---
schema_version: 1
id: PHYS-2026-08-16-01
date: 2026-08-16
updated_at: 2026-08-16
title: The One-Loop Running of a Gauge Coupling
summary: "Derive the one-loop flow of g-squared in Yang-Mills theory, add matter and thresholds, then apply the same machinery to the Standard Model SU(2)_L coupling, electroweak multiplets, unification, and weakly coupled fixed points."
language: en
entry_kind: daily
status: published
level: research
user_difficulty: unrated
domains:
  - quantum-field-theory
  - particle-physics
  - mathematical-physics
estimated_minutes: 190
---

Plain-text extraction hides a useful ambiguity: `g2` may denote $g^2(\mu)$, the square of a generic Yang–Mills coupling, or $g_2(\mu)$, the Standard Model coupling carrying an $SU(2)_L$ label. Both readings will be developed here, beginning with the general theory and ending with the weak coupling

$$
\mu\frac{\mathrm d g_2}{\mathrm d\mu}
=
-\frac{19}{6}\frac{g_2^3}{16\pi^2}
+O(g_2^5).
$$

The coefficient $19/6$ combines the $SU(2)_L$ gauge-and-ghost sector, three generations of left-handed quark and lepton doublets, and one complex Higgs doublet. Changing that content changes the coefficient. Crossing a mass threshold changes it again.

At one loop, the ultraviolet divergence maps a gauge group and its charged field content to a definite beta-function coefficient.

## Part I. The one-loop calculation

### 1. Why the coupling appears to move

Consider a compact simple gauge group $G$, with Hermitian generators $T^a$. It is convenient to place the coupling in front of the Yang–Mills action:

$$
S_{\mathrm{YM}}
=
\frac{1}{2g_0^2}
\int \mathrm d^4x\,
\operatorname{tr}
F_{\mu\nu}F^{\mu\nu},
$$

where

$$
F_{\mu\nu}
=
\partial_\mu A_\nu-\partial_\nu A_\mu-i[A_\mu,A_\nu].
$$

In this convention $A_\mu$ contains no explicit $g_0$. Rescale the field as

$$
A_\mu=g_0\mathcal A_\mu.
$$

Then

$$
F_{\mu\nu}
=
g_0
\left(
\partial_\mu\mathcal A_\nu-
\partial_\nu\mathcal A_\mu-
ig_0[\mathcal A_\mu,\mathcal A_\nu]
\right),
$$

and the action has a canonically normalized quadratic term, a cubic vertex proportional to $g_0$, and a quartic vertex proportional to $g_0^2$. A diagram with one additional loop carries an additional factor of $g_0^2/(16\pi^2)$, apart from group factors and logarithms. This is why $g^2$, rather than $g$, is the natural loop-counting parameter.

Classically, $g_0$ is a number written in the action. Quantum mechanically, loop integrals sample arbitrarily large momenta. A regulator introduces a scale, and the counterterm used to remove the regulator depends on the scale at which the renormalized coupling is defined. The bare parameter remains fixed while the renormalized parameter changes:

$$
g_0
=
\mu^\epsilon Z_g(g,\epsilon)g(\mu),
\qquad
d=4-2\epsilon.
$$

The beta function is the response required to keep $g_0$ fixed:

$$
\beta(g)
\equiv
\mu\frac{\mathrm d g}{\mathrm d\mu}
\bigg|_{g_0}.
$$

No physical law says that an observable changes because we renamed an arbitrary subtraction scale. A scattering amplitude depends on external momenta and masses. Its explicit logarithms of $p/\mu$ cancel the implicit $\mu$ dependence of $g(\mu)$ order by order. The renormalization group arranges that cancellation and resums large logarithms.

For an $n$-point renormalized Green function, the statement takes the Callan–Symanzik form

$$
\left[
\mu\frac{\partial}{\partial\mu}
+
\beta(g)\frac{\partial}{\partial g}
+
n\gamma(g)
\right]
\Gamma^{(n)}=0,
$$

where $\gamma$ is the field anomalous dimension. Only the combination of $g(\mu)$ with the explicit momentum logarithms in an observable is independent of $\mu$.

### 2. What one loop has to calculate

At one loop, the gauge coupling is determined by the ultraviolet divergence multiplying the gauge-invariant operator

$$
\frac{1}{2}\operatorname{tr}F_{\mu\nu}F^{\mu\nu}.
$$

One can compute several vertex and propagator counterterms and use Slavnov–Taylor identities to combine them. The background-field method is cleaner for our purpose because it keeps gauge invariance under transformations of a slowly varying background field manifest.

Split the gauge potential into a background and a fluctuation,

$$
A_\mu=\bar A_\mu+a_\mu,
$$

and define the background covariant derivative

$$
\bar D_\mu X
=
\partial_\mu X-i[\bar A_\mu,X].
$$

The background is held fixed and supplies the external legs; the path integral is performed over the quantum fluctuation $a_\mu$. Choose the background-covariant gauge

$$
\bar D^\mu a_\mu=0.
$$

After adding the gauge-fixing term, the quadratic Euclidean action has the schematic form

$$
S^{(2)}
=
\frac{1}{2g_0^2}
\int \mathrm d^dx\,
a_\mu
(\Delta_1)^{\mu\nu}
a_\nu
+
\frac{1}{g_0^2}
\int \mathrm d^dx\,
\bar c\,\Delta_0 c,
$$

with

$$
(\Delta_1)_{\mu\nu}
=
-\bar D^2\delta_{\mu\nu}
-2\,\operatorname{ad}(\bar F_{\mu\nu}),
\qquad
\Delta_0=-\bar D^2.
$$

Factors of $i$ in the spin term depend on whether the generators and Euclidean continuation are defined with Hermitian or anti-Hermitian conventions. The coefficient extracted from the final gauge-invariant divergence does not depend on that choice.

