N a m b u S y m p o s i u m O s a k a C i t y U n i v . 1 3 / 1 2 / 2 0 1 8 Symmetries and dynamics in Particle Physics: the legacy of Yoichiro Nambu K . K o n i s h i ( U n i v . P i s a / I N F N , P i s a )
Achievements by Y. Nambu Spontaneous symmetry breakdown 🔶 Chiral symmetry breaking and physics of massless pions /current algebra Color degrees of freedom 🔶 wA\ Quark model to Quantum Chromodynamics Strings from dual resonance model 🔶 Nambu-Goto action: birth of string theory Color (quark) confinement 🔶
Work and recollections by colleagues “Yoichiro Nambu: remembering an unusual physicist, a mentor, and a friend" 🔶 Giovanni Jona-Lasinio, Prog. Theor. Exp. Phys. 2016 , 07B102 “BCS, Nambu-Jona-Lasinio, and Han-Nambu - A sketch of Nambu’s works 🔶 in 1960-1965 ” Kazuo Fujikawa, arXiv:1602.08193 “Nambu at Work" 🔶 Peter G. Freund, arXiv:1511.06955 “Professor Nambu, String Theory and Moonshine Phenomenon" 🔶 Tohru Eguchi, arXiv:1608.06036 “Yoichiro Nambu" 🔶 Holger B Nielsen, Int. J. Mod. Pays. A (2016) “Birth of String Theory" 🔶 Hiroshi Itoyama arXiv:1604.03701
“Chiral symmetries and current algebra” Erice Summer School, 1972 Ettore Majorana Nambu’s lecture at the “Highlights in Particle Physics” Spontaneous symmetry breaking: non gauge theories 🔶 Nambu-Goldstone modes (pions), PCAC, soft-pion theorems Spontaneous symmetry breaking: gauge theories 🔶 wA\ Need of anomaly Englert-Brout-Higgs mechanism; Weinberg-Salam electroweak theory cancellation Bilinear quark algebra 🔶 A remark: 🔶 “ … I am convinced that the strong interactions are also 2 7 described by some sort of color gauge theory. ” 9 1 r e m m u S
In retrospect … J.C. Taylor Nambu was fully aware that the analogy with 🔶 superconductivity (with pion physics) was not perfect; Nambu played with the idea of giving mass (by the Higgs mechanism) 🔶 ρ ± ∗ , K ∗ . Freund to massive vectors, as P cfr. Weinberg-Salam Nambu was convinced that the strong interactions were described 🔶 by a gauge theory of color cfr. QCD of Fritzch-Gell-Mann-Leutwyler 3 more Nobel prizes within reach …
① ② ③ ④ ⑤ Symmetries and dynamics : Nambu’s legacy Color and QCD: confinement and XSB Magnetic monopole condensation: Puzzles and solutions Exact solutions in N=2 supersymmetric gauge theories NonAbelian vortices and monopoles topological solitons vs gauge dynamics Generalized symmetries, mixed ’t Hooft anomalies QCD; Chiral gauge theories
① Let’s start where Nambu has left out… Spontaneous chiral symmetry breaking XSB: (&) Y. Nambu and G. Jona-Lasinio, Phys. Rev. 1961 “The basic principle underlying the model is the idea that field theory may admit, as a result of dynamical instability, extraordinary (nontrivial) solutions that have less symmetries than are built into the Lagrangian. “ Massless Nambu-Goldstone bosons (pions) of broken SU A ( N f ) Color confinement ? (vs XSB) (&) Want a deeper understanding of
② Confinement = a dual superconductor ? Nambu, Mandelstam, ’t Hooft, ~‘80 SU (3) → U (1) 2 → 1 ’t Hooft-Polyakov monopoles (M) ! • Dual Abelian gauge theory h M i 6 = 0 Monopole condensation • q ¯ q (cfr. Cooper pair condensation) • Dual Meissner effect (color chromoelectric fields expelled) • Chromoelectric vortex Linearly rising q - q* potential !
D ¯ ψ � L Puzzles s × k ) = N f ) ( 2 ) • U 1 A No evidence from lattice GT S ( U ‘80~’18 ( π 1 = Confinement vs XSB ) 2 ) 1 ( U • / Doubling of the meson spectrum ) (*) 3 ( U S ( π 2 • XSB If Accidental SU(N F2 ) : too many NG bosons (**) Monopoles weak coupled • No dynamical mechanism for h M i 6 = 0 (***) Solution: • Non-Abelian monopole condensation Z = ) ) 1 ( U SU (3) → SU (2) × U (1) → 1 × ) 2 ( U S ( Π 1 (*), (**), (***) OK? no-go theorem ? Price: Non-Abelian monopoles strongly coupled ?
