Probing an additional bottom Yukawa coupling via $bg\to b A \to b Z H$ signature

TL;DR

This paper proposes a method to directly search for an additional bottom Yukawa coupling in a two Higgs doublet model at the LHC, which could have implications for electroweak baryogenesis.

Contribution

It introduces a novel search strategy for the extra bottom Yukawa coupling $ ho_{bb}$ via specific processes at the LHC, highlighting potential discovery prospects.

Findings

The $bg o b A o b Z H$ process could be discovered with ~300 fb$^{-1}$ at the LHC for $m_A ext{ around } 300$ GeV.

The $gg o b ar b A o b ar b Z H$ process may be observable at the high luminosity LHC.

Discovery of $ ho_{bb}$ could provide insights into electroweak baryogenesis.

Abstract

The recent discovery of the bottom quark Yukawa coupling ($hbb$) of the 125 GeV scalar motivates one to search for extra bottom Yukawa coupling that may exist in the nature. The two Higgs doublet model without a discrete $Z_2$ symmetry allows the possibility of additional bottom Yukawa coupling $\rho_{bb}$. We show that $\rho_{bb}$ can be searched directly at the LHC via $bg\to b A \to b Z H$ and $gg \to b \bar b A \to b \bar b Z H$ processes, where $A$ and $H$ are the CP-odd and CP-even scalars respectively. We find that the $bg\to b A \to b Z H$ process could be discovered with $\sim300$ fb$^{-1}$ integrated luminosity if $ m_A \sim 300$ GeV, while the latter process may emerge in the high luminosity LHC (HL-LHC) data. A discovery might touch upon the parameter space required for the electroweak baryogenesis.

Tables

Table 1. Table 1: Parameter values for the three benchmark points. See text for details.

BP	${\eta }_1$	${\eta }_2$	${\eta }_3$	${\eta }_4$	${\eta }_5$	${\eta }_6$	${\eta }_7$	$m_{H^{\pm }}$	$m_A$	$m_H$	$\frac{{\mu }_{22}^2}{v^2}$
								(GeV)	(GeV)	(GeV)
I	0.258	1.151	2.78	-1.557	-0.831	0	-0.236	335	301	200	0.46
II	0.258	2.155	2.496	-0.625	-1.885	0	0.375	349	400	214	0.76
III	0.258	2.63	2.142	-0.333	-2.436	0	-0.134	426	495	312	1.92

Table 2. Table 2: The branching ratios of A 𝐴 A for the three benchmark points given in Table  1 .

$\left|{\rho }_{bb}\right|$	BP	$ZH$	$b\overline{b}$
	I	0.618	0.382
0.1	II	0.047	0.953
	III	0.05	0.95

Table 3. Table 3: The cross sections (in fb) for the different background contributions of the b ​ Z ​ H 𝑏 𝑍 𝐻 bZH process after selection cuts at s = 14 𝑠 14 \sqrt{s}=14 TeV LHC. The subdominant backgrounds 4 ​ t 4 𝑡 4t , t ​ t ¯ ​ W 𝑡 ¯ 𝑡 𝑊 t\bar{t}W , t ​ W ​ h 𝑡 𝑊 ℎ tWh , t ​ W ​ Z 𝑡 𝑊 𝑍 tWZ and W ​ Z 𝑊 𝑍 WZ +jets are added together and denoted as “Others” in the second last column, while the last column conforms the total background (Total Bkg.) yield.

BP	$t\overline{t}+$	$DY+$	$Wt+$	$t\overline{t}Z$	$t\overline{t}h$	$tZ+$	Others	Total
	jets	jets	jets			jets		Bkg.
								(fb)
I	0.477	0.975	0.372	0.038	0.012	0.014	0.005	1.893
II	0.391	0.747	0.252	0.039	0.007	0.009	0.004	1.449
III	0.198	0.458	0.111	0.027	0.002	0.006	0.002	0.804

Table 4. Table 4: The signal cross sections after selection cuts of the b ​ Z ​ H 𝑏 𝑍 𝐻 bZH process for the three benchmark points are presented in the third column for | ρ b ​ b | = 0.1 subscript 𝜌 𝑏 𝑏 0.1 |\rho_{bb}|=0.1 . The corresponding significances are given in the fourth column.

BP	$\left|{\rho }_{bb}\right|$	Signal	Significance ($𝒵$)
		(fb)	300 fb-1
I		0.548	6.6
II	0.1	0.286	4.0
III		0.119	2.2

Table 5. Table 5: The cross sections (in fb) of the different backgrounds for the b ​ b ​ Z ​ H 𝑏 𝑏 𝑍 𝐻 bbZH process after selection cuts at s = 14 𝑠 14 \sqrt{s}=14 TeV.

BP	$t\overline{t}+$	$DY+$	$Wt+$	$t\overline{t}Z$	$t\overline{t}h$	$tZ+$	Others	Total
	jets	jets	jets			jets		Bkg.
								(fb)
I	0.013	0.065	0.031	0.002	0.003	0.0005	0.0002	0.115
II	0.016	0.064	0.018	0.001	0.002	0.0003	0.0002	0.102
III	0.011	0.039	0.018	0.001	0.0008	0.0002	0.0002	0.071

Table 6. Table 6: Same Table as in Table  4 however for the b ​ b ​ Z ​ H 𝑏 𝑏 𝑍 𝐻 bbZH process.

BP	$\left|{\rho }_{bb}\right|$	Signal	Significance ($𝒵$)
		(fb)	3000 fb-1
I		0.027	4.2
II	0.1	0.017	2.8
III		0.005	1.0

Equations

$$V(\Phi ,{\Phi }^{\prime })=m_{11}^2|\Phi {|}^2+m_{22}^2|{\Phi }^{\prime }{|}^2-(m_{12}^2{\Phi }^{†}{\Phi }^{\prime }+h.c.)$$

$$+\frac{{\lambda }_1}{2}|\Phi {|}^4+\frac{{\lambda }_2}{2}|{\Phi }^{\prime }{|}^4+{\lambda }_3|\Phi {|}^2|{\Phi }^{\prime }{|}^2+{\lambda }_4|{\Phi }^{†}{\Phi }^{\prime }{|}^2$$

$$\displaystyle+\bigg{[}\frac{\lambda_{5}}{2}(\Phi^{\dagger}\Phi^{\prime})^{2}+\left(\lambda_{6}|\Phi|^{2}+\lambda_{7}|\Phi^{\prime}|^{2}\right)\Phi^{\dagger}\Phi^{\prime}+h.c.\bigg{]},$$

$$c_{\beta -\alpha }^2=\frac{{\eta }_1{v}^2-m_h^2}{m_H^2-m_h^2},\sin 2(\beta -\alpha )=-\frac{2{\eta }_6{v}^2}{m_H^2-m_h^2},$$

$$m_{H^{\pm }}^2=\frac{1}{2}{\eta }_3{v}^2+{\mu }_{22}^2,$$

$$m_A^2=\frac{1}{2}({\eta }_3+{\eta }_4-{\eta }_5)v^2+{\mu }_{22}^2,$$

$$\displaystyle m_{h,H}^{2}=\frac{1}{2}\bigg{[}m_{A}^{2}+(\eta_{1}+\eta_{5})v^{2}$$

$$\displaystyle\quad\quad\quad\quad\mp\sqrt{\left(m_{A}^{2}+(\eta_{5}-\eta_{1})v^{2}\right)^{2}+4\eta_{6}^{2}v^{4}}\bigg{]}.$$

