THEORY OF THE MASS GAP IN QCD AND ITS VACUUM
V. Gogokhia
WIGNER RESEARCH CENTER FOR PHYSICS, HAS, BUDAPEST, HUNGARY
GGSWBS’12 (August 13-17) Batumi, Georgia 5-th ”Georgian-German School and Workshop in Basic Science
0-0 Lagrangian of QCD
LQCD = LYM + Lqg
1 L = − F a F aµν + L + L YM 4 µν g.f. gh.
a a a abc b c 2 Fµν = ∂µAν − ∂νAµ + gf AµAν, a = Nc − 1,Nc = 3
j j j j j Lqg = iq¯αDαβqβ +q ¯αm0qβ, α, β = 1, 2, 3, j = 1, 2, 3, ...Nf
j a a j Dαβqβ = (δαβ∂µ − ig(1/2)λαβAµ)γµqβ (cov. der.)
0-1 Properties of the QCD Lagrangian
QCD or, equivalently Quantum ChromoDynamics
1). Unit coupling constant g 2). g ∼ 1 while in QED (Quantum ElecroDynmics) it is g ¿ 1. 3). No mass scale parameter to which can be assigned a physical meaning.
Current quark mass m0 cannot be used, since the quark is a colored object QCD without quarks is called Yang-Mills, YM
0-2 The Jaffe-Witten (JW) theorem:
Yang-Mills Existence And Mass Gap: Prove that for any compact simple gauge group G, quantum Yang- Mills theory on R4 exists and has a mass gap ∆ > 0. (i). It must have a ”mass gap”. Every excitation of the vacuum has energy at least ∆ (to explain why the nuclear force is strong but short-range). (ii). It must have ”quark confinement” (why the physical particles are SU(3)-invariant). (iii). It must have ”chiral symmetry breaking” (to account for the ”current algebra” theory of soft pions). We need Mass Gap responsible for the NP dynamics
0-3 Gluon SD equation
0 0 Dµν(q) = Dµν(q) + Dµρ(q)iΠρσ(q; D)Dσν(q)
n o 1 D (q) = i T (q)d(q2) + ξL (q) µν µν µν q2
1 D0 (q) = i {T (q) + ξL (q)} µν µν µν q2
2 Tµν(q) = δµν − (qµqν/q ) = δµν − Lµν(q)
0-4 S D = + Γ D S
G D + GDµ + G D D D
+ T3 D + D T3 D D D T D D 3 + D T4 D D
q gh t (1) 2 Πρσ(q; D) = Πρσ(q) + Πρσ(q) + Πρσ(D) + Πρσ (q; D ) (2) 4 (20) 3 +Πρσ (q; D ) + Πρσ (q; D ).
0-5 The Mass Gap
s 2 Πρσ(q; D) = Πρσ(q; D) − Πρσ(0; D) = Πρσ(q; D) − δρσ∆ (D)
∆2(D) ≡ ∆2(λ, α; D)
Problems A. The first problem is how to satisfy the ST identity 2 qµqνDµν(q) = iξ but without going to ∆ (D) = 0 limit. B. The second problem is how to make the relevant gluon propagator purely transversal, since the ghosts will fail to do this when ∆2(D) will be explicitly present.
0-6 Methods
Theory of matrices Subtractions at the gluon propagator level Theory of generalized functions (distributions) Dimensional regularization method Theory of functions of complex variable The Weierstrass-Sokhotsky-Casorati theorem which describes the behavior of the Laurent expansions near their essential (zero) singularities in order to get finite results (Renormalization program) λ → ∞, α → 0
0-7 Solution
2 INP 2 PT Dµν(q; ∆ ) = Dµν (q; ∆R) + Dµν (q)
1 ∆2 DINP (q; ∆2 ) = iT (q)dINP (q2; ∆2 ) , dINP (q2; ∆2 ) = R µν R µν R q2 R q2
n o PT PT 2 2 Dµν (q) = i Tµν(q)d (q ) + ξLµν(q) (1/q )
PT 2 αs 1 2 d (q ) = 2 2 = 2 2 , q → ∞, AF 1 + αsb ln(q /ΛQCD) b ln(q /ΛQCD)
2 2 αs(mZ ) = 0.1184, ΛQCD = 0.09 GeV , b = (11/4π) for Y M
0-8 Some remarks
2 ∞←αs(λ) 2 αs(λ)→0 2 ΛINP ←−∞←λ ∆ (λ, αs(λ)) λ→∞ −→ ΛPT ,
2 2 2 2 ΛINP ≡ ∆R, ΛPT ≡ ΛQCD
INP QCD ⇐= QCD =⇒ P T QCD,
Intrinsically non-perturbative QCD (INP QCD) con- fines gluons (no free gluons, dressed gluons are sup- pressed in incoming and outgoing states) Perturbative QCD (PT QCD) is asymptotically free
0-9 VACUUM ENERGY DENSITY (VED) IN THE QUANTUM YM THEORY
The vacuum of QCD is a very complicated con- fining medium. Its dynamical and topological com- plexity means that its structure can be organized at various levels: classical and quantum, dynamical and topological. It is mainly NP by origin, character and magnitude, since the corresponding coupling constant is large. However, the virtual gluon field configura- tions and excitations of the PT origin, character and magnitude, due to AF, are also present there.
