We consider Kondo tunneling through a junction composed of two semi infinite carbon nanotubes (CNT) that serve as left and right leads (CNTL and CNTR, respectively) attached on both sides of a short CNT quantum dot with an atom A having an s-wave valence electron of spin 1/2 implanted on its axis (CNTQDA). The two wave numbers (valleys) K and K' (located on the two corners of the hexagonal Brillouin zone of the CNT) serve as two symmetry protected  flavor quantum numbers K and K'. The CNTQDA is gated such that its (neutral) ground state consists of the caged atom with spin up or down while its lowest excited (charged) states are singlet and triplet states. The Anderson model hybridizes lead and dot electrons with the same flavor and spin projection, and the Schrieffer-Wolf transformation, while mixing spin projections does not mix flavors, thereby realizing a two-channel Kondo physics.

Employing the poor man's scaling technique to the Kondo Hamiltonian, it is shown that when the ultraviolet cut off energy exceeds the Fermi energy measured from the bottom of the conduction band, there are two different regimes of renormalization depending either the effective bandwidth is above or below its critical value (equal to the Fermi energy). As a result, the effective couplings can be nonmonotonic functions of the effective bandwidth D. When D goes to zero, the effective coupling j corresponding to the spin-flipping exchange interaction renormalizes to the two-channel weak-coupling fixed point value j^*=1/2. The scaling invariant T_K (the Kondo temperature) changes in between 0.5 K and 5 K for reasonable parameter values. This behavior is unexpected, since in the standard two-channel Kondo model, the exchange coupling changes monotonically with D approaching j^* for D going to zero.

The non-monotonic behavior of the coupling results in the non-monotonic behavior of the conductance G as a function of the temperature T. In the standard two-channel Kondo effect (2CKE), G(T) is monotonic, depending on the bare value j₀ of j. If j₀<j^*, (j₀>j^*), the conductance increases (decreases) monotonically with reducing T. Non-monotonicity of G(T) exposed here is the result of the crossover between different scaling regimes. One of the paradigms of the 2CKE is that the physics related to over-screening  is exposed only in the strong coupling regime, where T<T_K. In this work we have demonstrated that the some physical phenomena related to over-screening can be exposed also in the weak coupling regime, where T>>T_K.