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The Hierarchy Problem and the Cosmological Constant Problem Revisited

Higgs Inflation and a New View on the SM of Particle Physics

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Abstract

We argue that the Standard Model (SM) in the Higgs phase does not suffer from a “hierarchy problem” and that similarly the “cosmological constant problem” resolves itself if we understand the SM as a low energy effective theory emerging from a cutoff-medium at the Planck scale. We actually take serious Veltman’s “The Infrared–Ultraviolet Connection” addressing the issue of quadratic divergences and the related huge radiative correction predicted by the SM in the relationship between the bare and the renormalized theory, usually called “the hierarchy problem” and claimed that this has to be cured. We discuss these issues under the condition of a stable Higgs vacuum, which allows extending the SM up to the Planck cutoff. The bare Higgs boson mass then changes sign below the Planck scale, such that the SM in the early universe is in the symmetric phase. The cutoff enhanced Higgs mass term as well as the quartically enhanced cosmological constant term provide a large positive dark energy that triggers the inflation of the early universe. Reheating follows via the decays of the four unstable heavy Higgs particles, predominantly into top–antitop pairs, which at this stage are massless. Preheating is suppressed in SM inflation since in the symmetric phase bosonic decay channels are absent at tree level. The coefficients of the shift between bare and renormalized Higgs mass as well as of the shift between bare and renormalized vacuum energy density exhibit close-by zeros at about \(7.7 \times 10^{14}\ \hbox {GeV}\) and \(3.1 \times 10^{15}\ \hbox {GeV}\), respectively. The zero of the Higgs mass counter term triggers the electroweak phase transition, from the low energy Higgs phase and to the symmetric phase above the transition point. Since inflation tunes the total energy density to take the critical value of a flat universe and all contributing components are positive, it is obvious that the cosmological constant today is naturally a substantial fraction of the total critical density. Thus taking cutoff enhanced corrections seriously the Higgs system provides besides the masses of particles in the Higgs phase also dark energy, inflation and reheating in the early universe. The main unsolved problem in our context remains the origin of dark matter. Higgs inflation is possible and likely even unavoidable provided new physics does not disturb the known relevant SM properties substantially. The scenario highly favors to understand the SM and its main properties as a natural structure emerging at long distance. This in particular concerns the SM symmetry patterns and their consequences.

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Notes

  1. The Planck medium, which we may call ether, somehow gets shaped by gravity and quasi-particle interactions emergent in the SM at low energies. It is characterized by the well known fundamental Planck cutoff \(\varLambda _{\mathrm{Pl}}\) or equivalently the Planck mass \(M_{\mathrm{Pl}}\), which derive from the basic fundamental constants, the speed of light \(c\) characterizing special relativity, the Planck constant \(\hbar \) intrinsic to quantum physics and Newton’s constant \(G_N\) the dimensionful key parameter of gravity. Unified they provide an intrinsic length \(\ell _{\mathrm {Pl}}\), the Planck length, which also translates into the Planck time \(t_{\mathrm{Pl}}\) and the Planck temperature \(T_{\mathrm{Pl}}\):

    Planck length: \(\ell _{\mathrm {Pl}}=\sqrt{\frac{\hbar G_N}{c^3}}=1.616252(81)\times 10^{-33}\mathrm {cm}\,,\)

    Planck time: \(t_{\mathrm {Pl}}=\ell _{\mathrm {Pl}}/c=5.4\times 10^{-44} \mathrm {s}\,,\)

    Planck (energy) scale: \(M_{\mathrm {Pl}}=\sqrt{\frac{c\hbar }{G_N}}=1.22\times 10^{19}~\text{ GeV }\,,\)

    Planck temperature: \(\frac{M_{\mathrm {Pl}}c^2}{k_{\mathrm {B}}}=\sqrt{\frac{\hbar c^5}{G_N k_{\mathrm {B}}^2}}=1.416786(71)\times 10^{32^{\circ }\mathrm {K}~}\,.\)

    In our context they define a shortest distance \(\ell _{\mathrm {Pl}}\) and a beginning of time \(t_{\mathrm{Pl}}\,.\) i.e. \(t>t_{\mathrm{Pl}}\,.\) The Planck era energy scale equivalently is set by \(E_{\mathrm {Pl}}=\varLambda _{\mathrm{Pl}}\equiv M_{\mathrm{Pl}}\) or temperature \(T_{\mathrm{Pl}}\), as for most time in the evolution of the early universe, when elementary particle physics is at work and before the epoch of formation of hadrons, particle processes are in thermal equilibrium, with well known exceptions during inflation and the electroweak phase transition.

