RG Flows and Stability in Defect Field Theories

We investigate defects in scalar field theories in four and six dimensions in a double-scaling (semiclassical) limit, where bulk loops are suppressed and quantum effects come from the defect coupling. We compute $\beta $-functions up to four loops and find that fixed points satisfy dimensional disentanglement -- i.e. their dependence on the space dimension is factorized from the coupling dependence -- and discuss some physical implications. We also give an alternative derivation of the $\beta$ functions by computing systematic logarithmic corrections to the Coulomb potential. In this natural scheme, $\beta $ functions turn out to be a gradient of a `Hamiltonian' function ${\cal H}$. We also obtain closed formulas for the dimension of scalar operators and show that instabilities do not occur for potentials bounded from below. The same formulas are reproduced using Rigid Holography.

An approach that has proven to be very useful in many instances is to search corners in the coupling parameter space in which to perform a controlled perturbative approximation. The semiclassical approximation itself is an example of this paradigm. Other examples include the large N approximation or the study of large spin sectors. A novel method introduced recently consists in the study of sectors of operators with large charge under a global symmetry (see [26] for a review and references). The method used in this paper is similar. This has been considered in [14,18] to study different aspects of (flat) defects in scalar field theories in d = 4 − ϵ and d = 6 − ϵ dimensions, by assuming a scaling limit of the couplings, where the defect couplings are large and the bulk couplings are small. As a result, quantum effects in the bulk vanish, while the defect still induces nontrivial quantum dynamics. In particular one can study the RG flow of the defect couplings and find interesting phenomena such as fixed point creation/annihilation. The results in [18] show, quite surprisingly, that the position of such fixed points is set by the oneloop approximation up to an overall scale that solely depends on ϵ. This separation of the dimension and coupling dependence is in general unexpected and it has been dubbed Dimensional Disentanglement (DD) in [18]. Additionally, the position of the fixed points can be dialed by tuning the bulk couplings, which act as knobs that can be adjusted.
In this paper we set out to study in more depth these aspects for flat defects in scalar field theories both in d = 4 − ϵ dimensions (where the defect is a line of codimension d T = 3 − ϵ) and in d = 6 − ϵ dimensions (where the defect is a surface of codimension d T = 4 − ϵ) . In particular, we extend the explicit two-loop computation of the defect β functions in [18] to four loops. This supports a conjecture that DD is actually a universal property holding for any theory in the double-scaling limit.
It was noticed in [18] that the two-loop β functions of the defect couplings are the gradient of a function H, where exp(H) matches the VEV of the circular defect. This has been proposed to reflect monotonic properties of the defect RG flow in [9]. Similar observations have been recently made in [24] for the 6d case, considering now a spherical two-dimensional defect.
Starting with three loops, the β functions contain scheme-dependent corrections. In the scheme of section 2 based on dimensional regularization, we find that the β-functions are no longer a gradient beyond two loops. The freedom left by the choice of scheme raises the question of whether there could be a scheme such that the β-functions are still a gradient of a function (as conjectured in [18]). This question is answered positively in section 3: an alternative calculation of the β function using the dressed Coulomb potential gives β i = 2 c ∂ i H up to four loop orders. We explicitly provide a formula for H for any 4d or 6d scalar field theory with general marginal potentials.
Using our results for the β functions, we construct theories in which, for ϵ = 0, both bulk and defect couplings are at a fixed point. These models thus define defect Conformal Field Theories (dCFT's). Given a dCFT, a problem of interest is to see if the theory may suffer from instabilities due to the presence of dangerously irrelevant operators. 1 Following [21], we study these possible instabilities in our theories, finding that they are absent provided that the potential is bounded from below. We also study a fermion-scalar theory with a Yukawa interaction in 4d, which perturbatively defines a dCFT, in search for such instabilities, finding also that they are absent.
When ϵ = 0, the double-scaling limit freezes the running of bulk couplings and the bulk theory becomes conformally invariant. In appendix C we make use of this property to engineer a setup suitable for holographic methods. As R d is conformal to H d T −1 ×S d T −1 , the theory can be directly put in H d T −1 × S d T −1 . Then the boundary of the H d T −1 is identified with the defect. This is similar in spirit to rigid holography [27] (for further developments along these lines, see e.g. [1, 2, 5-7, 10, 28, 29]). In our approach, we make use of this idea to compute defect β functions, finding a precise agreement with the field theory results.

