Editing Meson (section)

=== Spin, orbital angular momentum, and total angular momentum ===
{{Main|Spin (physics)|angular momentum operator|Total angular momentum|Quantum numbers}}
[[Spin (physics)|Spin]] (quantum number {{mvar|S}}) is a [[Euclidean vector|vector]] quantity that represents the "intrinsic" [[angular momentum]] of a particle. It comes in increments of {{sfrac|1|2}}&nbsp;[[Planck constant|{{mvar|ħ}}]].{{efn|
group=upper-alpha|
The {{mvar|ħ}} is often dropped because it is the "fundamental" unit of spin, and it is implied that "spin&nbsp;1" means "spin 1&nbsp;{{mvar|ħ}}". In some systems of [[natural units]], {{mvar|ħ}} is chosen to be 1, and therefore drops out of equations. The remainder of this article uses the "assume {{mvar|ħ}} units" convention for all types of spin.
}}

[[Quark]]s are [[fermion]]s—specifically in this case, particles having spin {{sfrac|1|2}} {{nobr|( {{mvar|S}} {{=}} {{sfrac|1|2}} ).}} Because spin projections vary in increments of 1 (that is 1&nbsp;{{mvar|ħ}}), a single quark has a spin vector of length {{sfrac|1|2}}, and has two spin projections, either {{nobr|( {{mvar|S}}{{sub|z}} {{=}} +{{sfrac|1|2}} }} or {{nobr| {{mvar|S}}{{sub|z}} {{=}} {{sfrac|&minus;|1|2}} ).}} Two quarks can have their spins aligned, in which case the two spin vectors add to make a vector of length {{nobr| {{mvar|S}} {{=}} 1 ,}} with three possible spin projections {{nobr|( {{mvar|S}}{{sub|z}} {{=}} +1,}} {{nobr| {{mvar|S}}{{sub|z}} {{=}} 0,}} and {{nobr| {{mvar|S}}{{sub|z}} {{=}} −1),}} and their combination is called a ''[[vector meson]]'' or [[Vector boson|spin-1]] triplet. If two quarks have oppositely aligned spins, the spin vectors add up to make a vector of length {{nobr| {{mvar|S}} {{=}} 0,}} and only one spin projection {{nobr|( {{mvar|S}}{{sub|z}} {{=}} 0 ),}} called a ''[[scalar meson]]'' or [[Scalar boson|spin-0]] singlet. Because mesons are made of one quark and one antiquark, they are found in triplet and singlet spin states. The latter are called [[scalar meson]]s or [[pseudoscalar meson]]s, depending on their parity (see below).

There is another quantity of quantized [[angular momentum]], called the [[angular momentum operator#Orbital angular momentum|orbital angular momentum]] (quantum number {{mvar|L}}), that is the angular momentum due to quarks orbiting each other, and also comes in increments of 1&nbsp;{{mvar|ħ}}. The total angular momentum (quantum number {{mvar|J}}) of a particle is the combination of the two intrinsic angular momentums (spin) and the orbital angular momentum. It can take any value from {{nobr| {{mvar|J}} {{=}} {{!}}{{mvar|L}} − {{mvar|S}}{{!}} }} up to {{nobr| {{mvar|J}} {{=}} {{!}}{{mvar|L}} + {{mvar|S}}{{!}} ,}} in increments of 1.

{|class="wikitable" style="margin-left:auto; margin-right:auto; text-align:center;"
|+ Meson angular momentum quantum numbers for {{mvar|L}} = 0, 1, 2, 3
|-
! style="width:30px;"| [[Spin (physics)|{{mvar|S}}]]
! style="width:30px;"| [[Angular momentum operator#Orbital angular momentum|{{mvar|L}}]]
! style="width:30px;"| [[#Parity|{{math|P}}]]
! style="width:50px;"| [[Total angular momentum|{{mvar|J}}]]
! style="width:70px;"| {{mvar|J}}{{sup|{{math|P}}}}
|-
|rowspan="4"| 0
| 0 || − || 0 || 0{{sup|−}}
|-
| 1 || + || 1 || 1{{sup|+}}
|-
| 2 || − || 2 || 2{{sup|−}}
|-
| 3 || + || 3 || 3{{sup|+}}
|-
|rowspan="4"| 1
| 0 || − || 1 || 1{{sup|−}}
|-
| 1 || + || 2,<!-- 1, --> 0 || 2{{sup|+}},<!-- 1{{sup|+}}, --> 0{{sup|+}}
|-
| 2 || − || 3,<!-- 2, --> 1 || 3{{sup|−}},<!-- 2{{sup|−}}, --> 1{{sup|−}}
|-
| 3 || + || 4,<!-- 3, --> 2 || 4{{sup|+}},<!-- 3{{sup|+}}, --> 2{{sup|+}}
|}

Particle physicists are most interested in mesons with no orbital angular momentum ({{mvar|L}}&nbsp;=&nbsp;0), therefore the two groups of mesons most studied are the {{mvar|S}}&nbsp;=&nbsp;1; {{mvar|L}}&nbsp;=&nbsp;0 and {{mvar|S}}&nbsp;=&nbsp;0; {{mvar|L}}&nbsp;=&nbsp;0, which corresponds to {{mvar|J}}&nbsp;=&nbsp;1 and {{mvar|J}}&nbsp;=&nbsp;0, although they are not the only ones. It is also possible to obtain {{mvar|J}}&nbsp;=&nbsp;1 particles from {{mvar|S}}&nbsp;=&nbsp;0 and {{mvar|L}}&nbsp;=&nbsp;1. How to distinguish between the {{mvar|S}}&nbsp;=&nbsp;1, {{mvar|L}}&nbsp;=&nbsp;0 and {{mvar|S}}&nbsp;=&nbsp;0, {{mvar|L}}&nbsp;=&nbsp;1 mesons is an active area of research in [[meson spectroscopy]].<ref>{{cite web |title=Particles of the Standard Model |url=https://pdfslide.net/documents/particles-of-the-standard-model.html |website=pdfslide.net |language=en |access-date=2020-05-24 |df=dmy-all}}</ref>