# Deconstructing the Gel’fand–Yaglom Method and Vacuum Energy from a Theory Space

Authors: Nahomi Kan; Kiyoshi Shiraishi
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

The discrete Gel’fand–Yaglom theorem was studied several years ago. In the present paper, we generalize the discrete Gel’fand–Yaglom method to obtain the determinants of mass matrices which appear in current works in particle physics, such as dimensional deconstruction and clockwork theory. Using the results, we show the expressions for vacuum energies in such various models.


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## Body

## 1. Introduction

The Gel’fand–Yaglom method [1] for obtaining functional determinants of differential operators with boundaries is widely known nowadays. For nice reviews, see [2, 3]. The applications of the Gel’fand–Yaglom method have been investigated quite recently, to evaluate one-loop vacuum energies in nontrivial boundary conditions [4–7].Among them, Altshuler examined vacuum energy in warped compactification [6, 7]. In recent years, it is supposed that extra dimensions of various types could play an important role in the hierarchy problem, and thus the study of physics in nontrivial background geometry is still advancing.The dimensional deconstruction has appeared as a new tool for understanding the properties of higher-dimensional field theories [8–10] more than a decade ago. In such a model of deconstruction, a “theory space” is considered, which consists of sites and links, to which four-dimensional fields are individually assigned. Theory spaces thus have the structures of graphs [11] and can be interpreted as the theory with discrete extra dimensions.Several years ago, the discrete Gel’fand–Yaglom method for difference operators was reviewed and studied by Dowker [12]. We generalize the discrete Gel’fand–Yaglom method for studying one-loop vacuum energies in extended deconstructed theories and models with discrete dimensions in the present paper. To this end, we develop the method of computing determinants of repetitive Hermitian matrices which correspond to mass matrices utilized in deconstructed theories.After completion of the first version of the manuscript of the present paper (arXiv:1711.06806), a paper which treats the determinants of discrete Laplace operators appeared [13]. Their method is substantially the same as ours, because the author also relies on the recurrence relation among three variables on a lattice (see Section 3 in the present paper and below). We recently become aware of another similar paper on the determinants of matrix differential operators [14]. They studied generalization of Gel’fand–Yaglom method to obtain the functional determinants. Their work differs essentially from ours because they considered differential operators while we treat matrices as operators. We also point out that they did not consider the matrices of large size which have certain continuum limits.The organization of this paper is as follows. In order to make the present paper self-contained, we show a short review of the Gel’fand–Yaglom method for a differential operator, along with Dunne’s review [2], in Section 2. In Section 3, we give the method to obtain determinants of tridiagonal matrices with repeated structure. This is a straightforward generalization of description in Ref. [12]. In Section 4, we give the method to obtain determinants of periodic tridiagonal matrices. Determinants of extended periodic tridiagonal matrices are obtained in Section 5. The rest of the present paper is devoted to applications to deconstructed theories and discrete systems. In Section 6, free energy on a graph is discussed by using the results of previous sections. In Section 7, we show the method of calculation for evaluating one-loop vacuum energy in deconstructed models from the determinants of mass matrices. In Section 8, we show a few more examples of one-loop vacuum energies for slightly complicated theory spaces. We give conclusions in the last section, Section 9.


## 2. Review of the Gel’fand–Yaglom Method [2]

Suppose that an eigenvalue equation(Δ+m2)ψλ(m2)(x)=λ(m2)ψλ(m2)(x) with Dirichlet-Dirichlet boundary conditions ψλ(m2)(0)=ψλ(m2)(L)=0 in one dimension is given.Then(1)det⁡Δ+m2det⁡Δ=ψ0m2Lψ00Lholds. Here ψ0(m2)(x) satisfies (Δ+m2)ψ0(m2)(x)=0 with a boundary condition ψ0(m2)(0)=0.Example.In the region 0≤x≤L, under the Dirichlet condition at the boundaries, we consider the functional determinant(2)det⁡-∂x2+m2det⁡-∂x2.The solution of (-∂x2+m2)ψ(x)=0 with ψ(0)=0 is sinh⁡mx. Thus according to the Gel’fand–Yaglom method, we obtain(3)det⁡-∂x2+m2det⁡-∂x2=sinh⁡mLmL=∏n=1∞n2π2/L2+m2n2π2/L2.Proof.
det ⁡ [ Δ + m 2 - λ ] is a function of λ and has zeros at λ=λ(m2). The function ψλ(x) which satisfies (Δ+m2-λ)ψλ(x)=0 and ψλ(0)=0 becomes the eigenfunction when λ=λ(m2). Then the boundary condition at x=L, i.e., ψλ(L)=0, is satisfied. In other words, ψλ(L) is a function of λ and has zeros at λ=λ(m2). Therefore det⁡[Δ+m2-λ]∝ψλ(L) holds.


## 3. Determinants of Tridiagonal Matrices

### 3.1. The Discrete Gel’fand–Yaglom Method for Tridiagonal Matrices

Now, we show the discrete Gel’fand–Yaglom method to obtain determinants of finite matrices. First, we consider the following Hermitian tridiagonal matrix ofN rows and columns:(4)H=c-b0⋯0-b∗a-b⋯00-b∗a⋯0⋮⋮⋮⋱⋮a-b000⋯-b∗d. In this case, the eigenvalue equation(5)Hv=λv, where vT=(v1,v2,…,vN), can be categorized into three parts:(6)∑j=1NH1jvj-λv1=0,(7)∑j=1NHkjvj-λvk=02≤k≤N-1,(8)∑j=1NHNjvj-λvN=0.Here, Eq. (7) is just the recurrence relation among three terms in vk as a sequence of numbers. In the present case, the general solution for the recurrence relation(9)bvk+1-a-λvk+b∗vk-1=02≤k≤N-1is(10)vk=Aαk-1+Bβk-1,where A and B are constants and(11)α=a-λ+a-λ2-4b22b,β=a-λ-a-λ2-4b22b.Note that α and β are roots of the second-order equation bx2-(a-λ)x+b∗=0 and αβ=b∗/b.The first row of the eigenvalue equation, Eq. (6), determines the relation between v1 and v2; in this case, that is (c-λ)v1-bv2=0. If we further choose(12)v1=A+B=1,the coefficients A and B are obtained as(13)A=Aλ=c-λ-bβbα-β=c-λ-bβa-λ2-4b2,(14)B=Bλ=-c-λ-bαbα-β=-c-λ-bαa-λ2-4b2.Substituting all of the results above into Eq. (8) in the present case, we get(15)Aλ-b∗+d-λααN-2+Bλ-b∗+d-λββN-2=0.Now, we set the left-hand side of Eq. (15) as D~(λ). D~(λ) is zero if λ is an eigenvalue of the matrix H in this case. By construction, D~(λ) should be an Nth order polynomial of λ. The reason is the following: v2=[(c-λ)/b]v1=(c-λ)/b, v3=[(a-λ)/b]v2-(b∗/b)v1=(a-λ)(c-λ)/b2-(b∗/b), and so on. This observation shows vN includes -λN-1/bN-1. Finally, since the left-hand side of Eq. (8) reads -b∗vN-1+(d-λ)vN in the present case, D~(λ) has the term -λN/bN-1 as the highest order term in λ. We can also directly confirm this by setting a=c=d=0 and the limit b→0 in the left-hand side of Eq. (15). We then verify D~(λ)→-λN/bN-1.Therefore, we conclude thatD(λ)=bN-1D~(λ)=∏p=1N(λp-λ) is the characteristic polynomial of H, where λp(p=1,2,…,N) are eigenvalues of H.The determinant ofH is given by D(0)=∏p=1Nλp. In the present case, we find(16)det⁡H=D0=bN-1a2-4b2c-bβ¯-b∗+dα¯α¯N-2-c-bα¯-b∗+dβ¯β¯N-2, where(17)α¯=a+a2-4b22b,β¯=a-a2-4b22b. After a lengthy calculation, we obtain(18)det⁡H=D0=12Na2-4b22cd-b2+a-c-da-a2-4b2a+a2-4b2N-1-2cd-b2+a-c-da+a2-4b2a-a2-4b2N-1foratridiagonalmatrix,in the present case. It is notable that the determinant depends only on |b| and does not depend on b∗/b in the present case. The reason is because the eigenvalues are unchanged under “gauge” transformation v→Pv and H→PHP-1, where P=diag.(1,eiχ,ei2χ,…,ei(N-1)χ) with an arbitrary real constant χ.The prescription of the above method to obtain the determinant is very similar to the Gel’fand–Yaglom method for differential operators. Namely, solving the differential equation corresponds to solving the recurrence relation, putting one of the boundary conditions corresponds to fixing the first term of the series of numbers, and obtaining the determinant at another boundary corresponds to obtaining the determinant as the equation of the last row in the eigenvalue equation. Note that, because we are treating a finite matrix, the idea of normalization becomes different from the functional determinant treated by the Gel’fand–Yaglom method.The method to obtain the determinant of tridiagonal matrices in this section is substantially equivalent to the method for difference operators described by Dowker [12], except for a specific choice for Hermitian matrix in the present section.


