deconstructing the gel’fand–yaglom method and vacuum energy...

Research ArticleDeconstructing the Gel’fand–Yaglom Method and VacuumEnergy from a Theory Space

Nahomi Kan1 and Kiyoshi Shiraishi 2

1National Institute of Technology, Gifu College, Motosu-shi, Gifu 501-0495, Japan2Graduate School of Sciences and Technology for Innovation, Yamaguchi University, Yamaguchi-shi, Yamaguchi 753-8512, Japan

Correspondence should be addressed to Kiyoshi Shiraishi; [email protected]

Received 5 March 2019; Revised 27 April 2019; Accepted 7 May 2019; Published 2 June 2019

Academic Editor: Antonio Scarfone

Copyright © 2019 Nahomi Kan and Kiyoshi Shiraishi. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

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

1. Introduction

The Gel’fand–Yaglom method [1] for obtaining functionaldeterminants of differential operators with boundaries iswidely known nowadays. For nice reviews, see [2, 3]. Theapplications of theGel’fand–Yaglommethod have been inves-tigated quite recently, to evaluate one-loop vacuum energiesin nontrivial boundary conditions [4–7].

Among them, Altshuler examined vacuum energy inwarped compactification [6, 7]. In recent years, it is supposedthat extra dimensions of various types could play an impor-tant role in the hierarchy problem, and thus the study ofphysics in nontrivial background geometry is still advancing.

The dimensional deconstruction has appeared as a newtool for understanding the properties of higher-dimensionalfield theories [8–10] more than a decade ago. In such amodel of deconstruction, a “theory space” is considered,which consists of sites and links, to which four-dimensionalfields are individually assigned. Theory spaces thus have thestructures of graphs [11] and can be interpreted as the theorywith discrete extra dimensions.

Several years ago, the discrete Gel’fand–Yaglom methodfor difference operators was reviewed and studied by Dowker[12]. We generalize the discrete Gel’fand–Yaglom methodfor studying one-loop vacuum energies in extended decon-structed theories and models with discrete dimensions in

the present paper. To this end, we develop the method ofcomputing determinants of repetitive Hermitian matriceswhich correspond tomass matrices utilized in deconstructedtheories.

After completion of the first version of the manuscriptof the present paper (arXiv:1711.06806), a paper which treatsthe determinants of discrete Laplace operators appeared [13].Their method is substantially the same as ours, because theauthor also relies on the recurrence relation among threevariables on a lattice (see Section 3 in the present paperand below). We recently become aware of another similarpaper on the determinants of matrix differential operators[14].They studied generalization of Gel’fand–Yaglommethodto obtain the functional determinants. Their work differsessentially from ours because they considered differentialoperators while we treat matrices as operators. We also pointout that they did not consider the matrices of large size whichhave certain continuum limits.

The organization of this paper is as follows. In orderto make the present paper self-contained, we show a shortreview of theGel’fand–Yaglommethod for a differential oper-ator, along with Dunne’s review [2], in Section 2. In Section 3,we give the method to obtain determinants of tridiagonalmatrices with repeated structure. This is a straightforwardgeneralization of description in Ref. [12]. In Section 4, we give

HindawiAdvances in Mathematical PhysicsVolume 2019, Article ID 6579187, 15 pageshttps://doi.org/10.1155/2019/6579187

http://orcid.org/0000-0002-9761-5137https://arxiv.org/abs/1711.06806https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/6579187

2 Advances in Mathematical Physics

the method to obtain determinants of periodic tridiagonalmatrices. Determinants of extended periodic tridiagonalmatrices are obtained in Section 5. The rest of the presentpaper is devoted to applications to deconstructed theoriesand discrete systems. In Section 6, free energy on a graphis discussed by using the results of previous sections. InSection 7, we show the method of calculation for evaluatingone-loop vacuum energy in deconstructed models from thedeterminants of mass matrices. In Section 8, we show afew more examples of one-loop vacuum energies for slightlycomplicated theory spaces. We give conclusions in the lastsection, Section 9.

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

Suppose that an eigenvalue equation (Δ + 𝑚2)𝜓(�푚2)�휆

(𝑥) =𝜆(�푚2)𝜓(�푚2)�휆

(𝑥) with Dirichlet-Dirichlet boundary conditions𝜓(�푚2)�휆

(0) = 𝜓(�푚2)�휆

(𝐿) = 0 in one dimension is given.Then

det [Δ + 𝑚2]det [Δ] = 𝜓(�푚2)0 (𝐿)𝜓(0)0 (𝐿) (1)

holds. Here 𝜓(�푚2)0 (𝑥) satisfies (Δ + 𝑚2)𝜓(�푚2)0 (𝑥) = 0 with aboundary condition 𝜓(�푚2)0 (0) = 0.Example. In the region 0 ≤ 𝑥 ≤ 𝐿, under the Dirichletcondition at the boundaries, we consider the functionaldeterminant

det [−𝜕2�푥 + 𝑚2]det [−𝜕2�푥] . (2)

The solution of (−𝜕2�푥 +𝑚2)𝜓(𝑥) = 0with 𝜓(0) = 0 is sinh𝑚𝑥.Thus according to the Gel’fand–Yaglommethod, we obtain

det [−𝜕2�푥 + 𝑚2]det [−𝜕2�푥] = sinh𝑚𝐿𝑚𝐿 = ∞∏�푛=1 𝑛2𝜋2/𝐿2 + 𝑚2𝑛2𝜋2/𝐿2 . (3)

Proof. det[Δ +𝑚2 − 𝜆] is a function of 𝜆 and has zeros at 𝜆 =𝜆(�푚2).The function𝜓�휆(𝑥)which satisfies (Δ+𝑚2−𝜆)𝜓�휆(𝑥) = 0and 𝜓�휆(0) = 0 becomes the eigenfunction when 𝜆 = 𝜆(�푚2).Then the boundary condition at 𝑥 = 𝐿, i.e., 𝜓�휆(𝐿) = 0, issatisfied. In otherwords,𝜓�휆 (𝐿) is a function of𝜆 and has zerosat 𝜆 = 𝜆(�푚2). Therefore det[Δ + 𝑚2 − 𝜆] ∝ 𝜓�휆(𝐿) holds.3. Determinants of Tridiagonal Matrices

3.1. The Discrete Gel’fand–Yaglom Method for Tridiago-nal Matrices. Now, we show the discrete Gel’fand–Yaglommethod to obtain determinants of finite matrices. First, weconsider the following Hermitian tridiagonal matrix of 𝑁rows and columns:

𝐻 = (((((((

𝑐 −𝑏 0 ⋅ ⋅ ⋅ 0−𝑏∗ 𝑎 −𝑏 ⋅ ⋅ ⋅ 00 −𝑏∗ 𝑎 ⋅ ⋅ ⋅ 0... ... ... d ...𝑎 −𝑏0 0 0 ⋅ ⋅ ⋅ −𝑏∗ 𝑑)))))))

. (4)In this case, the eigenvalue equation𝐻k = 𝜆k, (5)where k�푇 = (V1, V2, . . . , V�푁), can be categorized into threeparts:

�푁∑�푗=1

𝐻1�푗V�푗 − 𝜆V1 = 0, (6)�푁∑

�푗=1

𝐻�푘�푗V�푗 − 𝜆V�푘 = 0 (2 ≤ 𝑘 ≤ 𝑁 − 1) , (7)�푁∑

�푗=1

𝐻�푁�푗V�푗 − 𝜆V�푁 = 0. (8)Here, Eq. (7) is just the recurrence relation among three termsin V�푘 as a sequence of numbers. In the present case, the generalsolution for the recurrence relation𝑏V�푘+1 − (𝑎 − 𝜆) V�푘 + 𝑏∗V�푘−1 = 0 (2 ≤ 𝑘 ≤ 𝑁 − 1) (9)is

V�푘 = 𝐴𝛼�푘−1 + 𝐵𝛽�푘−1, (10)where 𝐴 and 𝐵 are constants and

𝛼 = 𝑎 − 𝜆 + √(𝑎 − 𝜆)2 − 4 |𝑏|22𝑏 ,𝛽 = 𝑎 − 𝜆 − √(𝑎 − 𝜆)2 − 4 |𝑏|22𝑏 .