  2. Einstein’s field equation for the metric tensor \(g_{{\mu \nu }}\), which incorporates the gravitational field, is given by \( G_{{\mu \nu }}= \kappa T_{{\mu \nu }}\) where \(\kappa =\frac{8\pi G_N}{c^2}\) is the effective interaction constant, \(G_{{\mu \nu }}=R_{{\mu \nu }}-\frac{1}{2} R\, g_{{\mu \nu }}\) is the Einstein curvature tensor (geometry) and \(T_{{\mu \nu }}\) is the energy-momentum tensor (matter and radiation). Cosmology is shaped by Einstein gravity, which together with Weyl’s postulate, that radiation and matter (galaxies etc.) on the cosmological scale behave like an ideal fluid, and the cosmological principle, assuming isotropy of space (implying homogeneity), fixes the form of the metric and of the energy-momentum tensor: (1) the metric (3-spaces of constant curvature \(k=\pm 1,0\)) takes the form \(\mathrm {d}s^2 = \left( c\mathrm {d}t\right) ^2-{a^2(t)}\,[{\mathrm {d}r^2}/({1-k r^2}) + r^2 \, \mathrm {d}\varOmega ^2]\), where in the comoving frame \(\mathrm {d}s =c\,\mathrm {d}t \) with \(t\) the cosmic time; (2) the energy-momentum tensor takes the form \(T^{{\mu \nu }}=\left( {\rho (t)}+{p(t)}\right) (t)\,u^{\mu } u^{\nu }-{p(t)}\,g^{\mu \nu }\,;\;\;u^{\mu }\doteq \frac{\mathrm {d}x^{\mu }}{\mathrm {d}s}\) where \(\rho (t)\) is the density and \(p(t)\) the pressure of the fluid. As a differential equation for the geometry factor a(t) one obtains Friedmann’s equations (3). One needs \(\rho (t)\) and \(p(t)\) (which are related by an equation of state characterizing the medium) in order to get the radius of the universe \(a(t)\) and its evolution in time. The Higgs potential contributes \(T_{{\mu \nu }} = \varTheta _{{\mu \nu }}=V(\phi )\,g_{{\mu \nu }}+ \mathrm { \ derivative \ terms}\), where \(\varTheta _{{\mu \nu }}\) is the symmetric energy momentum tensor of the SM (or extensions of it). Only a scalar potential can contribute a term proportional to \(g_{{\mu \nu }}\), which mimics a cosmological constant.

  3. Matter here includes dark matter (DM) and normal baryonic-matter (BM), the non-relativistic stuff; radiation includes all relativistic degrees of freedom: photons, neutrinos and at high energies other SM particles besides the Higgs bosons, which get boosted to be heavy because of their missing naturalness. Note that normal baryonic matter only emerges after the QCD [12, 13] hadronization phase transition, after protons and neutrons have been formed. In contrast cold dark matter looks must have existed much earlier not too long after Planck time.

  4. A complete two-loop calculation has been performed independently by Veretin and Kalmykov in the context of [80,81,82]. Two-loop integrals exhibiting Higgs propagators have been expanded in \(M_V^2/M_H^2\), assuming the Higgs to be heavier than the W and Z bosons. After the Higgs mass has been found this expansion turned out to be obsolete, and the relevant integrals had to be evaluated numerically. This has later been performed partially in  [83, 84] and “exact” in [87] such that a complete two-loop evaluation is available. For independent calculations of the matching conditions also see [88,89,90,91] and references therein.

  5. We denote on-shell masses by capital, \(\overline{\mathrm{MS}}\) masses by lower case letters as in [1].

  6. A LO estimate with \(M_W=80.385\pm 0.015~\text{ GeV }\), \( \varGamma _W=2.085\pm 0.042~\text{ GeV }\) and \( B(W\rightarrow \ell \nu _{\ell })=10.80\pm 0.09\%\) yields \(\hat{G}_{\mu }=1.15564(55)\times 10^{-5}~\text{ GeV }^{-2}\) to be compared with \(G_{\mu }=1.16637(1)\times 10^{-5}~\text{ GeV }^{-2}\), i.e. the on-shell Fermi constant at scale \(M_Z\) appears reduced by 0.92% relative to \(G_{\mu }\,.\)

  7. Obviously, in electroweak theory, what is called \(\overline{\mathrm{MS}}\) scheme may refer to two different versions, depending on whether tadpoles are dropped or not [101]. This only concerns calculations in the broken phase where mass effects play a role, as for the matching conditions (see [83, 84]). Note that RG coefficients are caclulated in the massless symmetric phase where they are unambiguous, since tadpoles are absent.