### 3.2. Examples

In this subsection, we show determinants of some simple tridiagonal matrices for example. For all the examples below, the eigenvalues are known and, then, one can find that the formulas1 for finite product including trigonometric functions are derived.Note that the determinantD(0) for the matrix H=Δ+m2I (where I is the identity matrix) is equivalent to D(-m2) for the matrix Δ, and we choose explicit expressions of D(0) for H here and hereafter.(i)a=c=d=2+4sinh2z/2andb=1.2In this case,H=ΔDD+4sinh2z/2IN, where(19)ΔDD≡2-10⋯0-12-1⋯00-12⋯0⋮⋮⋮⋱⋮2-1000⋯-12, and IN is the N×N identity matrix. Note that, for a=c=d, Eq. (18) becomes(20)D0=12N+1a2-4b2a+a2-4b2N+1-a-a2-4b2N+1. We now find(21)det⁡H=D0=sinh⁡N+1zsinh⁡zforH=ΔDD+4sinh2⁡z2IN.(ii)a=c=2+4sinh2⁡z/2, d=1+4sinh2⁡z/2,andb=1.3In this case,H=ΔDN+4sinh2⁡z/2IN, where(22)ΔDN≡2-10⋯0-12-1⋯00-12⋯0⋮⋮⋮⋱⋮2-1000⋯-1-1 We find(23)det⁡H=D0=sinh⁡N+1z-sinh⁡Nzsinh⁡z=cosh⁡Nz+2sinh⁡zsinh⁡NzforH=ΔDN+4sinh2z2IN.(iii) a=2+4sinh2(z/2), c=d=1+4sinh2(z/2),andb=1.4In this case,H=ΔNN+4sinh2(z/2)IN, where(24)ΔNN≡1-10⋯0-12-1⋯00-12⋯0⋮⋮⋮⋱⋮2-1000⋯-1-1=ΔPN, which is known as the graph Laplacian [15–18] for the path graph with N vertices (PN, see Figure 1).Figure 1
P 8 as an example of a path graph.We find(25)det⁡H=D0=2tanh⁡z2sinh⁡NzforH=ΔNN+4sinh2z2INin this case. Note that, since Δ(PN) has a zero mode, limz→0D(0)=0.(iv) Clockwork theory [19–22].5We considerH=Δq+l2IN, where Δq is the following N×N matrix:(26)Δq≡1-q0⋯0-q1+q2-q⋯00-q1+q2⋯0⋮⋮⋮⋱⋮1+q2-q000⋯-qq2. We find that the determinant of H can be written as(27)D0=l2·γ+N-γ-N1-q22+21+q2l2+l4forH=Δq+l2IN,where(28)γ±=121+q2+l2±1-q22+21+q2l2+l4.Of course, one can see that limq→1Δq=Δ(PN).


## 3.1. The Discrete Gel’fand–Yaglom Method for Tridiagonal Matrices

Now, we show the discrete Gel’fand–Yaglom method to obtain determinants of finite matrices. First, we consider the following Hermitian tridiagonal matrix ofN rows and columns:(4)H=c-b0⋯0-b∗a-b⋯00-b∗a⋯0⋮⋮⋮⋱⋮a-b000⋯-b∗d. In this case, the eigenvalue equation(5)Hv=λv, where vT=(v1,v2,…,vN), can be categorized into three parts:(6)∑j=1NH1jvj-λv1=0,(7)∑j=1NHkjvj-λvk=02≤k≤N-1,(8)∑j=1NHNjvj-λvN=0.Here, Eq. (7) is just the recurrence relation among three terms in vk as a sequence of numbers. In the present case, the general solution for the recurrence relation(9)bvk+1-a-λvk+b∗vk-1=02≤k≤N-1is(10)vk=Aαk-1+Bβk-1,where A and B are constants and(11)α=a-λ+a-λ2-4b22b,β=a-λ-a-λ2-4b22b.Note that α and β are roots of the second-order equation bx2-(a-λ)x+b∗=0 and αβ=b∗/b.The first row of the eigenvalue equation, Eq. (6), determines the relation between v1 and v2; in this case, that is (c-λ)v1-bv2=0. If we further choose(12)v1=A+B=1,the coefficients A and B are obtained as(13)A=Aλ=c-λ-bβbα-β=c-λ-bβa-λ2-4b2,(14)B=Bλ=-c-λ-bαbα-β=-c-λ-bαa-λ2-4b2.Substituting all of the results above into Eq. (8) in the present case, we get(15)Aλ-b∗+d-λααN-2+Bλ-b∗+d-λββN-2=0.Now, we set the left-hand side of Eq. (15) as D~(λ). D~(λ) is zero if λ is an eigenvalue of the matrix H in this case. By construction, D~(λ) should be an Nth order polynomial of λ. The reason is the following: v2=[(c-λ)/b]v1=(c-λ)/b, v3=[(a-λ)/b]v2-(b∗/b)v1=(a-λ)(c-λ)/b2-(b∗/b), and so on. This observation shows vN includes -λN-1/bN-1. Finally, since the left-hand side of Eq. (8) reads -b∗vN-1+(d-λ)vN in the present case, D~(λ) has the term -λN/bN-1 as the highest order term in λ. We can also directly confirm this by setting a=c=d=0 and the limit b→0 in the left-hand side of Eq. (15). We then verify D~(λ)→-λN/bN-1.Therefore, we conclude thatD(λ)=bN-1D~(λ)=∏p=1N(λp-λ) is the characteristic polynomial of H, where λp(p=1,2,…,N) are eigenvalues of H.The determinant ofH is given by D(0)=∏p=1Nλp. In the present case, we find(16)det⁡H=D0=bN-1a2-4b2c-bβ¯-b∗+dα¯α¯N-2-c-bα¯-b∗+dβ¯β¯N-2, where(17)α¯=a+a2-4b22b,β¯=a-a2-4b22b. After a lengthy calculation, we obtain(18)det⁡H=D0=12Na2-4b22cd-b2+a-c-da-a2-4b2a+a2-4b2N-1-2cd-b2+a-c-da+a2-4b2a-a2-4b2N-1foratridiagonalmatrix,in the present case. It is notable that the determinant depends only on |b| and does not depend on b∗/b in the present case. The reason is because the eigenvalues are unchanged under “gauge” transformation v→Pv and H→PHP-1, where P=diag.(1,eiχ,ei2χ,…,ei(N-1)χ) with an arbitrary real constant χ.The prescription of the above method to obtain the determinant is very similar to the Gel’fand–Yaglom method for differential operators. Namely, solving the differential equation corresponds to solving the recurrence relation, putting one of the boundary conditions corresponds to fixing the first term of the series of numbers, and obtaining the determinant at another boundary corresponds to obtaining the determinant as the equation of the last row in the eigenvalue equation. Note that, because we are treating a finite matrix, the idea of normalization becomes different from the functional determinant treated by the Gel’fand–Yaglom method.The method to obtain the determinant of tridiagonal matrices in this section is substantially equivalent to the method for difference operators described by Dowker [12], except for a specific choice for Hermitian matrix in the present section.


