Introduc+on to Par+cle Physics - Chapter 7 · PDF fileClaudio Luci – Introduc+on to Par+cle Physics – Chapter 7 10 4/4 The Sargent’s rule ... , the ones with L=2 second forbidden

WeakInterac+ons

ClaudioLuci–Introduc+ontoPar+clePhysics–Chapter7 1

Introduc+ontoPar+clePhysics-Chapter7-

ClaudioLuci

lastupdate:070117

•  Beta decay. •  Fermi’s theory of Beta decay •  Sargent’s rule •  Parity violation in the weak interactions. •  Two components theory of the neutrino. •  Goldhaber’s experiment. •  V-A interaction. •  Helicity eigenstates and chiral eigenstates. •  The W boson. •  The Weinberg’s angle. •  Charged pion decay. •  Charged K decay in the muon channel. •  The Cabibbo’s angle. •  Weak isospin doublets. •  GIM effect. •  The quark charm. •  CKM matrix.

2ClaudioLuci–Introduc+ontoPar+clePhysics–Chapter7

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Introduc/ontoweakinterac/ons

Δ++ à pπ ~10-23 s Σ0 à Λγ ~6·10-20 s π0 à γγ ~ 10-16 Σ à nπ ~10-10 s

π- à µ- νµ ~10-8 s µ- à e- νe νµ ~10-6 s n à p e- νe ~ 15 min

•  Let’srecallthelife+meofafewdecays:

1γ,e.m.int.2γ,e.m.int.

strongint.

weakint.

N.B. we observe the weak interact ions only when the strong and e.m. interactions are forbidden.

•  Weneedtoexplaintheenourmosrangeinthelife+megoingfrom10-12sun+ll15min.• Theweakinterac+onsarealsocharacterizedbycross-sec+onsextremelysmall(~10-39cm2=1R)

σ(π +N à N +π ) = 10-26 cm2 (10 mb) a 1 GeV

σ(νμ +N à N +π + μ) = 10-38 cm2 (10 fb) a 1 GeV

•  Theweakinterac+osviolatemanyconserva+onrules(parity,chargeconjuga+on,strangeness,etc...)• Becauseoftheir“weakness”,theweakinterac+onscanbeobservedinthe“standard”maYeronlyinthebetadecay,becausetheydonotgiveorigintoanyboundstates.Howevertheyarethebasefuelforthestarsfunc+oning,thereforewithouttheweakinterac+onswecoudnotexist

p+p → d+e++νe



Betadecay1/2

Mostofourknowledgeaboutthebaseprinciplesofbetadecayisbaseduponnucleibetadecays.

ν ν→ ⇒ →- -e en p+e + (A,Z) (A,Z+1)+e +

ν ν→ ⇒ →+ +e ep n+e + (A,Z) (A,Z-1)+e +

ν ν→⇒→- -e ee +p n+ (A,Z)+e (A,Z-1)+

•  Theexistenceoftheβ+decaywasestablishedin1934byCurieandJoliot.• In1919Chadwickdiscoveredthattheelectronintheβdecayhadacon+nuosspectrum.

Emax

•  The maximum energy of the spectrumcorresponds fairly well to the Q of thereac+on (Q=M(A,Z)-M(A,Z+1)),while for therestofthespectrumthereisaviola+onoftheenergyconserva+onrule.

•  Moreover there is also a viola+on of themomentum and angular momentumconserva+on rules (without the introduc+onoftheneutrino).



Neutrino”crea/on”2/2

•  Tore-establishthevariousconserva+onlaws,in1930Paulimadethehypothesisoftheexistenceofaverysmallneutralpar+cle:theneutron(laterrenamedneutrinobyFermi).

•  This leYer isvery important forPhysics…but it is also interes+ngfromasociologicalpointofviewJ•  The first theory of β decaywasdonebyFermiin1934.• TheneutrinowasdiscoveredbyReineseCowan“only”in1958• Wehavethreekindofneutrinos;recentlyhavebeenestablishedtheflavour neutrino oscilla+ons thatimply that neutrinos havemassesdifferent fromzero,althoughtheyare very small and not yetmeasured.



Fermi’stheoryofβdecay•  In1934Fermididthefirsttheoryofβdecay;hetookasamodeltheQEDdescrip+onoftheelectron-protonscaYering:

e− e−

p

2q

p

γ

µJ (e)

µJ (p)

Thematrixelementispropor+onalto:

µµ≈fi 2

1M - J (e)J (p) q ( ) ( )µν ναγ αγ≈ µ

fi e e p p2

gM u u u u

q

• Fermimadethehypothesisofapointlikeinterac+onlike:(thatislike) n+ν → p+e- n→ ν+p+e-

n p

eν-e

µJ (p)

µJ (e)

Thereisnopropagator

( ) ( )µµ νγ γ≈fi p n eM u u u u G vector-vectorinterac+on

TheGconstantisknownasFermi’sconstantanditisrelatedtothesquareofthe“weakcharge”.

interac+on between two (charged)currents:hadronicandleptoniccurrents.

ūpcreatesaproton(ordestroysanan+proton)undestroysaneutron(orcreatesanan+neutron)ūecreatesanelectron(ordestroysapositron)uνdestroysaneutrino(orcreatesanan+neutrino)



Nuclearβdecay1/4

•  Thetransi+onprobability(thedecayrateperunitof+me)canbefoundbyusingtheFermi’sgoldenrule:

πh

22

0

2W= dNG MdE

dNdE0

: phase space

isthematrixelementsquared.Itiscomputedbyintegra+ngoverallanglesofthefinalpar+cles,bysummingoverthefinalspinstatesandbyaveragingontheini+alspinstates.Itisaconstantoforderone.

2M

Fermi decays: Jleptoni=0 ⇒ M

2≈1

Gamow-Teller decays: Jleptoni=1 ⇒ M

2≈ 3

•  E0istheenergyavailableinthefinalstate(itisequaltotheQofthereac+on).TheenergyspreaddE0ispresentbecausetheenergyoftheini+alstateisnotpreciselyknownduetothefinitelife+me(Heisemberg’sprinciple).

!P+!q+!p=0 T+Eν+E=E0

•  InthenuclearβdecaysE0isoftheorderof1MeV.Theprotonkine+cenergyisoftheorderof10-3MeVandcanbeneglected.Theprotonistherejusttoensurethemomentumconserva+on.

ν 0 eq =E -E The energy is shared between the electron and the neutrino



Thephase space2/4

•  Thenumberofavailablestatesforanelectronwithmomentumbetweenpandp+dp,confinedinthevolumeV,withinthesolidangledΩ,is:

dN = VdΩ

(2π)3!3p2dp

•  Wenormalizethewavefunc+ontoV=1,wesumovertheen+resolidangleandweignoretheeffectofthespinontheangulardistribu+on.Wegetthefollowingphasespacefortheelectronandneutrino:

dNe = 4πp2dp

(2π)3!3 ; dNν =

4πqν2dqν

(2π)3!3

• Thetwophasespacefactorsareindependentbecausethereisnocorrela+onbetweenqandp,sinceit is a three bodies decay the protonwill absorb the remainingmomentum difference. The protonmomentumisfixed(givenqandp)sothereisnoprotonphasespacefactor.

•  The number of final states is: dN = dNe ⋅dNν = (4π)2

(2π)6!6p2qν

2dpdqν

•  ForagivenvalueoftheelectronenergyE,theneutrinoenergyEνisfixedaswellasitsmomentum:

qν ≡ Eν = (E0 − E) ; ⇒ dqν = dE0 ⇒ dN

dE0= dN

dqν= 1

4π 4!6p2(E0 − E)2dp

•  Oncewehave integratedthe transi+onprobabilityWover theen+resolidangle,M2 isequal toaconstant,thereforetheelectronenergyspectrumisen+relyduetothephacespaceform:

∝ −2 20( ) ( ) N p dp p E E dp



Kurieplot3/4

2( ) ( , )N p F Z p

p⋅ ± Tritium β decay.

Langer e Moffatt (1952)

∝ −2 20( ) ( ) N p dp p E E dp

( )eE keV=0 18.6 E keV

•  Ifweplot(N(p)/p2)½versustheelectronenergy,wegetastraightlinethatcrossesthex-axisatE=E0.ThisgraphusedtostudyβdecaywasdevelopedbyFranzN.D.Kurie.

•  Experimentallyweneedtoincludeacorrec+onfactorF(Z,p)totakeintoaccounttheCoulombinterac+onbetweentheelectronandthenucleus.

•  Iftheneutrinohasamass,itseffectwouldbetomodifythedistribu+oninthefollowingway:

ν⎛ ⎞∝ − − ⎜ ⎟−⎝ ⎠

22 2

00

( ) ( ) 1 mN p dp p E E dpE E

•  TheKurieplotismodifiedinawaythatthecurvecrossesthex-axisatE=E0-mν. Thisishowwetrytomeasuretheneutrino-emass.Unfortunatelyinthisregionthereareveryfeweventsanditisverydifficulttoperformtheexperiment.Atthemomentwehaveonlyanupperlimit.

mν

e

≤ 2.2 eV Mainzexp.;2000

The curves are plotted for several values of the neutrino mass.

T1

2= 12.3 years



TheSargent’s rule4/4

•  Thetotaldecayrateisobtainedbyintegra+ngthespectrumN(p)dp.Itcanbedoneanali+cally;howeverinthecaseswheretheelectronisrela+vis+cwecanusetheapproxima+onp≈Eandwegetaverysimpleformula:

0 2 2 50 00

( )E

N E E E dE E∝ − ∝∫•  Thedecayrateispropor+onaltothefi|hpoweroftheenergyavailableintheprocess.ThisistheSargent’srule.

•  Ifweconsiderallthenumericalfactorsintheprocess,weget:π h h

22 5

00

3 e6W= (per E 60 ( )

>> m )G M E

c

• TheFermi’sconstantGcanbefound,aswewillseelater,fromthelife+memeasurementsofafewβdecays(andwithsometheore+calspecula+ons,seeCabibbo’sangle)orinamoreprecisewayfromthemuonlife+me.

•  FromthePDGweget: 5 -23 1.16637(1) 10 GeV

( )Gc

−= ⋅h

•  Replacingthenumericalvaluesintheformulaweget: 2 50 0

-14

1 1.11 = W = (E in Ms e 1

V 0

)M Eτ

•  Forinstance,ifE0≈100MeVlikeinthemuondecayandM2=1,weget: -6 (1 10 s =2.2 s ) Wµ µτ µτ = ≈

N.B.itistheE05dependencethatexplainsthehugerangeinthelife+meofthedecaysmediatedbytheweakinterac+ons.



Thenuclearβdecay•  Thenuclearβdecayscanbediscriminatedasallowedtransi+onsandforbiddentransi+ons.•  TheallowedonesarethemostcommonandarecharacterizedbythefactthattheelectronandneutrinoemiYedDOESNOTcarryanyspa+alangularmomentum,thatistheyareintheS-state(L=0).Thisisjus+fiedbythefactthatthetwoleptonshaveenergiesoftheordera1MeV.

