A twist on broken U(3) $\times$ U(3) supersymmetry

Abstract

What symmetry breaking would be required for gauginos from a supersymmetric theory to behave like left-handed quarks of the Standard Model? Starting with a supersymmetric SU(3) $\times$ SU(3) $\times$ U(1) $\times$ U(1) gauge theory, the 18 adjoint-representation gauginos are replaced with 2 families of 9 gauginos in the (3,3*) representation of the group. After this explicit breaking of supersymmetry, two-loop quadratic divergences still cancel at a unification scale. Coupling constant unification is supported by deriving the theory from an SU(3) $\times$ SU(3) $\times$ SU(3) $\times$ SU(3) Grand Unified Theory (GUT). ${\hbox {Sin}}^2$ of the Weinberg angle for the GUT is 1/4 rather than 3/8, leading to a lower unification scale than usually contemplated, $\sim 10^9 \, \hbox {GeV}$. After spontaneous gauge symmetry breaking to SU(3) $\times$ SU(2) $\times$ U(1), the theory reproduces the main features of the Standard Model for two families of quarks and leptons, with gauginos playing the role of left-handed quarks and sleptons playing the role of the Higgs boson. An extension to the theory is sketched that incorporates the third family of quarks and leptons.

Appendices

Appendix

This Appendix shows the following about the SUSY-broken theory ${{\varvec{\mathcal {L}}}}$ compared to the supersymmetric theory ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ in the Landau gauge and at a unification scale where all coupling constants are equal: (a) both theories have the same one-loop wave function renormalization constants for all fields, (b) both theories have the same one-loop beta functions for their gauge couplings and for their supersymmetric d-term couplings, (c) a difference between the theories in the one-loop gaugino vertex is cancelled in quadratically divergent diagrams, and (d) two-loop quadratically divergent diagrams are the same in both theories. Since ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ is free of quadratic divergences, it means that ${{\varvec{\mathcal {L}}}}$ is also free of them to two loops at the unification scale. The analysis is simplified by the fact that ${{\varvec{\mathcal {L}}}}$ is constructed to be exactly the same as ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ except for terms involving gauginos. As a result, it is sufficient to just compare diagrams involving gauginos.

From a Feynman graph standpoint, the difference between the supersymmetric and SUSY-broken theories only shows up in the gaugino propagator and vertices:

(A1)

In these diagrams (and the ones below), scalars, fermions (from chiral multiplets), gauge bosons, and gauginos are represented by dashed lines, solid lines, wavy lines, and solid$+$wavy lines, respectively. The arrows point from $\phi ,\lambda ,\psi$ and toward $\phi ^{*}, {\bar{\lambda }}, {\bar{\psi }}$ and just the group structure is shown. For each of the diagrams showing a gaugino coupled to a scalar and fermion, there is another diagram (not shown) that has the arrows reversed. Hats have been added to the gaugino coupling constants of ${{\varvec{\mathcal {L}}}}$ to differentiate them from the gauge coupling constants. The reason to differentiate is because in ${{\varvec{\mathcal {L}}}}$ with broken supersymmetry, there is no guarantee that the running gaugino coupling will remain equal to the running gauge coupling as the scale is changed. That being said, it will be shown that before unification-scale symmetry breaking, they are indeed the same.

One-loop quadratic divergence cancellation and wave function renormalization

In both theories, there are four potentially quadratically divergent one-loop diagrams:

(A2)

where diagram (c) involves the supersymmetric 4-scalar interaction term (the d-term) that comes from solving the equations of motion for the auxiliary d field in the Lagrangian. In this Appendix, the Landau gauge is assumed. As a result, diagram (a) does not produce a quadratic divergence. It does produce a logarithmic divergence that contributes to the scalar wave function renormalization constant, but that contribution is the same in both theories by construction since the diagram has no gauginos. Similarly, diagrams (b) and (c) also produce the same result in both theories since they do not have gauginos. Since in ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$, the diagrams of (A2) all cancel, they will also cancel in ${{\varvec{\mathcal {L}}}}$ if diagram (d) produces the same result in both theories.

Diagram (d) is shown again below, along with the results produced from each theory.

