Vector-like Fermions and Higgs Effective Field Theory Revisited

TL;DR

This paper performs a comprehensive analysis of vector-like quark models, linking experimental constraints with Higgs physics and EFT parameters, revealing significant restrictions on these models.

Contribution

It provides a global fit of VLQ model parameters using precision data, explores their connection to Higgs and Zb interactions, and constrains these models within an EFT framework.

Findings

VLQ models are highly constrained by current data

Analytical formulas for S and T parameters are provided

Connection between Zb interactions and Higgs physics is analyzed

Abstract

Heavy vector-like quarks (VLQs) appear in many models of beyond the Standard Model physics. Direct experimental searches require these new quarks to be heavy, $\gsim$ 800-1000 GeV. We perform a global fit of the parameters of simple VLQ models in minimal representations of $SU(2)_L$ to precision data and Higgs rates. An interesting connection between anomalous $Z b {\overline {b}}$ interactions and Higgs physics in VLQ models is discussed. Finally, we present our analysis in an effective field theory (EFT) framework and show that the parameters of VLQ models are already highly constrained. Exact and approximate analytical formulas for the $S$ and $T$ parameters in the VLQ models we consider are posted at https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/vlq_17/ as Mathematica files.

Tables

Table 1. Table 1: Left-handed fermion –  W 𝑊 W couplings as defined in Eq. ( 2.14 ). We assume all couplings are real, and neglect the SM CKM angles.

Model	$A_{tb}^L$	$A_{tB}^L$	$A_{TB}^L$	$A_{Tb}^L$	$A_{XT}^L$	$A_{Xt}^L$	$A_{BY}^L$	$A_{bY}^L$
$T_s$	$c_L^t$			$s_L^t$
$B_s$	$c_L^b$	$s_L^b$
${\psi }_{XT}$	$c_L^t$			$s_L^t$	$c_L^t$	$-s_L^t$
${\psi }_{TB}$	$c_L^t{c}_L^b+s_L^t{s}_L^b$	$c_L^t{s}_L^b-s_L^t{c}_L^b$	$c_L^t{c}_L^b+s_L^t{s}_L^b$	$s_L^t{c}_L^b-c_L^t{s}_L^b$
${\psi }_{BY}$	$c_L^b$	$s_L^b$					$c_L^b$	$-s_L^b$
${\psi }_{XTB}$	$c_L^t{c}_L^b+\sqrt{2}{s}_L^t{s}_L^b$	$c_L^t{s}_L^b-\sqrt{2}{s}_L^t{c}_L^b$	$s_L^t{s}_L^b+\sqrt{2}{c}_L^t{c}_L^b$	$s_L^t{c}_L^b-\sqrt{2}{c}_L^t{s}_L^b$	$\sqrt{2}{c}_L^t$	$-\sqrt{2}{s}_L^t$
${\psi }_{TBY}$	$c_L^t{c}_L^b+\sqrt{2}{s}_L^t{s}_L^b$	$c_L^t{s}_L^b-\sqrt{2}{s}_L^t{c}_L^b$	$s_L^t{s}_L^b+\sqrt{2}{c}_L^t{c}_L^b$	$s_L^t{c}_L^b-\sqrt{2}{c}_L^t{s}_L^b$			$\sqrt{2}{c}_L^b$	$-\sqrt{2}{s}_L^b$

Table 2. Table 2: Right-handed fermion –  W 𝑊 W couplings as defined in Eq. ( 2.14 ). We assume all couplings are real.

Model	$A_{tb}^R$	$A_{tB}^R$	$A_{TB}^R$	$A_{Tb}^R$	$A_{XT}^R$	$A_{Xt}^R$	$A_{BY}^R$	$A_{bY}^R$
$T_s$
$B_s$
${\psi }_{XT}$					$c_R^t$	$-s_L^t$
${\psi }_{TB}$	$s_R^t{s}_R^b$	$-s_R^t{c}_R^b$	$c_R^t{c}_R^b$	$-c_R^t{s}_R^b$
${\psi }_{BY}$							$c_R^b$	$-s_R^b$
${\psi }_{XTB}$	$\sqrt{2}{s}_R^t{s}_R^b$	$-\sqrt{2}{s}_R^t{c}_R^b$	$\sqrt{2}{c}_R^t{c}_R^b$	$-\sqrt{2}{c}_R^t{s}_R^b$	$\sqrt{2}{c}_R^t$	$-\sqrt{2}{s}_R^t$
${\psi }_{TBY}$	$\sqrt{2}{s}_R^t{s}_R^b$	$-\sqrt{2}{s}_R^t{c}_R^b$	$\sqrt{2}{c}_R^t{c}_R^b$	$-\sqrt{2}{c}_R^t{s}_R^b$			$\sqrt{2}{c}_R^b$	$-\sqrt{2}{s}_R^b$

Table 3. Table 3: Diagonal Higgs couplings to fermions.

Model	$c_{bb}$	$c_{BB}$	$c_{tt}$	$c_{TT}$
$T_s$	$m_b$	-	$m_t{\left(c_L^t\right)}^2$	$M_T{\left(s_L^t\right)}^2$
$B_s$	$m_b{\left(c_L^b\right)}^2$	$M_B{\left(s_L^b\right)}^2$	$m_t$	-
${\psi }_{XT}$	$m_b$	-	$m_t{\left(c_R^t\right)}^2$	$M_T{\left(s_R^t\right)}^2$
${\psi }_{TB}$	$m_b{\left(c_R^b\right)}^2$	$M_B{\left(s_R^b\right)}^2$	$m_t{\left(c_R^t\right)}^2$	$M_T{\left(s_R^t\right)}^2$
${\psi }_{BY}$	$m_b{\left(c_R^b\right)}^2$	$M_B{\left(s_R^b\right)}^2$	$m_t$	-
${\psi }_{XTB}$	$m_b{\left(c_L^b\right)}^2$	$M_B{\left(s_L^b\right)}^2$	$m_t{\left(c_L^t\right)}^2$	$M_T{\left(s_L^t\right)}^2$
${\psi }_{TBY}$	$m_b{\left(c_L^b\right)}^2$	$M_B{\left(s_L^b\right)}^2$	$m_t{\left(c_L^t\right)}^2$	$M_T{\left(s_L^t\right)}^2$

Table 4. Table 4: EFT coefficients for VLQ models in the large VLQ mass limit.

