Solving Schwinger- Dyson Equations • we need to deal with divergent integrals • we need an expression for � . Approximating this as the bare vertex is a typical. • we need to evaluate some pretty nasty integrals • we need to solve a(many) nonlinear integral equation(s).
Solving Schwinger- Dyson Equations Wick rotate to Euclidean space q 0 → iq 4 d 4 q E 1 E − λ � − A ( − p 2 E ) p 2 E − B ( − p 2 E ) = − p 2 Aq 2 (2 π ) 4 2 E + B A = 1 � Λ q 3 E dq E 1 α = 2 π 2 E + m 2 + λα / 2 q 2 (2 π ) 4 0 1 2 λπ 2 Λ 2 π 2 α m 2 = m 2 Λ 2 = 0 + 1 − 1 1 − 1 2 λπ 2 2 λπ 2
Ladder QED − 1 − 1 = − � � − i g µ ν − (1 − � ) k µ k ν D µ ν = k 2 + i � k 2 i S ( p ) = A ( p 2 )/ p − B ( p 2 ) d 4 q γ µ D µ ν ( p + q )( A / q + B ) γ ν � p − m − e 2 A / p − B = / (2 π ) 4 A 2 q 2 − B 2
Ladder QED d 4 q γ µ D µ ν ( p + q )( A / q + B ) γ ν � p − m − e 2 A / p − B = / (2 π ) 4 A 2 q 2 − B 2 A ( ξ = 0) = 1 d 4 q (3 + 0) B ( q 2 ) � B ( p 2 ) = m − ie 2 (2 π ) 4 ( q 2 − B 2 )( p − q ) 2 � 1 � 1 q 2 θ ( q > p ) + 1 � ( p − q ) 2 = 2 π 2 p 2 θ ( p > q ) d Ω 4
ladder QED � p E � Λ � � E ) = 3 e 2 1 B ( q ) B ( q ) B ( − p 2 dqq 3 q 2 + B 2 + dqq q 2 + B 2 p 2 8 π 2 0 p E E � p E E B � = − 3 e 2 B d p 4 dqq 3 � = q 2 + B 2 8 π 2 dp 2 0 E E B � ) � = − 3 e 3 B ( p 4 16 π 2 p 2 E p 2 E + B 2
ladder QED E B � ) � = − 3 e 3 B ( p 4 16 π 2 p 2 E p 2 E + B 2 � p E � Λ � � E ) = 3 e 2 1 B ( q ) B ( q ) B ( − p 2 dqq 3 q 2 + B 2 + dqq q 2 + B 2 p 2 8 π 2 0 p E E ( p 4 B � ) | p =0 = 0 ( B + p 2 B � ) | p = Λ = 0 B → p − 1 ± √ α > α � ≡ π 1 − 3 e 2 / (4 π 2 ) 3
numerical methods expand A(p) in a convenient basis � A ( p ) = c i T i ( p ) i discretise A ( p ) → A i = A ( p i ) x i = f i ( { x } )
numerical methods x i = f i ( { x } ) (i) iterate x i | n +1 = f i ( { x } n ) (ii) iterative Newton-Raphson � � − δ ij + ∂ f i � | x i δ x j = x i − f i ( { x } ) ∂ x j j x i | n +1 = x i | n + δ x i (iii) minimise � ( x i − f i ( { x } )) 2 G ( { x } ) = i
final words Thus it is vital that the practitioner not abandon theoretical investigations too early. One must carefully track and deal with singularities in the equations, understand asymptotic behaviour, and develop decent analytic approximations to have any hope • All of the techniques discussed here will fail miserably unless one starts very close to the solution. • how does one truncate SD equations (beyond convenience)? • be prepared for heartbreak
QCD ghost quark - 1 - 1 - 1 - 1 = = - - gluon - 1 - 1 = - - - - - -
Exotic Theory: Schwinger-Dyson Equations J. Meyers, PhD Thesis, Pittsburgh, 2014.
