Renormalization and Running Couplings

A loop diagram is an instruction to sum over virtual fluctuations at all momenta allowed by the theory. The high-momentum part of that sum probes distances much shorter than the wavelength of the external particles. From far away, a tiny loop cannot be resolved: it looks like a point. This is the basic reason ultraviolet divergences are not arbitrary monsters. They are local.

The previous page used one-loop diagrams and power counting to identify which ultraviolet divergences can occur. This page turns that observation into the renormalization group. A local divergence can be absorbed into the coefficient of a local operator. Once the coefficient has been fixed at one scale, the same physical amplitude at another scale contains logarithms. Those logarithms are most cleanly described by allowing masses, fields, and couplings to depend on the scale at which they are defined.

Renormalization is therefore not merely a trick for subtracting infinity. It is the statement that a quantum field theory is specified by a choice of local operators and a rule for how their coefficients change when the resolution scale changes.

Two scale variables will appear. A Wilsonian cutoff $\Lambda$ describes which modes are still present in an effective action. A renormalization scale $\mu$ describes where finite parameters are defined after a subtraction prescription has been chosen. They are related, but not identical. Confusing them is one of the fastest ways to get the sign of an RG flow wrong.

Ultraviolet subgraphs are local

This lesson preserves the lecture sequence and develops renormalization from local ultraviolet subgraphs. Ultraviolet Sensitivity and the Renormalization Problem develops the canonical locality and input-setting treatment; Renormalization- Group Equations and Running separates subtraction-scale evolution from Wilsonian coarse graining.

Wilsonian cutoff versus subtraction scale

This page uses a UV cutoff $\Lambda$ to explain locality and running. The same physics can be expressed in dimensional regularization and minimal subtraction, where logarithms of $\Lambda$ are replaced by poles in $4-d$ and logarithms of the subtraction scale $\mu$. The regulator-dependent details change, but the local operator structure and the renormalization-group equations do not.

Consider a one-loop subgraph with external momenta $p_i$ and loop momentum $k$. Its typical structure is

$$I(p_i)=\int^\Lambda {d^4k\over(2\pi)^4}\, {N(k,p_i)\over \prod_a\left((k+P_a(p_i))^2-m_a^2+i0\right)}.$$

The ultraviolet region is $|k|\gg |p_i|,m_a$. In that region the integrand can be expanded in powers of the small ratios $p_i/k$ and $m_a/k$:

$${N(k,p_i)\over \prod_a\left((k+P_a)^2-m_a^2+i0\right)} =\sum_{r=0}^{R}{\mathcal P_r(p_i,m_a;\hat k)\over |k|^{\alpha_r}} +\text{terms convergent as } |k|\to\infty .$$

After angular integration, the divergent terms are polynomials in the external momenta and masses. A polynomial in momentum space is a finite number of derivatives in position space, so the divergent part has the form of a local operator:

$$I_{\mathrm{div}}(p_i) =c_0(\Lambda)+c_2(\Lambda)p^2+c_4(\Lambda)p^4+\cdots .$$

The coefficients $c_r(\Lambda)$ may diverge as powers of $\Lambda$ or as $\log \Lambda$, but the dependence on the slowly varying external fields is local. Nonlocal dependence, such as $\log(-p^2/m^2)$ or a threshold square root, comes from the momentum region where the loop is comparable to physical momenta. That part is not removed by counterterms; it is real long-distance physics.

When the loop momentum is much larger than all external momenta, the loop is unresolved by the external fields. Its divergent part collapses to a sum of local operators such as $|\phi|^4$, derivative corrections, and higher-dimensional terms.

This is the conceptual core of perturbative renormalization. A divergent subgraph cannot demand an arbitrary nonlocal modification of the theory. It can only demand that the coefficients of local terms in the Lagrangian be adjusted.

