Gluon and Ghost Propagators from Schwinger-Dyson Equation and - PowerPoint PPT Presentation

Gluon and Ghost Propagators from Schwinger-Dyson Equation and Lattice Simulations Joannis Papavassiliou Departament of Theoretical Physics and IFIC, University of Valencia – CSIC, Spain Approaches to QCD, Oberwölz, Austria, 7-13th September 2008

Outline of the talk Gauge-invariant gluon self-energy in perturbation theory Field theoretic framework: Pinch Technique Beyond perturbation theory: gauge-invariant truncation of Schwinger-Dyson equations Dynamical mass generation IR finite gluon propagator from SDE and comparison with the lattice simulations Obtaining physically meaningful quantities Conclusions

� ( q ) is independent of the gauge-fixing parameter to all �� Vacuum polarization in QED (prototype) �( q 2 ) = [ 1 + �( q 2 )℄ e Π µν ( q ) = � 1 = Z + �( q 2 ) = Z A [ 1 + � 0 ( q 2 )℄ q q � 1 = 2 e = Z 2 and Z e = Z � orders � ( q 2 ) = � 0 ( q 2 ) = e 2 �( q 2 ) = ) 1 +�( q 2 ) 1 q 2 e 0 and 1 e e From QED Ward identity follows Z 1 A RG-invariant combination e 2 0

� ( q ; � ) depends on the gauge-fixing parameter already �� Gluon self-energy in perturbation theory k + q k + q q q q q 1 � ( q ) = � � �� � + � 2 k k 6 = Z 2 in general) at one-loop (a) (b) Ward identities replaced by Slavnov-Taylor identities involving ghost Green’s functions. ( Z 1

� � ( q ) = 0 �� Difficulty with conventional SD series q The most fundamental statement at the level of Green’s functions that one can obtain from the BRST symmetry . It affirms the transversality of the gluon self-energy and is valid both perturbatively (to all orders) as well as non-perturbatively . Any good truncation scheme ought to respect this property Naive truncation violates it

� − 1 − 1 − 1 � ( q ) j 6 = 0 �� ∆ ( q ) ( a )+( b ) + 1 + 1 = = µν 2 2 µ ν ( b ) ( a ) � � ( q ) j 6 = 0 �� ( a )+( b )+( c ) + + 1 + 1 6 2 ( c ) ( d ) ( e ) q q Main reason : Full vertices satisfy complicated Slavnov-Taylor identities.

" # � k � ( 0 ) � ! � ( k ) = � ( 1 � � ) k Pinch Technique �� �� Diagrammatic rearrangement of perturbative " # expansion (to all orders) gives rise to effective + n ( 0 ) � k � � k � Green’s functions with special properties . � ! � ( k ) = � n �� �� J. M. Cornwall , Phys. Rev. D 26 , 1453 (1982) J. M. Cornwall and J.P. , Phys. Rev. D 40 , 3474 (1989) D. Binosi and J.P. , Phys. Rev. D 66 , 111901 (2002). � � = ( k = + p = � m ) � ( p = � m ) � 1 In covariant gauges: � 1 i � 1 g = ( k + p ) � S ( p ) ; k 2 k 2 1 In light cone gauges: i g k 2 nk k S 0 0

Pinch Technique rearrangement pinch ✲                                  pinch  ✲                                  pinch ✲ � ∆

Gauge-independent self-energy + + b h � �i �( q 2 ) = + + = b � ( q ) �� + bg 2 ln � 2 = 11 C A = 48 � 2 � -function � = � bg 3 ) in the absence of quark loops. 1 q 2 q 2 1 first coefficient of the QCD b (

� � � � 1 � 1 e � ( p 1 ; p 2 ) = ( p 2 ) � S ( p 1 ) � � � � � 1 � 1 e � ( q 1 ; q 2 ; q 3 ) = � ( q 2 ) � � ( q 3 ) ��� �� �� = ) easy to calculate Simple, QED-like Ward Identities , instead of Slavnov-Taylor Identities, to all orders q I g S abc gf abc I q 1 Profound connection with Background Field Method D. Binosi and J.P. , Phys. Rev. D 77 , 061702 (2008); arXiv:0805.3994 [hep-ph] � Π µν ( q ) = + q q q q Can move consistently from one gauge to another (Landau to Feynman, etc) A. Pilaftsis , Nucl. Phys. B 487 , 467 (1997)

� 1 = 2 b b b = ; Z g = b b = ) RG invariant combination � 0 ( q 2 ) = g 2 �( q 2 ) Restoration of: ( � ) = 4 � � ( q 2 ) = = + bg 2 ( � ) ln ( q 2 =� 2 ) � b ln ( q 2 = � 2 ) Abelian Ward identities Z 1 Z 2 Z A g 2 0 For large momenta q 2 , define the RG-invariant effective charge of QCD, g 2 1 1 4

Beyond perturbation theory ... ^ 2 d(q ) Non-perturbative effects Lattice, Schwinger-Dyson equations Asymptotic Freedom 2 q 2

New SD series The new Schwinger-Dyson series based on the pinch technique − 1 ˆ − 1 ∆ ( q ) + 1 + 1 = µν µ ν 2 2 ( a 2 ) ( a 1 ) + 1 + 1 + + 6 2 ( b 2 ) ( b 1 ) ( c 1 ) ( c 2 ) + + + + ( d 1 ) ( d 2 ) ( d 3 ) ( d 4 ) Transversality is enforced separately for gluon- and ghost-loops, and order-by-order in the “dressed-loop” expansion! A. C. Aguilar and J. P. , JHEP 0612 , 012 (2006) D. Binosi and J. P. , Phys. Rev. D 77 , 061702 (2008); arXiv:0805.3994 [hep-ph].

