Exact Renormalization Group for Point Interactions

Renormalization is one of the deepest ideas in physics, yet its exact implementation in any interesting problem is usually very hard. In the present work, following the approach by Glazek and Maslowski in the flat space, we will study the exact renormalization of the same problem in a nontrivial geometric setting, namely in the two dimensional hyperbolic space. Delta function potential is an asymptotically free quantum mechanical problem which makes it resemble non-abelian gauge theories, yet it can be treated exactly in this nontrivial geometry.


Introduction
Most problems of deep significance in the world of interacting many-particles are formulated by singular theories. Typically, these are plagued by divergences, which reflects our ignorance of the physics beyond the scales defined by our original theory.
A deep insight into this behavior came from Wilson [23,24,25,26,27,28]. He argued that the physics beyond the scales of interest should be incorporated into lower energies by some effective interactions. As we will show in Section 2.2, for a system defined by a Hamiltonian, if one calculates the form of the effective Hamiltonian at some energy scale Λ specifying the cutoff, the result will be where PHP is the projection of the Hamiltonian on the subspace where momentum eigenvalues are bounded by the cutoff Λ, and the operator X Λ depends on higher degrees of freedom. Since the cutoff is totally arbitrary, the effective Hamiltonian should not depend on it. The essence of Wilsonian Renormalization Group, or Exact Renormalization Group (ERG), is modifying the parameters of the theory without altering the energy eigenvalues, so that the effective Hamiltonian becomes cutoff-independent. This is done as follows: We start by specifying the system at some high energy scale Λ, called the bare scale. Then we introduce another scale λ, called the effective scale, such that 1 ≪ λ ≪ Λ. The ERG procedure consists of integrating out degrees of freedom between these two scales. This integrating out procedure is not performed in a single step. In each step one integrates over an infinitesimal momentum shell. This transformation, which is called a Renormalization Group (RG) transformation creates a trajectory, called RG trajectory in the space of theories, or in this particular case in the space of Hamiltonians. The Hamiltonians at different scales are related by the requirement that the eigenvalues do not change as one changes the scale. In other words the RG trajectory is determined by the condition that all the Hamiltonians on this trajectory give the same set of eigenvalues as the unrenormalized theory.
There is no systematic non-perturbative approach to implement this idea yet to this date, nevertheless many interesting problems can be solved by means of some approximation method. The literature in this direction is immense, and we don't feel competent enough to cite all the relevant works. We will mention just a few things related to the present work. A perturbative approach to the renormalization group for effective Hamiltonians in light-front field theory is given in [10]. Another renormalization procedure for light-front Hamiltonians is called Similarity Renormalization Group, where the bare Hamiltonian with an arbitrary large, but finite cutoff, is transformed by a similarity transformation which makes the Hamiltonian band diagonal [9,11]. A pedagogical treatment can be found in [7].
One of the main challenges is to understand renormalization in a system where the interactions lead to the appearance of bound states. Indeed quantum chromodynamics is the main example we have in mind, where the theory is formulated in terms of physically unobservable variables, in ordinary energy scales, as a result of interactions only their bound states become physical particles. Since one is interested in understanding the formation of these bound states and calculating the resulting masses, in principle, it is most natural to work with the Hamiltonian directly. Of course this is a very hard problem. As a result it is valuable and interesting to learn more about renormalization and its non-perturbative aspects even in very simple systems using the Hamiltonian formalism. This has been done by Glazek and Maslowski for the Dirac-delta function in two dimensions [8]. In the present work, we will consider the same problem on a nontrivial manifold, two dimensional hyperbolic space. This is interesting because the gauge theory problem also has a nontrivial geometry when it is formulated on the space of connections modulo gauge equivalent configurations [19]. Hence, it is a nice exercise to see that type of complications may arise when the underlying geometry is nontrivial.
We shall start by reviewing point interactions on the Euclidean Plane and answer why this problem requires renormalization. following [8], we will review the renormalization of point interactions in the Euclidean plane using the Wilsonian RG scheme and derive the flow equation. As an addendum to Glazek and Maslowski, we also investigate the range of renormalizability using the Banach Contraction principle. In the next section we shall analyze the same problem on the hyperbolic plane and show that the flow equation has the same form. Finally we will speak about a puzzle where this procedure fails at a technical level, if one study the same problem on a compact manifold, namely two-dimensional sphere, S 2 .
2 Exact Renormalization Group on the Euclidean Plane

