Threshold Corrections to the Bottom Quark Mass Revisited

We look first at the approximation made to the gluino-sbottom contribution. Gluinos couple to the down-type squarks and quarks proportional to the SU(3) gauge coupling $g_{3}$ and hence contribute large corrections to the bottom quark mass. The corrections are dominant when the squarks belong to the third family since the inter-generational mixings between the squarks are typically (and by assumption in this study) small. The detailed calculation can be found in the appendix. We quote the final, exact form here Pierce:1996zz .

	$\displaystyle\Delta m_{b}^{\tilde{g}}=\frac{8}{3}\frac{g_{3}^{2}}{16\pi^{2}}\left[\frac{\hskip 1.42262pt\text{sin}\hskip 1.42262pt2\theta_{b}M_{\tilde{g}}}{2}\left(B_{0}\left(p,M_{\tilde{g}},m_{\tilde{b}_{1}}\right)-B_{0}\left(p,M_{\tilde{g}},m_{\tilde{b}_{2}}\right)\right)\right.$
	$\displaystyle\left.-\frac{m_{b}}{2}\left(B_{1}\left(p,M_{\tilde{g}},m_{\tilde{b}_{1}}\right)+B_{1}\left(p,M_{\tilde{g}},m_{\tilde{b}_{2}}\right)\right)\right]\;,$		(3)

where the momentum of the bottom quark is given by $p$. In the limit $p\rightarrow 0$ (which is a good assumption here since $p^{2}=m_{b}^{2}$), the Passarino-Veltman functions can be written as

	$\displaystyle B_{0}(0,M_{\tilde{g}},m_{\tilde{b}})$	$\displaystyle=$	$\displaystyle-\ln\left(\frac{m_{\tilde{b}}^{2}}{Q^{2}}\right)+1+\left(\frac{1}{1-x}\right)\ln x$		(4)
	$\displaystyle B_{1}(0,M_{\tilde{g}},m_{\tilde{b}})$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left[-\ln\left(\frac{m_{\tilde{b}}^{2}}{Q^{2}}\right)+\frac{1}{2}+\frac{1}{1-x}+\frac{\ln x}{(1-x)^{2}}-\theta(1-x)\ln x\right]$		(5)

where $x=m_{\tilde{b}}^{2}/M_{\tilde{g}}^{2}$. The first term in the above expression simplifies to

			$\displaystyle\frac{\hskip 1.42262pt\text{sin}\hskip 1.42262pt2\theta_{b}M_{\tilde{g}}}{2}\left[B_{0}\left(p,M_{\tilde{g}},m_{\tilde{b}_{1}}\right)-B_{0}\left(p,M_{\tilde{g}},m_{\tilde{b}_{2}}\right)\right]$		(6)
		$\displaystyle=$	$\displaystyle\frac{\hskip 1.42262pt\text{sin}\hskip 1.42262pt2\theta_{b}M_{\tilde{g}}}{2}\left[\ln\left(\frac{m_{\tilde{b}_{2}}^{2}}{m_{\tilde{b}_{1}}^{2}}\right)+M_{\tilde{g}}^{2}\left(\frac{1}{M_{\tilde{g}}^{2}-m_{\tilde{b}_{1}}^{2}}\ln\left(\frac{m_{\tilde{b}_{1}}^{2}}{M_{\tilde{g}}^{2}}\right)-\frac{1}{M_{\tilde{g}}^{2}-m_{\tilde{b}_{2}}^{2}}\ln\left(\frac{m_{\tilde{b}_{2}}^{2}}{M_{\tilde{g}}^{2}}\right)\right)\right]\;.$

The angle $\hskip 1.42262pt\text{sin}\hskip 1.42262pt2\theta_{b}$ can be determined to be

$$\hskip 1.42262pt\text{sin}\hskip 1.42262pt2\theta_{b}=\frac{2m_{b}(\mu\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta-A_{b})}{\sqrt{(m_{\tilde{b}_{L}}^{2}-m_{\tilde{b}_{R}}^{2})^{2}+(2m_{b}(\mu\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta-A_{b}))^{2}}}=\frac{2m_{b}(\mu\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta-A_{b})}{m_{\tilde{b}_{2}}^{2}-m_{\tilde{b}_{1}}^{2}}\;,$$

