Spin-isospin Kondo effects for $\Sigma_{c}$ and $\Sigma_{c}^{\ast}$ baryons and $\bar{D}$ and $\bar{D}^{\ast}$ mesons

We study the Kondo effect for a $\Sigma_{c}$ ($\Sigma_{c}^{\ast}$) baryon in nuclear matter. In terms of the spin and isospin ($\mathrm{SU}(2)_{\mathrm{spin}} \times \mathrm{SU}(2)_{\mathrm{isospin}}$) symmetry, the heavy-quark spin symmetry and the S-wave interaction, we provide the general form of the Lagrangian for a $\Sigma_{c}$ ($\Sigma_{c}^{\ast}$) baryon and a nucleon. We analyze the renormalization equation at one-loop level, and find that the coexistence of spin exchange and isospin exchange magnifies the Kondo effect in comparison with the case where the spin-exchange interaction and the isospin-exchange interaction exist separately. We demonstrate that the solution exists for the ideal sets of the coupling constants, including the $\mathrm{SU}(4)$ symmetry as an extension of the $\mathrm{SU}(2)_{\mathrm{spin}} \times \mathrm{SU}(2)_{\mathrm{isospin}}$ symmetry. We also conduct a similar analysis for the Kondo effect of a $\bar{D}$ ($\bar{D}^{\ast}$) meson in nuclear matter. On the basis of the obtained result, we conjecture that there could exist a"mapping"from the heavy meson (baryon) in vacuum onto the heavy baryon (meson) in nuclear matter.

strange) quarks. Several heavy hadrons have been used in the previous studies: aD meson (D − orD 0 meson) and ā D * meson (D * − orD * 0 meson) [41,42], or a D − s meson and a D * − s meson [43], in charm flavor. It is certainly true that the heavy hadrons are not stable, because they can decay into the light hadrons via the weak interaction. Nevertheless, it is worth considering the heavy hadrons in the nuclear matter when we only consider the strong interaction or the electromagnetic interaction. The heavy hadrons may be produced in atomic nuclei experimentally at the high-energy accelerator facilities. Clearly, the conditions (i) and (ii) for the Kondo effect are met; the Fermi surface and the particle-hole creations exist in the nuclear matter at the low temperatures. When it comes to the condition (iii), the non-Abelian interaction is provided by the spin-exchange interaction and/or by the isospin-exchange interaction, both of which obey the SU(2) spin symmetry and/or the SU(2) isospin symmetry, respectively. The research on the Kondo effect for theD andD * mesons and the D − s and D * − s mesons in nuclear matter was conducted by using the perturbative calculation [41] and the mean-field approximation [43]. The Kondo effect for the heavy hadron in atomic nuclei was studied in terms of the mean-field approximation in the Lipkin model, in which the fluctuation effect was also considered [42].
From a QCD perspective, it is noteworthy that the Kondo effect was also studied for a charm or bottom quark in quark matter, where the non-Abelian interaction between the heavy quark and the itinerant light quark is provided by the color-exchange interaction in accordance with the SU(3) color symmetry [41,44]. This is called the QCD Kondo effect [44]. The QCD Kondo effect was studied in various theoretical methods: the simple perturbation [41], the (perturbative) renormalization group with gluon exchange [44], the mean-field approximation [45][46][47][48], the conformal boundary theory [49,50]. The competition between the QCD Kondo effect and the color superconductivity or the chiral condensate was analyzed [51,52]. In addition, the transport properties such as the electric conductivity and the shear viscosity were studied [48]. It is important to mention that the QCD Kondo effect in the quark matter with the light flavor N f ≥ 2 serves the overscreened Kondo effect instead of the normal Kondo effect with an exact screening, and it leads to the non-Fermi liquid behavior [49][50][51]. The heavy quark in strong magnetic field induces the QCD Kondo effect at the vanishing density (the magnetically-induced QCD Kondo effect), where the light quarks are confined with degeneracy in the lowest Landau level [53]. It was recently argued that the QCD Kondo effect occurs even in the absence of the heavy quark in quark matter: the color non-singlet gap in the two-flavor superconductivity (2SC) plays the role of the "heavy impurity", and it leads to the non-Abelian interaction with the light quarks which do not participate to form the 2SC gap [54].
