Weak gravity conjecture, multiple point principle and the standard model landscape

The measured value of the Higgs mass implies, in addition to the electroweak vacuum, the existence of a new vacuum where the Higgs field value is around the Planck scale.

“The measured value of the Higgs mass implies, in addition to the electroweak vacuum, the existence of a new vacuum where the Higgs field value is around the Planck scale.”

• Also stated in: § JHEP11(2017)043 1 Introduction · plain

  “Moreover, the measured value of the Higgs mass implies the existence of a new vacuum where the Higgs field value is around the Planck scale (see e.g.refs.[8][9][10]).”

• Also stated in: § 2 The SM vacua in four dimension · hedged

  “Interestingly, the current values of the SM parameter indicates the existence of a new vacuum at the high scale.3”

• cites [8] G. Degrassi et al., Higgs mass and vacuum stability in the Standard Model at NNLO, JHEP 08 (2012) 098 [arXiv:1205.6497] [INSPIRE]. sourceDisputed

  The paper discusses a second minimum near the Planck scale only as a conditional possibility requiring a positive, nearly zero Higgs self-coupling. It says this scenario is not favored by present data, while §4.1 reports a small negative best-fit coupling at the Planck scale. Thus it does not conclude that the measured Higgs mass implies such a vacuum. (LLM, conf=0.86)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured
• cites [9] Y. Hamada, H. Kawai and K.-y. Oda, Bare Higgs mass at Planck scale, Phys. Rev. D 87 (2013) 053009 [Erratum ibid. D 89 (2014) 059901] [arXiv:1210.2538] [INSPIRE]. sourceDisputed

  Section IV (page unavailable) says the metastability argument does not assume a true stable vacuum at very high field values. Section V describes an almost flat potential near the Planck scale as a possibility, not an additional vacuum implied by the measured Higgs mass. (LLM, conf=0.78)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured
• cites [10] D. Buttazzo et al., Investigating the near-criticality of the Higgs boson, JHEP 12 (2013) 089 [arXiv:1307.3536] [INSPIRE]. sourcePartly supported

  The Introduction and §6.3 describe the electroweak vacuum as metastable and not the true vacuum for the measured parameters. But §7 says metastability is preferred rather than conclusively established, owing mainly to uncertainty in the top-quark mass. The cited passages do not establish a new vacuum with a Higgs field value around the Planck scale, so the claim is more definite and specific than the paper supports. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

Because the lightest charged particle (electron) is much heavier than the neutrino, the potential of the Wilson line is exponentially suppressed.

“Because the lightest charged particle (electron) is much heavier than the neutrino, the potential of the Wilson line is exponentially suppressed[2].”

• cites [2] N. Arkani-Hamed, S. Dubovsky, A. Nicolis and G. Villadoro, Quantum Horizons of the Standard Model Landscape, JHEP 06 (2007) 078 [hep-th/0703067] [INSPIRE]. sourceSupported

• AssumptionRests on an assumption: The lightest charged particle (electron) is much heavier than the neutrino.

For a periodic Majorana neutrino, there are two solutions of V=0 for any value of mν,lightest allowed by experiment, mν,lightest ≳ 0.1 eV.

“Figure 14 corresponds to Majorana neutrino, and there are two solutions of V = 0 for any value of m ν,lightest allowed by the experiment, m ν,lightest 0.1 eV [50,51].”

• cites [50] Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589] [INSPIRE]. sourcePartly supported

  Section 3.2 (Neutrinos; page unavailable) reports Planck upper limits on neutrino mass, so the paper supports the general point that experiment constrains neutrino masses. It does not give the stated bound on the lightest neutrino mass or address periodic Majorana neutrinos and two solutions of V = 0. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [51] S.M. Bilenky and C. Giunti, Neutrinoless double-beta decay: A brief review, Mod. Phys. Lett. A 27 (2012) 1230015 [arXiv:1203.5250] [INSPIRE]. sourceCould not verify

  Section 4 (p. 12) discusses effective Majorana mass as a function of the lightest neutrino mass, and the Conclusions (p. 19) note that cancellations can occur in the normal mass scheme. Searches for periodic Majorana neutrinos and two solutions of V = 0 found no passage addressing those claims or the stated mν,lightest ≳ 0.1 eV condition. (LLM, conf=0.76)

  Experimental — accuracy not yet measured

Non-perturbative corrections can generate a potential for the moduli.