$${\mathcal{L}}_Y=$$

$$\displaystyle+\big{(}\lambda^{F}_{ij}c_{\beta-\alpha}-\rho^{F}_{ij}s_{\beta-\alpha}\big{)}H-i~{}{\rm sgn}(Q_{F})\rho^{F}_{ij}A\bigg{]}F_{jR}$$

$$-{\bar{U}}_i\left[(V{\rho }^D{)}_{ij}{P}_R-({\rho }^{U†}V{)}_{ij}{P}_L\right]D_j{H}^{+}$$

$$-{\bar{\nu }}_i{\rho }_{ij}^L{P}_R{L}_j{H}^{+}+h.c.,$$

$$\displaystyle\frac{g_{2}}{2c_{W}}Z_{\mu}\bigg{(}c_{\beta-\alpha}(h\partial^{\mu}A-\partial^{\mu}h\cdot A)$$

$$\displaystyle-s_{\beta-\alpha}(H\partial^{\mu}A-\partial^{\mu}H\cdot A)\bigg{)},$$

$${\mu }_i^f=\frac{{\sigma }_i{\mathcal{B}}^f}{({\sigma }_i{)}_{\text{SM}}({\mathcal{B}}^f{)}_{\text{SM}}}={\mu }_i{\mu }^f,$$

$$C_{7,8}^{(0)}(m_W)=F_{7,8}^{(1)}(x_t)+\delta C_{7,8}^{(0)}({\mu }_W),$$

$$\delta C_{7,8}^{(0)}(m_W)\simeq$$

$$R_{\text{exp}}=\frac{\mathcal{B}(B\to X_s\gamma {)}_{\text{exp}}}{\mathcal{B}(B\to X_s\gamma {)}_{\text{SM}}}.$$

$$R_{\text{theory}}=\frac{\mathcal{B}(B\to X_s\gamma {)}_{\text{G2HDM}}}{\mathcal{B}(B\to X_s\gamma {)}_{\text{SM}}},$$

$$\displaystyle\Delta\mathcal{A}_{\text{CP}}=\mathcal{A}_{B^{-}\to X_{s}^{-}\gamma}-\mathcal{A}_{B^{0}\to X_{s}^{0}\gamma}\approx 4\pi^{2}\alpha_{s}\frac{\tilde{\Lambda}_{78}}{m_{b}}\mbox{Im}\bigg{(}\frac{C_{8}}{C_{7}}\bigg{)},$$

$${\eta }_1=\frac{m_h^2{s}_{\beta -\alpha }^2+m_H^2{c}_{\beta -\alpha }^2}{v^2},$$

$${\eta }_3=\frac{2(m_{H^{\pm }}^2-{\mu }_{22}^2)}{v^2},$$

$${\eta }_4=\frac{m_h^2{c}_{\beta -\alpha }^2+m_H^2{s}_{\beta -\alpha }^2-2m_{H^{\pm }}^2+m_A^2}{v^2},$$

$${\eta }_5=\frac{m_H^2{s}_{\beta -\alpha }^2+m_h^2{c}_{\beta -\alpha }^2-m_A^2}{v^2},$$

$${\eta }_6=\frac{(m_h^2-m_H^2)s_{\beta -\alpha }{c}_{\beta -\alpha }}{v^2}.$$

Full text

I Introduction

The discovery of the 125 GeV scalar boson $h$ h125_discovery and its properties corroborate that the Standard Model (SM) is the correct effective theory at around weak scale. Even though no clear evidence of new physics (NP) has been found, the Run-2 era of LHC witnessed one of the most intriguing discovery, that is the bottom quark Yukawa coupling $hbb$ Aaboud:2018zhk ; Sirunyan:2018kst . The observation was announced simultaneously by the ATLAS and CMS collaborations. Both the experiments performed searches mainly in the process where $h$ is produced in association with a $Z$ or $W$ boson, followed by the $h\to b\bar{b}$ decay. When combined with the results from the other searches of Run-1 and Run-2, the observed signal strengths relative to the SM expectation were reported to be $1.01\pm 0.12(\text{stat.})^{+0.16}_{-0.15}(\text{syst.})$ at ATLAS Aaboud:2018zhk , while $1.04\pm 0.14(\text{stat.})\pm 0.14(\text{syst.})$ at CMS Sirunyan:2018kst . Although they are consistent with the SM prediction, both the measurements are quite accommodating for NP contribution. In the backdrop of these recent observations, it is timely to ask whether there exists any additional bottom Yukawa coupling in nature. In this article we explore the possibility of direct detection and identification of such extra bottom Yukawa coupling at the LHC.

The context is the two Higgs doublet model (2HDM). In the absence of discrete $Z_{2}$ symmetry, which was invoked to ensure Natural Flavor Conservation (NFC) Glashow:1976nt to forbid flavor changing neutral Higgs couplings, both the doublets couple to up- and down-type quarks. After the diagonalization of the fermion mass matrices two independent Yukawa matrices $\lambda_{ij}^{F}=({\sqrt{2}m_{i}^{F}}/{v})\,\delta_{ij}$ (with $v\simeq 246$ GeV) and $\rho_{ij}^{F}$ emerge, where $F$ denotes up- and down-type quarks and, leptons. The Yukawa matrices $\lambda_{ij}^{F}$ are real and diagonal, where as, $\rho_{ij}^{F}$ are in general non-diagonal and complex. Our focus of interest is the the extra bottom Yukawa coupling $\rho_{bb}$. In this paper, we analyze the prospect its direct detection at the LHC via $bg\to bA\to bZH$ and $gg\to b\bar{b}A\to b\bar{b}ZH$ processes (charge conjugate processes are implied) with $b$-tagging.

We investigate the discovery potential of $\rho_{bb}$ via $pp\to bA+X\to bZH+X$ ($X$ is inclusive activity) with $Z\to\ell^{+}\ell^{-}$ ($\ell=e,\mu$) and $H\to b\bar{b}$ (denoted as $bZH$ process) at the $14$ TeV LHC. In finding the discovery potential we assumed the extra top Yukawa coupling $\rho_{tt}$ to be relatively small to avoid the direct search constraints from $gg\to A/H$. A sizable $\rho_{bb}$ would also induce $gg\to b\bar{b}A\to b\bar{b}ZH$ which provides additional probe for $\rho_{bb}$. We study this process via $pp\to b\bar{b}A+X\to b\bar{b}ZH+X$ followed by $Z\to\ell^{+}\ell^{-}$ and $H\to b\bar{b}$ (denoted as $bbZH$ process). Recently, $\rho_{bb}$ received additional significance as it can drive electroweak baryogenesis (EWBG) rather efficiently Modak:2018csw . It was shown that $\mathcal{O}(0.1)$ imaginary $\rho_{bb}$ ($\mbox{Im}(\rho_{bb})$) can successfully generate the observed Baryon Asymmetry of the Universe Modak:2018csw . Although, the information of the phase could not be captured in the $bZH$ and $bbZH$ processes, however, a discovery might indicate $\rho_{bb}$ driven EWBG.

The paper is organized as follows. We outlined the formalism in the Sec. II, followed by discussion on the relevant constraints and available parameter space in the Sec. III. The Sec. IV is dedicated to the collider signatures of the $bZH$ and $bbZH$ processes respectively. We summarized our results with some discussions in the Sec. V.