Effective Lagrangians and changing the cutoff

Let $S_\Lambda$ denote an action defined with modes up to a cutoff $\Lambda$. Split a field into slow and fast modes,

$$\phi=\phi_{<}+\phi_{>},$$

where $\phi_{<}$ contains momenta $|p|<\Lambda'$ and $\phi_{>}$ contains the shell $\Lambda'<|p|<\Lambda$. The Wilsonian effective action at the lower cutoff is defined by

$$e^{iS_{\Lambda'}[\phi_{<}]} =\int_{\Lambda'<|p|<\Lambda}\mathcal D\phi_{>}\, e^{iS_\Lambda[\phi_{<}+\phi_{>}]}.$$

In Euclidean signature one replaces $e^{iS}$ by $e^{-S_E}$. The modes in the shell are not thrown away. Their effects are transferred into the coefficients of the lower-cutoff action.

Because the eliminated modes have short wavelengths compared with the remaining fields, their effect is captured by local operators:

$$S_{\Lambda'}[\phi] =\int d^4x\sum_i g_i(\Lambda')\,\mathcal O_i(x).$$

Equivalently,

$$\mathcal L_{\Lambda'} =\mathcal L_\Lambda+ \sum_i C_i(\Lambda,\Lambda')\mathcal O_i.$$

Lowering the cutoff from $\Lambda$ to $\Lambda'$ integrates out a shell of high-momentum modes. The result is a new effective Lagrangian with shifted coefficients $g_i(\Lambda')$ multiplying local operators $\mathcal O_i$.

The operators $\mathcal O_i$ are constrained by the symmetries of the theory. In a scalar theory with $\phi\to -\phi$, the allowed operators include $\phi^2$, $(\partial\phi)^2$, $\phi^4$, $\phi^6$, and so on. In scalar QED, the operators must be gauge invariant. This is why renormalization is never just dimensional analysis; it is dimensional analysis plus locality plus symmetry.

For a small shell, write

$$\Lambda'=\Lambda e^{-d\ell}.$$

The couplings obey differential flow equations

$${dg_i\over d\ell}=B_i(g_1,g_2,\ldots).$$

This is the Wilsonian renormalization group. The word “group” is historical: lowering the cutoff from $\Lambda$ to $\Lambda''$ can be done directly or by passing through an intermediate cutoff $\Lambda'$, and the result is the same after the appropriate rescaling and coupling redefinition. Strict coarse graining discards short-distance information and is therefore naturally a semigroup rather than an invertible group; the familiar name emphasizes the composition law of successive scale transformations.

Near the Gaussian fixed point, the leading part of this flow is dimensional. If $\mathcal O_i$ has engineering dimension $\Delta_i$ in four spacetime dimensions, then

$$[g_i]=4-\Delta_i.$$

After making couplings dimensionless, the linearized flow contains

$${d\bar g_i\over d\ell}=(4-\Delta_i)\bar g_i+\cdots$$

for the convention in which $\ell$ increases toward the infrared. Thus relevant operators with $\Delta_i<4$ grow under coarse graining, irrelevant operators with $\Delta_i>4$ shrink, and marginal operators require loop corrections to decide their fate.

This Wilsonian equation uses $\ell$ increasing toward the infrared. The beta functions later in the page use $\mu\,d/d\mu$, which increases toward the ultraviolet. The same physical flow can therefore look sign-reversed if one switches scale variables without saying so.

Scalar QED and the running set of couplings

A compact example that contains both matter self-interactions and gauge interactions is scalar QED. Its gauge-invariant Lagrangian is

$$\mathcal L =-{1\over4}F_{\mu\nu}F^{\mu\nu} +|D_\mu\phi|^2 -m^2|\phi|^2 -\lambda |\phi|^4,$$

with

$$D_\mu=\partial_\mu-ieA_\mu.$$

Here $\phi$ is taken to have charge $+e$, so $\phi\mapsto e^{ie\alpha}\phi$ and $A_\mu\mapsto A_\mu+\partial_\mu\alpha$. The conjugate field has the opposite charge.

A Wilsonian action at a sliding cutoff $\Lambda$ has coefficients

$$e(\Lambda), \qquad m^2(\Lambda), \qquad \lambda(\Lambda), \qquad Z_\phi(\Lambda), \qquad Z_A(\Lambda),$$

and, if we are honest in Wilsonian language, an infinite tower of higher-dimensional gauge-invariant operators suppressed by powers of the cutoff.

