Кварковая_структура_адронов

At1.1presentLeTheturelassesexis1. Partiofstrongpartiles,interalesthatandallthein tionsthetypesNatureandofis onstrusymmetriested from elementary parti les

and that allthere Na

ural

pro esses are used by the int rainteraof tionsthese

. Elementary

parti les are assumed today tobeliefgauge

quarks and Higgs s alar parti les. The

fund mental intera tions are assumed todaybosons,leptons,be strong,ele troweak

gravi ational intera tions.

So,

lly we an indi ate four lasses of elementary parti les andthreepartiypeslesof fund

intera tio .

are

leptons into three

families(τ ) and orresponding neutrino (ν , νµ, ντ ). It is onvenient to put six

theTheexisteonvention

γ W

+

, W −, Z

(g , = 1, ..., 8)

ains. Allphotonthese(parti), les transf-bosons,theinteraeighttions:gluons

s

and assummental

tion,

on

rsteinteraoflassgravi

gravitationalele Thetroweakseondeldlass. ontainsgluonsleptonstransfer.Atstrongpresentinterwea havetion andsixofgravitonsthem:γ, eleW +,ronWhypotheti−, Z- bosonsal quantatransfofr

ele tri

ele tron,muon

u, d, s, c, b,

and ea h quark

lepton

( ), muon (µ), tau-

th rdelearelasstroweakis representedandlly gravitationalneutral;byquarksintera.Todaytionsweand.knowtau-sixleptonquarkspossess-

ele tri harge. Leptons(1)

takeTheNeutrinospart

(0.511M eV )

µ(105.6M eV )

τ (1777M eV )

ν (< 3eV )

νµ(< 0.19M V )

ντ (< 18.2M eV )

hundreds

observed yet

onstituents

isastitionspartihereoisolatedmake.olorslesisofonarethe.phadronsLikeHrtieleiggsleptonslestroweak.s. HadronsTogetheralarquarks.partiTheintlikerawithrolelesaretiquarksofnotgluonsrepresentedfHiggsrenormalizablefromtheypartiwhiarebyleshthreetheexperimentallyin.theyInthepartifamiliesareNatureformedular,is.Inthemainlyoftakeminimalhadronsmassespartthe(2)

theoretisinandQuarksishemealloloredThetheretypesofareforthalinteraareonenotofonesomeinteralassandseentionsofofthree

(75 − 170M V )

(4.0 − 4.4G V )

(3 − 9M V )

(1 − 5M eV )

c(1.15 − 1.35G V )

(174G V )

all

pa

es are due to the ondensate of Higgs e d. Probably, the existe e of Higg

elds is ne

.

fo

so ving the fundamental problems

the homog neity and

ausality

of the Univ

symmetrionstruNormally,t ds fromindphysionsth ses quarksofelementarylaws.The. mainpar iattentiones the velo itiesbe payedofthe toorderlassiof -lightquarksationveloandofipartiyhadronsles,

Theelementaryfollowing le turesrvationquark stru ture of hadronswil osmology:onsider light u,

,

a tions and angular momenta of the order of Plank onstant

c the

the system of unitsarewhereenountered. It is natural

herefor

to use in physi s of elementary part mensionll

~

u, d, s-quarks and gluons is des ribed by the following Lagrangian

momentum, a tion

nd velo ity are

unit

ss:

~ = c = 1. Then, angular

have the dimensionality of mass

[J ] = [ ] = [v] = 1. The

nergydimensionalityandmentum

inverse mass

[E] = [p] = [m], and time a

d length have the

of

Strong1.2 Lagrangianintera[ltion] = [oft] =lightof[m−strong]. As theinteraof entions

1GeV

1

ergy itandis onveitsnsymmetriesient to hoose

.

1

Gµνa

Gµνa

+ q¯ i∂µ2+ gAµa

λa

γµq − qM¯ q .

(3)

L = −

4

2

interamassestion

with expligluonsitly,is diagonal; matrixare

isngtheon ouplingolor indexesonstantα whileof strongin interaindexesv tion;

the

M is a ting in avor spa e and is mptransformationssed from qua k

Here

are orresponddes ribingeld stLagrengths;rangian

SU (3)c-

are gau

po entials des ribing eight gluons elds ;

a

= ∂µAν

− ∂ν Aµ +

abc Aµ, c

= 1, ... 8

Gµν

gfDira AbispinorA

lds

matriu,esd, s - quarks (i = 1, 2, 3) of 3 olors (α = 1, 2, 3), the

α

are

µ

ν

are not written

i, α

λA

intera2 tion

invarian e of. theThe

underofquarkstheandfollowing lois alxed by loal gauge symmetry

whi h meansM = diagthe (mu, md, ms)

4)