•  Thetransi+onswithL=1arecalledfirstforbidden,theoneswithL=2secondforbiddenandsoon.Theyhavealife+meconsiderablylongerthattheallowedtransi+ons.

•  Sinceeandνhavespin½,thenucleusspinchangecanbeeither0or1.Thetransi+onswithΔJ=0arecalledFermitransi+onswhiletheoneswithΔJ=1arecalledGamow-Tellertransi+ons.

transitions Δ J nucleus Leptonic state

Fermi Δ J=0 singlet

Gamow-Teller

Δ J=1 Δ Jz=0,±1

triplet

•  Sincee-νhaveL=0,thereisnochangeinthespa+alangularmomentumofthenucleus,thereforeitsparitywillnotchange.ThenucluesundergoesaspinflipfortheG.T.transi+ons.

Fermi: G.-T.: Mixed: 0+ → 0+,Δ

!J = 0 1

+ → 0+,Δ!J = 1

12

+

→ 12

+

,Δ!J = 0,1

10 10 -eC B*e ν→

14 14 +eN*eO ν→

12 12 -eB Ce ν→

-en pe ν→

1/2



MeasurementoftheFermiconstant2/2

•  ThedecayratecanbewriYeninadifferentwaywithrespecttotheSargentrule.Wewriteexplicitlytheprotonmassm,thenweincludethephasespacefactorsandtheCoulombiancorrec+onF(±Z,p)inadimensionlessfunc+onf(±Z,E0/me)thatcanbecomputedanali+cally.

2 5 2203 6

1 ( ) = W = f( Z,E ) 2 (

)

mc G Mcτ π

±h h

E0 (MeV) 2 6MeV fm⋅

•  In spite of the big variations of lifetime due to the strong dependency of the function f from pemax,theproductG2M2isaboutthesameinalldecays.• Howeverweobserveasmalldifferenceduetothetypeofnucleartransi+on:Fermi,Gamow-Tellerormixedtransi+ons.

G2 M

2=constant

f ⋅ τ (constant=2π3

m5)

•  IfweconsiderapureFermitransi+on,weget: 5 -23 1.140(2) 10 GeV

( )Gc

−= ⋅h

thatisslightlydifferentfromtheonequotedbythePDGtakenfromthemuondecay.Wewillseelaterthereasonofsuchadiscrepancy(Cabibbo’sangle).



Generaliza/onoftheFermiLagrangian1/4

•  Thereisnoapriorireasonsthatjus+fyavectorialweakcurrentintheFermi’sLagrangian.•  ThesimplestLorentzinvariantLagrangianis:

† †( )( ) ( )( )i r p r n e r r n r p r er r

L C O O C O Oν νψ ψ ψ ψ ψ ψ ψ ψ= +∑ ∑ Hermitian conjugate

•  Crareconstantsthatdeterminetheintensityofthetheinterac+on.TheoperatorsOrare:

OS = 1 scalar ; OA=γ µγ 5 axial vector

OV = γ µ vector ; OP=iγ 5 pseudoscalar

OT=σ µν= i2

(γ µγ ν − γ νγ µ) antisymmetric tensor

•  Since the weak interac+ons do not conserve the parity, we will show later how to introduce thisfeatureintheLagrangian.Inwhatfollowsweassumethattheparityisconserved.

•  Inthepseudoscalartermthematrixelementismul+pliedbytheβ(=v/c)ofthenucleon,thereforethistermcanbeneglectedintheLagrangian,sinceβisoftheorder10-3.

•  Moreover,byexaminingtheelectronandneutrinospincorrela+ons,itturnsoutthatonlythevectororscalartermscancontributestothepureFermidecays(ΔJ=0),whileonlytheaxialortensorialtermscancontributetotheGamow-Tellerdecays.



Determina/onofthecoefficientsCr2/4

•  Ifwe consideronly theallowedβdecays (ΔL=0)where there isnoparity change,we canwrite theLagrangianinthefollowingway:

, ,( )( ) ( )( )i i p i n e i j p j n e j

i S V j T AL C O O C O Oν νψ ψ ψ ψ ψ ψ ψ ψ

= == +∑ ∑

Fermi Gamow-Teller

•  TheelectronorpositronenergyspectrumintheβdecayscanbewriYenas:

dn∓dEe

=PeEe

2π3(E0 − E)2 ⎡⎣ MF

2(CS

2 + CV2) + MGT

2(CT

2 + CA2) ± 2me

Ee(MF

2CsCV + MGT

2CTCA) ⎤⎦

MFandMGTaretheFermiandGamow-Tellermatrixelements;E0isthemaximumelectronenergy.

•  From this expression we see that there is no interference between Fermi and Gamow-Tellertransi+ons,whilethereisinterferencebetweenthetermsSandVandthetermsAandT.

• SincewehaveobservedpureFermitransi+onsorpureGamow-Tellertransi+ons,wecanNOThave:

CS = CV = 0 or CA=CT=0



Determina/onofthecoefficientsCr3/4

•  FromtheKurieplotoftheFermiorGamow-Tellertransi+ons,wedeterminethera+o:

⋅ ⋅= ± = ±

+ +2 2 2 20.00 0.15 ; 0.00 0.02 S V T A

S V T A

C C C CC C C C

thereforethedatasuggest(Michel,1957)thatCsorCVarezeroandCTorCAarezero.

N.B. the an+neutrinoisalwaysrighthanded

Angular distribution

Thefactor3comesfrom the threepossible total spinorienta+ons.

Dataare“peaked” atϑ=0,soCsiszero.Theelectronis“le|handed”

Data are “peaked” at ϑ=180, so CT is zero.

The electron is “lefthanded”

N.B.wemeasuretheangle between theelectron and thenucleus“recoil”.

Todeterminewhichtermiszero,we examine the correla+onbetween the electron andneutrinodirec+ons.



MeasurementofCVandCA4/4

•  Wecanwritedowntheelectronspectrumas:π

⎡ ⎤= − +⎢ ⎥⎣ ⎦2 22 2 2

0 F GT3e

( ) M MdE 2

e eV A

P Edn E E C C

•  Ifweintegrateovertheelectronspectrumwegetthenumberofcoun+ngperunitof+me:

τ π⎡ ⎤= = + −⎢ ⎥⎣ ⎦ ∫

02 22 2 2F GT 03

1 1 M M ( )2

e

E

V A e e em

n C C P E E E dE

m5·f

≈ ≈2 2F GT

M

1 ;

M 3

N.B.Inordertohavefdimensionless,wenormalizetheenergywiththemassm,thatcanbetheelectronmassortheprotonmass.N.N.B.ftakesintoaccountalsotheCoulombcorrec+on,thatitisdifferentforelec.andpositr.

•  TheparametersCVandCAareinferredfromthelife+meofsomenuclearβdecays.Actuallywhatismeasuredisthehalf-lifethatisrelatedtothemeanlife+meinthefollowingway:

N(t2) = 12

N0 = N0 ⋅e−t2

τ ⇒ t2 = τ ⋅ ln2

•  Fromtheneutrondecay(mixeddecay):π

− −⎡ ⎤= + ⋅ = ±⎣ ⎦⋅

52 2 1 1

31 3 (1080 16)

f t 2 ln2

V AmC C s

−−= = ⋅ = = =h

h

55 -2

3 2101.140(2) 10 GeV ( 1)

( )vp

GC cc M

•  FromthepureFermidecays(forinstance14O->14N*):

•  ComparingwiththeO14decay(2protondecays): ⇒+ ⋅ ±= = = ±

±

2 214

23( ) 3100 20 1.25 0.2

(ft)n 1080 162

V A A

VV

C C Cft OCC

FrompolarizedneutronsdecaywededucethatthesignofCAisnega+ve



Theτ-θpuzzle1/3

•  Thereweretwostrangepar+cles,withthesamemassandlife+me,thatdecayedintwofinalstateswithoppositeparity:

; τπ π π πθ π→→•  Parityofthemesonθ(Kàππ):

Pionshavespinzero,thereforeduetotheconserva+onofthetotalangularmomentum,theKspinmustbeequaltotherela+veorbitalangularmomentumofthetwopionssystem.Hencetheparityofthesystemisequalto: η = (−1) L

⇒ J p = 0+ , 1- , 2+ , 3- ! (natural spin parity)

If we consider the neutral K decaying into two π0 that are two identical bosons, the wave function must be symmetric, so they are allowe only the even L values:

⇒ J p = 0+ , 2+ ! ⇒ even K spin and positive parity

•  Parityofthemesonτ(Kàπππ):Wecanhandlethesystemasadi-pion(forinstancetwopionswiththesamechargeplusthethirdone).Let’scallltherela+veorbitalangularmomentumofthetwopionsandlet’scallLtheangularmomentumofthethirdpionwithrespecttothepionpair.Theparityofthethreepionssystemis:

3( 1) ( 1) ( 1) ( 1)l L Lη = − ⋅ − ⋅ − = − −

π+ π+

π-

l

L

N.B.lmustbeevenbecausethetwopionshavethesamecharge,thereforearetwoiden+calpar+cles.Moreoverrememberthatthepionhasnega+veintrinsicparity.



Theτ-θpuzzle2/3

•  ThetotalspinJofthethreepionssystemmustlieintheinterval: L l J L l− ≤ ≤ +thereforewehavethefollowingcombina+ons:

l L Jp

0 0 0-

0 1 1+

0 2 2-

2 0 2-

2 1 1+.2+,3+

2 2 0-,1-,2-,3-,4-

( 1)Lη = − −

Todeterminewhichistherightspinassignmentweneedtostudytheangulardistribu+onsofthedecayproductsasafunc+onofthevariousJcombina+ons(par+alwavesexpansionandDalitzplot)•  From these studies we deduce that thecombina+onmustbe:

Jp = 0− or 1+ but not 1-

Ifweincludealsotheeffectsofthephasespace,wehave:

0pJ −= (N.B.theKhasspin0)

•  Thereforetheτhadnega+veparitywhiletheθhadposi+veparity,hencetheso-calledτ-θpuzzle.T.D.LeeandC.N.Yangmadethehypothesisthattheweakinterac+onsviolatetheparityconserva+onandtheysuggestafewexperimentalcheckstoverifyit.



MadameWu’sexperiment3/3

•  AlsothisleYerisafondamentaloneforPhysics…and to understand the sociology of the physicistsandthescien+stsingeneral!

•  R.L.Garwin,L.M.Lederman,M.Weinrich Phys. Rev., 105, 1415 (1957)

• J.I.Friedman,V.L.Telegdi Phys. Rev., 106, 1290 (1957) (parity violation in the pion decay) Phys. Rev., 105, 1413 (1957)



Weylequa/on:twocomponentsneutrinotheory1/2

•  In1929, justa|erthepublica+onoftheDiracequa+on,Weylpublishedaverysimpleandeleganttheoryaboutmasslesspar+clesofspin½forwhichthehelicityisagoodquantumnumber.• Atthe+meofthepubblica+onthetheorydidn’thaveverymuchsuccesssincetherewerenoknownmasslesspar+clesofspin½• However,evena|ertheintroduc+onoftheneutrinobyPauli,PaulihimselfdisregardedtheWeyl’stheorybecauseitviolatedtheparity.• Onlya|er1957theWeyl’stheoryreceivedthedeservedcredit.

