Gluon and Ghost Propagators from Schwinger-Dyson Equation and Lattice Simulations

Home , Ghost (physics)

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 Vacuum polarization in QED (prototype)

Πµν(q) = q q

q is independent of the gauge-ﬁxing parameter to all orders

2 1

2 2

℄ q 1 q

1 2 2

℄

e Ze e0 and 1 q ZA 1 0 q =

1 2 From QED Ward identity follows Z1 Z2 and Ze ZA RG-invariant combination

2 2 2 2 2 «

«´ µ =

´ µ = ¡´ µ

e ¡0 q e q q 2 µ 0 1·¥´q

Gluon self-energy in perturbation theory

;

q depends on the gauge-ﬁxing parameter already

at one-loop

k · Õ k · Õ

Õ Õ Õ Õ

« «

¥ ´Õ µ =

¬ ¬

«¬

k k

´bµ ´aµ

Ward identities replaced by Slavnov-Taylor identities involving ghost Green’s functions. (Z1 6 Z2 in general)

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 −1 −1 −1 = = 1 1 ∆µν (q) +2 +2 µ ν (b)

(a)

+ 1 1 +6 +2

j 6

µ·´ µ

q q ´a b 0

j 6

µ·´ µ·´ µ q q ´a b c 0

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

" #

´0 k k 1

In covariant gauges: i k g 1

k 2 k 2

" #

0µ n k n k 1

! In light cone gauges: i k g

nk k 2

= = =

k k p m p m

1 1

; S0 k p S0 p Pinch Technique rearrangement

pinch -

               pinch   -                  

pinch -

∆ b

Gauge-independent self-energy

· ·

¥ ´Õ µ «¬

2 1

h i

q 2 2 2 q

q 1 bg ln 2

= ¬ b 11CA 48 ﬁrst coeﬃcient of the QCD -function

(¬ bg ) in the absence of quark loops. Simple, QED-like Ward Identities , instead of

Slavnov-Taylor Identities, to all orders

¢ £

1 1

;

q I p1 p2 g S p2 S p1 £

abc ¢

abc 1 1

; ;

q I «¬ q q q gf q q «¬ 1 1 2 3 «¬ 2 3

Profound connection with µ Background Field Method easy to calculate

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

Πµν(q) = + c q q q q

Can move consistently from one gauge to another (Landau to Feynman, etc) A. Pilaftsis , Nucl. Phys. B 487, 467 (1997)

Restoration of: =

1 2

b b b

; Abelian Ward identities Z1 Z2 Zg ZA

2 2 2 2

b b

RG invariant combination g0 0 q g q For large momenta q2, deﬁne the RG-invariant eﬀective charge of QCD,

2 g 4 1

« q

2 2 2 2 2

= = 1 bg ln q 4 b ln q Beyond perturbation theory ...

2 ^

d(q )

Non-perturbative effects

Lattice, Schwinger-Dyson equations

Asymptotic Freedom

q New SD series

The new Schwinger-Dyson series based on the pinch technique

ˆ −1 −1 ∆µν (q) = +1 +1 µ ν 2 2 (a2)

(a1)

1 1 + + +6 +2

(b2) (b1) (c1) (c2)

+ + + +

(d1) (d2) (d3) (d4)

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]. 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 IΓ 1 2 µ, a ν,b 2

e µ, a ν,b

´ µ q q a1 µ·´a2 0

α, c ρ, d (a2)

←k

(a1)

→k k→+q c ′ c The ghost contribution c d → → → q → q q q IΓ

µ, a µ, a

e ν,b ν,b

µ·´ µ q q ´b1 b2 0 (b2) x x′

←k

(b1) Dynamical mass generation: Schwinger mechanism in 4-d

2 1

2 2

℄ q 1 q

2 2

If q has a pole at q 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

2 = poles 1 q 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

= + + . . . + 1/q2 pole

Gauge-technique Ansatz for the full vertex:

;