The purpose of the present paper is to study the Kondo effect for a Σ c (Σ * c ) baryon in nuclear matter. The Σ c (Σ * c ) baryon has spin 1/2 (3/2) and isospin 1, and it can provide the non-Abelian interaction by the spin and isospinexchange with a nucleon. We consider the heavy mass limit for the heavy quark (a charm quark) [55][56][57], where the spin-flip and isospin-flip interactions work on the light component in the Σ c (Σ * c ) baryon, i.e., the light diquark (qq). Indeed, the spin-flip process for the heavy quark is suppressed by the factor Λ QCD /m Q with Λ QCD being the lowenergy scale of the QCD and m Q being the mass of the heavy quark. Thus, the spin of a heavy quark can be regarded as the conserved quantity in the heavy-quark mass limit. This is called the heavy-quark spin (HQS) symmetry [55][56][57] (see also Refs. [58]). In the present study, we consider only the leading-order term in the heavy-quark mass limit, and neglect the corrections at O(Λ QCD /m Q ). For example, the heavy quark symmetry is seen approximately in the small mass splitting between a Σ c baryon and a Σ * c baryon (about 65 MeV) which is much smaller than the baryon masses (2286 MeV and 2520 MeV). The HQS will provide us with a good approximation as the first step to investigate the Kondo effect for the Σ c (Σ * c ) baryon. The effective theory of the Σ c (Σ * c ) baryon can be constructed in a general form when we follow the HQS symmetry [59][60][61][62][63][64] (see also Refs. [58,65] for reviews), and this formalism will be applied to the interaction between a Σ c (Σ * c ) baryon and a nucleon. Given the fact that the Σ c (Σ * c ) baryon has two different non-Abelian interactions of spin and isospin, i.e., the SU(2) spin × SU(2) isospin symmetry, we will see that those two symmetries induce rich structures of the Kondo effect. As an ideal situation, for example, the SU(2) spin × SU(2) isospin symmetry will provide the SU(4) symmetry by tuning the coupling constants in the interaction term appropriately.
Throughout the present study, we will perform the analysis based on the renormalization group (RG) equation, namely the poor man's scaling method, as the simple perturbative method [5].
Several comments are in order. In the literature, the binding of a Σ c (Σ * c ) baryon in nuclear matter was estimated by the QCD sum rules [66,67]. The present discussion about the Kondo effect will be useful for further investigation on the binding energy. We notice that a Λ c baryon is not relevant to the Kondo effect in contrast to the Σ c (Σ * c ) baryon, because the light diquark (qq) in the Λ c baryon has spin 0 and isospin 0, and there is no exchange interaction of spin and isospin between the baryon and a nucleon, as it was analyzed in Ref. [68] (see also the recent work [69,70]). 2 Bottom hadrons, which are in general heavier than charm hadrons, could be more suitable for studying the Kondo effect; however, we will not repeat the same discussion for the bottom hadrons. Replacing a Σ c (Σ * c ) baryon by a Σ b (Σ * b ) baryon is a straightforward task, although it would provide more favorable conditions for greater accuracy of the HQS symmetry. The greater accuracy is seen directly in the mass splitting between a Σ b baryon and a Σ * b baryon (about 20 MeV) in comparison to their masses (5810 MeV and 5830 MeV, respectively).
The paper is organized as follows. In Sec. II, we introduce the Lagrangian for a Σ c (Σ * c ) baryon and a nucleon. We suppose the SU(2) spin × SU(2) isospin symmetry, the HQS symmetry, and the S-wave interaction. In Sec. III, we carefully investigate the solutions of the RG equation, and point out that the simultaneous flipping of the spin and the isospin is important for magnifying the Kondo effect. In Sec. IV, we revisit the Kondo effect for aD (D * ) meson in nuclear matter, where the similar analysis is applicable. In Sec. V, we surmise that the Kondo effect induces a mapping between the heavy meson (baryon) in vacuum and the heavy baryon (meson) in nuclear matter. The final section is devoted to the conclusion.

II. LAGRANGIAN FOR A Σc (Σ * c ) BARYON AND A NUCLEON
We begin by considering the nuclear matter in which a Σ c (Σ * c ) baryon exists as an impurity particle, assuming that the nuclear matter is approximately regarded as the free fermion gas where the nucleon is described by the nonrelativistic two-component spinor field ϕ(x). We follow the procedures for the construction of the field of the heavy hadron based on the HQS symmetry [59][60][61][62][63][64] (see also Refs. [58,65] for reviews), and apply this formalism to the interaction between a Σ c (Σ * c ) baryon and a nucleon. In this framework, the field of the Σ c (Σ * c ) baryon can be decomposed to the diquark part (qq) and the heavy quark part (Q), where the quantum number of the diquark is spin one and isospin one. We introduce the vector field A µ (x) (µ = 0, 1, 2, 3), which satisfies v µ A µ = 0, for the diquark part. We also introduce the effective heavy-quark field u v (x), which satisfies v ν γ ν u v = u v , for the heavy quark part. 2 Those studies rely on the ΛcN interaction strength estimated by the lattice QCD simulations [71] and the chiral extrapolations [72].
