Large-size expansion for triangular Wilson loops in confining gauge theories

The asymptotic behavior of Wilson loops in the large-size limit (L → ∞) in confining gauge theories with area law is controlled by effective string theory (EST). The L−2 term of the large-size expansion for the logarithm of Wilson loop appears within EST as a 2-loop correction. Ultraviolet divergences of this 2-loop correction for polygonal contours can be renormalized using an analytical regularization constructed in terms of Schwarz-Christoffel mapping. In the case of triangular Wilson loops this method leads to a simple final expression for the L−2 term.


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(with large flat contour C bounding a region with area S (C)) play an important role as models of the heavy-quark confinement and are often briefly called confining gauge theories (although the problem of confinement in real QCD is more complicated). In our modern understanding of confining gauge theories, the area law is considered as a secondary manifestation of a much deeper phenomenon of effective string formation. This phenomenon is described by effective string theory (EST) which provides a detailed information about the asymptotic behavior of Wilson loops in the limit of large size of contour C (denoted informally as |C| → ∞ in eq. (1.1)). EST assumes that the asymptotic behavior of Wilson loops in the limit of large size of contour C can be described by a functional integral over surfaces Σ bounded by contour C: 3) The idea that Wilson loops W (C) can be approximated by functional integral (1.3) has a long history. Starting from qualitative and heuristic arguments [2], one can try to justify the stringy approach to Wilson loops using various limits and expansions: large size, large number of colors [3][4][5][6], large number of space-time dimensions [7][8][9][10], Regge limit [10][11][12].
Here we are interested in EST understood as an effective theory describing the large-size limit. The first steps in this direction were made by M. Lüscher, G. Münster, K. Symanzik and P. Weisz [13][14][15]. In the computation of the first terms of the large-size expansion, one can approximate S EST [Σ] by Nambu action S Nambu (Σ) = σ 0 S (Σ) (1.4) where S (Σ) is the area of surface Σ and σ 0 is the bare string tension (different from the renormalized physical string tension σ appearing in area law (1.1)). If one wants to use EST for the calculation of higher-order terms of the large-size expansion then S EST [Σ] must be understood as an infinite series containing all possible terms compatible with the symmetries of the problem. The theoretical work in EST has been proceeding in various directions including • derivation of general constraints on terms appearing in EST action S EST [Σ] [16][17][18][19][20][21][22][23][24][25], • computation of loop corrections in EST for rectangular Wilson loops [24,[26][27][28][29] and for other closely related quantities like correlation functions of Polyakov lines and spectra of closed and open strings [16,[18][19][20][21][22][23], • analysis of string finite-width effects [15,30,31].
In the case when large-size limit |C| → ∞ is implemented as a uniform rescaling of contour C with a large factor λ, EST predicts the following structure of the large-size expansion for polygonal contours C: (1.5)

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Here f k (C) (k = ln, −2, −1, 0, . . .) are functionals of contour C. EST provides much information on f k (C) but some parameters controlling f k (C) depend on the underlying microscopic gauge theory (MGT). Two leading terms of expansion (1.5) are associated with area S (C) and perimeter L (C): Here σ is the string tension appearing in eq. (1.1). Parameters σ, ρ are determined by MGT and cannot be computed in EST. Parameter ρ depends on the renormalization scheme used for Wilson loops in MGT whereas string tension σ is renormalization invariant in MGT. Some properties of functionals f k (C) can be derived on general grounds without a direct involvement of EST. For example, the compatibility of expansion (1.5) with transitivity W ((λ 1 λ 2 ) C) = W (λ 1 (λ 2 C)) leads to relation which results in

Simple example
The main subject of this paper is functional f 2 (C). In the case of rectangular contours, f 2 (C) has been already computed (see section 1.3). Our aim is to provide a general computational scheme (including regularization and renormalization) for the case of f 2 (C) with arbitrary polygons C and to perform an explicit calculation for the simplest case of triangles C. Although term λ −2 f 2 (C) is strongly suppressed in expansion (1.5), this term plays a crucial role in lattice tests of EST. The computation of this correction for general polygons is also important for understanding the structure of renormalization in EST (in the sense of renormalization in effective theories).
As for more dominant terms of expansion (1.5) associated with f ln (C) and f 0 (C), these terms are well-known. A brief review of their properties can be found in appendices B, C.
In this introductory section we would like to concentrate on functional f 2 (C) and to minimize the involvement of terms f ln (C) and f 0 (C). Therefore it makes sense to start JHEP04(2020)204 from an expression combining several Wilson loops in such a way that the role of term f 2 (C) is enhanced whereas some of dominant terms f k (C) cancel. An instructive example is ratio taken in the limit λ → ∞ at fixed ν 1 , ν 2 . Using properties (1.9)-(1.12) of expansion (1.5), we find (1.14) The l.h.s. is a renormalization invariant combination of Wilson loops (which can be computed, e.g. by lattice Monte Carlo simulations). On the r.h.s. we have an asymptotic series in inverse powers of λ → ∞. Note that terms f −1 (C) and f 0 (C) cancel on the r.h.s. of (1.14). The explicit expression for f ln (C) is given by eq. (B.2). Expansion (1.14) provides a good illustration of the role of the λ −2 f 2 (C) correction. On the other hand, one can construct more complicated renormalization invariant ratios of Wilson loops whose large-size expansions involve also f 0 (C): see appendices B, C.

