---
title: Bottom-Quark Yukawa Coupling
url: https://www.emergentmind.com/topics/bottom-quark-yukawa-coupling
type: topic
---

# Bottom-Quark Yukawa Coupling

The bottom-quark Yukawa coupling is the Higgs-sector interaction that links electroweak symmetry breaking to the bottom-quark mass. In the Standard Model, a common normalization is \(y_b=\sqrt{2}\,m_b/v\) with \(v\simeq246\ \mathrm{GeV}\); using \(m_b(m_b)=4.2\ \mathrm{GeV}\) gives \(y_b^{\mathrm{SM}}(m_b)\simeq0.024\). In practice, the quantity is studied simultaneously as a low-energy running coupling, as the parameter controlling \(H\to b\bar b\), as an input to bottom-fusion and \(b\bar b H\) production, and as a probe of extended Higgs sectors, supersymmetric threshold effects, and high-scale flavor structure [1106.6208], [2307.08372].

## 1. Definition, normalization, and renormalization

In the mass-based normalization used in several precision-QCD analyses, the running bottom Yukawa coupling is
\[
y_b(\mu)=\frac{\sqrt{2}\,m_b^{\overline{\mathrm{MS}}}(\mu)}{v},
\]
with \(v\simeq246\ \mathrm{GeV}\). In the five-flavor scheme, both \(y_b\) and \(\alpha_s\) are renormalized in the \(\overline{\mathrm{MS}}\) scheme, and the Yukawa renormalization constant is identical to the quark-mass renormalization constant. Accordingly,
\[
\mu\frac{d}{d\mu}m_b(\mu)=-\gamma_m(\alpha_s)\,m_b(\mu),\qquad
\mu\frac{d}{d\mu}y_b(\mu)=-\gamma_m(\alpha_s)\,y_b(\mu),
\]
so the scale dependence of \(y_b\) is entirely inherited from the running mass [1904.09990], [1407.8114].

A distinct but equivalent interaction-level parameterization is often used when the Lorentz structure is under study. In that convention the Higgs interaction with bottom quarks is written as
\[
\mathcal{L}\supset -\,y_b\,h\,\bar b\,(\cos\alpha_b+i\gamma_5\sin\alpha_b)\,b,
\]
with \(g_S=y_b\cos\alpha_b\) and \(g_P=y_b\sin\alpha_b\). The Standard Model corresponds to \(\alpha_b=0\). This form is convenient because it separates magnitude and CP phase without committing to a specific effective-field-theory basis [2009.02000].

The relation between the running Yukawa coupling and pole-mass inputs receives electroweak threshold corrections. In the gaugeless-limit two-loop analysis of the Standard Model, it is useful to define the “Yukawa mass”
\[
m_Y(\mu)=2^{-3/4}G_F^{-1/2}y_b(\mu)=\frac{m_b(\mu)}{\sqrt{1+\Delta\bar r(\mu)}},
\]
because this suppresses numerically large tadpole contributions that otherwise appear in the \(\overline{\mathrm{MS}}\)-to-pole matching [1401.1844].

## 2. \(H\to b\bar b\) and precision theory

At tree level, \(H\to b\bar b\) is the dominant decay mode controlled by the bottom Yukawa coupling. The basic width formula may be written as
\[
\Gamma_0(H\to b\bar b)=\frac{N_c\,y_b^2\,m_H}{8\pi}\left(1-\frac{4m_b^2}{m_H^2}\right)^{3/2}
=\frac{3G_F m_H m_b^2}{4\sqrt{2}\pi}\left(1-\frac{4m_b^2}{m_H^2}\right)^{3/2}.
\]
For \(m_H\simeq125\ \mathrm{GeV}\), the Standard Model branching ratio is \(\mathrm{BR}(H\to b\bar b)\simeq58.1\%\) [1812.07811], [2307.08372].