The obtained binding energy for a Λc baryon is consistent with the results by the QCD sum rules [73].
and the heavy quark mass m Q , where u(x) is the original four-spinor heavy-quark field at x in the 4-dimensional coordinate system. We consider that the sum is taken over the repeated indices. The condition v ν γ ν u v = u v stems from the requirement to project the field u(x) to the positive-energy part. It is supposed that the heavy quark is at rest in the coordinate frame with the four-velocity v µ . In the following discussion, we consider the static frame by setting v µ = (1, 0). With this setup, we define the composite field for the Σ c (Σ * c ) baryon: Notice that Ψ µ v has only the off-mass-shell (residual) energy-momentum component with the energy scale smaller than the heavy-baryon mass, because the Σ c (Σ * c ) baryon is supposed to be at rest in the v-frame. We also notice that Ψ µ The former and latter properties are induced by v ν γ ν u v = u v and v µ A µ = 0, respectively. With those two conditions, the number of degrees of freedom in Ψ µ v is 3 × 2 = 6. In the above construction, Ψ µ v is a superposed state of the Σ c baryon (spin 1/2) and the Σ * c baryon (spin 3/2). This reflects the concept that the spin of the diquark and the spin of the heavy quark are good quantum numbers in the heavy-quark symmetry, and that the Σ c baryon and the Σ * c baryon can be superposed. In the physical space, it is convenient to introduce the fields of Σ c baryon and Σ * c baryon by projecting Ψ µ v to the Σ c baryon component and the Σ * c baryon component: for the Σ c baryon and for the Σ * c baryon. Equivalently, Ψ µ v is expressed as a sum of Ψ v1/2 and Ψ v3/2 , In the HQS formalism, the Σ c baryon and the Σ * c baryon are degenerate in mass and are interchangeable to each other by the HQS symmetry. For this reason, it is essential to consider a Σ c baryon and a Σ * c baryon to be the effective degrees of freedom. We will see that the heavy-quark-spin symmetry induces the mixing between the Σ c N state and the Σ * c N state (N for a nucleon) in the nuclear matter. With the above setup, we consider the Lagrangian in the case where a nucleon and a Σ c (Σ * c ) baryon interact with each other through the S-wave interaction on low-energy scale. The Σ c N (Σ * c N ) interaction was considered in the one-boson exchange model with a non-zero range [74,75]. In contrast to them, we suppose that the Σ c N (Σ * c N ) interaction is provided by the contact-type with a zero range. The contact-type interaction and the HQS symmetry allow us to have the most general form of the Lagrangian: with the kinetic term and interaction term with the coupling constants C A (A = 1, 2, 3, 4). We notice that the index µ in Ψ µ v is restricted to i = 1, 2, 3 in the rest frame. The above Lagrangian is invariant under the spin symmetry and the isospin symmetry, SU(2) spin ×SU(2) isospin .
In the operator A ⊗ B acting on the nucleon (ϕ), A and B are the operators for the spin and the isospin of a nucleon.
Similarly, in the operator A ⊗ B ⊗ C acting on the Σ c (Σ * c ) baryon (Ψ i v ), A and B are the operators for the spin of the light component (qq) and the spin of the heavy quark (Q), respectively, and C is the operator for the isospin of the light component (qq). 1 2 is the 2-by-2 identity matrix for spin or isospin, and 1 3 is the 3-by-3 identity matrix for isospin. We also use the notations σ ( = 1, 2, 3) and τ d (d = 1, 2, 3) for the Pauli matrices acting on the spin of a nucleon and the isospin of a nucleon or a Σ c (Σ * c ) baryon, respectively. We define ε ij (ε 123 = 1; i, j, = 1, 2, 3) as the anti-symmetric tensor for the spin of a Σ c baryon or a Σ * c baryon, and t d (d = 1, 2, 3) as the operator for the isospin of a Σ c (Σ * c ) baryon, whose explicit forms are given by They satisfy the following relations: and this will be used in later calculations. 3 With the basis in the isospin operator t a , the isospin components in Ψ v1/2 and Ψ µ v3/2 are expressed as We notice that this representation is not diagonal in the charge basis. The transformation to the diagonal form by the unitary transformation is shown in Appendix B. It is apparent that the Lagrangian (5) has the spin symmetry and the isospin symmetry, SU(2) spin × SU(2) isospin for both a nucleon and for a Σ c (Σ * c ) baryon. Although the numerical values of the coupling constants C A (A = 1, 2, 3, 4) have not been known, the discussion about the Kondo effect can proceed without the information about the specific value of C A as it will be presented later.
