An asymptotic safety scenario for gauged chiral Higgs–Yukawa models

Abstract

We investigate chiral Higgs–Yukawa models with a non-abelian gauged left-handed sector reminiscent to a sub-sector of the standard model. We discover a new weak-coupling fixed-point behavior that allows for ultraviolet complete RG trajectories which can be connected with a conventional long-range infrared behavior in the Higgs phase. This non-trivial ultraviolet behavior is characterized by asymptotic freedom in all interaction couplings, but a quasi conformal behavior in all mass-like parameters. The stable microscopic scalar potential asymptotically approaches flatness in the ultraviolet, however, with a non-vanishing minimum increasing inversely proportional to the asymptotically free gauge coupling. This gives rise to non-perturbative—though weak-coupling—threshold effects which induce ultraviolet stability along a line of fixed points. Despite the weak-coupling properties, the system exhibits non-Gaußian features which are distinctly different from its standard perturbative counterpart: e.g., on a branch of the line of fixed points, we find linear instead of quadratically running renormalization constants. Whereas the Fermi constant and the top mass are naturally of the same order of magnitude, our model generically allows for light Higgs boson masses. Realistic mass ratios are related to particular RG trajectories with a “walking” mid-momentum regime.

Appendices

Appendix A: Regulators and threshold functions

1.1 A.1 Regulators

We have to evaluate the r.h.s. of Eq. (5), for which we need the $\varGamma_{k}^{(2)}$ matrix. Let us consider the fields ϕ _i , ψ _L, ψ _R, W, c, $\bar{c}$ as column vectors, with a number of components, respectively, given by N _L, d _γ N _g N _L, d _γ N _g, dd _ad, d _ad, d _ad. Accordingly let us consider $\bar{\psi}_{\mathrm{L}}$ and $\bar{\psi}_{\mathrm{R}}$ as row vectors. Taking care of the partly Grassmann-valued field components and of the Fourier conventions, let us denote by Φ ^T(q) the row vector with components $\phi^{\mathrm{T}}_{1}(q)$, $\phi^{\mathrm{T}}_{2}(q)$, $\psi^{\mathrm{T}}_{\mathrm{L}}(q)$, $\bar{\psi}_{\mathrm{L}}(-q)$, $\psi^{\mathrm{T}}_{\mathrm{R}}(q)$, $\bar{\psi}_{\mathrm{R}}(-q)$, W ^T(q), c ^T(q), $\bar{c}^{\mathrm{T}}(q)$, and by Φ(p) the column vector given by its transposition. Then $\varGamma_{k}^{(2)}$ is computed as follows:

$$ \varGamma_k^{(2)}= \frac{\overrightarrow{\delta}}{\delta\varPhi^{\mathrm{T}}(-p)} \varGamma_k \frac{\overleftarrow{\delta}}{\delta\varPhi(q)}. $$

For a proper IR regularization, a regulator which is diagonal in field space is sufficient and convenient,

$$ R_k(q,p)=\delta(p-q) \begin{pmatrix} R_{\mathrm{B}} & 0 & 0 & 0 & 0\\ 0 & R_{\mathrm{L}} & 0 & 0 & 0\\ 0 & 0 & R_{\mathrm{R}} & 0 & 0\\ 0 & 0 & 0 & R_{\mathrm{G}} & 0\\ 0 & 0 & 0 & 0 & R_{\mathrm{gh}}\\ \end{pmatrix} (p) , $$

with a 2N _L×2N _L matrix for the scalar bosonic sector

$$ R_{\mathrm{B}}(p)= \begin{pmatrix} \delta^{ab} & 0\\ 0 & \delta^{ab}\\ \end{pmatrix} Z_{\phi}p^2r_{\mathrm{B}}\bigl(p^2\bigr), $$

an 2d _γ N _g N _L×2d _γ N _g N _L matrix for the left-handed spinor

an 2d _γ N _g×2d _γ N _g matrix for the right-handed spinor

a dd _ad×dd _ad matrix for the gauge vector boson

$$\begin{aligned} R_{\mathrm{G}}(p) =&Z_Wp^2 r_{\mathrm{GT}} \bigl(p^2\bigr) \varPi_{\mathrm {T}}^{\mu\nu}(p) \delta^{ij} \\ &{}+\frac{Z_\phi p^2 r_{\mathrm{GL}}(p^2)}{\alpha}\varPi_{\mathrm {L}}^{\mu\nu}(p) \delta^{ij}, \end{aligned}$$

where the Π’s are the usual longitudinal and transverse projectors with respect to p _μ , and a 2d _ad×2d _ad matrix for the ghosts

$$ R_{\mathrm{gh}}= \begin{pmatrix} 0 & \delta^{ij}\\ -\delta^{ij} & 0\\ \end{pmatrix} p^2r_{\mathrm{gh}}\bigl(p^2\bigr). $$

Notice that here and in the whole paper we set Z _gh=1 at any scale, that is: we neglect η _gh. Choosing different regulators for the scalar bosons (B), for the transverse gauge bosons (GT), for the longitudinal gauge boson (GL), for the ghosts (gh) and for the left-handed (L) as well as for the right-handed (R) spinors, allows one to write the flow equation in the form

$$ \partial_t \varGamma_k=\frac{1}{2}\tilde{\partial}_t{\rm STr}\log \bigl(\varGamma_{k}^{(2)}+R_k \bigr), $$

where

$$\begin{aligned} \tilde{\partial}_t \equiv& \frac{\partial_t(Z_\phi r_{\mathrm{B}})}{Z_\phi}\cdot \frac{\delta}{\delta r_{\mathrm{B}}} +\frac{\partial_t(Z_{\mathrm{L}} r_{\mathrm{L}})}{Z_{\mathrm{L}}}\cdot\frac {\delta}{\delta r_{\mathrm{L}}} \\ &{}+\frac{\partial_t(Z_{\mathrm{R}} r_{\mathrm{R}})}{Z_{\mathrm{R}}}\cdot\frac {\delta}{\delta r_{\mathrm{R}}} +\frac{\partial_t(Z_W r_{\mathrm{GT}})}{Z_W}\cdot \frac{\delta }{\delta r_{\mathrm{GT}}} \\ &{}+\frac{\partial_t(Z_\phi r_{\mathrm{GL}})}{Z_\phi}\cdot\frac {\delta}{\delta r_{\mathrm{GL}}} +\partial_t r_{\mathrm{gh}}\cdot\frac{\delta}{\delta r_{\mathrm{gh}}} \end{aligned}$$

(A.1)

and ⋅ denotes multiplication as well as integration over the common argument of the shape functions of the two factors. After having performed this differentiation we are free to specify the form of the shape functions r. See Appendix A.2 for an example of such a choice.

1.2 A.2 Threshold functions

Since in the SSB regime one of the left-handed Weyl fermions together with the right-handed one gets massive, it is useful to introduce a superscript (F) to denote the corresponding Dirac fermion. Then the regularized kinetic (or squared kinetic) terms are given by

$$\begin{aligned} &P_{\mathrm{B}/\mathrm{GT}/\mathrm{GL}/\mathrm{gh}} (x)= x\bigl(1+r_{\mathrm{B}/\mathrm{GT}/\mathrm{GL}/\mathrm{gh}}(x)\bigr), \\ &P_{\mathrm{L}}(x)= x\bigl(1 + r_{\mathrm{L}}(x)\bigr)^2, \\ &P_{\mathrm{F}}(x)= x\bigl(1 + r_{\mathrm{L}}(x)\bigr) \bigl(1 + r_{\mathrm{R}}(x)\bigr) . \end{aligned}$$

Accordingly, the loop momentum integrals appearing on the r.h.s. of the flow equation are classified, implicitly defining the corresponding threshold functions. In the following, the operator $\tilde{\partial}_{t}$ is the one defined in Eq. (A.1). We also use the abbreviations $\int_{p}\equiv\int \frac{d^{d}p}{(2\pi)^{d}}$ and v _d =1/(2^d+1 π ^d/2 Γ(d/2)), such that v ₄=1/(32π ²). Then, the threshold functions read