Defects in scalar field theories and dimensional disentanglement
We consider a general theory with N scalar fields in d = 4 − ϵ, d = 6 − ϵ dimensions. Denoting the fields Φ i , with i = 1, · · · , N , we consider the following action in Euclidean signature, where V is a generic homogeneous polynomial in the Φ i 's of strict degree n, with couplingŝ We now consider a trivial defect which is a line in d = 4 − ϵ dimensions and a surface in d = 6 − ϵ dimensions. Hence the dimension of the worldvolume is 1 in d = 4 − ϵ dimensions and 2 in d = 6 − ϵ dimensions, while the dimension of the transverse space is d T = 3 in the 4d theory and d T = 4 in the 6d theory. In both cases the defect admits a (slightly relevant for ϵ ̸ = 0) deformation by the Φ i 's. Thus, we are led to consider the defect theory with action where δ T denotes the Dirac delta function in the transverse space to the defect. We are now interested in a particular scaling limit of both the defect and bulk couplings (bulk couplings are collectively denoted byĝ α ). Specifically, we are interested in a situation where the defect couplings are very large and the bulk couplings are small, keepingĝ α h n−2 i fixed. In this limit, pure bulk loop corrections that do not involve h i couplings are suppressed, while quantum effects get organized in powers of this effective finite coupling (ĝ α h n−2 i ).
To implement this limit, one can formally introduce new variables as follows: This gives Thus, we see that a semiclassical limit exists where ℏ → 0 while ν i , g α are fixed. In the following we will take this limit and explore its consequences, specializing to flat defects.

Solving the saddle-point equation in perturbation theory
In the double-scaling limit introduced above, there is a semiclassical expansion for S eff . The corresponding equations of motion are where the subscript in V means derivative with respect to ϕ i . We will solve these equations in perturbation theory in the bulk couplings, extending a calculation done in [18] to higher orders. To that matter we write ϕ i = ϕ Here V and its derivatives are evaluated at ϕ (0) i . We can now solve order by order. Order 0: The equation is It will turn out convenient to introduce the function Using that ϕ (0) i = ν i ϕ, and the fact that V is a homogeneous degree n function, we have the identity, (2.8) Order 1: At this order we have Therefore, using (2.8), where V i 1 ···im refers now to V i 1 ···im (ν i ). To lighten the notation, let us define: so that Order 2: The equation is now where we have used (2.8) to write the result in terms of

Order 3: The equation is
and ϕ 3 . (2.17) Note that, once again, we have used (2.8) to write the result in terms of In addition

Order 4: The equation is
We recall that n = 3 in the 6d theory and n = 4 in the 4d theory. The integral I appears only in the 4d theory, since V ijkl vanishes in 6d. In fact, in the n = 3 case all V ijkl appearing from order 5 on will vanish, simplifying the expressions of ϕ (m) i . Putting everything together, we can write The integrals are computed in appendix A. Focusing in the d = 4 − ϵ case, we finally obtaiñ where the coefficients are defined in appendix A.

Renormalization and β functions
For the sake of clarity, in the following we will first describe the case of d = 4−ϵ dimensions in detail. Expanding theφ i in d T = 3 − ϵ, one finds The C i 's are divergent as ϵ → 0. These divergences can be renormalized introducing a renormalized coupling u i by demanding that C 0 is finite. Restoring the powers of the scale, one finds where the RHS is evaluated at u i , and Here, as in [18], Ω = 1 32π 2 . Demanding that the bare coupling is independent on the renormalization scale (and using that the β function for the bulk couplings is β gα = −ϵ g α ) we find the β functions for the defect couplings: A similar calculation in d = 6 − ϵ dimensions gives β functions with the same structure as in (2.26). To three-loop order, the coefficients are now given by where, for the 6d theory, we define Ω = 1 8π 2 . The coefficients β (1) and β (2) in (2.27) and (2.28) reproduce the two-loop terms previously computed in [18], and also agree with the earlier calculations in [11,12] once the double-scaling limit is taken (c.f. eq. (19) in [11] and eq. (3.17) in [12]).

