Planar RG Flows on Line Defects

We study a class of renormalization group flows on line defects that can be described by a generalized free field with ordered planar contractions on the line. They are realized, for example, in large $N$ gauge theories with matter in the fundamental representation and arise generically in non-relativistic CFTs. We analyze the flow exactly and compute the change in the $g$-function between the UV and IR fixed points. We relate the result to the change in the two-point function of the displacement operator and check the monotonicity of the defect entropy along the flow analytically. Finally, we give a general realization of this type of flow starting from the direct sum of the IR fixed point and a trivial line. This type of defect renormalization group flow parallels the well-studied case of double-trace flow.


Introduction
We study renormalization group (RG) flows that take place along a line defect in large N conformal field theories (CFTs) in d ≥ 2 dimensions.We focus on straight lines or circular defects.At a fixed point of these flows, the defect preserves an SL(2, R)×SO(d−1) subgroup of the d dimensional conformal symmetry.The combined system is called a defect CFT (DCFT).The defect RG flow can be triggered by deforming the action with a local relevant defect operator O(x) as Here, λ is the deformation parameter, M is the mass scale of the flow, and ∆ O is the dimension of the operator at λ = 0.The operator is relevant for ∆ O < 1.The RG flow can take place close to the defect without affecting the CFT far from it.In this case, it must end at a new fixed point.The flow can also destabilize the line and affect the infrared.
There is already extensive work on defect RG (DRG) flows; see .DRG flows that occur in large N CFTs are simpler to analyze due to their simplified diagrammatics.One class of such simple DRGs is triggered by a double-trace operator (1.2) Here, O is a single-trace operator.In this case ∆ O = 2∆ + O(1/N ), where ∆ is the dimension of O. What makes this flow particularly simple is the large N factorization of the correlators of the double-trace operator into products of two-point functions of its single-trace constituents; see figure 1. Double-trace flow was studied in [22][23][24][25], and the holographic dual of this flow was studied in [22,[24][25][26][27][28].They can be realized on defects of any dimension, not necessarily a line.The flow triggered by (1.2) depends only on ∆ and on the dimension of the defect, but not on the space-time dimension of the theory in which it is embedded; therefore, the results of these works apply equally well to defects.Our focus in this paper is on a second class of DRG flows.These are flows that appear in the large N limit of conformal gauge theories with matter in the fundamental representation of the gauge group.In such theories, a conformal defect in the fundamental representation can be deformed by an operator of the form where O is an operator that transforms in the fundamental representation of the gauge group and O in the anti-fundamental representation. 1 The operator O F F is in the adjoint representation and exists only on the defect.As for the double-trace deformation, ∆ O = 2∆ + O(1/N ), and the correlators of the operator in (1.3) factorize into a product of two-point functions of its constituents at large N .However, there are two key differences from the double-trace case (1.2).First, only products of two-point functions that are both planar and connected survive the large N limit (the products must be connected because additional disconnected components form additional fundamental loops).These are two-point functions between neighboring ordered points on the line, see figure 2. Second, the planar correlators of the operator (1.3) are of order one, while the expectation value of the undeformed line is of order N .This is because they form a fundamental loop.
Another class of examples where such a DRG flow is realized is non-relativistic theories.In these cases, the operator O annihilates a particle, while the operator O creates one.The planar factorization of the correlators of O F F on a timelike defect follows from the conservation of the particle number in the non-relativistic limit.This class of DRG flows also extends to higher-dimensional defects.
One characteristic of a defect is the defect entropy s.For relativistic theories it is given by2 s(RM where ⟨W ⟩ is the expectation value of a circular defect of radius R. In this paper we will consider the change in the defect entropy along the DRG where λ is the deformation parameter in (1.1).This quantity decreases monotonically along the flow [3,7,13].In particular, the change in the g-function, g ≡ ⟨W λ ⟩, between IR and UV fixed points (where the result loses its dependence on R), is negative, It can be thought of as a measure of the number of degrees of freedom that have decoupled along the flow.We compute δs and δg for the F F flow.The calculation is carried out by expanding ⟨W λ ⟩/⟨W 0 ⟩ in powers of λ and resuming all orders.We interpret our results, which are in agreement with the general g-theorem (1.6).One well-known example of a DCFT with O F F operators is obtained by coupling Chern-Simons theory to fermions in the fundamental representation.There, the conformal defect operator can be a standard Wilson line, and the operator O is a certain component of the fermion field [30,31].In this case, we relate the change in the g-function to the change in the two-point function of the displacement operator.
The paper is organized as follows.We begin in section 2 by analyzing the DRG on a straight line, and determine the fixed points.In section 3 we compute δg.In section 4 we interpret the result by embedding it in a flow that starts from the stable line plus a trivial line.In section 5 we relate it to the two-point function of the displacement operator in CS-matter theories.Finally, in section 6 we analyze the defect entropy along the DRG flow and check its monotonicity.

