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Κεφάλαιο 5: Η δομή των Αδρονίων

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H Dom twn AdronwnH skèdash qrhsimopoietai gia thn anagn¸rish th dom toustìqou. Qrhsimopoi¸nta thn gwniak katanom touskedazomènou bl mmato (p.q. hlektrìniou), mporoÔme naproume plhrofore gia thn dom .H genik idèa enai h eÔresh tou pargonta morf (form

factor) F (q)

dσ

dΩ=

dσ

dΩ

∣∣∣∣shmeiakì stìqo |F (q)|2me q h metaferìmenh orm metaxÔ tou prospptonto bl mmato e kai tou stìqou: q = ki − kf . Xekinme me thskèdash e se stìqo qwr spin, me forto Zeρ(x) ìpou

222

∫

ρ(x) d3x = 1Gia statikì stìqo èqoumeF (q) =

∫

ρ(x)e−iq·x d3xen¸

dσ

dΩ

∣∣∣∣shmeiakì stìqo =

dσ

dΩ

∣∣∣∣Mott

=(Zα)2E2

4k4 sin4 θ2

(

1− v2 sin2 θ

2

)

A doÔme p¸ ftnoume s' autì. Gia statikì forto me tim Zeja èqoume Aµ = (V, 0) me∇2V (x) = −Zeδ3(x) gia shmeiakì forto∇2V (x) = −Zeρ(x) gia forto me puknìthta ρ223

Opìte, gia to shmeikì forto

∫

e−iq·x∇2V (x) d3x = −∫

Zeδ3(x)e−iq·x d3x

(−iq)2

∫

e−iq·xV (x) d3x = −Ze∫

e−iq·xV (x) d3x =Ze

q2ìpou sthn deÔterh isìthta odhghj kame me dipl olokl rwshkat mèrh (kai mhdenismì tou epifaneiakoÔ oloklhr¸mato pouprokÔptei kje for). Gia to forto me puknìthta ρ èqoume

224

antstoiqa∫

e−iq·x∇2V (x) d3x = −∫

Zeρ(x)e−iq·x d3x

(−iq)2

∫

e−iq·xV (x) d3x = −ZeF (q)

∫

e−iq·xV (x) d3x =Ze

q2F (q)

T¸ra gnwrzoume ìti

Tfi = −ie∫

d4x ΨfγµΨiAµ = −ie

∫

d4x u(kf )γµu(ki)e

−i(ki−kf )xAµ

225

Gia Aµ = (V (x), 0) ja èqoume

Tfi = −ie(ufγ

0ui

)2πδ(Ei − Ef)

∫

d3x e−i(ki−kf )·x V (x)

= −ie(ufγ

0ui

)2πδ(Ei − Ef)

Ze

q2F (q)ìpou q = ki − kf kai F (q) = 1 gia shmeiakì forto. Gia na pmesthn diaforik energì diatom , parnoume

dσ =|Tfi|2T

d3kf

(2π)32Ef

1

2Ei

1

v=

= (e4Z2)

[1

2

∣∣ufγ

0ui

∣∣2](

F (q)

q2

)2

2πδ(Ei − Ef )d3kf

(2π)32Ef

1

2Ei

1

vìpou qrhsimopoi same ìti (2πδ)2 = T2πδ kai h agkÔlh uponoejroisma sta spin. 226

Gia statikì pedo, h enèrgeia tou hlektronou den allzei(δ(Ei − Ef )), opìte kai |kf | = |ki| kai E2 = k2 +m2 →EdE = kdk. 'Etsi, d3kf = k2

fdkfdΩ = kfEfdEfdΩ. To dσgrfetaidσ = Z2e4

(F (q)

q2

)2

2πδ(Ei − Ef )kfEfdEfdΩ

(2π)32Ef

1

2Ei

1

v

[1

2

∣∣ufγ

0ui

∣∣2]

=Z2α2

4(q2)2

[1

2

∣∣ufγ

0ui

∣∣2][F (q2)

