Hubble selection of the weak scale from QCD quantum critical point

There is growing evidence that the small weak scale may be related to self-organized criticality. In this regard, we note that if the strange quark were lighter, the QCD phase transition could have been first order, possibly exhibiting quantum critical points at zero temperature as a function of the Higgs vacuum expectation value $v_h$ smaller than (but near) the weak scale. We show that these quantum critical points allow a dynamical selection of the observed weak scale, via quantum-dominated stochastic evolutions of the value of $v_h$ during eternal inflation. Although the values of $v_h$ in different Hubble patches are described by a probability distribution in the multiverse, inflationary quantum dynamics ensures that the peak of the distribution evolves toward critical points (self-organized criticality), driven mainly by the largest Hubble expansion rate there -- the Hubble selection of the universe. To this end, we first explore the quantum critical points of the three-flavor QCD linear sigma model, parametrized by $v_h$ at zero temperature, and we present a relaxion model for the weak scale. Among the patches that have reached reheating, it results in a sharp probability distribution of $v_h$ near the observed weak scale, which is critical not to the crossover at $v_h=0$ but to the sharp transition at ${\sim}\Lambda_{\rm QCD}$.

There is growing evidence that the small weak scale may be related to self-organized criticality. In this regard, we note that if the strange quark were lighter, the QCD phase transition could have been first order, possibly exhibiting quantum critical points at zero temperature as a function of the Higgs vacuum expectation value v h smaller than (but near) the weak scale. We show that these quantum critical points allow a dynamical selection of the observed weak scale, via quantumdominated stochastic evolutions of the value of v h during eternal inflation. Although the values of v h in different Hubble patches are described by a probability distribution in the multiverse, inflationary quantum dynamics ensures that the peak of the distribution evolves toward critical points (self-organized criticality), driven mainly by the largest Hubble expansion rate there -the Hubble selection of the universe. To this end, we first explore the quantum critical points of the three-flavor QCD linear sigma model, parametrized by v h at zero temperature, and we present a relaxion model for the weak scale. Among the patches that have reached reheating, it results in a sharp probability distribution of v h near the observed weak scale, which is critical not to the crossover at v h = 0 but to the sharp transition at ∼ΛQCD.

I. INTRODUCTION
The Planck weak-scale hierarchy may be addressed by the near criticality of the Higgs mass parameter [1,2]. In this viewpoint, the small weak scale close to zero is special because the universe transitions between broken and unbroken phases of the electroweak symmetry at zero. The transition could generate various standard model (SM) backreactions that allow dynamical selection of the weak scale [3][4][5][6][7][8]. However, this transition is a second-order crossover in the SM, providing only relatively smooth selection rules. In addition, the SM Higgs potential, once renormalization-group evolved, was found to yield another almost degenerate vacuum near the Planck scale [9,10]. This surprising coincidence provides more evidence that the particular (seemingly unnatural) value of the weak scale might be related to criticality. This motivated ideas of multiple-point principle [11][12][13][14][15], classical scale invariance [16][17][18], as well as Higgs inflation [19][20][21]. Extremely small dark energy is also thought to be near the critical point. But yet, whether and how criticality plays a crucial role in naturalness remain unclear.
Recently, a cosmological selection mechanism for criticality was developed in Ref. [22], where inflationary quantum-dominated evolution of the relaxion inevitably drives a theory parameter close to a quantum critical point. In one setup, it is crucial that the critical point be the first-order separation between discrete phases with a significant energy difference, so that the Hubble rate can be sharply largest there. Then after long enough inflation (essentially eternal as will be discussed), such Hubble patches having a theory near the critical point * sunghoonj@snu.ac.kr † gimthcha@snu.ac.kr will dominate the multiverse, as they are expanding and are reproduced most rapidly -Hubble selection of the universe. This mechanism realizes self-organized criticality [22]; see also Refs. [23,24]. Then we ask the following: Why is the selection of criticality the selection of the observed small weak scale? To what first-order critical points does the Higgs mass have relevance? Reference [22] analyzed the aforementioned renormalization-scale dependence of the SM Higgs vacuum structure [9,10], but the critical scale was found to be far above the weak scale. References [5,6] studied a prototype model with multiple axions, where a QCD barrier trapping an axion disappears when v h turns off, so that the axion suddenly rolls down to the minimum, generating a large energy contrast necessary for Hubble selection. Critical changes of a theory could also induce the small weak scale in association with much smaller dark energy [7,8].