(11)

Note that𝛼 and𝛽 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), deter-

mines the relation between V1 and V2; in this case, that is(𝑐 − 𝜆)V1 − 𝑏V2 = 0. If we further chooseV1 = 𝐴 + 𝐵 = 1, (12)

the coefficients 𝐴 and 𝐵 are obtained as𝐴 = 𝐴 (𝜆) = 𝑐 − 𝜆 − 𝑏𝛽𝑏 (𝛼 − 𝛽) = 𝑐 − 𝜆 − 𝑏𝛽√(𝑎 − 𝜆)2 − 4 |𝑏|2 , (13)𝐵 = 𝐵 (𝜆) = − (𝑐 − 𝜆 − 𝑏𝛼)𝑏 (𝛼 − 𝛽) = − (𝑐 − 𝜆 − 𝑏𝛼)√(𝑎 − 𝜆)2 − 4 |𝑏|2 . (14)Substituting all of the results above into Eq. (8) in the

present case, we get

Advances in Mathematical Physics 3

𝐴 (𝜆) [−𝑏∗ + (𝑑 − 𝜆) 𝛼] 𝛼�푁−2+ 𝐵 (𝜆) [−𝑏∗ + (𝑑 − 𝜆) 𝛽] 𝛽�푁−2 = 0. (15)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. Byconstruction, 𝐷(𝜆) should be an𝑁th 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 V�푁 includes (−𝜆)�푁−1/𝑏�푁−1.Finally, since the left-hand side of Eq. (8) reads −𝑏∗V�푁−1+(𝑑−𝜆)V�푁 in the present case,𝐷(𝜆)has the term (−𝜆)�푁/𝑏�푁−1 as thehighest order term in 𝜆. We can also directly confirm this bysetting 𝑎 = 𝑐 = 𝑑 = 0 and the limit 𝑏 → 0 in the left-handside of Eq. (15). We then verify 𝐷(𝜆) → (−𝜆)�푁/𝑏�푁−1.

Therefore, we conclude that 𝐷(𝜆) = 𝑏�푁−1𝐷(𝜆) =∏�푁�푝=1(𝜆�푝 − 𝜆) is the characteristic polynomial of 𝐻, where𝜆�푝 (𝑝 = 1, 2, . . . , 𝑁) are eigenvalues of𝐻.The determinant of𝐻 is given by 𝐷(0) = ∏�푁�푝=1𝜆�푝. In the

present case, we find

det𝐻 = 𝐷(0)= 𝑏�푁−1√𝑎2 − 4 |𝑏|2 [(𝑐 − 𝑏𝛽) (−𝑏∗ + 𝑑𝛼) 𝛼�푁−2− (𝑐 − 𝑏𝛼) (−𝑏∗ + 𝑑𝛽)𝛽�푁−2] ,

(16)

where

𝛼 = 𝑎 + √𝑎2 − 4 |𝑏|22𝑏 ,𝛽 = 𝑎 − √𝑎2 − 4 |𝑏|22𝑏 .

(17)

After a lengthy calculation, we obtain

det𝐻 = 𝐷(0) = 12�푁√𝑎2 − 4 |𝑏|2 {[2 (𝑐𝑑 − |𝑏|2)+ (𝑎 − 𝑐 − 𝑑) (𝑎 − √𝑎2 − 4 |𝑏|2)](𝑎+ √𝑎2 − 4 |𝑏|2)�푁−1 − [2 (𝑐𝑑 − |𝑏|2)+ (𝑎 − 𝑐 − 𝑑) (𝑎 + √𝑎2 − 4 |𝑏|2)](𝑎− √𝑎2 − 4 |𝑏|2)�푁−1}

(for a tridiagonal matrix) ,

(18)

in the present case. It is notable that the determinant dependsonly on |𝑏| and does not depend on 𝑏∗/𝑏 in the present

case. The reason is because the eigenvalues are unchangedunder “gauge” transformation k → 𝑃k and 𝐻 → 𝑃𝐻𝑃−1,where𝑃 = diag.(1, 𝑒�푖�휒, 𝑒�푖2�휒, . . . , 𝑒�푖(�푁−1)�휒)with an arbitrary realconstant 𝜒.

The prescription of the above method to obtain thedeterminant is very similar to the Gel’fand–Yaglom methodfor differential operators. Namely, solving the differentialequation corresponds to solving the recurrence relation,putting one of the boundary conditions corresponds to fixingthe first term of the series of numbers, and obtaining thedeterminant at another boundary corresponds to obtainingthe determinant as the equation of the last row in theeigenvalue equation.Note that, becausewe are treating a finitematrix, the idea of normalization becomes different fromthe functional determinant treated by the Gel’fand–Yaglommethod.

The method to obtain the determinant of tridiagonalmatrices in this section is substantially equivalent to themethod for difference operators described by Dowker [12],except for a specific choice for Hermitian matrix in thepresent section.

3.2. Examples. In this subsection, we show determinantsof some simple tridiagonal matrices for example. For allthe examples below, the eigenvalues are known and, then,one can find that the formulas1 for finite product includingtrigonometric functions are derived.

Note that the determinant 𝐷(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)for𝐻 here and hereafter.

(i) 𝑎 = 𝑐 = 𝑑 = 2 + 4 sinh2(𝑧/2) and 𝑏 = 1.2In this case,𝐻 = Δ�퐷�퐷 + 4 sinh2(𝑧/2)𝐼�푁, where

Δ�퐷�퐷 ≡ (((((((

2 −1 0 ⋅ ⋅ ⋅ 0−1 2 −1 ⋅ ⋅ ⋅ 00 −1 2 ⋅ ⋅ ⋅ 0... ... ... d ...2 −10 0 0 ⋅ ⋅ ⋅ −1 2)))))))

, (19)and 𝐼�푁 is the𝑁×𝑁 identity matrix. Note that, for 𝑎 = 𝑐 = 𝑑,Eq. (18) becomes

𝐷(0) = 12�푁+1√𝑎2 − 4 |𝑏|2 [(𝑎 + √𝑎2 − 4 |𝑏|2)�푁+1− (𝑎 − √𝑎2 − 4 |𝑏|2)�푁+1] . (20)

We now find

det𝐻 = 𝐷(0) = sinh (𝑁 + 1) 𝑧sinh 𝑧

for 𝐻 = Δ�퐷�퐷 + 4 sinh2 𝑧2𝐼�푁. (21)


Figure 1: 𝑃8 as an example of a path graph.(ii) 𝑎 = 𝑐 = 2 + 4 sinh2(𝑧/2), 𝑑 = 1 + 4 sinh2(𝑧/2), and𝑏 = 1.3In this case,𝐻 = Δ�퐷�푁 + 4 sinh2(𝑧/2)𝐼�푁, where

Δ�퐷�푁 ≡ (((((((

2 −1 0 ⋅ ⋅ ⋅ 0−1 2 −1 ⋅ ⋅ ⋅ 00 −1 2 ⋅ ⋅ ⋅ 0... ... ... d ...2 −10 0 0 ⋅ ⋅ ⋅ −1 −1)))))))

(22)

We find

det𝐻 = 𝐷(0) = sinh (𝑁 + 1) 𝑧 − sinh𝑁𝑧sinh 𝑧= cosh𝑁𝑧 + 2 sinh 𝑧 sinh𝑁𝑧

for 𝐻 = Δ�퐷�푁 + 4 sinh2 𝑧2𝐼�푁.(23)

(iii) 𝑎 = 2+4 sinh2 (𝑧/2), 𝑐 = 𝑑 = 1+4 sinh2 (𝑧/2), and 𝑏 = 1.4In this case,𝐻 = Δ�푁�푁 + 4 sinh2 (𝑧/2)𝐼�푁, where

Δ�푁�푁 ≡ (((((((

1 −1 0 ⋅ ⋅ ⋅ 0−1 2 −1 ⋅ ⋅ ⋅ 00 −1 2 ⋅ ⋅ ⋅ 0... ... ... d ...2 −10 0 0 ⋅ ⋅ ⋅ −1 −1)))))))

= Δ(𝑃�푁) , (24)

which is known as the graph Laplacian [15–18] for the pathgraph with𝑁 vertices (𝑃�푁, see Figure 1).

We find

det𝐻 = 𝐷(0) = 2 tanh 𝑧2 sinh𝑁𝑧for 𝐻 = Δ�푁�푁 + 4 sinh2 𝑧2𝐼�푁 (25)

in this case. Note that, since Δ(𝑃�푁) has a zero mode,lim�푧�㨀→0𝐷(0) = 0.

