Meromorphic Continuation of Koba-Nielsen String Amplitudes

The string amplitudes at the tree level, also called Koba-Nielsen string amplitudes, were introduced and studied in the 60s by Veneziano, Virasoro, Koba and Nielsen, among others. Since then, theoretical physicists have used them as formal objects. In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes are bona fide integrals, which admit meromorphic continuations when considered as complex functions of the kinematic parameters. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities.


Introduction
In the recent years, scattering amplitudes, considered as mathematical structures, have been studied intensively, see e.g. [1], [18] and the references therein. The main motivations driving this research are, from one side, the development of more efficient methods to calculate amplitudes, and on the other side, the existence of deep connections with many mathematical areas, among them, algebraic geometry, combinatorics, number theory, p-adic analysis, etc., see e.g. [4], [7], [8], [11], [14], [30], [32], [33], and the references therein. The present work is framed in the 'emerging idea' that scattering amplitudes are local zeta functions in the sense of Gel'fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc., and also it continues our investigation of the connections between string amplitudes at the tree level and local zeta functions [7], [8].
In this article we establish, in a rigorous mathematical way, that the Koba-Nielsen string amplitudes are bona fide integrals, which admit meromorphic continuation, as complex functions, in the kinematic parameters. We express the Koba-Nielsen amplitudes as linear combinations of multivariate local zeta functions, and, by using embedded resolution of singularities (Hironaka's theorem [23]), we show that all these local zeta functions are holomorphic in a common domain, and then we use the fact that the local zeta functions admit meromorphic continuations. Since Hironaka's theorem is valid over any field of characteristic zero, we are able to regularize the Koba-Nielsen amplitudes defined over R, C, or Q p , the field of p-adic numbers, at the same time. The result in the p-adic case is already known, see [7]. Nowadays, there are algorithms for computing embedded resolution of singularities, see e.g. [9], but, in general, their complexity is too big for practical purposes. In the present article however, we work in the special framework of so-called hyperplane arrangements, for which the results presented could be transformed in an algorithm for computing Koba-Nielsen amplitudes.
We denote by K a local field of characteristic zero, and set f := (f 1 , . . . , f m ) and s := (s 1 , . . . , s m ) ∈ C m , where the f i (x) are non-constant polynomials in the variables x := (x 1 , . . . , x n ) with coefficients in K. The multivariate local zeta function attached to (f , Θ), where Θ is a test function, is defined as when Re(s i ) > 0 for all i, and where n i=1 dx i is the normalized Haar measure of (K n , +). These integrals admit meromorphic continuations to the whole C m , [25], [26], [31], see also [22], [27]. In the 60s, Weil studied local zeta functions, in the Archimedean and non-Archimedean settings, in connection with the Poisson-Siegel formula. In the 70s, Igusa developed a uniform theory for local zeta functions in characteristic zero [25], [26], see also [31], [40], [41]. In the p-adic setting, the local zeta functions are connected with the number of solutions of polynomial congruences mod p l and with exponential sums mod p l [15]. More recently, Denef and Loeser introduced the motivic zeta functions which constitute a vast generalization of p-adic local zeta functions [16].
In the case K = R and m = 1, the local zeta functions were introduced in the 50s by Gel'fand and Shilov. The main motivation was that the meromorphic continuation of Archimedean local zeta functions implies the existence of fundamental solutions (i.e. Green functions) for differential operators with constant coefficients. This fact was established, independently, by Atiyah [3] and Bernstein [6]. It is important to mention here that, in the p-adic framework, the existence of fundamental solutions for pseudodifferential operators is also a consequence of the fact that the Igusa local zeta functions admit a meromorphic continuation, see [42,Chapter 5], [29,Chapter 10]. This analogy turns out to be very important in the rigorous construction of quantum scalar fields in the p-adic setting, see [33] and the references therein.
The connections between Feynman amplitudes and local zeta functions are very old and deep. Let us mention that the works of Speer [34] and Bollini, Giambiagi and González Domínguez [11] on regularization of Feynman amplitudes in quantum field theory are based on the analytic continuation of distributions attached to complex powers of polynomial functions in the sense of Gel'fand and Shilov [22], see also [4], [5], [10], [32], among others. The book [22], which is one of the main sources for the 'iǫ regularization method' widely used, was written before the establishing of Hironaka's theorem [23]. After the work of Atiyah, Bernstein and Igusa, among others, the iǫ regularization method was substituted by the embedded resolution of singularities technique, see [25], [26]. However, this method is not widely used by theoretical physicists. In [39] Witten discusses the classical iǫ regularization method for string amplitudes, in this article, we present a rigorous regularization of the Koba-Nielsen string amplitudes using the 'modern iǫ regularization method'. Take N ≥ 4, and complex variables s 1j and s (N −1)j for 2 ≤ j ≤ N − 2 and s ij for denotes the total number of indices ij. In this article we introduce the multivariate local zeta function We have called integrals of type (1.1) Koba-Nielsen local zeta functions.
We will show that these functions are bona fide integrals, which are holomorphic in an open neighborhood of a part of the diagonal of C d , which is contained in the set −2 N −2 < Re(s ij ) < −2 N . Furthermore, they admit meromorphic continuations to the whole C d , see Theorems 5.1 and 3.1. We give a detailed proof in the case K = R, see Theorem 3.1 and Section 4; this proof can be easily extended to an arbitrary local field K of characteristic zero, see Section 5.
The Koba-Nielsen open string amplitudes for N-points over K are formally defined as A central problem is to know whether or not integrals of type (1.2) converge for some values k i k j ∈ C. Our Theorem 5.1 allows us to solve this problem. We use the integrals Z K (s) denotes the meromorphic continuation of (1.1). By Theorem 5.1, the A (N ) K (k) are well-defined meromorphic functions in the variables k i k j ∈ C, which agree with amplitudes (1.2) when they converge.
The string amplitudes were introduced by Veneziano in the 60s, [35], further generalizations were obtained by Virasoro [36], Koba and Nielsen [28], among others. The p-adic string amplitudes emerged in the 80s in the works of Freund and Olson [20], Freund and Witten [21], see also [12], Frampton and Okada [19], and Volovich [37]. Since the 60s the string amplitudes at the tree level have been used as formal objects in many physical calculations. In [7], it was established in the p-adic setting and by using techniques of Igusa's local zeta functions that the Koba-Nielsen amplitudes are bona fide integrals. In this article this result is extended to an arbitrary local field of characteristic zero. Finally, it is interesting to mention that these amplitudes can be studied in a uniform way on any local field of characteristic zero, see Theorem 5.1, this is consistent with Volovich's conjecture asserting that the mathematical description of physical reality must not depend on the background number field, see [38].