When the cutoff changes,

$$\Lambda\longrightarrow\Lambda',$$

these coefficients flow:

$$e(\Lambda)\longrightarrow e(\Lambda'), \qquad m^2(\Lambda)\longrightarrow m^2(\Lambda'), \qquad \lambda(\Lambda)\longrightarrow \lambda(\Lambda'), \qquad Z_\phi(\Lambda)\longrightarrow Z_\phi(\Lambda'), \qquad Z_A(\Lambda)\longrightarrow Z_A(\Lambda').$$

These are Wilsonian couplings along a trajectory of effective actions, not measured constants. It is useful to distinguish this from regulator language. If $\Lambda$ is a regulator that will be removed, the bare parameters $e_0(\Lambda)$, $m_0^2(\Lambda)$, and $\lambda_0(\Lambda)$ are tuned as the regulator changes so that observables stay fixed. If $\Lambda$ is a sliding Wilsonian resolution, the functions $g_i(\Lambda)$ record the effects of modes already integrated out. In a renormalized description, one instead varies $\mu$ while holding the bare theory fixed. All three descriptions encode the same scale dependence, but the quantities held fixed are different.

In scalar QED, the possible running terms are constrained by gauge invariance. For example,

$$F_{\mu\nu}F^{\mu\nu}, \qquad |D_\mu\phi|^2, \qquad |\phi|^2, \qquad |\phi|^4$$

are allowed local operators, while

$$A_\mu A^\mu$$

is not. A gauge-invariant renormalization of scalar QED can change the photon kinetic term, charge, scalar mass, scalar field normalization, and scalar self-coupling, but it cannot generate a photon mass.

The two marginal couplings $e$ and $\lambda$ generally run together:

$$\mu {de\over d\mu}=\beta_e(e,\lambda), \qquad \mu {d\lambda\over d\mu}=\beta_\lambda(e,\lambda).$$

At weak coupling the schematic form is

$$\beta_e=b_e e^3+\cdots,$$

and

$$\beta_\lambda =b_{\lambda\lambda}\lambda^2 +b_{e\lambda}e^2\lambda +b_{ee}e^4+\cdots .$$

The numerical coefficients depend on the normalization of $\lambda$ and on the charged matter content. For fixed normalizations, the leading nonzero coefficients are scheme independent under the usual analytic redefinitions, while higher-loop coefficients can be scheme dependent. The main lesson does not depend on those details: marginal couplings acquire logarithmic scale dependence once loops are included, and in a theory with several marginal couplings the RG flow is a vector field on coupling space. Never compare quoted coefficients without first checking the operator normalization, matter content, loop order, and scheme.

In scalar QED, short-distance loops can only renormalize gauge-invariant local operators. The coefficients of $F^2$, $|D\phi|^2$, $|\phi|^2$, and $|\phi|^4$ are part of the renormalizable data; higher operators are suppressed at low momentum but are naturally generated in the effective theory.

Renormalized couplings and logarithms

Suppose a dimensionless coupling $\lambda$ is measured by a four-point amplitude at a Euclidean momentum scale $\mu$. Call that measured value $\lambda(\mu)$. At one loop, the same amplitude at a different characteristic scale $q$ has the form

$$\lambda(q) =\lambda(\mu)+b\lambda(\mu)^2\log{q\over\mu}+O(\lambda^3),$$

where $b$ is a one-loop coefficient in the chosen convention. This is the simplest appearance of a running coupling.

Holding the bare theory fixed and changing the subtraction scale gives the same information in differential form. The expression can be written in a form that resums the leading logarithms generated by repeated one-loop insertions:

$$\lambda(q) ={\lambda(\mu) \over 1-b\lambda(\mu)\log(q/\mu)} +O(\text{nonleading logs}).$$

Expanding the denominator gives

$$\lambda(q) =\lambda(\mu) +b\lambda(\mu)^2\log{q\over\mu} +b^2\lambda(\mu)^3\log^2{q\over\mu} +\cdots .$$

Thus the one-loop beta function does more than reproduce the one-loop logarithm. It predicts an infinite tower of leading logarithms.