The one-loop effective action is therefore

$$
\Gamma^{(1)}[\bar A]
=
\frac12\operatorname{Tr}\ln\Delta_1
-
\operatorname{Tr}\ln\Delta_0.
$$

The factor $1/2$ belongs to a Gaussian integral over a real vector field. The minus sign belongs to the Grassmann ghost determinant. The trace runs over momentum, Lorentz, and adjoint group indices.

#### 2.1 The two pieces inside the vector operator

The term $-\bar D^2\delta_{\mu\nu}$ treats each Lorentz component like a covariant scalar. The second term couples the spin-one fluctuation directly to the background curvature. In a heat-kernel calculation these two structures generate contributions with opposite signs. Tong describes them as diamagnetic and paramagnetic pieces. The terminology gives useful intuition, provided it is not pushed too far.

The division of the answer among gauge-field diagrams, ghost diagrams, and individual polarizations depends on the gauge-fixing choice. The sum does not. What survives every correct calculation is

$$
b_0^{\mathrm{pure\ YM}}
=
\frac{11}{3}C_A,
$$

where $C_A=C_2(G)$ is the adjoint quadratic Casimir.

Equivalently, minimal subtraction gives

$$
Z_g
=
1-
\frac{1}{\epsilon}
\frac{11C_A}{6}
\frac{g^2}{16\pi^2}
+O(g^4).
$$

Differentiating $g_0=\mu^\epsilon Z_g g$ while holding $g_0$ fixed yields

$$
\beta(g)
=
-\frac{11C_A}{3}
\frac{g^3}{16\pi^2}
+O(g^5).
$$

The negative sign gives asymptotic freedom: the coupling decreases as the energy scale increases. Gross and Wilczek, and independently Politzer, established this behavior in 1973; their original papers are [Phys. Rev. Lett. 30, 1343](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1343) and [Phys. Rev. Lett. 30, 1346](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1346).

### 3. Adding matter

Matter fields screen gauge charge. The calculation again reduces to functional determinants.

A massless complex scalar in a representation $R_s$ contributes

$$
\Gamma_s^{(1)}
=
\operatorname{Tr}\ln(-D^2).
$$

A Dirac fermion in a representation $R_f$ gives

$$
\Gamma_f^{(1)}
=
-\ln\det(i\gamma^\mu D_\mu).
$$

Squaring the Dirac operator exposes its coupling to the field strength:

$$
(i\gamma^\mu D_\mu)^2
=
-D^2
-\frac{i}{4}[\gamma^\mu,\gamma^\nu]F_{\mu\nu}.
$$

The spin term changes the coefficient relative to a collection of scalar components. The Grassmann determinant supplies the overall fermion sign. After the traces are evaluated, the general one-loop result is

$$
\boxed{
b_0
=
\frac{11}{3}C_A
-
\frac{2}{3}
\sum_{\text{Weyl }f}T(R_f)
-
\frac{1}{3}
\sum_{\text{complex }s}T(R_s)
}
$$

and

$$
\boxed{
\beta(g)
=
-\frac{b_0}{16\pi^2}g^3
+O(g^5).
}
$$

The same formula can be written with Dirac fermions and real scalars:

$$
b_0
=
\frac{11}{3}C_A
-
\frac{4}{3}
\sum_{\text{Dirac }f}T(R_f)
-
\frac{1}{6}
\sum_{\text{real }s}T(R_s).
$$

One Dirac fermion equals two Weyl fermions. One complex scalar equals two real scalars. Mixing the counting conventions is a common source of factors of two.

Some particle-physics references define coefficients $b_a$ by

$$
\mu\frac{\mathrm d g_a}{\mathrm d\mu}
=
\frac{b_a}{16\pi^2}g_a^3.
$$

In that convention

$$
b_a=-b_{0,a}.
$$

Thus “$b_2=-19/6$” and “$b_{0,2}=19/6$” are the same claim. Any numerical beta coefficient should be accompanied by its defining equation.

### 4. The group theory carried by a loop

Let $T_R^a$ denote the generator in a representation $R$. We define the Dynkin index and the quadratic Casimir by

$$
\operatorname{tr}_R(T_R^aT_R^b)
=
T(R)\delta^{ab},
$$

and

$$
T_R^aT_R^a
=
C_2(R)\mathbf 1_R.
$$

Taking a trace of the second equation gives

$$
d(R)C_2(R)
=
d(G)T(R).
$$

For $SU(N)$ in the usual particle-physics normalization,

$$
T(\mathbf N)=\frac12,
\qquad
C_F=\frac{N^2-1}{2N},
\qquad
C_A=N.
$$

For an $SU(2)$ representation with isospin $j$,

$$
d(j)=2j+1,
\qquad
C_2(j)=j(j+1),
$$

and hence

$$
T(j)
=
\frac13j(j+1)(2j+1)
=
\frac{d(j)[d(j)^2-1]}{12}.
$$

This gives

$$
T(\mathbf2)=\frac12,
\qquad
T(\mathbf3)=2,
\qquad
T(\mathbf4)=5.
$$

For a product representation $R_a\otimes R_b$, a loop coupled to group factor $G_a$ contains a copy for every state in $R_b$:

$$
T_a(R_a\otimes R_b)
=
T_a(R_a)d(R_b).
$$

This is why a weak quark doublet carries a color multiplicity of three. One must not multiply once more by the two weak components. Their trace is already contained in $T(\mathbf2)=1/2$.

The general formula and its higher-loop extension are organized systematically in [Machacek and Vaughn, Nucl. Phys. B 222, 83 (1983)](https://doi.org/10.1016/0550-3213(83)90610-7).

