p-Adic open string amplitudes with Chan-Paton factors coupled to a constant B-field

Abstract We establish rigorously the regularization of the p-adic open string amplitudes, with Chan-Paton rules and a constant B-field, introduced by Goshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematics parameters, the B-field and the Chan-Paton factors. We show that these integrals admit meromorphic continuations in the kinematic parameters, this result allows us to regularize the Goshal-Kawano amplitudes, the regularized amplitudes do not have ultraviolet divergencies. Due to the need of a certain symmetry, the theory works only for prime numbers which are congruent to 3 modulo 4. We also discuss the limit p → 1 in the noncommutative effective field theory and in the GhoshalKawano amplitudes. We show that in the case of four points, the limit p → 1 of the regularized Ghoshal-Kawano amplitudes coincides with the Feynman amplitudes attached to the limit p → 1 of the noncommutative Gerasimov-Shatashvili Lagrangian.


I. INTRODUCTION
The deep connections between p-adic analysis and physics are a natural consequence of the emergence of ultrametricity in physics, which means the occurrence of ultrametric spaces in physical models, see e.g. [1][2][3][4][5][6][7][8][9] and the references therein. The existence of a Planck length implies that the spacetime considered as a topological space is completely disconnected, the points (which are the connected components) play the role of spacetime quanta. This is precisely the Volovich conjecture on the non-Archimedean nature of the spacetime below the Planck scale, [2,3], [8,Chapter 6]. On the other hand, the paradigm in physics of complex systems (for instance proteins) asserting that the dynamics of a such system can be modeled as a random walk on the leaves on a rooted tree (a finite ultrametric space), which is constructed from the energy landscape. Mean-field approximations of these models drive naturally to models involving p-adic numbers, see e.g. [7,[10][11][12], and the references therein.
In string theory, the scattering amplitudes are obtained integrating over the moduli space of Riemann surfaces. Even for tree-level amplitudes (on the sphere for closed strings and on the disk for open strings) N-point amplitudes are difficult to compute beyond four points.
Moreover, the convergence region of these integrals is not evident by itself [29,30]. Recently, in [31] was established 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.
String theory with a p-adic world-sheet was proposed and studied by the first time in [32]. Later this theory was formally known as p-adic string theory. The Adelic scattering amplitudes which are related with the Archimedean ones were studied in [33]. The treelevel string amplitudes were explicitly computed in the case of p-adic string world-sheet in [34] and [35]. These amplitudes can be formally obtained from a suitable action using general principles [36]. A general treatment starting by describing a discrete field theory on a Bruhat-Tits tree and obtaining the tree-level string amplitudes ( [34]) was established in [37]. Similarly as in the standard string theory, in p-adic string theories it is difficult to determine the convergence region in the momentum space, however this was done precisely for the N-point tree amplitudes in [38]. In this article we show (in a rigorous mathematical way) that the p-adic open string N−point tree amplitudes are bona fide integrals that admit meromorphic continuations as rational functions, by relating them with multivariate local zeta functions (also called multivariate Igusa local zeta functions [39][40][41]).
In p-adic string theory the limit p → 1 is very intriguing since it seems to be related to the real versions of these theories [36,42]. This limit is special since the effective theory shows that it is related to physical string theories as a boundary string field theory [43].
Another interpretation of the limit p → 1 was given in terms of the renormalization group scaling transformation in the Bruhat-Tits tree for some suitable p [44]. In the worldsheet theory we cannot forget the nature of p as a prime number, thus the analysis of the limit is more subtle. The correct way of taking the limit p → 1 involves the introduction of finite extensions of the p-adic field Q p . In [45] the limit p → 1 was discussed at the tree-level string amplitudes. We provided a rigorous definition of this limit using the theory of topological zeta functions due to Denef and Loeser [46,47].
In ordinary string theory the effective action for bosonic open strings in gauge field backgrounds was discussed many years ago in [48]. The analysis incorporating a Neveau-Schwarz B field in the target space leads to a noncommutative effective gauge theory on the world-volume of D-branes [49]. The study of the p-adic open string tree amplitudes including Chan-Paton factors was started in [34]. However the incorporation of a B-field in the p-adic context and the computation of the tree level string amplitudes was discussed in [50,51]. In these works was reported that the tree-level string amplitudes are affected by a noncommutative factor. In [50] Ghoshal and Kawano introduced new amplitudes involving multiplicative characters and a noncommutative factor, these amplitudes coincide with the ones obtained directly from the noncommutative effective action [52].