In this Letter, we present a cosmological account of the weak scale from possible first-order zero-temperature (hence, quantum) critical points of QCD. 1 We first point out that QCD may have built-in quantum critical points at some v * h v EW = 246 GeV; this has yet to be studied, and we initiate an exploration using the three-flavor linear sigma model (LSM) of low-energy QCD. Then we present a relaxion model that realizes Hubble selection of the QCD criticality and self-organizes v h close to the observed value. Then, v h is critical to ∼Λ QCD (not to the crossover at zero). An added benefit is that the weak scale and Λ QCD are generically close, which otherwise is accidental. Furthermore, building upon earlier works, we elaborate Hubble selection with different semi quantitative derivations.
We are inspired by observations that if the strange quark were slightly lighter, the (finite-temperature T ) QCD chiral phase transition could have been first order.
Although not yet firmly established [36][37][38][39][40][41][42][43], this possibility has been expected based on the (non)existence of infrared fixed points in the three-dimensional LSM [44,45]. In other words, QCD at T = 0 too (relevant during inflation) may have a rich vacuum structure, as a function of variable quark masses or v h . Our initial phenomenological exploration of the vacuum structure shall be verified by dedicated research.
The Letter discusses the basic model ingredients (Sec. II), exploration of QCD quantum critical points (Sec. III), Hubble selection (Sec. IV), realization of the weak-scale criticality (Sec. V), and conclusions with future improvements.

II. MODEL
The model consists of the relaxion φ [3], the Higgs h, and the meson field Σ: V tot = V φ +V h +V Σ . The relaxion couples only to the Higgs sector, scanning v h . But the change of v h induces changes in the Σ sector, developing the desired quantum criticality at v * h . Then the Hubble selection (acting on φ) self-organizes the universe to the critical point.
The real-scalar relaxion potential is axion like: For Hubble selection, its field range f φ shall exceed the Planck scale (see later), which is possible with multiple axions [46][47][48]. The Higgs potential takes the SM form (λ h 0.13) plus the coupling to the relaxion (see Ref. [3] for details) where h is the real Higgs field in unitary gauge. We shift φ such that the quadratic term µ 2 h = −gφ vanishes at φ=0. v 2 h ≡ −µ 2 h /λ h = gφ/λ h is used to label the relaxion scanning (v EW = 246 GeV). 2 The required field range of φ to scan µ 2 h up to the cutoff M 2 is δφ ∼ M 2 /g, thus we set f φ = M 2 /g. The dimensionful coupling g is a spurion of the relaxion shift symmetry, and thus can be small naturally.
Below Λ QCD = 200 MeV, meson fields Σ ij (x) are relevant degrees of freedom, whose condensates are order parameters for chiral symmetry breaking. This vacuum structure as a function of v h is what we want to explore. It can be conveniently described by the LSM with U (N f ) L × U (N f ) R symmetry of QCD [49][50][51], where fields and parameters are decomposed as Σ Without losing generality, λ 1,2 , h a are real, c > 0, and µ 2 can take either sign. Σ is bifundamental under the symmetry. The first line of Eq.
f ). One of the remaining U (1)'s is identified as the conserved baryon number U (1) V , simply omitted in our discussion. The other U (1) A is anomalous, broken by the instanton contribution c down to Z A (N f ) [52,53]. Symmetries are further broken by H, the leading chiral-symmetrybreaking mass term. We fix N f = 3 with the isospin symmetry m u = m d , as a first exploration; only h 0 , h 8 = 0.
It is worthwhile to note that the LSM indeed possesses necessary features for quantum critical points. For N f = 3, the instanton term is a cubic potential, possibly creating local vacua (even with µ 2 > 0). The linear term H can destabilize the local vacua at critical quark masses or v * h , just as the external magnetic field (the linear term) in ferromagnets flips higher-energy spin directions at a critical field strength.

III. QCD QUANTUM CRITICAL POINTS
To explore the vacuum structure as a function of v h , we first fix the benchmark "SM point" parameters of V Σ , reproducing a measured meson spectrum, and then we deduce how these parameters change with v h .
