Hadron Spectra in Strong Magnetic Fields M.A. Andreichikov, Yu.A. - PowerPoint PPT Presentation

Hadron Spectra in Strong Magnetic Fields M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP 2015-1-27 Talk by M.A. Andreichikov M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 1 / 47

QCD & QED in Strong Magnetic Field Strong magnetic fields - hadrons internal structure changes Spin and Isospin symmetries are broken eB ∼ σ ∼ 10 19 Gauss - string tension Strong Magnetic Fields in Nature(in Gauss): √ eB = a Bohr ) - 2 . 35 · 10 9 Atomic ( l B = 1 / Schwinger ( eB = m 2 e 3 ) - 4 . 4 · 10 13 Surface of magnetars - 10 14 RHIC and LHC - 10 18 − 10 20 M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 2 / 47

Systems in Mafnetic Field Positronium (Shabad, Usov) - Bethe-Salpeter approach Hydrogen (..., Khriplovich, Popov-Karnakov, Vysotsky-Godunov-Machet) ρ -meson (..., Mueller, Chernodub) Pion gas (Smilga, Agasian) Quark matter (Kharzeev-McLerran-Polikarpov-Zakharov..., Andreichikov-Kerbikov) Neutral and charged quark-antiquark systems ( ρ , π ) (Andreichikov-Kerbikov-Orlovsky-Simonov) Neutral and charged baryons (in progress) β -decay (Matese, O’Connel, Studenikin) Excitons (solid state physics) (Gorkov, Dzyaloshinskii) The common features: 3d → 1d dimension reduction Coulomb-type interaction screening Sphere transforms to ellipsoid for ground state. M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 3 / 47

Papers related to the Problem: Yu. Simonov “Relativistic path integral and relativistic Hamiltonians in QCD and QED” , PRD 88, 025028 (2013) Yu. Simonov “Spin interactions in mesons in strong magnetic field” , PRD 88, 053004 (2013) M. Andreichikov, B. Kerbikov, V. Orlovsky, Yu. Simonov “Meson spectrum in strong magnetic fields” , PRD 87, 094029 (2013) M. Andreichikov, V. Orlovsky, Yu. Simonov “Asymptotic Freedom in Strong Magnetic Fields” , PRL 110, 162002 (2013) A. Badalian, Yu. Simonov “Magnetic moments of mesons” , PRD 87, 074012 (2013) M. Andreichikov, B. Kerbikov, V. Orlovsky, Yu. Simonov “Neutron in strong magnetic field” , PRD 89, 074033 (2014) V. Orlovsky, Yu. Simonov “Nambu-Goldstone mesons in strong magnetic field” , JHEP 2013:9:136 (2013) Yu. Simonov “Magnetic focusing in atomic, nuclear and hadronic processes” , arXiv:1308.5553 (2013) M. Andreichikov, B. Kerbikov, Yu. Simonov “Magnetic Field Focusing of Hyperfine Interaction in Hydrogen” , arXiv:1304.2516 (2013) M. Andreichikov, B. Kerbikov, Yu. Simonov “Quark-Antiquark System in Ultra-Intense Magnetic Field” , arXiv:1210.0227 (2012) M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 4 / 47

Plan of the Talk Green’s functions : QCD & QED in magnetic field (for mesons), correlators and relativistic Hamiltonian. Relativistic Hamiltonians : ◮ dynamics, confinement, magnetic moments ◮ wave function factorization ◮ zero modes Perturbative corrections : ◮ self-energy corrections ◮ color Coulomb & q ¯ q screening in nagnetic field ◮ hyperfine interactions, magnetic “focusing” in Hydrgen atom and “magnetic collapse” Meson mass spectrum. Baryon features (OPE, spin-isospin splitings) Neutron mass spectrum. Conclusions and discussion. M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 5 / 47

Feynman-Fock-Schwinger Formalism: A ( e ) - EM field.) Single quark Green’s function ( ˆ A - gluon field, ˆ D i ( x , y ) = ( m i + ˆ ∂ − ig ˆ A − ie i ˆ A ( e ) ) − 1 xy = ( m i + ˆ D ( i ) ) − 1 xy Feynman-Fock-Schwinger representation � ∞ D i ( x , y ) = ( m i − ˆ σ ( x , y ) = ( m i − ˆ ds i ( Dz ) xy e − K i Φ ( i ) D i ) D i ) G i ( x , y ) 0 m i - quark mass, s i - proper time � 2 � s i � d z ( i ) i s i + 1 µ K i = m 2 d τ i 4 d τ i 0 M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 6 / 47

Gauge Fields in FSF Formalism:: Field-dependent term: � x � x � � Φ ( i ) A µ dz ( i ) A ( e ) µ dz ( i ) σ ( x , y ) = P A P F exp ig µ + ie i × µ y y �� s i � × exp d τ i σ µν ( gF µν + eB µν ) 0 (4 × 4) structures for gluon and EM field: � � � � 0 σ H σ E σ B σ µν F µν = , σ µν B µν = 0 σ E σ H σ B If only magnetic field B � = 0 - euclidean action - no negative eigenvalues (Stablisation theorem) M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 7 / 47

