Baryon Asymmetry with CP Violation
in the Standard Model
Md Irfan
Department of Physics, Indian Institute of Technology Guwahati
Roll Number: 242121029
Today
Abstract
The observed matter–antimatter asymmetry of the Universe,
\( \etaB \equiv (n_B - n_{\bar B})/s \approx 10^{-10} \), is one of the
outstanding puzzles of modern physics.
This article examines whether the Standard Model (SM), evaluated in
the vicinity of the electroweak phase transition at \(T \sim 100\,\mathrm{GeV}\),
can account for this asymmetry.
All three Sakharov conditions — baryon-number violation, CP violation,
and departure from thermal equilibrium — are present in the SM at the
electroweak scale.
The CP-odd source is identified by embedding the CKM phase into
the finite-temperature effective action; the thermal Higgs potential
supplies the non-equilibrium background.
A detailed numerical estimate yields
\( \eta_B^{\mathrm{SM}} \approx 10^{-33} \), many orders of magnitude below
the observed value.
The quantitative shortfall establishes that electroweak baryogenesis
within the SM alone is insufficient, motivating extensions with
additional CP violation and a strongly first-order electroweak transition.
Introduction
The visible Universe is composed almost entirely of matter rather than
antimatter.
Cosmological observations — most precisely the measurement of the
baryon-to-photon ratio from Big Bang nucleosynthesis and the CMB
angular power spectrum — pin down the baryon asymmetry to
\[
\etaB \;\equiv\; \frac{\nB - \nBbar}{s} \;\approx\; 10^{-10},
\]
where \(s\) is the entropy density.
This tiny but decisive excess of baryons over antibaryons is responsible
for all the structure we observe.
A compelling mechanism to generate \(\etaB\) dynamically from an
initially symmetric state must satisfy three necessary conditions,
identified by Sakharov in 1967:
- Baryon-number violation (\(\Delta B \neq 0\));
- CP violation (so that processes producing baryons and
antibaryons differ in rate);
- Departure from thermal equilibrium (to prevent
detailed balance from erasing any generated asymmetry).
All three conditions are, in principle, realised in the Standard Model
during the electroweak epoch at \(T \sim 100\,\mathrm{GeV}\).
The central question addressed here is quantitative:
\[
\eta_B^{\mathrm{SM}} \;\stackrel{?}{\sim}\; \eta_B^{\mathrm{obs}}
\;\approx\; 10^{-10}.
\]
The calculation proceeds in three steps:
(i) identify the anomalous baryon-number violation,
(ii) embed the CKM-phase CP violation into the effective action, and
(iii) extract numerical values from the finite-temperature Higgs potential.
CP Violation and Baryon Asymmetry
The Role of CP Symmetry
Under the combined charge-conjugation and parity transformation, a
left-handed particle state maps to a right-handed antiparticle state:
\[
\begin{aligned}
CP\,|\mathrm{Particle}_{LH}\rangle &\longrightarrow
|\mathrm{Antiparticle}_{RH}\rangle, \\
C\,|e^-(\vec p,s)\rangle &= |e^+(\vec p,s)\rangle, \\
P\,|\psi(\vec p,s)\rangle &= |\psi(-\vec p,s)\rangle, \qquad
P\,|\mathrm{helicity}\rangle \to -|\mathrm{helicity}\rangle.
\end{aligned}
\]
If CP were an exact symmetry, every process producing baryons would
occur at the same rate as the corresponding process producing antibaryons,
yielding \(n_B = n_{\bar B}\) and no net asymmetry.
CP violation is therefore the second Sakharov condition.
CP Violation in the Standard Model
In the SM, CP violation in the quark sector originates from the
irreducible complex phase \(\delta\) of the Cabibbo–Kobayashi–Maskawa
(CKM) matrix \(V_{\mathrm{CKM}}\), which governs flavour-changing
charged-current weak interactions.
While this phase makes CP violation possible, the resulting effect
is quantitatively very small.
As we will demonstrate, it is insufficient on its own to account
for the observed asymmetry.
Baryon-Number Violation via the Electroweak Anomaly
Baryon Current and Its Divergence
From Noether's theorem, the baryon current and baryon number are
\[
j_B^\mu = \tfrac{1}{2}\,\bar Q\,\gamma^\mu\, Q, \qquad
B(t) = \int d^3x\; j_B^0(\mathbf{x}, t).
\]
In classical field theory, \(\partial_\mu j_B^\mu = 0\).
