Editing Chiral anomaly (section)

==An example: baryon number non-conservation==
The Standard Model of [[electroweak]] interactions has all the necessary ingredients for successful [[baryogenesis]], although these interactions have never been observed<ref>{{cite journal | last1=Eidelman | first1=S. | last2=Hayes | first2=K.G. | last3=Olive | first3=K.A. | last4=Aguilar-Benitez | first4=M. | last5=Amsler | first5=C. | last6=Asner | first6=D. | last7=Babu | first7=K.S. | last8=Barnett | first8=R.M. | last9=Beringer | first9=J. | last10=Burchat | first10=P.R. | last11=Carone | first11=C.D. | last12=Caso | first12=S. | last13=Conforto | first13=G. | last14=Dahl | first14=O. | last15=D'Ambrosio | first15=G. | last16=Doser | first16=M. | last17=Feng | first17=J.L. | last18=Gherghetta | first18=T. | last19=Gibbons | first19=L. | last20=Goodman | first20=M. | display-authors=5|collaboration=Particle Data Group| title=Review of Particle Physics | journal=Physics Letters B | publisher=Elsevier BV | volume=592 | issue=1–4 | year=2004 | issn=0370-2693 | doi=10.1016/j.physletb.2004.06.001|arxiv=astro-ph/0406663 | pages=1–5| bibcode=2004PhLB..592....1P }}</ref> and may be insufficient to explain the total [[baryon number]] of the observed universe if the initial baryon number of the universe at the time of the Big Bang is zero. Beyond the violation of [[charge conjugation]] <math>C</math> and [[CP violation]] <math>CP</math> (charge+parity), baryonic charge violation appears through the '''Adler–Bell–Jackiw anomaly''' of the  <math>U(1)</math> group.

Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly. The classic electroweak [[Lagrangian (field theory)|Lagrangian]] conserves [[baryon]]ic charge. Quarks always enter in bilinear combinations <math>q\bar q</math>, so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current <math>J_\mu^B</math> is conserved:

:<math>\partial^\mu J_\mu^B = \sum_j \partial^\mu(\bar q_j \gamma_\mu q_j) = 0. </math>

However, quantum corrections known as the [[sphaleron]] destroy this [[conservation law]]:  instead of zero in the right hand side of this equation, there is a non-vanishing quantum term,
:<math>\partial^\mu J_\mu^B = \frac{g^2 C}{16\pi^2} G^{\mu\nu a} \tilde{G}_{\mu\nu}^a,</math>
where {{mvar|C}} is a numerical constant vanishing for ℏ =0,
:<math>\tilde{G}_{\mu\nu}^a = \frac{1}{2} \epsilon_{\mu\nu\alpha\beta} G^{\alpha\beta a},</math>
and the gauge field strength <math>G_{\mu\nu}^a</math> is given by the expression
:<math>G_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{a}_{bc} A_\mu^b A_\nu^c  ~. </math>

Electroweak sphalerons can only change the baryon and/or lepton number by 3 or multiples of 3 (collision of three baryons into three leptons/antileptons and vice versa).

An important fact is that the anomalous current non-conservation is proportional to the total derivative of a vector operator, <math>G^{\mu\nu a}\tilde{G}_{\mu\nu}^a = \partial^\mu K_\mu</math> (this is non-vanishing due to [[instanton]] configurations of the gauge field, which are [[pure gauge]] at the infinity), where the anomalous current <math>K_\mu</math> is
:<math>K_\mu = 2\epsilon_{\mu\nu\alpha\beta} \left( A^{\nu a} \partial^\alpha A^{\beta a} + \frac{1}{3} f^{abc} A^{\nu a} A^{\alpha b} A^{\beta c} \right),</math>
which is the [[Hodge star|Hodge dual]] of the [[Chern–Simons]] 3-form.

===Geometric form===

In the language of [[differential form]]s, to any self-dual curvature form <math>F_A</math> we may assign the abelian 4-form <math>\langle F_A\wedge F_A\rangle:=\operatorname{tr}\left(F_A\wedge F_A\right)</math>.  [[Chern–Weil theory]] shows that this 4-form is locally ''but not globally'' exact, with potential given by the [[Chern–Simons form|Chern–Simons 3-form]] locally:

:<math>d\mathrm{CS}(A)=\langle F_A\wedge F_A\rangle</math>.

Again, this is true only on a single chart, and is false for the global form <math>\langle F_\nabla\wedge F_\nabla\rangle</math> unless the instanton number vanishes.

To proceed further, we attach a "point at infinity" {{math|''k''}} onto <math>\mathbb{R}^4</math> to yield <math>S^4</math>, and use the [[clutching construction]] to chart principal A-bundles, with one chart on the neighborhood of {{math|''k''}} and a second on <math>S^4-k</math>.  The thickening around {{math|''k''}}, where these charts intersect, is trivial, so their intersection is essentially <math>S^3</math>.  Thus instantons are classified by the third [[homotopy group]] <math>\pi_3(A)</math>, which for <math>A = \mathrm{SU(2)}\cong S^3</math> is simply [[Homotopy groups of spheres|the third 3-sphere group]] <math>\pi_3(S^3)=\mathbb{Z}</math>.

The divergence of the baryon number current is (ignoring numerical constants)

:<math>\mathbf{d}\star j_b = \langle F_\nabla\wedge F_\nabla\rangle</math>,

and the instanton number is

:<math>\int_{S^4} \langle F_\nabla\wedge F_\nabla\rangle\in\mathbb{N}</math>.