Multiple Couplings and Coupled RG Flows

Most QFTs do not run along a single coupling axis. Every symmetry-allowed marginal or relevant parameter supplies a coordinate, and renormalization generally mixes them. The beta functions form a vector field; invariant subspaces identify consistent truncations, nullclines organize its direction, separatrices divide flow domains, and tangent evolution measures sensitivity to boundary data.

This page develops those tools and applies them to a real-scalar/Dirac-fermion Yukawa model continued to $d=4-\delta$. The one-loop system has a Gaussian point, a pure-scalar point, two scalar–Yukawa points, and analytic invariant rays. The positive ray is a heteroclinic trajectory from the Gaussian ultraviolet point to a perturbative infrared point when $0<\delta\ll1$. The calculation also shows why setting the quartic to zero while retaining the Yukawa coupling is not a closed truncation.

Required background. Beta Functions, Running Masses, and Field Anomalous Dimensions supplies the scalar–Yukawa coefficients and the fixed-bare sign conventions.

Helpful background. Operator Anomalous-Dimension Matrices gives the ordered matrix evolution used for masses and flavor tensors. Normal Forms, Spectra, and Projectors reviews the linear algebra of stability matrices.

RG flow as a vector field

Let

$$\mathbf g=(g^1,\ldots,g^n), \qquad t=\ln\frac{\mu}{\mu_0}.$$

The coupled RG equation is

$$\frac{dg^i}{dt}=\beta^i(\mathbf g).$$

One boundary point $\mathbf g(0)=\mathbf g_0$ selects a trajectory. A plot is meaningful only after declaring the coordinates, their normalizations, the orientation of $t$, the domain in which the beta functions are trusted, and the events that stop integration.

Several geometric objects serve different purposes:

Object	Definition	What it tells the reader
Nullcline for $g^i$	$\beta^i(\mathbf g)=0$	where the $i$th component of the arrows changes sign
Fixed point	$\boldsymbol\beta(\mathbf g_\star)=0$	where the entire flow vanishes
Invariant subspace	a surface $F^A(\mathbf g)=0$ with $\beta^i\partial_iF^A=0$ on the surface	which restricted coupling sets are closed under RG evolution
Separatrix	an invariant trajectory or surface separating distinct flow domains	which boundary data reach different endpoints or leave the validity region differently
Basin	boundary points sharing the same controlled endpoint	the domain of one asymptotic classification

A nullcline is not generally an invariant subspace. On $\beta^1=0$, the other components can move the trajectory to a point where $\beta^1\ne0$. By contrast, a coordinate plane $g^a=0$ is invariant exactly when

$$\beta^a(\mathbf g)\big|_{g^a=0}=0.$$

This is the practical closure test for a proposed one-coupling truncation. If a discarded coupling has a nonzero beta function on the proposed subspace, loops regenerate it and the truncation is inconsistent.

Collins derives the vector beta functions and the triangular dependence of dimensionless and dimensionful couplings in Collins 1984/2023, § 7.7, pp. 198–200.

Tangent evolution away from a fixed point

For two nearby trajectories, write $\mathbf g+\delta\mathbf g$. To first order,

$$\frac{d}{dt}\delta g^i = B^i{}_{j}(t)\,\delta g^j, \qquad B^i{}_{j}(t) = \left. \frac{\partial\beta^i}{\partial g^j} \right|_{\mathbf g(t)}.$$

The tangent propagator is

$$\delta\mathbf g(t) = M(t,t_0)\delta\mathbf g(t_0), \qquad M(t,t_0) = \mathcal P \exp\left[ \int_{t_0}^{t}B(s)\,ds \right].$$

Path ordering is required when $B(s_1)$ and $B(s_2)$ do not commute. Under a regular coordinate map $\mathbf g'=f(\mathbf g)$,

$$B' = JBJ^{-1} +\dot J J^{-1},$$

so the instantaneous eigenvalues of $B(t)$ away from a fixed point are not coordinate invariants. The full tangent map transforms covariantly:

$$\boxed{ M'(t,t_0) = J(t)M(t,t_0)J^{-1}(t_0). }$$

At a fixed point $\dot J=0$ and the familiar similarity relation returns. This distinction prevents an instantaneous arrow-field plot from being misread as a spectrum of physical scaling dimensions.

