The Physics of Mass Gap Problem in the General Theory Framework Elias Koorambas

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Elias Koorambas. The Physics of Mass Gap Problem in the General Field Theory Frame- work. International Journal of High Energy Physics, Science Publishing Group, 2015, Special Is- sue: Symmetries in Relativity, Quantum Theory, and Unified Theories., 2 ((4-1)), pp.104-111. ￿10.11648/j.ijhep.s.2015020401.18￿. ￿hal-01184168￿

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International Journal of High Energy Physics 2015; 2(4-1): 104-111 Published online August 6, 2015 (http://www.sciencepublishinggroup.com/j/ijhep) doi: 10.11648/j.ijhep.s.2015020401.18 ISSN: 2376-7405 (Print); ISSN: 2376-7448 (Online)

The Physics of Mass Gap Problem in the General Field Theory Framework

E. Koorambas

Physics Department, National Technical University, Zografou, Athens, Greece Email address: [email protected] To cite this article: E. Koorambas. The Physics of Mass Gap Problem in the General Field Theory Framework. International Journal of High Energy Physics . Special Issue: Symmetries in Relativity, Quantum Theory, and Unified Theories. Vol. 2, No. 4-1, 2015, pp. 104-111. doi: 10.11648/j.ijhep.s.2015020401.18

Abstract: We develop the introduced by Ning Wu with two Yang-Mills fields adjusted to make the mass term invariant. In the specific representation there arise quantum massive and classical massless no-Abelian vector modes and the gauge interaction terms. The suggested model will return into two different Yang-Mills gauge field models. Next, we focus on calculating `the meet of the ' of those quantum massive and classical massless vector fields with respects to the double Yang-Mills limit. We demonstrate that our proposed version of the (QCD) predicts mass gap ∆ > 0 for the compact simple gauge group SU (3). This provides a solution to the second part of the Yang-Mills problem. Keywords: Gauge field Theories, Quantum Chromodynamics, Yang-Mills Problem

dimensional space-time. 1. Introduction The YM theory is the key ingredient in the Standard The laws of quantum physics are for the world of Model of elementary particles and their interactions. A elementary particles what Newton’s laws of classical solution to the YM problem, therefore, would both place the mechanics are for the macroscopic world. Almost half a YM theory on a firm mathematical footing and demonstrate a century ago, Yang and Mills introduced a remarkable new key feature of the physics of strong interactions. framework to describe elementary particles by using The foundation of the YM theory and the mass gap geometrical structures. Since 1954, the Yang-Mills (YM) problem could be formulated as follows: Prove that, for any compact simple gauge group G, there exists a quantum YM theory [1] has been the foundation of contemporary 4 elementary particle theory. Although the predictions of the theory of R , and that this theory predicts a mass gap ∆ > 0. YM theory have been tested in many experiments [13], its Or, more explicitly, given a simple Lie group G (e.g. SU (2) mathematical foundations remain unclear. The success of the or SU (3)), show that A) there exists a full renormalized quantum version of YM YM theory in describing the strong interactions between 4 elementary particles depends on a subtle quantum- theory on R based on this group; mechanical property, termed the mass gap : quantum particles B) there is a number ∆ > 0 such that every state in the have positive masses even though classical waves travel at theory (except the vacuum) has energy at least ∆. In the speed of light. The mass gap has been discovered other words, there are no massless particles predicted experimentally and confirmed through computer simulations, by the theory (except the vacuum state) Assuming that the quantum chromodynamics (QCD) is the but is yet to be understood theoretically [2], [3], [4]. The [1] theoretical foundations of the YM theory and the mass gap valid theory , the first part of CMI problem description, the may require the introduction of new fundamental ideas in problem of the foundation of the YM theory, is a technical physics and mathematics [1]. As explained by Arthur Jaffe rather than a fundamental physical problem. The goal of this and Edward Witten in the Caly Mathematical Institute (CMI) paper is to investigate the possibility of a solution to the problem description [2], the YM theory is a generalization of second part of the official (CMI) problem description, Maxwell’s theory of , in which the basic namely the mass gap problem, by utilizing the General Field dynamical variable is a connection on a G-bundle over four- Theory (GFT) formulation of QCD ([5], [6], [7], [8], [9], [10], [11],[12]). International Journal of High Energy Physics 2015; 2(4-1): 104-111 105

