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We study five-dimensional N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} gauged supergravity coupled to five vector multiplets with SO(2)D×SO(3)×SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(2)_D\times SO(3)\times SO(3)$$\end{document} gauge group. 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The other two AdS5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_5$$\end{document} vacua preserve only N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} supersymmetry with SO(2)diag×SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(2)_{\text {diag}}\times SO(3)$$\end{document} and SO(2)diag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(2)_{\text {diag}}$$\end{document} symmetry. 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These vacua should be dual to N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} and N=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=1$$\end{document} superconformal field theories (SCFTs) in four dimensions with different flavour symmetries. We give the full scalar mass spectra at all of the AdS5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_5$$\end{document} critical points which provide information on conformal dimensions of the dual operators. Finally, we study holographic RG flows interpolating between these AdS5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AdS_5$$\end{document} vacua and find a new class of solutions. In addition to the RG flows from the trivial SO(2)D×SO(3)×SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(2)_D\times SO(3)\times SO(3)$$\end{document}N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} critical point, at the origin of the scalar manifold, to all the other critical points, there is a family of RG flows from the trivial N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} critical point to the new SO(2)diag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(2)_{\text {diag}}$$\end{document}N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2$$\end{document} critical point that pass arbitrarily close to the SO(2)D×SO(3)diag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(2)_D\times SO(3)_{\text {diag}}$$\end{document}N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=4$$\end{document} critical point.


Introduction
The study of holographic RG flows has attracted much attention since the proposal of the AdS/CFT correspondence in [1]. These solutions take the form of domain walls interpolating between Ad S vacua or between an Ad S vacuum and a singular geometry and holographically describe RG flows a e-mail: parinya.ka@hotmail.com (corresponding author) between conformal fixed points or from a conformal fixed point to a non-conformal phase in the dual field theories. In particular, holographic RG flows in AdS 5 /CFT 4 correspondence provide a very useful tool to understand strongly coupled dynamics of four-dimensional gauge theories. A number of solutions describing various deformations of the N = 4 Super Yang-Mills (SYM) theory, dual to type IIB theory on Ad S 5 × S 5 , has been constructed both in type IIB supergravity and in the effective N = 8 five-dimensional gauged supergravity, see for example [2][3][4][5][6][7]. The consistent truncation of type IIB supergravity on S 5 constructed recently in [8] allows solutions in the N = 8 gauged supergravity to be uplifted to ten dimensions.
On the other hand, similar solutions in gauged supergravities with N < 8 supersymmetry are less known than those of the maximal case. In particular, a number of RG flow solutions in half-maximal N = 4 gauged supergravity have appeared only very recently in [9,10], see also [11] for an earlier construction and [12] for domain wall solutions in N = 2 gauged supergravity. Furthermore, a number of RG flow solutions has been found in N = 4 gauged supergravity coupled to three vector multiplets obtained from a consistent truncation of the maximal SO (5) gauged supergravity in seven dimensions [13]. In this case, there is only one supersymmetric Ad S 5 vacuum, and the solutions describe RG flows from a conformal fixed point to non-conformal phases dual to singular geometries in the IR. In this work, we hope to fill this gap by adding a new family of holographic RG flow solutions between conformal fixed points to these results. We will work in the half-maximal gauged supergravity with a new gauge group that leads to more interesting Ad S 5 vacua and holographic RG flows, see [14][15][16][17][18][19][20][21][22][23] for similar studies in half-maximal gauged supergravity in other dimensions. N = 4 gauged supergravity coupled to an arbitrary number n of vector multiplets has SO(1, 1) × SO(5, n) global symmetry. Gaugings of a subgroup G 0 ⊂ SO(1, 1) × SO (5, n) can be implemented by using the embedding tensor formalism [24], see also [25]. We will mainly consider the case of n = 5 vector multiplets and SO (2) (5,5) gauge group. Unlike the previously studied SO(2) × SO(3) × SO (3) gauge group in [9,10] in which the SO (2) factor is a subgroup of the SO(5) R ∼ U Sp(4) R R-symmetry, the SO(2) D considered here is a diagonal subgroup of SO(2) ⊂ SO (5) R and SO(2) ⊂ SO (5) symmetry of the vector multiplets. We will look for supersymmetric Ad S 5 vacua dual to four-dimensional SCFTs and possible holographic RG flows between these SCFTs.
