Toward minimal composite Higgs models from regular geometries in bottom-up holography

Open Access

Toward minimal composite Higgs models from regular geometries in bottom-up holography

Daniel Elander, Ali Fatemiabhari, and Maurizio Piai

Phys. Rev. D 107, 115021 – Published 15 June 2023

Abstract

We study a bottom-up, holographic description of a field theory yielding the spontaneous breaking of an approximate $S O (5)$ global symmetry to its $S O (4)$ subgroup. The weakly coupled, six-dimensional gravity dual has regular geometry. One of the dimensions is compactified on a circle that shrinks smoothly to zero size at a finite value of the holographic direction, hence introducing a physical scale in a way that mimics the effect of confinement in the dual four-dimensional field theory. We study the spectrum of small fluctuations of the bulk fields carrying $S O (5)$ quantum numbers, which can be interpreted as spin-0 and spin-1 bound states in the dual field theory. This work supplements an earlier publication focused only on the $S O (5)$ singlet states. We explore the parameter space of the theory, paying particular attention to composite states that have the right quantum numbers to be identified as pseudo-Nambu-Goldstone bosons (PNGBs). We find that in this model the PNGBs are generally heavy, with masses of the same order as other bound states, indicating the presence of a sizeable amount of explicit symmetry breaking in the field-theory side. Nevertheless, we also find a qualitatively new, unexpected result. When the dimension of the field-theory operator inducing $S O (5)$ breaking is close to half the space-time dimensionality, there exists a region of parameter space in which the PNGBs and the lightest scalar are both parametrically light in comparison to all other bound states of the field theory. Although this region is known to yield metastable classical backgrounds, this finding might be relevant to model building in the composite Higgs context.

Received 28 March 2023
Accepted 3 May 2023

DOI:https://doi.org/10.1103/PhysRevD.107.115021

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Composite models Gauge-gravity dualities

Particles & Fields

Authors & Affiliations

Daniel Elander ¹, Ali Fatemiabhari ^2,*, and Maurizio Piai ²

¹Montpellier, France
²Department of Physics, Faculty of Science and Engineering, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, United Kingdom

