Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2018 / 3 / a

Because $({\gamma }_5{)}^2=1$, its even and odd powers are ${\gamma }_5^{2r}=1$ and ${\gamma }_5^{2r+1}={\gamma }_5$. Separate the matrix exponential into these two power series:No representation of the gamma matrices is needed. Hermiticity of ${\gamma }_5$ also makes this exponential unitary for real $\alpha$, while the projectors $(1\pm {\gamma }_5)/2$ acquire opposite phases.

Put $U=e^{-i\alpha {\gamma }_5/2}$. The transformed Dirac adjoint is ${\bar{\psi }}^{\prime }={\psi }^{†}{U}^{†}{\gamma }^0=\bar{\psi }U$, since ${\gamma }^0$ anticommutes with ${\gamma }_5$. It is therefore incorrect to transform the two sides of a Yukawa bilinear with mutually inverse matrices.

Write $M(\varphi )={\varphi }_1+i{\gamma }_5{\varphi }_2$. The transformed Yukawa interaction is invariant exactly whenSince $M$ commutes with ${\gamma }_5$, this is $M({\varphi }^{\prime })=e^{i\alpha {\gamma }_5}M(\varphi )$. Use i and equate the coefficients of $1$ and $i{\gamma }_5$:This is the global SO(2) rotation of a chiral scalar-doublet Yukawa model. Equivalently, $\Phi ={\varphi }_1+i{\varphi }_2$ transforms as ${\Phi }^{\prime }=e^{i\alpha }\Phi$, while the two chiral spinor components transform with opposite half-angle phases. The sign of the scalar rotation follows from the negative sign in the specified fermion transformation.

The scalar kinetic and potential terms depend only on the rotation-invariant combinations ${\partial }_{\mu }{\varphi }_1{\partial }^{\mu }{\varphi }_1+{\partial }_{\mu }{\varphi }_2{\partial }^{\mu }{\varphi }_2$ and ${\varphi }_1^2+{\varphi }_2^2$. They are unchanged by the constant SO(2) rotation. The Yukawa term is invariant by ii. Finally, anticommutation gives $U{\gamma }^{\mu }={\gamma }^{\mu }{U}^{-1}$, soNo derivative of $U$ appears because the parameter is global. Thus the entire classical Lagrangian density has the combined chiral symmetry.

An explicit Dirac mass term instead changes asIt is not invariant for arbitrary $\alpha$ unless $m=0$. For the concrete choice $\alpha =\pi$, the original mass term reverses sign. The invariant Yukawa term can nevertheless generate a mass after a vacuum is selected, because the scalar field transforms along with the fermions; this is spontaneous rather than explicit breaking.

The scalar potential isSince ${\mu }^2,\lambda >0$, all minima form the circle ${\varphi }_1^2+{\varphi }_2^2=v^2$, with $v=\mu /\sqrt{\lambda }>0$. Choose the representative vacuum $⟨{\varphi }_1⟩=v$, $⟨{\varphi }_2⟩=0$, and write ${\varphi }_1=v+h$, ${\varphi }_2=\pi$. A nontrivial scalar rotation moves this vacuum to another minimum, so the continuous symmetry undergoes spontaneous symmetry breaking. The invariant action and the selected ground state have different stabilizers; the circle of equivalent minima is not evidence that the action lost its symmetry.

The Yukawa terms becomeComparison with the free Dirac equation givesFor the usual choice $g>0$, this is simply $gv$. The absolute value matters if the real coupling is negative: the dispersion depends on $m_{\psi }^2$, and a field redefinition can change the mass coefficient's sign. There is no explicit symmetry-breaking fermion mass parameter in the original theory.

The scalar symmetry is broken completely. Because the spinor transformation uses a half angle, $\alpha =2\pi$ leaves the scalars fixed while sending $\psi \to -\psi$; this discrete fermion parity is unbroken in the full combined transformation group. It does not remove the broken continuous generator or the associated Goldstone boson.

Using ${\mu }^2=\lambda v^2$, expand the potential around the chosen vacuum:The linear term vanishes at the minimum. With canonical real-field kinetic terms, comparison with $m_h^2{h}^2/2+m_{\pi }^2{\pi }^2/2$ givesThe massive radial scalar changes the magnitude of the vacuum field; the massless Goldstone boson changes its orientation. The symmetry is global here, so no gauge field eats that mode.

The genuinely new scalar cubic interactions are $-\lambda v{h}^3-\lambda vh{\pi }^2$. Including identical-field factorials, their Feynman vertex factors areThe quartic interactions remain $-\lambda h^4/4-\lambda h^2{\pi }^2/2-\lambda {\pi }^4/4$, with respective vertex factors $-6i\lambda ,-2i\lambda ,-6i\lambda$. The shifted Yukawa terms from iv give $h\bar{\psi }\psi$ vertex $-ig$ and $\pi \bar{\psi }\psi$ vertex $g{\gamma }_5$. These Yukawa structures were present before the shift; the new fermion bilinear is the mass term, not an additional independent interaction coupling. There are no derivative interactions from the linear Cartesian field shift.