(7)

where we have ignored terms proportional to $M_{Z}$ or $m_{b}$. The trilinear coupling $A_{b}$ is often ignored since $\mu$ is enhanced by $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$.²²2We keep $A_{b}$ here in order to be consistent with the definitions of the squark masses. Similarly, the second term in LABEL:fullgluino is also neglected. Collecting terms, we arrive at the form in 1,

$$\frac{\Delta m_{b}^{\tilde{g}}}{m_{b}}\simeq\frac{8}{3}\frac{g_{3}^{2}}{16\pi^{2}}M_{\tilde{g}}(\mu\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta-A_{b})I(M_{\tilde{g}}^{2},m_{\tilde{b}_{1}}^{2},m_{\tilde{b}_{2}}^{2})\;,$$

(8)

where

$$I(a,b,c)=\frac{ab\ln\left(\frac{a}{b}\right)+bc\ln\left(\frac{b}{c}\right)+ac\ln\left(\frac{c}{a}\right)}{(a-b)(b-c)(a-c)}\;.$$

(9)

This is the expression that is typically used in most of the literature with large $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ models.

In 2, the exact, one-loop gluino-sbottom threshold correction to the bottom quark mass is compared to the approximate form of this correction given in 1. Darker shades of blue represent increasing squark masses from 1 TeV to $\geq$4 TeV. The black (lower) diagonal line represents where the exact and approximate forms would be equal. The red (upper) diagonal line represents where the correction from the exact form is $\sim$8% larger than the correction from the approximate form. Because the approximate form of the gluino-sbottom correction is equal to the terms in the exact form proportional to the $B_{0}$ Passarino-Veltman functions the discrepancy must be due to the terms in the exact form proportional to the $B_{1}$ Passarino-Veltman functions.

We refer to the term in 3 containing the $B_{0(1)}$ Passarino-Veltman functions and its prefactor as the $``B_{0(1)}^{\widetilde{g}}"$ term. In 3, the $B_{0}$ term is plotted against the $B_{1}^{\widetilde{g}}$ term. The color gradient from light to dark represents increasing sbottom masses from 1 TeV to $\geq$4 TeV. As the sbottom masses get pushed toward more than a few TeV, the $B_{0}^{\widetilde{g}}$ term decreases while the $B_{1}^{\widetilde{g}}$ term slightly increases, and the two terms are nearly the same magnitude. The increase in the $B_{1}^{\widetilde{g}}$ term can be understood by considering 5 in the limit of large sbottom masses. For a fixed gluino mass³³3In this analysis we consider gluinos to have mass $\leq$2 TeV. and in the limit of large $x$, one finds for the $B_{1}^{\widetilde{g}}$ term,

$$\frac{B_{1}^{\widetilde{g}}}{m_{b}}\simeq\frac{4}{3}\frac{g_{3}^{2}}{16\pi^{2}}\left[\ln\left(\frac{m_{\tilde{b}_{1}}m_{\tilde{b}_{2}}}{Q^{2}}\right)-\frac{1}{2}\right]\;.$$

(10)

Thus the $B_{1}^{\widetilde{g}}$ term grows logarithmically with increasing sbottom masses, which explains why there appears to be a constant vertical shift of $\sim$8% from the diagonal line along which the approximation is equal to the exact expression in 2. In this regime, where the $B_{0}^{\widetilde{g}}$ term is small, it is therefore important that the $B_{1}^{\widetilde{g}}$ term not be ignored. Finally, the points along the vertical line in 3 have $(\mu\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta-A_{b})\simeq 0$, and so one must be careful to check the size of $A_{b}$ relative to $\mu\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ also.

