Non-Perturbative Defects in Tensor Models from Melonic Trees

The Klebanov-Tarnopolsky tensor model is a quantum field theory for rank-three tensor scalar fields with certain quartic potential. The theory possesses an unusual large $N$ limit known as the melonic limit that is strongly coupled yet solvable, producing at large distance a rare example of non-perturbative non-supersymmetric conformal field theory that admits analytic solutions. We study the dynamics of defects in the tensor model defined by localized magnetic field couplings on a $p$-dimensional subspace in the $d$-dimensional spacetime. While we work with general $p$ and $d$, the physically interesting cases include line defects in $d=2,3$ and surface defects in $d=3$. By identifying a novel large $N$ limit that generalizes the melonic limit in the presence of defects, we prove that the defect one-point function of the scalar field only receives contributions from a subset of the Feynman diagrams in the shape of melonic trees. These diagrams can be resummed using a closed Schwinger-Dyson equation which enables us to determine non-perturbatively this defect one-point function. At large distance, the solutions we find describe nontrivial conformal defects and we discuss their defect renormalization group (RG) flows. In particular, for line defects, we solve the exact RG flow between the trivial and the conformal lines in $d=4-\epsilon$. We also compute the exact line defect entropy and verify the $g$-theorem. Furthermore we analyze the defect two-point function of the scalar field and its decomposition via the operator-product-expansion, providing explicit formulae for one-point functions of bilinear operators and the stress-energy tensor.

1 Introduction and Summary 1.1 Defects and their conformal limit  [1] and the Kondo impurity in lattice models [2]. More generally, a defect is characterized by its spacetime dimension p ≤ d −1 or codimension q = d −p and modifies the QFT path integral.
In recent years, the study of defects in QFT have been catalyzed by a combination of numerical and analytical tools that expand on the success in the case of conventional QFT observables such as local correlation functions. This is especially the case for defects in Conformal Field Theory (CFT) whose defect RG fixed points are described by Defect Conformal Field Theory (DCFT). These conformal defects (DCFTs) correspond to universality classes of defects in a given CFT and serve as the starting points to explore the full defect landscape.
If the defect has an explicit Lagrangian description, one can always compute the Feynman diagrams as in the standard perturbation theory to determine defect observables. However in practice what one obtains this way is at best an asymptotic series in the small couplings and it is generally difficult to extract non-perturbative results at finite couplings. Several powerful methods have been developed in the last few years to overcome this obstacle using large N [50][51][52][53][54][55], large charge [56][57][58][59], and integrability [60][61][62]. 1 In particular, perhaps the simplest interacting scalar field theory, the critical O(N) model (Wilson-Fisher fixed point in d = 4 − ǫ dimensions), is known to host a rich family of line defects (p = 1) connected by nontrivial defect RG flows, and similarly for boundaries (d − p = 1), which can be studied analytically in the large N limit [51][52][53]55] (see also related work in the large charge limit at finite N [56,58] and previous bootstrap studies at N = 1 [36]).