Multivariate Local Zeta Functions and Embedded Resolution of Singularities
. , x n ] be non-constant polynomials; we denote by D := ∪ m i=1 f −1 i (0) the divisor attached to them. We set f := (f 1 , . . . , f m ) and s := (s 1 , . . . , s m ) ∈ C m . For each Θ : R n → C smooth with compact support, the multivariate local zeta function attached to (f , Θ) is defined as when Re(s i ) > 0 for all i. Integrals of type (2.1) are analytic functions, and they admit meromorphic continuations to the whole C m , see [31], [26], [25]. By applying Hironaka's resolution of singularities theorem to D, the study of integrals of type (2.1) is reduced to the case of monomials integrals, which can be studied directly, see e.g. [31], [26], [25].
Theorem 2.1 (Hironaka, [23]). There exists an embedded resolution σ : where η (y) and the ε f j (y) belong to O × X,b , the group of units of the local ring of X at b.
There are two kinds of submanifolds E i , i ∈ T . Each blow-up creates an exceptional variety E i , the image by σ of any of these E i has codimension at least two in R n . The other E i are the so-called strict transforms of the irreducible components of D.
When using Hironaka's resolution theorem, we will identify the Lesbesgue measure n i=1 dx i with the measure induced by the top differential form dx 1 ∧ . . . ∧ dx n in R n . For a discussion on the basic aspects of analytic manifolds and resolution of singularities, the reader may consult [26,Chapter 2]. More generally, Hironaka's resolution theorem is valid over any field of characteristic zero, in particular over the local fields R, C, the field of p-adic numbers Q p , or a finite extension of Q p .
The resulting monomial integrals are then handled by the following lemma, which is an easy variation of well-known results, see e.g. [ where 1 ≤ r ≤ n, for each i the a j,i are integers (not all zero) and b i is an integer, and Φ (y, s 1 , . . . , s m ) is a smooth function with support in the polydisc Combining Theorem 2.1 and Lemma 2.1, the precise conclusion is as follows.
. , x n ] be non-constant polynomials and Θ : R n → C a smooth function with compact support, to which we associate the multivariate local zeta function Z Θ (f , s) as in (2.1). Fix an embedded resolution σ :

s) is convergent and defines a holomorphic function in the region
(ii) Z Θ (f , s) admits a meromorphic continuation to the whole C m , with poles belonging to

and the possible poles of its meromorphic continuation belong to the set
The above theorem is a consequence of the work of many people: Gel'fand (I. M. and S. I.), Berstein, Atiyah, Igusa, Loeser, as far as we know. We will use this result, as well as Lemma 2.1, along this article; the formulation that we are giving here is the one we require. The formulation of Lemma 2.1 will be crucial for dealing with certain non-classical local zeta functions that occur in Section 4.2.

