FYS3500 - spring 2020 Symmetry breaking in€¦ · FYS3500 Spring 2020 Alex Read, U. Oslo, Dept. Physics Introduction Symmetries are extremely important in many branches of physics,

FYS3500 - spring 2020

Symmetry breaking in the weak interaction*

Alex ReadUniversity Of OsloDepartment of Physics

*Martin and Shaw, Nuclear and Particle Physics, 3rd Ed., Chapter 7 (Last update 21.04.2020 11:48)

FYS3500 Spring 2020 Alex Read, U. Oslo, Dept. Physics

Introduction❖ Symmetries are extremely important in many branches of physics, including nuclear

and particle physics

❖ Conservation laws such as various charges, total angular momentum, parity, and C-parity allow us to distinguish between allowed and forbidden processes

❖ The breaking of symmetries is at least as interesting and revealing as the symmetries themselves

❖ e.g. breaking of gauge symmetry via the BEH mechanism

❖ e.g. the breaking of isospin symmetry by quark mass differences and electric charge differences

❖ Here we will how the (maximal) parity and C-parity violation in the (charged) weak interaction is restored by the product

❖ And how there are even small violations of CP-conservation

u, d

CP

2


Parity violation in 1956❖ In early 1950’s two particles with (experimentally) the same spins (0),

masses and longish lifetimes decayed into 2 different states of parity:

❖ (not the -lepton) and

❖ Two different particles (remember, this is before the quark model)?

❖ Or parity not conserved in weak interaction?

❖ Question: How could we sort out the parity states?

❖ Lee and Yang (theorists) surveyed experimental results and observed that there was no real test of parity conservation in weak interactions.

τ → ππ τ θ → πππ

3


Parity violation in 1957

❖ C.S. Wu et al tested parity conservation in polarized 60Co →60 Ni* + e− + νe

4

θπ − θ

Parity conservation: Γ(θ) = Γ(π − θ)C.S. Wu et al. Wolfgang Pauli: “I cannot believe God is a weak left-hander”.

r ⟶ − rp ⟶ − p

r × p ⟶ r × pL , S , J ⟶ L , S , J

P


Polarized muon decays, C and P❖ Parity changes to ( ) but spins unaffected.

❖ C-parity inverts identity of all particles but angles, spins unaffected:

❖ If C-parity is conserved then and

❖ Unless , parity is clearly violated

❖ Experiment consistent with and

❖ violation of both C and P- conservation!

❖ What about product ?

❖ and

❖ conserved (e.g. equal lifetimes )

θ π − θ cos θ → − cos θ

C(μ+ → e+νeνμ) = μ− → e−νeνμ

ξ+ = ξ− Γ+ = Γ−

ξ± = 0

Γ+ = Γ− ξ− = − ξ+ = + 1

∴

CP

cos θ ↔ − cos θ ξ+(−1) ↔ ξ−(+1)

∴ CP ℏ/Γ+ = ℏ/Γ−

5

Γμ±(cos θ) =12

Γ± (1 −ξ±

3cos θ)

-1 -0.5 0 0.5 1cos

0

0.2

0.4

0.6

0.8

1

(cos)/

Muon decay+

-


CP conservation❖ Of course conserved in reactions that separately conserve C and P (EM, strong)

❖ Conserved in nearly all weak interactions

❖ Remember symmetry breaking is interesting!

❖ Lots of evidence of small CP-violation in hadronic weak interactions (first time in 1964 in system)

❖ Open research question whether there is similar CP-violation in leptonic sector (neutrino-mixing)

❖ Neither of these is thought to be large enough to explain the CP-violation in the early universe hypothesized by A. Sakharov to explain the matter-antimatter asymmetry of the current universe:

❖ Baryon number violation, C-symmetry violation, CP-symmetry violation, and interactions out of thermal equilibrium

K0 − K0

6


Parity violation in neutral current interactions

❖ Remember, in any diagram with a we could just as easily write

❖ If the 4-momentum transfer is small compared to any effect will be tiny

❖ Challenging experiments, but even atomic physics is sensitive to the - atomic parity violation has been observed

