Nambu Slides

8/2/2019 Nambu Slides

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Spontaneous symmetry breaking in particlephysics: a case of cross fertilization

Yoichiro Nambulecture presented by Giovanni Jona-Lasinio

Nobel LectureDecember 8, 2008

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History repeats itself1960 Midwest Conference in Theoretical Physics, Purdue University

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Nambus backgroundY. Nambu, preliminary Notes for the Nobel Lecture

I will begin by a short story about my background. Istudied physics at the University of Tokyo. I wasattracted to particle physics because of the three famousnames, Nishina, Tomonaga and Yukawa, who were the

founders of particle physics in Japan. But these peoplewere at different institutions than mine. On the otherhand, condensed matter physics was pretty good atTokyo. I got into particle physics only when I came backto Tokyo after the war. In hindsight, though, I must say

that my early exposure to condensed matter physics hasbeen quite beneficial to me.

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Spontaneous (dynamical) symmetry breaking

Figure: Elastic rod compressed by a force of increasing strength

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Other examples

physical system broken symmetry

ferromagnets rotational invariancecrystals translational invariance

superconductors local gauge invariancesuperfluid 4He global gauge invariance

When spontaneous symmetry breaking takes place, the ground

state of the system is degenerate

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SuperconductivityAutobiography in Broken Symmetry: Selected Papers of Y. Nambu, World Scientific

One day before publication of the BCS paper, BobSchrieffer, still a student, came to Chicago to give aseminar on the BCS theory in progress. . . . I was verymuch disturbed by the fact that their wave function didnot conserve electron number. It did not make sense.. . . At the same time I was impressed by their boldnessand tried to understand the problem.

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The mechanism of BCS theory of superconductivityY. Nambu, J. Phys. Soc. Japan 76, 111002 (2007)

The BCS theory assumed a condensate of chargedpairs of electrons or holes, hence the medium was not

gauge invariant. There were found intrinsically masslesscollective excitations of pairs (Nambu-Goldstone modes)that restored broken symmetries, and they turned intothe plasmons by mixing with the Coulomb field.

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Quasi-particles in superconductivity

Electrons near the Fermi surface are described by the followingequation

Ep,+ = pp,+ + p,

Ep, =

p

p, + p,+

with eigenvalues

E = 2p +

2

Here, p,+ and p, are the wavefunctions for an electron and a

hole of momentum p and spin +

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Analogy with the Dirac equation

In the Weyl representation, the Dirac equations reads

E1 = ppp1 + m2E2 =

ppp2 + m1

with eigenvalues

E = p2 + m2

Here, 1 and 2 are the eigenstates of the chirality operator 5

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Nambu-Goldstone boson in superconductivityY. Nambu, Phys. Rev. 117, 648 (1960)

Approximate expressions for the charge density and the currentassociated to a quasi-particle in a BCS superconductor

(x, t) 0 + 12

tf

jjj(x, t) jjj0 fwhere 0 = e

3Z and jjj0 = e(ppp/m)Y with Y, Z and

constants and f satisfies the wave equation

2 12t2f 2e2Here, = (

1, 2)

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Plasmons

The Fourier transform of the wave equation for f gives

f 1q20 2q2

The pole at q20

= 2q2 describes the excitation spectrum of theNambu-Goldstone boson.

A better approximation reveals that, due to the Coulomb force,this spectrum is shifted to the plasma frequency e2n, where n is

the number of electrons per unit volume. In this way the field facquires a mass.

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The axial vector currentY. Nambu, Phys. Rev. Lett. 4, 380 (1960)

Electromagnetic current Axial current

5

The axial current is the analog of the electromagnetic current in

BCS theory. In the hypothesis of exact conservation, the matrixelements of the axial current between nucleon states offour-momentum p and p have the form

A

(p, p) = i5 2m5q/q2F(q2) q= p p

Conservation is compatible with a finite nucleon mass m providedthere exists a massless pseudoscalar particle, the Nambu-Goldstoneboson.

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In Nature, the axial current is only approximately conserved.Nambus hypothesis was that the small violation of axial current

conservation gives a mass to the N-G boson, which is thenidentified with the meson. Under this hypothesis, one can write

A

(p, p) i5 2m5q

q2 + m2

F(q2) q= p p

This expression implies a relationship between the pion nucleoncoupling constant G, the pion decay coupling g and the axialcurrent -decay constant gA

2mgA 2GgThis is the GoldbergerTreiman relation

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An encouraging calculationY. Nambu, G. Jona-Lasinio, Phys. Rev. 124, 246 (1961), Appendix

It was experimentally known that the ratio between the axialvector and vector -decay constants R = gA/gV was slightlygreater than 1 and about 1.25. The following two hypotheses werethen natural:

1. under strict axial current conservation there is no

renormalization of gA;

2. the violation of the conservation gives rise to the finite pionmass as well as to the ratio R > 1 so that there is somerelation between these quantities.

Under these assumptions a perturbative calculation gave a value ofR close to the experimental one. More important, therenormalization effect due to a positive pion mass went in the rightdirection.