•  Let’sstartfromtheDiracequa+oninthemomentumspace:

( ) ( ) 0p m pµµ µγ ψ− =

ifwesetm=0andwerememberthatγ0=βandγi=βαi,wehave:

γ 0E −

!γ ⋅!p( )ψ (pµ) = 0 ⇒ βE − β

!α ⋅!p( )ψ (pµ) ⇒ Hψ (p) ≡

!α ⋅!pψ (p) = Eψ (p)

•  TostudytheWeilequa+onispreferabletousetheWeilrepresenta+on(orchiralrepresenta+on),whereγ5isdiagonal,insteadoftheDirac-Paulirepresenta+onwhereγ0isdiagonal.

!α = −

!σ 00 !

σ

⎛

⎝⎜⎞

⎠⎟ ; β = 0 I

I 0

⎛

⎝⎜⎞

⎠⎟ γ 0 = β = 0 I

I 0

⎛

⎝⎜⎞

⎠⎟ ; !γ = 0 !

σ−!σ 0

⎛

⎝⎜⎞

⎠⎟ ; γ 5 = −I 0

0 I

⎛

⎝⎜⎞

⎠⎟



Weylequa/on2/2

!α ⋅!pψ = Eψ

•  Wecanwritedownthe4-compenentsspinorψas: ψ = χ

ϕ

⎛

⎝⎜⎞

⎠⎟ χ and ϕ are 2 components spinors

−!σ ⋅!p 0

0 !σ ⋅!p

⎛

⎝⎜⎜

⎞

⎠⎟⎟

χϕ

⎛

⎝⎜⎞

⎠⎟=E ⋅ χ

ϕ

⎛

⎝⎜⎞

⎠⎟ Wehavetwodecoupledequa+ons!

−!σ ⋅!pχ = Eχ

!σ ⋅!pϕ = Eϕ

⎧⎨⎪

⎩⎪

•  Sincetheneutrinoismassless,wehaveE2=P2.Foreachequa+onswehavetwosolu+ons,onewithposi+veenergyandanotheronewithnega+veenergy.• Thesolu+onswithposi+veenergycorrespondtoneutrinoswhile theoneswithnega+veenergycorrespondtoan+neutrinos.

•  posi+veenergysolu+ons:

E =

!p ⇒

!σ ⋅!p!p

χ = −χ ; !σ ⋅!p!p

ϕ = ϕ

(le|handedneutrino;righthandedneutrino)

!σ ⋅!p!p

⎛

⎝⎜⎜

is the helicity projector

⎞

⎠⎟

•  nega+veenergysolu+ons:

E = −

!p ⇒

!σ ⋅!p!p

χ = χ ; !σ ⋅!p!p

ϕ = −ϕ

(righthandedan+neutrino;le|handedan+neutrino)

N.B. The Weyl equa+on violates the parity because the le|handed neutrino and therighthandedneutrinoaredescribedbytwodifferentspinors(χandφ)thataredecoupled.

•  TheWeylequa+oncanbewriYenas:



helicityoftheneutrino1/6

•  According to theWeyl’s theory, theneutrino canexistonly inone stateofdefinitehelicity,eitherposi+veornega+ve.Thisisthemaximumpossibleparityviola+on.

• Itwasnecessaryanexperimenttoprovethattheneutrinohasonlyonehelicitystateandtodiscoverifitisaposi+veornega+vehelicitystate.Inthiscasethean+neutrinowillhaveahelicityopposedtotheneutrinoone.

• Theneutrinointeractsonlythroughweakinterac+onsanditisverydifficulttodetectit,andpra+callyimpossibletomeasureitshelicity.

•  However in 1958 Goldhaber, Grodzins and Sunyar designed and realized a veryingeniousexperimenttomeasuretheneutrinohelicity:theywillmeasurethehelicityofaphotonthathasthesamehelicityastheneutrinoone.

• Theresultoftheexperimentisthattheneutrinoisle|handedandthean+neutrinoisrighthanded.Sothe“true”wavefunc+onisdescribedbythespinorχ.

N.B.Sincethechargeconjuga+ontransformaneutrinoinanan+neutrino,butwithoutchangingthehelicity,soalsothechargeconjuga+onisviolatedbytheweakinterac+on.



1/3 Measurementoftheneutrinohelicity2/6

•  Goldhaberetal.foundthatthe152Euhadtherightfeaturesneededfortheexperiment:measurethephotonhelicityanddeducetheneutrinohelicity.

•  Agivenmetastabilestateofthe152Eu,throughaKelectroncapture,decaysin24%ofthecasesinanexcitedstateof152Sm*,whichinturndecaystothegroundstatebyemi�ngaphotonof961keV.

Tobeno+ced:1)  ThespinoftheEuropiumiszero;2)  TheexcitedSamariumdecaysinflight;3)  Theneutrinoandthephotonhavealmostthesameenergy.

a)  Electroncapture:b)  Radia+vedecay:

−TheSamariumdecaysinflight−photonenergy:Eγ=961keV

−itisatwobodyfinalstate−neutrinoenergy:Eν=840keV

152Sm* → 152Sm + γ

152 152 * eEu e Sm ν−+ → +



1/3 neutrinoandphotonhelicity3/6

1)  Due to the conserva+on of the total angularmomentum, the neutrino has the spin opposedto the one of the electron captured (that wedon’tknowwhatitis).

2)  TheexcitedSamariumandtheneutrinohavethesame helicity since it is a two body final state(TheEuropiumandthecapturedelectroncanbeconsideredatrest).

N.B.ONLYthephotonsgoingalongthe152Sm*lineofflighthavethesamehelicityoftheneutrino.

152 152 * eEu e Sm ν−+ → +

152Sm* → 152Sm + γ

1)  The life+meof the 152Sm* is about 10-14s, so itdecaysbeforecomingtorest.

2)  Since the Samariumhas spin zero, the photon“carries”awaythespinoftheexcitedSamarium.

3)  onlythephotonsdecayingalongthedirec+onofthe152Sm*lineofflighthavethesamehelicityofthe excited Samarium and therefore the samehelicityoftheneutrino.



Twoexperimentalproblems4/6

1)  Howtomeasurethephotonhelicity?itcandonebyexaminingthetransmissionofthephotonsthroughmagne+zediron.Thedominantinterac+onwithmaYerofphotonofenergy961keVistheComptoneffectandthemethodreliesonthefactthatthecross-sec/onforComptonscaUeringisspindependent.Thetransmissionisgreatestwhenthephotonspinisparalleltotheelectronspin.

2)  Howtoselectthephotonsdecayingalongthe152Sm*lineofflight?ThiscanbedoneusingthemethodofresonantscaUeringdevisedbyGoldhaberetal.§  Intheemissionofaphotonfromanexcitedstatewithenergyofexcita+onE0,amomentumE0/cmustbe

impartedtotheemi�ngnucleusandconsequentlytheenergyofthephotonisreducedbyanamountE02/2Mc2whereMisthemassofthenucleus(thenucleusisnotrela+vis+c).

§  Similarly,onabsorp+on,anextraenergyE02/2Mc2mustbesuppliedtocounteractthenuclearrecoil.§  Thisenergy,ΔE=E02/Mc2,lostbyrecoilinemissionandabsorp+on,isingeneralmuchgreaterthanthelevel

widthsothatresonantabsorp+onwilltakeplaceonlyifextraenergy,equaltotheenergylost,issuppliedtotheemiYedphoton.

§  InthisexperimentitispreciselythosephotonsemiYedinthedirec+onofrecoilofthe152Sm*whichhavethecorrectenergytoundergoresonantabsorp+on.Therecoiling152Sm*hasavelocityofEν/Mc2thatbyDopplereffectwillincreasetheenergyoftheemiYedphoton.Theresonantcondi+onrequirethatEνcosθ≈E0,soitisimportantthatEν≈E0;thermalmo+onwillsupplythesmallamountofenergys+llmissingthatpermitthattheresonancecondi+oncanbemetinprac+ce.

M γ

0Epc

= 0Epc

=

⇒ K = p2

2m=

E02

2Mc2

Eγ = E0 −E0

2

2Mc2

emission

Eγ = E0 +E0

2

2Mc2

absorption

Eν cosθ ≈ E0

resonant condition



TheGoldhaber’sexperiment5/6

γ

pSm* =

Eνc

≈E0c

152Sm*

ν TheexperimentmeasuresonlythephotonsemiYedalongthe152Sm*lineofflight

Itwillorienttheelectronspin.Thespinsaredirectedintheoppositedirec+onwithrespecttotheBfield.

Thetransmissionismaximalwhenthephotonandelectronspinsareparallel.

Select photons withnega+vehelicity.

OxideofSamarium.Itabsorbsthephotonsandcanre-emitonlythosethatmeettheresonantcondi+on.

PhotoMul+plier. ItdetectsthephotonsemiYedbythescin+llatorNaI

Leadshield.Itpreventsthatthephotons emiYedby the 152Sm*reachdirectlythePM



TheGoldhaber’s experiment: results6/6

Counts/ minute

Photon energy

•  The resonant peak (actually two peaks) isob ta i ned on l y f o r a g i ven B -fie ldconfigura+on selec+ng le|handed photons,thereforealsotheneutrinosarele|handed.•  The helicity of the an+neutrino has beenmeasuredbystudyingthedecayofpolarizedneutrons and it turns out that thean+neutrinosarerighthanded.