Known results for f 2 (C)
Dimensional arguments and scaling property (1.9) lead to the following structure of f 2 (C): where σ is the string tension appearing in area law (1.1) and g 2 (C) is a dimensionless function depending only on the geometry of polygon C.
For rectangular contours, f 2 (C) was computed in refs. [26,27] with an unfortunate arithmetic error which was later detected and corrected in refs. [28,29]. For rectangle C rectangle (L 1 , L 2 ) with sides L 1 , L 2 one has where D is the space-time dimension and E n (z) are Eisenstein series. In this work we compute f 2 (C) for the case of triangular loops C. For triangle C triangle (θ 1 , θ 2 , θ 3 ; S) with interior angles θ 1 , θ 2 , θ 3 and with area S we find (see section 7.4) According to eq. (1.15), the full explicit expression for f 2 (C triangle ) is

Outline of the work
The main part of this work concentrates on the computation of f 2 (C) in large-size expansion (1.5). Working with ratios (1.14) of Wilson loops, one can eliminate more dominant term f 0 (C) from the consideration. Nevertheless for completeness a review of well-known results for terms f ln (C) and f 0 (C) is provided in appendices A, B, C.
In section 2 we sketch basic formulas of the loop expansion in EST. This part of the work is common for our triangular case and for the rectangular case studied in refs. [26,27]. Correction f 2 (C) is given by a figure-eight-like Feynman diagram for a two-dimensional field theory of (∂φ) 2 + (∂φ) 4 type (see eq. (2.7)) in the region bounded by contour C with zero boundary conditions. We use notation S 4 (see eqs. (2.11), (2.17)) for the integral associated with this Feynman diagram. In principle, S 4 represents the result for f 2 (C) (up to the sign). However, this Feynman diagram has ultraviolet divergences.
The deviation of the current work from refs. [26,27] starts when it comes to the regularization of ultraviolet divergences in S 4 . Unfortunately the regularization used in refs. [26,27] can be applied only to the case of rectangular contours C. In our work we suggest another regularization which can be used for arbitrary polygons. In section 3 we describe the difference between the regularization used in refs. [26,27] and our regularization. Our renormalization procedure operationally consists of two steps: Here S 4 ren is completely free of ultraviolet divergences and provides the result for f 2 (C) (see eq. (2.24)) whereas the intermediate quantity S 4 ren,1 (introduced in eq. (3.12)) is only partly renormalized and still has ultraviolet divergences. Our calculation is based on Schwarz-Christoffel (SC) mapping for polygon C and on other powerful methods of complex analysis. In section 4 we introduce a complexified representation for S 4 and use it in order to perform the first step of renormalization S 4 → S 4 ren,1 (with technical details placed in appendix E).
In section 5 the derived expression for S 4 ren,1 is rewritten in terms of SC mapping.
In section 6 we describe the second step of renormalization procedure S 4 ren,1 → S ren based on the analytical regularization of the SC representation for S 4 ren,1 , assuming the general case of arbitrary polygons C.

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In section 7 we apply the general formula derived for S ren with an arbitrary polygon to the case of triangles. We show that in the triangular case all integrals involved in the computation of S ren can be computed explicitly and derive final expression (1.19) for f 2 (C) = − S ren . The technical details of the computation are placed in appendices F, G, H. For completeness we also provide information about f 0 (C) with triangular contours C in appendix C.2.

Expansion of Nambu action
The first steps of our work are identical to those of refs. [26,27]. For the computation of the first orders of the large-size expansion of Wilson loops (including f 2 (C)), one can replace full EST action S EST [Σ] in eq. (1.3) by Nambu action (non-Nambu terms of effective action S EST [Σ] become essential starting from the computation of correction f 3 (C) generated by the boundary action [16,19,24]). Thus for our aim we can approximate with S Nambu (Σ) given by eq. (1.4). Assuming parametrization X a x 1 , x 2 of surface Σ, one has In EST (like in the theory of fundamental boson strings) the reparametrization invariance must be treated as a gauge symmetry. In the case of flat Wilson contours C lying in the X 1 , X 2 plane, it is convenient to use the planar gauge (sometimes called static gauge): Expanding up to quartic terms, we find in this gauge We use notation U (C) for the flat region in x 1 , x 2 plane bounded by Wilson contour C.