	$C_{Ht}$	$C_{Hb}$	$C_{Hq}$	$C_{Hq}^s$	$C_{HY}^b$	$C_{HY}^t$	$C_{Htb}$
$T$	0	0	$\frac{{\lambda }_1^2}{4M_T^2}$	$-\frac{{\lambda }_1^2}{4M_T^2}$	0	$\frac{{\lambda }_t{\lambda }_1^2}{2M_T^2}$	0
$B$	0	0	$-\frac{{\lambda }_2^2}{4M_B^2}$	$-\frac{{\lambda }_2^2}{4M_B^2}$	$\frac{{\lambda }_b{\lambda }_2^2}{2M_B^2}$	0	0
$(T,B)$	$-\frac{{\lambda }_4^2}{2M_T^2}$	$\frac{{\lambda }_5^2}{2M_T^2}$	0	0	$\frac{{\lambda }_b{\lambda }_5^2}{2M_T^2}$	$\frac{{\lambda }_t{\lambda }_4^2}{2M_T^2}$	$\frac{{\lambda }_4{\lambda }_5}{M_T^2}$
$(X,T)$	$\frac{{\lambda }_3^2}{2M_X^2}$	0	0	0	0	$\frac{{\lambda }_t{\lambda }_3^2}{2M_X^2}$	0
$(B,Y)$	0	$-\frac{{\lambda }_6^2}{2M_B^2}$	0	0	$\frac{{\lambda }_b{\lambda }_6^2}{2M_B^2}$	0	0
$(X,T,B)$	0	0	$\frac{3{\lambda }_7^2}{4M_T^2}$	$\frac{{\lambda }_7^2}{4M_T^2}$	$\frac{{\lambda }_7^2{\lambda }_b}{M_T^2}$	$\frac{{\lambda }_7^2{\lambda }_t}{2M_T^2}$	0
$(T,B,Y)$	0	0	$-\frac{3{\lambda }_8^2}{4M_B^2}$	$-\frac{{\lambda }_8^2}{4M_B^2}$	$\frac{{\lambda }_8^2{\lambda }_b}{2M_B^2}$	$\frac{{\lambda }_8^2{\lambda }_t}{M_B^2}$	0

Equations

$${\psi }_L^0=\left(\begin{array}{c}{t}_L^0\\ b_L^0\end{array}\right),t_R^0,b_R^0,$$

$$H=\left(\begin{array}{c}{\varphi }^{+}\\ {\varphi }^0\end{array}\right),$$

$$-L_{Y,SM}={\lambda }_t{\overline{\psi }}_L^0\tilde{H}{t}_R^0+{\lambda }_b{\overline{\psi }}_L^{0}H{b}_R^0+h.c.,$$

$$\mathrm{Singlets}:$$

$$\mathrm{Doublets}:$$

$$\mathrm{Triplets}:$$

$${\mathcal{T}}_{L,R}^0=$$

$$-L_M={\overline{\mathcal{T}}}_L^0{M}^t{\mathcal{T}}_R^0+{\overline{\mathcal{B}}}_L^0{M}^b{\mathcal{B}}_R^0+M_Y{\overline{Y}}_{t,L}{Y}_{t,R}+M_X{\overline{X}}_{t,L}{X}_{t,R}.$$

$${\mathcal{T}}_{L,R}$$

$${\mathcal{B}}_{L,R}$$

$$V_{L,R}^{t,b}=\left(\begin{array}{cc}\cos {\theta }_{L,R}^{t,b}& -\sin {\theta }_{L,R}^{t,b}\\ \sin {\theta }_{L,R}^{t,b}& \cos {\theta }_{L,R}^{t,b}\end{array}\right),$$

$$M_{diag}^t=V_L^t{M}^t(V_R^t{)}^{†}=\left(\begin{array}{cc}{m}_t& 0\\ 0& M_T\end{array}\right),M_{diag}^b=V_L^b{M}^b(V_R^b{)}^{†}=\left(\begin{array}{cc}{m}_b& 0\\ 0& M_B\end{array}\right).$$

$$\mathrm{Doublets}$$

$$\mathrm{Triplets}$$

$$M_{T,B}\tan {\theta }_R^{t,b}$$

$$M_{T,B}\tan {\theta }_L^{t,b}$$

$$B\mathrm{singlet}:$$

$$L_W$$

$${\mathcal{L}}_Z$$

$$X_{ii}^L$$

$$\begin{array}[]{lcllcl}\delta X_{bb}^{L}&=&(s_{L}^{b})^{2}(I_{3}^{B}+1)&\qquad\delta X_{bb}^{R}&=&(s_{R}^{b})^{2}I_{3}^{B}\\ \delta X_{tt}^{L}&=&(s_{L}^{t})^{2}(I_{3}^{T}-1)&\qquad\delta X_{tt}^{R}&=&(s_{R}^{t})^{2}I_{3}^{T}\\ \delta X_{BB}^{L}&=&-1+(c_{L}^{b})^{2}(I_{3}^{B}+1)&\qquad\delta X_{BB}^{R}&=&(c_{R}^{b})^{2}I_{3}^{B}\\ \delta X_{TT}^{L}&=&1+(c_{L}^{t})^{2}(I_{3}^{T}-1)&\qquad\delta X_{TT}^{R}&=&(c_{R}^{t})^{2}I_{3}^{T}\\ \delta X_{XX}^{L}&=&\delta X_{XX}^{R}=I_{3}^{X}&\qquad\delta X_{YY}^{L}&=&\delta X_{YY}^{R}=I_{3}^{Y}\,.\end{array}$$

