Phenomenology of Strong Interactions — Towards an Effective Theory for Low Energy QCD

Generally, tachyons are classified as particles that travel faster than light or weakly interacting superluminal particles. Relativistically, single particle energy is expressed as $E^{2}=p^{2}c^{2}+m^{2}c^{4}$, where $p$ is the spatial momentum and $m$ is the mass of the particle. For a particle to be faster than light, the relativistic velocity $\beta=pc/E>1$. Hence, for tachyons with real $p$, $m$ must necessarily be imaginary Bilaniuk . However, this analogy does not make a strong and convincing case for the tachyons. Rather, Quantum Field Theory (QFT) provides some satisfactory explanation to the dynamics of tachyons. QFT suggests that particles that travel faster than light does not exist in nature. So tachyons are simply unstable particles that decay. Based on this understanding, we consider a scalar field, say $\phi$, with the usual kinetic term and a potential $V(\phi)$ whose extremum is at the origin. If one carries out perturbation quantization about $\phi=0$ up to the quadratic term and ignores the higher order terms in the action, we obtain a particle-like state at $V^{\prime\prime}(0)$. These results have two interesting interpretations; if $V^{\prime\prime}(0)>0$ we have a particle of mass $m^{2}_{\phi}$ but for $V^{\prime\prime}(0)<0$ we have a tachyonic state with a negative $m^{2}_{\phi}$. In this case, tachyons can be given a physical meaning. So far, we know that the tachyons have negative $m^{2}_{\phi}$ and potential whose maximum is at the origin, thus a small displacement of $\phi$ at the origin will cause it to grow exponentially with time towards the true minimum. By the above description, tachyons can be represented by a potential such as

$$V(\phi)=-\dfrac{1}{2}m_{\phi}^{2}\phi^{2}+c_{3}\phi^{3}+c_{4}\phi^{4}\cdots,$$

(1)

where $c_{3}\,\text{and}\,c_{4}$ are constants. Hence, tachyonic fields are associated with the ’usual’ Higgs fields where the Higgs particles acquire negative square mass at the true minimum of their potential. Accordingly, tachyons in QFT are related to instability that breaks down the perturbation theory in normal field theory. The usual quantization where the cubic and higher order terms are considered small corrections to the quadratic term is no longer tenable. Since $V(\phi)$ has its maximum at $\phi=0$, it renders it classically unstable at that point. So one cannot guarantee that the fluctuation at that point is small. This behavior comes with an inbuilt solution i.e. one can expand the potential about the true minima $\phi_{0}$ up to the quadratic term and proceed with the perturbation quantization about that point. Thus, the cubic and higher-order terms in the expansion can be discarded. This process will lead to the creation of a particle with a positive mass $m^{2}_{\phi}$ in the spectrum. This process removes the tachyonic modes in the spectrum. Additionally, in D$p$-brane systems, the theory must be invariant under $Z_{2}$ symmetry, i.e. $\phi\rightarrow-\phi$ and in the presence of brane-anti-brane, the theory must necessarily be invariant under phase symmetry $\phi\rightarrow e^{i\alpha}\phi$. Indeed, there are some benefits in working in tachyonic modes otherwise we can define a new field, $\eta=\phi-\phi_{0}$, and express the potential in terms of the new field,

$$V(\eta)=V(\eta+\phi_{0}).$$

(2)

Working with this potential from the onset will remove the tachyonic mode from the spectrum because $V^{\prime\prime}(\eta=0)$ will be positive. However, there are some benefits to working with tachyonic fields, for instance, they possess high symmetry. This symmetry might not be explicit in $V(\eta)$ as it is in $V(\phi)$. The high symmetry in tachyonic fields leads to the phenomenon of spontaneous symmetry breaking, where the potential has more than one minima i.e. $V(\phi)=V(-\phi)$ corresponding to $\pm\phi_{0}$. This phenomenon is well known in elementary particle physics Sent3 .

Recent advances in string theory have provided some explanations regarding nonperturbative features of QCD theory, thanks to the discovery of D$p$-branes. The study of D$p$-branes gives insight into physically relevant systems such as black holes, supersymmetric gauge theories, and the connection between Yang-Mills theories and quantum gravity. Upon the numerous progress, a consistent nonperturbative background-independent construction of the theory is yet to be developed. This poses a challenge in directly addressing cosmological problems. There are five known ways for which supersymmetric closed strings can be quantized, i.e. IIA, IIB, I, heterotic $\text{SO}(32)$ and heterotic $E_{8}\times E_{8}$ superstring theories. Individually, they give a perturbative description of quantum gravity. However, they are connected through duality symmetries Hull ; Witten . Indeed, some of these dualities are nonperturbative in nature because the string coupling $g_{s}$ in one theory may have an inverse relation $1/g_{s}$ with the other. The superstring theories together with M-theory (a theory that unifies superstring theories) consist of an extended object of higher dimensions, called D$p$-branes. Each of these theories contains branes but in different complements. Particularly, IIA/IIB superstring theories consist of even/odd D$p$-brane dimensional configurations. Using the appropriate duality transformations, one brane can be mapped onto the other even with the strings connecting them. Thus, none of the branes are seen to be more fundamental to others. This process is commonly referred to as ’brane democracy’.