2. The Mass Gap Problem 3. The General Field Theory Framework In the standard QCD model, processes that probe the As we have seen, the ([16], [17], [18], short-distance structure of hadrons predict that quarks inside [19], [20], [21], [22], [23]) is excluded in the QCD theory the hadrons interact weakly [13], [14], [15]. Since coupling, (gluons are massless because the QCD Lagrangian has no g, is small, the classical QCD analysis is a good first spinless fields and, therefore, no obvious possibility for approximation for these processes of interaction [13], [14], spontaneous -breaking). It follows that the only [15]. However, for YM theories in general, it is a general possible mechanism that can introduce the mass term in the requirement of the group equations of the gluon field while remaining consistent with the occurrence of quantum field that coupling, g s increases in reverse law to the mass gap is the Wu mass generator mechanism ([5-12],[41- hadrons’ momentum transfer, until momentum transfer 46]). Here, the Wu mass generator mechanism introduces becomes equal to the vector boson. Since spontaneous mass terms without violating the SU (3) gauge symmetry as symmetry-breaking via the Higgs mechanism to give the long as the second gauge field is a purely quantum gluons mass is not present in QCD ([16], [17], [18], [19], phenomenon [42]. [20], [21], [22], [23]), QCD contains no mechanism to stop The quark field is denoted as the increase of coupling, g . In consequence, quantum effects s ψ = become more and more dominant as distances increase. a (x ),( a 1,2,3) (1) Analysis of the behavior of QCD at long distances , which where α is color index and the flavor index is omitted. includes deriving the hadrons spectrum, requires foundation Ψ is defined as follows: of the full YM quantum theory. This analysis is proving to be very difficult ([2], [3], [4]). ψ  [1] Note: Lattice QCD is a non-perturbative approach to 1 ψ=  ψ  solving the QCD theory [27], [28]. Analytic solutions in low- 2  (2) ψ  energy QCD are hard due to the nonlinear nature of the 3  strong force. This formulation of QCD in discrete space -time introduces a momentum cut off at the order 1/s, where s is All ψα form the fundamental representative space of SU (3) the lattice spacing [29], [30]. c. a The mass gap problem can be expressed as follows: In this propose version of QCD, the gauge fields A1µ and Why are nuclear forces short-range? Why there are no a a A2µ are introduced .Gauge field A2µ is introduced to ensure massless gauge particles, even though current experiments the local gauge invariance of the theory. The generation of suggest that gluons have no mass? Aa Aa This contradiction is the physical basis of the mass gap gauge field 1µ is a purely quantum phenomenon: 1µ is problem. More explicitly, let us denote this energy minimum generated through non-smoothness of the scalar phase of the ′ fundamental spinor fields [42], [47]. (vacuum energy) of YM Hamiltonian H by Em , and let us a ϕ From the viewpoint of the gauge field A1µ generation denote the lowest energy state by m . By shifting original Hamiltonian H by −E , the new Hamiltonian H′ = H − E described here, the gauge principle is an “automatic” m m consequence of the non-smoothness of the field trajectory in has its minimum at E = 0 (massless state) and the first state ϕ the Feynman path integral [42], [47]. The Lagrangian of the m is the vacuum vector. We now notice that in space-time model is: the spectrum of Hamiltonian is not supported in region (0, ∆) , with ∆ > 0 .

5 1 µν ℑ=−ψ[( γ ∂−µigA µ )] + m ψ − TrAA ( µν ) 14K 1 1 2 (3) 1 µνm µ µ −TrAA(µν ) − Tr [(cos aA sin aA )(cos aAµ + sin aA µ )] 4K22 2 K 121 2 where = + G1µcos aA 1 µ sin aA 2 µ =∂ −∂ − (5) A1µν µν A 1 νµ A 1 igAA[ 11 µν , ] , = − + G2µsin aA 1 µ cos aA 2 µ =∂ −∂ + A2µν µννµ A 2 A 2 igta[ A 22 µν , A ] (4) After these, the Lagrangian given by equation (3) changes into This Lagrangian can be proved to have strict SU (3) c gauge symmetry. Since Aµ and A2µ are not eigenvectors of mass matrix, we can apply the following transformations:

106 E. Koorambas: The Physics of Mass Gap Problem in the General Field Theory Framework

5 1 µν i ℑ=−ψ[ γ ( ∂−µig cos aG µ + ig sin aG µ ) +− m ] ψ GG µν 1 24 10 10 2 , (6) 1 µνim i µ i −GGµν − GG µ +ℑ 420 20 2 1 1 gI

cos 2 θ ℑ=igψγµ (cos θ C − sin θµψ F ) − wu gfCCCijk iµν i k Ig2 wuµ wu θ 2 0 µ ν 2cos wu θ θ sinwu ijkiµν jk sin wu ijkiµν jk ijki µν jk +gfFFFνν − gfFCC µν + gsin θ fCCF µν 220 2 20 2wu 0

3 2 1− sin 2 θ 2 wu sin θ (6a) −4 gffCCCC2 ijkilm jklµ m ν − wu gff2 ijkilm Fj F k F lm F m ν 2 θ 2 µ ν 2 µ ν 4cos wu 2 + 2 θ θ ijk ilm j k lm m ν gtc2 wusin wu ffCCCFµ ν 2 θ sin wu 2 ijkilm jklµν m jkl µν m jkl µν m −gf f( CCFFµν + CFFC µν + CFCF µν ) 2 2 Where of freedom, because the temporal gauge condition is unchanged under the following local gauge transformation: =∂i −∂ i = = i λ Gm0µν µ G m ν ν Gm m µ ,( 1,2) , Gmµ G m µ i / 2 (7) →−1 +( θ ) ∂ − 1 G2 UGU 2 1/sin ig UUµ , (7b) ℑ The Lagrangian gI only contains interaction terms of where gauge fields. ℑ includes a quantum massive gluon field, G , gI 1 ∂U = 0, U= U( x ). with mass m ,and massless gluon field, G2 . t (7c) Since there exist two sets of gluons, there may exist three In order to make this remainder gauge transformation sets of in mass spectrum, (g1g1), (g1g2) and (g2g2), degree of freedom completely fixed, we have to select with the same spin- but different masses (see [12]). Transformation (5) is pure algebraic operations which do another gauge condition for gluons G1µ . For instance, we can not affect the gauge symmetry of the Lagrangian [5-12]. select the following gauge condition for gluons G1µ , They can, therefore, be regarded as redefinitions of gauge fields. The local gauge symmetry of the Lagrangian is still µ ∂G µ = 0. (7d) strictly preserved after field transformations. In other words, 1 the symmetry of the Lagrangian before transformations is If we select two gauge conditions simultaneously, when absolutely the same with the symmetry of the Lagrangian we quantize the theory in path integral formulation there will after transformations. We do not introduce any kind of be two terms in the effective Lagrangian. The symmetry breaking in this paper. effective Lagrangian can then be written as:

3.1. Renormalization of the Model 1 1 ℑ=ℑ−ffaa − ff aa +η M ηη + M η , eff 11 221f1 12 f 2 2 (7e) In order to quantize the suggested gauge field theory of 2a1 2 a 2 nuclear forces in the path integral formulation, we first have to select gauge conditions [9], [42]. To fix the degree of where freedom of the gauge transformation, we must select two a= a a= a gauge conditions simultaneously: one for the quantum f1 f 1( G 2 µ ), f2 f 2( G 1 µ ). (7δ) massive gluons G µ and another for the massless gluons G µ 1 2 If we select [9], [42] For instance, if we select temporal gauge condition a = ∂ µ α for massless gluons G2µ , f2 G 1 µ , (7ε)

= G2 0, (7a) then the for quantum massive gluons G1µ is:

there still exists a remainder gauge transformation degree

−∆ɶ αβ =−δ αβ 2 +−×−− 2 ε 2 + 2  iFµν () k i /( kmi g ) gµν (1)/( akkkam1 µ ν 2 g ).  (7µ)

If we let k approach infinity, then ɶ αβ 1 ∆µν (k ) = . (7θ) F k 2 International Journal of High Energy Physics 2015; 2(4-1): 104-111 107

