Charge quantization for dyons

Transcription

1 Charge quantization for dyons Electron monopole pair (e,g( e,g) There is an elementary electric charge eg =n n Z e 0 = 1 g, e=ne 0 Pair of dyons: (e 1,g 1 ), (e 2,g 2 ): e 1 g 2 e 2 g 1 =n, n Z There is an elementary magnetic charge (e 0,0)+(e,g)= e 0 g =n g = n e 0 = n n 0 g 0 Pair of dyons: (e 1,g 0 ), (e 2,g 0 ): e 1 e 2 = n g 0 = n e 0 =me 0 n 0 e i =e 0 n i + θ, θ [0.2π] 2π

2 Charge quantization and SL(2,) ) group Most general charge quantization condition: e+ig =e 0 n+m θ 2π +i m e 0 =e 0 (mτ +n) τ = θ 2π + i e 2 0 Transformations of SL(2,): τ aτ + b cτ +d Generators: T :τ τ +1; S :τ 1 τ

3 Non-Abelian monopoles Break-through of 1974: `t Hooft-Polyakov monopole solution While a Dirac monopole could be incorporated in an Abelian theory, some non-abelian models inevitably contain monopole solutions Non-Abelian monopole is a non-linear system of coupled gauge and scalar (Higgs) fields, its energy is finite and the fields are regular everywhere in space. The gauge symmetry is spontaneously broken via Higgs mechanism L= 1 2 Tr Fµ2ν + Tr (Dµ Φ)2 + V ( Φ )

4 Magnetic monopoles B =g r r 3, E =Q r r 3 Dirac monopole (1931) Non-Abelian monopole P.A.M.Dirac Wu-Yang monopole (1975) A.M.Polyakov *1945 A N = g 1 cosθ rsinθ êe A S = g 1+cosθ rsinθ êe Regular static configuration Gauge group SU(2) Magnetic charge is the topological number: Qg=n/2 The monopole is very heavy, M~m_v/e Gerard 't Hooft *1946

5 Properties of non-abelian monopoles [SU(5)] Monopole has a core of radius r m ~ m -1 x ~ 10 Monopole is superheavy: M ~ m x /α ~ cm 17 GeV ~ 10-7 g Magnetic charge of the monopole has topological roots: Φ vr S 2 S 2 Electromagnetic subgroup is associated with rotations about direction of the Higgs field Monopole solution mixes the spacial and group rotations: J = L+ T + S

6 Dyons Monopole has 4 collective coordinates: R k and χ(t) Electric charge of a dyon is Q=4π χ Charge quantization condition for a pair of dyons: Q 1 g 2 Q 2 g 1 = n 2 Consequence: Spin-statistic theorem admits both Bose-Einstein and Fermi-Dirac statistics.

7 Relic Monopoles Monopoles should have been produced in the very early Universe: SU(5) SU(3) SU(2) U(1) SU(3) U(1) em As T < T c ~ GeV the Higgs field acquires a non-zero v.e.v. V(Φ) V(Φ) Predictions of the Big Bang scenario (adiabatic expansion) 1 monopole per 10 4 nucleons! V(Φ) Inflation scenario: The potential is suffuciently flat at Φ=0, the phase transition occurs at T c ~ 10 9 GeV - only a few monopoles may survive the inflation!

8 Experimental search for monopoles Accelerator search (Fermilab,, CERN, DESY ) Superconducting coils Indirect limits (Parker( Parker s bound, neutron stars ) Search for monopole catalysis (IceCube,, Berkeley, Stanford, IBM ) Monopoles in cosmic rays (Scintillators and ionization detectors) No monopole detected yet! ( Picture by courtesy G.Giacomelli)

9 Fake monopoles Monopoles in spin-ice crystal structures (Castelnovo, C., R. Moessner, and S. L. Sondhi, Magnetic monopoles in spin ice, Nature, Vol. 451, 42-45, 2008; D.J.P. Morris et al, Dirac Strings and Magnetic Monopoles in the Spin Ice; Science, Vol. 326, , 2009) H =J ij S i S J +σ ij 3(êe i ˆr ij )(êe j ˆr ij ) (e i e j ) r 3 ij A sum of nearest-neighbor Ising model term and long range dipolar interactions (Mengotti et al, Nature Physics, 7 (2011) 68) Where is the cheat? Each dipole is replaced by a pair of equal and opposite magnetic charges