Fixed points and dimensional disentanglement
Let us consider the four-loop β functions (2.26). One can check that all solutions of β i = 0 are of the form that is, the ϵ-dependence factorizes, with This suggests that the location of the fixed points of the defect theory to all orders is determined by the vanishing of the one-loop β function up to a universal overall function f which entirely encodes the ϵ dependence (and hence the dimension), a phenomenon which was dubbed dimensional disentanglement (DD) in [18]. The F one−loop i (g α ) is given in terms of ratios of bulk couplings. These ratios are RG invariants, since all couplings g α have the same classical flow, β gα = −ϵg α .
The four-loop check extends the conjecture of [18] to the general class (2.2) of scalar field models with defects. Surprisingly, we find that the function f (ϵ) is universal: it is the same function for any 4d scalar field theory of the type (2.2). Although we have focused on the four-dimensional models, DD in the fixed points also occurs in 6d scalar field models with defect of the form (2.2), a property which, as shown below, only holds in the doublescaling limit. In the 6d case, the function f (ϵ) is different, and one finds an expansion of the form f 6d (ϵ) = ϵ 2 (1 + ϵ + · · · ).
To understand the origin of DD, it is useful to derive the solutions of β i = 0 in detail. Let us write Here everything is assumed to be evaluated at a i ϵ 1 2 . The vanishing of the β functions implies a cancellation order by order. A crucial property in the derivation is the homogeneity of V , which implies the general relation (2.32) From the leading term, we find Using this and the homogeneity of V , the second term gives the relation This can be easily solved by choosing b i = m a i for some m, since, by virtue of the homogeneity of V , Using again the leading order equation, we obtain This yields m = 3 4 , in agreement with the expansion of (2.30). As for the last term, let us also assume c i = κ a i . Then Using the leading order equation and the homogeneity of V we find Solving for κ we get κ = − 3 32 , once again in agreement with the expansion of (2.30). Even though we have so far explicitly shown dimensional disentanglement up to four loops in general theories, it is clear that the strategy extends to arbitrary orders. To further understand DD it is enlightening to study when it fails to hold, as happens upon including bulk loops. Focusing for definiteness on d = 4 − ϵ, where V is quartic, these enter to order O(V 2 ), with the diagrams in figure 1.
The first diagram contributes to the anomalous dimension of ϕ. The second diagram would produce a term in the β function of the form (2.40) Thus, to this order the full β functions would be We already see a crucial difference: while in the large charge limit, the β function to order O(V k ) contains a total of 2k − 1 derivatives, the corrections (coming from bulk loops) contain, to order O(V k ), more derivatives. For instance, the leading correction to order O(V k ) contains 2k + 1 derivatives. To see the implications of this, let us proceed as before and assume where again everything is evaluated at a i ϵ 1 2 . Grouping terms with the same dependence of ϵ we see that We now find the leading equation The crucial difference is that now there is an extra term with higher powers of the bulk coupling constant. We can solve this equation in perturbation theory, finding where a 0 i is the solution to ϵ 3 2 Thus we see that upon including loop corrections, the fixed points will be of the form where the {f i,k } are non-trivial functions of the bulk couplings. They are of the form In the double-scaling limit, 1 + O(g k+1 α ) → 1 and the function F i (g α ) factorizes, with giving rise to dimensional disentanglement. Summarizing, dimensional disentanglement is tied to the fact that, in the double scaling limit, to any given order in the bulk couplings only terms with the same number of derivatives of the potential with respect to the fields appear. This is no longer true in the full quantum theory once bulk loops are included. DD arises also thanks to the homogeneity of the potential (it is a degree n polynomial in the fields, linear in bulk couplings, with n = 4 in 4d and n = 3 in 6d).
In [18] it was shown that, up to two-loop order, the β functions can be obtained as a gradient from a function H, that is, β i = 2∂ i H. Although the four-loop β functions in (2.26) are not the gradient of any function, nevertheless dimensional disentanglement still holds, due to the structure of the corrections described above. Beyond two loops, the β-functions have a scheme-dependence and, as discussed below in section 3.1, it is possible to choose a scheme where they are still given as a gradient function.

Some physical implications of DD
DD implies that fixed points have the form (2.29). The main physical consequence is that the positions of fixed points in the defect coupling space do not depend on ϵ modulo an overall scale given by f (ϵ). In other words their relative position is independent of the dimension.
The RG flow, however, can have a dependence on the dimension, despite the fact that fixed points do not move when ϵ is varied, except for an overall scale. The way this happens can be illustrated by the twins model discussed in [18]. It is defined by the action (d = 4−ϵ) The β functions for the defect couplings can be read from (2.26). To quadratic order in the couplings, they are given by [18] (2.52) Defining In [18], IR stability was studied only to linear order in ϵ.
To understand to what extent quantitative and qualitative features of the RG flow can depend on ϵ, it is important to extend the stability analysis to order ϵ 2 . Consider the RG time variable t = − log µ.
Perturbing around the fixed points we find the following eigenvalues (λ 1 , λ 2 ) of the Hessian: (2.58) IR stability of a given fixed point requires that both eigenvalues are negative. We see that the ϵ dependence does not factorize. Stability properties change by varying ϵ at fixed couplings (ζ, η). For example, taking η ≫ 1, the b) fixed point is stable for sufficiently small ϵ, but it becomes unstable when ϵ > 2/η. This implies a drastic change in the RG flow, despite the fact that the relative positions of fixed points remain unchanged: an attractive fixed point becomes repulsive as ϵ is increased above a critical value (while keeping ϵ ≪ 1).
In conclusion, in the double-scaling limit, on one hand, fixed points satisfy the DD property, which allows one to determine them exactly (modulo the overall numerical constant f (ϵ)) by a one-loop calculation. On the other hand, β functions still describe extremely rich RG flows exhibiting phenomena such as fixed point creation/annihilation and non-trivial dynamics as ϵ is varied.

