ScientificIrfan

<!DOCTYPE html>

<head>

<title>Baryon Asymmetry with CP Violation in the Standard Model</title>

window.MathJax = {

tex: {

inlineMath: [['$', '$'], ['\$', '\$']],

displayMath: [['$$', '$$'], ['\\[', '\\]']],

packages: {'[+]': ['ams']}

svg: {

fontCache: 'global'

}

};

</script>

<style>

body {

font-family: Arial, sans-serif;

line-height: 1.7;

max-width: 900px;

margin: 0 auto;

padding: 24px;

color: #111;

background: #fff;

}

h1, h2, h3 {

line-height: 1.3;

}

h1 {

text-align: center;

margin-bottom: 6px;

}

.author {

text-align: center;

margin-top: 0;

margin-bottom: 18px;

}

.abstract-box {

border: 1px solid #ddd;

padding: 16px;

border-radius: 8px;

background: #fafafa;

margin: 20px 0;

}

.hr {

border-top: 1px solid #bbb;

margin: 18px 0;

}

.toc a {

text-decoration: none;

color: #0b57d0;

}

.bib li {

margin-bottom: 10px;

}

.eq {

overflow-x: auto;

margin: 16px 0;

}

</style>

</head>

<body>

<h1><strong>Baryon Asymmetry with CP Violation<br>in the Standard Model</strong></h1>

Md Irfan<br>

<small>Department of Physics, Indian Institute of Technology Guwahati</small><br>

<small>Roll Number: 242121029</small>

</p>

<p style="text-align:center;"><em>Today</em></p>

<h3>Abstract</h3>

<p>

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.

</p>

</div>

<h3>Contents</h3>

<ul>

<li><a href="#intro">Introduction</a></li>

<li><a href="#cpv">CP Violation and Baryon Asymmetry</a></li>

<li><a href="#anomaly">Baryon-Number Violation via the Electroweak Anomaly</a></li>

<li><a href="#effective_action">CP-Odd Part of the Effective Action</a></li>

<li><a href="#formula">CP Symmetry Breaking and the Baryon-Number Formula</a></li>

<li><a href="#potential">Finite-Temperature Effective Potential</a></li>

<li><a href="#numerics">Numerical Inputs and Results</a></li>

<li><a href="#conclusions">Conclusions</a></li>

<li><a href="#acknowledgements">Acknowledgements</a></li>

<li><a href="#bibliography">Bibliography</a></li>

</ul>

</div>

<h2 id="intro">Introduction</h2>

<p>

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

</p>

\etaB \;\equiv\; \frac{\nB - \nBbar}{s} \;\approx\; 10^{-10},

</div>

<p>

where $s$ is the entropy density.

This tiny but decisive excess of baryons over antibaryons is responsible

for all the structure we observe.

</p>

<p>

A compelling mechanism to generate $\etaB$ dynamically from an

initially symmetric state must satisfy three necessary conditions,

identified by Sakharov in 1967:

</p>

<ol>

<li><strong>Baryon-number violation</strong> ($\Delta B \neq 0$);</li>

<li><strong>CP violation</strong> (so that processes producing baryons and

antibaryons differ in rate);</li>

<li><strong>Departure from thermal equilibrium</strong> (to prevent

detailed balance from erasing any generated asymmetry).</li>

</ol>

<p>

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:

</p>

\eta_B^{\mathrm{SM}} \;\stackrel{?}{\sim}\; \eta_B^{\mathrm{obs}}

\;\approx\; 10^{-10}.

</div>

<p>

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.

</p>

<h2 id="cpv">CP Violation and Baryon Asymmetry</h2>

<h3>The Role of CP Symmetry</h3>

<p>

Under the combined charge-conjugation and parity transformation, a

left-handed particle state maps to a right-handed antiparticle state:

</p>

\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}

</div>

<p>

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.

</p>

<h3>CP Violation in the Standard Model</h3>

<p>

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.

