Universal Landau Pole and Physics below the 100 TeV Scale

TL;DR

This paper explores the idea that all Standard Model gauge couplings could unify at a Landau pole in the ultraviolet, analyzing supersymmetric models with additional matter fields and their potential detectability at future colliders.

Contribution

It provides a comprehensive numerical analysis of a unified Landau pole scenario with supersymmetry and extra matter, including extensions that could be tested at 100 TeV colliders.

Findings

Additional matter or supersymmetry breaking scales are below current LHC bounds.

Certain extensions can raise the new physics scale above LHC reach.

Some models involve new gauge groups that explain the multiplicity of heavy states.

Abstract

We reconsider the possibility that all standard model gauge couplings blow up at a common scale in the ultraviolet. The simplest implementation of this idea assumes supersymmetry and the addition of a single vector-like generation of matter fields around the TeV scale. We provide an up-to-date numerical study of this scenario and show that either the scale of the additional matter or the scale of supersymmetry breaking falls below potentially relevant LHC bounds. We then consider minimal extensions of the extra matter sector that raise its scale above the reach of the LHC, to determine whether there are cases that might be probed at a 100 TeV collider. We also consider the possibility that the heavy matter sector involves new gauge groups constrained by the same ultraviolet boundary condition, which in some cases can provide an explanation for the multiplicity of heavy states. We comment on the relevance of this framework to theories with dark and visible sectors.

Tables

Table 1. Table 1: Numerical results for m V subscript 𝑚 𝑉 m_{V} and Λ Λ \Lambda in the one-generation scenario, the (5,2,0,0) model, and a model whose vector-like sector consists of four 𝟓 + 𝟓 ¯ 5 ¯ 5 {\bf 5}+\overline{{\bf 5}} pairs, the (3,2,4,0) model. These models have the same one-loop beta functions, but differ at two-loop. Also shown are the theoretical error estimates as discussed in the text.

Model	$m_{\text{SUSY}}$ (TeV)	$m_V$ range (TeV)	$\Lambda$ range (GeV)	${\alpha }_3^{-1}\left(m_Z\right)\%$ error	${\sin }^2{\theta }_W\left(m_Z\right)\%$ error
$(5,2,0,0)$	$m_V$	$1.15-1.31$	$7.8-8.7\times {10}^{16}$	$+3.7\%,-2.1\%$	$+1.5\%,-1.5\%$
$(3,2,4,0)$	$m_V$	$0.66-1.16$	$6.9-11\times {10}^{16}$	$+2.8\%,-1.5\%$	$+1.4\%,-1.2\%$

Table 2. Table 2: Solutions for m V subscript 𝑚 𝑉 m_{V} and Λ Λ \Lambda , for a variety of next-to-minimal heavy matter sectors, for m s ​ u ​ s ​ y ≤ m V subscript 𝑚 𝑠 𝑢 𝑠 𝑦 subscript 𝑚 𝑉 m_{susy}\leq m_{V} .

Model	$m_{\text{SUSY}}$ (TeV)	$m_V$ range (TeV)	$\Lambda$ range (GeV)	${\alpha }_3^{-1}\left(m_Z\right)\%$ error	${\sin }^2{\theta }_W\left(m_Z\right)\%$ error
$(5,2,1,0)$	2	$95-260$	$4.9-8.2\times {10}^{16}$	$+4.2\%,-2.8\%$	$+1.5\%,-1.4\%$
	$m_V$	$13-28$	$3.2-5.9\times {10}^{16}$	$+4.0\%,-2.7\%$	$+1.5\%,-1.4\%$
$(3,2,5,0)$	2	$65-217$	$4.9-9.2\times {10}^{16}$	$+3.4\%,-2.2\%$	$+1.4\%,-1.2\%$
	10	$17-32$	$4.1-5.8\times {10}^{16}$	$+3.3\%,-2.2\%$	$+1.4\%,-1.2\%$
	$m_V$	$13-17$	$4.0-4.9\times {10}^{16}$	$+3.3\%,-2.2\%$	$+1.4\%,-1.2\%$
$(3,2,6,0)$	2	$3.8-13\times {10}^3$	$4.3-8.6\times {10}^{16}$	$+3.7\%,-2.7\%$	$+1.4\%,-1.2\%$
	10	$1.2-2.6\times {10}^3$	$3.6-5.6\times {10}^{16}$	$+3.7\%,-2.7\%$	$+1.4\%,-1.2\%$
	30	$522-794$	$3.1-4.0\times {10}^{16}$	$+3.6\%,-2.7\%$	$+1.4\%,-1.2\%$
$(3,2,0,2)$	2	$1.6-1.8\times {10}^4$	$7.1-7.6\times {10}^{16}$	$+5.6\%,-4.1\%$	$+1.5\%,-1.5\%$
	10	$3.0-5.3\times {10}^3$	$4.4-6.1\times {10}^{16}$	$+5.4\%,-3.9\%$	$+1.5\%,-1.5\%$
	100	$277-961$	$2.2-4.5\times {10}^{16}$	$+5.1\%,-3.8\%$	$+1.5\%,-1.5\%$
	$m_V$	$166-370$	$1.9-3.9\times {10}^{16}$	$+5.0\%,-3.7\%$	$+1.5\%,-1.5\%$

Table 3. Table 3: Solutions for m V subscript 𝑚 𝑉 m_{V} and Λ Λ \Lambda for m s ​ u ​ s ​ y > m V subscript 𝑚 𝑠 𝑢 𝑠 𝑦 subscript 𝑚 𝑉 m_{susy}>m_{V} . Of the models in Table  2 , only the ( 3 , 2 , 0 , 2 ) 3 2 0 2 (3,2,0,2) case provides viable solutions.