  8. String theory is motivated by the requirement to ’quantize gravity’. Note that in string theory the Planck energy level represents the ground state on top of which an infinite tower of harmonics at \(E_n=n\,M_{\mathrm{Pl}}\; (n=2,\ldots , \infty )\) resides. The algebra of the spectral raising and lowering operators can be closed to a Kac–Moody algebra (infinite dimensional analogs of semi-simple Lie algebra) only in a particular space–time dimension D. The unique supergravity string theory requires D = 11, where 7 of them are assumed to be compactified (see e.g. [108]). For me it is hard to beliefe that this is what shapes our real world.

  9. The unification paradigm celebrated its triumphal success in Maxwell’s electromagnetism, which unified electrical and magnetic laws and predicted electromagnetic waves. In contrast as we know the electroweak theory is not a true unification, it rather regulates the mixing of electromagnetic and weak interaction phenomena. At the heart is \(\gamma -Z\) mixing and Z resonance (a kind of “heavy-light”) physics, which manifests itself most convincingly in electron positron annihilation into Z bosons. All further unification attempts so far are missing confirmation.

  10. FCNCs are automatically absent in the SM by the GIM mechanism [109] , as it is also highly established by experiment.

  11. Similarly, a global lepton flavor symmetry \(U(1)_e\otimes U(1)_{\mu } \otimes U(1)_{\tau }\), which would imply exact lepton flavor conservation, is not emergent because renormalizability does not dependent on it. That it is a surprisingly accurate approximate symmetry is due to the smallness of the neutrino masses, which likely results as a consequence of a see-saw mechanism at work.

  12. Taming the infinities we encounter in the theory of elementary particles, i.e. of quantum field theories, by completing them with a cutoff, often called the UV-completion of a QFT, is as old as QFT itself. Actually, it took 20 years from Dirac 1928 (Dirac hole theory of relativistic electron–photon interaction [pre-QED]) to Feynman, Schwinger and Tomonaga in 1948 who found how to deal with the large cutoff limit and making QED a predictive theory. For non-Abelian gauge theories proposed by Yang and Mills in 1954 [116] it took another 17 years until a renormalizable formulation was found by ’t Hooft in 1971 [117,118,119] (actually by circumventing a cutoff regularization).

  13. It is interesting to note that statistical mechanical systems with short-range exchange and long-range multipole interactions exhibit vector bosons and graviton modes that follow from a multipole expansion of a static potential [126]. In this sense the emergence of gauge-bosons looks pretty natural.

  14. In the unitary gauge we can avoid problems related to Elitzur’s theorem [136], which claims that an order parameter cannot be associated with SSB of a non-Abelian gauge theory. In a physical gauge, on physical Hilbert space, Higgs ghost fields are absent and a Mexican hat potential is a phantom as it only exist if ghost space is taken into the display. A physical Mexican hat potential would imply the existence of three Nambu–Goldstone bosons.

  15. As an example we may consider an Ising ferromagnet in \(D=2\) dimensions, \(J\) is the nearest neighbor (n.n.) spin coupling between the spins on a lattice

    $$\begin{aligned} H(\sigma )= -J\sum _{<ij>}\sigma _{ i}\,\sigma _{ j}\,\,;\;\;P_{\beta }(\sigma )=\frac{\mathrm {e}^{-\beta H(\sigma )}}{Z_{\beta }}\,;\;\;Z_{\beta }=\sum _{\sigma } \mathrm {e}^{-\beta H(\sigma )}\,.\end{aligned}$$

    Here \( \beta =\frac{1}{k_B T}\) where \(k_B\) is the Boltzmann constant. The Onsager solution for the critical temperature reads \(\sinh ^2 \left( \frac{2J}{k_BT}\right) =1 \,;\;\;T_c=\frac{2J}{k_B\,\ln (1+\sqrt{2})}\) and the magnetization is given by \(M=\left( 1-\left[ \sinh 2 \beta J\right] ^{-4}\right) ^{\frac{1}{8}}\,,\) depending on temperature \(T\) and n.n. spin interaction strength \(J\). For more details see e.g. [113]

  16. Fig. 5 shows that the Higgs mass parameter m is varying little in the broken phase, while \(v=\sqrt{6/\lambda }\,m\) increases substantially because \(\lambda \) decreases rapidly.