</p>

<h2 id="anomaly">Baryon-Number Violation via the Electroweak Anomaly</h2>

<h3>Baryon Current and Its Divergence</h3>

<p>

From Noether's theorem, the baryon current and baryon number are

</p>

j_B^\mu = \tfrac{1}{2}\,\bar Q\,\gamma^\mu\, Q, \qquad

B(t) = \int d^3x\; j_B^0(\mathbf{x}, t).

</div>

<p>

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,

</p>

\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),

</div>

<p>

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.

</p>

<p>

The expectation value of this anomalous divergence is

</p>

\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].

</div>

<p>

This non-vanishing expectation value is the dynamical origin of

$\Delta B \neq 0$ in the SM.

</p>

<h3>Physical Picture</h3>

<p>

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.

</p>

<h2 id="effective_action">CP-Odd Part of the Effective Action</h2>

<h3>Net Baryon Production</h3>

<p>

The net baryon-number change $\Delta_B B$ is controlled by the difference

between the effective action $\Gamma(T)$ and its CP-conjugate

$\bar\Gamma(T)$:

</p>

\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}

</div>

<p>

After applying the CP operation to the Lagrangian, the two path integrals

take the form

</p>

\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}

</div>

<p>

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.

</p>

<h3>Embedding the CKM Phase</h3>

<p>

The Standard Model Lagrangian in the relevant sector is

</p>

\mathcal L = \mathcal L_{\mathrm{gauge}}

+ \mathcal L_{\mathrm{fermion}}

+ \mathcal L_{\mathrm{Higgs}}

+ \mathcal L_{\mathrm{Yukawa}}.

</div>

<p>

Rotating the down-quark mass matrix $M_d$ by the CKM matrix $U$,

</p>

U M_d U^\dagger = A + iB,

</div>

<p>

the Lagrangian splits into a CP-even and a CP-odd piece:

</p>

\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}

</div>

<p>

Here $A$ is real symmetric (CP-even) and $B$ is real antisymmetric

(CP-odd); the CP violation is entirely encoded in $B$.

</p>

<p>

With the effective potential $\Veff$, the effective action reads

</p>

\Gamma(\Phi) = -V\,t_f, \qquad

\Veff = -\frac{1}{T^4}\,\Veff.

</div>

<p>

One expands $e^{i\Gamma}$ in powers of $B$ to extract the leading

CP-odd contribution.

</p>

<h2 id="formula">CP Symmetry Breaking and the Baryon-Number Formula</h2>

<h3>The Baryon-Number Change</h3>

<p>

Collecting the expansion of the effective action in powers of $B$,

the baryon-number change becomes

</p>

\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)}.

}

</div>

<p>

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.

</p>

<h3>Non-Equilibrium Input from the Higgs Potential</h3>

<p>

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

</p>

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,

</div>

<p>

and $\Phimin(T)$ is determined by $\partial V/\partial\Phi_c = 0$.

Numerically, at $T \approx 100\,\mathrm{GeV}$,

</p>

\Phimin(T \approx 100\,\mathrm{GeV}) \approx 216.56\,\mathrm{GeV}.

</div>

<h2 id="potential">Finite-Temperature Effective Potential</h2>

<h3>Structure of the Effective Potential</h3>

<p>

At electroweak temperatures, the full effective potential is

</p>

\Veff(\Phi, T) = V_0(\Phi) + V_1(\Phi) + V_1^T(\Phi, T),

</div>

<p>

where the three contributions are

</p>

<p><strong>Tree-level potential:</strong></p>

V_0(\Phi) = -\frac{m^2}{2}\,\Phi^2 + \frac{\lambda}{4}\,\Phi^4.

</div>

<p><strong>Coleman–Weinberg one-loop potential:</strong></p>

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].

</div>

<p><strong>Thermal correction:</strong></p>

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),

</div>

<p>

where the thermal functions are

$J_{b/f}(x^2) = \int_0^\infty dk\,k^2\,\ln(1 \mp e^{-\sqrt{k^2+x^2}})$.

</p>

<p>

In Eqs. 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).

