The Nobel Prize in Physics 2008: Broken Symmetrieshome.thep.lu.se/~bijnens/talks/lund08.pdf · Y. Nambu, A ‘Superconductor’ Model of Elementary Particles and its Consequencies,

The Nobel Prize in Physics 2008: Broken SymmetriesJohan Bijnens

Lund University

[email protected]

http://www.thep.lu.se/∼bijnens

Lund 7/11/2008 The Nobel Prize in Physics 2008: Broken Symmetries Johan Bijnens p.1/44

Overview

The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics

for 2008 with one half to

Yoichiro Nambu, Enrico Fermi Institute, University of Chicago, IL, USA

"for the discovery of the mechanism of spontaneous brokensymmetry in subatomic physics"

and the other half jointly to

Makoto Kobayashi, High Energy Accelerator Research Organization (KEK), Tsukuba

Toshihide Maskawa,Yukawa Institute for Theoretical Physics (YITP), Kyoto University,

Japan

"for the discovery of the origin of the broken symmetrywhich predicts the existence of at least three families ofquarks in nature"

PS: Lots of information: http://nobelprize.org


Yoichiro Nambu

Enrico Fermi Institute,University of ChicagoChicago, IL, USAb. 1921 (in Tokyo, Japan)

D.Sc. 1952University of Tokyo, Japan


Makoto Kobayashi

High Energy AcceleratorResearch Organization(KEK)Tsukuba,b. 1944

Ph.D. 1972Nagoya University, Japan


Toshihide Maskawa

Kyoto Sangyo University;

Yukawa Institute for Theo-retical Physics (YITP)Kyoto UniversityKyoto, Japanb. 1940

Ph.D. 1967Nagoya University, Japan


Papers

Y. Nambu, Quasi-Particles and Gauge Invariance in the Theory ofSuperconductivity, Phys. Rev. 117 (1960) 648

Y. Nambu, Axial Vector Current Conservation in Weak Interactions, Phys. Rev.Lett. 4 (1960) 380

Y. Nambu, A ‘Superconductor’ Model of Elementary Particles and itsConsequencies, Talk given at a conference at Purdue (1960), reprinted in BrokenSymmetries, Selected Papers by Y. Nambu, ed:s T. Eguchi and K. Nishijima,World Scientific (1995).

Y. Nambu and G. Jona-Lasinio, A Dynamical Model of Elementary Particles basedon an Analogy with Superconductivity I, Phys. Rev. 122 (1961) 345

Y. Nambu and G. Jona-Lasinio, A Dynamical Model of Elementary Particles basedon an Analogy with Superconductivity II, Phys. Rev. 124 (1961) 246;

Y. Nambu and D. Lurié, Chirality Conservation and Soft Pion Production, Phys.Rev. 125 (1962) 1429

Y. Nambu and E. Shrauner, Soft Pion Emission Induced by Electromagnetic andWeak Interactions, Phys. Rev. 128 (1962) 862.

M. Kobayashi and K. Maskawa, CP Violation in the Renormalizable Theory of WeakInteractions, Progr. Theor. Phys. 49 (1973) 652.


Symmetry

nochangeif rotatedalongverticalaxis

InvariantunderrotationgroupU(1)


Symmetry

changesif rotatedalongvertical axis

But U(1) invariancegood approximation

Symmetry explicitly brokenby the picture


Spontaneously Broken Symmetry

Pencil balanced on end (on a table) has rotationalsymmetry about vertical axis.

Symmetry is broken when pencil falls over: lowerenergy state

Special direction is specified.

But, underlying law of gravity is still symmetrical.

Pencil could have fallen into any available direction


Spontaneously Broken Symmetry

http://nobelprize.org


Symmetry

Symmetries are very well known in physics (and definitelyin particle physics)

Heisenberg, Wigner, Eckart, Yang, Mills, Glashow,Weinberg, Salam,. . .

