Operator Anomalous-Dimension Matrices

An anomalous-dimension matrix is the connection induced on a renormalized operator basis when the subtraction scale changes at fixed bare data. Its sign and index direction are not universal typography: they follow from the declared equation relating bare and renormalized operators. With this chapter’s column convention $O_0=Z_OO$, the operator equation is $\mu\,dO/d\mu=-\gamma O$.

This page derives that equation, extracts $\gamma$ from minimal-subtraction poles, transforms it under finite basis changes, and identifies fixed-point scaling operators using the correct left eigenvectors. Wilson coefficients are mentioned only as an invariant check; their transpose equation and noncommuting path ordering belong to the next page.

Required background. Operator Mixing and Renormalization Matrices supplies a closed basis and $Z_O$. Renormalization Conditions, Schemes, and Finite Parts supplies finite scheme changes.

Helpful background. Commutators and Operator Exponentials is useful when matrices at different scales fail to commute.

Differentiating at fixed bare data

Let

$$O_0=Z_O(g,m/\mu,\epsilon)\,O(\mu)$$

for a closed column of local operators. The RG derivative at fixed bare fields and parameters is

$$\mathcal D \equiv \mu\frac{\partial}{\partial\mu} +\beta^a(g)\frac{\partial}{\partial g^a} -\gamma_m(g)m\frac{\partial}{\partial m} +\cdots,$$

where the omitted terms include any additional running renormalized coordinates. Bare insertions do not depend on the arbitrary subtraction scale:

$$0=\mathcal D O_0 =(\mathcal D Z_O)O+Z_O\mathcal D O.$$

Multiplication by $Z_O^{-1}$ gives

$$\boxed{ \gamma \equiv Z_O^{-1}\mathcal D Z_O, \qquad \mathcal D O=-\gamma O. }$$

The boxed definition fixes all three convention choices:

1. $O$ is a column;
2. $O_0=Z_OO$, not $O=Z_OO_0$;
3. the minus sign appears in the operator equation.

Writing indices removes any ambiguity:

$$\mathcal D O_i=-\gamma_{ij}O_j.$$

A reference using $O=Z_{\rm ref}O_0$ has $Z_{\rm ref}=Z_O^{-1}$. If it defines $\gamma_{\rm ref}=-(\mathcal D Z_{\rm ref})Z_{\rm ref}^{-1}$, then

$$\gamma_{\rm ref} = Z_O^{-1}(\mathcal D Z_O) = \gamma.$$

The formulas look different but describe the same operator flow. Comparing a sign before first translating the $Z$ direction is meaningless.

Collins derives the composite-operator RG equation from the scale independence of bare insertions and proves finiteness of its matrix coefficients Collins 1984/2023, § 7.12, pp. 219–221.

The figure keeps the dual transformations together. Read the upper map from bare to renormalized operators, then follow the inverse transpose on the source or Wilson-coefficient side. The same structure fixes both the sign convention above and the coefficient evolution derived on the next page.

Operator–coefficient duality for the convention $O_0=Z_OO$. Sources obey $s_0=Z_O^{-\mathsf T}s$, and ordered operator evolution $U_O$ is compensated by coefficient evolution $U_O^{-\mathsf T}$. The diagram is schematic and not to scale.

Extracting γ from minimal-subtraction poles

For one dimensionless coupling $g$, suppose

$$g_0 = \mu^{\kappa\epsilon} \bigl[g+\text{poles}\bigr],$$

so its $d$-dimensional beta function begins as

$$\beta_d(g) = \mathcal D g = -\kappa\epsilon g+\beta(g).$$

Write the operator matrix in a pure-pole scheme as

$$Z_O = \mathbf1 +\sum_{n=1}^{\infty} \frac{Z_O^{(n)}(g)}{\epsilon^n}.$$

At fixed bare data,

$$\mathcal D Z_O = \beta_d(g)\frac{\partial Z_O}{\partial g}.$$

The finite term produced when $-\kappa\epsilon g$ multiplies the simple pole is

$$\boxed{ \gamma(g) = -\kappa g \frac{\partial Z_O^{(1)}(g)}{\partial g}. }$$

This equation holds order by order after the higher-pole consistency relations are imposed. Products involving $Z_O^{-1}$ and $\beta(g)$ cancel all remaining poles. Failure of that cancellation signals a missing subdivergence, an incomplete operator sector, or inconsistent lower-order input.

For several couplings with $g_0^a=\mu^{\kappa_a\epsilon}[g^a+\cdots]$, the simple-pole formula becomes

$$\gamma(g) = -\sum_a \kappa_a g^a \frac{\partial Z_O^{(1)}}{\partial g^a}.$$

Matrix order matters in the full definition $Z_O^{-1}\mathcal D Z_O$, but the displayed simple-pole derivative acts entry by entry. A mass-dependent scheme adds explicit $m/\mu$ derivatives and generally cannot be reconstructed from a pure pole residue alone.