0-10 The bag constant I
B = VEDPT − VED
B = VEDPT −VED = VEDPT −[VED−VEDPT +VEDPT ] =
VEDPT − [VEDINP + VEDPT ] = −VEDINP > 0
The formalism which makes it possible to calculate the VED from first principles is the effective potential approach for composite operators J.M. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. D, 10 (1974) 2428.
0-11 The VED
S D
+ + + +... D T3 D T3 + D T4 D
D D S S
= + + + ...
D
i Z d4q n o V (D) = T r ln(D−1D) − (D−1D) + 1 ,V (D ) = 0 2 (2π)4 0 0 0
0-12 The bag constant II
BYM = −²YM Z q2 · ¸ 1 eff 3 = dq2 q2 ln[1 + 3αINP (q2)] − αINP (q2) , π2 0 4
2 2 INP 2 INP zc q ∆R d (q ) ≡ α (z) = , z = 2 zc = 2 z qeff qeff
4 BYM = qeff × ΩYM ,
Z 1 · ¸ 1 INP 3 INP ΩYM = dz z ln[1 + 3αs (z)] − αs (z) . π2 0 4
0-13 0.04
max zc =0.45
0.02 eff 4 0 zc =1.3786 ) / q
c 0 (z YM B
-0.02 Physical Region Non-Physical Region
-0.04
0 0.5 1 1.5 2 2.5
0-14 zc
4 Figure 1: The BYM /qeff effective potential vs. zc. The non-physical 0 region is zc ≥ zc , since BYM should be always positive. At zc = 0 the effective potential is also zero. 0.04
0.02 Physical Region Non-Physical Region eff 4 0 zc =1.3786 ) / q
c 0 (z YM ε
-0.02
min zc =0.45
-0.04
0 0.5 1 1.5 2 2.5
0-15 zc
4 Figure 2: The ²YM /qeff effective potential vs. zc. The non-physical 0 region is zc ≥ zc , since ²YM should be always negative. At zc = 0 the effective potential is also zero. ΩYM (zc)/∂zc = 0
2 max −1 2 4 qeff = (zc ) ∆R = 2.2∆R
4 4 BYM = −²YM = 0.1273 × ∆R = 0.0263 GeV
General properties of the bag constant being: • colorless (color-singlet); • electrically neutral; • transversal, i.e., depending only on ”physical” de- grees of freedom of gauge bosons;
0-16 • the explicit gauge invariance; • uniqueness, i.e., it is free of all the types of the PT contributions now; • finiteness; • positiveness; • no imaginary part (stable vacuum); • existence of the stationary state for the corre- sponding YM energy density (negative pressure); • the final dependence on the mass gap only; • a good numerical agreement with phenomenology (gluon condensate).
0-17 Contribution of BYM to the dark energy problem
Bag constant may also contribute to the so-called dark energy density. At least, from the qualitative point of view it satisfies almost all the criteria neces- sary for the dark energy/matter candidate. From the quantitative numerical point of view it is also much better than the estimate from the Higgs field’s contribution to the VED by S. Weinberg, arXiv:astroph/9610044
8 4 %H ∼ 10 GeV .
