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After electroweak symmetry breaking, the Higgs field has vacuum expectation value

⟨ H ⟩ = (0, v / 2)^{T}

. A Yukawa interaction

- y_{f} \overset{ˉ}{f}_{L} H f_{R} + h.c.

then becomes a fermion mass with

m_{f} = y_{f} v / 2

. The Higgs gauge-covariant kinetic term

∣ D_{μ} H ∣^{2}

gives

m_{W} = \frac{gv}{2}, m_{Z} = \frac{v}{2} g^{2} + g^{'2},

while the radial fluctuation is the massive Higgs boson. The photon and all eight gluons correspond to unbroken gauge generators and remain massless. In the minimal renormalizable Standard Model, which has no right-handed neutrinos, the neutrinos also remain massless; adding right-handed neutrinos permits Higgs-generated Dirac masses.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 2 / a / Solution Created 2026-09-28 Updated 2026-10-03

With

T^{a} = τ^{a} /2

, hypercharge

Y_{H} = 1/2

, and

D_{μ} H = (\partial_{μ} - i g W_{μ}^{a} T^{a} - \frac{i g ^{'}}{2} B_{μ}) H,

the most general local renormalizable bosonic Lagrangian is

L_{bos} = - \frac{1}{4} W_{μν}^{a} W^{a μν} - \frac{1}{4} B_{μν} B^{μν} + (D_{μ} H)^{†} D^{μ} H - m^{2} H^{†} H - λ (H^{†} H)^{2} .

Canonical normalization leaves four parameters:

g, g^{'}, m^{2}, λ .

(3)

Stability requires

λ > 0

. If

m^{2} > 0

, the minimum is

H = 0

and the electroweak symmetry is unbroken. If

m^{2} < 0

H^{†} H = \frac{v ^{2}}{2}, v^{2} = - \frac{m ^{2}}{λ},

(4)

and the electroweak symmetry breaking pattern is

S U (2)_{L} \times U (1)_{Y} ⟶ U (1)_{EM} .

(5)

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 4 / Solution Created 2026-09-24 Updated 2026-10-03

In the convention

(S U (3)_{c}, S U (2)_{L})_{Y}

, one Standard Model fermion generation including a right-handed neutrino is

Q_{L} : (3, 2)_{1/6}, u_{R} : (3, 1)_{2/3}, d_{R} : (3, 1)_{- 1/3},

L_{L} : (1, 2)_{- 1/2}, e_{R} : (1, 1)_{- 1}, ν_{R} : (1, 1)_{0} .

A bare Dirac mass term pairs left- and right-handed fields in the same gauge representation, but every charged left-handed Standard Model fermion is an

S U (2)_{L}

doublet while its right-handed partner is a singlet. The Higgs field

H : (1, 2)_{1/2}

and

H = i σ^{2} H^{*} : (1, 2)_{- 1/2}

permit the gauge-invariant Yukawa interactions

y_{d} \overset{ˉ}{Q}_{L} H d_{R} + y_{u} \overset{ˉ}{Q}_{L} H u_{R} + y_{e} \overset{ˉ}{L}_{L} H e_{R} + y_{ν} \overset{ˉ}{L}_{L} H ν_{R} + h.c.

(3)

The neutral Higgs vacuum expectation value turns them into masses after electroweak symmetry breaking.

Because

ν_{R}

is a gauge singlet, it may also have a large Majorana mass term

M

. Together with its Dirac mass

m_{D}

, the seesaw mechanism gives a light neutrino mass

m_{ν} ≃ m_{D}^{2} / M

. If no right-handed neutrino is retained, the same low-energy physics is encoded by the dimension-five Weinberg operator

(L_{L} H) (L_{L} H) /Λ

Now let

ϕ

be an electroweak scalar triplet with

Y = 1

. Its

T^{3}

weights are

1, 0, - 1

, and electric charge is

Q = T^{3} + Y

, so its components have charges

ϕ^{++} : Q = 2, ϕ^{+} : Q = 1, ϕ^{0} : Q = 0 .

(4)

In the given matrix convention,

ϕ^{++} = \frac{ϕ _{1} - i ϕ _{2}}{2}, ϕ^{+} = \frac{ϕ _{3}}{2}, ϕ^{0} = \frac{ϕ _{1} + i ϕ _{2}}{2} .