γ γ/Z

mZ

Z

7


Parity violation in e−e− → e−e−

8

e−

e−

e−

e−

⇐

e−

e−

e−

e−

⇒ P e−

e−

e−

e−

⇒

e−

e−

e−

e−

⇐

π

π

σR

σL

Parity violation: APV ≡σR − σL

σR + σL≠ 0 Question: What is the dominant

Feynman diagram?γ/Z


Helicity and chirality in the weak interaction

❖ Eigenstates of chirality take part in the weak interaction

❖ Think of chirality as a kind of conserved charge

❖ For massless particles (the photon and approximately the neutrinos) helicity is equivalent to chirality, where helicity is the particle spin projected on the direction of motion, i.e.

❖ For massless particles helicity is Lorentz-invariant (i.e. conserved)

❖ Question: Why not so for massive particles?

❖ Helicity states of electron are +1 (right-handed), -1 (left-handed), i.e. spin-up, spin-down in direction of motion

❖ Question: What are the helicity states of the photon (massless vector boson)?

❖ Question: What are the helicity states of massive vector bosons like the ?

h ≡J ⋅ p

| J | | p |

W±, and Z

9


The neutrinos❖ We assume here that the neutrinos are massless

and so are either left-handed or right-handed helicity (chiral) states, independent of Lorentz frame.

❖ Weak interaction data can be understood by postulating that only and take part in weak interactions!

❖ In particular, violation of both while conserving is consistent with the above.

❖ The bosons couple to chiral doublets

and singlets e.g.

νLνR

νL νR

C and PCP

W± and Z

(νeLe−

L ), e−R and (e+

RνeR), e+

L

10

P

C

P

C

⇐νL

⇐νR⇐νL

⇐νR

(see "A Model of Leptons", by Steven Weinberg)


InteractionV − A

❖ represents a vector interaction (a vector like changes sign under parity)

❖ A represents an axial-vector interaction (an axial vector like does not change sign under parity)

❖ Both would conserve parity but interference terms in violate it

❖ The data are in precise agreement with the theory for charged weak interactions

V r or p

L ≡ r × p

|ℳ |2 ∝ |V |2 or |A |2

|ℳ |2 ∝ |V − A |2

V − A

11


Mass and chirality❖ One of the consequences of is that “forbidden” helicity states of

massive fermions , e.g. , are suppressed by a factor

❖

❖ Excellent agreement between prediction and experiment when full calculation including the -values of the decays are taken into account.

❖ Question: What can we say about the degree of polarization of the muons in decays?

V − Af e−

R and e+L

(mfc2/Ef)2

Γ(π+ → e+ + νe)Γ(π+ → μ+ + νμ)

≈ (me

mπ /2mπ /2mμ

)2 = ( me

mμ )2

≈ 10−5

Q

π+ → μ+ + νμ

12


Muon decays (again)

❖ Consider only maximum electron energy (as a qualitative approximation)

❖ Question: Is the red curve at the left consistent with the argument above? How?

13

-1 -0.5 0 0.5 1cos

0

0.2

0.4

0.6

0.8

1

(cos)/

Muon decay+

-

Γμ±(cos θ) =12

Γ± (1 −ξ±

3cos θ)

Part II Neutral hadron mixing and -violationCP

14


Neutral kaon oscillations

❖ Two neutral kaons (e.g. produced in strong interactions)

❖ (S=+1) and (S=-1)

❖ Successive transitions allowed by second-order weak interaction. Lifetime is long enough that this must be taken into account.

❖ Question: Why not oscillations?