RG Flow on the Straight Line
Our starting point is a straight conformal line defect W that ends on a fundamental operator O of dimension ∆.We normalize the operator so that its two-point function on the straight defect of length L takes the standard form where i and j are color indices.We assume that ∆ ∈ [0, 1/2) and turn on the fundamental anti-fundamental ( F F ) deformation.Due to the planar structure of the large N factorization, the OPE of O F F with itself consists of O F F and its descendants only.Hence, no other operator can be generated along the flow.(This is in contrast to the double-trace deformation, where the OPE of two O DT operators includes the operator O 2 DT which also becomes relevant for ∆ < 1/4.)As a result, the two-point function changes to (2.2) where P stands for path ordering, and J n is the integrated n-point function of the operator O in (1.3).It is given by The series in (2.2) sums to where E α,β (z) is the Mittag-Leffler (E) function; see [32] for a review of its relevant properties.This result can also be obtained by solving the Schwinger-Dyson equation Because ∆ < 1/2, all integrals in (2.3) are finite and there is no need to introduce a new regularization scale.Hence, no new terms are generated.Here, we are working with the bare operator, and therefore the corresponding β-function is obtained by simple dimensional analysis from (1.1), β λ = β bare = −(1 − 2∆)λ.It is evident that λ = 0 is an unstable fixed point, with no other fixed point at finite λ.To analyze the IR limit, we study the asymptotic behavior of the two-point function (2.4) at large M L. For λ > 0 we have (2.6) It follows that λ → ∞ is a stable fixed point.The dimension of the fundamental operator Since the flow is super-renormalizable, the only dimensionful scale on which the wave function renormalization factor can depend is λM 1−2∆ .This factor is independent of the UV operator's normalization choice.It follows that the ratio is physical.In section 5 we will relate this quantity to the change in the two-point function of the displacement operator.For λ < 0 the two-point function grows exponentially in the IR, This deformation of the line therefore destabilizes the IR.It reflects an instability that can affect the theory far from the defect, and we cannot determine its fate on the basis of our assumptions.The same behavior is expected to hold for the double-trace deformation.
In several examples of double-trace deformations that have been studied in [33,34] it was found that a small correction to the factorization (which would be of order 1/N in our setup) is responsible for restoring conformal invariance in the IR.Assuming that this is the case, in section 4, we will evaluate the corresponding change in the defect entropy.

∆ = 1/2
The case where ∆ = 1/2 requires special care because O F F is classically marginal and the loop integrals in (2.3) have logarithmic divergences.We choose to work with a pointsplitting regularization, σ i+1 − σ i > ϵ/L.Working to second order in perturbation theory, we find To analyze the flow, we plug D into the corresponding Callan-Symanzik equation where γ O is the anomalous dimension matrix.From (2.10) we find that We see that , and so for λ < 0 the F F deformation is marginally relevant while for λ > 0 it is marginally irrelevant.

