Institut Non Linéaire de Nice

Non-Perturbative QCD Seen From theProperty of Effective LocalityICNAAM 2012, Kos Island, Greece, 19-25 September 2012

Thierry GRANDOUInstitut Non Linéaire de Nice - UMR-CNRS 7335

October 4, 2012Slide 1/26

The property of Effective Locality

A functional approach to Lagrangian QCD using exactFradkin’s representations for GF (x ,y |A) and L(A),functional differential identities, and linearization ofnon-abelian F2:

• Manifestly gauge invariant (MGI) and Lorentzcovariant (MLC)

• Non-Perturbative: Summing over all relevantFeynman graphs

• Displaying a remarkable property, dubbed “EffectiveLocality”, peculiar to the non-abelian structure of QCD

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 2/26

Reminder

• Covariant gauge-dependent gluon propagator,

Dab (ζ)F ,µν

(k) =iδab

k2 + iε

[gµν−ζkµ kν/k2] , ζ = λ/(1−λ)

• Fermionic (quark) propagator in an external gluonfield Aa

µ,

GF (x ,y |A) = 〈x |[iγµ (∂µ− i g Aaµ λa)−m]−1|y〉

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 3/26

Reminder• Closed-fermion loop functional,

L[A] = Tr ln[1− i g (γAλ)SF ] , SF = GF [gA = 0]

• Example of a functional differential identity

F [1i

δ

δj] e

i2

Rj·D(ζ)

F ·j = ei2

Rj·D(ζ)

F ·jeD(ζ)A F [A]|

A=R

D(ζ)F ·j

where D(ζ)A is the linkage operator

D(ζ)A =− i

2

Zd4x d4y

δ

δAaµ(x)

D(ζ)F

∣∣∣ab

µν

(x− y)δ

δAbν(y)

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 4/26

Reminder (Fradkin’s)

〈p|GF [A]|y〉=− 1(2π)2 e−ip·y i

Z∞

0ds e−ism2

e−12 Tr ln(2h)

×Z

d [u]m− iγ · [p−gA(y−u(s))] ei4

R s0 ds′ [u′(s′)]2eip·u(s)

×(

egR s

0 ds′σ·F(y−u(s′)) e−igR s

0 ds′ u′(s′)·A(y−u(s′)))

+

h(s1,s2) =R s

0 ds′Θ(s1− s′)Θ(s2− s′). Auxiliary functionalvariables, Ωa(s1), Ωb(s2), required to circumventSchwinger proper-time s′-ordering and take both GF [A]and L[A] to gaussian forms.

EL not readable on ZQCD[ j,η, η], but on its (even)fermionic momenta.Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 5/26

Reminder (Halpern’77)

The χaµν-field is a (real-valued) Halpern field introduced so

as to linearize the non-abelian F µνFµν dependence of theoriginal QCD Lagrangian density

e−i4

RFa

µνFµνa = Nχ

Zd[χ]e

i4

Rχa

µνχµνa + i

2

Rχa

µνFµνa

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 6/26

Current status

• Almost 3 years old an approach..

• Interesting, deep theoretical related issues, andconnexions with AdS/CFT, etc . . .

• And phenomenological applications:

• such as an analytic derivation of Q/Q BindingPotentials (relativistic!); estimates of “pion” and“nucleon” ground states.

• Attempt at reaching Nuclear Physics out of QCD firstprinciples .. encouraging 1st result .. BUT..

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 7/26

Current status

Some references of ongoing works:

1 arXiv:0903.2644v2 [hep-th], EPJC 65 (2010);

2 arXiv:1003.2936v2 [hep-th]

3 arXiv:1103.4179 [hep-th]

4 arXiv:1104.4663 [hep-th], Ann.Phys. (2012).

5 arXiv:1203.6137 [hep-ph], submitted

6 arXiv:1207.5017 [hep-th], to be submitted

Collaboration

H.M. Fried, Brown University (RI), USA

Y. Gabellini, Y-M Sheu, INLN, B. Candelpergher,Laboratoire de Mathématiques JAD. UNS, France.

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 8/26

EL functional statement in (very!) shortWith

FI[A] = exp

[i2

ZAK (2n)A + i

ZQ (n)A

], FII[A] = exp(L[A])

The functional statement of EL for 2n-points fermionicGreen’s functions can be read off

eDAFI[A]FII[A] = N exp

[− i

2

ZQ (n)K −1Q (n) +

12

Tr lnK]

× exp

[i2

Zδ

δAK −1 δ

δA−

ZQ (n)K −1 δ

δA

]× exp(L[A]) (1)

at

K (2n) = (D(ζ)F )−1 +K (2n), K (2n)

abµν

= ( KS (2n)+gfχ )abµν

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 9/26

EL functional statement in (very!) short

1 Because K = KS + g(f ·χ) is local

〈x |O|y〉= O(x)δ(4)(x− y)

as well as the extra contributions of L[A] to K and Q ,the contributions of (1) depend only on the Fradkinvariables ui(s′i ) and the space-time coordinates yi in aspecific but local way

2 Nothing in (1) ever refers to D(ζ)F : Gauge-Invariance is

rigorously achieved as a matter ofGauge-Independence! This is MGI in the most radicalsense .. hoped as such by R.P. Feynman in QED (cf.‘Quantum Field Theory In A Nutshell’, A. Zee)

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 10/26

Any Antecedents ?..Yes!

