Felipe J. Llanes-Estrada^,?, Alexandre Salas-Bernárdez^?Depto. Física Teórica, Universidad Complutense de Madrid, Plaza de las Ciencias 1, 28040 Madrid, Spain

Corresponding authors: ? E-mail:fllanes@fis.ucm.es

Received:2018-12-25Online:2019-04-1


Abstract
We reexamine the dynamical generation of mass for fermions charged under various Lie groups with equal charge and mass at a high Grand Unification scale, extending the Renormalization Group Equations in the perturbative regime to two loops and matching to the Dyson-Schwinger Equations in the strong coupling regime.
Keywords： standard model group;running fermion mass;grand unification scale


PDF (1389KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite
Cite this article
Felipe J. Llanes-Estrada, Alexandre Salas-Bernárdez. Chiral Symmetry Breaking for Fermions Charged under Large Lie Groups ^*. [J], 2019, 71(4): 410-416 doi:10.1088/0253-6102/71/4/410

1 Introduction

The Standard Model is a gauge field theory based on the gauged symmetry

(1)$ SU(3)_C\otimes SU(2)_L\otimes U(1)_Y\;. $

Here SU(3)$_C$ denotes the color interaction responsible for the strong force, SU(2)$_L$ the isospin coupling of left-handed fermions and U(1)$_Y$ the hypercharge group. The spontaneous breaking of the electroweak symmetry by the Higgs mechanism suggested the possibility of higher symmetries at yet higher scales that would also be spontaneously broken, providing strong and electroweak force unification at higher scales; these symmetries would also have to be spontaneously broken. ^?( ? It is usually and superficially stated that the gauge symmetry SU$(2)_L\times U(1)_Y$ is spontaneously broken; this, however cannot be, as dictated by Elitzur's theorem.^[1] Gauge symmetries must be broken explicitly by a gauge fixing term leaving only the global symmetry and then this remaining symmetry can be spontaneously broken. The modern viewpoint is that gauge symmetries are just a redundancy in the description of the theory on which expectation values of observables must not depend. The actual symmetry from which consecuences such as degeneracies in the spectrum, couplings or conserved currents appear is the true global symmetry. We will continue using "spontaneous symmetry breaking'' without specifying, though in the understanding that it is the global group, which is affected.)

In the SM, the Higgs vacuum expectation value breaks the global symmetry SU(2)$_L\times U(1)_Y$ of the Higgs sector in the SM down to U(1)$_{\rm em}$^[2] (or, considering the U(1) as a perturbation, and the approximate global custodial SU(2)$_c$, it breaks SU$(2)\times SU(2)\to SU(2)_c$). This generates masses for the $W^\pm$ and $Z$ bosons, and for fermions, leaving us the symmetry

(2)$ SU(3)_C\otimes U(1)_{\rm em}\,, $

(and the approximate custodial SU$(2)_c$). A feature of the symmetry group of the Standard Model that stands out is the small size of the numbers 1-2-3. Why are we confronted by such symmetry groups? Why not larger groups like SU(6) or Sp(10)?

To address these question we study in Sec. 2, how hypothetical quarks colored under different groups acquire masses from a Grand Unified Theory (GUT) scale where all groups under consideration are chosen to have the same couplings and quark masses, down to lower energies where the interaction becomes strong. For this task we will use the Renormalization Group Equations (RGE).

Then, Sec. 3 treats the Dyson-Schwinger Equations (DSE) for the lowest scales when the interactions become strong. Any workable truncation of the DSE typically fails to satisfy local gauge invariance, while respecting global symmetry. This is however enough to discuss its breaking in view of Elitzur's theorem. While realistic models^[3] that embed the SM such as SU(5) or SO(10) are often discussed, ^§(§ While the absence of proton decay rules out some classic implementations of the GUT idea, models keep being constructed that evade the constraints.^[4]) we are here less ambitious and keep the discussion at a general level, considering multiple groups.

In addition to a brief discussion in Sec. 4, the article has an appendix addressing the computation of color factors for almost all of the continuous Lie groups (results for E8 are not at hand). We have kept the article as short as is compatible with its being self-contained, since the theory behind our approach has already been laid out in a previous publication.^[5] We have striven to extend that calculation as explained next.

