research29 min studyLesson 684 in Physics

Scale compensation relates running parameters to field normalization

Derive the Callan–Symanzik equation with an explicit anomalous-dimension sign and distinguish scale variation from physical dilation.

Before you begin

On this page

The question for today

What you’ll understand

Derive the RG equation for renormalized connected correlators.
State the sign convention for the field anomalous dimension.
Relate a fixed-point anomalous dimension to spatial scaling.

Changing the arbitrary renormalization scale cannot change an exact physical prediction, but renormalized correlation functions carry field-normalization conventions and need not stay numerically fixed. The Callan–Symanzik equation organizes the compensation among explicit scale dependence, running parameters, and external fields. We will derive its sign from a bare correlator and then distinguish varying μ at fixed coordinates from physically separating two insertions at a scale-invariant fixed point.

Differentiate at fixed bare theory

Consider a massless theory with one dimensionless coupling g and renormalized connected n-point correlator GR at separated points. From φB=Zφ¹ᐟ²φ, the bare correlator equals Zφ to the power n/2 times GR. Differentiate that relation with respect to lnμ while holding bare parameters and the regulator prescription fixed. Define β as dg/dlnμ and γφ as one half the derivative of lnZφ. The derivative of the bare correlator vanishes, producing the displayed equation with a plus nγφ term.

This derivation fixes a convention rather than asserting that every reference uses the same sign or the same object. A one-particle-irreducible vertex uses inverse external-field normalization and has a different field term. A massive theory adds derivatives with respect to running masses and other dimensionful parameters. Coincident composite insertions can require extra contact counterterms. These extensions should be added from their definitions, not inferred by copying the massless connected-correlator equation into every amplitude.

G_B^{(n)}=Z_\phi^{n/2}G_R^{(n)},\qquad \gamma_\phi=\frac12\left.\frac{d\ln Z_\phi}{d\ln\mu}\right|_B,\qquad \left(\mu\partial_\mu+\beta(g)\partial_g+n\gamma_\phi(g)\right)G_R^{(n)}=0

The coupling and γφ are dimensionless, and μ is an energy scale. Partial μ differentiation holds g and insertion positions fixed; the total derivative additionally runs g. The equation concerns massless separated-point connected correlators with the specified normalization.↔ Scroll wide equations horizontally.

A fixed point changes scaling dimensions

At an RG fixed point β vanishes, but the field anomalous dimension need not. Combining the renormalization equation with canonical dimensional analysis yields a total scaling dimension equal to the canonical dimension plus γφ at the fixed point. For a scalar scaling operator, the Euclidean two-point function therefore decays as distance to minus twice that dimension. The coefficient depends on the operator normalization. At a conformal fixed point, this power law is a physical statement about the correlator’s dependence on separation after conventions are fixed.

Elementary gauge-field or gauge-dependent matter-field anomalous dimensions need not themselves be gauge-invariant observables. Physical gauge-invariant operator dimensions, and invariant critical exponents at a genuine fixed point, have a stronger interpretation. Away from a fixed point, γ generally runs and integrates along the flow rather than supplying one constant power at all distances. Mixing operators must also be treated as a matrix. A nonzero β signals additional scale dependence, so replacing the full solution by one guessed anomalous exponent can miss logarithmic running.

\Delta_\phi=\frac{d-2}{2}+\gamma_\phi^*,\qquad \langle\phi(x)\phi(0)\rangle=\frac{C}{r^{2\Delta_\phi}},\qquad \frac{G_R^{(n)}(\mu_2)}{G_R^{(n)}(\mu_1)}=\exp\!\left[-n\int_{\mu_1}^{\mu_2}\gamma_\phi(g(\mu))\,d\ln\mu\right]

The first two expressions concern a scalar scaling operator at an appropriate fixed point, r=|x|, and γ* its fixed-point value. The last expression follows along an RG characteristic at fixed insertion coordinates; it describes normalization running, not a physical change of separation.↔ Scroll wide equations horizontally.