Summary
By comparing the analysis at ∆ < 1/2 and ∆ = 1/2, we see that, as ∆ → 1/2 from below, the two fixed points collide.As ∆ crosses 1/2, they interchange with each other.To make this more manifest, we impose the renormalization condition that the two point function of the renormalized boundary operators at M L = 1 is finite as ∆ → 1/2.Using (2.2) and (2.3) we find that this renormalization condition is satisfied by the renormalized operators and coupling . (2.12) In terms of these, the two-point function evaluated at M L = 1 takes the form The corresponding β-function reads ( The two fixed points are f = 0 and f = 1 − 2∆.The latter (former) is the stable one for 3 The Change in the g-function Next, we turn to the computation of the change in the g-function between the UV and IR fixed points, (1.6).At any point along the flow, the change in the expectation value of the defect takes the form where I n is the integrated correlator of n insertions of O F F on the circle.It is given by a factorized product of two-point functions between neighboring ordered fundamental and anti-fundamental operators.Together, these form a planar connected contraction of O F F , as seen in figure 2. This integrated correlator is given by where θ i are angles on the circle, with θ n+1 ≡ θ 1 + 2π.Here, D arc 0 (θ) is the two-point function of the operator O on an arc of the circle with opening angle θ and radius R at λ = 0, The absence of a combinatorial factor on the right-hand side can be understood as the result of a cancellation between a factor of 1/n for the cyclic permutation of the points and a factor of n for the derivative of n contractions.
It can be obtained by a conformal transformation from the two-point function on a straight line (2.1).
Using the representation of δg as the IR value of the sum in (3.1) we arrive at (see figure 3) ) where ε is a point-splitting regulator on the propagator, and the (1 − R∂ R ) factor removes the local divergence at θ → 0, 2π.Instead, one can simply subtract a local counter term.After doing so, ∂ ∆ δg is independent of R. Here, D arc λ (θ) IR is the arc two-point function at the IR fixed point.To evaluate it, we note that the wave function renormalization factor in (2.6) is a local property of the operator.For ∆ < 1/2, there are no UV divergences on a line segment or an arc, so it is also scheme-independent.Therefore, its value at the IR fixed point is the same on the straight line and on the arc.We conclude that By plugging (3.3) and (3.5) into (3.4),we arrive at To obtain the change in the g-function, we note that at ∆ = 1 2 the deformation is marginally irrelevant, and therefore δg| ∆= 1 2 = 0. Since δg is continuous in ∆ at this point, we conclude that The functions δg and ∂ ∆ δg are plotted in figure 4. We note that ∂ ∆ δg is positive for 0 < ∆ < 1 2 and therefore δg increases monotonically in ∆.Since δg| ∆= 1 2 = 0, it follows that g UV > g IR for all 0 ≤ ∆ < 1 2 , in agreement with the g theorem (1.6). 3 Notice also that 0 ≤ g UV − g IR ≤ 1, with (g UV − g IR )| ∆=0 = 1.This fact is explained in the next section and is to be contrasted with the case of double-trace flow.There, the change in the defect entropy [25], is unbounded from below.As a result, the difference (g UV − g IR ) double-trace is only bounded at order N , 4 Interpretation of g UV − g IR We have found that δg ranges from δg = 0 for a marginally irrelevant deformation, 2∆ = 1, to δg = −1 for 2∆ = 0.In this subsection, we explain the physics that underlies this result.
To this end, we first construct the unstable line as the endpoint of a DRG flow from the direct sum of the stable line and the trivial (empty) line.We then use this realization of the unstable line to describe the DRG flow to the stable line that was considered in the previous section.The construction is general and mimics a perturbative procedure as laid out in [31].Let L 0 be the one-dimensional Lagrangian of the stable line.For example, in the case where the defect is a Wilson line, L 0 is proportional to the gauge connection in the direction of the line (which is an N × N matrix in color space).Using L 0 , the Lagrangian of the direct sum of the stable line and a trivial line can be written as It is evident that in the direct sum of lines, there is one more degree of freedom.Consequently, its g-function differs by one unit, To construct the unstable line, we deform the line Lagrangian as L → L + δL, with where M is an RG scale, and β c.t. is a local counter term, that we will fine tune and determine below so that the flow ends at the unstable fixed point.This deformation mixes between the stable and trivial lines.When the line is expanded in powers of δL, it decomposes into empty and filled segments, with δL transitioning between them.At small α the deformation is relevant because O has dimension 1 − ∆ < 1. Due to the large N factorization property of the correlators of O on the stable line, no new operator can be generated along the DRG flow. 4Assuming that the flow ends at a new conformal line, we can represent the conformal line in the IR as in (4.3) with some specific α = α * .
To find the new fixed point, we consider the expectation value of two mesonic lines.One is defined before the deformation,