Theneutrinoisle|handed

Thean+neutrinoisrighthanded



V-Ainterac/on1/2

•  Let’ssummarizewhatwehaveexperimentallyverifiedsofaraboutweakinterac+ons:

1.  IntheFermiinterac+onswehaveonlythevectorialterm(Oi=γμ)whileGamow-Tellerinterac+onswehaveonlytheaxialterm(Oj=iγμγ5);

2.  Theneutrinohasnega+vehelicity;3.  Theweakinterac+onsviolatetheparity.Inordertotakeintoaccountalsothisfeaturewehavetointroduce

intheLagrangianapseudoscalartermnexttothescalarone.Thisisdonewiththesubs+tu+on:

Ci → Ci + Ci

'γ 5( ) 12

The factor1/√2 isneeded to retain theoriginalvalueofG·CV(Fermiconstant)

•  TheFermiLagrangian,withtheparityviola+on,becomes: νψ ψ ψ γ ψ=

⎡ ⎤= +⎣ ⎦∑ ' 5

,

1 ( ) ( )2i p i n e i i

i V AL O O Ci C

•  Fromthehelicityneutrinoexperimentsweknowthat: = = −'1 ; 1i iC C

•  Let’swritedownexplici+lytheFermiconstant:

⇒ Li =

G2

CV (ψ pγµψ n) ψ eγ µ(1 − γ 5)ψν

⎡⎣⎢

⎤⎦⎥{ + CA(ψ piγ

µγ 5ψ n) ψ eiγ µγ 5(1 − γ 5)ψν⎡⎣⎢

⎤⎦⎥}

•  Byusingtheproper+esoftheγmatriceswehave:

µµ νψ γ γ ψ ψ γ γ ψ⎡ ⎤ ⎡ ⎤= + −⎣ ⎦ ⎣⇒ ⎦

5 5( ) (1 )2i p V A n eGL C C

γ 5( )2 = 1 ; γ µγ 5 = −γ 5γ µ



V-Ainterac/on2/2

•  Let’srecallthatinapureFermitransi+onwemeasuretheproductG·CV.IfwecomparethisnumberwiththevalueofGmeasuredinapureleptonicdecay,likeforinstancethemuondecaywherewedonothavethetermCV,wefindthatthetwovaluesareingoodagreement,thereforewededucethat:

= ⇒ = − ±1 1.26 0.02 V AC C

•  CA isnotequal to1because thestrong interac+onsmodify thehadronaxial currentwhile thevectorcurrentremainsunchanged.Ifwetakeotherhadronicweakdecays,besidestheneutron,wehave:

ν ν− − −Λ → + + = − Σ → + +⇒ ⇒+ = 0.72 ; 0.34A Ae e

V V

C Cp e n eC C

•  Howeverifweignorethestronginterac+onscorrec+onsontheaxialcurrent,wecanput:

= − = −1 A VC C

(thisse�nghasbeenvalidatedintheweakinterac+onsbetweenneutrinosandquarks)

•  ThereforewecanwriteagaintheLagrangianinthefollowingway:

µµ νψ γ γ ψ ψ γ γ ψ⎡ ⎤ ⎡ ⎤= − −⎣ ⎦ ⎣ ⎦

5 5(1 ) (1 )2i p n eGL

•  This is the so-called V-A interac+on. Besides the factor (1-γ5) is the same Lagrangian originallyproposedbyFermi.•  Thefactor(1-γ5)isveryimportantbecausebecause,aswewillseelater,selectsonlyonedefinedhelicity(actuallychirality)ofthefermionsthatpar+cipateintheweakinterac+ons.



TheuniversalFermiinterac/on•  Let’sconsiderthemuondecay: µν

eν-e

µµJ ( )

µJ (e)

µ−

µµ ν ν− −→ + +ee

•  TheLagrangiancanbewriYenas:µ

ρν µ ρ νψ γ γ ψ ψ γ γ ψ⎡ ⎤ ⎡ ⎤= − −⎣ ⎦⎣ ⎦

5 5(1 ) (1 )2 ei eGL

• ItisapureV-Ainterac+on:thevectorandaxialcurrentshavethesameintensityandopposedsign.• Themuonlife+me,takingintoaccountthephasespacefactor,canbewriYenas:

µ

µτ π= =

2 5

31

192

G mW

µ

µτ−

=

= ⋅ 6

105.658369 (9) MeV

(2.19703 (4)) 10 s

m

•  From the muon mass and lifetime we get: −= ⋅ ⋅62 3(1.4358 (1)) 10 J mG

•  From the pure Fermi β decay(0à0)wemeasure: −⋅ = ⋅ ⋅62 3(1.4116 (8)) 10 J mVG C

bycomparingthetwovalueswegetCV=0.98(seeCabibbo’sangle)

•  Thenear equality of the Fermi constantobtained froman analysis of nuclearβdecay, involvinghadrons as well as leptons, and that derived frommuon decay involving only leptons suggests auniversality of the weak charge; the value of the weak charge is the same for all par+cles whichposses it (it like theelectrical chargee). The so-calleduniversal Fermi interac+onassigns a singleglobalconstantGforthecouplingbetweenanyfourfermionfields.

N.B. the enormous span in lifetime is a kinematical effect



Thecurrent-currenthypothesis•  Theneutrondecayisdescribedbytheproductoftwocurrents:

Jnµ = ψ pγ

µ(1 − γ 5)ψ n (NeutroncurrentifCA=-1)µ µ

νψ γ γ ψ= − 5(1 )ee eJ (Electroncurrent)x

•  Themuondecayisdescribedbytheproductoftwoleptoncurrents,theelectronandthemuonones:

µ

ρ ρµ µ νψ γ γ ψ= − 5(1 )J (muoncurrent)

•  These are chargedcurrents,becausethereisachangebetweentheini+alandthefinalpar+clechargepresentinthecurrent.

•  Thisdescrip+onwasgeneralizedbyFeynmanandGell-Manntoincludeallweakprocesses(actuallyonlythechargedcurrentprocessesbecauseatthat+metheneutralcurrentswerenotyetknown).

•  Wedefinealeptonicweakcurrentthatisthesumofallleptoncurrrents:

µ µνψ γ γ ψ= − 5(1 )eelJ

thecurrentfortheotherleptons,withequalamplitudeduetoleptonicuniversality.+

andonehadroniccurrent: µ µψ γ γ ψ= − 5(1 )p nhJ similartermsforstrangepar+cles+ •  therefore all amplitudes of the weak processes can be written as:

µµ= ⋅ †

2G J JM due to electric charge conserva+on, in the amplitude must

appearsaraisingchargecurrentandaloweringchargecurrent.

•  N.B.Inthemodernformalism,wepreferetodefinethecurrentwiththefactor½(1-γ5)insteadoftheold(1-γ5),therefore:

µµ= ⋅ †

24 G J JM



Reminder:Diracequa/on1/6

•  Let’srecalltheDirac-Paulirepresenta+onoftheγmatrices,wheretheβmatrixisdiagonal:

!α = 0 !

σ!σ 0

⎛

⎝⎜⎞

⎠⎟ ; β = I 0

0 −I

⎛

⎝⎜⎞

⎠⎟ γ 0 = β = I 0

0 −I

⎛

⎝⎜⎞

⎠⎟ ; !γ = 0 !

σ−!σ 0

⎛

⎝⎜⎞

⎠⎟ ; γ 5 = 0 I

I 0

⎛

⎝⎜⎞

⎠⎟

•  originalformula+onoftheDiracequa+on:

Hu ≡ ( !α ⋅!p + βm)u = Eu

⇒ Hu ≡ m !

σ ⋅!p

!σ ⋅!p −m

⎛

⎝⎜⎜

⎞

⎠⎟⎟

uA

uB

⎛

⎝⎜⎜

⎞

⎠⎟⎟= E

uA

uB

⎛

⎝⎜⎜

⎞

⎠⎟⎟

two componentsspinors

⇒

!σ ⋅!puB = (E − m)uA

!σ ⋅!puA = (E + m)uB

⎧⎨⎪

⎩⎪

•  Solu+onwithposi+veenergy(E>0):

χ=( ) ( ) (s=1 ) ,2 s sAu

Wemakeuseofthespinorχχ χ⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠= =(1) (2)1 0

0 1 ; E>0

E<0

•  Solu+onwithnega+veenergy(E<0): χ=( ) ( )s sBu

⇒ uA

(s) =!σ ⋅!p

E − mχ(s) = −

!σ ⋅!p

E + mχ(s)

⇒ u(s+2) = N−!σ ⋅!p

E +mχ(s)

χ(s)

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⇒ uB

(s) =!σ ⋅!p

E + mχ(s)

⇒ u(s) = N χ(s)!σ ⋅!p

E+mχ(s)

⎛

⎝⎜⎜

⎞

⎠⎟⎟

normaliza+onfactor− ≡(3,4) (2,1)( ) ( )u p v p

•  Thesolu+onsu(1,2)withposi+veenergydescribetheelectronswhiletheu(3,4)withnega+veenergydescribethepositrons.

N.B. Parity : ψ (x) →ψ '(-x)=γ 0ψ⎡⎣⎢

⎤⎦⎥



Helicityoperator2/6

•  TheeigenstatesoftheDiracequa+onwithadefiniteenergyaredoublydegenerate(therearetwostateswiththesameenergy),thereforeitmustexistanotherobservablethatcommuteswiththeHamiltonian(thatiswiththemomentumoperatorsincewearedealingwithafreepar+cle)thatpermitstodis+nguishthetwostates.

•  Thefollowingoperatorfulfilthisrequirement:

!Σ ⋅ p̂ ≡

!σ ⋅ p̂ 0

0 !σ ⋅ p̂

⎛

⎝⎜⎜

⎞

⎠⎟⎟ ;

!Σ ≡

!σ 00 !

σ

⎛

⎝⎜⎞

⎠⎟Spin operator ;

p̂ =

!p!p

Momentum unit vector

•  is thespinprojec+onalongthemomentumdirec+on;it isagoodquantumnumbertodis+nguishthetwosolu+ons.• Thisquantumnumberiscalledhelicity.Itstwoeigenvaluesare:

!Σ ⋅ p̂

+

−

⎧= ⎨⎩

1

1h

•  Ifwechoosethez-axisalongthemomentumdirec+on,wehavep=(0,0,p),therefore:

!σ ⋅ p̂χ(s) = σ3χ

(s) = 1 00 −1

⎛

⎝⎜⎞

⎠⎟χ(s) = hχ(s) where h= ±1

Thespinorχ(s)iseigenstateofthehelicitywitheigenvalue±1(butonlywiththischoiseofthereferencesystem).(N.B.some+mesweintroducethefactor½inthehelicitydefini+on.)