Scalar quantum field theory and localized magnetic defects
Scalar quantum field theory is arguably the most intensively studied class of QFTs. Despite the general obstacle to obtain finite coupling results, perturbation series in these models can exhibit simplifying features in certain large N limits that make them more tractable. The aforementioned O(N) model is the familiar theory of N scalar fields φ a with a = 1, 2, . . . N and a quartic O(N)-invariant potential, also known as the O(N) vector model. The theory has a solvable large N limit that flows to an interacting but weakly coupled CFT for d < 4.
In particular, at infinite N the scalar fields have classical scaling dimensions (thus free) and the theory has a higher spin symmetry that is weakly broken by anomalous dimensions at and it is a unsolved problem to determine finite λ M dependence in matrix models at d > 0.
Given the incremental complexity from the vector to the matrix O(N) models, one would be surprised to find solvable yet non-trivial tensor models of rank k ≥ 3. Nonetheless, it has been shown that such a large N limit exists for tensor models at k = 3, where the perturbative expansion is governed by a single O(N) 3 -invariant quartic coupling known as the tetrahedral interaction and certain (fattened) Feynman diagrams of the melonic type dominate [70] (generalizing the d = 0 models in [71][72][73] and the d = 1 fermionic model in [74]). Thus this limit is commonly referred to as the melonic limit. The corresponding tensor model is known as the melonic tensor model which is described by the following action, The weakly broken higher spin symmetry in the O(N ) vector model is so constraining that the theory is essentially fixed at the first nontrivial order in the 1/N expansion [66].
3 See [68] for a review of the large N tensor models.
As in the matrix large N limit, there is a nontrivial 't Hooft coupling λ T (rescaled tetrahedral coupling) in the melonic limit. However, unlike in the matrix case, here miraculously, the infinite series in λ T can be resummed using an exact Schwinger-Dyson equation, which has lead to the discovery of strongly-coupled scalar CFTs in d < 4 with order N 3 degrees of freedom whose correlation functions can be determined exactly in the leading large N limit [70]. 4 Thus we see the melonic tensor model serves as an interesting middle ground in the world of scalar QFTs between the vector and matrix models, which is rich yet solvable. 5 Thus far the study of defects in the non-supersymmetric setting is largely restricted to weakly coupled bulk QFTs, which has already produced many interesting examples with intriguing features [55,58,[76][77][78][79][80][81]. 6 Bulk interactions will certainly modify the defect observables in nontrivial ways and it remains a challenge to incorporate their effects at finite coupling. Here we will capitalize on the advantageous features of the melonic tensor models mentioned above to study defects in strongly-coupled scalar QFTs.
More explicitly, we study the following defect in the melonic tensor model defined by a localized source for one component of the scalar field φ abc on a p-dimensional subspace, where we fix the flavor indices (abc) (no summation). 7 We refer to the coupled system as the defect tensor model. Here we have split the flat spacetime coordinates as x = (x , x ⊥ ) or with indices x µ = (x α , x i ⊥ ) where α = 1, 2 . . . , p and i = p + 1, . . . , d, so that the defect world-volume has coordinates x and locates at x ⊥ = 0. This is analogous to the pinning field defect (or localized magnetic defect) studied in the context of (2+1)-dimensional lattice models [82], where the defect is described by coupling the order parameter to a background magnetic field localized in space and extending in the time direction. The continuum field theory analysis of such defects was carried out in the vector O(N) model in [55]. We will follow suit and refer to the coupling in (1.3) as defining the localized magnetic defect D p in the tensor model. Note that contrary to previous studies of such defects in the vector O(N) model, we do not restrict to line defects (i.e. p is general). 8 4 One may wonder what happens for tensor models beyond rank k = 3. See [75] for a generalization of the solvable melonic limit to the higher rank tensor model at k ≥ 4, which is governed by a degree k + 1 "maximally single-trace" interaction that generalizes the tetrahedral coupling in (1.2). 5 At the diagrammatic level, one can see explicitly that the melonic diagrams constitute a special subset of the planar diagrams and this makes the resummation possible [68]. 6 In particular, even free theories can host nontrivial defects (see for example [76][77][78][79][80][81]). 7 It would be interesting to consider more general defect couplings. 8 The physically interesting cases correspond to d = 2 with p = 1 and d = 3 with p = 1, 2 but formally symmetry, the defect one-point function φ abc (x) Dp is zero unless a = b = c = 1. The main goal here is to study such nontrivial defect correlation functions at the non-perturbative level and we summarize the main results below. While it is straightforward to compute a few of these Feynman diagrams at leading orders in the λ T andJ 111 expansion for the one-point function φ 111 (x) Dp , it may seem a formidable task to obtain the nonperturbative answer which requires summing infinitely many diagrams. The success of the melonic tensor model without defects motivates us to look for a generalization of the large N limit thereof to the case with defects. Indeed, from we will work with general 1 ≤ p < d. This is possible because in our large N limit (1.4), the defect problem is reduced to solving a partial differential equation (from the Schwinger-Dyson equation) with a fractional Laplacian in the transverse directions of order d 2 .