Local Zeta Functions of Koba-Nielsen Type
We consider R N −3 as an R-analytic manifold, with N ≥ 4, and use {x 2 , . . . , x N −2 } as a coordinate system. In addition, we take and use N −2 i=2 dx i to denote the measure induced by the top differential form dx 2 ∧. . .∧dx N −2 .

Definition 1. A Koba-Nielsen local zeta function is defined to be an integral of the form
For later use in formulas, it will be convenient to put also s ij = s ji for any occurring {i, j}.
For simplicity of notation, we will put R N −3 instead of R N −3 D N in (3.2), and similarly in other such integrals. In order to regularize the integral (3.2), we will use a partition of R N −3 constructed using a smooth function χ : R → R satisfying for some fixed positive ǫ sufficiently small. The existence of such a function is well-known, see e.g. [ with the convention that i∈∅ · ≡ 1.
By using this partition of the unity, we have In the case I (s) is a classical multivariate Igusa local zeta function (since then ϕ I (x) has compact support). These integrals are holomorphic functions in a region including Re (s ij ) > 0 for all ij, and they admit meromorphic continuations to the whole C d , see Theorem 2.2.
In the case I = {2, . . . , N − 2}, by changing variables in (3.6) as , we have that supp χ ⊆ − 1 2 , 1 2 and χ ∈ C ∞ (R). Now setting ϕ I (x) := i ∈I χ (x i ) i∈I χ (x i ), and where D I is the divisor defined by the polynomial   Then a short argument will yield our main result.

