Кварковая_структура_адронов

Le ture 9. Chiral perturbation theory

9.1 Theoreti al motivations for Chiral perturbation theory

¯

SUA(3) -

is spontaneously

violated by quark ond nsates (< 0|uu¯ |0 >=<

ofrealized- t ansformbytheonepresenparti leesymmetryofinothet multof plet to anothermultiplet

SU (3)

massless

0broken|dd|0 >=< 0|¯ |0 > ) and massless pseudos alar mesons are Goldstone bosons of spontaneously

Inhiralthe limit of massless qu rks (

= 0, = 0,

= 0 ) the Lagrangian of QCD is invariant under

u

SUL(3)×SU (3)R

massesnsformationsonservation.ofAsve resulttorurrthentsotetmeansofve tor thereurrentsareand the

tet of axial

rents are onserved. -Thetr

multipletsof

of

generators

pa ti les wi degenerate

ins de ea h

nd ve t harges - the SU 3)

alar. mesonsTheonservation. The axial ofhargesxial - urrentsthegeneratorsan be

dos one

mesonSUA. (3)Axialtransform the one-parti le states to two-p rti les states with additional pseudos alar

PCAC

s

atofpse quarklowhniqueudosdivergennstrumomentaalarmatssomeiesmesonstum-..ofsymwasTheseFurther,etransferaxialemtivepowerfullassivealurrthe.modelulationsDireomntswithhypinitialmethodtwiththethimassesviolationweretopseudosforeldsalwaysnalsquaredoftheofpartialofstatealarpseudinal.onstruproportionalulatiSupplementedmesstatesnservation- alarsymmetrytedofdegreesofartmesonsamplitudandofbyapprofbyaxialfreadomurreisthepriatenzeunderstandablewiurrentoperatorhalgebraonlypseuquarklinear(thatPCACoslevelmassesPerturbationombinationalarwouldtheredualthough)mesonsrelatemakesiretiongiv

Theory SU (3)

SUA(3)

same resu ts as QCD ones. Su h

was

and is alled Chiral

(ChPT).

ChPT i volv s pseudos alar meson dmodelgres of freadom expli itely and has the same hiral SUL(3)×

SU (3)R

spontaneourresymmtualgebraslydosviolattryalarlikebutmesonsdQCDnowbyvathe.asSo,uumGoldstoneurrentstherexpeartationarebosonsonstrutetsvalue.ofDiretedofonservedpseudostfromvilationpseudosv torof eldsandalarandxialeldsthere.urrentsAxialisansymmetrywithotethe

masslesssame

elegantshouldaxialoversymmetry

SUL(3) × SU (3)R

of

alarQCDntamesonsanarerethattobeisnpresentormexpressanobtainedgaveandexalizableheserelationChPTinppossibilitytheoryparametersmuinandhamongChPTwhilmorweto

QCDssibility

sof

isalalsoTheulatequarkurrentsposexpewayremarkabletheviolationiblemasses.tamplitudandtheandsamandpseudoswhendi erenquarkrelationsresultswithomparedalarebetweenondenpseudosforameltheseates.ngQCDQCDSo,altheamplitudes.rInallrelationsmesonsandmassesChPTChPTdivergeningradientsatofPCACgivebutinitialpseudlowthenowthatiesmomhypothethis

ChPT is

lizable

. All

in QCD a

be bsorb

by (in nit

diverg n ies of diagrams initiat d by some

Lagrangian and

ntainmore

more loops

requirede nitionthenonrenormtrod tion of mortheoryand mo

startingw terms in the Lagrangian o

theseandivergen ies.

New

in the Lagrangian (with paultravioletam ters not xed by the theory

) ntain more and more

)

of

li ited number of parameters in

QCD Lagra

gian while in ChPT the

of pseudos alar

elds and lassi ation of terms by the numbabsorbof derivatives is alled

erivativesturbatermsive de om

osition in ChPT:

L = L2 + L4 + ...

. Let us des ribe the lowest order Lagrangian

amplitudes

L2 ,

pa

x

forameterstwopionsby atteringomparisonandto QCD and then onsider as an example the al ulations of

Φ

Φ2

U † = I − i

Φ

Φ2

and get the LagrangianU = I + i

−

+ ... ,

−

+ ...