“Even in this case, non-perturbative corrections can generate a potential for the moduli.”

• cites [32] U. Danielsson and G. Dibitetto, Fate of stringy AdS vacua and the weak gravity conjecture, Phys. Rev. D 96 (2017) 026020 [arXiv:1611.01395] [INSPIRE]. sourceCould not verify

  In “The brane and the flux pictures” (p. 5), the paper says that fluxes induce a potential for geometrical modes and can stabilize moduli. It does not say that non-perturbative corrections generate that potential; searches for that mechanism returned passages about non-perturbative stability and open-string instabilities instead. (LLM, conf=0.72)

  Experimental — accuracy not yet measured

The discussion in ref. [3] is applicable only to 2+ dimensions.

“The point is that the discussion in ref. [3] is applicable only to 2 + dimensions.”

• cites [3] J.M. Arnold, B. Fornal and M.B. Wise, Standard Model Vacua for Two-dimensional Compactifications, JHEP 12 (2010) 083 [arXiv:1010.4302] [INSPIRE]. sourceSupported

Current CMB measurements put a bound on the sum of the neutrino mass to be ∼0.2 eV.

“Current CMB measurements put a bound on the sum of the neutrino mass to be ∼ 0.2 eV [50].”

• cites [50] Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594 (2016) A13 [arXiv:1502.01589] [INSPIRE]. sourcePartly supported

  Section 3.2 (page not supplied) reports bounds of 0.21 eV and 0.17 eV when Planck CMB measurements are combined with BAO data. The cited copy gives weaker bounds of 0.72 eV and 0.49 eV for the corresponding CMB-only combinations, so the claim omits a necessary qualifier. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

Among τ, gαβ and ρ, only τ has dynamical degrees of freedom with a kinetic term in the action.

“It should be stressed that, among τ, g αβ and ρ, only τ has dynamical degrees of freedom with a kinetic term in the action.”

This vacuum is stable at least at tree level.

“This vacuum is stable at least at tree level.”

Depending on the mass of the top quark and the value of c6, the cosmological constant of the high scale vacuum can be positive, zero or negative.

“Depending on the mass of the top quark and the value of c 6 , the cosmological constant of the high scale vacuum can be positive, zero or negative.”

• Also stated in: § 2 The SM vacua in four dimension · plain

  “The potential has AdS, flat or dS vacua depending on the value of the top mass.”

Since the ρ field is not dynamical, it is fixed by the constraint equation V=0.

“Since the ρ field is not dynamical, it is fixed by the constraint equation V = 0.”

In the high scale vacuum, the QCD scale increases and becomes around 10^6 GeV.

“The QCD scale increases, and becomes around 10 6 GeV.”

In the high scale Higgs vacuum, a perturbative stable vacuum only appears for Λ4 = 0 and Dirac neutrino.

“It is found that a perturbative stable vacuum only appears for Λ 4 = 0 and Dirac neutrino.13”

• AssumptionThe authors state this as an assumption: In the high scale vacuum analysis for the Majorana neutrino case, heavy right-handed neutrinos exist with masses smaller than 10^16 GeV.

In T2 compactification of U(1) gauge theory with charged matter, no perturbatively stable solution can be found.

“As we will see, just as in the S 2 case, we can not find perturbatively stable solution of T 2 compactification.”

There are no vacua in S1 compactification of QED.

“Therefore, there are no vacua in S 1 compactification of QED.”

For Dirac neutrinos, mν,lightest = 8.4 meV gives a flat 3-dimensional vacuum with periodic neutrinos.

“On the other hand, m ν,lightest = 8.4 meV is taken for the Dirac neutrino case, which give a flat 3-dimensional vacuum with periodic neutrinos.”

The moduli potential in some cases develops a runaway behavior in the small compactification scale region (GeV^-1).

“We found that the moduli potential in some cases develops a runaway behavior in the small compactification scale region ( GeV -1 ).”

In S1 compactification of a U(1) gauge theory with neutral matter, if the boundary condition is close to periodic, a stable vacuum appears.

“We can see that, if the boundary condition is close to the periodic one, a stable vacuum appears.”

Within the numerical analysis, no positive energy minima in V^SM_S1,M=0 with respect to the Wilson line were found.