$$\begin{aligned} &l_0^{(\mathrm{B/F/L/gh})d}(\omega) = \frac{k^{-d}}{4 v_d}\int_p \tilde{\partial}_t \log \bigl(P_{\mathrm{B}/\mathrm {F}/\mathrm{L}/\mathrm{gh}}+\omega k^2 \bigr), \\ &l_{0 \mathrm{T/L}}^{(\mathrm{G})d}(\omega) = \frac {k^{-d}}{4 v_d}\int_p \tilde{\partial}_t \log \bigl(P_{\mathrm {GT}/\mathrm{GL}}+\omega k^2 \bigr), \\ &\begin{aligned}l_{n_1,n_2}^{(\mathrm{FB})d}(\omega_1, \omega_2) &=-\frac{k^{2(n_1+n_2)-d}}{4 v_d} \\ &\quad \times\int_p \tilde{\partial}_t \frac{1}{(P_{\mathrm{F}} + \omega_1 k^2)^{n_1}(P_{\mathrm{B}} + \omega_2 k^2)^{n_2}},\end{aligned} \\ &\begin{aligned} &l_{n_1,n_2}^{(\mathrm{BG})d}(\omega_1, \omega_2)\\ &\quad = -\frac{k^{2(n_1+n_2)-d}}{4 v_d} \int_p \tilde{\partial}_t \frac{1}{(P_{\mathrm{B}} + \omega_1 k^2)^{n_1}(P_{\mathrm{GT}} + \omega_2 k^2)^{n_2}},\end{aligned}\\ &m_2^{(\mathrm{F})d}(\omega) = - \frac{k^{6-d}}{4 v_d} \int_p p^2 \tilde{\partial}_t \biggl( \frac{\partial}{\partial p^2}\frac{1}{P_{\mathrm{F}} + \omega k^2} \biggr)^2, \\ &\begin{aligned} &m_4^{(\mathrm{F})d}(\omega) \\ &\quad = - \frac{k^{4-d}}{4 v_d} \int_p p^4 \tilde{\partial}_t \biggl( \frac {\partial}{\partial p^2} \frac{1+r_{\mathrm{L}}}{P_{\mathrm{F}} + \omega k^2} \biggr) \biggl( \frac{\partial}{\partial p^2} \frac{1+r_{\mathrm{R}}}{P_{\mathrm{F}} + \omega k^2} \biggr),\end{aligned} \\ &m_2^{(\mathrm{G})d}(\omega) = - \frac{k^{6-d}}{4v_d} \int_p p^2 \tilde{\partial}_t \biggl( \frac{\partial}{\partial p^2} \frac{1}{P_{\mathrm{GT}} + \omega k^2} \biggr)^2 ,\\ & \begin{aligned} &m_{2,2}^{(\mathrm{B})d}(\omega_1, \omega_2) \\ &\quad =-\frac {k^{6-d}}{4 v_d} \int_p p^2 \tilde{\partial}_t \biggl( \frac {\frac{\partial}{\partial p^2}P_{\mathrm{B}}}{(P_{\mathrm{B}} + \omega_1 k^2)^2} \frac{\frac{\partial}{\partial p^2}P_{\mathrm{B}}}{(P_{\mathrm{B}} + \omega_2 k^2)^2} \biggr),\end{aligned} \\ &\begin{aligned} &m_{1,2}^{(\mathrm{LB})d}(\omega_1, \omega_2)\\ &\quad =-\frac {k^{4-d}}{4 v_d} \int_p p^2 \tilde{\partial}_t \biggl( \frac {1+r_{\mathrm{R}}}{P_{\mathrm{F}} +\omega_1 k^2} \frac{\frac{\partial }{\partial p^2}P_{\mathrm{B}}}{(P_{\mathrm{B}} + \omega_2 k^2)^2} \biggr),\end{aligned} \\ &\begin{aligned} &m_{1,2}^{(\mathrm{RB})d}(\omega_1, \omega_2) \\ &\quad =-\frac {k^{4-d}}{4 v_d} \int_p p^2 \tilde{\partial}_t \biggl( \frac {1+r_{\mathrm{L}}}{P_{\mathrm{F}} +\omega_1 k^2} \frac{\frac{\partial }{\partial p^2}P_{\mathrm{B}}}{(P_{\mathrm{B}} + \omega_2 k^2)^2} \biggr),\end{aligned} \\ &\begin{aligned} &m_{1,2}^{(\mathrm{LG})d}(\omega_1, \omega_2)\\ &\quad =-\frac {k^{4-d}}{4 v_d} \int_p p^2 \tilde{\partial}_t \biggl( \frac {1+r_{\mathrm{R}}}{P_{\mathrm{F}} +\omega_1 k^2} \frac{\frac{\partial }{\partial p^2}P_\mathrm{GT}}{(P_\mathrm{GT} + \omega_2 k^2)^2} \biggr),\end{aligned}\\ & a_1^d(\omega) = - \frac{k^{6-d}}{16v_d} \int_p \frac {1}{p^2}\tilde{\partial}_t \biggl( \frac{1}{P_\mathrm{GT} + \omega k^2} \biggr)^2, \\ &a_3^d(\omega_1, \omega_2) = - \frac{k^{4-d}}{4v_d} \int _p \tilde{\partial}_t \biggl( \frac{1+r_{\mathrm{R}}}{P_{\mathrm{F}} +\omega_1 k^2} \frac{1}{P_\mathrm{GT} + \omega_2 k^2} \biggr) . \end{aligned}$$

For practical computations, we use the linear regulator for the scalar bosons, for the gauge bosons and for the ghosts

$$ x r_{\mathrm{B}/\mathrm{GT}/\mathrm{GL}/\mathrm{gh}}(x)=(1-x)\theta(1-x), $$

(A.2)

where x=q ²/k ². For the spinor fermions the linear regulator corresponds to a shape function r _L/R such that x(1+r _B(x))=x(1+r _L/R(x))². This regulator satisfies an optimization criterion within our present truncation and is technically advantageous, as we can perform all momentum integrations analytically, obtaining

$$\begin{aligned} &l_0^{(\mathrm{B})d}(\omega) = \frac{2}{d} \frac{1-\frac{\eta _\phi}{d+2} }{1+\omega} , \\ &l_0^{(\mathrm{F})d}(\omega) = \frac{2}{d} \frac{1-\frac{\eta _\mathrm{L}+\eta_\mathrm{R}}{2(d+1)}}{1+\omega} , \\ &l_0^{(\mathrm{L})d}(\omega) = \frac{2}{d} \frac{1-\frac{\eta _\mathrm{L}}{d+1}}{1+\omega} , \\ &l_{0\mathrm{T}}^{(\mathrm{G})d}(\omega) = \frac{2}{d} \frac {1-\frac{\eta_{W}}{d+2}}{1+\omega} , \\ &l_{0\mathrm{L}}^{(\mathrm{G})d}(\omega) = \frac{2}{d} \frac {1-\frac{\eta_\phi}{d+2}}{1+\omega} , \\ &l_0^{(\mathrm{gh})d}(\omega) = \frac{2}{d} \frac{1}{1+\omega} , \\ &\begin{aligned} l_{n_1,n_2}^{(\mathrm{FB})d}(\omega_1, \omega_2) &= \frac {2}{d} \biggl[ n_1 \frac{ 1-\frac{\eta_\mathrm{L}+\eta_\mathrm{R}}{2(d+1)} }{ (1+\omega_1)^{1+n_1}(1+\omega_2)^{n_2} } \\ &\quad + n_2\frac{ 1-\frac{\eta_\phi}{d+2} }{ (1+\omega _1)^{n_1}(1+\omega_2)^{1+n_2} } \biggr] ,\end{aligned} \\ &\begin{aligned} l_{n_1,n_2}^{(\mathrm{BG})d}(\omega_1, \omega_2) &= \frac {2}{d} \biggl[ n_1 \frac{1-\frac{\eta_\phi}{d+2}}{(1+\omega_1)^{1+n_1}(1+\omega _2)^{n_2}} \\ &\quad + n_2\frac{1-\frac{\eta_{W}}{d+2}}{(1+\omega _1)^{n_1}(1+\omega_2)^{1+n_2}} \biggr] , \end{aligned}\\ &m_2^{(\mathrm{F})d}(\omega) = \frac{1}{(1 + \omega)^4} , \\ &\begin{aligned} m_4^{(\mathrm{F})d}(\omega)& = \frac{1}{(1 + \omega)^4} + \frac{1-\frac{1}{2}(\eta_{\mathrm{L}} + \eta_{\mathrm{R}})}{(d-2)(1+\omega)^3} \\ &\quad - \biggl( \frac{1-\frac{1}{2}(\eta_{\mathrm{L}} + \eta_{\mathrm{R}})}{2d-4} + \frac{1}{4} \biggr)\frac{1}{(1 + \omega)^2} ,\end{aligned} \\ &m_2^{(\mathrm{G})d}(\omega) = \frac{1}{(1 + \omega)^4} ,\\ &m_{2,2}^{(\mathrm{B})d}(\omega_1, \omega_2) = \frac{1}{(1+\omega _1)^2(1+\omega_2)^2} , \\ &m_{1,2}^{(\mathrm{LB})d}(\omega_1, \omega_2) = \frac{1-\frac {\eta_\phi}{d+1}}{(1+\omega_1)(1+\omega_2)^2} , \\ &m_{1,2}^{(\mathrm{RB})d}(\omega_1, \omega_2) = \frac{1-\frac {\eta_\phi}{d+1}}{(1+\omega_1)(1+\omega_2)^2} , \\ &m_{1,2}^{(\mathrm{LG})d}(\omega_1, \omega_2) = \frac{1-\frac {\eta_{W}}{d+1}}{(1+\omega_1)(1+\omega_2)^2} , \\ &a_1^d(\omega) = \frac{1- \frac{\eta_{W}}{d}}{d-2} \frac {1}{(1 + \omega)^3} , \\ &\begin{aligned} a_3^d(\omega_1, \omega_2) &= \frac{2}{d-1}\frac{1-\frac{\eta _{W}}{d+1}}{(1+\omega_1)(1+\omega_2)^2} \\ &\quad + \frac{1}{d-1} \frac{ ( 1-\frac{\eta_{\mathrm{L}}}{d} ) - \omega_1 ( 1-\frac{\eta_{\mathrm{R}}}{d} )}{(1+\omega _1)^2(1+\omega_2)} .\end{aligned} \end{aligned}$$

Appendix B: Derivation of the flow equations for the matter sector

The computation of the RG flow of the matter sector inside the truncation (9) will be sketched. For further details see [105, 106].