We turn now to the approximation made to the chargino-stop contribution. The charginos couple to the up-type squarks and down-type quarks proportional to the SU(2) coupling $g_{2}$ and the Yukawa couplings $\lambda_{t,b}$ with strength depending upon their respective wino-higgsino composition. The corrections dominate when the squarks are from the third family due to CKM suppression of the contributions from the first two families of squarks. The calculation is presented in detail in the appendix. The exact closed form cannot be put into a simplified form as was the case for the gluino-sbottom contribution. This is due to the non-trivial convolution of the elements of the stop mixing matrix, the elements of the chargino mixing matrices, and the weak and Yukawa coupling constants obtained by summing over the left and right stops and the two charginos. We therefore list the exact results from the appendix and discuss the approximations made to obtain the form in 1.

The full expression is Pierce:1996zz

$\displaystyle\Delta m_{b}^{\tilde{\chi}^{\pm}_{i}}$

$\displaystyle=$

$\displaystyle\sum_{i=1}^{2}\sum_{x=1}^{2}B_{LR_{i}}^{x}+\frac{m_{b0}}{2}(A_{L_{i}}^{x}+A_{R_{i}}^{x})$

(11)

with

	$\displaystyle B_{LR_{i}}^{x}$	$\displaystyle=$	$\displaystyle-\frac{\bar{\Phi}^{x}_{i}\Phi^{x}_{i}M_{\widetilde{\chi}_{i}^{\pm}}}{16\pi^{2}}B_{0}(p,M_{\widetilde{\chi}_{i}^{\pm}},m_{\tilde{t}_{x}})$
	$\displaystyle A_{L_{i}}^{x}$	$\displaystyle=$	$\displaystyle-\frac{\left(\Phi^{x}_{i}\right)^{\dagger}\Phi^{x}_{i}}{16\pi^{2}}B_{1}(p,M_{\widetilde{\chi}_{i}^{\pm}},m_{\tilde{t}_{x}})$
	$\displaystyle A_{R_{i}}^{x}$	$\displaystyle=$	$\displaystyle-\frac{\left(\bar{\Phi}^{x}_{i}\right)^{\dagger}\bar{\Phi}^{x}_{i}}{16\pi^{2}}B_{1}(p,M_{\widetilde{\chi}_{i}^{\pm}},m_{\tilde{t}_{x}})\;.$		(12)

Here $i=1,2$ is the chargino index and $x=1,2$ is the stop index.

The couplings are given by

	$\displaystyle\Phi^{x}_{i}$	$\displaystyle=$	$\displaystyle\frac{\lambda_{t}}{\sqrt{2}}V^{\dagger}_{i2}\left(\Gamma^{x}_{R}\right)^{\dagger}-g_{2}V^{\dagger}_{i1}\left(\Gamma^{x}_{L}\right)^{\dagger}$
	$\displaystyle\bar{\Phi}^{x}_{i}$	$\displaystyle=$	$\displaystyle\frac{\lambda_{b}}{\sqrt{2}}U^{\dagger}_{i2}\Gamma^{x}_{L}\;,$		(13)

where $U,V$ are the chargino mixing matrices and $\Gamma_{L,R}$ are the columns of the stop mixing matrix. The momentum of the bottom quark is given by $p$.

The terms containing the $B_{1}$ functions are often neglected and so we focus on the $B_{LR_{i}}^{x}$ contributions. Setting $p=0$ and expanding these terms,

	$\displaystyle B_{LR_{i}}^{x}$	$\displaystyle=$	$\displaystyle-\frac{\bar{\Phi}^{x}_{i}\Phi^{x}_{i}M_{\widetilde{\chi}_{i}^{\pm}}}{16\pi^{2}}B_{0}(0,M_{\widetilde{\chi}_{i}^{\pm}},m_{\tilde{t}_{x}})$		(14)
		$\displaystyle=$	$\displaystyle\frac{-M_{\widetilde{\chi}_{i}^{\pm}}}{16\pi^{2}}\left[\frac{\lambda_{b}}{\sqrt{2}}U^{\dagger}_{i2}\Gamma^{x}_{L}\right]\left[\frac{\lambda_{t}}{\sqrt{2}}V^{\dagger}_{i2}\left(\Gamma^{x}_{R}\right)^{\dagger}-g_{2}V^{\dagger}_{i1}\left(\Gamma^{x}_{L}\right)^{\dagger}\right]B_{0}(0,M_{\widetilde{\chi}_{i}^{\pm}},m_{\tilde{t}_{x}})\;.$