For later convenience, we rewrite the interaction term of Eq. (5) in a compact form as where we introduce the following operators: for a nucleon (ϕ) and . The sum is taken over the repeated indices. Several comments are in order. Firstly, the heavy-quark-spin does not flip by the interaction with a nucleon in the HQS symmetry, and hence we have only the identity operator (1 2 ) for the heavy quark. This is because the spin for the heavy quark (c quark) in the Σ c (Σ * c ) is independent of the spin for the light diquark (qq). Thus, to be precise, the total symmetry should be given by SU(2) light spin × SU(2) heavy spin × SU(2) isospin including SU(2) heavy spin for the spin symmetry of the heavy quark.
Secondly, we remark that the propagator of the nucleon with an energy p 0 and a three-dimensional momentum p in nuclear matter with the chemical potential µ is given by with ε > 0 an infinitely small number. E p = p 2 /(2m) is the energy of the nucleon with a mass m, and µ is the chemical potential for the nucleon. Notice the difference in the pole positions between the particle component (E p > µ) and the hole component (E p < µ). The propagators of the Σ c and Σ * c baryons with an energy p 0 are given by in rest frame. Notice that the energy in the denominator (p 0 ) describes the residual momentum of the Σ c and Σ * c baryons.
Thirdly, we remark that the Σ c baryon and the Σ * c baryon can decay via Σ c → Λ c π and Σ * c → Λ c π, whose decay widths are around 2 MeV and 15 MeV, respectively [76]. In the present study, we consider that the Σ c and Σ * c baryons are in the quasi-stable states whose lifetimes are long enough. We also neglect the coupling between the Σ c N (Σ * c N ) and the Λ c N state. Those subjects are left for future work.

A. Derivation of the renormalization group equation
In the Kondo effect, the coupling constants in the medium are enhanced logarithmically in the low-energy region, and the system becomes a strongly-coupled one. In this situation, the coupling constants are not the constant values literally, but they should be regarded as the effective coupling constants whose property is dependent on the relevant energy scale in the medium. We therefore study how the coupling constant C A (A = 1, 2, 3, 4) in Eq. (11) is changed into the effective coupling constants in terms of the Kondo effect. We use the renormalization group (RG) equation.
Here we introduce the energy scale Λ, which is measured from the Fermi energy, and see the effective coupling constants for the small change of Λ. We estimate the coupling constants on the lower-energy scale Λ − dΛ by including the loop effect of the particle-hole creations with the energy shell between Λ − dΛ and Λ. The initial value of the coupling constant starting in the RG equation is assigned to the bare coupling constants in vacuum, i.e., the coupling constants in Eq. (11), where the relevant energy is denoted by Λ 0 . At the one-loop order, we find that the RG equation reads where the term on the left-hand side denotes the effective coupling-constants on the energy scale Λ − dΛ, and, on the right-hand side, the first term denotes the effective coupling-constant on the energy scale Λ, and the second (third) term denotes the loop-integrals with particle (hole) creation in the energy-shell between Λ − dΛ and Λ (Fig.1). In the above equation, the indices in the operator Γ A andΓ A (A = 1, 2, 3, 4) are shown as for the nucleon part, and for the Σ c (Σ * c ) baryon part. Here a, b = 1, 2 and α, β = 1, 2 are for the spin and the isospin of a nucleon, respectively, and i, j = 1, 2, 3 and µ, ν = 1, 2, 3 are for the spin and for the isospin of a diquark component (qq) in a Σ c (Σ * c ) baryon, respectively. We consider that the sum over the spin direction ( = 1, 2, 3) and the isospin direction (d = 1, 2, 3) is included if necessary. Taking into account that the integral region for the momentum is limited to the energy-shell, , we obtain the following approximated terms: and shell dp 0 2π where we leave only the leading terms for a small dΛ/Λ 1. We introduce ρ 0 ≡ m 3/2 √ µ for the state-number-density at the Fermi surface. Then, we rewrite the RG equation (16) as for each channel A = 1, 2, 3, 4. Here, we introduce the new variable λ ≡ − ln Λ/Λ 0 instead of the energy scale Λ.