Before moving into the specific kind of strings set out for this study, we will shed light on the features of open and closed strings. Closed strings are topologically the same as a circle $S^{1}$, i.e. $[0,2\pi]$. They give rise to a massless set of spacetime fields, identified as graviton $g_{\mu\nu}$, dilaton $\varphi$, antisymmetric two-form $B_{\mu\nu}$ and an infinite set of massive fields upon quantization. Supersymmetric closed strings are also related to massless fields identified with graviton supermultiplets such as Ramond-Ramond $p$-form fields $A^{(p)}_{\mu,\cdots,\mu_{p}}$ and gravitino $\psi_{\mu\alpha}$. Consequently, the quantum theory of closed strings is naturally related to the theory of gravity in spacetime. On the other hand, open strings are topologically similar to the interval $[0,\pi]$. They give rise to massless gauge field $A_{\mu}$ in spacetime upon quantization. Supersymmetric open strings are also associated with massless gaugino $\psi_{\alpha}$. Accordingly, open strings have their ends on Dirichlet $p$-branes (D$p$-brane) and the gauge field lives on the worldvolume of the D$p$-brane. Upon these differences, the physics of closed and open strings are related at the quantum level. Closed strings were first observed through the one-loop process of an open string. Under this process, close strings appeared as poles in nonplanar one-loop open string diagrams Gross1 ; Lovelace . The open strings have their endpoints on the D$p$-branes whilst close strings have no endpoints. The degree of freedom of the open strings is associated with standing wave modes on the fields while the close strings correspond to left-moving and right-moving waves. The boundary conditions for open strings on the bosonic field $X^{M}$ are Neumann (freely moving endpoints) and Dirichlet boundary conditions (fixed endpoints). The close string on the other hand corresponds to periodic and anti-periodic boundary conditions.

Some D$p$-branes have unstable configurations both in the supersymmetric or the nonsupersymmetric string theories. The instability is attributed to tachyonic modes with negative square mass $m^{2}_{\phi}\,<\,0$ in the open string spectrum, we are interested in investigating the effects of the tachyonic modes Sen ; Zwiebach . Some D$p$-brane configurations with open strings containing tachyons are:

Brane-antibrane; it is a type IIA or IIB string theory with parallel D$p$-branes separated by $d\,<\,l_{s}$ ($l_{s}$ is the string length scale). They also carry tachyons in their open string spectrum. The difference in their orientation leads to opposite Ramond-Ramond (R-R) charges Hughes . So the brane and anti-brane pair can annihilate leaving a neutral vacuum because the net R-R charge will be zero.

Wrong-dimension branes; the D$p$-brane with wrong dimension for type IIA/IIB with odd/even spatial dimension for $p$ instead of even/odd dimensions carry no charges under classical IIA/IIB supergravity fields. They have tachyons in their open string spectrum. Such branes can annihilate to form a vacuum without violating charge conservation.

Bosonic D$p$-branes; just as the wrong-dimension branes of type IIA/IIB string theory, the D$p$-brane of any particular dimension in the bosonic string theory have no conserve charge and has tachyons in their open string spectrum. Also, they can annihilate to form a neutral vacuum without violating charge conservation.

Again, even though the non-BPS D$p$-branes of type IIA/IIB string theory are unstable owing to the presence of tachyonic modes on their worldvolume Sen1 . We can obtain stable non-BPS branes by taking orientifolds/orbifolds of the theory which projects out the tachyonic modes Sen1 ; Bergman ; Sen2 .

Studying D$p$-branes under Yang-Mills theory enables us to understand the physics of D$p$-branes without necessarily applying any complex string theory artifacts. Detailed analyses show there is enough evidence that super Yang-Mills theory carries a lot of information concerning string theory than one may possibly imagine Banks . Besides, recent developments in high-energy physics investigations have shown that string theory gives insight into low-energy field theories in the nonperturbative region Taylor . This has been conjectured to be equivalent to QCD where confinement can be realized in color fluxtube picture. From Eq.(9), in the limit of vanishing $B_{\mu\nu}$ and further assuming that the D$p$-branes are almost flat, close to the hypersurface i.e. $X^{M}=0$, $M>p$. Again, we suppose that the fields are slowly varying such that $\partial_{\mu}X^{M}\partial^{\mu}X_{M}$ and $2\pi\alpha^{\prime}F_{\mu\nu}$ are in the same order. Therefore, the action can be expanded as

$$S=-\tau_{p}V_{p}-\dfrac{1}{4g^{2}_{YM}}\int d^{p+1}\xi\left(F_{\mu\nu}F^{\mu\nu}+\dfrac{2}{(2\pi\alpha^{\prime})^{2}}\partial_{\mu}X^{M}\partial^{\mu}X_{M}\right)+{{\cal O}((\partial_{\mu}X^{M})^{4},F^{4})}.$$