  17. Originally the Ginzburg–Landau theory of superconductivity has been proposed as a macroscopic phenomenological effective theory describing type-I superconductors without reference to microscopic properties. Later Bardeen–Cooper–Schrieffer could explain superconductivity from its microscopic structure in their BCS-theory [140, 141]. Afterward Gor’kov derived the GL-theory [142] showing that in some limit all GL parameters have a microscopic interpretation. In addition, Abrikosov showed that GL-theory also models type-II superconductors [143]. The effective GL-theory thus efficiently describes a rich variety if superconducting systems, without the need for a detailed microscopic understanding.

  18. Most groups are adopting essentially the same input parameters presented in [65, 68, 85] and a radiatively corrected effective potential and find the vacuum to lose stability at about a surprisingly low scale of about \(\mu \sim 10^{9}~\text{ GeV }\) [input not independent]. Keep in mind: the Higgs boson mass miraculously turns out to have a value very close to what was expected from vacuum stability. It looks like a tricky conspiracy with other couplings to reach this “purpose”. Assuming vacuum stability, the narrow stability window actually makes the Higgs mass to be a predictable quantity if we consider the other SM parameters as given. Also imposing Planck scale boundary conditions may be argued to fix the Higgs boson mass [66, 67]. If the Higgs boson misses to stabilize the vacuum, why does it just miss it almost not?

  19. As we have argued earlier we consider the bare Higgs potential to be the true potential, except that the bare parameters have to be calculated bottom-up from the known values at low energy. A low energy reparametrization also affects the form of the potential by radiative corrections as we know from Coleman–Weinberg [138]. The correspondingly modified effective potential plays a crucial role when the potential gets unstable and actually can turn instability into meta-stability [39, 65]. This will be discussed in Sect. 5.1 below. The Planck medium, from which the SM derives as a long distance tail, certainly exhibits a stable ground state. This we infer from our mere existence.

  20. Note that this is a finite prediction independent of quadratic cutoff effects. The transition point \(\mu _0\) is a matching point where bare and renormalized quantities at scale \(\mu _0\) agree, i.e. \(\lambda =\lambda (\mu _0)\) and \(v=v(\mu _0)\).

  21. Being a part of the SM Lagrangian the Higgs potential term considered so far gets reparametrized by a change of the effective parameters and the effective Higgs field and by appropriate counterterms only, as long as perturbation theory does not break down. All perturbative physics is obtained as usual by means of the renormalizable Lagrangian, written in terms of the quantized fields, and the corresponding Feynman rules. Also note that the Higgs contribution to the energy-momentum tensor of Einstein gravity is represented by the symmetric energy-momentum tensor

    $$\begin{aligned} \varTheta ^{\mu }_{\,\,\nu }=\frac{\partial {\mathcal {L}}}{\partial (\partial _{\mu } \phi )}\,\partial _{\nu }\phi -\delta ^{\mu }_{\,\,\nu }{\mathcal {L}}\,, \mathrm { \ where \ \ } {\mathcal {L}}(\phi )=\frac{1}{2}\,g^{\mu \nu }\,\partial _{\mu }\phi \, \partial _{\nu }\phi -V(\phi )\,, \end{aligned}$$

    in terms of the Higgs part of the bare SM Lagrangian.