(iv) Clockwork theory [19–22].5

We consider 𝐻 = Δ �푞 + 𝑙2𝐼�푁, where Δ �푞 is the following𝑁 ×𝑁matrix:

Δ �푞 ≡ ((((((((

1 −𝑞 0 ⋅ ⋅ ⋅ 0−𝑞 1 + 𝑞2 −𝑞 ⋅ ⋅ ⋅ 00 −𝑞 1 + 𝑞2 ⋅ ⋅ ⋅ 0... ... ... d ...1 + 𝑞2 −𝑞0 0 0 ⋅ ⋅ ⋅ −𝑞 𝑞2))))))))

. (26)

We find that the determinant of𝐻 can be written as𝐷(0) = 𝑙2 ⋅ (𝛾�푁+ − 𝛾�푁− )√(1 − 𝑞2)2 + 2 (1 + 𝑞2) 𝑙2 + 𝑙4

for 𝐻 = Δ �푞 + 𝑙2𝐼�푁, (27)where𝛾±= 12 [1 + 𝑞2 + 𝑙2 ± √(1 − 𝑞2)2 + 2 (1 + 𝑞2) 𝑙2 + 𝑙4] . (28)Of course, one can see that lim�푞�㨀→1Δ �푞 = Δ(𝑃�푁).4. Determinants of PeriodicTridiagonal Matrices

4.1. The Discrete Gel’fand–Yaglom Method for Periodic Tridi-agonalMatrices. In this section, we treat periodic tridiagonalmatrices, such as

𝐻 = (((((((

𝑎 −𝑏 0 ⋅ ⋅ ⋅ −𝑏∗−𝑏∗ 𝑎 −𝑏 ⋅ ⋅ ⋅ 00 −𝑏∗ 𝑎 ⋅ ⋅ ⋅ 0... ... ... d ...𝑎 −𝑏−𝑏 0 0 ⋅ ⋅ ⋅ −𝑏∗ 𝑎)))))))

. (29)

In this case, the recurrence relation is the same as in theprevious section. Therefore, we can write

V�푘 = 𝐴𝛼�푘−1 + 𝐵𝛽�푘−1, (30)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 rowsof the eigenvalue equation are also the relation among threeterms in the sequence of numbers. In the present case, theyare reduced to𝑏 [𝐴𝛽 (1 − 𝛼�푁) + 𝐵𝛼 (1 − 𝛽�푁)] = 0, (31)𝑏 [𝐴𝛼 (𝛼�푁 − 1) + 𝐵𝛽 (𝛽�푁 − 1)] = 0, (32)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 = 0requires 𝛽 (1 − 𝛼

�푁) 𝛼 (1 − 𝛽�푁)𝛼 (𝛼�푁 − 1) 𝛽 (𝛽�푁 − 1)= (𝛽2 − 𝛼2) (1 − 𝛼�푁) (𝛽�푁 − 1) = 0. (33)This equation is satisfied if 𝜆 is an eigenvalue of the matrix𝐻.In general, we suppose 𝛼 ̸= 𝛽 and the normalization can beknown from the limit 𝑎 = 0 and 𝑏 → 0. Then, we concludethat the characteristic polynomial 𝐷(𝜆) = ∏�푁�푝=1(𝜆�푝 − 𝜆)(where 𝜆�푝 (𝑝 = 1, . . . , 𝑁) are eigenvalues of𝐻) is written by𝐷 (𝜆) = 𝑏�푁 (1 − 𝛼�푁) (𝛽�푁 − 1)= 𝑏�푁 (𝛼�푁 + 𝛽�푁) − 𝑏�푁 − 𝑏∗�푁. (34)Therefore, the determinant of𝐻 in this case is given bydet𝐻 = 𝐷 (0)

= (𝑎 + √𝑎2 − 4 |𝑏|22 )�푁

+(𝑎 − √𝑎2 − 4 |𝑏|22 )�푁 − 𝑏�푁 − 𝑏∗�푁

(for a periodic tridiagonal matrix) .(35)

One may be aware of unnecessary arguments in abovediscussion. From the periodic structure, 𝛼�푁 = 1 or 𝛽�푁 =1 can be concluded. However, the discussion above can begeneralized to treat another type ofmatrix in the next section.

4.2. Example. (i) 𝑎 = 2 + 4 sinh2 (𝑧/2) and 𝑏 = 1.6In this case, 𝐻 = Δ(𝐶�푁) + 4 sinh2 (𝑧/2)𝐼�푁, where Δ(𝐶�푁)

is the graph Laplacian of the cycle graph with𝑁 vertices (seeFigure 2):

Δ (𝐶�푁) ≡ (((((((

2 −1 0 ⋅ ⋅ ⋅ −1−1 2 −1 ⋅ ⋅ ⋅ 00 −1 2 ⋅ ⋅ ⋅ 0... ... ... d ...2 −1−1 0 0 ⋅ ⋅ ⋅ −1 2)))))))

. (36)

We find

det𝐻 = 𝐷(0) = 4 sinh2𝑁𝑧2 = (𝑒�푁�푧/2 − 𝑒−�푁�푧/2)2for 𝐻 = Δ(𝐶�푁) + 4 sinh2 𝑧2𝐼�푁. (37)

Figure 2: 𝐶8 as an example of a cycle graph.Note that lim�푧�㨀→0𝐷(0) = 0 because of the zero mode ofΔ(𝐶�푁).

(ii) 𝑎 = 2 + 4 sinh2 (𝑧/2) and 𝑏 = 𝑒�푖�휒.7In this case,𝐻 = Δ(𝐶�푁, 𝜒) + 4 sinh2 (𝑧/2)𝐼�푁, whereΔ (𝐶�푁, 𝜒)

≡ ((((((((

2 −𝑒−�푖�휒 0 ⋅ ⋅ ⋅ −𝑒−�푖�휒−𝑒−�푖�휒 2 −𝑒−�푖�휒 ⋅ ⋅ ⋅ 00 −𝑒−�푖�휒 2 ⋅ ⋅ ⋅ 0... ... ... d ...2 −𝑒−�푖�휒−𝑒−�푖�휒 0 0 ⋅ ⋅ ⋅ −𝑒−�푖�휒 2

))))))))

. (38)

We find

det𝐻 = 𝐷(0) = 4 sinh2 𝑁𝑧2 + 4 sin2𝑁𝜒2= 𝑒�푁(�푧+�푖�휒)/2 − 𝑒−�푁(�푧+�푖�휒)/22for 𝐻 = Δ(𝐶�푁, 𝜒) + 4 sinh2 𝑧2𝐼�푁.

(39)

5. Determinants of Extended PeriodicTridiagonal Matrices

5.1. The Discrete Gel’fand–Yaglom Method For Extended Peri-odic Tridiagonal Matrices. In this section, we consider thefollowing (𝑁 + 1) × (𝑁 + 1)matrix:

𝐻 = ((((((((((

𝑎 −𝑏 0 ⋅ ⋅ ⋅ −𝑏∗ −𝑑−𝑏∗ 𝑎 −𝑏 ⋅ ⋅ ⋅ 0 −𝑑0 −𝑏∗ 𝑎 ⋅ ⋅ ⋅ 0 −𝑑... ... ... d ... ...𝑎 −𝑏 −𝑑−𝑏 0 0 ⋅ ⋅ ⋅ −𝑏∗ 𝑎 −𝑑−𝑑∗ −𝑑∗ −𝑑∗ ⋅ ⋅ ⋅ −𝑑∗ −𝑑∗ 𝑐

))))))))))

. (40)


The recurrence relation can be found as𝑏V�푘+1 − (𝑎 − 𝜆) V�푘 + 𝑏∗V�푘−1 = −𝑑V�푁+1(2 ≤ 𝑘 ≤ 𝑁 − 1) . (41)The general solution of this equation is

V�푘 = 𝐴𝛼�푘−1 + 𝐵𝛽�푘−1 − 𝑑V�푁+1𝑏 + 𝑏∗ − (𝑎 − 𝜆) , (42)where 𝛼 and 𝛽 are the same as Eq. (11).

The first row of the eigenvalue equation then becomes𝑏 [𝐴𝛽 (1 − 𝛼�푁) + 𝐵𝛼 (1 − 𝛽�푁)] = 0, (43)while the𝑁th row of the eigenvalue equation is𝑏 [𝐴𝛼 (𝛼�푁 − 1) + 𝐵𝛽 (𝛽�푁 − 1)] = 0. (44)The two equations are exactly the same as Eqs. (31) and (32).