The high-energy scale Λ 0 for which the RG equation starts is set to be equal to the chemical potential of the nuclear matter µ or the cutoff energy-scale D in the point-like interaction in Eq. (11). In the present discussion, however, there is no necessity to specify the value of Λ 0 explicitly. We notice that the variable λ changes from λ = 0 to λ → ∞ in correspondence to the change from the high-energy scale to the low-energy scale. As seen in Eq. (21), C 1 (λ) is not affected by the change of λ, and hence the spin and isospin-independent channel is not subject to the medium effect. Thus, we will consider only C 2 (λ), C 3 (λ), and C 4 (λ) in the following discussions. For convenience, we use the following dimensionless effective coupling constants instead of C 2 (λ), C 3 (λ), and C 4 (λ), and rewrite the RG equation (21) as Those are the basic equations used in the following discussions. Notice that we have added the minus sign for C 3 (λ) in Eq. (22) simply for the appearance of the equations. The initial conditions are given asC 2 (0) = 4ρ 0 C 2 ,C 3 (0) = −4ρ 0 C 3 , andC 4 (0) = 4ρ 0 C 4 with C 2 , C 3 , and C 4 being the coupling constants in the interaction Lagrangian (11). We plot the right-hand side of Eq. (23), i.e., the vector , and also show the stream lines for C 2 (λ),C 3 (λ),C 4 (λ) varying with λ and the several initial conditions (C 2 ,C 3 ,C 4 ) at λ = 0 as the solutions of Eq. (23). The initial conditions are plotted by the dots in the figure. We notice that, for the increasing λ, there are some initial conditions giving the stream lines convergent to zero and the other initial conditions giving the stream lines divergent. In the following subsections, we will investigate the solutions of Eq. (23) in detail. We will find that the C 4 term, i.e., the spin and isospin-dependent term in Eq. (11) plays an important role to extend the parameter region of the coupling constants in which the Kondo effect occurs.

B. Analytical solutions in special cases
Although Eq. (23) may look simple, it is difficult to obtain the analytic solution due to the nonlinearity of the equation. Therefore, we have to perform the numerical calculation. In order to understand roughly the properties of  the solutions before the numerical computing, we seek to obtain analytic solutions by restricting the parameter space to simpler subspaces and focusing on special cases: with |C 4 (λ)| 1, and (iv)C 2 (λ) =C 3 (λ) = ± 2/3C 4 (λ). We will show that, in the last case, the SU(4) symmetry is realized as an extension from the SU(2) spin × SU(2) isospin symmetry in the Lagrangian. The simple settings from (i) to (iv) will provide us with fresh insights about the Kondo effect for the Σ c (Σ * c ) baryon in the nuclear matter.
Then, the RG equation (23) is transformed to We find thatC 4 (λ) is constant, whileC 2 (λ) andC 3 (λ) change according to the change of the energy scale. Becausẽ withC 2 = 4ρ 0 C 2 andC 3 = −4ρ 0 C 3 as the initial condition at λ = 0. Thus, the three-dimensional parameter space is essentially reduced to the one-dimensional one. Let us consider the behavior of the solutionC 2 (λ) in detail in the energy scales from λ = 0 (high energy) to a larger value (low energy). For the positive value of C 2 (C 2 > 0), we notice thatC 2 (λ) becomes divergent at the end of the energy scale Λ = Λ K with Λ K = Λ 0 e −1/(4ρ0C2) . Λ K is called the Kondo scale (the Landau pole) which is quantity smaller than Λ 0 due to the exponential factor. At the Kondo scale, the coupling constant becomes sufficiently large. Thus, the system becomes a strongly coupled one and the non-perturbative analysis should be adopted. For the negative value of C 2 (C 2 < 0), the effective coupling constant becomes zero without divergence in the low-energy limit (λ → ∞), and hence such interaction disappears in the ground state.
A similar analysis is applied to the case ofC 3 (λ). We find that the effective coupling constant becomes divergent at the Kondo scale Λ K = Λ 0 e 1/(4ρ0C3) for the negative value of C 3 (C 3 < 0), while it disappears for the positive value of C 3 (C 3 > 0). Notice that the sign of C 3 for the Kondo effect is different from that of C 2 due to the definition in Eq. (22) and that the values of Λ K and Λ K can be different in general.
So far we have setC 4 (λ) = 0 (C 4 = 0) by neglecting the spin and isospin-dependent term in Eq. (11), and have seen that C 2 < 0 and C 3 > 0 lead to the absence of the Kondo effect. However, this is the case only forC 4 (λ) = 0 (C 4 = 0). In the following cases, we will demonstrate that the Kondo effect can occur even for C 2 < 0 and C 3 > 0 when a non-zero value ofC 4 (λ) is considered.
