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Violations of the lepton number conservation laws: The muon decay was described as 'rare' - and it's a pretty common one - I've tweaked some wording elsewhere in the paragraph
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{{Short description|Difference between number of leptons and antileptons}}
{{Flavour quantum numbers}}
{{Flavour quantum numbers}}
{{More footnotes|date=October 2014}}
In [[particle physics]], the '''lepton number''' is the number of [[lepton]]s minus the number of antileptons.
In [[particle physics]], '''lepton number''' (historically also called '''lepton charge''')<ref name=Gribov-Pontecorvo-1969-01-20-PhL-B/>
is a [[conservation law|conserved]] [[quantum number]] representing the difference between the number of [[lepton]]s and the number of [[Antiparticle|antileptons]] in an elementary particle reaction.<ref name=Griffiths-1987-txbk-Tipler-Llewellyn-2002-txbk/>
Lepton number is an additive [[quantum number]], so its sum is preserved in interactions (as opposed to multiplicative [[quantum number]]s such as parity, where the product is preserved instead). The lepton number <math>L</math> is defined by
<math display=block>L = n_\ell - n_{\overline\ell},</math>
where
* <math>n_\ell \quad </math> is the number of [[lepton]]s and
* <math>n_{\overline\ell } \quad </math> is the number of [[antilepton]]s.


Lepton number was introduced in 1953 to explain the absence of reactions such as
In equation form,
: {{math| {{subatomic particle|antineutrino}} + {{subatomic particle|neutron}} → {{subatomic particle|proton}} + {{subatomic particle|electron}} }}
in the [[Cowan–Reines neutrino experiment]], which instead observed
: {{math| {{subatomic particle|antineutrino}} + {{subatomic particle|proton}} → {{subatomic particle|neutron}} + {{subatomic particle|positron}} }}.<ref name=Konopinski-Mahmoud-1953-11-15-PhR/>
This process, [[inverse beta decay]], conserves lepton number, as the incoming [[antineutrino]] has lepton number&nbsp;−1, while the outgoing [[positron]] (antielectron) also has lepton number&nbsp;−1.


== Lepton flavor conservation ==
::<math>L = n_{\ell} - n_{\overline{\ell}}</math>
In addition to lepton number, lepton family numbers are defined as<ref name=Martin-2008-02-25-Phys3-Lect5/>
: <math>L_\mathrm{e}</math> the electron number''',''' for the [[electron]] and the [[electron neutrino]];
: <math>L_\mathrm{\mu}</math> the muon number, for the [[muon]] and the [[muon neutrino]]; and
: <math>L_\mathrm{\tau}</math> the tau number, for the [[tauon]] and the [[tau neutrino]].
Prominent examples of lepton flavor conservation are the [[muon decay]]s
: {{math| {{subatomic particle|link=y|muon}} → {{subatomic particle|link=y|electron}} + {{subatomic particle|link=y|electron antineutrino}} + {{subatomic particle|link=y|muon neutrino}} }}
and
: {{math| {{subatomic particle|link=y|antimuon}} → {{subatomic particle|link=y|positron}} + {{subatomic particle|link=y|electron neutrino}} + {{subatomic particle|link=y|muon antineutrino}} }}.


In these decay reactions, the creation of an [[electron]] is accompanied by the creation of an [[electron antineutrino]], and the creation of a positron is accompanied by the creation of an electron neutrino. Likewise, a decaying negative [[muon]] results in the creation of a [[muon neutrino]], while a decaying positive muon results in the creation of a [[muon antineutrino]].<ref name=Slansky-Raby-Goldman-etal-1997-i25/>
so all leptons have assigned a value of +1, antileptons &minus;1, and non-leptonic particles 0. Lepton number (sometimes also called lepton charge) is an additive [[quantum number]], which means that its sum is preserved in interactions (as opposed to multiplicative [[quantum number]]s such as parity, where the product is preserved instead).