(11)

Here, $V_{p}$ is the $p$-brane worldvolume and $g_{YM}$ is the Yang-Mills coupling given by,

$$g_{YM}^{2}=\dfrac{1}{(2\pi\alpha^{\prime})^{2}\tau_{p}}=\dfrac{g}{\sqrt{\alpha^{\prime}}}\left(2\pi\sqrt{\alpha^{\prime}}\right)^{p-2}.$$

(12)

The second term in Eq.(11) corresponds to $\text{U}(1)$ gauge theory in $p+1$ dimension coupled with $9-p$ scalar fields. Introducing fermions fields, as mentioned below Eq.(3) into the action, we recover the supersymmetric $\text{U}(1)$ Yang-Mills theory in $10\,d$ $\mathcal{N}=1$ Super Yang-Mills action,

$$S=\dfrac{1}{g_{YM}^{2}}\int d^{10}\xi\left(-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}+\dfrac{i}{2}\bar{\psi}\Gamma^{\mu}\partial_{\mu}\psi\right).$$

(13)

In the case of N parallel D$p$-branes, the p-dimensional branes must be distinctively labeled from $1\,\text{to}\,\text{N}$. Subsequently, the massless scalar fields living on the individual D$p$-brane worldvolume are related to $\text{U}(1)^{N}$ gauge group. The fields arising from the strings stretching from one brane to the other are labeled as $A^{\mu}_{i,j}$, where $i,j$ specifies the individual branes that carry the endpoints of the strings. The strings are oriented such that they consist of $N(N-1)$ fields, corresponding to $A_{\alpha}=(A_{\mu},\,X^{M})$ individual fields. The mass of the strings is proportional to the separation distances between the branes. So the strings become massless Witten1 ; Taylora when the D$p$-branes get very close to each other. Hence, the open strings can transform under the adjoint representation U(N). Thus, the corresponding fields can be decomposed similarly to adjoint supersymmetric gauge field $\text{U(N)}=\text{SU}(N)\times\text{U}(1)$ in $p+1$-dimensions. With the stacks of D$p$-branes crossing each other at some angles, one can also break the U(N) symmetry into SM particle physics i.e. $\text{SU}(3)\times\text{SU}(2)\times\text{U}(1)$ Lust . In sum, the DBI action can be generalized into a non-Abelian gauge group by considering stacks of D$p$-branes instead of a D$p$-brane.

A major motivation for string dual to QCD is derived from the ’t Hooft large $N_{c}$-limit Hoofta2 ; Witten4 . Though, $\text{SU}(N_{c})$ and $\text{U}(N_{c})$ have different representations, in the limit of $N_{c}\rightarrow\infty$, the difference can be overlooked. Thus, the number of gluons can be approximated as $~{}N_{c}^{2}$. This is more than the quark degree of freedom $N_{f}N_{c}$, therefore, we expect the dynamics of the gluons to dominate in this regime Mateos .

In this section, we introduce Dirac’s Lagrangian for free particles adopted by Maxwell in the unification of electric and magnetic field interactions. We introduce the dynamics of the fermions which were dropped in the previous discussions as stated below Sec. III. However, it will be introduced here through Dirac’s equation modified by $G(\phi)$ while taking into consideration all gauge invariance properties. We start with the well-known Dirac’s Lagrangian for free particles

$$\mathcal{L}_{0}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi.$$

(24)

Even though this Lagrangian is well known in QED, it is also used to describe free nucleons in terms of their composite fermions; protons, and neutrons in strong interaction. It is invariant under global phase rotation,

$$\psi\rightarrow\psi^{\prime}=e^{i\alpha}\psi.$$

(25)

Noticing that the fields in the Lagrangian are color neutral in nature, to apply them in studying color particles such as the ones considered here, we need to modify the fields. Hence, we will modify the bispinors with the $G(\phi)$ in order to give them some color features. Thus, we perform the transformations

$$\psi\rightarrow\psi^{\prime}\equiv G^{1/2}\psi\qquad\text{and}\qquad\bar{\psi}\rightarrow\bar{\psi}^{\prime}\equiv\bar{\psi}G^{1/2},$$

(26)

where $G$ is a function therefore, the gradient in Dirac’s equation will transform as

$$\partial_{\mu}\psi\rightarrow\partial_{\mu}\psi^{\prime}=\left[(\partial_{\mu}G^{1/2})+G^{1/2}\partial_{\mu}\right]\psi,$$