Bethe-Salpeter
2VPI = + regular = = 2VPI
Exotic Theory: Schwinger-Dyson Equations electron photon − 1 − 1 − 1 − 1 − 1 = = − − E.S. Swanson, arXiv 1008.4337
Results: QED3 P.M. Lo, E.S. Swanson, PRD83, 065006 (2011) P.M. Lo, E.S. Swanson, PRD81, 034030 (2010) BC+CP BC RL CBC 1 . 00 · N A N � (CBC) = � � � � 2 π � ¯ 1 . 10 · N A N � (RL) = ψψ � ( N f ) = aN f exp � � N � /N f � 1 N � (BC) = 1 . 21 · N A N � (CP) = �
Results: QED3 P.M. Lo, E.S. Swanson, PRD83, 065006 (2011) P.M. Lo, E.S. Swanson, PRD81, 034030 (2010) parity preserving maximal parity breaking RL parity symmetric: solns are (M,-M) maximally broken (M,M) A solution exists for eta<0.4, implying parity symmetry breaking! CBC solution no solution η = N + − N − N + + N −
Results: QED3, finite temperature and density P.M. Lo, E.S. Swanson, PRD89, 025015 (2014) P.M. Lo, E.S. Swanson, PLB697, 164 (2011) 0.2 symmetric First calculation with full frequency depencence in a gauge theory. First 0.15 calc in QED3 the treat the IR-div seriously. T c 0.1 broken 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 µ
Results: QCD propagators J. Meyers, E.S. Swanson, PRD90, 045037 (2014) 3.5 1 - 1 3 0.8 2.5 - M [GeV], 1/A 2 0.6 Z NOT TRUE that the mass of the visible 1.5 universe comes from the Higgs… it 0.4 comes from this —-> 1 0.2 0.5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8 p [GeV] p [GeV] 11 4 10 - 1 3.5 9 3 8 - 7 2.5 G [GeV -2 ] 6 2 h 5 1.5 4 3 1 2 0.5 1 0 0 0.01 0.1 1 10 100 0.01 0.1 1 10 100 p 2 [GeV 2 ] p 2 [GeV 2
Exotic Theory: Schwinger-Dyson and Bethe-Salpeter Equations J. Meyers, E.S. Swanson, PRD87, 036009 (2013) k + ; ¹ 2VPI P = k ¡ ;º = 2VPI + + +... +
Results: Glueballs d 4 q � (2 π ) 4 χ αβ ( q + , q − ) C αβ ig 2 N χ µ ν ( k + , k − ) = ..µ ν G ( q + ) G ( q − ) d 4 q � (2 π ) 4 χ αβ ( q + , q − ) T αβ + ig 2 N ..µ ν ( q + , q − , k + , k − ) G ( q + ) G ( q − ) G ( Q ) d 4 q � + ig 2 N (2 π ) 4 χ ( q + , q − ) G µ ν ( q + , q − , k + , k − ) H ( q + ) H ( q − ) H ( Q ) + ig 2 � tr [ γ µ S ( q + ) χ χ ( q + , q − ) S ( q − ) γ ν S ( Q )] 2 d 4 q � ig 2 N χ ( k + , k − ) = (2 π ) 4 χ ( q + , q − ) H ( q + , q − , k + , k − ) H ( q + ) H ( q − ) G ( Q ) d 4 q � (2 π ) 4 χ αβ ( q + , q − ) B αβ ( q + , q − , k + , k − ) G ( q + ) G ( q − ) H ( Q ) + ig 2 N � g 2 C F χ ( k + , k − ) = γ α S ( k − + q − ) γ β G ( q + ) G ( q − ) χ αβ ( q + , q − ) χ � + ig 2 C F γ µ S ( q + ) χ χ ( q + , q − ) S ( q − ) γ ν G ( Q ) P µ ν ( Q )
Results: Glueballs J. Meyers, E.S. Swanson, PRD87, 036009 (2013) 4 3.5 3 0 ++ 0 -+ 2.5 1/ � 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 |P| (GeV)
