Lattice QCD Method to Study Proton Spin Crisis

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1 The spin of the proton is 2 . Since the proton consists of quarks and gluons it was expected 1 that the spin of the quarks and gluons inside the proton add up to give spin 2 of the proton. The proton in motion high energy consists of quarks, antiquarks and gluons because sea quark-antiquark pairs and gluons are produced from the QCD vacuum at high energy. 1 For the proton in motion along z-axis the spin/helicity 2 was expected to be 1 =< p,s|Sˆz|p,s > + < p,s|Sˆz|p,s > (1) 2 q g

ˆz where < p,s|Sq |p,s > is the spin of the quarks plus antiquarks inside the proton and ˆz < p,s|Sg |p,s > is the spin of the gluons inside the proton. In eq. (1) the |p,s > is the energy-momentum eigenstate of the longitudinally polarized proton of momentum pµ and ˆz ˆz ~ˆ ~ˆ spin s and Sq (Sg ) is the z-component of the spin vector operator Sq (Sg) of the quarks plus antiquarks (gluons) inside the proton. However, the experimental data suggested otherwise. In 1988-89 the EMC collaboration [1] found that only a negligible fraction of the proton spin is carried by the spin of the quarks and antiquarks inside the proton which was also confirmed by the other experiments [2]. The addition of the spin of the gluons [3] inside the proton is not sufficient to explain 1 the spin 2 of the proton. At present the world data suggests that about fifty percent of the spin of the proton is due to the spin of the quarks plus antiquarks plus gluons inside the proton [4]. Since the remaining fifty percent spin of the proton is still missing it is known as the proton spin crisis. In order to solve this proton spin crisis it is suggested in the literature that the orbital angular momenta of the quarks, antiquarks and gluons inside the proton should be added. In the parton model it is suggested that [5]

1 ˜z ˜z =< p,s|Sˆz|p,s > + < p,s|Sˆz|p,s > + < p,s|Lˆ |p,s > + < p,s|Lˆ |p,s > (2) 2 q g q g z z ˜ˆ ˜ˆ where < p,s|Lq|p,s > and < p,s|Lg|p,s > are the orbital angular momenta of the quarks plus antiquarks and gluons respectively in the light-cone gauge inside the proton. In terms of the gauge invariant spin/angular momentum in QCD it is suggested that [6]

1 =< p,s|Sˆz|p,s > + < p,s|Lˆz|p,s > + < p,s|Jˆz|p,s > (3) 2 q q g

1 ˆz where < p,s|Lq|p,s > is the gauge invariant orbital angular momentum of the quarks ˆz plus antiquarks inside the proton and < p,s|Jg |p,s > is the gauge invariant total angular momentum of the gluons inside the proton. The eq. (3) is not correct because of the non-zero boundary surface term in QCD in the conservation equation of the angular momentum using the gauge invariant Noether’s theorem in QCD [7] due to the confinement of quarks and gluons inside the finite size proton [8]. Because of this the angular momentum sum rule as given by eq. (3) is violated in QCD [9]. By taking the confinement effect into account one finds in QCD that [9]

1 =< p,s|Sˆz|p,s > + < p,s|Lˆz|p,s > + < p,s|Jˆz|p,s > + < p,s|Jˆz |p,s > (4) 2 q q g B

ˆz where < p,s|JB|p,s > is the boundary surface term contribution to the total angular momentum in QCD due to the conﬁnement of quarks and gluons inside the ﬁnite size proton [8].

Note that < p,s|Sˆq|p,s >, < p,s|Lˆq|p,s >, < p,s|Jˆg|p,s > and < p,s|JˆB|p,s > in eq. (4) are non-perturbative quantities which cannot be calculated by using the perturbative QCD (pQCD) method. Hence the non-perturbative QCD is necessary to calculate the quantities

< p,s|Sˆq|p,s >, < p,s|Lˆq|p,s >, < p,s|Jˆg|p,s > and < p,s|JˆB|p,s > in eq. (4). However, the analytical solution of the non-perturbative QCD is not known yet. Hence the first principle method to study non-perturbative QCD is by using the lattice QCD method. Recently we have presented the formulation of the lattice QCD method to study the hadron formation from quarks and gluons by incorporating the non-zero boundary surface term in QCD due to the confinement of quarks and gluons inside the finite size proton [10]. In this paper we extend this to study the proton spin crisis and present the lattice QCD formulation to study the proton spin crisis by incorporating this non-zero boundary surface term in QCD due to confinement. We derive the non-perturbative formula of the