Neglecting terms proportional to $g_{2}$ and summing over the stops and charginos yields

	$\displaystyle\sum_{i=1}^{2}\sum_{x=1}^{2}B_{LR_{i}}^{x}\simeq$		$\displaystyle\frac{-M_{\widetilde{\chi}_{1}^{\pm}}}{16\pi^{2}}\left[\lambda_{b}\lambda_{t}U^{\dagger}_{12}V^{\dagger}_{12}\frac{\hskip 1.42262pt\text{sin}\hskip 1.42262pt2\theta_{t}}{2}\right]\left[B_{0}(0,M_{\widetilde{\chi}_{1}^{\pm}},m_{\tilde{t}_{1}})-B_{0}(0,M_{\widetilde{\chi}_{1}^{\pm}},m_{\tilde{t}_{2}})\right]$		(15)
		$\displaystyle+$	$\displaystyle\frac{-M_{\widetilde{\chi}_{2}^{\pm}}}{16\pi^{2}}\left[\lambda_{b}\lambda_{t}U^{\dagger}_{22}V^{\dagger}_{22}\frac{\hskip 1.42262pt\text{sin}\hskip 1.42262pt2\theta_{t}}{2}\right]\left[B_{0}(0,M_{\widetilde{\chi}_{2}^{\pm}},m_{\tilde{t}_{1}})-B_{0}(0,M_{\widetilde{\chi}_{2}^{\pm}},m_{\tilde{t}_{2}})\right]\;.$

For $|\mu|>|M_{2}|$, one finds that $U^{\dagger}_{12}V^{\dagger}_{12}\simeq 0$ and $U^{\dagger}_{22}V^{\dagger}_{22}\simeq 1$, whereas for $|\mu|<|M_{2}|$, one finds that $U^{\dagger}_{12}V^{\dagger}_{12}\simeq 1$ and $U^{\dagger}_{22}V^{\dagger}_{22}\simeq 0$. Furthermore, $\hskip 1.42262pt\text{sin}\hskip 1.42262pt2\theta_{t}=-2\lambda_{t}v_{d}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta(A_{t}-\frac{\mu}{\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta})/(m^{2}_{\tilde{t}_{2}}-m^{2}_{\tilde{t}_{1}})$ so that⁴⁴4The $\mu/\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ term is often neglected. We keep it here however in order to be consistent with the definitions of the squark masses.

$$\frac{\Delta m_{b}^{\tilde{\chi}^{\pm}_{i}}}{m_{b}}\simeq\frac{\lambda^{2}_{t}}{16\pi^{2}}\mu(A_{t}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta-\mu)I(\mu^{2},m_{\tilde{t}_{1}}^{2},m_{\tilde{t}_{2}}^{2})\;.$$

(16)

Among the two charginos, the dominant corrections are only from the Higgsino and are proportional to the Higgsino mass, $\mu$. Hence the chargino corrections tend be larger when $|\mu|>|M_{2}|$ (heavier Higgsino) and smaller when $|\mu|<|M_{2}|$ (lighter Higgsino) as shown in 4. We refer to the term in 11 containing the $B_{0(1)}$ Passarino-Veltman functions and its prefactor as the $``B_{0(1)}^{\widetilde{\chi}^{\pm}}"$ term.

In 5, the exact, one-loop chargino-stop threshold correction to the bottom quark mass is compared to the approximate form of this correction given in 1. Darker shades of blue represent increasing squark masses from 1 TeV to $\geq$4 TeV. It is clear that the chargino-stop approximation is a good approximation over all of the parameter space, particularly in the region in which the stops are heavy. We note that a nearly constant, positive contribution from the $B_{1}^{\widetilde{\chi}^{\pm}}$ term is present as in the gluino-sbottom case. Here however the contribution is $\lesssim 2$% and leaves the chargino-stop approximation as a good approximation.

