Running coupling

A running coupling is a renormalized coupling regarded as a function of the renormalization scale. Its scale derivative is its beta function.

For $\beta (\lambda )=-{\beta }_0{\lambda }^3/(4\pi {)}^2-{\beta }_1{\lambda }^5/(4\pi {)}^4+O({\lambda }^7)$, define $y=(4\pi {)}^2/{\lambda }^2$ and $L=\log ({\mu }^2/{\Lambda }^2)$. Then $dy/dL={\beta }_0+{\beta }_1/y+O(y^{-2})$. In the asymptotically free regime ${\beta }_0>0$, successive substitution gives the displayed expansion. If the beta function is exactly truncated to two loops, then $y-\frac{{\beta }_1}{{\beta }_0}\log [(y+{\beta }_1/{\beta }_0)/{\beta }_0]={\beta }_{0}L$, after choosing the integration constant. Its next term is ${\beta }_1^2(\log L+1)/({\beta }_0^{3}L)$. Thus the generic remainder after the first two terms is not $O(1/L)$ when ${\beta }_1\ne 0$. Higher-loop terms change the constant coefficient of $1/L$, but not the forced $\log L/L$ term.

From $\mu d{g}_i/d\mu =b_i{g}_i^3/(16{\pi }^2)$ and ${\alpha }_i=g_i^2/(4\pi )$, the chain rule gives $d{\alpha }_i/d\log \mu =b_i{\alpha }_i^2/(2\pi )$. Taking its reciprocal makes the differential equation linear, proving the displayed relation. It assumes a fixed set of active fields and constant one-loop coefficients between the two scales; mass thresholds require piecewise matching. A negative coefficient gives asymptotic freedom.

With ${\beta }_0>0$, differentiating ${\alpha }_s=g^2/(4\pi )$ under $\mu dg/d\mu =-{\beta }_0{g}^3/(16{\pi }^2)$ gives $d(1/{\alpha }_s)/d\ln \mu ={\beta }_0/(2\pi )$. Integration defines the strong-coupling scale $\Lambda$ and gives the displayed positive branch for $\mu >\Lambda$. The pole signals breakdown of perturbative quantum field theory. The equivalence between specifying ${\alpha }_s$ and specifying $\Lambda$ is dimensional transmutation.

For positive ${\beta }_0$, the one-loop running coupling tends to zero at high energy and grows as energy decreases. Near the strong-coupling scale, omitted perturbative terms cease to be small; the apparent infrared pole is not a controlled prediction of a physical infinity. Continuing the algebraic formula below $\Lambda$ gives a negative coupling and is not a valid physical continuation. Confinement and hadron formation are nonperturbative phenomena, not a theorem proved by this one-loop pole. The integration constant obeys $\Lambda ={\mu }_0\exp [-1/(2{\beta }_0{\alpha }_0)]$, so an order-one coefficient alone cannot determine an absolute mass scale; a reference coupling or empirical input is also necessary.

At a heavy-quark mass $m$, leading-order continuity gives ${\beta }_{-}\ln (m/{\Lambda }_{-})={\beta }_{+}\ln (m/{\Lambda }_{+})$. Therefore $\boxed{{\Lambda }_{+}=m({\Lambda }_{-}/m{)}^{{\beta }_{-}/{\beta }_{+}}}$. This is a change of the strong-coupling scale parametrization across a change of active flavour number, not a discontinuity of the matched QCD coupling. For four to five active flavours in $SU(3)$, ${\beta }_{-}=25/3$ and ${\beta }_{+}=23/3$, hence ${\Lambda }_5={\Lambda }_4(m_b/{\Lambda }_4{)}^{-2/23}$.

A Landau pole is a finite renormalization scale at which a perturbatively running coupling diverges. It signals that the perturbative description cannot be continued to that scale without new physics or a nonperturbative completion.