< p,s|Sˆq|p,s >, < p,s|Lˆq|p,s >, < p,s|Jˆg|p,s > and < p,s|JˆB|p,s > in eq. (4) from the first principle in QCD which can be calculated by using the lattice QCD method by incorporating this non-zero boundary surface term due to confinement. The paper is organized as follows. In section II we discuss the effect of confinement on angular momentum sum rule violation in QCD. In section III we discuss the proton formation from quarks and gluons by using the lattice QCD method. In section IV we present the

2 lattice QCD formulation to study the proton spin crisis and derive the non-perturbative formula of the spin and angular momentum of the partons inside the proton from the ﬁrst principle in QCD which can be calculated by using the lattice QCD method. Section V contains conclusions.

II. EFFECT OF CONFINEMENT ON ANGULAR MOMENTUM SUM RULE VIOLATION IN QCD

The conservation equation of the angular momentum in QCD can be obtained from the ﬁrst principle by using the gauge invariant Noether’s theorem in QCD under the combined gauge transformation plus rotation. The continuity equation obtained from the gauge invariant Noether’s theorem in QCD is given by [7]

νδσ ∂νJ (x)=0 (5)

where J νδσ(x) is the gauge invariant angular momentum tensor density in QCD which is related to the gauge invariant and symmetric energy-momentum tensor density T νδ(x) in QCD via the equation

J νδσ(x)= T νσ(x)xδ − T νδ(x)xσ. (6)

The gauge invariant and symmetric energy-momentum tensor density T νδ(x) in QCD is given by

1 i −→δ T νδ(x)= F νσb(x)F δb(x)+ gνδF σµb(x)F b (x)+ ψ¯ (x)[γν (δlk ∂ − igT b Aδb(x)) σ 4 σµ 4 l lk δ lk−→µ b νb ν lk←−δ b δb δ lk←−ν b νb +γ (δ ∂ − igTlkA (x)) − γ (δ ∂ + igTlkA (x)) − γ (δ ∂ + igTlkA (x))]ψk(x) (7)

d where ψj(x) is the quark ﬁeld with color index j =1, 2, 3, the Aδ (x) is the gluon ﬁeld with color index d =1, ..., 8, the Lorentz index δ =0, 1, 2, 3 and

c b b bca c a Fδσ(x)= ∂δAσ(x) − ∂σAδ(x)+ gf Aδ(x)Aσ(x). (8)

From eq. (5) we ﬁnd

d ~ ~ ~ ~ [< p,s|Sˆ |p,s > + < p,s|Lˆ |p,s > + < p,s|Jˆ |p,s > + < p,s|Jˆ |p,s >]=0 (9) dt q q g B

3 where

~ < p,s|Sˆ (t)|p,s >=< p,s| d3x ψ¯ (x)γ ~σψ (x)|p, s >, (10) q Z k 0 k

~ −→ ←− < p,s|Lˆ (t)|p,s >=< p,s| d3x ~x × ψ¯ (x)γ [−iD [A]+ iD [A]]ψ (x)|p, s >, (11) q Z l 0 lk lk k

~ < p,s|Jˆ (t)|p,s >=< p,s| d3x ~x × E~ d(x) × B~ d(x)|p,s > (12) g Z and

2 2 n nsj 4 s jc kc kj Ec (x) − Bc (x) s i < p,s|Jˆ (t)=< p,s|ǫ d x ∂ [x [E (x)E (x) − δ ] + [x ψ¯ ′ (x) B Z k 2 2 n k −→j c jc k ←−j c jc 1 n sj [γ (δ ′ ′′ ∂ − igT ′ ′′ A (x)) − γ (δ ′ ′′ ∂ + igT ′ ′′ A (x))]ψ ′′ (x)+ ψ¯ ′ (x){γ , σ }]]|p, s > . n n n n n n n n n 8 n (13)

The covariant derivative Dlk[A] in eq. (11) is deﬁned by

d d Dln[A]= δln(i6 ∂ − m) − igTlnA/ (x). (14)