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}: \quad \left( {2g_{m}^{2} \left( {t^{a}t^{a}} \right) _{ij} +g_{0}^{2} \left( {\left( {{\tilde{t}}_{R}^{3} } \right) ^{2}+\left( {{\tilde{t}}_{R}^{8} } \right) ^{2}} \right) \delta _{ij} } \right) \cdots \nonumber \\&\qquad \qquad \qquad \qquad =\left( {\tfrac{8}{3}g_{m}^{2} +\tfrac{1}{3}g_{0}^{2} } \right) \delta _{ij} \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}}\, \hbox {m}= 1: \quad \left( {{\hat{g}}_{1}^{2} \left( {\delta _{ik} \delta _{pl} -\tfrac{1}{3}\delta _{ip} \delta _{kl} } \right) \left( {\delta _{jk} \delta _{pl} -\tfrac{1}{3}\delta _{jp} \delta _{kl} } \right) +\tfrac{1}{9}{\hat{g}}_{0}^{2} \delta _{ip} \delta _{kl} \delta _{jp} \delta _{kl} } \right) \cdots \nonumber \\&\qquad \qquad \qquad \qquad =\left( {\tfrac{8}{3}{\hat{g}}_{1}^{2} +\tfrac{1}{3}{\hat{g}}_{0}^{2} } \right) \delta _{ij} \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}} \, \hbox {m}= 2: \quad \left( {\tfrac{8}{3}{\hat{g}}_{2}^{2} +\tfrac{1}{3}{\hat{g}}_{0}^{2} } \right) \delta _{ij} \cdots \end{aligned}$$

(A3)

In other words, at the unification scale where all coupling constants are the same (including unhatted and hatted), both theories give the same contribution to the one-loop diagram above (where “$\cdots$” stands for momentum and spin dependence which is the same in both theories). As described previously, it follows that ${{\varvec{\mathcal {L}}}}$ has no one-loop quadratic divergences at the unification scale. Furthermore, the one-loop scalar wave function renormalization constants for ${{\varvec{\mathcal {L}}}}$ and ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ are also the same.

By a similar calculation as above, results for the following diagram can be calculated.

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}: \quad \left( {\tfrac{8}{3}g_{m}^{2} +\tfrac{1}{3}g_{0}^{2} } \right) \delta _{ij} \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}}: \quad \left( {\tfrac{8}{3}{\hat{g}}_{{m}'}^{2} +\tfrac{1}{3}{\hat{g}}_{0}^{2} } \right) _{{m}'\ne m} \delta _{ij} \cdots \end{aligned}$$

(A4)

Again, at the unification scale, both theories produce the same correction. Since all other corrections are also the same by construction, the fermion wave function renormalization constants for ${{\varvec{\mathcal {L}}}}$ and ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ are the same.

The one-gaugino-loop correction to the gauge boson propagator is depicted by:

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}: \quad -g_{m}^{2} f^{acd}f^{bdc}\cdots =3g_{m}^{2} \delta ^{ab}\cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}}: \quad \delta _{jl} \delta _{jl} g_{m}^{2} \sum \limits _n {tr\left( {t^{a}t^{b}} \right) } \cdots =3g_{m}^{2} \delta ^{ab}\cdots . \end{aligned}$$

(A5)

Both theories again produce the same correction. Furthermore, there is no one-loop correction to the Abelian gauge bosons that involves gauginos, so the wave function renormalization constants for ${{\varvec{\mathcal {L}}}}$ and ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ are the same for all of the gauge bosons.

A one-loop correction to the gaugino propagator is depicted by:

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}} \, \hbox {nonAbelian gauginos:} \quad \sum \limits _{n,R} {2g_{m}^{2} tr\left( {t^{a}t^{b}} \right) } \cdots =\delta ^{ab}g_{m}^{2} \sum \limits _{n,R} \nonumber \\&\hbox {For}\, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}} \, \hbox {Abelian gauginos:} \quad g_{0}^{2} \sum \limits _{n,R} \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}} \, \hbox {traceless gauginos:} \quad \sum \limits _{m,R} {{\hat{g}}_{m}^{2} \left( {\delta _{ip} \delta _{jq} -\tfrac{1}{3}\delta _{ij} \delta _{pq} } \right) \left( {g_{m} \delta _{pk} \delta _{ql} -\tfrac{1}{3}\delta _{pq} \delta _{kl} } \right) } \cdots \nonumber \\&\quad =\left( {\delta _{ik} \delta _{jl} -\tfrac{1}{3}\delta _{ij} \delta _{kl} } \right) \sum \limits _{m,R} {{\hat{g}}_{m}^{2} } \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}} \, ``\hbox {Abelian}'' \, \hbox {gauginos:} \quad \tfrac{1}{3}\delta _{ij} \delta _{kl} {\hat{g}}_{0}^{2} \sum \limits _{m,R} \end{aligned}$$

(A6)

At the unification scale, all of the above corrections to gaugino propagators are the same.