]2dΩ

ìpou qrhsimopoi same ìti kf/Ef = v. A upologsoume t¸rathn agkÔlh (Ei = Ef = E kai |ki| = |kf | = k)1

2

∑

sf ,si

∣∣ufγ

0ui

∣∣2 =

1

2Tr [(k/i +m)γ0(k/f +m)γ0]

227

=1

2

[Tr [k/iγ0k/fγ0] +m2Tr

[γ2

0

])

= 2(2EiEf − kikf +m2

)

= 2(2E2 − (E2 − k2 cos θ) +m2

)

= 2(2E2 − (E2 − k2 cos θ) + E2 − k2

)

= 4E2

(

1− v2 sin2 θ

2

)

me v = k/E h taqÔthta tou hlektronou. Opìte,

dσ

dΩ=

Z2α2

4k4 sin4 θ2

E2

(

1− v2 sin2 θ

2

)

[F (q)]2 (23)

me q2 = (kf − ki)2 = 2k2(1− cos θ) = 4k2 sin2 θ

2

. Gia shmeiakì

228

stìqo, ìpw epame, F (q) = 1. 'Opìte, prgmati,

dσ

dΩ=

dσ

dΩ

∣∣∣∣shmeiakì stìqo [F (q)]2

'Askhsh 42 An to hlektrìnio eqe spin 0, dexte ìti h èkfrash

4E2

(

1− v2 sin2 θ

2

)

antikajstatai apì to 4E2 kai h Ex.23 gnetai

dσ

dΩ=

Z2α2

4k4 sin4 θ2

E2 [F (q)]2

Opìte, mpanei h er¸thsh: giat to mh sqetikistikì hlektrìnio,dhlad gia v → 0, me spin=1/2 den diafèrei apì to hlektrìniome spin=0? 229

To F (q = 0) = 1. Gia mikrè timè tou |q| mporoÔme naanaptÔxoume to ekjetikìF (q) =

∫ (

1− iq · x− (q · x)2

2+ ...

)

ρ(x) d3x

An ρ(x) = ρ(|x|), dhlad sfairik summetrik , o deÔtero ìro tou oloklhr¸mato mhdenzetai, giat epillègonta q = (0, 0, q),

q · x = qz = qr cos θ kai d3x = r2 sin θdθdφdr∫

q · xρ(x) d3x =

∫

qr cos θρ(r)r2 sin θdθdφdr

All ∫ π

0

sin θ cos θ dθ = 0

230

O trto ìro gnetai

−∫

(q · x)2

2ρ(r)d3x = −1

2

∫∑

i

(qixi)2ρ(r)d3x

Lìgw sfairik summetra 1

3

∫

(x21 + x2

2 + x23)ρ(r) d

3x =

∫

x2iρ(r) d

3x, i = 1, 2, 3opìte

− 1

2

∫∑

(q2i x

2i )ρ(r)d

3x = −1

2

∫

(q21x

21 + q2

2x21 + q2

3x21)ρ(r)d

3x =

− 1

2

(∑

q2i

)∫

x21ρ(r)d

3x = −1

2

(∑

q2i

)∫ 1

3

(∑

x2i

)

ρ(r)d3x =

− 1

6q2

∫

r2ρ(r)d3x = −1

6q2 < r2 >

231

ìpou < r2 > enai h mèsh tim tou r2. Opìte

F (q) = 1− 1

6q2 < r2 >Dhlad , gia mikrì |q|, h skèdash metr akrib¸ aut th mèshtim tou r2. To mikrì m ko kÔmato tou fwtonou mpore naxeqwrsei mìno ton sunolikì ìgko tou fortou ρ(r).'Askhsh 43 An h puknìthta fortou ρ(r) tan th morf

e−mr, dexte ìti o pargonta mof enaiF (|q|) ∝

(

1− q2

m2

)−2

Skèdash hlektronou-prwtonou. Pargonte morf tou prwtonouDÔo enai ta stoiqea pou diaforopoioÔn to prwtìnio-stìqo apì232

ta prohgoÔmena: to prwtìnio den enai statikì kai toprwtìnio èqei magnhtik rop . An, par' ìla aut, to prwtìnio tan shmeiakì me magnhtik rop a la Dirac sh me e/2M ,mporoÔme na qrhsimopoi soume to tÔpo pou eqame bre gia thnskèdash hlektronou-mionou, bzonta M thn mza touprwtonou ant tou mionoudσ

dΩ

∣∣∣∣lab

=α2

4E2 sin4 θ2

E ′

E

[

cos2 θ

2− q2

2M 2sin2 θ

2

]

