Cooper L.N., Feldman D. (Eds.) BCS: 50 Years

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Energy Gap, Mass Gap, and Spontaneous Symmetry Breaking 531

is formed, and charge conservation is broken. The two spinors of opposite

chirality get mixed by a mass term which thus breaks chirality. If this is an

SSB, the corresponding NG boson should be a pseudoscalar. I decided to

pursue this analogy since I had been always interested in the mass spectrum

of the various particles. (By the way, another analog of BCS would be the

Majorana mass, which is an issue for neutrinos.) A natural place to start

was the nucleon because there was indeed a pseudoscalar boson, i.e. the pion

with isospin one, coupled to the isodoublet proton and neutron. The isospin

symmetry was not exact, and the pion was not massless, but its mass was

small compared to the nucleon masses that were to be dynamically generated

by some interaction. The role of the pion in this connection was to restore

the conservation of the chirality after it was broken by the nucleon mass M.

The isovector current would take the form

5µ

= ψγ

τ ψ −

∂

π ,

where g is the π–N coupling constant. This is the analog of the London

current above. It is conserved, ∂

= 0 if the pion mass is put to zero. The

second term gave the correct value when it was used as the axial current for

the π− > µν decay rate. With the coeﬃcient 2M/g replaced by an empirical

number f ∼ 90 Mev, it was known as partial conservation of axial current

(PCAC). The particular relation between f and 2M/g had been noticed by

M. Goldberger and S. Treiman.

Thus the SSB idea provided theoretical

support for it.

I have to mention here some uneasiness that was felt. The mass of the

real pion is not zero although it is small compared to that of the nucleon.

Unlike the case of a gauge ﬁeld, there is no natural mechanism to shield the

pion ﬁeld, so to speak, and make it massive. How can I justify my claim

when I am following rigorous logic? Here I recalled an edict by S. Sakata.

He said that truth is not immutable but will evolve. What we think now as

truth is not absolute; it may be modiﬁed later as we learn more. Reassured

by Sakata’s words I decided to go ahead. Indeed we know now that there is

a quark substructure to the hadrons, and that the quarks’ own base masses

are responsible for the pion mass. We also know the so-called chiral anomaly

which intrinsically breaks chiral symmetry.

The basic ideas about SSB and mass generation were ﬁrst presented as a

remark at the 1959 High Energy Conference in Kiev,

then in the Physical

Review Letters.

A somewhat more detailed exposition was given by my

collaborator G. Jona-Lasinio at a regional conference at Purdue University,

followed by full papers with him.

We considered a model consisting of a

October 4, 2010 11:13 World Scientiﬁc Review Volume - 9.75in x 6.5in ch21

532 Y. Nambu

nucleon singlet having quadratic interaction terms (1 · 1, γ

· γ

) (U(1)

U(1)

), and one consisting of an isodoublet with terms (1 · 1, τ γ

· τ γ

)

(SU(2)

× SU(2)

). As a relativistic theory, it is non-renormalizable and

needs a quadratic cut-oﬀ, but I chose this BCS-like model for simplicity for

displaying the basic principles. The SSB actually generate a scalar meson in

addition to the pseudoscalar pion, corresponding to the composites (γ

, 1) ∼

(π, σ) or (τ γ

, 1) ∼ (π, σ). The σ meson is the analog of the τ

mode in the

BCS case, with mass 2M.

When these collective modes were treated as independent eﬀective ﬁelds,

this model turned out to be equivalent to the so-called sigma model of

M. Gell-Mann and M. L´evy,

just as the BCS theory gave a dynamical

basis to the Ginzburg–Landau theory. The diﬀerence from these eﬀective

theories is that the fermion and boson masses are related:

: m

= 0 : 1 : 2

Translating the NJL model to the quark level, this relation led to the propo-

sition that the observed broad enhancement in the π −π channel in hadron

reactions roughly over the region ∼2m

quark

∼ 700 Mev could be interpreted

as this σ, although the issue is not settled yet. The above formula was ac-

tually vindicated in the case of superconductivity when a collective mode at

∼2∆ was found by R. Sooryakumar, M. V. Klein.

I could have predicted

this explicitly, but I ﬁrst learned of their result and the relation to my work

from C. Varma.

