General Intrinsic Theory of General Large $ N {C} $ QCD, SU (3) QCD

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arXiv:1607.07810v1 [nucl-th] 26 Jul 2016 ngnrlsrn neatoso ur-lo systems, quark-gluon of research large for interactions useful and strong is physics QCD general for that on impacts well-known is huge It their Nobel science. of three because awarded prizes been have physics theories nuclear on tions strong the different is to states. relevant physics electricity states charged energy different high of hadron-dynamics theory in interaction SU(2) theory. theory it, as field field of viewed gauge development gauge is further Yang-Mills and a useful As developed and greatly powerful This be the [1]. has theory interaction now strong theory a SU(2) give non-Abelian to to theory theory gauge gauge U(1) Abelian the ized ur-lo neatos .. e.2 tde Nucleon- Large studied in Ref.[2] Annihilation Antinucleon e.g., interactions, quark-gluon te nenlfcoso oin:tesi ftegluons the of the spin [3– from the achieved spin motions: be nucleon or must the factors part internal of other other third the one therefore, give the 5], of the only of spin can spin total the quarks that the show to investigations Various contribution nucleon? the gluons and give quarks nucleon of motions in the do how problem: urgent ∗ † [email protected] [email protected] si elkont epe pt nw h investiga- the know, to up people, to known well is As general- time, first the for 1954 in Mills, and Yang 1.Introduction h elkonncensi rsssostekyand key the shows crisis spin nucleon well-known The arndnmc n ()QDguefil hoisi general in theories field gauge QED U(1) and hadron-dynamics N eea nrni hoyo eea large general of theory intrinsic General ulo pncii,ie,peet eea large general the presents field i.e., general in crisis, theories spin field nucleon gauge QED U(1) and dynamics edter.I h eea large general the In theory. field prtr fqakadguefilswith fields gauge and quark of operators eea edter,adntrlyddcstegueinvari gauge the deduces naturally and theory, quark-glu field in theorem general Coulomb corresponding new and momenta aallcniin aey o-bla iegneadc and divergence non-Abelian namely, ( conditions parallel ASnmes 11.q 23.t 14.20.Dh 12.38.-t, 11.15.-q, numbers: PACS exp physics particle current interact by etc. strong measured QCD properties exactly general be consisten the can are calculate w that inv to theories systems gauge utilized their interacting the be that can both strong so keeps of relations and lots commutation interesting, v on popularly is a researching difficulti which play of huge should some door the it a breakthroughs technique, paper new t this Specially, a Therefore, presents characteristics. theory symmetry potential intrinsic good has c hspprgvsgnrlitiscter fgnrllarge general of theory intrinsic general gives paper This C scretyuiie oinvestigate to utilized currently is QCD 2 nttt fTertclPyis ejn nvriyo Te of University Beijing Physics, Theoretical of Institute 3 eateto hsc,EoyUiest,Alna Georgia Atlanta, University, Emory Physics, of Department rgestwrssligtencensi crisis spin nucleon the solving towards progress 2 otwsenAeu,WLfyte N49723,USA 47907-2036, IN W.Lafayette, Avenue, Northwestern 525 1 .Huang, C. eateto hsc n srnm,Pru University, Purdue Astronomy, and Physics of Department N c QCD. 1, ∗ N c ogCagHuang, Yong-Chang C,w icvrntol h eea oain rnvrea transverse covariant general the only not discover we QCD, SU ( N N c c ag ymtyb ote hoe ngeneral in theorem Noether by symmetry gauge ) C’ ne tutrs ag nain angular invariant gauge structures, inner QCD’s aey ae ntesldbsso es[2 3 ta ) al at 13] Refs.[12, of bases ( respectively, solid [13], the QED theory on in gauge based the crisis, namely, isospin to SU(2) spin solutions nucleon and the to [12] generalize analogous to problems, only key This not problem. wants serious still the paper solved are Ref.