Summary of the main results
a topological argument, we prove in Section 2 that in the limit 4) and λ T fixed as in the melonic tensor model without defects, the dominant contributions to φ 111 (x) Dp come from melonic tree diagrams anchored on the defect with the exact propagator of φ abc on each edge (see Figure 1), Furthermore these infinitely many diagrams can be resummed thanks to an exact Schwinger-Dyson equation with the magnetic source, which determines φ 111 (x) Dp (as well as other defect correlation functions) non-perturbatively in λ T and J in the leading large N limit (1.4).
As we will explain in detail in Section 3, this compact looking Schwinger-Dyson equation is complicated by the presence of a fractional Laplacian operator ∆ For line defects in CFTs, it was recently proven that the defect RG obeys the g-theorem [13] generalizing previous results in d = 2 [6,9], which states that the g-function (its schemeindependent part) or equivalently the defect entropy must decrease monotonically under an RG flow triggered by a relevant deformation on the line defect. In Section 4, we study the line defect entropy in the tensor model (the p = 1 case of (1.3)). We compute the defect entropy exactly in the large N limit in Section 4.1 and observe that the g-theorem is satisfied for the localized magnetic defect. 9 For small ǫ = 4 − d, we also determine the defect entropy perturbatively in Section 4.2 from explicit Feynman diagrams, verifying our exact result in Section 4.1. Furthermore, we check that the gradient formula for line defect RG derived in [13] holds for our defect tensor model.
We expect the IR limit of the localized magnetic defect in the large N tensor model to be described by a full-fledged non-perturbative DCFT (at least in the leading large N limit). In particular this means that all correlation functions of local operators in the bulk are determined by their bulk operator-product-expansion (OPE) together with the defect one-point functions of general primary operators. The latter can also be determined by diagrammatic techniques in our solvable large N limit (1.4) of the defect tensor model.
With this in mind, in Section 5, we study two-point functions of φ abc in the DCFT, and provide formulae for one-point functions of bilinear operators in φ abc , including that of the stress-energy tensor T µν . We leave the more comprehensive analysis to the future.
Having explained the desirable features of the large N tensor model that enable us to solve the localized magnetic defects non-perturbatively, let us insert a word of caution.
The tensor model defined by the action (1.2) is not unitary. In particular, the tetrahedral coupling is not positive definite. Instead the theory is described by a complex CFT in the IR limit. Indeed, there exists primary operators of complex scaling dimensions which have been identified in the ǫ = 4 − d expansion and to reach the fixed point requires tuning certain couplings to small but complex values (suppressed in 1 N ) [83]. Nonetheless, there is substantial evidence that at least in the leading large N limit, the tensor model (1.2) is a non-perturbatively well-defined Euclidean CFT [84][85][86] and our results here lend further support by incorporating defects. 10 It was recently emphasized in [94] that complex CFTs are relevant for understanding subtle dynamics of unitary QFTs. For example the presence of complex RG fixed points (in the complexified coupling space) explains the weak first-order 9 The proof of [56] uses the locality and unitarity of the defect. More explicitly, the monotonicity of g hinges on the positivity of the two-point function of the trace of the bulk stress-energy tensor along the defect RG flow. Here we find this positivity property holds for the line defect in the melonic tensor model CFT despite the non-unitarity in the bulk operator spectrum (see around (4.27)). 10 Here by non-perturbatively well-defined we mean in the sense of having a spectrum of local operators and OPE coefficients that obey conformal bootstrap equations despite the non-unitarity. This is not to be confused with the non-perturbative instability of the conformal solutions discussed in [87][88][89]. It is possible to circumvent this instability (and the non-unitarity) by considering the long-ranged version of the tensor model [90] which in particular satisfies the F -theorem [91] but no longer has a finite canonical stress energy tensor (as is the case for generalized free fields [92,93]). phase transitions in statistical models and the walking behavior (slowly-running coupling) of four-dimensional gauge theories below the conformal window. In line with these applications, it would be interesting to understand the implications of the complex melonic CFT and its defects for the unitary tensor model in the neighboring coupling space. Furthermore, there are also generalizations of the large N melonic tensor model that are solvable and manifestly unitary, for example the prismatic theory of [95] in d ≤ 3 with a positive-definite sextic O(N) 3 symmetric interaction, as well as the supersymmetric version in [96]. It would be interesting to extend our analysis here to defects in those theories.