The initial values of (C 2 (λ),C 4 (λ)) are denoted by the points. When the initial points are in the gray region (the left panel), the effective coupling constants become zero at the end of the low-energy scale, which indicates that the Kondo effect does not occur. On the other hand, when the initial points are outside the gray region (the left panel), the effective coupling constants become infinity, and accordingly the Kondo effect occurs. Here the existence of the C 4 -term is important. In section III B 1, we showed that the negative value ofC 2 (λ) has not led to the Kondo effect, when the C 4 -term is absent (C 4 = 0). However, when the C 4 -term is present (C 4 = 0), the negative value ofC 2 (λ) can produce the Kondo effect as long asC 4 (λ) > −C 2 (λ)/ √ 2 orC 4 (λ) <C 2 (λ)/ √ 2 is satisfied. Therefore, we conclude that the non-zero value of |C 4 (λ)| is important to enhance the parameter region ofC 2 (λ) to realize the Kondo effect.
In Fig. 4, we plot the two-dimensional vector field C 23 (λ) 2 + 2C 4 (λ) 2 , 4C 23 (λ)C 4 (λ) , i.e., the right-hand side of Eq. (30). We also plot the solutions C 23 (λ),C 4 (λ) starting from λ = 0 by the streaming lines. It is shown that the solutions from the initial points with the negative value ofC 23 (λ) (C 23 (λ) < 0) and the small value of |C 4 (λ)| (|C 4 (λ)| 1) become convergent to zero for λ → ∞. From the numerical calculation, we find that the initial points in the gray region defined byC 23 (λ) >C 4 (λ) andC 23 (λ) < −C 4 (λ) do not lead to the divergence. The initial points outside this gray region can lead to the divergence and therefore can produce the Kondo effect. From the above analysis, we conclude that the non-zero value ofC 4 (λ) extends the parameter region of C 23 (λ),C 4 (λ) for the Kondo effect.

C. Flow diagrams in general cases
In the previous subsections, we highlighted special cases where the non-zero value of |C 4 |, i.e., the spin and isospindependent interaction in the Lagrangian (11), extends the parameter region ofC 2 ,C 3 , andC 4 and allows the Kondo effect to occur. As a summary, we consider the solutions C 2 (λ),C 3 (λ) which is projected to the two-dimensional surface with a constant value ofC 4 (λ). We suppose the initial conditions of |C 4 | = 0, 0.5, and 1 for the numerical demonstration. The results are shown in Fig. 5. For each C 4 , the initial conditions ofC 2 andC 3 are shown by the dots in the figure. Under the initial condition ofC 4 = 0, the Kondo effect occurs forC 2 > 0 orC 3 > 0 and does not for bothC 2 < 0 andC 3 < 0. This is confirmed directly in the figure, because the flows in the former is divergent toward largeC 2 (λ) andC 3 (λ), while the flows in the latter stops atC 2 (λ) =C 3 (λ) = 0. In contrast, ifC 4 has a non-zero value for the initial condition, the Kondo effect can occur even for bothC 2 < 0 andC 3 < 0.  Therefore, we understand numerically that the non-zero value ofC 4 helps to extend the region of the parameter space ofC 2 andC 3 for which the Kondo effect occurs.
Comparison of the Kondo scales allows us to grasp the importance of the C 4 term, and to do so we consider the Kondo scales for the SU(2) symmetry in C 4 = 0 and for the SU(4) symmetry in C 4 = 0. In the former case,  (2), because the C 2 term and the C 3 term are completely decoupled. In the latter case, assuming . Therefore, keeping the same coupling constants C 2 and C 3 , we find that the C 4 term, i.e., the mixing term of both spin and isospin, enhances the Kondo scale. Such enhancement makes the Kondo effect with the non-zero value of C 4 occur on higher energy scales than the case of C 4 = 0. This conclusion supports the argument that the C 4 term is important to magnify the Kondo effect for the Σ c (Σ * c ) baryon in nuclear matter.

IV. REVISITINGD ANDD * MESONS
Now we consider other systems where multiple number of non-Abelian symmetries exist, and here we focus on the Kondo effect for aD (D * ) meson in terms of the SU(2) spin symmetry and the SU(2) isospin symmetry. Although there have been many studies on theDN (D * N ) interaction [77][78][79][80][81][82][83][84] and the properties of aD (D * ) meson in nuclear systems  in the literature, there are only a few studies on the Kondo effect for aD (D * ) meson. In the previous studies, only the isospin symmetry was taken in Refs. [41,42], and only the spin symmetry was taken in Ref. [43]. In the present study, we extend their discussions to the case where both of them exist. We introduce H v defined by H v = γ µ P * vµ + iγ 5 P v 1+v / 2 with P * vµ ∼ (qQ) spin 1 (µ = 0, 1, 2, 3) for the vector meson and P v ∼ (qQ) spin 0 for a pseudoscalar meson. We note that the asterisk ( * ) denotes the vector field, not the complex conjugate. The vector field satisfies v µ P * vµ = 0 andH v = γ 0 H † v γ 0 . Under the spin and isospin-symmetries and the S-wave interaction at the low energies, we write the interaction Lagrangian as follows: with the kinetic term and the interaction term We define the Dirac matrices by Γ 1 = 1, Γ 2 = γ µ , Γ 3 = σ µν = (i/2) (γ µ γ µ − γ ν γ µ ), Γ 4 = γ µ γ 5 , Γ 5 = γ 5 , and here tr stands for the trace over the Dirac matrices. We introduce d s i and d t i (i = 1, 2, 3, 4) for the coupling constants in each isospin channel (singlet and triplet). The coefficient 1/2 is used for later convenience. In the interaction term, the relativistic field ψ for a nucleon is reduced to the non-relativistic form as ψ t = (ϕ, 0) t in the following discussion.