The masses of pions and kaons, being pseudo-Goldstones, are given by symmetry-breaking terms H, related by partially conserved axial-vector currents, where the last equality is obtained by the variation of Using measured values of m π,K (Table I), we fix the SMpoint value of h 0,8 [54], We proceed to fit masses of other pseudoscalar and scalar mesons to data. The minima of V Σ are numerically  (6) and (7)], compared with data from the Particle Data Group [55].
In units of MeV.
found by considering the stability along all 18 field directions. The N f = 3 LSM is known to have three types of vacua at H = 0 [56,57]: In particular, the global SU (3) V vacuum (that we live today) and the local SU [57]. Thus, this parameter space is our focus, that potentially exhibits first-order quantum critical points.
By scanning with these constraints, we found a range of good parameter space (see Appendix A in Supplemental Material [58]). The benchmark SM point is [with Eq. (6)] (7) yielding K = 47.8. Its goodness of fit to the meson spectrum (Table I) is χ 2 /degrees of freedom = 0.44 with the first seven observables and 3.11 including all. The first seven are the most reliable, while the last three are less precisely measured with unclear identities [55]. Here, 5% theoretical uncertainties are added as typical sizes of the perturbative corrections. Our benchmark is as good as existing benchmarks in the literature: Ref. [56] (with µ 2 > 0) yielded 0.20 and 2.84, respectively, and Ref. [59] (µ 2 < 0) yielded 1.22 and 4.64. 3 . But ours differs in that quantum critical points v * h may exist. We turn to discuss the vacuum structure away from the SM point for v h < v EW . How does V Σ , in particular H, depend on v h ? Since this is not known, we deduce it as follows. The current divergence [Eq. (4)] calculated from QCD or chiral Lagrangian yields m 2 π ∝ m q , which is also ∝ h 0,8 from Eq. (5). It suggests h 0,8 ∝ v h . Indeed, identifying the H term and the current mass term, L −m q (ūu +dd) − m ss s, yields MeV and m s = 93 MeV [55], same as the ratio from Eq. (6). Thus, we assume that H is linear to v h as Other LSM parameters and dimensionful factors could also depend on v h , either directly or indirectly, e.g. via  condensation or Λ QCD . Λ QCD depends on quark masses via renormalization running but only logarithmically, and instanton contributions on masses and condensates but complicated and nonperturbative [65]. In this initial exploration, Eq. (8) [58]). The energy difference of the coexisting vacua at the critical point is 93 MeV, comparable to Λ QCD ; the potential energies are parameterized in Eq. (14) and (15).
In all, we have shown that QCD may possess quantum critical points at v * h < v EW , which needs dedicated verification.

IV. HUBBLE SELECTION
Inflationary quantum fluctuations on the scalar field allow access to a higher potential regime, which is forbidden classically. Although the field in each Hubble patch always rolls down in average, larger Hubble rates at higher potentials can make a difference in the global fieldvalue distribution among patches, culminating in Hubble selection. This section reviews and supplements it [22].
The volume-weighted (global) distribution ρ(φ, t) of the field value φ obeys the modified Fokker-Planck equa-tion (FPV) [66][67][68][69] The first two terms represent the flow and diffusion, just as in the original Fokker-Planck equation which averages over Langevin motions. The variation of the Hubble rate 3H 0 accounts for volume weights within a distribution: M Pl = 2.4 × 10 18 GeV. The meanings become clearer if we look at a solution (for a linear potential without boundary conditions), where the exponent describes the motion of a peak. φ c = −V /3H is classical rolling. Remarkably, an additional velocityφ H = 3(∆H) σ φ (t) 2 with opposite sign arises from volume weights within the width σ φ , which grows in the beginning of FPV evolutions due to quantum diffusion σ 2 φ (t) = ( H 2π ) 2 Ht from the de Sitter temperature H/2π [70,71].
