Flavor Physics

Prerequisites: Standard Model of Particle Physics. Full path (32 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

The matter particles of the Standard Model come in three copies, called generations or flavors. The electron has two heavier versions, the muon and the tau. The up and down quarks are repeated as charm and strange, and as top and bottom. The copies have the same charges and feel the same forces. They differ only in mass, and the masses span a huge range: the top quark is nearly a hundred thousand times heavier than the up quark. Flavor physics studies these copies, how they differ, and how they turn into each other.

Such transitions happen through the weak interaction. A strange quark can turn into an up quark by emitting a W boson, which is why kaons decay. The strengths of these transitions are given by a small table of numbers, the CKM matrix, and its entries are far from democratic: transitions within one generation are strong, and transitions that jump between generations are suppressed. The same table contains a complex phase, which is the only established source of CP violation in the Standard Model, the difference between how matter and antimatter behave.

Some processes, such as a bottom quark turning into a strange quark and a photon, cannot happen directly in the Standard Model and only occur through quantum loops. Because they are so rare, a new heavy particle appearing in the loop could change their rate visibly. Precise measurements of rare decays and of meson-antimeson oscillations therefore test physics at energies far above what colliders reach directly.

Concrete

Flavor enters the Standard Model through the Yukawa couplings of the quarks to the Higgs doublet $H$, $$\mathcal L_Y = -Y^d_{ij}\,\overline{Q_L^i}\,H\,d_R^j - Y^u_{ij}\,\overline{Q_L^i}\,\tilde H\,u_R^j + \text{h.c.}, \qquad \tilde H = i\sigma_2 H^* ,$$ where $i,j = 1,2,3$ label generations, $Q_L^i$ are the left-handed quark doublets and $u_R^j, d_R^j$ the right-handed singlets. The Yukawa matrices $Y^u, Y^d$ are general complex $3\times 3$ matrices. After electroweak symmetry breaking, $\langle H\rangle = (0, v/\sqrt 2)^T$ with $v \approx 246\,\text{GeV}$, they give mass matrices $M^{u,d} = Y^{u,d}\,v/\sqrt2$.

Mass basis and the CKM matrix. Each mass matrix is diagonalized by two unitary matrices, $M^u = V^u_L\, \text{diag}(m_u, m_c, m_t)\, V^{u\dagger}_R$ and similarly for $M^d$. Rotating the fields to the mass basis leaves the neutral-current and kinetic terms unchanged, because they involve $\bar u_L \gamma^\mu u_L$ type combinations in which the unitary matrices cancel. The charged current mixes up and down sectors, and there the rotations do not cancel: $$\mathcal L_W = -\frac{g}{\sqrt 2}\,\overline{u_L}\,\gamma^\mu\, V\, d_L\, W^+_\mu + \text{h.c.}, \qquad V = V^{u\dagger}_L V^d_L .$$ $V$ is the unitary Cabibbo-Kobayashi-Maskawa (CKM) matrix. After removing unphysical phases by rephasing quark fields it has three mixing angles and one phase. In the Wolfenstein parameterization, with $\lambda \approx 0.22$, $$V = \begin{pmatrix} 1 - \lambda^2/2 & \lambda & A\lambda^3(\rho - i\eta) \\ -\lambda & 1-\lambda^2/2 & A\lambda^2 \\ A\lambda^3(1 - \rho - i\eta) & -A\lambda^2 & 1 \end{pmatrix} + \mathcal O(\lambda^4) ,$$ which makes the hierarchy of the matrix explicit.

Unitarity triangles and CP violation. Unitarity $V^\dagger V = 1$ gives relations such as $$V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0 ,$$ a triangle in the complex plane. Its angles are measured in $B$ meson decays, and its area is proportional to the rephasing-invariant Jarlskog invariant $J$, defined by $\text{Im}(V_{ij}V_{kl}V^*_{il}V^*_{kj}) = J\sum_{m,n}\epsilon_{ikm}\epsilon_{jln}$. CP violation in the quark sector of the Standard Model requires $J \neq 0$; experimentally $J \approx 3\times 10^{-5}$.