## 3.2. Examples

In this subsection, we show determinants of some simple tridiagonal matrices for example. For all the examples below, the eigenvalues are known and, then, one can find that the formulas1 for finite product including trigonometric functions are derived.Note that the determinantD(0) for the matrix H=Δ+m2I (where I is the identity matrix) is equivalent to D(-m2) for the matrix Δ, and we choose explicit expressions of D(0) for H here and hereafter.(i)a=c=d=2+4sinh2z/2andb=1.2In this case,H=ΔDD+4sinh2z/2IN, where(19)ΔDD≡2-10⋯0-12-1⋯00-12⋯0⋮⋮⋮⋱⋮2-1000⋯-12, and IN is the N×N identity matrix. Note that, for a=c=d, Eq. (18) becomes(20)D0=12N+1a2-4b2a+a2-4b2N+1-a-a2-4b2N+1. We now find(21)det⁡H=D0=sinh⁡N+1zsinh⁡zforH=ΔDD+4sinh2⁡z2IN.(ii)a=c=2+4sinh2⁡z/2, d=1+4sinh2⁡z/2,andb=1.3In this case,H=ΔDN+4sinh2⁡z/2IN, where(22)ΔDN≡2-10⋯0-12-1⋯00-12⋯0⋮⋮⋮⋱⋮2-1000⋯-1-1 We find(23)det⁡H=D0=sinh⁡N+1z-sinh⁡Nzsinh⁡z=cosh⁡Nz+2sinh⁡zsinh⁡NzforH=ΔDN+4sinh2z2IN.(iii) a=2+4sinh2(z/2), c=d=1+4sinh2(z/2),andb=1.4In this case,H=ΔNN+4sinh2(z/2)IN, where(24)ΔNN≡1-10⋯0-12-1⋯00-12⋯0⋮⋮⋮⋱⋮2-1000⋯-1-1=ΔPN, which is known as the graph Laplacian [15–18] for the path graph with N vertices (PN, see Figure 1).Figure 1
P 8 as an example of a path graph.We find(25)det⁡H=D0=2tanh⁡z2sinh⁡NzforH=ΔNN+4sinh2z2INin this case. Note that, since Δ(PN) has a zero mode, limz→0D(0)=0.(iv) Clockwork theory [19–22].5We considerH=Δq+l2IN, where Δq is the following N×N matrix:(26)Δq≡1-q0⋯0-q1+q2-q⋯00-q1+q2⋯0⋮⋮⋮⋱⋮1+q2-q000⋯-qq2. We find that the determinant of H can be written as(27)D0=l2·γ+N-γ-N1-q22+21+q2l2+l4forH=Δq+l2IN,where(28)γ±=121+q2+l2±1-q22+21+q2l2+l4.Of course, one can see that limq→1Δq=Δ(PN).


## 4. Determinants of Periodic Tridiagonal Matrices

### 4.1. The Discrete Gel’fand–Yaglom Method for Periodic Tridiagonal Matrices

In this section, we treat periodic tridiagonal matrices, such as(29)H=a-b0⋯-b∗-b∗a-b⋯00-b∗a⋯0⋮⋮⋮⋱⋮a-b-b00⋯-b∗a.In this case, the recurrence relation is the same as in the previous section. Therefore, we can write(30)vk=Aαk-1+Bβk-1, where α and β are the same as the previous ones, i.e., Eq. (11).In the periodic case, however, the first and the last rows of the eigenvalue equation are also the relation among three terms in the sequence of numbers. In the present case, they are reduced to(31)bAβ1-αN+Bα1-βN=0,(32)bAααN-1+BββN-1=0,where we used the fact that α and β are solutions of bx2-(a-λ)x+b∗=0 and αβ=b∗/b. The existence of A and B satisfying the above two equations and not being A=B=0 requires(33)β1-αNα1-βNααN-1ββN-1=β2-α21-αNβN-1=0. This equation is satisfied if λ is an eigenvalue of the matrix H. In general, we suppose α≠β and the normalization can be known from the limit a=0 and b→0. Then, we conclude that the characteristic polynomial D(λ)=∏p=1N(λp-λ) (where λp (p=1,…,N) are eigenvalues of H) is written by(34)Dλ=bN1-αNβN-1=bNαN+βN-bN-b∗N. Therefore, the determinant of H in this case is given by(35)det⁡H=D0=a+a2-4b22N+a-a2-4b22N-bN-b∗Nforaperiodictridiagonalmatrix.One may be aware of unnecessary arguments in above discussion. From the periodic structure,αN=1 or βN=1 can be concluded. However, the discussion above can be generalized to treat another type of matrix in the next section.


### 4.2. Example

(i)a=2+4sinh2(z/2)andb=1.6In this case,H=Δ(CN)+4sinh2(z/2)IN, where Δ(CN) is the graph Laplacian of the cycle graph with N vertices (see Figure 2):(36)ΔCN≡2-10⋯-1-12-1⋯00-12⋯0⋮⋮⋮⋱⋮2-1-100⋯-12.Figure 2
C 8 as an example of a cycle graph.We find(37)det⁡H=D0=4sinh2Nz2=eNz/2-e-Nz/22forH=ΔCN+4sinh2z2IN.Note that limz→0D(0)=0 because of the zero mode of Δ(CN).(ii)a=2+4sinh2(z/2)andb=eiχ.7In this case,H=Δ(CN,χ)+4sinh2(z/2)IN, where(38)ΔCN,χ≡2-e-iχ0⋯-e-iχ-e-iχ2-e-iχ⋯00-e-iχ2⋯0⋮⋮⋮⋱⋮2-e-iχ-e-iχ00⋯-e-iχ2. We find(39)det⁡H=D0=4sinh2Nz2+4sin2Nχ2=eNz+iχ/2-e-Nz+iχ/22forH=ΔCN,χ+4sinh2z2IN.


## 4.1. The Discrete Gel’fand–Yaglom Method for Periodic Tridiagonal Matrices

In this section, we treat periodic tridiagonal matrices, such as(29)H=a-b0⋯-b∗-b∗a-b⋯00-b∗a⋯0⋮⋮⋮⋱⋮a-b-b00⋯-b∗a.In this case, the recurrence relation is the same as in the previous section. Therefore, we can write(30)vk=Aαk-1+Bβk-1, where α and β are the same as the previous ones, i.e., Eq. (11).In the periodic case, however, the first and the last rows of the eigenvalue equation are also the relation among three terms in the sequence of numbers. In the present case, they are reduced to(31)bAβ1-αN+Bα1-βN=0,(32)bAααN-1+BββN-1=0,where we used the fact that α and β are solutions of bx2-(a-λ)x+b∗=0 and αβ=b∗/b. The existence of A and B satisfying the above two equations and not being A=B=0 requires(33)β1-αNα1-βNααN-1ββN-1=β2-α21-αNβN-1=0. This equation is satisfied if λ is an eigenvalue of the matrix H. In general, we suppose α≠β and the normalization can be known from the limit a=0 and b→0. Then, we conclude that the characteristic polynomial D(λ)=∏p=1N(λp-λ) (where λp (p=1,…,N) are eigenvalues of H) is written by(34)Dλ=bN1-αNβN-1=bNαN+βN-bN-b∗N. Therefore, the determinant of H in this case is given by(35)det⁡H=D0=a+a2-4b22N+a-a2-4b22N-bN-b∗Nforaperiodictridiagonalmatrix.One may be aware of unnecessary arguments in above discussion. From the periodic structure,αN=1 or βN=1 can be concluded. However, the discussion above can be generalized to treat another type of matrix in the next section.


## 4.2. Example

(i)a=2+4sinh2(z/2)andb=1.6In this case,H=Δ(CN)+4sinh2(z/2)IN, where Δ(CN) is the graph Laplacian of the cycle graph with N vertices (see Figure 2):(36)ΔCN≡2-10⋯-1-12-1⋯00-12⋯0⋮⋮⋮⋱⋮2-1-100⋯-12.Figure 2
C 8 as an example of a cycle graph.We find(37)det⁡H=D0=4sinh2Nz2=eNz/2-e-Nz/22forH=ΔCN+4sinh2z2IN.Note that limz→0D(0)=0 because of the zero mode of Δ(CN).(ii)a=2+4sinh2(z/2)andb=eiχ.7In this case,H=Δ(CN,χ)+4sinh2(z/2)IN, where(38)ΔCN,χ≡2-e-iχ0⋯-e-iχ-e-iχ2-e-iχ⋯00-e-iχ2⋯0⋮⋮⋮⋱⋮2-e-iχ-e-iχ00⋯-e-iχ2. We find(39)det⁡H=D0=4sinh2Nz2+4sin2Nχ2=eNz+iχ/2-e-Nz+iχ/22forH=ΔCN,χ+4sinh2z2IN.