For $\beta(\lambda)=b\lambda^2$ with $b>0$, the coupling grows toward the ultraviolet and decreases toward the infrared. The running is logarithmic, so the natural horizontal variable is $\log\mu$.

The corresponding beta function is

$$\beta_\lambda(\lambda) =\mu{d\lambda\over d\mu} =b\lambda^2+O(\lambda^3).$$

At this order the solution is

$${1\over\lambda(\mu)} ={1\over\lambda(\mu_0)}-b\log{\mu\over\mu_0},$$

or

$$\lambda(\mu) ={\lambda(\mu_0) \over 1-b\lambda(\mu_0)\log(\mu/\mu_0)}.$$

If $b>0$, this solution has a pole at

$$\mu_L=\mu_0\exp\left[{1\over b\lambda(\mu_0)}\right].$$

This is a Landau pole in the perturbative running. It should not be overinterpreted as an exact singularity of nature. It says that the weak-coupling description cannot be extrapolated indefinitely beyond that scale.

Callan–Symanzik equation

The same idea can be stated without a literal cutoff. Let $G_R^{(n)}(p_i;\mu,g,m)$ be a renormalized $n$-point function. The scale $\mu$ is arbitrary; it was introduced to define the renormalized parameters. The bare theory does not know about this arbitrary choice. Therefore the renormalized correlator satisfies a differential equation of the form

$$\left( \mu{\partial\over\partial\mu} +\beta_g{\partial\over\partial g} -\gamma_m m{\partial\over\partial m} +n\gamma_\phi \right)G_R^{(n)}(p_i;\mu,g,m)=0,$$

up to convention-dependent signs in the definitions of $\gamma_m$ and $\gamma_\phi$. This is the Callan–Symanzik equation.

For an amplitude dominated by a single external scale $q$, the equation says that large logarithms of $q/\mu$ can be avoided by choosing

$$\mu\sim q.$$

Then the amplitude is computed using the coupling appropriate to the momentum flowing through the process:

$$g\to g(q).$$

This is the precise meaning of the phrase “the coupling runs.” It is not that the fundamental rules change from place to place. It is that the best local parameters for a coarse-grained description depend on the scale at which the theory is probed.

Positive and negative beta functions

The sign of the leading beta function controls the qualitative physics. If

$$\beta(\lambda)=+b\lambda^2, \qquad b>0,$$

then the coupling grows toward the ultraviolet. QED has this qualitative behavior: charged matter screens electric charge at long distances, so the effective charge increases at shorter distances. Perturbation theory eventually predicts a Landau pole far outside the domain where ordinary QED is expected to be complete.

If instead a gauge coupling satisfies

$$\beta(g)=-b_0g^3+O(g^5), \qquad b_0>0,$$

then

$${1\over g^2(\mu)} ={1\over g^2(\mu_0)}+2b_0\log{\mu\over\mu_0}.$$

The coupling decreases at short distances. This is asymptotic freedom. The theory becomes weakly coupled in the ultraviolet and strongly coupled in the infrared.

Two one-loop possibilities. A positive beta function drives the coupling upward toward the ultraviolet. A negative gauge beta function, $\beta(g)=-b_0g^3$, drives the coupling to zero in the ultraviolet and produces asymptotic freedom.

For an asymptotically free theory, the running can be written as

$$g^2(\mu)={1\over 2b_0\log(\mu/\Lambda_{\mathrm{dyn}})},$$

where

$$\Lambda_{\mathrm{dyn}} =\mu\exp\left[-{1\over 2b_0g^2(\mu)}\right].$$

This is dimensional transmutation. A dimensionless coupling is traded for a dimensionful scale. The classical Lagrangian may have no mass scale, but the quantum theory generates one through logarithmic running.

This course stops at the doorway of that idea. In the next course, the same logic becomes central in non-Abelian gauge theory, sigma models, confinement, instantons, and the modern Wilsonian view of QFT.