Theorem 3.1. The Koba-Nielsen local zeta function Z (N ) (s) is a holomorphic function in an open neighborhood of a part of the diagonal of C d , which is contained in the set
By Theorem 2.2(iii), we have that Z (N ) I (s) is holomorphic in the half-space where {(N f,i , v i ) ; i ∈ T } are the numerical data of an embedded resolution σ of D N . We will explain how to construct such a resolution and obtain that this minimum value is 2 N −2 . Since D N is a so-called hyperplane arrangement (its irreducible components are all hyperplanes), there is well-known and straightforward way to construct an embedded resolution σ. First, note that the locus of D N where it is not a normal crossings divisor, i.e., not locally monomial as in Lemma 2.1, consists of the points with at least two coordinates equal to 0, at least two coordinates equal to 1, or at least three equal coordinates. The standard algorithm is to blow up consecutively in relevant centres of increasing dimension contained in that locus, until the total inverse image of D N becomes a normal crossings divisor.
For readers who are not familiar with these notions, we will treat explicitly the first blowups of such a resolution, presented as explicit change of variables operations, simplifying the original integral Z Proof. First, we consider an adequate partition of the unity subordinate to the compact set supp ϕ I . Let P be the set of 2 N −3 points p in R N −3 with all coordinates equal to 0 or 1. We take smooth functions Ω p , p ∈ P , such that each Ω p is supported in a neighborhood of p that is disjoint from some neighborhood of any other point of P , and such that ϕ I (x) = p∈P Ω p (x) for x ∈ supp ϕ I . Then (1) We start by improving the situation around the origin (p = 0). We remark that the factors |1 − x i | s in the integrand of Z Without loss of generality, we may assume that i 0 = 2. Then Then the contribution to Z (N ) Ω 0 (s) in the chart (4.1), with i 0 = 2, takes the form where the factor g(u, s) is invertible on the support of Ω 0 • σ 0 (and can be neglected from the point of view of convergence and holomorphy). Up to such negligible factors, further blowing-ups/change of variables, ultimately leading to monomial integrals as in Lemma 2.1, will not affect the variable u 2 anymore. The smooth hypersurface, given by u 2 = 0, corresponds to a submanifold E 0 (as in Theorem 2.1) with numerical data Important to note is that N f,0 is equal to the multiplicity of D N at the origin, which, in the case of a hyperplane arrangement, is just the number of hyperplanes containing the origin. Also, v 0 is equal to the codimension of the origin in R N −3 . This is a general fact: for any submanifold E i as in Theorem 2.1, created by a blow-up with centre Y , we have that N f,i is equal to the multiplicity of D N at (a generic point of ) Y , being the number of hyperplanes containing Y , and that v i is equal to the codimension of Y in the ambient space. (1.
2) The next blow-ups, in centres intersecting E 0 , are at those centres of dimension 1 whose image by σ 0 contains the origin. There are two such centres visible in the present chart. The first one is u 3 = . . . = u N −2 = 0 (this is the so-called strict transform of the line Then in this chart the contribution to Z (N ) Ω 0 (s) takes the form where the factor h(w, s) can be neglected from the point of view of convergence and holomorphy. The smooth hypersurface, given by w 3 = 0, corresponds to a submanifold E 1 with numerical data The second centre is 1 = u 3 = . . . = u N −2 (the strict transform of the line . After a change of variables u ′ i = u i − 1 for i = 3, . . . , N − 2, the calculation of this blow-up is the same as for the first centre. It gives rise to a submanifold E ′ 1 with the same numerical data ( We continue this way, blowing up in centres of increasing dimension, ending with blow-ups in centres of dimension N − 5 of two possible types, for instance corresponding to x N −3 = x N −2 = 0 and x N −4 = x N −3 = x N −2 , respectively, yielding submanifolds with numerical data (3,2).
Note that, up to now, the smallest quotient of numerical data that we obtained is indeed 2 N −2 .
(2) All other points p = (p 2 , . . . , p N −2 ) ∈ P , that are needed as centres of blow-ups, have at least one coordinate equal to 1 (and still at least two coordinates equal to 0 or at least two coordinates equal to 1), say p i = 1 for i ∈ J = ∅ and p i = 0 for i / ∈ J. For simplicity, we switch to the coordinate system y, given by y i = x i − 1 for i ∈ J and y i = x i for i / ∈ J, in order to view p as the new origin. Then where g p (y, s) is an invertible function on the support of Ω p (y), smooth in y and holomorphic in s. The divisor D p attached to (4.6) is given by the zero locus of It can be considered as a subarrangement of the arrangement D 0 . Hence, an embedded resolution of D p can be constructed by (part of) the same blow-ups we used to construct the embedded resolution of D 0 . Take any centre of blow-up Z i , of codimension v i , occurring in those resolutions, leading to the exceptional submanifold E i . Say n i and n ′ i are the number of hyperplanes in D 0 and D p , respectively, containing Z i ; then clearly n ′ i ≤ n i . Hence the numerical data of E i , considered in the embedded resolution of D 0 and D p , are (n i , v i ) and , all new quotients of numerical data are again at least 2 N −2 .
(3) The numerical data of (the strict transforms E i of) the components of D are all equal to (1,1) We note that, in the proof above, we assumed implicitly that N ≥ 6. When N = 4, the claim is trivial, and when N = 5, we only need to blow up at the points (0, 0) and (1, 1). A similar remark applies to the proof of the next case.
Next, we construct an embedded resolution of D I ∪ D I,0 . A crucial observation is that any blow-up with centre not contained in D I,0 will induce a condition that already appeared in the construction of the resolution σ in the proof of Proposition 4.1, i.e., a condition of the form Re(s) > − 2 N −2 or weaker. We could make this lower bound more precise, depending on the size of I, but this would not affect the end result of Theorem 3.1.
We now show that the blow-ups with centre in D I,0 induce as strongest condition Re(s) < − 2 N . In a small enough neighborhood of D I,0 , we can write the integrand of Z (N ) where the factor g(x, s) is invertible on the support of ϕ I (x) (and can be neglected from the point of view of convergence and holomorphy). After a permutation of the indices, we may assume that I = {2, . . . , N − 2} \ {2, . . . , l} with l ≥ 2. Then D I,0 is given by 2≤i≤l x i = 0. When l = 2, no blow-up with centre in D I,0 is needed. If l ≥ 3, we start by performing a blow-up τ with centre at x 2 = . . . = x l = 0, for instance in the chart x 2 = u 2 , x i = u i u 2 for i = 3, . . . , l, and x i = u i for i = l + 1, . . . , N − 2. This centre is contained in the l − 1 hyperplanes x i = 0, 2 ≤ i ≤ l, and in the corresponding Hence the power of |u 2 | in the pullback of the integrand is Next, we blow up with at centres of one dimension higher, being (the strict transforms of) the relevant linear spaces whose image by τ contains Continuing this way, we end with the condition Re(s) < − 2 2N −5 . The strongest of all conditions of this form occurs when l = N − 2 in (4.9), and is indeed Re(s) < − 2 N , which is a stronger condition than (4.7).

