Кварковая_структура_адронов

3.3

Isotopi states in

systems of two and three pions

expli it onstru tion

isotopi waveanfunhavetiontheoftotalthe

isospinsystem equalof twotopions:0, 1, 2 (3 3 = 1 + 3 + 5). An

ofTwothepions π1

, π2

InT the= 0previous, S = 1) leannturelookedweayhavetoseentwo thatpionstheduedeto

pions

S) for the total isospin

ynotherisospin

onserving part of strong intera tion.

T = 0

π1

· π2

= δij π1

π2j

S

angular momentum

T = 1

(π1 × π2)k

= ǫijk π1 π2j

A

f two pions (problem. Therefore,

2

a ordan e with Bose prin iple the orbital

shows that the isotopi waveT = fun2 tion ofπ1

π2j + π1j

π2i −

3

δij π1k

π2k

S

the syst m of two

is symmetri (

T = 0 2 and a isymmetri (A) for T = 1

symmetri

(

) an be even

T = 0, 2 and odd for T

= 1. So, ω - meson

parity,For nowthesystemwehaveof three pionsatthis

from a

ω → 2pointπ is .forbidden by the onservation of G -

we have one singlet

π1, π2, π3 where 3 3 3 = (1 + 3 + 5) 3 = 3 + (1 + 3 + 5) + (3 + 5 + 7)

three triplets one

of

an be hosen with

wave fun tion

whi h π1 ·

π2

×

π3

= ǫijk π1 π2j π3k

A ,

et . If

ne assumes

nserv tion of isotopi spin in threede ay

S

π1(π2 ·

π3) + π2

(π3

·

π1) + π3(π1 ·

π2)

violated in

180

G = CT2

- paritythis

oordinate/momentum wave fun tion of three pions( shouldviolation) threebantisymmetriinaordan e.

Thiswith Boseleadsprintotheiple theg tive

η → 3π

T = 0

inonsideredtheexample)ayde.thenIfayoisnesymmeassumesri-th(therepritytheleisofnoturetheoo eladinatsystemive/momentumorbitalofpionswaveofandpionsfuntoannotionduehe ofto smallofphaseionsinspaparitythe,

for

C

C

parity. Beforeis (seeonserved but2) wehe tothave isospinseenhatof pions

be equal toviolationzero.

isospin is not onservedC -

ssymmetryatteringhere tosiderationthepionso-nutoleonsay disensatteringangle.Therethemearehanism10proofesses

.

ηLet3.→4us3πPionapplydeay,-thenuisotopipresentedleon

isotopi

onne ted

180

π+p π+

π−

π−n

Theinolumn,seleftondso,olumnritsprootationamplitudeess.Theisaroundnotfourthisobservedtheequalpro2-ndtoesstheaxisoftheriainmentallyplilef udeolumnbeofspatheauseis .timeSo,fthof thewereversehavelak ofof(nototheonsidmatterfth proofnlywhiesstheofhprotheolumn)rightesses.

π0

→π0p

180

π0

→

π0n

π−

→

π−

T2

π+n

→π+n

π0 - beams. The remaining

thr e pro esses

exp

→

π0n

→

pro ess

π0p

→π+n

π−

bywherethe the p

→

0

n

+

→

0

,to the pro ess s in the left olumn

esses in the right olumnπ−

areπ

i allyπ

→

π

→

isotopi

π

p → π

p, π−p → π−p, π−p → π and

M+, M−, M0

+

+

0

have the amplitudes

symmetry. Considering the states of pion

nu leon with de nite total isospinonneone tedgetsby

π+p =11( 3

, +

3

)

2

2

4The.1 LeomparisonFundamentaltureof4. SUrepresentation(3) - symmetry

u, d, s - quark masses with hara teristi hadroni s ale (mu = 4M V, md =

7interaM eV,tionm

=under150M eV

<< 1GeV ) tells us that SU (3) - symmetry of the Lagrangian of strong

SU (3) - transformation

of quark elds

d

XP ( iω

λ

) d

.

2

→

−

violati

of

nonstrange

and d) quarks. The onsequen es of just this sour

mass

masswillan bedibe fundamentaleren SUredof(3)strangeasin -thesymmetryapproximate(next) andare the.Insy

the present leforture we willstateslassifyofmesonsthe and baryons that

metry(formulaeof stronggroundintera tion. Mainly it is viola ed by the

of mesonsTheonsiderand baryonsrepresentation.le tureof

SU (3) - multiplets

SU (3) - group is the triplet avor wave fun tion of quarks

The

qα =

.

s

SU (3) - transformations of the triplet are produ ed by the unitary matri es 3×3

where

U = exp(iωa

λa

),

2

SU (3) - group has two diagonal generators: the third omponent of isotopi spin and hyper harge,

λ1,2,3 =

0

0

; λ4,5

=

0

0

−

0

τ 1,2,3

0

0

0

1(

)

λ6,7 =

)

The

0

0 1(

; λ8 = √

0

1

0

.