The perturbative description is highly developed. The three-loop QCD \(Hb\bar b\) form factor in the massless-bottom limit provides a crucial ingredient for third-order QCD corrections to bottom-fusion Higgs production and to fully differential Higgs decay into bottom quarks, with infrared poles matching the universal QCD factorization structure [1407.8114]. Exact top-Yukawa-induced corrections to \(H\to b\bar b\) at \(\mathcal{O}(\alpha_s^2 y_t y_b)\) are sub-percent at physical masses, and the previously used heavy-top approximation agrees with the exact result at better than per-mill level. Their impact on \(p_T^{bb}\) and \(m_{bb}\) distributions relevant to \(VH(bb)\) analyses is correspondingly small [1812.07811].

At still higher order, the \(\mathcal{O}(\alpha_s^4)\) top-Yukawa-induced contribution with two top-Yukawa insertions and massive final-state bottom quarks increases the decay width by \(0.4\%\) relative to the \(\mathcal{O}(\alpha_s^3)\) result and reduces the scale dependence significantly down to \(0.4\%\). In the quoted \(\overline{\mathrm{MS}}\) setup near \(\mu=m_H\), the resulting prediction is
\[
\Gamma(H\to b\bar b)=2.421^{+0.008}_{-0.010}(\text{scale})^{+0.005}_{-0.005}(\alpha_s)\ \mathrm{MeV}.
\]
Because \(\Gamma\propto y_b^2\) at leading order, omitting this correction would bias an extracted \(y_b\) by about \(0.2\%\) [2603.18576].

## 3. Production channels and direct collider determinations

The most direct production observable proportional to \(y_b^2\) is bottom-quark fusion. In the five-flavor scheme, the leading partonic process is \(b\bar b\to H\), with
\[
\hat\sigma_{b\bar b}^{(0)}(z)=\frac{\pi}{6m_H^2}y_b^2(\mu_R)\,\delta(1-z).
\]
At N\(^3\)LO in perturbative QCD, using \((\mu_R,\mu_F)=(m_H,m_H/4)\), the inclusive prediction is \(\sigma_{bbH}^{\mathrm{N^3LO}}=0.542\ \mathrm{pb}\) at \(13\ \mathrm{TeV}\), with scale uncertainty \(+3.0\%/-4.8\%\), PDF\(+\alpha_s\) uncertainty \(\pm8.5\%\), \(m_b\) uncertainty \(+2.3\%/-1.7\%\), and an additional \(\pm2.5\%\) for the lack of N\(^3\)LO PDFs [1904.09990].

In \(pp\to b\bar bH\), the bottom Yukawa piece is not dominant in the Standard Model once top-Yukawa-induced contributions are included. In the four-flavor scheme at NLO QCD, the cross section decomposes into \(y_b^2\), \(y_b y_t\), and \(y_t^2\) terms, and the \(y_t^2\) component becomes the dominant production mechanism. The study identifies selection strategies that recover direct sensitivity to \(y_b\): requiring at least one \(b\)-jet, vetoing “\(bb\) jets,” and imposing a modest upper cut on \(p_T^H\). With a \(bb\)-jet veto and \(p_T^H<50\ \mathrm{GeV}\), the \(y_b^2\) share can be raised to about \(36\%\) while retaining about \(50\%\) of its rate [1808.01660].

Bottom-Yukawa-induced associated \(ZH\) production through \(b\bar b\to ZH\) is far smaller. The NNLO soft-virtual analysis of the \(t\)- and \(u\)-channel amplitudes proportional to \(y_b\) finds that the resulting cross section is three orders of magnitude smaller than the usual \(s\)-channel contribution, making this process unpromising as a standalone \(y_b\) measurement channel at the LHC [1912.06271].