The ~σ in eq. (10) is the Pauli spin matrix and the σνδ in eq. (13) is related to the Dirac matrices via the equation

i σsl = [γs,γl]. (15) 2

From eqs. (10), (11), (12), (13) and (9) we ﬁnd

d d [< p,s|Sˆz|p,s > + < p,s|Lˆz|p,s > + < p,s|Jˆz|p,s >]= − B (16) dt q q g dt ~ where the operator Oˆz means the z-component of the vector operator Oˆ where the expres- ~ ~ ~ ~ sions of the vector operators Sˆq(t), Lˆq(t), Jˆg(t) and JˆB(t) in terms of the quark field ψi(x) a and the gluons field Aµ(x) are given by eqs. (10), (11), (12) and (13) respectively. Due to the confinement of quarks and gluons inside the finite size proton we find that a the quark field ψi(t, r) and the gluon field Aµ(t, r) do not go to r →∞. Since the boundary surface is at the finite distance due to the finite size of the proton one finds that the boundary surface term in QCD is non-zero due to confinement of quarks and gluons inside the finite size proton irrespective of the forms of the r dependence of the quark field ψi(t, r) and the

4 a gluon field Aµ(t, r) [8]. Hence because of the non-zero boundary surface term in QCD due to the confinement of the quarks and gluons inside the finite size proton one finds that [9]

d B =06 . (17) dt From eqs. (17) and (16) we ﬁnd

d [< p,s|Sˆz|p,s > + < p,s|Lˆz|p,s > + < p,s|Jˆz|p,s >] =06 (18) dt q q g

ˆz ˆz ˆz which implies that < p,s|Sq |p,s > + < p,s|Lq|p,s > + < p,s|Jg |p,s > is time dependent. 1 ˆz ˆz Since the proton spin 2 is time independent and the < p,s|Sq |p,s > + < p,s|Lq|p,s > ˆz + < p,s|Jg |p,s > from eq. (18) is time dependent we find that 1 =6 < p,s|Sˆz|p,s > + < p,s|Lˆz|p,s > + < p,s|Jˆz|p,s > (19) 2 q q g which does not agree with eq. (3). From eq. (19) we find that the angular momentum sum rule in QCD as given by eq. (3) is violated due to the confinement of quarks and gluons inside the finite size proton.

III. FORMULATION OF THE LATTICE QCD METHOD TO STUDY PROTON FORMATION FROM QUARKS AND GLUONS

In this section we consider the formulation of the lattice QCD method to study the proton formation from quarks and gluons. In the next section we will extend this to study the proton spin crisis. For the proton formation the partonic operator is given by

ˆ u 5 d u Bp(x)= ǫijkψi (x)Cγ ψj (x)ψk (x) (20)

u d where ψi (x) is the Dirac ﬁeld of the up quark with color index i = 1, 2, 3, the ψi (x) is the Dirac ﬁeld of the down quark and C is the charge conjugation operator. The vacuum expectation value of the non-perturbative partonic correlation function in QCD is given by 1 δHa < 0|Bˆ (t′′,r′′)Bˆ (0)|0 >= [dψ¯u][dψu][dψ¯d][dψd][dA] Bˆ (t′′,r′′)Bˆ (0) × deu[ f ] p p Z[0] Z p p δωc i d4x[− 1 F c (x)F δσc(x)− 1 [Hc (x)]2+ψ¯l (x)[δlk(i6∂−mu)+gT c A/c(x)]ψk(x)+ψ¯l (x)[δlk(i6∂−md)+gT c A/c(x)]ψk(x)] ×e 4 δσ 2α f u lk u d lk d R (21)

5 where |0 > is the non-perturbative QCD vacuum state (i. e., the vacuum state of the full a QCD, not of pQCD), Hf (x) is the gauge ﬁxing term, α is the gauge ﬁxing parameter and δHa Z[0] = [dψ¯u][dψu][dψ¯d][dψd][dA] × deu[ f ] Z δωc i d4x[− 1 F c (x)F δσc(x)− 1 [Hc (x)]2+ψ¯l (x)[δlk(i6∂−mu)+gT c A/c(x)]ψk(x)+ψ¯l (x)[δlk(i6∂−md)+gT c A/c(x)]ψk(x)] ×e 4 δσ 2α f u lk u d lk d R (22) is the generating functional in QCD. The time evolution of the partonic operator in the Heisenberg representation is given by