0

0

0

1

0

0

ele tromagneti

1

0

1(+i)

−0

3

0

0

− 2

duerepresentationthatAntiquarkstoarethe

T3

λ3

1

8

,

= 2

Y = √3 λ

onserveditarityandareintheseoftransformedstrongthetransformationsgroupand.Unlikeasompl oinxaseonjugaideofinterawih toftionshethetransformations(seefundamentalleture 1).of( ontravarianttheovariantspinor)

the

equivalently)now equivalent.

sentationSU (2) (quarksgroup theand ovariantantiquarksspinortransforep mesentation

nois Theot

to the

ntravariant repr

SU (3) - group has two invariant tensors

δαβ → U αα′ δα′ β′ U −1β′ β = δαβ ,

αβγ

→ U

α

β

β′ U

γ

α′β′

γ′

αβγ

αβγ

ǫ

α′ U

γ′ ǫ 13

= ǫ

(detU ) = ǫ .

α

α

1

α

α

representasanexample):the o tet and singlet states. The basis states of the o tet are (we indi ate

isotopipseudostwo partstripletalarofwhimesonsofh

M

β = M0

β

+

δ

β M

(M0

α

= 0),

3

¯

1

¯

isotopi doublets of

ud, √

(uu¯ − dd), du¯;

2

K - mesons

¯

and isotopi singlet

us,¯ ds¯;

sd, su¯

η8- meson

1

¯

The basis state of the singlet is

√

(uu¯ + dd − 2 s¯).

6

η0- meson

1

¯

the o tet part of

Us the above o tet basis states η0 =

√3

(uu¯ + dd + s¯).

following form

tensor M αβ (M0αβ ) an be represented in the

π0

η8

+

+

√

+

√

π

K

2

6

η8

α

π0

0

quarks. The r wave

on presented by the ten or

Baryons4.3 Baryonsonsist of three M0

β =

π−

fun ti√2

+ √6

K

.

K−

−

K¯

0

2η8

− √6

Let us brie y des r be the de omposition of tensor

Bαβγ is redu ible.

identi ally divided in four parts as follows

Bαβγ into irredu ible parts. Tensor Bαβγ an be

where the

αβγsymbo1l

{αβγ}

1

αβγ′

′ ′

′

α′{β′γ}

1

α′βγ

′ ′

′

{αβ′}γ′

1

αβγ α′β′γ′

α′β′γ′

+

3 ǫ

B

+

3 ǫ

B

+

6 ǫ ǫ

B

,

B

= 6 B

ǫα β

γ

ǫα β

γ

{...} means symmetrization in permutation of indexes

B{αβγ} ≡ Bαβγ + Bβαγ + Bγβα + Bαγβ + Bβγα + Bγαβ

symmetri

α′ β′γ

}

α′β′γ

α′γβ′

αβ′

γ′

αβ′γ′

β′αγ′

{

{

}

part

≡ B

+ B

B

≡ B

+ B

.

B

represents the de upl

of

1 B{αβγ} of tensor Bαβγ ontains 3·4·5

= 10 independent omponents and

6

B′

, B014 are tra eless.

1·2·3

otally antisymmetri partSU (3) - group.

invariant under

61 ǫαβγ ǫα′β′γ′ Bα′β′γ′

of tensor Bαβγ with one independent omponent is

Two parts ofSUtensor(3) -transformations and represents

he singlet of SU (3) - group.

Bαβγ with mixed symmetry tha

are equivalent to tensors

represents two o tets be ause of tensors

′

′

γ}

′

}γ

′

′ γ

′ ′

γ

′

B

α {β

0

α

′ ′

′

B

{αβ

B0γ′ ≡ ǫα β

, B

α′

≡ ǫα β

γ

γ

α

0γ′

α′

and two o tets with mixed symmetry3/2 . Let us write theinresultsymmetriin the defollowinguplet, formantisymmetri singlet

Finally, we have divided the wave fun tionSBαβγ M

M A

fun

3 3 =

8′

+

of(deIf baryonsonesuuplettionh inomparesofompletenessisthebaryonssinglyfollowingthispresentedwithresultisformspinthewithandPaulitheandprinsingletexpotiplerimentallyof10bafobaryons+ onstituentisobservedabsentwith8 +spinquarksatmultipletsall11/2)..AsLetitweanusofwillbewritebaryonseenotedthebelowgroundthatbaryonthethereasonstaowavetet

