arXiv subjects

Hans Peter Nilles

Publications and source records attributed to Hans Peter Nilles.

At least 19 recordsLinked to original sources

arXiv2025

Demystifying stringy miracles with eclectic flavor symmetries

Effective field theories arising from string compactifications are subject to constraints originating from the duality transformations of string theory. Interpreting these so-called selection rules in terms of conventional symmetries has remained challenging. We show that particular selection rules in heterotic orbifolds can be explained from a subtle interplay between modular and traditional flavor symmetries within the eclectic flavor framework.

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arXiv2025

Flavor Symmetries and Winding Modes

Modular flavor symmetries have been proposed as a new way to address the flavor problem. It is known that they can emerge from string compactifications. We discuss this connection in detail, and show how the congruence subgroups of SL(2,Z), which underlie many modular flavor symmetries, emerge from stringy duality symmetries by orbifolding. This requires an analysis of massive states, which reveals a picture that is more intricate than the well-known situation on the torus. It involves towers of states of different quantum numbers, related by modular transformations. Members of different towers become massless at different points in moduli space. We also show that, at least in the Z_3 orbifold, the string selection rules can be understood as discrete remnants of continuous gauge symmetries. Non-Abelian discrete flavor symmetries arise as relics of various, relatively misaligned, continuous Abelian gauge symmetries. The generators of these U(1) symmetries give rise to CP-violating Clebsch-Gordan coefficients. If the modulus settles close to a critical point, the corresponding gauge bosons may be light enough to be searched for at future colliders.

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arXiv2024

The eclectic flavor symmetries of $\mathbb{T}^2/\mathbb{Z}_K$ orbifolds

Only four $\mathbb{T}^2/\mathbb{Z}_K$ orbifold building blocks are admissible in heterotic string compactifications. We investigate the flavor properties of all of these building blocks. In each case, we identify the traditional and modular flavor symmetries, and determine the corresponding representations and (fractional) modular weights of the available massless matter states. The resulting finite flavor symmetries include Abelian and non-Abelian traditional symmetries, discrete $R$ symmetries, as well as the double-covered finite modular groups $(S_3\times S_3)\rtimes\mathbb{Z}_4$, $T'$, $2D_3$ and $S_3\times T'$. Our findings provide restrictions for bottom-up model building with consistent ultraviolet embeddings.

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arXiv2024

Flavor's Delight

Discrete flavor symmetries provide a promising approach to understand the flavor sector of the standard model of particle physics. Top-down (TD) explanations from string theory reveal two different types of such flavor symmetries: traditional and modular flavor symmetries that combine to the eclectic flavor group. There have been many bottom-up (BU) constructions to fit experimental data within this scheme. We compare TD and BU constructions to identify the most promising groups and try to give a unified description. Although there is some progress in joining BU and TD approaches, we point out some gaps that have to be closed with future model building.

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arXiv2023

The Flavor Puzzle: Textures and Symmetries

We discuss aspects of a promising top-down origin of flavor symmetries in particle physics. Modular transformations originating from string theory dualities are shown to play a crucial role. We introduce the notion of an "eclectic" flavor scheme that unifies traditional flavor symmetries, modular symmetries and CP-transformations. It exhibits the phenomenon of "Local Flavor Unification" with enhanced flavor symmetries at fixed points or lines in moduli space. Successful fits of masses and mixing angles of quarks and leptons are found in the vicinity of these points and lines.

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arXiv2023

Matter matters in moduli fixing and modular flavor symmetries

Modular flavor symmetries provide us with a very compelling approach to the flavor problem. It has been argued that moduli values close to some special values like $τ=i$ or $τ=ω$ provide us with the best fits to data. We point out that the presence of hidden "matter" fields, needed to uplift symmetric AdS vacua, gives rise to a dynamical mechanism that leads to such values of $τ$.

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arXiv2022

The first string-derived eclectic flavor model with realistic phenomenology

Eclectic flavor groups arising from string compactifications combine the power of modular and traditional flavor symmetries to address the flavor puzzle. This top-down scheme determines the representations and modular weights of all matter fields, imposing strict constraints on the structure of the effective potential, which result in controlled corrections. We study the lepton and quark flavor phenomenology of an explicit, potentially realistic example model based on a T6/Z3xZ3 orbifold compactification of the heterotic string that gives rise to an Omega(2) eclectic flavor symmetry. We find that the interplay of flavon alignment and the localization of the modulus in the vicinity of a symmetry-enhanced point leads to naturally protected fermion mass hierarchies, favoring normal-ordered neutrino masses arising from a see-saw mechanism. We show that our model can reproduce all observables in the lepton sector with a small number of parameters and deliver predictions for so far undetermined neutrino observables. Furthermore, we extend the fit to quarks and find that Kaehler corrections are instrumental in obtaining a successful simultaneous fit to the quark and lepton sectors.