Nambu: Introduced the concept of spontaneouslybroken symmetries into particle physics

Kobayashi and Maskawa: Proposed the mechanism bywhich the combined charge-conjugation–parity (CP)symmetry is broken in the Standard Model


Spontaneous symmetry breaking

Magnets (e.g. in iron):

Symmetry: ≈ ≈

With interactions:

Low energy:

High energy:



a generic state



the ground state



No: A ground state



A ground state: choice of vacuum breaks the symmetry

Spontaneous symmetry breakingLund 7/11/2008 The Nobel Prize in Physics 2008: Broken Symmetries Johan Bijnens p.15/44


A very low energy excitation: spin wave



A very low energy excitation: spin waveA collective excitation: quasiparticle


Nambu and Superconductivity

Superconductivity: conductivity goes to zero below acritical temperature

Discovered by Kamerlingh-Onnes in 1911

Meisnner: expel magnetic fields 1933

Landau-Ginzburg-Abrikosov: Phenomenological Theory(scalar field)

Explained by Bardeen, Cooper and Schrieffer byCooper pair condensation 1957

Bogolyubov 1958, (Bogolyubov transformation)



Superconductivity: conductivity goes to zero below acritical temperature

Discovered by Kamerlingh-Onnes in 1911

Meisnner: expel magnetic fields 1933

Landau-Ginzburg-Abrikosov: Phenomenological Theory(scalar field)

Explained by Bardeen, Cooper and Schrieffer byCooper pair condensation 1957

Bogolyubov 1958, (Bogolyubov transformation)

Note the many other Nobel Prizes here



Condensed Cooper Pairs: Ground State is Charged

What about Gauge Invariance?



Condensed Cooper Pairs: Ground State is Charged

What about Gauge Invariance?

Electromagnetic Gauge Invariance

Wave function:ψ(x) → e−iα(x)ψ(x)

does not change |ψ(x)|2

~A(x) → ~A(x) − ~∇α(x) and V (x) → V (x) − ∂α(x)∂t

do not change ~E(x) and ~B(x)

Minimal coupling: both invariances belong together

Requiring gauge invariance determines the form ofthe interaction



He used the Feynman-Dyson methods forQuantumElectroDynamics (QED)

Reformulated BCS as a generalized Hartree-Fockground state

When deriving Green functions (which lead toamplitudes) you also need to introduce the loopgraphs both in the interaction vertex and in theselfenergiesWhen you do that all Ward identities are satisfiedHence everything is gauge invariant as should be





The photon “mixes” with the available electron/holestatesLongitudinal mode of the photon couples to thecollective excitationsIn effect produces a mass for the photon ⇒Meissner effect





The photon “mixes” with the available electron/holestatesLongitudinal mode of the photon couples to thecollective excitationsIn effect produces a mass for the photon

Brout-Englert, Higgs, Guralnik-Hagen-Kibble:generalize to Yang-MillsWeinberg, Salam apply to Glashow’s neutral currentmodel: the Standard Model


Nambu: SSB in the Strong Interaction

First an aside: The Weak Interaction and Parity

Parity (P ): ~x → −~x

related to mirror symmetry

Lee and Yang 1956: No experiment for P in weakinteraction and can solve τθ puzzle

Suggested 60Co and µ-decay: immediately seenLund 7/11/2008 The Nobel Prize in Physics 2008: Broken Symmetries Johan Bijnens p.20/44

Nambu: SSB in the Strong InteractionFirst an aside: The Electromagnetic and Weak Interaction

Electromagnetic coupling to electron:AµJ

µ = Aµeγµe (e ≡ ψe)

Current is conserved: ∂Jµ

∂xµ = 0 ⇒Matrix element of Jµ at q2 = 0 is ≡ 1.

Weak interaction is (leptonic µ decay)GF√

2J(µ)ρJ(e)ρ = GF√

2µγρ (1 − γ5) νµ νeγ

ρ (1 − γ5) e

Feynman-Gell-Mann, Marshak-Sudarshan: Vρ −Aρ

Vρ and Aρ conserved (CVC and CAC) for hadrons ⇒≡ 1 at q2 = 0.