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Here S (C) is the area of region U (C) bounded by flat contour C. One can rewrite this expansion in the form

Loop expansion in EST
The large-size expansion for Wilson loops is constructed by applying the steepest-descent method to functional integral (2.1). Expanding in small fluctuations of Σ near the minimal surface spanned on contour C, one finds in the 1-loop approximation [13][14][15]: Here Det reg [−∆ (C)] is the functional determinant of Laplace operator (referred as Laplace determinant for brevity) with zero (Dirichlet) boundary conditions on contour C and with properly regularized ultraviolet divergences. In this oversimplified approach, at intermediate stages of the calculation we formally deal with the loop expansion in small parameter 1/σ 0 → 0 at fixed contour C but in the end this formal 1/σ 0 expansion can be rearranged as a large-size expansion for Wilson loops at fixed physical (renormalized) string tension σ. The real situation is more complicated because • loop corrections of EST have ultraviolet divergences which must be renormalized (in the sense of renormalization in effective theories), • microscopic gauge theory (MGT) has its own ultraviolet divergences and renormalization.

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Fortunately, these complications play a limited role if one is interested in the EST computation of the first terms of large-size expansion (1.5).
For the computation of f 2 (C) one needs only the Nambu term of the full effective action. The loop expansion of EST with Nambu action naively goes in integer powers of 1/σ 0 which results in even terms λ −2n f 2n (C) of large-size expansion (1.5). This naive argument explains the vanishing of odd term f 1 (C) (1.12) but a serious proof of (1.12) requires more effort since divergences and their renormalization in EST may generate odd and non-analytical terms like f −1 (C) and f ln (C).
Anyway the concept of the naive and formal loop expansion in small parameter 1/σ 0 may be helpful (at least operationally) for the computation of f k (C) in EST and within this framework one has the following correspondence between loop counting and f k (C) computation: (2.14) The main subject of this paper is 2-loop correction f 2 (C). Technically (but not conceptually) the computation of f 2 (C) can be well separated from the computation of 1-loop terms f ln (C) ln λ + f 0 (C). Therefore in subsequent sections we concentrate on f 2 (C) whereas a detailed discussion of 1-loop terms is placed to appendices B, C.
In 1-loop approximation (2.12) there appears Laplace determinant which has area, perimeter and cusp divergences (see appendices A.1, A.2). Therefore the completion of the 1-loop computation (i.e. the computation of f ln (C), f 0 (C)) includes two parts: 1) renormalization of Laplace determinant (appendix A.2) using the heat-kernel expansion (appendix A.1), 2) computation of the renormalized Laplace determinant for arbitrary polygons (appendix C) using Schwarz-Christoffel (SC) mapping.
Both parts of this work are well described in literature so that we simply combine all known pieces together in appendices A, B, C.
An explicit expression for f ln (C) can be obtained already at the first step (renormalization of Laplace determinants) -see eq. (B.2). As for f 0 (C), the situation is more subtle because single Wilson loops W (C) and corresponding functionals f 0 (C) are not renormalization invariant quantities in MGT. EST provides unambiguous expressions only for renormalization invariant combinations

2-loop EST correction
Using action (2.8), we compute the functional integral of EST in the 2-loop approximation: (2.16) Here Thus in the 2-loop approximation Comparing this with the 1-loop approximation Using expression (2.11) for S 4 , we compute Gaussian integral (2.17)

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One should keep in mind that the r.h.s. is plagued by divergences. Indeed, nondiagonal matrix element (2.29) has a singularity at y → x so that quantity G µν (x) is divergent for any x. This means that on the r.h.s. of eq. (2.35) the integrand is divergent at any point x. Nevertheless eq. (2.35) provides a good starting point for the discussion of the renormalization.

