Running of gauge couplings in string theory

Open Access

Running of gauge couplings in string theory

Steven Abel, Keith R. Dienes, and Luca A. Nutricati

Phys. Rev. D 107, 126019 – Published 21 June 2023

Abstract

In this paper we conduct a general, model-independent analysis of the running of gauge couplings within closed string theories. Unlike previous discussions in the literature, our calculations fully respect the underlying modular invariance of the string and include the contributions from the infinite towers of string states which are ultimately responsible for many of the properties for which string theory is famous, including an enhanced degree of finiteness and $UV / IR$ mixing. In order to perform our calculations, we adopt a formalism that was recently developed for calculations of the Higgs mass within such theories, and demonstrate that this formalism can also be applied to calculations of gauge couplings. In general, this formalism gives rise to an “on-shell” effective field theory (EFT) description in which the final results are expressed in terms of supertraces over the physical string states, and in which these quantities exhibit an EFT-like “running” as a function of an effective spacetime mass scale. We find, however, that the calculation of the gauge couplings differs in one deep way from that of the Higgs mass: while the latter results depend on purely on-shell supertraces, the former results have a different modular structure which causes them to depend on off-shell supertraces as well. In some regions of parameter space, our results demonstrate how certain expected field-theoretic behaviors can emerge from the highly $UV / IR$ -mixed environment. In other situations, by contrast, our results give rise to a number of intrinsically stringy behaviors that transcend what might be expected within an effective field theory approach.

Received 26 March 2023
Accepted 27 April 2023

DOI:https://doi.org/10.1103/PhysRevD.107.126019

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)

String phenomenology

Particles & Fields

Authors & Affiliations

Steven Abel ^1,2,*, Keith R. Dienes ^3,4,†, and Luca A. Nutricati ^1,‡

¹IPPP and Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, United Kingdom
²CERN, Theoretical Physics Department, CH 1211 Geneva 23 Switzerland
³Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
⁴Department of Physics, University of Maryland, College Park, Maryland 20742, USA