−itH itH Bˆp(t, ~x)= e Bˆp(0, ~x)e (23) where H is the QCD Hamiltonian. Inserting the complete set of energy-momentum eigenstates of the proton

|pn′′′ >< pn′′′ | =1 (24) Xn′′′ in (21) and then using eq. (23) we ﬁnd in the Euclidean time that

′′ i~p·~r ′′ ′′ 2 − dtE ′′′ (t) e < 0|Bˆ (t ,r )Bˆ (0)|0 >= | < 0|Bˆ (0)|p ′′′ > | e n (25) p p p n R Xr′′ Xn′′′ where En′′′ (t) is the energy of all the quarks plus antiquarks plus gluons inside the proton and dt is an indeﬁnite integration. In the large Euclidean time t →∞ we ﬁnd R i~p·~r′′ ′′ ′′ 2 − dt′′E(t′′) e < 0|Bˆ (t ,r )Bˆ (0)|0 > | ′′ = | < 0|Bˆ (0)|p> | e (26) p p t →∞ p R Xr′′

The energy of the proton Ep is given by

Ep = E(t)+ EB(t) (27) where EB(t) is non-zero boundary surface term given by [10] 4 ′′ i~p·~r′′′ ˆ ′′′ ′′′ k0 ′′ ′′ ˆ ′′ d x r′′′ e < 0|Bp(t ,r )∂kT (t , ~x )Bp(0)|0 > EB(t )= |t′′′→∞ (28) R P i~p·~r′′′ ˆ ′′′ ′′′ ˆ r′′′ e < 0|Bp(t ,r )Bp(0)|0 > P where dt′′ is an indeﬁnite integration. Using eqs. (27) and (28) in (26) we ﬁnd R ei~p·~r < 0|Bˆ (t, r)Bˆ (0)|0 > ˆ 2 −tEp r p p | < 0|Bp(0)|p> | e = [ ′ |t→∞ (29) dP4x ei~p·~r <0|Bˆp(t′,r′)∂ T k0(t,r)Bˆp(0)|0> r′ k dt ′ |t′→∞ R P ei~p·~r <0|Bˆp(t′,r′)Bˆp(0)|0> eR r′ P which is the equation to study the proton formation from quarks and gluons by using the lattice QCD method. This formulation of the lattice QCD method used to study various non-perturbative quantities in QCD in vacuum [10, 11] and in QCD in medium [12] to study quark-gluon plasma at RHIC and LHC [13–16].