Formulation of the Problem
The Schrödinger equation for the simplest type of point interaction on a D-dimensional Euclidean space R D is given by, in units = 1 and 2m = 1, as where ∆ R D is the Laplacian operator on R D and g is real, positive parameter which determines the strength of the point interaction. If we parametrize the bound state energy by E = −ν 2 , then the Schrödinger equation for the bound state of the system becomes where φ(x) is the bound state wavefunction. This expression can be expressed in momentum space as whereφ(p, ω) is the Fourier Transform of φ(x). Note that we also switched to spherical coordinates in momentum space, where p and ω denote the radial and angular coordinates respectively. From (3), g −1 can be solved as where vol S D−1 denotes the volume of the unit sphere in D − 1 dimensions. It is easy to see that the integral diverges for D ≥ 2, so regularization and renormalization are needed to obtain physical results. Renormalization of point interactions has been studied by many authors; in position space [12,14,17], and in momentum space [2,6,5,18,20]. The renormalization group equations were derived in [1,2]. Instead of the conventional approach, we shall perform the renormalization using the Exact Renormalization Group (ERG) method.

Renormalization of Hamiltonians
In this section we perform the renormalization of a point interaction on the Euclidean plane, R 2 . This part will be mainly a review of the lecture notes given by G lazek and Maslowski [8] and we include it for the sake of completeness. However our approach will be slightly different and as an addendum we also investigate the range of renormalizability using the Banach Contraction Principle.
Before any kind of regularization or renormalization, the Schrödinger equation for the bound state can be written as We want to calculate the effective Hamiltonian H Λ eff at some energy scale Λ, where Λ ≫ 1. This is done by integrating out degrees of freedom above Λ. To this end we introduce the operators P and Q which are projections to the subspaces, where the momentum eigenvalue takes the values 0 ≤ p ≤ Λ and p > Λ respectively. Let us also define |φ P ≡ P |φ and |φ Q ≡ Q |φ . By using P + Q = I and PQ = 0 one can split (5) as From (7) we find If we substitute this result back into (6) we get and this implies that the effective Hamiltonian at the scale Λ is given by We note that, although we are working at the scale Λ, the effective Hamiltonian contains the X Λ term and this term, which is called counterterm, depends on the higher degrees of freedom. And as we shall see now, we will use this counterterm in order to define the effective coupling constant at the scale Λ. Let us write the Hamiltonian as H = H 0 + V where H 0 is the free Hamiltonian and V denotes the point interaction, i.e. x| V |φ = −gδ 2 (x)φ(x). (9) can be written in momentum space as whereφ P (p) = p φ P . By defining x Λ (p, p ′ ) ≡ (2π) 2 p| X Λ |p ′ and using we get where Θ Λ (p) is the step function. We see that the g−x Λ (p, p ′ ) term plays the role of the effective coupling constant. From now on we denote it by g Λ (p, p ′ ). The counterterm x Λ (p, p ′ ) acts like a correction to the initial theory and by using it we have defined the renormalized coupling constant g Λ (p, p ′ ) at the scale Λ.

