Landscape and anthropic inputs for the weak-scale hierarchy: the Higgs-VEV and Yukawa distributions and the weak-to-weakless habitable-universe ratio

If the electroweak scale is not stabilized by a dynamical mechanism, anthropic selection in a multiverse could in principle explain why a weak nuclear interaction exists at all. Gedalia, Jenkins and Perez (Phys. Rev. D 83, 115020, 2011) argued that if the Standard Model Yukawa couplings scan across vacua with an approximately scale-invariant distribution, as generic flavor dynamics suggests, then hospitable 'weakless' universes vastly outnumber universes like ours, so the gauge hierarchy problem calls for a dynamical solution. That conclusion rests on unverified inputs: the probability density function (PDF) of the Higgs vacuum expectation value in a real string-theory landscape, the PDF of the Yukawa couplings including possible lower cutoffs, the distribution of the underlying flavor parameters, and the observer-weighting factor comparing weak and weakless universes. The open problem is to determine these inputs in a controlled multiverse ensemble and thereby compute the ratio of hospitable weak to hospitable weakless universes, settling whether an anthropic solution of the gauge hierarchy problem is statistically viable or whether a dynamical solution is required.

Background

The gauge hierarchy problem asks why the scale of electroweak symmetry breaking, characterized by the Higgs vacuum expectation value (VEV) $v \approx 246$ GeV, is some 32 orders of magnitude (in the squared sense) below the Planck mass $M_{\rm Pl}$, when quantum corrections to the Higgs mass parameter would naturally drag it toward the cutoff scale. The Standard Model (SM) Higgs field obtains its VEV by spontaneous symmetry breaking, which sets the $W$ and $Z$ boson masses and, through the Yukawa couplings $y_f$, the fermion masses $m_f = y_f v/\sqrt{2}$.

The string landscape — the enormous ensemble of metastable vacua afforded by string compactifications — together with eternal inflation raises the possibility that the parameters of the SM vary from one region of the universe to another ('scan'), with observers measuring values selected by the conditions for life (the anthropic principle, as invoked by Weinberg for the cosmological constant). Agrawal, Barr, Donoghue and Seckel showed that when $v$ is varied while the Yukawa couplings are held fixed, atomic and nuclear chemistry restricts $v$ to within roughly an order of magnitude of its observed value, an early anthropic window on the weak scale.

Harnik, Kribs and Perez then constructed a 'weakless universe': a universe with the same gauge groups but $v^* \sim M_{\rm Pl}$ and rescaled Yukawas $y_f^* = y_f v / v^*$, so that the electron, up- and down-quark masses (and hence chemistry) are unchanged, all other quarks are heavier than $\Lambda_{\rm QCD}$ (the scale at which the strong interaction becomes non-perturbative), and a reduced baryon-to-photon ratio $\eta_b^* \sim 10^{-2}\eta_b$ leaves deuterium after Big Bang nucleosynthesis (BBN), allowing long-lived deuterium-burning stars and heavy-element production. If such universes are hospitable, the anthropic principle by itself does not fix the weak scale.

Gedalia, Jenkins and Perez (the source of this problem) observed that flavor dynamics changes the statistics. In classic explanations of the quark-mass hierarchy (Froggatt-Nielsen-type models, dynamical flavor, compositeness), effective Yukawa couplings arise as $y \propto \epsilon^Q$ with a small universal parameter $\epsilon$ and flavor charges $Q$, and are often the VEVs of dynamical fields — in which case the dimensionless Yukawas may scan along with the dimensionful parameters. For a power-law charge distribution $p(Q) \propto Q^n$ and arbitrary $p(\epsilon)$, the induced Yukawa PDF is $p(y) \propto \ln^n y / y$, which for $n = 0$ is the scale-invariant distribution $p(y) \propto 1/y$ (uniform in $\ln y$, the same statistical mechanism underlying Benford's law). Combining this with the conservative naturalness expectation $p(v) \propto v^2/M_{\rm Pl}^2$, they bounded the ratio $r$ of hospitable universes with $v = v_\oplus$ (weak universes, $\oplus$ denoting our values) to hospitable weakless universes with $v^* \sim M_{\rm Pl}$, finding $r \ll 1$ unless the charge distribution is extremely tilted ($n \lesssim -21$), itself a new hierarchy problem. Their conclusion: unless the landscape PDF has special features — for example a strong bias toward small Higgs self-coupling, as in Feldstein, Hall and Watari — the hierarchy problem demands a dynamical (not anthropic) solution.