Nambu: SSB in the Strong InteractionFirst an aside: The Electromagnetic and Weak Interaction

Electromagnetic coupling to electron:AµJ

µ = Aµeγµe (e ≡ ψe)

Current is conserved: ∂Jµ

∂xµ = 0 ⇒Matrix element of Jµ at q2 = 0 is ≡ 1.

Weak interaction is (leptonic µ decay)GF√

2J(µ)ρJ(e)ρ = GF√

2µγρ (1 − γ5) νµ νeγ

ρ (1 − γ5) e

Feynman-Gell-Mann, Marshak-Sudarshan: Vρ −Aρ

Vρ and Aρ conserved (CVC and CAC) for hadrons ⇒≡ 1 at q2 = 0.

Neutron decay:GF√

20.97 pγρ (1 − 1.27γ5)n eγ

ρ (1 − γ5) νeLund 7/11/2008 The Nobel Prize in Physics 2008: Broken Symmetries Johan Bijnens p.21/44


gA = 1.27 6= 1

Solved simultaneously by Gell-Mann-Levy and Nambu

PCAC: Partially Conserved Axial Current



gA = 1.27 6= 1



A CAC & gA 6= 0: p(

gAF1(q2))

(

γµγ5 − 2MN qµ

q2 γ5

)

n

But has pseudoscalar coupling and massless pole:ruled out



gA = 1.27 6= 1



A CAC & gA 6= 0: p(

gAF1(q2))

(

γµγ5 − 2MN qµ

q2 γ5

)

n

But has pseudoscalar coupling and massless pole:ruled out

p(

gAF1(q2))

(

γµγ5 − 2MN qµ

(q2+m2π)γ5

)

n


Nambu: SSB in the Strong InteractionConsequences:

PCAC:∂Aρ

∂xρ= im2

πFπφπ

Partially: conserved when m2π → 0

Gπ =√

2MN gA

Fπ, the pion coupling to the nucleon is

related to the pion decay constant and gAGolberger-Treiman relation


Nambu: SSB in the Strong InteractionConsequences:

PCAC:∂Aρ

∂xρ= im2

πFπφπ

Partially: conserved when m2π → 0

Gπ =√

2MN gA

Fπ, the pion coupling to the nucleon is

related to the pion decay constant and gAGolberger-Treiman relation

Similar suggestion also for the strangeness changinghyperon decays and the pion to Kaon

Both papers also suggested that there should be anaxial variant of isospin symmetry: exact for mπ → 0

Isospin: SU(2) symmetry from interchanging p and nWigner



Further developments

He realized that if an exact symmetry is spontaneouslybroken there must be a massless particle

Also suggested by Goldstone (Nambu-GoldstoneBosons)

Proven by Goldstone-Salam-Weinberg


Goldstone Modes

UNBROKEN: V (φ)

Only massive modesaround lowest energystate (=vacuum)

BROKEN: V (φ)

Need to pick a vacuum〈φ〉 6= 0: Breaks symmetryNo parity doubletsMassless mode along ridge


Goldstone Modes

Also here:low energy excita-tions


Goldstone Modes

Also here:low energy excita-tions

Overall direction isa symmetryθ(x) → θ(x) + Θ NoChangeInteractions mustvanish if θ(x) isconstantLow energy: nointeractions



Soft Pions and Current Algebra:

∂Aρ

∂xρ= im2

πFπφπ

This means that an axial symmetry rotation connectsprocesses with different numbes of pions



Soft Pions and Current Algebra:

∂Aρ

∂xρ= im2

πFπφπ

This means that an axial symmetry rotation connectsprocesses with different numbes of pions

This together with the small interactions at low energiesallows for a systematic expansion for processes onlyinvolving the low-energy excitations

Chiral Perturbation Theory is the modern way to usethis


How does this fit in with QCD

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: U(3)V

But LQCD =∑

q=u,d,s

[iq̄LD/ qL + iq̄RD/ qR − mq (q̄RqL + q̄LqR)]

So if mq = 0 then U(3)L × U(3)R.

uL

dL

sL

→ UL

uL

dL

sL

and

uR

dR

sR

→ UR

uR

dR

sR

Can also see that left right independent viav < c, mq 6= 0 =⇒

v = c, mq = 0 =⇒/


How does this fit in with QCD

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: U(3)V

But LQCD =∑

q=u,d,s

[iq̄LD/ qL + iq̄RD/ qR − mq (q̄RqL + q̄LqR)]

So if mq = 0 then U(3)L × U(3)R.

