Electric Dipole Moment

Transcription

1 Electric Dipole Moment Joshua Kames-King / 36

2 1 2 Standard Model CP / 36

3 The EDM and CP-Violation EDM: d = d 3 x x ρ( x) CPT T viol. CP viol. Non-zero EDM d 0 = CP 3 / 36

4 CP viol. in the SM CP-sources in SM Non-Pert.: θ-term in Pert.:CKM-matrix in 4 / 36

5 θ-term U(1) A -anomaly Massless limit: U L (N f ) U R (N f ) SU A (N f ) SU V (N f ) U V (1) U A (1) U V (1) : Symm. Bary.Numb.consv. observed SU V (3) : Symm. eightfoldway observed SU A (3) : SSB. GoldstoneBos. observed U A (1) Symm. : doubling of multiplets SSB. : pseudoscalar meson 5 / 36

6 θ-term U(1) A -anomaly Broken on QM-level at 1-loop: U(1)-Problem µ j µ 5 = N f /2 g 2 8π 2 GµνG a aµν However: µ K µ non-perturbative! Also: j µ 5 = j µ 5 2N g 2 f 16π 2 K µ 6 / 36

7 θ-term Instantons and -vacuum Consider: Demand finite energy: S E = 1 2g 2 d 4 x E Tr[F µν F µν ] F µν (x) r 0 = A µ (x) r U 1 µ U Euclidean space inf. S 3 ;also SU(2)(groupspace) S 3 since: U = u 0 + u σ = 1 u0 2 + u u = 1 S 3 S 3 n given by: n = d 4 xa w/a = 1 16π 2 Tr[ F µν F µν ] 7 / 36

8 Solutions action is finite: S E 8πn2 g 2 w/ F µν = ± F µν instanton solutions: A µ = ( r 2 r 2 +γ 2 )U µ U 1 U 1 ( r) = 1 r (x 0 + i x ) σ) w/ U n ( r) = (U 1 ( r)) n -vacuum Infinite number of vacua characterized by homotopy. True vacuum state: θ >= Σ n e inθ n > Have to include: L θ = θ 16π 2 Tr[ F µν F µν ] Breaks CP! 8 / 36

9 Handshaking: & Mass Matrix L Mq = ψ R Mψ L + ψ L M ψ R Now chiral trafo: ψ R(L) = U R(L)ψ R(L) gives: M ab = m a δ ab e iρ Now chiral trafo. : e iα Q 5 with α = θ/(2n f ): ρ = ρ θ N f Effective θ-vacuum θ = θ Arg Det[M] 9 / 36

10 CKM-matrix -Lagrangian L(f, φ) SSB quarks = Σ3 j,k=1 ( (m jk ) U ū j L uk R +(m jk) D d j L d k R with fermion mass matrices in flavour space: (m jk ) U/D = (Y U/D v ) jk 2 Diagonalization: V up L m UV up R = (m diag. ) U & VL down Charged current interaction: µ = ( u c t ) L γ µ V up J CC L V down L } {{ } :=V CKM m D V down R = (m diag. ) D d s b L ) ( 1+ φ ) +h.c. v 10 / 36

11 CKM-Matrix The charged current lagrangian: L cc Σ 3 i=1(ū i L γµ W + µ (V CKM ) ij d j L + d i L γµ W µ (V CKM ) iju j L ) CP d i L γµ W µ (V CKM ) ij u j L ) + ūi L γµ W + µ (V CKM ) ijd j L CKM-Matrix CP cons. would require V to be real Not the case for 3 families or more V CKM parametrized: 3 angles, 1 complex phase 11 / 36

12 Effective Lagrangian Other Techniques Need δl CP for pert.th.! Eff. Lagrangian for CP-violation Due to SU A (3)-SSB vacuum degenerate. = have to select from: L M = ψ R U R MU Lψ L + H.C. Demand: < Ω δl Ω >= min U R,L < Ω L M (U R,L ) Ω > Assume: 1 pure chiral: U R = U L 2 δl flavor-diagonal 12 / 36