</p>

<h2 id="numerics">Numerical Inputs and Results</h2>

<h3>Electroweak-Scale Parameters</h3>

<p>

The numerical evaluation uses the following electroweak-scale inputs:

</p>

\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}

</div>

<h3>Thermal Higgs Minimum</h3>

<p>

Minimising the effective potential at $T = 100\,\mathrm{GeV}$ gives

</p>

\Phimin(T = 100\,\mathrm{GeV}) \approx 216.6\,\mathrm{GeV}.

</div>

<p>

This value enters the CP-odd source formula through the thermally

corrected quark masses $m_{d_k}'(\Phimin)$.

</p>

<h3>Analytic Estimate of the Baryon-to-Entropy Ratio</h3>

<p>

The baryon-to-entropy ratio follows from Eq. above:

</p>

\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)].

</div>

<p>

Inserting the thermal minimum and CP-odd source gives the SM analytic

estimate

</p>

\eta_B^{\mathrm{analytic}} \approx 10^{-33}.

</div>

<h3>The Jarlskog Invariant and Dimensional Estimate</h3>

<p>

The physical CP-violating parameter is the rephasing-invariant

Jarlskog invariant $J$:

</p>

\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}

</div>

<p>

A dimensional analysis estimate of the full flavour contraction gives

</p>

\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}.

</div>

<h3>Comparison with Observation</h3>

<p>

The three estimates and the observed value are placed on a

logarithmic scale in the table below.

</p>

<tr>

<th>Quantity</th>

</tr>

<tr>

<td>Analytic estimate (this work)</td>

<td>$\approx -33$</td>

</tr>

<tr>

<td>Dimensional estimate (Jarlskog)</td>

<td>$\approx -19$</td>

</tr>

<tr>

<td>Observed value</td>

<td>$\approx -10$</td>

</tr>

</table>

<p>

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.

</p>

<h2 id="conclusions">Conclusions</h2>

<p>The analysis establishes the following points:</p>

<ol>

<li>

<strong>Baryon-number violation exists in the SM.</strong>

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.

</li>

<li>

<strong>A CP-odd source can be computed systematically.</strong>

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.

</li>

<li>

<strong>Thermal effects can be included consistently.</strong>

The finite-temperature effective potential provides the

non-equilibrium background via $\Phimin(T)$ and the thermally

corrected quark masses.

</li>

<li>

<strong>The SM prediction falls catastrophically short.</strong>

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:

<ul>

<li>The CKM CP violation is too weak, and the Jarlskog invariant

plus mass-splitting suppressions reduce $\delta_{\mathrm{CP}}$

to $\sim 10^{-19}$.</li>

<li>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.</li>

</ul>

</li>

</ol>

<p><strong>Final verdict.</strong></p>

\eta_B^{\mathrm{SM}} \;\ll\; \eta_B^{\mathrm{obs}}.

</div>

<p>

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.

</p>

<h2 id="acknowledgements">Acknowledgements</h2>

<p>

The author thanks the Department of Physics, IIT Guwahati, for

providing the research environment in which this work was carried out.

</p>

<h2 id="bibliography">Bibliography</h2>

<li>

R. Jora, <em>Baryon Asymmetry in the Standard Model Revisited</em>,

Int. J. Mod. Phys. A <strong>33</strong>, 1850199 (2018).

doi:10.1142/S0217751X18501992.

</li>

<li>

M. E. Peskin and D. V. Schroeder, <em>An Introduction to Quantum Field Theory</em>

(Addison-Wesley, Reading, Massachusetts, 1995). ISBN: 9780201503975.

</li>

<li>

M. Quirós, <em>Finite Temperature Field Theory and Phase Transitions</em>,

ICTP Summer School Lectures (1999), arXiv:hep-ph/9901312.

</li>

<li>

M. Trodden, <em>Electroweak Baryogenesis</em>, Rev. Mod. Phys. <strong>71</strong>, 1463 (1999).

doi:10.1103/RevModPhys.71.1463.

</li>

<li>

N. Tuning, <em>Lecture Notes on CP Violation</em> (2020).

</li>

</ol>

</body>

</html>

Search This Blog

ScientificIrfan

Comments

Post a Comment

Popular posts from this blog

Quantum Tunneling

Scientificirfan Biography

The Concept Of Schrodinger Equation