Since the ungauged N = 4 supergravity with n = 5 vector multiplets can be obtained from a T 5 reduction of N = 1 supergravity in ten dimensions, the N = 4 gauged supergravity studied here might be possibly embedded in string/Mtheory. However, it should be pointed out that this N = 4 gauged supergravity has currently no known higher dimensional origin. Therefore, a complete holographic description is not available at this stage. However, even without higherdimensional embedding, the results from the effective N = 4 gauged supergravity in five dimensions could still be useful in a holographic study of strongly coupled four-dimensional SCFTs.
Similar studies in the cases of SO(2) × SO(3) × SO(3) and SO(2) D ×SO(3) gauge groups have appeared recently in [9,10]. In the first case, there are two N = 4 supersymmetric Ad S 5 vacua while the second gauge group admits N = 4 and N = 2 Ad S 5 vacua. The holographic RG flows in both cases have also been given in [9,10]. In the SO(2) D × SO(3) × SO(3) gauge group considered in this paper, we find a number of more interesting results. First of all, we discover four supersymmetric Ad S 5 vacua within a truncation to SO(2) diag invariant sector of the SO(1, 1)× SO(5, 5)/SO(5)× SO(5) scalar manifold. Two of the critical points preserve the maximal N = 4 supersymmetry with SO(2) D × SO(3) × SO (3) and SO(2) D × SO(3) diag symmetries while the remaining two are only N = 2 supersymmetric with SO(2) diag ×SO (3) and SO(2) diag symmetries. The first three critical points have an analogue in the results of [9,10] although with some differences in scalar mass spectra. The N = 2 critical point with SO(2) diag symmetry is however entirely new.
Another interesting result of the present paper is that there exists a family of RG flows from the SO(2) D × SO(3) × SO(3) N = 4 critical point to the SO(2) diag N = 2 critical point that consists of a direct flow and flows that pass arbitrarily close to the SO(2) D × SO(3) diag N = 4 critical point. This is similar to the families of RG flows found in the maximal gauged supergravity in four dimensions [26][27][28]. To the best of the author's knowledge, the present result is the first example of families, or cones in the terminology of [26], of RG flows in the framework of five-dimensional gauged supergravity.
The paper is organized as follows. In Sect. 2, we review five-dimensional N = 4 gauged supergravity coupled to vector multiplets in the embedding tensor formalism. In Sect. 3, we consider the case of five vector multiplets and describe the embedding of SO(2) D × SO(3)× SO (3) gauge group in the SO (5,5) global symmetry. Supersymmetric Ad S 5 vacua and RG flows interpolating between these vacua are also given. Finally, we end the paper by giving some conclusions and comments in Sect. 4.

Five dimensional N = 4 gauged supergravity coupled to vector multiplets
In this section, we give a brief review of five-dimensional N = 4 gauged supergravity coupled to an arbitrary number n of vector multiplets. We mainly focus on the scalar potential and supersymmetry transformations of fermions which are relevant for finding supersymmetric Ad S 5 vacua and RG flow solutions. More details and the complete construction of N = 4 gauged supergravity can be found in [24,25]. The N = 4 supergravity multiplet consists of the graviton eμ μ , four gravitini ψ μi , six vectors (A 0 μ , A m μ ), four spin-1 2 fields χ i and one real scalar , the dilaton. Space-time and tangent space indices are denoted respectively by μ, ν, . . . = 0, 1, 2, 3, 4 andμ,ν, . . . = 0, 1, 2, 3, 4. The fundamental representation of SO(5) R ∼ U Sp(4) R R-symmetry is described by m, n = 1, . . . , 5 for the SO(5) R and i, j = 1, 2, 3, 4 for the U Sp(4) R . The latter also corresponds to spinor representation of SO(5) R . For the n vector multiplets, each multiplet contains a vector field A μ , four gaugini λ i and five scalars φ m . These multiplets will be labeled by indices a, b = 1, . . . , n. The corresponding component fields are accordingly denoted by (A a μ , λ a i , φ ma ). The 5n scalar fields parametrize the SO(5, n)/SO(5) × SO(n) coset.