However, the SM electroweak theory possesses a quantum anomaly: the
one-loop triangle diagram with one axial-vector baryon current and two
gauge bosons gives a non-zero divergence,
\[
\partial_\mu j_B^\mu = n_f \left(
\frac{g^2}{32\pi^2}\, W^a_{\mu\nu}\widetilde W^{a\,\mu\nu}
- \frac{g^{\prime\,2}}{32\pi^2}\, F_{\mu\nu}\widetilde F^{\mu\nu}
\right),
\]
where \(n_f = 3\) is the number of fermion generations, \(g\) and \(g'\) are
the \(\mathrm{SU(2)_L}\) and \(\mathrm{U(1)_Y}\) gauge couplings, and
\(\widetilde V^{\mu\nu} = \tfrac{1}{2}\varepsilon^{\mu\nu\rho\sigma}V_{\rho\sigma}\)
denotes the dual field strength.
The expectation value of this anomalous divergence is
\[
\langle \partial_\mu j_B^\mu \rangle
= \int \mathcal D A_\mu^a\,\mathcal D\bar\Psi\,\mathcal D\Psi\,
\mathcal D h\;\partial_\mu j_B^\mu\;
\exp\!\left[i\int d^4x\;\mathcal L\right].
\]
This non-vanishing expectation value is the dynamical origin of
\(\Delta B \neq 0\) in the SM.
Physical Picture
The \(W_{\mu\nu}\widetilde W^{\mu\nu}\) term is a total derivative and can
be written as the divergence of the Chern–Simons current.
A change in baryon number is therefore associated with a change in the
topological Chern–Simons number of the gauge field.
At zero temperature such transitions are exponentially suppressed
(instantons, rate \(\sim e^{-4\pi/\alpha_W}\)).
At \(T \sim \TEW\), however, classical thermal fluctuations — sphalerons —
drive these transitions at a rate \(\Gamma_{\mathrm{sph}} \sim \alpha_W^4 T^4\),
making baryon-number violation efficient.
CP-Odd Part of the Effective Action
Net Baryon Production
The net baryon-number change \(\Delta_B B\) is controlled by the difference
between the effective action \(\Gamma(T)\) and its CP-conjugate
\(\bar\Gamma(T)\):
\[
\begin{aligned}
\eta &= \frac{n_B}{n_\gamma}, \quad n_B = n_b - \bar n_b, \quad
\Delta_B B = \tfrac{1}{2}(\Delta B - \Delta\bar B), \\
\eta_1 &\equiv \frac{n_B}{s} \approx \frac{\Delta_B B}{g_*},
\quad s \approx g_* n_\gamma, \\
\Delta_B B &= 12\,V t_f\,\delta^{(0)}
\left[e^{i\Gamma(T)} - e^{i\bar\Gamma(T)}\right].
\end{aligned}
\]
After applying the CP operation to the Lagrangian, the two path integrals
take the form
\[
\begin{aligned}
e^{i\Gamma} &= \int \mathcal D A^a_\mu\,\mathcal Dh\,
\mathcal D\bar\Psi\,\mathcal D\Psi\;
\exp\!\left[i\int d^4x\,(\mathcal L_1 - \bar d\,Ad)\right]
\exp\!\left[\int d^4x\,\bar d\,Bd\right], \\
e^{i\bar\Gamma} &= \int \mathcal D A^a_\mu\,\mathcal Dh\,
\mathcal D\bar\Psi\,\mathcal D\Psi\;
\exp\!\left[i\int d^4x\,(\mathcal L_1 - \bar d\,Ad)\right]
\exp\!\left[-\int d^4x\,\bar d\,Bd\right].
\end{aligned}
\]
The difference \(e^{i\Gamma} - e^{i\bar\Gamma}\) is entirely sourced by
the matrix \(B\), which carries the CP-odd information from the CKM phase.
Embedding the CKM Phase
The Standard Model Lagrangian in the relevant sector is
\[
\mathcal L = \mathcal L_{\mathrm{gauge}}
+ \mathcal L_{\mathrm{fermion}}
+ \mathcal L_{\mathrm{Higgs}}
+ \mathcal L_{\mathrm{Yukawa}}.
\]
Rotating the down-quark mass matrix \(M_d\) by the CKM matrix \(U\),
\[
U M_d U^\dagger = A + iB,
\]
the Lagrangian splits into a CP-even and a CP-odd piece:
\[
\begin{aligned}
\mathcal L &= \mathcal L_1 - \bar d\,(A + iB)\,d, \\
\mathcal L^{(\mathrm{CP})} &= \mathcal L_1 - \bar d\,(A - iB)\,d.