Alternative calculation of β-functions
In this section -in which we will set ϵ = 0, that is, we shall compute the β-functions of the defect couplings for d = 4, 6 (this means that in our convention d T = 3, 4 respectively)we will show that the β functions can be computed in an elegant way from corrections to the Coulomb potential. In the appendix C a similar calculation of the β functions will be given using rigid holography.

β function for the defect couplings
Let us start with the action (2.4). Recall that V is a homogeneous polynomial of the fields ϕ i of degree n = 4 in the d = 4 theory, and n = 3 in the d = 6 theory.
We shall use spherical coordinates, and place the defect at r = 0. Explicitly For our purposes, it is sufficient to consider spherical symmetric solutions, where ϕ i only depends on r. Under this assumption, the equation of motion reads where V i = ∂V ∂ϕ i . Writing the equation of motion away from the source becomes where V i is now evaluated at u i . We can solve this equation in perturbation theory by setting where f (k) is of order g k α and s i is a constant. Up to order 3 where V and its derivatives are now evaluated at s i . It is straightforward to solve this equation order by order, finding The constants s i can be determined from the δ T source term on the right hand side of the equations of motion (3.2). They are given by The charges u i can be viewed as a "running" version of ν i . To third order where now it is understood that V and its derivatives are evaluated at s i . The numerical constant Ω was introduced in section 2.2 (Ω = 1 32π 2 in d = 4; Ω = 1 8π 2 in d = 6). Inverting this formula, we get (3.9) with V and its derivatives being evaluated at s i . Interpreting r −1 as the RG scale, we can compute the β function for u i by imposing the scale-independence (r-independence) of the "bare coupling" ν i . We obtain with c = 1/(d T − 2) (hence c = 1 in d = 4 and c = 1/2 in d = 6). Up to second order, this formula exactly matches the quantum field theory results given in (2.26), (2.27), (2.28), for the 4d and 6d theories. As we will shortly review, this is to be expected, since only the one-loop and two-loop terms of the β function are expected to be scheme-independent. Using this same method it is straightforward -albeit tedious-to go to higher loops. The four loop contribution is derived in appendix B. Remarkably, the β function is a gradient flow in the defect coupling space, where This supports the conjecture made in [18], albeit in a particular scheme which coincides with the one implicitly chosen by this alternative method.

Changing scheme
From 3-loops on, the coefficients in the β function obtained through the previous method fail to match the corresponding coefficients in the field-theoretic result in (2.26). For example, in the four-dimensional theory, in (2.27) and (2.28), β 1 = 16c 5 Ω 3 . We note that the coefficient β 2 is the same in both calculations. It is well known that β functions are scheme-dependent beyond two loops. To understand the origin of the discrepancy in more detail, let us study the effect of changing the scheme. To do this, we redefine our u i couplings in terms of new couplingsũ i . A natural ansatz is whereṼ means V evaluated on theũ R i 's. Then Inverting this matrix. we get In turn Thus we see that the coefficient β where we find a disagreement between (2.26) and (3.11) is precisely that altered by redefinition of couplings.