(5)

Thus the electromagnetic-neutral direction satisfies

ϕ_{3} = 0

and

ϕ_{1} - i ϕ_{2} = 0

The gauge-invariant type-II seesaw mechanism Yukawa interaction is

L_{Y} = - \frac{1}{2} y_{ϕ} L_{L}^{T} C i σ^{2} ϕ L_{L} + h.c. .

(6)

The two lepton doublets have total hypercharge

- 1

, which is cancelled by

Y (ϕ) = 1

, and the displayed

S U (2)

contraction is a singlet. Expanding it contains

- \frac{y _{ϕ}}{4} (ϕ_{1} + i ϕ_{2}) ν_{L}^{T} C ν_{L} + \dots .

(7)

Therefore a neutral condensate $⟨ ϕ^{0} ⟩ \neq = 0$ preserves electromagnetism and generates a left-handed-neutrino Majorana mass term proportional to $y_{ϕ} ⟨ ϕ^{0} ⟩$ .

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 305 / 4 / i / Solution Created 2026-09-24 Updated 2026-09-25

A Dirac mass term directly pairs a left-handed weak doublet with a right-handed weak singlet and is therefore not electroweak gauge invariant. Introduce the Higgs doublet

H : (1, 2)_{1/2}

and

H = i σ_{2} H^{*} : (1, 2)_{- 1/2}

. The gauge-invariant Yukawa terms are

- \overset{ˉ}{Q}_{L} Y_{d} H d_{R} - \overset{ˉ}{Q}_{L} Y_{u} H u_{R} - \overset{ˉ}{L}_{L} Y_{e} H e_{R} - \overset{ˉ}{L}_{L} Y_{ν} H ν_{R} + h.c.

When electroweak symmetry breaking gives

⟨ H ⟩ = (0, v_{H} / 2)^{T}

, these become Dirac mass matrices

M_{f} = Y_{f} v_{H} / 2

Past exam of the particle and nuclear physics course of the University of Cambridge / 2017 / 2 / c / Solution 2026-10-18

, the electroweak precision test with the W, top and Higgs masses uses

Production and identification. At a hadron collider, initial partons carry fractions of the proton momenta; observable production rates combine parton-level amplitudes with parton distribution functions.

Top quark pairs are produced through $gg \to t \overset{ˉ}{t}$ and $q \overset{q}{ˉ} \to g^{*} \to t \overset{ˉ}{t}$ . Single top production uses the weak charged current, for example $q b \to q^{'} t$ through a virtual W boson. The dominant decay is $t \to W^{+} b$ , and similarly for the antiquark. The subsequent W boson decays produce dilepton, lepton-plus-jets and all-hadronic channels. Reconstructing the W and b systems constrains the top mass. The top decays before forming an ordinary long-lived top hadron.
An on-shell W boson is produced by quark-antiquark annihilation, for example $u \overset{ˉ}{d} \to W^{+}$ . Other sources include top decay and boson-pair production; at a lepton collider, $e^{+} e^{-} \to W^{+} W^{-}$ provides a clean pair channel. The W boson decays to $ℓ ν_{ℓ}$ or to a kinematically accessible quark pair. Neglecting masses and strong corrections, the three lepton channels and two up-type quarks with three colors give approximately one-third leptonic and two-thirds hadronic branching. The leptonic modes combine a charged lepton with missing transverse momentum; the transverse-mass distribution constrains $m_{W}$ .
Higgs boson production includes gluon fusion $gg \to H$ through a heavy-quark loop, vector-boson fusion $qq \to qq H$ , associated $q \overset{q}{ˉ} \to W H, Z H$ , and $t \overset{ˉ}{t} H$ production. At a lepton collider, $e^{+} e^{-} \to Z H$ is another example. For a Higgs mass near $125 GeV$ , important decays include $b \overset{ˉ}{b}$ , $W W^{*}$ , $gg$ , $τ^{+} τ^{-}$ and $Z Z^{*}$ . The less frequent $γγ$ mode and $Z Z^{*} \to 4 ℓ$ mode give especially clean reconstructed mass peaks. The asterisk denotes an off-shell daughter boson; two on-shell W or Z bosons are inaccessible at that Higgs mass.