Proof of Theorem 3.1.
Proof. From Propositions 4.1 and 4.2, we already know that the Koba-Nielsen local zeta function Z (N ) (s) is holomorphic in some neighborhood of the points s = (s, . . . , s) contained in This is enough to imply meromorphic continuation to the whole C N −3 .

Local Zeta Functions of Koba-Nielsen type Over Local Fields
The Koba-Nielsen local zeta functions introduced in Definition 1 can be naturally defined over arbitrary local fields of characteristic zero, i.e., R, C, or finite extensions of Q p , the field of p-adic numbers, and the proof of the main theorem can be extended easily to the case of local fields different from R. We denote the corresponding local zeta functions as Z (N ) K (s) to emphasize the dependency on K. Note that the p-adic case was already treated in [7]- [8] through an alternative method, only available in that case, called Igusa's stationary phase formula.
5.1. Local fields. We take K to be a non-discrete locally compact field of characteristic zero. Then K is R, C, or a finite extension of Q p , the field of p-adic numbers. If K is R or C, we say that K is an R-field, otherwise we say that K is a p-field.
For a ∈ K, we define the modulus |a| K of a by the rate of change of the Haar measure in (K, +) under x → ax for a = 0, 0 for a = 0.
It is well-known that, if K is an R-field, then |a| R = |a| and |a| C = |a| 2 , where |·| denotes the usual absolute value in R or C, and, if K is a p-field, then |·| K is the normalized absolute value in K.
We now take K to be a p-field. Let R K be the valuation ring of K, P K the maximal ideal of R K , and K = R K /P K the residue field of K. The cardinality of the residue field of K is denoted by q, thus K = F q . For z ∈ K, ord (z) ∈ Z ∪ {+∞} denotes the valuation of z, and |z| K = q −ord(z) . We fix a uniformizing parameter p of R K , i.e., a generator of P K .
We fix a set S K ⊂ R K of representatives of the residue field K. We assume that 0 ∈ S K . Any z ∈ K\ {0} admits a power expansion of the form where m ∈ Z, the z k belong to S K , and z 0 = 0. The series (5.1) converges in the norm |·| K .

5.2.
Multivariate Local Zeta Functions: General Case. If K is a p-field, resp. an R-field, we denote by D(K n ) the C-vector space consisting of all C-valued locally constant functions, resp. all smooth functions, on K n , with compact support. An element of D(K n ) is called a test function. The multivariate local zeta function attached to (f , Θ), with Θ ∈ D(K n ), is defined as Integrals of type (5.2) are analytic functions, and they admit meromorphic continuations to the whole C m , see [25], [26], [27], [31]. By applying Hironaka's resolution of singularities theorem to D K , the study of integrals of type (5.2) is reduced to the case of monomials integrals, which can be studied directly, see e.g. [31], [26], [25].
Lemma 5.1. Let Φ (y, s 1 , . . . , s m ) be a test function with support in the polydisc {y ∈ K n ; |y i | < 1, for i = 1, . . . , n} , when K is an R-field, and with support p e R n K (e ∈ Z) when K is a p-field, which is holomorphic in s 1 , . . . , s m . Consider the integral where 1 ≤ r ≤ n, for each i the a j,i are integers (not all zero) and b i is an integer. Set K (s) is holomorphic in the solution set ∩ I H(I), see (3.8), in C d , and it has a meromorphic continuation to the whole C d . If K is an R-field, the poles belong to ∪ I P(I), see (3.9), where now t ∈ 1 2 N for K = C. If K is a p-field, then this meromorphic continuation is a rational function in the variables q −s ij , with poles having real parts belonging to The proof of Theorem 5.1 is a slight variation of the proof of Theorem 3.1. We just indicate the required modifications. The first step is to express Z (N ) K (s) as a finite sum of multivariate local zeta functions, see (3.5). This requires introducing an analogue of the functions χ, see (3.3), and ϕ I , see (3.4). We first define the analogue of χ in the complex case. We recall that an element of D(C n ) is a C ∞ function in the variables z 1 , z 1 , . . . , z n , z n (or in Re(z 1 ), Im (z 1 ) , . . . , Re(z n ), Im (z n )). We pick, for z = x + iy (x, y ∈ R), where χ is defined as in (3.3). Then χ C is a C ∞ function in the variables x, y satisfying We now define the function ϕ I as in (3.4). In the p-field case, we use Now the proof follows line by line the one given for Theorem 3.1. This is possible because, for any K, all the required blow-ups and centres are defined over the field of rational numbers.