M(L) ≡ ⟨O(L)Pe
where we have chosen a canonical normalization for O.The other mesonic line is defined after the line is deformed, From the structure of δL in (4.3) it follows that M (L) satisfies the differential equation where ϵ is a point-splitting regulator.Note that here we assume that a cutoff is only needed near x = L, and not near x = 0.That is, point-splitting is only necessary for M, but not M. Consequently, we shall only look for a solution that is consistent with this assumption.
We begin by treating 0 < ∆ < 1/2, excluding the points 0 and 1/2.For the divergence near x = L to cancel, we see that At the UV fixed point, the boundary operator (0, 1) ⊺ has the lowest dimension, equal to zero.Due to the gap, it cannot mix with (O, 0) ⊺ at small α.In fact, from (4.6) it follows that at the IR fixed point, (O, 0) ⊺ IR is the SL(2) descendant of (0, 1) ⊺ IR .In particular, at the IR fixed point (0, 1) ⊺ is an SL(2) primary of some dimension ∆ IR , lim For this behavior to be consistent with our assumption that the integral in (4.6) is finite near x = 0, we must have ∆ IR < 1/2.We plug the IR limit (4.8) into (4.6),where the normalization of M drops out because (4.6) is homogeneous.Taking then the limit ϵ/L → 0, we find (4.9)The local coefficient α * may depend on the RG and regularization scales, but it cannot depend on L. Terms with different powers of L must vanish independently.From the cancellation of the L independent terms we conclude that From the term of order (M L) 2∆ , upon restriction to ∆ IR < 1/2, we can solve for ∆ IR , Hence, the flow that was triggered by (4.3) has landed at the unstable fixed point of a single line operator.We can now use (4.3) to represent the flow back to the stable line that we have considered in the previous sections.The F F deformation takes the form The unstable line (4.3),(4.10), (4.11), can be thought of as a condensation of empty segments mixed within the stable line.The relevant F F deformation with positive coefficient leads to an exponential suppression of the empty segments.At the end of the flow that it triggers, the empty segments disappear, and we remain with the stable line only.The direct sum in (4.1), (4.3) can equivalently be realized using a worldline fermion, see appendix A for details.In this realization, the F F deformation (4.12) is a mass term for the worldlline fermion.It leads the fermion to decouple in the IR, leaving the stable line.
In summary, we have constructed the unstable line as the fixed point of a DRG from the stable line plus a trivial line.The stable line is then obtained as the fixed point of a second DRG flow that is triggered by (4.12).Therefore, the g-function of the unstable line must lie between that of the stable line and that of the stable plus trivial line, in agreement with (3.7).
The limit ∆ → 0 requires special care.In this limit, the deformation (4.3) becomes classically marginal and the solution to (4.6) needs to be reconsidered.We find that for ∆ = 0 the integral in (4.6) has a logarithmic divergence, which can only be eliminated by setting α 2 * (∆ = 0) = 0.It follows that δL is marginally irrelevant and ∆ IR = 0 = ∆.We conclude that in this limit g unstable → g stable + 1, in agreement with (3.7).

δg for the λ < 0 Flow
Realizing the unstable fixed point using (4.3) is also useful for understanding the flow with λ < 0, (2.8).For negative λ the F F deformation leads to an exponential enhancement of the empty segments.At the end of this flow, the empty segments dominate and we remain with the empty line only.If we assume that a small correction (of order 1/N ) is responsible for restoring conformality in the IR, then the change in the defect entropy is δg = 1 − g unstable + O(1/N ), which can be large due to the large value of g unstable .

Relation to the Two-Point Function of the Displacement Operator in CS-matter Theories
One characteristic of any conformal line operator is the two-point function of the displacement operator on the circle, which we denote by γ.The F F flow that we have studied here is realized in CS-matter theory.In that case, the two-point function of the displacement operator was computed in [31] and is given by Here, Λ is the two-point function on the mesonic line [31], and ⟨W 0 ⟩ ∝ N is the expectation value of the unknot Wilson loop in CS theory.In this section, we relate the changes between the two fixed points of the defect entropy, the two-point function of the displacement operator, and the wave function renormalization factor.