10 Fake monopoles Monopoles and low energy QCD Λ χ ~ 1 GeV Λ QCD ~ 180 MeV Perturbative QCD Low energy effective theory Hadrons (Quarks & gluons) (Pions and quarks) QCD confinement as dual Meissner effect: monopole condencation as a reason of formation of the chromoelectric flux tube and QCD is taking a form of the dual Ginzburg-Landau model (S.Mandelstam, G `t Hooft et al (1970s)) Where is the cheat? There is no monopoles in QCD!

11 Here we are: Yang-Mills Mills-Higgs Theory S = 1 2 d 4 x{f µν F µν + (D µ Φ)(D µ Φ) V(Φ)} F µν = µ A ν ν A µ +ie[a µ,a ν ] D µ Φ= µ Φ+ie[A µ,φ] V(Φ)=λ (Φ 2 a 2 ) 2 t Hooft-Polyakov static spherically symmetric solution φ a = ra er 2 H(ear) A a n =ε amn rm er 2 (1 K(ear))

12 t Hooft-Polyakov solution Topologically nontrivial asymptotic φ a Vacuum: r vr a r φ = a φ:s 2 S 2 The fox knows many things, but the hedgehog knows one big thing Archiolus D n φ a 0 n r a r eε abc A b n r c r =0 A a k(r) r U(1) U(1) t t Hooft Hooft tensor: 1 e ε r n ank r 2; Ba n r r a r n er 4 tensor: F µν = ˆφ a Fµν a + 1 e ǫ abcˆφ a D µˆφ b D νˆφ c µ Fµν =k ν ; k µ = 1 2 ε µνρσ ν F ρσ = 1 2v 3 e ε µνρσε abc ν φ a ρ φ b σ φ c Topological current: µ k µ 0

13 Magnetic charge: charge: g = t Hooft-Polyakov solution = 1 e d 3 xk 0 = 1 2ev 3 d 2 ξ g = 4πn e, n Z d 2 S n ε abc ε mnk φ a n φ b k φ c local coordinates on S 2 determinant of metric tensor on S 2 t Hooft-Polyakov ansatz: ξ =aer φ a = ra r m er 2H(ξ), Aa n =ε amn er 2[1 K(ξ)], Aa 0 =0 Field equations: ξ 2d2 K dξ 2 =KH2 +K(K 2 1), ξ 2d2 H dξ 2 =2K2 H + λ e 2H(H2 ξ 2 )

14 A a 0 = ra er 2J(r) q = 1 v J(r) Cr as r φ a A a 0 q = 4π e Dyons ds n E n = 1 v 0 dξ d dξ ds n E a nφ a = 1 v ξh d dξ JH ξ d 3 xe a nd n φ a = 4πC e =Cg E = = d 3 x 2 [Ba n Ba n +(D nφ a )(D n φ a )]+V(φ) d 3 x 1 d 3 x(b a 2 n D n φ a ) 2 + d 3 xbnd a n φ a + d 3 xv(φ) Note: The energy is minimal if 1 B a n =D n φ a Magnetic charge Bogomol nui-prasad-zommerfield equation 2 λ=0 (V(φ)=0) E =Q

15 BPS monopole B a n =D n φ a ξ dk dξ = KH, ξdh dξ =H+(1 K2 ) Note: This system has an analytical solution: K = ξ sinhξ, H =ξcothξ 1 Homework: Prove it! The BPS equation + Bianchi identity D n D n φ a =0 D n φ a D n φ a =( n φ a )( n φ a )+φ a ( n n φ a )= 1 2 n n (φ a φ a ) Long range asymptotic tail of the scalar field: φ a vˆr a ra er 2 The energy is completely defined by the scalar field: E = 1 2 d 3 x n n (φ a φ a )= 4πv e cothξ 1 ξ 1 ξ2 sinh 2 = 4πv ξ 0 e