Rectangular case: method of K. Dietz and T. Filk
The naive non-renormalized expression for the 2-loop correction (2.35) allows for arbitrary polygons C and in this sense is common for the current work concentrating on triangles and for refs. [26,27] devoted to rectangles. But starting from eq. (2.35) the paths of this work and refs. [26,27] diverge. The main reason is that the authors of refs. [26,27] use an ultraviolet regularization specific for the rectangular case which cannot be applied to other polygons. The regularization of refs. [26,27] is based on the explicit diagonalization of Laplace operator acting in rectangle with Dirichlet boundary conditions. The spectral problem has the obvious solution The method of refs. [26,27] is based on 1) formal spectral decomposition of G µν (2.28) in terms of λ mn , ψ mn ignoring the divergence of G µν , 2) insertion of this formal expression for G µν into the r.h.s. of eq. (2.35), 3) formal integration over x in eq. (2.35).
As a result, one obtains an expression for S 4 in terms of divergent series where R a (m 1 , m 2 , . . . , m n ) are some polynomials of integer m k (originating from parameters m, n of eigenvalues λ mn ). Formal series (3.5) are divergent. Refs. [26,27] use analytical regularization for numerators R 1 so that the problem reduces to the computation of sums

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in convergence region of α k -space with a subsequent analytical continuation in α k to integer values corresponding to monomials of the original numerator R 1 (m 1 , m 2 , . . . , m n ) in eq. (3.5). This approach allows for a complete analytical computation of f 2 (C) for rectangles C with result (1.16). However, the method is based on the explicit expressions for spectrum (3.3) and for eigenfunctions (3.4) and therefore is limited to the case of rectangles.
From this brief review of Dietz-Filk (DF) method it is clear that DF regularization belongs to the class of last-minute regularizations: one starts from divergent integrals and sums; after a naive formal algebra with these divergent expressions one introduces the regularization in the end of the calculation (the transition from eq. (3.5) to eq. (3.6)). From the rigorous point of view it would be preferable to use some other complete overall regularization of EST from the very beginning, covering all orders of the loop expansion in a consistent universal way (e.g. dimensional regularization). However, in the practical work a compromise between solid theoretical background and computational efficiency is often made in favor of DF method, see. e.g. refs. [18,19,24] where DF method was extended to higher orders of the large-size expansion (still as a last-minute regularization). As for tests of the consistency between the theoretically preferable dimensional regularization and the practically efficient DF method, one should mention the case of the correlation function of two parallel Polyakov lines where an explicit calculation of the 2-loop contribution (analog of f 2 (C) with rectangular contours C) in dimensional regularization [16] confirmed the original DF result [26,27].

General polygons: renormalization based on SC mapping
As discussed above, the analytical regularization of refs. [26,27] was designed for rectangles. Our aim is to construct a regularization (and renormalization) procedure for f 2 (C) with arbitrary polygons C. We also want this regularization method to be efficient for practical computations. The first hint for the construction of this regularization procedure comes from the lessons of 1-loop EST corrections. As discussed in appendix C, the explicit analytical computation of 1-loop correction f 0 (C) is based on SC mapping for polygon C. This observation suggests that we should try to construct the regularization and the renormalization for the 2-loop correction f 2 (C) also in terms of SC mapping.
The next observation is that the naive divergent expression (2.35) has two types of divergences: 1) The integrand is divergent at any point x because G µν (x) is divergent at any x.
Therefore first one must renormalize the integrand on the r.h.s. of (2.35).
2) After this step of the renormalization the integrand on the r.h.s. of (2.35) becomes finite in the internal part of the integration region but remains singular on the boundary C so that the integral is still divergent. At this moment we will need the second step of the renormalization.
This two-step renormalization was already announced -see eq. (1.20). Note that in the case of refs. [26,27] both types of divergences were renormalized using one renormalization procedure based on analytical regularization (3.6) of formal divergent series: discrete JHEP04(2020)204 summation (3.5) of the spectral representation keeps interior and boundary divergences in one pot. But our method based on SC representation requires a separate treatment of interior and boundary divergences.
Thus we have a two-stage renormalization procedure. A detailed description of the two steps will be given below. But already now it makes sense to announce that the second step (renormalization of boundary divergences) will be implemented via a sort of analytical regularization for divergent integrals in SC representation. This SC version of the analytical regularization (applicable to arbitrary polygons C) is different from the analytical regularization of refs. [26,27] designed for rectangles and dealing with discrete series.
Similarly to the case of DF method, the analytical regularization based on SC mapping belongs to the class of last-minute regularizations. The main aim is to develop an efficient computational method for the calculation of f 2 (C) with arbitrary polygonal contours postponing the problem of a rigorous justification for later. The detailed structure of the regularization and renormalization procedure described in this work is limited to the case of the 2-loop contribution f 2 (C). However, the main ingredient of the method, SC mapping, could be of use also for the computation of higher orders of the large-size expansion with arbitrary polygonal contours. Function G µν (2.28) is divergent. Indeed, matrix element (2.29) has a singularity in the diagonal limit y → x: This divergence is eliminated by a renormalization of string tension σ 0 . Operationally this renormalization corresponds to the replacement of G µν (2.28) by its finite part and by the replacement of bare string tension σ 0 with renormalized string tension σ appearing in area law (1.1). After the replacement eq. (2.35) takes the form We use label ren, 1 in S 4 ren,1 because the integral on the r.h.s. still contains additional divergences:

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• boundary divergences which appear in K µν (x) when x approaches boundary C of region U (C), • extra cusp divergences appearing in polygonal regions U (C) when x approaches a vertex of the polygon.
Thus, the first step S 4 → S 4 ren,1 of our renormalization program (1.20) has made the integrand finite in the internal part of U (C) on the r.h.s. of (3.12) but the integral is still divergent because of boundary and cusp singularities of the integrand.