−2 4 %our ∼ 10 GeV
0-18 exp −46 4 %vac ∼ 10 GeV
WMAP Collaboration, E. Komatsu et al., Astro- phys. J. Suppl. 180 (2009) 330 So relatively to the value inferred from the cosm. const. (experimental value
exp 54 %H /%vac ∼ 10 ,
exp 44 %our/%vac ∼ 10
0-19 Numerical value in different units
4 BYM = 0.0263 GeV = 3.4 GeV/fm3 = 3.4 × 1039 GeV/cm3.
1 GeV = 1.6 × 1010J = 4.45 × 10−23GW h,
1W = 10−3kW = 10−6MW = 10−9GW
17 3 BYM ∼ 10 GW h/cm
0-20 3 17 EYM = BYM cm ∼ 10 GW h
Total production of primary energy of the 25 EU countries in 2004 was
7 Et ∼ 10 GW h.
The number is taken from ”Energy FOR THE FU- TURE”, a position paper of the EPS, www.eps.org Approximately 1/3 of this energy was produced by nuclear power plants
0-21 Energy from the QCD vacuum
The bag constant is the energy density of the purely transversal severely infrared singular virtual gluon field configurations which are not only stable (no imagi- nary part), but are being in the stationary state as well, i.e., in the state with the minimum of energy. That is why it makes sense to discuss the ”releasing” of the bag energy EYM from the vacuum.
V E = −B V = −E ∼ −λ3, λ → ∞ vac YM YM cm3
Let us imagine now that we can release the finite portion EYM from the vacuum in k different places (different ”vacuum energy releasing facilities” (VERF)).
0-22 It can be done by nm times in each place, where m = 1, 2, 3...k. Then the releasing energy Er becomes
Xk Er = EYM nm = EYM (n1 + n2 + n3 + ... + nk) m=1
Xk 2 Er = EYM × lim nm ∼ λ , λ → ∞ (k,n )→∞ m m=1
ER = Evac − Er = Evac[1 + O(1/λ)], λ → ∞, i.e., the QCD vacuum is infinite and permanent reservoir of energy. The only problem is how to re- lease the finite portion – the bag energy, and whether
0-23 it will be profitable or not by introducing some type of cyclic process. ”Perpetuum mobile” does not exist, but ”perpetuum source” of energy does exist, and it is the QCD ground state. V. Gogokhia, G.G. Barnafoldi, ”The Mass Gap and its Application” (Word Scientific, 2012) Scientific problem what amount of energy can be released is resolved Technological problem how to release is not re- solved? Engineering roblem how to built VERF is not re- solved ?
0-24 Transversality of the full gluon self-energy
q g t Πρσ(q; D) = Πρσ(q) + Πρσ(q; D) + Πρσ(D)
g gh (1) 2 (2) 4 (20) 3 Πρσ(q; D) = Πρσ(q) + Πρσ (q; D ) + Πρσ (q; D ) + Πρσ (q; D ).
q g t qρΠρσ(q; D) = qρΠρσ(q) + qρΠρσ(q; D) + qρΠρσ(D)
0-25 The quark contribution
The color currents conservation condition implies
q q qρΠρσ(q) = qσΠρσ(q) = 0
q(s) q q q 2 Πρσ (q) = Πρσ(q) − Πρσ(0) = Πρσ(q) − δρσ∆q
q 2 q 2 q 2 Πρσ(q) = Tρσ(q)q Πt (q ) + qρqσΠl (q ),
q(s) 2 q(s) 2 q(s) 2 Πρσ (q) = Tρσ(q)q Πt (q ) + qρqσΠl (q ).
0-26 ∆2 Πq(q2) = Πq(s)(q2) + q , l l q2 ∆2 Πq(q2) = Πq(s)(q2) + q t t q2 ∆2 Πq(q2) = Πq(s)(q2) + q = 0 l l q2 ∆2 Πq(s)(q2) = − q l q2
2 q 2 q(s) 2 q 2 q(s) 2 ∆q = 0, Πl (q ) = Πl (q ) = 0, Πt (q ) = Πt (q )
q q(s) 2 q(s) 2 Πρσ(q) = Πρσ (q) = Tρσ(q)q Πt (q ).
0-27 QED
qµqνDµν(q) = iξ, qρΠρσ(q) = qσΠρσ(q) = 0
1 2 1 1 jµ(q)Dµν(q)jν , jµ(q)qµ = jν (q)qν = 0
0 s s 2 D(q) = D (q) + D0(q)iΠ (q)D(q), Π(q) → Π (q) ∼ O(q )
D0(q) D(q) = s = 1 − iΠ (q)D0(q)
0 s s s D (q)+D0(q)iΠ (q)D0(q)+D0(q)iΠ (q)D0(q)iΠ (q)D0(q)+...