Defects in the Large N limit and Tree Dominance
For later convenience, we write below the action describing the coupled system of the melonic tensor model with a localized magnetic defect at x ⊥ = 0, In this section, we investigate the perturbation theory from the above action. From the combinatorial properties of the Feynman diagrams, we will prove that in the large N limit (1.4), the dominant contributions to the nontrivial defect one-point function φ 111 Dp will come from the melonic trees (see Figure 1).
It is well-known that when J = 0 only specific types of diagrams, known as melonic diagrams (see Figure 3), contribute at the leading order in the melonic large N limit, which allows one to solve the theory (see [68] for a review). In the case when J = 0 we should take into account an additional type of diagrams that anchor on the defect world-volume at x ⊥ = 0. Nonetheless, we will show that again only a simple class of diagrams survive in the large N limit (1.4) and the coupled theory (2.1) can be solved analytically. The proof will be topological (it only depends on the O(N) 3 index structures) and therefore applicable in any dimensions and for any tensor model with the tetrahedral interaction but potentially different field content (e.g. the Gross-Neveu model of fermions ψ abc ). We also note that the combinatorial result in this section does not depend on the specific form of the source (e.g. localized as in (2.1) or not).
Let us consider an arbitrary Feynman diagram from the action (2.1), that contains V vertices with tetrahedral interactions, S > 0 sources and E propagators. Some examples of these diagrams are depicted in Figure 2. The task is to find the N scalings of the vertices Figure 2: On the left is an example of a planar diagram that is subleading in the large N limit (1.4). On the right is an example of the melonic tree diagrams that dominate in this limit.
and sources so as to have a smooth large N limit (which will be (1.4) as promised) and to identify the subset of the diagrams that dominate in this limit. We follow the standard strategy that is employed in the tensor models without defects (see [68] for a review). We make explicit the index structure for each Feynman diagram by resolved (stranded) graphs where each edge consists of three colored strands representing the propagation of the three indices of φ abc and the strands are joined one-by-one at the vertices preserving the color. We denote the number of index loops for each color by F i and the total number of loops by F = F 1 + F 2 + F 3 . The N and coupling dependence of the contribution from such a diagram is If we erase one of the colors from our diagram, we obtain a ribbon graph (fat graph), of the same type that appears in the matrix model. Consequently we can assign a genus g ij for each ribbon graph where i, j denote the colors of the remaining strands and F ij = F i + F j counts the number of index loops in the ribbon graph. Furthermore, since we are dealing with open ribbon graphs (due to the J sources), the corresponding surface has n ij boundary components. For instance, for the left diagram in the Figure 2 we have n ij = 2 and for the right diagram we have n ij = 1. We then arrive at the following combinatorial relation from the Euler characteristic, Since each propagator terminates either at a vertex or at the source term, we also have the following obvious relation Combining equations (2.3) and (2.4) we obtain, for a connected graph, where we have used g ≡ g 12 + g 23 + g 13 ≥ 0 and n ≡ n 12 + n 23 + n 13 ≥ 3. Comparing with (2.2), we thus find that to achieve a smooth large N limit we should keep λ T and J fixed as promised, and the dominant contributions can scale at most as N 3 2 for a connected graph. Now let us identify the maximal graphs, namely graphs whose connected components saturate the inequality (2.5). We focus on a connected maximal graph, which is achieved if and only if g = 0 and n = 3. In other words, all ribbon graphs obtained from erasing one color are planar and each has a single boundary. For example, the left diagram of Figure 2 is non-maximal because n bg = 2 for the blue and green colors, while n br = n gr = 1 for the other choices of colors (and so g = 0 but n = 4). For a resolved graph, a strand either forms a closed loop or connects two sources. For strands that pass through exactly m vertices, we denote the number of index loops by P m and count those ending on sources by L m . For a maximal graph, we have the following immediate relation, 11 (2.6) 11 We focus on connected graphs here so that any loop must pass through at least one vertex (i.e. P 0 = 0).
Since each tetrahedral vertex is passed through six times and each source has three outgoing strands, we also have As explained in [68], for the ribbon graphs (after erasing a color) to be planar, one must have P 1 = P 3 = 0. Then putting together (2.6) and (2.7), we find (2.8) Figure 4: An example of an empty graph, from a collection of disjoint empty trees that does not contain any interaction vertices but contribute to the leading order in the large N limit.
Below we prove that the maximal graph must be a melonic tree recursively. The strategy is to introduce two surgery operations on a resolved graph, trimming and reaping, as in Figure 5 and Figure 3 respectively. Intuitively, trimming cuts off branches of the graph that straddle neighboring sources, and reaping removes melonic sub-graphs, replacing them with bare branches. Importantly, these surgery operations reduce the graph by removing vertices and loops while preserving maximality. 12 We will argue below that any maximal graph can be reduced by these two operations to an empty (disconnected) graph as in Figure 4.
Undoing the surgeries, we then establish that all maximal graphs are (possibly disconnected) trees connecting the sources with melons on the tree branches (edges).
We start with the case of L 1 > 0. This requires a vertex with a pair of adjacent branches that anchor on the neighbouring sources. It is easy to see that we can perform a trimming 12 In particular, the N dependence is preserved if the reduced graph remains connected.  Figure 5: If we have a vertex such that it is connected to two sources J we can perform the trimming surgery and remove these two sources without spoiling maximality.
surgery as in Figure 5, removing this vertex and producing a disconnected empty sub-graph.
We can repeat this surgery operation until all such vertices are removed. In the case P 2 > 0, the situation is similar to the usual defect-less tensor model (i.e. graphs with S = 0). Here we have a subgraph that contains a pair of vertices connected by one loop. This loop separates the plane into two disconnected regions (see for example the left diagrams of Figure 2 and Figure 3). Since n ij = 1 (for the ribbon graph after erasing one color) one of these regions does not contain any sources (the left diagram of Figure 2 does not satisfy this condition and therefore is not maximal). In that region we can use the standard proof of melonic dominance [68,70,72] and conclude that it is an edge nested with melons, as in Figure 6. This means that the entire subgraph we start with is also a edge (branch) with melons. Therefore we can apply the reaping surgery to remove all melons as in Figure 3, leaving behind a bare branch.
Combining with trimming operation above, we can then reduce all maximal graphs to an empty graph. Tracing backward, we establish that all maximal graphs are melonic trees (e.g. the right diagram of Figure 2).
The tree dominance in the large N limit (1.4) dramatically simplifies the diagrammatics of the defect tensor model. In the following sections, we will exploit this feature to compute various observables for the localized magnetic defects (1.3) in the tensor model.
Before ending this section, let us comment on how the melonic tree dominance is affected if we consider localized magnetic defects in other generalizations of the tensor model, or more general types of defect couplings than those in (2.1).
There are generalizations of the tensor model with a single O(N) symmetry group where the field φ abc transforms in a non-trivial rank-three O(N) representation [97,98] which admit defects of the form (2.1). Even though we do not have a complete proof, we still expect that in these models, melonic tree diagrams dominate in the large N limit. Meanwhile, in other tensor-like models, such as the Gurau-Witten model [99] or SYK-like models [100], the non-trivial melonic tree diagrams that will play an important role here for defects would be suppressed in the large N limit. Nevertheless, it is curious to note that in the context of the SYK model, the melonic trees appear in the study of operator growth [101], where they encode the evolution of the complexity of the initial state with time.
One generalization of the defect coupling in (2.1) within the melonic tensor model is to turn on the O(N) 3 singlet operator φ 2 abc on the defect world-volume. This is a localized mass deformation that does not break any global symmetry. Instead of melonic trees, the large N contributions for defect observables (e.g. the one-point function of φ 2 abc ) now come from melonic ladder diagrams. The corresponding Schwinger-Dyson equation (see Section 3.1) becomes more complicated. We first note that all melon contributions (see Figure 6) to the exact propagator G(p) of φ abc can be resummed by the means of a Schwinger-Dyson equation that is closed in the melonic limit [70],