II Formalism

The most general $CP$-conserving two Higgs doublet potential is given in the general basis as Davidson:2005cw ; Hou:2017hiw

[TABLE]

where the parameters $m_{11}^{2}$, $m_{12}^{2}$, $m_{22}^{2}$ and $\lambda_{i}$s are all real. The vacuum expectation values of the doublets $\Phi$ and $\Phi^{\prime}$ are given as $\left\langle\Phi\right\rangle=(0,v_{1}/\sqrt{2})^{T}$ and $\left\langle\Phi^{\prime}\right\rangle=(0,v_{2}/\sqrt{2})^{T}$ respectively, such that $v^{2}=v_{1}^{2}+v_{2}^{2}=(246~{}\mbox{GeV})^{2}$ and $\tan\beta=v_{2}/v_{1}$. With no $Z_{2}$ symmetry in place to distinguish between $\Phi$ and $\Phi^{\prime}$, $\tan\beta$ becomes unphysical. We move to the Higgs basis through basis rotation such that $\left\langle\Phi\right\rangle=(0,v/\sqrt{2})^{T}$ and $\left\langle\Phi^{\prime}\right\rangle=(0,0)^{T}$, where the parameters in the Higgs basis can be identified by the replacements $m_{ij}^{2}\to\mu_{ij}^{2}$ and $\lambda_{i}\to\eta_{i}$  111The relations between the parameters in the two bases can be found out in Ref. Davidson:2005cw .. The scalar potential minimization conditions lead to $\mu_{11}^{2}=-\frac{1}{2}\eta_{1}v^{2}$ and $\mu_{12}^{2}=\frac{1}{2}\eta_{6}v^{2}$, with $\mu_{22}^{2}>0$. The mixing angle $\beta-\alpha$ between the $CP$ even bosons $h$ and $H$ satisfies the relations Hou:2017hiw

[TABLE]

with shorthand $\cos(\beta-\alpha)=c_{\beta-\alpha}$ and $\sin(\beta-\alpha)=s_{\beta-\alpha}$. In the alignment limit $c_{\beta-\alpha}=0$, which leads to $\eta_{1}v^{2}=m_{h}^{2}$. The masses of the charged scalar $m_{H^{\pm}}$ and the neutral scalars $A$, $h$ and $H$ can be expressed as:

[TABLE]

The $CP$-even scalars $h$, $H$, $CP$-odd scalar $A$ and, the charged scalar $H^{\pm}$ couple to the fermions by Davidson:2005cw

[TABLE]

where $i,j=1,2,3$ are the generation indices that are summed over, $P_{L,R}=\frac{1\mp\gamma_{5}}{2}$, $V$ is the CKM matrix, $\lambda^{F}_{ij}$ are real and diagonal and $\rho^{F}_{ij}$ are complex non-diagonal $3\times 3$ matrices.

Our process of interest is $bg\to bA\to bZH$ , where the production process is initiated by $\rho_{bb}$ and the decay $A\to ZH$ is conformed by

[TABLE]

with $g_{2}$ is the $SU(2)_{L}$ gauge coupling and $c_{W}$ is Weinberg’s angle. A search can be performed in $bg\to bA\to b\bar{b}b$ mode, but, the process suffers from the overwhelming QCD multijets backgrounds. For $m_{H^{\pm}}+m_{W}<m_{A}$ ($m_{W}$ is the $W$ boson mass) $bg\to bA\to bH^{+}W^{-}$ is possible, but if searched in $H^{+}\to t\bar{b}$ (induced by $\rho_{bb}$) with $t\to b\ell^{+}\nu_{\ell}$ one loses the mass reconstruction capability of $A$ and, hence, controlling of the $t\bar{t}$ background. In general, backgrounds are even higher if searched in the hadronically decaying $t$ mode. Notice that, the $\rho_{tt}$ coupling, which also can induce $H^{+}\to t\bar{b}$ decay, obfuscates the role of $\rho_{bb}$. We remark that the $bg\to bA\to bZH$ process, which can only be induced via $\rho_{bb}$, offers a unique probe for the $\rho_{bb}$ coupling 222In principle $qg\to bA\to bZH$ can replicate the same final state as in $bg\to bA\to bZH$ process in the $pp$ collision if $\rho_{bq}$ or $\rho_{qb}$ ($q=d$ and $s$) are sizable, however, they receive severe constraints from the $B_{q}$ mixing Chen:2018hqy ..

The process $bg\to bA\to bZh$ is indeed possible, however suppressed by the mixing angle $c_{\beta-\alpha}$. It should be clear from Eq. (7) that the decay $A\to ZH$ is proportional to $s_{\beta-\alpha}$. As a result, a discovery of $bg\to bA\to bZH$ process is possible even in the approximate alignment (i.e. for small $c_{\beta-\alpha}$), which is observed at the LHC approxalign . Further, $\rho_{bb}$ can initiate $b\bar{b}\to A\to ZH$ and loop induced $gg\to A\to ZH$ gg2ZH , however, the coupling information is lost in the $pp$ collision. Besides, as $\rho_{tt}$ can also get involved in the loop, the role of $\rho_{bb}$ is obscured in $gg\to A\to ZH$. One can also have $gg\to A\to Zh$ (see e.g. Refs. Aaboud:2017cxo ; Sirunyan:2019xls and references therein) and $gg\to b\bar{b}A\to b\bar{b}Zh$ (see e.g. Refs. Aaboud:2017cxo ; Sirunyan:2019xls ; Ferreira:2017bnx ; Coyle:2018ydo ), however, again both processes are suppressed by the mixing angle $c_{\beta-\alpha}$ .

III Allowed parameter space

Having already set up the formalism we now focus on the relevant constraints and the available parameter space for our study. We first scrutinize the constraints on $\rho_{bb}$. For simplicity, we set all $\rho_{ij}=0$ except for $\rho_{bb}$ and $\rho_{tt}$ in this section. We assume small $\rho_{tt}$ in order to avoid direct search limits from $gg\to A/H$. In particular we choose $\rho_{tt}=0.1$ for illustration. The most stringent constraints arise from the Higgs signal strength measurements, the branching ratio of $B\to X_{s}\gamma$ ($\mathcal{B}(B\to X_{s}\gamma)$), the asymmetry of the CP asymmetry between the charged and neutral $B\to X_{s}\gamma$ decays ($\Delta\mathcal{A}_{\text{CP}}$), electron electric dipole moment (EDM) measurement and the upper limit on the $h$ decay width.

The couplings $\rho_{ij}$ modify the $h$ boson couplings to the fermions for moderate values of $c_{\beta-\alpha}$, as can be seen from Eq.(6). Therefore, $\rho_{bb}$ receives meaningful constraint from the Higgs boson coupling measurements, unless $c_{\beta-\alpha}$ is vanishingly small. We utilized Run-2 ATLAS ATLAS:2019slw and CMS Sirunyan:2018koj measurements which are based on 80 fb*-1* and 35.9 fb*-1* data respectively. The results summarize the values of different signal strengths $\mu_{i}^{f}$ and corresponding errors to a particular decay mode $i\to h\to f$. Following the Refs. ATLAS:2019slw ; Sirunyan:2018koj , we define a signal strength $\mu_{i}^{f}$ as:

[TABLE]

where $\sigma_{i}$ is denoted as the production cross section of $i\to h$ and $\mathcal{B}^{f}$ is the branching ratio for $h\to f$. The production modes considered are $i=ggF~{}(\mbox{gluon-fusion})$,$~{}VBF~{}(\mbox{vector-boson-fusion})$, $Zh$, $Wh$, $tth$, and the branching ratios are $f=\gamma\gamma,~{}ZZ,~{}WW,~{}\tau\tau,~{}bb,~{}\mu\mu$. For simplicity we utilized the LO $\mu_{i}^{f}$ in our analysis and followed Refs. Djouadi:2005gi ; Branco:2011iw ; Fontes:2014xva ; Hou:2018uvr for their explicit expressions. In particular, we focused on two different production modes, the $ggF$ and the $VBF$ in our analysis. We find that for the $ggF$ category, the most relevant signal strengths for our analysis are $\mu_{ggF}^{ZZ}$, $\mu_{ggF}^{WW}$, $\mu_{ggF}^{\gamma\gamma}$ and $\mu_{ggF}^{\tau\tau}$, while in the $VBF$ category $\mu_{VBF}^{WW}$, $\mu_{VBF}^{\gamma\gamma}$ and $\mu_{VBF}^{\tau\tau}$. In addition, we further considered the recent observation of the $h\to b\bar{b}$ in the $Vh$ production by ATLAS Aaboud:2018zhk and CMS Sirunyan:2018kst . We referred them together as “Higgs signal strength measurements”. The parameter space excluded by the Higgs signal strength measurements are shown by the red (ATLAS) and green (CMS) shaded regions in Fig. 1 for $m_{H^{\pm}}=350$ GeV (left) and 450 GeV (right). In generating Fig. 1, we allowed $2\sigma$ errors on each signal strength measurements and, assumed $c_{\beta-\alpha}=0.5$.

The branching ratio measurement of $B\to X_{s}\gamma$ provide another stringent constraint on $\rho_{bb}$. The coupling $\rho_{bb}$ enters in the $\mathcal{B}(B\to X_{s}\gamma)$ via charged Higgs and top quark loop. At the matching scale $\mu=m_{W}$, the modified leading order (LO) Wilson coefficients $C^{(0)}_{7,8}$ are defined as

[TABLE]

with, $\overline{m}_{t}(m_{W})$ is the $\overline{\mbox{MS}}$ running mass of top quark at $m_{W}$, $x_{t}=(\overline{m}_{t}(m_{W})/m_{W})^{2}$, while the expression for $F^{(1)}_{7,8}(x)$ can be found out in the Refs. Ciuchini:1997xe ; Chetyrkin:1996vx . The second term in Eq.(9), which originates from the charged Higgs contribution, expressed at LO as Altunkaynak:2015twa

[TABLE]

where $y_{H^{+}}=(\overline{m}_{t}(m_{W})/m_{H^{+}})^{2}$. Here we have followed Ref. Ciuchini:1997xe , for the expression of $F^{(2)}_{7,8}(y_{H^{+}})$. The current world average of $\mathcal{B}(B\to X_{s}\gamma)_{\text{exp}}$, which is extrapolated to the photon energy cut $E_{0}=1.6$ GeV is found to be $(3.32\pm 0.15)\times 10^{-4}$ Amhis:2016xyh . The SM prediction of $\mathcal{B}(B\to X_{s}\gamma)$ at next-to-next-to LO (NNLO) for the same photon energy cut is $(3.36\pm 0.23)\times 10^{-4}$ Czakon:2015exa . In order to find the constraint on $\rho_{bb}$, we adopted the prescription of Ref. Crivellin:2013wna and defined

[TABLE]

Based on our LO calculation we further defined

[TABLE]

and took $m_{W}$ and $\overline{m}_{b}(m_{b})$ as the matching scale and low-energy scales respectively. We finally demanded $R_{\text{theory}}$ should not exceed $2\sigma$ error of $R_{\text{exp}}$. The excluded regions are displayed by the purple shaded regions in Fig. 1.

The direct CP asymmetry $\mathcal{A}_{\text{CP}}$ Kagan:1998bh of $B\to X_{s}\gamma$ is sensitive to $\mbox{Im}(\rho_{bb})$. However, it has been proposed Benzke:2010tq that $\Delta\mathcal{A}_{\text{CP}}$, defined as the asymmetry of the CP asymmetry between the charged and neutral $B\to X_{s}\gamma$ decay provides even more powerful probe for the CP violating effects. The $\Delta\mathcal{A}_{\text{CP}}$ is defined as Benzke:2010tq

[TABLE]

where, $\alpha_{s}$ is the strong coupling constant calculated at $\overline{m}_{b}(m_{b})$ and $\tilde{\Lambda}_{78}$ is a hadronic parameter. It is expected that hadronic parameter $\tilde{\Lambda}_{78}\sim\Lambda_{\text{QCD}}$ and estimated to be in the range of $17~{}\mbox{MeV}<\tilde{\Lambda}_{78}<190$ MeV Benzke:2010tq . We take the average value of $\tilde{\Lambda}_{78}=89$ MeV as a reference value for illustration. A recent Belle measurement report $\Delta\mathcal{A}_{\text{CP}}=(+3.69\pm 2.65\pm 0.76)\%$ Watanuki:2018xxg , where the first and second uncertainties are statistical and systematic respectively. Utilizing Eq. (13) and allowing $2\sigma$ error on the Belle measurement of $\Delta\mathcal{A}_{\text{CP}}$ we find the red dotted lines (the regions above are excluded) in Fig. 1 for $m_{H^{\pm}}=350$ GeV and 450 GeV. As a first approximation we have utilized the LO Wilson coefficients as in Eq. (9) in our analysis. We stress that the constraint heavily depends on the value of $\tilde{\Lambda}_{78}$ and becomes stronger for the larger values of $\tilde{\Lambda}_{78}$.

The most stringent constraint on $\mbox{Im}(\rho_{bb})$ comes from the electron EDM ($d_{e}$) measurements. The two-loop Barr-Zee diagrams Barr:1990vd , which is studied widely in the context of 2HDM EDM_2HDM , are the leading contributions to $d_{e}$. A recent result from ACME Collaboration finds $|d_{e}|<1.1\times 10^{-29}~{}e~{}\text{cm}$ Andreev:2018ayy , which excludes even the nominal value (i.e. $|\mbox{Im}(\rho_{bb})|\lesssim 0.058$) required for $\rho_{bb}$ driven EWBG Modak:2018csw . The constraint could be relaxed by either turning on $\rho_{ee}$, or even could vanish in the alignment limit. In the former scenario non-zero $\rho_{ee}$ and $\rho_{bb}$ induce other Barr-Zee diagrams with opposite sign, where as in the the latter case all the contributions to the EDM are simply decoupled. In particular, Ref. Modak:2018csw finds for $\text{Re}(\rho_{ee})=0$, $\mathcal{O}(0.1)$ $\mbox{Im}(\rho_{bb})$ is still allowed if $0.06\lesssim\text{Im}(\rho_{ee})/(\lambda_{e}\lambda_{b})\lesssim 0.3$.

The current upper limit of the $h$ boson decay width, which is extracted to be $<0.013$ GeV (95% CL) Tanabashi:2018oca , can provide some limit on $|\rho_{bb}|$ if $c_{\beta-\alpha}\neq 0$. Besides, the presence of the additional scalars modify the $Zbb$ vertex Haber:1999zh at one-loop and in principle can constrain the $\rho_{bb}$ coupling. However, we found these limits to be weaker and lie beyond the plotted ranges in Fig. 1.