In this case, and according to the power-counting law, the 4. The Physics of Mass Gap Problem in gauge field theory of nuclear forces with quantum massive gluons suggested in this paper is a kind of renormalizable the General Field Theory Framework theory [9], [42]. A strict proof on the renormalizability of the In (QFT) the number of particles is gauge field theory can be found in [9]. not constant, due to the particles’ creation and annihilation. 3.2. The Two Different Kinds of Yang-Mills QCD Limit Therefore, while we can define the mass-state of the system as the mass-state of all particles or equivalent, we cannot In a proper limit, GFT [5-12] gauge field theory can return define the mass-state of a specific particle. In propose version to YM gauge field theory. There are two kinds of YM limits. of QCD, nevertheless, the mass-state of the system has non- The first kind of YM limit corresponds to very small zero mass. A possible interpretation of the existence of both parameter a in the equation (5). Let: quantum massive and massless gluon states is that Wu’s mass generator mechanism introduces the phenomenon of mass → a 0 (7ι) gap. This is achieved in the following steps: First, in the Then proposed QCD version the non-zero gluon mass corresponds to the massive vector propagator [24] and the zero gluon cosθ= 1,sin θ = 0 (7ξ) mass corresponds to the massless vector propagator of the short-distance QCD [13], [14], [15]. For the given wave = = G1µ AG µ, 2 µ B µ (7φ) vectors k1 and k2 the propagators of massless and quantum massive gluons with respect to the two different YM gauge In this case, the Lagrangian density becomes field models are given by:

ℑ=−ψ γµ ( ∂−igGi T i ) + m  ψ g  µ1 µ  ∆αβ = − δ αβ 2µν 2µν (k 2 ) i 2  (8) 2 + ε (7λ) k2 i  1iiµν 1 ii µνm ii µ −CCµν − FF µν − CC µ 4 4 2 − δ αβ k k  ∆αβ =i + 1µ 1 ν 1µν(k 1 )  g 1 µν  (9) We could see that, only quantum massive gauge field kMi2+ 2 − ε  M 2  directly interacts with matter field. This limit corresponds to 1 the case that gauge interactions are mainly transmitted by where [2] quantum massive gauge field. But if a strictly vanishes, the Lagrangian does not have gauge symmetry and the theory is 1µ≠ 2,1 µν ≠ 2 ν , (10a) not renormalizable. The second kind of YM limit corresponds to very large parameter a in the equation (5). Let: = = g1µν( e 1 µ , e 1 ν ) , g2µν( e 2 µ , e 2 ν ) , (10b) a → ∞ (7ο) ∂ ∂ = e µ , = e µ (10c) then ∂x1µ 1 ∂x2µ 2 cosθ= 0,sin θ = 1 (7κ) g1µν , g2µν are metrics of the space-time regions and e2µ ,

= = − e2µ are vectors of the basis induced by the selection of local G1µ BG µµ, 2 A µ (7σ) coordinates x1µ , x2µ on the neighborhood of the given points Then, the Lagrangian density becomes corresponds to the two different YM gauge field models. k ℑ=−ψ γµ ( ∂+i i ) +  ψ Furthermore, due to equation (10a), the wave vectors 1µ µigG2 µ T m  and k2µ lie in different regions in the momentum space in 2 (7υ) 1iiµν 1 ii µνm ii µ −FFµν − CC µν − CC µ their own right. Here, we express the meet of the propagators 4 4 2 ∆αβ ∆αβ 1µν (k 1 ) and 2µν (k 2 ) of the two different YM gauge field In this case, only massless gauge field directly interacts models in the momentum space, in terms of its hypersurface with matter fields. This limit corresponds to the case that of the present ([25]) at the origin of the propagators paths as gauge interactions are mainly transmitted by massless gauge follows field. In the particles’ interaction model, the parameter a δ1µν∆ αβ ∧∆ δ 2 µν αβ = should be finite, αβ11 µν()k αβ 22 µν () k (11a) liminf {min(∆ (k ), ∆ ( k )} < < ∞ →′ ε → 11 2 2 0 a (7π) (,)kk1 2 Si 0

In this case, both quantum massive gauge field and where massless gauge field directly interact with matter fields, and gauge interactions are transmitted by both of them. 108 E. Koorambas: The Physics of Mass Gap Problem in the General Field Theory Framework