U (x) is 3 × 3 unitary matrix with determinant equal to one

q → U (x)

¯

¯

†( )

5

q → qU

x

invarpproxim

symmetri ribe

λ

transformation

λ

U †(x) + g U (x)∂µU †(x)

and

Aµ 2

→ U (x)Aµ 2

(6)

quarks we

therefore, the

an e under thediagonal

u, d, s -

λ

ave,sLagrangianareofexastrongt, sointfmstrongtionsnterais alsotions.Let isus des

underinthesequarkvarioussymmetavoglobalries.Intransformations:.thease of some(7)

ofThesymmetriFirst,Lagrangianth

U (

) =

EXP

(−

2 ω (x)) .

moreoSe

(see

u, d, s

onserved

arges

d

quarks

U, D, S

0

d

.

Asonsera onsequenationof

0

EXP (−iω )

XP (−iω )

0

0

u

thee numberlater)ofwe have

three

urrents

and

whi h represent

the(8)

→

0

0

XP (−iω )

Thisofsymmetry is exa t and,

vond,er,isasalsofarLagrangian,asexawet symmetryannegletoftheeleditromagnetierenor e b intwe -underrahargesthetionsmasses..

m =

=

in the Lagrangian,

is a symmetry under SU (3) - transformationquarks i.e. take

u, d - quarks i. . take

mu = m in the

there is an isotopi symmetry

isotopi transformation

s

→

0

1 s

as

u

EXP (

iωi

τ I )

0

u

.

(9)

Third,

far as we an negle tthere

es between the masses of

2

symmassesetries

themselves

di eren es

u, d,

introdu ing

left andofarisequarksrightwhen

d

XP ( iω

2

)

.

alsoNewthe

u

a λa

u

omponentswean negle. Toofquarktseenotitonlyleteldsustherewrite the Lagrangianbetween theofmassesstrongofinteraquarkstions(10)but

→

−

s

qL =

(1 + γ5)

q , qR =

(1 − γ5)

q

1)

2

2

1

λa

λa

L = −

Gµνa Gµνa

+ q¯L i∂µ + gAµa

γµqL + q¯R3 i∂µ + gAµa

γµqR − q¯LM qR − q¯RM qL .

4

2

2

(12)

Finally, if we negle t the

is

SU (2)L × SU (2)R

.

(13)

dL

−0iωL 2 )

1

dL

uL

→

i τ I

0

uL

L

symmetry

Lagrangian

EXP (

SU (3)L ×SU (3)R

strong intera tions

dR

−0

1

dR

.

uR

uR

R

→

τ

R

EXP ( iω

)

0

The orresponding symmetry

alled hiral

R 2

(14)

masses of

- symmetry.

largest symmetry of the

of

u, d,

- qu

rks

. .-takehiralmu = md = ms = 0 we will get t

transformations are

- symmetry. The

L

EXP (

iωLa

2

)

dL

(15)

uL

a

uL

and

harges

ransformations

nsformations

sL

→

−

λ

sL

The variation of theL(φ , ∂µφ ) anunderthen apply the results to the Lagrangian of strong intera ti .

generalofThe1.3orrespondinginvarianofConsLagrangian

R

XP ( iωR

2 )

dR

.

R

a λ

uR

tus

emosomeglobalstrateonservedthis trstatement (k. Chargesownof aseldsNoetherleadsas thetoheorem)thegeneratorsonservationfor(16)me

thee ofrvedsymmetrytheurrentsLagrangianurren. Let

→

−

R

R

Lagrangian L(φ , ∂µφi) under some variation of elds is equal to

δL

δL

Due to Lagrangian equations of motionδL =for δφeldsi +

δ∂µφi .

(17)

δφi

δ∂µφi

φi

δL

δL

Thus,

= ∂µ

.

(18)

δφi

δ∂µ

φi

If

transformation

Lagrangian then δL = 0 means the onservation of urrents

δL = ∂µ δ∂µφi

δφi

.

(19)

δL

is the symmetry transformation for the

δφ = −iωa

ija φj

(20)

J a

µ

∂µJµa = 0

(21)

momentum

( ,

) hange

signs, the axial ve tor of angu

parityandthat axialis(

lsoveuntorshanged underthe ispseudostheunhangedtransformaalar. thatTheionshangealarfspaproduthee inversionsignt ofundertwobutpol reveinversionalartor produforms.Thetheofspatialspoa

( , l

= ǫijk ipk )

spathe

forms

spa e inversionP ) of some.S alarobjeandt isaxialdeterminedinteravetionstareby its property to hange o

not to hange the sign under

i. . Theitis Lagrathe

. Toof strongseeitlet us write theis notPLagrangianevenhangedbutuonpsdeudrmtheores tralansformationand ofarespaP -e oddinversion.