(27)

and Lagrangian (24) becomes

	$\displaystyle\mathcal{L}$	$\displaystyle=\bar{\psi}^{\prime}\left[i\gamma^{\mu}\partial_{\mu}-m\right]\psi^{\prime}$
		$\displaystyle=\bar{\psi}\left[i\gamma^{\mu}G\partial_{\mu}+i\gamma^{\mu}G^{1/2}(\partial_{\mu}G^{1/2})-mG\right]\psi.$		(28)

The local gauge invariance is violated by $G(\phi)$ and the gradient function $(\partial_{\mu}G^{1/2})$. To ensure local gauge invariance is satisfied, we modify all variables involving derivatives, and color neutral fields such as the Dirac $\gamma$-matrix with $G(\phi)$ and also introduce the electromagnetic field $A_{\mu}(x)$ in other to make the equation gauge invariant. Consequently, we will adopt the transformations,

$$\gamma^{\mu}\rightarrow\gamma^{\prime\mu}=G^{-1}\gamma^{\mu}\qquad{\text{and}}\qquad\partial_{\mu}\rightarrow D_{\mu}\equiv\partial_{\mu}-iqA_{\mu},$$

(29)

where $D_{\mu}$ is the gauge-covariant derivative and $q$ is the electric charge. This type of derivative corresponds to momentum transformation, $p_{\mu}\rightarrow p_{\mu}-qA_{\mu}$. By this transformation, the gauge field also gets modified and enters the Lagrangian as $GA_{\mu}$. It also introduces a coupling between the electromagnetic field and matter in a form $D_{\mu}\psi$. So Eq.(V.1) becomes

	$\displaystyle\mathcal{L}$	$\displaystyle=\bar{\psi}\left[i\gamma^{\mu}\partial_{\mu}+i\gamma^{\mu}\partial_{\mu}(\ln G^{1/2})+qA_{\mu}\gamma^{\mu}-mG\right]\psi$
		$\displaystyle=\bar{\psi}\left[i\gamma^{\mu}\partial_{\mu}+q\gamma^{\mu}A_{\mu}-mG\right]\psi,$		(30)

consequently, the electromagnetic field transforms as

$$A_{\mu}\rightarrow A^{\prime}_{\mu}\equiv A_{\mu}+\dfrac{i}{q}\partial_{\mu}(\ln G^{1/2}).$$

(31)

Equation (V.1) looks similar to Dirac’s Lagrangian with interaction term that couples the gauge field $A_{\mu}(x)$ to the conserve current $j^{\mu}=q\bar{\psi}\gamma^{\mu}\psi$ with a modified mass term $M(r)=mG(\phi)$. Also, $\bar{\psi}D_{\mu}\psi$ becomes invariant under local phase rotation, granting interaction between $\bar{\psi}$, $\psi$ and $A_{\mu}$ with momentum $p_{\mu}\rightarrow i\partial_{\mu}$. In this way, the Lagrangian has been modified, but we ensure that all conservation laws are duly respected.

Now, to derive a complete Lagrangian that can mimic QCD theory, we include the kinetic energy term of the gauge field which has been derived in Eq.(18), thus,

$$\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}+q\gamma^{\mu}A_{\mu}-mG(\phi)\right)\psi+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)-\frac{1}{4}G(\phi)F_{\mu\nu}F^{\mu\nu}.$$

(32)

This expression looks similar to the usual QED Lagrangian with $\phi$ as intermediate field/particle and a mass term modified by color dielectric function, $G(\phi)$. As a result, this expression approximates the non-Abelian QCD theory with an Abelian one. This was motivated by the projection that the confining regime of QCD is mostly Abelian dominated Hoofta1 ; Ezawa ; Shiba ; Suzuki ; Sakumichi . The spinor fields $\psi$ and $\bar{\psi}$ represent the quarks and antiquarks and the modified gauge field $A_{\mu}$ also describes gluons. All the long-distance behavior of the gluons is absorbed in $G(\phi)$. The observed mass of the system $M(r)=mG(\phi)$ is expected to have a fixed value at the beginning of the interaction $r\rightarrow r_{0}$ and at the end of the interaction $r\rightarrow r_{*}$, so it can be measured precisely. That is, the dielectric function should be such that $G(\phi(r\rightarrow r_{0}))=G(\phi(r\rightarrow r_{*}))=\text{constant}$, where $1/r_{0}$ is the energy at the beginning of the interaction and $1/r_{*}$ is the energy at the end of the interaction. Thus, $M(r)=mG(\phi)$ is the constituent quark mass function of the system. We have presented a detailed study of this theory in Refs.Issifu1 ; Adamu

