We consider phenomenological aspects of a natural class of Standard Model-like supersymmetric F-theory vacua realized through flux breaking of rigid E 7 gauge factors. Three generations of Standard Model matter are realized in many of these vacua. We further find that many other Standard Model-like features are naturally compatible with these constructions. For example, dimension-4 and 5 terms associated with proton decay are ubiquitously suppressed. Many of these features are due to the group theoretical structure of E 7 and associated F-theory geometry. In particular, a set of approximate global symmetries descends from the E 7 group, leading to exponential suppression of undesired couplings.

United States

Shing Li

Washington Taylor

Towards natural and realistic E 7 GUTs in F-theory

Exponential resummation of the QCD finite-density Taylor series has been recently introduced as an alternative way of resumming the finite-density lattice QCD Taylor series. Unfortunately the usual exponential resummation formula suffers from stochastic bias which must be subtracted before identifying genuine higher-order contributions. In this paper, we present a new way of subtracting the stochastic bias at the level of each individual gauge configuration, up to a certain order of either the Taylor series or the cumulant expansion, by modifying the argument of the exponential. Retaining the exponential form of the resummation allows us to also calculate the phase factor of the fermion determinant on each gauge configuration. We present our results for the excess pressure, number density, and the average phase factor and show that the new results contain less stochastic bias and are in better agreement with the QCD Taylor series compared to the previous exponential resummation.

India

Sabarnya Mitra

Prasad Hegde

QCD equation of state at finite chemical potential from an unbiased exponential resummation of the lattice QCD Taylor series

We use the recently developed methods of arXiv:2108.07810 to analyze vertical flux backgrounds and associated chiral matter spectra in the 4D universal (SU(3) × SU(2) × U(1))/ℤ 6 model introduced in arXiv:1912.10991, which is believed to describe the most generic family of F-theory vacua with tuned (SU(3) × SU(2) × U(1))/ℤ 6 gauge symmetry. Our analysis focuses on a resolution of a particular presentation of the (SU(3) × SU(2) × U(1))/ℤ 6 model in which the elliptic fiber is realized as a cubic in ℙ 2 fibered over an arbitrary smooth threefold base. We show that vertical fluxes can produce nonzero multiplicities for all chiral matter families that satisfy 4D anomaly cancellation, which include as a special case the chiral matter families of the Minimal Supersymmetric Standard Model.

Patrick Jefferson

Andrew Turner

Chiral spectrum of the universal tuned (SU(3) × SU(2) × U(1))/ℤ 6 4D F-theory model

We study the massless charged spectrum of U(1) gauge fields in F-theory that arise from flux breaking of a nonabelian group. The U(1) charges that arise in this way can be very large. In particular, using vertical flux breaking, we construct an explicit 4D F-theory model with a U(1) decoupled from other gauge sectors, in which the massless/light fields have charges as large as 657. This result greatly exceeds prior results in the literature. We argue heuristically that this result may provide an upper bound on charges for light fields under decoupled U(1) factors in the F-theory landscape. We also show that the charges can be even larger when the U(1) is coupled to other gauge groups.

Large U(1) charges from flux breaking in 4D F-theory models

We give a general description of gauge symmetry breaking using vertical and remainder fluxes in 4D F-theory models. The fluxes can break a geometric gauge group to a smaller group and induce chiral matter, even when the larger group admits no chiral matter representations. We focus specifically on applications to realizations of the Standard Model gauge group and chiral matter spectrum through breaking of rigid exceptional gauge groups E 7 , E 6, which are ubiquitous in the 4D F-theory landscape. Supplemented by an intermediate SU(5) group, these large classes of models give natural constructions of Standard Model-like theories with small numbers of generations of matter in F-theory.

Gauge symmetry breaking with fluxes and natural Standard Model structure from exceptional GUTs in F-theory

We describe a broad class of 4D F-theory models in which an <math><msub><mi>E</mi><mn>7</mn></msub></math> gauge group is broken through fluxes to the standard model gauge group. These models are ubiquitous in the 4D F-theory landscape and can arise from flux breaking of most models with <math><msub><mi>E</mi><mn>7</mn></msub></math> factors. While in many cases the <math><msub><mi>E</mi><mn>7</mn></msub></math> breaking leads to exotic matter, there are large families of models in which the standard model gauge group and chiral matter representations are obtained through an intermediate SU(5) group. The number of generations of matter appearing in these models can easily be small. We demonstrate the possibility of getting three generations of chiral matter as the preferred matter content.

Shing Yan Li

Natural F-theory constructions of standard model structure from <math><msub><mi>E</mi><mn>7</mn></msub></math> flux breaking

Taylor expansion of the thermodynamic potential in powers of the (baryo)chemical potential <math><msub><mi>μ</mi><mi>B</mi></msub></math> is a well-known method to bypass the sign problem of lattice QCD. Due to the difficulty in calculating the higher order Taylor coefficients, various alternative expansion schemes as well as resummation techniques have been suggested to extend the Taylor series to larger values of <math><msub><mi>μ</mi><mi>B</mi></msub></math>. Recently, a way to resum the contribution of the first <math><mi>N</mi></math> charge density correlation functions <math><msub><mi>D</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>D</mi><mi>N</mi></msub></math> to the Taylor series to all orders in <math><msub><mi>μ</mi><mi>B</mi></msub></math> was proposed in [Phys. Rev. Lett. 128, 022001 (2022)]. The resummation takes the form of an exponential factor. Since the correlation functions are calculated stochastically, the exponential factor contains a bias which can be significant for large <math><mi>N</mi></math> and <math><msub><mi>μ</mi><mi>B</mi></msub></math>. In this paper, we present a new method to calculate the QCD equation of state based on the well-known cumulant expansion from statistics. By truncating the expansion at a maximum order <math><mi>M</mi></math>, we end up with only finite products of the correlation functions which can be evaluated in an unbiased manner. Although our formalism is also applicable for <math><msub><mi>μ</mi><mi>B</mi></msub><mo>≠</mo><mn>0</mn></math>, here we present it for the simpler case of a finite isospin chemical potential <math><msub><mi>μ</mi><mi>I</mi></msub></math> for which there is no sign problem. We present and compare results for the pressure and the isospin density obtained using Taylor expansion, exponential resummation and cumulant expansion, and provide evidence that the absence of bias in the latter actually improves the convergence.