Rela/onbetweenhelicityandγ53/6

•  Let’sapplythematrixγ5toaDiracspinor(weignorethenormaliza+onfactorN):

γ ⎛ ⎞⎜ ⎟⎝ ⎠

50

= 0I

I

u(s) = N χ(s)!σ ⋅!p

E+mχ(s)

⎛

⎝⎜⎜

⎞

⎠⎟⎟

u(s+2) = N

!σ ⋅!p

E−mχ(s)

χ(s)

⎛

⎝⎜⎜

⎞

⎠⎟⎟

•  If the particle has m=0 (massless particle) or E >> m, we have E=p, therefore :

γ5u(p) = (

!Σ ⋅ p̂) u(p)

γ5correspondstothehelicityoperatorformasslesspar+cles

χ χ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =(1) (2)1 00 1 ;

γ 5u

(1,2)= 0 II 0

⎛

⎝⎜⎞

⎠⎟ χ!σ ⋅!p

E+mχ

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

!σ ⋅!p

E+mχ

χ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

; γ 5u

(3,4)= 0 II 0

⎛

⎝⎜⎞

⎠⎟

!σ ⋅!p

E−mχ

χ

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

χ!σ ⋅!p

E−mχ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

•  Wemakeuseofthefollowingproperty:

!σ ⋅!p

E+m⋅!σ ⋅!p

E−m=!σ ⋅p̂pE+m

⋅!σ ⋅p̂pE−m

=!σ ⋅p̂( )2 p2

E2−m2 = 1

γ 5u(p) =

!σ ⋅!p

E+m0

0!σ ⋅!p

E−m

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟u(p)

γ 5u(1,2) =

!σ ⋅!p

E+mχ

!σ ⋅!p

E+m

!σ ⋅!p

E−mχ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

!σ ⋅!p

E+m0

0!σ ⋅!p

E−m

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

χ!σ ⋅!p

E+mχ

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

!σ ⋅!p

E+m0

0!σ ⋅!p

E−m

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟u(1,2)

γ 5u(3,4) =

!σ ⋅!p

E+m

!σ ⋅!p

E−m χ

!σ ⋅!p

E−mχ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=

!σ ⋅!p

E+m0

0!σ ⋅!p

E−m

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

!σ ⋅!p

E−mχ

χ

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

!σ ⋅!p

E+m0

0!σ ⋅!p

E−m

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟u(3,4)

σ i( )2 = 1

γ5v(p) = −(

!Σ ⋅ p̂) v(p)N.B. for a massless

antiparticle:



Helicityprojectorandchiraleigenstates4/6

•  Wecanverifythattheoperator½(1-γ5)behavesasahelicityprojector:

12

(1 − γ 5)u(p) = 0 if u(p) has helicity +1u(p) if u(p) has helicity −1⎧⎨⎪

⎩⎪(form=0)

µ µ µ µν νψ γ γ ψ ψ γ γ ψ= − = −5 † 51 1(1 ) ; (1 )

2 2e ee el lJ J

•  Let’s recall the form of the weak current:

•  Therefore in the w.i. contribute only the state with a definite helicity; in par+cular onlyle|handedneutrinosand,aswewillsee,righthandedan+neutrinos.Inthelimitofhighenergy(E>>m)alsoforthemassivefermionsonlythele|handedstateintervenesintheweakinterac+ons.

•  Wecandefinenowthechiraleigenstates(fromthegreekwordchiros,hand;theyarethestatesthatdis+nguishthele|handfromtherighthand).Thesestatescoincidewiththehelicityeigenstatesonlyformasslesspar+cles.Thishappensbecausethehelicityisagoodquantumnumberonlyformasslesspar+clesthatmoveatthespeedoflight,whileformassivepar+clesitisalwayspossibletofindanotherreferencesystemwherethehelicityflipsthesign.

•  Thechiraleigenstatesarecalledle|handedorrighthandedstates;theyhavehelicityequalto±1onlyformasslesspar+clesor,withagoodapproxima+on,forpar+cleswithE>>m.

γ γ

γ γ

− +≡ ≡

+ −≡ ≡

5 5

5 5

1 1( ) ( ) v ( ) ( )2 2

1 1( ) ( )

;

; v ( ) ( )2 2

L L

R R

u p u p p v p

u p u p p v pan+par+clesDefini+on:



Weakvectorcurrent5/6

γ γ

γ γ

− +≡ ≡

+ −≡ ≡

5 5

5 5

1 1( ) ( ) v ( ) ( )2 2

1 1( ) ( )

;

; v ( ) ( )2 2

L L

R R

u p u p p v p

u p u p p v p

•  Let’s see the projector for the adjoint spinor. Let’srecallthatγ5 ishermi+an(γ5=γ5†) anditan+commutewiththeotherγmatrices(γμγ5=-γ5γμ),therefore:

γ γγ γ γγ− += = = = +5 5† 0 † 0 † 0

51 12 2

12LL u uu uu γ γ γ− − +≡ ≡ ≡

5 5 51 1 1v ( ) ( ) ( ) ( ) v ( ) ( )2 2

; ;2L R Rp v p u p u p p v p

•  Afewproper+esoftheprojector:

1 − γ 5

2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

= 14

1 − 2γ 5 + (γ 5)2⎡⎣⎢

⎤⎦⎥= 1

41 − 2γ 5 +1⎡⎣⎢

⎤⎦⎥= 1 − γ 5

2Aprojectorappliedtwicegivesthesameresult

µµ µµµ γ γ γγ γγ γγγ γ γ γ+− − −− = −⇒+ 5 5 55 5 55 1 1 1= = 1 1 1 122

2 2 2 2 2

•  Let’srecalloneexampleofweakcurrent(vertexW-e-ν):

µ µγνγ− −=

5(1 ) 2

J e (destroyanelectronandcreateaneutrino)

µ µµ µ ν γγ γ γνγ ν γ− − + ⋅−= ⋅=5 5 5(1 ) (1 ) (1 ) =

2 2 2 L Le eeJ

Wegotapurevectorcurrentbetweentwole|handedpar+cles(eventuallyFermiwasrightJ)



Chiralsimmetry6/6

•  Aswehaveseenthe(charged)weakcurrentcouplesonlyle|handedelectronswithle|handedneutrinos(thisistheparityviola+onofthew.i.),whiletheelectromagne+ccurrentdoesnotdis+nguishthechiralityofthepar+clesinvolvedintheinterac+on.

•  HoweveralsoforQEDwecanmakeuseofthechiraleigenstates:

u = 1 − γ 5

2u + 1 + γ 5

2u = uL + uR (also u = uL + uR)

µ µ µ µ µγ γ γ γ⇒ = − = − + + − − ( ) ( )= emL R L R L L R RJ e e e e e e e e e e

•  thisistruebecausethe“crossed”termsarenotpresent:

µ µ µγ γ γ γγ γ γ+ + − += =

5 5 5 51 1 1 1 e = 0 2 2 2 2L Re e e e e

because: 1 − γ 5( ) 1 + γ 5( )=1- γ 5( )2=1 - 1 = 0

•  therefore the e.m. interac+ons conserve the chirality of the fermions involved. This is truebecauseitisavectorcurrent.Itcanbeprovedthatalsoanaxialcurrentconservethechirality.• Let’sseewhathappenstoascalarterm,likethemasstermappearingintheDiracLagrangian:

γ γ γ γ⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞− + − +⎢ ⎥= + = + = +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦

2 25 5 5 51 1 1 1 ( )2 2 2 2

R L L Rmee me e m e e e e m e e e e

•  Themass termsmix stateswith different chirality, therefore they break the chiral symmetry.ThiscausedalotofproblemstothefirstversionoftheelectroweaktheoryofGlashow,whereallfermionshadtobemassless. TheproblemwassolvedbyWeimbergeSalambyintoducinginthetheorytheHiggsmechanismofaspontaneousbreakingofalocalgaugesymmetry.



Unitarityviola/on1/3

•  Let’slookatthisprocess;intheFermitheorycanbewriYenas:ν ν− −+ → +e ee e

eν-e

µJ (e)

†µJ (e)

eν -eµ µ

ν νγ γγ γ

⎡ ⎤ ⎡ ⎤− −= ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

5 54 1 12 22 e ee e

G u u u uM

•  Withthisamplitude,andneglec+ngtheelectronmass,wegetthefollowingcross-sec+on:

σ ν νπ

− −+ → + =2

( )e eGe e s sisthecenterofmassenergysquared

• Fromtheformalismofthepar+alwavesexpansionweknowthatwehaveamaximumvaluefortheelas+cscaYeringcross-sec+on,compa+blewiththeconserva+onofunitarity:• Ifweignorethespin,themaximumcross-sec+onsis:

π π ππ −⇒ ⇒≤ ≤ = ≈

⋅

2

516 2 2

1870 Ge

10V

.67G s s

s G

π π πσ = + = =h h h2 2

max2 2 2

4 4 4(2 1) ( =1)elcm cm cm

lp p p

Scattering in S wave for pointlike particles

at high energy ν and e are lefthanded → J=0 ( S wave)

•  Therefore: ππ

≤2

24

cm

G sp

; let’srecallthat ν= + 2( )es p p

= =⇒2 2(2 ) 4cm cmss p p•  Inthelab.s=2meE0,whileinthecenterofmassframe:

includingthespin,theFermicross-sec+onviolatestheunitarityat√s≈√G≈300GeV



IntermediateVectorBoson2/3

•  Thedivergentbehaviourofthecross-sec+oncanbeavoidedif,inanalogywithQED,weintroduceanintermediatevectorbosonasapropagatoroftheweakinterac+ons.

•  ThediagramofthescaYeringprocessbecomes:µ ν

µν −

−

2

2 2w

w

q qgM

M qeν-e

eν -e

WThe propagator of amassivebosonofspin1is:

•  ThematrixelementcanbewriYenas: µν ν µ

γ γγ γ⎡ ⎤ ⎡ ⎤− −= ⎢ ⎥ ⎢ ⎥

−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

5 5

2 21 1 12 22 2e ee e

w

g gu u u uM q

M

•  gisadimensionlesscouplingconstant;• thefactors√2and½areintroducedtogettheconven+onaldefini+onofg;• Sincetherangeoftheweakinterac+onsisextremelyshort(oftheorder10-3fm),thenthemassoftheintermediatevectorbosonmustbeverybig;• forweakprocesseswhereq2transferredissmall,liketheβdecaysorthemuondecay,wehaveq2<<Mw

2,thereforewecanneglectq2withrespecttheWmassinthepropagatorexpression.

withtheonewiththeWboson,weget:

• IfwecomparetheFermimatrixelement: µµ νγ γ γ γ⎡ ⎤ ⎡ ⎤= − −⎣ ⎦ ⎣ ⎦

5 5(1 ) (1 )2 p n eG u u u uM

=2

22 8 w

G gM



Wbosonmass3/3

•  Fromthepreviousrela+onwegettheWbosonmass: =2

22 8 w

G gM

=2 28wgMG

• ifwedothehypothesisthatg≈e,wehave: παπ

⇒= = ≈2

2 21 4 g e =4 137 137e −

=5

21 0 ;p

GM

•  Pu�ngalltogetherweget:π

−= ⋅ ≈⋅ 5

4 2137 37.4 GeV8 10w pM M

•  actually: θθ

⇒= ≈ = ±37.4sin( ) Mw 80.425 0.038 GeVsin( )w

we g θWistheweakangle,

knownasWeinbergangle

Theweakinterac/onsare“weak”notbecauseofthe“weakness”ofthe couplingconstantbutbecauseofthehighvalueoftheWbosonmass.

•  Sinceg≈eitisnotnecessarytointroduceanewchargetounderstandtheweakinterac+ons.• Wehaveanewmassscale:theFermiscale,equaltotheWbosonmass≈100GeV

• Somethingsimilarhappensintheelectromagne+sm: F = e!E + em

!vx!B (e=em ⇒ unification)

•  Themagne+ceffectsbecomerelevantwhenvisbigandtheybecomecomparablewiththeelectricones.•  Wheneverthereisaunifica+onoftwodis+nctphenomenausuallyitappearsanewscale;intheelectromagne+smitisthespeedoflight.

Itisthescalethatdeterminestherela+vestrengthoftwoforces.



