Lattice QCD Method To Study Proton Spin Crisis

TL;DR

This paper introduces a lattice QCD approach to non-perturbatively calculate the spin and angular momentum of partons inside the proton, addressing the longstanding proton spin crisis in particle physics.

Contribution

It derives a first-principles non-perturbative formula for proton spin components using lattice QCD, providing a new computational framework.

Findings

Formulated a lattice QCD method for proton spin analysis

Derived non-perturbative formulas from first principles

Enables direct calculation of parton spin and angular momentum

Abstract

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 present the lattice QCD formulation to study the proton spin crisis. We 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.

Equations

$$\frac{1}{2}=<p,s|{\hat{S}}_q^z|p,s>+<p,s|{\hat{S}}_g^z|p,s>$$

$$\frac{1}{2}=<p,s|{\hat{S}}_q^z|p,s>+<p,s|{\hat{S}}_g^z|p,s>+<p,s|{\tilde{\hat{L}}}_q^z|p,s>+<p,s|{\tilde{\hat{L}}}_g^z|p,s>$$

$$\frac{1}{2}=<p,s|{\hat{S}}_q^z|p,s>+<p,s|{\hat{L}}_q^z|p,s>+<p,s|{\hat{J}}_g^z|p,s>$$

$$\frac{1}{2}=<p,s|{{\hat{S}}^z}_q|p,s>+<p,s|{\hat{L}}_q^z|p,s>+<p,s|{\hat{J}}_g^z|p,s>+<p,s|{\hat{J}}_B^z|p,s>$$

$${\partial }_{\nu }{J}^{\nu \delta \sigma }(x)=0$$

$$J^{\nu \delta \sigma }(x)=T^{\nu \sigma }(x)x^{\delta }-T^{\nu \delta }(x)x^{\sigma }.$$

$$T^{\nu \delta }(x)=F^{\nu \sigma b}(x)F_{\sigma }^{\delta b}(x)+\frac{1}{4}{g}^{\nu \delta }{F}^{\sigma \mu b}(x)F_{\sigma \mu }^b(x)+\frac{i}{4}{\bar{\psi }}_l(x)[{\gamma }^{\nu }({\delta }^{lk}{\overrightarrow{\partial }}^{\delta }-ig{T}_{lk}^b{A}^{\delta b}(x))$$

$$+{\gamma }^{\delta }({\delta }^{lk}{\overrightarrow{\partial }}^{\mu }-ig{T}_{lk}^b{A}^{\nu b}(x))-{\gamma }^{\nu }({\delta }^{lk}{\overleftarrow{\partial }}^{\delta }+ig{T}_{lk}^b{A}^{\delta b}(x))-{\gamma }^{\delta }({\delta }^{lk}{\overleftarrow{\partial }}^{\nu }+ig{T}_{lk}^b{A}^{\nu b}(x))]{\psi }_k(x)$$

$$F_{\delta \sigma }^c(x)={\partial }_{\delta }{A}_{\sigma }^b(x)-{\partial }_{\sigma }{A}_{\delta }^b(x)+g{f}^{bca}{A}_{\delta }^c(x)A_{\sigma }^a(x).$$

$$\frac{d}{dt}[<p,s|{\vec{\hat{S}}}_q|p,s>+<p,s|{\vec{\hat{L}}}_q|p,s>+<p,s|{\vec{\hat{J}}}_g|p,s>+<p,s|{\vec{\hat{J}}}_B|p,s>]=0$$

$$<p,s|{\vec{\hat{S}}}_q(t)|p,s>=<p,s|\int d^{3}x{\bar{\psi }}_k(x){\gamma }_0\vec{\sigma }{\psi }_k(x)|p,s>,$$

$$<p,s|{\vec{\hat{L}}}_q(t)|p,s>=<p,s|\int d^{3}x\vec{x}\times {\bar{\psi }}_l(x){\gamma }_0[-i{\overrightarrow{D}}_{lk}[A]+i{\overleftarrow{D}}_{lk}[A]]{\psi }_k(x)|p,s>,$$

$$<p,s|{\vec{\hat{J}}}_g(t)|p,s>=<p,s|\int d^{3}x\vec{x}\times {\vec{E}}^d(x)\times {\vec{B}}^d(x)|p,s>$$

$$<p,s|{\hat{J}}_B^n(t)=<p,s|{ϵ}^{nsj}\int d^{4}x{\partial }_k[x^s[E^{jc}(x)E^{kc}(x)-{\delta }^{kj}\frac{{E_c}^2(x)-{B_c}^2(x)}{2}]+[x^s\frac{i}{2}{\bar{\psi }}_{n^{\prime }}(x)$$

$$[{\gamma }^k({\delta }_{n^{\prime }{n}^{\prime \prime }}{\overrightarrow{\partial }}^j-ig{T}_{n^{\prime }{n}^{\prime \prime }}^c{A}^{jc}(x))-{\gamma }^k({\delta }_{n^{\prime }{n}^{\prime \prime }}{\overleftarrow{\partial }}^j+ig{T}_{n^{\prime }{n}^{\prime \prime }}^c{A}^{jc}(x))]{\psi }_{n^{\prime \prime }}(x)+\frac{1}{8}{\bar{\psi }}_{n^{\prime }}(x)\{{\gamma }^n,{\sigma }^{sj}\}]]|p,s>.$$

$$D_{ln}[A]={\delta }_{ln}(i/\partial -m)-ig{T}_{ln}^{d}A{/}^d(x).$$

$${\sigma }^{sl}=\frac{i}{2}[{\gamma }^s,{\gamma }^l].$$

$$\frac{d}{dt}[<p,s|{\hat{S}}_q^z|p,s>+<p,s|{\hat{L}}_q^z|p,s>+<p,s|{\hat{J}}_g^z|p,s>]=-\frac{d<p,s|{\hat{J}}_B^z|p,s>}{dt}$$

$$\frac{d<p,s|{\hat{J}}_B|p,s>}{dt}\ne 0.$$

$$\frac{d}{dt}[<p,s|{\hat{S}}_q^z|p,s>+<p,s|{\hat{L}}_q^z|p,s>+<p,s|{\hat{J}}_g^z|p,s>]\ne 0$$

$$\frac{1}{2}\ne <p,s|{\hat{S}}_q^z|p,s>+<p,s|{\hat{L}}_q^z|p,s>+<p,s|{\hat{J}}_g^z|p,s>$$

$${\hat{\mathcal{B}}}_p(x)={ϵ}_{ijk}{\psi }_i^u(x)C{\gamma }^5{\psi }_j^d(x){\psi }_k^u(x)$$

$$<0|{\hat{\mathcal{B}}}_p(t^{\prime \prime },r^{\prime \prime }){\hat{\mathcal{B}}}_p(0)|0>=\frac{1}{Z[0]}\int [d{\bar{\psi }}^u][d{\psi }^u][d{\bar{\psi }}^d][d{\psi }^d][dA]{\hat{\mathcal{B}}}_p(t^{\prime \prime },r^{\prime \prime }){\hat{\mathcal{B}}}_p(0)\times \mathrm{deu}[\frac{\delta H_f^a}{\delta {\omega }^c}]$$

$$\times e^{i\int d^{4}x[-\frac{1}{4}{F}_{\delta \sigma }^c(x)F^{\delta \sigma c}(x)-\frac{1}{2\alpha }[H_f^c(x){]}^2+{\bar{\psi }}_u^l(x)[{\delta }^{lk}(i/\partial -m^u)+g{T}_{lk}^{c}A{/}^c(x)]{\psi }_u^k(x)+{\bar{\psi }}_d^l(x)[{\delta }^{lk}(i/\partial -m^d)+g{T}_{lk}^{c}A{/}^c(x)]{\psi }_d^k(x)]}$$

$$Z[0]=\int [d{\bar{\psi }}^u][d{\psi }^u][d{\bar{\psi }}^d][d{\psi }^d][dA]\times \mathrm{deu}[\frac{\delta H_f^a}{\delta {\omega }^c}]$$

$$\times e^{i\int d^{4}x[-\frac{1}{4}{F}_{\delta \sigma }^c(x)F^{\delta \sigma c}(x)-\frac{1}{2\alpha }[H_f^c(x){]}^2+{\bar{\psi }}_u^l(x)[{\delta }^{lk}(i/\partial -m^u)+g{T}_{lk}^{c}A{/}^c(x)]{\psi }_u^k(x)+{\bar{\psi }}_d^l(x)[{\delta }^{lk}(i/\partial -m^d)+g{T}_{lk}^{c}A{/}^c(x)]{\psi }_d^k(x)]}$$

$${\hat{\mathcal{B}}}_p(t,\vec{x})=e^{-itH}{\hat{\mathcal{B}}}_p(0,\vec{x})e^{itH}$$

$$\sum _{n^{\prime \prime \prime }}|p_{n^{\prime \prime \prime }}><p_{n^{\prime \prime \prime }}|=1$$

$$\sum _{r^{\prime \prime }}{e}^{i\vec{p}\cdot {\vec{r}}^{\prime \prime }}<0|{\hat{\mathcal{B}}}_p(t^{\prime \prime },r^{\prime \prime }){\hat{\mathcal{B}}}_p(0)|0>=\sum _{n^{\prime \prime \prime }}|<0|{\hat{\mathcal{B}}}_p(0)|p_{n^{\prime \prime \prime }}>{|}^2{e}^{-\int dt{E}_{n^{\prime \prime \prime }}(t)}$$

$$\sum _{r^{\prime \prime }}{e}^{i\vec{p}\cdot {\vec{r}}^{\prime \prime }}<0|{\hat{\mathcal{B}}}_p(t^{\prime \prime },r^{\prime \prime }){\hat{\mathcal{B}}}_p(0)|0>{|}_{t^{\prime \prime }\to \infty }=|<0|{\hat{\mathcal{B}}}_p(0)|p>{|}^2{e}^{-\int d{t}^{\prime \prime }E(t^{\prime \prime })}$$

$$E_p=E(t)+E_B(t)$$

$$E_B(t^{\prime \prime })=\frac{\int d^4{x}^{\prime \prime }{\sum }_{r^{\prime \prime \prime }}{e}^{i\vec{p}\cdot {\vec{r}}^{\prime \prime \prime }}<0|{\hat{\mathcal{B}}}_p(t^{\prime \prime \prime },r^{\prime \prime \prime }){\partial }_k{T}^{k0}(t^{\prime \prime },{\vec{x}}^{\prime \prime }){\hat{\mathcal{B}}}_p(0)|0>}{{\sum }_{r^{\prime \prime \prime }}{e}^{i\vec{p}\cdot {\vec{r}}^{\prime \prime \prime }}<0|{\hat{\mathcal{B}}}_p(t^{\prime \prime \prime },r^{\prime \prime \prime }){\hat{\mathcal{B}}}_p(0)|0>}{|}_{t^{\prime \prime \prime }\to \infty }$$

Full text

I Introduction

The spin of the proton is $\frac{1}{2}$. Since the proton consists of quarks and gluons it was expected that the spin of the quarks and gluons inside the proton add up to give spin $\frac{1}{2}$ 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.

For the proton in motion along z-axis the spin/helicity $\frac{1}{2}$ was expected to be

[TABLE]

where $<p,s|{\hat{S}}^{z}_{q}|p,s>$ is the spin of the quarks plus antiquarks inside the proton and $<p,s|{\hat{S}}^{z}_{g}|p,s>$ 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 $p^{\mu}$ and spin $s$ and ${\hat{S}}^{z}_{q}$ (${\hat{S}}^{z}_{g}$) is the z-component of the spin vector operator ${\vec{\hat{S}}}_{q}$ (${\vec{\hat{S}}}_{g}$) of the quarks plus antiquarks (gluons) inside the proton.

However, the experimental data suggested otherwise. In 1988-89 the EMC collaboration emu 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 ohu . The addition of the spin of the gluons hcu inside the proton is not sufficient to explain the spin $\frac{1}{2}$ 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 wdu . 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 jfu

[TABLE]

where $<p,s|{\tilde{\hat{L}}}^{z}_{q}|p,s>$ and $<p,s|{\tilde{\hat{L}}}^{z}_{g}|p,s>$ 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 xfu

[TABLE]

where $<p,s|{{\hat{L}}}^{z}_{q}|p,s>$ is the gauge invariant orbital angular momentum of the quarks plus antiquarks inside the proton and $<p,s|{\hat{J}}^{z}_{g}|p,s>$ 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 nkymu due to the confinement of quarks and gluons inside the finite size proton nkbsu . Because of this the angular momentum sum rule as given by eq. (3) is violated in QCD nkagu .

By taking the confinement effect into account one finds in QCD that nkagu

[TABLE]

where $<p,s|{\hat{J}}^{z}_{B}|p,s>$ 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 nkbsu .

Note that $<p,s|{\hat{S}}_{q}|p,s>$, $<p,s|{{\hat{L}}}_{q}|p,s>$, $<p,s|{\hat{J}}_{g}|p,s>$ and $<p,s|{\hat{J}}_{B}|p,s>$ 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 $<p,s|{\hat{S}}_{q}|p,s>$, $<p,s|{{\hat{L}}}_{q}|p,s>$, $<p,s|{\hat{J}}_{g}|p,s>$ and $<p,s|{\hat{J}}_{B}|p,s>$ 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 nklqu .

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 $<p,s|{\hat{S}}_{q}|p,s>$, $<p,s|{{\hat{L}}}_{q}|p,s>$, $<p,s|{\hat{J}}_{g}|p,s>$ and $<p,s|{\hat{J}}_{B}|p,s>$ 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 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 nkymu

[TABLE]

where $J^{\nu\delta\sigma}(x)$ is the gauge invariant angular momentum tensor density in QCD which is related to the gauge invariant and symmetric energy-momentum tensor density $T^{\nu\delta}(x)$ in QCD via the equation

[TABLE]

The gauge invariant and symmetric energy-momentum tensor density $T^{\nu\delta}(x)$ in QCD is given by

[TABLE]

where $\psi_{j}(x)$ is the quark field with color index $j=1,2,3$, the $A_{\delta}^{d}(x)$ is the gluon field with color index $d=1,...,8$, the Lorentz index $\delta=0,1,2,3$ and

[TABLE]

From eq. (5) we find

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

The covariant derivative $D_{lk}[A]$ in eq. (11) is defined by

[TABLE]

The ${\vec{\sigma}}$ in eq. (10) is the Pauli spin matrix and the $\sigma^{\nu\delta}$ in eq. (13) is related to the Dirac matrices via the equation

[TABLE]

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

[TABLE]

where the operator ${\hat{\cal O}}^{z}$ means the z-component of the vector operator ${\vec{\hat{\cal O}}}$ where the expressions of the vector operators ${\vec{\hat{S}}}_{q}(t)$, ${\vec{\hat{L}}}_{q}(t)$, ${\vec{\hat{J}}}_{g}(t)$ and ${\vec{\hat{J}}}_{B}(t)$ in terms of the quark field $\psi_{i}(x)$ and the gluons field $A_{\mu}^{a}(x)$ 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 the quark field $\psi_{i}(t,r)$ and the gluon field $A_{\mu}^{a}(t,r)$ do not go to $r\rightarrow\infty$. 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 $\psi_{i}(t,r)$ and the gluon field $A_{\mu}^{a}(t,r)$ nkbsu . 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 nkagu

[TABLE]

From eqs. (17) and (16) we find

[TABLE]

which implies that $<p,s|{{\hat{S}}}^{z}_{q}|p,s>+<p,s|{{\hat{L}}}^{z}_{q}|p,s>+<p,s|{{\hat{J}}}^{z}_{g}|p,s>$ is time dependent.

Since the proton spin $\frac{1}{2}$ is time independent and the $<p,s|{{\hat{S}}}^{z}_{q}|p,s>+<p,s|{{\hat{L}}}^{z}_{q}|p,s>+<p,s|{{\hat{J}}}^{z}_{g}|p,s>$ from eq. (18) is time dependent we find that

[TABLE]

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

[TABLE]

where $\psi^{u}_{i}(x)$ is the Dirac field of the up quark with color index $i=1,2,3$, the $\psi^{d}_{i}(x)$ 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

[TABLE]

where $|0>$ is the non-perturbative QCD vacuum state (i. e., the vacuum state of the full QCD, not of pQCD), $H_{f}^{a}(x)$ is the gauge fixing term, $\alpha$ is the gauge fixing parameter and

[TABLE]

is the generating functional in QCD.

The time evolution of the partonic operator in the Heisenberg representation is given by

[TABLE]

where $H$ is the QCD Hamiltonian. Inserting the complete set of energy-momentum eigenstates of the proton

[TABLE]

in (21) and then using eq. (23) we find in the Euclidean time that

[TABLE]

where $E_{n^{\prime\prime\prime}}(t)$ is the energy of all the quarks plus antiquarks plus gluons inside the proton and $\int dt$ is an indefinite integration. In the large Euclidean time $t\rightarrow\infty$ we find

[TABLE]

The energy of the proton $E_{p}$ is given by

[TABLE]

where $E_{B}(t)$ is non-zero boundary surface term given by nklqu

[TABLE]

where $\int dt^{\prime\prime}$ is an indefinite integration. Using eqs. (27) and (28) in (26) we find

[TABLE]

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 nklqu ; nkalu and in QCD in medium nkalu1 to study quark-gluon plasma at RHIC and LHC qgu ; qgu1 ; qgu2 ; qgu3 .

