1

FisicaSperimentaleNucleareeSubnucleare

AA2019/2020,UniTS

Quark- VII

Prof.MarinaCobalUniversitàdiUdineeINFNTrieste

Resonances

StatewithenergyE0 ()andlifetimeτToallowfordecay,weneedtochangethetime-dependence:

Quantummechanicaldescriptionofdecay

Whatisthewavefunctionintermsofenergy(insteadoftime)?o Infinitesumofflatwaves,eachwithownenergyo Fouriertransformation:

( ) ÷øö

çèæ G--

Y=

2

1

0

0 iEEi

ProbabilitytofindparticlewithenergyE: Breit-Wigner

E0-Γ/2 E0 E0-Γ/2

Pmax

Pmax/2

Resonance-structurecontainsinformationon:§Mass§ Lifetime§ Decaypossibilities

Resonance

J/ψ

Z-boson

e+e- cross-section

e+e-→R→ e+e-

π+p→R→ π+pMoreresonances

Nr.of‘elementary’particles1932:electronprotonneutron

1936:electronprotonneutronmuon

1947:electronprotonneutronmuonpion

§ 1932:thepositronhadbeenobservedtoconfirmDirac’stheory,§ 1947:andthepionhadbeenidentifiedasYukawa’sstrongforcecarrier,

Ø So,thingsseemedundercontrol!?

§ Ok,themuon wasabitofamystery…

§ Rabi:“Whoorderedthat?”

1947

• Chargedpionsdecaymainlytothemuon-neutrinopair(BR~99.99%)havinglifetimesof2.6x10-8 s.

• Inquarkterms:

• Neutralpionsdecaymostlybytheelectro- magneticinteraction,havingshorterlifetimeof0.8x10-16 s

• Atthebeginningdiscoveredpionswerebelievedtoberesponsibleforthenuclearforces

• However,atrangescomparablewiththesizeofnucleonsthisdescriptionfails.

( ) µnµ +® +du

ggp +®0

• If crosssection formuon pairs isplotted one find the1/sdependence

• Inthehadronic final statethistrendis broken byvarious strongpeaks

• Resonances:shortlived states withfixed mass,andwell definedquantumnumbers® particles

• Theexponential timedependencegives theform oftheresonancelineshap

s1 10 100

s(c

m2 )

J/Yr,w

Muonpaircrosssection

Discoveryofthewmeson

Discoveryofthewmeson

Peak

Quarks• Quarksares=½fermions,subjecttoallkindofinteractions.

• Theyhavefractionalelectriccharges

• Quarksandtheirboundstatesaretheonlyparticleswhichinteractstrongly

• Likeleptons,quarksoccurin3generations:

• Correspondingantiquarksare:

÷ø

öçè

æ÷ø

öçè

æ÷ø

öçè

æbt

sc

du

÷ø

öçè

æ÷ø

öçè

æ÷ø

öçè

ætb

cs

ud

Thequarkmodel:

Baryons andantibaryons arebound states of3quarks

Mesons arebound states ofaquarkandananti-quark

Barions andMesons are:Hadrons

Hadrons

QuantumNumbers&flavours

• Strangeness:S=-1fors-quarkandS =1fortheantis-quark.Further,C=1forc-quark,B=-1forb-quarkandT=1fort-quark

• Since t-quarkis avery shortlivingone,there arenohadronscontaining top,i,e,T=0forall

• Quarknumbers foru- andd-quarks have noname,but they arealso conserved instrongandem interactions.• Baryons areassigned own quantumnumber B:• B=1forbaryons,B=-1forantibaryons,B=0formesons

Barions

• Theorypostulatedin1964(Gell-Mann)

ep

e’

• Inthe70’s,deepinelasticscatteringofelectronsonpandboundnshowevidenceforthequarkmodel

§ Instronginteractions theflavour quantumnumber is conserved

§ Quarks canchange flavours inweak interactions (DS=±1,DC=±1)

ParticlesandInteractions

Hadronsandlifetime

§ Majority ofhadrons areunstable andtend todecay bystronginteraction tothestatewiththelowest possible mass(t ~10-23 s)

§ Hadrons withthelowest possible massforeach quarknumber(S,C,etc.)may livemuch morebefore decaying weekly(t ~10-7- 10-13 s)orelectromagnetically (mesons,t~10-16- 10-21 s)Such hadrons arecalled stable particles

Mesons

• Resonances decay bystronginteractions (lifetimes ~10-23 s)

• If aground stateis amember ofanisospin multiplet,thenresonant states will form acorresponing multiplet too

• Since resonances have very shortlifetimes,they canonly bedetected through their decay products:

p- +p® n +XA+B

Resonancedecay

• Invariant massoftheparticle ismeasured viamasses ofits decayproducts:

• Atypical resonance peak intheK+K- invariant massdistribution

222

222 )()(

MpE

ppEEW BABA

=-

=+-+º!

!!