\end{aligned}
\]
Here \(A\) is real symmetric (CP-even) and \(B\) is real antisymmetric
(CP-odd); the CP violation is entirely encoded in \(B\).
With the effective potential \(\Veff\), the effective action reads
\[
\Gamma(\Phi) = -V\,t_f, \qquad
\Veff = -\frac{1}{T^4}\,\Veff.
\]
One expands \(e^{i\Gamma}\) in powers of \(B\) to extract the leading
CP-odd contribution.
The Baryon-Number Change
Collecting the expansion of the effective action in powers of \(B\),
the baryon-number change becomes
\[
\boxed{
\Delta_B B = i\,24\,\delta^{(0)}\,\frac{1}{T^4}\,
\sin\!\left[\frac{U_{ki}\,B_{ij}\,U^\dagger_{kj}}{T^4}
\left(-\frac{3}{4\pi^2}\,m_{d_k}^{\prime\,3}
\left(\ln\frac{m_{d_k}^{\prime\,2}}{\mu^2} - 1\right)
+ \frac{1}{2}\,m_{d_k}^{\prime}\,T^2\right)
\right]
e^{-\Geff(A)}.
}
\]
Here \(m_{d_k}^{\prime}\) are the thermally corrected down-quark masses,
\(\mu\) is the renormalisation scale, and \(e^{-\Geff(A)}\)
is the CP-even exponential suppression.
Non-Equilibrium Input from the Higgs Potential
The thermal minimum \(\Phimin(T)\) of the Higgs potential provides
the non-equilibrium background field.
At the electroweak scale, the high-temperature expansion of the potential is
\[
V(\Phi_c, T) = D(T^2 - T_0^2)\Phi_c^2
- E\,T\,\Phi_c^3
+ \frac{\lambda(T)}{4}\,\Phi_c^4,
\]
and \(\Phimin(T)\) is determined by \(\partial V/\partial\Phi_c = 0\).
Numerically, at \(T \approx 100\,\mathrm{GeV}\),
\[
\Phimin(T \approx 100\,\mathrm{GeV}) \approx 216.56\,\mathrm{GeV}.
\]
Finite-Temperature Effective Potential
Structure of the Effective Potential
At electroweak temperatures, the full effective potential is
\[
\Veff(\Phi, T) = V_0(\Phi) + V_1(\Phi) + V_1^T(\Phi, T),
\]
where the three contributions are
Tree-level potential:
\[
V_0(\Phi) = -\frac{m^2}{2}\,\Phi^2 + \frac{\lambda}{4}\,\Phi^4.
\]
Coleman–Weinberg one-loop potential:
\[
V_1(\Phi) = \sum_i n_i(-1)^{2s_i}
\frac{1}{64\pi^2}\,m_i^4(\Phi)
\left[\ln\frac{m_i^2(\Phi)}{\mu^2} - C_i\right].
\]
Thermal correction:
\[
V_1^T(\Phi, T) = \sum_{i=\mathrm{boson}} n_i\,\frac{T^4}{2\pi^2}\,
J_b\!\left(\frac{m_i^2}{T^2}\right)
- \sum_{j=\mathrm{fermion}} n_j\,\frac{T^4}{2\pi^2}\,
J_f\!\left(\frac{m_j^2}{T^2}\right),
\]
where the thermal functions are
\(J_{b/f}(x^2) = \int_0^\infty dk\,k^2\,\ln(1 \mp e^{-\sqrt{k^2+x^2}})\).
In the equations above, \(i\) labels the particle species, \(s_i\) its spin,
\(n_i\) its number of internal degrees of freedom, \(\mu\) is the
renormalisation scale, and \(C_i\) depends on the renormalisation scheme.
Representative values are: \(n_g = 3\) (gauge bosons),
\(n_q = 12\) (quarks), \(n_l = 4\) (leptons).
Numerical Inputs and Results
Electroweak-Scale Parameters
The numerical evaluation uses the following electroweak-scale inputs:
\[
\begin{aligned}
T &\approx 100\,\mathrm{GeV}, \qquad
\mu = m_t = 173.3\,\mathrm{GeV}, \qquad
\gstar = 106.75, \\
\lambda &= \frac{m_h^2}{2v^2} \approx 0.1291 \quad\text{(tree level)},\\
\lambda(T) &\approx 0.1317 \quad\text{(one-loop thermal)}.