### 5. Integrating the one-loop equation

The beta function becomes especially transparent when written for the inverse squared coupling:

$$
\frac{\mathrm d}{\mathrm d\ln\mu}
\left(\frac1{g^2}\right)
=
-\frac{2\beta(g)}{g^3}
=
\frac{b_0}{8\pi^2}.
$$

Therefore

$$
\boxed{
\frac1{g^2(\mu)}
=
\frac1{g^2(\mu_0)}
+
\frac{b_0}{8\pi^2}
\ln\frac{\mu}{\mu_0}
}
$$

or

$$
g^2(\mu)
=
\frac{g^2(\mu_0)}
{1+\dfrac{b_0g^2(\mu_0)}{8\pi^2}
\ln(\mu/\mu_0)}.
$$

If $\alpha=g^2/(4\pi)$, the same equation is

$$
\alpha^{-1}(\mu)
=
\alpha^{-1}(\mu_0)
+
\frac{b_0}{2\pi}
\ln\frac{\mu}{\mu_0}.
$$

This explains the familiar plot of inverse gauge couplings as straight lines against $\ln\mu$. The approximation remains one-loop even when the equation is solved exactly. The denominator resums the leading logarithms $(g^2\ln\mu)^n$ generated by repeated insertion of the one-loop result.

#### 5.1 Translating the cutoff formula

Take $\mu_0=\Lambda_{\mathrm{UV}}$ and call the coupling there $g_0$. Then

$$
\frac1{g^2(\mu)}
=
\frac1{g_0^2}
-
\frac{b_0}{16\pi^2}
\ln\frac{\Lambda_{\mathrm{UV}}^2}{\mu^2}.
$$

For pure Yang–Mills, $b_0=11C_A/3$. This is the formula whose superscript was flattened into “g2” by the PDF extractor in [Tong's Gauge Theory notes](https://davidtong.org/pdfs/teaching/gauge-theory/gauge2.pdf).

#### 5.2 The beta function of $g^2$

If the quantity of interest is literally $g^2$, then

$$
\beta_{g^2}
\equiv
\mu\frac{\mathrm d g^2}{\mathrm d\mu}
=
2g\beta(g)
=
-\frac{b_0}{8\pi^2}g^4
+O(g^6).
$$

The sign is unchanged. Squaring the coupling does not create a new running law; it changes the coordinate used to describe the same flow.

### 6. Dimensional transmutation

When $b_0>0$, the one-loop solution can be rewritten as

$$
\frac1{g^2(\mu)}
=
\frac{b_0}{8\pi^2}
\ln\frac{\mu}{\Lambda},
$$

where

$$
\boxed{
\Lambda
=
\mu
\exp\left[
-\frac{8\pi^2}{b_0g^2(\mu)}
\right].
}
$$

The dimensionless input $g(\mu)$ has been traded for a dimensionful scale $\Lambda$. This is dimensional transmutation. At one loop, differentiating the right-hand side with respect to $\mu$ gives zero.

The formal divergence at $\mu=\Lambda$ does not prove that the exact coupling has a pole. It announces that perturbation theory has reached its boundary. In QCD, the corresponding $\Lambda_{\overline{\mathrm{MS}}}$ is a useful scheme-dependent scale, while hadron masses and cross sections are physical. Confinement and chiral symmetry breaking require nonperturbative physics beyond the one-loop beta function.

## Part II. From model counting to electroweak running

### 7. Three quick theories

Pure $SU(N)$ Yang–Mills has

$$
b_0=\frac{11}{3}N.
$$

It is asymptotically free for every $N\ge2$. The coupling is weak at short distance and grows toward the infrared.

An $SU(N_c)$ theory with $N_f$ Dirac fermions in the fundamental has

$$
b_0
=
\frac{11}{3}N_c
-
\frac{4}{3}N_fT(\mathbf{N_c})
=
\frac{11}{3}N_c-
\frac{2}{3}N_f.
$$

Asymptotic freedom at one loop requires

$$
N_f<\frac{11}{2}N_c.
$$

For QCD, $N_c=3$. With six active quark flavors, $b_0=7$. At lower energies the top, bottom, and charm quarks successively decouple, so the slope changes across their thresholds.

QED has $C_A=0$. For $N_f$ Dirac fermions of unit charge,

$$
\beta(e)
=
\frac{N_f}{12\pi^2}e^3
+O(e^5).
$$

The coupling grows toward the ultraviolet. Matter screening is all that remains because an Abelian photon has no Yang–Mills self-interaction. The one-loop solution has a remote ultraviolet Landau pole. That pole says that QED alone cannot define a weakly coupled continuum theory to arbitrarily high energy; electroweak physics intervenes long before it becomes numerically relevant.

### 8. Computing the Standard Model weak coefficient

We now return to the genuinely subscripted coupling $g_2$. Above the electroweak scale, the Standard Model gauge group is

$$
SU(3)_c\times SU(2)_L\times U(1)_Y.
$$

Use the convention

$$
\mu\frac{\mathrm d g_i}{\mathrm d\mu}
=
\frac{b_i}{16\pi^2}g_i^3.
$$

For $SU(2)_L$, the gauge contribution is

$$
b_2^{\mathrm{gauge}}
=
-\frac{11}{3}C_A
=
-\frac{22}{3}.
$$

Only left-handed fermion doublets contribute. In each generation, the quark doublet $Q_L$ has three color copies and the lepton doublet $L_L$ has one:

$$
\sum_{\text{one generation}}T_2(R_f)
=
3\times\frac12
+
1\times\frac12
=
2.
$$

With three generations,

$$
\sum_{\text{Weyl }f}T_2(R_f)=6,
$$

so

$$
b_2^{\mathrm{fermions}}
=
\frac23\times6=4.
$$

The Standard Model Higgs is one complex doublet. It gives

$$
b_2^{\mathrm{Higgs}}
=
\frac13T(\mathbf2)
=
\frac16.
$$

Combining the sectors,

$$
\boxed{
b_2
=
-\frac{22}{3}+4+\frac16
=
-\frac{19}{6}.
}
$$

The count can be read at a glance:

| Standard Model sector | contribution to $b_2$ |
|---|---:|
| $SU(2)_L$ gauge field and ghosts | $-22/3$ |
| three generations of $Q_L$ | $+3$ |
| three generations of $L_L$ | $+1$ |
| one complex Higgs doublet | $+1/6$ |
| total | $-19/6$ |

Right-handed quarks and charged leptons are weak singlets. Gauge-singlet right-handed neutrinos would also leave $b_2$ unchanged. Their Yukawa couplings can affect gauge running only at higher loop order.