Hadrons do not come in parity doublets: symmetrymust be broken

A few very light hadrons: π0π+π− and also K, η

Both can be understood from spontaneous ChiralSymmetry Breaking

Anomaly: really SU(3)L × SU(3)RLund 7/11/2008 The Nobel Prize in Physics 2008: Broken Symmetries Johan Bijnens p.28/44

Kobayashi and MaskawaLet’s extend now P andGF√

20.97 pγρ (1 − 1.27γ5)n eγ

ρ (1 − γ5) νe



20.97 pγρ (1 − 1.27γ5)n eγ

ρ (1 − γ5) νe

P is broken by the weak interaction

CP is not

C: Charge conjugation, replace everyone by theirantiparticle

CP conservation still allows to solve the τθ puzzle



20.97 pγρ (1 − 1.27γ5)n eγ

ρ (1 − γ5) νe

P is broken by the weak interaction

CP is not

C: Charge conjugation, replace everyone by theirantiparticle

CP conservation still allows to solve the τθ puzzle

Experimentally CP is violated a little bit in KaonsCronin-Fitch 1964

K0(sd) and K0(ds) mix through the weak interaction,eigenstates KL and KS

A tiny but well measured mass difference is aconsequence


The 0.97

Suggestion: the weak current for hadrons really is the sameas for leptons but it mixes ∆S = 1 and ∆S = 0Gell-Mann-Levy 1960, Cabibbo 1963

Jρ = cos θCJ∆S=0ρ + sin θCJ

∆S=1ρ

∆S = 0 current: with p and n,

∆S = 1 current: with p and Λ

And the same for the coupling of pion to kaon

With sin θC ≈ 0.22 this explained the 0.97 and the by 1963

much better measured ∆S = 1 decays of Kaon and Hyper-

ons.Lund 7/11/2008 The Nobel Prize in Physics 2008: Broken Symmetries Johan Bijnens p.30/44

Quarks came

In quark language now weak interaction from W±

exchange and the vertex isW+µuγµ (1 − γ5) (cos θCd+ sin θCs)

A mechanism for getting the KL-KS mass difference is

K0 u

W

Wu K0

Result is way too large


Glashow-Iliopoulos-Maiani 1970

Add another quark

W+µuγµ (1 − γ5) (cos θCd+ sin θCs)+

W+µcγµ (1 − γ5) (− sin θCd+ cos θCs)

The mechanism is now

K0 u, c

W

Wu, c K0

Result is zero if mu = mc; the vertical lines havesin θC cos θC − cos θC sin θC (GIM mechanism)

Fits for mc ≈ 1.5GeV/c2 (Gaillard-Lee)

Charm discovered 1974 Richter-Ting


θC in the Standard Model

The standard model: interactions with W µ and the masses:

i, j, k, l = 1, 2, later 1, 2, 3

u1 = u, u2 = c, u3 = t, d1 = d, d2 = s, d3 = b

W Interactions: W+µ∑

i uWLiγµd

WLi + h.c.

Mass terms (v is the Higgs vacuum expectation value)

−v∑

ij dW

LiλDijd

WRj − v

∑

ij uWLiλ

Uiju

WRj+h.c.