2 From Grand Unification to Strong Inter-action Scale with the Renormalization Group Equations

We evolve the masses of one single color-charged fermion for the different color groups from the Grand Unification scale of $\mu_{{GUT}}=10^{15}$ GeV to the point where interactions become strong (at a scale $\sigma$) for each group, that is, when $C_F\,\alpha_s(\sigma)=0.4$ (a reasonable choice corresponding to about the charmonium scale in QCD). Once this happens we use Dyson-Schwinger equations for the non-perturbative regime in order to obtain the constituent masses for these fermions: this step is explained in Sec. 3.

An efficient way of keeping track of the parameter evolution needed for the physical predictions of a theory to be invariant under $\mu$ scale choice is the use of RGEs. We generalize those of Quantum Chromodynamics to an arbitrary gauge group $G$. The running of the coupling constant $g_s$ with $\mu$ is^[6] determined by the $\beta(g_s)$ function

(3)$ \beta(g_s)\equiv -\mu\frac{d g_s}{d\mu}=\beta_1g_s^3+\beta_2 g_s^5+\cdots $

The one-loop correction $\beta_1$ is

(4)$ \beta_1=\frac{1}{(4\pi)^2}\Big(\frac{11C_G-2T_RN_f}{3}\Big)\;, $

where $C_G$ is the adjoint Casimir,(For SU$(N)$, $C_G=N$, the group dimension. But in general, $C_G=aN+b$ with $a$, $b$ depending on the particular group, as listed in the appendix. This detail was in error in Ref. ^[5] and is being corrected.) $T_R$ the normalization of the generators $T^a$ of the group $G$ defined as $Tr(T^aT^b)\equiv T_R\delta^{ab}$ and $N_f$ the number of colored fermions.(In this article we take $N_f=1$, but a brief discussion in Ref. ^[5] reminds us that there is a critical number of colors $N_f^c$ that shuts off the vacuum antiscreening and thwarts spontaneous symmetry breaking.)

The two-loop contribution to the $\beta(g_s)$ function, $\beta_2$, entails a larger effort in perturbation theory, but can also be easily found in Ref. ^[6]

(5)$ \beta_2=\frac{1}{(4\pi)^4}\Big(\frac{34}{3}C_G^2-4\Big(\frac{5}{3}C_G+C_F\Big)T_RN_f\Big)\;, $

where $C_F$ is the Casimir of the fundamental representation (see Appendix). Using the color coefficients listed there, we obtain the running couplings of SU$(N)$, SO$(N)$, Sp$(N)$ and the exceptional groups G2, F4, E6, and E7, shown in Fig. 1.

Fig. 1

New window|Download| PPT slide
Fig. 1Running couplings for the families SU$(N)$, SO$(N)$, Sp$(N)$ and most of the Exceptional Lie groups.



The result of Ref. ^[5], that for small groups and one flavor $\sigma \propto e^{N}$ stands out. The very large groups have strongly interacting scales $\sigma$ clustering around the GUT scale, since they run very fast. We are then ready to start employing the DSEs down from the scale $\sigma$.

Simultaneously, running of the current mass $m_c$ is set by the self energy correction to the quark propagator that implies an anomalous mass dimension $\gamma_m$

(6)$ \gamma_m(g_s)\equiv -\frac{\mu}{m}\frac{d m}{d\mu}=\gamma_1g_s^2+\gamma_2 g_s^4+\cdots $

The one loop contribution to the $\gamma_2(g_s)$ function for the quarks, $\gamma_1$, amounts to

(7)$ \gamma_1=\frac{6C_F}{(4\pi)^2}\;. $

The two-loop contribution to $\gamma_m(g_s)$ (see Ref. ^[7]), $\gamma_2$, is

(8)$ \gamma_2=\frac{C_F}{(4\pi)^4}\Big(3C_F+\frac{97}{3}C_G-\frac{20}{3}T_R N_f\Big)\;. $

At the GUT starting scale of the RGEs we choose a fermion mass $m_{c}(\mu_{{GUT}})=1$ MeV and fix the coupling $\alpha_{s}(\mu_{{GUT}})\equiv g_s({\mu_{{GUT}}})^2/4\pi=0.0165$ to broadly reproduce the isospin average mass for the SU(3)$_C$ quarks of the first generation at the scale $\mu=2$ GeV

(9)$ \bar{m}(2~{\rm GeV})=\frac{m_u(2~{\rm GeV})+m_d(2~{\rm GeV})}{2}\simeq3.5~{\rm MeV}\;. $

These initial conditions are taken to be the same for all Lie groups, as suggested by the concept of GUT. Then, the mass running for the various Lie groups, with color factors taken from $A$ is plotted in Fig. 2.