Finally, the [[weak decay]] of a lepton into a lower-mass lepton always results in the production of a [[neutrino]]-[[antineutrino]] pair:
Beside the leptonic number, '''leptonic family numbers''' are also defined:
*{{math|''L''<sub>e</sub> }}, the '''electronic number''' for the [[electron]] and the [[electron neutrino]];
*{{math|''L''<sub>μ</sub> }}, the '''muonic number''' for the [[muon]] and the [[muon neutrino]];
*{{math|''L''<sub>τ</sub> }}, the '''tauonic number''' for the [[tau (particle)|tau]] and the [[tau neutrino]];
with the same assigning scheme as the leptonic number: +1 for particles of the corresponding family, &minus;1 for the antiparticles, and 0 for leptons of other families or non-leptonic particles.


: {{math| {{subatomic particle|link=y|tauon-}} → {{subatomic particle|link=y|muon-}} + {{subatomic particle|link=y|muon antineutrino}} + {{subatomic particle|link=y|tau neutrino}} }}.
==Violations of the lepton number conservation laws==
In the [[Standard Model]], leptonic family numbers (LF numbers) would be preserved if neutrinos were massless. Since [[neutrino oscillations]] have been observed, neutrinos do have a tiny nonzero mass and conservation laws for LF numbers are therefore only approximate. This means the conservation laws are violated, although because of the smallness of the neutrino mass they still hold to a very large degree for interactions containing charged leptons. The (total) lepton number conservation law still holds for most practical purposes (under the Standard Model). So, when dealing with lepton interactions, we can use the constancy of Lepton numbers to predict possible outcomes of interactions, such as:
{| cellpadding=4px
|- align=center style="font-size:160%"
|&nbsp; ||{{SubatomicParticle|Muon}} || &nbsp;→&nbsp; ||{{SubatomicParticle|Electron}}
| + ||{{SubatomicParticle|Electron Neutrino}} || + ||{{SubatomicParticle|Muon Antineutrino}}
|- align=center style="font-size:120%"
|{{mvar|L}}:&nbsp;&nbsp;&nbsp; ||1 || = ||1 || + ||1 || − ||1
|- align=center style="font-size:120%" bgcolor=#FF9999
|{{math|''L''<sub>e</sub> }}:&nbsp;&nbsp; ||0 || ≠ ||1 || + ||1 || + ||0
|- align=center style="font-size:120%" bgcolor=#FF9999
|{{math|''L''<sub>μ</sub> }}:&nbsp;&nbsp; ||1 || ≠ ||0 || + ||0 || − ||1
|}


One neutrino carries through the lepton number of the decaying heavy lepton, (a [[tauon]] in this example, whose faint residue is a [[tau neutrino]]) and an antineutrino that cancels the lepton number of the newly created, lighter lepton that replaced the original. (In this example, a muon antineutrino with <math>L_\mathrm{\mu} = -1</math> that cancels the muon's <math>L_\mathrm{\mu} = +1</math>.
Because the lepton number conservation law in fact is violated by [[chiral anomaly|chiral anomalies]], there are problems applying this symmetry universally over all energy scales. However, the quantum number [[B − L|''B'' − ''L'']] is much more likely to work and is seen in different models such as the [[Pati–Salam model]].


== Violations of the lepton number conservation laws ==
Experiments such as MEGA have searched for lepton number violation in muon decays to electrons; [[Mu to E Gamma|MEG]] set the current branching limit of order 10<sup>−13</sup>. Some BSM theories predict branching ratios of order 10<sup>-12 </sup>to 10<sup>−14</sup>.<ref>{{Cite journal|url = |title = New Limit on the Lepton-Flavor-Violating Decay mu to e+gamma|last = |first = |date =21 Oct 2011|journal = PRL|accessdate = |doi = 10.1103/PhysRevLett.107.171801|pmid = }}</ref>
Lepton flavor is only approximately conserved, and is notably not conserved in [[neutrino oscillation]].<ref name=Fukuda-Hayakawa-Ichihara-1998-08-24-PRL/>
However, both the total lepton number and lepton flavor are still conserved in the Standard Model.