"Hubble selection" starts to operate when the peak of a distribution starts to climb up the potential: Pl . The width at this moment is always Planckian, reflecting its quantum nature. The field excursion by this for a peak to climb, the field range δφ ∼ M 2 /g (needed to scan µ 2 h up to M 2 ) must accommodate both the field excursion ∆φ (stronger condition) and width σ φ , yielding respectively We call this condition global quantum beats classical (QBC). It is stronger than the usual local QBC, V H 3 , requiring g H H 2 HM Pl from condition 1 later. It also has different meanings as it involves the field range while the local one depends only on the potential slope. It turns out to be equivalent to the Quantum+Volume (QV) condition in Ref. [22] (see Appendix B in Supplemental Material [58]) which also accounted for volume effects. If it is not satisfied, ρ makes an equilibrium at the bottom of a potential, but with the sub-Planckian width consistently [72,73]. Thus, we require the global QBC Eq. (11) for Hubble selection.
The e-folding until this moment ∆N Pl H 2 , given by the de Sitter entropy [74][75][76]. Thus, Hubble selection needs eternal inflation, and the universe eventually reaches a stationary state [77][78][79][80]. Probability distributions are to be defined within an ensemble of Hubble patches that have reached reheating [81][82][83]. As the latest patches dominate the ensemble with an exponentially larger number, only stationary or equilibrium distributions matter; for landscapes, this can be different [84][85][86][87].
ρ(φ, t) makes an equilibrium somewhere near the top of a potential, which is the critical point φ * in this work. The distribution can be especially narrow [Planckian in the global QBC regime; see Eq.(B6) in Supplemental Material [58]] if energy drops sharply after φ * . This is how Hubble selection self-organizes the universe toward critical points [22].
The flatter the potential is (with stronger quantum effects), the closer to φ * is the equilibrium. The closest possible field distance is Planckian, again reflecting the uncertainty principle. For even flatter potentials, the equilibrium distribution rather spreads away from φ * , because distributions will be flat in the limit V → 0. The equilibrium near φ * is estimated as follows. The boundary condition ρ(φ ≥ φ * ) = 0 (discarding Hubble patches with φ ≥ φ * ) induces repulsive motioṅ which is the width in the Quantum 2 +Volume (Q 2 V ) regime [22]. The width indeed increases as V flattens; nevertheless, the v h distribution can be arbitrarily narrowed, as will be discussed. One also expects |φ peak −φ * | ∼ σ φ from dimensional ground. These heuristic discussions on Q 2 V are demonstrated with the method of images in Appendix B of Supplemental Material [58]. A theory enters the Q 2 V regime when the balance width becomes larger than M Pl (the width in the global QBC): This is equivalent to V H 3 H/M Pl [22], which is also derived from the local balance near φ * . Q 2 V is typically stronger than the global QBC and not absolutely needed for Hubble selection, but later will be useful for efficient localization of v h .

V. THE WEAK SCALE CRITICALITY
Finally, we come to calculate the equilibrium distribution of ρ(v h ) in our model. We first discuss conditions for the successful Hubble selection of v * h , and then present benchmark results.
The scanning of v h starts by φ rolling up its potential from φ < 0 to > 0. When φ < 0 (µ 2 h > 0), v h = 0 and V h = V Σ = 0 remain unchanged with φ. Thus φ simply keeps growing, driven by quantum effects. The   only constraint is that V φ must not affect the inflation dynamics (condition 1): Λ 4 φ H 2 M 2 Pl . As soon as φ > 0 (µ 2 h ≤ 0), the Higgs gets the vacuum expectation value v h > 0, and V h , V Σ minima now evolve with φ. V h = − λ h 4 v 4 h , and coexisting vacua of V Σ are, at leading orders in v h , where a 1 114, a 2 0.059, a 3 1.38 for the benchmark (Fig. 1). σ 0 (L×R) − σ 8 a 4 v h with a 4 0.025. Note that V h,Σ decrease with φ, which must be slower than the increase of V φ gΛ 4 φ M 2 φ, for Hubble selection. Which potential dominates the φ dynamics? Fig. 2 shows individual potential with φ, whose slope is The dominance of growing , decreasing V h begins to dominate and is prohibited from being Hubble selected again. So we need to make sure that V φ never compensates the energy drop in the intermediate region In all, V φ cannot be too flat or too steep (condition 2): In addition, h and Σ are required to sit in their respective minima, not quantum driven to overflow their potentials. Their equilibrium widths must be small enough: For numerical studies, we use the following benchmark satisfying the global QBC [Eq. (11)](and Q 2 V [Eq. (13)] marginally) and conditions 1-3. Potential energies near v * h are shown in Fig. 2. As desired, the total energy peaks sharply at v * h , drops significantly, and is never compensated afterwards; for much smaller or larger V φ , energy would not sharply peak.