Schwinger-Dyson Equations − 1 − 1 − 1 − 1 − 1 = = − − Vertex Ansa ̎ tze iS − 1 = A / p − B i Γ µ RL ( k, p ) = γ µ CBC ( k, p ) = 1 i Γ µ 2( A ( k ) + A ( p )) γ µ BC ( k, p ) = 1 2 ( A ( k ) + A ( p )) γ µ + 1 A ( k ) − A ( p ) p )( k µ + p µ ) − B ( k ) − B ( p ) ( k µ + p µ ) i Γ µ (/ k + / 2 k 2 − p 2 k 2 − p 2 CP ( k, p ) = 1 A ( k ) − A ( p ) γ µ ( k 2 − p 2 ) − ( k + p ) µ (/ i Γ µ � � k − / p ) 2 d ( k, p ) d ( k, p ) = ( k 2 − p 2 ) 2 + ( M ( k ) 2 + M ( p ) 2 ) 2 k 2 + p 2
k + ; ¹ P = + k ¡ ;º + = + = +
results: ghost 4 3.5 3 2.5 h 2 1.5 1 0 2 4 6 8 10 12 14 16 18 20 p 2 (GeV 2 ) I.L. Bogolubsky, E.M. Ilgenfritz, M. Muller-Preussker and A. Sternbeck, Phys. Lett. B676 , 69 (2009)
results: quark f π = 240 MeV � ¯ ψψ � (1 GeV) = ( � 251 MeV) 3 Z ( k ) S ( k ) = i 1 Z k − M ( k ) / 0.8 Z, M (GeV) 0.6 0.4 M 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 p (GeV) P.O. Bowman et al., Phys. Rev. D71 , 054507 (2005).
more on confinement D ( p ) = − 1 1 p 2 1 + u ( p ) u ( p ) → − 1 p → 0 Kugo-Ojima confinement criterion Alkofer, von Smekal, Fischer
Heavy Quarks modelling, quarks, heavy quarks
challenges QCD is • many body • relativistic • strong coupling (contrast to QED) • quantum • nonlinear
Modelling QCD ‣ physical pictures can change depending on the ‣ scale quark mass, glue ‣ observables ρ decay vs. ρ scattering ‣ gauge confinement in Coulomb gauge vs. Weyl gauge
Modelling QCD ‣ we seek to understand low energy hadronic physics -- we are fortunate that we have the theory, but it is not always helpful! h 2 − ¯ q i q j ‣ cf. the theory of DNA: � � 2 m ∇ 2 H = i + r ij i i<j ‣ to make progress we need to identify the appropriate effective degrees of freedom and their interactions ex: the bag model fermions current quarks bosons bag pressure (+ perturbative one gluon exchange) ex: the flux tube model fermions constituent quarks bosons flux tubes
Modelling QCD ‣ spontaneous chiral symmetry breaking implies both the existence of Goldstone bosons and constituent quarks current quarks evolve into constituent quarks at scales < Λ QCD ‣ it is the structure of the vacuum that gives chiral symmetry breaking and confinement it is desirable to incorporate the physics of the vacuum and chiral symmetry breaking into the model from the beginning ‣ effective degrees of freedom should be derived from QCD to the extent possible only in this way can we recover perturbative QCD in the high energy regime
