Spontaneous symmetry breaking in particle physics: a case of cross fertilization. Giovanni Jona-Lasinio

Transcription

1 Spontaneous symmetry breaking in particle physics: a case of cross fertilization Giovanni Jona-Lasinio QUARK MATTER ITALIA, aprile / 38

2 Spontaneous (dynamical) symmetry breaking Figure: Elastic rod compressed by a force of increasing strength 2 / 38

3 Other examples physical system ferromagnets crystals superconductors superfluid 4 He broken symmetry rotational invariance translational invariance local gauge invariance global gauge invariance When spontaneous symmetry breaking takes place, the ground state of the system is degenerate 3 / 38

4 Quasi-particles in superconductivity Electrons near the Fermi surface are described by the following equation Eψ p,+ = ɛ p ψ p,+ + φψ p, Eψ p, = ɛ pψ p, + φψ p,+ with eigenvalues E = ± ɛ 2 p + φ 2 Here, ψ p,+ and ψ p, are the wavefunctions for an electron and a hole of momentum p and spin + 4 / 38

5 Analogy with the Dirac equation In the Weyl representation, the Dirac equations reads Eψ 1 = σ pψ 1 + mψ 2 Eψ 2 = σ pψ 2 + mψ 1 with eigenvalues E = ± p 2 + m 2 Here, ψ 1 and ψ 2 are the eigenstates of the chirality operator γ 5 5 / 38

6 Nambu-Goldstone boson in superconductivity Y. Nambu, Phys. Rev. 117, 648 (1960) Approximate expressions for the charge density and the current associated to a quasi-particle in a BCS superconductor ρ(x, t) ρ α 2 tf j(x, t) j 0 f where ρ 0 = eψ σ 3 ZΨ and j 0 = eψ (p/m)y Ψ with Y, Z and α constants and f satisfies the wave equation ( 2 1α 2 t 2 ) f 2eΨ σ 2 φψ Here, Ψ = (ψ 1, ψ 2) 6 / 38

7 Plasmons The Fourier transform of the wave equation for f gives f 1 q 2 0 α2 q 2 The pole at q 2 0 = α2 q 2 describes the excitation spectrum of the Nambu-Goldstone boson. A better approximation reveals that, due to the Coulomb force, this spectrum is shifted to the plasma frequency e 2 n, where n is the number of electrons per unit volume. In this way the field f acquires a mass. 7 / 38

8 The Goldstone theorem J. Goldstone, Nuovo Cimento 19, 154 (1961) Whenever the original Lagrangian has a continuous symmetry group, which does not leave the ground state invariant, massless bosons appear in the spectrum of the theory. physical system broken symmetry massless bosons ferromagnets rotational invariance spin waves crystals translational invariance phonons 8 / 38

9 The axial vector current Y. Nambu, Phys. Rev. Lett. 4, 380 (1960) Electromagnetic current ψγ µ ψ Axial current ψγ5 γ µ ψ The axial current is the analog of the electromagnetic current in BCS theory. In the hypothesis of exact conservation, the matrix elements of the axial current between nucleon states of four-momentum p and p have the form Γ A µ (p, p) = ( iγ 5 γ µ 2mγ 5 q µ /q 2) F (q 2 ) q = p p Conservation is compatible with a finite nucleon mass m provided there exists a massless pseudoscalar particle, the Nambu-Goldstone boson. 9 / 38

10 In Nature, the axial current is only approximately conserved. Nambu s hypothesis was that the small violation of axial current conservation gives a mass to the N-G boson, which is then identified with the π meson. Under this hypothesis, one can write ( Γµ A (p, p) iγ 5 γ µ 2mγ ) 5q µ q 2 + m 2 F (q 2 ) q = p p π This expression implies a relationship between the pion nucleon coupling constant G π, the pion decay coupling g π and the axial current β-decay constant g A 2mg A 2G π g π This is the Goldberger Treiman relation 10 / 38

11 An encouraging calculation Y. Nambu, G. Jona-Lasinio, Phys. Rev. 124, 246 (1961), Appendix It was experimentally known that the ratio between the axial vector and vector β-decay constants R = g A /g V was slightly greater than 1 and about The following two hypotheses were then natural: 1. under strict axial current conservation there is no renormalization of g A ; 2. the violation of the conservation gives rise to the finite pion mass as well as to the ratio R > 1 so that there is some relation between these quantities. Under these assumptions a perturbative calculation gave a value of R close to the experimental one. More important, the renormalization effect due to a positive pion mass went in the right direction. 11 / 38

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13 The NJL model: an informal presentation 1960 Midwest Conference in Theoretical Physics, Purdue University 13 / 38

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15 The Nambu Jona-Lasinio (NJL) model Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345 (1961) The Lagrangian of the model is L = ψγ µ µ ψ + g [ ( ψψ) 2 ( ψγ 5 ψ) 2] It is invariant under ordinary and γ 5 gauge transformations ψ e iα ψ, ψ ψe iα ψ e iαγ 5 ψ, ψ ψe iαγ 5 15 / 38

16 Mean field approximation x = m = g 0mi 2π 4 d 4 p p 2 m 2 iε F (p, Λ) 16 / 38

17 The spectrum of the NJL model Mass equation 2π 2 ) (1 gλ 2 = 1 m2 Λ 2 ln + Λ2 m 2 where Λ is the invariant cut-off Spectrum of bound states nucleon mass µ spin-parity spectroscopic number notation S 0 0 2m P 0 0 µ 2 > 8 3 m2 1 ±2 µ 2 > 2m P 1 S 0 17 / 38