ìpou

E ′

E=

1

1 + 2EM

sin2 θ2Knonta thn dia doulei ìpw me thn skèdash hlektronou -mionou, ja grfame 233

Tfi = −i∫

jµ

(

− 1

q2

)

Jµ d4xìpou q = k − k′ = p′ − p kai jewr¸nta pia to prwtìnio w MHshmeiakì

jµ = −eu(k′)γµu(k)e−i(k−k′)x

Jµ = eu(p′)[...]u(p)e−i(p−p′)x

234

Akrib¸ , grfonta [...] deqnoume ìti to prwtìnio den enaishmeiakì kai den mporoÔme apl na gryoume γµ. Par' ìla aut,to Jµ ja prèpei na enai èna tetradinusma, kai epomènw japrèpei na qrhsimopoi soume thn pio genik morf enì tetradianÔsmato qrhsimopoi¸nta ti ormè p, p′ kai q kaj¸ kai tou γ pnake . MporoÔme, telik, na ftixoume dÔoanexrthte posìthte : mia anlogh tou γµ kai mia deÔterhanlogh tou iσµνqν .'Askhsh 44 H pio genik morf gia to Jµ enai (q = p′ − p)

F1γµ + F2iσ

µνqν + F3iσµν(p+ p′)ν + F4q

µ + F5(p+ p′)µ

Dexte ìti telik mènoun mìno dÔo anexrthtoi ìroi pouantistoiqoÔn sta F1 kai F2 235

Opìte, grfoume[...] = F1(q

2)γµ +κ

2MF2(q

2)iσµνqν (24)ìpou κ enai h an¸malh magnhtik rop tou prwtonou. Prosèxteìti to q2 enai h mình anexrthth bajmwt metablht sthnkoruf tou prwtonou (p2 = p′2 = M 2). Epsh , to p · q den enaianexrthto mia kai(q + p)2 = p′2 →M 2 + q2 + 2p · q = M 2 → 2p · q = −q2. An to

q2 → 0, dhlad ìtan to fwtìnio èqei meglo m ko kÔmato , denmporoÔme na diakrnoume dom sto prwtìnio kai parathroÔmeswmatdio me forto e kai magnhtik rop (1 + κ)/2M .Peiramatik, to κ = 1.79. Jumhjete ìtieu(p′)γµu(p) =

e

2Mu(p′)(p+ p′)µu(p) +

e

2Mu(p′)iσµνqνu(p)236

Opìte, o ìro γµ sthn Ex.24 perièqei thn kanonik rop en¸o llo ìro prosfèrei thn an¸malh rop tou prwtonou, kai japrèpei na epilèxoume, s' autì to ìrio,

F1(q2 → 0) = 1, F2(q

2 → 0) = 1Gia to netrìnio, oi antstoiqe timè enai F1(q2 → 0) = 0,

F2(q2 → 0) = 1 kai κn = −1.91. Qrhsimopoi¸nta loipìn, giaton upologismì th energoÔ diatom , thn Ex.(24), ja proume

dσ

dΩ

∣∣∣∣lab

=α2

4E2 sin4 θ2

E ′

E

[(

F 21 −

κ2q2

4M 2F 2

2

)

cos2 θ

2

− q2

2M 2(F1 + κF2)

2 sin2 θ

2

]

Aut enai h sqèsh Rosenbluth. Apotele mia parametropohsh237

th gnoi ma gia thn dom tou prwtonou. Parathr ste ìtigia F1 = 1 kai κ = 0 katal goume sthn skèdash apì shmeiakìstìqo. Peiramatik oi pargonte morf metrioÔntai seskèdash (sunrthsh th gwna skèdash tou hlektronou).Sthn prxh qrhsimopoioÔntai dÔo grammiko sunduasmo twn FGE = F1 +