2.1 B.1 Flow equation for the potential

The flow of the potential can be computed by setting the field ϕ ^a to a constant value and all the other fields to zero. This projects both sides of the flow onto the scalar potential. Then, in Landau gauge the matrix $\varGamma_{k}^{(2)}+R_{k}$ can be inverted easily. Multiplying with the derivative of the regulator, and taking the supertrace yields the result. This can be interpreted as an improved one-loop computation for a 0-point function, i.e. a sum over all the one-loop graphs with no external legs. The gauge contribution takes the form of a closed gauge boson propagator, and since it does not involve any vertex, it should not explicitly depend on $\bar{g}$. Indeed we get

$$\begin{aligned} \partial_tU&=\frac{1}{2}\int\frac{d^dp}{(2\pi)^d} \partial_t P_{\mathrm{B}} \biggl[ \frac{2N_{\text {L}}-1}{Z_\phi P_{\mathrm{B}} + U'}\\ &\quad + \frac {1}{Z_\phi P_{\mathrm{B}} + U' + 2\rho U''} \biggr] -d_{\gamma} N_{\text {g}}\int\frac{d^dp}{(2\pi)^d}\\ &\quad\times \biggl\{ \biggl[(N_{\text {L}}-1) + \frac{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}}}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}} + \rho\bar{h}^2} \biggr]\frac{\partial_t[Z_{\mathrm{L}} r_{\mathrm{L}}]}{Z_{\mathrm{L}} (1+r_{\mathrm{L}})} \\ &\quad + \frac{Z_{\mathrm{L}}Z_{\mathrm{R}} P_{\mathrm{F}}}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}} + \rho \bar{h}^2} \frac{\partial_t[Z_{\mathrm{R}} r_{\mathrm{R}}]}{Z_{\mathrm{R}} (1+r_{\mathrm{R}})} \biggr\} \\ &\quad + \frac{1}{2}\sum_{i=1}^{N_{\text {L}}^2-1}\int \frac{d^dp}{(2\pi )^d} \biggl[ (d-1)\frac{\partial_t(Z_W p^2r_{\mathrm{GT}})}{Z_W P_\mathrm{GT} + \bar{m}_{W,i}^2} \\ &\quad +\frac{\partial_t(Z_\phi p^2r_{\mathrm{GL}})}{Z_\phi P_\mathrm{GL}} \biggr] - \int\frac {d^dp}{(2\pi)^d}\frac{d_{\text {ad}}p^2\partial_t r_{\mathrm{gh}}}{P_{\mathrm{gh}}} \end{aligned}$$

that is, in terms of threshold functions

$$\begin{aligned} \partial_t U &= 2 v_d k^d \Biggl\{ (2 N_{\text {L}}-1)l_0^{(\mathrm{B})d} \biggl( \frac{U'}{Z_\phi k^2} \biggr)\\ &\quad + l_0^{(\mathrm{B})d} \biggl( \frac{U' + 2 \rho U''}{Z_\phi k^2} \biggr) - d_\gamma N_{\text {g}}\biggl[ (N_{\text {L}}-1)l_{0}^{(\mathrm{L})d} ( 0 )\\ &\quad + 2 l_{0}^{(\mathrm{F})d} \biggl( \frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2} \biggr) \biggr] - 2d_{\text {ad}}l_0^{(\mathrm{gh})d} ( 0 ) \\ &\quad + \sum _{i=1}^{d_{\text {ad}}} \biggl[ (d-1)l_{0\mathrm{T}}^{(\mathrm{G})d} \biggl( \frac{\bar {m}_{W,i}^2}{Z_W k^2} \biggr) + l_{0\mathrm{L}}^{(\mathrm{G})d} ( 0 ) \biggr] \Biggr\} \end{aligned}$$

where U is a function of ρ. Switching to dimensionless quantities this becomes Eq. (23).

2.2 B.2 Flow equation for the Yukawa coupling

For the derivation of the flow of the Yukawa coupling, we first separate the bosonic field into the vev and a purely radial deviation from the vev. This corresponds to setting Δϕ ₂=0 in Eq. (6). While this is irrelevant in the symmetric regime, it makes a difference in the SSB regime, as it projects onto the Yukawa coupling between the fermions and Higgs boson, being the radial mode. The projection of the flow equation onto such an operator reads

$$ \partial_t\bar{h}=-\frac{1}{\sqrt{2}} \frac{\overrightarrow{\delta }}{\delta\bar{\psi}_{\mathrm{L}}^{\hat{n}}(p)}\frac {\overrightarrow{\delta}}{\delta \varDelta \phi_1^{\hat{n}}(p')} \partial_t \varGamma_k \frac{\overleftarrow{\delta}}{\delta\psi _{\mathrm{R}}(q)} \bigg|_0. $$

(B.1)

The vertical line indicates that the equation is evaluated at vanishing momenta p′=p=q=0 and at vanishing fluctuation fields. Next, we can decompose the matrix $(\varGamma_{k}^{(2)}+R_{k})$ into two parts. One part, which we call $(\varGamma_{k,0}^{(2)}+R_{k})$, contains only v and is independent of the fluctuations. The remaining part, $\varDelta \varGamma_{k}^{(2)}$, contains all fluctuating fields. Using the $\tilde{\partial}_{t}$-notation of Appendix A.2 and expanding by means of the Mercator series, the flow equation can be written as

$$\begin{aligned} \partial_t \varGamma_k =& \frac{1}{2}\sum_{s=1}^{\infty} \frac {(-)^{s+1}}{s}{\rm STr}\tilde{\partial}_t \biggl[ \biggl( \frac{\varDelta \varGamma_k^{(2)}}{\varGamma_{k,0}^{(2)}+R_k} \biggr)^s \biggr] \\ &{}+\frac{1}{2}{\rm STr}\tilde{\partial}_t \log\bigl( \varGamma_{k,0}^{(2)}+R_k\bigr). \end{aligned}$$

(B.2)

Plugging this expression into Eq. (B.1), only the term to third power in $\varDelta \varGamma_{k}^{(2)}$ survives the projection. Since we took three derivatives of the Wetterich equation, the diagrammatic interpretation of the result is in terms of one-loop graphs with three external legs: two fermions of opposite chirality and one radial scalar. The gauge contribution comes from triangular loops with three different propagators: one scalar, one spinor and one gauge vector. It always involves the two-scalars-one-vector vertex. This vertex is proportional to the difference of incoming scalar momenta, while the gauge boson propagator in Landau gauge is transverse. These two facts plus conservation of momentum entail that the direct gauge contribution to the momentum-independent Yukawa coupling under consideration vanishes in our truncation. This formal argument can straightforwardly be verified by performing the matrix calculations and taking the supertrace, yielding

$$\begin{aligned} \partial_t \bar{h} =& - \frac{\bar{h}^3}{2}\int\frac{d^dp}{(2\pi )^d} \tilde{\partial}_t \biggl[ \frac{1}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}} + \rho\bar{h}^2} \biggl( \frac{2 \rho U''}{(Z_\phi P_{\mathrm{B}} + U')^2}\\ &{} - \frac{ 6\rho U'' + 4\rho^2 U'''}{(Z_\phi P_{\mathrm{B}} + U' + 2\rho U'')^2} \biggr) \\ &{}+\frac{2\rho\bar{h}^2}{(Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}} + \rho \bar{h}^2)^2} \biggl(\frac{1}{Z_\phi P_{\mathrm{B}} + U'} \\ &{}- \frac {1}{Z_\phi P_{\mathrm{B}} + U' + 2\rho U''} \biggr) \\ &{} - \frac{1}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}} + \rho\bar {h}^2} \biggl(\frac{1}{Z_\phi P_{\mathrm{B}} + U'}\\ &{} - \frac{1}{Z_\phi P_{\mathrm{B}} + U' + 2\rho U''} \biggr) \biggr] \end{aligned}$$

where the whole r.h.s. should be evaluated at the value $\rho=\frac {1}{2}{\bar{v}}^{2}$, which minimizes the potential U. In terms of the threshold functions as defined in Appendix A.2 this reads

$$\begin{aligned} \partial_t \bar{h}^2 &= \frac{4v_d \bar{h}^4}{Z_{\mathrm{L}} Z_{\mathrm{R}} Z_\phi k^{4-d}} \biggl[ \frac{2\rho U''}{Z_\phi k^2} l_{1,2}^{(\mathrm{FB})d} \biggl(\frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U'}{Z_\phi k^2} \biggr)\\ &\quad - \frac{6\rho U''+4\rho U'''}{Z_\phi k^2} l_{1,2}^{(\mathrm{FB})d} \biggl( \frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U'+2\rho U''}{Z_\phi k^2} \biggr) \\ &\quad + \frac{2\rho\bar{h}^2}{k^2} l_{2,1}^{(\mathrm{FB})d} \biggl( \frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U'}{Z_\phi k^2} \biggr)\\ &\quad - \frac{2\rho\bar{h}^2}{k^2} l_{2,1}^{(\mathrm{FB})d} \biggl( \frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U'+2\rho U''}{Z_\phi k^2} \biggr) \\ &\quad - l_{1,1}^{(\mathrm{FB})d} \biggl( \frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U'}{Z_\phi k^2} \biggr) \\ &\quad + l_{1,1}^{(\mathrm{FB})d} \biggl( \frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U'+2\rho U''}{Z_\phi k^2} \biggr) \biggr] . \end{aligned}$$

Switching over to dimensionless quantities, we end up with the representation (25) given in the main text.