• The wave function describing a decaying state is:

with ER = resonance energy and t = lifetime

• The Fourier transform gives:

• The amplitude as a function of E is then:

K= constant, ER = central value of the energy of the state

• But:

( )22 )0()0()(G+--- == RR

iEttti eeet yyy tw

( ) dtetg tiò¥

=0

)( wyw

( )

( )ò ò G--===

úû

ùêë

é -+÷øö

çèæ G-

2)0()()( 2

iEEKdtedtetER

EEitiEt R

yyc

)()(* EE cc

( ) úûù

êëé +-

=

4

422

2

maxG

G

REEss

• Supposetheinitial-stateparticles areunpolarised.• Totalnumber offinal spinsubstates available is:

gf =(2sc+1)(2sd+1)

• Totalnumber ofinitial spinsubstates:

gi =(2sa+1)(2sb+1)

• One has toaverage thetransition probability overall possibleinitial states,all equally probable,andsumoverall final states

ÞMultiply byfactor gf /gi

• All thecrossed reactions areallowed as well,anddescribed bythesame matrix-elements (but different kinematic constraints)

badcdcbabcdadbcadcba

+®+++®

+®+

+®+

+®+

Spin

• Thevalueofthepeakcross-sectionsmax canbefoundusingargumentsfromwaveoptics:

With=wavelenghtofscattered/scatteringparticleincms

• Includingspinmultiplicityfactors,onegetstheBreit-Wignerformula:

sa andsb:spinsoftheincidentandtargetparticlesJ:spinoftheresonantstate

( )( )( ) ( )[ ]4

41212124

22

22

G+-

G

+++

=Rba EEss

J!ps

( )124 2max += J!ps

!

• Theresonantstateccandecayinseveralmodes.

• “Elastic”channel:c®a+b(bywhichtheresonancewasformed)

• Ifstateisformedthroughchannelianddecaysthroughchannelj

• MeanvalueoftheBreit-Wignershapeisthemassoftheresonance:

M=ER..G isthewidthofaresonanceandisinversemeanlifetimeof

aparticleatrest:G =1/t

To get cross-section for both formation and decay, multiplyBreit-Wigner by a factor (Gel/G)2

To get cross-section for both formation and decay, multiplyBreit-Wigner by a factor (GiGj /G)2

4/)(4/)( 220

22 G+-=

G+-=

WWK

EEKN

R

• MeanvalueoftheBreit-Wignershapeistheresonancemass• M=ER.G isthewidthofaresonanceandisinversemeanlifetimeofaparticleatrest:G =1/t

• Internal quantum numbers are derived from resonance decay products:

X0 ® p+ + p-

• for X0: B = 0; S = C = = T = 0; Q = 0 Þ Y =0 and I3 = 0

• To determine whether I = 0, I =1 or I =2, searches for isospin multiplets

have to be done.

Example: r0(769) and r0(1700) both decay to p+p- pair and

have isospin partners r+ and r-:

p± + p ® p + r±

B~

p± + p0

• ForX0,bymeasuring angular distribution ofthep+p- pair,therelativeorbital angular momentum LcanbedeterminedÞJ=L;P =P2p(-1)L =(-1)L ;C=(-1)L

• Someexcited states ofpions:

• Resonances withB=0aremeson resonances,andwithB=1arebaryon resonances

• Baryon resonances canbeproduced inpion-nucleon scattering:

• Formation ofresonance R andits inclusivedecay into anucleon N

• Peaksintheobservedtotalcrosssectionofthep±preactioncorrespondstoresonancesformation

p± scatteringonproton

• All resonances produced inpion-nucleon scattering have thesame internal quantumnumbers as theinitial state:

B=1;S =C==T=0,andthus Y=1andQ =I3 +1/2

• Possible isospins areI=½orI=3/2,since forpion I=1andfornucleon I=½

I=½Þ N – resonances (N0,N+)I=3/2Þ D-resonances (D-,D0,D+,D++)

• Intheprevious figure,thepeak at ~1.2GeV/c2correspond toD0,D++ resonances:

p+ +p®D++® p+ +pp- +p®D0® p- +p

B~

p0 + n

• Fits bytheBreit-Wigner formulashowthat both D0 andD++have approximately same massof~1232MeV/c2 andwidth~120MeV/c2

• Studies ofangular distribution ofdecay products showthatI(JP)=3/2(3/2+)

• Remaining members ofthemultiplet arealso observed:D-,D+

• There is nolighter statewiththese quantumnumbersÞD is aground state,although aresonance

Dalitzplot

• TheZ0 intermediatevectorbosonisresponsibleformediatingtheneutralweakcurrentinteractions.

• MZ =91GeV,G=2.5GeV.

• TheZ0,candecaytohadronsviapairs,intochargedleptonse+e-,µ+µ-,t+t- orintoneutralleptonpairs:

• Thetotalwidthisthesumofthepartialwidthsforeachdecaymode.TheobservedGgivesforthenumberofflavours:

Nn = 2.99 ± 0.01

ttµµ nnnnnn ,,ee

Z0

qq

TheZ0 resonance