In the present article we study the p-adic string amplitudes, with Chan-Paton rules and a constant B-field, introduced in [50], by using techniques of multivariate local zeta functions.
The N-point, p-adic, open string amplitudes, with Chan-Paton rules in a constant B-field, have the form where H τ (x i ) = 1 2 (1 + sgn τ (x)), sgn τ (x) is a p-adic version of the sign function, θ is a fixed antisymmetric bilinear form, and N −2 i=2 dx i is the normalized Haar measure of Q N −3 p .
Unfortunately, this theory is not invariant under projective Möbius transformations and consequently the normalization x 1 = 0, x N −1 = 1, x N = ∞ can not be carried out. This is a consequence of the fact that Q p is not an ordered field. Anyway, in [50] the authors assumed a such normalization, which is equivalent to assume that the vertex operators are inserted in the boundary of the Bruhat-Tits tree at 'non-generic points', taking the normalization We have called such integrals Ghoshal-Kawano amplitudes. The main goal of this article is the study of the amplitude A (N ) (k, θ, τ ) using twisted multivariate Igusa's local zeta functions. We attach to A (N ) (k, θ, τ ) the following Igusa type integral: where the s ij are complex symmetric variables and the s ij are real antisymmetric variables.
We have called integrals Z (N ) (k, θ, τ ) Ghoshal-Kawano local zeta functions. As a consequence on the presence of the Chan-Paton factors and the normalization x 1 = 0, x N −1 = 1, . This fact implies that turning off the background B-field amplitude A (N ) (k, θ, τ ) does not reduce to the p-adic open string amplitude at the tree level. This fact was already noticed in [50] in the special case N = 4. We show that integrals Z (N ) (s, s, τ, θ) can be expressed as finite sums of twisted multivariate Igusa's local zeta functions, and by using the results of [39,53], we establish that Z (N ) (s, s, τ, θ) admits a meromorphic continuation as a rational function where N ij,k ∈ N, γ k ∈ N {0}, and M, T are finite sets, and the real parts of its poles belong to the finite union of hyperplanes of type We regularize the amplitude A (N ) (k, θ, τ ) by redefining it as in this way A (N ) (k, θ, τ ) is a well-defined meromorphic function of the kinematics parameters k i k j , which agrees with integral (3), if it exists. As a consequence of the description of the poles of Z (N ) (s, s, τ, θ), A (N ) (k, θ, τ ) is defined for arbitrary large momenta, since in (5) the values k i k j can take arbitrarily large values. This fact is not valid for the p-adic Koba-Nielsen amplitudes, see [38], and [31], since Ghoshal-Kawano amplitudes are supposed to be restrictions of the p-adic open amplitudes at the tree level, we conclude that the normalization x 1 = 0, x N −1 = 1, x N = ∞ is not possible in the presence of a background B-field. In a forthcoming article we expect to study amplitudes (1). It is worth to mention here that the Ghoshal-Kawano local zeta functions are new Igusa-type integrals coming from p-adic string theory.
The construction of a physical theory over a p−adic spacetime (worldsheet in our case) raises the question about the physical meaning of the prime p. The spacetime is a quadratic where q is a quadratic form, and consequently, the spacetime depends on the pair (p, q). In this article, we require p ≡ 3 mod 4 in order to have the symmetry This article is organized as follows. In section II, we study the limit p → 1 in the noncommutative version of the effective action discussed in [52]. We describe the noncommutative version of the Gerasimov-Shatashvili action and found explicitly its four-point amplitudes.
In section III, we review the basic aspects of the twisted, multivariate Igusa's local zeta functions. The local zeta functions required here are a variation of the ones considered in [53].
Sections IV-V are dedicated to establish the meromorphic continuation of Ghoshal-Kawano local zeta functions. Sections VI and VII are devoted to give the explicit calculation for the 4-point and 5-point amplitudes. The 4-point amplitude was already obtained by Ghoshal and Kawano in [50] under certain hypotheses and the 5-point amplitude is new. In section VIII, we compute the p → 1 limit of the p-adic 4-point and 5-point amplitudes. We verified that the p → 1 limit of 4-point amplitude coincides with the Feynman amplitude computed from the noncommutative Gerasimov-Shatashvili action in section II. The final remarks are collected in section IX. Finally in the Appendix, we review the basic aspects of the p-adic analysis, and introduce some notation and conventions used along this article.