IV Lattice QCD formulation of the Proton Spin Crisis

From eq. (16) we find

[TABLE]

which implies that $<p,s|{{\hat{S}}}^{z}_{q}|p,s>+<p,s|{{\hat{L}}}^{z}_{q}|p,s>+<p,s|{{\hat{J}}}^{z}_{g}|p,s>+<p,s|{\hat{J}}^{z}_{B}|p,s>$ is time independent.

Since the spin $\frac{1}{2}$ of the proton is time independent and $<p,s|{{\hat{S}}}^{z}_{q}|p,s>+<p,s|{{\hat{L}}}^{z}_{q}|p,s>+<p,s|{{\hat{J}}}^{z}_{g}|p,s>+<p,s|{\hat{J}}^{z}_{B}|p,s>$ from eq. (30) is time independent one finds, unlike eq. (19), that

[TABLE]

where $\frac{1}{2}$ in the left hand side is the proton spin and $<p,s|{{\hat{S}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{L}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{J}}}^{z}_{g}|p,s>$, $<p,s|{\hat{J}}^{z}_{B}|p,s>$ are the z-components of $<p,s|{\vec{\hat{S}}}_{q}|p,s>$, $<p,s|{\vec{\hat{L}}}_{q}|p,s>$, $<p,s|{\vec{\hat{J}}}_{g}|p,s>$, $<p,s|{\vec{\hat{J}}}_{B}|p,s>$ where the expressions of the spin/angular momentum operators ${\vec{\hat{S}}}_{q}$, ${\vec{\hat{L}}}_{q}$, ${\vec{\hat{J}}}_{g}$, ${\vec{\hat{J}}}_{B}$ in terms of quark field $\psi_{i}(x)$ and gluon field $A_{\mu}^{a}(x)$ are given by eqs. (10), (11), (12) and (13) respectively.

From eq. (31) we find that the spin $\frac{1}{2}$ of the proton is obtained from $<p,s|{{\hat{S}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{L}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{J}}}^{z}_{g}|p,s>$ and $<p,s|{\hat{J}}^{z}_{B}|p,s>$. However, these quantities $<p,s|{{\hat{S}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{L}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{J}}}^{z}_{g}|p,s>$ and $<p,s|{\hat{J}}^{z}_{B}|p,s>$ 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 these non-perturbative quantities $<p,s|{{\hat{S}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{L}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{J}}}^{z}_{g}|p,s>$ and $<p,s|{\hat{J}}^{z}_{B}|p,s>$ 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-perturbative quantities $<p,s|{{\hat{S}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{L}}}^{z}_{q}|p,s>$, $<p,s|{{\hat{J}}}^{z}_{g}|p,s>$ and $<p,s|{\hat{J}}^{z}_{B}|p,s>$ in QCD to study the proton spin crisis.

Let us first formulate the lattice QCD method to calculate the non-perturbative quantity $<p,s|{\vec{\hat{S}}}_{q}|p,s>$ in QCD where the spin operator ${\vec{\hat{S}}}_{q}$ is given by eq. (10) before proceeding to calculate the other non-perturbative quantities $<p,s|{\vec{\hat{L}}}_{q}|p,s>$, $<p,s|{\vec{\hat{J}}}_{g}|p,s>$ and $<p,s|{\vec{\hat{J}}}_{B}|p,s>$ in QCD where the angular momentum operators ${\vec{\hat{L}}}_{q}$, ${\vec{\hat{J}}}_{g}$ and ${\vec{\hat{J}}}_{B}$ in QCD are given by eqs. (11), (12) and (13) respectively.

The vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime\prime},r^{\prime\prime}){\vec{\hat{S}}}_{q}(t^{\prime}){\hat{\cal B}}_{p}(0)|0>$ in QCD is given by

[TABLE]

where the partonic operator ${\vec{\hat{S}}}_{q}(t)$ is given by eq. (10) and the partonic operator ${\hat{\cal B}}_{p}(t,r)$ is given by eq. (20). We evaluate the ratio of the vacuum expectation value of the three-point non-perturbative partonic correlation function

[TABLE]

to the vacuum expectation value of the two-point non-perturbative partonic correlation function

[TABLE]

where $<0|{\hat{\cal B}}_{p}(t^{\prime\prime},r^{\prime\prime}){\vec{\hat{S}}}_{q}(t^{\prime}){\hat{\cal B}}_{p}(0)|0>$ is given by eq. (32) and $<0|{\hat{\cal B}}_{p}(t^{\prime\prime},r^{\prime\prime}){\hat{\cal B}}_{p}(0)|0>$ is given by eq. (21).