Fig. 2

New window|Download| PPT slide
Fig. 2Running masses for the families SU$(N)$, SO$(N)$, Sp$(N)$, and four out of five Exceptional Lie groups.

3 Running at the Strong Interaction Scale with the Dyson-Schwinger Equations

Once the interactions become strong, perturbation theory breaks down and resummation becomes necessary: we thus adopt the simplest possible DSE for the quark self energy. The free propagator of a fermion with current mass $m_c$,^[6] $ S_0(p^2)={1}/({m_c-p})$, becomes a fully dressed one $ \tilde{S}(p^2)={1}/({B(p^2)-A(p^2)p})$. Being only interested in qualitative features of spontaneous mass generation, we can approximate $A(p^2)=1$ which leaves the physical mass as $M(p^2)\equiv B(p^2)$. Denoting $\Sigma(p)$ as the sum of all one-particle irreducible diagrams, the DSE takes the form (omitting the $p$ dependence)

(10)$ \tilde{S}(p^2)=S_0(p^2)\,(1-\Sigma(p) S_0(p^2))^{-1}\;. $

Inverting, we see that

$$\tilde{S}^{-1}(p^2)=S_0(p^2)^{-1}-\Sigma(p) \Rightarrow M(p^2)=m_c-\Sigma(p)\,. $$

To illustrate the possibilities, we will employ the \textit{rainbow truncation} that sums only "rainbow shaped'' diagrams, with great simplification (Fig. 3).

Fig. 3

New window|Download| PPT slide
Fig. 3Rainbow DSE for the full quark propagator (filled circle).



The one-loop self energy is then, passing to Euclidean space with $k^0\to i k^0$, $p^0\to i p^0$, given by

(11)$ \Sigma_{\rm rainbow}(p) =g^2_s\int\frac{d^4k}{i(2\pi)^4}\gamma^\mu(T^a) \qquad \times \frac{1}{M(k^2)-k}\gamma^\nu(T^a) \frac{\eta_{\mu\nu}}{(k-p)^2} \quad = C_Fg^2_s\int_0^\infty\frac{d k_{ E} k_{ E}^3}{\pi^3}\frac{-M(k^2)}{M^2(k^2)+k_{ E}^2} \qquad \times\int_{-1}^{+1}\sqrt{1-x^2}\frac{d x}{(k_{ E}^2-2|k_{ E}||p_{ E}|x+p^2_E)}\;. $

We define the last integral in $x$ as the averaged gluon propagator $D^0_{k-p}$ (in the Feynman Gauge) over the four-dimensional polar angle. Hence, we conclude that the Dyson-Schwinger equation in the rainbow approximation for the quark propagator is

(16)$ M(p^2)=m_c+C_Fg^2_s\int_0^\infty\frac{d q q^3}{\pi^3}\frac{M(q^2)}{M^2(q^2)+q^2}D^0_{q-p}\;. $

Note that the integral in Eq. (12) is divergent and must be regularized. We could employ a simple cutoff regularization cutting this integral at a scale $\Lambda$; instead we would like to preserve Lorentz invariance and exhibit renormalizability. Following again,^[5] we introduce renormalization constants $Z(\Lambda^2,\mu^2)$ to absorb infinities and any dependence on the cutoff $\Lambda$,

(13)$ \tilde{S}^{-1}(p^2,\mu^2)\equiv Z_2 S_0^{-1}(p^2)-\Sigma(p^2,\mu^2)\;, $

where the dependence of $\Sigma$ on $\mu$ is given by the fermion and gluon propagators. Apart from the wavefunction renormalization $Z_2$ we introduce $Z_m$ for the bare quark mass. The relation between the (cutoff dependent) unrenormalized mass $m_c(\Lambda^2)$ and the renormalized mass at the renormalization scale $\mu$, $m_{ R}(\mu^2)$, is