QQ Green’s Function: G q 1 ¯ q 2 ( x , y ) = � j Γ ( x ) j Γ ( y ) � And the path integral for it is ( ˆ T contains gamma matrices trace): � ∞ � ∞ ds 2 ( Dz (1) )( Dz (2) ) � ˆ q 2 ( x , y ) = TW σ ( A ) � A × G q 1 ¯ ds 1 0 0 � x � x � s 1 � s 2 � � A ( e ) µ dz (1) A ( e ) µ dz (2) × exp − ie 2 + e 1 d τ 1 ( σ B ) − e 2 d τ 2 ( σ B ) ie 1 µ µ y y 0 0 × exp( − K 1 − K 2 ) Gluon contribution( Wilson loop ) after averaging over stochastic vacuum background: � τ E � � σ | z 1 − z 2 | − 4 α s �� � W σ ( A ) � A = exp − dt E 3 | z 1 − z 2 | 0 Confinement + OGE (color Coulomb) (Minimal area for the Wilson loop) M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 8 / 47

Omega Formalism: Monotonous time t E ( τ ) = x 4 + τ s T provides: z 4 ( τ ) = t E ( τ ) + ∆ z 4 ( τ ) , ω i = T , T = | x − y | 2 s i Green’s function with omega-variables (after averaging) � ∞ d ω 1 d ω 2 q 2 ( x , y ) = T � � q 2 T � � � Tr ( ˆ Te − H q 1 ¯ G q 1 ¯ x � y � � ω 3 / 2 ω 3 / 2 8 π 0 1 2 Mass spectrum from Hamiltonian: (stationary point anaysis) H ψ = M n ψ, ∂ M n ( ω i ) ˆ = 0 ∂ω i Omegas are quark “dynamical” masses M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 9 / 47

Hamiltonian for Q ¯ Q Neutral Meson 1 1 ( p 1 − e A ( z 1 )) 2 + ( p 2 + e A ( z 2 )) 2 + σ | z 1 − z 2 | + H q ¯ q = 2 ω 1 2 ω 1 + m 2 1 + ω 2 1 − e σ (1) B + m 2 2 + ω 2 2 + e σ (2) B 2 ω 1 2 ω 2 Dynamics (mass & wave function) is defined by nonperturbative part: H q ¯ q Ψ n = M n Ψ n Perturbative effects are treated as corrections : M total = M n + � Ψ | V OGE | Ψ � + � Ψ | V SS | Ψ � + ∆ M SE Color Coulomb term V Coul , Self-Energy V SE (Wilson loop integration) and Spin-spin interaction V SS are treated as perturbation. M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 10 / 47

Insight from classical equations: m 1 = m 2 , q 1 = − q 2 domain (simplest case) m d 2 x 1 d t 2 = f 12 + q d x 1 d t × B ; m d 2 x 2 d t 2 = − f 12 − q d x 2 d t × B New coordinates R , η : R = x 1 + x 2 ; η = x 1 − x 2 2 Transformed equations: d � 2 m d R � d t − q η × B = 0; d t d � m d η � d t − q R × B = f 12 2 d t M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 11 / 47

Quantum case in m 1 = m 2 , q 1 = − q 2 domain (ansatz): Canonical transformation: R = x 1 + x 2 P = − i ∂ π = − i ∂ ∂ η = ˆ p 1 − ˆ p 2 ; η = x 1 − x 2 ; ˆ ∂ R = ˆ p 1 + ˆ p 2 ; ˆ 2 2 Hamiltonian: � 2 � 2 � � 1 P − 1 + 1 π − 1 ˆ ˆ H = 2 q B × η ˆ 2 q B × R + V ( η ) 4 m m Integral of motion: P + 1 ˆ I = ˆ 2 q B × η ; [ˆ I , ˆ H ] = 0 Wavefunction ansatz (Bilocal phase): Ψ( R , η ) = φ ( η ) e i PR − i 1 2 q ( B × η ) R References: J.E.Avron, I.W.Herbst, B.Simon, Ann. Phys. 114 , 431 (1978) D.Koller, M.Malvetti, H.Pilkuhn, Phys. Lett. A 132 , 5 (1988) M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 12 / 47

Quantum case in m 1 = m 2 , q 1 = − q 2 domain (solution): Harmonic oscillator problem: ∂ 2 φ 4 m ( P − q B × η ) 2 φ − 1 1 ∂ η 2 + V ( η ) φ = E φ m E η 2 – full factorization: External oscillator potential: V ( η ) = m ω 2 4 m + P 2 x + P 2 + P 2 � � n z + 1 1 + α + σ 1 y z E = Ω( n x + n y + 1) + ω E 2 4 m 4 β � qB z � 2 � Ω = ω E (1 + α ); α = 2 m ω E Remark: In Coloumb potential case V ( η ) = − 1 | η | the rotational symmetry References: M.Taut, Phys.Rev. A 48 , 5 (1994) (Isotropic oscillator + Coloumb) M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 13 / 47

More complicated case m 1 � = m 2 , q 1 = − q 2 domain (ansatz): Canonical transformation: R = m 1 x 1 + m 2 x 2 ; η = x 1 − x 2 ; m 1 + m 2 m 1 m 2 ; s = m 1 − m 2 µ = m 1 + m 2 m 1 + m 2 π + µ π + µ ˆ ˆ ˆ p 1 = ˆ P ; ˆ p 2 = − ˆ P m 2 m 1 Hamiltomian: � 2 � 2 H = 1 � π − 1 2 B × R + s 1 � P − 1 ˆ ˆ ˆ 2 B × η + 2 B × η + V ( η ) 2 µ 2 M Integral of motion: P + 1 ˆ I = ˆ 2 B × η Wavefunction ansatz: Ψ( R , η ) = φ ( η ) e i PR − i 1 2 ( B × η ) R References: J.E.Avron, I.W.Herbst, B.Simon, Ann. Phys. 114 , 431 (1978) M.A. Andreichikov, Yu.A. Simonov, B.O. Kerbikov, V.D. Orlovsky ITEP Hadron Spectra in Strong Magnetic Fields 2015-1-27 14 / 47