Why the three masses form a test. At tree level, electroweak symmetry breaking gives

m_{W} = gv /2

and

m_{Z} = g^{2} + g^{'2} v /2

. The Higgs mechanism relates masses to the vacuum field, but it does not by itself predict the top Yukawa coupling or the Higgs self-coupling, so the three masses are not fixed independently of measured parameters.

The stronger test comes from radiative corrections. With accurately measured

α

G_{F}

and

m_{Z}

m_{W}^{2} (1 - \frac{m _{W}^{2}}{m _{Z}^{2}}) = \frac{π α}{2 G _{F} ( 1 - Δ r )} .

The top-bottom loop contributes approximately

Δ ρ = 3 G_{F} m_{t}^{2} / (82 π^{2})

, while the Higgs contribution has a characteristic logarithmic mass dependence. Hence direct measurements of

m_{t}

and

m_{H}

turn the precision calculation into a prediction for

m_{W}

. Comparing that prediction with the direct W-mass measurement overconstrains the Standard Model; disagreement beyond experimental and theoretical uncertainties could indicate additional particles or interactions in the loops. The test concerns the relation among independently measured quantities, rather than merely the existence of the three particles.

Past exam of the particle Astrophysics course of the University of Cambridge / 2016 / Paper 1 / 1 / Solution 2026-10-20

The known active neutrinos are too light and too fast to supply the dominant cold dark matter. Their relativistic decoupling leaves substantial free streaming, which erases rather than seeds sufficiently small-scale structure; their allowed mass density is also inadequate for the dominant cold component. Changing a light active neutrino from Dirac to Majorana does not remove those problems. What can change the picture is an enlarged neutral-fermion sector containing sterile states with different masses, production and decay rates. A sufficiently long-lived sterile neutrino in a Majorana state can be a dark-matter candidate; Majorana character alone supplies neither stability nor the correct abundance.

For the spinor calculation, put

a = p_{z} / (E + m)

and define

A = N (1 - a) /2

B = N (1 + a) /2

. Applying the given chiral projectors gives

U_{1 L} = A 10 - 1 0, U_{1 R} = B 1010, V_{2 R} = B 1010 .

Consequently

\overline{U}_{1 R} U_{1 L} = 2 A B = \frac{E + m}{2} (1 - \frac{p _{z}^{2}}{( E + m ) ^{2}}) = m, \overline{V}_{2 R} U_{1 L} = 2 A B = m,

where the last equality uses

E^{2} - p_{z}^{2} = m^{2}

. Same-chirality Lorentz scalar products instead vanish: for example

\overline{U}_{1 L} U_{1 L} = 0

. The nonzero overlaps demonstrate the Lorentz scalar structures coupling left and right chirality. For

U

U

this is a particle-number-preserving Dirac coupling; for

U

to the charge-conjugate

V

sector it can describe a lepton-number-violating Majorana coupling.

There is a normalization subtlety: both displayed overlaps equal the on-shell mass parameter used to define the spinors. A chiral scalar spinor overlap verifies a coupling structure without fixing an independent Lagrangian coefficient

M_{L}

. Repeating the on-shell construction with mass parameter

M_{L}

would instead give the corresponding overlap

M_{L}

; this is a different input, rather than a newly derived parameter. The independent coefficients of a Dirac mass term and a Majorana mass term are defined by

L_{m} = - m (\overset{ν}{ˉ}_{R} ν_{L} + \overset{ν}{ˉ}_{L} ν_{R}) - \frac{1}{2} (M_{L} \overline{ν_{L}^{c}} ν_{L} + M_{R} \overline{ν_{R}^{c}} ν_{R} + h.c.) .

(3)

The Majorana terms connect a field with its charge conjugate and violate the corresponding continuous lepton phase symmetry by two units. With only unbroken Standard Model gauge symmetry, a left-handed Majorana mass also requires electroweak symmetry breaking or an appropriate higher-dimensional operator; it is not an arbitrary gauge-invariant bare mass for a lepton doublet.

Use an all-left-handed Weyl basis

n_{L} = (ν_{L}, ν_{R}^{c})^{T}

. For real parameters the one-family Dirac-Majorana mass matrix is

M = (M_{L} m m M_{R}), L_{m} = - \frac{1}{2} \overline{n_{L}^{c}} M n_{L} + h.c.