Applying the ERG Procedure
Now we are in a position to perform the ERG analysis of our theory. Since the original problem is rotationally symmetric, we want to keep rotational symmetry intact. Therefore we assume that the renormalized coupling constant g Λ does not depend on ω. At the bare scale Λ we can write the following equation: where We remark that we have switched to the unprojected wavefunctionφ(p, ω) and compensate this change by putting the step function Θ Λ (p) in front of the integral, which ensures that (14) is valid for p ≤ Λ. Following the ERG procedure, we write the analog of (14) at the infinitesimally lower scale Λ − dΛ.
We can rewrite (14) as For p = Λ this will give us and from this we can read ofφ(Λ, ω) as where we have ignored the term which is proportional to dΛ. If we substitute this result into (15) and perform the ω integral we find Finally we put this result into (17) to obtain Clearly, we can replace Θ Λ (p) by Θ Λ−dΛ (p) and write Now comparing this equation with (16) gives us an equation for the coupling constant which can be put into differential form as This equation determines the RG trajectory of the coupling constant. To find the effective coupling at the effective scale λ we integrate this from λ to Λ and find or Although this is an ordinary differential equation with three variables and we have one initial condition, there is also the requirement that g λ (p, p ′ ) should not depend on Λ when we take the Λ → ∞ limit. This can be satisfied by the appropriate choice of the counterterm x Λ (p, p ′ ). We try an iteration procedure to obtain a solution.
At the first order we choose After substituting these choices to (26) we get The integral diverges in the Λ → ∞ limit, therefore we choose the counterterm as where λ 0 is an another energy scale chosen such that 1 ≪ λ 0 < λ ≪ Λ. Now the effective coupling at the second order is finite and given by We note that it is independent of p and p ′ . If we repeat this procedure, then by induction it is straightforward to see that g (n) λ and x (n) Λ are independent of p and p ′ for all n. At the order n + 1, the effective coupling becomes We choose the counterterm as hence we find (33) It is not trivial to conclude that this iteration process has a limit. We shall deal with this later in this section and for now we assume that g and λ 0 are chosen such that the sequence {g (n) λ } ∞ n=1 has a limit given by After taking the limit we can write for the effective coupling which immediately implies g = g λ0 . This equation can be put into the following form which implies After integrating this equation from λ 0 to λ and solving for g λ , we obtain the final answer.
This result is in agreement with the one given in [1]. We also note that as λ → ∞, g λ → 0, so the theory is asymptotically free.

Estimating the Range of Renormalizability
Now we shall try to investigate under which conditions the sequence {g (n) λ } ∞ n=1 has a limit. However we don't have a closed form expression for g (n) λ , (31) tells us that g (n) λ depends on g (n−1) λ , hence we could not obtain a rigorous result for the convergence radius of the series {g (n) λ } ∞ n=1 in this way. An alternative way is to investigate under which circumstances does the integral equation given by has a unique solution. This can be done by using the theory of ordinary differential equations. We begin by defining a compact interval I = [λ 0 ,λ] ⊂ R where 1 ≪ λ 0 < λ <λ ≪ Λ. Let C(I) denote the space of continuous functions on I. It becomes a vector space if the vector space operations are defined pointwise. Moreover it is well known that it is a Banach Space if we define a norm on C(I) by We note that we use g, h as elements of C(I) to avoid confusion with the unrenormalized coupling constant g, that is, we made the definition g(s) ≡ g s . Now we introduce a map T : C(I) → C(I) defined by Then (39) can be expressed as g = T g, in other words the solution of (39) is also a fixed point of T . The existence and uniqueness of a solution to g = T g can be proved using Contraction Principle which can be stated as follows: Let D be a nonempty closed subset of a Banach Space B. If a map T : D → B is a contraction and maps D into itself, i.e. T (D) ⊆ D, then T has an exactly one fixed point g which is in D [22]. T is a contraction means that there exist a positive constant θ < 1 such that If T is a contraction, then the sequence {g (n) } ∞ n=1 defined by where g 0 is an arbitrary element of D, converges to the fixed point g, that is Therefore to estimate the range of renormalizability of our theory, we need to estimate under which cases the map T defined as in (41) is a contraction. First of all we need a closed subset of C(I). From (39) we can conclude that if g is a solution, then it should be monotone decreasing on I = [λ 0 ,λ]. Thus it is natural to choose our closed subset as Then thus T (D) ⊆ D. So it remains to show that T is a contraction. Let g, h ∈ D. Then we have the estimate By taking the supremum of both sides we find This tells us that T is a contraction if If we interpret the interval I = [λ 0 ,λ] as the range of renormalizability, then from (49) we can see that it is directly related to the coupling at the energy scale λ 0 . For a small coupling g λ0 ≪ 1, we can shift upλ considerably without breaking the contraction property of T , however for couplings g λ0 ∼ 1, the range is quite small or we may not even prove the existence of a solution by this approach.

Bound State Solution
We can check that with the coupling constant given as in (38) we get a finite answer for the bound state energy. For this we plug (38) into (14) to find By defining we obtainφ Substituting this result into (51) and diving both sides by N give us From this equation we can solve for ν 2 and in the Λ → ∞ limit and find which is finite.

