CHPR III: SYMMRIS Sytrs OFo CD CD CD s b on local SU( c gag sytry In aton: global sytrs. Nöthr s hor L CD ( ( [γ D ] ( 4 Ga ν(g ν a ( (a 8 whr D g ( ( a( λ a an (. s Lt L CD b nvarant nr a global transoraton o th qar ls: ( ( [Γ a Θ a ] ( Γ a Θ a ± wth Γ a : gnrators o U(N orsu(n Θ a nnnt o. Dn: Nöthr crrnt J a ( L CD ( Θ a (γ Γ a ( I L CD s nvarant L nr th global transoraton thn Nöthr crrnt J J a s consrv: J a ( I crrnt s localz n sac thn Nöthr crrnt h consrv charg. a Ja( (Γ a ( a a bc t J a J a a Global U( sytry:. Nbr an Flavor Crrnts ( θ ( J B ( (γ ( consrv charg: B (( nbr b Isosn crrnt: ( ( ( - ss: qal ss. Isosn oblt (N - SU( transoraton: ( ( θ - : SU( gnrators ( Pal atrcs - L CD wth q s nvarant nr SU( consrv crrnt: ( (γ ( consrv sosn charg: ( [ H ] c Flavor crrnt n SU( : (N s - ss: s - SU( transoraton: ( ( consrv crrnt: ( (γ λ ( consrv charg: ( λ ( Sytry brang: s L s ( s λ θ ( ( s ( ( ( 8
Start wth N :. CD wth Mslss ars: Chral Sytry L CD ( (γ D ( 4 Ga ν(g ν a ( L CD whr L s ( ( L s CD n th lt o slss qars: L CD (γ (L qar-glon Lt- an rght-han qar ls: R L L gl ( ± γ s ( N χ s ± hχ s hχ s χ s N ( ± hχ s ±( ± hχ s ( γ rocts on h (rght han. ( γ rocts on h (lt han. Mslss CD: L CD L (γ D L ( R (γ D R (L gl whr L γ D L 4 ( γ γ γ D ( γ R ( γ L ( γ whr γ γ γ γ γ γ wth { γ γ } γ ar l: ( s s ( N σ ( χ s σ χ s bs ( s ( s(v s ( N rght - h ± lt - han. χ s σ χ s 4 ( γ γ D ( γ γ D γ γ D R γ D R 4 ( γ γ γ D ( γ γ D γ γ D Global transoraton: chral SU( R SU( L sytry R ( L ( wth : Pal atrcs ( θ R θ L R ( L ( Ms tr br ths sytry lctly. L s ( ( R L L R ar s tr s lt- an rght-han qars.
In th lt : consrv crrnts: J R ( R (γ R( J L ( L (γ L( Convnnt to ntroc vctor an aal vctor crrnt: ( J R (J L ( (γ ( ( J R ( J L ( (γ γ ( ( Ms [G].. φ η ρ ω η K K N Psoscalar Msons (J Λ Σ Ga Λ G Consrv charg: ( ( t H ( ( (ctor charg (γ ( (al charg t H Gnralzaton to lavor (N SU( R SU( L sytry rlac λ : Gll-Mann atrcs ( 8 L algbra o th vctor an aal chargs: Sctr o stats n Wgnr-Wyl ralzaton Party oblts: or ach stat o ostv arty thr st b a stat o qal s wth ngatv arty. Bt: a nclon wth J thr s no qal s artnr wth J. b soscalar sons wth J thr s no chral artnr wth J. c ctor- an alvctor-sons: ctor sons: J al vctor sons: J wth : strctr constant o SU(. Crrnt corrlaton ncton: Π ν (q Π ν (q 4 q [ ( ν (] 4 q [ ( ν (] ( 4. Ralzatons o Chral Sytry Wgnr-Wyl ralzaton: Gron stat ( vac : otal sytry btwn ostv an ngatv arty. In Wgnr-Wyl ralzaton: Π ν (q q q ν q g ν Π (q ( Π (q Π (q ( Sctral nctons: η (s 4 IΠ (q s