The Schwinger-Dyson Equation and Defect One-point
where we are working with the momentum space and p 2 is the inverse of the bare propagator.
In the IR limit (at large distances) of this massless theory, we expect an emergent conformal symmetry and thus a nontrivial solution to (3.1) with definite scaling behavior given by, 13 where ∆ is the Laplacian and we have used the following relation For our purpose, we consider the one-point function as a function of J, which satisfies the following Schwinger-Dyson equation  can then be put into the following form, 14 15

Conformal defect at large distance
which now involves a fractional Laplacian which is naturally defined by Fourier transformation from the momentum space, as in

7)
14 There are two characteristic scales in the coupled system (2.1): one set by the UV cutoff in the bulk, and the other set by the bare defect coupling. We emphasize that we are studying defects in the melonic CFT where the UV cutoff has been sent to infinity. For that purpose, we are using the conformal solution (3.2) instead of the exact propagator that follows from the bulk Schwinger-Dyson equation (3.1). 15 Here and in the rest of the paper unless otherwise specified, λ T is the renormalized dimensionless coupling, φ abc is the renormalized operator that is a conformal primary and correspondingly J is the renormalized source.
for Laplacian ∆ on the entire spacetime and similarly for the transverse Laplacian ∆ ⊥ . The constant B d is defined in (3.2).
The magnetic defect induces a nontrivial one-point function for local operators, as for the classical case. However here φ 111 has a different spatial dependence due to the anomalous dimension of the operator ∆ φ = d 4 in the melonic CFT. While the Schwinger-Dyson equation (3.6) is difficult to solve, its IR limit simplifies and we expect its solution to describe a nontrivial DCFT. As is well-known, the one-point function of a bulk primary in DCFT is completely fixed up to an overall constant [38]. Here we have . (3.8) To determine the coefficient C d,p , it is convenient to work with the integral form of the Schwinger-Dyson equation (3.5), which simplifies in the IR limit to the following, This leads to the following integral which can be evaluated by standard methods and gives, (3.11) Note that the RHS above is manifestly positive for 1 ≤ p < d and p + 1 ≤ d < 4. It may seem that information of the source J has completely disappeared in the IR limit. However physically we expect the sign for the one-point function to be fixed by sgn(J) of the localized magnetic field. Before explaining how we derive this sign, let us state the result. We find the one-point function for the DCFT to be where we have normalized φ 111 by its two point function. As we will see, the overall sign is fixed by considering the case p = d − 1 in Section 3.3 and for line defects in Section 3.4.2.
We emphasize that the simplified Schwinger-Dyson equation (3.5) applies because the operator φ 111 (x , x ⊥ = 0) is a relevant deformation on the worldvolume of the trivial p ≥ 1dimensional defect, so that the part of (3.5) that depends explicitly on the source J becomes unimportant in the IR limit. Later we will study in detail the defect RG flow that interpolates between the UV coupling as in (2.1) which is irrelevant for sufficiently large n, the combinatorics is unaffected and thus the tree dominance proved in Section 2 holds, and consequently we arrive at a similar defect Schwinger-Dyson equation as in (3.5) where In this case, it is easy to see that F (x) is dominated at large x ⊥ by the classical contribution from the source term in (3.14) as expected, up to a constant that is independent of the couplings. This faster fall-off (than in (3.8)) is to be interpreted as a vanishing one-point function in the IR limit where the defect flows to the trivial one.
Finally we mention in passing that the conformal defect one-point function (3.8) in the rank-three tensor model has a simple generalization for general rank-k tensor models [75].
In this case, the coefficient C (3.17)

Analytic defect one-point functions for codimension-one defects
While in general it is difficult to solve for the complete profile of the one-point function φ 111 D d−1 from the Schwinger-Dyson equation (3.6) (or (3.5)), the equation simplifies for the codimension one defects (i.e. p = d − 1) and we will determine the exact solution here.
As a warm-up, let us consider the differential equation that describes a classical point-like (i.e. p = 0) defect at d = 1, which has the following simple solution for J > 0, and F (x) → −F (x) for J < 0. Note that the shift a is positive (for the solution to make sense) and determined by matching the strength of the singularity from both sides of (3.18).
Coming back to the interacting case, the equation we want to solve is A similar analysis as in the classical case above gives the solution for J > 0, where the shift can be determined by integrating (3.20) over x,

Defect RG flow and fixed point in the tensor model
Let us consider a general scalar QFT with a quartic potential in d = 4 − ǫ in the presence of a localized magnetic line defect. The coupled system is described by the follow action, where Y ijkl is a totally symmetric tensor of couplings. Using the ǫ-expansion we can compute the two-loop contributions to the bulk beta-function for Y ijkl and up to four-loop for the defect coupling constant h, for the defect-less theory in [83]. We are lead to the following action, where the quartic interactions are given by which correspond to the tetrahedral, pillow and double-trace operators respectively. We then find the following system of beta-functions for the defect tensor model using (3.25).
As expected, the beta-functions for the bulk interactions coincide with those derived in [83] and are given by, On the other hand, the beta function for the defect coupling is Applying the large N rescaling as in (1.4), and taking the limit N → ∞, we obtain The bulk fixed point couplings coincide with those found in [83] as expected. In particular, the couplings λ P and λ dt for the pillow and double-trace interactions are imaginary at the fixed point, which signals non-unitarity in the melonic CFT. We emphasize that since these couplings have been rescaled by large N factors as in (3.27), they are suppressed in the large N limit (1.4).

