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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
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