Bt rcally: Proo: Consr a stat o n Golston bosons (Φ n ( n. Η Ρson Ĥ (Φ n Ĥ ( ( Ĥ (Φ n n-ts ach Golston boson h nrgy-ont rlaton ε q. Snc n (slss a son Η Golston bosons ar gnrat wth vac t ollows Golston bosons o not ntract n th lt q...... Η s G Low nrgy CD s ralz n th or o an ctv l thory o waly ntractng Golston hs obsrvatons la to th Nab-Golston ralzaton o chral sytry: Gron stat os not hav all th sytrs o Lagrangan nsty. al sytry s sontanosly bron. (4 Isosn sytry. Golston s hor vry sontanosly bron global sytry thr sts a slss stat carrs th qant nbrs o th corrsonng sytry charg. Dn Φ Ĥ thn: Ĥ Φ Ĥ Ĥ Φ Φ nrgtcally gnrat wth gron stat (vac Mslss Golston Boson. Φ ar stats wth sn/arty J Psoscalar. N ; ; Isosn I Pons ( (Pons or N ; Psoscalar son octt ( K K η orn 6. Sontanos Sytry Brang nothr stanar al o sontanos sytry brang: Frro-agnt Sn syst: Haltonan H H G σ σ < Invarant nr rotatonal sytry n (O( sytry Low tratr: Magntzaton h non-zro ctaton val M rrr rcton n sac O( sytry s sontanosly bron (Nab- Golston ralzaton. Orr aratr: M Golston s thor: In th Nab-Golston ralzaton o (sontanosly bron chral sytry th Golston bosons ar waly ntractng at low nrgs. c t hgh tratr > c : O( sytry rstor n Wgnr-Wyl ralzaton.
Golston boson: Magnon Sn wav n CD: M Chral (qar connsaton qq Chral connsat qq s th orr aratr o sontanosly bron chral sytry n CD. Dnton o qar connsat: tr l SF ( y S F ( y y tr l y Ω : ( (y : Ω (8 7. Chral Connsat (ar Connsat Prtrbatv an non-rtrbatv vac ar l orator: ( ( b ( ( v( ( ( ( ( Prtrbatv vac: b Non-rtrbatv vac: Ω: b Ω Ω ( ( b ( v ( (6 ( ( Ω ( ( Ω Wc s thor: ( (y :( (y : ( (y noral roct S F ( y Dnton o noral roct: : b q : qb tc. (7 N lavor wth ū ; qq wth q ar Connsat an Sontanosly Bron Chral Sytry Sontanos brang o chral sytry (Nab-Golston ralzaton ls nontrval vac charactrz by non-vanshng chral connsat: Stch o roo: ntroc P ( (γ ( Rlaton: Us: ( t; P ( t (( (9 α ( t β (y t αβ ( y α ( t β (y t α ( t β (y t ( In th rtrbatv vac: : ( (y : an th stanar Fynan roagator s S F ( y ( (y. In th non-rtrbatv vac: Ω ( (y Ω Ω : ( (y : Ω S F ( y S F ( y a ctaton val o (9: P P ( I Chral connsat R L L R : Orr aratr o sontanosly bron chral sytry.