2.3 B.3 Flow of the scalar anomalous dimension

For the derivation of the flow of Z _ϕ , we decompose the bosonic field as in Appendix B.2. The projection of the Wetterich equation onto the massive scalar kinetic term leads us to

$$\begin{aligned} \partial_t Z_{\phi} =&-\frac{\partial}{\partial(p'^2)} \frac {\delta}{\delta \varDelta \phi_1^{\hat{n}}(p')} \frac{\delta}{\delta \varDelta \phi_1^{\hat{n}}(q')} \partial _t \varGamma_k \bigg|_0. \end{aligned}$$

As before the vertical line indicates that the equation is evaluated at vanishing momenta p′=q′=0 and at vanishing fluctuation fields. Expanding again the r.h.s. of the flow equation according to Eq. (B.2), this time only the second order term (s=2) contributes. Since two derivatives of the flow equation have to be taken, the result can diagrammatically be interpreted as one-loop graphs with two external scalar legs. From a one-loop analysis we expect two kinds of gauge contributions. One is due to the two-scalars-one-vector vertex and produces a loop containing one scalar and one gauge boson propagator. This is present in both the symmetric and in the spontaneously broken regimes. Another is due to the two-scalars-two-vectors vertex. If two external scalar legs are identified with the vev, the corresponding loop contains two gauge boson propagators. Therefore this contribution will be present only in the SSB regime. Indeed, performing the matrix calculations and taking the supertrace we find

$$\begin{aligned} \partial_t Z_\phi&= \frac{1}{d}\int \frac{d^dp}{(2\pi)^d}\tilde {\partial}_t \Biggl\{ \biggl[ \bigl(3\sqrt{2\rho}U'' +2\sqrt{2 \rho ^3}U''' \bigr)^2\\ &\quad \times p^2Z^2_\phi \biggl( \frac{\frac{\partial}{\partial p^2}P_{\mathrm{B}}}{(Z_\phi P_{\mathrm{B}}+U'+2\rho U'')^2} \biggr)^2 \\ &\quad +(2N_{\text {L}}-1)2\rho U''^2p^2Z^2_\phi \biggl( \frac{\frac{\partial }{\partial p^2}P_{\mathrm{B}}}{(Z_\phi P_{\mathrm{B}}+U')^2} \biggr)^2 \biggr] \\ &\quad + d_{\gamma} N_{\text {g}}\bar{h}^2 \biggl[ 2p^4Z_\mathrm{L} Z_\mathrm{R} \biggl( \frac{\partial}{\partial p^2}\frac{1+r_{\mathrm{L}}}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}}+\rho\bar{h}^2} \biggr)\\ &\quad \times \biggl( \frac{\partial}{\partial p^2}\frac{1+r_{\mathrm{R}}}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}}+\rho\bar{h}^2} \biggr)\\ &\quad -2\rho\bar{h}^2p^2 \biggl( \frac{\partial}{\partial p^2}\frac {1}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}}+\rho\bar{h}^2} \biggr)^2 \biggr] \\ &\quad -4(d-1) \bar{g}^2 Z^2_\phi\\ &\quad \times\sum _{a=1}^{N_{\text {L}}}\sum_{i=1}^{d_{\text {ad}}} \frac {T^i_{{\hat{n}}a}T^i_{a{\hat{n}}}}{(Z_\phi P_{\mathrm{B}} + U')(Z_W P_\mathrm{GT} + \bar{m}_{W,i}^2)} \\ &\quad + \frac{(d-1)}{\rho}\sum_{i=1}^{d_{\text {ad}}} \bar{m}_{W,i}^4 \biggl( \frac{1}{p^2(Z_W P_\mathrm{GT} + \bar{m}_{W,i}^2)^2} \\ &\quad + 2p^2 \biggl( \frac{\partial}{\partial p^2}\frac{1}{Z_W P_\mathrm {GT} + \bar{m}_{W,i}^2} \biggr)^2 \biggr) \Biggr\} . \end{aligned}$$

Again the whole r.h.s. should be evaluated at the value $\rho=\frac {1}{2}\bar{v}^{2}$, which minimizes the potential U. Translating this result in terms of threshold functions, yields

$$\begin{aligned} \partial_t Z_\phi&= -\frac{8v_d}{Z_\phi^2 k^{6-d}d} \biggl[ \bigl(3\sqrt{\rho}U''+2\sqrt{\rho^3}U''' \bigr)^2 m_{2,2}^{(\mathrm{B})d}\\ &\quad \times \biggl(\frac{U'+2\rho U''}{Z_\phi k^2}, \frac {U'+2\rho U''}{Z_\phi k^2} \biggr)\\ &\quad +(2N_{\text {L}}-1)\rho U''^2 m_{2,2}^{(\mathrm{B})d}\biggl(\frac{U'}{Z_\phi k^2},\frac{U'}{Z_\phi k^2} \biggr) \biggr] \\ &\quad -\frac{4v_d d_\gamma N_{\text {g}}}{d} \biggl[ \frac{2\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^{4-d}} m_4^{(\mathrm{F})d} \biggl(\frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2} \biggr)\\ &\quad - \frac{2\rho\bar{h}^4}{Z_{\mathrm{L}}^2 Z_{\mathrm{R}}^2k^{6-d}} m_2^{(\mathrm{F})d} \biggl( \frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2} \biggr) \biggr] \\ &\quad +\frac{16 v_d (d-1)}{d} \frac{\bar{g}^2 Z_\phi}{k^{4-d}Z_W}\\ &\quad \times \sum_{a=1}^{N_{\text {L}}} \sum_{i=1}^{d_{\text {ad}}} T^i_{{\hat{n}}a}T^i_{a{\hat{n}}} l_{1,1}^{(\mathrm{BG})d} \biggl( \frac{U'}{Z_\phi k^2}, \frac{\bar {m}_{W,i}^2}{Z_W k^2} \biggr) \\ &\quad -\frac{8v_d (d-1)}{d} \sum_{i=1}^{d_{\text {ad}}} \frac{\bar{m}_{W,i}^4}{Z_W^2 k^{6-d}\rho} \biggl[ 2 a_1^d \biggl( \frac{\bar{m}_{W,i}^2}{Z_W k^2} \biggr)\\ &\quad + m_2^{(\mathrm{G})d} \biggl( \frac{\bar{m}_{W,i}^2}{Z_W k^2} \biggr) \biggr]. \end{aligned}$$

In terms of dimensionless quantities, this leads to Eq. (26).

2.4 B.4 Flow of the spinor anomalous dimensions

For the anomalous dimensions of the spinors, the procedure is very similar to the one explained for the scalar. In the broken regime, the wave function renormalization of the left-handed spinors in principle splits into two functions. Here we concentrate only on the wave function renormalizations associated with the massive top quark, i.e. those for the $\hat{n}\mathrm{th}$ left-handed component and for the right-handed one. We start with the projection

$$\begin{aligned} \partial_tZ_{\mathrm{L/R},k}&=\frac{-1}{2v_dd_{\gamma} N_{\text {g}}}\mathrm{tr} \gamma^{\mu}\frac{\partial}{\partial p'^{\mu}}\frac {\overrightarrow{\delta}}{\delta\bar{\psi}_{\mathrm{L/R}}^{\hat{n}}(p')}\\ &\quad \times\partial_t \varGamma_k \frac{\overleftarrow{\delta}}{\delta\psi_{\mathrm{L/R}}^{\hat{n}}(q')}\bigg| _0 \end{aligned}$$

where the trace is over spinor and generation indices. As before, the vertical line denotes that the equation is evaluated at vanishing momenta p′=q′=0 and at vanishing fluctuation fields. Expanding the r.h.s. of the flow equation according to Eq. (B.2), only the second order term (s=2) contributes. Obviously, the right-handed fermion does not receive direct corrections from the gauge boson whereas the left-handed fermion does. The gauge-field-independent contributions differ from the results of [14, 15] by a factor of two due to a qualitatively irrelevant prefactor error in the earlier paper. For the right-handed spinor, the result is

$$\begin{aligned} \partial_t Z_{\mathrm{R}} &= \frac{\bar{h}^2}{d}\int \frac{d^dp}{(2\pi )^d} p^2\tilde{\partial}_t \biggl[ \frac{Z_{\mathrm{R}} (1+r_{\mathrm{R}})}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}}+\rho\bar{h}^2} \\ &\quad \times\biggl( \frac {Z_\phi\frac{\partial}{\partial p^2}P_{\mathrm{B}}}{(Z_\phi P_{\mathrm{B}}+U'+2\rho U'')^2}+\frac{Z_\phi\frac{\partial}{\partial p^2}P_{\mathrm{B}}}{(Z_\phi P_{\mathrm{B}}+U')^2} \biggr) \\ &\quad + 2(N_{\text {L}}-1)\frac{Z_{\mathrm{R}}(1+r_\mathrm{R})}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}}}\frac{Z_\phi\frac{\partial}{\partial p^2}P_{\mathrm{B}}}{(Z_\phi P_{\mathrm{B}}+U')^2} \biggr]. \end{aligned}$$