Let us understand Fig. 1. In generating Fig. 1 we set $c_{\beta-\alpha}=0.05$ and $\rho_{tt}=0.1$ for illustration. The constraint from the Higgs signal strength measurements depend primarily on the value of $c_{\beta-\alpha}$ and vanish in the alignment limit. Same is true for the constraint from the electron EDM, which also disappears in the alignment limit. On the other hand, bounds from $\mathcal{B}(B\to X_{s}\gamma)$ and $\Delta\mathcal{A}_{\text{CP}}$ alleviate for small $\rho_{tt}$, and/or for heavier $H^{\pm}$. It is clear from Fig. 1 that $|\rho_{bb}|\sim 0.1$ is still allowed, however, $c_{\beta-\alpha}$ and $\rho_{tt}$ should not be very large. In the following we would assume the alignment limit and set $\rho_{tt}=0$ for the sake of simplicity, however their impacts will be discussed in the latter part of the paper. In passing we remark that there exist several direct searches at the LHC which can also constrain $\rho_{bb}$, even for $c_{\beta-\alpha}=0$ and $\rho_{tt}=0$. We defer a detailed discussion of them for the next section.

For the dynamical parameters in Eq. (1), one needs to satisfy the perturbativity, positivity and tree-level unitarity conditions, for which we utilized 2HDMC Eriksson:2009ws . The quartic couplings $\eta_{1}$, $\eta_{3{\rm-}6}$ can be expressed in terms of $m_{h}$, $m_{A}$, $m_{H}$, $m_{H^{\pm}}$, $\mu_{22}$ and mixing angle ${\beta-\alpha}$, all normalized to $v$ Hou:2017hiw :

[TABLE]

The mixing angle ${\beta-\alpha}$ and the quartic couplings $\eta_{2}$ and $\eta_{7}$ are not related to masses. Hence, we take $v$, $m_{h}$, $m_{A}$, $m_{H}$, $m_{H^{\pm}}$, $c_{\beta-\alpha}$, $\eta_{2}$, $\eta_{7}$ and $\mu_{22}$ as the phenomenological parameters. However, in order to save computation time, we randomly generated these parameters in the following ranges: $\mu_{22}\in[0,700]$ GeV, $m_{A}\in[300,500]$ GeV, $m_{H^{\pm}}\in[300,800]$ GeV, $\eta_{2}\in[0,3]$, $\eta_{7}\in[-3,3]$, $m_{H}\in[200,m_{A}-m_{Z}]$ GeV, and $c_{\beta-\alpha}=0$ while satisfying $m_{h}=$ 125 GeV. Further, we demanded $m_{A}<m_{H^{\pm}}+m_{W}$ to forbid the $A\to H^{\pm}W^{\mp}$ decay for simplicity. In general, heavier $m_{A}$ is possible, however the discovery potential would alleviate due to the rapid fall in the parton luminosity. These randomly generated parameters are then passed to 2HDMC, which uses the input parameters Eriksson:2009ws $m_{H^{\pm}}$ and $\Lambda_{1-7}$ in the Higgs basis, and with $v$ as implicit parameter while scanning. We identify $\Lambda_{1-7}$ with $\eta_{1-7}$ and take $-\pi/2\leq\beta-\alpha\leq\pi/2$ to match the convention of 2HDMC. We further conservatively require all $|\eta_{i}|\leq 3$, however, $\eta_{2}>0$ is demanded by the positivity of the potential in Eq. (1), in addition to other involved conditions in 2HDMC.

We further imposed the stringent oblique $T$ parameter Peskin:1991sw constraint, which restricts the scalar masses $m_{H}$, $m_{A}$ and $m_{H^{\pm}}$ Froggatt:1991qw ; Haber:2015pua , and hence $\eta_{i}$s. Utilizing the expression given in Ref. Haber:2015pua the points that passed unitarity, perturbativity and positivity conditions from 2HDMC, were further required to satisfy the $T$ parameter constraint within the $2\sigma$ error Baak:2013ppa . These points are denoted as “scanned points”. Finally the scanned points are plotted as gray dots in Fig. 2 in the $|\eta_{3}+\eta_{4}-\eta_{5}|$ vs $m_{A}$ and $\mu_{22}^{2}/v^{2}$ vs $m_{A}$ plane. The Fig. 2 implies that finite parameter space exist for $300~{}\mbox{GeV}\lesssim m_{A}\lesssim 500$ GeV  333See also Ref. Hou:2019qqi for more on the parameter counting and scanning strategy..

IV Collider signatures

In this section we analyze the discovery potential of $pp\to bA+X\to bZH+X$ and $pp\to b\bar{b}A+X\to b\bar{b}ZH+X$ processes, followed by $H\to b\bar{b}$ and $Z\to\ell^{+}\ell^{-}$ decays. In general $Z\to\nu\bar{\nu}$ and $Z\to\tau^{+}\tau^{-}$ are possible, however, we found these modes to be not as promising. In order to illustrate the discovery potential we took three benchmark points (BPs) from the scanned points in Fig. 2, which are summarized in Table 1. As discussed earlier, the phase information of $\rho_{bb}$ is lost in the $bg\to bA\to bZH$ and $gg\to b\bar{b}A\to b\bar{b}ZH$ processes. Therefore, the only meaningful quantity in this section is the absolute value of $\rho_{bb}$ ($|\rho_{bb}|$). Unless otherwise specified we would only consider $|\rho_{bb}|$ from here on.

There exist several direct search limits from ATLAS and CMS that may restrict the parameter space of $\rho_{bb}$, even for $c_{\beta-\alpha}=0$ and $\rho_{tt}=0$. We find that the searches of heavy Higgs boson, in particular Refs. Sirunyan:2018taj ; ATLAS:2019jzx ; CMS:2018qbg ; Aaboud:2018cwk are the relevant ones for our study. The most stringent bound arises from the CMS search for a heavy Higgs boson production in association with least one additional $b$ quark and decaying into $b\bar{b}$ pair Sirunyan:2018taj . The search is performed with 13 TeV 35.7 fb*-1* data. It sets model-independent 95% CL upper limits on the $\sigma(pp\to bA/H+X)\cdot\mathcal{B}(A/H\to b\bar{b})$ for $m_{A}/m_{H}$ ranging from 300 to 1300 GeV. Utilizing this result we have extracted extrac 95% CL $\sigma(pp\to bA/H+X)\cdot\mathcal{B}(A/H\to b\bar{b})$ upper limit for BPI, BPII and BPIII. We then calculated the production cross sections ($pp\to bA/H+X$) at the leading order (LO) for the three BPs for a reference $|\rho_{bb}|$ value using Monte Carlo event generator MadGraph5_aMC@NLO Alwall:2014hca with the default NN23LO1 parton distribution function (PDF) set Ball:2013hta . Since, CMS does not veto additional activity in the event, we also included contributions from $gg\to b\bar{b}A/H$ along with $bg\to bA/H$ in the cross-section estimation. The cross sections are then rescaled by $|\rho_{bb}|^{2}\times\mathcal{B}(A/H\to b\bar{b})$ to get the corresponding 95% CL upper limits on $|\rho_{bb}|$. The upper limits for the BPI is $|\rho_{bb}|\lesssim 0.5$, where as $|\rho_{bb}|\lesssim 0.26$ and $|\rho_{bb}|\lesssim 0.24$ for BPII and BPIII respectively. A similar search has been performed by ATLAS ATLAS:2019jzx however the limits are somewhat weaker than that of Ref. Sirunyan:2018taj . The limits extracted from Ref. CMS:2018qbg , which searches for a light scalar decaying into $b\bar{b}$ pair, are weaker except for BPI, which we found also to be $|\rho_{bb}|\lesssim 0.5$ (at 95% CL). Moreover, ATLAS search for $H^{\pm}$ in association with a $t$ quark and a $b$ quark with $H^{+}/H^{-}\to t\bar{b}/\bar{t}b$ decay Aaboud:2018cwk is relevant, but the constraints are milder for all three BPs. The effective model is implemented in the FeynRules 2.0 Alloul:2013bka .