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The NambuJona-Lasinio (NJL) modelY. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961)

The Lagrangian of the model is

L = + g()2 (5)2

It is invariant under ordinary and 5 gauge transformations

e i, ei e i5, e i5

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Th f h NJL d l


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The spectrum of the NJL model

Mass equation

22

g2= 1 m

2

2ln

1 +

2

m2

where is the invariant cut-off

Spectrum of bound states

nucleon mass spin-parity spectroscopicnumber notation

0 0 0 1S00 2m 0+ 3P00 2 > 8

3m2 1 3P1

2 2 > 2m2 0+ 1S0

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O h l f BCS SSB
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Other examples of BCS type SSB

3He superfluidity

Nuleon pairing in nuclei

Fermion mass generation in the electro-weak sector of the

standard model

Nambu calls the last entry

my biased opinion, there being other interpretations as tothe nature of the Higgs field

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B k d h f
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Broken symmetry and the mass of gauge vector mesonsP. W. Anderson, Phys. Rev. 130, 439 (1963)F. Englert, R. Brout, Phys. Rev. Lett. 13, 321 (1964)P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964)

A simple example (Englert, Brout). Consider a complex scalar field = (1 + i2)/

2 interacting with an abelian gauge field A

Hint = ieA

e2AA

If the vacuum expectation value of is = 0, e.g. = 1/

2,the polarization loop for the field A in lowest orderperturbation theory is

(q) = (2)4ie212 g qq/q2Therefore the A field acquires a mass

2 = e212 and gaugeinvariance is preserved, q = 0.

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N b t
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Nambus commentY. Nambu, preliminary Notes for the Nobel Lecture

In hindsight I regret that I should have explored inmore detail the general mechanism of mass generation for

the gauge field. But I thought the plasma and theMeissner effect had already established it. I also shouldhave paid more attention to the Ginzburg-Landau theorywhich was a forerunner of the present Higgs description.

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El t k ifi ti
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Electroweak unificationS. Weinberg, Phys. Rev. Lett. 19, 1264 (1967)

Leptons interact only with photons, and with theintermediate bosons that presumably mediate weakinteraction. What could be more natural than to unitethese spin-one bosons into a multiplet of gauge fields?

Standing in the way of this synthesis are the obviousdifferences in the masses of the photon and intermediatemeson, and in their couplings. We might hope tounderstand these differences by imagining that thesymmetries relating the weak and the electromagnetic

interactions are exact symmetries of the Lagrangian butare broken by the vacuum.

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The NJL model as a lo energ effecti e theor of QCD


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The NJL model as a low-energy effective theory of QCDe.g. T. Hatsuda, T. Kunihiro, Phys. Rep. 247, 221 (1994)

The NJL model has been reinterpreted in terms of quark variables.

One is interested in the low energy degrees of freedom on a scalesmaller than some cut-off 1 Gev. The short distance dynamicsabove is dictated by perturbative QCD and is treated as a smallperturbation. Confinement is also treated as a small perturbation.The total Lagrangian is then

LQCD LNJL + LKMT + (Lconf+ LOGE)

where the KobayashiMaskawat Hooft term

LKMT = gD deti,j

[qi(1 5)qj + h.c.]

mimics the axial anomaly and LOGE is the one gluon exchangepotential.

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The mass hierarchy problem
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The mass hierarchy problemY. Nambu, Masses as a problem and as a clue, May 2004

Unlike the internal quantum numbers like charge and spin,mass is not quantized in regular manner

Mass receives contributions from interactions. In other words,it is dynamical.

The masses form hierarchies. Hierarchical structure is anoutstanding feature of the universe in terms of size as well ofmass. Elementary particles are no exception.

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Einstein used to express dissatisfaction with hisfamous equation of gravity

G = 8GT

His point was that, from an aesthetic point of view, theleft hand side of the equation which describes thegravitational field is based on a beautiful geometrical

principle, whereas the right hand side, which describeseverything else, . . . looks arbitrary and ugly.

. . . [today] Since gauge fields are based on a beautifulgeometrical principle, one may shift them to the left handside of Einsteins equation. What is left on the right are

the matter fields which act as the source for the gaugefields . . . Can one geometrize the matter fields and shifteverything to the left?

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Hierarchical spontaneous symmetry breaking


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Hierarchical spontaneous symmetry breakingY. Nambu, Masses as a problem and as a clue, May 2004

The BCS mechanism is most relevant to the mass

problem because introduces an energy (mass) gap forfermions, and the Goldstone and Higgs modes aslow-lying bosonic states. An interesting feature of theSSB is the possibility of hierarchical SSB or tumbling.Namely an SSB can be a cause for another SSB at lowerenergy scale.

. . . [examples are]1. the chain crystalphononsuperconductivity. . . . Its

NG mode is the phonon which then induces the Cooper

pairing of electrons to cause superconductivity.2. the chain QCDchiral SSB of quarks and

hadrons and mesonsnuclei formation and nucleonpairingnuclear and modesnuclear collective modes.

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Update of the NJL model


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Update of the NJL model

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