#
String Theory Bounds on the Cosmological Constant, the Higgs Mass, and the Quark and Lepton Masses^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Cosmological Constant

#### 2.1. The Problem

#### 2.1.1. Quantum Field Theory

#### 2.1.2. String Theory

#### 2.2. Resolving the Problem

#### 2.2.1. Phase Space

#### 2.2.2. Holography

- The relation (27) naturally gives ${\rho}_{0}\to 0$ when $l\to \infty $, and l is the IR length-scale.
- The relation (30) is radiatively stable since there is no UV dependence.
- Thus, essentially, the cosmological constant is small because the universe is filled with a large number of degrees of freedom: $N\sim {10}^{124}$.
- In turn, N is large because fluctuations scale as $\frac{1}{\sqrt{N}}$ and are small, indicating the stability of the universe.
- This estimates ${N}_{i}$ (where i is $t,x,y,z$) as ${N}^{1/4}\sim {10}^{31}$, which is not so unreasonable in comparison with Avogadro’s number, ${10}^{23}$, for matter degrees of freedom.

- First, $\u03f5$ is not a cut-off, since $\u03f5$ and $\lambda $ can be arbitrary, albeit related by $\lambda \phantom{\rule{0.166667em}{0ex}}\u03f5=\hslash $.
- Second, ${\u03f5}^{4}$ is effectively eliminated in favor of N, which is the new quantum number, and the size of spacetime, $l={r}_{CH}$, the cosmological horizon, i.e., the size of the observed classical spacetime.
- N is determined by the Bekenstein bound, (26) and is thereby related to l and ${l}_{P}$ (the ultimate IR and UV scales, respectively), which is where gravity enters, via ${G}_{N}\sim {l}_{P}^{2}$.

#### 2.3. The Cosmological Constant in QFT and Phase Space

#### 2.4. Realization in String Theory

- the concept of dual spacetime,
- the fundamental non-commutativity between spacetime and dual spacetime,
- the Heisenberg algebras: $[q,\tilde{q}]=i{l}_{P}^{2}$, $[q,p]=i$, $[\tilde{q},\tilde{p}]=i$, $[p,\tilde{p}]=0$.

## 3. The Cosmological Constant and the Higgs Mass

#### 3.1. Cosmological Scale

#### 3.2. Higgs Mass

#### 3.3. Summary

**1**) examining the modular spacetime and metastring formulation of string theory, (

**2**) combining the $[x,\tilde{x}]\ne 0$ non-commutativity and holography in x-space, and (

**3**) assuming that $\tilde{x}$ is of the Planck length size, leads in string theory to a seesaw formula also for the Higgs mass (61). Like the one for the vacuum energy, this formula represents a bound provided by the size of the phase space and the Bekenstein bound in which the effective length scale is associated with vacuum energy ${l}_{\Lambda}$. In this calculation, the two Heisenberg algebras in the metastring formulation ($[x,p]$ and $[x,\tilde{x}]$) are mutually consistent.

## 4. On the Masses and Mixing of Quarks and Leptons

#### 4.1. General Comments

#### 4.1.1. Criticality

#### 4.1.2. Seesaw Structure

**1**) for charged leptons, ${\mathcal{X}}_{\psi}\ne 0$, the second term in a (59)-like formula dominates, and formula (62) follows. (

**2**) For chargeless neutrinos, ${\mathcal{X}}_{\psi}=0$, only the first term in a (59)-like formula remains, and (64) follows.

#### 4.2. Masses

#### 4.2.1. The Bjorken–Zeldovich Scale

**1**) our N, (

**2**) the Bekenstein bound for gravitational degrees of freedom, (

**3**) the fact that in metastring theory the matter and spacetime degrees of freedom are “two sides of the same coin”, (

**4**) the extensive nature of entropy for the matter degrees of freedom

#### 4.2.2. Quarks

#### 4.2.3. Charged Leptons

#### 4.2.4. Neutrinos

#### 4.3. Fermion Mixing

#### 4.3.1. The CKM Matrix

#### 4.3.2. The PMNS Matrix

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Berglund, P.; Hübsch, T.; Minic, D.
String Theory Bounds on the Cosmological Constant, the Higgs Mass, and the Quark and Lepton Masses. *Symmetry* **2023**, *15*, 1660.
https://doi.org/10.3390/sym15091660

**AMA Style**

Berglund P, Hübsch T, Minic D.
String Theory Bounds on the Cosmological Constant, the Higgs Mass, and the Quark and Lepton Masses. *Symmetry*. 2023; 15(9):1660.
https://doi.org/10.3390/sym15091660

**Chicago/Turabian Style**

Berglund, Per, Tristan Hübsch, and Djordje Minic.
2023. "String Theory Bounds on the Cosmological Constant, the Higgs Mass, and the Quark and Lepton Masses" *Symmetry* 15, no. 9: 1660.
https://doi.org/10.3390/sym15091660