$$X_{ij}^L$$

$$\delta X_{bB}^L$$

$$\delta X_{tT}^L$$

$$L=-\frac{h}{v}{\overline{f}}_L^i{c}_{ij}{f}_R^j+h.c..$$

$$c_{ij}=V_L\mathcal{F}{V}_L^{†}{M}_{diag}$$

$$c_{ij}=M_{diag}{V}_R\mathcal{F}{V}_R^{†},$$

$$\mathcal{F}\equiv \left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right).$$

$$T$$

$${\theta }_{+}(y_1,y_2)$$

$${\theta }_{-}(y_1,y_2)$$

$$\Delta T$$

$$T_{SM}$$

Full text

I Introduction

The Standard Model (SM) has been remarkably successful at explaining both precision measurements and LHC data and so the possibilities for heavy, as yet unobserved particles are highly restricted by the experimental results. Here, we focus on new heavy quarks and their impact on electroweak scale physics. Heavy SM-like chiral fermions are excluded by the measured Higgs production rates Anastasiou:2011qw ; Anastasiou:2016cez . Therefore, we consider heavy vector-like quarks (VLQs), which are typically compatible with Higgs measurements. Motivated by the excellent agreement of Higgs measurements with SM predictions ATLAS-CONF-2015-044 , we assume that the observed Higgs boson is part of an $SU(2)_{L}$ doublet, $H$, and consider VLQs which can couple to $H$. This class of VLQs occurs in many composite Higgs models Anastasiou:2009rv ; Contino:2010rs ; Contino:2006qr ; Agashe:2005dk ; Gillioz:2012se ; Panico:2015jxa and little Higgs models ArkaniHamed:2002qy ; Han:2005ru ; Hubisz:2004ft and hence is well motivated phenomenologically. The phenomenology of VLQs has been considered in some detail in the literature Lavoura:1992np ; Burgess:1993vc ; delAguila:1998tp ; AguilarSaavedra:2002kr ; Cacciapaglia:2010vn ; Dawson:2012di ; Aguilar-Saavedra:2013qpa ; Ellis:2014dza ; Angelescu:2015kga ; Fajfer:2013wca ; Alok:2015iha ; Alok:2014yua and direct experimental searches ATLAS-CONF-2016-101 ; ATLAS-CONF-2016-102 ; ATLAS-CONF-2016-013 ; Aad:2016shx ; Aad:2016qpo ; Aad:2015voa ; Aad:2015kqa ; Aad:2015mba ; Khachatryan:2015gza ; Khachatryan:2015axa ; Chatrchyan:2013wfa ; Chatrchyan:2013uxa ; ATLAS-CONF-2017-015 require them to be heavy, with $M\gtrsim{\cal{O}}(800-1000)$ GeV.

We update previous fits Dawson:2012di ; Aguilar-Saavedra:2013qpa ; Ellis:2014dza to the parameters of VLQ models by performing a joint fit to the oblique parameters and asymmetries in the $b$ quark sector. The study is extended to include restrictions from Higgs coupling measurements with interesting results found in models containing a $B$ VLQ.

We briefly review the set-up of the VLQ models that we study in Section II. Section III reviews the contributions of VLQs to the oblique parameters and the $Zb{\overline{b}}$ coupling. We find that in some regions of parameter space the leading contributions to the oblique parameters can be quite small even with significant mass splittings between the VLQ multiplet members, due to numerical cancellations. We discuss the effects of these regions on the global fits to VLQ parameters. Section IV contains numerical fits and we present some conclusions in Section V. Appendix A contains a pedagogical description of the triplet models, which should be useful for model builders. The connection between our results in the full VLQ theories and in an EFT approach for heavy VLQ masses is given in Appendix B. Exact and approximate analytical formulas for the oblique parameters in the various models can be found at https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/vlq_17/.

II Vector-Like Quark Basics

II.1 Basics

In this section, we introduce our notation for VLQs. We consider the case where the VLQs interact only with the third generation quarks, since mixing with the first two generations is highly restricted by kaon and other low energy physics measurements AguilarSaavedra:2002kr . We indicate the SM weak eigenstate quarks as,

[TABLE]

and the Higgs doublet as,

[TABLE]

with $\phi^{0}={v+h\over\sqrt{2}}$. The SM Yukawa couplings are,

[TABLE]

where ${\tilde{H}}=i\sigma_{2}H^{*}$.

The models we consider have vector-like quarks in the $SU(2)_{L}$ representations,

[TABLE]

This is a complete set of VLQ representations that have renormalizable couplings to the SM Higgs doublet. The quarks have electric charge $Q_{T}={2\over 3},~{}Q_{B}=-{1\over 3},~{}Q_{X}={5\over 3}$, and $Q_{Y}=-{4\over 3}$. If there is only one VLQ representation, it is simple to write the most general CP conserving couplings between the SM fermions, the VLQs, and the Higgs boson,111This can be straightforwardly generalized to models with more than one VLQ representation Cacciapaglia:2015ixa ; Angelescu:2015kga ; Ellis:2014dza .

[TABLE]

Note that we do not include mixing between SM fermions and VLQs with identical quantum numbers since these terms can be rotated away by redefinitions of the fields. The singlet and doublet models have been extensively discussed in the literature Aguilar-Saavedra:2013qpa ; Ellis:2014dza ; Dawson:2012di ; Burgess:1993vc ; Lavoura:1992np ; Cacciapaglia:2010vn ; Angelescu:2015kga ; delAguila:1998tp ; AguilarSaavedra:2002kr , and we include a useful discussion of the details of the triplet model in Appendix A.