6 IV. LATTICE QCD FORMULATION OF THE PROTON SPIN CRISIS

From eq. (16) we find d [< p,s|Sˆz|p,s > + < p,s|Lˆz|p,s > + < p,s|Jˆz|p,s > + < p,s|Jˆz |p,s >]=0 (30) dt q q g B ˆz ˆz ˆz ˆz which implies that < p,s|Sq |p,s > + < p,s|Lq|p,s > + < p,s|Jg |p,s > + < p,s|JB|p,s > is time independent. 1 ˆz ˆz Since the spin 2 of the proton is time independent and < p,s|Sq |p,s > + < p,s|Lq|p,s > ˆz ˆz + < p,s|Jg |p,s > + < p,s|JB|p,s > from eq. (30) is time independent one finds, unlike eq. (19), that 1 =< p,s|Sˆz|p,s > + < p,s|Lˆz|p,s > + < p,s|Jˆz|p,s > + < p,s|Jˆz |p,s > (31) 2 q q g B 1 ˆz ˆz where 2 in the left hand side is the proton spin and < p,s|Sq |p,s >, < p,s|Lq|p,s >, ˆz ˆz ~ˆ ~ˆ < p,s|Jg |p,s >, < p,s|JB|p,s > are the z-components of < p,s|Sq|p,s >, < p,s|Lq|p,s >, ~ ~ < p,s|Jˆg|p,s >, < p,s|JˆB|p,s > where the expressions of the spin/angular momentum ~ˆ ~ˆ ~ˆ ~ˆ a operators Sq, Lq, J g, J B in terms of quark field ψi(x) and gluon field Aµ(x) are given by eqs. (10), (11), (12) and (13) respectively. 1 ˆz From eq. (31) we find that the spin 2 of the proton is obtained from < p,s|Sq |p,s >, ˆz ˆz ˆz < p,s|Lq|p,s >, < p,s|Jg |p,s > and < p,s|JB|p,s >. However, these quantities ˆz ˆz ˆz ˆz < p,s|Sq |p,s >, < p,s|Lq|p,s >, < p,s|Jg |p,s > and < p,s|JB|p,s > are non-perturbative quantities in QCD which can not be calculated by using pQCD. On the other hand the analytical solution of the non-perturbative QCD is not known. Hence one can calculate ˆz ˆz ˆz these non-perturbative quantities < p,s|Sq |p,s >, < p,s|Lq|p,s >, < p,s|Jg |p,s > and ˆz < p,s|JB|p,s > by using the lattice QCD method. In this section we will extend the lattice QCD formulation of the previous section to study the proton spin crisis, i. e., we will formulate the lattice QCD method to calculate the non- ˆz ˆz ˆz ˆz perturbative quantities < p,s|Sq |p,s >, < p,s|Lq|p,s >, < p,s|Jg |p,s > and < p,s|JB|p,s > in QCD to study the proton spin crisis. Let us first formulate the lattice QCD method to calculate the non-perturbative quantity ~ ~ < p,s|Sˆq|p,s > in QCD where the spin operator Sˆq is given by eq. (10) before proceeding ~ ~ to calculate the other non-perturbative quantities < p,s|Lˆq|p,s >, < p,s|Jˆg|p,s > and ~ ~ ~ ~ < p,s|JˆB|p,s > in QCD where the angular momentum operators Lˆq, Jˆg and JˆB in QCD are given by eqs. (11), (12) and (13) respectively.

7 The vacuum expectation value of the three-point non-perturbative partonic correlation ′′ ′′ ~ ′ function < 0|Bˆp(t ,r )Sˆq(t )Bˆp(0)|0 > in QCD is given by

a ~ 1 ~ δHf < 0|Bˆ (t′′,r′′)Sˆ (t′)Bˆ (0)|0 >= [dψ¯u][dψu][dψ¯d][dψd][dA] Bˆ (t′′,r′′)Sˆ (t′)Bˆ (0) × deu[ ] p q p Z[0] Z p q p δωc i d4x[− 1 F c (x)F δσc(x)− 1 [Hc (x)]2+ψ¯l (x)[δlk(i6∂−mu)+gT c A/c(x)]ψk(x)+ψ¯l (x)[δlk(i6∂−md)+gT c A/c(x)]ψk(x)] ×e 4 δσ 2α f u lk u d lk d (32) R ~ where the partonic operator Sˆq(t) is given by eq. (10) and the partonic operator Bˆp(t, r) is given by eq. (20). We evaluate the ratio of the vacuum expectation value of the three-point non-perturbative partonic correlation function

′ ′′ i~p·~r′′ ′′ ′′ ~ ′ C3(p, t , t )= e < 0|Bˆp(t ,r )Sˆq(t )Bˆp(0)|0 > (33) Xr′′ to the vacuum expectation value of the two-point non-perturbative partonic correlation function

′′ i~p·~r′′ ′′ ′′ C2(p, t )= e < 0|Bˆp(t ,r )Bˆp(0)|0 > (34) Xr′′

′′ ′′ ~ ′ ′′ ′′ where < 0|Bˆp(t ,r )Sˆq(t )Bˆp(0)|0 > is given by eq. (32) and < 0|Bˆp(t ,r )Bˆp(0)|0 > is given by eq. (21). The complete set of energy-momentum eigenstates of the proton with spin s is given by

|pn′′′ ,s>

Using eqs. (23) and (35) in

′ ′′ i~p·~r′′ ′′ ′′ ~ ′ C (p, t , t ) ′′ e < 0|Bˆ (t ,r )Sˆ (t )Bˆ (0)|0 > 3 = r p q p (36) ′′ P i~p·~r′′ ˆ ′′ ′′ ˆ C2(p, t ) r′′ e < 0|Bp(t ,r )Bp(0)|0 > P we ﬁnd in Euclidean time

