Calculating the Higgs mass in string theory

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Calculating the Higgs mass in string theory

Steven Abel and Keith R. Dienes

Phys. Rev. D 104, 126032 – Published 29 December 2021

Abstract

In this paper, we establish a fully string-theoretic framework for calculating one-loop Higgs masses directly from first principles in perturbative closed-string theories. Our framework makes no assumptions other than world sheet modular invariance and is therefore applicable to all closed strings, regardless of the specific string construction utilized. This framework can also be employed even when spacetime supersymmetry is broken (and even when this breaking occurs at the Planck scale), and can be utilized for all scalar Higgs fields, regardless of the particular gauge symmetries they break. This therefore includes the Higgs field responsible for electroweak symmetry breaking in the Standard Model. Notably, using our framework, we demonstrate that a gravitational modular anomaly generically relates the Higgs mass to the one-loop cosmological constant, thereby yielding a string-theoretic connection between the two fundamental quantities which are known to suffer from hierarchy problems in the absence of spacetime supersymmetry. We also discuss a number of crucial issues involving the use and interpretation of regulators in UV/IR-mixed theories such as string theory, and the manner in which one can extract an effective field theory (EFT) description from such theories. Finally, we analyze the running of the Higgs mass within such an EFT description, and uncover the existence of a “dual IR” region which emerges at high energies as the consequence of an intriguing scale-inversion duality symmetry. We also identify a generic stringy effective potential for the Higgs fields in such theories. Our results can therefore serve as the launching point for a rigorous investigation of gauge hierarchy problems in string theory.

Received 22 June 2021
Accepted 2 December 2021

DOI:https://doi.org/10.1103/PhysRevD.104.126032

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)

Compactification String phenomenology Strings & branes

Particles & Fields

Authors & Affiliations

Steven Abel^1,* and Keith R. Dienes ^2,3,†

¹IPPP, Durham University, Durham DH1 3LE, United Kingdom
²Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
³Department of Physics, University of Maryland, College Park, Maryland 20742, USA