+ 2 µν αβ 1 =′∩ ∈ ++>=+∞ δ 1 ∆ =∆ = U1{ S {( kk 11 , 12 ) Rak : 11 bk 12 c 0} (0, ), (11e) αβ µν ()()k k , (11b) 1 1 11 2+ 2 − ε k1 m i − =′ ∈2 ++<=−∞ U1{ S∩ {( kk 21 , 22 ) Rak : 21 bk 22 c 0} ( ,0) (11f) δ 2µν∆ αβ =∆ = 1 αβ2 µν ()k 2 22 () k 2 (11c) + + = − ε the r1 : akx bk y c 0 is line such that the points (k11 , k 12 ) k2 i + − and (k21 , k 22 ) of S lie on different sides of r1 and U1, U 2 are S′ = S\ r are functions in the momentum space. The subset 1 the propagator’s half-planes of the present. of the hypersurface of the present is disconnected, since it is Substituting equations (11d), (11c), (11d) to equation (11a), the disjoint union of the two propagator’s half-planes of the ∆αβ ∆αβ we calculate the meet of the 1µν (k 1 ) and 2µν (k 2 ) present that corresponds to the different 1µ and 2µ indices propagator in the momentum space of two different YM gauge field models. [2] Note: In the proposed version of QCD, the letters δ1µν∆ αβ ∧∆ δ 2 µν αβ = αβ11 µν()k αβ 22 µν ()0 k (11g) 1µ ,1 ν and 2µ ,2 ν denote the different space-time indices of the massive and massless propagators corresponding to Equation (11g) shows that the meet of the propagators ∆αβ ∆αβ different YM gauge field models 1µν (k 1 ) and 2µν (k 2 ) of the two different YM gauge field

+ − models is zero, because the propagator’s half-planes of the U∩ U = ∅ 1 2 present are disjoint in the momentum space. After some calculations, the Fourier transformation to the position space and of equation (11g) is given by ′ = =+ − = −∞ +∞ S Sr\1 U 1∪ U 2 ( , ) \{0} , (11d)

δδ(4)− 1µν ∆−∧ αβ δδ (4) − 2 µν ∆ αβ −= (xy22 )αβ 111 µν ()() xy xy 11αβ 222 µν ( xy )0 (12) where

+∞ 4 +∞ 4 d k − d k − δ (4) (x− y ) = 1 e ik1( x 1 y 1 ) , δ (4) (x− y ) = 2 e ik2( x 2 y 2 ) (13) 1 1 ∫ π 4 2 2 ∫ π 4 −∞ (2 ) −∞ (2 )

+∞ − 4 ik2( x 2 y 2 ) αβ d k e αβ ∆−=2 − δ 2µν (xy 2 2 )lim+ 4 2 gi2 µν () , (14) ε →0 ∫ π+ ε −∞ (2 ) k2 i

+∞ − 4 ik1( x 1 y 1 ) αβ d k e αβ kµ k ν  ∆−=(xy )lim1 () − igδ + 1 1 1µν 1 1 + 4 2 2 1 µν 2  (15) ε →0 ∫ π+ − ε −∞ (2 ) kMi1  M 

δ (4) − δ (4) − ∆αβ − ∆αβ − (x1 y 1 ) , (x2 y 2 ) are the Dirac delta functions and 1µν (x 1 y 1 ) , 2µν (x 2 y 2 ) are the two gluon propagators in the position space. 4 4 The integral of equation (12) over d x 1 , d x 2 is given as follows:

δδ(4)−∆− 1µν αβ 44 ∧ δδ (4) −∆− 2µν αβ 44 = ∫∫()()xy22αβ 11121 µν xydxdx ∫∫ ()() xy 11αβ 22212 µν xydxdx 0 (16)

Here, we integrate equation (16) in the following steps. (1)= 3 (2)= 3 V4 dxdt 1 , V4 dx dt 1 (19) First, by fixed y1, y 2 , the equation (16) becomes ∫1 ∫ ∫2 ∫

δ1µν∆− αβ 42 ∧∆− δ µν αβ 4 = (1) (2) ∫αβ1111 µν()xbdx ∫ αβ 2222 µν ()0 xbdx (17) V4 , V4 are the space-time volumes. Equation (18) is written in style of (11g) as follows Next, by considering an insulating color source, in δ1µν∆ αβ ∧∆ δ 2 µν αβ = Feynman gauge, the propagators magnitudes are constant in αβ11 µν()x αβ 22 µν () x the position space. Therefore the integral of the equation (12) over d4 x , d4 x , gives liminf{min(∆ (x ), ∆ ( x )} = 0 1 2 → ɶ′ 11 2 2 (20a) (x1 , x 2 ) S δ1µν∆− αβ (1) ∧∆ δ 2 µν αβ − (2) = αβ1114 µν()xbV αβ 2224 µν ()0 xbV (18) where where International Journal of High Energy Physics 2015; 2(4-1): 104-111 109

− mx 1 ɶ ɶ µν αβ e ′ = of δ 1 ∆ =∆ = are functions in the position space. The subset S S\ r 2 αβ1 µν ()x 1 11 () x , (20b) x1 the hypersurface of the present is disconnected, since it is the disjoint union of the two propagator’s half-planes of the µ µ δ 2µν∆ αβ =∆ = 1 present that corresponds to the different 1 and 2 indices αβ2 µν ()x 2 22 () x (20c) x2 of two different YM gauge field models.