K0(498) = ds K0(498) = ds

|ΔS | = 2

n ↔ n

15

K0 → K0 → K0 → . . .S = + 1 S = − 1


-eigenstates of the weak interactionK0

❖ Assume that CP is conserved (must test this)

❖ Assume that neutral kaons are eigenstates of CP

❖

❖ Choose phases of C-parity such that

❖ Intrinsic parity is -1:

❖ satisfies all of the above

|K01,2 >

C P |K01,2 > ≡ (+1, − 1) |K0

1,2 >

C |K0, p > = − | K0, p > and C | K0, p > = − |K0, p >

P |K0, 0 > = − |K0, 0 > and P | K0, 0 > = − | K0, 0 >

|K01,2, 0 > ≡

1

2[ |K0, 0 > ± | K0, 0 > ]

16


CP of K0x → ππ

❖ Decay of spin-0 particle to 2 spin-0 ’s

❖ Orbital angular momentum of must be 0

❖ = +1

❖

❖

❖ for

❖ Identify

π0

π0π0

P = (Pπ)2(−1)L

C = (Cπ0)2 = + 1

C |π+π−, L > = (−1)L |π+π−, L > = + 1

∴ CP = + 1 K → ππ

K01 → π0π0, π+π−

17


CP of K0x → πππ

❖ Decay of spin-0 particle to 3 spin-0 ’s

❖ only possible for

❖

❖ C-parity of so

❖ for ( and by isospin symmetry)

❖ Identify

π0

L12 + L3 = 0 L12 = L3

P = (Pπ)3(−1)L12+L3 = − 1(−1)2L = − 1

π0 = + 1 (Cπ0)3 = + 1

∴ CP = − 1 K0x → π0π0π0 π+π−π0

K02 → π0π0π0, π+π−π0

18


K0S , K0

L

❖ Two states with almost equal mass 499 MeV/c2

❖ ,

❖ ,

❖ Tempting to identify ,

❖ 1964: CP violation observed!

❖ Question: How do we make a sample of ?

K0S → ππ τS ≈ 9 × 10−11 s

K0L → πππ τL ≈ 5 × 10−8 s

K01 = K0

S K02 = K0

L

B(K0L → π+π−) ≈ 10−3 ⟹

K0L

19

Stopped here 14.04.20


Semi-leptonic decaysK0

❖ Study and

❖If CP is conserved then

❖ and

❖ Question: How we know which is which?

❖ So if CP is conserved we expect equal numbers of …

K0L → π−e+νe K0

L → π+e−νe

K0L = K2 =

1

2[ |K0 > + | K0 > ]

K0(ds) → π−e+νe K0(ds) → π+e−νe

π−e+νe and π+e−νe

20


Asymmetry in semi-leptonic decaysK0L

21

Question: What is going on before ~10-10 s?


Key question about CP-violation ( )CP❖ Is in the -system due to mixing or direct ?

❖ Mixing - the and are (non-orthogonal!) mixtures of CP-eigenstates ( ) and ( )

❖

❖ Direct - the and are pure CP-eigenstates and , but the latter decay to forbidden CP-states with a small

probability.

CP K0 CP

K0S K0

L K01

CP = + 1 K02 CP = − 1

|K0S , 0 > =

1

1 + |ϵ |2( |K0

1 , 0 > + ϵ |K02 , 0 > )

|K0L, 0 > =

1

1 + |ϵ |2(ϵ |K0

1 , 0 > + |K02 , 0 > )

K0S K0

L |K0S , 0 > = |K0

1 , 0 >|K0

L, 0 > = |K02 , 0 >

22


Asymmetry in semi-leptonic decays IIK0L

❖Let’s calculate in the mixing scenario

❖

❖

❖

❖

❖ Also

❖

A =N+ − N−

N+ + N−

N+ ∝ | < K0 |K0L > |2 , N− ∝ | < K0 |K0

L > |2

|K0L > ∝ ϵ |K0

1 > + |K02 >

|K01 > ∝ ( |K0 > + | K0 > ), |K0

2 > ∝ ( |K0 > − | K0 > )

|K0L > ∝ (1 + ϵ) |K0 > − (1 − ϵ) | K0 >

|K0S > ∝ (1 + ϵ) |K0 > + (1 − ϵ) | K0 >

N+ ∝ (1 + ϵ)2, N− ∝ (1 − ϵ)2

23


Asymmetry in semi-leptonic decays IIK0L

24

❖

❖ The observation of means that and consist of more matter ( ) than antimatter ( )!!