Experimentally, \(H\to b\bar b\) remains the central direct handle. ATLAS, using the full \(139\ \mathrm{fb}^{-1}\) Run-2 dataset at \(13\ \mathrm{TeV}\), reports for resolved \(VH,H\to b\bar b\)
\[
\mu^{b\bar b}_{VH}=1.02^{+0.12}_{-0.11}(\text{stat.})^{+0.14}_{-0.13}(\text{syst.}),
\]
with observed significance \(6.7\sigma\); the boosted analysis gives
\[
\mu^{b\bar b}_{VH}=0.72^{+0.29}_{-0.28}(\text{stat.})^{+0.26}_{-0.22}(\text{syst.}),
\]
with \(2.1\sigma\) significance. In \(ttH\), ATLAS reports \(\mu_{ttH}=0.35\pm0.20(\text{stat.})^{+0.30}_{-0.28}(\text{syst.})\) with \(1.0\sigma\) observed significance [2307.08372]. CMS, in a simultaneous \(ttH(H\to b\bar b)\) and \(ttH(H\to c\bar c)\) analysis with \(138\ \mathrm{fb}^{-1}\), measures
\[
\mu_{ttH,bb}=0.91^{+0.26}_{-0.22},
\]
with \(4.4\sigma\) observed significance; within the specific \(\kappa\)-framework used there, fixing \(\kappa_c=1\) yields \(|\kappa_b|<3.0\) at \(95\%\) CL [2509.22535].

Global coupling fits sharpen this picture. In a broken-phase effective-coupling analysis of Run-2 data, the allowed \(95\%\) CL range is \(c_b/c_V\in[0.81,1.14]\), while a universal third-family rescaling gives \(c_f\in[0.88,1.14]\). The same study projects \(c_b/c_V\in[0.95,1.05]\) at the HL-LHC and sub-percent sensitivity at future Higgs factories [2006.01164].

## 4. Lorentz structure, CP phase, and sign

The bottom Yukawa interaction need not be purely scalar. A general spin-zero coupling can be written as
\[
\mathcal{L}_Y=-\,X\,\bar b\,(y_S+i\,y_P\gamma_5)\,b
=-\,y_b\,X\,\bar b\,(\cos\varphi+i\sin\varphi\,\gamma_5)\,b.
\]
An axial field redefinition,
\[
b\to e^{i\alpha\gamma_5}b,\qquad \bar b\to \bar b\,e^{i\alpha\gamma_5},
\]
rotates scalar and pseudoscalar pieces into one another while leaving the gauge interactions invariant. As a result, any observable distinction between scalar and pseudoscalar bottom Yukawa couplings vanishes in the \(m_b\to0\) limit and is strongly suppressed when the bottom quarks are relativistic [1904.09895].

This suppression explains why the inclusive \(h\to b\bar b\) width has almost no sensitivity to the CP phase. In the explicit Higgs-factory analysis,
\[
\Gamma(h\to b\bar b)=\Gamma^{\mathrm{SM}}(h\to b\bar b)\left(\frac{y_b}{y_b^{\mathrm{SM}}}\right)^2
\left(\cos^2\alpha_b+\beta_b^{-2}\sin^2\alpha_b\right),
\]
and the \(\alpha_b\)-dependent correction reduces to a factor \(1+0.0058\sin^2\alpha_b\). Even for \(y_b=y_b^{\mathrm{SM}}\), this corresponds only to \(\Gamma(h\to b\bar b)\simeq\Gamma^{\mathrm{SM}}(1.0029\pm0.29\%)\), which is beyond ordinary rate-based sensitivity [2009.02000].

Differential information can recover direct sensitivity. The proposed Higgs-factory method exploits interference in \(h\to b\bar b g\) between amplitudes containing the \(hb\bar b\) vertex and those containing an effective \(hgg\) interaction, with
\[
\mathcal{M}=e^{\pm i\alpha_b}\mathcal{M}_1+\mathcal{M}_2.
\]
The key rest-frame observable is
\[
\zeta_H\equiv \frac{2E_{b_1}E_{b_2}}{E_{b_1}^2+E_{b_2}^2}\cos\theta_{b_1b_2},
\]
which becomes most sensitive in the nearly collinear \(b\bar b\) region. The projected precision is \(\delta(\cos\alpha_b)\simeq\pm0.23\) at \(\sqrt{s}=240\ \mathrm{GeV}\) with \(5.6\ \mathrm{ab}^{-1}\), improving to \(\pm0.17\) when combined with a \(365\ \mathrm{GeV}\), \(1.5\ \mathrm{ab}^{-1}\) run [2009.02000].