(4)

Writing

(ν_{1 L}, ν_{2 L})^{T} = (cos θ sin θ - sin θ cos θ) (ν_{L}, ν_{R}^{c})^{T}

gives an orthogonal diagonalization, with

tan 2 θ = 2 m / (M_{R} - M_{L})

up to the rotation-sign convention. Massive-particle chirality and helicity are not identical; the familiar left/right helicity language becomes an approximation in the relativistic regime.

The characteristic polynomial is

det (M - λ 1) = λ^{2} - (M_{L} + M_{R}) λ + M_{L} M_{R} - m^{2} .

(5)

Its roots are

λ_{\pm} = \frac{1}{2} [M_{R} + M_{L} \pm (M_{R} - M_{L})^{2} + 4 m^{2}] .

(6)

For Majorana fermions the nonnegative physical masses, as explained by physical Majorana mass and signed eigenvalue, are

∣ λ_{\pm} ∣

after a possible field rephasing; a negative real eigenvalue is not a negative physical rest mass. When

M_{R} ≫ ∣ m ∣, ∣ M_{L} ∣

λ_{light} ≃ M_{L} - \frac{m ^{2}}{M _{R}}, λ_{heavy} ≃ M_{R} + \frac{m ^{2}}{M _{R}} .

(7)

The light state is predominantly

ν_{L} - (m / M_{R}) ν_{R}^{c}

, while the heavy state is predominantly

ν_{R}^{c} + (m / M_{R}) ν_{L}

. Each mass eigenstate combines its Weyl component and charge conjugate to form a Majorana field.

For

M_{R} = 1 0^{12} GeV

, the heavy mass is approximately $1 0^{12} GeV$ . In the type-I limit

M_{L} ≃ 0

m_{light} ≃ 1 0^{- 3} (\frac{m}{1 GeV})^{2} eV .

(8)

The seesaw light-mass estimate needs a Dirac scale: no unique numerical light mass follows because the Dirac mass and

M_{L}

were not supplied. As an explicitly conditional illustration, choosing

m = 100 GeV

and

M_{L} = 0

gives

10 eV

; this is an assumed Dirac scale, not additional exam data or a prediction from

M_{R}

alone. A heavy seesaw state is also not automatically stable enough to be dark matter.

Figure 1.
A large right-handed mass produces reciprocal light and heavy scales
. Original exact one-family eigenvalues for $M_{L} = 0$ , expressed relative to an unspecified fixed Dirac mass. Negative signed eigenvalues are shown by their positive physical masses; no numerical Dirac mass is assumed. **Details**.

The conventional minimal Pati-Salam model gauge group is

S U (4)_{C} \times S U (2)_{L} \times S U (2)_{R} .

(9)

The group printed in this paper is different: it contains two

S U (4)

factors and only one

S U (2)

. This is a discrepancy in the original PDF, not just in the transcription. The standard-model discussion below uses the conventional intended Pati-Salam model model. A literal theory with the printed product would require additional fermion representations and symmetry-breaking data that are not specified.

The Pati-Salam model unifies the three quark color charges with lepton number as a fourth color charge. A family forms

(4, 2, 1)

left-handed and

(4, 1, 2)

right-handed multiplets, including a right-handed neutrino. The extended color charge group contains quark-lepton gauge bosons, while the right-handed weak factor supplies right-handed charged currents after symmetry breaking. The Pati-Salam charge embedding is

Q = T_{3 L} + T_{3 R} + \frac{B - L}{2} .

(10)

The original paper explicitly develops the smaller local subgroup

S U (2)_{L} \times S U (2)_{R} \times S U (4)_{C}

, as well as a larger underlying global structure; neither distinction licenses silently replacing the printed group.^[ref] The product group is a quark-lepton unification scheme rather than a simple gauge group with one automatically common coupling. Breaking the right-handed and

B - L

symmetries can generate a large

M_{R}

and light/heavy Majorana neutrinos through the seesaw. The gauge group alone does not uniquely fix their masses or guarantee a dark-matter lifetime.

The most important laboratory test of Majorana character is neutrinoless double beta decay,

(A, Z) ⟶ (A, Z + 2) + e^{-} + e^{-},

(11)

with no outgoing neutrinos. Under light-neutrino exchange, the amplitude involves

m_{ββ} = \sum_{i} U_{e i}^{2} m_{i}, [T_{1/2}^{0 ν}]^{- 1} = G_{0 ν} ∣ M_{0 ν} ∣^{2} \frac{m _{ββ}}{m _{e}}^{2} .