Instabilities in defect field theories
In the previous sections we have computed the β functions for the defect couplings assuming a double-scaling limit. In particular, the effect of such limit is to freeze the running of the bulk couplings, in such a way that the bulk theory is effectively a CFT if we set ϵ = 0 (so that the classical running is also frozen). Thus, armed with our previous results, we will now study cases where also the defect β functions vanish, so that we have a defect CFT (dCFT). It is of interest to investigate if these potential dCFT's may have further instabilities triggered by condensates of marginal or relevant operators, just as it happens in the scalar QED example of [21] (see also appendix C.2), where the dCFT ceases to exist beyond certain critical values of the couplings.
To make this concrete, let us consider a model with two fields ρ and ⃗ ϕ, being ⃗ ϕ an O(N ) vector. We choose the potential to be of the form ρ n−2 ⃗ ϕ 2 , with n = 4 in d = 4 and n = 3 in d = 6. Introducing now a defect to which in general both ρ and ϕ i couple, the βfunctions of the defect couplings can be computed from (2.26). Denoting the corresponding renormalized defect couplings by u ρ and u ϕ i in the obvious way, it is straightforward to check that the model has a fixed point at u ϕ i = 0 for arbitrary u ρ . 2 The action is given by (we denote the only bare defect coupling simply by ν) which describes, in principle, a dCFT. We wish to study whether, similarly to the QED case in [21], there are other instabilities triggered by relevant operators.

One-loop considerations in field theory
In this subsection we shall analyze the stability of the fixed points in perturbation theory. Let us first consider the theory in the absence of the defect (the bulk theory). Prior to the scaling limit in (2.3) theĝ coupling has a β function which reads βĝ = bĝ a + · · · (a = 2 in d = 4, a = 3 in d = 6). Then, upon taking the limit (2.3) β g = ℏ a 2 (n−2) b g a + · · · . (4.2) Therefore, in the limit ℏ → 0 with fixed g, β g vanishes and hence the bulk theory is classical (and conformal). Note in particular that all bulk loops vanish: a diagram with L bulk loops (and no interaction with the defect) is proportional toĝ L = ℏ a 2 L(n−2) g L , which vanishes in this limit.
Let us now turn to the defect. One way to search for instabilities is to look for marginal/relevant operators in the defect. Such information is encoded in the correlation functions of the defect operators. Denoting generically the fields by Φ i , one would generically be interested in ⟨Φ I 1 (x 1 ) · · · Φ In (x n )⟩, whose path-integral representation is However, in the double-scaling limit this integral simplifies to where ⟨Φ I (x)⟩ is the field evaluated in the semiclassical solution obtained in section 2, which can be identified with the one-point function of Φ I . Thus, in this limit, the correlator is completely dominated by the disconnected piece. 3 The disconnected piece is non-vanishing due to defect interactions. The one-point function in the presence of the defect is given in eq. (4.3) in [18]. This can be cast as Using this formula for the ϕ i fields in the models at hand, we obtain (4.7) From (4.4), it also follows that ∆(ϕ i 1 ...ϕ ir ) = r ∆(ϕ i ). Provided Q, P ≥ 0, the ϕ i 's and all operators made with ϕ i are irrelevant in perturbation theory, since ∆(ϕ i ) ≥ d T − 2. On the other hand, applying the formula (4.6), one gets ∆(ρ) = d T − 2, so the ρ deformation is marginal. We shall discuss more aspects of stability in the next subsection.