In terms of threshold functions, this reads

$$\begin{aligned} \partial_t Z_{\mathrm{R}} &= -\frac{4v_d \bar{h}^2}{d Z_\phi Z_{\mathrm{L}} k^{4-d}} \biggl[ m_{1,2}^{(\mathrm{LB})d}\biggl(\frac{\rho\bar {h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U' + 2\rho U''}{Z_\phi k^2} \biggr) \\ &\quad + m_{1,2}^{(\mathrm{LB})d}\biggl(\frac{\rho\bar{h}^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U'}{Z_\phi k^2}\biggr)\\ &\quad + 2(N_{\text {L}}-1)m_{1,2}^{(\mathrm{LB})d}\biggl(0, \frac{U'}{Z_\phi k^2}\biggr) \biggr], \end{aligned}$$

which leads to Eq. (27) in terms of dimensionless quantities. For the left-handed fermion, the result is

$$\begin{aligned} \partial_t Z_{\mathrm{L}} &= \int\frac{d^dp}{(2\pi)^d} \tilde {\partial}_t \Biggl\{ \frac{\bar{h}^2}{d}\frac{Z_{\mathrm{L}} p^2(1+r_{\mathrm{L}})}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}}+\rho\bar {h}^2}\\ &\quad \times \biggl( \frac{Z_\phi\frac{\partial}{\partial p^2}P_{\mathrm{B}}}{(Z_\phi P_{\mathrm{B}}+U'+2\rho U'')^2}+\frac{Z_\phi\frac{\partial }{\partial p^2}P_{\mathrm{B}}}{(Z_\phi P_{\mathrm{B}}+U')^2} \biggr) \\ &\quad -\frac{(d-1)}{d}\bar{g}^2 Z^2_{\mathrm{L}} \sum_{i=1}^{d_{\text {ad}}} \bigl(T^i_{{\hat{n}}{\hat{n}}} \bigr)^2 \biggl[ \biggl( \frac{2}{Z_W P_\mathrm{GT} + \bar{m}_{W,i}^2}\\ &\quad + 2p^2 \frac{\partial}{\partial p^2}\frac{1}{Z_W P_\mathrm{GT} + \bar{m}_{W,i}^2} \biggr)\\ &\quad \times \biggl(\frac{Z_{\mathrm{R}}(1 + r_{\mathrm{R}})}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}} + \rho\bar{h}^2}- \frac{Z_{\mathrm{R}}(1 + r_{\mathrm{R}})}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}} } \biggr) \biggr] \\ &\quad -\frac{(d-1)}{d}\bar{g}^2 Z^2_{\mathrm{L}} \sum_{a=1}^{N_{\text {L}}}\sum _{i=1}^{d_{\text {ad}}} T^i_{{\hat{n}}a}T^i_{a{\hat{n}}} \biggl[ \biggl( \frac{2}{Z_W P_\mathrm{GT} + \bar{m}_{W,i}^2} \\ &\quad + 2p^2\frac{\partial}{\partial p^2} \frac{1}{Z_W P_\mathrm{GT} + \bar{m}_{W,i}^2} \biggr) \frac{Z_{\mathrm{R}}(1 + r_{\mathrm{R}})}{Z_{\mathrm{L}} Z_{\mathrm{R}} P_{\mathrm{F}}} \biggr] \Biggr\} . \end{aligned}$$

In terms of threshold functions, we obtain

$$\begin{aligned} \partial_t Z_{\mathrm{L}} &=-\frac{4v_d \bar{h}^2}{d Z_\phi Z_{\mathrm{R}} k^{4-d}} \biggl[ m_{1,2}^{(\mathrm{RB})d} \biggl(\frac{\rho h^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U' + 2\rho U''}{Z_\phi k^2} \biggr)\\ &\quad +m_{1,2}^{(\mathrm{RB})d} \biggl(\frac{\rho h^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{U'}{Z_\phi k^2} \biggr) \biggr]\\ &\quad -\frac{8 v_d(d-1)}{d}\frac{\bar{g}^2 Z_{\mathrm{L}}}{Z_W k^{4-d}} \\ &\quad \times\Biggl\{ \sum_{i=1}^{d_{\text {ad}}} \bigl(T^i_{{\hat{n}}{\hat{n}}} \bigr)^2 \biggl[ m_{1,2}^{(\mathrm{LG})d} \biggl( \frac{\rho h^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac{\bar{m}_{W,i}^2}{Z_W k^2} \biggr)\\ &\quad -m_{1,2}^{(\mathrm{LG})d} \biggl(0, \frac{\bar{m}_{W,i}^2}{Z_W k^2} \biggr) - a_3^d \biggl( \frac{\rho h^2}{Z_{\mathrm{L}} Z_{\mathrm{R}} k^2}, \frac {\bar{m}_{W,i}^2}{Z_W k^2} \biggr)\\ &\quad + a_3^d \biggl(0, \frac{\bar{m}_{W,i}^2}{Z_W k^2} \biggr) \biggr] \\ &\quad +\sum_{a=1}^{N_{\text {L}}}\sum _{i=1}^{d_{\text {ad}}} T^i_{{\hat{n}}a}T^i_{a{\hat{n}}} \biggl[ m_{1,2}^{(\mathrm{LG})d} \biggl(0, \frac{\bar {m}_{W,i}^2}{Z_W k^2} \biggr)\\ &\quad - a_3^d \biggl(0, \frac{\bar {m}_{W,i}^2}{Z_W k^2} \biggr) \biggr] \Biggr\} , \end{aligned}$$

the translation into dimensionless quantities of which agrees with Eq. (28). Upon a global color rotation, we can choose (without loss of generality) the direction of the vev ${\hat{n}}$ to point along a single color axis, i.e. ${\hat{n}}^{a}\propto\delta^{aA}$. Then, this anomalous dimension takes a simpler form, given in Eq. (29).

Appendix C: Flow equation for the gauge coupling

In this appendix, we set the spacetime dimension to d=4, and we focus on the gauge group SU(N _L). Since we will be satisfied with the one-loop beta function we set all the wave function renormalizations to one (terms of order $\mathcal {O}(\partial_{t} Z)$ on the r.h.s. of the flow equation lead to higher-loop corrections of the $\beta_{g^{2}}$ function). Still, relevant non-perturbative information will arise from the threshold behavior describing the decoupling of massive modes.

3.1 C.1 Contribution from the gauge modes

For the gauge contribution, the relevant part of the effective Lagrangian is

$$\begin{aligned} \mathcal{L}_k \ni&\frac{1}{4} \bigl(F^i_{\mu\nu} \bigr)^2 +\bar{g}^2 W^i_\mu W^j_\mu\phi^{\dagger a}T^i_{ab}T^j_{bc} \phi^c +\mathcal{L}_{k,\text{gf}}+\mathcal{L}_{k,\text{gh}} \\ =&\frac{1}{4} \bigl(F^i_{\mu\nu}\bigr)^2 + \frac{\bar{g}^2 \bar{v}^2}{2} \hat{n}^{\dagger a}T^i_{ab}T^j_{bc} \hat{n}^c W^i_\mu W^j_\mu +\cdots. \end{aligned}$$

This defines the mass matrix for the gauge bosons, as given in Eq. (10). As the generators are real, the mass matrix has real eigenvalues. In order to compute the running coupling, we use the background-field method and project on the operator F ²/4. To the present one-loop order, no distinction between the background field and the fluctuation field has to be made [67, 68, 70–73], such that it suffices to compute the Hessian $\bar{\varGamma}_{k}^{(2)}$ at zero fluctuation field. Here and in the following, we use the notation of [67, 68, 74]. This Hessian for the W-boson reads

$$\begin{aligned} \bar{\varGamma}^{(2)\, ij}_{k\ \ \mu\nu} \big|_W=\mathcal {D}_{\mathrm{T}\, \mu\nu}^{\phantom{T}\, ij}+ \biggl(1-\frac{1}{\alpha} \biggr)D_\mu^{il}D_\nu^{lj} + \bar{m}^{2\ \ ij}_W\delta_{\mu\nu}, \end{aligned}$$

where $\mathcal{D}_{\mathrm{T}\, \mu\nu} = -D^{2} \delta_{\mu\nu} + 2 i \bar{g} F_{\mu\nu}$. The contributions from ghost fluctuations are

$$\begin{aligned} \bar{\varGamma}^{(2)\, ij}_k \big|_{\mathrm{gh}}=-D_\mu^{il}D_\mu ^{lj}+O\bigl(\alpha^2\bigr) \equiv -\bigl(D^2 \bigr)^{ij}+O\bigl(\alpha^2\bigr). \end{aligned}$$

As we will focus on Landau gauge (α→0), we ignore from now on the ghost-Higgs contributions ∼O(α ²). For a covariantly constant background field, projectors onto the longitudinal and transverse subspaces w.r.t. the background field exist,

with $\varPi_{\text{L} \mu\nu}=-(\mathcal{D}^{-1}_{\text {T}})_{\mu\lambda} D_{\lambda}D_{\nu}$ and , such that

$$\begin{aligned} \bar{\varGamma}^{(2)\, ij}_{k\ \ \mu\nu} \big|_W&= \varPi_{\text {T}\, \mu\lambda}^{\phantom{T}\, il} \bigl[ \mathcal{D}_{\mathrm{T}\, \lambda\nu}^{\phantom{T}\, lj}+ \bar{m}^{2\ \ ij}_W\delta _{\lambda\nu} \bigr] \\ &\quad +\varPi_{\text{L}\, \mu\lambda}^{\phantom{T}\, il} \biggl[\frac {1}{\alpha} \mathcal{D}_{\mathrm{T}\, \lambda\nu}^{\phantom{T}\, lj}+\bar{m}^{2\ \ ij}_W \delta_{\lambda\nu} \biggr]. \end{aligned}$$