The Change in the Displacement Operator
The DRG is triggered in the UV by deforming the line action with the F F operator (1.3).The corresponding UV change in the displacement operator that follows is where δ ± O stands for the path derivative of O, see [30,31,35] for details.Consider first the deformation of D + .We have δ + O = ϵ 1−2∆ O + , where ϵ is a UV cutoff renormalization scale, and O + is an operator of dimension ∆ + = 2 − ∆.This term, with a positive power of ϵ, vanishes as we remove the cutoff ϵ → 0. In the second term, δ + O = O + is an operator of dimension ∆ + = ∆ + 1.In the IR, M is large and the second term in (5.2) dominates the displacement operator, Similarly, The operators O + and O − carry a transverse spin that is different from that of O and O respectively.As a result, their two-point function is blind to the F F deformation and does not change along the flow.In the UV, this two-point function obeys (see [30,31,35]) The operators O and O are normalized as in (2.1).Therefore, On the other hand, the two-point function of O and O changes along the flow.In the IR it is given in (2.6), which we repeat here for convenience, (5.7) It follows that the two-point function of the displacement operator in the IR is equal to6 in agreement with the expression (5.1) and a flow from ∆ and 1 − ∆.

The Defect Entropy Along the RG flow
We now turn to the computation of the change in the defect entropy (1.5) along the F F DRG flow.We will then use this result to check the monotonicity of the defect entropy along the flow explicitly.
The change in the expectation value of the defect δ⟨W ⟩ is of order one, while ⟨W 0 ⟩ is of order N .Therefore, The expansion of δ⟨W ⟩ in powers of λ is given in (3.1).To evaluate I n in (3.2), it will be useful to first relax the condition that θ n+1 = θ 1 + 2π, and let θ 2 , . . ., θ n+1 run on the universal cover of the circle, Here, we have dressed D arc 0 (θ j+1 −θ j ) by e ia(θ j+1 −θ j ) .These factors telescope to e ia(θ n+1 −θ 1 ) .We then use this phase factor to impose the condition θ n+1 ≡ θ 1 + 2π as where the combinatorial factor of 1/n accounts for cyclic permutations of the n points on the circle (there was no such combinatorial factor in (3.2) because the first point was specified).

Divergences and Regularization
The loop integrals in (3.2) have potential divergences from the integration regions where some of the insertion points collide.Consider first the region where only a subset of m < n consecutive points collide.In this region, the integral behaves in the same way as the line integral, J n in (2.3), behaves in the region where m insertion points collide.Because J n is finite, this region does not cause a divergence.To see this more explicitly, we denote by ρ the common relative distance between these insertion points.Near ρ = 0, the integral behaves as ρ m−2 dρ/ρ 2(m−1)∆ .Here, the numerator comes from the measure after factoring out the center of mass of these m points.The denominator comes from the m − 1 contractions between the m insertion points.This integral is finite for ∆ < 1/2.Therefore, the only potential divergence comes from the integration region where all n points collide.In this region, all m = n contractions are singular (instead of only m−1 for m < n).Consequently, the integral behaves as ρ n−2 dρ/ρ 2n∆ ∝ ϵ n(1−2∆) /ϵ, where ϵ is a UV cutoff on ρ.We conclude the following: • For any ∆ < 1/2, only finitely many I n 's are divergent.
• The divergences are linear in R and can be removed by a local counter term.They are also removed by the factor (1 − R∂ R ) in the defect entropy (1.5).
• To regulate these divergences, it suffices to regulate one of the n insertions.
In the following, we choose to regulate the I n integrals by point-splitting the insertion at θ 1 as where ε = ϵ/R is the regulator.This is achieved by modifying (6.3) to Ĩn (a) .(6.5)

Evaluation of δs
The integrals in Ĩn , (6.2), are ordered convolutions.They are therefore diagonalized by a Laplace transform The inverse transform reads where the path C is parallel to the imaginary axis at some positive real value, and is oriented towards the positive imaginary s axis.Here, for convenience, we have factored out the dependence on R so that L(s) is dimensionless.
Using the convolution relation the regulated integral (6.5) takes the form In this fashion, the regulated partition function resums to The renormalized operator is obtained from (6.10) by subtracting the local divergent contributions.For ∆ < 1/4 the only divergent contribution is the term linear in λ, which is due to the self-contractions of the two operators in O F F , (1.3).After subtracting it, one can take ε → 0. Other values of ∆ can be obtained by analytically continuing the result in ∆ or, equivalently, by subtracting additional divergent terms.These local divergent terms drop out from the change in the defect entropy (6.1).