The one-loop Standard Model coefficients are

$$
(b_1,b_2,b_3)
=
\left(
\frac{41}{10},
-\frac{19}{6},
-7
\right),
$$

where

$$
g_1=\sqrt{\frac53}\,g_Y.
$$

Without the grand-unified normalization, the hypercharge coefficient is $b_Y=41/6$. The $U(1)$ normalization must be stated whenever different references are compared. The current [Particle Data Group review of grand unification](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-guts.pdf) uses the GUT-normalized convention above.

### 9. How fast does $g_2$ actually run?

The 2026 Particle Data Group grand-unification review uses the illustrative $\overline{\mathrm{MS}}$ inputs

$$
\hat\alpha^{-1}(M_Z)=127.930,
\qquad
\sin^2\hat\theta_W(M_Z)=0.23122.
$$

Using the corresponding $\overline{\mathrm{MS}}$ definitions at $M_Z$, the leading relation is

$$
\alpha_2
=
\frac{\hat\alpha}{\sin^2\hat\theta_W},
$$

so

$$
\alpha_2^{-1}(M_Z)
=
\hat\alpha^{-1}(M_Z)
\sin^2\hat\theta_W(M_Z)
\simeq29.580,
$$

and

$$
g_2(M_Z)
=
\sqrt{4\pi\alpha_2(M_Z)}
\simeq0.6518.
$$

With $M_Z=91.1879\,\mathrm{GeV}$, the one-loop solution is

$$
\alpha_2^{-1}(\mu)
=
29.580
+
\frac{19}{12\pi}
\ln\frac{\mu}{M_Z}.
$$

It gives the following teaching-level extrapolation:

| $\mu$ | $\alpha_2^{-1}(\mu)$ | $g_2(\mu)$ |
|---:|---:|---:|
| $M_Z$ | $29.580$ | $0.6518$ |
| $1\,\mathrm{TeV}$ | $30.787$ | $0.6389$ |
| $10^6\,\mathrm{GeV}$ | $34.268$ | $0.6056$ |
| $10^{10}\,\mathrm{GeV}$ | $38.910$ | $0.5683$ |
| $10^{16}\,\mathrm{GeV}$ | $45.873$ | $0.5234$ |
| $2.435\times10^{18}\,\mathrm{GeV}$ | $48.643$ | $0.5083$ |

Across the table, the modest one-loop slope raises $\alpha_2^{-1}$ by about $19$. A logarithm needs many decades of energy to make an order-one change in $g_2$, but a small change in the slope sustained over that interval produces a visible shift in a unification test.

The final row should not be treated as a precision Standard Model prediction at the reduced Planck scale. Two- and three-loop terms, electroweak matching, quark thresholds, and any new particles modify the curve. Quantum gravity may invalidate the field theory extrapolation near that scale. Precision studies use multi-loop running and matching, as in [Buttazzo et al.](https://arxiv.org/abs/1307.3536), rather than the single straight line above.

There is a further convention trap. The on-shell weak angle, the effective leptonic angle measured at the $Z$ pole, and $\sin^2\hat\theta_W(M_Z)$ in $\overline{\mathrm{MS}}$ are different renormalized quantities. One cannot insert whichever value is most familiar into the equation for an $\overline{\mathrm{MS}}$ coupling. The distinctions are tabulated in the current [PDG electroweak review](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-standard-model.pdf).

### 10. A massive field does not run forever

The formula for $b_0$ was derived as if every field were massless. A particle of mass $M$ contributes fully when the external scale is far above $M$ and decouples from low-energy amplitudes when the scale is far below $M$. The original general statement is the [Appelquist–Carazzone decoupling theorem](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.11.2856).

There is a technical qualification. The $\overline{\mathrm{MS}}$ counterterm is mass independent, so a heavy field does not automatically disappear from an $\overline{\mathrm{MS}}$ beta function. One constructs an effective field theory (EFT) without the heavy field and matches its coupling to the full-theory coupling near $\mu=M$.

Let

$$
\Delta b
=
b_{\mathrm{full}}-b_{\mathrm{EFT}}.
$$

At leading logarithmic order, the relation can be written as

$$
\alpha_{\mathrm{EFT}}^{-1}(\mu)
=
\alpha_{\mathrm{full}}^{-1}(\mu)
+
\frac{\Delta b}{2\pi}
\ln\frac{\mu}{M}
+
\delta_{\mathrm{finite}}.
$$

For a simple fermion or scalar threshold, choosing $\mu=M$ removes the one-loop logarithm. Finite matching corrections can remain for broken gauge theories, nondegenerate spectra, particular mass definitions, and higher orders.

If $\mu_0$ lies below all new thresholds and $\mu$ lies above some masses $M_a$, the leading-log result is

$$
\alpha_i^{-1}(\mu)
=
\alpha_i^{-1}(\mu_0)
-
\frac{b_i^{\mathrm{light}}}{2\pi}
\ln\frac{\mu}{\mu_0}
-
\sum_{M_a<\mu}
\frac{\Delta b_i^a}{2\pi}
\ln\frac{\mu}{M_a}.
$$

Every threshold changes the slope of the inverse coupling. The exact crossover is smooth in a physical amplitude; the stepwise EFT description reproduces it as a controlled expansion in powers of external momentum over $M_a$.