## 5. Determinants of Extended Periodic Tridiagonal Matrices

### 5.1. The Discrete Gel’fand–Yaglom Method For Extended Periodic Tridiagonal Matrices

In this section, we consider the following(N+1)×(N+1) matrix:(40)H=a-b0⋯-b∗-d-b∗a-b⋯0-d0-b∗a⋯0-d⋮⋮⋮⋱⋮⋮a-b-d-b00⋯-b∗a-d-d∗-d∗-d∗⋯-d∗-d∗c. The recurrence relation can be found as(41)bvk+1-a-λvk+b∗vk-1=-dvN+12≤k≤N-1. The general solution of this equation is(42)vk=Aαk-1+Bβk-1-dvN+1b+b∗-a-λ, where α and β are the same as Eq. (11).The first row of the eigenvalue equation then becomes(43)bAβ1-αN+Bα1-βN=0, while the Nth row of the eigenvalue equation is(44)bAααN-1+BββN-1=0. The two equations are exactly the same as Eqs. (31) and (32).Now, in addition, the (N+1)st row of the eigenvalue equation reads(45)-d∗v1+v2+⋯+vN+c-λvN+1=0,and, by using the general solution, this can be reduced to(46)-d∗A1-αN1-α+B1-βN1-β-NdvN+1b+b∗-a-λ+c-λvN+1=0.As in the previous section, we require that a nontrivial set of(A,B,vN+1) exists. This leads to the following equation:(47)β1-αNα1-βN0ααN-1ββN-10-d∗1-αN1-α-d∗1-βN1-βNd2b+b∗-a-λ+c-λ=β2-α21-αNβN-1Nd2b+b∗-a-λ+c-λ=0. The second left-hand side of the equation should be proportional to D(λ), as for discussion in the previous section. Because we have already known the normalization of bN(1-αN)(βN-1), we conclude that the characteristic polynomial in the present case is written by(48)Dλ=bN1-αNβN-1Nd2b+b∗-a-λ+c-λ. Thus, the determinant of H in this section is given by(49)D0=c-Nd2a-b-b∗a+a2-4b22N+a-a2-4b22N-bN-b∗Nforanextendedperiodictridiagonalmatrix.


### 5.2. Examples

(i)a=2r+s+l2, c=Ns+l2, d=s,andb=r [23, 24].8Suppose the(N+1)×(N+1) matrix [23, 24],(50)Δr,s=r2-10⋯-10-12-1⋯000-12⋯00⋮⋮⋮⋱⋮⋮2-10-100⋯-120000⋯000+s100⋯0-1010⋯0-1001⋯0-1⋮⋮⋮⋱⋮⋮000⋯1-1-1-1-1⋯-1N, where r and s are constants. The determinant of H=Δ(r,s)+l2IN+1 is(51)det⁡H=D0=l21+Nss+l2η+N+η-N-2rN,where(52)η±≡122r+s+l2±2r+s+l22-4r2.(ii) a=3+l2, c=N+l2,andb=d=1.This is the previous case withr=s=1. In this case, H=Δ(WN+1)+l2IN+1, where(53)ΔWN+1≡3-10⋯-1-1-13-1⋯0-10-13⋯0-1⋮⋮⋮⋱⋮⋮3-1-1-100⋯-13-1-1-1-1⋯-1-1N is the graph Laplacian of the wheel graph (see Figure 3) with N+1 vertices. We do not repeat writing the expression of H, which is given by Eq. (51) with Eq. (52) when r=s=1.Figure 3
W 9 as an example of a wheel graph.(iii)b=0.In this case, the determinant simply becomes as(54)det⁡H=D0=aNc-NaN-1d2. Particularly, Δ(0,1) is in this category and can be written as(55)Δ0,1=100⋯0-1010⋯0-1001⋯0-1⋮⋮⋮⋱⋮⋮000⋯1-1-1-1-1⋯-1N=ΔK1,N. This is the graph Laplacian of a star graph K1,N (Figure 4). The eigenvalues of Δ(K1,N) are known as(56)λ0=0,λp=1p=1,2,…,N-1,λN=N+1. The determinant of H=Δ(K1,N)+l2IN+1 is(57)det⁡H=D0=l21+N1+l21+l2N.Figure 4
K 1,8 as an example of a star graph.


## 5.1. The Discrete Gel’fand–Yaglom Method For Extended Periodic Tridiagonal Matrices

In this section, we consider the following(N+1)×(N+1) matrix:(40)H=a-b0⋯-b∗-d-b∗a-b⋯0-d0-b∗a⋯0-d⋮⋮⋮⋱⋮⋮a-b-d-b00⋯-b∗a-d-d∗-d∗-d∗⋯-d∗-d∗c. The recurrence relation can be found as(41)bvk+1-a-λvk+b∗vk-1=-dvN+12≤k≤N-1. The general solution of this equation is(42)vk=Aαk-1+Bβk-1-dvN+1b+b∗-a-λ, where α and β are the same as Eq. (11).The first row of the eigenvalue equation then becomes(43)bAβ1-αN+Bα1-βN=0, while the Nth row of the eigenvalue equation is(44)bAααN-1+BββN-1=0. The two equations are exactly the same as Eqs. (31) and (32).Now, in addition, the (N+1)st row of the eigenvalue equation reads(45)-d∗v1+v2+⋯+vN+c-λvN+1=0,and, by using the general solution, this can be reduced to(46)-d∗A1-αN1-α+B1-βN1-β-NdvN+1b+b∗-a-λ+c-λvN+1=0.As in the previous section, we require that a nontrivial set of(A,B,vN+1) exists. This leads to the following equation:(47)β1-αNα1-βN0ααN-1ββN-10-d∗1-αN1-α-d∗1-βN1-βNd2b+b∗-a-λ+c-λ=β2-α21-αNβN-1Nd2b+b∗-a-λ+c-λ=0. The second left-hand side of the equation should be proportional to D(λ), as for discussion in the previous section. Because we have already known the normalization of bN(1-αN)(βN-1), we conclude that the characteristic polynomial in the present case is written by(48)Dλ=bN1-αNβN-1Nd2b+b∗-a-λ+c-λ. Thus, the determinant of H in this section is given by(49)D0=c-Nd2a-b-b∗a+a2-4b22N+a-a2-4b22N-bN-b∗Nforanextendedperiodictridiagonalmatrix.


## 5.2. Examples

(i)a=2r+s+l2, c=Ns+l2, d=s,andb=r [23, 24].8Suppose the(N+1)×(N+1) matrix [23, 24],(50)Δr,s=r2-10⋯-10-12-1⋯000-12⋯00⋮⋮⋮⋱⋮⋮2-10-100⋯-120000⋯000+s100⋯0-1010⋯0-1001⋯0-1⋮⋮⋮⋱⋮⋮000⋯1-1-1-1-1⋯-1N, where r and s are constants. The determinant of H=Δ(r,s)+l2IN+1 is(51)det⁡H=D0=l21+Nss+l2η+N+η-N-2rN,where(52)η±≡122r+s+l2±2r+s+l22-4r2.(ii) a=3+l2, c=N+l2,andb=d=1.This is the previous case withr=s=1. In this case, H=Δ(WN+1)+l2IN+1, where(53)ΔWN+1≡3-10⋯-1-1-13-1⋯0-10-13⋯0-1⋮⋮⋮⋱⋮⋮3-1-1-100⋯-13-1-1-1-1⋯-1-1N is the graph Laplacian of the wheel graph (see Figure 3) with N+1 vertices. We do not repeat writing the expression of H, which is given by Eq. (51) with Eq. (52) when r=s=1.Figure 3
W 9 as an example of a wheel graph.(iii)b=0.In this case, the determinant simply becomes as(54)det⁡H=D0=aNc-NaN-1d2. Particularly, Δ(0,1) is in this category and can be written as(55)Δ0,1=100⋯0-1010⋯0-1001⋯0-1⋮⋮⋮⋱⋮⋮000⋯1-1-1-1-1⋯-1N=ΔK1,N. This is the graph Laplacian of a star graph K1,N (Figure 4). The eigenvalues of Δ(K1,N) are known as(56)λ0=0,λp=1p=1,2,…,N-1,λN=N+1. The determinant of H=Δ(K1,N)+l2IN+1 is(57)det⁡H=D0=l21+N1+l21+l2N.Figure 4
K 1,8 as an example of a star graph.