Scalar–Yukawa flow in 4 − δ dimensions

Use the model and normalization from the beta-function page: one real scalar, one massless Dirac fermion, Yukawa coupling $y$, and scalar interaction $\lambda\phi^4/4!$. Define

$$Y\equiv y^2, \qquad d=4-\delta, \qquad 0<\delta\ll1.$$

Here $\delta$ is the codimension below four dimensions, not the dimensional-regularization pole parameter used during subtraction. Adding the canonical terms to the one-loop four-dimensional coefficients gives the mass-independent flow

$$\boxed{ \begin{aligned} \dot Y &= -\delta Y +\frac{5Y^2}{8\pi^2}, \\ \dot\lambda &= -\delta\lambda +\frac{ 3\lambda^2+8\lambda Y-48Y^2 }{16\pi^2}. \end{aligned} }$$

Only the displayed one-loop polynomial is retained. The loop coefficients follow from Toms 2018, §§ 5.1–5.2, pp. 14–18; the continuation adds the engineering dimensions $[Y]=[\lambda]=\delta$. Control requires $Y/(16\pi^2)$, $|\lambda|/(16\pi^2)$, and $\delta$ to be small.

Closure and nullclines

The $Y$ equation factorizes:

$$\dot Y = \frac{5}{8\pi^2}Y(Y-Y_\star), \qquad Y_\star=\frac{8\pi^2}{5}\delta.$$

Hence $Y=0$ is invariant. In contrast,

$$\dot\lambda\big|_{\lambda=0} = -\frac{3Y^2}{\pi^2},$$

so $\lambda=0$ is not invariant when $Y\ne0$. Fermion loops generate the quartic counterterm. A “pure Yukawa” running calculation that discards $\lambda$ is therefore not closed.

The $\lambda$ nullclines are

$$\lambda_\pm(Y) = \frac{ 16\pi^2\delta-8Y \pm \sqrt{ (8Y-16\pi^2\delta)^2+576Y^2 } }{6}.$$

Together with $Y=0$ and $Y=Y_\star$, their intersections give all real fixed points of the displayed system.

Fixed points and the coupling-space stability matrix

For $Y>0$, introduce

$$r\equiv\frac{\lambda}{Y}, \qquad r_\pm=\frac{1\pm\sqrt{145}}{3}.$$

The four fixed points are:

Point	$(Y_\star^{\rm point},\lambda_\star^{\rm point})$	Stability eigenvalues for increasing $t=\ln\mu/\mu_0$	Controlled interpretation
Gaussian $G$	$(0,0)$	$(-\delta,-\delta)$	ultraviolet-attractive in the two-coupling plane
Pure scalar $S$	$(0,16\pi^2\delta/3)$	$(-\delta,+\delta)$	saddle; infrared-attractive only inside the invariant $Y=0$ subspace
Yukawa $Y_-$	$(8\pi^2\delta/5,r_-\,8\pi^2\delta/5)$	$(+\delta,-\sqrt{145}\,\delta/5)$	saddle with $\lambda_\star<0$; excluded if a stable classical quartic is required
Yukawa $Y_+$	$(8\pi^2\delta/5,r_+\,8\pi^2\delta/5)$	$(+\delta,+\sqrt{145}\,\delta/5)$	infrared-attractive within the coupling plane for $0<\delta\ll1$

The Jacobian at a general point is

$$B(Y,\lambda) = \begin{pmatrix} -\delta+\dfrac{5Y}{4\pi^2} & 0 \\[6pt] \dfrac{8\lambda-96Y}{16\pi^2} & -\delta+\dfrac{6\lambda+8Y}{16\pi^2} \end{pmatrix}.$$

At the positive scalar–Yukawa point,

$$B_{Y_+} = \delta \begin{pmatrix} 1 & 0 \\[4pt] \dfrac{4r_+-48}{5} & \dfrac{\sqrt{145}}{5} \end{pmatrix},$$

which directly gives the two positive eigenvalues in the table. Positive eigenvalues mean that perturbations shrink when $t$ decreases toward the infrared. The statement is confined to this two-coupling truncation; the interpretation of eigenoperators and critical exponents belongs to Chapter 5.