F

2F 2

F

2F 2

L2 at the se ond oder in elds

relations among

masses

= 1

pseudos alar

π

η8

2

(

0

same

+

− + 2

+

0

+

L

2

∂µπ

∂µπ

∂µπ

∂µπ

∂µK

∂µK

∂µK

∂µK

∂µη ∂µη ...

mixing angle

B

symmetry

d)π+π− +

u - and d - quark masses. The

− 2 ((mu

+

d)π0π0

+ 2(

u

+

θ de ed by the mixing formulae

+2(mu

+ m )K+K−

+ 2(

d

+

s)K0K0 +

1

(mu

+ md + 4

)η8η8

+

1

(mu

0

η8 + η8

0

) + ...) .

symmetryFrom here

violatiowe see +th3

√3

− m )(π

π

met rs(46)of

natrethethe

as in QCDthe.Also, thereof

mixing massmesonstermandfor the par

due to the

of iso opi

by nonzero di eren e of

0 and

esons

π0 ≈ π˜0 + θη˜8

where tilde stays to

denote mass

eigenstates, is a small0 number

η8

≈ η˜8

− θπ˜

,

osition

de opm

matrix

U

mπ2˜

(mu

)B (mu

md)

0

0.3

θpions

s −

−

0.012 .

9.3

Two

≈ √

2

2

≈

(m + ) √

2

2

≈ − √

≈ −

attering and

3(mη˜8 − mπ˜0 )

3(mη˜8 − π˜0 )

u

d

15 3

Now let us go to the

osition f η → 3π de ay

de opm

prof essesLagrangian

up to the forth oder in lds followed by the

spe i

- two pion L2aonsidered

in two

tteringtheforthand oder in elds. Be ause of we are inter sted only

matrix

η → 3π de ay - we will put to zero other elds in

Ina)

twotheU irrelevanttwopionspionsatteringsforatteringtheproweessetake

.

K, η8 = 0 and negle t small π0η8 mixing. Then

F42 h∂µU ∂µU

†i is equal to

U = EXP (iπ · τ /F )

U = I + i

π · τ

π2

i

π · τ π2

+

π2π2

π

F

−

2F 2 −

π

6F 3

24F 4

Forth oder in elds

ontribution

·

τ

π2

·

τ π2

π2π2

U † = fromI i

−

+

+

.

F

2F 2

6F 3

24F 4

−

F 2

2

∂µπ2∂µπ2 33

∂µπ

∂µ(ππ2)

− 2 · 2

·

4

4F 4

6F 4

1

2

2

m2

2

2

π

where the total derivative terms are=

skeeped and equation of motion

8F 2

∂µ

π

∂µπ

−

6F

2 π π

,

∂µ2 π = −mπ2 π for pion eld is

used. Forth oder in elds ontribution from F22 BhM (U + U †)

F 2

π2π2

m2

2

is equal to

so, in total the part of Lagrangian

BhM 2

=

π

2

,

π

π

2

24F 4

24F

2

L2

responsible

for two

pion s attering is equal to

R membering th

1

∂µπ2∂µπ2 − mπ2 π2π2 .

8F 2

reate nal

π2 = 2π+π− + π0π0 and that π+ eld opera or annihilate initial π+state and

amplitudes

be easily al ulated

three

πan− stat

while π−

eld operator annihilate initial π− state and reate nal π+

1

2

2

1

1

1

Mπ+π−→π+π− =

2 · 2 · 2 ( − mπ ) + ( − mπ )

=

M2 +

M1 +

M0

8F 2

6

2

3

M 0

0

0

0

1

2

2

2 (s

m2 ) + (

m2 ) + (u

m2 ) =

2

M2 +

1

M0

π

π

→π

π

= 8F 2

·

·

1−

π

−

π

1

−

π

3

3

Here we have us d the de +omposition− 0 0

of two pion

2

π ) = 3

(M0 − M2) .

Mπ π →π π

=

8F 2 2 · 2 · 2( −

total isospin and

states

π+π− and π0π0 to the states with de nite

Extra ting

further

amplitudesdenote the

s attering amplitudes

for total isospin

I = 0, 1, 2

respe tively.

M0, M1

, M2

M0, M1, M2 we obtain

M0 =

1

3(s − mπ2 ) + ( − mπ2 ) + (u − mπ2 )