0-28 The gluon contribution
g qρΠρσ(q; D) = h i gh (1) 2 (2) 4 (20) 3 qρ Πρσ(q) + Πρσ (q; D ) + Πρσ (q; D ) + Πρσ (q; D ) = 0,
g(s) g g g 2 Πρσ (q; D) = Πρσ(q; D) − Πρσ(0; D) = Πρσ(q; D) − δρσ∆g(D)
X X 2 g 2 0 ∆g(D) ≡ Π (0; D) = Πa(0; D) = ∆a(D), a = gh, (1), (2), (2 ) a a
2 2 ∆g(D) ≡ ∆g(λ, α; D)
0-29 g 2 g 2 g 2 Πρσ(q; D) = Tρσ(q)q Πt (q ; D) + qρqσΠl (q ; D)
g(s) 2 g(s) 2 g(s) 2 Πρσ (q; D) = Tρσ(q)q Πt (q ; D) + qρqσΠt (q ; D)
∆2(D) Πg(q2; D) = Πg(s)(q2; D) + g . t t q2
∆2(D) Πg(q2; D) = Πg(s)(q2; D) + g . l l q2
0-30 ∆2(D) Πg(q2; D) = Πg(s)(q2; D) + g = 0 l l q2
∆2(D) Πg(s)(q2; D) = − g l q2
2 g 2 g(s) 2 g 2 g(s) 2 ∆g(D) = 0, Πl (q ; D) = Πl (q ; D) = 0, Πt (q ; D) = Πt (q ; D)
g g(s) 2 g(s) 2 Πρσ(q; D) = Πρσ (q; D) = Tρσ(q)q Πt (q ; D).
0-31 The tadpole term contribution
g t 2 2 qρΠρσ(q; D) = qρΠρσ(D) = qρδρσ∆t (D) = qσ∆t (D) 6= 0
2 f 2 f 2 Πρσ(q; D) = Tρσ(q)q Πt (q ; D) + qρqσΠl (q ; D)
∆2(D) Πf (q2; D) = t l q2
2 f 2 2 Πρσ(q; D) = Tρσ(q)q Πt (q ; D) + Lρσ∆t (D)
0-32 q g 2 Πρσ(q; D) = Πρσ(q) + Πρσ(q; D) + δρσ∆t (D)
2 f 2 2 q(s) 2 g(s) 2 2 q Πt (q ; D) = q [Πt (q ) + Πt (q ; D)] + ∆t (D)
2 2 2 2 Πρσ(q; D) = Tρσ(q)[q Π(q ; D) + ∆t (D)] + Lρσ∆t (D)
2 q(s) 2 g(s) 2 Π(q ; D) = [Πt (q ) + Πt (q ; D)].
2 2 ∆t (D) = ∆ (D)
0-33 ST identity
qµqνDµν(q) = iξ
n o 1 D (q) = i T (q)d(q2) + ξL (q) µν µν µν q2
0 0 Dµν(q) = Dµν(q) + Dµρ(q)iΠρσ(q; D)Dσν(q) =
0 0 2 2 2 Dµν(q) + Dµρ(q)iTρσ(q)[q Π(q ; D) + ∆ (D)]Dσν(q)
0 2 +Dµρ(q)iLρσ(q)∆ (D)Dσν(q)
0-34 ∆2(D) q q D (q) = iξ − iξ2 µ ν µν q2
∆2(D) = 0
∆2(D) = 0, −→ D = DPT
PT 2 2 PT Πρσ(q; D ) = Tρσ(q)q Π(q ; D )
PT qρΠρσ(q; D ) = 0
0-35 PT qµqνDµν (q) = iξ,
n o 1 DPT (q) = i T (q)dPT (q2) + ξL (q) µν µν µν q2
PT 0 0 2 2 PT PT Dµν (q) = Dµν(q) + Dµρ(q)iTρσ(q)q Π(q ; D )Dσν (q)
1 dPT (q2) = 1 + Π(q2; DPT )
0-36 On the other hand
n o 1 D (q) = i T (q)d(q2) + ξL (q) µν µν µν q2
0 0 2 s 2 2 Dµν(q) = Dµν(q)+Dµρ(q)iTρσ(q)[q Π (q ; D)+∆ (D)]Dσν(q)
0 2 ˜ 2 +Dµρ(q)iLρσ(q)q Π(q ; D)Dσν(q)
1 d(q2) = 1 + Πs(q2; D) + (∆2(D)/q2)
Π(˜ q2; D) = 0, → ∆2(D) = 0
0-37 Preliminary discussion
The formal ∆2(D) = 0 limit is a real way how to pre- serve the color gauge invaraince/symmetry in QCD. Why does ∆2(D) (which is nothing but the tadpole term) exist in this theory at all? There is no doubt that this symmetry should be maintained at non-zero ∆2(D) as well.