- In the pure YM case, early 90’s, H. Reinhardt, K.Langfeld, L.v. Smekal discover a surprising effective localinteractionZ

d4z ∂λχ

aλµ(z)

([(gfχ)]−1)µν

ab (z) ∂ρχ

bρν(z)

- H.M. Fried himself in ‘Functional Methods and EikonalModels’ (Eds. Frontières, 1990)

- EL ‘made easy’ to discover within functional differentiationidentities; very difficult within functional integrations.

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 11/26

On some structural EL outputs

• Spin from Isospin (K.Huang, D.R. Stump, PRL(1977))

• Lorentz scalar character of the confining force (A.V.Nefediev, Y.A. Simonov, PRD76(2007))

• No theory dual to QCD (Supersymmetric caseexcluded)

• Interaction of the contact-type in the confining phase(R. Hofmann et al., Pure YM case, 2006)

• How QCD differs from pure YM

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 12/26

On some formal striking aspects of EL

? An intriguing factor of δ(2)(~b)

As a typical and most important part of a 4-point fermionicfunction at g >> 1, one gets in an exponential theargument

+i2

gZ

d4w1

Z s

0ds1

Z s

0ds2 u′µ(s1) u′ν(s2)

×Ωa(s1) Ωb(s2) (f ·χ(w1))−1∣∣µν

ab

×δ(4)(w1− y1 + u(s1))δ

(4)(w1− y2 + u(s2))

But how to think of

δ(4)(w1− y1 + u(s1))δ

(4)(y1− y2 + u(s2)−u(s1)) ?

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 13/26

An intriguing factor..That is, how to interpret

δ(4)(u(s2)−u(s1)) ?

Skipping to the Wiener functional space, it can be provenrigorously (Theorem) that

δ(u0(s2)−u0(s1))δ(u3(s2)−u3(s1)) =12

δ(s1)δ(s2)

|u′3(s1)||u′0(s2)|

+δ(s1)δ(s2)

|u′0(s1)||u′3(s2)|

The first of the two previous δ(4) fixes the unique point ofinteraction; the second δ(4) is proportional to

δ(2)(~y1⊥−~y2⊥)≡ δ

(2)(~b)Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 14/26

An intriguing factor..

δ(2)(~b), where~b is the impact parameter, or transversedistance between the two scattering quarks: Not due toany artefact and/or approximation scheme!

Being confined, Quarks cannot be dealt with as ordinary(abelian) particles.. only their longitudinal components canbe measured .. if not solely estimated

Several ways to justify a replacement of δ(2)(~b) by someCst exp−µ2~b2 [V.A. Matveev et al , Theor. Math. Phys 132(2002) 1119] ...

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 15/26

An intriguing factor..But in order to preserve a truly confining Q/Q potential,

V (r)' ξ µ(µr)1+ξ

one must proceed to a deformation of the Gaussiandistribution into

δ(2)(~b)→ ϕ(b) =

µ2

π

1 + ξ/2

Γ( 11+ξ/2)

e−(µb)2+ξ

, |ξ| 1

The deformed Gaussian is a (characteristic function of a)Lévy Flight distribution (LFs: Stable probabilitydistributions. Including and generalizing Gaussians, theycomply with a generalized central limit theorem ofstatistical physics).Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 16/26

A (possible) history of ξ

In the derivation of the Deuteron potential instead,

V (r)' c

(g2

R4π

)µ[2−µ2r2] e−

µ2r2

2 ,

ξ can be safely taken to zero, while essential to colorconfinement! The following properties appear to be closelyrelated

• Q/Q confining potential: V (r)' ξ µ(µr)1+ξ

• LF propagation modes of confined Qs• Non-commutative geometrical aspects of the (C2

n )scattering transverse planes (de Moyal planes,[x1,x2] = Θ)

• (D,E,S)χSB. µ'< ψψ >, and F(Θ,µ,ξ) = 0Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 17/26

History of µ? EL introduces mass scale(s) .. not surprising!

Formally imposed by the interaction locality, through thecanonical GF-construction itself!

ei4

Rd4w χµνχµν(w)→ e

i4 ∆2 χµνχµν(w0)

Here ∆ must be thought of in terms of the ‘probingenergies’ s = (P1 + P2)2, or sij ' xixjs ' (Pi + Pj)

2,i, j = 1,2, ..,n. QMs: ∆2 ' sµ2, Ann. Phys. (2012).