Combining the gravity and vector multiplets, we have 6+n vector fields denoted collectively by A M μ = (A 0 μ , A m μ , A a μ ) and 5n + 1 scalars in the R + × SO(5, n)/SO(5) × SO(n) coset. All fermionic fields are symplectic Majorana spinors subject to the condition with C and i j being the charge conjugation matrix and U Sp(4) symplectic matrix, respectively.
In addition, the matrix V M A satisfies the relation with η M N = diag(−1, −1, −1, −1, −1, 1, . . . , 1) being the SO(5, n) invariant tensor. Furthermore, the SO(5, n)/SO (5) × SO(n) coset can also be described in terms of a symmetric matrix which is manifestly SO(5) × SO(n) invariant. Gaugings of N = 4 supergravity can be efficiently described by using the embedding tensor which determines the embedding of admissible gauge groups in the global symmetry SO(1, 1) × SO (5, n). Supersymmetry allows for three components of the embedding tensor of the form ξ M , . These components also need to satisfy a set of quadratic constraints. Furthermore, the existence of supersymmetric Ad S 5 vacua requires ξ M = 0 [29]. Therefore, we will consider only gaugings with ξ M = 0. In this case, the gauge group is entirely embedded in SO (5, n), and the quadratic constraints reduce to With ξ M = 0, the gauge generators in the fundamental representation of SO(5, n) can be written as (5, n) generators. The gauge covariant derivative reads where ∇ μ is the usual space-time covariant derivative. We use the definition of ξ M N and f M N P that includes the gauge coupling constants. Note also that SO(5, n) indices M, N , . . . are lowered and raised by η M N and its inverse η M N . In this paper, we are mainly interested in Ad S 5 vacua and holographic RG flows in the form of domain walls that involve only the metric and scalar fields. For Ad S 5 vacua with all scalars constant, this is always a consistent truncation. However, for holographic RG flows given by domain wall solutions with non-constant scalars, some of the scalars can be charged under the gauge group and couple to the gauge fields via the following Yang-Mills equation, see [24] for more detail, with the covariant field strengths defined by It should be noted that the two-form fields are introduced in the embedding tensor formalism for the requirement of gauge covariance for the gauge field strengths. However, these are auxiliary fields since they do not have kinetic terms. In all the truncations we will consider here, it turns out that the Yang-Mills currents given by the right hand side of (8) vanish. Therefore, it is consistent to set all the gauge fields to zero in the RG flow solutions. With all vector fields vanishing, it is also consistent to set all the two-form fields to zero as can be seen from the corresponding field equation given in [24]. Accordingly, for simplicity of various expressions, we will set all the fields but the metric and scalar fields to zero from now on. The bosonic Lagrangian of a general gauged N = 4 supergravity coupled to n vector multiplets can be written as where e is the vielbein determinant. The scalar potential reads where M M N is the inverse of M M N , and M M N P Q R is obtained from by raising indices with η M N . Fermionic supersymmetry transformations are given by The fermion shift matrices are in turn defined by V M i j is defined in terms of V M m and SO(5) gamma matrices mi j as In this paper, we will use the following representation of SO(5) gamma matrices with σ i , i = 1, 2, 3, being the Pauli matrices. The covariant derivative on i is given by with the composite connection defined by Finally, we note the relation between the scalar potential and the fermion shift matrices A 1 and A 2 which is a consequence of the quadratic constraints (5). Note also that raising and lowering of i, j, . . . indices by i j and i j are related to complex conjugation.

N = 4 gauged supergravity with SO(2) D × SO(3) × SO(3) gauge group
We now consider N = 4 gauged supergravity coupled to n = 5 vector multiplets with the global symmetry group SO(1, 1) × SO (5,5). We are interested in a compact gauge group of the form SO(2) D ×SO (3)×SO (3) with the embedding tensor given by where g 1 , g 2 , h 1 and h 2 are the corresponding coupling constants.
The form of ξ M N implies that the SO(2) D is a diagonal subgroup of SO(2) R ⊂ SO(5) R and SO(2) ⊂ SO(5) generated by the SO(5, 5) generators t 12 and t 9,10 . This factor is gauged by the vector field A 0 μ in the supergravity multiplet, see [29] for more detail. Since ξ M N and f M N P have no indices in common, the second quadratic condition in (5) is identically satisfied while the first condition holds by virtue of the Jacobi's identity for the two SO(3) factors. Therefore, this is an admissible gauge group. Similar gauge groups with g 2 = 0 or h 2 = 0 have already been considered in [9,10].