Wilson and Kogut explain how fixed points, invariant subspaces, and flow domains organize renormalized trajectories in Wilson and Kogut 1974, §§ 12.2–12.4, pp. 166–172.

Invariant rays and connecting trajectories

For $Y>0$, the ratio obeys

$$\begin{aligned} \dot r &= \frac{\dot\lambda Y-\lambda\dot Y}{Y^2} \\ &= \frac{Y}{16\pi^2} \left(3r^2-2r-48\right) \\ &= \frac{3Y}{16\pi^2} (r-r_+)(r-r_-). \end{aligned}$$

The canonical $-\delta$ terms cancel, so

$$\lambda=r_+Y \qquad\text{and}\qquad \lambda=r_-Y$$

are invariant rays. The positive ray lies in the stable-potential half-plane and connects $G$ to $Y_+$. For a boundary value $0<Y_0<Y_\star$ at $t=0$,

$$\boxed{ Y(t) = \frac{Y_\star}{ 1+\left(Y_\star/Y_0-1\right)e^{\delta t} }, \qquad \lambda(t)=r_+Y(t). }$$

Thus

$$\begin{aligned} t\to+\infty &: (Y,\lambda)\to(0,0), \\ t\to-\infty &: (Y,\lambda)\to(Y_\star,r_+Y_\star). \end{aligned}$$

This is a heteroclinic trajectory of the truncated system. The negative ray similarly connects $G$ to the saddle $Y_-$; because ratios below and above $r_-$ have different infrared behavior, that saddle trajectory is a separatrix. The exact positive-ray formula is useful for testing an integrator, but extrapolating it to $\delta=1$ without higher orders or nonperturbative evidence is not justified.

A relevant mass deformation

Let

$$u\equiv\frac{m^2}{\mu^2}.$$

Using the one-loop mass running from the same model,

$$\dot u = \left[ -2+ \frac{\lambda+4Y}{16\pi^2} \right]u.$$

At $Y_+$ this becomes

$$\dot u=-y_m u, \qquad y_m = 2-\frac{r_++4}{10}\delta.$$

For small positive $\delta$, $y_m>0$: as the trajectory runs toward the infrared, a nonzero $u$ grows and eventually invalidates the massless fixed-point flow. Reaching $Y_+$ requires tuning onto the massless critical surface $u=0$. This identifies the relevant deformation without attempting the full eigenoperator analysis.

Mass ratios, matrices, and flavor covariance

Several masses or relevant couplings generally obey a matrix equation,

$$\frac{d\mathbf m}{dt} = \Gamma_m\bigl(\mathbf g(t)\bigr)\mathbf m.$$

Its solution is path ordered:

$$\mathbf m(t) = \mathcal P \exp\left[ \int_{t_0}^{t} \Gamma_m\bigl(\mathbf g(s)\bigr)\,ds \right] \mathbf m(t_0).$$

Even if $\Gamma_m$ is diagonal, a ratio follows the complete coupling path:

$$\frac{m_1(t)/m_2(t)}{m_1(t_0)/m_2(t_0)} = \exp\left[ \int_{t_0}^{t} \bigl(\eta_1(\mathbf g(s))-\eta_2(\mathbf g(s))\bigr) ds \right].$$

There is no universal single-coupling formula unless the trajectory lies on a proven invariant subspace or ray.

Flavor couplings add another covariance. Under constant unitary basis changes,

$$Y_f' = U_LY_fU_R^\dagger, \qquad \beta_{Y_f}' = U_L\beta_{Y_f}U_R^\dagger.$$

Individual matrix entries depend on basis. Singular values, traces of consistently transformed products, and complete amplitudes are the meaningful comparison data. If the flavor basis itself runs, derivatives of $U_L(t)$ and $U_R(t)$ add connection terms, just as $\dot J J^{-1}$ enters tangent flow. Model-specific Standard Model matrix running is outside this page’s scope.

Numerical flow portraits and event checks

A reliable two-dimensional flow portrait follows a reproducible order:

1. declare $t$, coupling normalizations, loop order, scheme, and the trusted coupling box;
2. solve each nullcline analytically when possible and find their intersections;
3. evaluate $\boldsymbol\beta$ in every cell between nullclines to orient the arrows;
4. test proposed invariant subspaces by substituting their defining equations into the normal beta components;
5. integrate representative trajectories in both RG directions with threshold, strong-coupling, instability, and noninvertible-map events;
6. compare numerical fixed-point residuals and stability data with analytic values;
7. vary order and coordinates before assigning physical meaning to a finite-order endpoint.

A reproducible calculation uses the dimensionless benchmark

$$\beta_x=-x+x^2, \qquad \beta_y=-2y+\frac12xy+y^2.$$

Its nullclines are $x=0$, $x=1$, $y=0$, and $y=2-x/2$. The complete no-JavaScript reference is:

Fixed point	Stability matrix	Eigenvalues	Local type for increasing $t$
$(0,0)$	$\begin{pmatrix}-1&0\\0&-2\end{pmatrix}$	$(-1,-2)$	attractive
$(0,2)$	$\begin{pmatrix}-1&0\\1&2\end{pmatrix}$	$(-1,2)$	saddle
$(1,0)$	$\begin{pmatrix}1&0\\0&-3/2\end{pmatrix}$	$(1,-3/2)$	saddle
$(1,3/2)$	$\begin{pmatrix}1&0\\3/4&3/2\end{pmatrix}$	$(1,3/2)$	repulsive, hence infrared-attractive when RG time is reversed

This polynomial system is a deterministic numerical fixture, not by itself a claim that a continuum QFT realizes all four points.

For the regular map

$$u=x, \qquad v=y+\frac14x^2,$$

the interacting point maps to $(u_\star,v_\star)=(1,7/4)$. With

$$J_\star = \begin{pmatrix} 1&0 \\[3pt] 1/2&1 \end{pmatrix},$$

the transformed stability matrix is

$$B'_\star = J_\star B_\star J_\star^{-1} = \begin{pmatrix} 1&0 \\[3pt] 1/2&3/2 \end{pmatrix}.$$

The matrix entries change while the eigenvalues remain $(1,3/2)$. A numerical implementation should preserve that spectrum to relative tolerance $10^{-10}$. By contrast, $v=(y-3/2)^3$ has zero Jacobian at the fixed point and must produce an explicit noninvertible-map failure.

Coordinate-dependent data and invariant claims

The same comparison table used on the scheme-transformation page applies to multidimensional flows:

Item	What may change	What survives a consistent translation	Required qualification or check
Renormalized $g^i$, masses, and field normalizations	Numerical values under finite scheme or basis changes	A prediction expressed in the same physical inputs	Translate every parameter and field factor through the retained order
Beta function away from a fixed point	Components and higher-order coefficients	The integral curves as geometric trajectories under a nonsingular coordinate map	Compare transformed vector fields, not coefficients at equal numerical coupling
Elementary-field anomalous dimension	Finite field rescaling; gauge parameter in a gauge theory	Scaling of a gauge-invariant observable after all factors are combined	Never identify a gauge-dependent elementary-field exponent with an observable
Exact fixed point	Coordinate location $g_\star^i$	Existence of the zero under a regular map	Exclude singular redefinitions and verify the fixed point lies in the method’s domain
Fixed-point stability data	Matrix representation and basis	Eigenvalues in a closed physical sector	Include operator mixing and redundant directions before diagonalizing
Transmuted scale	Its conventional normalization	Matched dimensionless ratios or predictions	State the scheme and reference condition defining the scale
Zero or singularity of a truncated beta function	Location and even apparent existence at insufficient order	Only the demonstrated breakdown of the stated approximation	Vary scheme/order and stop before couplings become large
Wilson coefficient versus power correction	Factorization scheme and, for an asymptotic series, summation prescription	Their consistently defined sum in an observable	Match the ambiguity of the perturbative term to the operator matrix element; developed on the renormalon page
Residual $\mu$ or scheme dependence	Numerical size at finite order	Vanishing in the exact consistently matched prediction	Treat the residual as a diagnostic, not a universal probability law

For coupled flows, the second and fifth rows require special care: the whole curve and the full similarity class of the fixed-point matrix are covariant, while individual components, slopes in a chosen plot, and instantaneous eigenvalues away from a fixed point are not.