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arXiv2022

Top-Down Anatomy of Flavor Symmetry Breakdown

A top-down approach to the flavor puzzle leads to eclectic flavor groups which include modular and traditional flavor symmetries. Based on examples of semirealistic T2/Z3 orbifold compactifications of heterotic string theory, we discuss the breakdown patterns of the eclectic flavor group via the interplay of vacuum expectation values (vevs) of moduli and flavon fields. This leads to an attractive flavor scheme with various possibilities to obtain "flavor hierarchies" through the alignment of these vevs. Despite the fact that the top-down approach gives strong restrictions for bottom-up flavor model building, it seems to be well suited to provide a realistic flavor pattern for quarks and leptons.

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arXiv2021

Orbifolds from $\boldsymbol{\mathrm{Sp}(4,\mathbb Z)}$ and their modular symmetries

The incorporation of Wilson lines leads to an extension of the modular symmetries of string compactification beyond $\mathrm{SL}(2,\mathbb Z)$. In the simplest case with one Wilson line $Z$, Kähler modulus $T$ and complex structure modulus $U$, we are led to the Siegel modular group $\mathrm{Sp}(4,\mathbb Z)$. It includes $\mathrm{SL}(2,\mathbb Z)_T\times\mathrm{SL}(2,\mathbb Z)_U$ as well as $\mathbb Z_2$ mirror symmetry, which interchanges $T$ and $U$. Possible applications to flavor physics of the Standard Model require the study of orbifolds of $\mathrm{Sp}(4,\mathbb Z)$ to obtain chiral fermions. We identify the 13 possible orbifolds and determine their modular flavor symmetries as subgroups of $\mathrm{Sp}(4,\mathbb Z)$. Some cases correspond to symmetric orbifolds that extend previously discussed cases of $\mathrm{SL}(2,\mathbb Z)$. Others are based on asymmetric orbifold twists (including mirror symmetry) that do no longer allow for a simple intuitive geometrical interpretation and require further study. Sometimes they can be mapped back to symmetric orbifolds with quantized Wilson lines. The symmetries of $\mathrm{Sp}(4,\mathbb Z)$ reveal exciting new aspects of modular symmetries with promising applications to flavor model building.

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arXiv2021

Flavor and CP from String Theory

Modular transformations of string theory are shown to play a crucial role in the discussion of discrete flavor symmetries in the Standard Model. They include CP transformations and provide a unification of CP with traditional flavor symmetries within the framework of the "eclectic flavor" scheme. The unified flavor group is non-universal in moduli space and exhibits the phenomenon of "Local Flavor Unification", where different sectors of the theory (like quarks and leptons) can be subject to different flavor structures.

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arXiv2021

Eclectic flavor scheme from ten-dimensional string theory -- II. Detailed technical analysis

String theory leads to a flavor scheme where modular symmetries play a crucial role. Together with the traditional flavor symmetries they combine to an eclectic flavor group, which we determine via outer automorphisms of the Narain space group. Unbroken flavor symmetries are subgroups of this eclectic group and their size depends on the location in moduli space. This mechanism of local flavor unification allows a different flavor structure for different sectors of the theory (such as quarks and leptons) and also explains the spontaneous breakdown of flavor- and CP-symmetries (via a motion in moduli space). We derive the modular groups, including CP and R-symmetries, for different sub-sectors of six-dimensional string compactifications and determine the general properties of the allowed flavor groups from this top-down perspective. It leads to a very predictive flavor scheme that should be confronted with the variety of existing bottom-up constructions of flavor symmetry in order to clarify which of them could have a consistent top-down completion.

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arXiv2021

Completing the eclectic flavor scheme of the $\boldsymbol{\mathbb Z_2}$ orbifold

We present a detailed analysis of the eclectic flavor structure of the two-dimensional $\mathbb Z_2$ orbifold with its two unconstrained moduli $T$ and $U$ as well as $\mathrm{SL}(2,\mathbb Z)_T\times \mathrm{SL}(2,\mathbb Z)_U$ modular symmetry. This provides a thorough understanding of mirror symmetry as well as the $R$-symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the $(T,U)$ modulus landscape. In view of applications towards the flavor structure of particle physics models, we are led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry, CP and $R$-symmetries on the superpotential and Kaehler potential of the theory.