#### 10.1 The electroweak threshold is structural

Below the weak scale, the Higgs expectation value breaks

$$
SU(2)_L\times U(1)_Y
\longrightarrow
U(1)_{\mathrm{EM}}.
$$

The $W$ and $Z$ are massive and are eventually integrated out. The appropriate low-energy theory contains QCD and QED, plus higher-dimensional four-fermion and other operators. It does not contain an independent unbroken $SU(2)_L$ gauge field whose coupling can be run unchanged toward zero energy.

One can still quote a renormalized $g_2(\mu)$ within the full Standard Model at a chosen scale. Connecting it to low-energy observables requires electroweak matching. Extrapolating the coefficient $-19/6$ below every weak threshold would confuse a convenient full-theory parameter with the active coupling of the low-energy EFT.

### 11. New electroweak multiplets

The one-loop effect of new matter can be read directly from its $SU(2)$ isospin. For a colorless isospin-$j$ multiplet,

$$
\Delta b_2^{\mathrm{Weyl}}
=
\frac23T(j),
\qquad
\Delta b_2^{\mathrm{Dirac}}
=
\frac43T(j),
$$

and

$$
\Delta b_2^{\mathrm{complex\ scalar}}
=
\frac13T(j),
\qquad
\Delta b_2^{\mathrm{real\ scalar}}
=
\frac16T(j).
$$

If the multiplet also transforms under color, multiply by the dimension of its color representation. Some examples are:

| new colorless field | $T_2$ | $\Delta b_2$ |
|---|---:|---:|
| complex scalar doublet | $1/2$ | $1/6$ |
| Weyl fermion doublet | $1/2$ | $1/3$ |
| Dirac fermion doublet | $1/2$ | $2/3$ |
| real scalar triplet | $2$ | $1/3$ |
| complex scalar triplet | $2$ | $2/3$ |
| Majorana fermion triplet | $2$ | $4/3$ |
| Dirac fermion triplet | $2$ | $8/3$ |
| complex scalar quartet | $5$ | $5/3$ |

Matter makes $b_2$ less negative. Five additional colorless Dirac doublets give

$$
b_2
=
-\frac{19}{6}
+5\left(\frac23\right)
=
\frac16.
$$

The weak coupling then loses asymptotic freedom at one loop. Nineteen additional complex scalar doublets make $b_2=0$ exactly at this order.

For $b_2>0$, the formal one-loop Landau scale is

$$
\Lambda_L
=
\mu_0
\exp\left[
\frac{8\pi^2}{b_2g_2^2(\mu_0)}
\right].
$$

This estimate becomes unreliable if a new threshold, a sizeable two-loop term, a Yukawa coupling, or a fixed point appears first. Large representations introduce a second warning sign: the relevant expansion parameter can be $g_2^2C_2(R)/(16\pi^2)$, which may cease to be small well below the naive pole.

#### 11.1 A beta function does not certify a consistent spectrum

The table accepts any representation as bookkeeping input. A chiral gauge theory must also satisfy its anomaly constraints. Local cubic and mixed anomalies restrict hypercharges and representations. $SU(2)$ has an additional global obstruction: an odd number of left-handed fundamental doublets is inconsistent, even though the perturbative cubic $SU(2)$ anomaly vanishes. More generally, for the original Witten anomaly on a spin manifold, the fermion representations must obey

$$
\sum_f 2T(R_f)=0\pmod 2.
$$

The fundamental has $2T(\mathbf 2)=1$, which gives the familiar even-doublet rule. This is the [Witten $SU(2)$ anomaly](https://doi.org/10.1016/0370-2693(82)90728-6).

The Standard Model has twelve left-handed weak doublets when color copies are counted, so the number is even. Adding one isolated Weyl doublet would spoil that consistency. A vectorlike Dirac doublet contains two Weyl doublets and avoids the global anomaly, though its hypercharge assignments must still be checked.

Running tells us how an allowed theory changes with scale. It does not decide whether the proposed field content defines an allowed quantum theory in the first place.

### 12. Two Higgs doublets and supersymmetry

An additional complex Higgs doublet contributes $1/6$ to $b_2$. With GUT-normalized hypercharge, the two-Higgs-doublet model (2HDM) has

$$
(b_1,b_2,b_3)_{\mathrm{2HDM}}
=
\left(
\frac{21}{5},
-3,
-7
\right).
$$

The one-loop gauge coefficients do not care whether the Yukawa sector is type I, II, X, or Y. Those variants use the same gauge representations. Their different Yukawa assignments enter the gauge beta functions at two loops.

The minimal supersymmetric Standard Model (MSSM) changes the counting more substantially. For an $\mathcal N=1$ supersymmetric gauge theory,

$$
b=-3C_A+
\sum_{\text{chiral multiplets}}T(R).
$$

For $SU(2)_L$, the vector multiplet gives $-6$. Each of three generations contributes

$$
3\times T(\mathbf2)+T(\mathbf2)=2
$$

from $Q$ and $L$, and the two Higgs chiral multiplets contribute one more unit in total. Thus

$$
b_2^{\mathrm{MSSM}}
=
-6+3\times2+1=1.
$$

The full MSSM vector is

$$
(b_1,b_2,b_3)_{\mathrm{MSSM}}
=
\left(
\frac{33}{5},
1,
-3
\right).
$$

Above the superpartner thresholds, $g_2$ grows rather than shrinks toward the ultraviolet. The change in sign is modest enough that the coupling remains perturbative up to the usual grand-unification scale. A detailed supersymmetric calculation is normally performed in $\overline{\mathrm{DR}}$ rather than $\overline{\mathrm{MS}}$; precision work includes the scheme conversion and nondegenerate superpartner thresholds. A useful entry point is [Martin's Supersymmetry Primer](https://arxiv.org/abs/hep-ph/9709356).