Chargedpiondecay1/3

π −

d

u −Wµ−

µν

p

kqπ⋅q f

π −(q) → µ−(p) + νµ(k)

•  Theamplitudehastheform: M = G

2!( )µ u(p)γ µ(1 − γ 5)v(k)( )

; !( )µ : quark weak current

•  Ifthequarkswerefreepar+cle,wewouldhave: !( )µ = udγ

µ(1 − γ 5)vu( )thisisnotcorrectbecausethequarksūanddarenotfreebuttheyareboundwithintheπ-meson.

•  However:1.  MisaLorentzinvariant,therefore(···)μmustbeavectororanaxialvector;2.  theπ-hasspinzero,thereforethequadrimomentumqμistheonlyquadrimomentum

availabletobuild(···)μ

⇒ !( )µ = qµ ⋅ f (q2) = qµ ⋅ fπ ; fπ is a constant

[fisfunc+ononlyofq2becausethereisnootherscalarthatcanbeconstructed] ⇒ q2 = mπ2 ⇒ f (mπ

2) ≡ fπ

( ) ( )µ µπ µγ γ⇒ = + − 5( ) (1 ) ( )

2G p k f u p v kM ( )µ µ µ= +q p k

( ) ( )µ µµ µγ γ− = + = ; 0 0p m u p m v•  memento: eq. di Dirac

⇒ u(p)γ µpµ = mµu(p) ; γ µkµv(k) = 0

( )π µ γ= ⋅ ⋅ −⇒ 5( )(1 ) ( )2

mG f u p v kM



Chargedpiondecay2/3

( )π µ γ= ⋅ ⋅ −⇒ 5( )(1 ) ( )2

mG f u p v kM

•  Inthepioncenterofmassreferenceframe,thetransi+onprobabilityperunitof+meisequalto:

ππ δ

π π ωΓ = − −

3 32 4

3 31 (2 ) ( )2 (2 ) 2 (2 ) 2

d p d kd q p km E

M

Sum on the final spin and average on the initial spin Muon phase space Neutrino phase space

Quadrimomentum conservation

•  Thepionhasspinzero,thereforenoaverageontheini+alspin;thesumonthemuonandneutrinospin isdonewiththe“traceology”mechanismoftheγmatrix.

π µ= ⋅22 2 (

2G f m Tr p2M µ γ+ − 5)(1 )m k π µγ⎡ ⎤+ = ⋅⎣ ⎦

5 2 2 2(1 ) 4 ( )G f m p k

•  Inthepioncenterofmasswehave,therefore: !k = −

!p p ⋅ k = Eω −

!k ⋅!p = Eω + k2 = ω(E +ω)

•  pu�ngalltogetherwehave: Γ =

G2fπ2mµ

2

(2π)22mπ

d3pd3kEω∫ ω(E +ω)δ(mπ − E − ω)δ (3)(

!k +!p)

•  Theintegra+onind3pistakenintoaccountbytheδ(3),andsincethereisnoangulardependency,weareonlyle|withtheintegra+onindω:

•  Thefinalresultis: µπ π µ

πτ π

⎛ ⎞⎜ ⎟Γ = = −⎜ ⎟⎝ ⎠

2222 2

21 18

mG f m mm

N.B.actuallywehavecomputedthepar+alwidthofthepiondecayinthemuon-neutrinochannel,butsincethisisthedominantchannel,itisalmostequaltothetotalwidth,thentotheinverseofthelife+me.

τΓ = Γ =Γ

⇒∑ .1

tot parztot



Chargedpiondecay3/3

µπ π µ

πτ π

⎛ ⎞⎜ ⎟Γ = = −⎜ ⎟⎝ ⎠

2222 2

21 18

mG f m mm

•  IfwetakethevalueofGmeasuredintheβdecayorinthemuondecayandweassumethatfπ=mπ(atleastfordimensionalreasons)wefindthepionlife+me:2.6·10-8s

•  Thisisnotarealtestofthetheorysincetheassump+onfπ=mπisnotjus+fied.Howeverwecandoaquan+ta+vetestbycomparingtheB.R.ofthedecayinthemuonchannelwiththeoneinelectron::p-→e-νe.Thecomputa+onisiden+cal,exceptthatwehavetoreplacethemuonmasswiththeelectronmass:

π

µµ π µ

π νπ µ ν

− −−

− −

⎛ ⎞⎛ ⎞Γ → −⎜ ⎟= =⎜ ⎟⎜ ⎟ ⎜ ⎟Γ → −⎝ ⎠ ⎝ ⎠

22 2 24

2 2( ) 1.2 10( )

e e e xe m m m

m m m

•  Thenumericalvalueobtainedinser+ngthemassesintheformulacoincideswiththeoneobtainedwiththemeasuredB.R.• The pionpreferstodecay intomuonsratherthaninelectrons.This isnotwhatonewouldexpectbasedonphasespaceconsidera+ons,sincetheoneoftheelectronchannelismuchbiggerthattheoneofthemuon. Ontheotherhandthecouplingconstantisthesameforelectronandmuon(universalityofweakinterac+ons).Theexplana+ondependsonthehelicityofthetwopar+cles.• Thepionhasspinzero;inthedecaywemustconservthetotalangularmomentum,therefore:

π −

νe−e The antineutrino must be righthanded The electron is forced to be righthanded

•  This is the state of “wrong” helicity of the electron, because in the limit of zero mass it would have beenle|handed(andthedecaywouldnotbepossible).• Sincethemuonhasamassbiggerthantheelectronone,itiseasierthatitgoesinthestateof“wrong”helicity.



DecayofthechargedK±1/2

−K

s

u −Wµ−

µν

p

kq⋅ kq f

µµ ν− −→ +K

The B.R. is 64%Itissimilartothepiondecay,withaquarksreplacingaquarkd

( ) ( )µ µµ µγ γ γ= + − = −5 5( ) (1 ) ( ) ( )(1 ) ( )

2 2k kG Gp k f u p v k f m u p v kM

µµ µµ ν

π− −

⎛ ⎞⎜ ⎟Γ → + = −⎜ ⎟⎝ ⎠

2222 2

2 ( ) 1 8 k k

k

mGK f m mm

WereplacethepionmasswiththeKmass,andtheconstantfπwithfk

•  SinceπandKbelongtothesameSU(3)octet,ifthiswouldbeanexactsimmetryàfπ=fk• Sincethesimmetryisbroken(butnottoomuch),theyarediffent,butnotsodifferent:fπ=130MeV,fk=160MeV.• Ifwedothera+oofthetransi+onprobabilityK/π,assumingfπ=fk,wehave:

µ

µ

πµ µ

π

µ ν

π µ ν

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟−− − ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

− − ⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Γ →= =

Γ →

22

1

2

1

( )17.67

( )

m

mkkm

m

K mm

•  ThediscrepancycannotbeexplainedbythedifferencebetweenfπefkduetothebrokenSU(3)simmetry.

•  whiletheexperimentalvalueis:µ

µ

µ ν

π µ ν

− −

− −

Γ →= ±

Γ →

( )1.336 0.004

( )

K



DecayofthechargedK±2/2

•  Thediscrepancycanbeexplainedbyadifferentcouplingconstantforthehadroncurrentthatchangesstrangeness.•  Let’scallGstheFermicouplingconstantwiththequarksandGdtheFermicouplingwiththequarkd.

µ

µ

µ ν

π µ ν

− −

− −

Γ → ⎛ ⎞± = = ⋅ ⎜ ⎟Γ → ⎝ ⎠

22

2

( ) 1601.336 0.004 (17.67)130( )

s

d

K GG

k

π

ff

GsGd

= 0.223

•  Thisimpliesabreakingoftheweakinterac/onsuniversality.

• Anexplana+onofthisphenomenonthatpreservestheweakinterac+onsuniversalitywasgivenbyCabibboin1963.

•  BesidesthechargdKdecays,wecanalsotakeintoaccountotherweakdecays:

eν -e

µ−µν

W −

eν -e

n p

W −

eν -e

0Λ p

W −

muon decay neutron decay Λ0 decay

Δ =⎛ ⎞⎜ ⎟⎜ ⎟Δ =⎜ ⎟⎝ ⎠

1 1

2

S

I

•  In the three cases we are dealing with weak charged currents. In the first two cases does not change thestrangeness(ΔS=0)whileinthethirdcasethereisachangeinthestrangeness(ΔS=1).

• Tobeno+cedthatthehadroniccurrentwithΔS=0isslightly“smaller”thantheleptoniccurrent(CV=0.98)whileitisabout5+meshigherofthehadroniccurrentwithastrangenesschange.



Parenthesis:doubletstructureofthew.i.•  In1962Schwartz,LedermaneSteimbergerfoundthatintheinterac+onofaneutrinobeam,obtainedfromthepionandKdecays,theresultwas:

νµ + N → µ− + N but never νµ + N → e− + N

•  Thisexperimentprovedtheexistenceofasecondtypeofnutrino,associatedwiththemuondecay,differentfromtheonepresentintheβdecay• Weintroducetheconserva+onoftheleptonnumberseparetelyforeachlepton.Inthiswayweexplaintheabsenceofthemuondecayinelectronplusphoton.• Leptonsareorganizedinadoubletstructure:

µ τνν ν

τµ− −−

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

e

e+ the antiparticles To be fair,we should put in the doublets

onlythele|handedstateofthefermions.

•  Thechargedweakinterac+ons(thatiswithaWexchange)makethetransi+onbetweenthetwocomponentsofadoubletbutneverfromadoublettotheotherone(leptonnumberconserva+on).N.B.sincetheneutrinoshaveamass,thisisnolongertrue(flavouroscilla+ons)buttheprobabilityissosmallthatasamaYeroffactitneverhappens.

• Asfarasthequarksareconcerned,thepicturewaslessclear,becausewedidobservetransi/onsofthequarkdtowardthequarku,aswellastransi/onofthequarkstowardthequarku.Thereforeitwasnotevidentwhatisthedoubletinvolvedinthetransi/on.

Tobeno+cedthatwedonotobservetransi+onsbetweenthequarkdandthequarks(flavourchangingneutralcurrent–FCNC).ThiswasexplainedbyGlashow-IliopouloseMaiani(GIM)nel1970.



TheCabibboangle1/4

•  Thesolu/ontorecovertheweakinterac/onuniversalitywasfoundbyCabibboin1963.Heproposedthatthemasseigenstates,thatarealsoeigenstatesofthestronginterac/ons,areNOTeigenstatesoftheweakinterac/ons.

•  Theoriginalpublica/onofCabibbowasbasedonthecurrent-currentinterac/onmodeloftheweakinterac/ons,butinwhatfollowswepresenta“modern”versionofthetheorybasedonquarks,thatitiseasiertounderstand(in1963thequarkswerenotyet“invented”).

•  Werecallthatexperimentallyweobservepar+cleswithadefinitemassandlife+me,inotherwordsweobserveonlythemasseigenstates.