A =(1 + ϵ*)(1 + ϵ) − (1 − ϵ*)(1 − ϵ)(1 + ϵ*)(1 + ϵ) + (1 − ϵ*)(1 − ϵ)

=1 + |ϵ |2 + ϵ * +ϵ − 1 − |ϵ |2 + ϵ * +ϵ1 + |ϵ |2 + ϵ * +ϵ + 1 + |ϵ |2 − ϵ * −ϵ

=2(ϵ * +ϵ)

2(1 + |ϵ |2 )=

2ℜ(ϵ)1 + |ϵ |2 ≈ 2ℜ(ϵ)

2ℜ(ϵ) ≈ 2.3 × 10−3 K0L K0

SK0 K0


by mixingCP

❖We can define and

❖ If there is only mixing ( ) then

❖ Measured: and

❖ Difference is , consistent with 0, however, ….

❖ If direct ( ) is included (in detailed calculations) then and

❖ Combination of all available results gives and

❖ in the system is dominated by mixing, but there is also direct

η00 ≡< π0π0 |K0

L >< π0π0 |K0

S >η+− ≡

< π+π− |K0L >

< π+π− |K0S >

B(K02 → ππ) = 0 η00 = η+− = ϵ

η00 = (2.220 ± 0.011) × 10−3 η+− = (2.232 ± 0.011) × 10−3

η+− − η00 = (0.012 ± 0.026) × 10−3

CP B(K02 → ππ) ≠ 0 η+− = ϵ + ϵ′

η00 = ϵ − 2ϵ′

|ϵ | = (2.228 ± 0.011) × 10−3 |ϵ′ | = (3.69 ± 0.50) × 10−6

∴ CP K0 CP

25

|K0S , 0 > =

1

1 + |ϵ |2( |K0

1 , 0 > + ϵ |K02 , 0 > )

|K0L, 0 > =

1

1 + |ϵ |2(ϵ |K0

1 , 0 > + |K02 , 0 > )


Flavor oscillations ( )K0 ↔ K0

❖ Let’s neglect for a minute…

❖ The lifetime of the -quark is long enough that we have to consider second-order weak interactions, such as

❖

❖ We can produce a pure state of in a strong interaction

❖ What happens as the propagates?

❖ Mathematics comparable to neutrino-flavor oscillations!

CP

s

K0(ds)

K0

26



❖ The initial strong state ( ) is (approximately, due to ) a mixture of weak eigenstates

❖ If then only the fact that changes the strangeness of the initial state

❖ In general we have

❖ where

❖ We decompose the and into the and components:

❖ , where and

t = 0 CP|K0, 0 > = ( |K0

S , 0 > + |K0L, 0 > )/ 2

m(K0L) ≡ mL = m(K0

s ) ≡ mS τ(K0S) ≪ τ(K0

L)

A(t) = (as(t) |K0S , 0 > + aL(t) |K0

L, 0 > )/ 2

aα(t) ≡ e−imαc2t/ℏe−Γαt/(2ℏ)

|K0S > |K0

L > |K0 > | K0 >

|A(t) > ≡ A0(t) |K0, 0 > + A0(t) | K0, 0 > A0(t) ≡ (aS(t) + aL(t))/2A0(t) ≡ (aS(t) − aL(t))/2

27



❖ (copied) , where and

❖

❖

❖ where

❖ Repeating much of the work, one can show that and

❖ This is not quite true if

|A(t) > ≡ A0(t) |K0, 0 > + A0(t) | K0, 0 >A0(t) ≡ (aS(t) + aL(t))/2 A0(t) ≡ (aS(t) − aL(t))/2

I(K0 → K0) = | < K0 |A(t) > |2 = |A0(t) |2 = . . .= [e−ΓSt/ℏ + e−ΓLt/ℏ + 2e−(ΓS+ΓL)t/ℏ cos(Δmc2t/ℏ)]/4

I(K0 → K0) = | < K0 |A(t) > |2 = | A0(t) |2 = . . .= [e−ΓSt/ℏ + e−ΓLt/ℏ − 2e−(ΓS+ΓL)t/ℏ cos(Δmc2t/ℏ)]/4

Δm ≡ |mS − mL |

I(K0 → K0) = I(K0 → K0)I(K0 → K0) = I(K0 → K0)

CP!