### 13. Why three straight lines became evidence for unification

At one loop,

$$
\alpha_i^{-1}(\mu)
=
\alpha_i^{-1}(M_Z)
-
\frac{b_i}{2\pi}
\ln\frac{\mu}{M_Z}.
$$

If the three couplings meet at one scale, their measured separations and their slopes must obey

$$
R_{\mathrm{exp}}
\equiv
\frac{\alpha_2^{-1}-\alpha_3^{-1}}
{\alpha_1^{-1}-\alpha_2^{-1}}
=
\frac{b_2-b_3}{b_1-b_2}.
$$

Using $\alpha_s(M_Z)\simeq0.1180$ together with the electroweak inputs above gives

$$
R_{\mathrm{exp}}\simeq0.717.
$$

The slope ratios are

| theory above the weak threshold | $(b_2-b_3)/(b_1-b_2)$ |
|---|---:|
| Standard Model | $0.528$ |
| two-Higgs-doublet model | $0.556$ |
| MSSM | $0.714$ |

One additional Higgs doublet moves the slopes toward the measured ratio but does not make them meet. The MSSM value $0.714$ lies close to $R_{\mathrm{exp}}\simeq0.717$ in this one-loop treatment. Its couplings meet near

$$
M_G\simeq2\times10^{16}\,\mathrm{GeV},
\qquad
\alpha_G^{-1}\simeq24.3.
$$

This is evidence about relative slopes, not a direct observation of a grand-unified gauge boson. The current [PDG GUT review](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-guts.pdf) emphasizes the required qualifications: two-loop evolution, weak-scale and supersymmetric thresholds, $\overline{\mathrm{MS}}$ to $\overline{\mathrm{DR}}$ conversion, and GUT-scale thresholds all matter in a precision fit. The minimal Standard Model misses exact unification; supersymmetric spectra can achieve it accurately, but the result depends on their masses.

A degenerate complete GUT multiplet gives equal one-loop shifts to the canonically normalized $b_i$. It changes the unified coupling while leaving $b_i-b_j$ and the one-loop meeting scale unchanged. Splitting the multiplet masses generates unequal threshold logarithms and can move the apparent intersection.

## Part III. Beyond the isolated one-loop coefficient

### 14. At two loops, $g_2$ stops running alone

The independence of the three gauge equations is a one-loop property. In the Standard Model, with $g_1=\sqrt{5/3}g_Y$, the next term for $g_2$ is

$$
\begin{aligned}
\beta_{g_2}
={}&
-\frac{19}{6}
\frac{g_2^3}{16\pi^2}
\\[4pt]
&+
\frac{g_2^3}{(16\pi^2)^2}
\bigg[
\frac9{10}g_1^2
+\frac{35}{6}g_2^2
+12g_3^2
\\[2pt]
&\hspace{36mm}
-\frac32\operatorname{Tr}(Y_u^\dagger Y_u)
-\frac32\operatorname{Tr}(Y_d^\dagger Y_d)
-\frac12\operatorname{Tr}(Y_e^\dagger Y_e)
\bigg]
+\cdots .
\end{aligned}
$$

The strong coupling enters through quarks carrying both color and weak isospin. Hypercharge enters through fields carrying both $Y$ and weak isospin. Yukawa couplings enter because the same matter propagators are dressed by Higgs interactions. The Higgs quartic does not enter a gauge beta function at two loops; it first appears at three loops.

Once these terms are retained, all gauge, Yukawa, and scalar equations must be solved as a coupled system. The general two-loop organization goes back to [Machacek and Vaughn](https://doi.org/10.1016/0550-3213(83)90610-7). The Standard Model gauge functions are known through higher orders; [Mihaila, Salomon, and Steinhauser](https://arxiv.org/abs/1201.5868) computed the three-loop gauge contributions including Yukawa and Higgs-sector effects.

Two-loop corrections are small over a short interval when every coupling is perturbative. Across many decades, small local corrections accumulate. A unification analysis that quotes percent-level intersections while using only one-loop slopes has assigned more precision than the calculation contains.

### 15. When two loops create a fixed point

Consider $SU(N_c)$ with $N_f$ Dirac fundamental fermions and write

$$
\beta(g)
=
-\frac{g^3}{16\pi^2}\beta_0
-
\frac{g^5}{(16\pi^2)^2}\beta_1
+O(g^7),
$$

where

$$
\beta_0
=
\frac{11}{3}N_c-
\frac{2}{3}N_f,
$$

and

$$
\beta_1
=
\frac{34}{3}N_c^2
-
\left(
\frac{10}{3}N_c+2C_F
\right)N_f.
$$

Take $N_f$ just below $11N_c/2$. Then $\beta_0$ is positive and small, while $\beta_1$ is negative. The two terms balance at

$$
\frac{g_*^2}{16\pi^2}
=
-\frac{\beta_0}{\beta_1}.
$$

The flow approaches this value in the infrared. Because $g_*^2$ can be made parametrically small by choosing $N_f$ close to the loss of asymptotic freedom, the fixed point is perturbatively controlled. This is the Banks–Zaks mechanism, building on the two-loop analysis of [Caswell](https://doi.org/10.1103/PhysRevLett.33.244) and developed in [Banks and Zaks](https://doi.org/10.1016/0550-3213(82)90035-9).

For $SU(3)$ with $N_f=16$,

$$
\beta_0=\frac13,
\qquad
\beta_1=-\frac{302}{3},
$$

so

$$
\alpha_*
=
\frac{g_*^2}{4\pi}
\simeq0.0416.
$$

At this order, the zero is the perturbative signal of an interacting infrared conformal field theory. Moving to smaller $N_f$ strengthens the fixed point until perturbation theory can no longer locate the lower edge of the conformal window reliably. A one-loop coefficient can tell us where asymptotic freedom ends; it cannot determine the entire infrared phase diagram.

### 16. What is universal, and what belongs to a scheme

The one-loop coefficient $b_0$ is independent of the gauge-fixing parameter and of ordinary analytic changes of renormalization scheme. It is also insensitive to Yukawa and scalar self-couplings. Those facts make it unusually robust.