Since the renormalization theory remains the systematic approach for which UV divergences can be resolved Neubert , we will compare the result with the renormalized QED Lagrangian to properly identify the nature of the dielectric function in the context of renormalization factor $Z(\mu)$, where $\mu$ is a scale that comes from the dimensional regularization scheme Bollini ; Hoofta . The objective is to ensure that the result obtained in Eq.(32) does not pose any UV divergences. When the real QED Lagrangian is written in terms of the renormalized factors, it takes the form,

$$\mathcal{L}_{\text{QED}}=Z_{\psi}\bar{\psi}(i\gamma_{\mu}\partial^{\mu}-m_{0})\psi-\dfrac{Z_{A}}{4}F^{\mu\nu}F_{\mu\nu}-Z_{1}e\bar{\psi}\gamma_{\mu}A^{\mu}\psi,$$

(33)

where

$$\psi_{0}=Z_{\psi}^{1/2}\psi\quad\text{,}\quad A_{0}^{\mu}=Z_{A}^{1/2}A^{\mu}\quad\text{,}\quad m_{0}=\dfrac{Z^{\prime}_{\psi}}{Z_{\psi}}m\quad\text{and}\quad e=\dfrac{Z_{\psi}}{Z_{1}}Z_{A}^{1/2}e_{0},$$

(34)

the two equations bear some resemblance, so we can compare them. With $Z_{3}$ the gluon propagator renormalization factor, $Z_{1}$ the quark-quark-gluon vertex renormalization, and $Z_{2}$ the quark self-energy renormalization factor. Additionally, the covariant derivative can be expressed in terms of the renormalized factors as $D^{ren}_{\mu}\,=\,\partial_{\mu}-ie(Z_{1}/Z_{\psi})A_{\mu}$, gauge invariance requires that $Z_{1}\,=\,Z_{\psi}$. We have substituted the ’conventional’ representation of the renormalization factors $Z_{2}$ and $Z_{3}$ with $Z_{\psi}$ and $Z_{A}$ respectively, and $Z_{1}\,=\,Z_{\psi}Z_{A}^{1/2}$ to make the distinction more obvious relative to fermions and the gauge field. The renormalized fields are without subscript ’0’ Itzykson ; Pascual ; Peskin ; Collins ; Weinberg ; Weinberg1 ; Schwartz . Comparing the results in Eqs.(32) and (33), we identify $Z_{A}\,=\,Z_{\psi}\,=\,G$ and the gauge invariance warrant that, $Z_{A}\,=\,1$. Thus, in addition to the color properties carried by the color dielectric function, it also absorbs the UV divergences. Consequently, the dielectric function follows the restrictions,

$$G\left(\phi(r)\right)\rightarrow\begin{cases}1&\text{for}\,\;r\,\rightarrow\,0,\;\text{deconfinement/Coulombian regime}\\ 0&\text{for}\,\;r\,\rightarrow\,r_{*},\;\text{confinement regime}\end{cases}.$$

(35)

From the gauge conditions adopted above, a gluon mass term,

$$\mathcal{L}_{\gamma}=\dfrac{1}{2}M^{2}(r)A_{\mu}A^{\mu},$$

(36)

will not be invariant under the local gauge transformation in Eq.(31) because

$$A_{\mu}A^{\mu}\rightarrow A^{\prime}_{\mu}A^{\prime\mu}\equiv\left(A_{\mu}+\dfrac{i}{q}\partial_{\mu}(\ln G^{1/2})\right)\left(A^{\mu}+\dfrac{i}{q}\partial^{\mu}(\ln G^{1/2})\right)\neq A_{\mu}A^{\mu}.$$

(37)

Consequently, local gauge invariance accounts for the existence of massless photons Quigg .

In this section, we will focus on constructing a non-Abelian gauge theory traditionally used for describing the strong nuclear force. Additionally, we buy into the projection that proton and neutron have the same mass and are charge independent due to the strong nuclear force. Therefore, there is an agreement that isospin is conserved in strong interactions. As in the case of QED theory, we will base the discussions on $\text{SU}(2)$-isospin gauge theory introduced by Yang and Mills Yang , elaborated by Shaw Shaw . Here, we will consider Eq.(24) as the Lagrangian for free nucleons with composite fermions, protons (p), and neutrons (n)

$$\psi\equiv\begin{pmatrix}p\\ n\end{pmatrix}.$$

(38)

The Lagrangian is invariant under global spin rotation

$$\psi\rightarrow\psi^{\prime}=e^{i\vec{\tau}\cdot\vec{\alpha}/2}\psi,$$

(39)

also, isospin current, $j^{\mu}=\bar{\psi}\gamma^{\mu}({\tau}/{2})\psi$, is conserved therefore, proton and neutron can be treated symmetrically in the absence of electromagnetic interactions. In that case, the distinction between proton and neutron is arbitrary and conventional under this representation.