[13] momenta serious 16], [15, angular the missing invariant is gauge still that there problem symmetry, interaction strong isospin in with Because, systems SU(2) theory. in of field strength gauge properties field isospin gauge the the show of the components conditions that different consistent found four further con- Ref.[13] other the of while two ex- given conditions, had been sistent Ref.[14] had Ref.[12]. that in parts shown actly corresponding generaliz- Abelian properly their by condi- achieved ing consistent impressively non-Abelian be Furthermore, the can all tions that consistent. see and achieved will invariant people new gauge some the is be that must theory so there is conditions that generalizations additional find of required will kind This people decomposi- and fields. nontrivial, intrinsic gauge the color under of tion fields systems quark-gluon interaction to the the order of of in structure [13] theory inner field the gauge investigate isospin and [12] QED in e.g.,Refs.[6–11]. crisis, the have solving endeavors representing for good fully made of of been kinds operators Various work the movements. original the these define the give to spin, motions generally and nucleon these is quarks total how the the uncover to to of contribution order movements In angular gluons. orbital the and e nti ae,gnrlz h eeatinvestigations relevant the generalize paper, this in We, 2, i ae’ eeal eopsn gauge decomposing generally paper’s his N † n pnadobtlaglrmomentum angular orbital and spin ant .Epcal,teahee eut here results achieved the Especially, t. r ,btas htti eea system general this that also but ), url osadt ietepeiepredictions precise the give to and ions si h ulo pncii n opens and crisis spin nucleon the in es N c n .H Zhou H. B. and rac n hi nua momentum angular their and ariance rmnsdet hi ag invariant gauge their to due eriments hooy ejn,102,China 100124, Beijing, chnology, c C,S()QD SU(2) QCD, SU(3) QCD, tlrl nftr hsc research. physics future in role otal r n rgestwrssligthe solving towards progress and ory nfil neato ytm ae on based systems interaction field on t ieetsmercproperties, symmetric different ith C,S()QD U2 hadron- SU(2) QCD, SU(3) QCD, 02-40 USA 30322-2430, 3 edter and theory field nd 2 in order to make essential steps towards solving nucleon and an equation spin crisis, but also to give their consistent unification description theory. dabef cde +2dacef bde − f abedecd + f adedebc + dadef ebc =0 The arrangement of this paper: Sect. 2 is SU(Nc) gauge field inner structures and new Jacobi identity; Sect. (2.7) Using Jacobi identity dabef cde +dacef bde +f adedebc =0 3 is general decompositions of SU(Nc) gauge fields; Sect. [16] to simplify Eq.(2.7), we deduce a new Jacobi identity 4 shows gauge invariant angular momenta of SU(Nc) quark-gluon field interaction systems and gives a new of structure constants viewpoint on the resolution of the nucleon spin crisis; Sect. 5 is discussion; Sect. 6 gives summary and conclu- dacef bde − f abedecd + dadef ebc =0, (2.8) sion. 2. SU(Nc) gauge field inner structures and new Jacobi because Eq.(2.8) can directly be deduced by identity a For SU(Nc) gauge field generators T ,a = a b c b c a c a b 2 {T , [T ,T ]}−[T , {T ,T }]−{T , [T ,T ]} = 0 (2.9) 1, 2, ..., Nc − 1; Nc may be enough large, we have Eq.(2.9) is just a new Jacobi identity, which can be 1 1 T aT b = (T aT b + T bT a)+ (T aT b − T bT a) proved by directly expanding Eq.(2.9). Eq.(2.8) gives 2 2 a new relation between symmetric and antisymmetric structure constants. 1 ab 1 abe abe e The other more investigations on inner structures of = δ + (d + if )T ,N1 = N,N2 = M, (2.1) 2Nc 2 SU(Nc) gauge fields will be naturally given in following sections. T a,T b 1 δab 1 dabeT e, T a,T b because { } = 2Nc + 2 [ ] = 3. General decompositions of SU(Nc) gauge fields if abeT e. Using Eq.(2.1), we have As a general generalization of the QED and the isospin cases, we investigate general decompositions of SU(Nc) 1 1 δabT cT d = [T aT b − (dabe + if abe)T e]T cT d. (2.2) gauge fields by the general method of field theory. The 2Nc 2 projection operators are defined as [12, 13]