To parametrize the coset representative for SO(5, 5)/ SO(5) × SO(5), we first identify the SO(5, 5) non-compact generators Dealing with all 25 scalars in this coset is not practically possible, we will truncate to a smaller submanifold of (3). For later convenience, we will denote the gauge group by SO (2) (3) corresponding to the components (ξ 12 , ξ 9,10 ), fmñp and fãbc of the embedding tensor, respectively. SO(3) R and SO (3) are subgroups of the SO(5) R R-symmetry and the SO(5) symmetry of the five vector multiplets. A simple truncation that is more manageable and still gives interesting results is given by scalars that are singlet under (3). In this truncation, SO(2) diag , generated by a linear combination of t 12 , t 45 and t 9,10 , is There are nine singlet scalars under SO(2) diag corresponding to the non-compact generatorŝ The coset representative can then be written as (28) It turns out that the resulting scalar potential and fermion shift matrices computed from this coset representative are still highly complicated. However, it can be straightforwardly verified that setting φ i = 0, for i = 4, . . . , 9, is a consistent truncation. In this case, the analysis simplifies considerably. In particular, the A i j 1 tensor is diagonal in this subtruncation. Furthermore, non-vanishing φ i , i = 4, . . . , 9, do not give rise to any new Ad S 5 vacua other than those given below. We then make a subtruncation by setting φ i = 0, i = 4, . . . , 9, for simplicity.
Using (28) with φ 4 = · · · = φ 9 = 0, we find the scalar potential +2 cosh 2φ 1 cosh 2φ 3 sinh 4 φ 2 The A i j 1 tensor is given by with Either of α 1 or α 2 leads to the superpotential in terms of which the scalar potential can be written as for W = W 1 = 2 3 |α 1 | or W = W 2 = 2 3 |α 2 |. In this paper, we are only intested in supersymmetric Ad S 5 vacua which are critical points of both the scalar potential and the superpotential with negative cosmological constants. Before giving these critical points, we first note that in general, the two eigenvalues of A i j 1 are not equal but related by We see that either W 1 or W 2 corresponds to N = 2 supersymmetry but for g 2 = 0 or φ 3 = 0, we find, with α 1 = −α 2 , W 1 = W 2 leading to N = 4 supersymmetry as studied in [9,10]. In particular, for g 2 = 0, non-vanishing φ 3 breaks supersymmetry from N = 4 to N = 2.

Supersymmetric Ad S 5 vacua
From the scalar potential (29), there are four supersymmetric Ad S 5 critical points: • I. The first critical point is given by This critical point preserves SO(2) D × SO(3) diag symmetry. Both signs of φ 2 lead to equivalent critical points with the same cosmological constant and scalar masses. • III. The next critical point is given by , Since φ 3 = 0, this critical point is N = 2 supersymmetric with SO (2) The two sign choices in (36) give equivalent critical points. • IV. The last critical point is also N = 2 supersymmetric given by At this critical point, the gauge group is broken down to just SO(2) diag with SO(2) diag being the diagonal subgroup of SO(2) D × SO(2) R × SO (2). As in critical point III, the four sign choices lead to equivalent critical points.
We note that there are various possible values of the coupling constants h 1 and h 2 in order for critical points II and IV to exist. For definiteness, we will choose both h 1 and h 2 to be positive and take h 2 > h 1 . We will also choose the upper sign choice for critical points II, III and IV. Scalar masses at these critical points are given in Tables 1, 2, 3 and 4. To simplify the expressions, we have followed the notation of [9] by redefining the coupling constants as We also note that the existence of the N = 2 Ad S 5 critical point III requires ρ > 1 as pointed out in [9]. In the tables, we have given conformal dimensions of the operators dual to the scalar fields of N = 4 gauged supergravity by using the relation m 2 L 2 = ( − 4) with the Ad S 5 radius given by For some values of m 2 L 2 , we have given only one root of since the other root violates the unitarity bound > 1 for all values of ρ > 1. For g 2 = 0 or ρ → ∞, scalar masses for critical points I and II reduce exactly to those given in [10] for SO (2) 3 − 2 ρ , 1 + 2 ρ = 3 + 25 − 72 2+ρ , 1 + 25 − 72 2+ρ at critical points I and III are in agreement with those of the two scalar modes considered in [9]. However, the full scalar masses have not been given in [9]. We hope the present results fill this gap and could be useful in other holographic study.