II. THE LIMIT p → 1 IN THE EFFECTIVE ACTION WITH A B-FIELD
A. The limit p → 1 in the noncommutative effective action In [52], it was considered a noncommutative action as the effective action of the theory of p-adic open strings with a B-field. The corresponding action in the D-dimensional spacetime is given by where g and ∆ are the coupling constant and the Laplacian, respectively, and (⋆φ) p is defined by φ ⋆ φ ⋆ · · · ⋆ φ p-times. Here ⋆ is the Moyal star product, which is defined for any suitable pair of smooth functions f and g as .
The corresponding equation of motion is given by The solutions of this equation are defined in the target space R D , where p plays the role of a real parameter. In particular the limit p approaches to one makes sense.
Now following [42], by considering the Taylor expansion of exp(− 1 2 ∆ log p) and exp(p log(⋆φ)) at p = 1, and keeping only the linear term, we get where log(⋆φ) = φ − 1 2 φ ⋆ φ + 1 3 φ ⋆ φ ⋆ φ − · · · . Thus the heuristic p → 1 limit leads to a noncommutative version of the Gerasimov and Shatashvili Lagrangian: where In noncommutative field theory, it is well known that the nontrivial noncommutative effect comes from the potential energy of the Lagrangian. The propagators associated to the kinetic energy of the Lagrangian are the same as the ones of the commutative theory.
Thus the free Lagrangian with an external source J(x) is The propagators are given by x ij = 1 k i ·k j +1 , where k i with i = 1, . . . , N are the external momenta of the particles. The Feynman rule for the interaction vertex can be obtained in the noncommutative theory by considering the cubic, quartic, etc. interaction terms and computing the correlation functions, see for instance, [54,55].

B. Amplitudes from the noncommutative Gerasimov-Shatashvili Lagrangian
In this subsection we show how to extract the four-point amplitudes from the noncommutative Gerasimov-Shatashvili Lagrangian (9). In order to do that, we first require to study the interacting theory. The generating functional of the correlation function for the free theory is given by where U(⋆φ) = 2(⋆φ) 2 ⋆ log(⋆φ). We expand U(⋆φ) in Taylor series as follows: where A, B and C are certain real constants.
The generating Z[J] functional incorporating the interaction is given by We are interested in checking whether connected tree-level scattering amplitudes of this theory match exactly with the corresponding p-adic amplitudes in the limit when p tends to one.
The computation of the field theory performed here will be compared to the computation of the p-adic string amplitudes at section VIII.

C. Four-point amplitudes
In this subsection we consider the quartic term from the potential (11). The expansion of the exponential function of this term in the interacting generating functional is expressed as A straightforward computation of the 4-point vertex yields where G F (x − y) is the propagator and ∂ 1,2,3,4 are the partial derivative with respect to the coordinates x 1 , x 2 , x 3 and x 4 respectively.
The interaction term B(⋆φ) 3 in the Lagrangian has also a non-vanishing contribution to the 4-points tree amplitudes at the second order in perturbation theory. They are described by Feynman diagrams with two vertices located at points y and z connected by a propagator G F (y − z) and with two external legs attached to each vertex. In this case the amplitude is computed from the relevant part of the generating functional This expression can be written explicitly in terms of the Moyal product as The connected 4-point amplitudes at the second order from the cubic interaction Bφ 3 yields This total amplitude corresponds exactly to the sum of the partial amplitudes associated to the channels s, t and u. Expression (16) and the 4-point vertex (14) constitute the tree-level amplitudes arising in the 4-point p-adic amplitudes in the limit p → 1. In section VIII we show that the limit p → 1 in the p-adic Ghoshal-Kawano amplitudes exists and it is given precisely by this heuristic limit. Moreover, five-point non-commutative amplitudes (and higher-order amplitudes) in the limit p → 1, can be computed following a similar procedure but it will not be performed here.