The BCS type system characterized by weak short range interaction

yields simple mass relations among fermions and bosons. This is because

the Yukawa coupling g and the self-coupling λ of the σ–π ﬁeld in an eﬀective

Gell-Mann–L´evy Hamiltonian are determined from the original theory and

are related as g

= λ. In the case of

He B phase there are collective modes

with total angular momentum j = 0.1, 2 including two NG modes with j = 0

and 1 corresponding to the SSB of particle number and spin, respectively.

For each j, the mass relations satisfy a generalization of the above form

∆

+ ∆

= 4∆

The subsequent developments in particle physics followed the GMLH line

because of its generality and renormalizability. I do not have time to elab-

orate on them, but the eﬀorts to go back to the BCS level of underlying

dynamics have also been around from time to time. But it is only a vague

hope that we may eventually understand the pattern of the masses of the

fermions even if the existence of the Higgs boson is conﬁrmed.

October 4, 2010 11:13 World Scientiﬁc Review Volume - 9.75in x 6.5in ch21

Energy Gap, Mass Gap, and Spontaneous Symmetry Breaking 533

References

1. C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).

2. J. J. Sakurai, Ann. Phys. 11, 1 (1960).

3. R. P. Feynman and M. Gell-Mann, Phys. Rev. 103, 193 (1958).

4. E. C. G. Sudarshan and R. E. Marshak, Phys. Rev. 109, 1880 (1958).

5. I. Langmuir and L. Tonks, PR 33, 195 (1929).

6. D. Bohm and E. Gross, Phys. Rev. 75, 1851, 1864 (1949); D. Bohm and

D. Pines, Phys. Rev. 82, 625 (1951); 85, 338 (1952); 92, 609 (1953).

7. S. Tomonaga, Prog. Theoret. Phys. 5, 544 (1950).

8. M. R. Schafroth, Phys. 100, 463 (1955).

9. J. Bardeen, L. N. Cooper and J. R. Schrieﬀer, Phys. Rev. 106, 162 (1957).

10. N. N. Bogoliubov, J. Expl. Theoret. Phys. (USSR) 34, 58, 73 (Soviet Phys.

JETP 34, 41, 51) (1958).

11. J. G. Valatin, Nuovo Cimento 7, 843 (1958).

12. Y. Nambu, Phys. Rev. 117, 648 (1960).

13. J. Bardeen, Nuovo Cimento 5, 1765 (1957); P. W. Anderson, Phys. Rev. 110,

827 (1958); 110, 1900 (1958); G. Rickaysen, Phys. Rev. 111, 817 (1958); Phys.

Rev. Lett. 2, 91 (1959); and others.

14. L. P. Gor’kov, Soviet Phys. JETP 9, 1364 (1959); 10, 998 (1960).

15. M. Baker and S. L. Glashow, Phys. Rev. 128, 2462 (1962).

16. W. Heisenberg, H. P. Duerr, S. Schlieder and K. Yamazaki, Zeits. f. Naturf.

14a, 441 (1959).

17. J. Goldstone, Nuovo Cimento 19, 154 (1961); F. Englert and R. Brout, Phys.

Rev. Lett. 13, 321 (1964); P. Higgs, Phys. Rev. Lett. 13, 508 (1964); G. E.

Guralnik, C. R. Hagen and T. W. B. Kibble, Phys. Rev. Lett. 12, 585 (1964);

and others.

18. V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).

19. M. L. Goldberger and S. B. Treiman, Phys. Rev. 110, 1178 (1958).

20. Proc. Int. High En. Conf. in Kiev, 1959, a remark after a talk by B. F. Toushek.

21. Y. Nambu, Phys. Rev. Lett. 4, 380 (1959).

22. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961).

23. M. Gell-Mann and M. L´evy, Nuovo Cimento 16, 705 (1958).

24. R. Sooryakumar and M. V. Klein, Phys. Rev. Lett. 45, 660 (1980).

25. C. A. Balseiro and L. M. Falicov, Phys. Rev. Lett. 45, 662 (1980); P. B. Little-

wood and C. M. Varma, Phys. Rev. B 26, 4883 (1982).

26. Y. Nambu, Physica 15D, 147 (1985).

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October 4, 2010 10:21 World Scientiﬁc Review Volume - 9.75in x 6.5in ch22

BCS AS FOUNDATION AND INSPIRATION: THE

TRANSMUTATION OF SYMMETRY

Frank Wilczek

Center for Theoretical Physics, Department of Physics,

Massachusetts Institute of Technology,

Cambridge, MA 02139, USA

The BCS theory injected two powerful ideas into the collective conscious-

ness of theoretical physics: pairing and spontaneous symmetry breaking.