  22. As shown in [138], the potential satisfies the RG equation

    $$\begin{aligned} \left( \mu \,\frac{\partial }{\partial \mu }+\sum _i \beta _i \frac{\partial }{\partial \lambda _i} +\gamma \,\phi \,\frac{\partial }{\partial \phi }\right) \,V=0 \end{aligned}$$

    where \(\lambda _i=m^2,\lambda ,g'=g_1,g=g_2,g_s=g_3,y_t\) with corresponding beta-functions \(\beta _i\) and \(\gamma \) the anomalous dimension of the Higgs field. The RG as usual is solved along characteristic curves where t parametrizes the position on the curve. The solution reads

    $$\begin{aligned} V(\mu ,\lambda _i,\phi )=Z^4_{\phi }(t)\,V(\mu (t),\lambda _i(t),\phi )\,, \end{aligned}$$

    with \(Z_{\phi }(0)=1,\,\lambda _i=\lambda _i(0)\) and \(\phi =\phi (0)\,.\)

  23. This is the Horizon problem: the finite age \(t\) of the universe together with the finite speed of light \(c\) allows us to see to distances \(D_{\mathrm{hor}}=c\,t\) at most. The CMB sky is much larger [\(d_{t_{\mathrm{CMB}}}\simeq 4\cdot 10^{7}~\ell \mathrm {y}\)] than the causally connected patch [\(D_{\mathrm {CMB}}\simeq 4 \cdot 10^5~\ell \mathrm {y}\)] at the time of last scattering \(t_{\mathrm{CMB}}\simeq 380 000~\mathrm{yrs}\) when the CMB decoupled from matter. As we know, no \(D_{\mathrm {CMB}}\) spot shadow is distinguishable at the full CMB sky.

  24. Unbroken SUSY would require a perfect cancellation to happen at all scales. Broken SUSY would largely diminish the quadratic and quartic enhancements which are key effects in our scenery.

  25. The appearance of an non-vanishing v provides a large negative contribution, which however by far does not compensate the large positive offset \(\langle V(\phi )\rangle \) we have from the symmetric phase. A more accurate analysis would have to take into account subleading effects from the chiral phase transition of QCD as well.

  26. We note that, in contrast to claims that the SM cannot explain baryogenesis, the latter looks to be well possible within the LEESM scenario, provided the EW phase transition happens not too far below the Planck scale, at a scale \(\mu _0\) where the mentioned dimension 6 four-fermion operators can be sufficiently effective i.e. \((\mu _0/\varLambda _{\mathrm{Pl}})^2\sim 1.4\times 10^{-6}\) is not too small. The observed baryon asymmetry is \(\eta _B \sim 10^{-10}\). Remains the question of whether CP violation as given by the SM is big enough and sufficiently efficient in the new context.

  27. As mentioned earlier, the constructive understanding of LEETs is Wilson’s renormalization semi-group, based on integrating out short distance fluctuations. This produces all kinds, mostly of irrelevant higher order interactions. A typical example is the Ising model, which by itself seen as the basic microscopic system has simple nearest neighbor interactions only and by the low energy expansion develops a tower of higher order operators, which at the short distance scale are simply absent altogether. Such operators don’t do any harm at the intrinsic short distance scale. As a minimal fairly realistic Planck system representative in the universality class of the SM, we may consider the lattice SM, a SM generalization of lattice QCD, which also is expected to behave decently at the short distance scale.

  28. Note that the locations of the nearby zeros of \(\delta m_H^2\) Eq. (19) and \(\delta \rho _{\varLambda }\) Eq. (39) are independent of the value of the cutoff \(\varLambda \), but they depend very sensitively on the input parameters specified in Table 1.

  29. We note that right-handed neutrinos have been expected to exist from the point of view of the SM gauge symmetry. They have been excluded by imposing additional global flavor conservation, because the latter had not been observed within the achieved experimental accuracy. In fact flavor violation is so tiny that a direct observation still is missing. The corresponding smallness of neutrino masses very likely is a consequence of a see-saw mechanism, which could be triggered within the SM when the singlets would be of Majorana type. Majorana mass terms are not protected by any symmetry and therefore would be expected to be related to the Planck scale in a similar way as the Higgs field mass in the symmetric phase.

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Acknowledgements

I thank the organizers of the Naturalness, Hierarchy and Fine Tuning Workshop, at the RWTH Aachen, for the kind invitation and the support.

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  1. Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489, Berlin, Germany

    Fred Jegerlehner

  2. Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, 15738, Zeuthen, Germany

    Fred Jegerlehner

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Appendix: How Natural is the Minimal SM?

Appendix: How Natural is the Minimal SM?