Due to weaker coupling strengths compared to $g_{3}$ and $\lambda_{t}$, the contributions to the threshold correction of the bottom quark mass from $W$, $Z$, Higgses, and neutralinos are often neglected. It is possible that while the gluino and chargino contributions may each be of much greater magnitude than these other contributions, a cancellation occurs such that their sum is of the same magnitude as the other contributions. Since these terms are dropped altogether, the validity of this approximation is simply based on the magnitude of their contribution compared to the total approximate correction as given in 1.

6 shows the size of these quantities relative to the total approximate correction. We find that in the heavy squark regime the neutralino contribution is typically $\leq$1%. Furthermore, the $W$ and $Z$ contributions are very close to 0 for all points. This leaves the contributions from the Higgses, which give a correction of $\sim$4% for all points. Thus, in the heavy squark regime in which the correction to the bottom quark mass given by 1 is small, the contribution from the Higgses should not be ignored. Note that the contributions from the Higgses are not $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$-enhanced contributions Pierce:1996zz . The implications of this statement will be discussed in 5.

A good choice of scale to integrate out the massive SUSY particles is the $M_{Z}$ scale. At the $M_{Z}$ threshold one then has to match the value of $m_{b}$ before and after integrating out the massive states. This leads to the relation

$$m_{b}(M_{Z})^{\rm SM}=m_{b}(M_{Z})^{\rm MSSM}\left(1+\Delta m_{b}/m_{b}\right)\;.$$

(17)

$m_{b}(M_{Z})^{\rm below}$ can be determined by taking the value of $m_{b}(m_{b})=4.19$ GeV and running it to the $M_{Z}$ scale. This is evaluated using the RunDec package to be $m_{b}(M_{Z})^{\rm below}=2.69$ GeV. The hope then is that the right choice of bottom Yukawa coupling and the appropriate set of SUSY boundary conditions at some UV scale will give rise to the necessary $m_{b}(M_{Z})^{\rm above}$ and $\Delta m_{b}/m_{b}$ to satisfy 17.

When fitting the bottom quark mass, it is common to use the full, exact one-loop correction. This is done in most numerical spectrum calculators, such as SOFTSUSY Allanach:2014nba and SPheno Porod:2003um . Physical interpretations are often based however on the approximate formula given in 1. As was shown in the previous section, additional terms, namely the $B_{1}^{\widetilde{g}}$ terms from the gluino-sbottom contributions and the contributions from the Higgses, should also be included for a full description. These “missing” terms contribute $\sim$12% to the correction. In 1 the conditions for obtaining an appropriate SUSY threshold correction to the bottom quark mass in models with third family Yukawa unification were determined by an interpretation of the approximate formula given in 1. We revisit this scenario here to offer a more accurate interpretation.

In models with third family Yukawa unification, the SUSY threshold corrections to the bottom quark typically need to be $-\mathcal{O}$(few)%. For $\mu>0$, the common interpretation is $A_{t}$ needs to be large and negative in order for the chargino-stop contribution to overcome the $B_{0}^{\widetilde{g}}$ term from the gluino-sbottom contribution. By including the “missing” terms, which are positive, we see that the size of $A_{t}$ is underestimated when the approximate form of the corrections is used to interpret the size of the parameters. This is particularly true when the squarks are heavy. In this regime, the chargino-stop term is suppressed but the “missing” terms are not and so $A_{t}$ must be quite large to overcome both the suppression by the heavy stops and also the positive contribution from the missing terms. It has been pointed out in earlier works that light Higgsinos are disfavoured in Yukawa unified GUTs Baer:2012jp ; Anandakrishnan:2013tqa , and this can be traced back to 4, where we see that the corrections from the chargino are small for small $\mu$ and do not compensate for the large gluino corrections.

For $\mu<0$, the common interpretation is that the parameters need not be large since the terms in 1 already have the needed minus sign. Including the “missing” terms introduces a positive contribution (these terms are not proportional to $\mu$) that is relatively large and so $A_{t}$ and/or $M_{\widetilde{g}}$ must be larger than expected in order to overcome the additional contributions.