We choose a similar decomposition for the regulator

$$\begin{aligned} R_k |_W=\varPi_{\text{T}} \mathcal{D}_{\mathrm{T}} r_k \biggl(\frac {\mathcal{D}_{\mathrm{T}}}{k^2} \biggr)+\varPi_{\text{L}} \frac {1}{\alpha} \mathcal{D}_{\mathrm{T}} r_k \biggl( \frac{\mathcal {D}_{\mathrm{T}}}{k^2} \biggr) \end{aligned}$$

hence also the functional trace on the r.h.s. of the flow equation decomposes into these two sectors. Using the important property that

$$\begin{aligned} {\rm Tr} \bigl[\varPi_{\text{L}} f(\mathcal{D}_{\mathrm{T}}) \bigr]={ \rm Tr} \bigl[f\bigl(-D^2\bigr) \bigr], \end{aligned}$$

we get

$$\begin{aligned} {\rm Tr} \biggl[\frac{\partial_t{R}_k}{\varGamma_k^{(2)}+R_k} \biggr]_W&= {\rm Tr} \biggl[ \varPi_{\text{T}}\frac{\mathcal{D}_{\mathrm{T}}\partial _t{r}_k (\frac{\mathcal{D}_{\mathrm{T}}}{k^2} )}{\mathcal {D}_{\mathrm{T}} (1+r (\frac{\mathcal{D}_{\mathrm{T}}}{k^2} ) )+\bar{m}^2_W} \biggr] \\ &\quad +{\rm Tr} \biggl[\frac{ (-D^2)\partial_t{r}_k (\frac {-D^2}{k^2} )}{(-D^2) (1+r (\frac{-D^2}{k^2} ) )+\alpha\bar{m}^2_W} \biggr] \end{aligned}$$

and writing Π _T=1−Π _L in the first term we obtain two unconstrained traces for different differential operators. The ghost contribution gives

$$\begin{aligned} {\rm Tr} \biggl[\frac{\partial_t{R}_k}{\varGamma_k^{(2)}+R_k} \biggr]_{\text{gh}}= -2{\rm Tr} \biggl[ \frac{(-D^2)\partial_t{r}_k (\frac {-D^2}{k^2} )}{(-D^2) (1+r (\frac{-D^2}{k^2} ) )} \biggr] \end{aligned}$$

such that the total contribution reads in the Landau gauge α→0

$$\begin{aligned} {\rm STr} \biggl[\frac{\partial_t{R}_k}{\varGamma_k^{(2)}+R_k} \biggr]_{W}&= { \rm Tr} \biggl[\frac{\mathcal{D}_{\mathrm{T}}\partial_t{r}_k (\frac{\mathcal{D}_{\mathrm{T}}}{k^2} )}{\mathcal{D}_{\mathrm{T}} (1+r (\frac{\mathcal{D}_{\mathrm{T}}}{k^2} ) )+\bar{m}^2_W} \biggr] \\ &\quad -{\rm Tr} \biggl[\frac{(-D^2)\partial_t{r}_k (\frac {-D^2}{k^2} )}{(-D^2) (1+r (\frac{-D^2}{k^2} ) )+\bar{m}^2_W} \biggr] \\ &\quad -{\rm Tr} \biggl[\frac{(-D^2)\partial_t{r}_k (\frac {-D^2}{k^2} )}{(-D^2) (1+r (\frac{-D^2}{k^2} ) )} \biggr] . \end{aligned}$$

(C.1)

To simplify the calculation, we choose a basis in adjoint color space where the gauge boson mass matrix is diagonal, as in Eq. (11), and we also specify a constant pseudo-abelian magnetic background field

$$\begin{aligned} F^i_{\mu\nu}=\hat{m}^i F_{\mu\nu},\qquad \hat{m}_i\hat {m}^i=1, \qquad F_{\mu\nu}=B \epsilon^\perp_{\mu\nu} \end{aligned}$$

where $\hat{m}$ is a unit vector pointing into a direction in the Cartan of the algebra. The constant antisymmetric tensor ϵ characterizes the space directions which are affected by the constant magnetic field upon the Lorentz force. Recalling that the adjoint generators are (τ ^l) _ij =if ^ilj, we choose a basis in adjoint color space, such that $if^{ilj}\hat{m}^{l}$ is diagonal with eigenvalues ν _i . Then, the covariant derivative

$$\begin{aligned} D_\mu^{ij}=(\partial_\mu-i \nu_i)\delta^{ij} \quad\text{(no sum over i)} \end{aligned}$$

(C.2)

is also diagonal, and so are D ² and $\mathcal{D}_{\mathrm{T}}$. Hence $\mathcal{D}_{\mathrm{T}}$ and $\bar{m}_{W}^{2}$ commute, as well as D ² and $\bar{m}_{W}^{2}$. Equation (C.1) can thus be brought into proper-time form:

$$\begin{aligned} &{\rm STr} \biggl[\frac{\partial_t{R}_k}{\varGamma_k^{(2)}+R_k} \biggr]_{W}\\ &\quad = -\int _0^\infty \,ds\,\tilde{h}(s,0){\rm Tr} \bigl[e^{-s\frac {-D^2}{k^2}} \bigr] \\ &\qquad +\int_0^\infty ds\,{\rm Tr} \bigl[\tilde{h} \bigl(s,m_W^2 \bigr) \bigl(e^{-s\frac{\mathcal{D}_{\mathrm{T}}}{k^2}}-e^{-s\frac{- D^2}{k^2}} \bigr) \bigr] \end{aligned}$$

where $\tilde{h}$ is the Laplace transform of the function

$$\begin{aligned} h \bigl(y,m_W^2 \bigr)=\frac{y\partial_t{r}_k(y)}{y(1+r_k(y))+m_W^2} \end{aligned}$$

with respect to y, that is,

$$\begin{aligned} h \bigl(y,m_W^2 \bigr)=\int_0^\infty ds\,\tilde{h} \bigl(s,m_W^2 \bigr)e^{-s y}, \end{aligned}$$

where, as before, $m_{W}^{2}=\bar{m}^{2}_{W}/k^{2}$. The heat-kernel traces are known, see [74]

$$\begin{aligned} \begin{aligned} &{\rm Tr} \bigl[\tilde{h} \bigl(s,m_W^2 \bigr)e^{-s\frac{\mathcal {D}_{\mathrm{T}}}{k^2}} \bigr]\\ &\quad = \frac{\varOmega k^4}{4\pi^2s^2} \sum _{i=1}^{d_{\text {ad}}} \tilde{h} \bigl(s,m_{W,i}^2 \bigr) \\ &\qquad \times \biggl\{ \frac{\frac{s b_i}{k^2}}{\sinh (\frac{s b_i}{k^2} )}+ \frac{s b_i}{k^2}\sinh \biggl( \frac{s b_i}{k^2} \biggr) \biggr\} , \\ &{\rm Tr} \bigl[\tilde{h} \bigl( s,m_W^2 \bigr) e^{ -s\frac {-D^2}{k^2} } \bigr]\\ &\quad = \frac{\varOmega k^4}{16\pi^2s^2} \sum_{i=1}^{d_{\text {ad}}} \tilde{h} \bigl(s,m_{W,i}^2 \bigr) \frac{\frac{s b_i}{k^2}}{\sinh (\frac{s b_i}{k^2} )}, \\ &{\rm Tr} \bigl[\tilde{h}(s,0)e^{-s\frac{-D^2}{k^2}} \bigr]= \frac{\varOmega k^4}{16\pi^2s^2} \sum _{i=1}^{d_{\text {ad}}} \tilde{h}(s,0) \frac{\frac{s b_i}{k^2}}{\sinh (\frac{s b_i}{k^2} )}\end{aligned} \end{aligned}$$

(C.3)

where $b_{i}=\bar{g} |\nu_{i}| B$ and Ω is the spacetime volume. The first trace above is over spacetime and Lorentz and color indices, the other two only over spacetime and color indices. For the running gauge coupling we just need the terms of order $b_{i}^{2}$, since the relevant term on the l.h.s. of the flow equation is ∂ _t Γ _k ∋ΩB ² ∂ _t Z _W /2. Using that the running of the renormalized coupling g ² is given in terms of the anomalous dimension, $\partial_{t} g^{2}=\beta_{g^{2}}=\eta_{W} g^{2}$, we find

$$\begin{aligned} \eta_W |_W =\frac{-g^2}{32\pi^2} \sum _{i=1}^{d_{\text {ad}}} \bigl[21h \bigl(0,m_{W,i}^2 \bigr)+h(0,0) \bigr] \frac{|\nu_i|^2}{3}. \end{aligned}$$

(C.4)

In the background-field method, the y→0 limit of the regulator is constrained [70–74]; the only regulators permitted must satisfy h(y→0,0)=2. In the massless limit we thus obtain

$$\begin{aligned} \eta_W |_W=-\frac{1}{16\pi^2}\frac{22}{3}g^2 \sum_{i=1}^{d_{\text {ad}}}|\nu_i|^2=- \frac{1}{16\pi^2}\frac{22}{3}g^2 N_{\text {L}}, \end{aligned}$$

which agrees with standard perturbation theory. Let us work out the massive case using the linear regulator (A.2). In this case, h(y,x)=2(1+x)⁻¹ θ(1−y), such that the gauge contribution to the gauge $\beta_{g^{2}}$ function reads

$$\begin{aligned} \beta_{g^2} |_W=\eta_W |_W g^2=-\frac{g^4}{16\pi^2} \Biggl[\frac{21}{3}\sum _{i=1}^{d_{\text {ad}}}\frac{|\nu_i|^2}{1+m_{W,i}^2}+\frac {N_{\text {L}}}{3} \Biggr]. \end{aligned}$$

The first term now depends on the choice of $\hat{n}^{a}$ the direction of the vev in fundamental color space. This is expected, as for higher gauge groups different breaking patterns and gauge masses can arise. This term also depends in general on |ν _i |², i.e. on $\hat{m}^{i}$. This is also plausible, as the directions of the vev implicitly also allows for the definition of different couplings: depending on the relative direction of the gauge fluctuation w.r.t. the vev, the fluctuations can couple differently to matter.