In the 50 years since the seminal work of Bardeen, Cooper and Schrief-

fer, those ideas have found important use in areas quite remote from the

stem application to metallic superconductivity. This is a brief and eclectic

sketch of some highlights, emphasizing relatively recent developments in

QCD and in the theory of quantum statistics, and including a few thoughts

about future directions. A common theme is the importance of symmetry

transmutation, as opposed to the simple breaking of electromagnetic U(1)

symmetry in classic metallic superconductors.

The Bardeen-Cooper-Schrieﬀer (BCS) theory of superconductivity

has been

fruitful in many ways. Most obviously, of course, it provided profound and

at many points surprising concrete insights into the superconducting state

of solids, right from the start. It predicted, for instance, the very diﬀerent

eﬀect of the onset of superconductivity on acoustic versus electromagnetic

relaxation, due to the diﬀerent signs in coherence factors, which is a delicate

quantum-mechanical eﬀect. And it provided the intellectual foundation for

such wonders as the Josephson eﬀects and Andreev reﬂection.

The inﬂuence of BCS theory on the broader discipline of theoretical

physics has been no less profound. Two key ideas abstracted from BCS

theory, that have been widely transplanted and borne abundant fruit, are

pairing and dynamical symmetry breaking. Pairing was an essentially new

idea, introduced by Cooper and brought to fruition by BCS. The symmetry

breaking aspect was mostly implicit in the original BCS work, and in ear-

lier ideas of Fritz London and Landau–Ginzburg; but the depth and success

535

October 4, 2010 10:21 World Scientiﬁc Review Volume - 9.75in x 6.5in ch22

536 F. Wilczek

of the BCS theory seized the imagination of the theoretical physics com-

munity, and catalyzed an intellectual ferment. The concept of spontaneous

symmetry breaking was promptly made explicit, generalized, and put to use

by several physicists including Anderson, Josephson, Nambu and Goldstone.

The ﬂexibility and transformative power of these ideas revealed itself grad-

ually, in applications to phenomena that at ﬁrst sight appear to have little

or nothing in common with superconductivity.

From a wealth of possible material, I have chosen to discuss some rela-

tively recent developments close to my own work, that I think well illustrate

how naturally the basic BCS concepts combine with other ideas of fundamen-

tal and emergent symmetry, often with dramatic consequences. A common

theme is that symmetry breaking forms a special case of a more general

phenomenon: symmetry transmutation.

1. QCD Meets BCS: Color-ﬂavor Locking, Conﬁnement and

Chiral Symmetry Breaking

Quantum chromodynamics or QCD, having run the gauntlet of many

exquisite quantitative confrontations with observation, is now established

as the fundamental theory of the strong interaction.

QCD is a challenging

theory to understand, however, and not primarily because of its technical

complexity.

The real challenge comes in relating the wonderfully “trivial”

basic equations to observed reality. The primary ingredients of QCD are

massless gluons and nearly massless quarks (u, d, s; the heavy quarks c, b,

t are a separate, and much easier, study). The observed hadrons, of course,

are neither massless like gluons nor fractionally charged like quarks. Many

techniques have been deployed to bridge the chasm separating the theory

world from the physical world, but in my opinion none is clearer nor more

elegant than the straightforward application of BCS ideas to the regime of

high density. (Here by “high density” I intend large baryon number density,

at low temperature).

1.1. QCD meets BCS

A wise principle states “It is more blessed to ask forgiveness than permis-

sion.” In that spirit, we consider the possibility of constructing a description

“Technical complexity” is a time-dependent concept. I’ve heard graduate students

accustomed to a diet of high supersymmetry, Calabi–Yau manifolds and intersecting

D-branes refer to QCD as “trivial”, with no evident ironic intent.

October 4, 2010 10:21 World Scientiﬁc Review Volume - 9.75in x 6.5in ch22

BCS as Foundation and Inspiration: The Transmutation of Symmetry 537

of high-density QCD based on its elementary degrees of freedom, quarks and

gluons.