## 6. Free Energy on a Graph

In this section, we consider applications of the results on determinants for studying discrete systems.We first considerN scalar degrees of freedom and define the action as follows:(58)S=12N2L2∑k=1N∑k′=1NϕkΔCNkk′ϕk′+μ2∑k=1Nϕk2,where Δ(CN) is the graph Laplacian for CN and(59)μ≡2NLsinh⁡mL2N,where L and m are constants.Then, the Gaussian free energy onCN [25] is obtained using Eq. (37) as(60)FCN=12lndet⁡ΔCN+4sinh2mL2NIN=ln2sinh⁡mL2+const.. This is interesting because the action (58) can be rewritten as(61)S=12N2L2∑k=1Nϕk-ϕk+12+μ2∑k=1Nϕk2, under the “periodic” condition, ϕN+1≡ϕ1. A continuum limit, a0≡L/N→0, enforces (ϕk+1-ϕk)/a0→∂xϕ, where x is a coordinate of one dimension with periodicity x+L~x. Therefore, we can find that the one-loop free energy of a real scalar field ϕ(x) with mass m on a circle (S1) with circumference L governed by the action(62)S=12∫dx∂xϕ2+m2ϕ2takes the form(63)FS1=ln⁡2sinh⁡mL2, after some regularization [25, 26]. Note that, since(64)2sinh⁡mL2=mL∏n=1∞4π2n2/L2+m2∏n=1∞4π2n2/L2, we find that the eigenvalues of -∂x2+m2, where -∂x2 is the one-dimensional Laplacian on S1, are shown by(65)4π2n2L2+m2nisaninteger.Similarly, we can consider the other matrices. For example, the action for complex scalar fields, defined as(66)S=12N2L2∑k=1N∑k′=1Nϕk†ΔCN,χkk′ϕk′+μ2∑k=1Nϕk2=12N2L2∑k=1Nϕk-eiχϕk+12+μ2∑k=1Nϕk2, leads to the free energy(67)FCN,χ=lndet⁡ΔCN,χ+4sinh2mL2NIN=ln⁡4sinh2Nz2+4sin2Nχ2+const.. Here we will avoid repeated discussion and only note that the eigenvalue spectrum of the continuum limit of this case is given by(68)2πn-χ2L2+m2nisaninteger.Continuum limits exist also in other some cases.The largeN limit of the determinant of ΔDD+μ2IN (according to Eq. (21)) becomes(69)sinh⁡mLsinh⁡mL/N→Nsinh⁡mLmL, which coincides with the result of the example stated in Section 2 up to the constant. We find that the continuum limit corresponds to the system of massive scalar field in a line 0≤x≤L with Dirichlet-Dirichlet boundary conditions at its ends.The largeN limit of the determinant of ΔDN+μ2IN (according to Eq. (23)) becomes simply(70)cosh⁡mL+2sinh⁡mLNsinh⁡mL→cosh⁡mL. A comparison to a known mathematical relation(71)cosh⁡mL=∏n=0∞π2/L2n+1/22+m2∏n=0∞π2/L2n+1/22 leads to the conclusion that the continuum limit of spectrum is given by (π2/L2)n+1/22+m2; thus the boundary condition of the system is Dirichlet-Neumann condition.Finally, the determinant ofΔ(PN)+μ2IN (according to Eq. (25)) is(72)2tanh⁡mL2Nsinh⁡mL. Since the free energy is proportional to the logarithm of this, we drop the N dependent term (which is log divergent if N→∞). The boundary condition of the continuum system is Neumann-Neumann condition (which can be judged from the existence of a zero mode).In the next section, we will consider the way to obtain one-loop vacuum energy of scalar field theory with mass matrix required by structure of a theory space with four-dimensional spacetime.


## 7. Vacuum Energy from a Theory Space

### 7.1. Formulation

One-loop vacuum energy density in quantum field theory can be derived from the functional determinants [2]. In the present paper, we only consider scalar field theories for simplicity. As seen in the previous section, N-scalar field theory can resemble compactification of a dimension. This is the key idea of the dimensional deconstruction [8–10]. The structure of the theory space is determined by the quadratic term of fields, i.e., the mass matrix. Suppose that a mass matrix (precisely, the mass2 matrix) Δ/a02 is given (in other words, a theory space is given). The eigenvalues of Δ are denoted by λp, as previously. Then, using the characteristic polynomial D(λ)=∏p(λp-λ), one-loop vacuum energy density for real scalar fields is calculated by(73)V=12∫d4l2π4ln⁡det⁡Δ/a02+l2det⁡Δ/a02+l2+M2=12a042π2∫0∞l3dl2π4ln⁡D-l2D-l2-M2a02,where we used M of the Pauli-Villars regularization, which is considered to be M→∞. The constant a0 illustrates an overall scale in the theory space, i.e., related to mass scale of new physics via a0~mnewphysics-1.In practice, regularization is an art of assembly of mathematical techniques. We adopt here the following approach. A physical value of the vacuum energy should be determined independently of the unphysicalM and the UV divergence must be subtracted in the expression of it. Thus, we consider, in the denominator in log in Eq. (73), as (74)D-l2-M2a02⇒AsymptoticformofD-l2whenl2→∞.Further, if the theory contains the N (scalar) fields, the integrand of the most divergent part should be proportional to N. Thus, we extract the part of ∝l2N⊂D(-l2) for large l2.


### 7.2. Dimensional Deconstruction of a Circle

A concrete example is in order. We consider a theory space associated withΔ(CN,χ). This model has widely been studied by many authors [8–10, 27]. We have already obtained detΔ(CN,χ)+4sinh2(z/2)IN (∝D(-l2), in the present case) in Eq. (39). The asymptotic behavior can be found as(75)detΔCN,χ+4sinh2z2IN=eNz+iχ/2-e-Nz+iχ/22→eNz+iχ/22=eNzz→∞. Thus, in our regularization scheme,9(76)Vχ=1a042π2∫0∞dz2π48sinh3z2cosh⁡z2ln⁡eNz+iχ/2-e-Nz+iχ/22eNz+iχ/22, where we set l=2sinh⁡(z/2). Now, the integration can be done by elementary methods as(77)Vχ=2π2a04∫0∞dz2π44sinh2z2sinh⁡zln⁡1-e-Nz+iχ+ln⁡1-e-Nz-iχ=-116π2a04∑n=1∞∫0∞dzez/2-e-z/22ez-e-ze-Nnz+iχn+e-Nnz-iχn=-18π2a04∑n=1∞cos⁡Nnχn∫0∞dze2z-2ez+2e-z-e-2ze-Nnz=-18π2a04∑n=1∞cos⁡Nnχn1Nn-2-2Nn-1+2Nn+1-1Nn+2=-18π2a04∑n=1∞cos⁡Nnχn4N2n2-4-4N2n2-1=-32π2a04∑n=1∞cos⁡NnχnN2n2-1N2n2-4. This result exactly coincides with the known result [8–10, 27].10 Incidentally, for large N,(78)Vχ~-32π2Na04∑n=1∞cos⁡Nnχn5,V0~-3ζ52π2Na04, where ζ(z) is Riemann’s zeta function. We find that there exists a “continuum limit,” N→∞ as Na0 and Nχ are fixed.


### 7.3. The Clockwork Theory

Next, we turn to consider the theory space of the clockwork theory [19–22] for real scalar fields. The action is(79)S=12∫d4x∑k=1N∂μϕk2+m2∑k=1N-1ϕk-qϕk+12, where m=a0-1. Thus, the relevant matrix determinant is given as Eq. (27). The subtraction of UV divergence is subtle because of the complicated form of the determinant in this case. We separate the vacuum energy density into three parts, such as V=VN(q)+V0+NV1. Here, VN(q) is a finite part,(80)VNq=12a042π2∫0∞l3dl2π4ln⁡1-γ-γ+N, where γ± is given by Eq. (28). This will be of order of O(N-4) as in the previous case and thus will have a continuum limit in vacuum energy density.The change of the integration variablecosh⁡y=(l2+1+q2)/2q makes the integration simple. Then, we can rewrite VN(q) as(81)VNq=12a042π2∫ln⁡q∞dy2π4qsinh⁡y2qcosh⁡y-1-q2ln⁡1-e-2Ny, and we get the form with infinite summations,(82)VNq=-q216π2a04∑n=1∞q-2Nn2nq22Nn-22Nn-1-212Nn-12Nn+1+q-22Nn+12Nn+2q≥1,(83)VNq=-q216π2a04∑n=1∞q2Nn2nq-22Nn-22Nn-1-212Nn-12Nn+1+q22Nn+12Nn+2q<1. Note that q-2VN(q)=q2VN(q-1). The numerical results for (Na0)4q-2(5/N)VN(q5/N) are shown in Figure 5 for N=3,4,…,10. These curves indicate that there is a continuum limit N→∞, while Na0 and q1/N are fixed constants. If we can treat q as a dynamical variable, the effective potential of q seems to have a minimum at q~1 for large N, where the mass matrix simply becomes the graph Laplacian of PN. Note also that VN(q)→0 both for q→0 and for q→∞.Figure 5
The numerical results for(Na0)4q-2(5/N)VN(q5/N) as functions of q for N=3,4,…,10, from lower to upper curves.We now estimate the separated contributions. They are written as(84)V0=12a042π2∫0∞l3dl2π4ln⁡l21-q22+21+q2l2+l4, and(85)V1=12a042π2∫0∞l3dl2π4ln⁡γ+=12a042π2∫0∞l3dl2π4ln⁡1+q2+l2+1-q22+21+q2l2+l42.As forV0, if we use the standard formula of derivation of the Coleman-Weinberg potential(86)12∫d4l2π4ln⁡l2+M2regularized=M464π2ln⁡M2, to regularize V0, aside from the contribution of a zero mode (as ln⁡l2 in the integrand), we find(87)V0=-1264π2a041-q4ln⁡1-q2+1+q4ln⁡1+q2. It is notable that this contribution is equivalent to subtraction of the half of vacuum energy densities due to scalar fields with mass squared 1-q2/a02 and 1+q2/a02. The UV divergence of this part can be regarded to be canceled by the zero-mode contribution.On the other hand, for the complicated form of a genuine divergent contribution ofNV1, we introduce a cut-off Λ in the integration over l and find(88)NV1=N64π2a04Λ4ln⁡Λ2-12+21+q2Λ2-1+4q2+q4ln⁡2Λ21+q2+1-q2+1+q22-321+q21-q2. The quartic divergence seems to be independent of the structure of the mass matrix and the quadratic divergence is proportional to the trace of the mass matrix.