Often it is considered that it would be more natural to have a left-right symmetric world including mirror fermions. The following consideration, which goes back to Veltman [181] (see also [182,183,184,185]), is instructive as it helps to understand why parity violation is quite natural and why QED conserves parity. It has a lot to do with the assumption of a minimal Higgs system. I reproduce a version, which I had presented in [186] some time ago. Actually within the context of our LEESM scenario we gain a much deeper insight, because the assumptions made are now emergent properties resulting from the low energy expansion.

In order to try to derive the SM let us make the following assumptions:

  1. (1)

    local field theory

  2. (2)

    interactions follow from a local gauge principle

  3. (3)

    renormalizability

  4. (4)

    masses derive from the minimal Higgs system

  5. (5)

    \(\nu _R\) which we know must exist does not carry hypercharge. Note that all points besides the last one are emergent structures in a LEESM as we may learn from [126,127,128,129,130,131,132] (see also Sect. 3). We admit that the last assumption looks somewhat ad hoc, but nevertheless we make it. From the above assumptions the following picture develops:

    • For the gauge interactions the simplest non-trivial possibility is that the fundamental massless matter fields group according to the simplest possibilities, into doublets and triplets, which are the fundamental representations of \( SU (2)\) and \( SU (3)\), besides possible singlets.

    • Since fields are massless all fields can be chosen left-handed. Left-handed particles and left-handed antiparticles at this stage are uncorrelated.

    • We must have pairing for particles that are going to be massive, since a mass term (we ignore the possibility to have Majorana fields here) has the form \(\bar{\psi } \psi = \bar{\psi }_L \psi _R + \bar{\psi }_R \psi _L\). Notice that for massive particles only, we know which left-handed antiparticle belongs to which left-handed particle to form a Dirac field.

    • For \( SU (3)_c\) triplets we must have pairing in order to avoid axial anomalies. \( SU (3)\) is the simplest group having complex representations. This allows to put particles in 3 and antiparticles in the inequivalent \(3^*\). As a consequence a rich color singlet structure (\(\equiv \) hadron spectrum) results. Furthermore, confinement requires \( SU (3)_c\) to be unbroken !

    • \( SU (2)_L\) is anomaly free and hence there is no anomaly condition associated with this group. To generate mass we have to break \( SU (2)_L\) by a Higgs mechanism. The simplest and natural possibility is to choose one Higgs field in the fundamental representation of \( SU (2)_L\). There is no hypercharge for the moment. The Higgs field may be written in the form

      $$\begin{aligned} \varPhi _b = \widetilde{\varPhi }\, \chi _b \; ; \; \chi _b = \left( \begin{array}{c} 0 \\ 1 \end{array} \right) \end{aligned}$$

      in terms of a \(2 \times 2\) matrix field (\(\tau _i\,,\,\, i=1,2,3\,\) the Pauli matrices)

      $$\begin{aligned} \widetilde{\varPhi } = \frac{1}{\sqrt{2}}(H_s +\mathrm {i}\tau _i \phi _i)\;. \end{aligned}$$

      The covariant derivative being given by

      $$\begin{aligned} D_{\mu } \varPhi _b = (\partial _{\mu } -\mathrm {i}\frac{g}{2} \tau _a W_{\mu a})\, \varPhi _b \; , \end{aligned}$$

      and the \( SU (2)\) invariant renormalizable Higgs system

      $$\begin{aligned} {\mathcal {L}}_{\mathrm{Higgs}}= \left( D_{\mu } \varPhi _b \right) ^+ \left( D^{\mu } \varPhi _b \right) - \lambda \left( \varPhi _b^+ \varPhi _b \right) ^2 + \mu ^2 \left( \varPhi _b^+ \varPhi _b \right) \;, \end{aligned}$$
      (44)

      exhibits an extra global \( SU (2)_R\)-symmetry \(\chi _b \rightarrow V^+ \chi _b\). One easily checks that the transformation

      $$\begin{aligned} \widetilde{\varPhi } \rightarrow U(x)\, \widetilde{\varPhi } V^+\,, \end{aligned}$$

      with \(U(x)\in SU (2)_{L,\mathrm{local}}, V \in SU (2)_{R,\mathrm{global}}\) leaves the Higgs Lagrangian invariant. This implies that the fields (\(W^+, W_3, W^-\)) form a weak isospin triplet with \(M_Z=M_{W^{\pm }}\).