Model	$m_{\text{SUSY}}$ (TeV)	$m_V$ range (TeV)	$\Lambda$ range (GeV)	${\alpha }_3^{-1}\left(m_Z\right)\%$ error	${\sin }^2{\theta }_W\left(m_Z\right)\%$ error
$(3,2,0,2)$	250	$71-250$	$1.7-2.8\times {10}^{16}$	$+5.0\%,-3.7\%$	$+1.5\%,-1.5\%$
	500	$22-216$	$1.5-3.6\times {10}^{16}$	$+4.9\%,-3.6\%$	$+1.5\%,-1.5\%$
	1000	$7-64$	$1.3-3.1\times {10}^{16}$	$+4.8\%,-3.5\%$	$+1.5\%,-1.5\%$
	1500	$3-31$	$1.2-2.8\times {10}^{16}$	$+4.7\%,-3.5\%$	$+1.5\%,-1.5\%$

Table 4. Table 4: Results for the ( 3 , 2 , 0 , 2 ) 3 2 0 2 (3,2,0,2) scenario with m V = m s ​ u ​ s ​ y subscript 𝑚 𝑉 subscript 𝑚 𝑠 𝑢 𝑠 𝑦 m_{V}=m_{susy} taking into account the effect of the SU(2) X gauge group.

Model	$m_{\text{SUSY}}$ (TeV)	$m_V$ range (TeV)	$\Lambda$ range (GeV)	${\alpha }_3^{-1}\left(m_Z\right)\%$ error	${\sin }^2{\theta }_W\left(m_Z\right)\%$ error
$(3,2,0,2)$	$m_V$	$198-497$	$1.6-3.6\times {10}^{16}$	$+5.6\%,-4.1\%$	$+1.6\%,-1.5\%$

Equations

$${\alpha }_1^{-1}(\Lambda )={\alpha }_2^{-1}(\Lambda )={\alpha }_3^{-1}(\Lambda )=0.$$

$$\frac{d{g}_i}{dt}=\frac{g_i}{16{\pi }^2}\left[b_i{g}_i^2+\frac{1}{16{\pi }^2}\left(\sum _{j=1}^3{b}_{ij}{g}_i^2{g}_j^2-\sum _{j=U,D,E}{a}_{ij}{g}_i^2\text{Tr}[Y_j{Y}_j^{†}]\right)\right],$$

$$b_{i}=\left(\begin{array}[]{c}\frac{53}{5}\\ 5\\ 1\end{array}\right)\,\,\,\,\,\mbox{ and }\,\,\,\,\,b_{ij}=\left(\begin{array}[]{ccc}\frac{977}{75}&\frac{39}{5}&\frac{88}{3}\\ \frac{13}{5}&53&40\\ \frac{11}{3}&15&\frac{178}{3}\end{array}\right),$$

$$b_{i}^{NS}=\left(\begin{array}[]{c}\frac{41}{10}\\ -\frac{19}{6}\\ -7\end{array}\right)\,\,\,\,\,\mbox{ and }\,\,\,\,\,b_{ij}^{NS}=\left(\begin{array}[]{ccc}\frac{199}{50}&\frac{27}{10}&\frac{44}{5}\\ \frac{9}{10}&\frac{35}{6}&12\\ \frac{11}{10}&\frac{9}{2}&-26\end{array}\right).$$

$$\frac{4\pi }{{\alpha }_i^{\overline{\text{MS}}}}=\frac{4\pi }{{\alpha }_i^{\overline{\text{DR}}}}+\frac{1}{3}(C_A{)}_i,$$

$$a_{ij}=\left(\begin{array}{ccc}\frac{26}{5}& \frac{14}{5}& \frac{18}{5}\\ 6& 6& 2\\ 4& 4& 0\end{array}\right)\text{mbox}and{a}_{ij}^{NS}=\left(\begin{array}{ccc}\frac{17}{10}& \frac{1}{2}& \frac{3}{2}\\ \frac{3}{2}& \frac{3}{2}& \frac{1}{2}\\ 2& 2& 0\end{array}\right),$$

$$\frac{d{y}_t}{dt}=\frac{y_t}{16{\pi }^2}\left(-\sum c_i{g}_i^2+6y_t^2\right),c_i=\left(\frac{13}{15},3,\frac{16}{3}\right),$$

$$\frac{d{y}_t}{dt}=\frac{y_t}{16{\pi }^2}\left(-\sum c_i^{\text{SM}}{g}_i^2+\frac{9}{2}{y}_t^2\right),c_i^{\text{SM}}=\left(\frac{17}{20},\frac{9}{4},8\right).$$

$${\alpha }_{\text{EM}}^{-1}(m_Z)=\frac{5}{3}{\alpha }_1^{-1}(m_Z)+{\alpha }_2^{-1}(m_Z),$$

$${\sin }^2{\theta }_W(m_Z)=\frac{3{\alpha }_1(m_Z)}{3{\alpha }_1(m_Z)+5{\alpha }_2(m_Z)}.$$

$${\sin }^2{\theta }_W=0.23129\pm 5\times 10^{-5},{\alpha }_3^{-1}(m_Z)=8.4674\pm 0.0789,$$

$$b_i=\left(\begin{array}{c}2\\ 2\\ 2\end{array}\right)n_g+\left(\begin{array}{c}\frac{3}{10}\\ \frac{1}{2}\\ 0\end{array}\right)n_h+\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)n_5+\left(\begin{array}{c}3\\ 3\\ 3\end{array}\right)n_{10}+\left(\begin{array}{c}0\\ -6\\ -9\end{array}\right),$$

$$b_{ij}=\left(\begin{array}{ccc}\frac{38}{15}& \frac{6}{5}& \frac{88}{15}\\ \frac{2}{5}& 14& 8\\ \frac{11}{15}& 3& \frac{68}{3}\end{array}\right)n_g+\left(\begin{array}{ccc}\frac{9}{50}& \frac{9}{10}& 0\\ \frac{3}{10}& \frac{7}{2}& 0\\ 0& 0& 0\end{array}\right)n_h$$