′′′ ′′′ ′ ′′ ~ ′ − dt En′′′ (t ) C (p, t , t ) ′′ ′′′ < 0|Bˆp(0)|pn′′′ ,s> e 3 n ,n R = P ′′′ ′′′ . ′′ 2 − dt En′′′ (t ) C2(p, t ) ′′′ | < 0|Bˆp(0)|pn′′′ ,s> | e n R P (37)

Neglecting the higher energy level contributions at the large time limit t′′ →∞ we ﬁnd

′ ′′ 2 ~ ′ − dt′′′E(t′′′) C3(p, t , t ) | < 0|Bˆp(0)|p,s > | < p,s|Sˆq(t )|p,s>e |t′′→∞ = R (38) ′′ 2 − dt′′′E(t′′′) C2(p, t ) | < 0|Bˆp(0)|p,s > | e R

8 where for the ground state n′′′ = n′′ = 0 of the proton we have

|p0,s>= |p,s>, E0(t)= E(t). (39)

From eq. (38) and (36) we ﬁnd

i~p·~r′ ˆ ′ ′ ~ˆ ˆ ~ r′ e < 0|Bp(t ,r )Sq(t)Bp(0)|0 > < p,s|Sˆq(t)|p,s >= |t′→∞ (40) P i~p·~r′ ˆ ′ ′ ˆ r′ e < 0|Bp(t ,r )Bp(0)|0 > P ~ where the partonic operator Sˆq(t) is given by eq. (10), the partonic operator Bˆp(t, r) is given by eq. (20), the vacuum expectation value of the two-point non-perturbative partonic corre- ′ ′ lation function < 0|Bˆp(t ,r )Bˆp(0)|0 > is given by eq. (21) and the vacuum expectation value ′ ′ ~ of the three-point non-perturbative partonic correlation function < 0|Bˆp(t ,r )Sˆq(t)Bˆp(0)|0 > is given by eq. (32). Since the vacuum expectation value of the two-point non-perturbative partonic correla- ′ ′ tion function < 0|Bˆp(t ,r )Bˆp(0)|0 > in eq. (21) and the vacuum expectation value of the ′ ′ ~ three-point non-perturbative partonic correlation function < 0|Bˆp(t ,r )Sˆq(t)Bˆp(0)|0 > in eq. (32) can be calculated by using the lattice QCD method we ﬁnd that the non-perturbative ~ quantity < p,s|Sˆq(t)|p,s > in eq. (31) to study the proton spin crisis can be calculated from eq. (40) by using the lattice QCD method. Following the similar procedure we ﬁnd

i~p·~r′ ˆ ′ ′ ~ˆ ˆ ~ r′ e < 0|Bp(t ,r )Lq(t)Bp(0)|0 > < p,s|Lˆq(t)|p,s >= |t′→∞ (41) P i~p·~r′ ˆ ′ ′ ˆ r′ e < 0|Bp(t ,r )Bp(0)|0 > P ~ where the partonic operator Lˆq(t) is given by eq. (11), the partonic operator Bˆp(t, r) is given by eq. (20), the vacuum expectation value of the two-point non-perturbative partonic corre- ′ ′ lation function < 0|Bˆp(t ,r )Bˆp(0)|0 > is given by eq. (21) and the vacuum expectation value ′ ′ ~ of the three-point non-perturbative partonic correlation function < 0|Bˆp(t ,r )Lˆq(t)Bˆp(0)|0 > is given by

a ~ 1 ~ δHf < 0|Bˆ (t′′,r′′)Lˆ (t′)Bˆ (0)|0 >= [dψ¯u][dψu][dψ¯d][dψd][dA] Bˆ (t′′,r′′)Lˆ (t′)Bˆ (0) × deu[ ] p q p Z[0] Z p q p δωc i d4x[− 1 F c (x)F δσc(x)− 1 [Hc (x)]2+ψ¯l (x)[δlk(i6∂−mu)+gT c A/c(x)]ψk(x)+ψ¯l (x)[δlk(i6∂−md)+gT c A/c(x)]ψk(x)] ×e 4 δσ 2α f u lk u d lk d . (42) R

