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

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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

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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

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Vacuum polarization in QED (prototype)

Πµν(q) =

q q e e

(q

) is independent of the gauge-fixing parameter to all

rders

(q2 ) =

1 q2

[1 + (q2 )℄

e

= Z 1

e

e0 and 1

+ (q2 ) = ZA [1 + (q2 )℄

From QED Ward identity follows Z1

= Z2 and Ze = Z 1 =2

A

RG-invariant combination

e2

(q2 ) = e2 (q2 ) = ) (q2 ) =

1+(q2

)

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Gluon self-energy in perturbation theory

(q

;

) depends on the gauge-fixing parameter already

at one-loop

(q

) = 1 2

(a)
k

+ q k q q +

k

+ q k q q (b)

Ward identities replaced by Slavnov-Taylor identities involving ghost Green’s functions. (Z1

6= Z2 in general)

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Difficulty with conventional SD series

q

(q

) = 0

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

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(a) (b) (d) (e)

+1

2

+1

6

+1

2

+1

2

−1

+

(c)

∆ (q)

µν −1 = = −1 ν µ

q

(q

)j (a )+(b ) 6= 0

q

(q

)j (a )+(b )+(c ) 6= 0

Main reason : Full vertices satisfy complicated Slavnov-Taylor identities.

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Pinch Technique

Diagrammatic rearrangement of perturbative expansion (to all orders) gives rise to effective Green’s functions with special properties .

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).

In covariant gauges:

!

i

(0

)

(k

) = "

g

(1
)k

k

k 2

#

1 k 2

In light cone gauges:

!

i

(0

)

(k

) = "

g

n

k

+ n

k

nk

#

1 k 2 k

=

(k = + p = m )

(p

= m ) =

S

1 (k + p ) S 1 (p ) ;

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Pinch Technique rearrangement

pinch ✲                                                                    pinch ✲ pinch ✲

∆

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Gauge-independent self-energy

+ + + + = b

(q

) b (q2 ) =

1 q2

h

1

+ bg2 ln

q2

2 i

b

= 11CA =48 2

first coefficient of the QCD

function

(

=

bg3) in the absence of quark loops.

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Simple, QED-like Ward Identities , instead of Slavnov-Taylor Identities, to all orders

q

e

I

(p1

; p2 ) =

g

S

1 (p2 ) S 1 (p1 )

q
1

e

I

abc
(q1

; q2 ; q3 ) =

gf abc

1
(q2

)

1
(q3

)

Profound connection with

Background Field Method

= ) easy to calculate

D. Binosi and J.P. , Phys. Rev. D 77, 061702 (2008); arXiv:0805.3994 [hep-ph]
Πµν(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)

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Restoration of: Abelian Ward identities

b

Z1

= b

Z2

; Zg = b

Z

1 =2

A

= ) RG invariant combination

g2

b (q2 ) = g2 b (q2 )

For large momenta q2, define the RG-invariant effective charge of QCD,

(q2

) =

g2

()=4

1

+ bg2 () ln (q2 =2 ) =

1 4

b ln (q2 =2 )

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Beyond perturbation theory ...

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

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New SD series

The new Schwinger-Dyson series based on the pinch technique

(a1) (a2) (b2) (c1) (c2) (b1) (d1) (d2) (d3) (d4)

ˆ ∆ (q)

+1

2

+ + +1

6

+1

2

+ + + + +1

2

= µ ν µν −1 −1

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].

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Transversality enforced loop-wise in SD equations

µ, a ν, b α, c ρ, d β, x σ, e

→

q (a1)

k+q

→

q

k

←

1 2

→

q µ, a

→

q ν, b ρ, c σ, d

k

→ (a2)

I

Γ

1 2

The gluonic contribution q

(q

)j (a1 )+(a2 ) = 0

The ghost contribution q

(q

)j (b1 )+(b2 ) = 0

µ, a ν, b

→

q (b1)

→

q c c′ x′

k+q

→ x

k

←

→

q

→

q

k

→ µ, a ν, b c d (b2)

I

Γ

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Dynamical mass generation: Schwinger mechanism in 4-d

(q2 ) =

1 q2

[1 + (q2 )℄

If

(q2 ) has a pole at q2 = 0 the vector meson is massive ,

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

1 =q2

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)

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Ansatz for the vertex

= + + . . . + 1/q2 pole

Gauge-technique Ansatz for the full vertex:

e

I

=
+ i q
q2
(k

+ q )

(k

)

;

Satisfies the correct Ward identity

q

1

e

I

abc
(q1

; q2 ; q3 ) = gf abc

1
(q2

)

1
(q3

)

Contains longitudinally coupled massless bound-state

poles

1 =q2 , instrumental for

1

(0 ) 6= 0

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System of coupled SD equations

1

(q2 ) =

q2

+ c1 Z

k

(k )(k + q )f1 (q ; k ) + c2 Z

k

(k )f2 (q ; k )

D

1 (p2 ) =

p2

+ c3 Z

k

p2
(p

k )2

k 2

(k

) D (p + k ) ;

Infrared finite

1

(0 ) 6= 0

Renormalize Solve numerically

A. C. Aguilar, D. Binosi and J. P. , Phys. Rev. D 78, 025010 (2008).

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Numerical results and comparison with lattice

Use lattice to calibrate the SDE solution.