The numerical value of a renormalized coupling at a quoted scale is scheme dependent. A finite redefinition

$$
g'
=
g+c\frac{g^3}{16\pi^2}+O(g^5)
$$

changes higher-order beta coefficients while leaving the one-loop term intact. For a single coupling, the two-loop coefficient is also unchanged under such an analytic redefinition. With several coupled gauge and Yukawa parameters, scheme changes act on the full vector field of beta functions, and comparisons require all conventions to be transformed consistently.

The scale $\Lambda$ obtained by dimensional transmutation is scheme dependent by a multiplicative constant. A physical mass gap or cross section is not. The relation between the two contains nonperturbative information.

The running coupling itself is therefore not an observable in isolation. Experiments determine cross sections, decay rates, asymmetries, and potential energies. A renormalization prescription extracts $g(\mu)$ from those data. Another prescription extracts a different number and supplies different perturbative coefficients, while predicting the same observable when calculated consistently.

For electroweak theory, the tree-level identities

$$
e=g_2\sin\theta_W=g_Y\cos\theta_W
$$

and

$$
\frac1{e^2}
=
\frac1{g_2^2}
+
\frac1{g_Y^2}
$$

are useful definitions at leading order. Beyond tree level, the weak angle and the couplings must share a stated scheme and matching prescription. Mixing an on-shell angle with $\overline{\mathrm{MS}}$ couplings breaks that consistency.

### 17. Domain of validity

The one-loop equation is controlled only while every relevant loop parameter remains small, the field content is fixed, and the symmetry phase has been stated. A threshold requires matching to a new effective theory. A long extrapolation may require higher-loop gauge, Yukawa, and scalar running even when the coupling never becomes large.

Within those limits, the calculation establishes four precise facts. A non-Abelian gauge field contributes with the sign required for ultraviolet asymptotic freedom. Matter screens that effect in amounts fixed by Dynkin indices. The inverse squared coupling runs linearly with $\ln\mu$ between thresholds. A dimensionless coupling can generate a physical scale through dimensional transmutation.

The same calculation does not establish confinement, a mass gap, exact gauge unification, or a physical Landau singularity. It does not make $g(\mu)$ scheme independent. Those questions require EFT matching, higher loops, nonperturbative methods, or experimental input.

## Part IV. Problems and references

### 18. Problems

#### Problem 1: recover the cutoff expression

Start from

$$
\beta(g)=-\frac{b_0}{16\pi^2}g^3.
$$

Derive the beta function for $g^2$, integrate it between $\Lambda_{\mathrm{UV}}$ and $\mu$, and recover the logarithm of $\Lambda_{\mathrm{UV}}^2/\mu^2$. Explain why a factor of two moves when the logarithm is squared.

<details>
<summary>Solution</summary>

Since $\mu\,\mathrm d g^2/\mathrm d\mu=2g\beta(g)$,

$$
\mu\frac{\mathrm d g^2}{\mathrm d\mu}
=
-\frac{b_0}{8\pi^2}g^4.
$$

It follows that

$$
\frac{\mathrm d}{\mathrm d\ln\mu}\frac1{g^2}
=
\frac{b_0}{8\pi^2}.
$$

Integrating from $\Lambda_{\mathrm{UV}}$ to $\mu$ gives

$$
\frac1{g^2(\mu)}
=
\frac1{g^2(\Lambda_{\mathrm{UV}})}
-
\frac{b_0}{8\pi^2}
\ln\frac{\Lambda_{\mathrm{UV}}}{\mu}.
$$

Because $\ln(\Lambda^2/\mu^2)=2\ln(\Lambda/\mu)$, this is

$$
\frac1{g^2(\mu)}
=
\frac1{g_0^2}
-
\frac{b_0}{16\pi^2}
\ln\frac{\Lambda_{\mathrm{UV}}^2}{\mu^2}.
$$

</details>

#### Problem 2: count $b_2$ without double counting

Compute the Standard Model $b_2$ using Weyl fermions and complex scalars. Repeat the calculation by grouping the fermions as much as possible into Dirac fields. Identify why the second route is awkward before electroweak symmetry breaking.

<details>
<summary>Solution</summary>

The Weyl count is

$$
b_2
=
-\frac{11}{3}(2)
+
\frac23
\left[
3_{\mathrm{gen}}
\left(
3_{\mathrm{color}}\frac12+\frac12
\right)
\right]
+
\frac13\frac12
=
-\frac{19}{6}.
$$

The unbroken Standard Model is chiral. A left-handed weak doublet and its right-handed partners do not transform in the same $SU(2)_L$ representation, so they do not form a Dirac representation of the weak gauge group. Grouping particles into massive Dirac fermions is natural only after symmetry breaking, precisely where the unbroken-theory beta coefficient must be connected to a different EFT by matching. The Weyl formula respects the ultraviolet gauge representations directly.

</details>

#### Problem 3: insert one threshold

Add a colorless vectorlike Dirac weak doublet of mass $M=10\,\mathrm{TeV}$. Use $\alpha_2^{-1}(M_Z)=29.580$ and stepwise one-loop running to estimate $g_2(10^{10}\,\mathrm{GeV})$. Compare with the Standard Model value in the table.

<details>
<summary>Solution</summary>

Below $M$, $b_2=-19/6$. Above $M$, the Dirac doublet adds $2/3$, so

$$
b_2^{\mathrm{high}}
=
-\frac{19}{6}+\frac23
=
-\frac52.
$$

Therefore

$$
\begin{aligned}
\alpha_2^{-1}(10^{10}\,\mathrm{GeV})
={}&29.580
-\frac{-19/6}{2\pi}
\ln\frac{10^4}{M_Z}
\\
&-
\frac{-5/2}{2\pi}
\ln\frac{10^{10}}{10^4}
\\
\simeq{}&37.44.
\end{aligned}
$$

Thus

$$
g_2(10^{10}\,\mathrm{GeV})
\simeq
\sqrt{\frac{4\pi}{37.44}}
\simeq0.579.
$$

The Standard Model-only value is about $0.568$. The extra matter screens the gauge contribution, so $g_2$ decreases more slowly toward the ultraviolet.