(12)

The coherent squared mixing elements allow phase cancellations. A null result constrains this combination within the assumed mechanism, rather than proving that all neutrinos are Dirac.

The observable is a peak in summed electron energy at the decay Q value, unlike the continuous two-neutrino spectrum. Enriched source material, large exposure, good energy resolution, underground operation and radiopure construction suppress backgrounds. germanium arrays use the source as a semiconductor detector; xenon time-projection chambers combine ionization, light and event-position information; bolometers measure deposited energy through a temperature rise. The GERDA collaboration's Phase-I study is a concrete example of an enriched-germanium source-detector search with background suppression.^[ref] energy calibration, resolution tails, radon and material backgrounds, cosmic-ray vetoes, signal-like event topology and sideband fits all need independent checks before a Q-value excess is convincing. Collider same-sign dilepton processes can test heavier Majorana states, but require a specified production and decay mechanism. Lepton-number violation, absolute-mass measurements and oscillation data provide complementary information; oscillations alone do not settle Majorana character.

Past exam of the particle physics course of the University of Oxford / 2022 / 5 / c / Solution 2026-10-25

Write

f = (v + h) / 2

. The Pauli matrices give the weak isospin operators

τ^{+} = (0010), τ^{-} = (0100), τ_{3} = \frac{1}{2} (10 0 - 1) .

For the Higgs doublet

Φ = (0, f)^{T}

with weak hypercharge

Y = 1

τ^{+} Φ = (f 0), τ^{-} Φ = 0, τ_{3} Φ = (0 - f /2), Q Φ = (τ_{3} + \frac{1}{2} I) Φ = 0.

The Higgs field direction, including its vacuum expectation value, is electrically neutral. Consequently the electromagnetic U(1) leaves the vacuum invariant: electroweak symmetry breaking preserves electromagnetism and the photon remains massless. The result concerns this neutral vacuum direction; it does not say that both components of the full Higgs doublet have zero electric charge.

Standard Model Created 2026-09-24 Updated 2026-11-05

The Standard Model is the quantum field theory of quarks, leptons and their strong, weak and electromagnetic interactions, organized by the gauge group

S U (3)_{c} \times S U (2)_{L} \times U (1)_{Y}

. Quantum chromodynamics describes the color sector; the electroweak interaction combines the remaining gauge fields with the Higgs field.

Electroweak symmetry breaking leaves the electromagnetic

U (1)_{Q}

unbroken. The W bosons and Z boson become massive, while the photon stays massless; Yukawa interactions generate charged-fermion masses. This structure relates scattering, decay and particle-production processes using a common set of couplings. Predictions require specifying parameters and the approximation used, not only listing the group.

The minimal renormalizable field content has massless neutrinos. Observed neutrino oscillations require masses and mixing, motivating extensions such as the Weinberg operator or additional neutrino fields. The Standard Model also does not supply a quantum description of gravity.

SU(2) gauge theory with an adjoint Higgs field Created 2026-10-05 Updated 2026-10-06

A real scalar field in the Adjoint representation can have a nonzero vacuum expectation value, leaving one massless and two massive gauge bosons. This breaking differs from electroweak symmetry breaking by a complex Higgs doublet.

Unbroken neutral gauge group of a zero-hypercharge real triplet Created 2026-10-25 Updated 2026-11-05

A real scalar triplet of weak hypercharge zero with vacuum expectation value

(0, 0, v)

breaks

T^{1}, T^{2}

but preserves both

T^{3}

and the independent weak hypercharge generator. With canonical kinetic terms its gauge-boson mass matrix in

(W^{1}, W^{2}, W^{3}, B)

g^{2} v^{2} diag (1, 1, 0, 0)

. The two massless neutral gauge bosons prevent this field alone from reproducing Standard Model electroweak symmetry breaking.

Wino 2026-10-14

A wino is a spin-one-half gaugino of the weak

S U (2)_{L}

gauge group. Before electroweak symmetry breaking, its three adjoint components transform as

(1, 3, 0)

and can be described by a triplet of Majorana spinors. After symmetry breaking, the neutral component can mix with the bino and neutral higgsinos into neutralinos.