ɶ+ ɶ − = ∅ ɶ′ = ɶ =ɶ+ ɶ − =−∞+∞ U1∩ U 2 and S Sr\2 U 1∪ U 2 ( , ) \{0} (20d)

ɶ + =ɶ′ ∈2 ++>=+∞ U1{ S∩ {(,) xx 1112 Rax : 11 bx 12 c 0}(0,), (20e)

ɶ − =ɶ′ ∈2 ++<=−∞ U1{ S∩ {(,) xx 2122 Rax : 21 bx 22 c 0}(,0) (20f) + + = the r2 : ax by c 0 is line such that the points (x11 , x 12 ) Next, a basic principle in nuclear physics which states that + − the combined operations of charge conjugation (C), time and (x , x ) of ɶ lie on different sides of r and Uɶ, U ɶ 21 22 S 2 1 2 reversal (T), space inversion (P) in any order is an exact are the propagator’s half-planes of the present. symmetry of the strong force [31], [32]. The CPT symmetry Equation (20a) shows that the propagator’s half-planes of is conserved only if our theory respects the Lorentz invariant the present are disjoint in the position space, because the and the microcasuality principle [33], [34]. ∆αβ ∆αβ meet of 1µν (x 1 ) and 2µν (x 2 ) of the two different YM Gluon of complex mass exchanged between quarks affect αβ αβ our massless YM light cone limit, respecting the gauge field models is zero. ∆µν (x − b ) and ∆µν (x − b ) 1 1 1 2 2 2 microcasuality principle by the following postulates: lie outside each other’s light cone [25]; these propagators, A) The proposed gluons of complex mass µg+iµg lie outside therefore, are not casually connected [26]. Propagators of our massless YM light cone limit. Therefore such gluons ∆αβ − ∆αβ − 1µν (x 1 b 1 ) and 2µν (x 2 b 2 ) are separated from each are not an observable mass state for an observer that lies other in a (space-like) direction of distance inside our massless YM light cone limit. B) In the low scale of energies quarks and gluons will 2=η 11µν − η 22 µν S|1µν xx 2 µν xx | (20g) arrange themselves into a strong bound state in where at least two quarks and gluons must lie outside each other’s YM light η η cone limit. Furthermore, for a multi quarks and gluons states Where 1µν , 2µν are the metrics of the disjoint space- that lie inside our massless YM light cone limit, there exists time regions of the two different YM gauge field models. The 2 at least one quark and gluon state that lies inside the massive space-like region S corresponds to an energy scale YM light cone limit. 24= 2 2 ( mg cℏ / S ). Here, the gluon mass is complex number A gluon of complex mass which is exchanged between quarks is equivalent to the massive Abelian gluon ∆µ by µg+iµg, the imaginary part iµg corresponds to the space-like region between the two YM light cone limits and the real part postulate (B), whose absolute value of mass m g is not an observable mass state in our massless YM light cone limit by µg is obtained by the mass definition within the space-time regions of the two different YM gauge field models, which the postulate (A). must be real. The m is the absolute value of the gluon mass The postulates predict the quantum massive gap Abelian g gluon ∆µ whose absolute value of mass affects our massless mg = (µg+iµg)(µg-iµg) that is constrained to the above space- like portion of space-time [37],[38],[39],[40]. This massive YM limit in a way that makes nuclear forces short-range at gluon either travels forward in time with complex mass low scale of energies [1], [2]. The Lagrangian density (7υ) becomes µg+iµg or travels backward in time with complex conjugate mass µg-iµg [37], [38].