Exact dimensions and instabilities
The analysis above is just the statement that the theory indeed perturbatively defines a dCFT. However, one may fear that this is a statement only holding in perturbation theory. Indeed, in view of the dimensions above, one may worry that for large enough Q, P one may find operators going towards marginality. Let us first consider the four-dimensional case. We note that Q > 0 for g > 0, that is, when the potential is positive. Thus, ∆( ⃗ ϕ) > 1 and, in perturbation theory, the theory is stable provided the potential is bounded from below. As a consequence, in this regime we indeed have a dCFT as anticipated, recovering exactly the same conclusions as those drawn originally from the fixed points of the β functions. However, a question of interest is what happens for finite values of Q; in particular, whether instabilities as those appearing in scalar QED can appear in field theories containing only scalar fields. An exact formula for finite Q in the double scaling limit can be obtained by solving saddle-point equations, which effectively resums the perturbative series. For the model we are studying, the equations of motion are (we now assume "mostly minus" Minkowski signature) Here we have used the O(N ) symmetry to align the ⃗ ϕ along some direction. These equations are solved by ϕ = 0 and Computing the integral one finds Let us now consider time-independent fluctuations around the background. In polar coordinates as above, the eom for the ϕ fluctuation is where Q is precisely the same Q as introduced above. The general solution is given by where Y lm are the spherical harmonics on the S 2 , while , (4.13) with ∆ ± l = 1 ± 2 1 4 + l(l + 1) + Q . (4.14) Note that time-dependent fluctuations have the same behavior in the vicinity of r = 0. This is seen by adding a factor e iEt in the ansatz for ϕ. In (4.11) this gives rise to a new term E 2 ϕ, which can be neglected in the vicinity of r = 0.
The two sets of solutions with coefficientsφ + lm andφ − lm in principle define two different dCFT's according to the choice of boundary conditions. Settingφ − lm = 0 leaves a set of defect operatorsφ + lm whose dimension can be read from (4.13) using the fact that r has dimension -1 and the bulk field ϕ (hence R lm ) has dimension 1 in 4d (recall that there is no bulk anomalous dimension as bulk loops are suppressed). This gives ∆(φ + lm ) = 1 2 ∆ + l or, for the alternative boundary condition, ∆(φ − lm ) = 1 2 ∆ − l . For g > 0, corresponding to a potential bounded from below, Q > 0. It then follows that, since ∆(φ − lm ) < 0, the alternative boundary conditions are not allowed. Consider now ∆(φ + lm ). It follows that ∆((φ + lm ) 2 ) = ∆ + l . Expanding at small Q, we verify that the first corrections in Q for ∆ + l=0 matches the perturbative formula (4.7). The formula for ∆ + l also shows that ∆(φ + lm ) > 1, which implies that {φ + lm } correspond to irrelevant operators. General possible deformations are composites of the schematic form ∂ l T ϕ k , where ∂ l T represents the action of l derivatives with respect to the transverse coordinates. They are all irrelevant operators. Therefore in this theory there are no instabilities at any finite Q and the model indeed describes a dCFT.
Instabilities appear for the theory with g < 0, which corresponds to an unbounded potential. In this case, Q < 0 and alreadyφ + l=0 -corresponding to ϕ itself-becomes relevant (of dimension ∆ = 1 2 (1 + 1 − |Q|) < 1), and must be added to the defect. However, here we will not consider theories with unbounded potentials.
Let us now comment on the 6d theory. With no loss of generality, we may assume g > 0, since the sign of g can be flipped by a redefinition ρ → −ρ, ν → −ν. In this case the potential is unbounded in the negative ρ direction. A similar calculation as above, leads to a background for ρ given by (4.10) with d T = 4. Then, one finds the following formula for the dimension of operatorsφ ± lm (x || ) on the defect, The expansion at small P of the branch with + sign reproduces the perturbative result in (4.7). We note that P > 0 if and only if gν > 0. This is precisely the case where the term of the potential gρ|ϕ| 2 is positive in the background provided by ρ, for either sign of ν. In this case there is a dCFT defined by choosing the boundary conditionφ − lm (x || ) = 0. As before, the {φ + lm (x || )} correspond to operators of the schematic form ∂ l T ϕ n , which are all irrelevant.

A glimpse into fermion models
We now consider the possibility of constructing dCFT's involving scalar and fermion fields using the double-scaling limit (see [19,20] for other interesting studies of fermion dCFT's and boundary CFT's). This is feasible in four dimensions, where the Yukawa interaction is classically marginal.
Let us consider a Dirac fermion coupled to a real scalar field with a Yukawa interaction, with the action (using 'mostly minus' Minkowski signature) In this model the trivial line defect along x 0 is deformed by a classically marginal deformation provided by the scalar itself. The double-scaling limit corresponds to consider the case of a large defect coupling h and small bulk couplingĝ; specifically,ĝ → 0 and h → ∞ with hĝ fixed. This is formally implemented by the scaling h = ℏ −1 ν ,ĝ = ℏ g . (4.17) Upon appropriately rescaling the fields, the action becomes In the ℏ → 0 limit with g and ν fixed, bulk loops are suppressed and quantum effects arise due to the interaction with the defect in terms of the effective coupling gν. Let us first consider the bulk theory by itself, i.e. let us momentarily set ν = 0. Prior to the scaling limit, the β function for theĝ coupling is βĝ = bĝ 3 + · · · . Therefore β g = ℏ 2 b g 3 + · · · → 0 and there is no RG flow in the bulk, as expected since bulk loops are suppressed. The bulk theory is a CFT. We now turn to the defect. In the absence of bulk loops, no diagram can correct the ρ one-point function. 4 As a result, ν does not run and the theory seems to be indeed a dCFT. This is basically a consequence of the fact that the theory contains, at least in perturbation theory, no other operator close to marginality on the line.
Besides the ρ operator, the lowest scalar operator that the theory contains isψψ. Classically this has dimension 3, and therefore it is safely irrelevant in perturbation theory. However, it is important to understand whether for large values of the couplings this operator can hit marginality and eventually become a relevant operator of dimension less than 1. To study the problem, we proceed as before. Like in the previous scalar model, the defect induces a background for ρ given by It remains to study the fermion fluctuations in this background. Using the Dirac representation for the γ matrices, the equation of motion for the fermion fluctuations decomposes into two equations for the Weyl spinor components χ and ξ We are assuming time-independent fluctuations (time dependence can be incorporated by a factor e iEt , which leads to a subleading dependence near r = 0 and it is thus unimportant for the determination of the dimension). Solving for ξ in the second equation and substituting it in the first equation, we find (4.21) The solution to this is being χ 0 a constant spinor. Now, given that the dimension of a bulk fermion is 3/2, we can write In order to avoid negative dimensions, we must impose boundary conditions that keep the branch with the '+' sign. Therefore In the free-field limit, ∆(ψ + 0 ) → 3/2, as expected. It then follows that the dimension ofψψ is 2∆(ψ + 0 ). Since Q F > 0 (at least for unitary theories),ψψ is an irrelevant operator in the whole region Q F ∈ (0, ∞) allowed by unitarity. This supports the fact that the line defect in the double-scaling limit indeed defines a dCFT. Finally, as a curiosity, let us comment that had we considered the case of a parity breaking theory with a potential iĝρψγ 5 ψ we would have obtained exactly the same results.