c (an 9. 9. ts ralzaton n Lattc CD hroynacs o th Chral Connsat hroynacs o th Chral Connsat 9. hroynacs o th Chral Connsat c a ctaton val:. Pon Dcay Constant. Pon Dcay Constant. Pon DcayConstant. Pon Dcay Constant CD (an at tratr vol : ts ralzaton n Lattc CD. Pon Dcay Constant (an ts ralzaton n Lattc n CD (an ts ralzaton Lattc CD Startng ont: SU (R SU (L chral sytry SU Startng ont: SUL( ( sytry R SU L chral S DDD Startng ont: SU ( chral ( (ont: ( R ss on Golston boson Startng SU (R SU (sytry sytry (y L chral Startng SU SU ( sytry : tr l ( vol CD at tratr ( : R L chral sontanosly bron ( ( bron ons t tratr tratr vol ont: :vol Golston ctatons sontanosly ( ons Golston S y DDD sontanosly bron ( ons ons Golston sontanosly bron ( Golston S ( S S ( ( Golston (y DDD ( (y DDD ( (y sontanosly bron ( ons DDD ( Introc ( qant stat o on tr l ( Introc ( qant stat o on tr l ( ( Introc ( qant stat o on S ( tr l ( y S DDD ( ( Introc ( qant stat o on y S DDD y DDDβ Introc ( qant stat o on ( whr ( Noralzaton Noralzaton ( ( ( ( ( whr whr ( Noralzaton ( ( ( ( ( clan acton: S ( L β β CD Noralzaton ( ( ( ( whr β clan acton: ( L ( ( CD an acton: S ( Noralzaton S ( L ( whr. an ( CD ( PCC. an acton: S ( LCD. PCC an th Gll-Mann O Rnnr Rlaton ββ N a (N al L a N N a L (N a (N a. PCC an th Gll-Mann O Rnnr Rlaton ( ( Gll-Mann th O (. ( PCC an Rnnr R ( ( ( Sall qar ( ss lct brang o chral sytry ( Sall o chralqar N a L (N a ss Sall qar ss lct brang sytry ( ( ( Rslt or tratr nnc o or tratr nnc o Rslt or tratr nnc o ΛCD or tratr nnc o M Crtcal tratr c 9 oynacs o th Chral Connsat tratr c 9 M ΛCD tratr ts CD c 9 M ΛCD!!ralzaton ΨCrtcal Ψ#! tratr cn Lattc 9 M ΛCD Sall qar ss al-ctor Crrnt lct brang o chral sy Partally Consrv (PCC Partally Consrv al-ctor Crrnt (PCC Partally Consrv8 Partally Consrv al-ctor Crrnt (PCC ( ( γ ( ( ( γ ( (8 87 ( ( γ( 87 Consr th c ; ( γ or q (chral lt: n orr h 87!! Ψ Ψ#! Consr th c ; γ or q (chral lt: n orr h ( ol : 87 transton (N 87 Cobn transton (N Consr thwth c(9: ; ( γ (chral S or lt: n orr h ( q DDD ((y Consr th c or( q (chral lt: orr h Cobn n wth (9: or q : crossovr transton tr l tγ ( t (( S q % ( ( y q $ DDD (N transton or transton q : crossovr tγ ( t (( q % ( wth Cobn (9: transton (N q $ β wth ((8 c Cobn or q : crossovr transton % ( tγ ( t ( L q a ctaton val: CD q $ c or q : crossovr transton. PCC an thval: Gll-Mann a ctaton O Rnnr Rlaton % q q $ (. Pon Dcay Constant (N a c ( (8. Pon Dcay Constant Sall qar ss lct brang o chral sytry ss on Golston boson onats sctr o soscal a ctaton val: SU (Rc SU (L chral sytry Startng nnc o ont: ctatons ss on Golston boson(pcc onats sctr o soscalar sov Partally Consrv al-ctor Crrnt ( ( g ont: SU (R SU (L chral a ctaton sontanosly bron ( Dcay sytry Constant ons Golston ( ctatons. Pon 9 M ΛCD ( ( ( γ ( ((8 nosly (( qant ons o Golston ( ( bron Introc stat on ss on Golston boson o ( onats ( sctr ( ( goc ont: SU ( (Lochral sytry R SU. Constant Noralzaton ( qant stat on ctatons. (Pon ( Dcay whr ( ( an ( ( ( O Rnnr rla Gll-Mann ( (