Physical scale dependence remains after arbitrary-scale cancellation

An observable can depend on an actual momentum Q even when its dependence on arbitrary μ cancels exactly. Moving Q changes an experiment; moving μ reorganizes the calculation of the same experiment. Field-normalization changes cancel against external-leg factors or operator coefficients in complete physical quantities. At finite perturbative order the cancellation is incomplete by higher-order terms. Residual μ dependence can help diagnose missing orders, but it is not an exact error bar and need not reveal every important correction.

The heat lab illustrates a multiplicative exponential with a known rate. Reset and set both controls to mode n=1 and diffusivity κ=0.5, then Step twice to t=1 second; predict amplitude exp(−0.5)≈0.607. Reset and set n=2 and κ=0.5, then Step twice; predict exp(−2)≈0.135. This is one analytic periodic heat mode. Its time can be used only as an analogy for a logarithmic scale parameter; its decay is not a field anomalous dimension, quantum RG flow, or physical loss of probability.

Work it through

Compute a fixed-point spatial decay

A scalar scaling operator in four dimensions has γφ*=0.10. Normalize its Euclidean two-point function to one at separation r₀. Find its total dimension and the correlator at 2r₀. Use 2⁰·²≈1.149.

The canonical scalar dimension is (4−2)/2=1. Adding the anomalous part gives Δφ=1.10.
The two-point power is 2Δφ=2.20. Doubling separation therefore multiplies the normalized correlator by 2⁻²·².
Using 2²·²=4×1.149≈4.596 gives a ratio about 0.218. The free canonical result would instead be 1/4, showing the effect of the anomalous exponent.

The result

The total dimension is 1.10 and the normalized correlator at doubled separation is approximately 0.218.

The living notebook

Explore the model

Interactive lab

Interactive lab · heat

Inspect multiplicative exponential factors

Reset and set BOTH controls to n=1 and κ=0.5, then Step twice to t=1 s: predict amplitude e⁻⁰·⁵≈0.607. Reset and set n=2 and κ=0.5, then Step twice: predict e⁻²≈0.135. This periodic heat mode is an exponential analogy, not QFT evolution.

Current temperature modeInitial sineCurrent amplitude

Current amplitude: 1
Spatial L² norm: 1.772
Decay rate · κn²: 3.6 /s

Spatial mode n3Diffusivity κ0.4

t = 0 s

The exact periodic heat-equation mode is u(x,t) = exp(−κn²t) sin(nx), on [0,2π]. Playback advances physical time. The initial sine is shown separately, and the unnormalized L² norm is √π exp(−κn²t). This analytic instrument does not run a finite-difference solver or test discretization stability. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA scale-normalization factor involves an integrated rate, while the physical interpretation of that rate belongs to the underlying theory.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Q labels physical kinematics, while μ organizes a calculation. RG running compensates explicit μ dependence; it does not remove genuine momentum dependence from observables.

Your turn

Try it yourself

At a fixed coupling, suppose Zφ(μ)/Zφ(μ₀)=(μ/μ₀)^0.4. Compute γφ in the stated convention and the change of a two-point renormalized correlator when μ doubles at fixed physical coordinates. Use 2⁰·⁴≈1.320.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Differentiate lnZφ, divide by two, and remember that GR carries Zφ⁻¹ for two fields.

Compare with the solution

γφ=0.4/2=0.20. The two-point correlator changes by 2⁻⁰·⁴≈1/1.320≈0.758 at fixed coordinates. This is a change of renormalized field normalization, not the response to doubling separation. Suitable external-leg or coefficient factors cancel it in a complete physical observable.

Keep the idea

Take it with you

The anomalous-dimension sign follows from the chosen field-renormalization relation.
Physical dilation and arbitrary subtraction-scale variation are different operations.
Exact observables are μ independent; truncated predictions retain higher-order scale dependence.

Read further

Original sources and deeper explanations to follow at your own pace.

Niklas Beisert — Quantum Field Theory II, Lecture Notes, section 5.3: Callan–Symanzik Equation and Running Coupling, printed pp. 5.15–5.18

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.