In the self-coupling-dominated approximation $d{\alpha }_H/d\ln {\mu }^2=b{\alpha }_H^2$, $b>0$, the running coupling reaches a perturbative Landau pole at the displayed scale. With the conventional Higgs potential, $m_H^2=2{\lambda }_H{v}^2$ and one can normalize ${\alpha }_H={\lambda }_H/(4\pi )$. A larger Higgs boson mass therefore decreases the logarithmic interval before the self-coupling becomes strong. Requiring perturbative validity up to a fixed high cutoff gives an upper Higgs-mass bound in this approximation. The full Standard Model beta function also contains gauge and Yukawa couplings; its sign is not universally positive. A perturbative pole is not a theorem excluding every nonperturbative ultraviolet completion.

The beta function of a coupling $g$ isIt describes how the running coupling changes when the renormalization scale changes.

For a field in a tensor-product representation of a product gauge group, its contribution to the beta function of factor $i$ includes the dimensions of all other factors. Those indices label independent copies for that factor. For an Abelian factor, replace its Dynkin index by the squared charge. Thus a quark weak doublet contributes three copies to the weak beta function and two copies to the colour beta function; hypercharge sums include all six components. Chirality conjugation changes a representation and the sign of an Abelian charge, but not its quadratic index.

For a four-dimensional dimensionless gauge coupling, the one-loop correction is cubic in $g$. With $\alpha =g^2/(4\pi )$, the chain rule and $d/d\ln {\mu }^2=(1/2)d/d\ln \mu$ give the displayed conversion. In QCD, $c=-(11N-2n_f)/(48{\pi }^2)$, hence $d{\alpha }_s/d\ln {\mu }^2=-(11N-2n_f){\alpha }_s^2/(12\pi )+O({\alpha }_s^3)$. Other types of coupling or several interacting couplings can have different leading expansions; the squared-scale convention must not be mixed with the unsquared one.

In $d=4-ϵ$, the quartic scalar coupling has engineering dimension $ϵ$. Factoring out ${\mu }^{ϵ}$ makes the renormalized running coupling dimensionless. Differentiating its bare expression at fixed bare parameters generates the linear term $-ϵ\lambda$, even when the quantum contribution to the renormalization-group beta function begins quadratically. The simple-pole formulas for minimal-subtraction renormalization group functions determine the remaining contribution from the simple pole of $F$.

For three quark generations, their corresponding leptons, and one complex Higgs doublet, use $Q=T_3+Y$ and $D_{\mu }\supset i{g}_{Y}Y{B}_{\mu }$. In $\mu d{g}_i/d\mu =b_i{g}_i^3/(16{\pi }^2)$ the coefficients are the displayed values. Per generation, the fermionic Dynkin index sums are two for both non-Abelian factors and the sum of squared hypercharges is $10/3$. The Higgs index for $SU(2)$ and its sum of squared hypercharges are each $1/2$. Gauge terms are $-22/3,-11$ for $SU(2),SU(3)$, respectively. For the grand-unified normalization $g_1=\sqrt{5/3}{g}_Y$, the Abelian coefficient instead is $41/10$.

For $A,B>0$, the nonzero zero of the renormalization-group beta function has slope ${\beta }^{\prime }(g_{\ast })=-2B^2/A<0$ and is ultraviolet attractive. The origin is infrared attractive. A fixed-point field-renormalization anomalous dimension gives a power-law quantum field theory propagator factor $d(z)\sim C{z}^{{\gamma }_{\varphi }(g_{\ast })}$. A weak fixed point is needed for control by a perturbative truncation.

For the asymptotically free two-loop running coupling $da/d\log \mu =-{\beta }_0{a}^2-{\beta }_1{a}^3$, with ${\beta }_0>0$, let $y=1/a$, $c={\beta }_1/{\beta }_0$, and $L=\log (\mu /\Lambda )$. A choice of the scale $\Lambda$ gives the exact implicit equation $y-c\log ((y+c)/{\beta }_0)={\beta }_{0}L$ on the ultraviolet branch. Its expansion isThe remainder after the first two terms is therefore $O(\log L/L)$, and is not $O(1/L)$ when ${\beta }_1\ne 0$. A fixed rescaling of $\Lambda$ cannot change the coefficient of $\log L/L$. This distinction matters when quoting errors for asymptotic expansions.

An asymptotically free theory has a running coupling that tends to zero at arbitrarily high energy. Its negative weak-coupling beta function makes short-distance perturbation theory increasingly accurate.