Numerous searches for [[physics beyond the Standard Model]] incorporate searches for lepton number or lepton flavor violation, such as the hypothetical decay<ref name=Adam-Bai-Baldini-etal-2011-10-21/>
==References==
: {{math| {{subatomic particle|link=y|muon}} → {{subatomic particle|link=y|electron}} + {{subatomic particle|photon|link=y}} }}.
{{Reflist}}

*{{cite book | author=Griffiths, David J. | title=Introduction to Elementary Particles | publisher=Wiley, John & Sons, Inc | year=1987 | isbn=0-471-60386-4}}
Experiments such as MEGA and SINDRUM have searched for lepton number violation in muon decays to electrons; [[Mu to E Gamma|MEG]] set the current branching limit of order {{10^|−13}} and plans to lower to limit to {{10^|−14}} after 2016.<ref name=Baldini-2016-05-arXiv/>
*{{cite book | author=Tipler, Paul; Llewellyn, Ralph | title=Modern Physics (4th ed.) | publisher=W. H. Freeman | year=2002 | isbn=0-7167-4345-0}}
Some theories beyond the Standard Model, such as [[supersymmetry]], predict branching ratios of order {{10^|−12}} to {{10^|−14}}.<ref name=Adam-Bai-Baldini-etal-2011-10-21/> The [[Mu2e]] experiment, in construction as of 2017, has a planned sensitivity of order {{10^|−17}}.<ref Name=Kwon-2015-04-21-Fermilab-PR/>
*{{cite journal | author =M. Raidal et al. | year = 2008 | journal =Eur. Phys. J. C 57, 13.}}

Because the lepton number conservation law in fact is violated by [[chiral anomaly|chiral anomalies]], there are problems applying this symmetry universally over all energy scales. However, the quantum number [[B − L|{{mvar|B − L}}]] is commonly conserved in [[Grand Unified Theory]] models.

If neutrinos turn out to be [[Majorana fermions]], neither individual lepton numbers, nor the total lepton number
<math display=block>L \equiv L_\mathrm{e} + L_\mathrm{\mu} + L_\mathrm{\tau},</math>
nor
: [[B − L|{{math|''B'' − ''L''}}]]
would be conserved, e.g. in neutrinoless [[double beta decay]], where two neutrinos colliding head-on might actually annihilate, similar to the (never observed) collision of a neutrino and antineutrino.

== Reversed signs convention ==
Some authors prefer to use lepton numbers that match the signs of the charges of the leptons involved, following the convention in use for the sign of [[weak isospin]] and the sign of [[strangeness]] quantum number ([[strange quark|for quarks]]), both of which conventionally have the otherwise arbitrary sign of the [[quantum number]] match the sign of the particles' electric charges.

When following the electric-charge-sign convention, the lepton number (shown with an over-bar here, to reduce confusion) of an [[electron]], [[muon]], [[tauon]], and any [[neutrino]] counts as <math>\bar{L} = -1;</math> the lepton number of the [[positron]], [[antimuon]], [[antitauon]], and any [[antineutrino]] counts as <math>\bar{L} = +1.</math> When this reversed-sign convention is observed, the [[baryon number]] is left unchanged, but the difference [[B − L|{{mvar|B − L}}]] is replaced with a sum: {{mvar|[[baryon number|B]] + {{overline|L}} }}, whose number value remains unchanged, since
: {{mvar|1= {{overline|L}} = −L,}}
and
: {{mvar|1= [[baryon number|B]] + {{overline|L}} = [[baryon number|B]] − L.}}

== See also ==
* [[Baryon number]]