Similar to deduction of Eq.(2.2), we get j j 1 j j j L = ∂ ∂ ,T = δ − L , (∆ = ∂ ∂k) (3.1) k ∆ k k k k k 1 ad b c a d 1 ade ade e b c δ T T = [T T − (d + if )T ]T T , (2.3) and the gauge potential Aia(i = 1, 2, 3; a = 2Nc 2 2 1, 2, 3, ..., Nc − 1) are intrinsically decomposed as Using Eqs.(2.2) and (2.3), we can have ja j ka ja j ka A⊥ = Tk A , Ak = LkA . (3.2) 1 ab cd ad bc a b c d 1 abe abe 1 (δ δ −δ δ )= tr(T [T T ,T ])− (d +if ) ( SU N 4Nc 2 4 The general action for the ( c) quark-gluon ﬁeld interaction system is [16]

ecd ecd 1 ade ade 1 ecd ecd 4 1 aµν µ d + if )+ (d + if ) (d + if ) (2.4) S = d x[− F Faµν + ψ (iγ Dµ − m) ψ], (3.3) 2 4 Z 4 where we have used tr(T aT b)= 1 δab and tr(T aT bT c)= where Dµ = ∂µ − igAµ are the covariant derivative 2 a a a 1 (dabc + if abc) [16]. with gauge fields Aµ = AµT and T the generator of 4 the SU(N ) group, and ψ stands for the quark field. Using [T bT c,T d]= T b[T c,T d]+[T b,T d]T c and further c When Dµ acts on any vector in Lie algebra space, simplifying Eq.(2.4), we deduce a a a a a we have DµB = ∂µB T − ig[Aµ,B T ] = (∂µB + abc b c a gf AµB )T . In SU(Nc) gauge field theory, the gauge 2 field strength is (δabδcd − δadδbc)+ dabedcde = f acef bde + dadedbce+ N i F aµν = ∂µAaν − ∂ν Aaµ + gf abcAbµAcν . (3.4) We can decompose the spacial part of the gauge po- abe cde ace bde abe ecd ade ebc ade ebc i(d f +2d f − f d + f d + d f ) tential Aak into two components (2.5) ak ak ak Using Eq.(2.5), we achieve both a general contraction A = A⊥ + Ak . (3.5) expression of structure constants of SU(Nc) gauge group Analogous to QED [12], under a SU(Nc) gauge transfor- k ak a k ak a mation U, A⊥ = A⊥ T and Ak = Ak T transform as 2 f acef bde = (δabδcd − δadδbc)+ dabedcde − dadedbce [13] Nc ′ k k † (2.6) A⊥ = UA⊥U , (3.6) 3

′ i A k = UAkU † − U∂kU †, (3.7) k k g + dx3gf abc∂ AakAb Ac0 = 0 (3.14) respectively, which make Z 0 ⊥ kk

′ i where the divergence term has been neglected. A k = UAkU † − U∂kU †. (3.8) g We further utilize the consistent conditions (whose physics meanings will be explained latter) The expressions from Eq.(3.5) to Eq.(3.8) simplify the relevant expressions in Ref.[13]. Using Eqs.(3.2) and (3.5), the general action (3.3) can d3x∂0Aak(−∂ Aa + gf abcAb Ac0)=0, (3.15) Z k 0 ⊥k ⊥k be rewritten as

abc ak b c0 k a0 b c 4 1 k a0 a 0 ak a 0 ak a gf (∂0Ak A⊥kA − ∂ A A⊥kA0) S = d x[− (∂ A ∂kA +∂ A ∂0A +∂ A ∂0A ) Z 2 0 k kk ⊥ ⊥k 2 g 2 bk c0 b c bk c0 c b − (Ak A A⊥kA0 − Ak A A⊥kA0) Nc ∂kAa0∂ Aa ∂kAa0∂ Aa ∂0Aak∂ Aa ∂kAa0 + 0 kk + 0 ⊥k + k 0 ⊥k − ( ′ ′ ′ ′ ′ ′ 2 abb acc abc acb bk c0 b c b c b c abc 0 ak abc b c b c −g d d − d d Ak A A⊥kA0 =0, (3.16) AkkA0 + A⊥kA0)gf + ∂ Ak gf (AkkA0 + A⊥kA0)+ 0 ak abc b c b c 1 abc bk c0 bk c0 and substitute the contraction expression (2.6) of struc- ∂ A⊥ gf (AkkA0 + A⊥kA0) − gf (Ak A + A⊥ A )( 2 ture constants of SU(Nc) gauge group into Eq.(3.9), we achieve

′ ′ ′ ′ ′ ′ ajk a b c b c ab c F Fjk µ AkkA0 +A⊥kA0 )gf − +ψ (iγ Dµ − m) ψ] 4 1 k a0 a 0 ak a 0 ak a 4 S = d x{− (∂ A ∂kA0 + ∂ Ak ∂0Akk + ∂ A⊥ ∂0A⊥k) (3.9) Z 2 k a0 a k a0 abc b c abc b c In order to cancel the non-independent terms in +∂ A ∂0Akk − ∂ A (gf AkkA0 + gf A⊥kA0)+ Eq.(3.9), we need to use the basic consistent conditions 2 0 ak abc b c 0 ak abc b c g bk b c0 c in SU(Nc) gauge theory to simplify Eq.(3.9). ∂ Ak gf AkkA0 + ∂ A⊥ gf A⊥kA0 − (Ak AkkA A0 Nc Because the nonlinearity of the non-Abelian SU(Nc) bk b c c0 bk b c0 c bk b c c0 gauge ﬁeld system, the non-Abelian system is much more −Ak A0AkkA + A⊥ A⊥kA A0 − A⊥ A0A⊥kA ) − complex than the Abelian one. We now generalize the ′ ′ ′ ′ ′ ′ 1 2 abb acc abc acb bk c0 b c two well known Abelian gauge potential (or ﬁeld) consis- g d d − d d (Ak A AkkA0 + 2 tence conditions in QED [12]