For SU(2), these issues simplify as

$$ \bar{m}_W^{2\ ij}=\frac{\bar{g}^2 \bar{v}^2}{4} \hat{n}^{\dagger a}\sigma^i_{ab}\sigma^j_{bc} \hat{n}^c =\frac{\bar{g}^2 \bar{v}^2}{4}\bigl(\delta^{ij}+i \epsilon^{ijl}\bigl(\hat {n}^\dagger\sigma^l\hat{n} \bigr)\bigr), $$

(C.5)

such that ${\rm tr}\,\bar{m}_{W}^{2 \ ij}=3\bar{g}^{2} \bar{v}^{2}/4$. Let us denote $c^{l}=(\hat{n}^{\dagger}\sigma^{l}\hat{n})$. This is a vector in adjoint space which is an eigenvector of the mass matrix, with eigenvalue $\bar{g}^{2} \bar{v}^{2}/4$. One can choose a diagonalizing orthonormal basis {e ₁,e ₂=c/|c|,e ₃} in adjoint space such that the mass matrix takes the form

$$\begin{aligned} \bar{m}_W^{2}=\frac{\bar{g}^2 \bar{v}^2}{4} \begin{pmatrix} 2&0&0\\0&1&0\\0&0&0\\ \end{pmatrix} . \end{aligned}$$

Now recall that the ν _i denote the eigenvalues of $(-if^{ijl}\hat{m}^{l})$, which for SU(2) simply are $(-i\epsilon^{ijl}\hat{m}^{l})$. Therefore in SU(2) the eigenvalues are (1,−1,0) for any choice of $\hat{m}$. However, depending on the direction of $\hat{m}$ w.r.t. the basis defined above, the ν _i could be {ν ₁=1,ν ₂=−1,ν ₃=0} or possibly permutations thereof. The two extreme cases for SU(2) are maximal or minimal decoupling. Maximal decoupling happens if |ν ₁|=|ν ₂|=1 and ν ₃=0, and in this case

$$\begin{aligned} \beta_{g^2} |_W=-\frac{g^4}{16\pi^2} \biggl[\frac{21}{3} \biggl(\frac{1}{1+\frac{g^2 v^2}{2k^2}}+\frac{1}{1+\frac{g^2 v^2}{4k^2}} \biggr)+ \frac{2}{3} \biggr] \end{aligned}$$

(C.6)

while minimal decoupling happens if ν ₁=0 and |ν ₂|=|ν ₃|=1, and correspondingly

$$\begin{aligned} \beta_{g^2} |_W=-\frac{g^4}{16\pi^2} \biggl[\frac{21}{3} \biggl(1+\frac{1}{1+\frac{g^2 v^2}{4k^2}} \biggr)+ \frac{2}{3} \biggr]. \end{aligned}$$

(C.7)

For SU(2) the ambiguity of the β-function arises solely from the ambiguity of defining a coupling in the presence of a vev. In fact, there are more quadratic invariants than the only F ², such as for example $\hat{n}^{\dagger a}F^{i}_{\mu\nu}T^{i}_{ab}T^{j}_{bc}F^{j}_{\mu\nu}\hat{n}^{c}$. For higher groups, even the mass matrix depends on the choice of $\hat{n}^{a}$.

3.2 C.2 Contribution from scalar modes

The contribution from scalar fluctuations to the gauge β function arises from the scalar kinetic term. The calculation is very similar to that of the longitudinal gauge modes with two differences: the field is complex and lives in the fundamental representation. Moreover the dimensionless scalar mass matrix in the broken regime reads $m_{\phi}^{2\,ab}=(\lambda_{2} v^{2}/2k^{2})\hat{n}^{a}\hat{n}^{\dagger b}$. Here, we do not attempt to solve the problem in full generality as for the gauge modes, but confine ourselves to a simple choice of backgrounds. Most importantly, we choose the direction of the pseudo-abelian background to satisfy

$$ W^i_\mu=\hat{m}^i W_\mu ,\qquad\hat{m}_i\hat{m}^i=1 ,\qquad \bigl[\bigl(\hat {m}_iT^i\bigr),\hat{n}\otimes \hat{n}^\dagger\bigr]=0. $$

(C.8)

It is important to note that this does not constrain the choice of the vev direction $\hat{n}^{a}$. This is because we can always choose a basis in fundamental color space such that the projector $P_{\hat{n}}=\hat{n}\otimes\hat{n}^{\dagger}$ is diagonal. Then the commutation relation (C.8) can be satisfied by choosing $(\hat{m}_{i}T^{i})^{ab}$ to be in the Cartan, i.e. by choosing it to be diagonal in that basis.

Let’s consider SU(2) as an example. Let $\hat{n}=(0,1)$. Then we choose $\hat{m}=(0,0,1)$ such that

$$\begin{aligned} \bigl(\hat{m}_iT^i\bigr)^{ab}= \frac{1}{2}\sigma^3=\frac{1}{2} \begin{pmatrix}1&0\\0&-1\\ \end{pmatrix} , \end{aligned}$$

(C.9)

obviously satisfying Eq. (C.8). Before we continue with the scalar fluctuations, let us work out the consequences of this choice for the gauge modes of the preceding section. The vector c for this choice becomes c=(0,0,−1) and the mass matrix for the gauge modes, given by (C.5), is

$$\begin{aligned} \bar{m}_W^{2 \ ij}=\frac{\bar{g}^2 \bar{v}^2}{4}\bigl( \delta^{ij}-i\epsilon^{ij3}\bigr) =\frac{\bar{g}^2 \bar{v}^2}{4} \begin{pmatrix} 1&-i&0\\ i&1&0\\0&0&1\\ \end{pmatrix} . \end{aligned}$$

(C.10)

The definition of ν _i , right above (C.2), combined with the choice $\hat{m}=(0,0,1)$ requires us to compute the eigenvalues of

$$\begin{aligned} -i\epsilon^{ij3}=-i \begin{pmatrix} 0&1&0\\-1&0&0\\0&0&0\\ \end{pmatrix} . \end{aligned}$$

The simultaneous eigenvectors of this matrix and of $\bar{m}_{W}^{2}$ are given by

$$\begin{aligned} v_1= \begin{pmatrix} 0\\0\\1\\ \end{pmatrix} ,\qquad v_2= \begin{pmatrix} 1\\i\\0\\ \end{pmatrix} ,\qquad v_3= \begin{pmatrix} 1\\-i\\0\\ \end{pmatrix} \end{aligned}$$

with the corresponding set of eigenvalues: $\{\bar{m}_{W,1}^{2}=\frac{\bar{g}^{2} \bar{v}^{2}}{4}, \nu_{1}=0\}$, $\{\bar{m}_{W,2}^{2}=\frac{\bar{g}^{2} \bar{v}^{2}}{2},\nu_{2}=1\}$, $\{\bar{m}_{W,3}^{2}=0,\nu_{3}=-1\}$. This choice of $\hat{m}$ corresponds to the minimal decoupling case of Eq. (C.7). These considerations tell us that the maximal decoupling solution of Eq. (C.6) might not be permitted, as it would not correspond to a legitimate choice of $\hat{m}$ with $\hat{m}_{i}T^{i}$ in the Cartan (which we had also assumed in the gluonic case in Eq. (C.2)). The choice (C.8) for defining $\hat{m}$ therefore is related to defining the coupling with respect to the unbroken part of the gauge group.