κq2

4M 2F2, GM = F1 + κF2kai h energì diatom grfetai

dσ

dΩ

∣∣∣∣lab

=α2

4E2 sin4 θ2

E ′

E

[G2

E + τG2M

1 + τcos2 θ

2+ 2τG2

M sin2 θ

2

]

me τ = −q2/2M . Prosèxte ìti me aut n thn allag denuprqoun ìroi anlogoi tou GMGE.238

Anelastik Skèdash ep→ eXTi gnetai ìtan megal¸sei h enèrgeia pou qnei to hlektrìnio?Dhlad ìtan to −q2 enai meglo. Gia mesaa −q2parousizontai difora swmatdia-suntonismo (resonances):

ep→ e∆+ → epπ0. S' aut n thn perptwsh h anallowth mzatwn proðìntwn W 2 ≃M 2∆. Gia pio meglh metaferìmenh enèrgeiato prwtìnio spei kai qreiazìmaste èna kainoÔrgioformalismì gia na perigryoume to gegonì .

239

Sthn ellastik skèdash antikatast same, sto anallowtoplto , to uγµu tou mionou me uΓµu kai qrhsimopoi same thnpio genik morf gia to Γµ. T¸ra oÔte autì gnetai. Ja prèpeina pme mesa sthn energì diatom (dhlad sto tetragwnismènoanallowto plto ) kai antdσ = L(e)

µν

(L(µ)

)µν

pou isqÔei gia thn perptwsh tou mionou, na gryoume

dσ = L(e)µν W

µν

To leptonikì kommti paramènei to dio. To W µν parametropoiethn sunolik ma gnoia gia thn morf tou reÔmato sthn meritou prwtonou. Kai pli ja prèpei na gryoume to W µν me thnpio genik morf qrhsimopoi¸nta ta pµ, qµ kai to gµν . Prosèxte240

ìti p′ = p+ q kai epsh den mporoÔme na qrhsimopoi soume γpnake mia kai grfoume to tetrgwno tou anallowtoupltou ìpou èqoume ajrosei sta spin. Epomènw grfoume

W µν = −W1gµν +W2

M 2pµpν +

W4

M 2qµqν +

W5

M 2(pµqν + qµpν)

Krat same to W µν summetrikì stou dekte mia kai to L(e)µνenai summetrikì. Kje mh summetrikì kommti tou W µν den jasuneisèfere sto dσ. Ta W ja exart¸ntai apì ta mìno dÔobajmwt megèjh pou sqetzontai me thn koruf tou prwtonou.MporoÔme na epilèxoume ta

q2 kai ν =p · qMH anallowth mza th telik katstash sundèetai me ti dÔo241

parapnw metablhtè W 2 = (p+ q)2 = p2 + q2 + 2p · q = M 2 + 2Mν + q2

H diat rhsh tou reÔmato odhge sti sqèsei

qµL(e)µν = qνL(e)

µν = 0, kai qµW (e)µν = qνW (e)

µν = 0Oi dÔo teleutae sqèsei odhgoÔn se susqètish metaxÔ twntessrwn diaforetik¸n W . Opìte, xanagrfoume

W µν = W1

(

−gµν +qµqν

q2

)

+W21

M 2

(

pµ − p · qq2

qµ

)(

pν − p · qq2

qν

)

'Askhsh 45 Dexte ìti h diat rhsh tou reÔmato sthn

242

adronik koruf (dhlad qνWµν = 0) odhge sti sqèsei

W5 = −p · qq2

W2, W4 =M 2

q2W1 +

(p · qq2

)2

W2

Upenjumzoume ìti ta W1 kai W2 exart¸ntai apì ta q2 kai ν.Sun jw qrhsimopoioÔntai, ant' aut¸n, oi x kai yx =

−q2

2p · q =−q2

2Mνy =

p · qp · kìpou k h (tetr)orm tou eiserqìmenou hlektronou.Upologzoume t¸ra thn energì diatom ep→ eX.