$$+\left(\begin{array}{ccc}\frac{21}{45}& \frac{9}{5}& \frac{32}{15}\\ \frac{3}{5}& 7& 0\\ \frac{4}{15}& 0& \frac{34}{3}\end{array}\right)n_5+\left(\begin{array}{ccc}\frac{23}{5}& \frac{3}{5}& \frac{48}{5}\\ \frac{1}{5}& 21& 16\\ \frac{6}{5}& 6& 34\end{array}\right)n_{10}+\left(\begin{array}{ccc}0& 0& 0\\ 0& -24& 0\\ 0& 0& -54\end{array}\right),$$

$$b_i^{NS}=\left(\begin{array}{c}\frac{4}{3}\\ \frac{4}{3}\\ \frac{4}{3}\end{array}\right)n_g+\left(\begin{array}{c}\frac{1}{10}\\ \frac{1}{6}\\ 0\end{array}\right)n_h+\left(\begin{array}{c}\frac{2}{3}\\ \frac{2}{3}\\ \frac{2}{3}\end{array}\right)n_5+\left(\begin{array}{c}2\\ 2\\ 2\end{array}\right)n_{10}+\left(\begin{array}{c}0\\ -\frac{22}{3}\\ -11\end{array}\right),$$

$$b_{ij}^{NS}=\left(\begin{array}{ccc}\frac{19}{15}& \frac{3}{5}& \frac{44}{15}\\ \frac{1}{5}& \frac{49}{3}& 4\\ \frac{11}{30}& \frac{3}{2}& \frac{76}{3}\end{array}\right)n_g+\left(\begin{array}{ccc}\frac{9}{50}& \frac{9}{10}& 0\\ \frac{3}{10}& \frac{13}{16}& 0\\ 0& 0& 0\end{array}\right)n_h+\left(\begin{array}{ccc}\frac{7}{30}& \frac{9}{10}& \frac{16}{15}\\ \frac{3}{10}& \frac{49}{6}& 0\\ \frac{2}{15}& 0& \frac{38}{3}\end{array}\right)n_5$$

$$+\left(\begin{array}{ccc}\frac{23}{10}& \frac{3}{10}& \frac{24}{5}\\ \frac{1}{10}& \frac{49}{2}& 8\\ \frac{3}{5}& 3& 38\end{array}\right)n_{10}+\left(\begin{array}{ccc}0& 0& 0\\ 0& -\frac{136}{3}& 0\\ 0& 0& -102\end{array}\right).$$

$$\psi \sim (10,2)\text{mbox}and\overline{\psi }\sim (\overline{10},2).$$

$${\varphi }_1\sim (1,2)\text{mbox}and{\varphi }_2\sim (1,2).$$

$$b_{i}=\left(\begin{array}[]{cccc}\frac{63}{5}&7&3&5\end{array}\right)\,\,\,,$$

$$b_{ij}=\left(\begin{array}[]{cccc}\frac{429}{25}&\frac{33}{5}&\frac{184}{5}&18\\ \frac{11}{5}&67&56&18\\ \frac{23}{5}&21&82&18\\ 6&18&48&53\end{array}\right)\,\,\,.$$

$${\overline{\psi }}_1\sim (\overline{10},1)\text{mbox}and{\overline{\psi }}_2\sim (\overline{10},1).$$

$${\varphi }_1^{\prime }\sim (1,2)\text{mbox}and{\varphi }_2^{\prime }\sim (1,2).$$

Full text

I Introduction

The idea that the three gauge couplings of the standard model may assume a common value at a high energy scale has motivated a vast literature on grand unified theories Langacker:1980js . The particle content of the minimal supersymmetric standard model (MSSM) is consistent with such a unification, with a perturbative unified gauge coupling obtained around $2\times 10^{16}$ GeV. However, it was pointed out long ago Maiani:1977cg ; Cabibbo:1982hy that a different framework also leads to the correct predictions for the gauge couplings at observable energies, namely one in which the gauge couplings blow up at a common scale $\Lambda$ in the ultraviolet (UV):

[TABLE]

Since the SU(3) coupling is asymptotically free, this boundary condition can only be obtained via the introduction of extra matter Cabibbo:1982hy ; Maiani:1986cp ; Moroi:1993zj ; Andrianov:2013irn . Supersymmetric models offer the simplest possibility, a single vector-like generation of mass $m_{V}$ Cabibbo:1982hy ; Maiani:1986cp ; Moroi:1993zj . For a chosen value of $m_{V}$, one may fix the scale $\Lambda$ by the requirement that the low-energy value of the fine structure constant $\alpha_{EM}$ is reproduced; the values of $\sin^{2}\theta_{W}$ and $\alpha_{3}^{-1}$ are then predicted at any chosen renormalization scale $\mu$, up to theoretical uncertainties. If a value of $m_{V}$ can be found in which both $\sin^{2}\theta_{W}(m_{Z})$ and $\alpha_{3}^{-1}(m_{Z})$ are consistent with the data, then a viable solution is obtained. This approach, followed in Ref. Moroi:1993zj , found $m_{V}$ around the TeV scale, assuming that $m_{V}$ is also the scale of supersymmetry breaking (which we call $m_{susy}$ below).