16 BPS monopole: gauge zero mode Kinetic energy: Static configuration, A 0 =0 0 =0 T = d 3 x (E n E n +D 0 φd 0 φ) Gauss law: D n E n ie[φ,d 0 φ]=0 E n =D 0 φ=0 Stationary configuration, A Time-dependent gauge transformations: A n (r,t)=u(r,t)a n (r,0)u 1 (r,t) i e U(r,t) nu 1 (r,t) A n (r,δt) A n (r)+(ie[ω,a n (r)] n ω)δt U(r,t)=e ieω(r)t 1+ieω(r)δt A n =ie[ω,a n (r)] n ω = D n ω 0 φ=ie[ω,φ] Is everything correct? Gauge zero mode

17 BPS monopole: gauge zero mode 0 A n =ie[ω,a n (r)] n ω = D n ω 0 φ=ie[ω,φ] But: But: E a n = 0 A n D n A 0 = D n ω+d n ω 0, D 0 φ= 0 φ+ie[a 0,φ]=ie[ω,φ] ie[ω,φ] 0 The Gauss law is satisfied trivially Background gauge: D n ( 0 A n ) ie[φ,( 0 φ)] = 0 Now: Now: ω = Υ(t)φ, U(r,t) =exp{ieυ(t)}φ(r) 1+ie Υφδt 0 A n = ΥD n φ; 0 φ=0 E n = 0 A n = Υ(t)D n φ = Υ(t)B n, D 0 φ=0 Kinetic energy: T = Υ 2 d 3 xd n φ a D n φ a = Υ 2 d 3 xb a n Ba n =2πvg Υ 2 =M Υ 2

18 Self-dual monopoles vs non-self dual monopoles BPS monopoles: In the limit h=0 the energy becomes: The first order Bogomol'nyi equations yield absolute minimum: M = 4πg E =Tr d 3 x 1 4 (ε ijkf ij ±D i Φ) ε ijkf ij D k Φ B k =±D k Φ No net interaction between the BPS monopoles: the electromagnetic repulsion is compensated by the long-range scalar interaction. Non self-dual monopoles: They are solutions of the second order Yang-Mills equations: µ F µν =0 E>M BPS even if g=0 (deformations of the topologically trivial sector) The constituents are non BPS monopoles and/or vortices in a static equilibrium; separation is relatively small, there are no long-range forces

19 Self-dual monopoles vs non-self dual monopoles BPS monopoles: Integrability of the BPS equations: there is a correspondence to the reduced selfduality equations of the Yang-Mills theory: D k Φ a D k A a 0 Fa 0k B a k 1 2 ε kmnf mn F a 0k Fa 0k Properties of the BPS monopole are completely defined by the Higgs field: D k Φ a D k Φ a = 1 2 k k Φ 2 E = 1 2 d 3 x k k Φ 2 =4πg An infinite chain of YM instantons along Euclidean time axis A a k =ε akn n lnρ+δ ak 0 lnρ; A a 0 = alnρ where (z =r+iτ) is equal to the BPS monopole : ρ= 1 2r n= n= 1 z iω n + n= n= 1 z +iω n = 1 2r coth z 2 +coth z 2 1 r 2 +(τ 2πn) 2 n= ρ= = 1 2r sinhr coshr cosτ.

20 MA pair: magnetic dipole dipole (Taubes, Nahm, Rüber, R Kleihaus,Kunz & Shnir) Magnetic dipole A µ dx µ = K 1 r dr+(1 K 2)dθ τ ϕ (n) 2e nsinθ Φ=Φ aτa 2 =a τ H (n,m) r τ 1 2 +H (n,m) θ 2 2. τ r K (n,m) 3 2e +(1 K 4 ) τ(n,m) θ 2e dϕ; Magnetic charge: Q= 1 2π (ΦdΦ dφ) S 2 = 1 2 n[1 ( 1)m ]

21 Rational map monopoles There is a transformation of a monopole into a rational map from the Riemanian sphere to inself: R: S 2 # S 2 (P. Sutcliffe, N.Manton et al) R(z)= a(z) b(z) = a 1z n 1 + +a n z n +b 1 z n 1 + +b n, z =x 1 +ix 2 Construction of the rational maps monopoles: Represent BPS equation in spherical coordinates r,z,z Impose a complex gauge Φ= ia r = i 2 U 1 r U, A z =U 1 z U, A z =0 Construct the monopoles using U exp 2r 1+ R 2 R 2 1 2R 2R 1 R 2