Conclusions
In this paper we have considered generic -that is, with an arbitrary number of scalar fields and an arbitrary marginal potential-d dimensional scalar field theories with defect deformations, in d = 4 − ϵ and d = 6 − ϵ as well as a scalar-fermion theory with a Yukawa interaction in d = 4. All calculations are performed in a double-scaling limit (2.3), where the defect couplings go to infinity and the bulk couplings go to 0.
We summarize the main results of this paper.
• β functions for the defect couplings have been computed up to four loops using dimensional regularization and standard perturbation theory of the QFT.
• The fixed points exhibit the property of dimensional disentanglement, namely the dependence on the dimension appears through a universal function, which is factorized from the coupling dependence. The universal function is the same for all fixed points, and for all models, independent of the number of scalar fields and independent of the potential. We showed that the DD property is a peculiar feature of the double-scaling limit and that it is not expected to hold once the full quantum effects are taken into account.
• DD implies that, modulo an overall scale, the location of fixed points remains unaltered as the dimension is varied. However, the RG flow depends on the dimension in a non-trivial way. In particular, an IR stable fixed point can become unstable by varying ϵ while keeping ϵ ≪ 1.
• In section 3 we provide an alternative calculation of the defect β functions from the dressed Coulomb potential. In this scheme, (at least up to four loops) the β functions are a gradient, β i = 2 c ∂ i H, where ∂ i stands for derivatives with respect to the couplings ν i of the defect deformations ν i ϕ i .
• We have considered a few concrete examples of dCFT's and computed the dimension of operators that could lead to instabilities. The first examples are 4d and 6d scalar field models obtained by sitting on particular fixed points. In all cases we showed that all potentially dangerous operators are irrelevant even for finite values of the coupling, therefore the dCFT's are stable. The results also show that instabilities may appear if one considers potentials that are unbounded from below, beyond some critical coupling.
• In addition, we computed the dimension of the bilinear fermion operatorψψ in fermion-scalar theory with Yukawa interactions, showing that it is always irrelevant (for a real Yukawa coupling, where the theory is unitary).
• In the appendix C we provide a practical framework for rigid holography, by which one can compute β functions to all loop orders. The approach is essentially equivalent to the field theory calculation of section 3, being related through the conformal map. The double scaling limit leads to effects that are analogous to the effects produced by the large N limit in standard holography: it suppresses bulk loops and makes the correlation functions dominated by the disconnected term.
• Using as an example the scalar QED model studied in [21], we also show that rigid holography can be used to compute the dimension ofφϕ.
There remain many open questions and many interesting aspects of defect theories, which are worth of further investigation. In particular, it would be interesting to establish if exp(H) is related to the VEV of the circular defect (in the 4d theory) or to the VEV of the spherical defect (in the 6d theory) to any order in the loop expansion. Another interesting problem is extending the application of rigid holography to theories on spaces H p+1 × S d−p−1 and codimension p defects for other values of p. Other very interesting problems include understanding if dimensional disentanglement also arises in theories with fermions or vector fields, or the role of unbroken global symmetries and conformal manifolds along the lines of [15].

A The integrals
In this appendix we collect technical details of the evaluation of the integrals, borrowing results from [32]. First, we compile formulas for several integrals that appear repeatedly.
Consider the following integral Explicit evaluation gives the formula .
Another useful integral is This gives It will turn out to be convenient to introduce Let us now compile the results for the relevant integrals for n = 4.