III. MULTIVARIATE LOCAL ZETA FUNCTIONS
For the notation and the definition of basic objects such as multiplicative characters, sign functions, Haar measure, etc., the reader may consult the Appendix. In this section we review some basic aspects of the twisted multivariate local zeta functions. The meromorphic continuation of the local zeta functions play a central role in sections IV and V. Let the divisor attached to them. Let χ 1 , . . . , χ m be multiplicative characters. We set f := (f 1 , . . . , f m ) , χ := (χ 1 , . . . , χ m ) , and s := (s 1 , . . . , s m ) ∈ C m . The multivariate local zeta function attached to (f , χ, Θ), with Θ a test function (i.e. a locally constant function with compact support), is defined as with Re(s i ) > 0 for all i. Integrals of type (17)  ], see also [39]. More precisely, the integral Z Θ (f , χ, s) admits a meromorphic continuation as a rational function in p −s 1 , . . . , p −sm . Let us emphasize that the notation χ i (ac(x)), x = 0, means that character χ i depends only on the angular component of x, see Appendix.
We need a slightly variation of the Loeser result [53, Théorème 1.1.4.], more precisely, when each χ i • ac is the trivial character χ triv (x) or sgn τ (x). We denote by χ i one of these characters. This last function is a multiplicative character on Q × p , but it depends on the angular component of x and on the order of x. By using Hironaka's resolution of singularities theorem, Z Θ (f , χ, s) can be written as as linear combination of integrals of type is an m-tuple of nonnegative integers, v j a positive integer, for j running through a finite set T , see proof of [ Then, we have to study the meromorphic continuation of an integral of type since the one corresponding to the trivial character is already known, see e.g. [ In the case c j ∈ p e Z p , we have where y j = p l u.
In conclusion, since Z Θ (f , χ, s) is a finite linear combination of products of integrals of type I(s), then Z Θ (f , χ, s) admits a meromorphic continuation as a rational function in the variables p −s 1 , . . . , p −sm . More precisely, where L Θ,χ (s) is a polynomial in the variables p −s 1 , . . . , p −sm , and the real parts of its poles belong to the finite union of hyperplanes This result is a slightly variation of [ . . , k l,i ), i = 1, . . . , N, is the momentum vector of the i-th tachyon (with Minkowski product k i k j = −k 0,i k 0,j +k 1,i k 1,j +· · ·+k l,i k l,j ) obeying to (2) and N −2 i=2 dx i is the normalized Haar measure of Q N −3 p , and In the bosonic string theory l = 26, however, this dimension does not play any role in our calculations.
In order to study amplitude A (N ) (k, θ, τ ), we introduce We assume that s ij = k i k j in R, and s ij = s ji for any i and j. Furthermore, we set Later on, we will use the convention x 1 = 0, x N −1 = 1 and x N = ∞. Now, we define the Ghoshal-Kawano local zeta function as For the sake of simplicity, from now on, we will use Q N −3 p as domain of integration in (20).
where Z (N ) (s) is the Koba-Nielsen string amplitude studied in [38], see also [31]. Since this last integral is holomorphic in an open set K ⊂ C d , we conclude that Z (N ) (s, s, τ, θ) is holomorphic in s ∈ K for any s, τ , θ.
We set T := {2, . . . , N − 2}, and define for I ⊆ T , the sector attached to I as where Z (N ) In the case in which I c = T I = ∅, by using that H τ (x)H τ (−x) = 0, we have and consequently Z By using this formula, and the convention x 1 = 0, x N −1 = 1, we obtain that with the convention that j∈∅ ≡ 1. In conclusion, In a similar way, we obtain that where the e I,J,K s are constants.

C. Further remarks
The Ghoshal-Kawano local zeta function depends on x 1 , Z (N ) (s, s, τ, θ, x 1 , x N −1 ). In [50], the corresponding amplitude was considered in the case x 1 = 0, x N −1 = 0, x N = ∞. Our result about the meromorphic continuation of Z (N ) (s, s, τ, θ) is also valid for Z (N ) (s, s, τ, θ, x 1 , x N −1 ). Indeed, by using that where a i ∈ Z N −3 p and L is a positive integer sufficiently large, we have and E(x, s, τ ) depends on x 1 , x N −1 , (19). The meromorphic continuation of

can be obtained by the methods presented in Sections (V A)-(V B), by giving a local description polynomial
We illustrate the technique in a particular but relevant case. We consider the case, x 1 , . . , l, and b i = x N −1 for i = l + 1, . . . , N − 2, for some 2 ≤ l < N − 2. Now, we change variables as x i = x 1 + p L y i , i = 2, . . . , l, and x i = x N −1 + p L y i , for i = l + 1, . . . , N − 2, then Consequently, for L sufficiently large. Now for L sufficiently large. Consequently, Now, by using formulae (26) In this section we compute the Ghoshal-Kawano local zeta function for four points: We recall that Ghoshal and Kawano take x 1 = 0, x 3 = 1, x 4 = ∞. By using the fact that sgn τ (y) ∈ {1, −1} and H τ (y) ∈ {0, 1}, one verifies that and consequently We first compute some p-adic integrals needed in this section.
A. Some p-adic integrals This formula follows from changing variables as x 2 = −y and using the fact that sgn τ (−y) = −sgn τ (y).

