Flavour puzzle

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

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

Intuitive

The flavor puzzle is the question why do the particles in the standard model of particle physics have the masses they have, and why do the mixing angles in the PMNS and CKM matrix have the values they have.

Concrete

In the standard model the fermion masses and mixings all come from the Yukawa couplings between the fermions and the Higgs doublet $H$,

$$-\mathscr L_Y = Y^u_{ij}\,\bar Q_i\,\tilde H\,u_j + Y^d_{ij}\,\bar Q_i\,H\,d_j + Y^e_{ij}\,\bar L_i\,H\,e_j + \text{h.c.},$$

where $i,j = 1,2,3$ label the three generations, $Q_i$ and $L_i$ are the left-handed quark and lepton doublets, $u_j, d_j, e_j$ the right-handed singlets, $\tilde H = i\sigma_2 H^*$, and the $Y$ are arbitrary complex $3\times3$ matrices. When the Higgs field gets its vacuum expectation value $\langle H\rangle = (0, v/\sqrt2)$ with $v \approx 246\,\mathrm{GeV}$, these terms become mass matrices $M^f = Y^f v/\sqrt2$.

Each mass matrix is diagonalized by two unitary matrices, $M^f = V^f_L\,\mathrm{diag}(m_1,m_2,m_3)\,V^{f\dagger}_R$. The $W$ boson couples up-type to down-type quarks, so the mismatch between the up and down rotations becomes physical. It is the CKM matrix

$$V_{\text{CKM}} = V^{u\dagger}_L V^d_L .$$

In the quark sector this leaves ten physical parameters: six masses, three mixing angles and one CP-violating phase. In the lepton sector, the analogous mixing matrix for neutrinos is the PMNS matrix.

The puzzle is the pattern of these numbers:

• The masses span many orders of magnitude. The Yukawa coupling of the top quark is close to one, $y_t = \sqrt2 m_t/v \approx 1$ with $m_t \approx 173\,\mathrm{GeV}$, while the electron has $y_e \approx 3\times10^{-6}$. Within each type the masses grow strongly from one generation to the next.
• The CKM matrix is close to the unit matrix, with a hierarchy of small off-diagonal elements. With $\lambda \approx 0.22$, roughly $|V_{us}| \sim \lambda$, $|V_{cb}| \sim \lambda^2$ and $|V_{ub}| \sim \lambda^3$.
• The PMNS matrix looks quite different: two of its mixing angles are large (about $33^\circ$ and about $45^\circ$) and one is smaller (about $8.5^\circ$).
• Neutrino masses are tiny compared with all other fermion masses. Oscillation experiments measure only the differences of their squares, $\Delta m^2_{21} \approx 7.4\times10^{-5}\,\mathrm{eV}^2$ and $|\Delta m^2_{31}| \approx 2.5\times10^{-3}\,\mathrm{eV}^2$.

In the standard model none of these numbers is predicted. They are inputs. The flavour puzzle asks whether there is a theory in which they follow from a smaller set of parameters or from a symmetry, and why there are exactly three generations.

• The Fermion Mass Problem by K.S. Babu
• A Model-Building Approach to the Origin of Flavor by E. Schuhmacher
• Pieces of the Flavour Puzzle by Ferruccio Feruglio

Abstract

Without the Yukawa couplings, the gauge-invariant kinetic terms of the standard model fermions are invariant under the flavour group

$$G_F = U(3)_Q\times U(3)_u\times U(3)_d\times U(3)_L\times U(3)_e ,$$

which acts on the generation index of each of the five gauge multiplets. The Yukawa matrices break $G_F$. They can be treated as spurions that transform as $Y^u \sim (\mathbf 3,\bar{\mathbf 3},\mathbf 1)$ and $Y^d \sim (\mathbf 3,\mathbf 1,\bar{\mathbf 3})$ under $U(3)_Q\times U(3)_u\times U(3)_d$ (and analogously $Y^e$), which makes $\mathscr L_Y$ formally invariant.

The number of physical parameters follows from this. The two quark Yukawa matrices have $2\times 18 = 36$ real parameters, 18 moduli and 18 phases. A $G_F$ transformation can remove as many parameters as there are broken generators. $U(3)^3$ has 27 generators (9 rotation angles and 18 phases), and the only unbroken subgroup is baryon number $U(1)_B$, so 26 are broken. This leaves $36 - 26 = 10$ physical parameters, $18 - 9 = 9$ moduli (six masses and three angles) and $18 - 17 = 1$ phase. With $n$ generations the same counting gives $(n-1)(n-2)/2$ CP phases, so CP violation through quark mixing requires at least three generations.

Approaches to the puzzle explain the small parameters through approximate symmetries:

• In the Froggatt-Nielsen mechanism, a $U(1)$ flavour symmetry under which the generations carry different charges is broken by the expectation value of a field $\phi$ at a high scale $M$. A Yukawa coupling allowed only with $n$ insertions of $\phi$ is suppressed by $\epsilon^n$ with $\epsilon = \langle\phi\rangle/M$, so hierarchies become powers of one small number, for example $\epsilon \sim \lambda$.
• Non-abelian discrete or continuous flavour groups acting on the generations aim at specific mixing patterns, mainly for the lepton sector.
• Minimal flavour violation is the assumption that the Yukawa matrices are the only spurions that break $G_F$, also in new physics. Under this assumption new physics near the TeV scale is compatible with the strong suppression of flavour-changing neutral current processes, but the Yukawas themselves remain unexplained.

For neutrinos, the smallness of the masses may come from the Weinberg operator $(LH)(LH)/\Lambda$, the only gauge-invariant operator of dimension five built from standard model fields, which gives Majorana masses $m_\nu \sim v^2/\Lambda$ and arises in seesaw models from heavy right-handed neutrinos.

Research

The Yukawa couplings express the flavour problem, i.e. our inability to compute quark and lepton masses and mixing angles. […]

no significant progress has ever been made in computing quark and lepton masses and mixing angles or understanding the dynamical origin of their pattern. […] Given our failure in cracking the flavour problem, it is unlikely that its solution is simple and that it manifests itself in simple ways. Only crazy results in flavour physics have a chance to be true.

https://arxiv.org/pdf/1710.07663.pdf#page23

Why is it interesting?

"In a recent interview published in CERN COURIER, Steven Weinberg was asked what single open question he would like to see answered in his lifetime, and Weinberg replied that it is only the mystery of the observed pattern of quark and lepton masses[1]. " https://arxiv.org/pdf/1712.08052.pdf

Path to this page

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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

Leads to: Flavor Democracy.