Exact Renormalization Group on the Hyperbolic Plane
We will begin this section by constructing the spectral representation of the Laplacian on the hyperbolic plane H 2 . By using this construction we shall perform the ERG analysis of a point interaction on the hyperbolic plane.

The Geometry and Spectra of the Hyperbolic Plane
We shall do the construction by using ideas given in [15] and [21]. There are various models for the hyperbolic plane. We will use the upper half-plane model, where H 2 is realized as the set with the Riemannian metric g H 2 given by where −R −2 is the constant sectional curvature. The Riemannian volume element is given by and the Laplacian is The eigenfunctions can be found by solving the closed eigenvalue problem on L 2 (H 2 , dV H 2 ) expressed by where λ ∈ R. For notational simplicity, let us define∆ H 2 ≡ R 2 ∆ H 2 andλ ≡ R 2 λ. Then (59) will be equivalent to Since∆ H 2 is separable in (x, y) coordinates we can use separation of variables. So we choose f (z) = v(x)w(y) and put this into (60) to obtain This implies that there is a constant ξ 2 such that The x-part can be solved easily as v(x) = e iξx . To solve the y-part we introduce a new function by u(y) ≡ y −1/2 w(y). After substituting this into the y-part of (62) and making some rearrangements we get The eigenvalues of the Laplacian on H 2 starts withλ 0 = 1 4 [4]. Therefore 1 4 −λ ≤ 0 so we introduce a new variable τ ∈ [0, ∞) such that 1 4 −λ = (iτ ) 2 . Then (63) becomes There are two linearly independent solutions which are the modified Bessel Functions I iτ (|yξ|) and K iτ (|yξ|). Since I iτ (|yξ|) is singular at infinity, we exclude it from our solution space. Moreover K iτ (|yξ|) has a singularity at ξ = 0 given by [21] K iτ (|yξ|) So we choose u(y) = |ξ| iτ K iτ (|yξ|) as a solution to (64) and write the eigenfunctions of ∆ H 2 as where we have put an extra (2π) −1/2 to simplify our construction of the spectral representation. We note that this does not alter the spectrum of the eigenvalues. To obtain the spectral representation of ∆ H 2 we introduce the following transform, which is a combination of the Fourier Transform in (x, ξ) variables and Kontorovich-Lebedev Transform in (y, τ ) variables [16]. The range of K can be formulated by considering L 2 (R, dξ) as the Hilbert Space corresponding to ξ, and then taking a direct integral of it with respect to the measure space (0, ∞). So the map K can be expressed formally as To prove that K provide a spectral representation of ∆ H 2 , we need two identities involving modified Bessel Functions which are given by [15] 2 Moreover we shall also use the property K iτ (u) = K −iτ (u). Using (70) one can show that K diagonalizes the Laplacian, that is for allψ ∈ and therefore So the spectrum of ∆ H 2 is given by By using (69) it is straightforward to show that K is an isomorphism, i.e. for all ψ ∈ L 2 (H 2 , dV H 2 ) we have Therefore, the transformation K with the eigenfunctions given as in (66) provide a complete spectral representation of ∆ H 2 .

