Physics:Weak hypercharge
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In the Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently denoted [math]\displaystyle{ Y_\mathsf{W} }[/math] and corresponds to the gauge symmetry U(1).^{[1]}^{[2]}
It is conserved (only terms that are overall weakhypercharge neutral are allowed in the Lagrangian). However, one of the interactions is with the Higgs field. Since the Higgs field vacuum expectation value is nonzero, particles interact with this field all the time even in vacuum. This changes their weak hypercharge (and weak isospin T_{3}). Only a specific combination of them, [math]\displaystyle{ ~Q = T_3 + \tfrac{1}{2}\, Y_\mathsf{W} }[/math] (electric charge), is conserved.
Mathematically, weak hypercharge appears similar to the GellMann–Nishijima formula 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 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.^{[3]}^{[4]}
Weak hypercharge was first introduced by Sheldon Glashow in 1961.^{[4]}^{[5]}^{[6]}
Definition
Weak hypercharge is the generator of the U(1) component of the electroweak gauge group, SU(2)×U(1) and its associated quantum field B mixes with the W^{3} electroweak quantum field to produce the observed Z boson gauge boson and the photon of quantum electrodynamics.
The weak hypercharge satisfies the relation
 [math]\displaystyle{ Q = T_3 + \tfrac{1}{2} Y_\text{W} ~, }[/math]
where Q is the electric charge (in elementary charge units) and T_{3} is the third component of weak isospin (the SU(2) component).
Rearranging, the weak hypercharge can be explicitly defined as:
 [math]\displaystyle{ Y_{\rm W} = 2(Q  T_3) }[/math]
Fermion family 
Leftchiral fermions  Rightchiral fermions  

Electric charge Q 
Weak isospin T_{3} 
Weak hyper charge Y_{W} 
Electric charge Q 
Weak isospin T_{3} 
Weak hyper charge Y_{W}  
Leptons  electron neutrino, muon neutrino, tau neutrino  0  +1/2  −1  ν_{R May not exist }  0  0  0 
_{}e^{−}, muon, tau  −1  −1/2  −1  _{}e−R, _{}μ−R, _{}τ−R  −1  0  −2  
Quarks  up quark, charm quark, top quark  +2/3  +1/2  +1/3  _{}u_{R}, _{}c_{R}, _{}t_{R}  +2/3  0  +4/3 
d, s, b  −1/3  −1/2  +1/3  _{}d_{R}, _{}s_{R}, _{}b_{R}  −1/3  0  −2/3 
where "left" and "right"handed here are left and right chirality, respectively (distinct from helicity). The weak hypercharge for an antifermion 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.
Interaction mediated 
Boson  Electric charge Q 
Weak isospin T_{3} 
Weak hypercharge Y_{W} 

Weak  _{}W^{±}  ±1  ±1  0 
Z boson0  0  0  0  
Electromagnetic  Photon0  0  0  0 
Strong  Gluon  0  0  0 
Higgs  _{}H^{0}  0  −1/2  +1 
The sum of −isospin and +charge is zero for each of the gauge bosons; consequently, all the electroweak gauge bosons have
 [math]\displaystyle{ \, Y_\text{W} = 0 ~. }[/math]
Hypercharge assignments in the Standard Model are determined up to a twofold ambiguity by requiring cancellation of all anomalies.
Alternative halfscale
For convenience, weak hypercharge is often represented at halfscale, so that
 [math]\displaystyle{ \, Y_{\rm W} = Q  T_3 ~, }[/math]
which is equal to just the average electric charge of the particles in the isospin multiplet.^{[8]}^{[9]}
Baryon and lepton number
Weak hypercharge is related to baryon number minus lepton number via:
 [math]\displaystyle{ \tfrac{1}{2}X + Y_{\rm W} = \tfrac{5}{2}(B  L) \, }[/math]
where X is a conserved quantum number in GUT. Since weak hypercharge is always conserved within the Standard Model and most extensions, this implies that baryon number minus lepton number is also always conserved.
Neutron decay
 neutron → proton + _{}e^{−} + _{}ν_{e}
Hence neutron decay conserves baryon number B and lepton number L separately, so also the difference B − L is conserved.
Proton decay
Proton decay is a prediction of many grand unification theories.
_{}p^{+} → Positron + Pion0 └→ 2Gamma
Hence this hypothetical proton decay would conserve B − L , even though it would individually violate conservation of both lepton number and baryon number.
See also
 Standard Model (mathematical formulation)
 weak charge
References
 ↑ Donoghue, J.F.; Golowich, E.; Holstein, B.R. (1994). Dynamics of the Standard Model. Cambridge University Press. p. 52. ISBN 0521476526. https://archive.org/details/dynamicsstandard00jfdo.
 ↑ Cheng, T.P.; Li, L.F. (2006). Gauge Theory of Elementary Particle Physics. Oxford University Press. ISBN 0198519613.
 ↑ Tully, Christopher G. (2012). Elementary Particle Physics in a Nutshell. Princeton University Press. p. 87. doi:10.1515/9781400839353. ISBN 9781400839353. https://princetonup.degruyter.com/view/title/563410.
 ↑ ^{4.0} ^{4.1} Glashow, Sheldon L. (February 1961). "Partialsymmetries of weak interactions". Nuclear Physics 22 (4): 579–588. doi:10.1016/00295582(61)904692. https://linkinghub.elsevier.com/retrieve/pii/0029558261904692.
 ↑ The rise of the Standard Model: A history of particle physics from 1964 to 1979 (1st ed.). Cambridge University Press. 19971113. p. 14. doi:10.1017/cbo9780511471094. ISBN 9780521570824. https://www.cambridge.org/core/product/identifier/9780511471094/type/book.
 ↑ Quigg, Chris (20151019). "Electroweak symmetry breaking in historical perspective". Annual Review of Nuclear and Particle Science 65 (1): 25–42. doi:10.1146/annurevnucl102313025537. ISSN 01638998. http://www.annualreviews.org/doi/10.1146/annurevnucl102313025537.
 ↑ Lee, T.D. (1981). Particle Physics and Introduction to Field Theory. Boca Raton, FL / New York, NY: CRC Press / Harwood Academic Publishers. ISBN 9783718600335. https://archive.org/details/particlephysicsi0000leet.
 ↑ Peskin, Michael E.; Schroeder, Daniel V. (1995). An Introduction to Quantum Field Theory. AddisonWesley Publishing Company. ISBN 9780201503975. https://archive.org/details/introductiontoqu0000pesk.
 ↑ Anderson, M.R. (2003). The Mathematical Theory of Cosmic Strings. CRC Press. p. 12. ISBN 0750301600.
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