To maintain the differences between the Abelian theory treated in Sec. V.1 and the non-Abelian theory intended for this section, we will replace $A_{\mu}$ from the Abelian covariant derivative, $D_{\mu}$, expressed in Eq.(29) with its non-Abelian counterpart $B_{\mu}$ i.e.,

$$D_{\mu}\equiv\partial_{\mu}-igB_{\mu}.$$

(40)

We have also replaced $q$ with $g$, the strong coupling constant to make the analysis distinct from Sec. V.1 and befitting for describing strong interactions. Despite the similar global gauge invariance satisfied by both theories, they also exhibit some major differences; the algebra of the non-Abelian group is more complex and the associated gauge bosons self-interact due to the structure of the non-Abelian group. Accordingly, the field can be expressed as,

$$B_{\mu}=\dfrac{1}{2}\vec{\tau}\cdot\vec{b}_{\mu}=\dfrac{1}{2}\tau^{a}b^{a}_{\mu}=\dfrac{1}{2}\left({\begin{array}[]{cc}b_{\mu}^{3}&b^{1}_{\mu}-ib^{2}_{\mu}\\ b^{1}_{\mu}+ib^{2}_{\mu}&-b^{3}_{\mu}\\ \end{array}}\right)=\dfrac{1}{2}\left({\begin{array}[]{cc}b_{\mu}^{3}&\sqrt{2}\,b^{+}_{\mu}\\ \sqrt{2}\,b^{-}_{\mu}&-b^{3}_{\mu}\\ \end{array}}\right),$$

(41)

where the three non-Abelian gauge fields are $\vec{b}_{\mu}=(b_{\mu}^{1},b_{\mu}^{2},b_{\mu}^{3})$. The generators $\tau^{a}$ $(a=1,\,2$ and $3$) are associated with Pauli’s matrices, $b_{\mu}^{\pm}=(b_{\mu}^{1}\mp ib_{\mu}^{2})/\sqrt{2}$ are the charged gauge bosons and the isospin step-up and step-down operators $1/2(\tau_{1}\pm i\tau_{2})$ exchange $p\leftrightarrow n$ following the absorption of $b_{\mu}^{\pm}$ boson. The gradient in Eq.(24) will then transform as,

	$\displaystyle D_{\mu}\psi\rightarrow D\psi^{\prime}$	$\displaystyle=G^{1/2}\partial_{\mu}\psi+(\partial_{\mu}G^{1/2})\psi-igB_{\mu}(G^{1/2}\psi)$
		$\displaystyle=G^{1/2}\left(\partial_{\mu}-igB^{\prime}_{\mu}\right)\psi,$		(42)

thus,

	$\displaystyle-igG^{1/2}B^{\prime}_{\mu}$	$\displaystyle=-igB_{\mu}G^{1/2}+(\partial_{\mu}G^{1/2})\rightarrow$
	$\displaystyle B^{\prime}_{\mu}$	$\displaystyle=G^{1/2}B_{\mu}G^{-1/2}+\dfrac{i}{g}(\partial_{\mu}G^{1/2})G^{-1/2}$
		$\displaystyle=G^{1/2}\left(B_{\mu}+\dfrac{i}{g}G^{-1/2}(\partial_{\mu}G^{1/2})\right)G^{-1/2}.$		(43)

Again, following similar procedure as adopted for Eq.(V.1) using the necessary transformations, we obtain

$\displaystyle\mathcal{L}^{\prime}=\bar{\psi}\left[i\gamma^{\mu}\partial_{\mu}+gB_{\mu}\gamma^{\mu}-mG\right]\psi,$

(44)

with gauge invariant transformation similar to Eq.(31),

$$B_{\mu}\rightarrow B^{\prime}_{\mu}\equiv B_{\mu}+\dfrac{i}{g}(\partial_{\mu}\ln G^{1/2}).$$

(45)

By comparing Eq.(V.2) and Eq.(45) we can deduce,

$$B_{\mu}\rightarrow B^{\prime}_{\mu}=G^{1/2}B_{\mu}G^{-1/2}.$$

(46)

The $\text{SU}(2)$-isospin generators can be expressed in terms of commutator relation,

$$\left[\tau^{j},\tau^{k}\right]=2i\varepsilon_{jkl}\tau^{l}.$$

(47)

Conventionally, $\tau_{i}$ do not commute with others in different spatial directions, so only one component can be measured at a time and this is taken to be the third component $T_{3}=1/2\tau_{3}$, generally in $z$-direction.