⇀ ′ ′ 1 ∇ · A⊥ =0, (3.10) AbkAc0Ab Ac )− F ajkF a +ψ (iγµD − m) ψ} (3.17) ⊥ ⊥k 0 4 jk µ One can see that there is no the cross terms between ⇀ parallel and vertical gauge potentials in Eq.(3.17), which A , ∇× k =0 (3.11) just shows that the general system has very good symmetry properties, i.e., we discover that this general system respectively to non-Abelian SU(Nc) corresponding expressions has very good intrinsic symmetry characteristics. Therefore, we can calculate the corresponding canon- ak abc b ck ical momenta conjugate to the two components of the ∂kA⊥ + gf AkkA⊥ =0. (3.12) gauge potential, respectively,

j k k j a j ak k aj abc bj ck (D Dk −Dk D ) =0, i.e.∂ Ak −∂ A +gf A Ak =0 δL k k k k πak = = −∂0Aak + ∂kAa0 + gf abcAbkAc0 (3.13) k a k k δ(∂0Akk) One can easily prove that Eqs.(3.12) and (3.13) are (3.18) SU(Nc) gauge covariant, and they are similar to the isospin relevant expressions in Ref.[13] except the structure constants and superscripts’ taking values of SU(Nc) ak δL 0 ak abc bk c0 π⊥ = a = −∂ A⊥ + gf A⊥ A (3.19) gauge group and being related to gauge ﬁelds [16]. δ(∂0A⊥k) Using Eq.(3.12) we can have Eqs.(3.18) and (3.19) are the direct generalization of the 3 k a0 a 3 abc a bk c0 isospin relevant expressions in Ref.[13] with the struc- dx ∂ A ∂0A k − dx gf ∂0A A A Z ⊥ Z kk ⊥ ture constants and superscripts’ taking values of SU(Nc) 4

⇀ ⇀ ⇀ ⇀ gauge group and are related to gauge fields [14]. It is = Sq + Lq + Sg + Lg, (4.3) apparent that Eqs.(3.18) and (3.19) are both gauge co- 2 2 2 variant and their summation is exactly the conventional ⇀ ⇀ πak, which show that the above investigations are con- in which, because of Eq.(3.7), Dk = ∇− igAk is the new ⇀ ⇀ ⇀ ⇀ sistent. q q g g covariant derivative. S , L2, S2 and L2 are the spin and Consequently, we need further to generalize, in QED orbital angular momenta of the quark field and gauge respectively, the well-known Abelian gauge field strength field, respectively. Eq.(4.3) is just a direct generaliza- consistent conditions [14] tion of the relevant expression of SU(2) isospin symmetry theory in Ref.[13] because of their similar mathematical ⇀ structure. ∇ · E⊥ =0, (3.20) ak Due to the properties of Ak shown in Eqs.(3.7) and ⇀ ⇀ ⇀ ⇀ q q q q (3.13), L2 satisfies the commutation law L2 × L2 = iL2 ⇀ ⇀ q ∇× Ek =0, (3.21) and is gauge invariant [6]. The gauge invariance of S ⇀ g ak and S2 is obvious; furthermore, A⊥ is gauge covariant ⇀ ⇀ g ∇ · Ek = ρe, (3.22) in Eq.(4.3), which means that L2 is gauge invariant. Under the general Lorentz transformation of the fun- to non-Abelian SU(N ) corresponding expressions c damental fields, Noether’s theorem requires the invari- ak abc b ck ance of the Lagrangian and Hamiltonian densities of the ∂kπ⊥ + gf Akkπ⊥ =0. (3.23) general system [17, 18], e.g., this system is invariant under the Lorentz transformations of Aµ’s all components, Dj πak − Dkπaj =0, (3.24) µ k k although Lorentz transformation of A⊥ is complex, the relevant complex terms of the transformations of all their components can be canceled each other in the whole sys- ∂ πak + gf abcAb πck = ρa = gψ†T aψ, (3.25) k k kk k tem according to the symmetric invariance property of where ρa is the non-Abelian charge density. this system, which is the general rule, see Refs.[17, 18]. 4. Gauge invariant angular momenta of SU(Nc) quark- Furthermore, the frame-dependence issue has important gluon field interaction systems and a new viewpoint on physics meaning, Refs.[9–11] have given the very good the resolution of the nucleon spin crisis investigations on the issue etc. Using Noether theorem, the action (3.17) results in Next, we will derive some useful relations from the the angular momentum of the SU(Nc) quark-gluon field consistency conditions for field strength. Making gauge interaction system as follows transformation of Eq.(4.3)’s the last term, we have