1E-3 0,01 0,1 1 10 100 1000 2 4 6 8 =5.7 L=64 =5.7 L=72 =5.7 L=80 SDE solution = 4.5 GeV (q 2 ) q 2 [GeV 2 ]

I. L. Bogolubsky, et al , PoS LAT2007, 290 (2007)

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Ghost propagator

0,01 0,1 1 10 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 Ghost dressing function lattice =5.7 =m b p 2 D(p 2 ) p 2 [GeV 2 ] 0,01 0,1 1 10 100 1000 0,8 0,9 1,0 1,1 1,2 1,3 p 2 D(p 2 ) p 2 [GeV 2 ] Ghost dressing function SDE solution =m b

In the deep IR p2D

(p2 ) ! constant )

No power-law enhancement

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Making contact with physical quantities

The conventional

(q2 ) and the PT-BFM b (q2 ) are

related by

(q2 ) =

1

+ G (q2 ) 2 b (q2 )

Formal relation derived within Batalin Vilkovisky formalism

D. Binosi and J. P., Phys. Rev. D 66, 025024 (2002) .

G(q) =

D ∆

Auxiliary Green’s function related to the full gluon-ghost vertex

G

(q2 ) = CAg2

3

Z

k

2

+ (k q )2

k 2q2

(k

)D (k + q ) :

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Enforces

function coefficient in front of UV logarithm.

1

+ G (q2 ) = 1 + 9

4 CAg2 48 2 ln

(q2 =2 )

1

(q2 ) = q2

1

+ 13

2 CAg2 48 2 ln

(q2 =2 )

+

b

1

(q2 ) = q2

1

+ 11CAg2

48 2 ln

(q2 =2 )

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Numerical Results

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

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Numerical Results

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

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Physically motivated fit: Cornwall’s massive propagator The RG invariant quantity,

b

d

(q2 ) = g2 b (q2 ), has the form: b

d

(q2 ) =

g2

(q2 )

q2

+ m2 (q2 )

where the running charge is g2

(q2 ) =

1 b ln

q2 +4m2 (q2 ) 2

and the running mass

m2

(q2 ) = m2 "

ln

q2

+ 4 m2 2

.

ln

4 m2

2

#

12 =11

J. M. Cornwall , Phys. Rev. D 26, 1453 (1982)

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Phenomenological studies

A. A. Natale , Braz. J. Phys. 37, 306 (2007).
E. G. S. Luna and A. A. Natale , Phys. Rev. D 73, 074019 (2006).
A. C. Aguilar, A. Mihara and A. A. Natale , Int. J. Mod. Phys. A 19, 249 (2004).
S. Bar-Shalom, G. Eilam and Y. D. Yang , Phys. Rev. D 67, 014007 (2003).

A.C.Aguilar, A.Mihara and A.A.Natale,

Phys. Rev. D 65, 054011 (2002).

F.Halzen, G.I.Krein and A.A.Natale,

Phys. Rev. D 47, 295 (1993).

m

(0 ) = 500 200 MeV (0 ) =

1 4

b ln

4m2

(0 ) 2

Freezes at a finite value in the deep IR

(0 ) = 0 :7 :3

SLIDE 26 1E-3 0,01 0,1 1 10 100 1000 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,55 Running Charge m(0)=500 MeV m(0)=478 MeV (q 2 ) q 2 [GeV 2 ] 1E-3 0,01 0,1 1 10 100 1000 0,20 0,25 0,30 0,35 0,40 0,45 0,50 m(q 2 ) q 2 [GeV 2 ] Running Mass m(0)= 500 MeV m(0)= 478 MeV 1E-3 0,01 0,1 1 10 100 1000 5 10 15 20 25 30 d(q 2 ) q 2 [GeV 2 ] Numerical Solution Cornwall's propagator with m(0) = 478 MeV ^ 1E-3 0,01 0,1 1 10 100 1000 2 4 6 8 10 12 14 16 18 20 22 24 26 d(q 2 ) q 2 [GeV 2 ] Numerical solution Cornwall's propagator with m(0)=500 MeV ^

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Conclusions

Self-consistent description of the non-perturbative QCD dynamics in terms of an IR finite gluon propagator appears to be within our reach. Gauge invariant truncation of SD equations furnishes reliable non-perturbative information and strengthens the synergy with the lattice community. Meaningful contact with phenomenological studies