Example: resumming a leading logarithm

Suppose a four-point amplitude in a marginal scalar theory is known at one loop to be

$$\Gamma^{(4)}(q) =-\lambda(\mu) \left[1+b\lambda(\mu)\log{q\over\mu}+O(\lambda^2)\right].$$

The renormalization group says that, to leading-log accuracy, this is improved by replacing $\lambda(\mu)$ with the running coupling $\lambda(q)$:

$$\Gamma^{(4)}(q)\simeq -\lambda(q).$$

Using the one-loop solution,

$$\Gamma^{(4)}(q) \simeq -{\lambda(\mu)\over 1-b\lambda(\mu)\log(q/\mu)}.$$

Expanding the denominator reproduces the leading logarithms:

$$\Gamma^{(4)}(q) \simeq -\lambda(\mu) \left[1+b\lambda(\mu)\log{q\over\mu} +b^2\lambda(\mu)^2\log^2{q\over\mu}+\cdots\right].$$

A finite-order loop calculation gives the first few logarithms. The beta function organizes the infinite tower.

Summary

Renormalization is the statement that the effect of unresolved short-distance physics can be absorbed into local operators. At a fixed cutoff, loop diagrams produce divergent local terms. When the cutoff is changed, the coefficients of those local terms must change so that long-distance observables remain fixed.

For relevant operators, renormalization changes masses and vacuum energies. For marginal operators, the ultraviolet sensitivity is logarithmic. These logarithms become beta functions, and beta functions define running couplings. The one-loop equation $\beta(\lambda)=b\lambda^2$ resums powers of $\lambda\log(q/\mu)$; the sign of $b$ decides whether the coupling grows or decreases toward short distances.

The deepest lesson is that a quantum field theory is not defined only by a Lagrangian written at one scale. It is defined by a trajectory in the space of local actions. Symmetry restricts the allowed trajectory; locality makes the trajectory possible; logarithms make it visible.

Common pitfalls

Do not call every cutoff-dependent quantity unphysical. Bare parameters are cutoff dependent precisely so that physical quantities are cutoff independent.

Do not confuse the cutoff $\Lambda$ with the renormalization scale $\mu$. The cutoff is a regulator or Wilsonian resolution scale. The renormalization scale is a convention used to define finite couplings. In Wilsonian language they are related, but not identical; in minimal subtraction $\mu$ can appear even after the regulator has been removed.

Do not treat a Landau pole as an exact physical singularity unless the approximations are under control. It usually signals that the weak-coupling description has reached the edge of its validity.

Do not forget that couplings mix. In a theory with several marginal operators, the beta function is a vector field on coupling space, not a single number. Scalar QED already shows this: $e$ and $\lambda$ cannot be understood as completely independent one-coupling problems once loops are included.

Do not compare RG signs without checking the direction of the flow parameter. Wilsonian coarse graining often uses a variable that increases toward the infrared, while $\mu d/d\mu$ increases toward the ultraviolet.

Do not ignore symmetries. A divergent loop can only renormalize operators allowed by the symmetries of the regulated theory. Gauge invariance is especially restrictive.

Exercises

Exercise 1 — Solving a one-loop beta function

Let

$$\mu{d\lambda\over d\mu}=b\lambda^2, \qquad b>0.$$

Solve for $\lambda(\mu)$ in terms of $\lambda(\mu_0)$ and find the scale where the one-loop solution has a pole.

Solution

Separate variables:

$${d\lambda\over\lambda^2}=b{d\mu\over\mu}=b\,d\log\mu.$$

Integrating from $\mu_0$ to $\mu$ gives

$$-{1\over\lambda(\mu)}+{1\over\lambda(\mu_0)} =b\log{\mu\over\mu_0}.$$

Thus

$${1\over\lambda(\mu)} ={1\over\lambda(\mu_0)}-b\log{\mu\over\mu_0},$$

or

$$\lambda(\mu) ={\lambda(\mu_0)\over 1-b\lambda(\mu_0)\log(\mu/\mu_0)}.$$

The denominator vanishes at

$$\mu_L=\mu_0\exp\left[{1\over b\lambda(\mu_0)}\right].$$

This is the one-loop Landau-pole scale.