A non-Abelian gauge theory can have a weak infrared fixed point when the one-loop coefficient of its renormalization-group beta function is small and positive while the two-loop coefficient is negative. For $a=g^2/(16{\pi }^2)$ and evolution in $\log {\mu }^2$, $\beta (a)=-{\beta }_0{a}^2-{\beta }_1{a}^3$ has the displayed zero and slope $-{\beta }_0^2/{\beta }_1>0$. Fundamental-fermion flavor number just below the asymptotic-freedom boundary produces this regime.

For $\beta =-b{g}^3$ and $\gamma =c{g}^2$, integrate the running coupling equation and then the anomalous-dimension multiplier. The correlator correction grows as a power $c/b$ of the logarithm of energy, rather than as a fixed power of energy. Higher perturbative orders change subleading terms while retaining this leading logarithmic exponent.

The strong-coupling scale of an asymptotically free gauge theory is the renormalization-group invariant scale at which the one-loop running coupling formally diverges. It replaces a dimensionless ultraviolet coupling through dimensional transmutation.

Dimensional transmutation replaces a dimensionless coupling by a dimensionful renormalization-group invariant scale. In an asymptotically free theory, specifying the coupling at one scale is equivalent to specifying its strong-coupling scale.

Gauge coupling unification is the convergence of several low-energy gauge couplings to one coupling at a high scale because the gauge factors embed into one simple group.

The normalization of an Abelian gauge coupling matters in testing gauge coupling unification. If $\mu d{g}_i/d\mu =b_i{g}_i^3$, then ${\alpha }_i^{-1}(\mu )={\alpha }_i^{-1}({\mu }_0)-8\pi b_i\log (\mu /{\mu }_0)$. For normalized hypercharge ${\alpha }_Y^{\mathrm{GUT}}=(5/3){\alpha }_Y$, the inverse slope is $(3/5)b_Y$. Subtracting pairs of inverse couplings at the common scale and eliminating its logarithm gives a low-energy consistency relation. Division by the difference of two slopes requires that difference to be nonzero; a scale above ${\mu }_0$ additionally requires a positive inferred logarithm.

An SU(5) grand unified theory embeds $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$ into $SU(5)$. Canonical normalization of its hypercharge generator predicts $g_s^2=g_w^2=(5/3)g_Y^2$ at the unification scale.

For simply connected space $SU(5)$ broken to the connected Standard Model subgroup, the exact homotopy sequence gives ${\pi }_2(G/H)\cong {\pi }_1(H)\cong \mathbb{Z}$. Stable magnetic monopoles form. Their gauge and scalar cores have scales $(g\eta {)}^{-1}$ and $(\sqrt{\lambda }\eta {)}^{-1}$; their masses are $M_M\sim 4\pi \eta /g$ times a dimensionless profile factor. The physical subgroup is $(SU(3)\times SU(2)\times U(1))/{\mathbb{Z}}_6$, which retains an infinite-cyclic fundamental group.

// Target: cosmology.bigb

Normalize the Abelian coupling by $g_1^2=(5/3)g^{\prime 2}$ and match $g_1=g_2=g_3$ in magnitude at scale $M_X$. With common-flavor coefficients $b_3=(33-2n_f)/3$, $b_2=(22-2n_f)/3$ and $b_1=-2n_f/3$, the one-loop running of the strong coupling extends to $1/{\alpha }_i(\mu )=1/{\alpha }_5-b_i\log (M_X^2/{\mu }^2)/(4\pi )$. The differences obey ${\alpha }_1^{-1}-{\alpha }_2^{-1}=2({\alpha }_2^{-1}-{\alpha }_s^{-1})$. Using ${\alpha }_2^{-1}={\sin }^2{\theta }_W/\alpha$ and ${\alpha }_1^{-1}=3{\cos }^2{\theta }_W/(5\alpha )$ gives the displayed prediction and $\log (M_X/\mu )=\frac{\pi }{11}({\alpha }^{-1}-8/(3{\alpha }_s))$. These simplified coefficients omit the full matter and scalar counting of a realistic unified model; threshold changes require piecewise running with the actual coefficients.