“Although we do not exclude this possibility completely, within our numerical analysis, we do not find positive energy minima in V SM S 1 ,M =0 with respect to the Wilson line.”

For charged matter, the potential is minimized when the Wilson line is at q_e A1 + (1-z1e)/2 = 1/2 and q_e A2 + (1-z2e)/2 = 1/2.

“we can see that the potential is minimized when the Wilson line is at q e A 1 + 1-z 1e 2 = 1 2 and q e A 2 + 1-z 2e 2 = 1 2 .”

The curvature of 2d, R(2), is not determined by the height of the potential, but by R(2)=2∂ρ V / M_P^2.

“The curvature of 2d, R (2) , is not determined by the height of the potential, but by R (2) = 2∂ ρ V /M 2 P .”

If the high scale vacuum has Λ4 < 0, there are many AdS3 × S1 minima corresponding to w.

“If we consider the high scale vacuum with Λ 4 < 0, we have many AdS 3 × S 1 minima corresponding to w.”

In the Standard Model on S1, if the boundary condition is close to periodic, the potential has a minimum at around the neutrino mass scale.

“We can see that, if the boundary condition is close to the periodic one, the potential has a minimum at around the neutrino mass scale”

The minimum around τ=e^{iπ/3} survives as a local minimum if the boundary condition is close to periodic.

“The minimum around τ = e iπ/3 survives as a local minimum if the boundary condition is close to periodic.”

For T2 compactification of U(1) gauge theory with neutral matter, perturbatively stable dS2, M2 or AdS2 vacua can be obtained.

“We show that perturbatively stable dS 2 , M 2 or AdS 2 vacua can be obtained.”

If Λ4 < 0, all moduli can be fixed at the tree level.

“This shows that if Λ 4 < 0, we can fix all moduli at the tree level.”

Numerically, local minima exist for 0.9 ≲ z1,2 ≲ 1.1.

“Numerically, we have checked that the local minima exists for 0.9 z 1,2 1.1, see figure 21.”

The corresponding two-dimensional vacuum is AdS2 × T2.

“The corresponding two-dimensional vacua is AdS 2 × T 2 .”

If two axion-like particles are added, the ρ and τ moduli can be fixed at the tree level if Λ4 < 0.

“If we add two axion-like particles, we can fix the ρ and τ moduli at the tree level if Λ 4 < 0.”

For the periodic Majorana neutrino solutions, only the AdS2 vacuum corresponding to τ=e^{iπ/3} is stable.

“In the right panel of figure 15, the perturbative stability of each solution is investigated, from which it is concluded that only the AdS 2 vacuum corresponding to τ = e iπ/3 is stable.”

For neutral matter, depending on the value of Λ4, there exist two, one or zero solutions of the Hamiltonian constraint V=0.

“Depending on the value of Λ 4 , there exists two, one or zero solution(s) of the Hamiltonian constraint V = 0.”

If Λ4 is close to 10^{-2} M_e^4, both the τ=e^{iπ/3} and e^{iπ/2} dS2 vacua can be stable.

“Furthermore, if Λ 4 is close to 10 -2 M 4 e both the τ = e iπ/3 and the e iπ/2 dS 2 vacua can be stable.”

New dS2 × T2 vacua were found which were overlooked before.

“As a result, we found new dS 2 × T 2 vacua which were overlooked before.”

In the massless limit, V_T2(ρ,τ,0,0) reduces to eq. (B.30).

“Moreover, in the massless limit, we have”

In the limit where one radius of T2 becomes infinite, the potential becomes that of the S1 case.

“If we take the limit where the one of the radius of T 2 becomes infinite, the potential should becomes that of the S 1 case.”

In the AdS3 vacuum, the BF bound reads -1/L_AdS^2 = R(3)/6 ≤ M^2.

“In the case of the AdS 3 vacuum, the BF bound reads”

For the parameter set (m1,n1,m2,n2)=(1,1,1,2), the minimum is around (τ1,τ2)=(0.6,0.3).

“For example, if we take the parameter set (m 1 , n 1 , m 2 , n 2 ) = (1, 1, 1, 2), we can fix the τ moduli as shown in figure 22.”

• Also stated in: § JHEP11(2017)043 · plain

  “We can see that the minimum is around (τ 1 , τ 2 ) = (0.6, 0.3).”