Definitions and conventions
In order to proceed, it is convenient to pass from Cartesian 2D coordinates x to complex coordinates ζ. Our conventions for this complexification are We use the metric tensor with components g ζζ = g ζ * ζ * = 0, (4.8) For symmetric tensor we have Now eq. (3.12) becomes In real and complex representations for the Green function we assume the equivalence of various notations representing the same coordinate state.
With the above conventions we can rewrite eq. (3.9) in the complex form: |ζ . (4.23)

Conformal mapping to semiplane
If one knows a conformal mapping of region U (bounded by Wilson contour C) to the upper complex semiplane then functions K ζζ , K ζζ * can be expressed via this conformal mapping. The computation is done in appendix E. The result is {z, ζ}, (4.24) (4.25) Here ζ is a complex coordinate in region U and z (ζ) is the conformal mapping from U to the upper complex z-semiplane. We use notation {z, ζ} for Schwarz derivative (D.1).

Reduction of S
where (4.27)

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Next we change the integration variable from ζ to z in integrals (4.27), (4.28). In expression (4.28) for I 2 we use property (D.3) of Schwarz derivative (4.29) Thus Here the integration runs over upper complex semiplane

SC mapping with finite vertices
In the case of polygons, conformal mapping ζ (z) is known as SC mapping. SC mapping ζ (z) for a polygon with n v vertices is defined by differential equation Parameters β k determine interior angles of the polygon: Angles θ k satisfy geometric conditions so that parameters β k obey constraints Sometimes it is convenient to use parameters For SC mapping (5.1), Schwarz derivative {ζ, z} (D.1) can be easily computed: (5.8) Now eqs. (4.30), (4.31) lead to (5.10) Thus (5.13)

SL(2,R) symmetry
It is well-known that linear fractional transformations with unit determinant (more exactly, with SL(2,R)/Z 2 ). For a given polygon its SC mapping is defined up to the freedom of these linear fractional transformations. In particular, this allows for the freedom of changing SC vertices z k z k = az k + b cz k + d . JHEP04(2020)204

SC mapping with one vertex at infinity
The freedom of linear fractional transformations (5.18) allows for taking one SC vertex to infinity. Let us put vertex z nv to infinity: When taking this limit, one should tune parameter A = A (z nv ) in eq. (5.1) so that Then SC equation (5.1) reduces to This equation does not contain parameter β nv but it still can be defined by eq. (5.6) and obeys condition (5.5) together with other parameters β k : The expression for Schwarz derivative (5.8) of mapping (5.21) can be obtained by taking limit (5.19) in eq. (5.8): (5.24) Now we apply general formulas (5.11)-(5.12) to SC mapping (5.21): (5.27) JHEP04(2020)204

SC vertex at infinity: pro and contra
Thus we have two versions of SC representation for I a . Representation (5.11) is based on SC mapping with finite vertices z k whereas representation (5.25) assumes that SC vertex z nv is taken to infinity. Both integrals (5.11), (5.25) are divergent. These integrals must be renormalized (by the second step S 4 ren,1 → S 4 ren of our renormalization program (1.20)) and our plan is to use analytical regularization. In principle, our procedure of analytical regularization can be formulated for both versions of SC mapping with the same final renormalized result for I ren a . But when it comes to the practical work, the SC representation with one vertex at infinity seems to be preferable. One advantage of this approach is obvious: keeping one SC vertex at infinity we significantly simplify all integrals. The second argument against SC mapping with finite vertices is that parameters β k must obey constraint (5.6). This constraint is critical at some steps of analytical regularization so that one meets the problem of analytical continuation in β k under constraint (5.6) which leads to some technical (solvable but annoying) problems.
As was discussed in section 5.2, in the case of SC mapping with all SC vertices being finite we have a powerful constraint of SL(2,R) symmetry on allowed renormalization procedures. Our choice to construct renormalization in terms of SC mapping with one vertex at infinity makes SL(2, R) symmetry rather implicit. We are left with the freedom of choice: which vertex of the polygon is associated with the SC vertex taken to infinity. The result of the renormalization must be independent of this choice.
where n ≥ 2, (6.2) P (z, z * ) is a polynomial and z k (1 ≤ k ≤ n) are different real numbers: We use label nonren in M (n)nonren P because the integral on the r.h.s. of eq. (6.1) may be divergent.
Expression (5.25) can be rewritten for I 1 in the form where M (n)nonren 1 is M (n)nonren P with trivial constant polynomial P (z, z * ) ≡ 1.