The gauge eigenstate fields can be written in general as,

[TABLE]

where $T^{0}=T^{0}_{s},T^{0}_{d}$ or $T^{0}_{t}$ and $B^{0}=B^{0}_{s},B^{0}_{d}$ or $B^{0}_{t}$ (the $X$ and $Y$ fields do not mix with the other fermions and are therefore also mass eigenstates). The terms contributing to the mass matrices are found from Eq. 2.5 and we write them as,

[TABLE]

We denote the mass eigenstate fields as $(t,T)$ and $(b,B)$ and they are found through bi-unitary transformations,

[TABLE]

where

[TABLE]

For simplicity of notation we abbreviate $\cos\theta_{L}^{t}\equiv c_{L}^{t}$, etc. Through these rotations we obtain the diagonal mass matrices

[TABLE]

There are relationships between the angles and mass eigenstates that depend on the representation (see for example Aguilar-Saavedra:2013qpa ),

[TABLE]

and

[TABLE]

Examples of the derivation of these relations are given in Appendix A for the case of vector triplets.

Except for the $(TB)$ doublet model, there are sufficient relationships that the results can always be expressed in terms of two parameters. For our numerical fits, we take as input parameters,

[TABLE]

The couplings to the $W$ boson are,

[TABLE]

where $q^{i},q^{j}$ are any two quarks in the model for which $Q(q^{i})-Q(q^{j})=1$. The values of $A_{ij}^{L,R}$ in the VLQ models we consider are reported in Tabs. 1 and 2.

The neutral current couplings to the $Z$ boson are also modified. The couplings for $f_{i,j}=t,b,T,B,X,Y$ are,

[TABLE]

where $s_{W}=\sin\theta_{W}$ is the weak mixing angle. The SM couplings are normalized such that $X_{ij}^{L}=\delta_{ij}$ for $i=t$ and $X_{ij}^{L}=-\delta_{ij}$ for $i=b$ , with all other $X$ equal to [math]. For multiplets containing a heavy charge $-{1\over 3}$ quark with isospin $I_{3}^{B}$ that mixes with the SM-like $b$ quark or a heavy charge ${2\over 3}$ quark with isospin $I_{3}^{T}$ that mixes with the SM-like $t$ quark, the diagonal fermion couplings to the $Z$ are222$I_{3}=(2,0,-2)$ for triplets, $(1,-1)$ for doublets, [math] for singlets.,

[TABLE]

where the $I^{3}_{i}$ term in the left-handed couplings survive only for the top and bottom quarks, and

[TABLE]

The off-diagonal couplings to the $Z$ boson are,

[TABLE]

where

[TABLE]

Finally, the couplings to the Higgs boson can be parameterized as,

[TABLE]

The flavor non-diagonal fermion-Higgs couplings are important for double Higgs production Dawson:2012mk ; Gillioz:2012se and can be found in Ref. Aguilar-Saavedra:2013qpa . For models with a singlet or triplet VLQ,

[TABLE]

and for models with a doublet VLQ,

[TABLE]

where

[TABLE]

These formulae hold for both the charge ${2\over 3}$ and charge $-{1\over 3}$ sectors. The $X$ and $Y$ fermions do not couple to the Higgs. The diagonal Higgs couplings are given in Table 3.

III VLQ Contributions to Precision Measurements

Electroweak precision data place strong restrictions on the parameters of models with VLQs. In this section, we review the contributions to the oblique parameters and the $Zb{\overline{b}}$ couplings in the VLQ models introduced in the previous section.

III.1 Oblique Parameters

The general expression for the contribution to the $T$ parameter from fermions is Lavoura:1992np ; Anastasiou:2009rv ; Chen:2003fm

[TABLE]

where $N_{c}=3$, $y_{i}\equiv{M_{Fi}^{2}\over M_{Z}^{2}}$, $M_{F_{i}}$ are the fermion masses, and $A^{L,R}_{ij}$, $X^{L,R}_{ij}$ are defined in Eqs. 2.14 and 2.15 respectively. For the input parameters we use Olive:2016xmw $m_{t}=173.5$ GeV, $m_{b}=4.2$ GeV, $m_{Z}=91.1876~{}{\rm GeV},m_{W}=80.385~{}{\rm GeV}$ and define the weak angle through $c_{W}={m_{W}\over m_{Z}}$.

The functions $\theta_{\pm}(y_{i},y_{j})$ are,

[TABLE]

We note that $\theta_{+}(y,y)=\theta_{-}(y,y)=0$. When $y_{1}>>y_{2}$, $\theta_{+}(y_{1},y_{2})\xrightarrow[]{y_{1}>>y_{2}}y_{1}$ and $\theta_{-}(y_{1},y_{2})\xrightarrow[]{y_{1}>>y_{2}}0$. We will make use of these properties as we compute all electroweak parameters in the limit $m_{b}<<m_{t}$.

As customary, we subtract the SM top-bottom contribution,

[TABLE]

where

[TABLE]

For the top and bottom singlet partner models, the exact results are simple Lavoura:1992np ; Dawson:2012di

[TABLE]

where $r_{F}\equiv{M_{F}^{2}\over m_{t}^{2}}$. The contribution to the $T$ parameter in the $(BY)$ doublet model also has a simple expression:

[TABLE]

In the large VLQ mass and small mixing angle limits, we obtain simple approximate expressions for the $T$ parameter for all the VLQ representations333 Exact results are posted at https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/vlq_17/ as a mathematica notebook.,444In all our studies we will use the exact expressions for the Peskin-Takeuchi parameters, retaining the full dependence on the VLQs masses and on the mixing angles.

[TABLE]

[TABLE]

The contributions to $\Delta T$ in the various VLQ models are shown in Fig. 1 (Fig. 2 for the $(TB)$ doublet, which has two mixing angles as free parameters). Here we use the exact expressions for the $T$ parameter. For small mixing, the contribution to $\Delta T$ is positive in the $T$ singlet and $(XTB)$ triplet models, negative in the $B$ singlet, $(XT)$ doublet and $(TBY)$ triplet models, and extremely small in the $(BY)$ doublet model (RHS of Fig. 1), as one could expect from the approximate results in Eq. LABEL:tdef_approx. In all the models where the $T$ parameter is negative for small mixing $\Delta T$ changes sign at an intermediate value of $\sin\theta$ and therefore vanishes again for non-small mixing. In the case of the $(XT)$ doublet, $\Delta T^{XT}\sim 0$ even for $s_{R}^{t}\sim 1$, due to a numerical cancellation. Therefore, in these models there could be regions of parameter space with quite sizeable mixing that are allowed by precision tests. We will explore this possibility in Sec. IV.