Now, in addition, the (𝑁 + 1)st row of the eigenvalueequation reads−𝑑∗ (V1 + V2 + ⋅ ⋅ ⋅ + V�푁) + (𝑐 − 𝜆) V�푁+1 = 0, (45)

and, by using the general solution, this can be reduced to

− 𝑑∗ (𝐴1 − 𝛼�푁1 − 𝛼 + 𝐵1 − 𝛽�푁1 − 𝛽 − 𝑁𝑑V�푁+1𝑏 + 𝑏∗ − (𝑎 − 𝜆))+ (𝑐 − 𝜆) V�푁+1 = 0. (46)As in the previous section, we require that a nontrivial set

of (𝐴, 𝐵, V�푁+1) exists. This leads to the following equation:𝛽 (1 − 𝛼�푁) 𝛼 (1 − 𝛽�푁) 0𝛼 (𝛼�푁 − 1) 𝛽 (𝛽�푁 − 1) 0−𝑑∗ 1 − 𝛼�푁1 − 𝛼 −𝑑∗ 1 − 𝛽�푁1 − 𝛽 𝑁 |𝑑|2𝑏 + 𝑏∗ − (𝑎 − 𝜆) + 𝑐 − 𝜆

= (𝛽2 − 𝛼2) (1 − 𝛼�푁) (𝛽�푁 − 1)⋅ [ 𝑁 |𝑑|2𝑏 + 𝑏∗ − (𝑎 − 𝜆) + 𝑐 − 𝜆] = 0.

(47)

The second left-hand side of the equation should be pro-portional to 𝐷(𝜆), as for discussion in the previous section.Because we have already known the normalization of 𝑏�푁(1 −𝛼�푁)(𝛽�푁 − 1), we conclude that the characteristic polynomialin the present case is written by𝐷(𝜆) = 𝑏�푁 (1 − 𝛼�푁) (𝛽�푁 − 1)

⋅ [ 𝑁 |𝑑|2𝑏 + 𝑏∗ − (𝑎 − 𝜆) + 𝑐 − 𝜆] . (48)

Thus, the determinant of𝐻 in this section is given by𝐷(0) = (𝑐 − 𝑁 |𝑑|2𝑎 − 𝑏 − 𝑏∗)[[[(

𝑎 + √𝑎2 − 4 |𝑏|22 )�푁

+(𝑎 − √𝑎2 − 4 |𝑏|22 )�푁 − 𝑏�푁 − 𝑏∗�푁]]](for an extended periodic tridiagonal matrix) .

(49)

5.2. Examples. (i) 𝑎 = 2𝑟+ 𝑠 + 𝑙2, 𝑐 = 𝑁𝑠+ 𝑙2, 𝑑 = 𝑠, and 𝑏 = 𝑟[23, 24].8

Suppose the (𝑁 + 1) × (𝑁 + 1)matrix [23, 24],

Δ (𝑟, 𝑠) = 𝑟((((((((((

2 −1 0 ⋅ ⋅ ⋅ −1 0−1 2 −1 ⋅ ⋅ ⋅ 0 00 −1 2 ⋅ ⋅ ⋅ 0 0... ... ... d ... ...2 −1 0−1 0 0 ⋅ ⋅ ⋅ −1 2 00 0 0 ⋅ ⋅ ⋅ 0 0 0

))))))))))

+ 𝑠((((((((((

1 0 0 ⋅ ⋅ ⋅ 0 −10 1 0 ⋅ ⋅ ⋅ 0 −10 0 1 ⋅ ⋅ ⋅ 0 −1... ... ... d ... ...0 0 0 ⋅ ⋅ ⋅ 1 −1−1 −1 −1 ⋅ ⋅ ⋅ −1 𝑁

))))))))))

,(50)

where 𝑟 and 𝑠 are constants. The determinant of𝐻 = Δ(𝑟, 𝑠)+𝑙2𝐼�푁+1 isdet𝐻 = 𝐷 (0) = 𝑙2 (1 + 𝑁𝑠𝑠 + 𝑙2 ) (𝜂�푁+ + 𝜂�푁− − 2𝑟�푁) , (51)

where

𝜂± ≡ 12 [2𝑟 + 𝑠 + 𝑙2 ± √(2𝑟 + 𝑠 + 𝑙2)2 − 4𝑟2] . (52)(ii) 𝑎 = 3 + 𝑙2, 𝑐 = 𝑁 + 𝑙2, and 𝑏 = 𝑑 = 1.

This is the previous case with 𝑟 = 𝑠 = 1. In this case,𝐻 = Δ(𝑊�푁+1) + 𝑙2𝐼�푁+1, where


Figure 3:𝑊9 as an example of a wheel graph.

Δ (𝑊�푁+1) ≡((((((((((

3 −1 0 ⋅ ⋅ ⋅ −1 −1−1 3 −1 ⋅ ⋅ ⋅ 0 −10 −1 3 ⋅ ⋅ ⋅ 0 −1... ... ... d ... ...3 −1 −1−1 0 0 ⋅ ⋅ ⋅ −1 3 −1−1 −1 −1 ⋅ ⋅ ⋅ −1 −1 𝑁

))))))))))

(53)

is the graph Laplacian of the wheel graph (see Figure 3) with𝑁+ 1 vertices. We do not repeat writing the expression of𝐻,which is given by Eq. (51) with Eq. (52) when 𝑟 = 𝑠 = 1.

(iii) 𝑏 = 0.In this case, the determinant simply becomes as

det𝐻 = 𝐷(0) = 𝑎�푁𝑐 − 𝑁𝑎�푁−1 |𝑑|2 . (54)Particularly, Δ(0, 1) is in this category and can be written as

Δ (0, 1) = ((((((((((

1 0 0 ⋅ ⋅ ⋅ 0 −10 1 0 ⋅ ⋅ ⋅ 0 −10 0 1 ⋅ ⋅ ⋅ 0 −1... ... ... d ... ...0 0 0 ⋅ ⋅ ⋅ 1 −1−1 −1 −1 ⋅ ⋅ ⋅ −1 𝑁

))))))))))= Δ(𝐾1,�푁) .

(55)

This is the graph Laplacian of a star graph 𝐾1,�푁 (Figure 4).The eigenvalues of Δ(𝐾1,�푁) are known as𝜆0 = 0,𝜆�푝 = 1 (𝑝 = 1, 2, . . . , 𝑁 − 1) ,𝜆�푁 = 𝑁 + 1. (56)The determinant of𝐻 = Δ(𝐾1,�푁) + 𝑙2𝐼�푁+1 is

det𝐻 = 𝐷(0) = 𝑙2 (1 + 𝑁1 + 𝑙2) (1 + 𝑙2)�푁 . (57)

Figure 4: 𝐾1,8 as an example of a star graph.6. Free Energy on a Graph

In this section, we consider applications of the results ondeterminants for studying discrete systems.

We first consider 𝑁 scalar degrees of freedom and definethe action as follows:

𝑆 = 12 (𝑁2𝐿2 �푁∑�푘=1

�푁∑�푘=1

𝜙�푘Δ (𝐶�푁)�푘�푘 𝜙�푘 + 𝜇2 �푁∑�푘=1

𝜙2�푘) , (58)where Δ(𝐶�푁) is the graph Laplacian for 𝐶�푁 and𝜇 ≡ 2𝑁𝐿 sinh 𝑚𝐿2𝑁, (59)where 𝐿 and 𝑚 are constants.

Then, the Gaussian free energy on 𝐶�푁 [25] is obtainedusing Eq. (37) as

𝐹�퐶𝑁 = 12 ln [det(Δ (𝐶�푁) + 4 sinh2 𝑚𝐿2𝑁𝐼�푁)]= ln(2 sinh 𝑚𝐿2 ) + 𝑐𝑜𝑛𝑠𝑡.. (60)This is interesting because the action (58) can be rewritten as

𝑆 = 12 [𝑁2𝐿2 �푁∑�푘=1

(𝜙�푘 − 𝜙�푘+1)2 + 𝜇2 �푁∑�푘=1

𝜙2�푘] , (61)under the “periodic” condition, 𝜙�푁+1 ≡ 𝜙1. A continuumlimit, 𝑎0 ≡ 𝐿/𝑁 → 0, enforces (𝜙�푘+1 − 𝜙�푘)/𝑎0 → 𝜕�푥𝜙,where 𝑥 is a coordinate of one dimension with periodicity𝑥+𝐿 ∼ 𝑥.Therefore, we can find that the one-loop free energyof a real scalar field 𝜙(𝑥) with mass 𝑚 on a circle (𝑆1) withcircumference 𝐿 governed by the action

𝑆 = 12 ∫𝑑𝑥 [(𝜕�푥𝜙)2 + 𝑚2𝜙2] (62)takes the form 𝐹�푆1 = ln (2 sinh 𝑚𝐿2 ) , (63)


after some regularization [25, 26]. Note that, since

2 sinh 𝑚𝐿2 = 𝑚𝐿∏∞�푛=1 (4𝜋2𝑛2/𝐿2 + 𝑚2)∏∞�푛=1 (4𝜋2𝑛2/𝐿2) , (64)we find that the eigenvalues of −𝜕2�푥 + 𝑚2, where −𝜕2�푥 is theone-dimensional Laplacian on 𝑆1, are shown by4𝜋2𝑛2𝐿2 + 𝑚2 (𝑛 is an integer) . (65)