\end{aligned}
\]
Thermal Higgs Minimum
Minimising the effective potential at \(T = 100\,\mathrm{GeV}\) gives
\[
\Phimin(T = 100\,\mathrm{GeV}) \approx 216.6\,\mathrm{GeV}.
\]
This value enters the CP-odd source formula through the thermally
corrected quark masses \(m_{d_k}'(\Phimin)\).
Analytic Estimate of the Baryon-to-Entropy Ratio
The baryon-to-entropy ratio follows from the baryon-number formula above:
\[
\eta_1 = \frac{\Delta_B B}{\gstar}
= \frac{24i\,\delta^{(0)}_{\mathrm{reg}}}{\gstar\,T^4}
\times \sin(\mathrm{arg}_b)
\times \exp[-\Geff(A)].
\]
Inserting the thermal minimum and CP-odd source gives the SM analytic
estimate
\[
\eta_B^{\mathrm{analytic}} \approx 10^{-33}.
\]
The Jarlskog Invariant and Dimensional Estimate
The physical CP-violating parameter is the rephasing-invariant
Jarlskog invariant \(J\):
\[
\begin{aligned}
J &\approx c_{12}\,s_{12}\,s_{23}\,s_{13}\,\sin\delta \notag\\
&\approx (0.9745)(0.2245)(0.0410)(0.0037)(0.93) \notag\\
&\approx 3.0 \times 10^{-5}.
\end{aligned}
\]
A dimensional analysis estimate of the full flavour contraction gives
\[
\delta_{\mathrm{CP}} \approx J\,
\frac{(m_t^2-m_c^2)(m_t^2-m_u^2)(m_c^2-m_u^2)}{T_{\mathrm{EW}}^{12}}
\times (m_b^2-m_s^2)(m_b^2-m_d^2)(m_s^2-m_d^2)
\approx 10^{-19}.
\]
Comparison with Observation
The three estimates and the observed value are placed on a
logarithmic scale in the table below.
| Quantity |
\(\log_{10}|n_B/s|\) |
| Analytic estimate (this work) |
\(\approx -33\) |
| Dimensional estimate (Jarlskog) |
\(\approx -19\) |
| Observed value |
\(\approx -10\) |
The full flavour contraction confirms that the SM result lies many
orders of magnitude below the observed asymmetry, regardless of the
level of approximation used.
Conclusions
The analysis establishes the following points:
-
Baryon-number violation exists in the SM.
The electroweak anomaly produces \(\partial_\mu j_B^\mu \neq 0\), and
sphaleron processes change the baryon number efficiently at
\(T \sim \TEW\) via anomaly-driven transitions in the Chern–Simons number.
-
A CP-odd source can be computed systematically.
Rotating the down-quark mass matrix by the CKM matrix and
decomposing it as \(UM_d U^\dagger = A + iB\) makes the CP-odd piece
\(B\) explicit in the effective action.
The resulting formula captures the CP-violating contribution to
\(\Delta_B B\) at leading order.
-
Thermal effects can be included consistently.
The finite-temperature effective potential provides the
non-equilibrium background via \(\Phimin(T)\) and the thermally
corrected quark masses.
-
The SM prediction falls catastrophically short.
The numerical result is \(\eta_B^{\mathrm{SM}} \approx 10^{-33}\),
compared to \(\eta_B^{\mathrm{obs}} \approx 10^{-10}\), a shortfall
of 23 orders of magnitude.
Two separate deficiencies are responsible:
- The CKM CP violation is too weak, and the Jarlskog invariant
plus mass-splitting suppressions reduce \(\delta_{\mathrm{CP}}\)
to \(\sim 10^{-19}\).
- The electroweak transition is not strongly first-order for
\(m_h \approx 125\,\mathrm{GeV}\); perturbative analyses show
it is at most a smooth crossover, removing the necessary
departure from equilibrium.
Final verdict.
\[
\eta_B^{\mathrm{SM}} \;\ll\; \eta_B^{\mathrm{obs}}.
\]
So the Standard Model cannot explain the observed matter–antimatter
asymmetry of the Universe.
Successful electroweak baryogenesis therefore requires physics beyond
the Standard Model: additional CP-violating phases and/or new dynamics
that drive the electroweak transition strongly first-order.
Candidate extensions include the MSSM, singlet extensions of the Higgs
sector, and other BSM scenarios that enrich both the CP structure and
the electroweak potential.
Acknowledgements
The author thanks the Department of Physics, IIT Guwahati, for
providing the research environment in which this work was carried out.
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