Let us now return to the scalar fluctuations; Eq. (C.8) ensures that the covariant derivative in the fundamental representation satisfies

$$\begin{aligned} \bigl[D_\mu,\hat{n}\otimes\hat{n}^\dagger\bigr]=0 \end{aligned}$$

for our choice of the background field. Then also $[-D^{2}, \hat {n}\otimes\hat{n}^{\dagger}]=0$ and thus $[-D^{2},m_{\phi}^{2}]=0$ follow, such that −D ² and $m_{\phi}^{2}$ can be simultaneously diagonalized. Therefore

$$\begin{aligned} {\rm Tr} \biggl[\frac{\partial_t{R}_k}{\varGamma^{(2)}+R_k} \biggr]_{\phi}= {\rm Tr} \biggl[ \frac{\frac{-D^2}{k^2}\partial_t{r}_k (\frac {-D^2}{k^2} )}{\frac{-D^2}{k^2} (1+r_k (\frac {-D^2}{k^2} ) )+m_\phi^2} \biggr]. \end{aligned}$$

Because of the above considerations, we can rewrite the previous expression in the proper-time form:

$$\begin{aligned} {\rm Tr} \biggl[\frac{\partial_t{R}_k}{\varGamma^{(2)}+R_k} \biggr]_{\phi}&= \int _0^\infty ds\, {\rm Tr} \bigl[\tilde{h} \bigl(s,m_\phi^2\bigr)e^{-s\frac {-D^2}{k^2}} \bigr] \\ &=\frac{\varOmega}{16\pi^2} \int_0^\infty ds\, \sum _{a=1}^{N_{\text {L}}}\tilde{h}\bigl(s,m_{\phi,a}^2 \bigr) \biggl(-\frac{1}{6}b_a^2 \biggr), \end{aligned}$$

where we have retained only the term of order $b_{a}^{2}$ (compare with the second equation of (C.3)). We have denoted the eigenvalues of the mass matrix by $m_{\phi,a}^{2}$ (there is only one non-vanishing eigenvalue for the radial mode). Furthermore, $b_{a}=\bar{g}|\nu_{a}|B$, where ν _a now are the eigenvalues of $(\hat{m}_{i}T^{i} )^{ab}$ related to the fundamental representation. Using the standard normalization for the generators of the fundamental representation, we have

$$\begin{aligned} \sum_{a=1}^{N_{\text {L}}}|\nu_a|^2={ \rm tr} \bigl[ \bigl(\hat{m}_iT^i \bigr)^2 \bigr] =\hat{m}_i\hat{m}_j\frac{1}{2} \delta^{ij}=\frac{1}{2}. \end{aligned}$$

Another difference from the gauge case is that the scalar field is complex and thus there is no factor 1/2 in front of the trace on the r.h.s. of the flow equation. Hence, analogous to (C.4), the contribution of the scalar to the flow of Z _W reads

$$\begin{aligned} \eta_W |_{\phi}=\frac{g^2}{16\pi^2}\sum _{a=1}^{N_{\text {L}}}h\bigl(0,m_{\phi,a}^2 \bigr)\frac{|\nu_a|^2}{3}. \end{aligned}$$

(C.11)

In the massless case, since $h(0,0)=2, \eta_{W}|_{\phi}=\frac{g^{2}}{16\pi^{2}}\frac{1}{3}\nonumber$ in agreement with perturbation theory. In the general massive case and using the linear regulator, we get

$$\begin{aligned} \eta_W |_{\phi}=\frac{g^2}{16\pi^2}\frac{2}{3} \sum_{a=1}^{N_{\text {L}}}\frac{1}{1+m_{\phi,a}^2}| \nu_a|^2. \end{aligned}$$

Generically, only one particular component of $m_{\phi,a}^{2}$ is non-vanishing and equal to 2λ ₂ κ. For SU(2) the ν _a are unique and equal to $\{\frac{1}{2},-\frac{1}{2}\}$. Therefore in this case

$$\begin{aligned} \eta_W |_{\phi}=\frac{g^2}{16\pi^2}\frac{1}{3} \biggl[\frac {1}{2}+\frac{1}{2}\frac{1}{1+2\lambda_2\kappa} \biggr]. \end{aligned}$$

3.3 C.3 Contribution from fermion modes

The relevant part of the effective Lagrangian is

where again we have set any wave function renormalization to one. For Eq. (6), we can choose a gauge background field such that $D_{\mu}^{ab}$ and $P_{\hat{n}}=\hat{n}\otimes\hat{n}^{\dagger}$ as well as $P_{(1-\hat{n})}=1-P_{\hat{n}}$ commute and the above parts of $\mathcal{L}_{k}$ can be written as

(C.12)

Here, the in the second term is projected along ${\hat{n}}$. The first line corresponds to the massless bottom-type fermions. Their contribution is the standard perturbative contribution weighted by eigenvalues ν _a in the orthogonal complement. Let $\hat{n}$ point into the A-direction: $\hat{n}^{a}=\delta^{aA}$. Then the contribution of the massless fermions to the running coupling is

$$\begin{aligned} \partial_tg^2 \big|_{\psi_{(1-\hat{n})}}=\frac{g^4}{16\pi^2} \frac {2d_\gamma N_{\text {g}}}{3}\sum_{a=1,a\neq A}^{N_{\text {L}}} | \nu_a|^2. \end{aligned}$$

If the sum ran over all a’s we would get $\sum_{a=1}^{N_{\text {L}}} |\nu_{a}|^{2}=1/2$ leading to the correct perturbative result. Combining $\psi_{\mathrm{L}}^{\hat{n}}$ and ψ _R into a Dirac spinor , the second line of (C.12) can be written

where , with the usual definition of the left projector $P_{\mathrm{L}}=\frac{1}{2}(1-\gamma_{5})$. We have also introduced the “top mass” $\bar{m}_{t}$ as defined in Eq. (12). Since the regularized fluctuation operator for Ψ satisfies and since , we get

(C.13)

Here we need to know the spectrum of

(C.14)

where we have denoted $\partial_{\mathrm{L/R}}^{\mu}=\partial^{\mu}P_{\mathrm{L/R}}$ and took advantage of: and $\partial_{\mathrm{L}}^{\mu}\partial_{\mathrm{R}}^{\nu}=0$. The determination of this spectrum probably is an analytically soluble problem for a constant magnetic field background, as the differential operator is of harmonic oscillator type, however, with an involved Dirac structure.

As we are mainly interested in the decoupling of massive modes in the flow of the gauge coupling, let us simply take a shortcut at this point. We already know that the contribution of Eq. (C.13) to the β-function in the massless limit must be of the form

$$\begin{aligned} \partial_tg^2 \big|_{\psi_{\hat{n}}}= \frac{g^4}{16\pi^2}\frac {2d_\gamma N_{\text {g}}}{3} |\nu_A|^2. \end{aligned}$$

This fixes the $\mathcal{O}(s^{0})$-term in to be the same as the $\mathcal{O}(s^{0})$-term in . These heat-kernel traces could still differ to higher orders in s, due to the two extra terms in (C.14). These higher-order terms could (unlike as for ) in principle contain terms of order B ² and thus contribute to the beta function via functions of the form

$$\begin{aligned} f_p\bigl(m_\mathrm{t}^2\bigr)=\int _0^\infty ds\, \tilde{h}\bigl(s,m_\mathrm{t}^2 \bigr)s^p= \biggl[ \biggl(-\frac{\partial\ }{\partial y} \biggr)^p h\bigl(y,m_\mathrm{t}^2\bigr) \biggr]_{y=0} \end{aligned}$$

where $m_{\mathrm{t}}^{2}$ is the dimensionless top mass squared: $m_{\mathrm{t}}^{2}=h^{2}\kappa$. Because of the constraints from the massless limit discussed before, we must have f _p (0)=0. Furthermore f _p also has to exhibit a generic threshold behavior, that is: $f_{p}(m_{\mathrm{t}}^{2}\rightarrow\infty)\rightarrow0$. As the precise dependence of $h(0,m_{\mathrm{t}}^{2})$ is anyway regulator-dependent, we simply ignore potentially non-vanishing contributions of $f_{p}(m_{\mathrm{t}}^{2})$ for all practical discussions in the main text. Therefore, without any further explicit calculation, we approximate the threshold behavior of the massive fermion mode by the same form as for the other modes

$$\begin{aligned} \partial_tg^2\big|_{\psi}=\frac{g^4}{16\pi^2} \frac{2d_\gamma N_{\text {g}}}{3}\sum_{a=1}^{N_{\text {L}}} \frac{1}{1+h^2\kappa\delta^{aA}}|\nu_a|^2. \end{aligned}$$

For SU(2) this implies

$$\begin{aligned} \partial_tg^2\big|_{\psi}=\frac{g^4}{16\pi^2} \frac{d_\gamma N_{\text {g}}}{3} \biggl(\frac{1}{2}+\frac{1}{2} \frac{1}{1+h^2\kappa} \biggr). \end{aligned}$$

To summarize, we can write the gauge one-loop β-function approximately as given in the main text in Eqs. (30, 31).

Appendix D: Results for the anomaly-free two-generation model N _g=2

In this appendix, we verify explicitly that the properties of the anomaly-free SU(N _L=2) model with two left-handed generations N _g=2 are essentially identical to the results for the one-generation model discussed in the main text. This can be seen manifestly by comparing Figs. 6, 7, where N _g=2, to Figs. 2, 3, where N _g=1. These explicit solutions verify that the transition from one generation to two generations induces only small quantitative differences in the fixed-point values as well as in the values for the critical exponents.

Fig. 6

Fixed-point values for $\mu_{W}^{2}$ (left panel) and χ (right panel) as a function of the fixed-point value of $\mu_{\mathrm{t}}^{2}$ for N _L=2 and N _g=2. This shows the similarity to the one-generation case depicted in Fig. 2

Fig. 7

Critical exponents for the line of fixed points computed in the mass parametrization as a function of the fixed-point top-mass parameter $\mu_{\text{t}}^{\ast2}$ for N _L=2 and N _g=2; left panel: real parts, right panel: imaginary parts. Again the similarity to the one-generation case, cf. Fig. 3, is obvious

We emphasize again that the N _g=1 model discussed in the main text for phenomenological reasons has a Witten anomaly and thus should be considered as embedded into a larger anomaly-free model, such as the standard model. By contrast, with the results of this appendix, we conclude that the N _g=2 model as it stands can be a consistent UV-complete quantum field theory for all trajectories emanating from the line of non-Gaußian fixed points.