The mass splitting between the VLQ multiplet components, $\delta_{Q_{1}Q_{2}}\equiv M_{Q_{1}}-M_{Q_{2}}$, is fixed by the mixing angles (Eq. 2.11). In the large VLQ mass and small mixing angle approximation, and in the limit for massless bottom quark,

[TABLE]

[TABLE]

From Eq. 3.11, in all cases ${\delta\over M}\sim\sin^{2}\theta_{i}$, so for small angles $\Delta T$ grows with the mixing between the SM fermions and the VLQs. For large masses, the mixing goes to zero for fixed Yukawa couplings (see Table 3), and decoupling is recovered Dawson:2012di .

The expression for the contributions to the $S$ parameter from fermions is Lavoura:1992np ; Carena:2007ua ; Chen:2003fm

[TABLE]

where we subtract the SM top-bottom contribution,

[TABLE]

The functions appearing in $S$ are Lavoura:1992np ,

[TABLE]

where $\chi_{+}(y,y)=\chi_{-}(y,y)=0$ and in the limit $y_{1}>>y_{2}$,

[TABLE]

For the singlet bottom and top VLQ models, the exact results (full mass and angle dependence) are

[TABLE]

As in the case of the $T$ parameter, the $(BY)$ doublet model also has a simple exact expression for $S$,

[TABLE]

The contributions to $\Delta S$ are shown in Figs. 3 and 4 using the exact results (full mass and angle dependence) in all the models.

In the heavy VLQ mass limit (and assuming small mixings between the doublet and triplet components),555 Exact results are posted at https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/vlq_17/ as a mathematica notebook.

[TABLE]

[TABLE]

The $B$ singlet, $(XT)$ doublet, and $(TBY)$ triplet models have the interesting feature that $\Delta T$ vanishes for particular fine-tuned choices of the parameters with non-zero mass splittings between the members of the VLQ multiplets. In the left panel of Fig. 5 we show the VLQ mass and mixing angle for which $\Delta T=0$ and in the right panel we show $\Delta S$ for these parameters.

The oblique fit, ignoring correlations, requires $\Delta S<0.3$ at $95~{}\%$ confidence level deBlas:2016ojx ; Baak:2014ora , so there are regions where both $\Delta T$ and $\Delta S$ can escape the oblique constraints in these models. These fined-tuned regions will have important impacts on the global fits in the next section and we note that the mass splittings between VLQ multiplet members can be significant for these choices of parameters. Fig. 6 shows the mass difference for the points where $\Delta T$ is fine-tuned to be zero, corresponding to the mixing angles of Fig. 5.

III.2 Contributions to $Zb{\overline{b}}$

In the VLQ models where $\delta X_{bb}^{L}$ vanishes at tree level, we will use the one-loop contributions to the left-handed $Zb{\overline{b}}$ coupling for our fit to electroweak precision data. This occurs in the $(TB)$ doublet model (Eq. 2.17), as well as in the $T$ singlet and $(XT)$ doublet models. The one-loop corrections from $t-T$ mixing to $\delta X_{bb}^{L}$ are Burgess:1993vc ; Anastasiou:2009rv ,

[TABLE]

where $x=m_{t}^{2}/M_{W}^{2}$, $x^{\prime}=M_{T}^{2}/M_{W}^{2}$ and the SM contribution has been subtracted. In the limit $x,x^{\prime}>>1$,

[TABLE]

IV Numerical Restrictions on VLQs

The properties of VLQ models are restricted by $Zb{\overline{b}}$, oblique parameter, and Higgs coupling measurements. In this section, we perform global fits to $Zb{\overline{b}}$ and oblique parameter data and demonstrate that Higgs coupling measurements are not competitive with the limits from the electroweak parameters.

The experimental constraints from the right-handed $Zb{\overline{b}}$ coupling are considerably weaker than those from the left-handed coupling, so we consider only right-handed couplings, $\delta X_{bb}^{R}$, that arise at tree level. On the other hand, if the left-handed coupling is zero at tree level, we include the loop corrections from $t-T$ mixing using the $1-$loop results of Refs. Burgess:1993vc ; Anastasiou:2009rv reported in Eq. 3.21. In the VLQ models where $\delta X_{bb}^{R}=0$ at tree level, ($T$ and $B$ singlet, $(XT)$ doublet, $(TBY)$ triplet), we use the $3-$parameter fit to $\Delta S$, $\Delta T$ and $\delta X_{bb}^{L}$ from Ref. deBlas:2016ojx 666$\delta X_{bb}^{L}=2\delta g_{L}^{b}$ in the notation of Ref. deBlas:2016ojx .. In addition, $\delta X_{bb}^{R}\sim 0$ at tree level also in the $(XTB)$ triplet model in the limit $m_{b}<<M_{B}$ (Eq. 2.12), and it vanishes in the $(TB)$ doublet model when we fix $s_{R}^{b}=0$. In all these cases we use the $3-$parameter fit,

[TABLE]

with the correlation matrix,

[TABLE]

In the $(BY)$ model we have non-zero values for $\Delta S$, $\Delta T$ and $\delta X_{bb}^{R}$. In the small bottom-mass limit $\delta X_{bb}^{L}$ will be extremely suppressed and one can neglect it. Indeed, for $m_{b}\to 0$ the left-handed $Zb\bar{b}$ coupling is zero at tree level. The one-loop contributions vanish as well, since all the electroweak couplings of the bottom quark are proportional to $s_{L}^{b}$, which in this limit goes to zero (Eq. 2.12). The $3-$parameter fit we use in this case is,