13 Effective Lagrangian Other Techniques δl = ψ a R (µ a + iω)ψ a L e iδ m 2 a = µ 2 a + ω 2 θ = 1 3 Π a(µ a + iω) CP-Langrangian In the limit θ << 1: δl(cp) = Source of CP-effects in hadr. int. 3m u m d m s m u m d + m u m s + m d m s θ( ψiγ 5 ψ) 13 / 36

14 Calculation Other Techniques Idea: CP-pion-nucleon coupling ḡ πnn generates contribution to EDM N via π-loop: Correlator: T < n(p f ) J em. µ (0)i d 4 xδl CP n(p i ) >= d n (0)ū f σ µν k ν γ 5 u i +O(k 2 ) 14 / 36

15 Calculation Other Techniques Calc. of correlator: Σ X < n(p f ) J µ X > < X δl CP n(p i ) > with X >= { N >, πn >, ππn >,...} = πn > Full pion-nucleon eff. interactions: L πnn = π N σ(ig πnn γ 5 + ḡ πnn )N Determine ḡ πnn directly by < π a N δl CP N >: ḡ πnn θ = d n (0) = ḡπnng πnn ln( M 4π 2 N M N M π ) θ ecm 15 / 36

16 Other Techniques Techniques Quark models: 1 Bag model of confinement 2 Constituent quark model: 3 constituent q with 2 spin states Lattice : Compute EDM form factor in leading non-triv. θ-order Chiral pert. th.: chiral symmetry for low energies Naive dimensional analysis(nda): match loop corrections to tree level at Λ coeff. of operators sum rules 16 / 36

17 Sum Rules Other Techniques Idea Calculate hadronic correlator at high virtuality with OPE and match with phenomenological ansatz for low virtuality Relevant correlator: Π(Q 2 ) = i d 4 xe ipx < 0 T (η N (x) η N (0)) 0 > CP,F,π w/ linear comb. of two sources : η N = η 1 + βη 2 η 1 = 2ɛ abc (d T a Cγ 5 u b )d c & η 1 = 2ɛ abc (d T a Cu b )γ 5 d c η 1 dominant projection on neutron state. 17 / 36

18 Sum Rules Other Techniques Chiral inv. implies ansatz: Π(Q 2 ) pheno = 1 2 f d(q 2 ){ F σ, /q} f π(q 2 )/q Have to match w/ OPE evaluation: Π(q 2 ) = Σ d C d (q 2 ) < 0 O d 0 > CP,F,π Calculate generalized q-propagator in terms of: background field, CP-odd sources, vacuum condensates Compute relevant contractions d n = θecm 18 / 36

19 Strong CP-problem Other Techniques Discussion d n,exp ecm θ < Requires finetuning: θ and M q Possible Solutions Relaxation: θ dynamically zero Cancellation: Appropriate choice of M q,s.t. Det[M u M d ] real Handshaking lost Θ 0 19 / 36

20 Calculation Other Techniques Idea 1 Calculate Quark EDM at W-boson level 2 Move to nucleon level by RGE Figure: 1-loop and 2-loop level However: No contribution to quark EDM up to 3-loop level Need realistic approach 20 / 36

21 Diquark Moment Consider all 3 quarks simultaneously: 2 interacting q and 1 spectator quark Other Techniques 21 / 36

22 Calculation Other Techniques All 3 diagrams give amp.s of the form: (formfactor) ( /Qγ µ /p /pγ µ /Q)P L = Get 5-particle amplitude Γ µ ((V ) dt V ts (V ) su V ud ) }{{} :=V 22 / 36

23 Calculation Other Techniques EDM EDM given by: Γ 0 QA 0 Q 0 A using non-rel. formalism of 2 comp. spin states and sph. symm. of nucleon wf: Γ 0 2iIm(V) Integrating over quark phase space :d n (2iIm(V )) From Kaon-system: c 1 s 2 1 c 2s 2 c 3 s 3 sinδ CKM Therefore: d n ecm 10 6 smaller than exp. upper bound 23 / 36

24 Discussion Other Techniques Discussion rough estimates like hadron wf. However: gap of 10 6! strong exp. constraints can t be closed in SM 24 / 36

25 Experimental procedure General Procedure Larmor precession: hν = 2µ n B + 2d n E hν = 2µ n B 2d n E = ν = 4dE h 25 / 36