⇀ ⇀ ⇀ ⇀ 3 † 1 3 †⇀ 1 3 a a J1 = d xψ Σψ+ d xψ x × ∇ψ+ d xπ⊥×A⊥+ ′ ⇀ ′k ⇀ k Z 2 Z i Z tr(π⊥k x × ∇A⊥ )= tr(π⊥k x × ∇A⊥)

⇀ ⇀ ⇀ ⇀ d3xπa×Aa+ d3xπa x ×∇Aak+ d3xπa x×∇Aak. Z k k Z ⊥k ⊥ Z kk k + k + tr(x × U ∇U[A⊥k, π ]) = 0, (4.4) (4.1) ⊥ Actually, Eq.(4.1) is a straightforward generalization thus gauge invariance of the last term of Eq.(4.3) de- that in Ref.[13]. mands In order to simplify Eq.(4.1), using Eq.(3.13) we have

⇀ ⇀ ⇀ ⇀ abc b ck a a ⇀ a a a ⇀ ai f A⊥kπ⊥ =0, (4.5) ∇ · [πk (Ak × x)] = −πk × Ak − πki x × ∇Ak consequently we naturally deduce a new consistent condition [6]. ⇀ ak abc b ck ⇀ a Then, with the well known Coulomb law in QCD [16] − (∂kπk + gf Akkπk )(x × Ak). (4.2) ak abc b ck a Utilizing the divergence consistency condition for par- ∂kπ + gf Akπ = ρ , (4.6) allel components (3.25) of gauge ﬁeld strength and the identity (4.2), we can simplify Eq.(4.1) as and in terms of Eqs.(3.23), (3.25) and (4.6), we get ⇀ ⇀ ⇀ 3 † 1 3 †⇀ 1 abc b ck J2 = d xψ Σψ + d xψ x × D ψ gf A⊥kπ =0. (4.7) Z 2 Z i k ⇀ ⇀ ⇀ Using the consistent condition (4.5) of gauge invariance + d3xπa × Aa + d3xπa x × ∇Aak Z ⊥ ⊥ Z ⊥k ⊥ of the last term of Eq.(4.3), we can simplify Eq.(4.7) as 5

⇀ ⇀ ⇀ ⇀ q q g g = S + L3 + S3 + L3. (4.12) abc b ck f A⊥kπk =0, (4.8) Therefore, in general ﬁeld theory, we prove that Eq. (4.12) can strictly and naturally be given out from the ac- which can be used to cancel some extra freedoms of this tion (3.17) by Noether theorem. Analogous to the deriva- system. ⇀ g Furthermore, in order to illustrate the meanings of the tion of Eq. (4.5), the gauge invariance of L3 demands the two consistent conditions (3.15) and (3.16) in simplifying condition (4.7) (or Eqs. (4.6) and (3.23)), which shows a the original action, substituting Eq.(3.19) into Eq.(3.15) key role in transforming Eq. (4.12) to the conventional and using Eq.(3.18), we show that Eq.(3.15) is just the form [6] orthogonal relation between the two components of the ⇀ ⇀ ⇀ 3 † 1 3 †⇀ 1 canonical momentum J4 = d xψ Σψ + d xψ x × ∇ψ Z 2 Z i ⇀ ⇀ ⇀ + d3xπa × Aa + d3xπa x × ∇Aak d3xπakπa =0. (4.9) Z Z k Z k ⊥k