From divergent integrals to meromorphic functions
The suggested scheme of analytical renormalization raises many questions if one wants a mathematically impeccable implementation of this program. On the other hand, in practical calculations one can typically (but not always) perform formal mathematical operations with divergent integrals without wasting time for a careful justification of these naive formal manipulations. A careful mathematical theory of functions M (n) P for arbitrary n and arbitrary polynomials P can be constructed. But in the framework of the current work devoted to triangular Wilson loops we need functions M (n) P only for the special case n = 2 and for rather simple polynomials P . In this simple case the integrals can be computed explicitly and the problem of analytical continuation can be solved by using these explicit expressions rather than invoking the general theory of functions M (n) P with arbitrary n and arbitrary P .
Therefore the complete description of the theory of functions M (n) P will be presented in a separate work devoted to Wilson loops with arbitrary polygonal contours. Here only a list of main results will be given without proofs and derivations.
When one starts with the construction of the rigorous theory of functions M The brief answers are: 1) The region of parameters {γ k } n k=0 where integral (6.9) is convergent is non-empty for any fixed {z k } n k=1 and for any fixed P .
2) Starting from function M (n) P ({γ k } n k=0 , {z k } n k=1 ) defined by integral (6.9) in the convergence region and performing analytical continuation in C n+1 space of parameters {γ k } n k=0 (at fixed {z k } n k=1 and fixed P ), one arrives at meromorphic function M  A careful proof of these statements for arbitrary n ≥ 2 (i.e. for polygons with arbitrary number of vertices n v ≥ 3) requires some effort. We postpone the proof till a separate work JHEP04(2020)204 devoted to arbitrary polygons. In the case of triangles, i.e. for n = n v − 1 = 2 many of the above general properties can be seen directly from explicit expressions computed in appendices G, H.

Functions Π (n) P
The above discussion of the analytical regularization proceeded in terms of functions M ) . (6.14) Using eq. (6.9), we find that functions Π (n) P are given by analytical continuation in 7 Calculation of f 2 (C) for triangles

Case of triangles
In the case of triangles eqs. (6.16), (6.17) take the form

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Below we will see that function Π (2) 1 (α, {γ 1 , γ 2 } , {z 1 , z 2 }) with arbitrary arguments can be easily computed analytically, which will immediately lead to the result for I ren 1 . As for I ren 2 , one can compute Π In appendix G we derive expression (G.10) for function Π 1 . Using this result in our expression (7.1) for I ren 1 , we find Next we express parameter Ã via area S (C) of the triangle using eq. (F.11) and simplify the result using expression (F.6) for β 3 :

Calculation of I ren 2
Now we turn to the computation of I ren 2 , using its expression (7.2) via Π (2) P 2 . Remember that polynomial P 2 can be expressed via polynomial T 2 according to eq. (6.6). Let us compute polynomial T 2 (z) (6.6) for triangles (n v = 3) with z k given by eq. (7.3). We have

Final result
Our final results for I ren 1 (7.6) and I ren 2 (7.23) are symmetric with respect to permutations of parameters β 1 , β 2 , β 3 whereas the intermediate steps of the computation were asymmetric. The symmetry of the final result is a good test of the consistency of our renormalization procedure. In fact, it is a test of the compatibility of our analytical renormalization with SL(2, R) symmetry discussed in section 5.4. In our case of triangular contours, the compatibility of the renormalization procedure with SL(2, R) symmetry was demonstrated via the explicit computation of I ren 1 and I ren 2 . In principle, one can prove that our analytical renormalization respects SL(2, R) symmetry for arbitrary polygons but this is a subject of a separate work devoted to arbitrary polygons.
Note that in the case of triangles general polygonal constraint (5.5) is enhanced to 0 < β k < 1.

A.1 Heat-kernel expansion for Laplace operator
For Laplace operator acting in the region bounded by contour C one has the following heat-kernel expansion [13,[40][41][42][43][44]: Here θ k are interior angles of polygon C, n v (C) is the number of vertices and For function ξ(θ), an integral representation was derived by M. Kac in ref. [41]. Explicit expression (A.3) for ξ(θ) was obtained by D.B. Ray (the derivation is described in ref. [45]).