      Now consider the fermions (still no hypercharge). Since \(L_f\) and \(\varPhi _b\) are \( SU (2)\) doublets there also must exist singlet fermions \(R_f\), otherwise we would not be able to write down an invariant and renormalizable fermion–Higgs coupling. Therefore \( SU (2)_L\)must be parity violating of V-A-type! The Yukawa term has the general form

      $$\begin{aligned} {\mathcal {L}} _{\mathrm{Yukawa}} = - \bar{L}_f \widetilde{\varPhi } \left( \begin{array}{c} g_{11}~~g_{12} \\ g_{21}~~g_{22} \end{array} \right) R_f + h.c.\,, \end{aligned}$$

      with 4 complex couplings \(g_{ij}\) and \(R_f\) a “doublet” having two right-handed singlets as entries. Although we have not used hypercharge to restrict these couplings the existence of a global \( SU (2)_R\)-symmetry of the Higgs system allows us to transform the Yukawa couplings

      $$\begin{aligned} \widetilde{\varPhi }\, (\cdot )\, R_f \rightarrow \widetilde{\varPhi } V^+ (\cdot ) W R_f \end{aligned}$$

      to standard form, \(V^+ (\cdot ) W\)= real diagonal. Since \(V \in SU (2)_R\) has 3 parameters and W is an arbitrary unitary matrix with 4 parameters we end up with one free parameter such that the system exhibits a global U(1) invariance. This is not surprising since in the unitary gauge we always can end up only with \({\mathcal {L}}_{\mathrm{Yukawa}}\) in the simple standard form

      $$\begin{aligned} {\mathcal {L}}_{\mathrm{Yukawa}}= & {} - \sum _{f} m_f \bar{\psi }_f \psi _f \;\left( 1 + \frac{H}{v}\right) \,.\end{aligned}$$
      (45)

    • The global U(1) which is a consequence of the minimal Higgs mechanism may be interpreted as a global \(U(1)_Y\). We are free to assign \(Y=1\) to \(\varPhi _b\), which means nothing else than that we measure Y in units of the \(\varPhi _b\)- hypercharge. Then

      $$\begin{aligned} \varPhi _t = \widetilde{\varPhi }\, \chi _t \; ; \; \chi _t = \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \end{aligned}$$

      has \(Y=-1\) , and we may write \(\widetilde{\varPhi }=(\varPhi _b,\varPhi _t)\). Since we have the global \(U(1)_Y\) for free, we may assume this symmetry to be local. The covariant derivative for \(\widetilde{\varPhi }\) now reads

      $$\begin{aligned} D_{\mu } \widetilde{\varPhi }= \partial _{\mu } \widetilde{\varPhi }+\mathrm {i}\frac{g'}{2}B_{\mu } \widetilde{\varPhi } \,\tau _3 -\mathrm {i}\frac{g}{2} \tau _a W_{\mu a}\widetilde{\varPhi } \end{aligned}$$

      and we find back the usual Higgs Lagrangian

      $$\begin{aligned} {\mathcal {L}}_{\mathrm{Higgs}}= & {} \frac{1}{2}(\partial _{\mu } H \partial ^{\mu } H) +\frac{(H+v)^2}{2 v^2}(M_Z^2 Z_{\mu } Z^{\mu } +2 M_W^2 W_{\mu }^+ W^{- \mu }) \nonumber \\&-\frac{\lambda }{4} H^4 -\lambda v H^3 -\frac{1}{2} m_H^2 H^2\,.\end{aligned}$$
      (46)

      The 3 real fields \(\phi _a \; a=1,2,3 \) could and have been gauged away and only 3 out of 4 gauge fields can acquire a mass. Hence there must exist one massless field, the photon! Evidently we obtain the relations \(g'=g \tan \varTheta _W\) and \(\rho =M_W^2/ (M_Z^2\cos ^2 \varTheta _W)=1\) ! instead of \(M_Z=M_{W^{\pm }}\) when \(g'=0\).

      Now, what can we say about the hypercharge of the fermions?