It isn’t the sole mass scale µ introduced in the replacement

δ(2)(~b)→ ϕ(b) =

µ2

π

1 + ξ/2

Γ( 11+ξ/2)

e−(µb)2+ξ

, |ξ| 1

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 18/26

History of µStill, µ-physics are controlled by ∆ (i.e. by s) as will bedisplayed shortly.

Confinement + Q/Q-Helicity Conservation⇒ SχSB (A.Casher’79, S. Brodsky’09, etc..): At relevant ∆ range, theprobed vacuum is made out of overlapping QQ pairs ..

.. whose necessary non-zero averaged inter-pairseparation is ∼< b >∼ 1/µThat is: µ must be on the order of the SχSB scale,< ψψ > and/or the pion mass mπ, G.O.R.- related

m2π =− 1

f 2π

limmQ=0

mQ < ψψ >

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 19/26

History of µ

That estimate of µ: Confirmed by a (non-relativistic) modelpion of an equal mass Q− Q system with Hamiltonian

H = 2m +1m

p2 +(

VB(r) = ξ µ(µr)1+ξ

)where m is the system reduced mass. Minimizationtechniques then give for the system ground state energy

E0 = µξ1/2 2−1/4 [1 + 3],

Then, at E0 'mπ, ξ =√

2/16 as expected (O(0.1)) Ann.Phys. 2012.

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 20/26

History of µThat b-variable, the impact parameter , (< b >∼ 1/µ) ‘is’the ζ Lorentz-invariant separation of the hadronic valencequark constituents at equal light-front time (ctlf ≡ x0± x3)in LF- quantized QCD (P. Dirac, S. Brodsky et al.)..... that allows a direct/precise holographic connection to anAdS5-space:

ds2 = (R2/z2)(ηµνdxµdxν−dz2)

in the framework of an AdS/QCD-correspondence. Thenz←→ ζ and things somehow culminate into (for a 2massless parton hadronic bound state, L = |Lz |)(

− d2

dζ2 −1−4L2

4ζ2 + V (ζ)

)φ(ζ) = M2

φ(ζ)

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 21/26

AdS5/QCD insightsWith V (ζ) relativistic ! and instantaneous .. in LF-time !

But contrarily to the EL-approach,

AdS5/QCD cannot calculate V (ζ) !

As the holographic mapping is made between AdS5 andLF- quantized QCD, then the AdS5/QCD approach claimsto incorporate in a single framework, both

1 the long-range confining hadronic domain, and,

2 the constituent quark, conformal short-distanceparticle limit.

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 22/26

What EL can do ..EL allows one to define the infinite dimensional functionalintegrations over Halpern’s field χa

µν configuration space:Zd[χ] = ∏

i∈M

N2−1

∏a=1

3

∏0=µ<ν

Zd[χa

µν](wi)

reducing it to ordinary Lebesgue integrations overfinite-dimensional Rn spaces (not possible in general)

A consequence of EL, through the measure-imagetheorem. Then χa

µν can be SU(Nc)- Lie-algebra valuated(adjoint representation):

χaµν→

N2−1

∑a=1

χaµνT a ≡M

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 23/26

What EL can do ..

.. and the full power of ‘Random Matrix ’ used, withH ∈MN(C), algebra of hermitian N×N traceless randommatrices at N ≡ D× (N2

c −1)

d [H] = dΘ1 .. dΘN ∏1≤i<j≤N

(Θi −Θj)2

× f (p) dp1 .. dpl

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 24/26

.. so as to get at eikonal and quenching approximations afull 2-body scattering amplitude with generic structure, atgϕ(b) = (µ/

√s) g exp−(µb)2+ζ

N

− s

m2Q

√1−

m2Q

s

N(64π2

g2Nc

)N4

(8i)N4 (−8i)

N22 N

N22 +1

c

× ∑monomials

(±1)1≤j≤N

∏∑qj =N(N−1)

[1− i(−1)qj ]

( √8Nc

gϕ(b)√

i

)×

G4236

gϕ(b)√

i√128Nc

m2Q/s√

1−m2Q/s

2

| 1, 1, 12

1, 1, 1, 2qi +34 , 1

2 , 12

where partonic (s) and non-perturbative physics (gϕ(b))show up in one and the same expression! ...Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 25/26

That amplitude ..

• ξ isn’t ruled out: VB(r) = ξ µ(µr)1+ξ

• Non-perturbative (hadronic) physics disappears atg→ 0, as it must (AF)

• and atµm2

Q/s3/2(=−m2πmQ/s3/2,at µ' f−2

π < ψψ >)→ 0,as known from Nuclear Physics

• Complies with a general conjecture (D.D. Ferrante,G.S. and Z. Guralnik, C. Pehlevan. S. Gukov and E.Witten, etc., 2008-2011) so far illustrated on scalarfield models, that QFT’s GF are expandable in termsof Gmn

pq - Meijer’s special functions.

Thierry GRANDOU (INLN) — Non-Perturbative QCD Seen From the Property of Effective Locality — October 4, 2012Slide 26/26