Lµν(e)Wµν = 4W1(k · k′) +

2W2

M 2

(2(p · k)(p · k′)−M 2k · k′

)

243

pou sto sÔsthma ergasthrou gnetai (me = 0)

Lµν(e)Wµν

∣∣∣lab

= 4EE ′[

W2 cos2 θ

2+ 2W1 sin2 θ

2

]

'Askhsh 46 Dexte thn parapnw sqèsh.Telik h energì diatom grfetaid2σ

dΩdE ′

∣∣∣∣lab

=α2

4E2 sin4 θ2

[

W2(ν, q2) cos2 θ

2+ 2W1(ν, q

2) sin2 θ

2

]

(25)

244

Perlhyh gia ton formalismì skèdash epH teleutaa sqèsh grfetaid2σ

dΩdE ′

∣∣∣∣lab

=α2

q4

E ′

ELµν

(e)Wµν (26)

ìpou

q2 = (k − k′)2 = −2k · k′ = −2EE ′(1− cos θ) = −4EE ′ sin2 θ2

.Jumìmaste ìti gia eµ→ eµ eqamedσ =

1

4ME

d3k′

(2π)32E ′d3p′

(2π3)2p′0

[e4

q4Lµν

(e)L(µ)µν

]

(2π)4δ(4)(p+ q − p′)245

Mpore na xanagrafte me th morf (26) an

Wµν =1

4ME

1

2

∑

s

∑

s′

∫d3p′

(2π)32E ′ (2π)4δ(4)(p+ q − p′)

× < p, s|J†µ|p′, s′ >< p′, s′|Jν |p, s >ìpou

< p′, s′|Jν |p, s >= u(s′)p′ γνu

(s)pAn ant autoÔ gryoume

< p′, s′|Jν|p, s >= F1(q2)γν +

κ

2MF2(q

2)iσµνqνja proume ton tÔpo th elastik skèdash ep. GenikeÔonta ,loipìn, mporoÔme na gryoume gia thn perptwsh pou to246

prwtìnio spei kai èqoume parousa N swmatidwn

Wµν =1

4ME

∑

N

(

1

2

∑

s

)∫ N∏

n=1

d3p′n(2π)32E ′

n

×∑

sn

< p, s|J†µ|X >< X|Jν |p, s > (2π)4δ(4)(p+ q −

∑

n

p′n)

'Etsi, h energì diatom grfetai genikd2σ

dΩdE ′ =α2

q4

E ′

E4EE ′ [...] =

4α2E ′2

q4[...]

247

ìpou[...] =

(

cos2 θ

2− q2

2Msin2 θ

2

)

δ

(

ν +q2

2M

)

, gia e−µ− → e−µ−

[...] =

(G2

E +G2M

1 + τcos2 θ

2+ 2τG2

M sin2 θ

2

)

δ

(

ν +q2

2M

)

,gia e−p→ e−p

[...] = W2(ν, q2) cos2 θ

2+ 2W1(ν, q

2) sin2 θ

2, gia e−p→ e−X

ìpou τ = −q2/4M . An oloklhr¸soume ti dÔo pr¸te w pro

E ′ qrhsimopoi¸nta thn sunrthsh δ, ja proumedσ

dΩ=

α2

4E2 sin4 θ2

E ′

E[...]

248

Bjmish Bjorken kai to prìtupo twn partonwnQrhsimopoi¸nta ti nèe metablhtè

Q2 = −q2 = 4EE ′ sin2 θ

2= 2EE ′(1− cos θ), ν = E − E ′

èqoume (jewr¸nta azimoujiak summetra, opìte ∫ dφ = 2π)

dΩ = 2πd(cos θ), (cos θ = (−1, 1))