We will now develop the field strength tensor that will form the kinetic term of the gauge field. Starting with the electromagnetism gauge, we can construct,

$$F_{\mu\nu}=\dfrac{1}{2}{\bf F}_{\mu\nu}\cdot{\bf\tau}=\dfrac{1}{2}F^{a}_{\mu\nu}\tau^{a}\qquad{\text{where}}\qquad tr\left(\tau^{a}\tau^{b}\right)=2\delta^{ab}.$$

(48)

From these transformations, we can develop a gauge invariant field,

$$\mathcal{L}^{\prime}_{gauge}=-\dfrac{1}{4}{\bf F}_{\mu\nu}\cdot{\bf F}^{\mu\nu}=-\dfrac{1}{2}tr\left(F_{\mu\nu}F^{\mu\nu}\right),$$

(49)

where the field strength tensor transforms under the dielectric function as

$$F_{\mu\nu}\rightarrow F^{\prime}_{\mu\nu}\equiv G^{1/2}F_{\mu\nu}G^{-1/2}.$$

(50)

We know from the QED relation constructed in Sec. V.1 that the field strength can be expressed as

	$\displaystyle F^{\prime}_{\mu\nu}$	$\displaystyle=\partial_{\nu}B^{\prime}_{\mu}-\partial_{\mu}B^{\prime}_{\nu}=\partial_{\nu}\left[G^{1/2}B_{\mu}G^{-1/2}+\dfrac{i}{g}(\partial_{\mu}G^{1/2})G^{-1/2}\right]-\partial_{\mu}\left[G^{1/2}B_{\nu}G^{-1/2}+\dfrac{i}{g}(\partial_{\nu}G^{1/2})G^{-1/2}\right]$
		$\displaystyle=\left\{(\partial_{\nu}G^{1/2})B_{\mu}G^{-1/2}+G^{1/2}(\partial_{\nu}B_{\mu})G^{-1/2}+G^{1/2}B_{\mu}(\partial_{\nu}G^{-1/2})+\dfrac{i}{g}\left[\partial_{\nu}(\partial_{\mu}G^{1/2})G^{-1/2}+(\partial_{\mu}G^{1/2})(\partial_{\nu}G^{-1/2})\right]\right\}$
		$\displaystyle-\left\{(\partial_{\mu}G^{1/2})B_{\nu}G^{-1/2}+G^{1/2}(\partial_{\mu}B_{\nu})G^{-1/2}+G^{1/2}B_{\nu}(\partial_{\mu}G^{-1/2})+\dfrac{i}{g}\left[\partial_{\mu}(\partial_{\nu}G^{1/2})G^{-1/2}+(\partial_{\nu}G^{1/2})(\partial_{\mu}G^{-1/2})\right]\right\}.$		(51)

We adopted the expression for $B^{\prime}_{\mu}$ defined in Eq.(V.2), regrouping the terms in the above equation yields,

	$\displaystyle F^{\prime}_{\mu\nu}$	$\displaystyle=G^{1/2}\left[\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}\right]G^{-1/2}+\left[(\partial_{\nu}G^{1/2})B_{\mu}-(\partial_{\mu}G^{1/2})B_{\nu}\right]G^{-1/2}+G^{1/2}\left[B_{\mu}(\partial_{\nu}G^{-1/2})-B_{\nu}(\partial_{\mu}G^{-1/2})\right]$
		$\displaystyle+\dfrac{i}{g}\left[(\partial_{\mu}G^{1/2})(\partial_{\nu}G^{-1/2})-(\partial_{\nu}G^{1/2})(\partial_{\mu}G^{-1/2})\right]$
		$\displaystyle\neq G^{1/2}\left[\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}\right]G^{-1/2},$		(52)

higher derivative terms were discarded. We can cast this result in a slightly symmetric form by using the identity $G^{1/2}G^{-1/2}=\mathbb{I}$, so

	$\displaystyle\partial_{\mu}(G^{1/2}G^{-1/2})$	$\displaystyle=(\partial_{\mu}G^{1/2})G^{-1/2}+G^{1/2}(\partial_{\mu}G^{-1/2})=0\rightarrow$
	$\displaystyle G^{1/2}(\partial_{\mu}G^{-1/2})$	$\displaystyle=-(\partial_{\mu}G^{1/2})G^{-1/2}.$		(53)

Using this identity appropriately leads to,

	$\displaystyle F^{\prime}_{\mu\nu}$	$\displaystyle=G^{1/2}\left(\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}\right)G^{-1/2}+\left((\partial_{\nu}G^{1/2})B_{\mu}-B_{\mu}(\partial_{\nu}G^{1/2})\right)G^{-1/2}+\left(B_{\nu}(\partial_{\mu}G^{1/2})-(\partial_{\mu}G^{1/2})B_{\nu}\right)G^{-1/2}$
		$\displaystyle+\dfrac{i}{g}\left((\partial_{\nu}G^{1/2})G^{-1/2}(\partial_{\mu}G^{1/2})-(\partial_{\mu}G^{1/2})G^{-1/2}(\partial_{\nu}G^{1/2})\right)G^{-1/2}$
		$\displaystyle=G^{1/2}\left(\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}\right)G^{-1/2}+G^{1/2}\left\{\left[G^{-1/2}(\partial_{\nu}G^{1/2}),B_{\mu}\right]-\left[G^{-1/2}(\partial_{\mu}G^{1/2}),B_{\nu}\right]\right\}G^{-1/2}$
		$\displaystyle+\dfrac{i}{g}G^{1/2}\left[G^{-1/2}(\partial_{\nu}G^{1/2}),G^{-1/2}(\partial_{\mu}G^{1/2})\right]G^{-1/2}.$		(54)