9 can be calculated by using the lattice QCD method we ﬁnd that the non-perturbative ~ quantity < p,s|Lˆq(t)|p,s > in eq. (31) to study the proton spin crisis can be calculated from eq. (41) by using the lattice QCD method. Similarly we ﬁnd

i~p·~r′ ˆ ′ ′ ~ˆ ˆ ~ r′ e < 0|Bp(t ,r )J g(t)Bp(0)|0 > < p,s|Jˆg(t)|p,s >= |t′→∞ (43) P i~p·~r′ ˆ ′ ′ ˆ r′ e < 0|Bp(t ,r )Bp(0)|0 > P ~ where the partonic operator Jˆg(t) is given by eq. (12), the partonic operator Bˆp(t, r) is given by eq. (20), the vacuum expectation value of the two-point non-perturbative partonic corre- ′ ′ lation function < 0|Bˆp(t ,r )Bˆp(0)|0 > is given by eq. (21) and the vacuum expectation value ′ ′ ~ of the three-point non-perturbative partonic correlation function < 0|Bˆp(t ,r )Jˆg(t)Bˆp(0)|0 > is given by

a ~ 1 ~ δHf < 0|Bˆ (t′′,r′′)Jˆ (t′)Bˆ (0)|0 >= [dψ¯u][dψu][dψ¯d][dψd][dA] Bˆ (t′′,r′′)Jˆ (t′)Bˆ (0) × deu[ ] p g p Z[0] Z p g p δωc i d4x[− 1 F c (x)F δσc(x)− 1 [Hc (x)]2+ψ¯l (x)[δlk(i6∂−mu)+gT c A/c(x)]ψk(x)+ψ¯l (x)[δlk(i6∂−md)+gT c A/c(x)]ψk(x)] ×e 4 δσ 2α f u lk u d lk d . (44) R

Since the vacuum expectation value of the two-point non-perturbative partonic correlation ′ ′ function < 0|Bˆp(t ,r )Bˆp(0)|0 > in eq. (21) and the vacuum expectation value of the three- ′ ′ ~ point non-perturbative partonic correlation function < 0|Bˆp(t ,r )Jˆg(t)Bˆp(0)|0 > in eq. (44) can be calculated by using the lattice QCD method we ﬁnd that the non-perturbative ~ quantity < p,s|Jˆg(t)|p,s > in eq. (31) to study the proton spin crisis can be calculated from eq. (43) by using the lattice QCD method. Finally we ﬁnd [17]

i~p·~r′ ˆ ′ ′ ~ˆ ˆ ~ r′ e < 0|Bp(t ,r )J B(t)Bp(0)|0 > < p,s|JˆB(t)|p,s >= |t′→∞ (45) P i~p·~r′ ˆ ′ ′ ˆ r′ e < 0|Bp(t ,r )Bp(0)|0 > P ~ where the partonic operator JˆB(t) is given by eq. (13), the partonic operator Bˆp(t, r) is given by eq. (20), the vacuum expectation value of the two-point non-perturbative ′ ′ partonic correlation function < 0|Bˆp(t ,r )Bˆp(0)|0 > is given by eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function ′ ′ ~ < 0|Bˆp(t ,r )JˆB(t)Bˆp(0)|0 > is given by a ~ 1 ~ δHf < 0|Bˆ (t′′,r′′)Jˆ (t′)Bˆ (0)|0 >= [dψ¯u][dψu][dψ¯d][dψd][dA] Bˆ (t′′,r′′)Jˆ (t′)Bˆ (0) × deu[ ] p B p Z[0] Z p B p δωc i d4x[− 1 F c (x)F δσc(x)− 1 [Hc (x)]2+ψ¯l (x)[δlk(i6∂−mu)+gT c A/c(x)]ψk(x)+ψ¯l (x)[δlk(i6∂−md)+gT c A/c(x)]ψk(x)] ×e 4 δσ 2α f u lk u d lk d . (46) R 10 Since the vacuum expectation value of the two-point non-perturbative partonic correlation ′ ′ function < 0|Bˆp(t ,r )Bˆp(0)|0 > in eq. (21) and the vacuum expectation value of the three- ′ ′ ~ point non-perturbative partonic correlation function < 0|Bˆp(t ,r )JˆB(t)Bˆp(0)|0 > in eq. (46) can be calculated by using the lattice QCD method we ﬁnd that the non-perturbative ~ quantity < p,s|JˆB(t)|p,s > in eq. (31) to study the proton spin crisis can be calculated from eq. (45) by using the lattice QCD method. 1 Hence we ﬁnd that the spin 2 of the proton in eq. (31) can be calculated by using the lattice QCD method using the non-perturbative formulas derived in eqs. (40), (41), (43), (45) where the spin/angular momentum operators of the partons in QCD are given by eqs.