28



❖

❖

❖

❖ A fit to the data allows us to extract the and decay rates (lifetimes) and most importantly

❖ Question: Does everybody remember how to separate and decays?

I(K0 → K0) = [e−ΓSt/ℏ + e−ΓLt/ℏ + 2e−(ΓS+ΓL)t/ℏ cos(Δmc2t/ℏ)]/4

I(K0 → K0) = [e−ΓLt/ℏ + e−ΓSt/ℏ − 2e−(ΓS+ΓL)t/ℏ cos(Δmc2t/ℏ)]/4

αK(t) =I(K0 → K0) + I(K0 → K0) − I(K0 → K0) − I(K0 → K0)I(K0 → K0) + I(K0 → K0) + I(K0 → K0) + I(K0 → K0)

= . . .

=2e−(ΓS+ΓL)t/(2ℏ) cos(Δmc2t/ℏ)

e−ΓSt/ℏ + e−ΓLt/ℏ

K0S K0

LΔm = (3.483 ± 0.006) × 10−12 MeV/c2

K0 K0

29


CPT invariance❖ Relativistic quantum field theory predicts that is conserved, i.e., that

processes that also but conserve

❖ This has actually been tested at CERN (CPLEAR experiment)

❖ A fundamental prediction of CPT-invariance is that masses of particles and anti-particles should be identical, i.e.

❖ One can show that the measured is consistent with to better than 1 in 1018!

❖ are only tested to 1 part in 108-109

❖ Many tests of CPT-invariance - so far no exceptions

CPTCP T CPT

me+ = me−, mp = mp, mK0 = mK0, etc.

ΔmmK0 = mK0

me+ = me− and mp = mp

30

https://link.springer.com/article/10.1007/s100520100793


and -hadron decaysCP b

❖ Some similarity to , but direct- is much larger

❖ in mixing, direct, and interference

❖ Observable in charged -decays

❖ Question: Why isn’t in mixing not seen with charged- decays?

K0 − K0 CP

CP

B

CPB

31


Direct in -decaysCP B❖ CP conserved:

❖CP-violation:

❖ e.g.

❖ Example in charged sector:

❖ Question: What are we comparing in this ?

Γ(A → f ) = Γ(A → f )

ACP ≡Γ(A → f ) − Γ(A → f )Γ(A → f ) + Γ(A → f )

≠ 0

ACP(B0 → K+π−) = − 0.082 ± 0.006

ACP(B+ → K+η) = − 0.37 ± 0.08

ACP

32


oscillations in a nutshellB0 − B0

❖ but (where heavy, light)

❖

❖ occurs when

❖ We still get flavor oscillations just like and can determine

❖ By the way, oscillations are now also observed!

τ(K0S) ≪ τ(K0

L) τ(B0H) ≈ τ(B0

L) H, L =

|B0L > = [ |B0 > + ξ | B0 > ]/ 2

|B0H > = [ |B0 > − ξ | B0 > ]/ 2

ξ ≈ e−2iβ

CP |ξ | ≠ 1

K0 − K0

mH − mL = (3.337 ± 0.033) × 10−10 MeV/c2

D0(cu) ↔ D0(cu)

33

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.241801


in interference in a nutshellCP❖ If then the amplitude has two terms

❖ Since the two amplitudes can interfere and give us in interference

❖ This is observed in experiments that can measure time-dependent

asymmetry

❖ (BaBar-PEP II and Belle-Tristian II), (Tevatron), and (LHC)

❖ Question: How can we tag a or if both can decay to ?