This directly contradicts Donoghue, Dutta, Ross and Tegmark, who, using a scale-invariant Yukawa distribution with cutoffs plus anthropic constraints from atoms, found the observed $v$ favored; Gedalia et al. claim their own bound is cutoff-independent — a claim about the form of their bound, whose Yukawa integrals are dominated by the upper limits — distinct from the separate question, item (ii) below, of whether a lower cutoff imposed by the landscape would forbid the weakless couplings altogether. The disagreement is not a matter of calculation but of unknown inputs, each explicitly flagged in the source: (i) the true PDF of $v$ in a realistic string-theory ensemble is unknown ('It is unclear at this point how $v$ might be distributed in the landscape of string theory'); (ii) whether the landscape or selection effects impose a lower cutoff on Yukawas, which would forbid the $y_f^* \sim 10^{-21}$ needed for weakless constructions, is unknown, with 'no known dynamical reason' for such values to be forbidden; (iii) the distribution of flavor charges (the exponent $n$) and of $\epsilon$ in actual constructions is unknown; (iv) the observer-weighting factor $p(\text{observer} | \{\alpha_i\})$ for widely varying SM parameters (correlated changes in nuclear physics, baryon asymmetry, stellar evolution) is assumed comparable for weak and weakless universes but not computed; and (v) no explicit landscape realization (string vacua ensembles with computed Yukawa spectra and Higgs-parameter distributions) exists in which these assumptions could be checked.

Problem Statement

Determine the landscape and anthropic inputs that Gedalia, Jenkins and Perez left unspecified, and thereby decide whether their conclusion stands for a realistic multiverse. Concretely: (1) compute or constrain, in an explicit or controlled string-landscape ensemble, the probability density function of the Higgs VEV $v$ (equivalently the electroweak scale) relative to $M_{\rm Pl}$, testing whether $p(v) \propto v^2/M_{\rm Pl}^2$ or a distribution with special features (such as a bias toward small Higgs self-coupling) is realized; (2) determine the joint PDF of the Standard Model Yukawa couplings over the ensemble — in particular whether a lower cutoff exists that would forbid the $y_f^* \sim 10^{-21}$ couplings required by weakless constructions, and what the distribution of the underlying flavor parameters ($\epsilon$, the charges $Q$, or the exponent $n$ in $p(Q) \propto Q^n$) actually is; and (3) establish the observer-weighting factor $p(\text{observer}|\{\alpha_i\})$ comparing universes like ours with hospitable weakless universes, including the correlated variations of nuclear physics, baryon asymmetry, and stellar evolution. Evaluate the ratio $r$ of hospitable weak universes to hospitable weakless universes in that ensemble and settle whether $r \ll 1$ — so that the gauge hierarchy problem cannot be solved anthropically and requires a dynamical mechanism — or whether landscape features or selection effects rescue an anthropic explanation of the observed weak scale. Here $r$ follows the source's Bayesian convention, $r \equiv p_{\rm measured}\{\alpha_i^\oplus\}/p_{\rm measured}\{\alpha_i^*\}$ with $p_{\rm measured} \propto p(\text{observer}|\{\alpha_i\})\,p(\{\alpha_i\})$: it is an observer-weighted probability ratio rather than a raw vacuum-counting ratio, and 'hospitable' weakless configurations are those satisfying the criteria summarized in the Background ($v^* \sim M_{\rm Pl}$; rescaled $y_f^*$ leaving the electron and up- and down-quark masses unchanged; $m_f^* \gtrsim \Lambda_{\rm QCD}$ for the other flavors; and $\eta_b^* \sim 10^{-2}\eta_b$ leaving deuterium after BBN).

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