Formulation of the Problem
Just like the flat case we consider a particle of mass m interacting with a Dirac-Delta potential on the hyperbolic plane H 2 . Let z 0 = (x 0 , y 0 ), y 0 = 0 denote the location of the Dirac-Delta potential. The corresponding Schrödinger equation for the bound state is written in coordinates = 1, 2m = 1 as On a Riemannian manifold (M, g) the Dirac Delta function δ g (z, z 0 ) is defined such that M dV g (z) δ g (z, z 0 ) = 1 for all z 0 ∈ M.
Thus δ H 2 (z, z 0 ) is given by Since K is an isomorphism we can write (75) as First term can be read directly from (71), which is Second term can be easily calculated using δ H 2 (z, z 0 ). The result is If we put (79) and (80) into the (75) and rearrange the terms we obtain where we made the definition a 2 ≡ 1 4 + ν 2 R 2 . Our next goal is to determine the type and the cause of the divergence. To this end we make an attempt to solve (81). We define We put this result back into (82) and find Let us denote the ξ-integral by Υ(τ ′ ). Using the explicit form of the eigenfunctions as given in (66) we can write Υ(τ ′ ) as To evaluate the integral we will use the following integral representation of the modified Bessel functions [3]: The dv integral can be evaluated using the definite integral [13] ∞ 0 In our case a = τ ′ , b = 1, β = 1, c = cosh u and cosh u ≥ 1 > 0, for all u ∈ [0, ∞) so we can use (88). Hence dv integral becomes Then we have Finally we will use [13] ∞ 0 dx sin(αx) cos(βx) sinh(γx) = π sinh πa γ 2γ cosh απ γ + cosh βπ γ (91) for Im (α + β) < Re γ. In our case α = β = τ ′ , γ = 1 and τ ′ ∈ R, therefore Im (α + β) = 0 < 1. Thus and Υ(τ ′ ) becomes Finally we put this result into (84) and obtain For large values of τ ′ , tanh(πτ ′ ) ≈ 1 and the integrand behaves as 1/τ ′ so just like the flat case we face with a logarithmic divergence. This analysis also shows us that there is no divergence in the ξ term. Therefore we only need to concern with the renormalization of τ .

Applying the ERG Procedure
We start by writing the eigenvalue equation at the bare scale Λ.
where g Λ (τ, τ ′ ) = g − x Λ (τ, τ ′ ) and At an infinitesimally lower scale Λ − dΛ we write We can rewrite (95) as and for τ = Λ we obtain Thenφ(Λ, ξ) becomes where we have again ignored the term proportional to dΛ. From this result we can find ϑ(τ, Λ; ξ) as By putting the explicit expression for ϑ(Λ, τ ′ ; ξ ′ ) we get If we put this result back into (98) we find By comparing this with (97) we arrive to an equation for the coupling constant which can be put into differential form as and by integrating from λ to Λ we find Λ λ ds s s 2 + a 2 tanh(πs)g s (s, τ ′ )g s (τ, s).
To obtain a solution we shall use the same procedure we used in the flat case. We begin with Then g (2) λ becomes We choose the counterterm as so that the effective coupling at the second order is now finite and given by Just like the flat case, g (n) λ (τ, τ ′ ) is independent of τ and τ ′ for all n. At the order n + 1, the effective coupling becomes By choosing the counterterm as we find If we assume the existence of we can write for the effective coupling which again implies g = g λ0 . This expression can be put into the following form By evaluating the integral on the LHS and expressing tanh(πs) as tanh(πs) = 1 − 2 e 2πs + 1 , we obtain Let us define a function α(s) by Then (116) becomes By redefining the coupling constant byg −1 s ≡ g −1 s + α −1 s and evaluating the integral in (118) we obtain which can be solved asg We see that by slightly modifying the coupling constant we can obtain the same solution as the flat case.
To show that this modification brings no problems at high energies let us try to estimate α −1 (λ).
Using Cauchy-Schwartz inequality we find We see that for high energies λ ≫ 1, this correction term behaves like a constant.

Estimating the Range of Renormalizability
Investigating that under which conditions does the sequence {g Just like the flat case, T is a contraction if

Bound State Solution
To find the bound state energy we rewrite (94) in terms of renormalized coupling.
Thus (126) becomes The integrals on both sides vanish so we are left with with the solution in the Λ → ∞ limit By plugging this result into (134) we find g −1 as where we have used By using Maclaurin-Cauchy integral test we can see that the l ′ sum in (136) is logarithmically divergent and the divergence is caused by the large l ′ values.
By putting this result back into (141) we get C m l [l(l + 1) + ν 2 R 2 ] = Θ Λ (l) and by comparing this result with (140) we obtain a recursion relation for the effective coupling constant.
By applying the same iteration procedure as in the previous cases we can obtain This is a very complicated recursive relation, and unlike previous cases, we could not convert it to a differential equation, from which we can solve for g λ in the n → ∞ limit.

Conclusion
In this paper we investigated a non-perturbative renormalization of point interactions on the twodimensional hyperbolic space, using the Exact Renormalization Group method. We showed that the theory is asymptotically free and the flow equations has the same form as in the flat case.