As a check, the preceding insertion page found

$$Z_{\phi^2} = 1-\frac{\lambda}{32\pi^2\epsilon}+\cdots$$

for $d=4-2\epsilon$, where $\kappa=2$. Therefore

$$\gamma_{\phi^2} = -2\lambda \frac{\partial}{\partial\lambda} \left(-\frac{\lambda}{32\pi^2}\right) = \frac{\lambda}{16\pi^2} +\cdots.$$

Direct differentiation of the full $Z_{\phi^2}$ gives the same finite result. The positive entry appears in $\mathcal D[\phi^2]=-\gamma_{\phi^2}[\phi^2]$; changing the sign in the operator equation would also require changing the definition of $\gamma$.

A two-operator pole benchmark

Let a dimensionless coupling satisfy $a_0=\mu^\epsilon[a+\cdots]$, so $\kappa=1$. Suppose a normalized two-operator calculation gives the simple-pole matrix

$$Z_O = \mathbf1-\frac{a}{\epsilon} \underbrace{ \begin{pmatrix} 1&2\\ 0&-1 \end{pmatrix}}_{M} +\mathcal O(a^2).$$

The simple-pole formula gives

$$\gamma(a) = aM+\mathcal O(a^2) = a \begin{pmatrix} 1&2\\ 0&-1 \end{pmatrix} +\mathcal O(a^2).$$

The lower-left zero is inherited from a closed triangular physical–redundant sector. A missing off-diagonal subtraction would instead appear as an uncancelled pole in $Z_O^{-1}M_{\rm bare}$.

At a fixed point $a=a_\star$, assume both operators have the same engineering dimension $\Delta_0$. A scaling operator is a linear combination

$$\widehat O=w^{\mathsf T}O.$$

Its RG equation is diagonal only when $w^{\mathsf T}$ is a left eigenvector:

$$w^{\mathsf T}\gamma_\star = \gamma_\star^{(w)}w^{\mathsf T}, \qquad \mathcal D\widehat O = -\gamma_\star^{(w)}\widehat O.$$

For $M$ above, the left eigenvectors and eigenvalues are

$$\begin{array}{c|c|c} w^{\mathsf T}&\text{eigenvalue of }M&\text{scaling operator}\\ \hline (1,1)&+1&O_1+O_2\\ (0,1)&-1&O_2 \end{array}$$

so the full scaling dimensions in this algebra benchmark are

$$\Delta_\pm=\Delta_0\pm a_\star+\mathcal O(a_\star^2).$$

Right eigenvectors would diagonalize the evolution of coordinate columns, not the linear combinations of basis operators written as $w^{\mathsf T}O$. Confusing the two transposes gives the wrong scaling operators even though the eigenvalue set happens to agree.

The numerical matrix $M$ is the first segment of a reproducible calculation. It is a deterministic algebra and ordering benchmark, not a claim about the spectrum of a named QFT; the benchmark absorbs the overall coupling into its RG-time interval.

Fixed points, degeneracy, and Jordan blocks

At a fixed point, $\beta^a(g_\star)=0$ and $\gamma_\star$ is constant in a fixed basis. If it is diagonalizable, left eigenoperators scale with definite anomalous dimensions. If two eigenvalues coincide, diagonalizability must still be checked. A Jordan block

$$\gamma_\star = \begin{pmatrix} \gamma_\star&1\\ 0&\gamma_\star \end{pmatrix}$$

produces logarithmic mixing:

$$e^{-\gamma_\star t} \begin{pmatrix} 1&-t\\ 0&1 \end{pmatrix}.$$

There is then no basis of two independent ordinary scaling eigenoperators. The extra power of $t=\ln(\mu/\mu_0)$ is physical within the stated sector and cannot be removed by pretending the matrix is diagonal.

Away from a fixed point, even a diagonalizable $\gamma(g(\mu))$ need not admit one scale-independent eigenbasis. Matrices at different scales can fail to commute:

$$[\gamma(\mu_1),\gamma(\mu_2)]\ne0.$$

Pointwise diagonalization introduces derivatives of the basis matrix, so simply integrating instantaneous eigenvalues is generally wrong. Ordered evolution is developed on the coefficient-evolution page.

Finite basis covariance

Let

$$O'=B(g,\mu)O$$

be an invertible finite basis change. Differentiating and comparing

$$\mathcal D O'=-\gamma'O'$$

with $\mathcal D O=-\gamma O$ gives

$$\boxed{ \gamma' = B\gamma B^{-1} -(\mathcal D B)B^{-1}. }$$

For constant $B$, this is a similarity transformation. At a fixed point, a nonsingular coupling-dependent $B(g)$ also has $\mathcal D B=\beta^a\partial_aB=0$, so the eigenvalue spectrum of $\gamma_\star$ is invariant. Away from a fixed point, the connection term is essential and the eigenvalues of the instantaneous matrix are not scheme-invariant observables.