We choose $|\rho_{bb}|=0.1$ as a representative value for illustration in this section. Since our working assumptions are alignment limit with all $\rho_{ij}=0$ except $\rho_{bb}$, the total decay width of $A$ is nicely approximated as the sum of the partial widths of $A\to b\bar{b}$ and $A\to ZH$, while $H$ only decays to $b\bar{b}$. The corresponding branching ratios of $A$ for the three BPs are given in Table 1, where as $\mathcal{B}(H\to b\bar{b})\approx 100\%$ for all three BPs. Note, non-zero $\rho_{bb}$ induces $A/H\to\gamma\gamma$ and $A/H\to gg$ decays at one loop. These branching ratios are negligibly small, and hence neglected.

IV.1 The $bZH$ process

There exist several SM backgrounds for the $bZH$ process. The dominant backgrounds are $t\bar{t}+$jets, Drell-Yan+jets (DY+jets), $Wt+$jets, $t\bar{t}Z$+jets, $t\bar{t}h$, $tZ$+jets, with subdominant contributions arise from four-top ($4t$), $t\bar{t}W$, $tWh$, $tWZ$ and $WZ$+jets. Backgrounds from $WW$+jets and $ZZ$+jets are negligibly small and hence not included. The signal and background event samples are generated at LO, utilizing MadGraph5_aMC@NLO for $pp$ collisions at $\sqrt{s}=14$ TeV with the PDF set NN23LO1 and then interfaced with PYTHIA 6.4 Sjostrand:2006za for showering and hadronization. We adopted MLM matching scheme Alwall:2007fs for matrix element and parton shower merging. The event samples are finally fed into fast detector simulator Delphes 3.4.0 deFavereau:2013fsa for detector effects. Here we have incorporated default ATLAS based detector card available within Delphes framework. We do not include backgrounds from the fake and non-prompt sources. Such backgrounds are not properly modeled in Monte Carlo simulations and requires data to estimate such contributions.

The LO $t\bar{t}+$jets and $Wt+$jets cross sections are normalized to NNLO+NNLL cross sections by factors 1.84 twiki and $1.35$ Kidonakis:2010ux respectively. The DY+jets background cross section is adjusted to the NNLO QCD+NLO EW one by a factor 1.2, which is obtained utilizing FEWZ 3.1 Li:2012wna . The $t\bar{t}Z$, $\bar{t}Z+$ jets, $t\bar{t}h$, $4t$ and $t\bar{t}W^{-}$ ($t\bar{t}W^{+}$) cross sections at LO are normalized to NLO ones by the $K$-factors 1.56 Campbell:2013yla , 1.44 Alwall:2014hca , 1.27 twikittbarh , 2.04 Alwall:2014hca and 1.35 (1.27) Campbell:2012dh respectively, while $tWh$ and $tWZ$ are kept at LO. Further, the $W^{-}Z+$jets background is normalized to NNLO cross section by factor 2.07 Grazzini:2016swo . For simplicity, we assumed the QCD correction factors for the $tZj$ and $W^{+}Z+$jets to be the same as their respective charge conjugate processes. The signal cross sections are kept at LO.

In order to distinguish the signal from the background processes, we have applied following event selection criteria: Each event should contain two same flavor opposite sign leptons ($e$ and $\mu$), at least three jets with at least three of them are $b$-tagged. The minimum transverse momenta ($p_{T}$) of the leading and subleading leptons are required to be $>28$ GeV and $>25$ GeV respectively, where as the $p_{T}$ of all three $b$-jets should be $>20$ GeV. The absolute value of the pseudo-rapidity ($|\eta|$) of the leptons and all three $b$-jets are needed to be $<2.5$. The jets are reconstructed by anti-$k_{T}$ algorithm with radius parameter $R=0.6$. The separations $\Delta R$ between any two $b$-jets, any two leptons and, any $b$-jet and lepton in an event are required be $>0.4$. In order to reduce the $t\bar{t}$+jets background we vetoed events with missing transverse energy ($E_{T}^{\mbox{miss}}$) $>35$ GeV. The invariant mass of the two same flavor opposite charge leptons ($m_{\ell\ell}$) is required to be within the $Z$ boson mass window i.e. $76<m_{\ell\ell}<100$ GeV. To reduce backgrounds further, we finally demanded the invariant mass of the two same flavor opposite charge leptons and two leading $b$-jets ($m_{\ell\ell bb}$) to remain within $|m_{A}-m_{\ell\ell bb}|<100$ GeV. The normalized $m_{\ell\ell}$ and $m_{\ell\ell bb}$ distributions before application of any selection cuts are presented in Appendix. We adopted the $\eta$ and $p_{T}$ dependent $b$-tagging efficiency and, $c$- and light-jets misidentification efficiencies of Delphes. The background cross sections of the three benchmark points after selection cuts are summarized in Table. 3, while the signal cross sections along with their corresponding significances with the integrated luminosity $\mathcal{L}=300$ fb*-1* are given in Table 4 for $|\rho_{bb}|=0.1$. We remark that in our exploratory study we have not optimized the selection cuts such as $m_{\ell\ell}$ and $m_{\ell\ell bb}$, and kept them unchanged for all three benchmark points, however impact of changing them will be discussed in the following.

The statistical significances in Table 4 are determined by using $\mathcal{Z}=\sqrt{2[(S+B)\ln(1+S/B)-S]}$ Cowan:2010js , where $S$ and $B$ are the number of signal and background events after selection cuts. The achievable significances for BPI, BPII and BPIII are $\sim 6.6\sigma$, $\sim 4.0\sigma$ and $\sim 2.2\sigma$ with 300 fb*-1* integrated luminosity. We find that even the collected Run-2 data ($\sim 150$ fb*-1* ) would lead to $\sim 4.7\sigma$, $\sim 2.8\sigma$ significances for the BPI and BPII respectively, where as lower than $2\sigma$ for BPIII. As for the parameter space of EWBG, $|\mbox{Im}(\rho_{bb})|$ should be $\gtrsim 0.058$ Modak:2018csw , which leads to $\sim 12.1\sigma$, $\sim 4.5\sigma$ and $\sim 2.5\sigma$ significances for the BPI, BPII and BPIII with the full HL-LHC dataset (i.e. 3000 fb*-1* integrated luminosity). This implies that the $bZH$ process can fully probe the parameter space required for $\rho_{bb}$ driven EWBG if $m_{A}\lesssim 300$ GeV, where as evidence ($3\sigma$) could be found for $300~{}\mbox{GeV}\lesssim m_{A}\lesssim 400$ GeV all three BPs. Here, we have kept the $Z$-pole invariant mass cut fixed to $76~{}\mbox{GeV}<m_{\ell\ell}<100$ GeV. The significances change negligibly up to $76~{}\mbox{GeV}<m_{\ell\ell}<110$ GeV for all three BPs, however, reduce if broadened further. On the other hand, we have kept $m_{\ell\ell bb}$ cut fixed to $|m_{A}-m_{\ell\ell bb}|<100$ GeV for all three BPs. Significances do change with the change in $|m_{A}-m_{\ell\ell bb}|$ cut, however, mildly. We have checked that, e.g., if the cut $|m_{A}-m_{\ell\ell bb}|$ is set to $<80$ GeV the significances enhance by $\sim 1\%$, $\sim 5\%$ and $\sim 3\%$ for BPI, BPII and BPIII respectively, while reduce by $\sim 2\%$, $\sim 3\%$ and $\sim 2\%$ if set to $<120$ GeV. A narrower $m_{\ell\ell bb}$ cut than 80 GeV again reduces the significance for all three BPs.