Similarly, we can consider the other matrices. For exam-ple, the action for complex scalar fields, defined as

𝑆 = 12 (𝑁2𝐿2 �푁∑�푘=1

�푁∑�푘=1

𝜙†�푘Δ (𝐶�푁, 𝜒)�푘�푘 𝜙�푘 + 𝜇2 �푁∑�푘=1

𝜙�푘2)= 12 [𝑁2𝐿2 �푁∑

�푘=1

𝜙�푘 − 𝑒�푖�휒𝜙�푘+12 + 𝜇2 �푁∑�푘=1

𝜙�푘2] , (66)leads to the free energy𝐹�퐶𝑁,�휒 = ln [det (Δ (𝐶�푁, 𝜒) + 4 sinh2 𝑚𝐿2𝑁𝐼�푁)]= ln(4 sinh2 𝑁𝑧2 + 4 sin2𝑁𝜒2 ) + 𝑐𝑜𝑛𝑠𝑡.. (67)Here we will avoid repeated discussion and only note that theeigenvalue spectrum of the continuum limit of this case isgiven by (2𝜋𝑛 − 𝜒)2𝐿2 + 𝑚2 (𝑛 is an integer) . (68)

Continuum limits exist also in other some cases.The large 𝑁 limit of the determinant of Δ�퐷�퐷 + 𝜇2𝐼�푁

(according to Eq. (21)) becomes

sinh𝑚𝐿sinh (𝑚𝐿/𝑁) → 𝑁 sinh𝑚𝐿𝑚𝐿 , (69)

which coincides with the result of the example stated inSection 2 up to the constant. We find that the continuumlimit corresponds to the system of massive scalar field in aline 0 ≤ 𝑥 ≤ 𝐿 with Dirichlet-Dirichlet boundary conditionsat its ends.

The large 𝑁 limit of the determinant of Δ�퐷�푁 + 𝜇2𝐼�푁(according to Eq. (23)) becomes simply

cosh𝑚𝐿 + 2 sinh 𝑚𝐿𝑁 sinh𝑚𝐿 → cosh𝑚𝐿. (70)A comparison to a known mathematical relation

cosh𝑚𝐿 = ∏∞�푛=0 [(𝜋2/𝐿2) (𝑛 + 1/2)2 + 𝑚2]∏∞�푛=0 [(𝜋2/𝐿2) (𝑛 + 1/2)2] (71)leads to the conclusion that the continuum limit of spectrumis given by (𝜋2/𝐿2)(𝑛 + 1/2)2 + 𝑚2; thus the boundarycondition of the system is Dirichlet-Neumann condition.

Finally, the determinant ofΔ(𝑃�푁)+𝜇2𝐼�푁 (according to Eq.(25)) is

2 tanh 𝑚𝐿2𝑁 sinh𝑚𝐿. (72)Since the free energy is proportional to the logarithm of this,we drop the 𝑁 dependent term (which is log divergent if𝑁 → ∞).The boundary condition of the continuum systemis Neumann-Neumann condition (which can be judged fromthe existence of a zero mode).

In the next section, we will consider the way to obtainone-loop vacuum energy of scalar field theory with massmatrix 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 quan-tum field theory can be derived from the functional determi-nants [2]. In the present paper, we only consider scalar fieldtheories for simplicity. As seen in the previous section, 𝑁-scalar field theory can resemble compactification of a dimen-sion. This is the key idea of the dimensional deconstruction[8–10].The structure of the theory space is determined by thequadratic term of fields, i.e., the mass matrix. Suppose that amass matrix (precisely, the (𝑚𝑎𝑠𝑠)2 matrix) Δ/𝑎20 is given (inother words, a theory space is given).The eigenvalues ofΔ aredenoted by 𝜆�푝, as previously. Then, using the characteristicpolynomial 𝐷(𝜆) = ∏�푝(𝜆�푝 − 𝜆), one-loop vacuum energydensity for real scalar fields is calculated by

𝑉 = 12 ∫ 𝑑4𝑙(2𝜋)4 ln det [Δ/𝑎20 + 𝑙2]det [Δ/𝑎20 + 𝑙2 +𝑀2]= 12𝑎40 (2𝜋2) ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln 𝐷(−𝑙2)𝐷 (−𝑙2 −𝑀2𝑎20) ,

(73)

where we used 𝑀 of the Pauli-Villars regularization, whichis considered to be 𝑀 → ∞. The constant 𝑎0 illustrates anoverall scale in the theory space, i.e., related to mass scale ofnew physics via 𝑎0 ∼ 𝑚−1�푛�푒�푤 �푝ℎ�푦�푠i�푐�푠.

In practice, regularization is an art of assembly of math-ematical techniques. We adopt here the following approach.A physical value of the vacuum energy should be determinedindependently of the unphysical 𝑀 and the UV divergencemust be subtracted in the expression of it. Thus, we consider,in the denominator in log in Eq. (73), as𝐷(−𝑙2 −𝑀2𝑎20) ⇒ Asymptotic form of 𝐷(−𝑙2)

when 𝑙2 → ∞. (74)Further, if the theory contains the 𝑁 (scalar) fields, theintegrand of the most divergent part should be proportionalto𝑁. Thus, we extract the part of∝ (𝑙2)�푁 ⊂ 𝐷(−𝑙2) for large𝑙2.


7.2. DimensionalDeconstruction of a Circle. Aconcrete exam-ple is in order. We consider a theory space associated withΔ(𝐶�푁, 𝜒). This model has widely been studied by manyauthors [8–10, 27]. We have already obtained det[Δ(𝐶�푁, 𝜒) +4 sinh2 (𝑧/2)𝐼�푁] (∝ 𝐷(−𝑙2), in the present case) in Eq. (39).The asymptotic behavior can be found as

det [Δ (𝐶�푁, 𝜒) + 4 sinh2 𝑧2𝐼�푁]= 𝑒�푁(�푧+�푖�휒)/2 − 𝑒−�푁(�푧+�푖�휒)/22 →𝑒�푁(�푧+�푖�휒)/22 (= 𝑒�푁�푧) (𝑧 → ∞) .(75)

Thus, in our regularization scheme,9𝑉 (𝜒) = 1𝑎40 (2𝜋2) ∫∞0 𝑑𝑧(2𝜋)4 8 sinh3 𝑧2 cosh 𝑧2⋅ ln 𝑒�푁(�푧+�푖�휒)/2 − 𝑒−�푁(�푧+�푖�휒)/22𝑒�푁(�푧+�푖�휒)/22 ,

(76)

where we set 𝑙 = 2 sinh(𝑧/2). Now, the integration can bedone by elementary methods as𝑉(𝜒) = 2𝜋2𝑎40 ∫∞0 𝑑𝑧(2𝜋)4 4 sinh2 𝑧2 sinh 𝑧⋅ [ln (1 − 𝑒−�푁(�푧+�푖�휒)) + ln (1 − 𝑒−�푁(�푧−�푖�휒))]

= − 116𝜋2𝑎40 ∞∑�푛=1 ∫∞0 𝑑𝑧 (𝑒�푧/2 − 𝑒−�푧/2)2 (𝑒�푧 − 𝑒−�푧)⋅ [ 𝑒−�푁�푛(�푧+�푖�휒)𝑛 + 𝑒−�푁�푛(�푧−�푖�휒)𝑛 ] = − 18𝜋2𝑎40 ∞∑�푛=1cos𝑁𝑛𝜒𝑛⋅ ∫∞

0𝑑𝑧 (𝑒2�푧 − 2𝑒�푧 + 2𝑒−�푧 − 𝑒−2�푧) 𝑒−�푁�푛�푧

= − 18𝜋2𝑎40 ∞∑�푛=1cos𝑁𝑛𝜒𝑛 ( 1𝑁𝑛 − 2 − 2𝑁𝑛 − 1+ 2𝑁𝑛 + 1 − 1𝑁𝑛 + 2) = − 18𝜋2𝑎40 ∞∑�푛=1cos𝑁𝑛𝜒𝑛⋅ ( 4𝑁2𝑛2 − 4 − 4𝑁2𝑛2 − 1) = − 32𝜋2𝑎40⋅ ∞∑�푛=1

cos𝑁𝑛𝜒𝑛 (𝑁2𝑛2 − 1) (𝑁2𝑛2 − 4) .