</details>

#### Problem 4: test one-loop unification without finding the meeting scale

Show that exact one-loop unification implies

$$
\frac{\alpha_2^{-1}-\alpha_3^{-1}}
{\alpha_1^{-1}-\alpha_2^{-1}}
=
\frac{b_2-b_3}{b_1-b_2}.
$$

Evaluate the right-hand side for the Standard Model and the MSSM. Why is this ratio insensitive to the assumed unified coupling and unification scale?

<details>
<summary>Solution</summary>

If all three couplings equal $\alpha_G$ at $M_G$, then

$$
\alpha_i^{-1}(M_Z)
=
\alpha_G^{-1}
+
\frac{b_i}{2\pi}
\ln\frac{M_G}{M_Z}.
$$

Subtracting two equations eliminates $\alpha_G^{-1}$. Taking a ratio of two differences also eliminates $\ln(M_G/M_Z)$. For the Standard Model,

$$
\frac{-19/6+7}{41/10+19/6}
=
\frac{115/30}{218/30}
\simeq0.528.
$$

For the MSSM,

$$
\frac{1+3}{33/5-1}
=
\frac4{28/5}
=
\frac57
\simeq0.714.
$$

The experimental ratio is about $0.717$, before precision threshold and higher-loop corrections.

</details>

#### Problem 5: locate a perturbative Banks–Zaks point

For $SU(3)$ with $N_f=16$ fundamental Dirac fermions, compute $b_0$, $b_1$, and $\alpha_*$. Then estimate the expansion parameter and explain why the result is more trustworthy than the same formula at $N_f=9$.

<details>
<summary>Solution</summary>

For $SU(3)$, $C_F=4/3$, so

$$
b_0=11-\frac23N_f,
\qquad
b_1=102-\frac{38}{3}N_f.
$$

At $N_f=16$,

$$
b_0=\frac13,
\qquad
b_1=-\frac{302}{3}.
$$

The fixed point satisfies

$$
\frac{g_*^2}{16\pi^2}
=
-\frac{b_0}{b_1}
=
\frac1{302},
$$

and therefore

$$
\alpha_*
=
\frac{g_*^2}{4\pi}
=
\frac{4\pi}{302}
\simeq0.0416.
$$

The loop parameter is about $0.0033$, so higher loops are plausibly controlled. At $N_f=9$, the candidate fixed point is much stronger. Higher-loop and nonperturbative effects can shift or remove a fixed point inferred from a low-order truncation, so the two-loop answer alone cannot locate the lower edge of the conformal window.

</details>

### 19. Further reading

1. [David Tong, *Gauge Theory*, Chapter 2](https://davidtong.org/pdfs/teaching/gauge-theory/gauge2.pdf). Sections 2.4 and 2.7 motivate the pure Yang–Mills result, the background-field computation, matter contributions, and QCD-like infrared phases.
2. [D. J. Gross and F. Wilczek, “Ultraviolet Behavior of Non-Abelian Gauge Theories”](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1343). One of the two original 1973 discoveries of asymptotic freedom.
3. [H. D. Politzer, “Reliable Perturbative Results for Strong Interactions?”](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1346). The independent companion result.
4. [M. E. Machacek and M. T. Vaughn, “Two-Loop Renormalization Group Equations in a General Quantum Field Theory”](https://doi.org/10.1016/0550-3213(83)90610-7). The systematic general framework for gauge theories with fermions and scalars.
5. [Particle Data Group, “Grand Unified Theories”](https://pdg.lbl.gov/2026/reviews/rpp2026-rev-guts.pdf). Current conventions, Standard Model and MSSM coefficients, thresholds, and the status of precision unification.
6. [T. Appelquist and J. Carazzone, “Infrared Singularities and Massive Fields”](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.11.2856). The foundational decoupling theorem behind threshold EFTs.
7. [L. N. Mihaila, J. Salomon, and M. Steinhauser, “Gauge Coupling Beta Functions in the Standard Model to Three Loops”](https://arxiv.org/abs/1201.5868). A concise route from the one-loop coefficients to modern high-order running.
8. [J. Brod and Z. Polonsky, “Two-loop Beta Function for Complex Scalar Electroweak Multiplets”](https://arxiv.org/abs/2007.13755). A useful extension when large $SU(2)$ scalar representations or minimal-dark-matter multiplets are added.
9. [T. Banks and A. Zaks, “On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions”](https://doi.org/10.1016/0550-3213(82)90035-9). The perturbative infrared fixed point near the loss of asymptotic freedom.

### 20. The compact answer

For a simple gauge group with Weyl fermions and complex scalars,

$$
\beta(g)
=
-\frac{g^3}{16\pi^2}
\left[
\frac{11}{3}C_A
-\frac{2}{3}\sum_fT(R_f)
-\frac{1}{3}\sum_sT(R_s)
\right]
+O(g^5).
$$

The corresponding running of the squared coupling is

$$
\mu\frac{\mathrm d g^2}{\mathrm d\mu}
=
-\frac{b_0}{8\pi^2}g^4
+O(g^6),
$$

with solution

$$
\frac1{g^2(\mu)}
=
\frac1{g^2(\mu_0)}
+
\frac{b_0}{8\pi^2}
\ln\frac{\mu}{\mu_0}.
$$

For the Standard Model $SU(2)_L$ coupling,

$$
b_{0,2}
=
\frac{22}{3}-4-\frac16
=
\frac{19}{6},
$$

so $g_2$ decreases logarithmically toward the ultraviolet. The coefficient remains $19/6$ only while the active theory is the unbroken Standard Model with its stated field content. Threshold matching is required whenever the active field content changes.