A numerical renormalization group analysis cannot directly encode the boundary condition in Eq. (1) since the gauge couplings are in the non-perturbative regime, where the renormalization group equations (RGEs) cannot be trusted. In Ref. Moroi:1993zj , the boundary condition studied was $\alpha_{1}(\Lambda)=\alpha_{2}(\Lambda)=\alpha_{3}(\Lambda)=10$, values that are barely perturbative. Since the couplings are rapidly increasing as the renormalization scale is increased, one makes the reasonable assumption that the value of $\Lambda$ that satisfies this boundary condition is very close to the one given by Eq. (1). On the other hand, as the renormalization scale is decreased, the couplings become increasingly perturbative. Of particular importance is that the results are insensitive to the precise choice of boundary condition as long as each of the couplings is large Parisi:1974cf . It was shown in Ref. Moroi:1993zj , that varying the $\alpha_{i}(\Lambda)$ by an order of magnitude in either direction has only a small effect on the final results. We will see this explicitly in our study of the one-vector-like-generation scenario in Sec. II. The insensitivity of the predicted values of $\sin^{2}\theta_{W}(m_{Z})$ and $\alpha_{3}^{-1}(m_{Z})$ to the choice of boundary conditions is due to the existence of an infrared fixed point in the renormalization group equation for the ratios of the gauge couplings Ghilencea:1997yr . Note that this insensitivity includes the case where the $\alpha_{i}(\Lambda)$ are taken to be large but not strictly identical at a common high scale.

The possibility that the gauge couplings may have large values in the UV is interesting from a variety of perspectives. Large couplings may arise in strongly coupled heterotic string theories, which often also provide the additional vector-like states necessary to drive the gauge couplings to large values Ghilencea:1997yr . On the other hand, a universal Landau pole, as defined by Eq. (1), may arise in models with composite gauge bosons: compositeness implies the vanishing of the gauge fields’ wave-function renormalization factors at the compositeness scale, where the gauge fields become non-dynamical Eguchi:1974cg . Redefining fields and couplings so that the gauge fields’ kinetic terms are always kept in canonical form, one finds that the vanishing wave-function renormalization factors translate into the blow-up of the gauge couplings at the same scale. Thus, the framework we study may be consistent with a wider range of possible ultraviolet completions than a conventional grand unified theory (GUT) with a large unified gauge coupling, though it is not necessary to commit ourselves to any one of them in order to study the consequences at low energies.

An additional motivation relevant to the present work is that the assumption of a universal Landau pole leads to the expectation of new physics at a calculable energy scale, $m_{V}$, that is above the weak scale but potentially within the reach of future collider experiments111This, of course, assumes that the vector-like matter occurs at a single common scale. This assumption is relaxed in Ref. Andrianov:2013irn .. In Sec. II, we show that the minimal scenario, involving one vector-like generation of additional matter, requires values of either $m_{V}$ or $m_{susy}$ that are below some of the current LHC bounds on vector-like quarks or colored superparticles, respectively. Although experimental bounds come with model-specific assumptions that are usually easy to evade, we pursue an alternative possibility. We show that there are small extensions of the new matter sector that successfully reproduce the correct values of the gauge couplings at $m_{Z}$ while predicting values of $m_{V}$ that are above the reach of the LHC, but below $100$ TeV for some choices of $m_{susy}$. In some cases, $m_{V}$ may be light enough for the vector-like states to be explored at a $100$ TeV hadron collider, which makes study of this sector more interesting. Aside from the presence of the heavy matter fields, one possibility that we also discuss in the present work is that these fields may transform under an additional gauge group factor. The motivation is two-fold: (1) By placing the additional matter fields into irreducible representations of a new gauge group, we might provide an explanation for the multiplicity of states needed to achieve the desired UV boundary condition. In the case where the heavy matter remains vector-like, the new gauge group can be broken at a much lower scale. The resulting low-energy theory is that of a “dark” sector consisting of the new gauge and symmetry breaking fields; the heavy matter provides for communication between the dark and visible sectors, via a “portal” of higher-dimension operators that are induced when the heavy fields are integrated out. The gauge coupling of the dark gauge boson is predicted from a boundary condition analogous to Eq. (1) and the magnitude of the portal couplings are set by the value of $m_{V}$ obtained in the RGE analysis. This presents a simpler framework for constraining some of the otherwise free parameters of a dark sector than, for example, attempting to embed both dark and visible sectors in a conventional GUT. (2) The heavy matter may be chiral under the new gauge group. The structure of the new sector is then more analogous to the the electroweak sector of the MSSM, and the scale $m_{V}$ is associated with one or more massive gauge bosons that may have observable consequences.

Our paper is organized as follows: In Sec. II, we consider the consequences of a universal Landau pole in the minimal case where the MSSM is augmented by a single vector-like generation. The study presented in this section differs from the past literature not only in our use of up-to-date experimental errors for our input parameters, but also in that we allow the scales $m_{V}$ and $m_{susy}$ to vary independently. In addition, we consider an alternative choice for the vector-like matter that contributes the same amount to the beta functions at one loop, but differs from the one-generation scenario at two loops. In Sec. III, we consider extensions of these minimal scenarios, in particular, including a small number of additional complete SU(5) multiplets of vector-like matter. We focus on finding solutions in which $m_{V}$ is less than $100$ TeV, with a special interest in cases where the vector-like matter is light enough to be detected at a future hadron collider. In Sec. IV we consider model building issues associated with the physics at the scale $m_{V}$, focusing on the implication of additional gauge groups. In Sec. V, we summarize our conclusions.

II One vector-like generation

In this section, we consider a minimal scenario studied in the past literature Cabibbo:1982hy ; Maiani:1986cp ; Moroi:1993zj , the MSSM augmented by an additional vector-like generation of matter fields. We denote the scale of the vector-like matter $m_{V}$ and we impose the same boundary conditions as in Ref. Moroi:1993zj , namely $\alpha_{1}(\Lambda)=\alpha_{2}(\Lambda)=\alpha_{3}(\Lambda)=10$ as an approximation to Eq. (1). Taking $m_{V}$ as an input, we determine $\Lambda$ by the condition that the weak scale value of the fine structure constant $\alpha_{EM}(m_{Z})$ is reproduced. With $\Lambda$ fixed, we are now able to determine the gauge couplings at any lower scale, as a function of our choice for $m_{V}$. Above the scale $m_{susy}$, we use the two-loop supersymmetric RGEs for the gauge couplings. Below $m_{susy}$, we do the same using the two-loop nonsupersymmetric RGEs, aside from running between the top quark mass and $m_{Z}$ which we treat as a threshold correction and include at one loop. We assume the presence of the second Higgs doublet required by supersymmetry above the scale $m_{susy}$. Expanding on the approach of Ref. Moroi:1993zj , we do not assume that the scales $m_{V}$ and $m_{susy}$ are the same, though the relaxation of that requirement will only be important in Sec. III.