•  Togofromthemasseingenstatebasetotheweakinterac+onbaseweneedaunitarytransforma+onthatpreservesthevectornormaliza+on.Inatwodimensionalspaceitissufficienta“rota+on”matrixcharacterizedonlybyoneparameter,thatisoneangle:theCabibboangleθC

•  Theeigenstateoftheweakinterac+onsisalinearcombina+onofthemasseigenstates:

θ θ= + ' cos sin c cd d s

•  wecanthenconstructaweakisospindoublet(thathasthesamealgebraofthestrongisospinbutitisacompletelydifferentconcept):

θ θ⎛ ⎞⎛ ⎞

= ⎜ ⎟⎜ ⎟ +⎝ ⎠ ⎝ ⎠

cos sin' c c

uud sd TheWcouplesthestated’ withthequarku

Wu d '



Cabibbohadroniccurrent2/4

•  ThestructureoftheCabibbohadroniccurrentthat“raisesthecharge”(itmakesthetransi+onfromthelowercomponentofthedoublettothetopone)isofthistype(wedonotwritethefactor½(1-γ5):

Jµ+(quarks) ≈ g u,d cosθc + s sinθc( ) 0 1

0 0

⎛

⎝⎜⎞

⎠⎟ ud cosθc + ssinθc

⎛

⎝⎜

⎞

⎠⎟ = g(ud cosθc + ussinθc)

•  Thematrix ( ) ( )τ τ τ+= + =0 11 20 0

1 12 2

i isthechargeraisingoperatorforaweakisospindoublet

•  Inacompactwaywecanwritetheraisingcurrentasfollows:

Jµ+(q) ≈ gqLτ+qL , where qL = u

d '⎛

⎝⎜⎞

⎠⎟We consider only the quark lefthanded components

•  Inasimilarwaywecanwritethe“lowering”currentJ-(withaW+exchange):

Jµ−(quarks) ≈ g u,d cosθc + s sinθc( ) 0 0

1 0

⎛

⎝⎜⎞


⎛

⎝⎜

⎞

⎠⎟ = g(ducosθc + susinθc)

µ τ−−≈( ) L LJ q gq q•  inacompactway: ( ) ( )τ τ τ−

⎛ ⎞= − =⎜ ⎟⎝ ⎠0 0

1 2 1 01 12 2

i eν +e

p n

W +

duudd

u



Quarksβdecays3/4

•  Let’sexaminethehadronicβdecays,takingintoaccountthathadronsarenotelementarypar+cles;atamorefundamentalleveltheβdecayinvolvesthequarksthatcons+tutethehadrons:

eν -e

µ−µν

W

muon decay neutron decay Λ0 decay

0Λ

eν -e

n p

Wuuudd

d

eν -e

p

Wuuudd

s

IneveryvertextheWmustconservetheelectriccharge.TheWonlycouplestole|handedfermionsandtorighthandedan+fermions.

weakinterac+onuniversality: gisthesamecouplingconstanteverywhere.

Mµ→eν

eν

µ

= Jlepton1

Mw2 − q2

Jlepton

Jhadron = g

2uuγ

µ 1 − γ 5

2ud

⎡

⎣⎢⎢

⎤

⎦⎥⎥cosθC + g

2uuγ

µ 1 − γ 5

2us

⎡

⎣⎢⎢

⎤

⎦⎥⎥sinθC

Jlepton = g

2uν

e

γ µ1 − γ 5

2ue

•  Let’swritetheleptoncurrentandthehadroncurrent,thenwecomputethematrixelementandthedecayrateforthethreeprocesses:

Mn→peν

e

= Jd→u1

Mw2 − q2

Jlepton MΛ0→peν

e

= Js→u1

Mw2 − q2

Jlepton

Γ(µ− → e−νeνµ) ∝ g4

Γ(n → pe−νe) ∝ g4cos2θc Γ(Λ0 → pe−νe) ∝ g4 sin2θc



TheCabibboangle:thevalue4/4

•  IntheCabibbo’stheoryallpar+cles(quarksandleptons)carryaweakchargeg,butthequarksaremixed:

Jµ+(q) ∝ g cosθc for the current where ΔS=0 ; Jµ

+(q) ∝ gsinθc for the current where ΔS=1

•  therefore:

Γ(µ− → e−νeνµ) ∝ g4 pure lepton current

Γ(n → pe−νe) ∝ g4cos2θc ΔS = 0 semi-leptonic

Γ(Λ0 → pe−νe) ∝ g4 sin2θc ΔS = 1 semi-leptonic Γ(Λ0 → pe−νe)

Γ(n → pe−νe)=tan2 θc

DataareconsistentwithaCabibboangleofθc≈13°

•  Processespropor+onaltocos2θcarecalled“Cabibbofavored” whiletheonespropor+onaltosin2θc arecalled“Cabibbosuppressed”

•  Tobeno+cedthatcos13°=0.974andindeedexperimentallyithasbeenfoundCV≈0.98.

µ

µ

µ νθ θ

π µ ν

− −

− −

Γ⇒ ⇒

→= = = °

Γ →

( )0.223 tag 12.57

( )s

c cd

K GG

cos2θc = 0.949 ; sin2θc = 0.050 ; tag2θc = 0.053

•  IfwegobacktothecomparisonbetweentheKdecayandthepiondecay,wehave:thatitisconsistentwiththevalueobtainedfromthecomparisonbetweentheneutronandΛ0decays.OnlyoneCabibboangleis“sufficient”forallweakprocesses.



AbsenceofFCNC1/5

•  Experimentallyweobservethatdonotexistneutralweakcurrentsthatchangetheflavourofthequarks.Thisstatementcanbeexplained,forinstance,bylookingattwoKdecays,oneaboutthechargedKandtheotheroneabouttheneutralK.

K+

u

sW+ µ+

µν

K µµ ν+ +→ +

( )B.R. 63.51 0.19 %= ±

0K

d

s Zµ+

µ−

0K µ µ+ −→ +

( ) 9B.R. 7.27 0.14 10−= ± ⋅

N.B. actually this graph does notexist.Wehaveanempiricalrulethatsaysthat,atthefirstorder,ΔS=ΔQ.InthiscasewehaveΔS=-1;ΔQ=0

•  TheW± is a charged boson (nega+ve and posi+ve) that acts as amediator in theweak interac+ons due tochargedcurrents.• Forotherreasonsduetounitarityviola+onoftheweakinterac+ons,inthiscaseintheprocessofWpairproduc+on(uū−>W+W-),weneedtointroduceanotherintermediatevectorboson,neutral,calledZ,thatisthemediatorofweakinterac+onswithneutralcurrents.(TheZbosonwillbeabyproductoftheunifica+onofweakinterac+onsandelectromagne+cinterac+ons,aswewillseelaterwhenwewillstudytheStandardModel.Inanycaselet’sstatesincenowthattheZcouplingisdifferentfromtheWcoupling).• Basedonwhatwesaidweshouldexpectthatthesecondprocessshouldexistwitharatecomparablewiththefirstone,butasamaUeroffactitisnotso.Itishighlysuppressed:why?



Neutralweakcurrents2/5

•  Let’srecalltheraisingandloweringchargedcurrents(weomitthefactorsγμ(1-γ5)andthecouplingconstant):

Jµ+(q) ≈ g u,d cosθc + s sinθc( ) 0 1

0 0

⎛

⎝⎜⎞


⎛

⎝⎜

⎞

⎠⎟ = g(ud cosθc + ussinθc) ( ) ( )τ τ τ+= + =0 1

1 20 01 12 2

i

Jµ−(q) ≈ g u,d cosθc + s sinθc( ) 0 0

1 0

⎛

⎝⎜⎞


⎛

⎝⎜

⎞

⎠⎟ = g(ducosθc + susinθc)

( ) ( )τ τ τ−⎛ ⎞= − =⎜ ⎟⎝ ⎠

0 01 2 1 0

1 12 2

i

•  Formallyitshouldexistalsothethirdcomponentofthecurrents,duetoaZexchange:

0 2 2( ) cos sin (sd+sd)sin cosc c c cJ q uu dd ssµ θ θ θ θ−⇒ ≈ − −

ΔS = 0 ΔS = 1

Thelasttermisresponsibleoftheflavourchangingneutralcurrentprocesses,withamplitudepropor+onaltosinθccosθc.ThistermishighlysuppressedinNature.

03( ) ' 'J q gq q uu d dµ τ≈ ≈ − ( )1 0

3 0 1τ

−=

'u

qd⎛ ⎞

= ⎜ ⎟⎝ ⎠

where and θ θ= + ' cos sin c cd d s

u Z

u

' cos sinc cd d sθ θ= +

Z


+cos w

gθ

(N.B. the vertex is more complicated than the W one; θW = Weinberg angle)

•  Ifwewritethecurrentintermsofmasseigenstatesweget:



TheGIMeffect(thequarkcharm)3/5

•  Theexplana+onofthesuppressionoftheflavourchangingneutralcurrentwasproposedbyGlashow,IliopoulosandMaiani(GIM)in1970.

•  Theyintroducedanewquark,thecharm,thathasthesamechargeofthequarku.Thentheyproposedaseconddoubletofquarkweakisospin:

c

s '⎛

⎝⎜⎞

⎠⎟ TheWconnectss’ withc

•  Itwasfullyrestoredthesymmetrywiththeleptondoubletsknownin1970:

νe

e−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

νµ

µ−

⎛

⎝⎜⎜

⎞

⎠⎟⎟ u

d '⎛

⎝⎜⎞

⎠⎟ c

s '⎛

⎝⎜⎞

⎠⎟Thechargedifferencebetweentheupperandthelowercomponentsis+1.

•  Theweakeigenstates’canbefoundusingtheCabibbotheorythatconnectsthemasseigenstatestotheweakeigenstatesbyaunitarymatrix

cos sin'

sin cos'c c

c c

d ds s

θ θθ θ

⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠⎝ ⎠

s' = scosθc − d sinθc

Byconven+onwerotatethedown-typequarksandweleaveunchangedtheup-typequarks.Itwouldbeabsoluteyequivalenttheotherchoice,namelytorotatetheup-typequarks.



Zcouplings4/5

•  Let’sseehowtheintroduc+onofcands’willsolveourproblems.WesawthecouplingoftheZwiththeuandd’:

u Z

u


Z


+ Jµ0(q) ≈ uu − dd cos2θc − sssin2θc − (sd+sd)sinθc cosθc

ΔS = 0 ΔS = 1

• NowweneedsimilargraphsforthecouplingoftheZwiththequarkscands’:

c Z

c

' cos sinc cs s dθ θ= −

Z

' cos sinc cs s dθ θ= −

+ ⇒ Jµ

0(q) ≈ cc − dd sin2θc − sscos2θc + (sd+sd)sinθc cosθc

ΔS = 0 ΔS = 1

• Ifwesumuptheamplitudesofthefourgraphsweget:

0( ) ' ' ' 'J q uu d d cc s sµ ≈ − + − = uu + cc − (dd + ss)cos2θc − (dd + ss)sin2θc + (sd+sd-sd-sd)sinθc cosθc

ΔS = 0 ΔS = 1

Jµ

0(q) ≈ uu − dd − ss + cc The Z couples to the mass eigenstates.NoneedoftheCabibboangle.

