Effective gluon mass and freezing of the QCD coupling J. Papavassiliou Department of Theoretical Physics and IFIC, University of Valencia–CSIC Based on: A. C. Aguilar and J. Papavassiliou, In preparation A. C. Aguilar and J. Papavassiliou “Gluon mass generation in the PT-BFM scheme,” JHEP 0612 , 012 (2006) [arXiv:hep-ph/0610040] J. Papavassiliou Effective gluon mass and freezing of the QCD
� ( q ) �� � � � q � � q � � ( q ) = � q �( q 2 ) + q : �� �� � 1 � ( 0 ) 6 = 0. Acts as an effective “mass” for General Considerations Gluon “propagator” i g q 2 q 4 Dynamical generation of an infrared cutoff. The QCD dynamics allow for the gluons. Cornwall, Phys. Rev. D 26 , 1453 (1982) 24 20 16 ) 2 (q 12 8 4 0 - 3 - 1 1 3 10 10 10 10 q 2 [GeV 2 ] J. Papavassiliou Effective gluon mass and freezing of the QCD
� in the QCD = m ( q 2 ) . = ) QCD remains Does not correspond to a term m 2 A 2 Lagrangian. The local gauge symmetry remains exact. Not hard but momentum dependent mass: m Drops off in the UV “sufficiently fast”. renormalizable Purely non-perturbative effect: Lattice (discretized space-time) Schwinger-Dyson equations (continuum). J. Papavassiliou Effective gluon mass and freezing of the QCD
Schwinger-Dyson Equation � � � 1 ( a 1 ) ( a 2 ) � ( q 2 ) P ( q ) = q 2 P ( q ) + i � ( q ) + � �� �� �� �� Z 0 0 ( a 1 ) �� � � e 0 0 � ( q ) = [ dk ℄� � ( k ) � � ( k + q ) �� ��� � � � Z ( a 2 ) � = � C A g 2 g [ dk ℄�( k ) �� �� 1 2 C A g 2 I J. Papavassiliou Effective gluon mass and freezing of the QCD
��� ��� ��� ��� e � = L + T + T � Vertex ��� ��� �� q e ( q ; p 1 ; p 2 ) = � ( q ; p 1 ; p 2 ) + ig [�( p 2 ) � �( p 1 )℄ � � ��� � g �� � g �� The expression for the vertex that we will use is given by ( q ; p 1 ; p 2 ) = � i c 1 � q [�( p 1 ) + �( p 2 )℄ " # � � �( p 1 ) �( p 2 ) ��� � g �� � g �� ( q ; p 1 ; p 2 ) = � ic 2 � q + I 1 2 with e � ( q ; p 1 ; p 2 ) = ( p 1 � p 2 ) + 2 q � 2 q ��� � g �� � g �� � g �� L q 2 T q 1 q 2 T q 2 p 2 p 2 1 2 J. Papavassiliou Effective gluon mass and freezing of the QCD
X � 1 � 1 � ( x ) = Kx + bg 2 ( x ) + � ( 0 ) = 1 Z Z x 1 SD equation ( x ) = �( x ) �( y ) ( x ) = � 2 ( y ) Z Z x 1 ( x ) = �( y ) ( x ) = �( y ) 8 Z x Z x a i A i �( x ) ( x ) = �( x ) �( y ) ( x ) = �( y ) i Z x Z x ( x ) = � 2 ( y ) ( x ) = � 2 ( y ) � 1 dyy 2 � ( � 2 ) = � 2 A 1 a 1 x dyy A 5 a 5 0 x A 6 a 6 dyy A 2 a 2 x dy 0 x dyy 3 A 3 a 3 x dyy A 7 a 7 x 0 0 1 dyy 2 dyy 3 A 4 a 4 A 8 a 8 x 0 0 The renormalization condition K is fixed by J. Papavassiliou Effective gluon mass and freezing of the QCD
Z x Z x � 2 � 2 � 1 ~ ~ ( x ) ln x = g � ( 0 ) + � 1 ( y ) � ( y ) + ( y ) � ( y ) ~ � ( q 2 ) = ; The UV behavior of effective gluon mass + m 2 ( q 2 ) � 4 m 2 dy m 2 dy ym 2 � 2 � 1 �� ( x ) = ( ln x ) = ) h G a i �� G x 0 0 with 1 q 2 The asymptotic solutions: m 2 2 a x J. Papavassiliou Effective gluon mass and freezing of the QCD
( q 2 ) = g 2 �( q 2 ) , has the general form: ( q 2 ) ( q 2 ) = ; + m 2 ( q 2 ) ( q 2 ) and effective charge g 2 ( q 2 ) are Propagator and Running Masses � � q 2 � � �� The RG quantity, d � 2 � 1 . + � m 2 � m 2 ( q 2 ) = + m 2 � 2 � 2 g 2 d q 2 � � q 2 � � � 1 + � m 2 ( q 2 ) ( q 2 ) = where the dynamical mass m 2 � 2 m 4 m 2 0 0 0 ln ln q 2 0 g 2 b ln J. Papavassiliou Effective gluon mass and freezing of the QCD
Numerical Results J. Papavassiliou Effective gluon mass and freezing of the QCD
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