Common pitfalls

Calling every nullcline invariant. A nullcline freezes one coordinate only at that point. Test the normal component of the vector field on the entire proposed subspace.

Discarding a generated coupling. In the scalar–Yukawa example, $\dot\lambda|_{\lambda=0}\ne0$ for $Y\ne0$. The missing quartic is a required counterterm, not an optional refinement.

Reversing ultraviolet and infrared stability. Eigenvalues in this page use increasing $t=\ln\mu/\mu_0$. Positive eigenvalues repel toward the ultraviolet and attract when the flow is followed toward the infrared.

Diagonalizing $B(t)$ along a trajectory and calling the result critical exponents. Away from a fixed point, $B$ has an inhomogeneous coordinate-transformation term. Use the full tangent propagator for sensitivity and reserve critical exponents for controlled fixed-point analysis.

Extending an epsilon expansion to order-one $\delta$ without an error study. The interacting coordinates are perturbative because their loop-normalized values scale with $\delta$. That control is lost when $\delta$ is set to one without higher-order or nonperturbative evidence.

Letting an integrator cross a validity boundary. Dense output beyond a strong-coupling, threshold, instability, or singular-map event is a numerical continuation, not a QFT prediction.

Exercises

Verify that the positive scalar–Yukawa ray is invariant and derive its analytic trajectory.

Solution

For $r=\lambda/Y$,

$$\dot r = \frac{Y}{16\pi^2} (3r^2-2r-48).$$

The roots are $r_\pm=(1\pm\sqrt{145})/3$, so $\dot r=0$ identically on either ray. On $r=r_+$, the remaining equation is

$$\dot Y=-\delta Y\left(1-\frac{Y}{Y_\star}\right).$$

Separating variables and imposing $Y(0)=Y_0$ gives

$$Y(t) = \frac{Y_\star}{1+(Y_\star/Y_0-1)e^{\delta t}}, \qquad \lambda(t)=r_+Y(t).$$

Derive the transformed benchmark stability matrix for $u=x$, $v=y+x^2/4$.

Solution

At $(x,y)=(1,3/2)$,

$$B_\star = \begin{pmatrix} 1&0 \\[3pt] 3/4&3/2 \end{pmatrix}, \qquad J_\star = \begin{pmatrix} 1&0 \\[3pt] 1/2&1 \end{pmatrix}.$$

Since the point is fixed, the $\dot J J^{-1}$ term vanishes. Multiplication gives

$$J_\star B_\star J_\star^{-1} = \begin{pmatrix} 1&0 \\[3pt] 1/2&3/2 \end{pmatrix}.$$

Both matrices are triangular and have eigenvalues $1$ and $3/2$.

Where to continue

• Large Logarithms and RG Improvement uses coupled evolution to transport boundary functions between natural scales.
• Fixed Points and Linearized RG Flow develops eigenoperators, critical surfaces, and scaling fields at fixed points.
• Scheme Transformations and RG Invariants supplies the coordinate-covariance proof used in the tangent-flow and calculation checks.

References

• Collins, John C. Renormalization: An Introduction to Renormalization, the Renormalization Group, and the Operator-Product Expansion. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press, 1984; open-access digital edition, 2023. DOI. Open PDF.
• Toms, David J. “Effective Action for the Yukawa Model in Curved Spacetime.” Journal of High Energy Physics 2018, no. 5 (2018): 139. DOI. Open PDF.
• Wilson, Kenneth G., and J. Kogut. “The Renormalization Group and the ε Expansion.” Physics Reports 12, no. 2 (1974): 75–200. DOI.