^*s.a.abel@durham.ac.uk
^†dienes@arizona.edu

Article Text

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Issue

Vol. 104, Iss. 12 — 15 December 2021

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Images

Figure 1
Left panel: the modular-invariant regulator function $G_{2} (a, τ_{2})$ , plotted within the $(a, τ_{2})$ plane for $a ≪ 1$ and $τ_{2} \geq 1$ . Right panel: the modular-invariant regulator $G_{2} (a, τ_{2})$ , plotted as a function of $τ_{2}$ for $a = 0.05$ (blue), $a = 0.1$ (orange), and $a = 0.3$ (green). In all cases we see that $G_{2} (a, τ_{2}) \to 0$ for all $a > 0$ as $τ_{2} \to \infty$ , while $G_{2} (a, τ_{2}) \to 1$ for all $τ_{2} \geq 1$ as $a \to 0$ . Indeed, for $a = 0.05$ , we see that $G_{2} (a, τ_{2}) \approx 1$ for all $τ_{2} ≲ 100$ . Thus for small nonzero $a$ this regulator succeeds in suppressing the divergences that might otherwise arise as $τ_{2} \to \infty$ while nevertheless having little effect throughout the rest of the fundamental domain.
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Figure 2
The expression in Eq. (4.39), plotted as a function of $μ / M$ . This quantity is the Bessel-function contribution per unit $X_{2}$ charge to the running of the regulated Higgs mass ${\hat{m}}_{ϕ}^{2} (μ) |_{X} / M^{2}$ from a bosonic state of nonzero mass $M$ . This contribution is universal for all $μ / M$ and assumes only that $μ ≪ M_{s}$ . When $μ ≫ M$ , the state is fully dynamical and produces a running which is effectively logarithmic. By contrast, when $μ ≪ M$ , the state is heavier than the scale $μ$ and is effectively integrated out, thereby suppressing any contributions to the running. Finally, within the intermediate $μ \approx M$ region, the Bessel-function expression in Eq. (4.39) provides a smooth connection between these two asymptotic behaviors and even gives rise to a transient “dip” in the overall running. Note that for a fixed scale $μ$ , adjusting the mass $M$ of the relevant state upward or downward simply corresponds to shifting this curve rigidly to the right or left, respectively. In this way one can imagine summing over all such contributions to the running as one takes the supertrace over the entire $X_{2}$ -charged string spectrum.
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Figure 3
The “running” of the regulated Higgs mass ${\hat{m}}_{ϕ}^{2} (μ)$ , as calculated from first principles in a fully modular-invariant string framework. As discussed in the text, this running (green curve) exhibits a rather complicated anatomy. In the deep infrared (as $μ \to 0$ ), the Higgs mass approaches an asymptotic value which depends on the cosmological constant of the theory as well as on the supertrace over the $X_{1}$ charges of all of the string states. Moving toward higher values of $μ$ , a nontrivial scale-dependence does not emerge until $μ \sim M_{lightest}$ , where $M_{lightest}$ collectively represents the masses of the lightest massive $X_{2}$ -charged states. The “dip” that is observed in this region is a string-theoretic transient effect which smoothly connects the asymptotic deep-IR region ( $μ ≪ M_{lightest}$ ) to an EFT-like region ( $M_{lightest} ≲ μ ≪ M_{s}$ ). Moving beyond this dip region, the theory then enters the EFT-like region in which the Higgs mass experiences a running which is either logarithmic or power-law, depending on the density of string states with masses in the range $M_{lightest} ≲ M ≪ M_{s}$ . Note that this logarithmic/power-law running behavior will actually persist all the way into the deep infrared (dashed blue curve) if $M_{lightest}$ is exceedingly small. Finally, as $μ$ approaches $M_{s}$ , the regulator we have employed is no longer valid and thus we cannot explicitly calculate the running. One possible running is sketched (dashed green curve). However, as a general principle, modular invariance requires that the running of the Higgs mass (and indeed the running of any physical quantity) exhibit an invariance under $μ \to M_{s}^{2} / μ$ . Thus, as $μ$ increases beyond $M_{s}$ , we inevitably begin to reenter an IR-like regime which we may associate with a “dual” EFT. The background colors of this sketch indicate the transition from the deep IR (red) to the UV (blue) and then back to IR (red). This symmetry under $μ \to M_{s}^{2} / μ$ implies that the self-dual scale $μ \sim M_{s}$ exhibits the “maximum possible” UV behavior, in the sense that further increases in $μ$ only serve to push the theory back toward IR behavior. Of course, as discussed in Sec. 4a, a fully modular-invariant string theory does not distinguish between the UV and the IR. Indeed, the UV and IR labels in this figure only arise upon attempting to extract a field-theoretic description of our theory, as we are implicitly doing when discussing the “running” of the Higgs mass.
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Figure 4
The scale structure of physical quantities in a modular-invariant string theory. For the modular-invariant regulator function ${\hat{G}}_{ρ} (a, τ_{2})$ discussed in this paper, the mapping between the regulator parameter $ρ a^{2}$ and the physical spacetime scale $μ$ has two distinct branches. The traditional branch (shown in blue) identifies $μ^{2} / M_{s}^{2} = ρ a^{2}$ , but modular invariance implies the existence of an invariance under the scale-inversion duality symmetry $μ \to M_{s}^{2} / μ$ . This in turn implies the existence of a second branch (shown in green) on which we can alternatively identify $μ^{2} / M_{s}^{2} = 1 / (ρ a^{2})$ . Although ${\hat{G}}_{ρ} (a, τ_{2})$ functions as a regulator for $ρ a^{2} < 1$ , its symmetry under $ρ a^{2} \to 1 / (ρ a^{2})$ implies that this function also acts as a regulator when extended into the $ρ a^{2} > 1$ region. This then allows us to see the full fourfold modular structure of the theory. The Higgs-mass plot shown in Fig. 3 can now be understood as following the $μ^{2} / M_{s}^{2} = ρ a^{2}$ branch from the lower-left corner of this figure inward toward the central location at which $μ = M_{s}$ , after which it then follows the $μ^{2} / M_{s}^{2} = 1 / (ρ a^{2})$ branch outward toward the upper-left corner. However, in a modular-invariant theory, all four quadrants of this figure are equivalent and describe the same physics. Likewise, in such theories there is no distinction between IR and UV. Thus one can exchange “IR” $\leftrightarrow$ “UV” within all labels of this sketch, and we have simply chosen to show those labels that have the most natural interpretations within the lower-left quadrant. Finally, regions with beige shading indicate locations where EFT descriptions exist, while stringy effects dominate in the yellow central region. As a result, focusing on any one of the four EFT regions by itself necessarily breaks modular invariance because the choice of EFT region is tantamount to picking a preferred direction for the flow of the scale $μ$ relative to the underlying string-theoretic regulator parameter $ρ a^{2}$ . However, even within the EFT regions, string states at all mass scales contribute nontrivially. Thus even these EFT regions differ from what might be expected within quantum field theory.
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