The complete set of energy-momentum eigenstates of the proton with spin $s$ is given by

[TABLE]

Using eqs. (23) and (35) in

[TABLE]

we find in Euclidean time

[TABLE]

Neglecting the higher energy level contributions at the large time limit $t^{\prime\prime}\rightarrow\infty$ we find

[TABLE]

where for the ground state $n^{\prime\prime\prime}=n^{\prime\prime}=0$ of the proton we have

[TABLE]

From eq. (38) and (36) we find

[TABLE]

where the partonic operator ${\vec{\hat{S}}}_{q}(t)$ is given by eq. (10), the partonic operator ${\hat{\cal B}}_{p}(t,r)$ is given by eq. (20), the vacuum expectation value of the two-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\hat{\cal B}}_{p}(0)|0>$ is given by eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\vec{\hat{S}}}_{q}(t){\hat{\cal B}}_{p}(0)|0>$ is given by eq. (32).

Since the vacuum expectation value of the two-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\hat{\cal B}}_{p}(0)|0>$ in eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\vec{\hat{S}}}_{q}(t){\hat{\cal B}}_{p}(0)|0>$ in eq. (32) can be calculated by using the lattice QCD method we find that the non-perturbative quantity $<p,s|{\vec{\hat{S}}}_{q}(t)|p,s>$ 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

[TABLE]

where the partonic operator ${\vec{\hat{L}}}_{q}(t)$ is given by eq. (11), the partonic operator ${\hat{\cal B}}_{p}(t,r)$ is given by eq. (20), the vacuum expectation value of the two-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\hat{\cal B}}_{p}(0)|0>$ is given by eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\vec{\hat{L}}}_{q}(t){\hat{\cal B}}_{p}(0)|0>$ is given by

[TABLE]

Since the vacuum expectation value of the two-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\hat{\cal B}}_{p}(0)|0>$ in eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\vec{\hat{L}}}_{q}(t){\hat{\cal B}}_{p}(0)|0>$ in eq. (42) can be calculated by using the lattice QCD method we find that the non-perturbative quantity $<p,s|{\vec{\hat{L}}}_{q}(t)|p,s>$ in eq. (31) to study the proton spin crisis can be calculated from eq. (41) by using the lattice QCD method.

Similarly we find

[TABLE]

where the partonic operator ${\vec{\hat{J}}}_{g}(t)$ is given by eq. (12), the partonic operator ${\hat{\cal B}}_{p}(t,r)$ is given by eq. (20), the vacuum expectation value of the two-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\hat{\cal B}}_{p}(0)|0>$ is given by eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\vec{\hat{J}}}_{g}(t){\hat{\cal B}}_{p}(0)|0>$ is given by

[TABLE]

Since the vacuum expectation value of the two-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\hat{\cal B}}_{p}(0)|0>$ in eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\vec{\hat{J}}}_{g}(t){\hat{\cal B}}_{p}(0)|0>$ in eq. (44) can be calculated by using the lattice QCD method we find that the non-perturbative quantity $<p,s|{\vec{\hat{J}}}_{g}(t)|p,s>$ in eq. (31) to study the proton spin crisis can be calculated from eq. (43) by using the lattice QCD method.

Finally we find nkncp

[TABLE]

where the partonic operator ${\vec{\hat{J}}}_{B}(t)$ is given by eq. (13), the partonic operator ${\hat{\cal B}}_{p}(t,r)$ is given by eq. (20), the vacuum expectation value of the two-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\hat{\cal B}}_{p}(0)|0>$ is given by eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\vec{\hat{J}}}_{B}(t){\hat{\cal B}}_{p}(0)|0>$ is given by

[TABLE]

Since the vacuum expectation value of the two-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\hat{\cal B}}_{p}(0)|0>$ in eq. (21) and the vacuum expectation value of the three-point non-perturbative partonic correlation function $<0|{\hat{\cal B}}_{p}(t^{\prime},r^{\prime}){\vec{\hat{J}}}_{B}(t){\hat{\cal B}}_{p}(0)|0>$ in eq. (46) can be calculated by using the lattice QCD method we find that the non-perturbative quantity $<p,s|{\vec{\hat{J}}}_{B}(t)|p,s>$ in eq. (31) to study the proton spin crisis can be calculated from eq. (45) by using the lattice QCD method.

Hence we find that the spin $\frac{1}{2}$ 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 ${\hat{\cal B}}_{p}(x)$ given by $(\ref{pru})$.

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.