[TABLE]

with the correlation matrix,

[TABLE]

When both $\delta X_{bb}^{L}$ and $\delta X_{bb}^{R}$ are non-zero, we use the $4-$parameter fit of Ref. deBlas:2016ojx to $\Delta S$, $\Delta T$, $\delta X_{bb}^{R}$ and $\delta X_{bb}^{L}$. For a massless $b$ quark, this case only occurs in the $(TB)$ doublet model, where $\delta X_{bb}^{L}$ arises at one loop. The $4-$parameter fit is,

[TABLE]

with the correlation matrix,

[TABLE]

We perform a $\chi^{2}$ fit,

[TABLE]

where $O_{i}$ are the measured observables ($\Delta S$, $\Delta T$, $\delta X_{bb}^{L}$, $\delta X_{bb}^{R}$), $O_{i}^{fit}$ are their predicted values in the different VLQ models, and $\sigma^{2}_{ij}=\sigma_{i}\rho_{ij}\sigma_{j}$, where $\sigma_{i}$ are the uncertainties in Eqs. 4.1, 4.3 and 4.5. The correlation matrices are given in Eqs. 4.2, 4.4 and 4.6. In each model, we scan over the parameters to obtain the $95\%$ confidence level limits. All the models but the $(TB)$ doublet have two independent degrees of freedom (see Eq. 2.13). Also in the $(TB)$ model we will analyse two specific scenarios, one with $s_{R}^{b}$ fixed, and one with $M_{T}$ fixed. Therefore, in all cases the number of degrees of freedom is two, and we require $\Delta\chi^{2}<5.99$.

Our results for the regions of parameter space allowed by the electroweak precision observables are shown in Figs. 7, 8 and 9, where we use the exact results for the oblique parameters777The exact result for $\Delta S$ in the $(XT)$ doublet model shows numerical instabilities in the small angle region. Hence, we have used an expansion up to ${\cal O}\left((s_{R}^{t})^{16}\right)$ for $s_{R}^{t}<$ 0.2. At the matching point, the exact result is stable and the difference with the expanded one is below the percent level..

Ref. Aguilar-Saavedra:2013qpa showed limits from the oblique parameters and the $Zb{{\overline{b}}}$ couplings separately, and our fit results are roughly consistent with theirs, although the experimental constraints have tightened somewhat. For the $T$, $B$ singlet models, the $(BY)$ doublet model and the triplet models, the limits on the mixing angles are quite stringent and for large VLQ masses relatively independent of the VLQ mass itself (Fig. 7).

The $(XT)$ doublet model has an interesting region seen in Fig. 8 (black dotted area), where the contribution to $\Delta T$ vanishes, allowing relatively large values of the mixing angle. This region is consistent with the $\Delta T\sim 0$ region of Fig. 1 for $M_{T}=1~{}$TeV.

Also in the $(TB)$ doublet model (Fig. 9) we find an interesting region with relative large mixings both in the top and bottom sectors allowed by the fit. Doing a global fit strengthens the bounds in the $(TB)$ and $(XTB)$ models relative to those of Ref. Aguilar-Saavedra:2013qpa . Models with $B$ VLQs Gillioz:2013pba are allowed by the fits, with a relatively large mixing angle permitted in the $(BY)$ doublet model. The strongest limit on models with $B$ VLQs occurs in the $B$ singlet case, where for all $M_{B}$, the global fit requires $s_{L}^{b}<0.04$ due to the strong dependence of $\delta X^{L}_{bb}$ on the mixing angle (Eq. 2.17). We note that for large VLQ masses, the fits asymptote to an approximately constant mixing angle in each case. This suggests that the value of the VLQ mass is not critical and that an effective field theory (EFT) approach is warranted. We discuss the EFT approach for heavy VLQs in Appendix B.

We have presented our results in terms of the masses and mixing angles given in Eq. 2.13. Using Eq. 3.11, we redisplay our fit results in terms of the allowed mass differences between members of the VLQ multiplets. In Fig. 10, we demonstrate that the maximum allowed mass differences are of ${\cal{O}}(1-3~{}{\rm GeV})$, except for the $(BY)$ and $(TB)$ doublet models, where mass differences of ${\cal{O}}(5-10~{}{\rm GeV})$ are consistent with precision electroweak data.

The mixing of SM and vector-like quarks also changes the $tbW$ couplings, $A_{tb}^{L,R}$. The limits from 7 and 8 TeV data from $t-$channel single top production Khachatryan:2014iya ; Aguilar-Saavedra:2013qpa are however not yet competitive with the precision electroweak limits.

Finally, in the VLQ models Higgs production and decay rates are modified. The Higgs signal strengths for the gluon fusion production channel using the $95\%$ confidence level results at 8 TeV are ATLAS-CONF-2015-044 ,

[TABLE]

The production rate $gg\rightarrow h$ and the decays $h\rightarrow gg$ and $h\rightarrow\gamma\gamma$ are affected by the VLQ contributions through loops of heavy quarks and changes in the SM quarks Yukawa couplings, while the $h\rightarrow b\bar{b}$ decay is modified at tree level.

The contribution to the Higgs signal strength from colored fermions is well known. At leading order Gunion:1989we ,

[TABLE]

where $c_{ff}$ are the Higgs-fermion couplings defined in Eq. 2.20, $m_{f}$ is the mass of the corresponding quark, $c_{ff}^{SM}=m_{f}$, $\tau_{f}={m_{h}^{2}\over 4M_{f}^{2}}$, and

[TABLE]

In the heavy fermion mass limit $F_{F}(x)\rightarrow{4\over 3}$, while for light quarks $F_{F}(\tau_{b})\rightarrow 0$. Therefore in the limit of massless $b$ quark and infinitely heavy $(t,T,B)$ quarks, the leading order Higgs production rate is independent of the fermion masses,

[TABLE]

The deviations of the gluon fusion production rate, $\mu_{ggF}$, are directly related to deviations in the $b$ couplings,

[TABLE]

We observe that in all cases the presence of heavy $B$ VLQs ${\it{increases}}$ the Higgs signal strength.