== References ==
{{Reflist|25em|refs=

<ref name=Adam-Bai-Baldini-etal-2011-10-21>
{{cite journal
|first1=J. |last1=Adam |first2=X. |last2=Bai
|first3=A.M. |last3=Baldini |first4=E. |last4=Baracchini
|first5=C. |last5=Bemporad |first6=G. |last6=Boca
|display-authors = etal
|collaboration = MEG Collaboration
|s2cid = 119278774
|date = 21 Oct 2011
|title = New limit on the lepton-flavor-violating decay mu+ to e+ gamma
|journal = Physical Review Letters
|volume = 107 |issue=17 |pages=171801
|arxiv =1107.5547 |bibcode=2011PhRvL.107q1801A
|doi=10.1103/PhysRevLett.107.171801 |pmid = 22107507
}}
</ref>

<ref name=Baldini-2016-05-arXiv>
{{cite arXiv
|first1=A.M. |last1=Baldini
|display-authors=etal
|collaboration=MEG collaboration
|title=Search for the lepton flavour violating decay {{math|μ{{sup|+}} → e{{sup|+}} γ}} with the full dataset of the MEG Experiment
|date=May 2016
|eprint=1605.05081 |class=hep-ex
}}
</ref>

<ref name=Fukuda-Hayakawa-Ichihara-1998-08-24-PRL>
{{cite journal
|last1=Fukuda |first1=Y. |last2=Hayakawa |first2=T.
|last3=Ichihara |first3=E. |last4=Inoue |first4=K.
|last5=Ishihara |first5=K. |last6=Ishino |first6=H.
|display-authors=etal
|collaboration=Super-Kamiokande collaboration
|date=1998-08-24
|title=Evidence for oscillation of atmospheric neutrinos
|journal=Physical Review Letters
|volume=81 |issue=8 |pages=1562–1567
|doi=10.1103/PhysRevLett.81.1562
|arxiv=hep-ex/9807003
|bibcode=1998PhRvL..81.1562F
|s2cid=7102535 }}
</ref>

<ref name=Gribov-Pontecorvo-1969-01-20-PhL-B>
{{cite journal
|last1=Gribov |first1=V.
|last2=Pontecorvo |first2=B.
|date=1969-01-20
|title=Neutrino astronomy and lepton charge
|journal=Physics Letters B
|volume=28 |issue=7 |pages=493–496
|doi=10.1016/0370-2693(69)90525-5
|issn=0370-2693
|bibcode=1969PhLB...28..493G
}}
</ref>

<ref name=Griffiths-1987-txbk-Tipler-Llewellyn-2002-txbk>
{{cite book
| last=Griffiths |first=David J.
| year=1987
| title=Introduction to Elementary Particles
| publisher=Wiley, John & Sons, Inc
| isbn=978-0-471-60386-3
| postscript=;
}}
{{cite book
| last1=Tipler |first1=Paul
| last2=Llewellyn |first2=Ralph
| year=2002
| title=Modern Physics
| edition=4th
| publisher=W.H. Freeman
| isbn=978-0-7167-4345-3
}}
</ref>

<ref name=Konopinski-Mahmoud-1953-11-15-PhR>
{{cite journal
|last1=Konopinski |first1=E.J.
|last2=Mahmoud |first2=H.M.
|date=1953-11-15
|title=The universal Fermi interaction
|journal=Physical Review
|volume=92 |issue=4 |pages=1045–1049
|doi=10.1103/physrev.92.1045
|bibcode=1953PhRv...92.1045K
}}
</ref>

<ref Name=Kwon-2015-04-21-Fermilab-PR>
{{cite press release
|last=Kwon |first=Diana
|date=2015-04-21
|title=Mu2e breaks ground on experiment seeking new physics
|publisher=[[Fermi National Accelerator Laboratory]]
|lang=en-US
|url=http://news.fnal.gov/2015/04/mu2e-breaks-ground-on-experiment-seeking-new-physics/
|access-date=2017-12-08
}}
</ref>

<ref name=Martin-2008-02-25-Phys3-Lect5>
{{cite report
|first=Victoria J., Professor |last=Martin
|date=25 February 2008
|title=Quarks & leptons, mesons, & baryons
|volume=Lecture&nbsp;5
|type=lecture notes
|series=Physics&nbsp;3
|page=2
|publisher=University of Edinburgh
|url=https://www2.ph.ed.ac.uk/~vjm/Lectures/ParticlePhysics2008_files/Feb25th2008.pdf
|access-date=May 23, 2021
}}
</ref>