IR Limit
We can now check that the change in the g-function (3.7) is reproduced from (6.10) upon taking the IR limit, M R → ∞.Recalling that ∆ < 1 2 and λ > 0, we see that in this limit the one inside the logarithm in (6.10) is negligible.Using all the s-independent terms from (6.6) drop out and we remain with To evaluate this integral, we take a derivative with respect to ∆ and then use the sum representation of the digamma function.After dropping the sindependent terms using (6.11) and interchanging the order of summation and integration, we remain with By closing the contour towards the negative real axis and picking the two conjugate poles these integrals evaluate to The sum in (6.where in the second step we have expanded the result at small ε.The 1/ε = R/ϵ divergence is linear in R and is removed by a local counter term.We thus find in agreement with (3.6).

Monotonicity of the Defect Entropy
The monotonicity of the defect entropy has been proven in [3] by relating its derivative with respect to the RG scale to the two-point function of the defect stress tensor T D , .18)In this section, we explicitly verify this relation for the F F flow.This serves as a consistency check and also as a more direct proof of the monotonicity in M R of our result for δs, (6.10).
For the purpose of checking this relation, we take 0 < ∆ < 1/4 and subtract the only divergent contribution in δ⟨W ⟩ reg in this range, which is Other values of ∆ ∈ [0, 1/2] are obtained from ∆ ∈ (0, 1/4) by analytic continuation.When working with δ⟨W ⟩ ren we can set ε = 0 because all I n 's are finite.The defect stress tensor along a DRG flow is given by T D = i β i O i , where the sum is over all defect primary operators O i that participate in the flow, and β i are the beta functions of their couplings.In our case, due to the large N factorization, the only primary operator that appears in the two-point function of the F F operator (1.3) is the F F operator itself.Using the result for the beta function found previously, we conclude that Its Laplace transform can be evaluated as a convolution as was done for I n (6.9), and takes the form The integral on the right hand side of (6.25) is convergent for every ∆ ∈ (0, 1/2).For ∆ ∈ (0, 1/4) it converges for each of the five terms in (6.26) separately.This allows us to shift the integration variable in the inverse Laplace transform of the terms with L n (s ± i) by one imaginary unit, such that (6.25) holds with C ds e 2πs L n (s) .
(6.28)For the left hand side of (6.21), by substituting (6.9) into (6.19)we can express the term of order −λ(M R) 1−2∆ n as C ds e 2πs L n (s) , (6.29) which concurs exactly with (6.28).This concludes the verification of (6.18) for our case of F F flow.We note that an analogous derivation also applies to the monotonicity of the defect entropy under a double-trace deformation (1.2).

Figure 1 .
Figure 1.Four-point correlation function of the double-trace operator on the circle.At leading order in the large N limit it factorizes into a product of the two-point functions of the single-trace operators.All possible contractions of the single-trace operators contribute.

Figure 2 .
Figure 2. Four-point correlation function of the fundamental-anti-fundamental operator on the circle.At leading order in the large N limit it factorizes into a product of the two-point functions between the fundamental and anti-fundamental operators.Only connected planar contractions contribute, joining together into a single fundamental loop.

Figure 3 .
Figure 3.At order λ n , the expectation value of the circular loop consists of n ordered contractions between the fundamental and anti-fundamental operators.The insertion points of these operators are integrated around the circle and are labeled by 1, . . ., n in the figure.Upon differentiating by ∆, we get the derivative of a single contraction, times n − 1 ordered contractions on an arc.The absence of a combinatorial factor on the right-hand side can be understood as the result of a cancellation between a factor of 1/n for the cyclic permutation of the points and a factor of n for the derivative of n contractions.

Figure 4 .
Figure 4.The graph of δg and ∂ ∆ δg as a function of ∆.Note that ∂ ∆ δg is always positive and, therefore, δg increases monotonically.