Only the τ=e^{iπ/3} AdS2 vacuum is stable for Λ4 ≳ 10^{-2} M_e^4.

“It can be seen that only the τ = e iπ/3 AdS 2 vacuum is stable for Λ 4 10 -2 M 4 e .”

Perturbatively stable vacua can be found for boundary conditions with 0.9 ≲ z1,2 ≲ 1.1.

“Indeed, we can find perturbatively stable vacua for this range of boundary conditions.”

To obtain the 2d spacetime independent solution of T2 compactification, one must solve V=0 and ∂τa V=0.

“To summarize, we need to solve”

If the SM is supplemented with an additional axion, there are a lot of flux vacua from compactifications of the high scale vacuum with a negative 4D cosmological constant.

“However, if the SM is supplemented with an additional axion, we found a lot of flux vacua from compactifications of the high scale vacuum with a negative 4D cosmological constant.”

A necessary condition for the existence of a global minimum is the positivity of eq.(4.15).

“A necessary condition for the existence of a global minimum is the positivity of eq.(4.15), which we will check below.”

For sufficiently small ρ, the potential for τ moduli is unbounded.

“we have found that for sufficiently small ρ, the potential for τ moduli is unbounded.”

The T2 potential in that limit is the same as eq. (3.9) under the identification ρ/τ2 ∼ L2.

“which is the same as eq.(3.9) under the identification ρ/τ 2 ∼ L 2 .”

In two dimensions, a magnetic field can be introduced without violating Lorentz symmetry.

“One crucial difference from the S 1 compactification case is that we can introduce a magnetic field without violating the Lorentz symmetry in 2 dimensions”

The neutrinos have a Dirac mass.

“we predict that the neutrinos have a Dirac mass, with the mass of the lightest neutrino ∼ O(1-10) meV.”

• AssumptionThe authors state this as an assumption: The multiple point principle is valid and applicable to the standard model landscape.

The two- and three-dimensional AdS critical points of the Standard Model landscape may be subject to quantum tunneling instabilities.

“Therefore, even though there exist two-and three-dimensional AdS critical points of the SM landscape, we argue that these candidate vacua may be subject to quantum tunneling instabilities.”

The SM supplemented by an axion (and nothing else) seems to be at odds with the conjecture [30,31] if the high scale vacuum has a negative cosmological constant.

“Thus, the SM supplemented by an axion (and nothing else) seems to be at odd with the conjecture [30,31] if the high scale vacuum has negative cosmological constant.”

• cites [30] H. Ooguri and C. Vafa, Non-supersymmetric AdS and the Swampland, arXiv:1610.01533 [INSPIRE].Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [31] B. Freivogel and M. Kleban, Vacua Morghulis, arXiv:1610.04564 [INSPIRE].Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If the high scale vacuum of the Standard Model has a negative cosmological constant and is stable, it would be at odds with the conjecture that all non-supersymmetric AdS vacua are unstable.

“If the high scale vacuum of the SM has a negative cosmological constant and is stable, it would seem to be at odd with the weak gravity conjecture.”

• Also stated in: § Abstract · hedged

  “We also point out that a high scale AdS 4 vacuum of the Standard Model, if exists, would be at odd with the conjecture that all non-supersymmetric AdS vacua are unstable.”

• cites [30] H. Ooguri and C. Vafa, Non-supersymmetric AdS and the Swampland, arXiv:1610.01533 [INSPIRE].Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [31] B. Freivogel and M. Kleban, Vacua Morghulis, arXiv:1610.04564 [INSPIRE].Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In the high scale vacuum, the Dirac neutrino mass yν H can become larger than the Majorana mass.

“The Dirac neutrino mass, y ν H , can become larger than the Majorana mass.”

There is a constraint between the neutrino mass and the observed cosmological constant based on the multiple point criticality principle.

“we still find an intriguing constraint between the neutrino mass and the observed cosmological constant based on a rather general principle that had some success in a different particle physics context.”

Even if the neutrinos are of the Dirac type, the neutrino vacuum does not appear if the absolute value of the cosmological constant is large compared with the mass of the neutrino.

“Even if the neutrinos are of the Dirac type, the neutrino vacuum does not appear if the absolute value of the cosmological constant is large compared with the mass of the neutrino.”