Exercise 2 — A negative beta function and dimensional transmutation

Let

$$\mu{dg\over d\mu}=-b_0g^3, \qquad b_0>0.$$

Solve the equation and show how a dimensionful scale $\Lambda_{\mathrm{dyn}}$ appears.

Solution

It is easiest to differentiate $1/g^2$:

$$\mu{d\over d\mu}{1\over g^2} =-{2\over g^3}\mu{dg\over d\mu} =2b_0.$$

Thus

$${1\over g^2(\mu)} ={1\over g^2(\mu_0)} +2b_0\log{\mu\over\mu_0}.$$

Define $\Lambda_{\mathrm{dyn}}$ by absorbing the integration constant:

$${1\over g^2(\mu)} =2b_0\log{\mu\over\Lambda_{\mathrm{dyn}}}.$$

Solving gives

$$\Lambda_{\mathrm{dyn}} =\mu\exp\left[-{1\over 2b_0g^2(\mu)}\right].$$

The dimensionless coupling has been traded for a dimensionful scale. This is dimensional transmutation.

Exercise 3 — Classifying scalar-QED operators

In four spacetime dimensions, classify the following scalar-QED operators as relevant, marginal, or irrelevant by engineering dimension:

$$\phi^\dagger\phi, \qquad (D_\mu\phi)^\dagger D^\mu\phi, \qquad F_{\mu\nu}F^{\mu\nu}, \qquad (\phi^\dagger\phi)^2, \qquad (\phi^\dagger\phi)^3, \qquad (\phi^\dagger\phi)F_{\mu\nu}F^{\mu\nu}.$$

Use $[\phi]=1$, $[A_\mu]=1$, $[D_\mu]=1$, and $[F_{\mu\nu}]=2$.

Solution

The dimensions are

$$[\phi^\dagger\phi]=2,$$

so $\phi^\dagger\phi$ is relevant.

Next,

$$[(D_\mu\phi)^\dagger D^\mu\phi]=2+2[\phi]=4,$$

so the scalar kinetic term is marginal by engineering dimension.

Also,

$$[F_{\mu\nu}F^{\mu\nu}]=4,$$

so the gauge kinetic term is marginal.

For the quartic interaction,

$$[(\phi^\dagger\phi)^2]=4,$$

so it is marginal in four dimensions.

For the sextic interaction,

$$[(\phi^\dagger\phi)^3]=6,$$

so it is irrelevant.

Finally,

$$[(\phi^\dagger\phi)F_{\mu\nu}F^{\mu\nu}]=2+4=6,$$

so this operator is also irrelevant and appears with a coefficient of order $1/\Lambda^2$ in a four-dimensional effective field theory.

Exercise 4 — A logarithmic shell integral

Evaluate the Euclidean shell integral

$$I(\Lambda,\Lambda')= \int_{\Lambda'<|k|<\Lambda}{d^4k\over(2\pi)^4}\,{1\over k^4}.$$

Solution

The area of the unit three-sphere is $\Omega_3=2\pi^2$. Therefore

$$I(\Lambda,\Lambda') ={2\pi^2\over(2\pi)^4} \int_{\Lambda'}^\Lambda dk\,{k^3\over k^4}.$$

Thus

$$I(\Lambda,\Lambda') ={1\over8\pi^2}\int_{\Lambda'}^\Lambda {dk\over k} ={1\over8\pi^2}\log{\Lambda\over\Lambda'}.$$

This is the basic logarithm produced by a four-dimensional loop whose ultraviolet integrand behaves as $1/k^4$.

Further reading

• Sidney Coleman, Lectures on Quantum Field Theory, edited by Bryan Gin-ge Chen et al., World Scientific, 2019, lectures on divergences, counterterms, QED renormalization, and the renormalization group.
• A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, 1987, chapter 2.
• Mark Srednicki, Quantum Field Theory, Cambridge University Press, 2007, chapters 18–21, 27–28, and 29–31.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, chapters 11–12; and Volume II: Modern Applications, Cambridge University Press, 1996, chapter 18.
• A. Zee, Quantum Field Theory in a Nutshell, 2nd edition, Princeton University Press, 2010, part III.