^*a.fatemiabhari.2127756@swansea.ac.uk

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 107, Iss. 11 — 1 June 2023

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Mass spectrum $\frac{M^{2}}{| M |}$ of fluctuations computed for confining backgrounds, with various choices of $Δ$ , as a function of the IR parameter $ϕ_{I}$ for $g = 5$ (left) and as a function of $g$ for $ϕ_{I} = ϕ_{I} (c)$ (right). For each $Δ$ , we show the spectrum of scalar (blue), pseudo-scalar (purple), vector (black), and tensor (red) states. The values of the IR and UV cutoffs in the calculations are, respectively, given by $ρ_{1} - ρ_{o} = 10^{- 9}$ and $ρ_{2} - ρ_{o} = 5$ in all of the cases. The different symbols refer to the quantum numbers with respect to the unbroken $S O (4)$ symmetry: Disks are used for singlets and have already been reported in Ref. [120]; diamonds represent the 6 of $S O (4)$ , and crosses the 4 of $S O (4)$ . All masses are normalized to the mass of the lightest spin-2 state. Because the masses of the $S O (5)$ singlets do not depend on $g$ , we do not repeat them in the right panels, which display only nontrivial $S O (5)$ multiplets.
Reuse & Permissions
Figure 2
Mass spectrum $\frac{M^{2}}{| M |}$ of fluctuations computed for confining backgrounds, with various choices of $Δ$ , as a function of the IR parameter $ϕ_{I}$ for $g = 5$ (left) and as a function of $g$ for $ϕ_{I} = ϕ_{I} (c)$ (right). For each $Δ$ , we show the spectrum of scalar (blue), pseudo-scalar (purple), vector (black), and tensor (red) states. The values of the IR and UV cutoffs in the calculations are, respectively, given by $ρ_{1} - ρ_{o} = 10^{- 9}$ and $ρ_{2} - ρ_{o} = 5$ in all of the cases. The different symbols refer to the quantum numbers with respect to the unbroken $S O (4)$ symmetry: Disks are used for singlets and have already been reported in Ref. [120]; diamonds represent the 6 of $S O (4)$ , and crosses the 4 of $S O (4)$ . All masses are normalized to the mass of the lightest spin-2 state. Because the masses of the $S O (5)$ singlets do not depend on $g$ , we do not repeat them in the right panels, which display only nontrivial $S O (5)$ multiplets.
Reuse & Permissions
Figure 3
Mass spectrum $\frac{M^{2}}{| M |}$ of fluctuations computed for confining backgrounds for $Δ = 2.35$ (top), $Δ = 2.40$ (middle), and $Δ = 2.45$ (bottom), and $ϕ_{I} = 3$ as a function of $g$ (left), and free energy $\hat{F}$ a function of the normalized source ${\hat{ϕ}}_{J}$ (right). For the spectrum, scalar (blue), pseudo-scalar (purple), vector (black), and tensor (red) states are displayed as disks for $S O (4)$ singlets, as diamonds to represent the 6 of $S O (4)$ , and crosses for the 4 of $S O (4)$ . The values of the IR and UV cutoffs in the calculations are, respectively, given by $ρ_{1} - ρ_{o} = 10^{- 9}$ and $ρ_{2} - ρ_{o} = 5$ . All masses are normalized to the mass of the lightest spin-2 state. The definition of (normalized) free energy $\hat{F}$ and (normalized) source ${\hat{ϕ}}_{J}$ can be found in Ref. [120]. The black (stable), gray (metastable), and dashed black (tachyonic) regions of the curve refer to the confining solutions of interest, while red and blue curves refer to singular solutions. The black dots on the right panels denote the solutions with $ϕ_{I} = 3$ .
Reuse & Permissions
Figure 4
Mass spectrum $\frac{M^{2}}{| M |}$ of fluctuations computed for confining backgrounds with various choices of $Δ$ , as a function of the IR parameter $ϕ_{I}$ for $g = 5$ . For each $Δ$ , we show the spectrum of scalar (blue), pseudo-scalar (purple), vector (black), and tensor (red) states. The values of the IR and UV cutoffs in the calculations are, respectively, given by $ρ_{1} - ρ_{o} = 10^{- 9}$ and $ρ_{2} - ρ_{o} = 5$ in all of the cases. The different symbols refer to the quantum numbers with respect to the unbroken $S O (4)$ symmetry: Disks are used for singlets and have already been reported in Ref. [120], diamonds represent the 6 of $S O (4)$ , and crosses the 4 of $S O (4)$ . All masses are normalized to the mass of the lightest spin-2 state.
Reuse & Permissions
Figure 5
Mass spectrum $\frac{M^{2}}{| M |}$ of fluctuations computed for confining backgrounds with various choices of $Δ$ , as a function of the IR parameter $ϕ_{I}$ for $g = 5$ . For each $Δ$ , we show the spectrum of scalar (blue), pseudo-scalar (purple), vector (black), and tensor (red) states. The values of the IR and UV cutoffs in the calculations are, respectively, given by $ρ_{1} - ρ_{o} = 10^{- 9}$ and $ρ_{2} - ρ_{o} = 5$ in all of the cases. The different symbols refer to the quantum numbers with respect to the unbroken $S O (4)$ symmetry: Disks are used for singlets and have already been reported in Ref. [120], diamonds represent the 6 of $S O (4)$ , and crosses the 4 of $S O (4)$ . All masses are normalized to the mass of the lightest spin-2 state.
Reuse & Permissions
Figure 6
Mass spectrum $\frac{M^{2}}{| M |}$ of fluctuations computed for confining backgrounds with various choices of $Δ$ , as a function of the IR parameter $ϕ_{I}$ for $g = 5$ . For each $Δ$ , we show the spectrum of scalar (blue), pseudo-scalar (purple), vector (black), and tensor (red) states. The values of the IR and UV cutoffs in the calculations are, respectively, given by $ρ_{1} - ρ_{o} = 10^{- 9}$ and $ρ_{2} - ρ_{o} = 5$ in all of the cases. The different symbols refer to the quantum numbers with respect to the unbroken $S O (4)$ symmetry: Disks are used for singlets and have already been reported in Ref. [120], diamonds represent the 6 of $S O (4)$ , and crosses the 4 of $S O (4)$ . All masses are normalized to the mass of the lightest spin-2 state.
Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology