Lattice QCD Method To Study Proton Decay

TL;DR

This paper develops a lattice QCD approach to calculate the non-perturbative proton decay matrix element, a key quantity for understanding proton decay, which has not yet been observed experimentally.

Contribution

The paper formulates a first-principles lattice QCD method to compute the proton decay matrix element, enabling non-perturbative calculations of this important quantity.

Findings

Derived a non-perturbative formula for proton decay matrix element

Established a lattice QCD framework for proton decay studies

Provided a basis for future numerical calculations

Abstract

The proton decay has not been experimentally observed with the lower limit of the proton lifetime being $\gtrsim 10^{34}$ years which is more than the age of the universe. One of the important quantity that appears in the study of the proton decay is the proton decay matrix element which is a non-perturbative quantity in QCD which cannot be calculated by using the perturbative QCD (pQCD) method but it can be calculated by using the lattice QCD method. In this paper we formulate the lattice QCD method to study the proton decay matrix element. We derive the non-perturbative formula of the proton decay matrix element from the first principle in QCD which can be calculated by using the lattice QCD method.

Equations

$$p\to {\pi }_0+e^{+},p\to {\pi }_0+{\mu }^{+}$$

$$p\to K^{+}+\nu$$

$${\mathcal{O}}_p(x)={ϵ}_{ijk}{u}_i^T(x)C{\gamma }_5{d}_j(x)u_k(x)$$

$${\mathcal{O}}_p(r,t)=e^{-iHt}{\mathcal{O}}_p(r,0)e^{iHt}$$

$$<0|{\mathcal{O}}_p(r^{\prime },t^{\prime }){\mathcal{O}}_p(0)|0>=\frac{1}{Z[0]}\int [d\bar{u}][du][d\bar{d}][dd]{\mathcal{O}}_p(r^{\prime },t^{\prime }){\mathcal{O}}_p(0)\det [\frac{\delta G_f^c}{\delta {\omega }^a}]$$

$$\times \exp [i\int d^{4}x[-\frac{1}{4}{F}_{\nu \lambda }^d(x)F^{\nu \lambda d}(x)-\frac{1}{2\alpha }[G_f^c(x){]}^2+{\bar{u}}_k(x)[{\delta }^{kj}(i/\partial -m_u)+g{T}_{kj}^{d}A{/}^d(x)]u_j(x)$$

$$+{\bar{d}}_k(x)[{\delta }^{kj}(i/\partial -m_d)+g{T}_{kj}^{d}A{/}^d(x)]d_j(x)]]$$

$$Z[0]=\int [d\bar{u}][du][d\bar{d}][dd]\det [\frac{\delta G_f^c}{\delta {\omega }^a}]\times \exp [i\int d^{4}x[-\frac{1}{4}{F}_{\nu \lambda }^d(x)F^{\nu \lambda d}(x)-\frac{1}{2\alpha }[G_f^c(x){]}^2$$

$$+{\bar{u}}_k(x)[{\delta }^{kj}(i/\partial -m_u)+g{T}_{kj}^{d}A{/}^d(x)]u_j(x)+{\bar{d}}_k(x)[{\delta }^{kj}(i/\partial -m_d)+g{T}_{kj}^{d}A{/}^d(x)]d_j(x)]]$$

$$F_{\nu \lambda }^s(x)={\partial }_{\nu }{A}_{\lambda }^s(x)-{\partial }_{\lambda }{A}_{\nu }^s(x)+g{f}^{sab}{A}_{\nu }^a(x)A_{\lambda }^b(x).$$

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

$$\sum _r<0|{\mathcal{O}}_p^{†}(r,t){\mathcal{O}}_p(0)|0>=\sum _{n^{\prime }}|<H_{n^{\prime }}|{\mathcal{O}}_p(0)|0>{|}^2{e}^{-\int dt{E}_{n^{\prime }}(t)}$$

$$H|H_{n^{\prime }}>=E_{n^{\prime }}(t)|H_{n^{\prime }}>.$$

$$[\sum _r<0|{\mathcal{O}}_p^{†}(r,t){\mathcal{O}}_p(0)|0>{]}_{t\to \infty }=|<p|{\mathcal{O}}_p(0)|0>{|}^2{e}^{-\int dt{E}_p(t)}$$

$$H|H_0>=E_0(t)|H_0>,|p>=|H_0>,E_p(t)=E_0(t).$$

$$E_p(t)=<p|\sum _{q,\bar{q},g}\int d^{3}r{T}^{00}(r,t)|p>$$

$$T^{\nu \lambda }(x)=F^{\nu \mu d}(x)F_{\mu }^{\lambda d}(x)+\frac{1}{4}{g}^{\nu \lambda }{F}_{\mu \sigma }^d(x)F^{\mu \sigma d}(x)+{\bar{u}}_k(x){\gamma }^{\nu }[i{\partial }^{\lambda }-ig{T}_{kj}^d{A}^{\lambda d}(x)]u_j(x)$$

$$+{\bar{d}}_k(x){\gamma }^{\nu }[i{\partial }^{\lambda }-ig{T}_{kj}^d{A}^{\lambda d}(x)]d_j(x).$$

$$\frac{d}{dt}<p|\int d^{3}r{T}^{00}(r,t)|p>=-<p|\int d^{3}r{\partial }_k{T}^{k0}(r,t)|p>.$$

$$<p|\int d^{3}r{\partial }_k{T}^{k0}(r,t)|p>\ne 0$$

$$\frac{d{E}_p(t)}{dt}\ne 0.$$

$$E_p\ne E_p(t)$$

$$\frac{d}{dt}[E_p(t)+E_B(t)]=0$$

$$\frac{d{E}_B(t)}{dt}=<p|\sum _{q,\bar{q},g}\int d^{3}r{\partial }_k{T}^{k0}(r,t)|p>\ne 0$$

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

$$|<p|{\mathcal{O}}_p(0)|0>{|}^2{e}^{-m_{p}t}=[\frac{{\sum }_r<0|{\mathcal{O}}_p^{†}(r,t){\mathcal{O}}_p(0)|0>}{e^{\int dt{E}_B(t)}}{]}_{t\to \infty }$$

$$<0|{\mathcal{O}}_p(r^{\prime \prime },t^{\prime \prime })[\int d^3{r}^{\prime }{\partial }_k{T}^{k0}(r^{\prime },t^{\prime })]{\mathcal{O}}_p(0)|0>=\frac{1}{Z[0]}\int [d\bar{u}][du][d\bar{d}][dd]{\mathcal{O}}_p(r^{\prime \prime },t^{\prime \prime })$$

$$\times [\int d^3{r}^{\prime }{\partial }_k{T}^{k0}(r^{\prime },t^{\prime })]\times {\mathcal{O}}_p(0)\det [\frac{\delta G_f^c}{\delta {\omega }^a}]\times \exp [i\int d^{4}x[-\frac{1}{4}{F}_{\nu \lambda }^d(x)F^{\nu \lambda d}(x)-\frac{1}{2\alpha }[G_f^c(x){]}^2$$

$$+{\bar{u}}_k(x)[{\delta }^{kj}(i/\partial -m_u)+g{T}_{kj}^{d}A{/}^d(x)]u_j(x)+{\bar{d}}_k(x)[{\delta }^{kj}(i/\partial -m_d)+g{T}_{kj}^{d}A{/}^d(x)]d_j(x)]].$$

$$\sum _{r^{\prime }}<0|{\mathcal{O}}_p^{†}(r^{\prime },t^{\prime })[\sum _{q,\bar{q},g}\int d^{3}r{\partial }_k{T}^{k0}(r,t)]{\mathcal{O}}_p(0)|0>=\sum _{n^{\prime },n^{\prime \prime }}<0|{\mathcal{O}}_p^{†}(0)|H_{n^{\prime }}>$$

$$<H_{n^{\prime }}|[\sum _{q,\bar{q},g}\int d^{3}r{\partial }_k{T}^{k0}(r,t)]|H_{n^{\prime \prime }}><H_{n^{\prime \prime }}|{\mathcal{O}}_p(0)|0>{|}^2{e}^{-\int d{t}^{\prime }{E}_{n^{\prime }}(t^{\prime })}.$$

$$[\sum _{r^{\prime }}<0|{\mathcal{O}}_p^{†}(r^{\prime },t^{\prime })[\sum _{q,\bar{q},g}\int d^{3}r{\partial }_k{T}^{k0}(r,t)]{\mathcal{O}}_p(0)|0>{]}_{t^{\prime }\to \infty }$$

$$=<p|\sum _{q,\bar{q},g}\int d^{3}r{\partial }_k{T}^{k0}(r,t)|p>|<p|{\mathcal{O}}_p(0)|0>{|}^2{e}^{-\int d{t}^{\prime }E(t^{\prime })}.$$

$$\frac{d{E}_B(t)}{dt}=[\frac{{\sum }_{r^{\prime }}<0|{\mathcal{O}}_p^{†}(r^{\prime },t^{\prime })[{\sum }_{q,\bar{q},g}\int d^{3}r{\partial }_k{T}^{k0}(r,t)]{\mathcal{O}}_p(0)|0>}{{\sum }_{r^{\prime }}<0|{\mathcal{O}}_p^{†}(r^{\prime },t^{\prime }){\mathcal{O}}_p(0)|0>}{]}_{t^{\prime }\to \infty }.$$

Full text

I Introduction

The decay of the proton has not been experimentally observed since the universe was created almost 14 billion years ago. On the other hand the other hadrons such as neutron, pion and kaon etc. decay with finite lifetime. For example, the lifetime of the neutron is $\sim$ 880 seconds, the lifetime of neutral pion is $\sim 10^{-16}$ seconds and the lifetime of charged kaon is $\sim 10^{-8}$ seconds. In comparison to this the lower limit of the proton lifetime is $\gtrsim 10^{34}$ years which is more than the age of the universe.

The main reason why the proton decay has not been observed is due to the baryon number conservation. Note that in the standard model of physics the baryon number is conserved. For example, the baryon number is conserved in the (free) neutron decay $n~{}\rightarrow~{}p+e+{\bar{\nu}}_{e}$ where $n$ is the neutron, $p$ is the proton, $e$ is the electron and ${\bar{\nu}}_{e}$ is the electron type antineutrino. However, the (free) proton decay processes such as

[TABLE]

are not allowed because the baryon number is not conserved for the processes in eq. (1) in the standard model of physics where $\pi_{0}$ is the neutral pion, $e^{+}$ is the positron and $\mu^{+}$ is the muon. It is well known that the (free) proton decay process $p~{}\rightarrow~{}n+e^{+}+\nu_{e}$ is not allowed even if the baryon number is conserved because the neutron mass is larger than the proton mass where $\nu_{e}$ is the electron type neutrino. In this paper we refer the (free) proton decay as the proton decay.

In the beyond standard model of physics the baryon number violation can occur which can lead to the proton decay. For example, the beyond standard model of physics such as the grand unified theories (GUTs) and the supersymmetry grand unified theories (SUSY-GUTs) predict the proton decay.

Note that even if the beyond standard model of physics predicts the proton decay but as mentioned above the proton decay has not been experimentally observed since the universe was created almost 14 billion years ago. Over the several decades various experiments have searched for the proton decay although these experiments have not found any clear evidence of the proton decay. By comparing these experimental searches with the parameter spaces of the GUTs and SUSY-GUTs these experimental searches have imposed tight constraints into the parameter spaces of the GUTs and SUSY-GUTs.

For the proton decay channels $p\rightarrow\pi_{0}+e^{+}$ and $p\rightarrow\pi_{0}+\mu^{+}$ in eq. (1) the Super-Kamiokande experiment sk has imposed the lower limit of the proton decay lifetime to be $t_{p\rightarrow\pi_{0}+e^{+}}>8.2\times 10^{33}$ years and $t_{p\rightarrow\pi_{0}+\mu^{+}}>6.6\times 10^{33}$ years respectively. For the proton decay channel

[TABLE]

the Super-Kamiokande experiment sk1 has imposed the lower limit of the proton decay lifetime to be $t_{p\rightarrow K^{+}+\nu}>5.9\times 10^{33}$ years where $K^{+}$ is the positively charged kaon.

The initial state for the proton decay channels in eq. (1) is $|p>$ and the final states are $|\pi_{0}e^{+}>$ and $|\pi_{0}\mu^{+}>$ respectively. Since the leptons $e^{+}$ and $\mu^{+}$ in the final states can be treated trivially one needs to calculate the matrix element $<\pi_{0}|{\cal O}^{\not B}|p>$ to study the proton decay where ${\cal O}^{\not B}$ is the three-quark operator violating the baryon number [see eq. (28)].

The proton and pion consist of quarks, antiquarks and gluons which are described by the quantum chromodynamics (QCD) ymk which is a fundamental theory of the nature. The partonic cross section at the short distance can be calculated by using the perturbative QCD (pQCD) due to asymptotic freedom in QCD gwk . The factorization theorem in QCD fck ; fck1 ; fck2 plays a central role to calculate the hadron cross section from the parton cross section at the high energy colliders.