There is another one-loop correction to the gaugino propagator involving a gauge boson loop. In the Landau gauge that correction is finite. This section is only considering divergent wave function renormalization constants (assuming a minimal renormalization scheme), so that correction is not addressed here. However, it is addressed in (A12) below. Equation (A6) has shown that the gaugino wave function renormalization constant is the same in both theories at the unification scale.

More generally, it has now been shown that the one-loop wave function renormalization constants for all fields are the same in both ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$ at the unification scale.

One-loop vertex corrections

In ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$, the cancellation of diagrams in (A2) persists as the scale is changed. This is due to the fact that supersymmetry ensures that the couplings involved in diagrams c and d (the “d-term coupling” and the “gaugino coupling”) have the same scale dependence as the gauge couplings in diagrams a and b. For ${{\varvec{\mathcal {L}}}}$, supersymmetry is broken, so it is not assured that the d-term and gaugino couplings will have the same scale dependence as the gauge coupling. This section will show two things at the unification scale. First, the d-term coupling does indeed have the same one-loop scale dependence as the gauge coupling. Second, a difference between ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$ in the scale dependence of the gaugino coupling cancels in the context of quadratically divergent diagrams.

Since both ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$ are gauge theories, gauge invariance ensures that for nonAbelian fields ($\hbox {m} = 1,2$), the gauge couplings to scalars in diagrams a) and b) of (A2) have the same beta function as the 3-gauge coupling. The one-gaugino-loop correction to the latter is depicted by the following diagram:

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}: \quad g_{m}^{3} f^{ade}f^{bef}f^{cfd}\cdots =\left( {\tfrac{3}{2}g_{m}^{2} } \right) \left( {g_{m} f^{abc}} \right) \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}}: \quad ig_{m}^{3} \sum \limits _{n=1,2} {\delta _{jl} \delta _{jl} tr\left( {t^{a}t^{c}t^{b}} \right) } \cdots =\left( {\tfrac{3}{2}g_{m}^{2} } \right) \left( {g_{m} f^{abc}} \right) \cdots \end{aligned}$$

(A7)

Both Lagrangians have the same one-loop contribution. Since all other one-loop corrections to the 3-gauge vertex are the same in both theories by construction, and since Eq. (A5) shows that gauge wave function renormalization constants are the same in both theories, it follows that the gauge coupling constants $g_{1}$ and $g_{2}$ have the same one-loop beta functions in both ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$.

The one-loop beta function for the Abelian gauge coupling $g_{0}$ requires evaluation of the following diagram:

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}: \quad g_{0} \left( {i\tfrac{1}{\sqrt{2} }\left( {-1} \right) ^{m}{\tilde{t}}_{R}^{3;8} } \right) \left( {\tfrac{8}{3}g_{m}^{2} +\tfrac{1}{3}g_{0}^{2} } \right) \delta _{j{j}'} \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}}: \quad g_{0} \left( {-i\tfrac{1}{\sqrt{2} }\left( {-1} \right) ^{m}{\tilde{t}}_{R}^{3;8} } \right) \left( {\tfrac{8}{3}{\hat{g}}_{m}^{2} +\tfrac{1}{3}{\hat{g}}_{0}^{2} } \right) \delta _{j{j}'} \cdots \end{aligned}$$

(A8)

Diagram (A8) shows coupling of Abelian gauge fields to fermions which has the same factor of $\left( {-1} \right) ^{m}$ in ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$. The minus sign for ${{\varvec{\mathcal {L}}}}$ is due to the fact that in ${{\varvec{\mathcal {L}}}}$, the fermion in the loop has ${m}'\ne m$. But the theories have opposite signs for their couplings to scalars (see Eq. (8)), so the vertex renormalization constant is the same in both theories. Since the wave function renormalization constants for all fields involved in the vertex are also the same in both theories, and all other relevant diagrams are the same by construction, the Abelian gauge coupling beta functions and scaling behavior are the same in both theories at the unification scale.