∂ν

∂E ′ = −1,∂ν

∂ cos θ= 0

∂Q2

∂E ′ = 2E(1− cos θ),∂Q2

∂ cos θ= −2EE ′

dνdQ2 =

∣∣∣∣∣∣

∂ν∂E′

∂ν∂ cos θ

∂Q2

∂E′

∂Q2

∂ cos θ

∣∣∣∣∣∣

dE ′d(cos θ) = 2EE ′dE ′d(cos θ)

249

h Ex.(25) grfetaid2σ

dνdQ2=

πα2

4E2 sin4 θ2

1

EE ′

[

W2 cos2 θ

2+ 2W1 sin2 θ

2

]

O Bjorken prìteine ìti sto ìrioQ2 →∞ν →∞

me x =Q2

2Mν= stajerì

oi sunart sei W gnontaiMW1(Q

2, ν)→ F1(x), νW2(Q2, ν)→ F2(x)prgma pou ta peiramatik dedomèna to epibebai¸noun.Shmantikì stoiqeo th upìjesh tou Bjorken enai ìti sto ìrioautì, oi sunart sei F1 kai F2 enai peperasmène .250

P¸ katalabanoume aut n thn bjmish?O Feynman prìteine na jewr soume elastik skèdash meshmeiak forta (partìnia) pou brskontai mèsa sto prwtìnio.To fwtìnio mpanei baji kai blèpei eswterik dom stoprwtìnio.An gryoume pµi = xP µ (kai mi ≃ xM), dhlad ìti to partìnio ièqei kpoio klsma th orm tou prwtonou, elastik skèdashtou hlektronou me to partìnio ja dnei

d2σ

dνdQ2=

πα2

4E2 sin4 θ2

1

EE ′

[

e2i cos2 θ

2+ 2e2

i

Q2

4m2i

sin2 θ

2

]

δ

(

ν − Q2

2mi

)

pou ja prèpei na sugkrije me thnd2σ

dνdQ2=

πα2

4E2 sin4 θ2

1

EE ′

[

W2 cos2 θ

2+ 2W1 sin2 θ

2

]

251

'Ara ja prèpei h suneisfor sta W1 kai W2 apì èna edo partonou na enaiW i

1 = e2i

Q2

4xM 2δ

(

ν − Q2

2xM

)

W i2 = e2

i δ

(

ν − Q2

2xM

)

Gia Q2, ν →∞ jewroÔme ìti oi suneisforè twn partonwnajrozontai asÔmfwna (incoherently). 'Ara, ajrozoume gia ìlata edh twn partonwn kai oloklhr¸noume gia ìla ta x = (0, 1).Bèbaia, to olokl rwma sta x ja prèpei na èqei kai kpoiasunrthsh brou fi(x) gia kje edo partonou. Autè oisunart sei , pou kaloÔntai katanomè pijanìthta , den252

problèpontai apì autì to prìtupo. Epomènw ,

W2(ν,Q2) =

∑

i

∫ 1

0

dxfi(x)e2i δ

(

ν − Q2

2xM

)

kai epeid δ(g(x)) =

δ(x− x0)∣∣ dgdx

∣∣x=x0

, me g(x0) = 0

ja èqoume

δ

(

ν − Q2

2xM

)

= δ

(

x− Q2

2Mν

)(Q2

2Mx20

)−1

= δ

(

x− Q2

2Mν

)(x

ν

)

Epomènw ,

νW2(ν,Q2) =

∑

i

e2ixfi(x) ≡ F2(x)

253

ìpou x = Q2

2Mν

.Anloga parnoumeMW1(ν,Q

2) =1

2

∑

i

e2i fi(x) ≡ F(x)

opìte

F2(x) = 2xF1(x)H teleutaa sqèsh, sqèsh Callan-Gross enai mesa sundedemènhme to ìti ta partìnia èqoun spin=1/2.To prìtupo twn kourk-partonwnA upojèsoume ìti ta partìnia enai ta kourk tou Gell-Mann meti gnwstè idiìthte (forto, tim tou barionikoÔ arijmoÔ, th paradoxìthta k.lp.). Tìte gia thn allhlepdrash fwtonou me254

ta kourk, ja èqoumeF ep

2 (x) = x

[4

9(u(x) + u(x)) +

1

9

(d(x) + d(x) + s(x) + s(x)