Three of the eight massless scalars at critical point II are Goldstone bosons for the symmetry breaking SO (2)

Holographic RG flows
We now look for holographic RG flow solutions interpolating between supersymmetric Ad S 5 vacua identified in the previous section. These solutions take the form of domain walls in N = 4 gauged supergravity described by the metric ansatz ds 2 = e 2 A(r ) dx 2 1,3 + dr 2 (39)    with dx 2 1,3 = η αβ dx α dx β , α, β = 0, 1, 2, 3, being the fourdimensional Minkowski metric. All the scalar fields and the Killing spinors i are functions of only the radial coordinate r .
Before analysing the BPS equations, we first note the scalar kinetic terms in which we have denoted the r -derivatives by . We now consider supersymmetry transformations of ψ μi , χ i and λ a i . By setting these to zero with the metric ansatz (39), we obtain the corresponding BPS equations for the RG flow solutions of interest. The analysis is essentially the same as in [9,10], so we will mainly give the results with some detail omitted.
From the δψα i = 0 conditions, we find Multiply by A γ r and use (41) again, we find where W is the superpotential identified with the eigenvalues α 1 or α 2 of the A i j 1 tensor. Recall that choosing one of these eigenvalues breaks half of the N = 4 supersymmetry. By choosing W = 2 3 α 1 , we find that the corresponding Killing spinors are given by 1,3 while supersymmetry associated with 2,4 is broken. We then set 2 = 4 = 0 or equivalently impose the following projector for P = diag(1, 0, 1, 0). Using all these results in Eqs. (41) and (42), we find the flow equation for the metric function together with the γ r -projector with I i j defined by It should be noted that for φ 3 = 0, we have α 1 = −α 2 leading to N = 4 supersymmetry. In this case, the projector (44) is not needed. As expected, the condition δψr i = 0 gives the rdependent Killing spinors of the form i = e A 2 0i for constant spinors 0i satisfying (44) and (46). Finally, using the projector (46) in δχ i = 0 and δλ a i = 0 equations, we find the first-order flow equations for scalar fields.
By the procedure described above, we obtain the following BPS equations 164 Page 8 of 13 Eur. Phys. J. C (2023) 83 :164 which can be rewritten more compactly in terms of the superpotential as In deriving these equations we have chosen the upper sign choice in (45) and (46). This also allows for identifying the UV and IR fixed points as the Ad S 5 critical points at r → ∞ and r → −∞, respectively. It can also be verified that the BPS equations are compatible with the second-order field equations obtained from the Lagrangian (10). For g 2 = 0, the BPS equations reduce to those of N = 4 RG flows studied in [9,10] while for h 2 = 0, we recover the BPS equations for N = 2 RG flows in [9]. The former describes holographic RG flows between N = 4 critical points I and II and can be obtained analytically. The latter corresponding to holographic RG flows from an N = 2 SCFT in the UV to an N = 1 SCFT in the IR have been obtained numerically. Although the N = 4 Ad S 5 vacua in the SO(2) D × SO(3) × SO(3) gauge group considered here have some of the scalar masses different from those in the SO(2) × SO(3) × SO(3) gauge group studied in [9,10] as seen from Table 1, it turns out that setting φ 3 = 0 and φ 1 = φ 2 leads to the same BPS equations as those in [9,10]. Therefore, the flow solutions are the same and will not be repeated here. For other possible RG flows, we are not able to find any analytic solutions. Accordingly, we will rely on a numerical analysis for obtaining relevant solutions.
We first look at asymptotic behaviors of scalar fields near all of the Ad S 5 critical points. These behaviors give information about possible types of deformations for the SCFT dual to each Ad S 5 critical point. It is convenient to redefine the coupling constants as in (38) together with for 0 < ζ < 1 in order to have h 2 > h 1 > 0. Linearizing the BPS equations leads to the following results: Critical point I Near critical point I, the BPS equations give, for ρ = 2, with L I = 2 g . We see that , φ 1 and φ 2 are dual to operators of dimension = 2 while φ 3 is dual to an operator of dimension = 2 + 2 ρ . As pointed out in [9], , φ 1 and φ 2 correspond to the vev of relevant operators with dimension 2 while φ 3 leads to a source term of an operator with dimension 2 + 2 ρ in the dual N = 2 SCFT. For ρ = 2, however, the cubic term of the form φ 2 3 of the expansion of the superpotential is of the same order as the quadratic term 2 as pointed out in [3]. In this case, the asymptotic behaviors are given by which indicates that and φ 3 corresponds to source terms of dimension-2 and -3 operators as also shown in [9].