26 Ramsey Technique General Procedure 1 Inject polarized neutrons 2 Spin flip π/2 3 Free precession 4 Spin flip π/2 For d n 0 : = spin not flipped over. Count: n &n for E &E 26 / 36

27 Ultracold Neutrons General Procedure Why UCN? T ande Creation of UCN: Neutron turbine Polarization: magnetically saturated 1µm iron-cobalt foil. Filling chambers: 40s Confinement: 100s Discharge and counting: 40s E-polarity switched in 40s interval betweeen storage times 27 / 36

28 Atomic EDMs General Procedure Schiff s Theorem In neutral atomic system of,charged nonrel. point particles interacting (only) electromagnetically the shielding of the nucleus is complete, such that the net atomic EDM is zero Figure: Charged constituent particles rearrange 28 / 36

29 BUT General Procedure Loopholes relativistic effects for electrons finite size of nucleus magnetic interactions (primarily nucleus-electron) Paramag. Atoms: heavily enhanced atomic EDM by d e due to magnetic effects Z Diamag. Atoms: rel. and finite size effects weaken shielding; EDM associated with nuclear spin 29 / 36

30 SUSY SUSY Additional Dimensions Splitting of SM and sup.part masses: L soft = 1 2 (M 3 g g + M 2 W W + M 3 B B) + c.c. ( ūa u QHu dad QHd ēa e QHe ) + c.c. Q m 2 Q Q L m 2 L L ū m 2 ū ū d m 2 d d ē m 2 ē ē m H 2 u H uh u m H 2 d H d H d (bh u H d + c.c.) CPV a f = A F Y f suppresses FCNC L soft and µ-term give 40(!) CPV-phases not suppressed by Jarlskog invariant 30 / 36

31 SUSY CP-Problem SUSY Additional Dimensions Solutions Heavy sfermions: 1-loop EDMs suppressed by M ferm.sup.partners > 1TeV Cancellation: different CPV phases from different CPV ops. cancel CP-nonviolating SUSY-breaking: (approximate)cp symm. of soft-breaking sector 31 / 36

32 Additional Dimensions RS1 SUSY Additional Dimensions Solve hierarchy problem: 2 (3+1)branes of different energy Higgs located at TeV-brane gravity at Planck-brane Only gravity can propagate in bulk of 5th dim. between branes (ds) 2 = 1 (kz) 2 [η µν x µ x ν (dz) 2 ] 32 / 36

33 Additonal Dimensions SUSY Additional Dimensions CPV 5-dim. Yukawa int.: L = hδ(z z 0 )λ 5D Q(u, u,d d) 4-dim eff. theory on Tev-brane: m q vf Q λ 5D u,d F u.d Coupling of KK and fermions: λ 5D u,d F Q doesn t align with m q CPV contributions 33 / 36

34 Additional Dimensions-EDM SUSY Additional Dimensions EDM d d (gluon, KK) kv[d L F Qλ 5D d F dd R ] 11 [diag(m d, m s, m b )] 23 = 0 d d (Higgs, KK) 2k 3 v[f Q (λ 5D u λ 5D u + λ 5D d λ5d d )λ 5D d d n = (10 11 efm)( 2kλ5D 4 ) 2 ( 3TeV m kk ) 2 34 / 36 Fd]

35 35 / 36

36 G. Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D 14 (1976) 3432, [Erratum Phys. Rev. D 18 (1978) 2199]; Phys. Rep. 142 (1986) 357. M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. E. Witten, Nucl. Phys. B 156 (1979) 269 Roger Dashen Phys. Rev. D3,1879(1971) M. Pospelov, A. Ritz / Annals of Physics 318(2005) V. Baluni, Phys. Rev. D 19 (1979) / 36

37 J. Engel,M. J. Ramsey-Musolf, U. van Kolck;Electric dipole moments of nucleons, nuclei, and atoms: The Standard Model and beyond D. V. NANOPOULOS, A.YILDIZANNALS OF PHYSICS 127, ( 1980) I.B. KHRIPLOVICH; THE QUARK ELECTRIC DIPOLE MOMENT AND THE INDUCED 0-TERM IN THE KOBAYASHI-MASKAWA MODEL 37 / 36