(14)$ m_c(\Lambda^2)=Z_m(\Lambda^2,\mu^2)m_{ R}(\mu^2)\,. $

Since we will maintain the restriction $A(p^2)=1$, renormalization of the quark wavefunction is not necessary, therefore $Z_2=1$. The only renormalization condition is to fix the mass function at $p^2=\mu^2$. The DSE is then

(15)$ M(p^2)=Z_m m_{ R}(\mu^2)-\Sigma(p^2,\mu^2)\,. $

Evaluating Eq. (15) at $p^2=\mu^2$ and subtracting it again to Eq. (15) we obtain

(16)$ M(p^2)=M(\mu^2) +C_Fg^2_s\int_0^\infty\frac{d q q^3}{\pi^3}\frac{M(q^2)}{M^2(q^2)+q^2} \\\ \times(D^0_{q-p}-D^0_{q-\mu})\;. $

It is easy to show, taking $\mu$ parallel to $p$, that asympto\-tically^[5]

(17)$ \frac{\partial M(p^2;\Lambda)}{\partial \Lambda}\propto\frac{M((p-\mu)^2; \Lambda)}{\Lambda^2}\;. $

Therefore, for large $\Lambda$, $M(p^2)$ stops depending on the cutoff, which can be taken e.g. to $\Lambda=10^{10}$ GeV and renormalization is achieved.

Now we are ready to obtain the quark constituent masses for all the groups studied. We match the RGE solution (high scales) to the DSE solution (low scales) at the matching energy $\sigma$ where interactions become strong, $C_F\alpha_s(\sigma)=0.4$ for each group, as advertised.

For SU(3) ($C_F={4}/{3}$), the scale where $\alpha_s(\sigma)=0.3$ is $\sigma=2.09$ GeV. From this point down in scale we freeze $\alpha_s$. A constant vertex factor of order 7 is applied to the DSE to guarantee sufficient chiral symmetry breaking at low scales, requiring the constituent quark mass $M(0)$ to be close to 300~MeV using the substracted DSE (16). This is supposed to mock up the effect of vertex-corrections not included, and is known to scale with $N$^[8] for large $N$, the group's fundamental dimension.

Finally, the $M(p^2)$ obtained is plotted in Fig. 4.

Fig. 4

New window|Download| PPT slide
Fig. 4Matching of RGE and DSE solutions of the Mass Running for SU(3).



To obtain the constituent fermion masses for the different Lie Groups we use a trick presented in Ref. ^[5]: to perform a scale transformation

(18)$ p^2\to \lambda^2p^2 \,,\quad \sigma^2\to \lambda^2\sigma^2\,, $

on the DSE (16). Changing the integration variable $q^2\to \lambda^2q^2$, giving $d^4q\to \lambda^4d^4q$,

the modified DSE equation is satisfied by a modified $\tilde{M}$ and the relation between the constituent masses is simply $ M(0)={\tilde{M}(0)}/{\lambda}$.

Now, taking $\lambda$ as the ratio of the scales where interactions become strong for SU(3) and another group,

(19)$ \frac{\sigma_{\rm group}}{\sigma_{{SU(3)}}}=\lambda\;, $

the mass function scales in the same way

(20)$ \frac{M_{\rm group}(0)}{M_{\rm SU(3)}(0)}=\lambda\;. $

Hence, eliminating the auxiliary $\lambda$, we find

(21)$ \frac{M_{\rm group}(0)}{M_{\rm SU(3)}(0)}=\frac{\sigma_{\rm group}}{\sigma_{{SU(3)}}}\,. $

Using these results we compute the constituent masses for the quarks charged under the different groups (Fig. 5).

Fig. 5

New window|Download| PPT slide
Fig. 5Constituent Masses for the groups, which break chiral symmetry in RGE before $10^{-5}$ GeV.



The outcome is that the special Lie groups examined do not spontaneously generate fermion mass at a high scale: their interactions, running at two loops from the GUT, are too weak to do so. This is because the $C_G$ Casimir of the adjoint representation, though proportional to the group dimension, carries a small numerical factor that reduces the intensity of coupling running.