For the triangular scalar sector of the previous page,

$$Z_O= \begin{pmatrix} A^{-1}&-A^{-1}B_E\\ 0&1 \end{pmatrix},$$

where $B_E$ denotes the field-coordinate derivative there. Direct multiplication gives

$$\gamma = \begin{pmatrix} -\mathcal D\ln A& -\mathcal D B_E+B_E\mathcal D\ln A\\ 0&0 \end{pmatrix}.$$

The equation-of-motion direction has zero diagonal entry in its contact normalization, while the physical representative can still shift by it. A finite redefinition $P\to P+r(g)E$ changes the off-diagonal entry by the derivative term in the boxed covariance law and leaves the physical quotient unchanged.

Checks before interpreting eigenvalues

Check	What to calculate	Failure diagnosed
Sign	Differentiate $O_0=Z_OO$ explicitly	A convention imported from the inverse $Z$ relation
Finiteness	Evaluate $Z_O^{-1}\mathcal D Z_O$ including higher poles	Missing subdivergence or incomplete lower-order counterterms
Closure	Project every pole onto the enlarged basis	Omitted EOM, total-derivative, BRST-exact, identity, or evanescent operator
Basis covariance	Apply a finite $B$ and include $-(\mathcal DB)B^{-1}$	Spurious scheme dependence or a missing connection term
Eigenoperator orientation	Test $w^{\mathsf T}\gamma=\gamma_w w^{\mathsf T}$	Right eigenvectors mistaken for operator combinations
Fixed-point status	Verify all beta functions vanish and $B$ is nonsingular	Running matrix eigenvalues misreported as scaling dimensions
Jordan structure	Compare algebraic and geometric multiplicities	Logarithmic mixing hidden by an invalid diagonalization
Physical quotient	Retain the redundant block until after renormalization	On-shell projection used to conceal uncancelled off-shell poles

These entries specialize the chapter’s mixing convention record.

Common pitfalls

Defining $\gamma$ without stating $Z_O$. The sign is inseparable from whether bare operators equal $Z_O$ times renormalized operators or the inverse relation.

Keeping only the simple pole but skipping pole consistency. The simple-pole formula is the finite result of cancellations involving higher poles, the beta function, and $Z_O^{-1}$. Those cancellations must be verified.

Diagonalizing with right eigenvectors. With basis operators stored as a column, a linear combination is $w^{\mathsf T}O$ and uses a left eigenvector.

Calling running eigenvalues critical exponents. Scaling dimensions are fixed-point data. Away from a fixed point they depend on basis and receive a connection term under $g$-dependent finite changes.

Exercises

1. Derive the operator RG equation from $O_0=Z_OO$ without assuming that $Z_O$ commutes with its derivative.

Solution

At fixed bare data,

$$0=(\mathcal D Z_O)O+Z_O\mathcal D O.$$

Multiplying from the left by $Z_O^{-1}$ gives

$$\mathcal D O = -Z_O^{-1}(\mathcal D Z_O)O.$$

Therefore $\gamma=Z_O^{-1}\mathcal D Z_O$. Reversing the factors is not allowed for a noncommuting matrix.

2. Let $g_0=\mu^{2\epsilon}[g+\cdots]$ and $Z_O^{(1)}=-gM/3$. Find the leading anomalous-dimension matrix.

Solution

Here $\kappa=2$, so

$$\gamma = -2g\frac{\partial}{\partial g} \left(-\frac g3M\right) = \frac{2g}{3}M.$$

The answer changes if the bare coupling carries $\mu^\epsilon$ instead; the engineering exponent must be declared.

3. Verify the left eigenvectors of the benchmark matrix and show that the right eigenvector with eigenvalue $-1$ does not give the operator $O_2$.

Solution

For

$$M= \begin{pmatrix} 1&2\\ 0&-1 \end{pmatrix},$$

one finds

$$(1,1)M=(1,1), \qquad (0,1)M=-(0,1).$$

Thus the operator combinations are $O_1+O_2$ and $O_2$. The right eigenvector at eigenvalue $-1$ is proportional to $(1,-1)^{\mathsf T}$; treating it as coefficients of a row combination would incorrectly produce $O_1-O_2$.

4. Derive the finite-basis transformation of $\gamma$.

Solution

Differentiate $O'=BO$:

$$\mathcal D O' =(\mathcal DB)O-B\gamma O = \left[ (\mathcal DB)B^{-1}-B\gamma B^{-1} \right]O'.$$

Comparing with $\mathcal D O'=-\gamma'O'$ gives

$$\gamma'=B\gamma B^{-1}-(\mathcal DB)B^{-1}.$$

Continue to Dual Evolution of Operators and Wilson Coefficients to solve noncommuting scale evolution and derive the coefficient transpose from invariance of $C^{\mathsf T}O$. Continue to Symmetry-Protected Operators, Currents, and Improvement to determine when a zero anomalous dimension follows from an exact identity rather than a basis choice.

References

• Collins, John C. 1984; open-access reissue 2023. Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion. Cambridge Monographs on Mathematical Physics. Cambridge University Press. DOI and Open PDF.

• Joglekar, Satish D., and Benjamin W. Lee. 1976. “General Theory of Renormalization of Gauge Invariant Operators.” Annals of Physics 97 (1): 160–215. DOI.