A. The first problem is how to satisfy the ST iden- tity but without going to ∆2(D) = 0 limit.
B. The second problem is how to make the rel- evant gluon propagator purely transversal, since the ghosts will fail to do this when ∆2(D) will be explicitly present.
0-38 The spurious mechanism
0 0 2 2 2 Dµν(q) = Dµν(q) + Dµρ(q)iTρσ(q)[q Π(q ; D) + ∆ (D)]Dσν(q)
0 2 +Dµρ(q)iLρσ(q)∆ (D)Dσν(q)
1 D0 (q) → D0 (q; ∆2(D)) = D0 (q)+iξL (q)d (q2; ∆2(D)) µν µν µν µν 0 q2
0 0 2 2 2 Dµν(q) = Dµν(q) + Dµρ(q)iTρσ(q)[q Π(q ; D) + ∆ (D)]Dσν(q)
2 2 +Iµν(q ; ∆ (D))
0-39 2 2 2 Iµν(q ; ∆ (D)) = Iµν(q )
" # ∆2(D) 1 I (q2) = iξL (q) d (q2; ∆2(D)) − ξ[1 + d (q2; ∆2(D)] µν µν 0 0 q2 q2
∆2(D) d (q2; ∆2(D)) = ξ[1 + d (q2; ∆2(D))] 0 0 q2
2 Iµν(q; ∆ (D)) = 0
0-40 NP QCD
2 Dµν(q) = Dµν(q; ∆ (D)) =
0 0 2 2 2 2 Dµν(q) + Dµρ(q)iTρσ(q)[q Π(q ; D) + ∆ (D)]Dσν(q; ∆ (D))
1 D (q; ∆2(D)) = i{T (q)d(q2; ∆2(D) + ξL (q)} µν µν µν q2
1 d(q2; ∆2(D)) = . 1 + Π(q2; D) + (∆2(D)/q2)
0-41 PT QCD
formal PT ∆2(D) = 0 limit
PT 0 0 2 2 PT PT Dµν (q) = Dµν(q) + Dµρ(q)iTρσ(q)q Π(q ; D )Dσν (q)
1 DPT (q) = i{T (q)dPT (q2) + ξL (q)} µν µν µν q2
1 dPT (q2) = . 1 + Π(q2; DPT )
0-42 The Jaffe-Witten mass gap
" # ∆2(d) d(q2) = 1 − Π(q2; d) + d(q2) q2
∆2(d) = ∆2c(d)
∆2 ≡ ∆2(λ, α, ξ, g2)
2 2 2 2 ∆R = Z(λ, α, ξ, g )∆ (λ, α, ξ, g ).
0-43 B. Restoration of transversality of the gluon propagator
TNP 2 2 2 2 PT Dµν (q; ∆ ) = Dµν(q; ∆ )−Dµν(q; ∆ = 0) = Dµν(q; ∆ )−Dµν (q)
1 DTNP (q; ∆2) = iT (q)dTNP (q2; ∆2) µν µν q2
n o PT PT 2 2 Dµν (q) = i Tµν(q)d (q ) + ξLµν(q) (1/q )
n o 2 2 Dµν(q) = i Tµν(q)d(q ) + ξLµν(q) (1/q )
0-44 1 d(q2; ∆2) = 1 + Πs(q2; D) + (∆2/q2)
[q2Πs(q2; DPT ) − q2Πs(q2; D) − ∆2] dTNP (q2; ∆2) = [q2 + q2Πs(q2; D) + ∆2][1 + Πs(q2; DPT )]
1 dPT (q2) = 1 + Πs(q2; DPT )
n o 2 2 2 2 Dµν(q; ∆ ) = i Tµν(q)d(q ; ∆ ) + ξLµν(q) (1/q )
PT 2 2 PT 2 2 = −iTµν(q)d (q )(1/q ) + iTµν(q)d (q )(1/q )
TNP 2 PT = Dµν (q; ∆ ) + Dµν (q)
0-45 Prescription
2 TNP 2 PT Dµν(q; ∆ ) = Dµν (q; ∆ ) + Dµν (q) ——————————————————————— ——
2 TNP 2 2 PT Dµν(q; ∆ ) → Dµν (q; ∆ ) = Dµν(q; ∆ ) − Dµν (q)
d(q2; ∆2) → dTNP (q2; ∆2) = d(q2; ∆2) − dPT (q2). ——————————————————————— ——-
2 PT 2 TNP 2 Dµν(q; ∆ ) → Dµν (q) = Dµν(q; ∆ ) − Dµν (q; ∆ )
0-46 d(q2; ∆2) → dPT (q2) = d(q2; ∆2) − dTNP (q2; ∆2).