As indicated in the introduction, the ratios of the gauge couplings are driven towards infrared fixed point values, so that predictions for $\sin^{2}\theta_{W}$ and $\alpha_{3}^{-1}$ at $m_{Z}$ are relatively insensitive to the choice of boundary conditions at the scale $\Lambda$. For example, allowing the $\alpha_{i}(\Lambda)$ to vary independently between 1 and 100, we find that the their weak-scale values scatter within roughly $2\%$ for $\alpha_{1}(m_{Z})$ and $\alpha_{2}(m_{Z})$ and $5\%$ for $\alpha_{3}(m_{Z})$. Given the same variation of boundary conditions, we take the resulting scatter in the values of $\sin^{2}\theta_{W}(m_{Z})$ and $\alpha_{3}^{-1}(m_{Z})$ as a measure of the theoretical uncertainty in our output predictions. We include these estimates with our numerical results.

The RGEs that we use above the top mass have the form

[TABLE]

where $t=\ln\mu$ is the log of the renormalization scale, $\alpha_{i}=g_{i}^{2}/4\pi$, and the $Y_{i}$ are Yukawa matrices. The beta function coefficients $b_{i}$ and $b_{ij}$ can be determined using general formulae Yamada:1993ga ; Machacek:1983tz . For example, in the case of one vector-like generation with $m_{V}=m_{susy}$, one finds for $\mu>m_{V}$

[TABLE]

while for $m_{t}<\mu<m_{V}$ we have the nonsupersymmetric beta functions

[TABLE]

More general forms for the one- and two-loop beta functions that take into account the possibility of additional matter are presented in Sec. III. Note that the gauge couplings for $\mu>m_{susy}$ are defined in the dimensional reduction ($\overline{\text{DR}}$) scheme, which preserves supersymmetry; the couplings are converted to the modified minimal subtraction scheme ($\overline{\text{MS}}$) at the matching scale $\mu=m_{susy}$ before they are run to lower energies. The gauge couplings in the two schemes are related by Antoniadis:1982qw

[TABLE]

where $C_{A}=\{0,2,3\}\text{ for }i=1,2,3$.

The coefficients for the terms that depend on the Yukawa couplings in Eq. (2) are given by

[TABLE]

for $\mu>m_{susy}$ and $\mu<m_{susy}$, respectively. In practice, we only need to take the top quark Yukawa coupling $y_{t}$ into account, since it is significantly larger than the other Yukawa couplings. Since $y_{t}$ affects the running of the gauge couplings only through a two-loop term, we need only include its running at one-loop. For $\mu>m_{susy}$ we have rge

[TABLE]

while for $\mu<m_{susy}$ rge ,

[TABLE]

For definiteness, we assume $\tan\beta=2$, and compute the weak scale value of $y_{t}$ via $y_{t}(m_{Z})=\frac{\sqrt{2}\,m_{t}}{v\sin\beta}$, using the $\overline{{\rm MS}}$ value of the top quark mass, $160^{+5}_{-4}$ GeV pdg , and $v=246$ GeV. The value $y_{t}(\Lambda)$ is computed numerically so that we obtain the desired $y_{t}(m_{Z})$ value for a given set of input parameters. While this approach is sufficient to determine the representative impact of including the top quark Yukawa coupling in our RGE analysis, it turns out to be overkill: in models where the gauge couplings blow up in the UV, the top quark Yukawa coupling is rapidly driven to zero in the same limit. Hence, its effect on the values of $m_{V}$ and $\Lambda$ determined in our numerical analysis turns out to be small, less than the estimates of theoretical uncertainty that we build into the analysis. Although we include it, ignoring $y_{t}$ altogether does not affect our results qualitatively and can be a useful approach for speeding up numerical cross-checks.

For a given choice of $m_{V}$ and $m_{susy}$, the blow-up scale $\Lambda$ is chosen to yield the correct value of the fine structure constant at the weak scale,

[TABLE]

where the factor of $5/3$ comes from the fact that we assume SU(5) normalization g1renorm of the U(1) gauge coupling, as in Ref. Moroi:1993zj . While this makes the analysis compatible with a conventional SU(5) GUT at large coupling, this normalization can also arise directly in string theory without an SU(5) GUT Dienes:1996du . Other normalizations of the U(1) factor are certainly possible, depending on the UV completion. However, we do not consider other possibilities here and adopt the normalization that has been assumed almost uniformly in the past literature. For our numerical study, we take the target central value of $\alpha_{\text{EM}}^{-1}(m_{Z})=127.95$ pdg . With $\Lambda$ determined in this way, we compute $\alpha_{3}(m_{Z})^{-1}$ and the Weinberg angle $\sin^{2}\theta_{W}(m_{Z})$, which is determined by $\alpha_{1}(m_{Z})$ and $\alpha_{2}(m_{Z})$:

[TABLE]

We compare the output predictions of $\alpha_{3}(m_{Z})^{-1}$ and $\sin^{2}\theta_{W}(m_{Z})$, including the theoretical uncertainty that we discussed earlier, to the experimentally measured values pdg

[TABLE]

both given in the $\overline{{\rm MS}}$ scheme. A previous study of the one vector-like generation scenario found viable solutions with $m_{V}=m_{susy}\approx 1$ TeV Moroi:1993zj . Since the time of that work, the experimental errors in $\sin^{2}\theta_{W}(m_{Z})$ and $\alpha_{3}^{-1}(m_{Z})$ have decreased substantially. Nevertheless, as indicated in Table 1, we find $m_{V}=m_{susy}\approx 1.2$ TeV, assuming $\pm 2$ standard deviation experimental error bands and using our protocol for determining theoretical error bands; those bands are both displayed in Fig. 1.