Order 0
To order zero (A.7)

Order 1
We now need to compute I 1 . After some manipulations (A.8) Using the formulas above, we see that (A.9)

Order 2
We now need I 2 , which can be re-written as (A.10) with Using the results for the integrals above Order 3 Now we have two integrals 3 : after some tedious but straightforward manipulations, one can show that Using the results above This can be written as 3 : in this case, one finds Using the results above This can be rewritten as It then follows that 3 . (A.21)

Order 4
Now we have 4 integrals 4 : we have I which reduces to the simpler form where the V are evaluated at h i . Inverting this formula we get −p 3 (log r) 4 + 4p 2 (log r) 3 − 12p(log r) 2 + 24 log r .

(B.3)
Now we interpret once again r −1 as the RG scale and differentiate both sides of the equation with respect to log r to get the β function for u i . We obtain where c = 1/p = (d T − 2) −1 . From this formula, we compute the function H found in (3.12) up to fourth loop order.

C Rigid holography
In this section we will show that the β functions and dimensions can also be computed using holographic techniques. As it is well known, R d can be conformally mapped to H a+1 × S b+1 with a + b + 2 = d. To see this, we start with the R d metric, written as ds 2 R d = dr 2 1 + r 2 1 ds 2 S a + dr 2 2 + r 2 2 ds 2 S b , a + b + 2 = d . (C.1) Next, we perform the following change of coordinates r 1 = sinh ρ cosh ρ − cos ψ , r 2 = sin ψ cosh ρ − cos ψ . (C.2) Then, the metric becomes ds R d = F 2 dρ 2 + sinh 2 ρ ds 2 S a + dψ 2 + sin 2 ψ ds 2 S b , F = 1 cosh ρ − cos ψ . (C.3)

C.2 Dimension of gauge-invariant operators in scalar QED
In a recent paper [21], Aharony et al. studied phase transitions in scalar QED with a Wilson line, by computing the dimension of scalar operators on the defect. Rigid holography can also be used to reproduce the results in [21]. We consider a Wilson line along, say, x 0 in R 1,3 . Assuming mostly minus signature, the action is (we follow the conventions in [21]) Introducing nowλ = λ e 2 andq = q e −2 and appropriately rescaling the fields, we can write We now take the semiclassical limit e → 0 with q and λ fixed. In this limit bulk loops are suppressed. Since q cannot run due to gauge invariance, we again find, naively, a dCFT. Let us now map the problem to AdS 2 × S 2 , with metric (C.4). Assuming an ansatz A 0 = A 0 (z) and ϕ = ϕ(z), the bulk equations of motion become ∂ z (z 2 ∂ z A 0 ) + 2 e 2 A 0 |ϕ| 2 = 0 , ∂ 2 z ϕ + e 2 A 2 0 ϕ − z −2 λ |ϕ| 2 ϕ = 0 . (C.10) Let us now look for the appropriate holographic configuration representing charge source. The general background solution for A 0 with z-dependence and ϕ = 0 is given by A 0 = a 0 + j 0 z −1 .
We will choose the boundary condition a 0 = 0, which corresponds to a current -as opposed to a dynamical gauge field-in the boundary [33]. Moreover, just as in the scalar case, the j 0 constant is fixed by the Coulomb law as Turning now to the equation for the ϕ fluctuations in this background, to quadratic order one finds The solution to this equation is Note that in terms of the original variables Q = e 4q2 4π 2 . (C.14) This reproduces the results in [21], now by using holography.
As discussed in [21], there are two possible quantizations, corresponding to the two possible boundary conditions C + = 0 or C − = 0. In one quantization, the Wilson line defines a stable dCFT with |ϕ| 2 being an irrelevant deformation of dimension ∆ + = 1 + √ 1 − Q. The other quantization defines an unstable dCFT where |ϕ| 2 is a relevant deformation of dimension ∆ − = 1 − √ 1 − Q. As Q is increased, at Q = 1 these two branches approach and merge, resulting in fixed point annihilation and conformality loss. From the viewpoint of the stable dCFT, the naively irrelevant operator |ϕ| 2 decreases its dimension and eventually becomes marginal at Q = 1 (in standard terminology, it is a dangerously irrelevant operator).
It is instructive to compare with the scalar field models, where instabilities only appeared for potentials with the wrong sign. This is consistent with the fact that in scalar QED the term A 2 0φ ϕ contributes with negative sign to the effective potential. As a result, the effective charge Q appears with negative sign inside the square root, leading to instabilities at critical values.