Before closing, we remark that the scope for discovery of the $bZh$ process (i.e. $pp\to bA+X\to bZh+X$ with $Z\to\ell^{+}\ell^{-}$, $h\to b\bar{b}$) is limited if $c_{\beta-\alpha}$ is small. E.g., if $|\rho_{bb}|=0.1$, $c_{\beta-\alpha}=0.05$, $m_{A}=388$ and $m_{H}=236$ 444Corresponding parameters for this scanned point are: $\eta_{1}=0.259,~{}\eta_{2}=0.838,~{}\eta_{3}=1.633,~{}\eta_{4}=-0.044,~{}\eta_{5}=-1.569,~{}\eta_{6}=-0.033,~{}m_{H^{\pm}}=323~{}\mbox{and}~{}\eta_{7}=0.130$., the significance lies below $\sim 1\sigma$, even for the full HL-LHC dataset 555 In evaluating the significance we assumed the same cut based analysis as in $bZH$, except the application of an additional $|m_{h}-m_{bb}|<25$ GeV cut.. The significance improves substantially if $m_{A}<m_{H}+m_{Z}$ and/or $\rho_{bb}$ is large. A larger $c_{\beta-\alpha}$ would also help, however in such cases the significance would be balanced by more severe bounds from Higgs signal strength measurements. For $c_{\beta-\alpha}\sim 0.05$, $H\to ZZ$, $H\to W^{+}W^{-}$ would open up, but we do not find them to be very promising even for HL-LHC.

IV.2 The $bbZH$ process

As for the $bbZH$ process, the SM backgrounds are essentially the same as in the preceding subsection, however with one extra $b$-jet in the final state. We have adopted similar procedure for the signal and background events generation and, followed the event selection cuts as in the $bZH$ process except the additional $b$-jet is required to have $p_{T}>20$ GeV and $|\eta|<2.5$. The separation $\Delta R$ between any two $b$-jets, any two leptons and, any $b$-jet and lepton should be $>0.4$. All other cuts are kept same as in $bZH$. Finally, we applied the $m_{\ell\ell}$ and $m_{\ell\ell bb}$ selection cuts as before. The background and signal cross sections after the selection cuts are summarized in Table 5 and Table 6 respectively. We assumed the QCD correction factors for the different backgrounds as in the $bZH$ process and kept the signal cross sections at LO. Therefore, we remark that, there are slightly greater uncertainties involved in the background cross sections.

As can be seen from Table 6, the cross sections of the $bbZH$ process is suppressed due its to $2\to 4$ body nature. Hence, the significances are provided only for 3000 fb*-1* integrated luminosity, which can reach up to $\sim 4.2\sigma$, $\sim 2.8\sigma$ and $\sim 1\sigma$ for the BPI, BPII and BPIII respectively. Hence, a discovery is beyond the HL-LHC, that is unless $\rho_{bb}$ is large. The significances can be higher if the upper limits of $|\rho_{bb}|$ for the corresponding BPs are saturated, which can rise up to $\sim 6.5\sigma$, $\sim 13.8\sigma$ and $\sim 4.5\sigma$ for BPI, BPII and BPIII respectively with the full HL-LHC data. The impacts of other choices of $m_{\ell\ell}$ are similar as in $bZH$ process. However, if $|m_{A}-m_{\ell\ell bb}|$ cut is set to $<80$ GeV ($<120$ GeV), the significances enhance (reduce) by $\sim 2\%$ ($\sim 4\%$), $\sim 11\%$ ($\sim 5\%$) and $\sim 4\%$ ($\sim 2\%$) for BPI, BPII and BPII respectively. As in before, $pp\to b\bar{b}A+X\to b\bar{b}Zh+X$ is possible, however, the significances are even smaller than the $bZh$ process.

V Discussion and summary

Motivated by the recent observation of $hbb$ coupling, we have investigated the possibility of probing extra bottom Yukawa coupling $\rho_{bb}$ at the LHC. We first looked for the existing constraints on $\rho_{bb}$, mainly from the Higgs signal strength measurements, $\mathcal{B}(B\to X_{s}\gamma)$, $\Delta\mathcal{A}_{\text{CP}}$ of $B\to X_{s}\gamma$, electron EDM, as well as several direct searches at the LHC. We found that $\mathcal{O}(0.1)$ $|\rho_{bb}|$ is allowed by the current data, however $c_{\beta-\alpha}$ and $\rho_{tt}$ should not be large. We remark that additional constraints can come from the $\mathcal{A}_{\text{CP}}$ and isospin violating asymmetry ($\Delta_{0+}$) of $B\to K^{*}\gamma$ measurement by Belle Horiguchi:2017ntw , and could be comparable to the inclusive one, however, they both suffer from sizable uncertainties in their theoretical predictions Hurth:2010tk .

We have shown that $bg\to bA\to bZH$ with $Z\to\ell^{+}\ell^{-}$ and $H\to b\bar{b}$ offers excellent probe for $\rho_{bb}$. Discovery seems plausible with 300 fb*-1* integrated luminosity for $\mathcal{O}(0.1)$ $\rho_{bb}$ however $m_{A}$ needs to be $\lesssim 300$ GeV. For $400~{}\mbox{GeV}\lesssim m_{A}\lesssim 500$ GeV one may need the HL-LHC data. The process could be followed by $gg\to b\bar{b}A\to b\bar{b}ZH$, however, we find that a discovery is unlikely even with the full HL-LHC dataset if $|\rho_{bb}|\simeq 0.1$. We focused on the scenario where $m_{H}+m_{Z}<m_{A}$. However, for $m_{A}+m_{Z}<m_{H}$ our study can be extended to $bg\to bH\to bAZ$ (and $gg\to b\bar{b}H\to b\bar{b}ZA$) process where a complementary search strategy as in $bZH$ (and $bbZH$) can be adopted. Note that, $\rho_{bb}$ also invokes $gg\to\bar{t}bH^{+}\to\bar{t}bt\bar{b}$, which we leave out for future study. We have not included QCD corrections for the signal and neglected systematic uncertainties in our analysis.

As a first estimate we have not included uncertainties arising from factorization scale ($\mu_{F}$) and renormalization scale ($\mu_{R}$) dependences in our LO cross section calculation. In general, LO $bg\to bA/bH$ cross section has sizable scale uncertainties ($\sim 25-30\%$) in the mass range $m_{A/H}\sim 300-400$ GeV for bottom quark $p_{T}>15-30$ GeV and $|\eta|<2.5$ Campbell:2002zm (see also Dicus:1998hs ; Maltoni:2005wd ). The uncertainties are much higher for $gg\to b\bar{b}A/H$. E.g., the scale uncertainties could be as large as $\sim 40\%$ at LO Harlander:2003ai depending on the choices of $\mu_{R}$ and $\mu_{F}$. For both these processes the uncertainties are much smaller at NLO (or at NNLO)Campbell:2002zm ; Harlander:2003ai . It was discussed in Ref. Maltoni:2003pn that the LO cross sections estimated with LO PDF set CTEQ6L1 Pumplin:2002vw have large factorization scale dependence. We remark that our LO cross sections, which are determined utilizing LO PDF set NN23LO1, may have similar level of uncertainties. It is found that at $\mu_{F}\approx m_{A}~{}(\mbox{or}~{}m_{H})$ the corrections are large negative ($\sim-70\%$) Maltoni:2003pn , while at $\mu_{F}\approx m_{A}/4~{}(\mbox{or}~{}m_{H}/4)$ the corrections are small; indicating $\mu_{F}\approx m_{A}/4$ is the relevant factorization scale. In particular, the cross section uncertainties from renormalization and factorization scale choices are found to be small for $\mu_{R}=m_{A}$ and varied between $\mu_{R}=m_{A}/2$ to $\mu_{R}=2m_{A}$, along with $\mu_{F}=m_{A}/4$ and varied between $\mu_{F}=m_{A}/8$ to $\mu_{F}=m_{A}/2$ Maltoni:2003pn . Furthermore, we have not included PDF uncertainties, which could be substantial for the heavy quark initiated process, as discussed e.g. in Ref. Maltoni:2012pa . A discussion on PDF choices and their uncertainties for LHC can be found out in Ref. Butterworth:2015oua . These would induce some uncertainties in our results. A detailed estimation of such uncertainties we leave out for future.