At ﬁrst sight this approach looks extremely promising. High density

means large fermi surfaces. Neglecting interactions, the low-energy excita-

tions are associated with modes near the fermi surface: a mode just above

the fermi surface, empty in the ground state, becomes occupied, or a mode

just below becomes empty. Since the fermi surface is large, all modes near

the fermi surface carry large momentum and energy. Thus scattering among

the low-energy excitations will either involve only small angles, and leave

the distribution of particles over modes nearly unchanged, or else bring in

large momentum transfers, and therefore weak coupling (asymptotic free-

dom). It appears, therefore, that perturbation theory should be a good

approximation.

But when one actually does the calculations, one ﬁnds infrared diver-

gences. They arise from two sources:

• The preceding argument only concerns the quarks. Its central point is that

Pauli blocking removes the infrared divergences that usually arise through

low-virtuality quarks. Gluons, however, are not subject to any such eﬀect.

Color electric forces are screened by the quark medium, but color magnetic

forces remain long-ranged, and lead to infrared divergences.

• Interacting fermions are subject to the Cooper instability. One has many

near-zero energy excitations at zero momentum, associated with particle–

particle or hole–hole pairs carrying equal and opposite three-momenta ±~p.

Thus in perturbing around the many-body state which is the ground state

of the non-interacting theory, that is the fully occupied fermi sphere, one

is engaging in highly degenerate perturbation theory. As a general mat-

ter, degenerate perturbation theory can result in signiﬁcant restructuring

of the ground state. In this speciﬁc context, Bardeen, Cooper and Schri-

eﬀer (BCS) taught us that even a small attractive interaction will lead

to a drastic re-arrangement of the ground state, by inducing pairing and

superﬂuidity.

In conventional superconductors it is quite subtle to ﬁnd an eﬀective

attractive interaction between electrons. The primary interaction between

electrons is the Coulomb interaction, and it is of course repulsive. To ﬁnd an

attractive interaction one must bring in phonons, retardation, and screening,

and concentrate on modes within a thin shell around the Fermi surface. For

many “unconventional” superconductors, famously including the cuprates,

the mechanism of attraction remains unclear at present, despite much eﬀort

October 4, 2010 10:21 World Scientiﬁc Review Volume - 9.75in x 6.5in ch22

538 F. Wilczek

to identify it. But in all known cases the superconducting transition tem-

perature (which reﬂects the attractive dynamics) is far below the melting

temperature (which reﬂects the primary dynamics).

In QCD the situation is more straightforward, because the primary inter-

action — the QCD analogue of the Coulomb interaction — can already be

attractive. Two separated quarks, each in the triplet 3 representation, can

be brought together in the antisymmetric

3. The disturbance in the gluon

ﬁeld due to color charge is then half what it was before. Since the energy

has decreased, the force is attractive. Nothing like this can happen when

there is just one type of charge, of course. The existence of three diﬀerent

color charges is crucial here. (On the other hand, with larger numbers of

colors antisymmetrization yields relatively less reduction in ﬂux, and so the

attractive force is relatively weaker.)

By zeroing the spin — that is, once again, choosing the antisymmetric

channel — we also remove the sources of magnetic disturbance. Thus, on

very general grounds we expect a powerful attractive interaction between

quarks in the channel where both colors and spins are antisymmetric. This

intuition is borne out by calculations using one-gluon exchange, instanton

models, and direct numerical simulations, though those simulations could

and should be sharpened.

Thus color superconductivity occurs straightforwardly, and should be ro-

bust physically, at high density. What does it mean? Here is a list of physi-

cal eﬀects we can anticipate, by translating intuition from superconductivity

into the language of particle physics:

• Gluons acquire mass — that is a way to state the equations of the Meissner

eﬀect. If all the gluons acquire mass, their exchange will no longer produce

infrared divergences.

• Quarks acquire mass — that is a way to state the equations of the energy

gap. If all the quarks acquire mass, Cooper’s infrared divergence will be

removed.

• Thus we construct a new ground state, around which our weak-coupling

expansion works.

• This ground state does not contain massless gluons nor exhibit long-range

forces. In that sense, it exhibits conﬁnement. We also have the classic

phenomenon of conﬁnement — that is, absence of fractional electric charge

in the spectrum, as I will explain shortly.

• The energy gap for quarks suggests that chiral symmetry, which is asso-

ciated with massless quarks, may be broken.