For the decay width to photons we have,

[TABLE]

where $f_{SM}$ includes all SM fermions, $Q_{f}$ and $N_{c}$ are charge and color of the fermion, $F_{F}(x)$ is defined in Eq. 4.10 and

[TABLE]

Modifications of Higgs signal strength for the various VLQs are shown in Figs. 11, 12, 13 and 14, where we define,

[TABLE]

The $B$ and $T$ singlet and $(BY)$ $XT$ doublet models are so highly constrained by the electroweak fits, that the deviations in Higgs production are too small to be observed. The $(TB)$ doublet model can have modest increases in Higgs signal strengths when the mixing in the right-handed $b$ sector is allowed to be significant (RHS of Fig. 13). In the $(XTB)$ triplet triplet model, an increase of about $10\%$ in the $h\rightarrow\gamma\gamma$ signal strength is consistent with the results from the electroweak fits, while the other Higgs decay channels are constrained to be within about $4\%$ of the SM predictions.

V conclusions

We have considered restrictions on the parameters of models with vector-like quarks and updated electroweak fits to the parameters of these models. The constraints on VLQ masses and mixings are strengthened from previous fits. Mixing in the $B$ VLQ sector is highly constrained due to the tree-level effect on the $Zb\bar{b}$ coupling, while mixings up to $s_{R}^{t}\sim 0.2$ are allowed in the $T$ singlet case. In the doublet models mixings up to $s_{R}\sim 0.1-0.15$ are allowed, with an interesting region of $0.3\lesssim s_{R}^{t}\lesssim 0.6$ in the $(XT)$ doublet scenario and non-zero mixing allowed in both the top and bottom sectors in the $(TB)$ doublet model. In the triplet models the mixing is somewhat more constrained, $s_{L}^{t}\lesssim 0.7$. Finally, we show that in order for Higgs coupling measurements to probe regions beyond those excluded by precision fits, measurements of a few $\%$ will be required.

Acknowledgements

We would like to thank G. Panico for useful discussion. SD thanks B. Jaeger and the University of Tuebingen where this work was begun and Fermilab where it was finally completed for their hospitality. This work is supported by the U.S. Department of Energy under grant DE-SC0012704, by the Swiss National Science Foundation (SNF) under contracts 200021-165772 and 200021-160814, and by the Advanced ERC Grant Pert QCD. The work of C.-Y.C is supported by NSERC, Canada. Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT. Digital data related to our results can be found at  https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/vlq_17/

Appendix A VLQ Triplet Lagrangian

In the following, the fields are the current eigenstates, but for simplicity of notation we shall omit the superscript “0”.

To establish our normalization convention, we shall use

[TABLE]

where $\psi_{L}$ is the top-bottom left-handed doublet, $q_{R}=\{t_{R},b_{R}\}$ are the right-handed singlets, $W_{\mu}=W_{\mu}^{a}\frac{\sigma^{a}}{2}$ ($a=\{1,2,3\}$) and $\sigma^{a}$ are the Pauli matrices. They satisfy

[TABLE]

The gauge boson interactions therefore read

[TABLE]

where $I^{3(SM)}_{q},Y^{(SM)}_{q}$ are the eigenvalues of the quark $q=\{t_{L,R},b_{L,R}\}$ under $\sigma_{3}$ and the $U(1)_{Y}$ generator respectively,

[TABLE]

This yields the correct charges for the top and bottom quarks (first line of eq. (A.3)). In the second line we replaced the hypercharge quantum number with the corresponding charge of the quark as derived from the first line.

As a reminder, the physical gauge bosons are defined by

[TABLE]

and $e=gs_{W}=g^{\prime}c_{W}$.

A.1 Vector Triplets

Introducing the fields

Let $\rho^{0}$ be a fermionic field that transforms as a triplet under $SU(2)_{L}$, i.e.888 Recall that the triplet vector field $W_{\mu}$ transforms as

$W_{\mu}\rightarrow U\left(W_{\mu}+\frac{i}{g}\partial_{\mu}\right)U^{-1}\;.$

A triplet fermionic field has a similar transformation law, up to the shift term.

[TABLE]

Here $\alpha^{a}(x)$ are the gauge transformation parameters and $\tau^{a}=\frac{\sigma^{a}}{2}$. For simplicity of notation, from now on we will omit the subscript“0” in the fermionic fields and use $U\equiv U(x)$.

Note that we can decompose fermions in a $SU(2)_{L}$ triplet on the basis of Pauli matrices as

[TABLE]

As for the charged gauge bosons, we introduce

[TABLE]

and we can use that

[TABLE]

Let us remind ourselves that the charges of $\tau^{1}\pm i\tau^{2}$ (i.e. $\rho^{\pm}$ and $W^{\pm}$) are $\pm 1$ respectively,

[TABLE]

Guage invariance and the Lagrangian

From the transformation laws of the gauge and fermion fields, the gauge-invariant covariant derivative must be defined as999We want $D_{\mu}\rho\rightarrow U\left(D_{\mu}\rho\right)U^{-1}$, and we need to pick the same normalization conventions as those for the Standard Model.