<ref name=Slansky-Raby-Goldman-etal-1997-i25>
{{cite web
|first1=Richard |last1=Slansky
|first2=Stuart |last2=Raby
|first3=Terry |last3=Goldman
|first4=Gerry |last4=Garvey
|editor-first=Necia Grant |editor-last=Cooper
|title=The Oscillating Neutrino: An introduction to neutrino masses and mixing
|year=1997
|issue=25 |pages=10–56
|publisher=[[Los Alamos National Laboratory]]
|magazine=Los Alamos Science
|url=https://library.lanl.gov/cgi-bin/getfile?00326609.pdf
|access-date=23 May 2021
}}
</ref>

}} <!-- end "refs=" -->

{{Authority control}}


[[Category:Conservation laws]]
[[Category:Conservation laws]]
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[[Category:Leptons]]
[[Category:Leptons]]
[[Category:Flavour (particle physics)]]
[[Category:Flavour (particle physics)]]

[[he:מספר לפטוני]]

Latest revision as of 04:16, 13 August 2024

In particle physics, lepton number (historically also called lepton charge)[1] is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction.[2] Lepton number is an additive quantum number, so its sum is preserved in interactions (as opposed to multiplicative quantum numbers such as parity, where the product is preserved instead). The lepton number is defined by where

  • is the number of leptons and
  • is the number of antileptons.

Lepton number was introduced in 1953 to explain the absence of reactions such as


ν
+
n

p
+
e

in the Cowan–Reines neutrino experiment, which instead observed


ν
+
p

n
+
e+
.[3]

This process, inverse beta decay, conserves lepton number, as the incoming antineutrino has lepton number −1, while the outgoing positron (antielectron) also has lepton number −1.

Lepton flavor conservation

[edit]

In addition to lepton number, lepton family numbers are defined as[4]

the electron number, for the electron and the electron neutrino;
the muon number, for the muon and the muon neutrino; and
the tau number, for the tauon and the tau neutrino.

Prominent examples of lepton flavor conservation are the muon decays


μ

e
+
ν
e
+
ν
μ

and


μ+

e+
+
ν
e
+
ν
μ
.

In these decay reactions, the creation of an electron is accompanied by the creation of an electron antineutrino, and the creation of a positron is accompanied by the creation of an electron neutrino. Likewise, a decaying negative muon results in the creation of a muon neutrino, while a decaying positive muon results in the creation of a muon antineutrino.[5]

Finally, the weak decay of a lepton into a lower-mass lepton always results in the production of a neutrino-antineutrino pair:


τ

μ
+
ν
μ
+
ν
τ
.

One neutrino carries through the lepton number of the decaying heavy lepton, (a tauon in this example, whose faint residue is a tau neutrino) and an antineutrino that cancels the lepton number of the newly created, lighter lepton that replaced the original. (In this example, a muon antineutrino with that cancels the muon's .

Violations of the lepton number conservation laws

[edit]

Lepton flavor is only approximately conserved, and is notably not conserved in neutrino oscillation.[6] However, both the total lepton number and lepton flavor are still conserved in the Standard Model.

Numerous searches for physics beyond the Standard Model incorporate searches for lepton number or lepton flavor violation, such as the hypothetical decay[7]


μ

e
+
γ
.

Experiments such as MEGA and SINDRUM have searched for lepton number violation in muon decays to electrons; MEG set the current branching limit of order 10−13 and plans to lower to limit to 10−14 after 2016.[8] Some theories beyond the Standard Model, such as supersymmetry, predict branching ratios of order 10−12 to 10−14.[7] The Mu2e experiment, in construction as of 2017, has a planned sensitivity of order 10−17.[9]

Because the lepton number conservation law in fact is violated by chiral anomalies, there are problems applying this symmetry universally over all energy scales. However, the quantum number B − L is commonly conserved in Grand Unified Theory models.