∆2 µ i i 001i 20µν i 1 00 µν 00 µ ℑ=−ψ γ() ∂+µigGT µ +∆ g δ µ T + m  ψ − G G µν −∆∆−∆∆ µν µ (20υ) 2 1  420 4 2

2 where ∆µ quantum massive gap Abelian gluon, G2 massless not supported in an space-like region S∆ that corresponds to gluon and (∆) is the mass gap: ∆=(µg+iµg)(µg-iµg)= mg. ∆24 = ℏ 2 2 Where g and gδ are the massless gluon coupling and an energy scale ( c/ S ∆ ). Therefore, the Hamiltonian quantum massive gap Abelian gluon coupling respectively. in space-time is bounded by ∆. Here, the quantum massive gap Abelian gluon is defined in Finally, in the proposed version of QCD, the vacuum state the Abelian subgroup U (1) ∆ of the SU(3)c gauge group for has zero energy (E=0), corresponding to G2 (massless gluon). details (see [48]). The excitation state above the vacuum energy has non-zero For the Lagrangian density (20υ) we construct the energy Em, corresponding to G1 (quantum massive gluon). Hamiltonian of the system for details (see [49]). The latter predicts that the Hamiltonian is not supported in Equations (11g) and (20a) predict that the Hamiltonian is the region (0, ∆) with ∆ > 0 and is bounded below by ∆ . 110 E. Koorambas: The Physics of Mass Gap Problem in the General Field Theory Framework

= ϕ 5. Discussion E 0 . Τhe first state m is the vacuum vector. We now notice that, in space-time, the spectrum of the YM Hadron colliders, such as the Tevatron, can place the ∆ strongest limits on the new colored particles (squarks and Hamiltonian is not supported in region (0, ) , with ∆ > 0 . gluinos in or KK excitations of quarks and According to the YM theory, therefore, over compact ϕ gluons in models with universal extra dimensions [35–36]. gauge groups SU (3) there is always a vacuum vector m and Typically, limits of ∼ 200 GeV are obtained for new colored a mass gap ∆. Wu’s mass generator mechanism can always be particles. Here, the predicted gluon mass gap state ∆ is not applied to any YM type of action. Thus, any YM theory close to the gluon mass ∼ 200 GeV , thus gluon mass gap is accounts for a mass gap. not subject to the Hadron collider’s limits [35-36]. The proposed model is a massive theory without a In the Lorentz gauge, the free Lagrangian of QCD field symmetry breaking mechanism and is therefore a model with ℑ = theory (equation (6), with gI 0 ) yields to the following a quantum mass gap. Thus, the massless YM theory is altered field equations: with a quantum massive model and the original problem could be solved. ∂2 1G1µ 2 1 = ∇Grµ( ) − Gr µ ( ), c2∂ t 21 L 2 1 (30) 6. Conclusion ∂2 1 G µ 2 = ∇ 2 The general gauge theory with quantum massive and 2 2 G2µ ( r ) c∂ t classical massless no-Abelian vector modes and the gauge interaction terms is developed. The suggested gauge field L the length of the system given by model will return to the two different YM gauge field models. ℏ `The meet of the propagators' of those quantum massive and L = (31) mc classical massless vector fields with respects to the double Yang-Mills limit is calculated. Where ℏ the Planck constant, c is the speed of light, We have shown that the proposed gauge model can ∆=(µg+iµg)(µg-µg)= m g the mass gap of the system. If the introduce mass terms while being consistent with the existing source is a point color at the origin, only the time- mass gap ∆. In this model the mass gap is introduced by Wu’s = Φ = Φ mass generator mechanism. componentsC02 2 ,C01 1 are nonvanishing. The short- distance solution of the eq (30) that corresponds to the Since the gluon quantum field G1 is massive, whereas the massless gluons is given as follows: gluon field G2 is massless, we conclude that system’s free energies in the proposed version of QCD are bounded in a space-time. G( r ) = s (32) 20 4π |r | We also notice that, in space-time, the spectrum of the YM Hamiltonian is not supported in region (0, ∆) with ∆ > 0 . where as is the quark-gluon coupling. Wu’s mass generator mechanism can always be applied to Gluons that respect the solution (32) are mediators of the any YM type of action. Therefore any YM theory accounts long-range nuclear force at high energy scale. The long- for a mass gap. distance solution of eq (30) corresponds to the massive The proposed model is a massive theory without a gluons and is given as follows: symmetry breaking mechanism and is therefore a model with a quantum mass gap. Thus, the massless YM theory is altered as −|r |/ L with a quantum massive model and the original problem G( r ) = e (33) 10 4π |r | could be solved.

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