0-47 The Bag constant
One of the important characteristics of the QCD ground state is the Bag constant.
B = VEDPT − VED, VED is the NP but ”contaminated” by the PT con- tributions (i.e., it is a full VED like the full gluon propagator).
B = VEDPT −VED = VEDPT −[VED−VEDPT +VEDPT ] = VEDPT − [VEDTNP + VEDPT ] = −VEDTNP > 0,
Thus the Bag constant is completely free of the PT ”contaminations”.
0-48 The SDE for the TNP gluon propagator
TNP 2 0 2 s 2 2 s 2 PT 2 PT Dµν (q; ∆ ) = Dµρ(q)iTρσ(q)[q Π (q ; D)−q Π (q ; D )+∆ ]Dσν (q)
h i 0 2 s 2 2 TNP 2 +Dµρ(q)iTρσ(q) q Π (q ; D) + ∆ Dσν (q; ∆ )
0 0 2 s 2 2 Dµν(q) = Dµν(q)+Dµρ(q)iTρσ(q)[q Π (q ; D)+∆ (D)]Dσν(q)
PT 0 0 2 s 2 PT PT Dµν (q) = Dµν(q) + Dµρ(q)iTρσ(q)q Π (q ; D )Dσν (q),
No free gluons in TNP QCD
0-49 The necessity of the proposed subtractions
The first subtraction at the level of the full gluon self-energy makes it possible to introduce the mass gap The second subtraction at the level of the full gluon propagator makes it possible:
2 TNP 2 PT Dµν(q; ∆ ) = Dµν (q; ∆ ) + Dµν (q)
A. To achieve transversality in a gauge invariant way B. Do not affect its true NP structure Exact separation between the NP and PT dynamics
0-50 C. Exclude the free gluons from the theory
0-51 Conclusions
The mass gap is generated in the gluon sector of QCD mainly due to the self-interaction of massless gluon modes. It is defined as the difference between the full gluon self-energy and its value at some point, so it is not introduced by hand. No any truncations/approximations/asumptions, no special gauge choice, only algebraic (i.e., exact) deriva- tions have been done. The common belief (coming from PT) that mass gap contradicts the color gauge invariance/symmetry of QCD is false.
0-52 This fundamental symmetry is maintained/preserved at non-zero mass gap as well.
0-53 We distinguish between NP and PT QCD by the explicit presence of the mass gap, and not by the strength of the coupling constant. It plays no role when the mass gap is kept ”alive”.
Proposed subtraction makes it possible to make the relevant gluon propagator to become purely transver- sal in a gauge invariant way and to remove free gluons from the theory at the same time.
So unitarity of the S-matrix in TNP QCD is main- tained.
No free gluons in TNP QCD.
Emission and absorbtion of ”dressed” gluons will be
0-54 suppressed by the renormalization of the mass gap.
0-55 QED vs QCD
QED. We cannot release the mass gap from the QED vacuum, while we can release the photons and the electron-positron pars from it.
QCD. We can release the mass gap from the QCD vacuum, as it has been described in this talk. But we cannot release the gluons and the quarks/antiquarks from its vacuum because of the color confinement phe- nomenon.
The next step is to find a formal solution(s) for the full gluon propagator as a function of the regularized mass gap and its renormalization.
0-56 Renormalization of the mass gap
1. The two independent types of general solutions (a). Smooth (Massive) (b). Singular (NL iteration) 2. Renormalization of the relevant gluon propaga- tor 3. Gluon and Quark confinement criteria 4. Asymptotic Freedom from the mass gap 5. The JW theorem is not completely correctly formulated 6. INP QCD ⇐= QCD =⇒ PT QCD
0-57