To determine the theoretical error band, we find the maximum and minimum values of $\sin^{2}\theta_{W}(m_{Z})$ and $\alpha_{3}^{-1}(m_{Z})$ that are obtained by varying the $\alpha_{i}$ independently between 1 and 100 at the blow-up scale. In particular, we find that $\sin^{2}\theta_{W}(m_{Z})$ is maximum when $\{\alpha_{1}(\Lambda),\,\alpha_{2}(\Lambda),\,\alpha_{3}(\Lambda)\}=\{100,\,1,\,100\}$ and minimum when the boundary condition set is $\{1,\,100,\,1\}$; $\alpha_{3}^{-1}(m_{Z})$ is maximized and minimized for the sets $\{100,\,100,\,1\}$ and $\{1,\,1,\,100\}$, respectively. We quote the variation in the output predictions as a percentage relative to the value obtained when the $\alpha_{i}(\Lambda)=10$, for $i=1\ldots 3$, in Table 1. For the values of $m_{V}$ that yield viable predictions for $\sin^{2}\theta_{W}(m_{Z})$ and $\alpha_{3}^{-1}(m_{Z})$, we find that the scale $\Lambda$ is around $8\times 10^{16}$ GeV.

The value of $m_{susy}$ for this solution can be compared to recent bounds on gluinos from the LHC, which now exceed $2$ TeV (for example, see Ref. ATLAS:2017cjl ). These bounds generally make assumptions about the supersymmetric particle spectrum (for example, light neutralinos) and one can always play the game of making model-specific adjustments to evade the assumptions of any given experimental exclusion limit. We will not pursue that approach here. We instead consider the possibility that $m_{V}$ and $m_{susy}$ are not identical, so that $m_{susy}$ can be raised unambiguously above the LHC reach. In this case, however, we obtain lower values of $m_{V}$, which in this model would place an entire vector-like generation below $1$ TeV. As a point of comparison, current LHC bounds on a charge-$2/3$ vector-like quark that decays 100% of the time to $bW$ is $1.295$ TeV at the 95% CL CMS:2017wdp . The same comment regarding the limitations of experimental exclusion limits applies here as well; we will be content simply to point out that the one-generation model will become less plausible as time goes on given the increasing reach of LHC searches for superparticles and vector-like quarks.222Unless, of course, some of these particles are discovered.

This result motivates the topic of the next section, extensions of this minimal sector that include sets of new particles that fill complete SU(5) multiplets. We find that these lead to larger values of $m_{V}$. In studies of perturbative gauge coupling unification, it is well known that adding additional matter in complete SU(5) multiplets preserves successful unification. In the present framework, we find viable solutions for $m_{V}$ are also obtained when complete SU(5) multiplets are added. To study the effect on $m_{V}$ and $\Lambda$, we consider adding the smallest SU(5) representations, with dimensions five and ten, allowing for multiple copies. We label models by four numbers $(n_{g},n_{h},n_{5},n_{10})$ which represent the number of chiral generations, complex Higgs doublets, $\bf{5+\overline{5}}$ pairs and $\bf{10+\overline{10}}$ pairs.333It is interesting to note that in level-one string theories with Wilson line symmetry breaking, extra vector-like matter will naturally appear in $\bf{5+\overline{5}}$ and $\bf{10+\overline{10}}$ pairs, since these are representations found in the ${\bf 27+\overline{27}}$ of $E_{6}$ Ghilencea:1997yr . In this notation, the one-vector-like-generation scenario that we have discussed in this section will be called the $(5,2,0,0)$ model henceforth. We note that a model with four $\bf{5+\overline{5}}$ pairs added to the MSSM, the $(3,2,4,0)$ model, has the same one-loop beta functions as the $(5,2,0,0)$ model, and could be considered an equally minimal alternative. Results for the $(3,2,4,0)$ model are also shown in Table 1, and are useful for illustrating the effect of different two-loop beta functions. The preferred range of $m_{V}$ in the $(3,2,4,0)$ model is slightly below that of the $(5,2,0,0)$ model, again pointing to the need for alternative choices for the new matter sector to avoid potential phenomenological difficulties.

III Next-to-minimal possibilities

In this section, we consider vector-like matter sectors that are consistent with values of $m_{susy}$ and $m_{V}$ that are no smaller than $2$ TeV. We look at next-to-minimal scenarios, i.e. ones with a small number of additional ${\bf 5}+\overline{{\bf 5}}$ and ${\bf 10}+\overline{{\bf 10}}$ pairs, for the reasons discussed at the end of the previous section. We have particular interest in solutions that may be plausible for exploration at a $100$ TeV hadron collider. To proceed, we use the results for the one- and two-loop beta functions, derived from the general formulae in Refs. Yamada:1993ga and Machacek:1983tz . In the supersymmetric case, we find

[TABLE]

[TABLE]

while in the nonsupersymmetric case,

[TABLE]

[TABLE]

As indicated earlier, $n_{g},\,n_{h},\,n_{5}\text{ and }n_{10}$ represent the number of chiral generations, Higgs doublets, $\bf{5+\overline{5}}$ and $\bf{10+\overline{10}}$ pairs, respectively. One can check that these formulae reduce to the expected results for the MSSM, where $n_{g}=3,\,n_{h}=2,\,n_{5}=n_{10}=0$ in Eqs. (12) and (13), and for the standard model, where $n_{g}=3,\,n_{h}=1,\,n_{5}=n_{10}=0$ in Eqs. (14) and (15).