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arXiv2021

The eclectic flavor symmetry of the $\boldsymbol{\mathbb{Z}_2}$ orbifold

Modular symmetries naturally combine with traditional flavor symmetries and $\mathcal{CP}$, giving rise to the so-called eclectic flavor symmetry. We apply this scheme to the two-dimensional $\mathbb{Z}_2$ orbifold, which is equipped with two modular symmetries $\mathrm{SL}(2,\mathbb{Z})_T$ and $\mathrm{SL}(2,\mathbb{Z})_U$ associated with two moduli: the Kähler modulus $T$ and the complex structure modulus $U$. The resulting finite modular group is $((S_3\times S_3)\rtimes \mathbb{Z}_4)\times\mathbb{Z}_2$ including mirror symmetry (that exchanges $T$ and $U$) and a generalized $\mathcal{CP}$-transformation. Together with the traditional flavor symmetry $(D_8\times D_8)/\mathbb{Z}_2$, this leads to a huge eclectic flavor group with 4608 elements. At specific regions in moduli space we observe enhanced unified flavor symmetries with as many as 1152 elements for the tetrahedral shaped orbifold and $\langle T \rangle = \langle U \rangle = \exp(π\,\mathrm{i}\,/\,3)$. This rich eclectic structure implies interesting (modular) flavor groups for particle physics models derived form string theory.

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arXiv2021

Siegel modular flavor group and CP from string theory

We derive the potential modular symmetries of heterotic string theory. For a toroidal compactification with Wilson line modulus, we obtain the Siegel modular group $\mathrm{Sp}(4,\mathbb{Z})$ that includes the modular symmetries $\mathrm{SL}(2,\mathbb{Z})_T$ and $\mathrm{SL}(2,\mathbb{Z})_U$ (of the "geometric" moduli $T$ and $U$) as well as mirror symmetry. In addition, string theory provides a candidate for a CP-like symmetry that enhances the Siegel modular group to $\mathrm{GSp}(4,\mathbb{Z})$.

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arXiv2020

Lessons from eclectic flavor symmetries

A top-down approach to the flavor problem motivated from string theory leads to the concept of eclectic flavor groups that combine traditional and modular flavor symmetries. To make contact with models constructed in the bottom-up approach, we analyze a specific example based on the eclectic flavor group Omega(1) (a nontrivial combination of the traditional flavor group Delta(54) and the finite modular group T') in order to extract general lessons from the eclectic scheme. We observe that this scheme is highly predictive since it severely restricts the possible group representations and modular weights of matter fields. Thereby, it controls the structure of the Kaehler potential and the superpotential, which we discuss explicitly. In particular, both Kaehler potential and superpotential are shown to transform nontrivially, but combine to an invariant action. Finally, we find that discrete R-symmetries are intrinsic to eclectic flavor groups.

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arXiv2020

Eclectic flavor scheme from ten-dimensional string theory -- I. Basic results

In a consistent top-down approach based on orbifold compactifications, modular and traditional flavor symmetries combine nontrivially to the so-called eclectic flavor symmetry. We extend this scheme from two extra dimensions, discussed previously, to the six extra dimensions of string theory. By doing so, new insights on the nature of CP and its spontaneous breaking emerge. Moreover, we identify a new interpretation of R-symmetries as unbroken remnants from modular symmetries that are associated with geometrically stabilized complex structure moduli. Hence, all symmetries (i.e. modular, traditional flavor, CP and R) share a common origin in string theory: on a technical level, they are given by outer automorphisms of the Narain space group. The eclectic top-down approach leads to a very restrictive scheme with high predictive power. It remains a challenge to connect this with existing bottom-up constructions of modular flavor symmetry.

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arXiv2020

Eclectic Flavor Groups

The simultaneous study of top-down and bottom-up approaches to modular flavor symmetry leads necessarily to the concept of eclectic flavor groups. These are nontrivial products of modular and traditional flavor symmetries that exhibit the phenomenon of local flavor enhancement in moduli space. We develop methods to determine the eclectic flavor groups that can be consistently associated with a given traditional flavor symmetry. Applying these methods to a large family of prominent traditional flavor symmetries, we try to identify potential candidates for realistic eclectic flavor groups and show that they are relatively rare. Model building with finite modular flavor symmetries thus appears to be much more restrictive than previously thought.

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arXiv2019

Axion clockworks from heterotic M-theory: the QCD-axion and its ultra-light companion

A previously discussed clockwork mechanism within heterotic M-theory is applied to its axion landscape. We identify a unique candidate for a QCD-axion with a decay constant in the preferred "axion window" around $10^{11}\,$GeV. It is accompanied by at least one ultra-light axion that couples predominantly to hidden sector gauge groups.

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