«¬ «¬ «¬ I i «¬ k q k q 2

Satisﬁes the correct Ward identity £

abc ¢

abc 1 1

; ;

q I «¬ q q q gf q q «¬ 1 1 2 3 «¬ 2 3

Contains longitudinally coupled massless bound-state

2 1

= 6 poles 1 q , instrumental for 0 0

System of coupled SD equations

Z Z

1 2 2

´ µ = · ¡´ µ¡´ · µ ´ ; µ · ¡´ µ ´ ; µ ¡ q q c1 k k q f1 q k c2 k f2 q k

k k

Z 2

¡ µ

´p k

1 2 2 2

¡´ µ = · µ ´ · µ ; D ´p p c3 p 2 k D p k k k

Infrared ﬁnite

6 0 0

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

=5.7 L=72

=5.7 L=80

SDE solution

= 4.5 GeV

6 ) 2

(q 4

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

Ghost dressing function 2,8

lattice =5.7

SDE solution

2,6 b =m

1,2 b

2,4

2,2

1,1 ) 2

2,0 ) 2 D(p

1,8 p D(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

2 2 2 2

p [GeV ] p [GeV ]

2 2

! µ In the deep IR p D p constant No power-law enhancement Making contact with physical quantities

2 b 2

The conventional q and the PT-BFM q are

related by £

2 ¢ 2 2 2

q 1 G q q

Formal relation derived within Batalin Vilkovisky 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

2 Z 2

´ µ

2 CAg k ¡ q

¡´ µ = · µ ´ · µ : G ´q 2 2 2 k D k q 3 k k q Enforces ¬ function coeﬃcient in front of UV logarithm.

2 9 CAg 2 2

´ µ = · ´ = µ 1 · G q 1 2 ln q

4 48 2

13 CAg

1 2 2 2 2

´ µ = · ´ = µ ¡ q q 1 2 ln q

2 48

· 2

CAg

1 2 2 2 2

¡ µ = · ´ = µ ´q q 1 11 2 ln q

48 Numerical Results

1,2

1,1 ) 2

1,0

1+G(q

0,9

0,8

1E-3 0,01 0,1 1 10 100 1000

2 2

q [GeV ] Numerical Results

^ 2 2 2 ^

28 d(q )= g (q )

= M

24 b

20 )

16 2

^ d(q

1E-3 0,01 0,1 1 10 100 1000

2 2

q [GeV ]

Physically motivated ﬁt: Cornwall’s massive propagator b

2 2 b 2

The RG invariant quantity, d q g q , has the form:

2 2 g q

b 2

d q 2 2 2

q m q

where the running charge is

2 2 1

g q 2 2 2

´ µ q ·4m q

b ln 2 £

and the running mass

" #

12 11

2 2 º 2

2 2 2 q 4 m0 4 m0

m q m

0 ln 2 ln 2

J. M. Cornwall , Phys. Rev. D 26, 1453 (1982) 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 2 µ 4m ´0

4 b ln 2 £

Freezes at a ﬁnite value in the deep IR

: ¦ : « 0 0 7 0 3

26 30

Numerical solution

24 Numerical Solution

Cornwall's propagator

22 with m(0)=500 M eV 25

with m(0) = 478 M eV

18 20

16 ) )

14 2 2 15

^ ^ d(q d(q 12

10 10

6 5

2 0

1E-3 0,01 0,1 1 10 100 1000 1E-3 0,01 0,1 1 10 100 1000

2 2 2 2

q [GeV ] q [GeV ]

0,55

0,50 Running M ass

Running Charge

0,50

m(0)= 500 MeV

m(0)=500 MeV

m(0)= 478 MeV

m(0)=478 MeV

0,45

0,40

0,35 ) 2 )

0,30 2

m(q 0,35 (q

0,25

0,30 0,20

0,15

0,25

0,10

0,20 0,05

1E-3 0,01 0,1 1 10 100 1000 1E-3 0,01 0,1 1 10 100 1000

2 2 2 2

q [GeV ] q [GeV ] 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