### 7.4. Latticization of a Disk

The matrixΔ(r,s) is used in [23, 24] as a latticization of a disk. Using the result of Eqs. (51) and (52), one-loop vacuum energy density of scalar field theory with mass matrix Δ(r,s)/a02 can be written formally as(89)V=2π22a04∫0∞l3dl2π4ln⁡l21+Nss+l2η+N+η-N-2rN=VNr,s+V0+NV1, where(90)VN=2π22a04∫0∞l3dl2π4ln1+η-η+N-2rη+N=2π22a04∫0∞l3dl2π42ln⁡1-rη+N, with(91)η+≡122r+s+l2+2r+s+l22-4r2,(92)V0=2π22a04∫0∞l3dl2π4ln⁡l21+Nss+l2, and(93)V1=2π22a04∫0∞l3dl2π4ln⁡η+=2π22a04∫0∞l3dl2π4ln⁡2r+s+l2+2r+s+l22-4r22.A finite partVN can be rewritten as(94)VNsr=4π2r22a04∫0∞l3dl2π4ln1-22+s/r+l2+2+s/r+l22-4N. Furthermore, introducing new variables s/r=2sin2(u/2) and cosh⁡y=l2/2+cosh⁡u, we find(95)VNu=4π2r22a04∫u∞dy2π42sinh⁡ycosh⁡y-cosh⁡uln⁡1-e-Ny=-r28π2a04∑n=1∞e-Nnune2uNn-2Nn-1-21Nn-1Nn+1+e-2uNn+1Nn+2.The numerical result of(Na0)4r-2VN(u/N) is plotted as a function of N and u in Figure 6, where we treat N as a continuous parameter. We find that limN→∞(Na0)4r-2VN(0)=-3ζ(5)/4π2, while we find no other limiting cases for general r and s; i.e., no precise continuum limit exists in general cases.Figure 6
The numerical results for(Na0)4r-2VN(u/N) as a function of N and u.We now turn to consider the other part of the vacuum energy. ForV0, using similar estimation as in the previous subsection, we obtain, up to the zero-mode contribution,(96)V0=164π2a04s2N+12lnN+12s2⁡-ln⁡s2, which is equivalent to the contribution of a scalar field with mass squared (N+1)2s2/a02 minus the contribution of a scalar field with mass squared s2/a02. The UV divergence is canceled in these two contributions.The divergent part is analyzed by using the cut-offΛ and is found to be(97)NV1=N64π2a04Λ4ln⁡Λ2-12+22r+sΛ2-6r2+4rs+s2ln⁡2Λ22r+s+s4r+s+2r+s2-322r+ss4r+s. Again, we find that the quartic divergence is independent of the mass matrix and the quadratic divergence is proportional to the trace of the mass matrix.11In the next section, we will exhibit one more example of calculation of one-loop vacuum energy density for a slightly complicated theory space.


## 7.1. Formulation

One-loop vacuum energy density in quantum field theory can be derived from the functional determinants [2]. In the present paper, we only consider scalar field theories for simplicity. As seen in the previous section, N-scalar field theory can resemble compactification of a dimension. This is the key idea of the dimensional deconstruction [8–10]. The structure of the theory space is determined by the quadratic term of fields, i.e., the mass matrix. Suppose that a mass matrix (precisely, the mass2 matrix) Δ/a02 is given (in other words, a theory space is given). The eigenvalues of Δ are denoted by λp, as previously. Then, using the characteristic polynomial D(λ)=∏p(λp-λ), one-loop vacuum energy density for real scalar fields is calculated by(73)V=12∫d4l2π4ln⁡det⁡Δ/a02+l2det⁡Δ/a02+l2+M2=12a042π2∫0∞l3dl2π4ln⁡D-l2D-l2-M2a02,where we used M of the Pauli-Villars regularization, which is considered to be M→∞. The constant a0 illustrates an overall scale in the theory space, i.e., related to mass scale of new physics via a0~mnewphysics-1.In practice, regularization is an art of assembly of mathematical techniques. We adopt here the following approach. A physical value of the vacuum energy should be determined independently of the unphysicalM and the UV divergence must be subtracted in the expression of it. Thus, we consider, in the denominator in log in Eq. (73), as (74)D-l2-M2a02⇒AsymptoticformofD-l2whenl2→∞.Further, if the theory contains the N (scalar) fields, the integrand of the most divergent part should be proportional to N. Thus, we extract the part of ∝l2N⊂D(-l2) for large l2.


## 7.2. Dimensional Deconstruction of a Circle

A concrete example is in order. We consider a theory space associated withΔ(CN,χ). This model has widely been studied by many authors [8–10, 27]. We have already obtained detΔ(CN,χ)+4sinh2(z/2)IN (∝D(-l2), in the present case) in Eq. (39). The asymptotic behavior can be found as(75)detΔCN,χ+4sinh2z2IN=eNz+iχ/2-e-Nz+iχ/22→eNz+iχ/22=eNzz→∞. Thus, in our regularization scheme,9(76)Vχ=1a042π2∫0∞dz2π48sinh3z2cosh⁡z2ln⁡eNz+iχ/2-e-Nz+iχ/22eNz+iχ/22, where we set l=2sinh⁡(z/2). Now, the integration can be done by elementary methods as(77)Vχ=2π2a04∫0∞dz2π44sinh2z2sinh⁡zln⁡1-e-Nz+iχ+ln⁡1-e-Nz-iχ=-116π2a04∑n=1∞∫0∞dzez/2-e-z/22ez-e-ze-Nnz+iχn+e-Nnz-iχn=-18π2a04∑n=1∞cos⁡Nnχn∫0∞dze2z-2ez+2e-z-e-2ze-Nnz=-18π2a04∑n=1∞cos⁡Nnχn1Nn-2-2Nn-1+2Nn+1-1Nn+2=-18π2a04∑n=1∞cos⁡Nnχn4N2n2-4-4N2n2-1=-32π2a04∑n=1∞cos⁡NnχnN2n2-1N2n2-4. This result exactly coincides with the known result [8–10, 27].10 Incidentally, for large N,(78)Vχ~-32π2Na04∑n=1∞cos⁡Nnχn5,V0~-3ζ52π2Na04, where ζ(z) is Riemann’s zeta function. We find that there exists a “continuum limit,” N→∞ as Na0 and Nχ are fixed.


