In the

Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...

of electroweak interactions of

particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...

, the weak hypercharge is a

quantum number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can ...

relating the

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...

and the third component of

weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It c ...

. It is frequently denoted

Y_\mathsf

and corresponds to the

gauge symmetry In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...

U(1) In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...

. It is conserved (only terms that are overall weak-hypercharge neutral are allowed in the Lagrangian). However, one of the interactions is with the

Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the St ...

. Since the Higgs field

vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...

is nonzero, particles interact with this field all the time even in vacuum. This changes their weak hypercharge (and weak isospin ). Only a specific combination of them,

~Q = T_3 + \tfrac\, Y_\mathsf

(electric charge), is conserved. Mathematically, weak hypercharge appears similar to the

Gell-Mann–Nishijima formula The Gell-Mann–Nishijima formula (sometimes known as the NNG formula) relates the baryon number ''B'', the strangeness ''S'', the isospin ''I3'' of quarks and hadrons to the electric charge ''Q''. It was originally given by Kazuhiko Nishijima and ...

for the hypercharge of strong interactions (which is not conserved in weak interactions and is zero for leptons). In the electroweak theory SU(2) transformations

commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...

with U(1) transformations by definition and therefore U(1) charges for the elements of the SU(2) doublet (for example lefthanded up and down quarks) have to be equal. This is why U(1) cannot be identified with U(1)_em and weak hypercharge has to be introduced. Weak hypercharge was first introduced by

Sheldon Glashow Sheldon Lee Glashow (, ; born December 5, 1932) is a Nobel Prize-winning American theoretical physicist. He is the Metcalf Professor of Mathematics and Physics at Boston University and Eugene Higgins Professor of Physics, Emeritus, at Harvard U ...

in 1961.

Definition

Weinberg angle (relation between coupling constants)

Weak hypercharge is the generator of the U(1) component of the

electroweak In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...

gauge group, and its associated

quantum field In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...

mixes with the electroweak quantum field to produce the observed gauge boson and the

photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...

quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...

. The weak hypercharge satisfies the relation ::

Q = T_3 + \tfrac Y_\text ~,

where is the electric charge (in

elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundam ...

units) and is the third component of

(the SU(2) component). Rearranging, the weak hypercharge can be explicitly defined as: ::

Y_ = 2(Q - T_3)

where "left"- and "right"-handed here are left and right

chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

, respectively (distinct from helicity). The weak hypercharge for an anti-fermion is the opposite of that of the corresponding fermion because the electric charge and the third component of the weak isospin reverse sign under

charge conjugation In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-sy ...

The sum of −isospin and +charge is zero for each of the gauge bosons; consequently, all the electroweak gauge bosons have ::

\, Y_\text = 0 ~.

Hypercharge assignments in the

are determined up to a twofold ambiguity by requiring cancellation of all anomalies.

Alternative half-scale

For convenience, weak hypercharge is often represented at half-scale, so that ::

\, Y_ = Q - T_3 ~,

which is equal to just ''the average electric charge of the particles in the isospin multiplet''.

Baryon and lepton number

Weak hypercharge is related to baryon number minus lepton number via: :

\tfracX + Y_ = \tfrac(B - L) \,

where '' X'' is a conserved quantum number in GUT. Since weak hypercharge is always conserved within the

and most extensions, this implies that baryon number minus lepton number is also always conserved.

Neutron decay

: Hence neutron decay conserves

baryon number In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as ::B = \frac\left(n_\text - n_\bar\right), where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Baryo ...

and

lepton number In particle physics, lepton number (historically also called lepton charge) is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction. Lepton number ...

separately, so also the difference is conserved.

Proton decay

Proton decay In particle physics, proton decay is a hypothetical form of particle decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron. The proton decay hypothesis was first formulated by Andrei Sakharov ...

is a prediction of many

grand unification theories A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this ...

. : Hence this hypothetical proton decay would conserve , even though it would individually violate conservation of both

and

References