Critical point II
with L II = 2(1−ζ 2 ) 1 3 g . At this critical point, and φ 3 are still dual respectively to operators of dimensions = 2 and = 2 + 2 ρ as in the previous case. On the other hand, φ 1 and φ 2 are now dual to irrelevant operators of dimension = 6.
As in critical point I, these behaviors suggest that and φ 3 correspond respectively to a vev and a source term of operators with dimensions 2 and 2 + 2 ρ . For ρ = 2, the asymptotic expansion gives which again implies source terms for operators of dimension 2 and 3 dual to and φ 3 , respectively.

Critical point IV
Finally, we find the behaviors near critical point IV g(2+ρ) . and φ 3 are still dual to combinations of operators of dimensions = 1 + 25ρ−22 2+ρ and = 3 + 25ρ−22 2+ρ as in the case of critical point III while φ 1 and φ 2 are dual to combinations of operators of dimensions = 1 + ρ+26 2+ρ and = 3 + ρ+26 2+ρ . We note that these behaviors give conformal dimensions consistent with the scalar masses given in the previous section. We are now in a position to give numerical RG flow solutions to the BPS equations. First of all, we take the numerical values of the coupling constants to be There is also a direct RG flow from critical point II to critical point IV as shown in Fig. 2. We have consistently set φ 1 = φ 2 = 1 2 ln 1−ζ 1+ζ along the flow. Although this simplifies the BPS equations considerably, we are not able to find an analytic flow solution. As in Fig. 1, the dashed lines refer to the values at the critical points. This RG flow is driven by relevant operators of dimensions 2 and 2 + 2 ρ dual respectively to and φ 3 . The solution is similar to the N = 2 RG flow from N = 4 critical point I to N = 2 critical
In addition, we have found a genuinely new N = 2 Ad S 5 vacuum with SO(2) diag symmetry. We have also computed the full scalar masses at all of the aforementioned Ad S 5 vacua and studied holographic RG flows interpolating among these critical points. Apart from the known N = 4 RG flows between N = 4 critical points and the N = 2 RG flow from the trivial N = 4 critical point to SO(2) diag × SO(3) N = 2 critical point, we have found a family of RG flows between the trivial N = 4 critical point to the new SO(2) diag N = 2 critical point. Some of these RG flows pass arbitrarily close to the SO(2) D × SO(3) diag N = 4 critical point in addition to the direct RG flow from the trivial N = 4 critical point to the SO(2) diag N = 2 critical point. We have numerically given all of these RG flows. The results could give a holographic description of the possible RG flows between various conformal phases of strongly coupled N = 1 and N = 2 SCFTs in four dimensions and might be useful in other related study.
It would be interesting to identify precisely the dual N = 1 and N = 2 SCFTs dual to all the above Ad S 5 vacua and the corresponding RG flows. Since the SO(2) D × SO(3) × SO(3) gauged supergravity under consideration here has currently no known higher-dimensional origin, it is of particular interest to find the embedding of this N = 4 gauged supergravity in string/M-theory in which a complete holographic description can be obtained. This might be achieved by using recent results in exceptional field theories and double field theories. In particular, in [30], consistent truncations of N = 4 gauged supergravity with n ≤ 3 vector multiplets from eleven-dimensional supergravity have been shown. In this case, the N = 4 Ad S 5 vacua are embeddable in the maximal SO(5) gauged supergravity in seven dimensions. The extension of this result to include Ad S 5 vacua that cannot be embedded in the seven-dimensional theory might possibly lead to the embedding of the N = 4 gauged supergravity with SO(2) D × SO(3) × SO(3) gauge group. Finally, other types of holographic solutions such as black strings, black holes and Janus solutions within the N = 4 gauged supergravity studied here are also worth consideration. This could be done along the same line as in the recent results [10,31,32].