Exact RG for line defect at small ǫ = 4 − d
Here we start by studying the behavior of the one-point function φ 111 D 1 in the region close to the line defect described by (2.1) with p = 1. This corresponds to the UV limit of the defect tensor model and is controlled by a perturbation series in the defect coupling J.
The coefficients in the J expansion have specific dependence on the transverse distance x ⊥ that is dictated by the perturbative contributions at each order and multiplied by certain constants. By consistency, these constants are constrained by the defect Schwinger-Dyson equation (3.6). By solving these constraints, we will determine the defect one-point function along the entire defect RG flow for small ǫ = 4 − d. In particular, we will see explicitly the DCFT solution found in Section 3.2 emerges in the IR limit.
For convenience, we introduce here the rescaled one-point function of φ 111 and defect (3.30) such that the defect Schwinger-Dyson equation (3.6) can be written as We expect the following perturbative expansion for small x ⊥ (equivalently large k ⊥ in the Fourier space), where the coefficients are related bỹ a n = π 4 a n . (3.33) In particular the first coefficient a 0 is fixed by the source term in (3.31) to be In addition, the higher order coefficients obey the following recursive relation from (3.31), This equation simplifies when we expand to leading order in ǫ = 4 − d, a n = − 2 nǫ n 1 +n 2 +n 3 +1=nã n 1ã n 2ã n 3 , which can be solved as follows. We introduce the generating function α(t), which satisfies the following differential equation due to (3.36), The solution satisfying the initial condition α(0) =ã 0 is given by Consequently, we have up to O(ǫ) corrections, where we have used that (1 + O(ǫ)) ,ã 0 = 32ǫ Beyond small ǫ, we can also solve the recursive relation (3.35) numerically. The numerical solutions we find forã n agree with (3.40) for small ǫ, providing a consistency check.
Let us now discuss some features of our solution (3.40) which is non-perturbative in the defect couplingJ . We see thatJ 4/ǫ sets the characteristic scale of the coupled system. The solution (3.40) has simple behaviors at small and large distance relative to this scale,

Defect Entropy and g-Function
Conformal defects of odd dimensions (i.e. p odd) have a universal observable known as the defect entropy which we denote as s(D p ). It is defined as the finite piece in the free energy of the defect placed on a sphere S p of size R, Here R i denotes schematically the degree i Riemann curvature invariants (which scales as R −2i ) and Λ is the UV cutoff. While the coefficients α i are scheme-dependent, the finite term s(D p ) is unambiguous. For line defects, the defect entropy is related to the defect g-function by s = log g which was first studied in the context of conformal boundaries in 2d CFTs [3].
In this case, the only possible divergence in log D p is a cosmological constant along the line and the scheme-independent defect entropy can be obtained as follows [6], at large R. The defect entropy (for general odd p) is expected to provide a measure for the degrees of freedom on the defect, playing a similar role as the sphere free energy (finite part thereof) for CFTs. In particular, the defect entropy can be defined along defect RG flows (away from the fixed points) and has been conjectured to be monotonically decreasing under defect RG flows (see [11] for a recent summary). This was proven recently for line defects (p = J Figure 7: Two infinite families of diagrams contributing to the line defect g-function.
Their contributions are related by a factor of two as a consequence of the Schwinger-Dyson equations.

Exact defect entropy from melonic trees
We start with the line defect stretched along the x 2 direction at, for which we have already determined the defect one-point function φ 111 D 1 in the IR DCFT (see (3.8) and (3.12)). By performing an inversion transformation x i → x i x 2 followed by a translation x 1 → x 1 + R , we map this line to the circle located at, Accordingly, the defect one-point function in (3.8) transforms to, 17 The defect partition function D 1 can be computed diagrammatically by summing over melonic trees (see for example Figure 2) anchored on the defect. In this case there are two infinite families of diagrams that contribute to the defect partition function, as shown in 17 For a review on the conformal structures of correlation functions in DCFT, we refer the readers to [38].
where z(ϕ) parameterize the defect loop (4.4) with ϕ ∈ [0, 2π) as usual. Here G 0 (x) is the bare propagator and J 0 is the bare defect coupling. While the first few integrals are easy to do which we will study in Section 4.2, it is clear that this is not feasible to obtain results at finite (renormalized) coupling. Instead here we will take advantage the defect Schwinger-Dyson equation (3.5) derived in Section 3.1 to resum all these diagrams in one shot in the large N limit (1.4).
Indeed, the contributions from the two infinite families of diagrams in Figure 7 have the following compact expressions, (4.7) The Schwinger-Dyson equations (3.1) and (3.9) further imply that the second term above is proportional to the first one by a factor of two. Therefore the total contribution can be rewritten as To compute this integral it is convenient to work with the following "toroidal" coordinates, wheren i is a unit vector parameterizing the unit S d−3 . The ranges of these coordinates are τ ∈ [0, ∞), σ ∈ [0, π] and ϕ ∈ [0, 2π]. The defect loop now locates at τ = ∞. At large τ (i.e. a "toroidal" neighbourhood of the defect loop), the above coordinates take the following form, We see ϕ, σ together with the unit vectorn i are precisely the coordinates on the S 1 × S d−2 boundary of this "toroidal" neighbourhood.
The defect one-point function (4.5) takes a simple form in the toroidal coordinates (4.9), . (4.9) The line defect free energy is then given by the following integral, where we have used that the volume form in the toroidal coordinates is The integral (4.10) is UV divergent due to contributions near the defect. This can be regularized by introducing a UV cutoff Λ that truncates the τ integral up to e τ * = RΛ. We then obtain at large Λ, We see the divergent term is proportional to the radius of the defect loop as expected, and the scheme-independent defect entropy can be extracted using (4.2),