Threshold behavior supplies a second discriminator. For \(e^+e^-\to b\bar b X\) through a virtual \(Z\), a scalar coupling gives
\[
\sigma(e^+e^-\to b\bar b\,h)\sim(\sqrt{s}-2m_b-m_h)^2,
\]
whereas a pseudoscalar gives
\[
\sigma(e^+e^-\to b\bar b\,A)\sim(\sqrt{s}-2m_b-m_A)^3.
\]
This distinction follows from CP and angular-momentum selection rules, but it is useful only very near threshold and for sufficiently large \(Xb\bar b\) coupling [1904.09895].

A separate issue is the sign of the bottom Yukawa coupling. In type-II 2HDM language,
\[
k_b=\sin(\beta-\alpha)-\tan\beta\cos(\beta-\alpha),
\]
and a wrong-sign coupling corresponds approximately to \(\tan\beta\,\cos(\beta-\alpha)\simeq2\). In the MSSM this regime requires extreme \(\tan\beta\) and is strongly disfavored by heavy-Higgs searches and perturbativity, whereas the NMSSM can realize \(k_b\simeq-1\) for \(\tan\beta\simeq6\)–10, \(\lambda\simeq\mathcal{O}(1)\), \(m_A\simeq300\)–\(500\ \mathrm{GeV}\), and \(\mu\simeq-(0.8\)–\(1.3)\ \mathrm{TeV}\), with correlated signatures such as \(A\to hZ\) and \(H/A\to H^\pm W^\mp\) [1802.09122].

## 5. Supersymmetric threshold effects and high-scale relations

In supersymmetric models the bottom Yukawa coupling is not determined by \(m_b\) and \(v\) alone. In the MSSM,
\[
y_b^{\mathrm{MSSM}}(M_Z)=\frac{\sqrt{2}\,m_b}{v\cos\beta},
\]
so large \(\tan\beta\) enhances the tree-level coupling by \(1/\cos\beta\). More importantly, finite threshold corrections modify the relation between the measured mass and the effective Yukawa coupling:
\[
m_b=y_b\,v\cos\beta\,(1+\Delta_b),\qquad
y_b^{\mathrm{eff}}\simeq\frac{\sqrt{2}\,m_b}{v\cos\beta}\,\frac{1}{1+\Delta_b}.
\]
At large \(\tan\beta\), the dominant one-loop contributions are approximately
\[
\Delta_b \approx \frac{2\alpha_s}{3\pi}\frac{\mu m_{\tilde g}\tan\beta}{m_{\tilde b}^2}
+\frac{y_t^2}{16\pi^2}\frac{\mu A_t\tan\beta}{m_{\tilde t}^2},
\]
arising from gluino–sbottom and chargino–stop loops [1106.6208].

These threshold effects can be resummed in an effective Lagrangian. For the neutral MSSM Higgs bosons,
\[
\tilde g_b^h=\frac{g_b^h}{1+\Delta_b}\bigl[1-\Delta_b\cot\alpha\cot\beta\bigr],\qquad
\tilde g_b^H=\frac{g_b^H}{1+\Delta_b}\bigl[1+\Delta_b\tan\alpha\cot\beta\bigr],\qquad
\tilde g_b^A=\frac{g_b^A}{1+\Delta_b}\bigl[1-\Delta_b/\tan^2\beta\bigr].
\]
The two-loop SUSY-QCD calculation reduces the residual theoretical uncertainty from \(\mathcal{O}(10\%)\) at one loop to the per-cent level [1001.1935].