Flavor-changing neutral currents. Since the $Z$ and photon couplings stay flavor diagonal in the mass basis, transitions like $b \to s$ or $s \to d$ have no tree-level contribution. They arise from loops with $W$ bosons and up-type quarks, and the result is proportional to sums like $\sum_{q=u,c,t} V_{qb}^* V_{qs}\, F(m_q^2/m_W^2)$. Unitarity makes this sum vanish if all up-type masses were equal (the GIM mechanism), so these processes are suppressed by mass differences and small CKM elements. This is why rare decays such as $B_s \to \mu^+\mu^-$ and neutral meson mixing ($K^0$-$\bar K^0$, $B^0$-$\bar B^0$) are sensitive probes of new particles in the loops.

• A great introduction is Just a Taste: Lectures on Flavor Physics by Yuval Grossman, Philip Tanedo
• See also Flavor Physics Constraints for Physics Beyond the Standard Model by Gino Isidori et. al.

Abstract

Without Yukawa couplings the quark sector of the Standard Model has a global flavor symmetry $$G_F = U(3)_Q \times U(3)_u \times U(3)_d ,$$ acting on the three generations of $Q_L$, $u_R$ and $d_R$. The Yukawa terms are invariant if the couplings are promoted to spurions transforming as $Y^u \sim (3, \bar 3, 1)$ and $Y^d \sim (3, 1, \bar 3)$, that is $Y^u \to U_Q Y^u U_u^\dagger$ and $Y^d \to U_Q Y^d U_d^\dagger$. Physical observables are functions of $Y^u, Y^d$ that are invariant under $G_F$.

This gives a clean count of physical parameters. Two general complex $3\times 3$ matrices have $18$ real moduli and $18$ phases. The group $G_F$ has $3\times 3 = 9$ real rotation angles and $3\times 6 = 18$ phases. With nonzero Yukawas $G_F$ is broken to baryon number $U(1)_B$, which removes one phase from the count of usable transformations. The physical parameters are $18 - 9 = 9$ moduli (six quark masses and three mixing angles) and $18 - 17 = 1$ phase, the CKM phase. The same counting with $n$ generations gives $n(n-1)/2$ angles and $(n-1)(n-2)/2$ phases, so CP violation through quark mixing needs at least three generations.

The invariant description of CP violation is $$\det\left[\,Y^u Y^{u\dagger},\; Y^d Y^{d\dagger}\,\right] \propto J \prod_{i<j}(m_{u_i}^2 - m_{u_j}^2)\prod_{i<j}(m_{d_i}^2 - m_{d_j}^2) ,$$ which vanishes if any two up-type or any two down-type masses are degenerate, since then the remaining phase can be rotated away.

The spurion language also organizes extensions of the Standard Model. A new operator $\frac{c_{ij}}{\Lambda^2}\mathcal O_{ij}$ with generic flavor structure $c_{ij}\sim 1$ is constrained by kaon mixing to have $\Lambda$ far above the TeV scale. Minimal flavor violation is the hypothesis that $Y^u$ and $Y^d$ are the only sources of $G_F$ breaking also in the new physics, so every new flavor-violating coefficient is a $G_F$-covariant polynomial in the spurions. New effects then inherit the CKM suppression and the GIM structure, which relaxes the bounds on $\Lambda$ considerably. In the language of effective field theory, flavor physics measures the flavor structure of the Wilson coefficients of higher-dimensional operators.

Why is it interesting?

Through flavor physics it is in principle possible to probe physics at energy scales much higher than given by the collider in question.

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Group Theory
6. Hilbert Space
7. Legendre Transformation
8. Manifold
9. Newton's Second Law
10. Vector Calculus
11. Coulomb's law
12. Fiber Bundles
13. Newtonian Mechanics
14. Symmetry
15. Variational Calculus
16. Configuration Space
17. Euler-Lagrange Equations
18. Maxwell Equations
19. Special Relativity
20. Lagrangian Formalism
21. Phase Space
22. Classical Field Theory
23. Gauge Symmetry
24. Hamiltonian Formalism
25. Symmetry Breaking
26. Gauge Models
27. Goldstone's theorem
28. Schrödinger Equation
29. Canonical Quantum Mechanics
30. Higgs Mechanism
31. Canonical Quantum Field Theory
32. Standard Model of Particle Physics