Defect entropy from ǫ-expansion and gradient formula
As a consistency check, we also determine the defect entropy in the ǫ = 4 − d expansion. 18 We focus on the leading answer at small ǫ and large N, which receives contributions up to two-loops, as in (4.6), 19 1 where J 0 is the bare defect coupling. The first term in (4.14) is the contribution from a bulk propagator anchored on the defect loop, while the second and third terms correspond to the diagrams in Figure 7 at leading order in the perturbative expansion at small ǫ. Explicitly, their contributions are captured by the following integrals, with the bare propagator (4. 16) where the coefficient f (d, a) is defined in (3.3).
It is straightforward to evaluate the above integrals in dimensional regularization, which gives 20 (4.17) For example, we perform the third integral here explicitly using (3.3) and the following 18 See [55] for similar computations in the O(N ) vector model. 19 The higher order terms in the couplings will contribute corrections at higher orders in ǫ to the defect free energy. This can be inferred from the counting J ∼ ǫ 1 4 and λ T ∼ ǫ 1 2 . 20 The integrals for I 1 and I 2 can also be found in [13].
integration identity, and find to this order, Then from the one-point function of the renormalized field φ 111 , we arrive at the following counter-term coefficients, Putting together (4.14), (4.17), (4.20) and (4.24), we obtain This matches with (4.13) in the small ǫ limit.
It is recently proven in [13] that the RG flow of a line defect loop of radius R satisfy a gradient formula (generalizing [6]), which states the change of the defect entropy in R is controlled by the connected two-point function of the defect stress tensor T D . Here for the localized magnetic defect (see also [55]), the defect stress tensor is We can check the gradient formula (4.27) for our defect flow. Using the Callan-Symanzik equation, 29) it follows that the gradient formula (4.27) becomes at this order in the ǫ-expansion, which is indeed satisfied by the second line of (4.25).

Towards Defect Two-point Functions
Two-point functions of bulk operators in DCFT are important observables that connect bulk and defect data by bootstrap equations [38,102]. The two-point function O 1 (x)O 2 (y) Dp in the presence of a defect D p is no longer fixed (up to a constant) by the conformal symmetry alone. Rather it depends nontrivially on two conformally invariant cross-ratios ξ 1 and ξ 2 , Note that ξ 2 is trivial for p = d − 1. There are two OPE channels for O 1 (x)O 2 (y) Dp . In the bulk channel, we take OPE of the bulk operators and decompose the two-point function into a sum over one-point functions multiplied by the bulk three-point OPE coefficients.
In the defect OPE channel, we instead expand the bulk operators into local operators on the defect multiplied by bulk-defect OPE coefficients and then sum over the exchanged defect local operators. Associativity of the DCFT operator algebra demands that the two OPE decompositions must produce the same two-point function, which implies nontrivial constraints between local operator data on the defect worldvolume and OPE data in the bulk CFT.
It is generally very difficult to obtain defect two-point functions in strongly coupled CFTs which exhibit the above properties. In this section, we will explain how to obtain such twopoint functions for the primary operators φ abc in the defect tensor model. Furthermore we discuss how to extract one-point functions of bilinear operators in φ abc from expanding the two-point function in the bulk OPE channel.

Two-point functions of φ abc
Having understood how the non-vanishing one-point functions φ abc Dp arises through the melonic tree diagrams in the defect tensor model, here we study the O(N) 3 singlet two-point by the same diagrammatic method. In this case in addition to the melonic tree diagrams in Figure 2, there are other contributions of the same order in N from the ladders diagrams in Figure 9. 21 While the tree contributions to (5.2) are rather simple, and accounted for by the product of two one-point functions φ 111 Dp , the ladder part is much more complicated. We note that the problem of the resummation of ladder contributions also arises when computing four-point functions in the tensor models or the Sachdev-Ye-Kitaev (SYK) model [84,103]. The resummation of ladders for the two-point function (5.2) can be performed using the following kernel, Consequently, the two-point function takes the following form, where F (x) is defined in (3.4). Formally these nested integrals represent a geometric progression and can be resummed as We note that F s (x 1 , x 2 ; x 3 , x 4 ) is related to the four-point functions in the (defect-less) mel- 21 We emphasize that for general two-point functions of φ abc that are not O(N ) 3 singlets, the ladder contributions will be suppressed by 1 N 2 due to non-planarity.
onic CFT [84,100,104], We thus arrive at the following relation after using the Schwinger-Dyson equation (3.5),  We now apply the bulk OPE in the two-point function (5.6) to compute one-point functions of the singlet bilinear operators φ abc ∂ µ 1 . . . ∂ µs h φ abc .