At the unification scale, the bottom Yukawa becomes a probe of GUT boundary conditions. In the CMSSM, exact \(b\)–\(\tau\) unification,
\[
y_b(M_{\mathrm{GUT}})=y_\tau(M_{\mathrm{GUT}}),
\]
is possible only for very particular parameter choices. Over most viable large-\(\tan\beta\) parameter space, the ratio is shifted above unity. The quoted scan finds \(y_\tau/y_b(M_{\mathrm{GUT}})\) roughly between \(1.3\) and \(1.6\) for \(\tan\beta=30\), and between \(1.4\) and \(1.9\) for \(\tan\beta=50\), with experimentally preferred regions naturally yielding
\[
\frac{y_\tau}{y_b}(M_{\mathrm{GUT}})\approx\frac{3}{2}.
\]
This makes the “\(3/2\)” scenario more generic than exact \(b\)–\(\tau\) unification in the CMSSM [1106.6208].

## 6. Ultraviolet completions and nonminimal bottom Yukawa structures

Several ultraviolet constructions use the bottom Yukawa coupling as a structural diagnostic rather than merely a fit parameter. In an \(SU(5)\times SU(5)_\perp\) F-theory GUT, the third-family bottom Yukawa arises from a renormalizable \(10\cdot\overline{5}\cdot\overline{5}_H\) operator localized at a matter-curve intersection. The local overlap integral gives
\[
y_b(M_{\mathrm{GUT}})\simeq0.29,
\]
very close to the corresponding top value \(y_t\simeq0.31\), which points to a large-\(\tan\beta\) regime. In the symmetry limit the same operator implies \(y_b=y_\tau\), while threshold corrections and hypercharge-flux effects can split the lighter-family down-quark and charged-lepton relations without spoiling the third-family one [1009.6000].

A different realization appears in the toy \(SU(6)\) model with an intermediate \(331\) stage. There, only one electroweak doublet gets the dominant vacuum expectation value, while bottom and tau masses are generated through small induced doublet VEVs in additional multiplets. The SM-like Higgs coupling to bottoms obeys
\[
\kappa_b\simeq \frac{v^2}{v_\phi V_{331}},
\]
and requiring \(\kappa_b\simeq1\) suggests \(V_{331}\sim\mathcal{O}(10)\ \mathrm{TeV}\) for \(v_\phi\sim\mathrm{GeV}\) [2112.14509].

The bottom Yukawa can also be reduced through fermion mixing. In the vector-like quark doublet model with a new \((b',p')\) doublet of hypercharge \(Y/2=-5/6\), right-handed \(b\)–\(b'\) mixing gives
\[
\kappa_b=\frac{g_{hbb}}{g_{hbb}^{\mathrm{SM}}}=\frac{1}{1+\delta^2},
\qquad \delta\equiv \Delta/M.
\]
The combined Higgs and \(Z\)-pole fits quoted in the analysis prefer moderate suppression, for example \(\kappa_b=0.9826^{+0.0066}_{-0.0063}\) or \(\kappa_b=0.9868^{+0.0055}_{-0.0099}\), while simultaneously increasing the right-handed \(Zb\bar b\) coupling and reducing the long-standing \(A_{FB}^b\) tension [1901.05626].

In the general 2HDM without a \(Z_2\) symmetry, the bottom Yukawa sector contains an additional coupling \(\rho_{bb}\). In the alignment limit, this coupling controls processes such as
\[
bg\to bA\to bZH,\qquad gg\to b\bar bA\to b\bar bZH.
\]
The dedicated collider study finds that \(bg\to bA\to bZH\) could be discovered with \(\sim300\ \mathrm{fb}^{-1}\) if \(m_A\sim300\ \mathrm{GeV}\), while the \(gg\to b\bar bA\) mode becomes relevant at the HL-LHC. The same parameter space overlaps with the region \(|\mathrm{Im}\,\rho_{bb}|\gtrsim0.058\) highlighted for electroweak baryogenesis [1905.02137].

Taken together, these constructions show that the bottom-quark Yukawa coupling is unusually sensitive to threshold corrections, vacuum-alignment structure, fermion mixing, and GUT-scale operator selection. Its measured near-Standard-Model value constrains each of these mechanisms differently, but in every case \(y_b\) remains one of the most incisive probes of whether the Higgs sector is minimal or only effectively so.

Source: https://www.emergentmind.com/topics/bottom-quark-yukawa-coupling