Germany

Christian Schmidt

New way to resum the lattice QCD Taylor series equation of state at finite chemical potential

Using high statistics datasets generated in (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD calculations at finite temperature, we present results for low-order cumulants of net-baryon-number fluctuations at nonzero values of the baryon chemical potential. We calculate Taylor expansions for the pressure (zeroth-order cumulant), the net-baryon-number density (first-order cumulant), and the variance of the distribution on net-baryon-number fluctuations (second-order cumulant). We obtain series expansions from an eighth-order expansion of the pressure and compare these to diagonal Padé approximants. This allows us to estimate the range of values for the baryon chemical potential in which these expansions are reliable. We find <math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>/</mo><mi>T</mi><mo>≤</mo><mn>2.5</mn></mrow></math>, 2.0, and 1.5 for the zeroth-, first-, and second-order cumulants, respectively. We, furthermore, construct estimators for the radius of convergence of the Taylor series of the pressure. In the vicinity of the pseudocritical temperature <math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi><mi>c</mi></mrow></msub><mo>≃</mo><mn>156.5</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, we find <math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>/</mo><mi>T</mi><mo>≳</mo><mn>2.9</mn></mrow></math> at vanishing strangeness chemical potential and somewhat larger values for strangeness neutral matter. These estimates are temperature dependent and range from <math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>/</mo><mi>T</mi><mo>≳</mo><mn>2.2</mn></mrow></math> at <math><mrow><mi>T</mi><mo>=</mo><mn>135</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> to <math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>/</mo><mi>T</mi><mo>≳</mo><mn>3.2</mn></mrow></math> at <math><mi>T</mi><mo>=</mo><mn>165</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. The estimated radius of convergences is the same for any higher-order cumulant.

D. Bollweg

J. Goswami

O. Kaczmarek

F. Karsch

Swagato Mukherjee

P. Petreczky

C. Schmidt

P. Scior

Taylor expansions and Padé approximants for cumulants of conserved charge fluctuations at nonvanishing chemical potentials

Taylor expansion in powers of baryon chemical potential (<math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></math>) is an oft-used method in lattice QCD to compute QCD thermodynamics for <math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></mrow></math>. Based only upon the few known lowest order Taylor coefficients, it is difficult to discern the range of <math><msub><mi>μ</mi><mi>B</mi></msub></math> where such an expansion around <math><msub><mi>μ</mi><mi>B</mi></msub><mo>=</mo><mn>0</mn></math> can be trusted. We introduce a resummation scheme for the Taylor expansion of the QCD equation of state in <math><msub><mi>μ</mi><mi>B</mi></msub></math> that is based on the <math><mi>n</mi></math>-point correlation functions of the conserved current (<math><msub><mi>D</mi><mi>n</mi></msub></math>). The method resums the contributions of the first <math><mi>N</mi></math> correlation function <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math> to the Taylor expansion of the QCD partition function to all orders in <math><msub><mi>μ</mi><mi>B</mi></msub></math>. We show that the resummed partition function is an approximation to the reweighted partition function at <math><msub><mi>μ</mi><mi>B</mi></msub><mo>≠</mo><mn>0</mn></math>. We apply the proposed approach to high-statistics lattice QCD calculations using <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavors of Highly Improved Staggered Quarks with physical quark masses on <math><mrow><msup><mrow><mn>32</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><mn>8</mn></mrow></math> lattices and for temperatures <math><mrow><mi>T</mi><mo>≈</mo><mn>145</mn><mi>–</mi><mn>176</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>. We demonstrate that, as opposed to the Taylor expansion, the resummed version not only leads to improved convergence but also reflects the zeros of the resummed partition function and severity of the sign problem, leading to its eventual breakdown. We also provide a generalization of scheme to include resummation of powers of temperature and quark masses in addition to <math><msub><mi>μ</mi><mi>B</mi></msub></math>, and show that the alternative expansion scheme of [S. Borsányi et al., Phys. Rev. Lett. 126, 232001 (2021).] is a special case of this generalized resummation.

Sourav Mondal

Lattice QCD Equation of State for Nonvanishing Chemical Potential by Resumming Taylor Expansions

We prove that, for every 6D supergravity theory that has an F-theory description, the property of charge completeness for the connected component of the gauge group (meaning that all charges in the corresponding charge lattice are realized by massive or massless states in the theory) is equivalent to a standard assumption made in F-theory for how geometry encodes the global gauge theory by means of the Mordell-Weil group of the elliptic fibration. This result also holds in 4D F-theory constructions for the parts of the gauge group that come from sections and from 7-branes. We find that in many 6D F-theory models the full charge lattice of the theory is generated by massless charged states; this occurs for each gauge factor where the associated anomaly coefficient satisfies a simple positivity condition. We describe many of the cases where this massless charge sufficiency condition holds, as well as exceptions where the positivity condition fails, and analyze the related global structure of the gauge group and associated Mordell-Weil torsion in explicit F-theory models.