The large-time equilibrium distribution of v h is shown in Fig. 3; see Appendix C in Supplemental Material [58] for details. The width  [58].
arbitrarily narrower at the price of arbitrarily smaller g or larger φ range, moving into a deeper Q 2 V regime; only the resulting hierarchy f φ = M 2 /g M Pl needs to be generated consistently in field theory [46][47][48]. On the other limit, unwanted σ v h v * h is resulted for 10-100 times larger g, where Q 2 V is not even marginally satisfied. Postinflationary dynamics is model dependent but such that φ slow rolls to the today's v EW . Today, φ could be still safely slow rolling or trapped by SM backreactions. Signals from time-dependent v h or phase transitions could be produced.

VI. DISCUSSION
In this Letter, we have discussed the self-organized criticality of the weak scale, by exploiting possible first-order quantum critical points of QCD. Although we saw some success, our exploration of critical points is much simplified and far from conclusive. We have used only LSM with N f = 3 at the tree level with a simplified dependence on v h ≤ v EW in Eq. (8). They shall be verified and generalized by lattice calculations [36][37][38][39][40][41][42][43], incorporating higher-order and nonperturbative effects [60][61][62][63][64], not only for the SM point but also away from it with v h ≤ v EW . N f > 3 likely yields a richer vacuum structure but needs a dedicated calculation. Theoretical intuitions from confining gauge theories might also be useful.
If such a critical point is indeed built in QCD, it would shed significant light on the role of near criticality of the SM.
The proposed scenario makes an advancement on the hierarchy problem, albeit not yet completely solving it. It is not complete because Hubble selection requires a mild separation of scales Λ φ M from Eq. (17) (if M ∼ Λ φ strictly, M Λ QCD is too low) but this is not quantum stable (Higgs loop diagrams with external relaxion legs yield Λ φ ∼ cutoff M [4,47]). Thus, a little hierarchy remains; with the fine-tuning ≡ Λ φ /M < 1, the cutoff can be as high as M Λ QCD / 2 . Another advancement is that choosing Λ φ can be translated to a dynamical problem of choosing dimensionless parameters of the extended relaxion sector, such as in Ref. [4]. Further explorations will be enlightening.
The near criticality and naturalness of nature may be intimately connected by quantum cosmology, with necessary criticality perhaps built in just around the SM. Further theoretical and experimental studies are encouraged to unveil this connection.
We present our initial exploration of quantum critical points v * h of the N f = 3 LSM at tree-level. In Sec. III, we presented the benchmark SM point, Eqs. (6) and (7), that best fits the meson spectrum in Table I. Here we discuss further details of our search and the resulting ranges of best-fit parameters and v * h . It turns out that λ 1 and µ 2 are least constrained by meson spectrum, as the last three meson observables in Table I have large uncertainties; without such freedom, the tree-level LSM would have been over-constrained. So we vary these two parameters while fixing all others to the benchmark values. We should also focus on the parameter space with K > 4.5 so that coexisting vacua are present at H = 0; this roughly requires µ 2 10 times the benchmark value. Fig. 4 shows the numerical results of the critical point v * h (upper panel) and χ 2 /dof (lower) as a function of µ 2 and λ 1 ; we denote the values of µ 2 and λ 1 by the ratio (scale factors) relative to the benchmark values, asμ 2 andλ 1 . We found that v * h is sensitive mostly only to µ 2 , while χ 2 only to λ 1 . Thus, a wide range of v * h is found to be consistent.
The energy contrast at the critical point does not vary much, 70∼120 MeV close to Λ QCD , in the parameter space shown in the figure.  Table 1.
In this appendix, we discuss further details. We first introduce the QV and Q 2 V regimes derived in [22] and show their equivalence to our results; then we discuss the scaling behaviors of equilibrium solutions, which turn out to provide further intuition as well as useful handles in numerical calculations (see App. C); finally we derive equilibrium properties of Q 2 V near a boundary using the method of images. We also refer to [7,22,72] and references therein for other details on FPV solutions.