(77)

This result exactly coincides with the known result [8–10,27].10 Incidentally, for large𝑁,𝑉 (𝜒) ∼ − 32𝜋2 (𝑁𝑎0)4 ∞∑�푛=1cos𝑁𝑛𝜒𝑛5 ,𝑉 (0) ∼ − 3𝜁 (5)2𝜋2 (𝑁𝑎0)4 ,

(78)

where 𝜁(𝑧) is Riemann’s zeta function. We find that thereexists a “continuum limit,”𝑁 → ∞ as𝑁𝑎0 and𝑁𝜒 are fixed.7.3. The Clockwork Theory. Next, we turn to consider thetheory space of the clockwork theory [19–22] for real scalarfields. The action is

𝑆 = 12 ∫ 𝑑4𝑥 [ �푁∑�푘=1

(𝜕�휇𝜙�푘)2 + 𝑚2�푁−1∑�푘=1

(𝜙�푘 − 𝑞𝜙�푘+1)2] , (79)where 𝑚 = 𝑎−10 . Thus, the relevant matrix determinant isgiven as Eq. (27). The subtraction of UV divergence is subtlebecause of the complicated form of the determinant in thiscase. We separate the vacuum energy density into three parts,such as 𝑉 = 𝑉�푁(𝑞) + 𝑉0 + 𝑁𝑉1. Here, 𝑉�푁(𝑞) is a finite part,

𝑉�푁 (𝑞) = 12𝑎40 (2𝜋2)∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln[1 − (𝛾−𝛾+)�푁] , (80)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 limitin vacuum energy density.

The change of the integration variable cosh 𝑦 = (𝑙2 + 1 +𝑞2)/2𝑞 makes the integration simple. Then, we can rewrite𝑉�푁(𝑞) as𝑉�푁 (𝑞) = 12𝑎40 (2𝜋2) ∫∞| ln �푞| 𝑑𝑦(2𝜋)4 𝑞 sinh 𝑦⋅ (2𝑞 cosh 𝑦 − 1 − 𝑞2) ln (1 − 𝑒−2�푁�푦) , (81)and we get the form with infinite summations,

𝑉�푁 (𝑞) = − 𝑞216𝜋2𝑎40 ∞∑�푛=1𝑞−2�푁�푛2𝑛 [ 𝑞2(2𝑁𝑛 − 2) (2𝑁𝑛 − 1)− 2 1(2𝑁𝑛 − 1) (2𝑁𝑛 + 1) + 𝑞−2(2𝑁𝑛 + 1) (2𝑁𝑛 + 2)(𝑞 ≥ 1) ,(82)

𝑉�푁 (𝑞) = − 𝑞216𝜋2𝑎40 ∞∑�푛=1𝑞2�푁�푛2𝑛 [ 𝑞−2(2𝑁𝑛 − 2) (2𝑁𝑛 − 1)− 2 1(2𝑁𝑛 − 1) (2𝑁𝑛 + 1)+ 𝑞2(2𝑁𝑛 + 1) (2𝑁𝑛 + 2)] (𝑞 < 1) .

(83)

Note that 𝑞−2𝑉�푁(𝑞) = 𝑞2𝑉�푁(𝑞−1). The numerical resultsfor (𝑁𝑎0)4𝑞−2(5/�푁)𝑉�푁(𝑞5/�푁) are shown in Figure 5 for 𝑁 =3, 4, . . . , 10. These curves indicate that there is a continuumlimit𝑁 → ∞, while𝑁𝑎0 and 𝑞1/�푁 are fixed constants. If wecan treat 𝑞 as a dynamical variable, the effective potential of𝑞 seems to have a minimum at 𝑞 ∼ 1 for large 𝑁, where themass matrix simply becomes the graph Laplacian of 𝑃�푁. Notealso that 𝑉�푁(𝑞) → 0 both for 𝑞 → 0 and for 𝑞 → ∞.


0.6 0.8 1.2 1.4 1.6 1.8 2

−0.0025

−0.002

−0.0015

−0.001

−0.0005

(．；0)4Ｋ−2(5/．)６．(Ｋ

5/．)q

Figure 5:The numerical results for (𝑁𝑎0)4𝑞−2(5/�푁)𝑉�푁(𝑞5/�푁) as func-tions of 𝑞 for𝑁 = 3, 4, . . . , 10, from lower to upper curves.

We now estimate the separated contributions. They arewritten as

𝑉0 = 12𝑎40 (2𝜋2) ∫∞0 𝑙3𝑑𝑙(2𝜋)4⋅ ln 𝑙2√(1 − 𝑞2)2 + 2 (1 + 𝑞2) 𝑙2 + 𝑙4 ,(84)

and

𝑉1 = 12𝑎40 (2𝜋2)∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln 𝛾+= 12𝑎40 (2𝜋2) ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln[[[

1 + 𝑞2 + 𝑙2 + √(1 − 𝑞2)2 + 2 (1 + 𝑞2) 𝑙2 + 𝑙42 ]]] .(85)

As for 𝑉0, if we use the standard formula of derivation ofthe Coleman-Weinberg potential12 ∫ 𝑑4𝑙(2𝜋)4 ln (𝑙2 +𝑀2)�푟�푒�푔�푢�푙�푎�푟�푖�푧�푒�푑 = 𝑀464𝜋2 ln𝑀2, (86)to regularize 𝑉0, aside from the contribution of a zero mode(as ln 𝑙2 in the integrand), we find

𝑉0 = − 12 (64𝜋2) 𝑎40 [(1 − 𝑞)4 ln (1 − 𝑞)2+ (1 + 𝑞)4 ln (1 + 𝑞)2] . (87)It is notable that this contribution is equivalent to subtractionof the half of vacuum energy densities due to scalar fields withmass squared (1 − 𝑞)2/𝑎20 and (1 + 𝑞)2/𝑎20 . The UV divergenceof this part can be regarded to be canceled by the zero-modecontribution.

On the other hand, for the complicated form of a genuinedivergent contribution of𝑁𝑉1 , we introduce a cut-offΛ in theintegration over 𝑙 and find

𝑁𝑉1 = 𝑁64𝜋2𝑎40 [Λ4 (lnΛ2 − 12) + 2 (1 + 𝑞2) Λ2− (1 + 4𝑞2 + 𝑞4) ln 2Λ21 + 𝑞2 + 1 − 𝑞2 + (1 + 𝑞2)2− 32 (1 + 𝑞2) 1 − 𝑞2] .(88)

The quartic divergence seems to be independent of thestructure of the mass matrix and the quadratic divergence isproportional to the trace of the mass matrix.

7.4. Latticization of a Disk. The matrix Δ(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 theorywith mass matrix Δ(𝑟, 𝑠)/𝑎20 can be written formally as𝑉 = 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4⋅ ln [𝑙2 (1 + 𝑁𝑠𝑠 + 𝑙2 ) (𝜂�푁+ + 𝜂�푁− − 2𝑟�푁)] = 𝑉�푁 (𝑟, 𝑠)+ 𝑉0 + 𝑁𝑉1,

(89)

where

𝑉�푁 = 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln [1 + (𝜂−𝜂+)�푁 − 2( 𝑟𝜂+ )�푁]= 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 2 ln[1 − ( 𝑟𝜂+)�푁] ,(90)

with 𝜂+ ≡ 12 [2𝑟 + 𝑠 + 𝑙2 + √(2𝑟 + 𝑠 + 𝑙2)2 − 4𝑟2] , (91)𝑉0 = 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln [𝑙2 (1 + 𝑁𝑠𝑠 + 𝑙2 )] , (92)and

𝑉1 = 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln 𝜂+ = 2𝜋22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4⋅ ln 2𝑟 + 𝑠 + 𝑙2 + √(2𝑟 + 𝑠 + 𝑙2)2 − 4𝑟22 .

(93)

A finite part 𝑉�푁 can be rewritten as𝑉�푁 ( 𝑠𝑟) = 4𝜋2𝑟22𝑎40 ∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln[[[1

−( 22 + 𝑠/𝑟 + 𝑙2 + √(2 + 𝑠/𝑟 + 𝑙2)2 − 4)�푁]]] .