−v∑

ij dW

LiλDijd

WRj − v

∑

ij uWLiλ

Uiju

WRj+h.c.

vλU and vΛD are complex 2 by 2 (3 by 3) matrices

Diagonalization is always possible by two complexmatrices each

V UL vλ

UVU†R = diag(mu,mc,mt)

V DL vλ

DVD†R = diag(md,ms,mb)



−v∑

ij dW

LiλDijd

WRj − v

∑

ij uWLiλ

Uiju

WRj+h.c.

vλU and vΛD are complex 2 by 2 (3 by 3) matrices

Diagonalization is always possible by two complexmatrices each

V UL vλ

UVU†R = diag(mu,mc,mt)

V DL vλ

DVD†R = diag(md,ms,mb)

Define: uWLi =∑

j VU†LijuLj and uWRi =

∑

j VU†RijuRj ,

dWLi =∑

j VD†Lij dLj , d

WRi =

∑

j VD†RijdRj

mass eigenstates

and substitute



−v∑

ijkl

dLiVDLijλ

DjkV

D†RkldRk − v

∑

ijkl

uLiVULijλ

UjkV

U†RkluRk

+h.c. = −∑

i

dLimdidRi −

∑

i

uLimuiuRi + h.c.

Phase invariance left:

dLi → eiαidLi, dRi → e−iαidRi

uLi → eiβiuLi, uRi → e−iβiuRi



−v∑

ijkl

dLiVDLijλ

DjkV

D†RkldRk − v

∑

ijkl

uLiVULijλ

UjkV

U†RkluRk

+h.c. = −∑

i

dLimdidRi −

∑

i

uLimuiuRi + h.c.

The W Interactions: W+µ∑

i uWLiγµd

WLi + h.c. become

W+µ∑

ijkl uLiγµVULijV

D†

LjkdLk + h.c.



−v∑

ijkl

dLiVDLijλ

DjkV

D†RkldRk − v

∑

ijkl

uLiVULijλ

UjkV

U†RkluRk

+h.c. = −∑

i

dLimdidRi −

∑

i

uLimuiuRi + h.c.

The W Interactions: W+µ∑

i uWLiγµd

WLi + h.c. become

W+µ∑

ijkl uLiγµVULijV

D†

LjkdLk + h.c.

VCKM = V UL V

D†

L is a unitary matrix

All other terms in the Lagrangian get V UL V

U†L = 1, . . .



The two generation case (VCKM = V )

Flavours only:(

uL cL

)

V

(

dL

sL

)

V is a general 2 by 2 unitary matrix.




Flavours only:(

uL cL

)

V

(

dL

sL

)


Use αi and βi to make V11, V12 and V21 real :(

e−iβ1uL e−iβ2cL

)

(

V11 V12

V21 V22

)(

eiα1dL

eiα2sL

)

=

(

uL cL

)

(

ei(α1−β1)V11 ei(α2−β1)V12

ei(α1−β2)V21 ei(α2−β2)V22

)(

dL

sL

)

α2 − β2 = (α2 − β1) + (α1 − β2) − (α1 − β1)




Flavours only:(

uL cL

)

V

(

dL

sL

)


Use αi and βi to make V11, V12 and V21 real

Unitary implies∑

i V∗ikVil = δil

V =

(

cos θC sin θC

− sin θC cos θC

)

This gives exactly what we had before


Kobayashi and Maskawa 1972

Bold suggestion: add a third generation (remember charmnot discovered)

Flavours only:(

uL cL tL

)

V

dL

sL

bL




Bold suggestion: add a third generation (remember charmnot discovered)

Flavours only:(

uL cL tL

)

V

dL

sL

bL


Use αi and βi to make V11, V12, V13, V21 and V31 real

Unitary implies∑

i V∗ikVil = δil

V =

c1 − s1c3 − s1s3

s1c2 c1c2c3 − s2s3eiδ c1c2s3 + s2c3e

iδ

s1s2 c1s2c3 + s2s3eiδ c1s2s3 − c2c3e

iδ

ci = cos θ1, si = sin θiLund 7/11/2008 The Nobel Prize in Physics 2008: Broken Symmetries Johan Bijnens p.37/44


This extra eiδ makes the Lagrangian intrinsicallycomplex (i.e. not removable by phase redefinitions)

This implies CP violation

can explain CP violation that was seen in kaon decays

via box diagrams K0 u, c, t

W

Wu, c, t K0



This extra eiδ makes the Lagrangian intrinsicallycomplex (i.e. not removable by phase redefinitions)