Table 2 displays results analogous to those presented for the minimal scenario in Table 1, for a variety of heavy matter sectors, with $m_{susy}\leq m_{V}$. The cases considered fall into pairs that have the same one-loop beta functions; for example, adding one additional ${\bf 5}+\overline{{\bf 5}}$ pair to the one-vector-like generation scenario gives us the $(5,2,1,0)$ model, which has the same $b_{i}$ as a model with five ${\bf 5}+\overline{{\bf 5}}$ pairs, namely $(3,2,5,0)$. The same can be said for the remaining two models, involving six ${\bf 5}+\overline{{\bf 5}}$ and two ${\bf 10}+\overline{{\bf 10}}$ pairs, respectively. Results are shown for values of $m_{susy}$ ranging from $2$ TeV to $m_{V}$. We see that solutions for $m_{V}$ decrease as $m_{susy}$ is increased. Holding $m_{susy}$ fixed, heavy matter sectors that give larger contributions to the one-loop beta functions tend to have larger values of $m_{V}$. Larger collections of heavy matter do not provide additional solutions with $m_{susy}\leq m_{V}$ and $m_{V}<100$ TeV.

Of the cases shown in Table 2, the lowest values of the vector-like matter scale, $m_{V}\approx 13$ TeV, are obtained in the $(5,2,1,0)$ and $(3,2,5,0)$ scenarios, for $m_{V}=m_{susy}$. While vector-like quarks with this mass are within the kinematic reach of a $100$ TeV hadron collider, their detectability is a separate question. Assuming that a $100$ TeV collider has a discovery reach that is greater than that of the LHC by a factor of 5 Arkani-Hamed:2015vfh , and that the LHC’s ultimate sensitivity to vector-like quarks is just below $2$ TeV Bhattacharya:2013iea , one might roughly expect a discovery reach for vector-like quarks at a $100$ TeV hadron collider just below $\sim 10$ TeV. This rough estimate is consistent with the $9$ TeV reach projected in Ref. Golling:2016gvc for fermionic top quark partners, which are also color triplet fermions. These statements are very rough, and a detailed collider study would be required to determine whether the $13$ TeV vector-like quarks in the $(5,2,1,0)$ and $(3,2,5,0)$ models would have observable consequences at a $100$ TeV machine.

Fortunately, we find that if the supersymmetry-breaking scale is raised above the scale $m_{V}$, the reduction in $m_{V}$ continues. Interestingly, however, we only find the correct predictions for the gauge couplings at the weak scale in the $(3,2,0,2)$ model. Although a higher $m_{susy}$ indicates that supersymmetry is less effective at addressing the hierarchy problem, one could still argue that this case has its merits: (1) supersymmetry still ameliorates the hierarchy problem between $m_{susy}$ and $\Lambda$, which are the scales with the widest separation in the models that we consider, and (2) supersymmetry may be expected if string theory is the UV completion, whether or not supersymmetry has anything to do with solving the hierarchy problem. From a purely phenomenological perspective, taking $m_{susy}>m_{V}$ brings the $(3,2,0,2)$ heavy matter sector down into the range where it might be directly probed. In Table 3, we present numerical results for that case. As the supersymmetry breaking scale increases from $250$ TeV to $1500$ TeV, the minimum allowed values of $m_{V}$ decrease from $71$ TeV to $3$ TeV. It seems more likely in this case that the vector-like matter could be within the discovery reach of a $100$ TeV hadron collider, while all the superpartners remain undetectable. It is interesting to note that it is easiest in the $(3,2,0,2)$ model to incorporate an additional gauge group that acts on the heavy matter sector, a topic we turn to in the next section.

IV Model building issues

The results of the previous section indicate that there are values of $m_{V}$ implied by Eq. (1) that are beyond the reach of the LHC, but may be within the reach of future collider experiments, particularly in the case where the supersymmetry breaking scale exceeds the scale $m_{V}$. Aside from the extra matter fields, other physics associated with this sector might also be experimentally probed. In this section, we consider two motivations for including an extra gauge group that only affects the heavy fields: (1) The heavy fields may fall in irreducible representations of the new gauge group, explaining the multiplicity of new particles required to achieve the blow up of the couplings at the scale $\Lambda$, and (2) the new sector may be chiral under the new gauge groups, rendering it more analogous in structure to the matter sector of the MSSM. Although there are a large number of ways in which either possibility might arise, we consider one example here, based on the $(3,2,0,2)$ model discussed in the previous section.

Regarding the first motivation, we consider the possibility that the duplication of vector-like ${\bf 10}+\overline{{\bf 10}}$ pairs in the $(3,2,0,2)$ model is a result of their embedding into a two-dimensional representation of an additional gauge group, which is necessarily non-Abelian. The simplest possibility for the gauge group structure of the model is $G_{SM}\times$SU(2)X, where $G_{SM}$ represents the standard model gauge factors. As before, we indicate the standard model charge assignments implicitly and compactly by displaying the SU(5) multiplets that the heavy matter fields would occupy in a conventional unified theory, even though that is not our assumption. Hence under SU(5)$\times$SU(2)X, we now assume that the extra matter is given by

[TABLE]

We also introduce two SU(2)X doublet Higgs fields that will be responsible for spontaneously breaking the new gauge group factor

[TABLE]

The matter fields in Eq. (16) and the new Higgs fields in Eq. (17) are separately vector-like, so that these fields may be made massive at any desired scale; it also follows that all chiral gauge anomalies are canceled. Note that the multiplicity of SU(2) doublets in Eqs. (16) and (17) is even, which implies that the SU(2)X Witten anomaly is absent. Given these assignments, the one-loop beta function for the new gauge factor is positive, allowing for straightforward implementation of the UV boundary condition in Eq. (1).