(94)


46

8

10N

0

0.5

1

1.5

2

u

−0.15−0.125−0.1

−0.075−0.05

(．；0)4Ｌ−2６．(u/N)

Figure 6:Thenumerical results for (𝑁𝑎0)4𝑟−2𝑉�푁(𝑢/𝑁) as a functionof𝑁 and 𝑢.Furthermore, introducing new variables 𝑠/𝑟 = 2 sin2 (𝑢/2)and cosh 𝑦 = 𝑙2/2 + cosh 𝑢, we find𝑉�푁 (𝑢) = 4𝜋2𝑟22𝑎40 ∫∞�푢 𝑑𝑦(2𝜋)4 2 sinh 𝑦 (cosh 𝑦 − cosh 𝑢)⋅ ln [1 − 𝑒−�푁�푦] = − 𝑟28𝜋2𝑎40 ∞∑�푛=1𝑒−�푁�푛�푢𝑛⋅ [ 𝑒2�푢(𝑁𝑛 − 2) (𝑁𝑛 − 1) − 2 1(𝑁𝑛 − 1) (𝑁𝑛 + 1)

+ 𝑒−2�푢(𝑁𝑛 + 1) (𝑁𝑛 + 2)] .(95)

The numerical result of (𝑁𝑎0)4𝑟−2𝑉�푁(𝑢/𝑁) is plotted as afunction of 𝑁 and 𝑢 in Figure 6, where we treat 𝑁 as a con-tinuous parameter. We find that lim�푁�㨀→∞(𝑁𝑎0)4𝑟−2𝑉�푁(0) =−3𝜁(5)/4𝜋2, while we find no other limiting cases for general𝑟 and 𝑠; i.e., no precise continuum limit exists in general cases.

We now turn to consider the other part of the vacuumenergy. For 𝑉0, using similar estimation as in the previoussubsection, we obtain, up to the zero-mode contribution,

𝑉0 = 164𝜋2𝑎40 𝑠2 {(𝑁 + 1)2 ln [(𝑁 + 1)2 𝑠2] − ln 𝑠2} , (96)which is equivalent to the contribution of a scalar field withmass squared (𝑁+1)2𝑠2/𝑎20 minus the contribution of a scalarfield with mass squared 𝑠2/𝑎20 .The UV divergence is canceledin these two contributions.

The divergent part is analyzed by using the cut-off Λ andis found to be

𝑁𝑉1 = 𝑁64𝜋2𝑎40 [Λ4 (lnΛ2 − 12) + 2 (2𝑟 + 𝑠) Λ2− (6𝑟2 + 4𝑟𝑠 + 𝑠2) ln 2Λ22𝑟 + 𝑠 + √𝑠 (4𝑟 + 𝑠)+ (2𝑟 + 𝑠)2 − 32 (2𝑟 + 𝑠)√𝑠 (4𝑟 + 𝑠)] .

(97)

Again, we find that the quartic divergence is independent ofthe mass matrix and the quadratic divergence is proportionalto the trace of the mass matrix.11

In the next section, we will exhibit one more example ofcalculation of one-loop vacuum energy density for a slightlycomplicated theory space.

8. Some Other Examples of Vacuum Energy

Using the additional formulas on determinants, we canfurther obtain determinants of various matrices. In thissection, we show some other examples below.

8.1. Adding an Edge with a Vertex to Each Vertex of a Graph.Let Δ�푁 be an𝑁×𝑁Hermitian matrix and define a 2𝑁× 2𝑁matrix Δ 2�푁 as follows:Δ 2�푁 ≡ (Δ�푁 + 𝐼�푁 −𝐼�푁−𝐼�푁 𝐼�푁 ) , (98)where 𝐼�푁 is an 𝑁 dimensional identity matrix. In particular,if Δ�푁 is the graph Laplacian of a graph 𝐺, Δ 2�푁 is the graphLaplacian of the graph generated by adding an edge with avertex to every vertex of 𝐺.

Then, the formula on determinants

det(𝐴 𝐵𝐶 𝐷) = det (𝐴 − 𝐵𝐷−1𝐶) det (𝐷) (99)tells us that

det (Δ 2�푁 − 𝜆𝐼2�푁)= (1 − 𝜆)�푁 det(Δ�푁 − 1 − (1 − 𝜆)21 − 𝜆 𝐼�푁) . (100)Therefore, if 𝐷�푁(𝜆) ≡ ∏�푁�푝=1(𝜆�푝 − 𝜆) = det(Δ�푁 − 𝜆𝐼�푁),

where {𝜆�푝} are eigenvalues of Δ�푁, is known, 𝐷2�푁(𝜆) ≡det(Δ 2�푁 − 𝜆𝐼2�푁) is obtained as𝐷2�푁 (𝜆) = (1 − 𝜆)�푁 𝐷�푁 (1 − (1 − 𝜆)21 − 𝜆 ) . (101)

For example, we will calculate vacuum energy density ofthe scalar field theorywithmassmatrixΔ 2�푁/𝑎20 , whereΔ 2�푁 isgenerated fromΔ�푁 = Δ(𝐶�푁); i.e., Δ 2�푁 is the graph Laplacianof the graph shown in Figure 7.

In this case, after some manipulation, we get

𝐷2�푁 (𝜆) = 𝛼 (𝜆)�푁 [1 − (1 − 𝜆𝛼 (𝜆) )�푁]2 , (102)


Figure 7: The graph generated by adding an edge with a vertex toevery vertex of 𝐶9.where

𝛼 (𝜆) ≡ 𝜆2 − 4𝜆 + 2 + √𝜆 (𝜆3 − 8𝜆2 + 16𝜆 − 8)2 . (103)Now, we can obtain the vacuum energy density in this

theory by utilizing ln𝐷2�푁(−𝑙2), as in the previous section.We separate the finite and divergent parts of vacuum energydensity as

𝑉�푁 = 12𝑎40 (2𝜋2) ∫∞0 𝑙3𝑑𝑙(2𝜋)4 2 ln[1 − ( 1 + 𝑙2𝛼 (−𝑙2))�푁] , (104)and

𝑉1 = 12𝑎40 (2𝜋2)∫∞0 𝑙3𝑑𝑙(2𝜋)4 ln𝛼 (−𝑙2) . (105)The numerical result of (𝑁𝑎0)4𝑉�푁 in the present case

is shown in Figure 8, where 𝑁 is treated as a continuousparameter. In the limit of 𝑁 → ∞, (𝑁𝑎0)4𝑉�푁 approaches−3𝜁(5)/16𝜋2, which is quarter of the value of the large𝑁 limitof (𝑁𝑎0)4𝑉�푁 in the case of the real scalar theory based onthe graph Laplacian Δ(𝐶�푁). The divergent part 𝑁𝑉1 can beestimated, because 𝛼(−𝑙2) ∼ 𝑙4 for large 𝑙, as

𝑁𝑉1 = 2𝑁64𝜋2𝑎40 Λ4 (lnΛ2 − 12) + 𝑁Λ28𝜋2 + ⋅ ⋅ ⋅ . (106)The leading term is proportional to the number of real scalarfields, as expected. The quadratic divergence is proportionalto the trace of the mass matrix.

8.2.The Graph Cartesian Products 𝐺 × 𝑃2. Let Δ�푁 be an𝑁×𝑁 Hermitian matrix and define a 2𝑁 × 2𝑁 matrix Δ̂ 2�푁 asfollows:

Δ̂ 2�푁 ≡ (Δ�푁 + 𝐼�푁 −𝐼�푁−𝐼�푁 Δ�푁 + 𝐼�푁) . (107)In particular, ifΔ�푁 is the graph Laplacian of a graph𝐺, Δ̂ 2�푁 isthe graph Laplacian of the graph Cartesian product 𝐺 × 𝑃2.12

6 8 10 12 14N

−0.05

−0.04

−0.03

−0.02(．；0)

4６．

Figure 8: The numerical value of (𝑁𝑎0)4𝑉�푁 for the model inthis subsection as a function of 𝑁. The dotted lines indicate(1/4)(𝑁𝑎0)4𝑉�푁, where 𝑉�푁 is the vacuum energy density in the realscalar theory whose mass matrix is Δ(𝐶�푁)/𝑎20 , and the constant−3𝜁(5)/16𝜋2.

Then, the use of the formula on determinants

det(𝐴 𝐵𝐶 𝐷) = det (𝐴 − 𝐵𝐷−1𝐶) det (𝐷)= det (𝐴𝐷 − 𝐵𝐶) , (108)provided that [𝐶,𝐷] = 0, leads to

det (Δ̂ 2�푁 − 𝜆𝐼2�푁) = det ([Δ�푁 + (1 − 𝜆) 𝐼�푁]2 − 𝐼�푁)= det (Δ�푁 − 𝜆𝐼�푁)⋅ det (Δ�푁 + (2 − 𝜆) 𝐼�푁) . (109)Therefore, if𝐷�푁(𝜆) ≡ ∏�푁�푝=1(𝜆�푝 − 𝜆) = det(Δ�푁 − 𝜆𝐼�푁), where{𝜆�푝} are eigenvalues of Δ�푁, is known, 𝐷2�푁(𝜆) ≡ det(Δ̂ 2�푁 −𝜆𝐼2�푁) is obtained as𝐷2�푁 (𝜆) = 𝐷�푁 (𝜆)𝐷�푁 (𝜆 − 2) . (110)

The vacuum energy density of the scalar field theory withmass matrix Δ̂ 2�푁/𝑎20 , where Δ̂ 2�푁 is generated from Δ�푁 =Δ(𝐶�푁), i.e., Δ̂ 2�푁 is the graph Laplacian of the graph Cartesianproduct 𝐶�푁 × 𝑃2, called as the prism graph 𝑌�푁. We show thegraph 𝑌9 in Figure 9.

