable transformation. Density depends on coordinate volume; probability does not.

EFT operator reduction can go further, using field redefinitions and equations of motion to remove operators redundant for the chosen scattering process. That requires consistent truncation and transformation of all relevant terms. Reducing redundant operators requires its own order-consistent field-theory argument.

Source: Aneesh V. Manohar

Use the check in a fit

Choose a physical scattering energy and compare the prediction there. Change the basis and check both center and width. Then deliberately omit the covariance transformation to locate the error. This is more informative than comparing isolated coefficient error bars.

A coordinate change also cannot repair omitted physics. An energy range outside the low-energy expansion can produce a biased prediction that remains equally biased in every correctly transformed basis. The next chapter addresses truncation and the additional assumptions behind positivity.

Try the reasoning in an experiment

Follow one physical prediction

Predict whether shearing the error ellipse should change the prediction band at fixed energy.
Drag the basis parameter and record the physical prediction and its standard deviation. Save two clearly different bases.
Compare with the deliberately omitted covariance transformation. Identify whether the difference comes from coordinates or changed statistical assumptions, and write an equality that another reader can check.

Exercises and explanations

At $x=0$ , what is the new expression for the same observable?

Read the explanation

The original observable $c_0$ becomes $c_0^\prime-c_1^\prime$ . Its variance is $0.13+0.09-2(0.09)=0.04$ , unchanged.

A change of units reduces a coefficient and its quoted error by a factor of one hundred. Is the constraint stronger?

Read the explanation

No. The corresponding prediction-vector component increases by one hundred. The physical prediction and its uncertainty stay fixed.

Two fits in different bases have the same central prediction but different bands. What information should you request first?

Read the explanation

Request the full covariance, basis transformation, design matrix and prior definitions. The full matrix supplies the correlations needed for the prediction band.

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.

David Tong, Gauge Theory (2018 lecture notes)

Sections 2.1–2.1.3 (connections and Wilson loops), 2.7.3 (Higgs phase), 3.1.1–3.2.2 (two-dimensional anomaly and Ward identities), 4.2.1 (link endpoints) and 4.3.1–4.3.3 (lattice fermions). Printed pages 26–37, 105–108, 122–139, 207–210, 220–228.

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.