A.2 Renormalization of Laplace determinants
The determinant of Laplace operator in the proper-time regularization has a divergence in the limit τ → +0 which is controlled by heat-kernel expansion (A.1): In ζ-regularization method, one first defines the regularizing ζ-function at Re s > 1 Then one performs the analytical continuation in s and computes the derivative Laplace determinant in the ζ-regularization is given by and has the property where δ (C) is given by eq. (A.2).

B 1-loop EST corrections
According to (2.13) terms f ln (C) ln λ + f 0 (C) of expansion (1.5) arise from the 1-loop EST contribution. These terms are generated by Laplace determinant coming from Gaussian integral (2.19) so that roughly speaking However, precise expressions are sensitive to certain technical subtleties because both MGT and EST have ultraviolet divergences which must be renormalized in a consistent way matching MGT and EST. The explicit expression for f ln (C) can be extracted from the naive result (B.1) by comparing eq. (B.1) with eq. (A.9): where ξ (θ) given by eq. (A.3).

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Now let us turn to f 0 (C). Wilson loop W (C) is not a renormalization invariant quantity so that the single functional f 0 (C) is not a physical quantity and any attempt to write an explicit expression for single functional f 0 (C) will result in an ugly combination of scheme dependent parameters. However, simple elegant formulas can be written for linear combinations (2.15) associated with renormalization invariant combinations of Wilson loops -see eqs. (B.13) and (C.29).
The simplest examples of renormalization invariant ratios are combination (1.13) and Creutz ratios Here L (C a ) is the perimeter of contour C a . In addition, one must obey a vertex-balance condition. Roughly speaking, each vertex angle value must appear in the numerator of (B.4) as many times as in the denominator. In order to make a more careful formulation of this constraint, let us first define Θ = {θ A } as a set of all vertex angle values appearing in polygons C a so that each θ A is included in Θ only once (whatever often it may appear in polygons C a ): (B.7) Next, let p aA be the number of occurrences of angle value θ A among vertices of polygon C a (so that possible values are p aA = 0, 1, 2, . . .). Then the vertex-balance constraint reads where θ ai with i = 1, 2, 3, . . . , n v (C a ) are interior angles of polygon C a and φ (θ) is an arbitrary function of angle θ.

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Sometimes one also needs area-balance condition It is well-known that Laplace determinants in 2D regions can be computed using the conformal anomaly [46][47][48][49]. In order to compute Laplace determinant Det (−∆ (C)) for JHEP04(2020)204 the region bounded by contour C, one has to construct a conformal mapping of this region to some standard region (semiplane or circle). In the case of polygons C this conformal mapping is known as Schwarz-Christoffel (SC) mapping. Thus combining the conformal anomaly and SC mapping, one can compute Laplace determinants for arbitrary polygons. However, on this way one must solve two problems: • the anomaly-based representation for Laplace determinants has an integral form and this integral must be computed, • this integral representation has cusp divergences which must be renormalized.
These two problems were successfully solved by E. Aurell and P. Salomonson in ref. [39] where the renormalized functional determinant (in ζ-regularization) of two-dimensional Laplace operator with Dirichlet boundary condition for an arbitrary polygonal region was computed and expressed via parameters of SC mapping.
In In order to simplify the extraction of final expressions for Laplace determinants from ref. [39], we provide a small dictionary relating the notation used in the current work (l.h.s.) and the notation adopted in ref. [39] (r.h.s. with label AS).
Polygonal boundary C of the region where Laplace operator acts: Area of the region bounded by contour C: Complex coordinate in the polygon plane: Function Z 1−β (0) (do not confuse it with quantity Z P (0) appearing in eq. (C.5)) playing an important role in ref. [39] but irrelevant for our work: For angles of polygons ref. [39] uses the same parameters θ k , β k , α k as in this work (but with Greek indices) -see eqs. (5.2), (5.7).

C.1.2 SC mapping from unit circle to polygon
One should keep in mind that ref. [39] uses Schwarz-Christoffel (SC) mapping of the unit circle |u| ≤ 1 in complex u-plane to the polygon in complex ζ-plane (the original notation of ref. [39] uses z instead of ζ). Here e iφ k are points on the boundary of the unit circle which are mapped to vertices ζ k of polygon C. λ 0 is a real constant controlling the size of the polygon.

C.1.3 Result for Laplace determinant
The general result for the determinant of Laplace operator defined in polygon C with Dirichlet boundary condition in ζ-regularization (A.6)-(A.8) is given by eq. (62) of ref. [39]: The r.h.s. contains circular SC parameters λ 0 , {φ k }, {β k } and function h (β) (C.7) whose explicit form plays no role for our work.