The hadron formation from the quarks and gluons is a long distance phenomena in QCD which cannot be studied by using the pQCD but can be studied by using the non-perturbative QCD. Hence the proton decay matrix element $<\pi_{0}|{\cal O}^{\not B}|p>$ is a non-perturbative matrix element in QCD which cannot be calculated by using perturbative QCD but can be calculated by using the non-perturbative QCD. On the other hand the analytical solution of the non-perturbative QCD is not known yet. Hence the lattice QCD method can be used to calculate the proton decay matrix element $<\pi_{0}|{\cal O}^{\not B}|p>$.

Recently we have presented the lattice QCD method to study the proton formation from the quarks and gluons pqg and to study the proton spin crisis psc by implementing the non-zero boundary surface term in QCD due to the confinement of quarks and gluons inside the finite size proton nkbs .

In this paper we extend this to study the proton decay matrix element and present the lattice QCD formulation to study the proton decay matrix element by implementing this non-zero boundary surface term in QCD due to confinement. We derive the non-perturbative formula of the proton decay matrix element $<\pi_{0}|{\cal O}^{\not B}|p>$ from the first principle in QCD at all orders in coupling constant which can be calculated by using the lattice QCD method by implementing this non-zero boundary surface term in QCD due to confinement. Extension of this procedure to calculate the other proton decay matrix elements such as $<K^{+}|{\cal O}^{\not B}|p>$ is straightforward.

The paper is organized as follows. In section II we describe the lattice QCD method to study the proton formation from quarks and gluons by implementing the non-zero boundary surface term in QCD due to confinement. In section III we present the formulation of the lattice QCD method to study the proton decay matrix element by implementing this non-zero boundary surface term in QCD due to confinement. Section IV contains conclusions.

II Proton formation from quarks and gluons using lattice QCD Method

We denote the up and down quark fields by $u_{i}(x)$ and $d_{i}(x)$ respectively where $i=1,2,3$ is the color index. The partonic operator to study the proton formation from the partons is given by

[TABLE]

where $C$ is the charge conjugation operator. The time evolution of the partonic operator ${\cal O}_{p}(x)$ is given by

[TABLE]

where $H$ is the QCD hamiltonian of the partons.

The vacuum expectation value of the two point correlation function of the partonic operators in QCD is given by

[TABLE]

where $|0>$ is the vacuum state of the full QCD (not pQCD), $A_{\mu}^{a}(x)$ is the gluon field, $G_{f}^{c}(x)$ is the gauge fixing term, $\alpha$ is the gauge fixing parameter and

[TABLE]

is the generating functional in QCD with

[TABLE]

Note that the ghost fields are absent in eq. (5) because we are directly dealing with the ghost determinant ${\rm det}[\frac{\delta G_{f}^{c}}{\delta\omega^{a}}]$ in this paper.

The complete set of hadronic energy-momentum eigenstates is given by

[TABLE]

Using eqs. (4) and (8) in (5) we find in the Euclidean time

[TABLE]

where $\int dt$ is an indefinite integration and $E_{n}(t)$ is the energy of all the partons inside the proton in its $n$th energy level state which is time dependent [see eq. (17)] given by

[TABLE]

Neglecting the higher energy level contributions at the large time we find

[TABLE]

where $|p>$ is the energy-momentum eigenstate of the proton $p$, the $E_{p}(t)$ is the energy of all the partons inside the proton given by

[TABLE]

In terms of the energy-momentum tensor of the partons inside the proton we find

[TABLE]

where $T^{\mu\nu}(x)$ is the energy-momentum tensor density in QCD given by

[TABLE]

From the continuity equation $\partial_{\nu}T^{\nu\lambda}(x)=0$ we obtain

[TABLE]

Due to the confinement of quarks and gluons inside the finite size proton we find the non-zero boundary surface term in QCD nkbs

[TABLE]

which from eqs. (13) and (15) gives

[TABLE]

Hence from eq. (17) we find that the energy $E_{p}(t)$ of all the quarks, antiquarks and gluons inside the proton is not constant but is time dependent. Since the energy $E_{p}$ of the proton is constant (time independent) we find that

[TABLE]

where $E_{p}(t)$ is the energy of all the partons inside the protopn $p$ and $E_{p}$ is the energy of the proton $p$. From eqs. (13), (15) and (16) we obtain

[TABLE]

where

[TABLE]

Hence, unlike eqs. (17) and (18), we find from eq. (19) that

[TABLE]

where $E_{p}(t)$ is given by eq. (13) and $E_{B}(t)$ is given by eq. (20).