NonAbelian magnetic monopoles, gauge dynamics and confinement: theoretical developments ’94 - ‘18 P isa, Titech, Seiberg-Witten • Exact solutions of N=2 Minnesota, • NonAbelian vortex Cambridge, Keio supersymmetric gauge theories • NonAbelian monopoles • Exact quantum mech. monopoles • V ortices in A r g y G r e a s i , o S High density QCD / cold atoms t … e t o i b • N=2 SCFT’s and dualities , e T … r a g c , h i k a w • CP N-1 in finite width strip a • RG and IRFP and confinement
③ Seiberg-Witten solutions in N=2 supersymmetric theories ’94 - SW curves: all pert/nonpertive effects encoded 🔶 Exact low-energy dual Abelian or nonAbelian L eff 🔶 n a i l e b A n o n d n a n a i l e b Analytic demonstration of confinement A ; r K e O n 🔶 s s t i n e e M m l e a n u fi d n o D c C f o Q e r o u t t c r i a p l i m Confinement vs XSB i s T O 🔶 N t u B Argyres-Douglas vacua 🔶 ’ ０６ ’09 Argyres-Seiberg S duality, Gaiotto and SCFT : t 🔶 l u c fi f i d t u b g n i t s e ? r D e t C n i Q t s o Confining vacuum near strongly coupled SCFT o t M y 🔶 g o l a n a e m o s
Confinement and RG flow red curves= deformations by some relevant operators
RG flows Bolognesi, Giacomelli, KK ‘16 Real-world QCD N =2 SCFT N =0 SCFT SU(N), N F =2N-1 a UV = 7 N 2 − N − 5 a UV = 11 N f N c + 31 180( N 2 c − 1) 360 � 24 20 N f N c + N 2 c UV = 4 N 2 − N − 2 c UV = 1 c � 1 ; 10 12 N f < 11 2 N c a N =2 SCF T = 7 N ( N − 1) 24 c N =2 SCF T = N ( N − 1) 3 − a IR = (2 N − 1) 2 − 1 a IR = N 2 f − 1 48 . in adjoint repr of G F 360 c IR = (2 N − 1) 2 − 1 c IR = N 2 f � 1 ; 24 120
③ Summary of deep insights into quantum behaviors of monopoles ♦ in the infrared and their role in confinement/XSB ♦ not quite a good model for QCD, ♦ except for the confinement vacua “near” strongly coupled CFT Abelian and nonAbelian monopoles = topological solitons ? ♦
④ Auzzi et. al, Hanany-Tong ’03- Shifman-Yung, Nitta-Ohashi-Sakai … NonAbelian vortices and monopoles NonAbelian vortex = 🔶 ANO vortex embedded in CF locked vacuum Internal orientational zeromodes ~ 🔶 Fluctuations: CP N-1 sigma model in 2D vortex world sheet Non-Abelian monopoles are the endpoints of the 🔶 nonAbelian vortices non-Abelian monopoles Notorious problems avoided 🔶 Bolognesi, Gudnason, Ohashi, KK ’16-‘18 Quantum physics of CP N-1 sigma model on finite-width 🔶 vortex world sheet ; Unbroken isometry SU(N) cfr. Gorsky, Pikalov, Standard CP N-1 model at L Vainshtein’ nov 18
N onAbelian monopoles and duality v 1 v 2 SU ( N + 1) color ⊗ SU ( N ) flavor → ( SU ( N ) × U (1)) color ⊗ SU ( N ) flavor → SU ( N ) C + F − − 3 0 ’ g n u Y , 3 , 0 K ’ K , g n n , o l i T s v h , E a n i , s n e a n H g o l 4 o 0 B ’ , 3 . , z i 0 K z ’ . u , K A g n n , u l i Y s v n , E a m i , s e i f h n g S o Monopole Vortex l o B , z i z u A Vortices with fluctuating orientational CP N-1 modes • The monopole ~ N of a new (dual) SU(N) --- isometry group of CP N-1 = Origin of the dual SU(N) gauge group 1 1 0 2 i , n l i e h c M i K . K . n , o s a n d u G i , n o g r i o D , n i a 4 r i 1 p 0 i 2 C K . . K , e e r j e t a t h C
Summary of ③ , ④ Quantum behavior of Abelian and nonAbelian monopoles 🔶 in the IR fairly well (in N=2, N=1, susy gauge theories) understood; The topological soliton monopoles and vortices 🔶 (many applications): origin of the nonAbelian monopoles Understanding of confinement /chiral symmetry breaking 🔶 in the real-world (non supersymmetric) strong-interaction theory however remains a holy grail Some new ideas ? 🔶
⑤ Generalized symmetries, line and surface operators; , o t t o i a G y , n o ’05-‘18 Mixed ’t Hooft anomalies; r . a . , h i k A s d , n o i g t s r u a p m a o K K , g , r t e e l b l i i W e S Symmetry protected topological order (SPT) , , a i a w k a a k S i h i , c k a a T z i n a T , i h c u k i K , z a , t r i p u p k o e n P o Y , u z m i i 3 h 8 S ’ e n a d l a H 9 8 ‘ , n From 0-form symmetries (acting on local operators) e W 🔶 to k-form symmetries (acting on line, surface, etc operators) e.g. the center symmetry in SU(N) YM (k=1) ♦ p o o p l o n o o l Criteria for different phases s v l i o W k a y l o P H γ A → Ω N e i H e i γ A , Ω N = e 2 π i/N ∈ N Phases / symmetry realization of the system not described by the vacuum expectation values Nambu-Jona Lasinio of a local operator g r u b z n G i - u a d n a L h ¯ h φ ( x ) i , ψ ( x ) ψ ( x ) i : a germ for change of PARADIGM (growing out of Nambu’s teaching…)
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