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

Article Text

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Issue

Vol. 107, Iss. 12 — 15 June 2023

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Figure 1
Distinct steps associated with our analysis of an arbitrary physical quantity, as discussed in the text. The particular sequence of steps to be followed depends on the goal of the analysis. Particularly relevant results are those in the lower portions of this sketch, in which our physical quantity is expressed purely in terms of contributions from on-shell, physical string states and also in a form which runs as a function of a spacetime mass scale $μ$ .
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Figure 2
The infinite sums $f_{β} (τ_{2})$ in Eq. (3.64), plotted as functions of $τ_{2}$ for $0 \leq β \leq 6$ . The $β = 0$ curve is the upper blue curve which asymptotes to 1 as $τ_{2} \to \infty$ , while the $β = 1, 2, \dots, 6$ curves all asymptote to 0 as $τ_{2} \to \infty$ and can be identified according to the increasing values that they have at any fixed large value of $τ_{2}$ (with $n = 1$ orange, $n = 2$ green, $n = 3$ red, etc.). As $τ_{2}$ drops to zero from a large value, we see that our functions $f_{β} (τ_{2})$ do not diverge (as would have been expected as we slowly remove the $τ_{2}$ cutoff), but instead remain within the bounds indicated in Eq. (3.62), leading to finite values for $f_{β} (τ_{2})$ as $τ_{2} \to 0$ .
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Figure 3
String states arranged as a matrix according to their right-moving (vertical) and left-moving (horizontal) worldsheet energies $(m, n) \equiv (p - r, p)$ , respectively. The states with $r = 0$ lie along the principal diagonal, while the states with $r = 1, 2, \dots$ lie along successive shifted diagonals. The requirement $r \geq 0$ selects only those string states along or above the principal diagonal, and the $τ \to τ + 1$ invariance of the partition function ensures that only those “squares” shaded in green with $m - n \in Z$ can be populated. The pink square is necessarily empty in any tachyon-free theory, and the row shaded in orange is excluded for heterotic strings because such strings have right-moving worldsheet energies $\geq - 1 / 2$ . In drawing this figure we have assumed that states populate only integer or half-integer mass levels, but in general the spectrum of states can be far denser and may even approach a continuum in $(m, n)$ [or equivalently in $p$ ] for exceedingly large or small compactification radii.
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Figure 4
The function $ϕ_{5} (M, μ)$ in Eq. (3.124) and the corresponding contribution to the beta function $β_{Δ}$ in Eq. (3.137), plotted as functions of $\log_{10} (μ / M)$ (green and blue, respectively). Note that $ϕ_{5} (M, μ)$ is the Bessel-function contribution to ${\hat{Δ}}_{G} (μ)$ per unit $A^{(2)}$ charge from a given physical bosonic string state of nonzero mass $M$ . When $μ ≪ M$ , the state is effectively integrated out of the theory, whereupon the running contribution $ϕ_{5} (M, μ)$ asymptotes to a constant. However, at larger energy scales $μ ≫ M$ , the state is fully dynamical and produces a running which is effectively logarithmic. Finally, within the intermediate $μ \sim M$ region, the Bessel-function expression for $ϕ_{5} (M, μ)$ in Eq. (3.124) provides a smooth connection between these two asymptotic behaviors and even gives rise to a transient “hump” in the value of ${\hat{Δ}}_{G} (μ)$ , or equivalently a “dip” in the value of the running coupling $g_{G} (μ)$ .
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Figure 5
The one-loop running of the inverse gauge coupling ${\hat{Δ}}_{G} = 16 π^{2} / g_{G}^{2}$ for any gauge group $G$ , as calculated from first principles in a fully modular-invariant string framework. The tree-level contribution is sketched in red, and the total one-loop coupling is sketched in green. In the deep IR, the coupling approaches an asymptotic value which receives contributions from all of the states in the string spectrum which carry nontrivial helicity $A^{(1)} \sim {\bar{Q}}_{H}^{2}$ charges. This assumes that our theory contains no net massless states charged under $A^{(2)} + B^{(2)} \sim ({\bar{Q}}_{H}^{2} - 1 / 12) Q_{G}^{2}$ where $Q_{G}^{2}$ is the sum of the squares of the charges in the Cartan subalgebra of $G$ ; otherwise ${\hat{Δ}}_{G} (μ)$ diverges in the IR limit. Moving towards higher values of $μ$ , we see that a nontrivial scale dependence does not emerge until $μ \sim M_{lightest}$ , where $M_{lightest}$ collectively represents the masses of the lightest massive states which are charged under $A^{(2)} + B^{(2)}$ . The nonmonotonic “dip” in $g_{G}$ (or “hump” in ${\hat{Δ}}_{G}$ ) that is observed in this region is a transient effect which smoothly connects the asymptotic deep-IR region $μ ≪ M_{lightest}$ to an EFT-like region $M_{lightest} ≲ μ ≪ M_{s}$ . Beyond the dip region, the theory then enters an EFT-like region in which the gauge coupling experiences a logarithmic running. As $μ \to M_{s}$ , it is possible that we might cross the energy threshold $R^{- 1}$ associated with large compactification radii. In such cases, this logarithmic running can be modified by the appearance of Kaluza-Klein and winding states which might appear at mass scales significantly below $M_{s}$ and which might tend to cancel this logarithmic running, leading to the existence of a higher-dimensional fixed-point regime, as shown. The subtleties involved in this behavior will be discussed further in Ref. [21]. However, as a general principle, modular invariance requires that the running of ${\hat{Δ}}_{G}$ exhibit an invariance under $μ \to M_{s}^{2} / μ$ . Thus, as $μ$ increases beyond $M_{s}$ , the theory inevitably begins to reenter an IR-like regime which we may associate with a “dual” EFT, followed by a dual dip region and then a dual deep-IR region. The background colors of this sketch indicate the transition from the deep IR (red) to the UV (blue) and then back to IR (red). As such, there is a maximum degree to which our theory can approach the UV: once the energy scale $μ$ passes the self-dual point $μ \sim M_{s}$ , further increases in $μ$ only push us towards increasingly IR behavior. The quantity $κ$ is defined in Eq. (3.139).
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