18 Other examples of BCS type SSB 3 He 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 to the nature of the Higgs field 18 / 38

19 Broken symmetry and the mass of gauge vector mesons P. 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 ϕ = (ϕ 1 + iϕ 2 )/ 2 interacting with an abelian gauge field A µ H int = iea µ ϕ µ ϕ e 2 ϕ ϕa µ A µ If the vacuum expectation value of ϕ is 0, e.g. ϕ = ϕ 1 / 2, the polarization loop Π µν for the field A µ in lowest order perturbation theory is Π µν (q) = (2π) 4 ie 2 ϕ 1 2 [ g µν ( q µ q ν /q 2)] Therefore the A µ field acquires a mass µ 2 = e 2 ϕ 1 2 and gauge invariance is preserved, q µ Π µν = / 38

20 Electroweak unification S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967) Leptons interact only with photons, and with the intermediate bosons that presumably mediate weak interaction. What could be more natural than to unite these spin-one bosons into a multiplet of gauge fields? Standing in the way of this synthesis are the obvious differences in the masses of the photon and intermediate meson, and in their couplings. We might hope to understand these differences by imagining that the symmetries relating the weak and the electromagnetic interactions are exact symmetries of the Lagrangian but are broken by the vacuum. 20 / 38

21 The NJL model as a low-energy effective theory of QCD e.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 scale smaller than some cut-off Λ 1 Gev. The short distance dynamics above Λ is dictated by perturbative QCD and is treated as a small perturbation. Confinement is also treated as a small perturbation. The total Lagrangian is then L QCD L NJL + L KMT + ε (L conf + L OGE ) where the Kobayashi Maskawa t Hooft term L KMT = g D det i,j [ q i(1 γ 5 )q j + h.c.] mimics the axial anomaly and L OGE is the one gluon exchange potential. 21 / 38

22 Analysis of the mean field approximation x = m = g 0mi 2π 4 d 4 p p 2 m 2 iε F (p, Λ) This equation has the obvious solution m = 0 but if 2π 2 g 0 Λ 2 < 1 there is a second non zero solution which lowers the energy of the vacuum. 22 / 38

23 The Bogolubov-Valatin transformation 23 / 38

24 Structure of the vacuum Since the sum in expoment is negative and divergent we have (Ω (0), Ω (m) ) = 0 24 / 38

25 Chiral transformations of the vacuum the sum in the exponent is negative and divergent so that (Ω (m) α, Ω (m) α ) = 0 25 / 38

26 Nucleon-nucleon scattering 26 / 38

27 Due to the mass self-consistency equation J P (0) = 1 27 / 38

28 Summary 28 / 38

29 Chirality conservation and soft pion production Y. Nambu, D. Lurie, Phys. Rev. 125, 1429 (1962). An effective model consisting of a nucleon field ψ of mass m and a massless pseudoscalar field varφ(pion) coupled through 29 / 38

30 Chirality is defined by One verifies that < in χ in >=< out χ out > 30 / 38

31 A small fermion bare mass For the observed value of µ 2 /4m 2 1 1/200 we have m0 1 5Mev. 31 / 38

32 The effective action G. Jona-Lasinio, Nuovo Cimento 34, 1790 (1964) Define the partition function Z[J] = 0 T exp i[ dx(l I + J i Φ i )] 0 where the fields Φ i transform, e.g., according to the fundamental representation of the orthogonal group. Then G[J] = i log Z[J] is the generator of the time ordered vacuum expectation values (in statistical mechanics G is the free energy in the presence of an external field J) δg δj = Φ = φ The effective action is the dual functional Γ[φ] defined by the Legendre transformation δγ δφ = J 32 / 38

33 The vacuum of the theory is defined by the variational principle δγ δφ = 0 Γ[φ] is the generator of the vertex functions and can be constructed by simple diagrammatic rules. Its general form is Γ[φ] = L cl [φ] + Q[φ] 33 / 38

34 Proof of the Goldstone theorem Consider an infinitesimal transformation of the group δφ = t ij φ j. Due to the invariance of the effective action Γ we find which implies 1 ij (q = 0)t jkφ k = 0 det[ 1 ij (q = 0)] = 0 34 / 38

35 The mass hierarchy problem Y. 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 an outstanding feature of the universe in terms of size as well of mass. Elementary particles are no exception. 35 / 38

36 Einstein used to express dissatisfaction with his famous equation of gravity G µν = 8πGT µν His point was that, from an aesthetic point of view, the left hand side of the equation which describes the gravitational field is based on a beautiful geometrical principle, whereas the right hand side, which describes everything else,... looks arbitrary and ugly.... [today] Since gauge fields are based on a beautiful geometrical principle, one may shift them to the left hand side of Einstein s equation. What is left on the right are the matter fields which act as the source for the gauge fields... Can one geometrize the matter fields and shift everything to the left? 36 / 38

37 Hierarchical spontaneous symmetry breaking Y. 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 for fermions, and the Goldstone and Higgs modes as low-lying bosonic states. An interesting feature of the SSB is the possibility of hierarchical SSB or tumbling. Namely an SSB can be a cause for another SSB at lower energy scale.... [examples are] 1. the chain crystal phonon superconductivity.... Its NG mode is the phonon which then induces the Cooper pairing of electrons to cause superconductivity. 2. the chain QCD chiral SSB of quarks and hadrons π and σ mesons nuclei formation and nucleon pairing nuclear π and σ modes nuclear collective modes. 37 / 38

38 Chiral molecules 3 ed ) 0.8 ND 3 a- an 4) t), le a- h ν (cm 1 ) NH 3 0 c / 38