❖ Under the interference hypothesis

Γ(B0 → f ) ≠ 0 and Γ(B0 → f ) ≠ 0ℳ(B0 → B0 → f ) and ℳ(B0 → B0 → f )

I ∝ ℳ2 CP

αfCP(t) ≡I (B0(t) → f) − I (B0(t) → f)I (B0(t) → f) + I (B0(t) → f)

e+e− pp pp

B0 B0 f

αfCP(t) = − ηf sin(2β)sin(Δmc2t/ℏ)

34


Observation of in interferenceCP

❖

❖

❖ Requires , , where is

a CP-eigenstate, and assumes there is no direct here

αfCP(t) = − ηf sin(2β)sin(Δmc2t/ℏ)

sin(2β) = 0.682 ± 0.019

Δm ≠ 0B0 → f and B0 → f f

CP

35

f = J/Ψ + K0S (CP = − 1) f = J/Ψ + K0

L (CP = + 1)

Belle experiment at KEK

?!


and the CKM matrixCP

❖

Recall that

❖ may be complex.

❖ It can be shown (a fun project) that real parameters can be reduced by 9 unitary conditions ( ) and 5 arbitrary quark phases to 3 real angles and a complex

phase, e.g.

❖ P.S. It is less work to show that for 2 generations there is no need of a complex phase.

❖

d′

s′

b′

= VCKM (dsb) ≡

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb(

dsb)

Vij

2 × 3 × 3 = 18V†V = I

d′

s′

b′

=1 0 00 cos β sin β0 −sin β cos β

cos α 0 e−iδ sin α0 1 0

−eiδ sin α 0 cos α

cos θ sin θ 0−sin θ cos θ 0

0 0 1 (dsb)

VCKM =cos θ cos α sin θ cos α sin αe−iδ

−sin θ cos β − cos θ sin α sin βeiδ cos θ cos β − sin θ sin α sin βeiδ cos α sin βsin θ sin β − cos θ sin α cos βeiδ −cos θ sin β − sin θ sin α cos βeiδ cos α cos β

36


and the CKM matrixCP❖ Time-reversal operator (M&S 1.23)

❖ Since not all there will be and by CPT therefore also .

❖ None of the CKM angles are predicted by theory.

❖ However, the prediction of the CKM matrix is that there can be and all such phenomena can be parametrized by a single phase !

❖ So far all experimental data support this…

❖ …apart from the need for a much larger in the early universe.

TΨ( x , t) = Ψ * ( x , − t)

Vij * = Vij TCP

CPδ

CP

37


and Wolfenstein parameterizationCP❖ The Wolfenstein parameterization is an approximation of that highlights

the size of -effects in different parts of the matrix.

❖

❖ We aren’t going to go into the details, but one can show that this is consistent with:

❖ is small in strange and charm sectors and dominated by mixing

❖ in bottom sector is relatively large and direct is important in decays to final states with no charm nor strange

❖ By the way, has now also been observed in charm decays!

VCKMCP

VWP =

1 − λ2/2 − λ4/8 λ Aλ3(ρ − iη)−λ + A2λ5 [1 − 2(ρ + iη)]/2 1 − λ2/2 − λ4(1 + 4A2)/8 Aλ2

Aλ3 [1 − (1 − λ2/2)(ρ + iη)] −Aλ2 + Aλ4 [1 − 2(ρ + iη)]/2 1 − A2λ4/2+ O(λ6)

CP

CP CPB

CP

38

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.211803


and 3 generations of leptonsCP❖ Similar consideration of 3 neutrino

generations allows for in the lepton sector.

CP

39

(sorry about the paywall)Question: How did T2K identify the neutrino-type in their detector?


Lists of concepts

❖ Parity violation

❖ C-violation

❖ CP-conservation

❖ Helicity

❖ Chirality

40

❖ Right-handed

❖ Left-handed

❖

❖ V-A

❖ Polarization

ν

ν

l−L , l−

R

❖ oscillations

❖ Semi-leptonic decays

❖

❖ CP eigenstates

❖ CP violation ( )

❖ by mixing

❖ by interference

K0 − K0

K0

K0S , K0

L

CP

CP

CP

❖ Direct

❖ Flavor oscillations

❖ Flavor tagging

❖ CPT invariance

❖ CKM matrix

❖ Wolfenstein parameterization

❖ in lepton sector

CP

CP

Documents

FYS3500 - spring 2020 Symmetry breaking in€¦ · FYS3500 Spring 2020 Alex Read, U. Oslo, Dept. Physics Introduction Symmetries are extremely important in many branches of physics,