Large SU$(N)$ and Sp$(N)$ groups, on the other hand, behave as advanced in Ref. ^[5], and generate a mass for the fermions that puts them beyond reach of past accelerators. The exceptions are Sp$(4)$, for which the mass generation is similar to QCD; and Sp(2), which again yields too weak a chiral symmetry breaking.

As for the special orthogonal groups, for SO$\;(N>10)$, once more the fermion mass generated is too large to be accessible at accelerators.

4 Conclusions and Outlook

We have examined mass generation for different Lie groups with an arbitrary number of colours. As a definite starting point, we have adopted the philosophy of Grand Unification in which fermion masses as well as coupling constants, for all groups, coincide at a high scale, namely $10^{15}$ GeV.

We have run the couplings and masses for each group to lower scales employing two-loop Renormalization Group Equations, using as an input the Cuadratic Casimirs obtained in Appendix A.

We chose the initial conditions at $\mu=\mu_{{GUT}}$ to be the same for all groups and selected so that SU(3)$_C$ at the scale of 2 GeV yields a rough approximation of the strong force coupling and first-generation isospin-averaged quark mass.

Typically, for all but the smallest groups, a scale arises where interactions become strong (discernable as a Landau pole in perturbation theory). We stop running at the scale $\mu$ such that $\alpha_s(\mu)C_F=0.4$; below that, we employ a non-perturbative treatment, namely Dyson-Schwinger Equations in the rainbow approximation to assess the masses down to yet lower scales.

Combining the methods of RGE and DSE and requiring that the constituent masses of SU(3)$_C$ colored quarks to be 300 MeV has allowed us to obtain the constituent masses of hypothetical fermions charged under different groups from a Grand Unification Scale of $10^{15}$ GeV.

From this treatment we can conclude that groups belonging to the SU$(N)$ and Sp$(N)$ families, with $N>4$, generate masses of order or above the few TeV. Notwithstanding the crude approximations we have employed, our computation gives about 5 TeV to SU(4)-charged fermions, which would not be far out of reach of mid-future experiments provided the GUT conditions apply. It appears from our simple work that larger groups (except the Exceptional Groups and SO$(N)$ with $N<10$) might endow fermions with a mass too high to make them detectable in the foreseeable future.

In case these superheavy fermions would have been coupled to the Standard Model, they would have long decayed in the early universe due to the enormous phase space available. If they existed and be decoupled from the SM, they would appear to be some form of dark matter. We have also provided a partial answer to the question "Why the symmetry group of the Standard Model, SU(3)$_C \otimes SU(2)_L \otimes U(1)_Y$, contains only small-dimensional subgroups?'' It happens that, upon equal conditions at a large Grand Unification scale, large-dimensioned groups in the classical SO$(N)$, SU$(N)$ and Sp$(N)$ families force dynamical mass generation at higher scales because their coupling runs faster. Should fermions charged under these groups exist, they would appear in the spectrum at much higher energies than hitherto

explored.^[9]

Interestingly for collider phenomenology, we find the masses of fermions charged under the following groups are within reach of the energy frontier:

$M_{\rm SU(4)}\simeq 5$ TeV; $M_{\rm Sp(6)}\simeq4.4$ TeV; $M_{\rm SO(10)}\simeq 7$ TeV.

The LHC might be able to exclude those.\footnote^[7]{For comparison, the one-loop results are

$M_{\rm SU(4)}\simeq 2$ TeV, $M_{\rm Sp(6)}\simeq1.5$ TeV, $M_{\rm SO(10)}\simeq 3$ TeV, which indicates fair convergence.

However, the groups SO$(N<10)$, $E_6$, $E_7$, $G2$, and $F4$ yield masses below the TeV scale and should already have been seen if they coupled to the rest of the Standard Model (one could argue that among these, the ones isomorphic to groups present in the SM cannot be distinguished from their SM counterparts and thus have already been accounted for). Their absence from phenomenology thus suggests that fermions charged under any of those groups, if at all existing, belong to a decoupled dark sector.

Appendix A Color Factors

We here present some of the calculations carried out to obtain the quadratic Casimirs needed in both RGE and DSE. Such quadratic Casimirs are elements in the Lie Algebra which commute with all the other elements (See Refs. ^[10-12] for the necessary group theory).