gian

s alar

Under spa e inversion quarkL =

1

µνa

a

λ

µ

(32)

elds transform like Dira bispinors

−4 Gµν G + q¯

i∂µ

+ gAµ

2

γ q − qM¯ q .

q → p γ0q

,

q is the internal spatial parity of the quark

elds and an take the values ±1, ±i. Then

nd

qq¯ → qq¯

qγ¯ µq → qγ¯ µq; ∂µ → ∂µ;

Aµ → Aµa;

Gµν → Gµνa

(34)

Invarianofethewithprorespeess andt tothespaamplitudee inversionof the.onspatheequantuminverted-prome haniess oinallevidel means that the

a

R

dXL( , X) → R

dXL( , −X) = R

XL( , X)

(33)

wheremplitude

where

< bP |S|

P >=< b|P −1SP |

>=< b|S|a > ,

(35)

is the s atte respondingmatrix

de nite

4

and

is the part of the Lagrangian of

strongIftheinterainitialtionsand ornal states to

of elds.

S

S =interaT EXPtion( R

xLint)

Lint

have

P - parity,

P |

>= pa|a >

6)

then

P |b >= pb|

>,

(37

StrongP - intera tionsorof the partionserv. . le-stheparity is

.

p

·

b

= 1

p

= b

P

P - parityonservedse ve also U, D, S - harges i.e. the numbers of

parities

that

u,Therefore,d, s - quarksthe.quark

parities pu, pd, ps an be hosen arbitrarily. It is onvenient to take them equal

to 1:

pu

is equal to -.1:As we will see later the produ t of the internal parities of fermion and

= pd

= ps

= 1

= −1 for

antifermionour onvention

pupu¯ = pdpd¯ = psps¯ = −1 . This means that pu¯ =

d¯ = ps¯

pu = pd = ps = 1. The internal parity6 of the system of two parti les (of the meson

onsisting from quark and antiquark, for example) is equal to the pr du t of the internal parities of

the onstituent parti les and of the orbital parity of their relative motion

orbital

ground states

relative

quarks and

summation

q , q¯

is assumed. For

= 0

P (M (qq¯)) = −1

(pseudos alar

produ t

+

of mesonsti les (of

baryon

+

+

(rq¯)|0 >

P

+

(− q¯)|0 >

Ψ( q − q¯)aq (rq)

q¯

→ pq pq¯Ψ(rq¯ − rq )

q (−rq ) q¯

wherei. .

=

q pq¯(−1) Ψ(rq − q¯)aq+( q )aq+¯ ( q¯)|0 >,

orbital

is the

momentum of

motionparityof

the

over the variables

third

P (M (qq¯)) =

−1)l+1,

third

is the. orbitalForgroundmomentumsatesofofbaryonsthepair, L is

the

relative orbital momentum of the pair and

for

example)Themesons)internalonmotion.istituentequalparityofanytoofthepairthe ofsystemonstituentsofthreetheinteandparthenal

ritiesandtheofofrelativethe onstorbionsistingaluents,motionfromtheofpathhosenityeequaandofpairrelativeks,veandt

parti le

l = L = 0

P (B(qqq)) = 1

where

parti le

p(qqq) = q q pq (−1) (−1)L,

.les (nothargesonlyof whiele trihhaveally

2e geentfunChargeosigntionderjugation,withonjugationrespebydet tonition,thetransformh rgesnsformsof

. Trulytoandantiparneutrali le,partill the

diCha2.

neutral)

parity. F

r example,C - waveonjugationfuntion (or the state)to themselvesofsome partiwhat lesallows( to introdu e the notion of C -

harge

njugation

γ, ρ0, ω, φ) hanges the sign under

wave

of other parti les (

Cγ = −γ, Cγ = −1, ...,

η, η′, π0) does not hange the sign under harge onjugation

hangedlike Diraunderbispinorsthetransformation of harge

onjugation.

UnderThe hargeL grangianonjugationofstrongquarkinteraeldstionstransformCηis not= η, Cη

= 1, ....

quark

invariant

internal

parities

transformations

−1 in the produ t pq · p¯ = −1. It an be seen from the following

hain

Then

q → qc = Cq¯T

, C = iγ0

γ2 .

(38)

qq¯

→

qq¯

qγ¯

µ λ q

qγ¯

µ −λ T q;

A λ

→

A

−λ T ;

G

λ

→

G

−λ T

(39)

is

2

→. Let us2

that2

spatial2

2

2and

giveand the Lagrangian

antiquark

show

µ

spin s

µ

µν

of

µν

antiquark

Cmomentumparity ofof mesonsquarkandonst u ted

fromand

theiquarktotaland antiquark

is determined by

the relative orbital

q

c P

γ

0