)]

me u(x), d(x), s(x), ... h katanom pijanìthta gia kje èna apìaut. 'Isw fanetai ìti antikatast same mia gnwsth posìthta,

F2, apì èxi gnwste posìthte ! All, oi die posìthte parousizontai, me diaforetikoÔ sunduasmoÔ bèbaia, gia stìqonetronwn (ant prwtonwn) , akìma, gia qr sh netrnwn kaiantinetrnwn ant fwtonwn. Gia pardeigma, gia stìqo netronwn(qrhsimopoi¸nta thn diat rhsh tou isospin) mporoÔme nagryoume gia to netrìnio

u(n)(x) = d(p)(x) ≡ d(x), d(n)(x) = u(p)(x) ≡ u(x)255

opìteF en

2 (x) = x

[4

9

(d(x) + d(x)

)+

1

9(u(x) + u(x) + s(x) + s(x))

]Mia kai ìle oi sunart sei pijanìthta prèpei na enai jetikè ,apodeiknÔetai ìti

1

4≤ F en

2

F ep2

≤ 4sqèsh pou epibebai¸netai kai peiramatik.

'Askhsh 47 Apodexte thn parapnw sqèsh.Epsh , gia to prwtìnio kai to netrìnio me paradoxìthta 0, jaisqÔei

∫ 1

0

dx [s(x)− s(x)] = 0

256

Apì to forto tou prwtonou kai tou netronou èqoume ti sqèsei ∫ 1

0

dx

[2

3(u− u)− 1

3(d− d)

]

= 1, gia to prwtìnio

∫ 1

0

dx

[2

3(d− d)− 1

3(u− u)

]

= 0, gia to netrìnioApì ti parapnw dÔo sqèsei parnoume ti

∫ 1

0

dx [u− u] = 2

∫ 1

0

dx[d− d

]= 1pou akrib¸ deqnei thn persseia twn kourk u kai twn kourk

d se sqèsh me ta anti-kourk. 257

Akìma mia endiafèrousa sqèsh phgzei apì to gegonì ìti xfi(x)enai to klsma th orm pou metafèrei to kourk i. Opìte

∫ 1

0

dx x[u+ u+ d+ d+ s+ s

]= 1− ǫ

ìpou me ǫ dhl¸noume to klsma th orm tou prwtonou pou denmetafèretai apì ta kourk. Peiramatik to ǫ ∼ 1/2, pouupodhl¸nei ìti meglo klsma th orm metafèretai apìafìrtista antikemena. Kat thn Kbantik Qrwmodunamik , taantikemena aut enai ta gklouìnia.MporoÔme na proume kai llou tètoiou kanìne anproqwr soume se jewrhtik prìtupa gia ti katanomè twnkourk. 'Etsi, eisgoume thn ènnoia gia ta kourk sjènou kai ta kourk jlassa . Gia pardeigma, gia to prwtìnio oi258

katanomè twn kourk u kai d parametropoioÔntai

u = uv + qs, d = dv + qsen¸ gia ta kourk s kai ta anti-kourk

u = d = s = s = qs'Etsi, oi èxi gnwste sunart sei antikajstantai apì trei .Fusik, uprqoun pareklsei apì thn bjmish Bjorken kai toaplì prìtupo pou perigryame parapnw, all h KQD dneiapant sei .

259

Χρηματοδότηση

Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί στα πλαίσια του

εκπαιδευτικού έργου του διδάσκοντα. Το έργο «Ανοικτά Ακαδημαϊκά

Μαθήματα» του ΕΜΠ έχει χρηματοδοτήσει μόνο την αναδιαμόρφωση του

υλικού. Το έργο υλοποιείται στο πλαίσιο του Επιχειρησιακού

Προγράμματος «Εκπαίδευση και Δια Βίου Μάθηση» και

συγχρηματοδοτείται από την Ευρωπαϊκή ΄Ενωση (Ευρωπαϊκό Κοινωνικό

Ταμείο) και από εθνικούς πόρους.