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BCS as Foundation and Inspiration: The Transmutation of Symmetry 539

In short, we have the prospect of a phase that exhibits the main non-

perturbative features of QCD — conﬁnement and chiral symmetry breaking

— in a transparent, fully controlled theoretical framework. Let me empha-

size that here I am speaking of a phase of QCD itself, not of some idealization

of a model of a caricature of QCD. Now let us discuss how it is realized, more

concretely.

1.2. Color-ﬂavor locking

For concreteness, and because my emphasis here is on QCD rather than as-

trophysics, I will assume as the initial default that all quarks are massless,

that they are subject to a common chemical potential, and that electromag-

netism can be treated as a perturbation. I will circle back to revisit these

assumptions, and ask your forgiveness, in due course.

Because the most attractive channel for quarks is antisymmetric both in

color and spin, Fermi statistics requires another source of antisymmetry. One

possibility is antisymmetry in the spatial wave-function of the quark pairs.

For example, we might have p-wave pairing. But for simple, purely attractive

interaction potentials, s-wave tends to be favored, because it allows pairs

from all directions over the Fermi surface to act in phase. So s-wave pairing,

if possible, is likely to be favorable.

The remaining possible source of antisymmetry is ﬂavor. Thus we must

pair oﬀ diﬀerent ﬂavors of quarks to take best advantage of the attractive

interaction between quarks. This requirement brings in some signiﬁcant

complications. Obviously, it means that the one-ﬂavor case is not represen-

tative, and that we cannot build up the analysis one ﬂavor at a time. The

two-ﬂavor case also does not go smoothly. Antisymmetry in ﬂavor and spin

(and lack of orbital structure) reduces the quark–quark channel to a sin-

gle vector in color space. Therefore condensation in this channel can break

color symmetry only partially, in the pattern SU(3) → SU(2). Some gluons

remain massless, and some quarks remain gapless, so infrared divergences

remain.

1.2.1. Ground state

Simplicity and self-consistency (that is, consistent use of weak coupling) ﬁrst

arrive when we consider three ﬂavors.

I will describe the full structure of the condensate momentarily, but since

that is a little intimidating, let me begin with a sketch. Since the spin

(singlet) and spatial (s-wave) structures are unremarkable I will suppress

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540 F. Wilczek

them, and also chirality. The favored condensate should be antisymmetric

in color and in ﬂavor, which suggests the form

i ∼ 

αβ∗



ab∗

where the Greek indices are for color, the Latin indices are for ﬂavor, and

* is a wildcard. Now by setting the wild cards equal, and contracting, we

maintain as much residual symmetry as possible. Any ﬁxed choices for the

wildcards will break both color and ﬂavor symmetries. But by locking color

to ﬂavor we maintain symmetry under the combined (so-called diagonal)

symmetry group. Thus, we arrive at

i ∼ 

αβ∗



ab∗

→ 

αβi



abi

∝ (δ

− δ

) (1)

This condensate breaks local color times global ﬂavor SU (3) × SU(3) to a

diagonal,“modiﬁed ﬂavor” global SU (3).

It also spontaneously breaks baryon number symmetry. To a particle

physicist encountering these ideas for the ﬁrst time, that might sound dra-

matic — and it is, but not in the sense that it allows the material to decay.

With the sample enclosed in a ﬁnite volume, outside of which the order pa-

rameter vanishes, there is a strict conservation law for the integrated baryon

number. As in the theory of liquid helium 4, where one speaks of a con-

densate of helium atoms, the true implication is that there is easy transport

of baryon number within the sample. More speciﬁcally, there is a massless

Nambu–Goldstone ﬁeld, which supports the supercurrents characteristic of

superﬂuidity.

Now comes the full structure, in all its glory:

h1|(q

)

(

k)(q

)

(−

k)|1i

= 



k|)(δ

− δ

) + v

k|)(δ

+ δ

)



= −(L ↔ R) . (2)

Here some further words of explanation are in order. The mid-Latin

indices i, j are for spin. The “L” and “R” are for left and right chirality.

The relative sign between left and right condensates reﬂects conservation of

parity. The functions v

k|), v

k|) are, for weak coupling, peaked near

the Fermi surface. Our preceding discussion anticipated the v

term, but

the v

term is also allowed by the residual symmetry. That latter term

indeed emerges from calculations based on the microscopic theory, though

with v

 v