## 7.3. The Clockwork Theory

Next, we turn to consider the theory space of the clockwork theory [19–22] for real scalar fields. The action is(79)S=12∫d4x∑k=1N∂μϕk2+m2∑k=1N-1ϕk-qϕk+12, where m=a0-1. Thus, the relevant matrix determinant is given as Eq. (27). The subtraction of UV divergence is subtle because of the complicated form of the determinant in this case. We separate the vacuum energy density into three parts, such as V=VN(q)+V0+NV1. Here, VN(q) is a finite part,(80)VNq=12a042π2∫0∞l3dl2π4ln⁡1-γ-γ+N, where γ± is given by Eq. (28). This will be of order of O(N-4) as in the previous case and thus will have a continuum limit in vacuum energy density.The change of the integration variablecosh⁡y=(l2+1+q2)/2q makes the integration simple. Then, we can rewrite VN(q) as(81)VNq=12a042π2∫ln⁡q∞dy2π4qsinh⁡y2qcosh⁡y-1-q2ln⁡1-e-2Ny, and we get the form with infinite summations,(82)VNq=-q216π2a04∑n=1∞q-2Nn2nq22Nn-22Nn-1-212Nn-12Nn+1+q-22Nn+12Nn+2q≥1,(83)VNq=-q216π2a04∑n=1∞q2Nn2nq-22Nn-22Nn-1-212Nn-12Nn+1+q22Nn+12Nn+2q<1. Note that q-2VN(q)=q2VN(q-1). The numerical results for (Na0)4q-2(5/N)VN(q5/N) are shown in Figure 5 for N=3,4,…,10. These curves indicate that there is a continuum limit N→∞, while Na0 and q1/N are fixed constants. If we can treat q as a dynamical variable, the effective potential of q seems to have a minimum at q~1 for large N, where the mass matrix simply becomes the graph Laplacian of PN. Note also that VN(q)→0 both for q→0 and for q→∞.Figure 5
The numerical results for(Na0)4q-2(5/N)VN(q5/N) as functions of q for N=3,4,…,10, from lower to upper curves.We now estimate the separated contributions. They are written as(84)V0=12a042π2∫0∞l3dl2π4ln⁡l21-q22+21+q2l2+l4, and(85)V1=12a042π2∫0∞l3dl2π4ln⁡γ+=12a042π2∫0∞l3dl2π4ln⁡1+q2+l2+1-q22+21+q2l2+l42.As forV0, if we use the standard formula of derivation of the Coleman-Weinberg potential(86)12∫d4l2π4ln⁡l2+M2regularized=M464π2ln⁡M2, to regularize V0, aside from the contribution of a zero mode (as ln⁡l2 in the integrand), we find(87)V0=-1264π2a041-q4ln⁡1-q2+1+q4ln⁡1+q2. It is notable that this contribution is equivalent to subtraction of the half of vacuum energy densities due to scalar fields with mass squared 1-q2/a02 and 1+q2/a02. The UV divergence of this part can be regarded to be canceled by the zero-mode contribution.On the other hand, for the complicated form of a genuine divergent contribution ofNV1, we introduce a cut-off Λ in the integration over l and find(88)NV1=N64π2a04Λ4ln⁡Λ2-12+21+q2Λ2-1+4q2+q4ln⁡2Λ21+q2+1-q2+1+q22-321+q21-q2. The quartic divergence seems to be independent of the structure of the mass matrix and the quadratic divergence is proportional to the trace of the mass matrix.


## 7.4. Latticization of a Disk

The matrixΔ(r,s) is used in [23, 24] as a latticization of a disk. Using the result of Eqs. (51) and (52), one-loop vacuum energy density of scalar field theory with mass matrix Δ(r,s)/a02 can be written formally as(89)V=2π22a04∫0∞l3dl2π4ln⁡l21+Nss+l2η+N+η-N-2rN=VNr,s+V0+NV1, where(90)VN=2π22a04∫0∞l3dl2π4ln1+η-η+N-2rη+N=2π22a04∫0∞l3dl2π42ln⁡1-rη+N, with(91)η+≡122r+s+l2+2r+s+l22-4r2,(92)V0=2π22a04∫0∞l3dl2π4ln⁡l21+Nss+l2, and(93)V1=2π22a04∫0∞l3dl2π4ln⁡η+=2π22a04∫0∞l3dl2π4ln⁡2r+s+l2+2r+s+l22-4r22.A finite partVN can be rewritten as(94)VNsr=4π2r22a04∫0∞l3dl2π4ln1-22+s/r+l2+2+s/r+l22-4N. Furthermore, introducing new variables s/r=2sin2(u/2) and cosh⁡y=l2/2+cosh⁡u, we find(95)VNu=4π2r22a04∫u∞dy2π42sinh⁡ycosh⁡y-cosh⁡uln⁡1-e-Ny=-r28π2a04∑n=1∞e-Nnune2uNn-2Nn-1-21Nn-1Nn+1+e-2uNn+1Nn+2.The numerical result of(Na0)4r-2VN(u/N) is plotted as a function of N and u in Figure 6, where we treat N as a continuous parameter. We find that limN→∞(Na0)4r-2VN(0)=-3ζ(5)/4π2, while we find no other limiting cases for general r and s; i.e., no precise continuum limit exists in general cases.Figure 6
The numerical results for(Na0)4r-2VN(u/N) as a function of N and u.We now turn to consider the other part of the vacuum energy. ForV0, using similar estimation as in the previous subsection, we obtain, up to the zero-mode contribution,(96)V0=164π2a04s2N+12lnN+12s2⁡-ln⁡s2, which is equivalent to the contribution of a scalar field with mass squared (N+1)2s2/a02 minus the contribution of a scalar field with mass squared s2/a02. The UV divergence is canceled in these two contributions.The divergent part is analyzed by using the cut-offΛ and is found to be(97)NV1=N64π2a04Λ4ln⁡Λ2-12+22r+sΛ2-6r2+4rs+s2ln⁡2Λ22r+s+s4r+s+2r+s2-322r+ss4r+s. Again, we find that the quartic divergence is independent of the mass matrix and the quadratic divergence is proportional to the trace of the mass matrix.11In the next section, we will exhibit one more example of calculation of one-loop vacuum energy density for a slightly complicated theory space.


## 8. Some Other Examples of Vacuum Energy

Using the additional formulas on determinants, we can further obtain determinants of various matrices. In this section, we show some other examples below.


### 8.1. Adding an Edge with a Vertex to Each Vertex of a Graph

LetΔN be an N×N Hermitian matrix and define a 2N×2N matrix Δ2N as follows:(98)Δ2N≡ΔN+IN-IN-ININ, where IN is an N dimensional identity matrix. In particular, if ΔN is the graph Laplacian of a graph G, Δ2N is the graph Laplacian of the graph generated by adding an edge with a vertex to every vertex of G.Then, the formula on determinants(99)det⁡ABCD=det⁡A-BD-1Cdet⁡D tells us that(100)det⁡Δ2N-λI2N=1-λNdet⁡ΔN-1-1-λ21-λIN.Therefore, ifDN(λ)≡∏p=1Nλp-λ=det⁡(ΔN-λIN), where {λp} are eigenvalues of ΔN, is known, D2N(λ)≡det⁡(Δ2N-λI2N) is obtained as(101)D2Nλ=1-λNDN1-1-λ21-λ.For example, we will calculate vacuum energy density of the scalar field theory with mass matrixΔ2N/a02, where Δ2N is generated from ΔN=Δ(CN); i.e., Δ2N is the graph Laplacian of the graph shown in Figure 7.Figure 7
The graph generated by adding an edge with a vertex to every vertex ofC9.In this case, after some manipulation, we get(102)D2Nλ=αλN1-1-λαλN2, where(103)αλ≡λ2-4λ+2+λλ3-8λ2+16λ-82.Now, we can obtain the vacuum energy density in this theory by utilizingln⁡D2N(-l2), as in the previous section. We separate the finite and divergent parts of vacuum energy density as(104)VN=12a042π2∫0∞l3dl2π42ln⁡1-1+l2α-l2N, and(105)V1=12a042π2∫0∞l3dl2π4ln⁡α-l2.The numerical result of(Na0)4VN in the present case is shown in Figure 8, where N is treated as a continuous parameter. In the limit of N→∞, (Na0)4VN approaches -3ζ(5)/16π2, which is quarter of the value of the large N limit of (Na0)4VN in the case of the real scalar theory based on the graph Laplacian Δ(CN). The divergent part NV1 can be estimated, because α(-l2)~l4 for large l, as(106)NV1=2N64π2a04Λ4ln⁡Λ2-12+NΛ28π2+⋯. The leading term is proportional to the number of real scalar fields, as expected. The quadratic divergence is proportional to the trace of the mass matrix.Figure 8
The numerical value of(Na0)4VN for the model in this subsection as a function of N. The dotted lines indicate (1/4)Na04VN, where VN is the vacuum energy density in the real scalar theory whose mass matrix is Δ(CN)/a02, and the constant -3ζ(5)/16π2.