If neutrinos turn out to be Majorana fermions, neither individual lepton numbers, nor the total lepton number nor

BL

would be conserved, e.g. in neutrinoless double beta decay, where two neutrinos colliding head-on might actually annihilate, similar to the (never observed) collision of a neutrino and antineutrino.

Reversed signs convention

[edit]

Some authors prefer to use lepton numbers that match the signs of the charges of the leptons involved, following the convention in use for the sign of weak isospin and the sign of strangeness quantum number (for quarks), both of which conventionally have the otherwise arbitrary sign of the quantum number match the sign of the particles' electric charges.

When following the electric-charge-sign convention, the lepton number (shown with an over-bar here, to reduce confusion) of an electron, muon, tauon, and any neutrino counts as the lepton number of the positron, antimuon, antitauon, and any antineutrino counts as When this reversed-sign convention is observed, the baryon number is left unchanged, but the difference B − L is replaced with a sum: B + L , whose number value remains unchanged, since

L = −L,

and

B + L = B − L.

See also

[edit]

References

[edit]
  1. ^ Gribov, V.; Pontecorvo, B. (1969-01-20). "Neutrino astronomy and lepton charge". Physics Letters B. 28 (7): 493–496. Bibcode:1969PhLB...28..493G. doi:10.1016/0370-2693(69)90525-5. ISSN 0370-2693.
  2. ^ Griffiths, David J. (1987). Introduction to Elementary Particles. Wiley, John & Sons, Inc. ISBN 978-0-471-60386-3; Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W.H. Freeman. ISBN 978-0-7167-4345-3.
  3. ^ Konopinski, E.J.; Mahmoud, H.M. (1953-11-15). "The universal Fermi interaction". Physical Review. 92 (4): 1045–1049. Bibcode:1953PhRv...92.1045K. doi:10.1103/physrev.92.1045.
  4. ^ Martin, Victoria J., Professor (25 February 2008). Quarks & leptons, mesons, & baryons (PDF) (lecture notes). Physics 3. Vol. Lecture 5. University of Edinburgh. p. 2. Retrieved May 23, 2021.{{cite report}}: CS1 maint: multiple names: authors list (link)
  5. ^ Slansky, Richard; Raby, Stuart; Goldman, Terry; Garvey, Gerry (1997). Cooper, Necia Grant (ed.). "The Oscillating Neutrino: An introduction to neutrino masses and mixing" (PDF). Los Alamos Science. Los Alamos National Laboratory. pp. 10–56. Retrieved 23 May 2021.
  6. ^ Fukuda, Y.; Hayakawa, T.; Ichihara, E.; Inoue, K.; Ishihara, K.; Ishino, H.; et al. (Super-Kamiokande collaboration) (1998-08-24). "Evidence for oscillation of atmospheric neutrinos". Physical Review Letters. 81 (8): 1562–1567. arXiv:hep-ex/9807003. Bibcode:1998PhRvL..81.1562F. doi:10.1103/PhysRevLett.81.1562. S2CID 7102535.
  7. ^ a b Adam, J.; Bai, X.; Baldini, A.M.; Baracchini, E.; Bemporad, C.; Boca, G.; et al. (MEG Collaboration) (21 Oct 2011). "New limit on the lepton-flavor-violating decay mu+ to e+ gamma". Physical Review Letters. 107 (17): 171801. arXiv:1107.5547. Bibcode:2011PhRvL.107q1801A. doi:10.1103/PhysRevLett.107.171801. PMID 22107507. S2CID 119278774.
  8. ^ Baldini, A.M.; et al. (MEG collaboration) (May 2016). "Search for the lepton flavour violating decay μ+ → e+ γ with the full dataset of the MEG Experiment". arXiv:1605.05081 [hep-ex].
  9. ^ Kwon, Diana (2015-04-21). "Mu2e breaks ground on experiment seeking new physics" (Press release). Fermi National Accelerator Laboratory. Retrieved 2017-12-08.