A discovery might indicate EWBG driven by $\rho_{bb}$. With the full HL-LHC dataset the $bZH$ process can probe the entire parameter space required for the EWBG if $m_{A}\lesssim 300$ GeV. Although, a discovery would be intriguing, however it would not be sufficient to establish it to the EWBG without the information of the phase of $\rho_{bb}$. This would need further scrutiny and perhaps angular analysis of the $bZH$ (or $bbZH$) process would be indicative. Information of the phase can also be extracted from the future measurement of $\Delta\mathcal{A}_{\text{CP}}$ of $B\to X_{s}\gamma$ at Belle-II, if $H^{\pm}$ is not too heavy.

In principle, $\rho_{bd}$, $\rho_{db}$, $\rho_{bs}$ and $\rho_{sb}$ all can replicate $bZH$ and $bbZH$ signatures at the LHC, however, their impacts are inconsequential due to severe bounds from $B_{d}$ and $B_{s}$ mixings. If the charm quark gets misidentified as $b$-jet, a sizable $\rho_{cc}$ can mimic $bZH$ signature in $pp$ collision via $cg\to cA\to cZH$. However, such possibilities can be excluded with the simultaneous application of $c$- and $b$-tagging on the final state event topologies Hou:2018npi .

While determining the discovery potential we set all $\rho_{ij}=0$ except for $\rho_{bb}$ for the sake of simplicity. In general, non-zero $\rho_{ij}$s suppress $\mathcal{B}(A/H\to b\bar{b})$ and hence the discovery potential of the $bZH$ and $bbZH$. It is plausible that $\rho_{ij}$ couplings might share the same flavor organization patterns of the SM, i.e. trickling down off-diagonal elements as observed in the quark masses and mixings. If so, $\rho_{ii}$ could be $\sim\lambda_{i}$ and, similarly $\rho_{\tau\tau}\sim\lambda_{\tau}$ Hou:2017hiw . In demonstrating the discovery potential of $bZH$ and $bbZH$ processes, we have chosen $|\rho_{bb}|=0.1$ for illustration, however, $|\rho_{bb}|$ could be $\sim\lambda_{b}$. For $|\rho_{bb}|=\lambda_{b}$, one has $\sim 3\sigma$ for BPI with the full HL-LHC data for the $bZH$ process, where as, significances lie below $1\sigma$ for the BPII and BPIII. We remark, $\rho_{\tau\tau}\sim\lambda_{\tau}$ impacts negligibly on the discovery potential. Throughout we assumed the flavor changing neutral Higgs coupling $\rho_{tc}$ to be small, however, a $\mathcal{O}(1)$ value is still allowed by the current data Kohda:2017fkn ; Hou:2018zmg (see also Ref. Altmannshofer:2019ogm ), and could potentially reduce the significances of both the processes.

We assumed small $\rho_{tt}$ in order to avoid strong constraints arising from the $gg\to A/H$ searches. Notwithstanding, $\mathcal{O}(1)$ $\rho_{tt}$ with complex phase provides another robust mechanism for EWBG Fuyuto:2017ewj (see also Ref. deVries:2017ncy ), which can be probed by the conventional search programs such as $gg\to A/H\to t\bar{t}$ or $gg\to A/Ht\bar{t}\to t\bar{t}t\bar{t}$ Craig:2016ygr . The former process suffers from large interference Carena:2016npr with the overwhelming $gg\to t\bar{t}$ background, however recent ATLAS Aaboud:2017hnm and CMS CMS:2019lei studies found some sensitivity. We remark that $\rho_{tt}$ could be $\sim\lambda_{t}$. Utilizing results from Refs. CMS:2019nig ; Aaboud:2018cwk ; CMS:1900zym , Ref. Hou:2019gpn found $|\rho_{tt}|\gtrsim 0.8-0.9$ are excluded (at 95% CL) if $m_{A}/m_{H}\approx 400-500$ GeV, or $m_{H^{\pm}}\sim 300-400$ GeV. These are roughly the ballpark values of $m_{A}$ and $m_{H^{\pm}}$ of the three BPs, as can be seen from Table 1. Non-zero $\rho_{tt}$, alleviates the discovery potential of both $bZH$ and $bbZH$ processes via $\mathcal{B}(A/H\to t\bar{t})$, for $m_{A}/m_{H}>2m_{t}$. For example, taking $|\rho_{tt}|=0.8$ as yardstick and by a simple rescaling of the numbers in Table 4 and Table 6, we find that the significances for BPII and BPIII drop by $\sim 60\%$ and $\sim 70\%$ respectively. Therefore, discovery of $bZH$ is possible only for BPII (and BPI since $m_{A}$ lies below $2m_{t}$ threshold), but would require full HL-LHC data. In such cases, discovery is not possible for BPII and BPIII for $bbZH$ process, even with full HL-LHC data. For sizable $\rho_{bb}$ and $\rho_{tt}$, $bg\to bA/H\to bt\bar{t}$ as well as $gg\to b\bar{b}A/H\to b\bar{b}t\bar{t}$ ttbb both are possible, and would provide complementary information.

In summary, we have explored the possibility of discovery and identification of additional bottom Yukawa coupling that might exist in nature via $bg\to bA\to bZH$ and $gg\to b\bar{b}A\to b\bar{b}ZH$ processes at $\sqrt{s}=14$ TeV LHC. We found that the former process could be discovered with 300 fb*-1* integrated luminosity if $m_{A}\sim 300$ GeV, which could be extend up to $\sim 500$ GeV but the full HL-LHC dataset would be required. The latter process could also be discovered at the HL-LHC, however $\rho_{bb}$ needs to be large. A discovery would not only confirm physics beyond the Standard Model, but may also indicate the EWBG driven by $\rho_{bb}$.

Acknowledgements.

We thank W.-S Hou, M. Kohda and E. Senaha for many fruitful discussions. We also thank U.K. Dey for comments. This research is supported by grant MOST-107-2811-M-002-3069.

Appendix A Invariant Mass Distributions

The normalized invariant mass distributions $m_{\ell\ell}$ and $m_{\ell\ell bb}$ for the signal and backgrounds of the $bZH$ and $bbZH$ processes are presented in Fig. 3. These figures are generated without any selection cuts. The events for the signal and background processes are generated with the default cuts of MadGraph5_aMC@NLO with minimal modifications, followed by showering and hadronization via PYTHIA 6.4 and, with ATLAS based Delphes card.