This equation shows the additional terms that come from the non-vanishing commutators due to the non-Abelian group structure. By this expression, we deduce that a term can be added to $\partial_{\nu}B_{\mu}-\partial_{\mu}B_{\nu}$ to modify $F_{\mu\nu}$ to achieve the desired transformation property we require. With this inspiration, the observed electromagnetic field strength tensor can be modified to read,

$$F_{\mu\nu}=\dfrac{1}{iq}\left[D_{\nu},D_{\mu}\right].$$

(55)

Substituting the definition for $D_{\mu}$ in Eq.(29) into the above expression, we obtain

$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+iq\left[A_{\nu},A_{\mu}\right].$$

(56)

The commutator vanishes for Abelian theories. Applying this transformation to Eq.(50) yields,

$$F^{\prime}_{\mu\nu}=\partial_{\nu}B^{\prime}_{\mu}-\partial_{\mu}B^{\prime}_{\nu}+ig\left[B^{\prime}_{\nu},B^{\prime}_{\mu}\right].$$

(57)

Expanding the commutator using the transformation Eq.(V.2) to enable us to compare the outcome to the nonvanishing commutator relations in Eq.(V.2) leads to,

	$\displaystyle ig\left[B^{\prime}_{\nu},B^{\prime}_{\mu}\right]$	$\displaystyle=ig\left[\left(G^{1/2}B_{\nu}G^{-1/2}+\dfrac{i}{g}(\partial_{\nu}G^{1/2})G^{-1/2}\right),\left(G^{1/2}B_{\mu}G^{-1/2}+\dfrac{i}{g}(\partial_{\mu}G^{1/2})G^{-1/2}\right)\right]$
		$\displaystyle=igG^{1/2}\left[B_{\nu},B_{\mu}\right]G^{-1/2}-G^{1/2}\left\{\left[G^{-1/2}(\partial_{\nu}G^{1/2}),B_{\mu}\right]-\left[G^{-1/2}(\partial_{\mu}G^{1/2}),B_{\nu}\right]\right\}G^{-1/2}$
		$\displaystyle-\dfrac{i}{g}G^{1/2}\left[G^{-1/2}(\partial_{\nu}G^{1/2}),G^{-1/2}(\partial_{\mu}G^{/2})\right]G^{-1/2}.$		(58)

The commutator relations that come after the first term in the last step are the exact terms required to cancel the extra terms in Eq.(V.2). Therefore, the field strength tensor expressed in Eq.(57) has the required structure under local gauge transformation. Combining Eqs.(44) and (49) gives rise to modified Yang-Mills Lagrangian Quigg ; Yang ,

$$\mathcal{L}_{\text{YM}}=\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}+g\gamma^{\mu}B_{\mu}-mG\right)\psi-\dfrac{1}{2}tr\left(F_{\mu\nu}F^{\mu\nu}\right),$$

(59)

where $M(r)=mG(\phi)$ is the modified nucleon mass. This Lagrangian is also invariant under local gauge transformations and does not permit the existence of mass term $M^{2}(r)B_{\mu}B^{\mu}$. It should also be noted that introducing the non-Abelian gauge leads to the automatic cancellation of the color dielectric function. Hence, we can infer that the color dielectric function attached to the Abelian gauge induces strong interaction properties. Using the expression in Eq.(41) and the commutator relation in Eq.(47), we can rewrite the transformed version of Eq.(57) as

	$\displaystyle F_{\mu\nu}$	$\displaystyle=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}+ig\left[B_{\mu},B_{\nu}\right]\rightarrow$
	$\displaystyle F^{l}_{\mu\nu}$	$\displaystyle=\dfrac{1}{2}\left(\partial_{\mu}(\tau^{l}b^{l}_{\nu})-\partial_{\nu}(\tau^{l}b^{l}_{\mu})\right)+\dfrac{ig}{4}\left(-2i\varepsilon_{jkl}(b^{j}_{\nu}b^{k}_{\mu}\tau^{l})\right)$
		$\displaystyle=\partial_{\mu}b^{l}_{\nu}-\partial_{\nu}b^{l}_{\mu}+g\varepsilon_{jkl}b^{j}_{\nu}b^{k}_{\mu},$		(60)

in the last step we have dropped the three isospin generators $\tau^{l}$ of the gauge field because they are linearly independent. Generally, non-Abelian gauge groups that do not fall under $\text{SU}(2)$, the Levi-Civitá symbol $\varepsilon_{jkl}$ is replaced with the antisymmetric structure constant $f_{jkl}$.