C.1.4 SC mapping: from unit circle to semiplane
One can easily establish a connection between semiplane version of SC mapping (5.1) and unit-circle version of SC mapping (C.8). Indeed, one can map the unit circle in the u-plane to the upper semiplane of complex variable z using linear-fractional transformation with the inverse mapping Transformation (C.10) maps points e iφ k on the boundary of the unit circle in the u-plane to points z k on the real axis of the z-plane 14) one finds that in terms of the z-semiplane parametrization, SC mapping (C.8) takes form (5.1).
(C. 15) It follows from eq. (C.14) that Combining the last two equations and using constraint (5.6), one can derive relation Now eq. (C.9) takes the form This provides an expression for Laplace determinant via parameters z k , β k , A of SC mapping in semiplane form (5.1). The r.h.s. of (C.18) contains function h (β) (C.7) which is computed in ref. [39] but cancels in our final formulas.

C.1.6 Taking one SC vertex to infinity
Sometimes it is convenient to keep one SC vertex at infinity and to work with SC representation (5.21). Taking  h (β j ) + 1 12 In the case of SC mapping (5.21) with vertex z nv at infinity we derive from eq. (C. 19) (C.23) Now we can compute the combination of Laplace determinants appearing on the r.h.s. of eq. (B.13). For each polygon C a appearing in eq. (B.13) we introduce SC mapping (5.1) Remember that this result was derived assuming vertex-balance condition (B.8). The primary subject of this paper is the computation of 2-loop correction f 2 (C) for triangles. For completeness it makes sense to compute also 1-loop terms f ln (C) and f 0 (C) for triangles. Quantity f ln (C) is given by simple formula (B.2) which can be used for any polygon including triangles. As for f 0 (C), we must compute the r.h.s. of eq. (C.28) assuming balance conditions. If all polygons C a are triangles and vertex-balance condition (B.8) holds then one can derive relation

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One can wonder whether using only triangle contours C a we can satisfy balance condition in a non-trivial way so that N a=1 m a f 0 (C a ) = 0 (C. 35) i.e. without using similar triangles like in eq. (1.13). The answer to this question is positive but the examples are rather exotic because vertex-balance constraint (B.8) must be combined with the angle sum constraint (5.4) for each triangle C a . An example of the solution of this problem can be constructed using Wilson loops w (n a1 , n a2 , n a3 ) for triangles C a with interior angles θ ak = π 9 n ak (C. 37) with angle sum rule n a1 + n a2 + n a3 = 9.

C.2.3 Derivation
Now we turn to the derivation of eq. (C.30). In principle, one can derive eq. (C.30) from general equation (C.28). On the other hand, one can profit from the explicit expression computed in ref. [39] for Laplace determinant with triangle contour C. One can read this explicit expression from eqs. (68) and (69) of ref. [39]. In the original notation of ref. [39] the result reads where parameters α k are defined by eq. (5.7) and label T stands for triangle. Using relations of appendix C.1.1 (with P = T ), we can translate eq. (C.40) to the notation adopted in the current work: . (C.42)

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Green function for the semiplane can be constructed by the well-known image method from the Green function on the plane: Here w R is the reflection of point w with respect to w 1 -axis: This result can be rewritten using complex coordinates on the upper semiplane of complex z and where ζ = f (z) , (E.10) and f is the conformal mapping from the semiplane to U corresponding to mappingf (E.1) formulated in terms of points with real coordinates. Let z = φ (ζ) (E.12) be the inverse mapping. Then eq. (E.9) takes the form

F SC mapping for triangles
In this appendix we apply SC equation (5.21) with one vertex at infinity to the case of triangles and derive some useful relations between geometric parameters and SC parameters. The case of triangles corresponds to setting n v = 3 in eq. This result can be easily generalized to the case of arbitrary z 1 , z 2 : . Integrals of this type appear in various problems of mathematical physics. In appendix B of ref. [50] this integral was computed (in a slightly different notation assuming z 2 = 1) using advanced techniques including a reduction to higher hypergeometric function 3 F 2 .
Below an elementary calculation of integral (G.2) is sketched. Using the representation

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H Calculation of Π (2) P In this appendix we compute function Π (2) P (α, {γ 1 , γ 2 } , {z 1 , z 2 }) for the special value of argument α = 1. For simplicity we choose z 1 = 0, z 2 = 1. We will perform a formal calculation ignoring divergences of some intermediate integrals but still leading to the correct final result. A rigorous justification of this formal calculation requires methods whose discussion makes sense in the context of the theory of functions Π (n) P with arbitrary n, i.e. in the case of the theory of analytical regularization for arbitrary polygons C.
We start from the integral over complex plane C Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.