View source for B − L

{{Redirect|B-L|other uses|BL (disambiguation)}}
{{DISPLAYTITLE:''B'' &minus; ''L''}}
{{short description|Quantum number; the difference between the baryon and lepton numbers}}
{{More citations needed|date=January 2021}}

{{Flavour quantum numbers}}

In [[particle physics]], '''''B''&nbsp;−&nbsp;''L''''' (pronounced "bee minus ell") is a [[quantum number]] which is the difference between the [[baryon number]] ({{mvar|B}}) and the [[lepton number]] ({{mvar|L}}) of a [[quantum system]].

==Details==
This [[quantum number]] is the charge of a [[global symmetry|global]]/[[gauge symmetry|gauge]] [[U(1)]] symmetry in some [[Grand Unified Theory]] models, called {{math|U(1)<sub>''B''&minus;''L''</sub>}}. Unlike baryon number alone or lepton number alone, this hypothetical symmetry would not be broken by [[chiral anomaly|chiral anomalies]] or [[gravitational anomaly|gravitational anomalies]], as long as this symmetry is global, which is why this symmetry is often invoked.

If {{math|''B'' – ''L''}} exists as a symmetry, then for the [[seesaw mechanism]] to work {{math|''B'' – ''L''}} has to be [[spontaneously broken]] to give the [[neutrino]]s a nonzero mass.

The [[Anomaly (physics)|anomalies]] that would break baryon number conservation and lepton number conservation individually cancel in such a way that {{math|''B'' – ''L''}} is always [[conservation (physics)|conserved]]. One hypothetical example is [[proton decay]] where a [[proton]] ({{math|1=''B'' = 1, ''L'' = 0}}) would decay into a [[pion]] ({{math|1=''B'' = 0, ''L'' = 0}}) and [[positron]] ({{math|1=''B'' = 0, ''L'' = –1}}).

The [[weak hypercharge]] {{math|''Y''{{sub|W}}}} is related to {{math|''B'' – ''L''}} via

<math display=block>X + 2\,Y_\text{W} = 5\,( B - L ),</math>

where [[X (charge)|''X'' charge]] (not to be confused with the [[X and Y bosons|X boson]]) is the conserved [[quantum number]] associated with the global [[U(1)]] symmetry [[Grand Unified Theory]].<ref>{{cite journal |last1=Wilczek |first1=Frank |last2=Zee |first2=A. |year=1979 |title=Operator analysis of nucleon decay |journal=Physical Review Letters |volume=43 |issue=21 |pages=1571–1573 |bibcode=1979PhRvL..43.1571W |doi=10.1103/PhysRevLett.43.1571}}</ref>

==See also==
*[[Baryogenesis]]
*[[Leptogenesis (physics)|Leptogenesis]]
*[[Majoron]]
*[[Proton decay]]
*[[X and Y bosons]]
*[[X (charge)]]
*[[Leptoquark]]

==References==
{{reflist}}

{{Proton decay experiments}}

{{DEFAULTSORT:B-L}}
[[Category:Conservation laws]]
[[Category:Flavour (particle physics)]]


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