Since the neutrino vacuum requires a non-trivial spin structure of the fermion, it does not decay by Witten's bubble of nothing.

“Note that, since this vacuum requires a non-trivial spin structure of the fermion, it does not decay by the Witten's bubble of nothing [45].”

• cites [45] E. Witten, Instability of the Kaluza-Klein Vacuum, Nucl. Phys. B 195 (1982) 481 [INSPIRE]. sourceNot checked — no full text of the source

In AdS2 vacua, some amount of tachyonic mass is not in contradiction with the stability condition, known as the BF bound.

“Notice that, in the case of AdS 2 vacua, some amount of tachyonic mass is not in contradiction with the stability condition, known as BF bound [48,49].”

• cites [48] P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-de Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE]. sourceNot checked — no full text of the source
• cites [49] P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE]. sourceNot checked — no full text of the source

The mass of the lightest neutrino is predicted to be O(1-10) meV.

“the mass of the lightest neutrino to be O(1-10) meV.”

• Also stated in: § 4.3 SM on T 2 · plain

  “the lightest neutrino mass is predicted to be around O(1-10) meV where the T 2 vacuum has a curvature close to the our four-dimensional vacuum.”

• Also stated in: § JHEP11(2017)043 · plain

  “we predict that the neutrinos have a Dirac mass, with the mass of the lightest neutrino ∼ O(1-10) meV.”

• AssumptionThe authors state this as an assumption: The multiple point principle is valid and applicable to the standard model landscape.

The potential conflict with the weak gravity conjecture found for the SM with high scale AdS4 vacuum can also be found in S1 compactification.

“Here we point out that this potential conflict with the conjecture can also be found in S 1 compactification.”

Our results seem to be consistent with a recent conjecture that all non-supersymmetric AdS solutions are unstable.

“In the case of S 1 compactification with Casimir energy, our results seem to be consistent with a recent conjecture that all non-supersymmetric AdS solution are unstable [30,31].”

• cites [30] H. Ooguri and C. Vafa, Non-supersymmetric AdS and the Swampland, arXiv:1610.01533 [INSPIRE].Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [31] B. Freivogel and M. Kleban, Vacua Morghulis, arXiv:1610.04564 [INSPIRE].Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There is also a possibility of moduli stabilization thanks to both the tree level potential and the one-loop Casimir energy.

“There is also a possibility of moduli stabilization thanks to both the tree level potential and the one-loop Casimir energy.”

The non-perturbative stability of T2 vacuum is more subtle than the S1 case.

“The non-perturbative stability of T 2 vacuum is more subtle than the S 1 case.”

The criteria for a perturbatively stable vacuum upon compactifying on T2 were clarified.

“We have clarified the criteria for a perturbatively stable vacuum upon compactifiying on an T 2 .”

For T2 compactification, the Casimir energy was calculated for general boundary conditions of fields in the compact space.

“For a T 2 compactification, we have calculated the Casimir energy for general boundary conditions of fields in the compact space.”

Unlike in other dimensions, one cannot go to the Einstein frame in T2 compactification because of Weyl invariance of the Einstein-Hilbert action.

“Unlike in other dimensions, we cannot go to the Einstein frame because of Weyl invariance of the Einstein Hilbert action.”

This may imply that a metastable electroweak vacuum with the addition of axions cannot be consistently embedded into a quantum theory of gravity.

“As discussed in section 3.6, this may imply that metastable electroweak vacuum with the addition of axions cannot be consistently embedded into a quantum theory of gravity.”

This prediction implies the absence of neutrino-less double beta decay.

“This prediction implies the absence of the neutrino-less double beta decay.”

This work considered compactifications not only of the electroweak vacuum but also of the high scale Higgs vacuum.

“In this work, we have considered compactifications not only of the electroweak vacuum but also of the high scale Higgs vacuum.”

When the neutrino term is dominant in the potential, lower dimensional vacua may exist according to the discussion of section 4.2.2.

“When the neutrino term is dominant in the potential, we may have lower dimensional vacua according to the discussion of section 4.2.2.”

The subtlety of the Witten bubble of nothing does not change the arguments because neutrinos are uncharged under the Wilson line.

“However, this subtlety does not change the arguments that follow as the neutrinos are uncharged under the Wilson line.”

• AssumptionRests on an assumption: Neutrinos are uncharged under the Wilson line.