The one-gaugino-loop correction to the four-scalar d-term vertex is represented by the following diagram:

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}: \quad \delta _{{i}'{i}''} \delta _{{i}'''i} \left( {2t_{ij}^{c} t_{{j}'{i}'}^{c} g_{m}^{2} +\left( {\left( {{\tilde{t}}_{R}^{3} } \right) ^{2}+\left( {{\tilde{t}}_{R}^{8} } \right) ^{2}} \right) g_{0}^{2} \delta _{ij} \delta _{{i}'{j}'} } \right) \nonumber \\&\qquad \left( {2t_{{i}''{j}''}^{b} t_{{j}'''{i}'''}^{b} g_{m}^{2} +\left( {\left( {{\tilde{t}}_{R}^{3} } \right) ^{2}+\left( {{\tilde{t}}_{R}^{8} } \right) ^{2}} \right) g_{0}^{2} \delta _{{i}''{j}''} \delta _{{j}'''{i}'''} } \right) \cdots \nonumber \\&\quad =\left( {\left( {\delta _{i{i}'} \delta _{j{j}'} -\tfrac{1}{3}\delta _{ij} \delta _{{i}'{j}'} } \right) g_{m}^{2} +\tfrac{1}{3}g_{0}^{2} \delta _{ij} \delta _{{i}'{j}'} } \right) \left( {\left( {\delta _{{i}'i} \delta _{{j}''{j}'''} -\tfrac{1}{3}\delta _{{i}'{j}''} \delta _{{j}'''i} } \right) g_{m}^{2} +\tfrac{1}{3}g_{0}^{2} \delta _{{i}'{j}''} \delta _{{j}'''i} } \right) \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}}: \quad \left( {\left( {\delta _{i{i}'} \delta _{j{j}'} -\tfrac{1}{3}\delta _{ij} \delta _{{i}'{j}'} } \right) {\hat{g}}_{m}^{2} +\tfrac{1}{3}{\hat{g}}_{0}^{2} \delta _{ij} \delta _{{i}'{j}'} } \right) \left( {\left( {\delta _{{i}'i} \delta _{{j}''{j}'''} -\tfrac{1}{3}\delta _{{i}'{j}''} \delta _{{j}'''i} } \right) {\hat{g}}_{m}^{2} +\tfrac{1}{3}{\hat{g}}_{0}^{2} \delta _{{i}'{j}''} \delta _{{j}'''i} } \right) . \end{aligned}$$

(A9)

At the unification scale, both theories produce the same one-loop correction. Since all other one-loop corrections to the d-term vertex are the same in both theories by construction, and since Eq. (A3) shows that scalar wave function renormalization constants are the same, it follows that the d-term coupling constants have the same one-loop beta functions and one-loop scaling behavior in both theories at the unification scale. As a consequence, at the unification scale, the one-loop beta functions and scaling behavior of the d-term couplings in ${{\varvec{\mathcal {L}}}}$ are the same as those of the gauge couplings, since both are equal to their ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ counterparts.

In the Landau gauge, the only divergent one-loop correction to the gaugino coupling vertex is given by the diagram below:

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}: \quad \sqrt{2} g_{m} t_{{i}'j}^{b} \left( {g_{m} f^{cab}} \right) \left( {ig_{m} t_{{i}'i}^{Tc} } \right) \cdots =\sqrt{2} g_{m} t_{ij}^{a} \left( {\tfrac{3}{2}g_{m}^{2}} \right) \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}} \, \hbox {m}=1: \quad \left( {{\hat{g}}_{1} \left( {\delta _{j{k}'} \delta _{{i}'{l}'} -\tfrac{1}{3}\delta _{{i}'j} \delta _{{k}'{l}'} } \right) +\tfrac{1}{3}{\hat{g}}_{0} \delta _{{i}'j} \delta _{{k}'{l}'} } \right) \left( {-ig_{2} t_{{l}'l}^{c} \delta _{{k}'k} } \right) \left( {ig_{2} t_{{i}'i}^{Tc} } \right) \cdots \nonumber \\&\quad =\left( {\left( {\delta _{jk} \delta _{il} -\tfrac{1}{3}\delta _{ij} \delta _{kl} } \right) \left( {\tfrac{7}{6}{\hat{g}}_{1} +\tfrac{1}{6}{\hat{g}}_{0} } \right) +\tfrac{1}{3}\delta _{ij} \delta _{kl} \tfrac{4}{3}{\hat{g}}_{1} } \right) g_{2}^{2} \cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}} \, \hbox {m}= 2: \quad \left( {\left( {\delta _{jk} \delta _{il} -\tfrac{1}{3}\delta _{ij} \delta _{kl} } \right) \left( {\tfrac{7}{6}{\hat{g}}_{2} +\tfrac{1}{6}{\hat{g}}_{0} } \right) +\tfrac{1}{3}\delta _{ij} \delta _{kl} \tfrac{4}{3}{\hat{g}}_{2} } \right) g_{1}^{2} \cdots . \end{aligned}$$

(A10)

There is no one-loop correction with a scalar in the loop since it is not possible to draw such a diagram in which the arrows of the last vertex in (A1) are all consistent. At the unification scale, the one-loop gaugino vertex correction for ${{\varvec{\mathcal {L}}}}$ is $\tfrac{8}{9}$ that of ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$. But the vertex correction for ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ only applies to vertices involving the 8 nonAbelian gauginos in each family, whereas the correction for ${{\varvec{\mathcal {L}}}}$ applies to vertices involving all 9 of the gauginos. As a result, in any diagram where gauginos appear only in loops, the vertex correction for ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ will be applied $\tfrac{8}{9}$ of the times that it will be applied for ${{\varvec{\mathcal {L}}}}$. In other words, ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$ give the same result for these one-loop vertex corrections in the context of any diagram that has no external gaugino lines. Since no quadratically divergent diagrams have external gaugino lines, in the context of those diagrams, the one-loop beta function and scaling behavior of the gaugino coupling is the same in both theories.

More generally, it has now been shown that in the context of diagrams with no external gaugino lines, the one-loop beta functions for all coupling constants are the same in both ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$ at the unification scale.

Two-loop quadratic divergence cancellation

There are three types of quadratically divergent two-loop diagrams: (i) diagrams with no gauginos, (ii) diagrams that modify those of (A2) by incorporating the one-loop wave function or vertex corrections considered previously in this Appendix, and (iii) other diagrams. Type (i) diagrams give the same result in ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$ by construction. The previous sections of this Appendix have shown that type (ii) diagrams also give the same result.

The only diagram in the “other” category is the following one:

(A11)

This diagram involves a one-loop correction to the gaugino propagator that was not considered previously since it is not divergent in the Landau gauge. However, the finite propagator correction still leads to a quadratic divergence in the above diagram.

The relevant gaugino propagator correction is depicted by:

$$\begin{aligned}&\hbox {For} \, {{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}: \quad -g_{m}^{2} f^{acd}f^{bdc}\cdots =3g_{m}^{2} \delta ^{ab}\cdots \nonumber \\&\hbox {For} \, {{\varvec{\mathcal {L}}}}: \quad \left( {g_{1}^{2} \left( {t^{a}t^{b}} \right) _{ik} \delta _{jl} +g_{2}^{2} \left( {t^{aT}t^{bT}} \right) _{jl} \delta _{ik} } \right) \cdots =\tfrac{4}{3}\delta _{ik} \delta _{jl} \sum \limits _m {g_{m}^{2} } \cdots . \end{aligned}$$

(A12)

Just as in (A10), the correction for ${{\varvec{\mathcal {L}}}}$ at the unification scale is $\tfrac{8}{9}$ the one for ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$. But the correction for ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ only applies to nonAbelian gauginos, whereas the one for ${{\varvec{\mathcal {L}}}}$ applies to all gauginos. This means that ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$ generate the same correction for diagram (A12). More generally, it has been shown that ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$ and ${{\varvec{\mathcal {L}}}}$ generate the same result for all quadratically divergent two-loop diagrams. Since all quadratic divergences cancel in ${{\varvec{\mathcal {L}}}}_\mathrm{{SUSY}}$, all two-loop quadratic divergences cancel in ${{\varvec{\mathcal {L}}}}$ at the unification scale.