� � ( q ) j = 0 �� ( a 1 )+( a 2 ) Transversality enforced loop-wise in SD equations k k + q → → β, x σ, e ρ, c σ, d The gluonic contribution → → → → q q q q 1 1 � I Γ 2 2 µ, a ν, b µ, a ν, b q � � ( q ) j = 0 α, c �� ( b 1 )+( b 2 ρ, d ) ( a 2 ) k ← ( a 1 ) k → k + q → c c c ′ d The ghost contribution → → q → → q q q � I Γ µ, a µ, a ν, b ν, b q ( b 2 ) x x ′ k ← ( b 1 )

�( q 2 ) = [ 1 + �( q 2 )℄ �( q 2 ) has a pole at q 2 = 0 the vector meson is massive , Dynamical mass generation: Schwinger mechanism in 4-d � 1 = q 2 1 q 2 If even though it is massless in the absence of interactions. J. S. Schwinger, Phys. Rev. 125 , 397 (1962); Phys. Rev. 128 , 2425 (1962). Requires massless, longitudinally coupled , Goldstone-like poles Such poles can occur dynamically , even in the absence of canonical scalar fields. Composite excitations in a strongly-coupled gauge theory. R. Jackiw and K. Johnson, Phys. Rev. D 8 , 2386 (1973) J. M. Cornwall and R. E. Norton, Phys. Rev. D 8 (1973) 3338 E. Eichten and F. Feinberg, Phys. Rev. D 10 , 3254 (1974)

Ansatz for the vertex � � � e � = � + i q � ( k + q ) � � ( k ) ; ��� ��� �� �� = + + . . . + 1 /q 2 pole � � � � 1 � 1 e � ( q 1 ; q 2 ; q 3 ) = gf abc � ( q 2 ) � � ( q 3 ) ��� �� �� Gauge-technique Ansatz for the full vertex: � 1 � 1 = q 2 , instrumental for � ( 0 ) 6 = 0 I q 2 Satisfies the correct Ward identity abc q I 1 Contains longitudinally coupled massless bound-state poles

Z Z � 1 � ( q 2 ) = + c 1 �( k )�( k + q ) f 1 ( q ; k ) + c 2 �( k ) f 2 ( q ; k ) � � Z ( p � k ) 2 � 1 ( p 2 ) = + c 3 � �( k ) D ( p + k ) ; System of coupled SD equations � 1 � ( 0 ) 6 = 0 q 2 k k p 2 p 2 D k 2 k Infrared finite Renormalize Solve numerically A. C. Aguilar, D. Binosi and J. P. , Phys. Rev. D 78, 025010 (2008) .

Numerical results and comparison with lattice Use lattice to calibrate the SDE solution. =5.7 L=64 8 =5.7 L=72 =5.7 L=80 SDE solution = 4.5 GeV 6 ) 2 (q 4 2 0 1E-3 0,01 0,1 1 10 100 1000 2 2 q [GeV ] I. L. Bogolubsky, et al , PoS LAT2007 , 290 (2007)

Ghost propagator 3,0 1,3 Ghost dressing function 2,8 Ghost dressing function lattice =5.7 SDE solution =m 2,6 =m b 1,2 b 2,4 2,2 1,1 ( p 2 ) ! constant ) ) 2,0 2 D(p ) 2 D(p 1,8 2 p 2 1,0 p 1,6 1,4 0,9 1,2 1,0 0,8 0,8 0,01 0,1 1 10 0,01 0,1 1 10 100 1000 p 2 [GeV 2 ] p 2 [GeV 2 ] In the deep IR p 2 D No power-law enhancement

b �( q 2 ) and the PT-BFM �( q 2 ) are � � 2 b �( q 2 ) = + G ( q 2 ) �( q 2 ) Making contact with physical quantities The conventional related by 1 Z � � ( k � q ) 2 Formal relation derived within Batalin Vilkovisky ( q 2 ) = � C A g 2 + �( k ) D ( k + q ) : formalism D. Binosi and J. P., Phys. Rev. D 66, 025024 (2002) . ∆ Auxiliary Green’s function related G ( q ) = to the full gluon-ghost vertex D 2 G 3 k 2 q 2 k

� function coefficient in front of UV logarithm. + G ( q 2 ) = 1 + 9 ( q 2 =� 2 ) 48 � 2 ln � � � 1 � ( q 2 ) = q 2 + 13 ( q 2 =� 2 ) 48 � 2 ln Enforces + C A g 2 1 4 � � � 1 b � ( q 2 ) = q 2 + 11 C A g 2 ( q 2 =� 2 ) 48 � 2 ln C A g 2 1 2 1

Numerical Results 1,2 1,1 ) 2 1+G(q 1,0 0,9 0,8 1E-3 0,01 0,1 1 10 100 1000 q 2 [GeV 2 ]

Numerical Results ^ ^ 28 2 2 2 d(q )= g (q ) = M Z = M 24 b 20 16 ) 2 d(q ^ 12 8 4 0 1E-3 0,01 0,1 1 10 100 1000 q 2 [GeV 2 ]

b b ( q 2 ) = g 2 �( q 2 ) , has the form: ( q 2 ) b ( q 2 ) = + m 2 ( q 2 ) Physically motivated fit: Cornwall’s massive propagator � q 2 � ( q 2 ) = + 4 m 2 ( q 2 ) The RG invariant quantity, d � 2 g 2 d q 2 " # � 12 = 11 � � � � . + 4 m 2 ( q 2 ) = m 2 � 2 � 2 where the running charge is 1 g 2 b ln and the running mass q 2 4 m 2 m 2 0 0 ln ln 0 J. M. Cornwall , Phys. Rev. D 26 , 1453 (1982)