Withtheintroduc+onofthequarkcaredisappearedtheneutralcurrentwithaflavourchanging.AswesawtheZcouplesonlytoquark-an+quarkpairsofthesameflavour.



FCNC:thetruegraphs5/5

•  ExperimentallywehaveseenthattheFCNCarehighlysuppressedbutneverthelesstheydoexist.TheycannotbeduetoaZexchangebuttheyareduetoasecondorderWexchanges(boxdiagrams):

d

s

µ+

µν

cos cθ

u0( )K ds W +

µ−

W−

sin cθ0K µ µ+ −→ •  Theamplitudeispropor+onaltosinθCcosθC

•  Thequarkuintervenesasavirtualpar+cleinthebox

FromthemeasuredB.R.ofthisdecay,G.I.andM.predictedthanthemassofthecharmshouldlayintherange1−3GeV.Thehuntforthequarkcharmwasofficiallylaunched.

•  Withthe“inven+on”ofthequarkcharmwehaveanothergraph:

d

s

µ+

µν

sin cθ−

c0( )K ds W +

µ−

W−

cos cθ •  Theamplitudeispropor+onalto-sinθCcosθC

•  Ifthemassofthequarkcwasequaltotheoneofquarkup,therewouldbeanexactcancella+onbetweenthetwographsandtheB.R.ofthisdecaywouldhavebeenzero.

ThesearchforhighlysuppressedFCNCdecays,likeforinstanceiss+llapowerfultoolinthesearchfornewphysics,becausewecouldhavenewpar+clesintheloopthatwillenhancetheStandardModelBranchingRa+o.

Bs0 → µ+µ−



J/Ψdiscovery1/2

•  InNovember1974therewasthe“Novemberrevolu+on”withthediscoveryofaresonanceverypeculiar,becauseitslife+mewasaboutthreeorderofmagnitudebiggerthanwhatonewouldexpect.

•  ThediscoverywasdoneinanindependentmannerbytheTing’sgroupatBrookhavenandbytheRichter’soneatSlac,andjusta|erwardalsoattheAdonee+e-colliderintheFrasca+INFNLaboratory.

3.10 3.12 3.14

Brookhaven

p Be J anything+ → +

Ting was looking for a peak in theinvariant mass of the e+e- pairsproduced in the collision of 28 GeVprotonsonaberilliumtarget.

He foundaverynarrowpeakat≈3.1GeV

Ting called this resonance J

mJ /Ψ = 3096.900 ± 0.006 MeVPDG 2016

SLAC

Richteretal.usedthee+e-colliderSpearat SLAC. They measured the cross-sec+onoftheprocesse+e-àhadronsasafunc+onofthecenterofmassenergy.

Theyalsofoundapeakat≈3.1GeV

Richter called this resonance Ψ

EverybodyelsecallitJ/Ψ

Richter → Ting e+e- → J/Ψ → hadronsRichter ← Ting



Whatwas“wrong”withtheJ/Ψ?2/2

/ 2191.0 3.2 keV 7 1 0J

tot sτΨ −= ≈ ⋅Γ

= ± ⇒Γ h

•  Onewouldexpectalife+metypicalofstronginterac+ons(10-23),similartootherstrongdecays,like:

s

s s

us

uΦ

K-

K+

m~1.02 GeV Γ~0.004 GeV

•  TheJ/ΨcannotdothisdecaybecausethelightestcharmedmesonD0has: 0 1864.6 0.5 MeVDm = ±

/ 0 2Jm m Dψ < ⋅

π0

π+

d

u u

dd

dρ+

m~0.77 GeV Γ~.15 GeV c

c c

uc

uJ/Ψ

D0

D0

mJ /Ψ = 3096.900 ± 0.006 MeV

•  Thenthestrongdecaymustgothroughthethreegluonschannel(OZIrule)andbecomesofthesameorderofthee.m.decaychannels:

c

c

e+ ; μ+

e- ; μ-

Γ(ee)~5 KeV; Γ(µµ)~5 KeV

c

chadrons

Γ~70keV•  Ofcoursewhatwesaiditisonlytrueifthequarksofthisnewresonancecarryanewquantumnumberthatcannotbeviolatedbythestronginterac+ons(otherwisetherewerealotofmesonslighterthantheJ/Ψ)• Sincethedecaywasnotduetoweakinterac+ons(thelife+mewouldhavebeenmuchlonger)wasanindica+onthattheresonanceitselfwasnotcarryingthisnewquantumnumber,hencethehypothesthattheJ/Ψwasamesoncomposedbyacharm-an+charmpair.



Cabibbo-Kobayashi-Maskawa(CKM)matrix1/5

•  In1973KobayashiandMaskawawantedtointroducetheCPviola+onintheStandardModel.ForthisitwasnecessarytointroduceacomplexnumberintheHamiltonian;let’srecallthatifH*≠HthentheTimeReversalTisviolatedwhileCPTisalwaystrue.

•  Thesimplestwaywastointroduceaphaseinthequarkmixingmatrix.

•  ANxNunitarymatrixhas:

12

N(N-1) real parameters (Euler angles)

12

(N-1)(N-2) non trivial phases (you can't get rid of by changing the phase of the quarks)

•  WithN=2onecannotintroduceanyphases,thereforeK.andM.proposedin1973thatshouldexistathirdfamilyofquarks,becausewithN=3wehavethreeanglesandonephase:

'' = '

ud us ub

cd cs cb

td ts tb

d V V V ds V V V sb V V V b

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

where:

Eigenstates of theweakinterac+ons CMK matrix of

thequarkmixing Masseigenstates

Example

CBV

c W−

b

' ' 'u c td s b⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

+theiran+par+cles

TheCKMmatrixisunitaryanditcanbeparametrizedinseveralways.Itsparametermustbedeterminedexperimetally.Thereisnotheory(yet)thatcanforeseetheCKMvalues.

s '=Vcd ⋅d +Vcs ⋅s +Vcb ⋅b



CKMmatrix2/5

•  AswesaidtheCKMmatrixcanbewriYeninseveralways,forinstance:

1.  Intermsof3anglesand1phase:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−−−−−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

′′′ −

bsd

ccescsscesccsscsesssccessccsescscc

bsd

ii

ii

i

132313231223121323122312

132313231223121323122312

1313121312

1313

1313

13

δδ

δδ

δ

Thefourrealparametersare:δ13,θ12,θ23,andθ13.Theabbrevia+onis:s=sin,c=cosandthenumbersrefertothequarkgenera+on.Forinstances12=sinθ12.

2. Intermsofquarkcouplings(thisisthebestonetounderstandthe“physics”)

d's'b'

= Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

dsb

3. IntermsofaseriesexpansionoftheCabibbloangleθ12(explo+ngthefactthats12>>s23>>s13)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−

−−=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

′′′

bsd

AiAA

iA

bsd

1)1(2/1

)(2/1

23

22

32

ληρλλλλ

ηρλλλ λ=sinθ12,whileA,ρandηarerealandcloseto1.Thisparameteriza+onissuitabletorelatetheCPviola+on to some specific processes and theirdecayrate.

“Wolfenstein” representation

Example

CBV

c W−

b

From PDG



CKMmatrix3/5

•  Let’sseethevalueofmatrixelementstakenfromthePDG2016:

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=

•  Lookingatthematrixnumericalvalueswecanpointoutafewthings:1.  TheCKMmatrixisalmostdiagonal(theoffdiagonalelementsaresmall)2.  Themoreoffwearefromafamily,thesmallerthematrixelementsis(forinstanceVub<<Vud)3.  Bymakinguseof points 1. and2.wededuce that somedecays arepreferredwith respect to some

otherones,forinstance:

4.  Sincethematrixissupposedtobeaunitarymatrix,wehaveseveralconstraintsamongtheelements,forinstance:

so far theexperimental resultsareconsistentwithaunitaryCKM;howeverwecon+nue to look for

devia+onfromunitaryasanindica+onfornewphysicssignalwithrespecttowhatisforeseenbytheStandardModel.

0

1***

***

=++

=++

tdtbcdcbudub

tdtdcdcdudud

VVVVVV

VVVVVV

c → s over c → d D0 → K−π + over D0 → π−π + (exp. find 3.8% vs 0.15%)

b → c over b → u B0 → D−π + over B0 → π−π + (exp. find 3 ×10−3 vs 1 ×10−5)



MeasurementoftheCKMmatrixelements4/5

•  AtthemomentthereisnotheorythatisabletopredictthevaluesoftheCKMmatrixelements,thereforetheymustbedeterminedexperimentally.

•  The“cleanest”waytodoitistousedecaysorprocesseswherearepresentleptons,sothattheCKMmatrixintervenesonlyatonevertex.Forinstance:

Vud:neutrondecay: nàpev dàuevVus:kaondecay: K0àπ+e-ve sàuevVbu:B-mesondecay: B-à(ρ0orπ0)e-vebàuevVbc:B-mesondecay: B-àD0e-ve bàcevVcs:charmdecay: D0àK-e+ve càsevVcd:neutrinointerac+ons: νμdàμ-c dàc

D0=cū;B-=ūb

c

u

s

u

e+,µ+

νe,νµW+

K- D0 Vcs

It is called “spectator”model because only onequarkpar+cipatestothedecay,whiletheothersjust“goround,lookanddonothing”.

N.B.Forthemassiveneutrinosexistamatrixsimilar to theCKMonecalledPMNSmatrix.Iftheneutrinosweremasslessitwouldhavebeendiagonal.

“Spectator”modeloftheD0àK-e+vedecay

Amplitude∝ VcsDecayrate∝ Vcs

2



DiscoveryoftheYupsilon(Υ)5/5

•  In1977atFermilab(FNAL)inChicagowerediscoveredothernarrowresonanceswithmassesbetween9e10.5GeVthatshowedthesamepeculiari+esoftheJ/Ψ,thatisalife+metoolongwithrespecttotheoneexpected.Thiswasthehintofanewquantumnumber:thebeauty.

Lederman, in a similar manner to the Ting’s experiment, waslooking for resonances in the invariantmass of themuon pairsproducedinthereac+on:

p(400 GeV) + nucleus → µ+µ- + anything

Thereareseveralpeaks.TheY(1S),Y(2S)andY(3S)arebelowthethresholdtodecayinpairsofmesonwithopenbeauty.

Backgroundsubtructed

•  In1975atSlacwasdiscoveredthethirdlepton,thetau: 1776.99 0.39 MeVmτ = ±

•  In1994aFNALwasdiscoveredthequarktop: 174 5 GeVtm = ±

•  Andfinally,againFNAL,in2000wasdetectedinadirectwaytheneutrinotau.Thethreefermionfamiliesarecomplete!


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Introduc+on to Par+cle Physics - Chapter 7 · PDF fileClaudio Luci – Introduc+on to Par+cle Physics – Chapter 7 10 4/4 The Sargent’s rule ... , the ones with L=2 second forbidden