Using eq. (21) in (11) we find

[TABLE]

where $m_{p}$ is the mass of the proton.

The vacuum expectation value of the three point correlation function of the partonic operators in QCD is given by

[TABLE]

Using eqs. (4) and (8) in (23) we find in the Euclidean time

[TABLE]

Neglecting the higher energy level contributions at the large time we obtain

[TABLE]

From eqs. (11), (25) and (20) we find

[TABLE]

Using eq. (26) in (22) we obtain

[TABLE]

where $\int dt$ is indefinite integration.

Eq. (27) is the non-perturbative formula to study the proton formation from quarks, antiquarks and gluons by implementing the non-zero boundary surface term in QCD due to confinement which can be calculated by using the lattice QCD method.

III Lattice QCD Method To Study Proton Decay

In this section we will extend the procedure of the previous section to derive the non-perturbative formula of the proton decay matrix element $<\pi_{0}|{\cal O}^{\not B}|p>$ by implementing the non-zero boundary surface term in QCD due to confinement which can be calculated by using the lattice QCD method. This procedure is also applied to study various non-perturbative quantities in QCD in vacuum psc ; allg and in QCD in medium allgm to study the quark-gluon plasma at RHIC and LHC qgk ; qgk1 ; qgk2 .

The baryon number violating three-quark operator ${\cal O}^{\not B}(x)$ is given by

[TABLE]

where $P_{R/L}$ means right or left projection matrix respectively given by

[TABLE]

The partonic operator ${\cal O}_{p}(x)$ for the proton formation is given by eq. (3) and the patonic operator ${\cal O}_{\pi_{0}}(x)$ for the pion formation is given by

[TABLE]

The vacuum expectation value of the three point non-perturbative partonic correlation function $<0|{\cal O}_{\pi_{0}}(r^{\prime},t^{\prime}){\cal O}^{\not B}(r,t){\cal O}_{p}(0)|0>$ is given by

[TABLE]

Similar to eq. (8) for the proton the complete set of energy-momentum eigenstates of the pion $\pi_{0}$ is given by

[TABLE]

Using eqs. (4), (8) and (32) in (31) we find in the Euclidean time

[TABLE]

where $\int dt^{\prime}$ and $\int dt^{\prime\prime}$ are indefinite integrations, ${\vec{p}}_{\pi_{0}}$ is the momentum of the pion $\pi_{0}$ and the proton is at rest.

In the limit $t^{\prime\prime}>>>t^{\prime},~{}~{}t^{\prime}\rightarrow\infty$ we find by neglecting the higher energy level contributions

[TABLE]

where $E_{\pi_{0}}(t)$ is the energy of all the partons inside the pion $\pi_{0}$ and $E_{p}(t)$ is the energy of all the partons inside the proton $p$.

From eq. (11) we find for the proton formation

[TABLE]

Similarly for the pion $\pi_{0}$ formation we find

[TABLE]

From eqs. (34), (35) and (36) we find

[TABLE]

From eq. (27) we find

[TABLE]

and similarly for the pion we find

[TABLE]

where $E_{\pi_{0}}$ is the energy of the pion $\pi_{0}$.

Using eqs. (38) and (39) in (37) we find

[TABLE]

which can be calculated by using the lattice QCD method.

Eq. (40) is the non-perturbative formula of the proton decay matrix element $<\pi_{0}|{\cal O}^{\not B}|p>$ derived from the first principle in QCD which can be calculated by using the lattice QCD method. Extension of eq. (40) to study other proton decay matrix elements such as                   $<K^{+}|{\cal O}^{\not B}|p>$ is straightforward.

IV Conclusions

The proton decay has not been experimentally observed with the lower limit of the proton lifetime being $\gtrsim 10^{34}$ years which is more than the age of the universe. One of the important quantity that appears in the study of the proton decay is the proton decay matrix element which is a non-perturbative quantity in QCD which cannot be calculated by using the perturbative QCD (pQCD) method but it can be calculated by using the lattice QCD method. In this paper we have formulated the lattice QCD method to study the proton decay matrix element. We have derived the non-perturbative formula of the proton decay matrix element from the first principle in QCD which can be calculated by using the lattice QCD method.