in the rest frame v µ = (1, 0), where we define the new coupling constants by (42) is invariant under the flavor symmetry for the light quark and under the spin symmetries for the spin of the light quark and the heavy antiquark. In terms of the spin symmetry, the transformation of P v and P * i v is given by for the small rotation angle θ i (i = 1, 2, 3).
For the coupling constants in the Lagrangian (42), we consider the effective coupling constants D a (λ) (a = 1, 2, 3,4) which follows the RG equation, as we have considered for a Σ c (Σ * c ) baryon in section III. Referring the similar diagram in Fig. 1 and using the momentum integrals (19) and (20) as well as the identities (A1), we obtain the RG equations d dλD with λ = − ln Λ/Λ 0 , where we define the dimensionless quantities bỹ Here, the minus sign is put by convention. The RG equation (44) is essentially the same as the RG equation for a Σ c (Σ * c ) baryon, Eq. (21) or Eq. (23). Thus, we obtain the similar behavior for the Kondo effect which indicates the importance of the spin and isospin-dependent term with D 4 .
It is known that one of the interesting low-energy properties in the Kondo effect is the formation of the singlet pairing in the ground state [2][3][4]. Here the singlet pairing indicates the bound state where an itinerant fermion is bound to an impurity particle and the total spin of the bound state is singlet. In other words, this is the dressed state surrounded by of particles and holes around the impurity site (exact screening). The dressed state is also known as the Kondo cloud. The singlet pairing was studied for D − s and D * − s mesons in nuclear matter [43] and for aD meson in an atomic nucleus [42]. It is also possible that the singlet pairing exists for theD andD * mesons. In such a situation, the singlet pairing should be composed of a nucleon (N ) and a light quark (q = u, d) in theD (D * ) meson, i.e., the heavy hadron dressed state (mapped) screening type Ref.

D,D * meson
Λc baryon-like exact screening - Ξc baryon-like exact screening [43] Σc, Σ * c baryonD,D * meson-like underscreening -  In contrast, the Σ c (Σ * c ) baryon cannot have the singlet pairing. In fact it is known that the singlet pairing is not formed when the dimensions of the representations (fundamental, adjoint, etc.) in SU(N ) are different in the itinerant fermion and the impurity particle. Let us consider the itinerant fermion with spin 1/2 and the impurity particle with spin S. We observe that, for S > 1/2, the spin of the impurity particle cannot be screened by the spin of one itinerant fermion, and that there remains an unscreened spin S * = S − 1/2 for the impurity site. This is called the underscreening Kondo effect [112]. A similar situation arises for a Σ c (Σ * c ) baryon in nuclear matter. That is, the spin S = 1 and the isospin I = 1 of the diquark (qq) in the Σ c (Σ * c ) baryon would lead to the unscreened Kondo effect, making the N qq state with the spin S * = 1/2 and the isospin I * = 1/2 as the dressed state by particles and holes. Furthermore, we argue that it would lead to the composite state of N qqQ with spin 0 or 1 and isospin 1/2, i.e. the same spin and isospin as the qQ meson such as aD andD * meson. Therefore, it is thought that the Σ c (Σ * c ) meson in nuclear matter should behave as the composite state (N qqQ), which has the same spin and isospin as aD (D * ) meson.