×

×

assumed

SU (3) olor

S

A

αβγ

S

antisymmetri

theolor waveoordinatefun tionwavethatfunistionassumedthat isto be

to be symmetriin for the

indexesgroundwhere theondstatemultiplierrstof multiplierthebaryon;isΨ the= isΨ(x1, x2

, x3) × Ψ(c1

, c2, c3) × B

× Ψ( 1,

2, 3) ,

symmetry

S

M

M

antisy

metry(baryof lornslikewaveallfunhadrtionsforare assumed to be single s of

SU (3)

- olor gr

up; mp re

c1

, c2

, c3

fun tion for

SU (3) olor singlet with the antisymmetry of avabove;r wav

third SU (3) avor singlet);

the fourth

Bαβγ is avor wave fun tion the symmetry of whi h was just des ri

d

bytheTheommonavorthdantisymmetryandfollowupletindexsp

2

2

2 =

2′

+

.

inofgwillwavespinsimpleofak3/2funthe6argument.tiovalues:Anothertals iswave.symmIfway,onefuntheritiononsiders.oOnetet(Pauli4+wayofthespinprintovormake1/2,iple)2andisthwspinsnotllprodubesovariablesaobvioushievedtsymme.imultaneouslyifLetrihe produisobvious:larifyttheofit

× ×

As4on

sible

16 = 8 · 2

F rty of them o respond to

u, d,

6·7·8

= 56 independent

mponents.

de uplet of spin 3/2 (

avor-spin indexesesentsor ( avor-spin wavequarksfun withtion)upontainsand down

. The symmetri in ommon

1·2·3

annot not toprinrep iple theplathe of

pin 1/2 (

10 · 4). There remain 16 omp

that

So,.4weethemorehaveTwoPaulithatsepresentationsthereintheisselenoase tsof

esonsofforpobaryonthevorosingletavortetoofmultipletstetamongpart)les.theandangroundspinslemdesstateofribedgroundbaryonsbythestate.trabaryonsless . Note

matrixputt

ompared

be

matwaythirdtheix omponentoantetbeofsolvedbaryonsofisotopiuniquely. Thespinbyrobandthe thewhatomparisontielet lewitharges ouldmesonf

3partiandin×one3by.Letlesortheshouldanotherusbsesvationberibeplaequalinethasuin(hthe

we have the

e

and

Q = T3 + Y /2

are both the generators of

SU (3)

- gr up). Hen e

orresponden T3

π0

π−

η8

√2

+ √6

K

15

Σ−

√2

+ √6

n

√

2

+

√

6

π

η8

K

√

2

+

√

6

Σ

p

−

π0

0

→

−

Σ0

Λ

K−

K¯

− √6

Ξ−

Ξ

−√6

0

2η8

0

2Λ

and de upletqqq. Thof baryonsbese were. Ifhe o tets and singlets of pseudos alar and ve tor

and the o tet

multiplet would

equal toSUea(3)h othersymme. In

ry werereal worldexa t thesymmetry the

asses of parti les in ea h

symmetry and the masses

f parti les in the multiplets are essentiallySU (3) - disymerenmet:ry is an approximate

π

K

η

η′

P αβ ( 140

490

550

960 )

ρ

K

ω

φ

V αβ ( 770

890

780

1020 )

N

Λ

Σ

Ξ

Bαβ ( 940

1116

1190

1320 )

Σ

Ξ

Ω

The vi lation of

Dαβγ ( 1230

1380

1530

1670 )

) andSU (3)strange- symmetry is mass Lagrangian of quarks

isotopi

di eren e of nons rangeSU (3) - symmetryviolatingtiates on the quark level and is onne ted to the mass

Onsymmof justth

u, d - quarks (4, 7M eV

tiesthemofLagrangianofquarksymmetquarksangian( onlythe.are the mass formulae- q arkthat( an be). Thededu onsequenedfromthee

trythisquarkpropermelevelhani

u, d

4, 7M eV

150M eV

(singlet underLagrangian

presented

SU (3)

metri al term

omponent of theSUtensor(3) - groupunder ansformation) and SU (3) violatingterms:

(transforming like (3,3)

gletromagnetited the v olation¯

of

=

symmetry¯

on e tedisotopi the mass di eren e of

nonstrangeHere we have ele−

m = mq (¯uu + dd

+ m ¯