We will focus on the Casimir invariant in the fundamental representation of the group $G$, $C_F\delta^{ij}=(T^a T^a)^{ij}$, and the Casimir invariant in the adjoint representation, $C_G\delta^{ab}=f^{acd}f^{bcd}$. Normalization of the algebra generators is chosen as Tr$(T^aT^b)=({1}/{2})\delta^{ab}$.

(i) Special Unitary Groups SU$(N)$

We start with the special unitary family SU$(N)$. Its generators $T^{a}$ are traceless hermitian. Therefore every Hermitean $N\times N$ matrix $A$ can be written as,

(A1)$ A=A^\dagger=c_0\mathbb{I}+c_a T^a\,. $

From this we find

(A2)$ c_0=\frac{(A)}{N}\;\, \quad c_a=2\, (AT^a)\,. $

Having then

(A3)$ A_{ij}=A_{lk}\delta^{li}\delta^{kj}=A_{lk}\Big(2(T^a)_{ij}(T^a)^{kl}+\frac{1}{N}\delta^{kl}\delta_{ij}\Big) $

(A4)$ \hphantom{A_{ij}=} \Rightarrow A_{lk}\Big(2(T^a)_{ij}(T^a)^{kl}+\frac{1}{N}\delta^{kl}\delta_{ij}-\delta^{li}\delta^{kj}\Big)=0\;. $

Since $A$ is arbitrary, we find for the generators the useful relation

(A5)$(T^a)_{ij}(T^a)_{kl}=\frac{1}{2}\Big(\delta_{li}\delta_{kj}-\frac{1}{N}\delta_{kl}\delta_{ij}\Big)\;. $

Contracting $j$ and $k$ we obtain the fundamental representation Casimir or Color Factor

(A6)$(T^aT^a)_{ij}=\frac{1}{2}\Big(\frac{N^2-1}{N}\Big)\delta_{ij}=C_F\delta_{ij}\;. $

Now we compute the following combination,

(A7)$(T^a)_{i}^j(T^b)_{jk}(T^a)^k_{l}= \frac{1}{2}\Big((T^b)_{jk}\delta_{li}\delta^{kj}-\frac{1}{N}(T^b)_{jk}\delta^k_{l}\delta_{i}^j\Big) \\\ =-\frac{1}{2N}(T^b)_{il}\;. $

Noting the following identity and using the results already computed, we obtain the adjoint Casimir for SU$(N)$,

(A8)$ f^{acd}f^{bcd}=-2\,([T^a,T^c][T^b,T^c]) \\\ =-2 (2T^aT^cT^bT^c-(T^aT^b+T^bT^a)T^cT^c) \\\ =N\delta^{ab}=C_G\delta^{ab}\,. $

(ii) Special Orthogonal Groups SO$(N)$}

We will follow now the same steps for the Special Orthogonal family SO$(N)$. Its generators are antisymmetric and traceless and they form a basis for the antisymmetric $N\times N$ matrices. Thus, taking an antisymmetric matrix $A$, we have

(A9)$ A=-A^{\rm T}=c_a T^a\Rightarrow c_a=2 (A T^a)\;. $

Then we have

(A10)$ A_{ij}=A_{kl} \delta^k_i\delta^l_j =\frac{1}{2}A_{kl}( \delta^k_i\delta^l_j-\delta^k_j\delta^l_i) \\\ =A_{lk}(2 (T^a)_{ij}(T^a)^{kl})\;. $

Finding

(A11)$ A_{kl}\Big[ \frac{1}{2}( \delta^k_i\delta^l_j-\delta^k_j\delta^l_i)+(2 (T^a)_{ij}(T^a)^{kl})\Big]=0\;, $

and since $A$ is an arbitrary antisymmetric matrix we get

(A12)$(T^a)_{ij}(T^a)^{kl}=\frac{1}{4}(\delta^k_j\delta^l_i- \delta^k_i\delta^l_j)\;. $

Here, as the group is real, we need no distinction between upper and lower indices. Contracting in the previous expression $j$ with $k$ we obtain the Color Factor