⇀ ⇀ ⇀ ⇀ which is just a generalization of Abelian orthogonal re- q q g g 3 k = S + L4 + S4 + L4, (4.13) lation d xE⊥Ekk = 0 (in the sense of the whole space) to a non-AbelianR case. Consequently, under certain conditions, we can see that Then, substituting Eq.(3.19) into Eq.(3.23) and using ⇀ ⇀ ⇀ J2, J3 and J4 present the same value of the total angular Eqs.(2.6) and (3.12), we can prove that ⇀ momenta. However, the last three terms of J4 are appar- ⇀ ently all gauge dependent, thus J4 is not properly defined. abc 0 b ck abc bk c0 2 2 b bk a0 ⇀ ⇀ gf ∂ AkkA⊥ + gf A⊥ ∂kA − g (AkkA⊥ A + Nc In comparison, both J2 and J3 are more physical. Fur- ⇀ thermore, J2 is the simplest and the most rational one ′ ′ ′ ′ ′ ′ ⇀ a ck c0 2 abc b a c ab c ba c b a k b 0 AkkA⊥ A )+ g (d d − d d )AkkA⊥ A =0 for the sake of simplicity, because J2 eliminates the terms (4.10) that don’t contribute to the total angular momentum for a Multiplying its right side by A0 and rearranging the large Nc gauge field theory, as Eq. (4.11) shows. SU(Nc) group indices, Eq.(4.10) results in Eq.(3.16). 5. Discussion Consequently, all the assumptions being made here are The nonlinear properties of interaction systems of consistent, namely, Eqs.(3.15), (3.16) and (4.5) can be the non-Abelian quark-gluon fields produce considerable given out from Eqs.(3.12), (3.23), (3.25) and (4.9), which complex properties, and the consistency and gauge in- are just directly generalized from the Abelian conditions variance of the quark-gluon interaction system theory to non-Abelian cases in order to keep their gauge covari- naturally require to establish the six equations: (3.12), ant way and their consistent properties. (3.13), (3.23-3.25) and (4.9), which are directly uni- Furthermore, we will show the relation between our formly generalized from their Abelian expressions to their result and the previous ones. To do this, we need to non-Abelian corresponding expressions according to their give another expression of angular momenta J2. Using gauge covariance and consistent properties, thus they can Eqs.(3.12), (3.13) and (4.8), we achieve the surface term be directly simplified as the Abelian corresponding expressions when their structure constants f abc are taken

⇀ ⇀ ⇀ ⇀ as zero. In order to give very obviously direct physics a a ⇀ a a ak⇀ a ∇ · [A⊥(πk × x)] = πk × A⊥ + πk x × ∇A⊥k meanings of the six equations we rewrite them as the direct compact expressions

⇀ + Aa εlmnx e (Dkπa − D πak), (4.11) ⊥k n l k km km k −→ −→a D k · A ⊥ =0, (5.1) ⇀ k a where el is the spatial unit vector and Dk πkm = ∂kπa + gf abcAbkπc ,because there is generalizing the −→ −→ a km k km (D k × D k) =0, (5.2) Abelian Eq.(3.21) to non-Abelian gauge expression (3.24). Adding Eqs.(3.17) and (3.18) to Eq.(4.3), we de- −→ −→a duce Chen et al’s useful separation of the angular mo- D k · π ⊥ =0, (5.3) mentum [6]

⇀ ⇀ ⇀ −→ −→a 3 † 1 3 †⇀ 1 D × π =0, (5.4) J3 = d xψ Σψ + d xψ x × D ψ k k Z 2 Z i k ⇀ ⇀ 3 a a 3 a⇀ ak −→ + d xπ × A⊥ + d xπk x × ∇A⊥ −→a a Z Z D k · π k = ρ , (5.5) 6