Constituent Quarks pre-QCD quarks: m~ 5 GeV Copley, Karl, & Obryk: m ~ 330 MeV QCD: m(2 GeV) ~ 4 MeV but recall that quarks are not observable ⇒ different kinds of quark masses exist: current/constituent
��.��1���������������������������������� 4 3 2 ~ ~ ~ �����((������((�������((((������� 4 2 3 ✝ ✝ ✝ ✝ ✝ ✝ ������������� * �-����������-� k,m -k,m k,m -k,m ��� ��1����)+������ ����1����)+� ��) ��1����)+������ ����1����)+�
Cb pm C † = η C d pm Cd † pm C † = η C b † pm
(2 S +1) L J example L S J P C 0 0 0 − + 1 S 0 π 3 S 1 0 1 1 −− ρ 1 0 1 + − 1 P 1 h 1 1 1 (0 , 1 , 2) ++ 3 P (0 , 1 , 2) a 0 , a 1 , a 2 2 0 2 − + 1 D 2 π 2 2 1 (1 , 2 , 3) −− 3 D (1 , 2 , 3) ρ , ρ 2 , ρ 3 -+ +- -- not in the list: 0 , (even) , (odd) ‘(quantum number) exotics’ discovering such a state would be the first time a meson has been observed with no qq content
Heavy Quarkonia ψ , Υ • Bohr levels with a Bohr radius (2/3m α ) ~ 0.01 s fm 3 3 3 1 • L , L , L & L splittings are due to tensor L+1 L-1 L L and spin-orbit interactions 3 1 • S - S splittings are due to the contact 1 0 interaction
Constituent Quarks (heavy) spatial regimes: r ∼ 1 / 2 fm r < 1 / 10 fm r > 1 fm confinement one gluon exchange one pion exchange Lorentz structure: sources of spin-dependence are (i) gluon exchange (ii) corrections to the static potential (iii) meson exchange (Fock sector mixing) (iv) instanton forces
make a (field-theoretic) Foldy-Wouthuysen transformation
Interactions sig1_i*sig1_j = del_ij+ eps_ijk sig k so = B^2 so not spin- dependent σ .B σ .B σ .B σ .B (b) (a) zero hyperfine + tensor D 2 σ .B D 2 σ .B (d) (c) V V 1 2
spin-dependence in the confinement potential examine in Coulomb gauge via the Foldy - W outhuysen transformation
model building — more later V SI ( r ) = − 3 α s r + br 4 � σ q � � 1 + 2 � � σ ¯ q + σ q � � 1 � dV conf dV 1 dV 2 + σ ¯ q V SD ( r ) = + · L · L 4 m 2 4 m 2 2 m q m ¯ r dr r dr r dr q q q ¯ 1 1 � � + 3 σ q · ˆ q · ˆ V 3 + q V 4 r σ ¯ r − σ q · σ ¯ σ q · σ ¯ q 12 m q m ¯ 12 m q m ¯ q q �� σ q � � σ q − σ ¯ � � +1 − σ ¯ q q · L + (1) · L V 5 . m 2 2 m 2 m q m ¯ q q ¯ q Eichten & Feinberg Ng, Pantaleone, & Tye
oge approximation/model spin dependence ⃗ 2 · − ⃗ ∗ λ 1 λ 2 U = ( V C + V so + V hyp ) 2 2 ( 1 1 V C = α r − απ 1 + 2 ) δ ( ⃗ r ) m 2 m 2 � � − 3( ⃗ r )( ⃗ r ) σ 1 · ⃗ σ 2 · ⃗ ⃗ σ 1 · ⃗ σ 2 − 8 π α V hyp = σ 2 δ ( ⃗ r ) 3 ⃗ σ 1 · ⃗ r 3 r 5 4 m 1 m 2 � � r · ( ⃗ p 1 ) ⃗ p 2 + ⃗ r · ⃗ p 2 4 r 3 ( ⃗ r × ⃗ p 1 · ⃗ σ 1 − ⃗ r × ⃗ p 2 · ⃗ σ 2 α α V so = − ) ⃗ p 1 · ⃗ − r 2 m 2 m 2 2 m 1 m 2 r 1 2 α 2 m 1 m 2 r 3 ( ⃗ σ 1 ) r × ⃗ p 1 · ⃗ σ 2 − ⃗ r × ⃗ p 2 · ⃗ −
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