The MSSM predicts four new physical Higgs states in addition to the light CP-even (SM like) Higgs boson. The coupling of the Higgs bosons to the bottom quark depends on the MSSM parameters, particularly, $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$. In addition, the couplings also depend on the bottom quark threshold corrections and the effect of these corrections have been the subject of many works especially in the large $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta\$ regime Carena:1999py ; Guasch:2001wv ; Belyaev:2002eq ; Guasch:2003cv ; Crivellin:2009ar ; Crivellin:2010er ; Crivellin:2012zz ; Baer:2011af ; Carena:2012rw ; Dittmaier:2014sva ; Carena:2013qia . The low energy effective Lagrangian coupling the bottom quark with the up- and the down-type Higgs bosons in the MSSM including the supersymmetric threshold corrections can be written as

$${\mathcal{L}}_{\rm eff}=-\lambda_{b}^{0}\bar{b}_{R}^{0}\left[(1+\Delta_{1})\phi^{0}_{d}+\Delta_{2}\phi_{u}^{0*}\right]b_{L}^{0}+\text{h.c.}\;,$$

(18)

where

	$\displaystyle\phi_{d}^{0}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}}\left(v_{d}+H\hskip 1.42262pt\text{cos}\hskip 1.42262pt\alpha-h\hskip 1.42262pt\text{sin}\hskip 1.42262pt\alpha+iA\hskip 1.42262pt\text{sin}\hskip 1.42262pt\beta-iG^{0}\hskip 1.42262pt\text{cos}\hskip 1.42262pt\beta\right)$		(19)
	$\displaystyle\phi_{u}^{0}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}}\left(v_{u}+H\hskip 1.42262pt\text{sin}\hskip 1.42262pt\alpha+h\hskip 1.42262pt\text{cos}\hskip 1.42262pt\alpha+iA\hskip 1.42262pt\text{cos}\hskip 1.42262pt\beta+iG^{0}\hskip 1.42262pt\text{sin}\hskip 1.42262pt\beta\right)\;.$		(20)

Here $\Delta_{2}$ represents the coupling of the bottom quark to the “wrong” Higgs, which is generated by the radiative effects discussed in this paper. The corrections to the coupling of the bottom quark to the down-type Higgs are represented by $\Delta_{1}$. The $\Delta_{2}$ interactions are $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$-enhanced while the $\Delta_{1}$ corrections are not. The expression in 20 must be matched to the renormalized Lagrangian given by Guasch:2003cv

$${\mathcal{L}}_{\rm eff}=-\lambda_{b}\bar{b}_{R}\left[\phi^{0}_{d}+\frac{\Delta_{b}}{\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta}\phi_{u}^{0*}\right]b_{L}+\text{h.c.}\;,$$

(21)

yielding the relations

	$\displaystyle\lambda_{b}=\lambda_{b}^{0}(1+\Delta_{1})$		(22)
	$\displaystyle\frac{\Delta_{b}}{\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta}=\frac{\Delta_{2}}{1+\Delta_{1}}\;.$		(23)

Consider the gluino contribution in the approximate form of the threshold corrections given by 1. The $\mu$-term, which is proportional to $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$, is included in $\Delta_{2}$ while the $A_{b}$-term is included in $\Delta_{1}$. The $\Delta_{1}$ correction is typically found to be $\mathcal{O}$(1)% and is therefore often neglected Guasch:2003cv . We point out here that neither the $B_{1}^{\widetilde{g}}$ terms from the gluino-sbottom contribution nor the contributions from the Higgses are proportional to $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$, and therefore they enhance $\Delta_{1}$ by $\sim$12%. By considering the forms of the effective couplings of the Higgses to the bottom quark, we can determine if this enhancement translates to a nonnegligible correction. The effective couplings are given by Carena:1999py