d3x−→π a · −→π a =0. (5.6) the resolution of the nucleon spin crisis in a general field Z k ⊥ theory. The gauge invariance and angular momenta commutation relation are not satisfied simultaneously in the Given an arbitrary gauge field configuration Aν (x), one −→ ⇀ ⇀ −→ frame of general field theory of QCD in the past, how- can really find an explicit solution A ⊥ (Ak = A − A ⊥ ever, this paper, for the first time, makes these satisfied in is not independent), which satisfies all the six consistent the same time in a general field theory, without artificial conditions (5.1-5.6), which gurrantee the consistency of choice. the achieved theory in this paper. Otherwise any new When the Nc =2, Eq.(2.6) is reduced as theory cannot return to the well-known Abelian theory [18], which leads to no consistent theory. ace bde ab cd ad bc Generalization from Abelian Eqs.(3.10) and (3.11) f f = δ δ − δ δ , (5.8) to non-Abelian Eq.(3.12) and ∂j Aak − ∂kAaj + k k because dabe can only be equivalently taken as d123 abc bj ck gf Ak Ak = 0 and their important physics meanings, here, then replacing the quark-gluon interaction with respectively, are given by Wang et al [14]. This paper the fermi-gauge interaction, the above theory is simpli- also gives the better and more direct expression (3.13) fied as the Ref.[13]’s investigations and which are the or (5.2) for non-Abelian quark-gluon interaction system exact same as Ref.[13] results, i.e., the current hadron- in general case than that in Ref.[13]. Consequently, the dynamics gauge field theory about strong interaction compact Eqs.(5.2-5.6) with direct physics meanings for with different charged electricity systems [13] is achieved. a general large Nc QCD system in a general field theory When the Nc =1,people usually take U(1) gauge sym- are discovered in this paper. The non-Abelian Eqs.(3.23) metry, i.e., the Abelian gauge field is considered, its struc- and (3.24), generalized from the Abelian Eqs. (3.20) ture constant is zero, thus replacing the quark-gluon in- and (3.21), are the corresponding non-Abelian divergence teraction with the fermi-gauge interaction, the above the- and curl generalizations, respectively (namely, the non- ory is simplified as the Ref.[12]’s investigations and which Abelian transverse and parallel conditions); Eq. (5.5) are the exact same as Ref.[12]’s results. is generalized from Eq.(3.22) and is the new SU(Nc) 6. Summary and conclusion Coulomb law; the orthogonal relation (5.6) is the gen- Consequently, the above theory not only gives the gen- −→ −→ eralized expression for the Abelian d3xE a · E a = 0. eral intrinsic theory of general large Nc QCD, SU(3) k ⊥ QCD, SU(2) hadron-dynamics and U(1) QED gauge field Furthermore, we show also that Eq.(4.7)R with a similar theories in general field theory, but also and specially expression, imposed by Chen et al [6], which is key to ⇀ meaningfully makes essential steps towards solving nu- g maintain L3 gauge invariant, is just a direct result of cleon spin crisis in the general field theory, i.e., presents Eqs.(3.23), (3.25) and (4.6). Under the six consistent progress towards solving the nucleon spin crisis. The conditions, we naturally achieve the simplest gauge in- discoveries of the inner-structure similarity among the variant expression (4.3) of the total angular momentum above theories, their gauge invariant angular momenta of a general QCD system with a general Nc—the general and the corresponding new Coulomb theorem etc of the quark-gluon field interaction system by Noether theorem non-Abelian fermi-gauge field interaction systems with in general field theory. a general large Nc vividly uncover the consistency and For all the above general investigations with a general gauge invariance of the gauge field theory and will fur- Nc, we get a general theory of general large Nc QCD’s in- ther deepen our understanding on some basic questions ner structure and its gauge invariant angular momentum in physics. and the corresponding new general Coulomb theorem etc Because of the universality of Eqs. (5.1)-(5.6), they can so that we give the new viewpoint on the resolution of be applied to treat the inner structure and other aspects the nucleon spin crisis relative to the large Nc in a gen- of any fermi-gauge interaction system, including SU(N) eral situation. Specially, this theory is also suitable for gauge field theory and different gauge theories for the investigating lattice QCD. For instance, one can use this fundamental interactions of the universe. Their aspects method to discuss the theory of the lattice QCD with like Euler-Lagrange equations, corresponding conserva- large Nc [16]. tion quantities and relevant dynamics etc. need to be When the Nc =3, it follows from Eq.(2.6) that revised and improved by using the new general theory of the decomposed non-Abelian gauge fields. Due to the 2 length limit of the paper, lots of applications of this pa- f acef bde = (δabδcd −δadδbc)+dabedcde −dadedbce (5.7) per’s theory will be written in our following works. 3 This paper not only presents the simplest expression i.e., the current QCD [16] is considered. Thus substitut- (4.3) but also gives the current expression (4.12), and ing Eq.(5.7) into all its relevant expressions in all the discovers their relation between Eqs.(4.3) and (4.12). above investigations, then we get a general theory of This paper decomposes the gluon fields into transverse QCD’s inner structure and its gauge invariant angular and longitudinal parts in general field theory, which are momentum as well as the corresponding new Coulomb then used to obtain a general expression for the total theorem etc so that we give out the new viewpoint on angular momentum J of a general quark-gluon system in 7 general field theory. This expression divides J into quark i.e. succeeding in figuring out a natural way to construct, spins and gluon spins and their orbital angular momen- for the first time, the consistent angular momentum op- tum parts in general field theory, with gauge invariant erators with both their commutation relation and gauge expressions for each of the contributions. invariance satisfied in general field theory, and further a This paper is one in a long history of attempting to new door is eventually opened to investigate lots of strong decompose the nucelon’s spin into various pieces which interacting systems with different symmetric properties, together must sum to one half. Such an effort is really which is very important in understanding the nature and of general interest because if the individual parts are ei- thus popularly interesting, and their theories are consis- ther measurable experimentally or directly computable tent. These interesting works here will be extensively in the physics. This paper just gives the theory as how applied and largely cited, and further be written into rel- the individual parts of the spin in the decomposition can evant books because of their generalities and being able be obtained theoretically and experimentally because all to be extensively applied. the achieved quantities are exactly calculated and gauge Specially, we want to stress that the achieved theory in invariant, which then results in that they are measurable, this paper can be utilized to calculate the general QCD because the quantities without gauge invariance cannot strong interactions and give the precise predictions, fur- be exactly measured [6]. ther the achieved predictions in the calculations can be This paper’s generally decomposing gauge potential exactly measured by current particle physics experiments theory also presents a new technique, or methodology, due to their gauge invariant properties etc, because any it should play a votal role in future physics research, and physical quantity ( relative to gauge transformations ) make apparent its immediate consequences for physicists, without gauge invariant property cannot be exactly mea- because the generally decomposing gauge potential the- sured [6]. ory in general field theory can be directly and effectively ACKNOWLEDGMENT: This work is partly sup- applied to arbitrary SU(N), SO(N) etc gauge interaction ported by NSF through a CAREER Award PHY-0952630 systems with different Fermi and the other matter fields. and by DOE through Grant de-sc0007884 in USA and the Therefore, this paper makes a breakthrough progress in National Natural Science Foundation of China (Grants overcoming the key huge difficulty of nucleon spin crisis, No. 11275017 and No. 11173028).

