Infrared Finite Effective Charge of QCD Joannis Papavassiliou - PowerPoint PPT Presentation

Infrared Finite Effective Charge of QCD Joannis Papavassiliou Departament of Theoretical Physics and IFIC, University of Valencia – CSIC, Spain Light Cone 2008, Mulhouse, France 7-11th July 2008 Joannis Papavassiliou Light Cone 2008 1/ 27

Outline of the talk Effective charge of QED (prototype) QCD effective charge in perturbation theory Field theoretic framework: Pinch Technique Beyond perturbation theory: Schwinger-Dyson equations and lattice Dynamical mass generation IR finite gluon propagator and effective charge The role of the quarks Conclusions Joannis Papavassiliou Light Cone 2008 2/ 27

� ( q 2 � ) is defined from the vacuum �( q ) . � q � ( q ) = g � q �� �� � ( q ) = � i P ( q )�( q 2 ) �� �� Effective charge of QED (prototype) Textbook construction: �( q 2 ) = [ 1 + �( q 2 )℄ polarization 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 P q 2 � � ( q 2 ) = � 0 ( q 2 ) = e 2 �( q 2 ) = ) +�( q 2 ) 1 q 2 e 0 and 1 e e From QED Ward identity follows Z 1 A RG-invariant combination e 2 1 0 Joannis Papavassiliou Light Cone 2008 3/ 27

Properties Gauge-independent (to all orders) Renormalization group invariant, � m 2 Universal (process-independent) 2 p p p 1 1 1 k 1 Non-trivial dependence on the masses m i of the particles in = the loop. Reconstruction from physical amplitudes, using p p p 2 2 2 k 2 = 4 � optical theorem and dispersion relations. � ( q 2 ) ! � e 2 b log ( q 2 = m 2 ) = � 2 n f [ n f = number of fermion flavors]. For q 2 i , the effectice charge coincides with the running coupling (solution of RG equation). e 2 1 f 1 where b 6 Joannis Papavassiliou Light Cone 2008 4/ 27

6 = Z 2 in general) � ( q ) depends on the gauge-fixing parameter already at �� QCD effective charge in perturbation theory Ward identities replaced by Slavnov-Taylor identities k + q k + q involving ghost Green’s functions. ( Z 1 q q q q 1 � ( q ) = � � � � �� + 2 k k one-loop (a) (b) 2 p 1 p 1 p 1 k 1 Optical theorem does not hold for individual Green’s 6 = p p p 2 2 2 k 2 functions Joannis Papavassiliou Light Cone 2008 5/ 27

" # � 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 Joannis Papavassiliou Light Cone 2008 6/ 27

Pinch Technique rearrangement pinch ✲                                 pinch   ✲                                  pinch ✲ � ∆ Joannis Papavassiliou Light Cone 2008 7/ 27

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 ( Joannis Papavassiliou Light Cone 2008 8/ 27

� � � � 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 q I 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 Joannis Papavassiliou Light Cone 2008 9/ 27

� 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 Strong version of optical theorem J.P., E. de Rafael and N.J.Watson, Nucl. Phys. B 503 , 79 (1997) Joannis Papavassiliou Light Cone 2008 10/ 27

Beyond perturbation theory ... Joannis Papavassiliou Light Cone 2008 11/ 27

Non-perturbative tools Lattice QCD (discretization of space-time) Schwinger-Dyson equations (continuous approach) k + q q ( a 2 ) ( a 4 ) ( a 1 ) ( a 3 ) k k, σ k ) − 1 = ( H σν ( k, q ) = H (0) q, ν σν + ( ) − 1 + p p p p + k k + q A.C. Aguilar, D. Binosi, J. P. , arXiv:0802.1870 [hep-ph], Phys. Rev. D (in press) Joannis Papavassiliou Light Cone 2008 12/ 27

� � ( 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 ) Joannis Papavassiliou Light Cone 2008 13/ 27

�( 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) Joannis Papavassiliou Light Cone 2008 14/ 27

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 Joannis Papavassiliou Light Cone 2008 15/ 27

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 System of coupled SD equations � 1 ( p 2 ) = + c 3 � �( k ) D ( p + k ) ; q 2 k k p 2 p 2 D k 2 k Renormalize Solve numerically Joannis Papavassiliou Light Cone 2008 16/ 27

Gluon propagator (Landau gauge) I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preussker and A. Sternbeck, PoS LATTICE, 290 (2007). P. O. Bowman et al., Phys. Rev. D 76 , 094505 (2007) A. Cucchieri and T. Mendes, PoS LATTICE, 297 (2007). Joannis Papavassiliou Light Cone 2008 17/ 27

Ghost propagator No power-law enhancement Joannis Papavassiliou Light Cone 2008 18/ 27

The physical picture = m ( q 2 ) Dynamical generation of an infrared cutoff . L QCD . � in J. M. Cornwall, Phys. Rev. D 26 , 1453 (1982); A.C.Aguilar, A.A.Natale and P.S.R. da Silva, Phys. Rev. Lett. 90 , 152001 (2003); A. C.Aguilar and J.P., JHEP 0612 , 012 (2006);. A.C.Aguilar, D. Binosi, J.P., arXiv:0802.1870 [hep-ph], Phys. Rev. D (in press). Acts as an effective “mass” for the gluons. Not hard but momentum dependent mass m Drops off “sufficiently fast” in the UV. Eur.Phys.J.A35:189-205 (2008) . A. C. Aguilar and J.P., Does not induce to a term m 2 A 2 The local gauge symmetry remains exact . Joannis Papavassiliou Light Cone 2008 19/ 27

The physical picture ... = ) Finite threshold for popping The “mass” is not directly measurable. Must be related to glueball masses, string tension, and condensates . J. M. Cornwall, Phys. Rev. D 26 , 1453 (1982) Phys. Rev. D 44 , 26 (1991) . M.Lavelle, = ) m ( 0 ) = 500 � 200 MeV Potential energy of a pair of heavy, static sources in the adjoint (adjoint Wilson Loop ). Flux tube formed dynamical gluons out of the vacuum. C. Bernard , Nucl. Phys. B 219 :341,1983 Bag Model : Gluon production requires a net energy cost because of confinement. Acts like a constituent quark mass John F. Donoghue , Phys.Rev.D29:2559,1984 Phenomenological studies F.Halzen, G.I.Krein and A.A.Natale, Phys. Rev. D 47 , 295 (1993). G.Parisi and R.Petronzio, Phys. Lett. B 94 , 51 (1980). A.C.Aguilar, A.Mihara and A.A.Natale, Phys. Rev. D 65 , 054011 (2002). Joannis Papavassiliou Light Cone 2008 20/ 27