One-point functions of bilinear operators
In the limit x 1 → x 2 , we expand φ abc (x 1 )φ abc (x 2 ) into a sequence of bilinear singlet local operators. Taking this OPE limit on both sides of (5.6), we find for a singlet operator O(x): where O(x) represents a general O(N) 3 singlet bilinear operator in φ abc .
If the operator O(x) is a primary operator with dimension ∆, we know that the three-point function is completely fixed up to an overall constant, and it follows that where we have defined 22 . (5.10)

One-point function of the stress-energy tensor
Here we study the defect one-point function of the stress-energy tensor T µν in the defect tensor model. We note that the stress tensor has in addition to a bilinear part also a quartic interaction part, (5.11) Following a similar analysis as in the last section, we find that the stress tensor one-point function is given by the following expression, g µν F 4 (y) .

(5.12)
We emphasize that the second term on the RHS is crucial for conformal Ward identities, ensuring T µν is traceless and conserved. Indeed the trace is given by, (5.13) 22 As usual these integrals may contain divergences that need to be regularized.
where in the last line we have used the Schwinger-Dyson equation (3.5). More explicitly, we for the scalar field one-point function captures the exact defect RG flow from the trivial transparent defect in the UV to the nontrivial conformal defect in the IR. We also describe how to compute the defect two-point functions, as well as one-point functions of bilinear operators and the stress-energy tensor by taking the bulk OPE limit. Below we discuss some future directions.
In this paper we have focused on defect observables that only involve possible local operator insertions in the bulk. An immediate generalization is to consider correlation functions that also contain nontrivial local operators on the defect world-volume. There is a set of distinguished defect local operators that are associated with bulk symmetries broken by the defect insertion. This includes the displacement operators D i for each transverse direction corresponding to the broken translation symmetries [38]. Here for the localized magnetic defect, the broken part of the bulk O(N) 3 symmetry also give rise to 3(N − 1) "tilt" operators t a (m) with a = 2, 3, . . . , N and m = 1, 2, 3, analogous to those studied in [105] for the critical O(N) model with a symmetry breaking boundary. These defect local operators have protected spacetime quantum numbers thanks to Ward identities for bulk currents in the presence of a conformal defect. Here the displacement operators D i appear in the divergence of the bulk stress energy tensor Consequently, D i and t a (m) are both defect scalar operators (e.g. invariant under SO(p)) and have scaling dimensions ∆(D i ) = p + 1 and ∆(t a (m) ) = p. It would be interesting to study correlation functions involving these defect local operators in our defect tensor model.
As already emphasized in the Introduction (see also around (3.29)), a somewhat exotic feature of the melonic CFT is its non-unitarity, which is evident from the complex scaling dimension of φ 2 abc and the complex fixed point couplings in (3.29) [83]. Nevertheless the nonunitarity is rather restricted in the leading large N limit. Indeed the complex couplings in (3.29) are all suppressed by higher powers of 1 N , and φ 2 abc is the only singlet primary operator of complex scaling dimension in the OPE of φ abc with itself [83]. This restricted non-unitarity could be related to the fact that all defect observables that we compute explicitly here appear to respect constraints from unitarity. It would be interesting to understand whether there is a unitary subsector in the melonic DCFT in the sense of [106] (generalized to DCFT).
There are close cousins of the melonic tensor models that are manifestly unitary already at finite N. Examples include the prismatic tensor model of [95] and the supersymmetric generalization in [96]. We expect the diagrammatic methods developed here can be generalized to study defects in those strongly coupled theories.
Finally the tensor models with N 3 degrees of freedom are reminiscent of QFTs constructed in M-theory from M5-branes [107]. In particular at d = 3, it is known that a stack of N M5-branes compactified on a hyperbolic three-manifold with certain topological twist produces an N = 2 supersymmetric QFT that generally flows to an interacting superconformal CFT in the IR [108,109]. 24 Such an SCFT has a sphere free energy that scales as N 3 at large N [111]. It would be interesting to look for a M-theory embedding of the d = 3 tensor model (and its cousins 25 ) including defects therein, by considering a non-supersymmetric three-manifold compactification. Relatedly, it is also interesting to realize the d = 3 tensor models via (supersymmetry breaking) deformations of 3d N = 2 theories constructed in [108,109]. For this purpose, the N = 1 supersymmetric tensor model in [96] could be a natural starting point. Such an embedding of tensor models in M-theory will provide much needed insights on their holographic dual. Indeed, the AdS 4 duals for the 3d N = 2 SCFTs from N wrapped M5-branes have been identified using M-theory [111]. It would be very interesting to see if suitable modifications in the bulk would produce the holographic duals for the tensor models.