22 R = 1/z: one spherically symmetric monopole centered at the origin; R(z) = a 1z+a 2 z 2 +b 1 z+b 2 : two monopoles R(z) = i 3z 2 1 z(z 2 i : Tetrahedral monopoles (degree 3 map) 3) R(z) = z4 +2i 3z 2 +1 z 4 2iz 2 : Octahedral monopoles (degree 4 map) 3+1 By courtesy of P Sutcliffe

23 The Nahm transform Duality: Bogomolny equation Nahm equation Reduction of the Bogomolny equation to a system of coupled nonlinear ordinary differential equations for a triplet of Hermitian matrix valued functions T i (s) k k T i (s)=t i(s); η/2 s η/2 dt i ds = i 2 ε ijk[t j,t k ] T i (s) have simple poles on the boundaries: T i (s)= L± i s η/2 +O(1) The matrix residues of T i (s) at each pole form the irreducible k-dim representation of su(2): L ± i,l± j =iεijk L ± k

24 Inverse Nahm transform The construction equation = I 2k d ds +(r ii k T i ) σ i ; ω(s,r)=0 Normalizable solutions form a complete linearly independent set ω n (s,r) η/2 dsω n (s,r)ω m(s,r)=δ nm dim ker =2 η/2 Reconstructing the fields: η/2 Φ nm = dssω n(s,r)ω m (s,r); A k nm = i η/2 ds ω n(s,r) k ω m (s,r) η/2 η/2 U(k) gauge symmetry on the Nahm data vs SU(2) symmetry of the BPS model 1:1 Duality between the U(k) Nahm system in R 1 and the SU(2) BPS k-monopole k system in R 3

25 Reconstructing k=1 monopole 1 1 Nahm data: T i (s)=c i, c i = const Weyl equation d ds σ i r i ω(s)=0 ω 1 = ω 2 = r 2sinhr r 2sinhr cosh sr 2 +sinhsr 2 σ i r i 1 0 cosh sr 2 +sinhsr 2 σ i r i 0 1 Higgs field: Φ nm = η/2 dssω n (s,r)ω m(s,r)=i cothr 1 σ i r i r η/2

26 Reconstructing k=2 monopole 2 2 Nahm data: T i (s) = 1 2 f i(s) σ Nahm equations: df i ds = 1 2 ε ijkf j f k Euler top functions: f 1 = D cn kd(s+v/2) sn k D(s+v/2) ; f 2 = D dn kd(s+v/2) sn k D(s+v/2) ; f 3 = D sn k D(s + v/2) Weyl equation 0 k 1; asd 1, k 1 16e Ds cn k (s) dn k (s) sech(s); sn k (s) tanh(s) d ds (ri +De z /2) σ i 0 0 (r i De z /2) σ i ω(s)=0 Two widely separated monopoles

27 Monopole catalysis of proton decay Naive question: What happened when a fermion collides with a monopole? = +? There are zero-energy energy solutions of the Dirac equation for a massless fermion coupled to a monopole; γ µ ( µ +ea µ )ψ = 0; J = L+ T + S; L =0, T + S =0 The ground state of a monopole becomes two-fold degenerated: (i) Ω> (no fermions; Q F = -1/2) (ii) Ω> (zero mode); Q F = 1/2) Ω> (ii) a Ω Spin-flip? Charge conjugation? Chirality? There are non-suppressed fermionic condensates on the monopole background: (e + e d 3 d 3 )(ūu 1 u 1 ūu 2 u 2 )) r 6

28 Rubakov-Callan effect SU(5) model with massless fermions in s-waves Monopole could catalyse baryon number violating processes like P + monopole e + + mesons + monopole Caution: the effect is model-dependent! dependent!

29 To conclude: Topology is power! Even Abelian Dirac monopole has topological roots. There are various monopoles in many theories. Consistent description of monopoles is related with topological arguments, it needs a non-abelian theory with spontaneousy broken symmetry. Rubakov-Callan effect is strongly model-dependent. Fake monopoles are just fakes A monopole beyond the horizon?