[TABLE]

The Lagrangian, imposing also the correct normalization of the kinetic term and adding the $U(1)_{Y}$ part, is then

[TABLE]

Couplings of the fermions to the EW gauge bosons

Let us recall that in the normalization we chose the $\{X,T,B\}$ triplet has isospin $I^{3}_{\rho_{XTB}}=\{2,0,-2\}$, $Y_{\rho_{XTB}}=\frac{2}{3}$, and the $\{T,B,Y\}$ triplet has $I^{3}_{\rho_{TBY}}=\{2,0,-2\}$, $Y_{\rho_{TBY}}=-\frac{1}{3}$. The couplings to the electroweak gauge bosons are easily derived from eq. (Guage invariance and the Lagrangian) via the replacements

[TABLE]

Neutral couplings: the photon

The photon couplings allow us to determine the electric charge of the three quarks. From the Lagrangian (Guage invariance and the Lagrangian), with the definition of the gauge boson fields of the Standard Model, one gets

[TABLE]

Hence

• •

for the triplet of hypercharge $Y=\frac{2}{3}$, $\rho^{+}\equiv X$ has charge 5/3, $\rho^{3}\equiv T$ has charge 2/3, and $\rho^{-}\equiv B$ has charge -1/3;

• •

for the triplet of hypercharge $Y=-\frac{1}{3}$, $\rho^{+}\equiv T$ has charge 2/3, $\rho^{3}\equiv B$ has charge -1/3, and $\rho^{-}\equiv Y$ has charge -4/3,

as we expect.

Neutral couplings: the Z boson

The couplings of the quarks in the vector triplet to the $Z$ boson are

[TABLE]

This is perfectly consistent with what one expects from the Standard Model case (hypercharge minus twice the electric charge).

Charged couplings

The couplings to the charged gauge bosons are

[TABLE]

and similarly for the $Y=-\frac{1}{3}$ triplet. Notice that these couplings are a factor $\sqrt{2}$ larger than the Standard Model couplings (eq. (A.3)).

A.2 Physical couplings

To obtain the physical fields we follow the procedure described in eqs. (2.6 - 2.9)

$Y=\frac{2}{3}$ triplet $(X,T,B)$

The physical couplings to the electroweak gauge bosons read (cfr. eqs. (A.3), (Neutral couplings: the Z boson), (Charged couplings))

[TABLE]

This is in agreement with the results of Ref. (Aguilar-Saavedra:2013qpa, ).

We also show here how one derives the relations among masses and angles of eq. (2.11) for the $(X,T,B)$ triplet. Starting from the Lagrangian (2.5), the bare mass matrices are

[TABLE]

Let us notice that

[TABLE]

The condition

[TABLE]

yields

[TABLE]

Next, one can “reconstruct” the square bare mass matrix, both in the top and bottom sector, inverting eq. (A.18). Imposing that the entry $(2,2)$ is the same and equals $M_{XTB}^{2}=M_{X}^{2}$, we get

[TABLE]

Finally, using

[TABLE]

and noticing that the entries $(1,2)$ of these matrices are related,

[TABLE]

yields

[TABLE]

$Y=-\frac{1}{3}$ triplet $(T,B,Y)$

The physical couplings to the electroweak gauge bosons read (cfr. eqs. (A.3), (Neutral couplings: the Z boson), (Charged couplings))

[TABLE]

The top and bottom mass matrices are (eq. (2.5))

[TABLE]

Following the same proof that lead to eqs. (A.19) and (A.20), we obtain

[TABLE]

Similarly, the equivalent of eq. (A.23) is

[TABLE]

Appendix B EFT Coefficients and limits from $b$ and Higgs Couplings

Searches for VLQs at the LHC suggest that the masses are relatively heavy, $M\gtrsim{\cal O}(800-1000)$ GeV ATLAS-CONF-2016-101 ; ATLAS-CONF-2016-102 ; ATLAS-CONF-2016-013 ; Aad:2016shx ; Aad:2016qpo ; Aad:2015voa ; Aad:2015kqa ; Aad:2015mba ; Khachatryan:2015gza ; Khachatryan:2015axa ; Chatrchyan:2013wfa ; Chatrchyan:2013uxa ; ATLAS-CONF-2017-015 . This means that we are always in the regime,

[TABLE]

where an effective field theory approach is warranted. The Lagrangian involving third generation SM quarks and VLQs can be written as,

[TABLE]

where $L_{Y,SM}$ is defined in Eq. 2.3, $L_{Q}$ contains the VLQ interactions given in Eq. 2.5 and $L_{KE}$ is the kinetic energy term. At tree level, the heavy VLQs can be integrated out using the equations of motion AguilarSaavedra:2009mx ; Chen:2014xwa ; Delaunay:2013iia , generating an effective low-energy Lagrangian that only contains SM fields,

[TABLE]

where $L_{KE}^{\prime}$ now includes only the SM quarks and

[TABLE]

contains the effective interactions of the SM quarks with the gauge and Higgs boson through higher dimensional operators (we restrict to dimension-6 operators). We have normalized the coefficients of these operators to be ${\cal{O}}\left({1\over M^{2}}\right)$. The new Higgs-fermion dimension-6 operators are delAguila:2000aa :

[TABLE]

To ${\cal{O}}({1\over M^{2}})$, the coefficients of Eq. B.4 are given in Table 4 in terms of the Yukawa couplings and we assume the splitting between the VLQ masses in a given representation are small, corresponding to small mixing angles. The different VLQ representations have quite different patterns for the coefficients delAguila:2000aa ; Chen:2014xwa ; Batell:2012ca .

These operators generate non-SM interactions of the fermions with the gauge and Higgs bosons. The interactions with the $W$ boson defined in Eq. 2.14 become in the EFT limit,

[TABLE]

and the couplings to the $Z$ boson defined in Eq. 2.15 are,

[TABLE]

From Table 4, we see that right-handed $W$ couplings are only generated in the $(TB)$ model, while non-standard $W_{L}$ couplings arise in the singlet and triplet models. In a similar fashion, the doublet VLQ models have SM couplings of the $Z$ boson to the top and bottom quarks. Measuring the gauge boson fermion couplings puts strong constraints on the possible VLQ representations.

Finally, the $t,b$ couplings to the Higgs boson are also modified,

[TABLE]

corresponding to,

[TABLE]

For the singlet and triplet models, we have the interesting relation between the $Zf{\overline{f}}$ couplings and the Higgs Yukawa coupling,

[TABLE]

For non-zero $C_{HY}^{f}$, the Yukawa coupling is no longer proportional to the mass, leading to flavor non-diagonal Higgs-fermion interactions Harnik:2012pb ; Altmannshofer:2015esa .