(A13)$ (T^a)_{ij}(T^a)^{jl}=\frac{1}{4}(\delta^j_j\delta^l_i- \delta^j_i\delta^l_j)=\frac{N-1}{4}\delta_i^l=C_F\delta_i^l\;. $

As before, we compute

(A14)$ (T^a)_{ij}(T^b)^{jk}(T^a)_{kl}=\frac{1}{4}((T^b)^{jk}\delta_{il}\delta_{kj}-(T^b)^{jk}\delta_{ik}\delta_{lj}) \\\ \quad =-\frac{1}{4}(T^b)_{li}=\frac{1}{4}(T^b)_{il}\;. $

We are able now to obtain the adjoint Casimir for SO$(N)$. Similar to Eq. (A8)

(A15)$ f^{acd}f^{bcd}=2\Big(\frac{1}{2}T^aT^b-(T^aT^b+T^bT^a)\frac{N-1}{4}\Big) \\\ =\frac{1}{2}(N-2)\delta^{ab}=C_G\delta^{ab}\,. $

(iii) Simplectic Groups Sp(N)

The elements $M\in {\rm Sp}(N)$ (with $N$ even) are $N\times N$ matrices, which preserve the antisymmetric tensor

(A16)$ \Omega= \begin{pmatrix} 0 & \mathbb{I}_{{N}{2}\times{N}{2}}\\\ -\mathbb{I}_{{{N}{2}\times{N}{2}}} & 0 \end{pmatrix}, $

in the sense

(A17)$ \Omega=M^{\rm T}\Omega \,M\Rightarrow\,M^{-1}=\Omega^{\rm T}M^{\rm T}\Omega\,. $

Using this relation it is possible to prove that the generators of the group take the form

(A18)$ -T^a=\Omega^{\rm T}(T^a)^{\rm T}\Omega\;\Rightarrow\;T^a= \begin{pmatrix} A B \\ C -A^{\rm T} \end{pmatrix}, $

where $B$ and $C$ are symmetric matrices. It is now possible to show that the generators satisfy

(A19)$(T^a)_{ij}(T^a)_{kl}=\frac{1}{4}(\delta_{il}\delta_{jk}+\Omega_{ik}(\Omega^{-1})_{jl})\;. $

Therefore

(A20)$(T^a)_{ij}(T^a)^j_{l}=\frac{1}{4}(N+1)\delta_{ij}=C_F\delta_{ij}\;. $

Noticing $\Omega^{\rm T}=\Omega^{-1}=-\Omega$, we compute the usual combination

(A21)$ (T^a)_{ij}(T^b)^{jk}(T^a)_{kl} =\frac{1}{4}(\delta_{il}(T^b)^{jk}\delta_{jk}+\Omega_{ik}(T^b)^{jk}(\Omega^{\rm T})_{jl}) \quad =\frac{1}{4}(\Omega^{\rm T})_{ik}((T^b)^{\rm T})^{kj}\Omega_{jl}=-\frac{1}{4}(T^b)_{il}\;, $

where in the last equality we have used Eq. (A18). The adjoint Casimir now falls down easily

(A22)$ f^{acd}f^{bcd} =-2\Big(-\frac{1}{2}T^aT^b-(T^aT^b+T^bT^a)\frac{N+1}{4}\Big) \\\ \quad =\frac{1}{2}(N+2)\delta^{ab}=C_G\delta^{ab}\,. $

To obtain the Color Factors and adjoint Casimirs for G2, F4, E6, and E7 we refer to $P$. Cvitanovi\'c's.^[10] The results obtained are presented in Table 1.


Table 1
Table 1Cuadratic casimirs for different Lie group.

New window|CSV

The authors have declared that no competing interests exist.

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1] S. Elitzur , Phys. Rev. D 12 (1975) 3978.
DOI:10.1103/PhysRevD.12.3978URL [Cited within: 1]
It is argued that a spontaneous breaking of local symmetry for a symmetrical gauge theory without gauge fixing is impossible. The argument is demonstrated in a simple system of Abelian gauge fields on a lattice. (AIP)

[2] S. Willenbrock , Symmetries of the Standerd Model, Physics in $D = 4$ Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics 2004, Boulder, USA, June 6-July 2, (2004), p. 3-38, World Scientific, Hackensack, USA, hep-ph/0410370..
[Cited within: 1]

[3] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32(1974) 438, doi: 10.1103/PhysRevLett.32.438.
URL [Cited within: 1]
Strong, electromagnetic, and weak forces are conjectured to arise from a single fundamental interaction based on the gauge group SU(5).