[1] C. N. Yang and R. Mills, Phys. Rev. 96 (1): 191. no. 8, 085030 (2014) [arXiv:1310.4263 [hep-ph]]; X. Ji, [2] Thomas D. Cohen, Brian Mc Peak, Bendeguz Oﬀertaler, J. H. Zhang and Y. Zhao, Phys. Lett. B 743, 180 (2015) Phys. Rev. C 92, 015204 (2015). [arXiv:1409.6329 [hep-ph]]. [3] E. Ageev et al. (COMPASS Collaboration), Physics Let- [12] B. H. Zhou and Y. C. Huang, Phys. Rev. D84, 047701 ters B 612, 154 (2005). (2011); Phys. Rev. A84, 032505 (2011) . [4] V. Alexakhin et al. (COMPASS Collaboration), Physics [13] C. Huang, Y. C. Huang and B. H. Zhou, Phys. Rev. D Letters B 647, 8 (2007). 92, 056003 (2015). [5] A. Airapetian et al. (HERMES Collaboration), Phys. [14] F. Wang, X. S. Chen, X. F. Lu, W. M. Sun, and T. Rev. D 75, 012007 (2007). Goldman, (2009), arXiv:0909.0798 [hep-ph]. [6] X. S. Chen, X. F. Lu, W. M. Sun, F. Wang, and T. [15] S. Lerma H., B. Errea, J. Dukelsky et al, Phys. Rev. Goldman, Phys. Rev. Lett. 100, 232002 (2008). Lett., 99, 032501 (2007); Sz. Borsanyi, S. D¨urr, Z. Fodor [7] X. Ji, Phys. Rev. Lett. 78, 610 (1997). et al, Phys. Rev. Lett., 111, 252001 (2013). [8] M. Wakamatsu, Phys. Rev. D81, 114010 (2010), arXiv: [16] Review of Particle Physics, Phys. Rev. D 86 (2012) 1004.0268 [hep-ph]. 010001. [9] X. Ji, Phys. Rev. Lett. 104, 039101 (2010); X. Ji, Phys. [17] W. Greiner and J. Reinhardt, Field Quantization (Bei- Rev. Lett. 106, 259101 (2011) [arXiv:0910.5022 [hep-ph]] jing World Publishing Corp., 2003). [10] X. Ji, J. H. Zhang and Y. Zhao, Phys. Rev. Lett. 111, [18] M. E. Peskin and D. V. Schroeder, An Introduction to 112002 (2013) [arXiv:1304.6708 [hep-ph]]. Quantum Field Theory (Beijing World Publishing Corp., [11] Y. Hatta, X. Ji and Y. Zhao, Phys. Rev. D 89, 2006).