One issue that needs to be addressed in a model like this one is the stability of the extra matter fields. Vector-like ${\bf 5}+\overline{{\bf 5}}$ and ${\bf 10}+\overline{{\bf 10}}$ pairs have the appropriate electroweak and color quantum numbers to participate in mass mixing with standard model matter fields. The amount of such mixing is arbitrary, and only a small amount is necessary so that the heavy states are rendered unstable, avoiding any cosmological complications. Assigning the matter fields of the heavy sector to multiplets of a new gauge group can have unwanted consequences if these states are rendered exactly stable (or extremely long lived). In the present model, this problem does not arise provided that the new gauge group is spontaneously broken, since mass mixing is generated via renormalizable couplings involving $\psi$, the $\phi_{i}$, and the standard model fields identified with a ${\bf 10}$. If embedding in an additional gauge group is used to account for the multiplicity of states in some of the other models that we have considered, the model must also provide for the decay of the heavy states; the $(3,2,0,2)$ models seem to naturally avoid this problem with smallest field content and the potentially simplest symmetry-breaking sector, which is one reason why we focus on this example here.

Note that the numerical results for the $(3,2,0,2)$ model described in Sec. III must be adjusted to take into account the presence of the SU(2)X gauge group, whose coupling blows up at the same scale as the other gauge couplings and affects their renormalization group running. However, since the effect is only via two-loop terms, we don’t expect a dramatic change in our qualitative conclusions. To support this statement, we consider the case where $m_{susy}=m_{V}$ and take into account the effect of the new gauge group by modifying the supersymmetric RGEs for running between the scales $\Lambda$ and $m_{V}$. In this case, the supersymmetric beta functions become

[TABLE]

[TABLE]

Repeating the analysis of Sec. III, we find only a modest adjustment in the ranges for $m_{V}$ and $\Lambda$, as shown in Table 4 below.

It is interesting to note that SU(2)X breaking scale is not tied to the value of $m_{V}$ in this model, which means it could in principal be much lower. For example, with $\langle\phi\rangle\sim 1$ GeV, the resulting low-energy effective theory would be that of a non-Abelian dark sector with a one- or two-Higgs doublet symmetry-breaking sector. Communication between the visible and dark sectors would follow from operators generated when the $m_{V}$-scale physics is integrated out, suggesting that this sector may have other interesting consequences besides its effect on gauge coupling running. Whether phenomenologically interesting models of this type can be constructed remains an open question.

Finally, we note that a different motivation for an extra gauge factor is to render the $m_{V}$-scale physics chiral, so that the structure of the new matter sector is more similar to the rest of the MSSM. In the previous example, we could simply change the charge assignment of $\overline{\psi}$ to

[TABLE]

Now the mass terms for the extra matter are generated via Yukawa couplings involving $\psi$, $\overline{\psi}$ and the $\phi_{i}$; the vacuum expectation value $\langle\phi\rangle$ is now associated with the scale $m_{V}$ determined in the RGE analysis. We make one additional modification to the theory, which is to add an additional pair of Higgs fields

[TABLE]

The modification in Eq. (20) leads to the vanishing of the one-loop beta function for SU(2)X, while Eq. (21) restores the desired asymptotic non-freedom. Based on our earlier observations, it is clear that the numerical values for $m_{V}$ and $\Lambda$ in this model will be qualitatively similar to those of the other $(3,2,0,2)$ models that we have considered, and we leave further numerical study for the interested reader.

V Conclusions

In this paper, we have revisited the possibility that the standard model gauge couplings reach a common Landau pole in the ultraviolet. This provides a predictive framework for relating the values of the gauge couplings at the weak scale, without the necessary assumption of conventional grand unification. To implement this framework, all the gauge couplings must be asymptotically non-free, which implies that new matter must be included in the theory. We have numerically explored the possibility that this new matter appears at two scales, the scale of supersymmetry breaking, $m_{susy}$, and the scale where additional vector-like states appear, $m_{V}$. We have revisited a scenario considered in the past in which the minimal supersymmetric standard model is enlarged by a single vector-like generation and found that either $m_{susy}$ or $m_{V}$ falls below potentially relevant LHC lower bounds on colored MSSM superparticles or vector-like quarks. Although one cannot rule out the possibility that these states are present and have evaded detection for model-specific reasons, we are motivated to consider a safer possibility: we include a relatively small additional amount of extra heavy matter, which leads to solutions for $m_{V}$ that are beyond the reach of the LHC, but potentially within the reach of a higher-energy hadron collider. For example, given a heavy sector consisting in total of five ${\bf 5}+\overline{{\bf 5}}$ pairs, we obtain successful gauge coupling predictions for $m_{susy}=m_{V}\approx 13$ TeV. For a heavy sector of two ${\bf 10}+\overline{{\bf 10}}$ pairs, we can achieve $m_{V}$ as low as $3$ TeV, if we allow higher values of $m_{susy}\approx 1500$ TeV.

We also considered whether the size of the new matter sector could be related to its embedding into the irreducible representation of an additional non-Abelian gauge group. We presented the simplest model that was consistent with our numerical solutions, a model with two ${\bf 10}+\overline{{\bf 10}}$ pairs, in which this duplication is due to their embedding in the fundamental representation of a new SU(2) gauge group. In the case where the heavy matter sector is vector-like under the new SU(2), the new gauge group can be broken at a much lower scale and the effective theory is that of a spontaneously broken non-Abelian dark sector. In the case where the heavy matter sector is chiral under the new SU(2), $m_{V}$ is associated with the symmetry breaking scale. In this case, new heavy gauge bosons would be among the spectrum of particles that might be sought at a future collider with a suitable reach.

Acknowledgements.

This work was supported by the NSF under Grant PHY-1519644.