### 8.2. The Graph Cartesian ProductsG×P2

LetΔN be an N×N Hermitian matrix and define a 2N×2N matrix Δ^2N as follows:(107)Δ^2N≡ΔN+IN-IN-INΔN+IN. In particular, if ΔN is the graph Laplacian of a graph G, Δ^2N is the graph Laplacian of the graph Cartesian product G×P2.12Then, the use of the formula on determinants(108)det⁡ABCD=det⁡A-BD-1Cdet⁡D=det⁡AD-BC, provided that [C,D]=0, leads to(109)det⁡Δ^2N-λI2N=det⁡ΔN+1-λIN2-IN=det⁡ΔN-λIN·det⁡ΔN+2-λIN. Therefore, if DN(λ)≡∏p=1Nλp-λ=det⁡(ΔN-λIN), where {λp} are eigenvalues of ΔN, is known, D^2N(λ)≡det⁡(Δ^2N-λI2N) is obtained as(110)D^2Nλ=DNλDNλ-2.The vacuum energy density of the scalar field theory with mass matrixΔ^2N/a02, where Δ^2N is generated from ΔN=Δ(CN), i.e., Δ^2N is the graph Laplacian of the graph Cartesian product CN×P2, called as the prism graph YN. We show the graph Y9 in Figure 9.Figure 9
The graph Cartesian productC9×P2, or the prism graph Y9.We can obtain the vacuum energy density in this theory by utilizingln⁡D^2N(-l2). We separate the finite and divergent parts of vacuum energy density as(111)VN=116π2a04∫0∞l3dl·2ln1-22+l2+2+l22-4N+ln1-24+l2+4+l22-4N, and(112)V1=116π2a04∫0∞l3dlln2+l2+2+l22-42+ln4+l2+4+l22-42.The numerical result of(Na0)4VN in the present case is shown in Figure 10, where N is treated as a continuous parameter. In the limit of N→∞, (Na0)4VN approaches the vacuum energy density of the model associated with CN and -3ζ(5)/4π2. The divergent part NV1 can be estimated as(113)NV1=2N64π2a04Λ4ln⁡Λ2-12+3NΛ216π2+⋯. The leading term is proportional to the number of real scalar fields, as expected. The quadratic divergence is proportional to the trace of the mass matrix.Figure 10
The numerical value of(Na0)4VN for the model whose theory space is associated with YN as a function of N. The dotted lines indicate (Na0)4VN, where VN is the vacuum energy density in the real scalar theory whose mass matrix is Δ(CN)/a02, and the constant -3ζ(5)/4π2.


## 8.1. Adding an Edge with a Vertex to Each Vertex of a Graph

LetΔN be an N×N Hermitian matrix and define a 2N×2N matrix Δ2N as follows:(98)Δ2N≡ΔN+IN-IN-ININ, where IN is an N dimensional identity matrix. In particular, if ΔN is the graph Laplacian of a graph G, Δ2N is the graph Laplacian of the graph generated by adding an edge with a vertex to every vertex of G.Then, the formula on determinants(99)det⁡ABCD=det⁡A-BD-1Cdet⁡D tells us that(100)det⁡Δ2N-λI2N=1-λNdet⁡ΔN-1-1-λ21-λIN.Therefore, ifDN(λ)≡∏p=1Nλp-λ=det⁡(ΔN-λIN), where {λp} are eigenvalues of ΔN, is known, D2N(λ)≡det⁡(Δ2N-λI2N) is obtained as(101)D2Nλ=1-λNDN1-1-λ21-λ.For example, we will calculate vacuum energy density of the scalar field theory with mass matrixΔ2N/a02, where Δ2N is generated from ΔN=Δ(CN); i.e., Δ2N is the graph Laplacian of the graph shown in Figure 7.Figure 7
The graph generated by adding an edge with a vertex to every vertex ofC9.In this case, after some manipulation, we get(102)D2Nλ=αλN1-1-λαλN2, where(103)αλ≡λ2-4λ+2+λλ3-8λ2+16λ-82.Now, we can obtain the vacuum energy density in this theory by utilizingln⁡D2N(-l2), as in the previous section. We separate the finite and divergent parts of vacuum energy density as(104)VN=12a042π2∫0∞l3dl2π42ln⁡1-1+l2α-l2N, and(105)V1=12a042π2∫0∞l3dl2π4ln⁡α-l2.The numerical result of(Na0)4VN in the present case is shown in Figure 8, where N is treated as a continuous parameter. In the limit of N→∞, (Na0)4VN approaches -3ζ(5)/16π2, which is quarter of the value of the large N limit of (Na0)4VN in the case of the real scalar theory based on the graph Laplacian Δ(CN). The divergent part NV1 can be estimated, because α(-l2)~l4 for large l, as(106)NV1=2N64π2a04Λ4ln⁡Λ2-12+NΛ28π2+⋯. The leading term is proportional to the number of real scalar fields, as expected. The quadratic divergence is proportional to the trace of the mass matrix.Figure 8
The numerical value of(Na0)4VN for the model in this subsection as a function of N. The dotted lines indicate (1/4)Na04VN, where VN is the vacuum energy density in the real scalar theory whose mass matrix is Δ(CN)/a02, and the constant -3ζ(5)/16π2.


## 8.2. The Graph Cartesian ProductsG×P2

LetΔN be an N×N Hermitian matrix and define a 2N×2N matrix Δ^2N as follows:(107)Δ^2N≡ΔN+IN-IN-INΔN+IN. In particular, if ΔN is the graph Laplacian of a graph G, Δ^2N is the graph Laplacian of the graph Cartesian product G×P2.12Then, the use of the formula on determinants(108)det⁡ABCD=det⁡A-BD-1Cdet⁡D=det⁡AD-BC, provided that [C,D]=0, leads to(109)det⁡Δ^2N-λI2N=det⁡ΔN+1-λIN2-IN=det⁡ΔN-λIN·det⁡ΔN+2-λIN. Therefore, if DN(λ)≡∏p=1Nλp-λ=det⁡(ΔN-λIN), where {λp} are eigenvalues of ΔN, is known, D^2N(λ)≡det⁡(Δ^2N-λI2N) is obtained as(110)D^2Nλ=DNλDNλ-2.The vacuum energy density of the scalar field theory with mass matrixΔ^2N/a02, where Δ^2N is generated from ΔN=Δ(CN), i.e., Δ^2N is the graph Laplacian of the graph Cartesian product CN×P2, called as the prism graph YN. We show the graph Y9 in Figure 9.Figure 9
The graph Cartesian productC9×P2, or the prism graph Y9.We can obtain the vacuum energy density in this theory by utilizingln⁡D^2N(-l2). We separate the finite and divergent parts of vacuum energy density as(111)VN=116π2a04∫0∞l3dl·2ln1-22+l2+2+l22-4N+ln1-24+l2+4+l22-4N, and(112)V1=116π2a04∫0∞l3dlln2+l2+2+l22-42+ln4+l2+4+l22-42.The numerical result of(Na0)4VN in the present case is shown in Figure 10, where N is treated as a continuous parameter. In the limit of N→∞, (Na0)4VN approaches the vacuum energy density of the model associated with CN and -3ζ(5)/4π2. The divergent part NV1 can be estimated as(113)NV1=2N64π2a04Λ4ln⁡Λ2-12+3NΛ216π2+⋯. The leading term is proportional to the number of real scalar fields, as expected. The quadratic divergence is proportional to the trace of the mass matrix.Figure 10
The numerical value of(Na0)4VN for the model whose theory space is associated with YN as a function of N. The dotted lines indicate (Na0)4VN, where VN is the vacuum energy density in the real scalar theory whose mass matrix is Δ(CN)/a02, and the constant -3ζ(5)/4π2.


## 9. Conclusion

In the present paper, we showed the method of obtaining the determinant of repetitive tridiagonal matrices with concrete examples. The concept of the method is similar to the Gel’fand–Yaglom method of obtaining functional determinants for differential operators.The repetitive matrices as mass matrices are widely considered in modern models in particle physics, in order to attack the hierarchy problem by adopting a theory space. We showed one-loop vacuum energies of such models can be evaluated by using the determinant of the mass matrices obtained by our method stated in earlier sections.We have seen that there are not always genuine continuum limits in largeN for general theory spaces. In Section 7, we have also found that contributions of V0 expressed in logarithmic functions remain in general. They can be compensated by addition of bosonic or fermionic free fields with appropriate mass in some cases.13In future work, we wish to study one-loop energy density in models of deconstructed warped (theory) space [28–33]. Although it is difficult to evaluate the determinants in a closed form in such a model, calculation based on recurrence relations would be suitable for a computer. It is also interesting to investigate the recent model of deconstruction of torus with magnetic flux [34].14If we would like to deal with the matrices related with more complicated graphs or higher dimensional lattices, we confront other difficulties.15 The graph Laplacians of generic graphs cannot be expressed by tridiagonal matrices, though, fortunately, it is known that arbitrary square matrices can be systematically tridiagonalized by the Householder method [35] (see also Refs. [36, 37]). Thus, in principle, our Gel’fand–Yaglom-type method can be applied to the matrix with the general graph structure.Finally, we add a comment on exclusion of zero modes. Zero modes of operators which appear in quantum field theory have crucial meanings related with nonperturbative aspects of the theory (see, for example, the first section of Ref. [14]). In our present paper, we considered mass terms in almost all examples and the cases with zero modes can be considered as the limit that the value of mass goes to zero. Because we considered the vacuum energies and their dependence on the parameters in this paper, the analysis is just sound. Moreover, it is known that, if a matrix is expressed as a graph Laplacian of a simple graph (as in each example in this paper), the matrix has a single zero mode. Therefore, further analysis on zero modes, if necessary, could be fulfilled appropriately.



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