	$\displaystyle\tilde{g}_{b}^{h}$	$\displaystyle=$	$\displaystyle\frac{g^{h}_{b}}{(1+\Delta_{b})}\left(1-\frac{\Delta_{b}}{\hskip 1.42262pt\text{tan}\hskip 1.42262pt\alpha\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta}\right)$		(24)
	$\displaystyle\tilde{g}_{b}^{H}$	$\displaystyle=$	$\displaystyle\frac{g^{H}_{b}}{(1+\Delta_{b})}\left(1+\frac{\Delta_{b}}{\hskip 1.42262pt\text{cot}\hskip 1.42262pt\alpha\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta}\right)$		(25)
	$\displaystyle\tilde{g}_{b}^{A}$	$\displaystyle=$	$\displaystyle\frac{g^{A}_{b}}{(1+\Delta_{b})}\left(1-\frac{\Delta_{b}}{\hskip 1.42262pt\text{tan}\hskip 1.42262pt^{2}\beta}\right)\;,$		(26)

where $g^{h,H,A}_{b}$ are the tree level couplings. In the decoupling limit, $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\alpha\rightarrow-\hskip 1.42262pt\text{cot}\hskip 1.42262pt\beta$ and we obtain

	$\displaystyle\tilde{g}_{b}^{h}$	$\displaystyle=$	$\displaystyle g^{h}_{b}$		(27)
	$\displaystyle\tilde{g}_{b}^{H}$	$\displaystyle=$	$\displaystyle\frac{g^{H}_{b}}{(1+\Delta_{b})}\left(1-\frac{\Delta_{b}}{\hskip 1.42262pt\text{tan}\hskip 1.42262pt^{2}\beta}\right)\simeq\frac{g^{H}_{b}}{(1+\Delta_{b})}$		(28)
	$\displaystyle\tilde{g}_{b}^{A}$	$\displaystyle=$	$\displaystyle\frac{g^{A}_{b}}{(1+\Delta_{b})}\left(1-\frac{\Delta_{b}}{\hskip 1.42262pt\text{tan}\hskip 1.42262pt^{2}\beta}\right)\simeq\frac{g^{A}_{b}}{(1+\Delta_{b})}\;.$		(29)

We therefore only need to determine the extent to which $\Delta_{1}$ affects the size of the factor $(1+\Delta_{b})^{-1}$. From 23, the factor may be written as

$$\frac{1}{1+\Delta_{b}}=\frac{1+\Delta_{1}}{1+\Delta_{1}+\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta}\equiv\delta_{12}\;.$$

(30)

Let us define $\delta_{2}\equiv(1+\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta)^{-1}$ and $\delta_{\Phi}$ to be the relative change between ignoring $\Delta_{1}$ and including it,

$$\delta_{\Phi}\equiv\frac{\delta_{12}-\delta_{2}}{\delta_{2}}\;.$$

(31)

By setting $\Delta_{1}=0.12$, $\delta_{\Phi}$ can be plotted as a function of $\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ as shown in 7. For positive values of $\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$, the relative change is never more than 6%. Unless $\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ is $\mathcal{O}(1)$, the relative correction to the heavy Higgs couplings is only a few percent. The effect of including $\Delta_{1}$ can be more drastic if $\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ is negative. As $\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ approaches $-\mathcal{O}(1)$, the relative change increases quickly to the nearly the same magnitude. Such large, negative values of $\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ may be a more extreme case however. For most values of $\Delta_{2}\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$ obtained in the parameter scan $(<40\%)$, the relative change is again only a few percent. Thus unless the magnitude of the $\hskip 1.42262pt\text{tan}\hskip 1.42262pt\beta$-enhanced corrections to the bottom quark mass are $\mathcal{O}(1)$ it is safe to neglect the $\Delta_{1}$ correction to the couplings of the bottom quark with the heavy Higgses.⁵⁵5It is expected that the LHC and ILC will be able to measure Higgs couplings to within a few percent Peskin:2013xra ; Han:2013kya . It will then be necessary to include the $\Delta_{1}$ corrections. Note that in calculating the $B_{1}^{\widetilde{g}}$ contribution to $\Delta_{1}$ we take $Q=M_{Z}$. If the scale is chosen to be higher, then $B_{1}^{\widetilde{g}}$ would be smaller and the relative change, $\delta_{\Phi}$, would be more suppressed.