The above consideration helps us introduce the concept of the "meson-baryon mapping" induced by the Kondo effect. As we have discussed, aD (D * ) meson or a D − s (D * − s ) meson in nuclear matter can be regarded as a Λ c baryon or a Ξ c baryon, and a Σ c (Σ * c ) baryon in nuclear matter can be regarded as aD (D * ) meson (table I). Thus, the heavy meson is "baryonized" and the heavy baryon is "mesonized" due to the Kondo effect. Such a meson-baryon mapping may cast new light on the properties and the dynamics of heavy hadrons in nuclear matter. We comment that the simple correspondence between the composite state (N qQ or N qqQ) and the hadron-like state (Λ c -like orD(D * )-like) holds only when both spin and isospin are subject to the Kondo effect. When only spin (isospin) is subject to the Kondo effect and isospin (spin) is not, there should arise an additional degeneracy by isospin (spin) leading to the hadron-like state whose quantum number is not realized in vacuum. A more detailed investigation of these things must await another occasion.

VI. CONCLUSION
We have studied the Kondo effect for a Σ c (Σ * c ) baryon in nuclear matter. By virtue of the SU(2) spin × SU(2) isospin symmetry, the HQS symmetry, and the S-wave interaction, we have provided the spin-exchange (or spin-nonexchange) and isospin-exchange (or isospin-nonexchange) interactions between the Σ c (Σ * c ) baryon and the nucleon. By adopting the RG equation at one-loop order, we have found that the coexistence of the spin exchange and the isospin exchange magnifies the Kondo effect. We have extensively investigated the RG equation for several cases in terms of the coupling constants, including the SU(4)-limit case. We have also conducted the analysis for theD (D * ) meson with the SU(2) spin × SU(2) isospin symmetry, and have shown the solution in the SU(4)-limit. In addition, we have ventured to develop the concept of the "meson-baryon mapping" for the Σ c (Σ * c ) baryon, theD (D * ) meson, and the D − s (D * − s ) meson in the Kondo effect. It is straightforward to apply the mapping to other heavy hadrons when the light component in the heavy hadron has the spin interaction with a nucleon which flips the spin and/or the isospin.
Also, we mention that several issues are left unanswered: the corrections at O(1/m Q ) (beyond the heavy-quark mass limit); applying the Σ c N (Σ * c N ) interaction to many-body problems [74,75]; discussing the "meson-baryon mapping" within the non-perturbative framework; the production mechanisms of the heavy hadrons in atomic nuclei; applications to atomic nuclei; and (as a more advanced topic) the continuity of the Kondo effect between the hadronic phase and the quark phase (see the discussions in Ref. [44]). The continuity, which was proposed for the color-flavor locked color superconductivity in Refs. [113,114], is now studied intensively in view of topological objects [115][116][117][118][119][120].
It is worthwhile to study how the Kondo cloud changes from the hadronic matter to the quark matter. Simulations of the Kondo effect with SU(3) symmetry in cold atomic gases are also important [121]. It remains unclear as to how the Kondo effect with SU(4) symmetry for a Σ c (Σ * c ) baryon is related to the Kondo effect with SU(4) symmetry in condensed matter systems, such as quantum dots, which has been studied theoretically [122][123][124][125][126][127][128][129] and experimentally [130][131][132][133]. Those issues need to be addressed in future work. We consider the Kondo effect with one single non-Abelian interaction for an itinerant fermion in the fermi gas and the heavy impurity. We suppose that they belong to the fundamental representation of the SU(N ) symmetry, and the interaction between the itinerant fermion and the heavy impurity is provided by the non-Abelian interaction L int = g(λ a ) ij (λ a ) k with the coupling constant g and the Gell-Mann matrices λ a (a = 1, 2, . . . , N 2 − 1) in the SU(N ) symmetry. For example, the case of N = 2 is the spin-exchange interaction, in which an attraction between the itinerant fermion and the heavy impurity is provided in the spin-antiparallel channel for g < 0 and in the spin-parallel channel for g > 0. We consider the RG group equation at one-loop level. Utilizing the momentum integrals (19) and (20) and the relationships for the Gell-Mann matrices we obtain the RG equation with the energy scale λ = − ln(Λ/Λ 0 ). Here Λ is the energy scale moving from high-energy to the low-energy region, and Λ 0 is the ultraviolet-energy scale as the initial point. The solution of the above RG equation is found to be g(λ) = g 1 + 2ρ 0 N gλ , with g being the coupling constant in vacuum or in the interaction Lagrangian. Noting that the energy scale runs from λ = 0 (the high-energy scale) to λ → ∞ (the low-energy scale), we find that negative coupling constant (g < 0) leads to divergence of the coupling constant g(λ) at λ K = −1/(2ρ 0 N g) or Λ K = e 1/(2ρ0N g) and that the positive coupling constant (g > 0) leads to the vanishing coupling constant (g(λ) → 0). The relevant fixed point in the former case gives the Kondo effect, while the irrelevant fixed point in the latter does not. Thus, the coupling strength in the spin-antiparallel channel is enhanced, while that in the spin-parallel channel is suppressed.