We can obtain the vacuum energy density in this theoryby utilizing ln𝐷2�푁(−𝑙2). We separate the finite and divergentparts of vacuum energy density as𝑉�푁 = 116𝜋2𝑎40 ∫∞0 𝑙3𝑑𝑙

⋅ 2{{{{{ln[[[1 −( 22 + 𝑙2 + √(2 + 𝑙2)2 − 4)�푁]]]

+ ln[[[1 −( 24 + 𝑙2 + √(4 + l2)2 − 4)�푁]]]

}}}}} ,(111)


Figure 9: The graph Cartesian product 𝐶9 × 𝑃2, or the prism graph𝑌9.and

𝑉1 = 116𝜋2𝑎40 ∫∞0 𝑙3𝑑𝑙 [[[ln2 + 𝑙2 + √(2 + 𝑙2)2 − 42

+ ln4 + 𝑙2 + √(4 + 𝑙2)2 − 42 ]]] .(112)

The numerical result of (𝑁𝑎0)4𝑉�푁 in the present caseis shown in Figure 10, where 𝑁 is treated as a continuousparameter. In the limit of 𝑁 → ∞, (𝑁𝑎0)4𝑉�푁 approachesthe vacuum energy density of the model associated with 𝐶�푁and −3𝜁(5)/4𝜋2. The divergent part𝑁𝑉1 can be estimated as

𝑁𝑉1 = 2𝑁64𝜋2𝑎40 Λ4 (lnΛ2 − 12) + 3𝑁Λ216𝜋2 + ⋅ ⋅ ⋅ . (113)The leading term is proportional to the number of real scalarfields, as expected. The quadratic divergence is proportionalto the trace of the mass matrix.

9. Conclusion

In the present paper, we showed the method of obtaining thedeterminant of repetitive tridiagonal matrices with concreteexamples. The concept of the method is similar to theGel’fand–Yaglom method of obtaining functional determi-nants for differential operators.

The repetitive matrices as mass matrices are widelyconsidered in modern models in particle physics, in orderto attack the hierarchy problem by adopting a theory space.We showed one-loop vacuum energies of such models canbe evaluated by using the determinant of the mass matricesobtained by our method stated in earlier sections.

We have seen that there are not always genuine contin-uum limits in large𝑁 for general theory spaces. In Section 7,we have also found that contributions of 𝑉0 expressed inlogarithmic functions remain in general. They can be com-pensated by addition of bosonic or fermionic free fields withappropriate mass in some cases.13

6 8 10 12 14N

(．；0)4６．

−0.08

−0.09

−0.11

−0.12

−0.13

Figure 10: The numerical value of (𝑁𝑎0)4𝑉�푁 for the model whosetheory space is associated with 𝑌�푁 as a function of 𝑁. The dottedlines indicate (𝑁𝑎0)4𝑉�푁, where 𝑉�푁 is the vacuum energy densityin the real scalar theory whose mass matrix is Δ(𝐶�푁)/𝑎20 , and theconstant −3𝜁(5)/4𝜋2.

In future work, we wish to study one-loop energy densityin models of deconstructed warped (theory) space [28–33].Although it is difficult to evaluate the determinants in a closedform in such a model, calculation based on recurrence rela-tions would be suitable for a computer. It is also interesting toinvestigate the recent model of deconstruction of torus withmagnetic flux [34].14

If we would like to deal with the matrices related withmore complicated graphs or higher dimensional lattices,we confront other difficulties.15 The graph Laplacians ofgeneric graphs cannot be expressed by tridiagonal matrices,though, fortunately, it is known that arbitrary square matricescan be systematically tridiagonalized by the Householdermethod [35] (see also Refs. [36, 37]). Thus, in principle, ourGel’fand–Yaglom-type method can be applied to the matrixwith the general graph structure.

Finally, we add a comment on exclusion of zero modes.Zero modes of operators which appear in quantum fieldtheory have crucial meanings related with nonperturbativeaspects of the theory (see, for example, the first section ofRef. [14]). In our present paper, we considered mass termsin almost all examples and the cases with zero modes canbe considered as the limit that the value of mass goes tozero. Because we considered the vacuum energies and theirdependence on the parameters in this paper, the analysis isjust sound.Moreover, it is known that, if amatrix is expressedas a graph Laplacian of a simple graph (as in each examplein this paper), the matrix has a single zero mode. Therefore,further analysis on zeromodes, if necessary, could be fulfilledappropriately.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.


Endnotes

1. For example, see [38].2. In this simple case, eigenvalues are known as𝜆�푝 = 4 sin2 𝜋𝑝2 (𝑁 + 1) + 4 sinh2 𝑧2 (𝑝 = 1, 2, . . . , 𝑁) . (∗)3. In this case,

𝜆�푝 = 4 sin2 𝜋 (𝑝 − 1/2)2 (𝑁 + 1/2) + 4 sinh2 𝑧2 (𝑝 = 1, 2, . . . , 𝑁) . (∗∗)4. In this case, the eigenvalues are𝜆�푝 = 4 sin2 𝜋𝑝2𝑁 + 4 sinh2 𝑧2(𝑝 = 0, 1, 2, . . . , 𝑁 − 1) .(∗ ∗ ∗)5. In this case,𝜆0 = 𝑙2,𝜆�푝 = 𝑞2 + 1 − 2𝑞 cos𝜋𝑝𝑁 + 𝑙2(𝑝 = 1, 2, . . . , 𝑁 − 1) .(∗ ∗ ∗∗)6. In this case, the eigenvalues are𝜆�푝 = 4 sin2𝜋𝑝𝑁 + 4 sinh2 𝑧2(𝑝 = 0, 1, 2, . . . , 𝑁 − 1) .(∗ ∗ ∗ ∗ ∗)

Note that degeneracy occurs.7. In this case, the eigenvalues are

𝜆�푝 = 4 sin2 (𝜋𝑝𝑁 + 𝜒2 ) + 4 sinh2 𝑧2(𝑝 = 0, 1, 2, . . . , 𝑁 − 1) .(∗ ∗ ∗ ∗ ∗∗)8. In this case,𝜆0 = 𝑙2,𝜆�푝 = 𝑠 + 𝑟 sin2 𝜋𝑝𝑁 + 𝑙2(𝑝 = 1, 2, . . . , 𝑁 − 1) ,𝜆�푁 = (𝑁 + 1) 𝑠 + 𝑙2.

(∗ ∗ ∗ ∗ ∗ ∗ ∗)9. We now consider complex scalar fields.10. It is notable that the regularized vacuum energy is𝑂(𝑁−4), while the subtracted part whose integrand

including ln 𝑒�푁�푧 = 𝑁𝑧 is proportional to𝑁.

11. Note that, remembering the one zero-mode contributionseparated from 𝑉0, the quartic divergence is found to beproportional to𝑁 + 1.

12. The graph Cartesian product of 𝐺1 and 𝐺2 is also oftenwritten as 𝐺1◻G2.

13. In order to apply the models to the hierarchy problem,there must be other matter fields coupled to the fieldsin the theory space. Therefore, it may not be a greatdifficulty in the model.

14. A partially deconstructed model with flux has beenconsidered more than a decade ago [39].

15. Even in the case with differential operators in higerdimensions, there is a problem of degeneracy, whichbecomes an origin of another divergence [40].

References

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[2] G. V. Dunne, “Functional determinants in quantum fieldtheory,” Journal of Physics A: Mathematical andTheoretical, vol.41, no. 30, Article ID 304006, 2008.

[3] S. Coleman, Aspects of Symmetry, Cambridge University Press,Cambridge, UK, 1988.

[4] C. Ccapa Ttira, C. D. Fosco, and F. D. Mazzitelli, “Lifshitz for-mula for the Casimir force and the Gelfand–Yaglom theorem,”Journal of Physics A: Mathematical and Theoretical, vol. 44, no.46, artilce 465403, 2011.

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