      A left-handed doublet transforms like

      $$\begin{aligned} L \rightarrow e^{\mathrm {i}\frac{g'}{2}Y_L} L\,, \end{aligned}$$

      where \(Y_L\) is arbitrary. By inspection of \({\mathcal {L}}_{\mathrm{Yukawa}}\) we find for the hypercharges of the singlets: \(\psi _{1R}\) must have \(Y_{1R}=Y_L+1\) and \(\psi _{2R}\) must have \(Y_{2R}=Y_L-1\). One consequence is that \(U(1)_Y\)must violate parity. The astonishing thing is that the fermion current which couples to the photon preserves parity. By inspection we find

      $$\begin{aligned} D_{\mu } L_f= & {} \left( \partial _{\mu } -\mathrm {i}\,\frac{g'}{2}Y_L B-\mu -\mathrm {i}\frac{g}{2} \tau _3 W_{\mu 3}-\cdots \right) \, L_f \,,\\ D_{\mu } R_f= & {} \left( \partial _{\mu } -\mathrm {i}\,\frac{g'}{2}Y_L B-\mu -\mathrm {i}\frac{g}{2} \tau _3 B_{\mu }-\cdots \right) \, R_f\,, \end{aligned}$$

      and the couplings of \(L_f\) and \(R_f\) to \(A_{\mu }\) read

      $$\begin{aligned}&L_f \;: -\mathrm {i}\,\left( g \sin \varTheta _W\ \frac{\tau _3}{2} + g' \cos \varTheta _W\frac{Y_L}{2}\right) \, A_{\mu }\\&R_f \;: -\mathrm {i}\,\left( g' \cos \varTheta _W\ \frac{\tau _3}{2} + g' \cos \varTheta _W\frac{Y_L}{2}\right) \, A_{\mu } \;. \end{aligned}$$

      Because we have \(g' \cos \varTheta _W= g \sin \varTheta _W= e\) we find the Gell-Mann-Nishijima relation

      $$\begin{aligned} Q=T_3+\frac{Y}{2} \end{aligned}$$

      as a consequence of a minimal Higgs structure! What we find is that, whatever the hypercharge of \(L_f\) is, \(L_f\) and \(R_f\) must couple identically to photons. Thus QED must be parity conserving! Furthermore, the charges of the upper (1) and lower (2) components of the doublets satisfy

      $$\begin{aligned} Q_{Li}=Q_{Ri}\; ,\; Q_1-Q_2=1 \;\mathrm {and} \; Q_1+Q_2=Y_L \;. \end{aligned}$$

      So far we have no charge quantization. Here we need the last assumption.

    • If \(\nu _R\) does not couple to the U(1) gauge field, we have to set \(Y_{\nu R}=0\) and consequently we must have \(Y_{\nu L}=-1=Y_{\ell L}=0\) and \(Q_{\nu }=0\), \(Q_{\ell }=-1\). For the \(U(1)_Y\) anomaly cancellation we need lepton-quark duality and the charges of the quarks must have their known values if they appear in three colors. One thus must have the usual charge quantization.

    We finally summarize the consequences of the assumptions stated above:

    • Breaking \( SU (2)_L\) by a minimal Higgs automatically leads to a global \(U(1)_Y\), which can be gauged

    • parity violation of \( SU (2)_L\)

    • \(\rho =M_W^2/(M_Z^2\cos ^2 \varTheta _W)=1\)

    • the existence of the photon

    • parity conservation of QED

    • the validity of the Gell–Mann–Nishijima relation

    • family structure

    • charge quantization.

    We do not know why right-handed neutrinos are sterile i.e. do not couple to gauge bosons. In the SM of electroweak interactions, neutrinos originally were assumed to be massless i.e. that right-handed neutrinos did not exist. This is definitely ruled out by the observation of neutrino oscillations.Footnote 29

I think this reasoning is able to help understanding how various excitations in the chaotic Planck medium develop a pattern like the SM as a low energy effective structure. Renormalizability as a consequence of the low energy expansion and the very large gap between the EW and the Planck scales plus a certain minimality (not too little but not too much e.g. only up to symmetry triplets) determines the SM structure without much freedom. After all, minimality is not a new concept in physic as we know from the principle of least action. Three fermion families are required in order CP violation emerges in a natural way, and to make baryogenesis eventually possible within the LEESM scenario as addressed in Sect. 7. We have been emphasizing the high self-consistency of the SM where all essential structures look to be emergent properties in the low energy effective viewport of a cutoff-system residing at the Planck scale. “What is not capable of surviving at long distances does not exist there” (Darwin revisited).

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Jegerlehner, F. The Hierarchy Problem and the Cosmological Constant Problem Revisited. Found Phys 49, 915–971 (2019). https://doi.org/10.1007/s10701-019-00262-2

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