( SU (L chral sytry S ( ctatons ( ons ( Golston. ctatons PCC an th Gll-Mann O Rnnr Rlaton. PCC an th Gll-Mann O Rnnr Rlaton stat o on ( ( (7 ( ( ( ( ( ( (. PCC an th Gll-Mann O Rnnr Rlaton ( ( ( ll qar ss lct brang o chral sytry. lct brang.sytry Sall ss chral nr Rnnr whr ( Rlaton ( ( th whr PCC Rnnr o Gll-Mann qar an ( Rlaton an Gll-Mann rlaton: O Rnnr rlat. an th Rlaton ( Gll-Mann Rnnr O. PCC an Gll-Mann O ( ( O Rnnr. PCC an th Gll-Mann O Rlaton Partally Consrv al-ctor Crrnt (PCC SallConsrv qar ss Crrnt lct brang o chral sytry Partally al-ctor (PCC l sytry lct brang o Sall qar ss chral sytry Sall qar ss lct osytry sytry (4 chral chral(pcc ( ( brang γcrrnt qar ss lct brang o (8 ( Partally Consrv al-ctor ( ( ( ( γ ( (4 ( al-ctor Partally ( ( (Consrv Consrv al-ctor Crrnt (PCC Partally Crrnt (PCC Crrnt rtally Consrv al-ctor (PCC ( ( (4 ( γ sr th c (4 ; ( γ ( (4 ( ( 88 ( ( γγ (4 ( 88 ( ( ( Consr th c ; γ ( γ (4 M scal G M (at rnoralzaton G (at rnoralzaton scal Consr th c ; ( γ Cobn wth (9: s orr th Consr th c ; ( Consr c ; ( t: h n γ wth (9: γ γ Cobn ± r th c ; ( γ 9.6 M or ± scal G tγ ( t (. 9.6 M or M (at rnoralzaton G (( ( G ( tγ (M t (at rnoralzaton (( 9.4 M ro ν (. Cobn wth (9: 87 Cobn ( scal G Cobn wth (9: wth ± (9: 9.6 9.4 M ro M or ν ( Cobn wth (9: (. G transton M (at rnoralzaton scal G ( tγ ± (( M 9.6 ( t (( ((( ( tγ or tγ t t ( ( ro ν(.4 9.4 M G.8 (( (. G ( ( tγ ( t (( a ctatonval: ( (.4 G.8 or ( ± 9.6 M a ctaton val: ( 9.4 M ro ν (. G ( nbr 9.4 M ro ν (.4 G.8o to Coar agnt baryon nsty n cntr at ( (6 a ( (6 a ctaton val: val: ctaton Coar agnt to baryon cntr o atoc nc a ctaton val: (.4 G.8 nbr nstyz n N nbr nsty cntr o a ctaton val: (.4 Coar agnt to baryon n G.8 ρ.6 on Golston boson onats sctr o soscalar sovctor ( (6 boson ( (6 ss on Golston onats o soscalar sovctor sctr Z N ( (6 ncls:.6z tatons baryon nbr nsty n cntrρo atoc (6 Coar ( (6 N to ctatons agnt Coar agnt to baryon nbr nsty n cntr o atoc ncls: ρ.6 ss on Golston boson onats sctr o soscalar sovctor ss on Golston boson onats sctr o soscalar sovctor ( ( ( (7(7 ( ss on Golston boson onats sctr o soscalar sovctor Z N ( ctatons o soscalar sovctor ctatons ( on Golston boson onats sctr o soscalar sovctor.6 Z N ρ ρ.6 ( ctatons ( (7 ( (7 ( ( atons ( ( ( ( ( ( ( (7 (7 ( ( (7 Rnnr ( ( an ( Gll-Mann O rlaton: ( ( ( ( ( ( ( ( ( ( Gll-Mann O Rnnr rlaton: ( hr. an ( O Gll-Mann rlaton: ( ( Gll-Mann ( an ( ( ( O Rnnr rlaton: ( Rnnr ( ( ( (8(8 ( ( rlaton:. ( an Rnnr ( ( O rlaton: O Rnnr rlaton: ( Gll-Mann Gll-Mann Rnnr (8 ( (8 ( ( 88 88