[4] B. Fornal and B. Grinstein , arXiv: 1808. 00953 [hep-ph].
[Cited within: 1]

[5] G. García Fernández, J. Guerrero Rojas, F. J. Llanes-Estrada , Nucl. Phys. B 915 (2017) 262, doi: https://ctp.itp.ac.cn/article/2019/0253-6102/10.1016/j.nuclphysb.2016.12.010, [arXiv:1507.08143 [hep-ph]]. ^[6] T. Muta, Foundations of Quantum Chromodynamics, World Scientific Lecture Notes in Physics, Vol. 57, World Scientific, New Jersey (1984).
URL [Cited within: 8]
The result follows from strong antiscreening of the running coupling for those larger groups (with an appropriately small number of flavors) together with scaling properties of the Dyson–Schwinger equation for the fermion mass.

[7] R. Tarrach , Nucl. Phys. B 183 (1981) 384, doi: https://ctp.itp.ac.cn/article/2019/0253-6102/10.1016/%200550-3213(81)90140-1.
[Cited within: 2]

[8] R. Alkofer, et al., Annals Phys. 324(2009) 106, doi: https://ctp.itp.ac.cn/article/2019/0253-6102/10.1016/j.aop.2008.07.001; A. Kizilersu, it et al., Eur. Phys. J. C 50 (2007) 871, doi https://ctp.itp.ac.cn/article/2019/0253-6102/10.1140/epjc/s10052-007-0250-6; A. C. Aguilar, it et al., Phys. Rev. D 90 (2014) 065027, doi https://ctp.itp.ac.cn/article/2019/0253-6102/10.1103/PhysRevD.90.065027.
URL [Cited within: 1]
Abstract The infrared behavior of the quark–gluon vertex of quenched Landau gauge QCD is studied by analyzing its Dyson–Schwinger equation. Building on previously obtained results for Green functions in the Yang–Mills sector, we analytically derive the existence of power-law infrared singularities for this vertex. We establish that dynamical chiral symmetry breaking leads to the self-consistent generation of components of the quark–gluon vertex forbidden when chiral symmetry is forced to stay in the Wigner–Weyl mode. In the latter case the running strong coupling assumes an infrared fixed point. If chiral symmetry is broken, either dynamically or explicitly, the running coupling is infrared divergent. Based on a truncation for the quark–gluon vertex Dyson–Schwinger equation which respects the analytically determined infrared behavior, numerical results for the coupled system of the quark propagator and vertex Dyson–Schwinger equation are presented. The resulting quark mass function as well as the vertex function show only a very weak dependence on the current quark mass in the deep infrared. From this we infer by an analysis of the quark–quark scattering kernel a linearly rising quark potential with an almost mass independent string tension in the case of broken chiral symmetry. Enforcing chiral symmetry does lead to a Coulomb type potential. Therefore, we conclude that chiral symmetry breaking and confinement are closely related. Furthermore, we discuss aspects of confinement as the absence of long-range van der Waals forces and Casimir scaling. An examination of experimental data for quarkonia provides further evidence for the viability of the presented mechanism for quark confinement in the Landau gauge.

[9] This Work was Presented at the Odense $CP^3$ Origin of Mass at the High Intensity Frontier Conference in May 2018, .
URL [Cited within: 1]

[10] P. Cvitanovic , Phys. Rev. D 14 (1976) 1536, doi: https://ctp.itp.ac.cn/article/2019/0253-6102/10.1103/PhysRevD.14.1536.
URL [Cited within: 2]
A simple and systematic method for the calculation of group-theoretic weights associated with Feynman diagrams in non-Abelian gauge theories is presented. Both classical and exceptional groups are discussed.

[11] A. Zee , Group Theory in a Nutshell for Physicists, Princeton University Press, New Jersey( 2016).


[12] F. Wilzeck and A. Zee , Phys. Rev. D 25 (1982) 553.
DOI:10.1103/PhysRevD.25.553URL [Cited within: 1]