(10), (11), (12), (13) respectively with the partonic operator Bˆp(x) given by (20).

V. CONCLUSIONS

The proton spin crisis remains an unsolved problem in particle physics. The spin and angular momentum of the partons inside the proton are non-perturbative quantities in QCD which cannot be calculated by using the perturbative QCD (pQCD). In this paper we have presented the lattice QCD formulation to study the proton spin crisis. We have derived the non-perturbative formula of the spin and angular momentum of the partons inside the proton from the ﬁrst principle in QCD which can be calculated by using the lattice QCD method.

[1] J. Ashman et al, European Muon Collaboration, Phys. Lett. B 206 (1988) 364; Nucl. Phys. B 328 (1989) 1. [2] See for example, C. A. Aidala, S. D. Bass, D. Hasch and G. K. Mallot, Rev. Mod. Phys. 85 (2013) 655. [3] A. Adare et al., PHENIX Collaboration, Phys. Rev. D 93 (2016) 011501; A. Adare et al., PHENIX Collaboration, Phys. Rev. D 94 (2016) 112008. [4] A. Bazilevsky, J. Phys. Conf. Ser. 678 (2016) 012059. [5] J. R. Ellis and R. L. Jaﬀe, Phys. Rev. D 9 (1974) 1444; Phys. Rev. D 10 (1974) 1669; R. L. Jaﬀe and A. Manohar, Nucl. Phys. B337 (1990) 509; A. V. Manohar, Phys. Rev. Lett. 66

11 (1991) 289. [6] X.-D. Ji, Phys. Rev. Lett. 78 (1997) 610. [7] G. C. Nayak, arXiv:1802.07825. [8] G. C. Nayak, arXiv:1807.09158. [9] G. C. Nayak, arXiv:1803.08371 [hep-ph]. [10] G. C. Nayak, arXiv:1811.09685. [11] G. C. Nayak, arXiv:1810.12088; arXiv:1904.03998. [12] G. C. Nayak, . arXiv:1902.10522; arXiv:1904.05376. [13] G. C. Nayak and P. van Nieuwenhuizen, Phys. Rev. D 71 (2005) 125001; G. C. Nayak, Phys. Rev. D 72 (2005) 125010; F. Cooper and G. C. Nayak, Phys. Rev. D73 (2006) 065005; D. Dietrich, G. C. Nayak and W. Greiner, Phys. Rev. D64 (2001) 074006; M. C. Birse, C-W. Kao and G. C. Nayak, Phys. Lett. B570 (2003) 171. [14] G. C. Nayak et al., Nucl. Phys. A687 (2001) 457; G. C. Nayak and R. S. Bhalerao, Phys. Rev. C 61 (2000) 054907; G. C. Nayak and V. Ravishankar, Phys. Rev. C 58 (1998) 356; Phys. Rev. D 55 (1997) 6877; F. Cooper, E. Mottola and G. C. Nayak, Phys. Lett. B555 (2003) 181. [15] G. C. Nayak, Annals Phys. 325 (2010) 682; Phys. Lett. B442 (1998) 427; JHEP 9802 (1998) 005; Eur. Phys. J. C64 (2009) 73; JHEP 0906 (2009) 071; Annals Phys. 324 (2009) 2579; Annals Phys. 325 (2010) 514; Eur. Phys. J.C59 (2009) 715. [16] G. C. Nayak, Eur. Phys. J. Plus 133 (2018) 52; arXiv:1506.02651 [hep-ph]; J. Theor. Appl. Phys. 11 (2017) 275; arXiv:1705.07913 [hep-ph]; G. C. Nayak, JHEP 1709 (2017) 090; arXiv:1808.01937; Eur. Phys. J. C76 (2016) 448. [17] G. C. Nayak, arXiv:1905.03717 [hep-ph].