This implies CP violation

can explain CP violation that was seen in kaon decaysvia box diagrams

What has happened since

Third generation discovered

C. Jarlskog: easy way to see from λU and λD if CPviolation is there

Many more predictions have been seen


The predictions: 1

There is another type of CP violation in Kaon decays:direct CP violation from Penguin diagrams:

K0

W

π+π−, π0π0γ, Z, g

1


The predictions: 1

There is another type of CP violation in Kaon decays:direct CP violation from Penguin diagrams:

K0

W

π+π−, π0π0γ, Z, g

1Seen in 1999 byKTeV at Fermilaband NA48 at CERN(earlier at NA31CERN in 1988 notconfirmed by Fermi-lab experiment)


The origin of penguinsTold by John Ellis:

Mary K. [Gaillard], Dimitri [Nanopoulos], and I first got interestedin what are now called penguin diagrams while we were studyingCP violation in the Standard Model in 1976. The penguin namecame in 1977, as follows.In the spring of 1977, Mike Chanowitz, Mary K. and I wrote apaper on GUTs [Grand Unified Theories] predicting the b quarkmass before it was found. When it was found a few weeks later,Mary K., Dimitri, Serge Rudaz and I immediately started workingon its phenomenology.

from symmetrymagazine

That summer, there was a student at CERN, Melissa Franklin, who is now an experimentalistat Harvard. One evening, she, I, and Serge went to a pub, and she and I started a game ofdarts. We made a bet that if I lost I had to put the word penguin into my next paper. Sheactually left the darts game before the end, and was replaced by Serge, who beat me.Nevertheless, I felt obligated to carry out the conditions of the bet.For some time, it was not clear to me how to get the word into this b quark paper that wewere writing at the time. Later, I had a sudden flash that the famous diagrams look likepenguins. So we put the name into our paper, and the rest, as they say, is history.John Ellis in Mikhail Shifman, ITEP Lectures in Particle Physics and Field Theory,hep-ph/9510397


More predictions

In B meson decays you can have all three generationsat tree level in a process. CP violations can (and are)much larger

B0

b

d

c

cs

J/ψ = cc

KS

W

Observed at the predicted level in many processes


More predictions

In B meson decays you can have all three generationsat tree level in a process. CP violations can (and are)much larger

B0

b

d

c

cs

J/ψ = cc

KS

W

Observed at the predicted level in many processes

Penguins also contribute and again in many places

has led to great confidence in CKM picture both for theangles and the CP violation part


Results

Parametrization of VCKM :

Vud Vus Vub

Vcd Vcs VcbVtd Vts Vtb

The Unitarity triangle of three complex numbers

VudV∗ub + VcdV

∗cb + VtdV

∗tb = 0 (more exist)


Results

Parametrization of VCKM :

Vud Vus Vub

Vcd Vcs VcbVtd Vts Vtb

The Unitarity triangle of three complex numbers

VudV∗ub + VcdV

∗cb + VtdV

∗tb = 0 (more exist)

|Vus| ≈ 0.2; |Vcd| ≈ 0.04 and |Vub| ∼ 0.003

Approximate Wolfenstein parametrization:

1 − λ2

2 λ Aλ3(ρ− iη)

−λ 1 − λ2

2 Aλ2

Aλ3(1 − ρ− iη) −Aλ2 1

All three side are of order λ3


Results

γ

γ

α

α

dm∆

Kε

Kε

sm∆ & dm∆

ubV

βsin 2

(excl. at CL > 0.95) < 0βsol. w/ cos 2

excluded at CL > 0.95

α

βγ

ρ-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

η

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5excluded area has CL > 0.95

ICHEP 08

CKMf i t t e r

Each num-ber is itselfan averageof severalmeasure-ments


Conclusion

I hope I have given you a feeling for what these people have

accomplished and what the physics behind is


Documents

The Nobel Prize in Physics 2008: Broken Symmetrieshome.thep.lu.se/~bijnens/talks/lund08.pdf · Y. Nambu, A ‘Superconductor’ Model of Elementary Particles and its Consequencies,