Now by using that
Then By using the partition 30) and the fact that we have where If j = 0, 1, then If j = 0, then by using Formula 1, The case j = 1 is similar to the case j = 0, Formula (29) follows from (32) by using (33)- (35) and Formula 2.
In this section we compute the amplitude for five points: and Taking x 1 = 0, x 4 = 1 and x 5 = ∞, and using the reasoning given at the beginning of the previous section we have where First we give some formulae needed in the following calculations.
A. More p-adic Sums and Integrals

Formula 6
We set for a, b, c ∈ C, Then L 00 (a, b, c) = 1 8 In order to compute L 00 (a, b, c), we introduce the following subsets: and L 00 (a, b, c) = L We compute first L (A) 00 (a, b, c), by using the following change of variables: Then dx 2 dx 3 = |u| p dudv and By using partition (30), c) .
and the contribution of all these integrals is For i = 0, For i = 1, Therefore, from (42)- (45), 00 (a, b, c), by using the following change of variables: Then dx 2 dx 3 = |t| p dzdt and

Formula 8
For a, b, c ∈ C, we set Then L This identity is obtained by changing variables as u = x 2 , v = x 2 − x 3 , and using Formula 7.

Formula 9
For a, b, c ∈ C, we set The computation of Z (5) (s, s, τ, θ) is reduced to the computation of integral L(s, τ ), see (38)- (39). By using the partition We have The calculation of these integrals is achieved by considering several cases.
In this case, by using that Now by using Formula 5, the contribution of all these integrals is Case i, j ∈ {2, 3, . . . , p − 1} and i = j.
In this case, by using (34), Now, by using that p ≡ 3 mod 4, τ = ε, and Formula 2, the contribution of all these integrals Case i = 1 and j = 0.
In this case by using Formula 1, Case i = 0 and j = 1. Since we have L 01 (s, τ ) = 0.
In this case,

By using that
and the notation introduced in Formulae 6 to 9, we have L 00 (s, τ ) = L 00 (s 12 , s 13 , s 23 ) + L Case i = j = 1.
In this case, In these cases, The vanishing of the integral L 0j (s, τ ) follows from The other case is treated in a similar way.
Case i ∈ {2, 3, . . . , p − 1} and j = 0. zeta functions [46]. Notice that the calculations involving the limit p → 1 in the case of the effective action are performed in R D , meanwhile the calculations involving the limit p → 1 in the case of p-adic string amplitudes are performed in Q D p , and in the p-adic topology the limit p → 1 does not make sense. However, surprisingly, the computation of the limit p → 1 (considering p as a real parameter) of the p-adic open string amplitudes gives the right answer! In this subsection we compute limit p → 1 (considering p as a real parameter) in the cases N = 4, 5. The computation of the limit p → 1 in the general case require the so called explicit formulas, see [45] for further details.
In the case N = 4, the amplitude agrees with the amplitude the Feynman obtained from the noncommutative version of the Gerasimov-Shatashvili action with a logarithmic potential (9).

IX. FINAL REMARKS
In the present article, we study the Ghoshal-Kawano amplitudes for p-adic open strings at tree-level level [50]. These amplitudes include Chan-Paton factors and an external B-field.
In section III starting from the noncommutative effective action (6) discussed in [50,52], in the present article, we obtain the corresponding tree-level four-point amplitudes (36) in the limit p → 1. This result was achieved by adapting the heuristic approach given in [42] for the noncommutative case. By an explicit computation using the noncommutative field theory [54,55], we determine the four-point amplitude at the tree level coming from the noncommutative Gerasimov-Shatashvili Lagrangian. This amplitude is the sum of the expressions (14) and (16). The first one represents the noncommutative vertex four-point function and the second one is the superposition of the amplitudes corresponding to the noncommutative channels s, t and u. The calculated tree-level amplitude is completely described by planar Feynman diagrams and consequently the noncommutativity effect arises as a global phase factor in front of the amplitude. Five-point amplitudes (or higher-order amplitudes) can be also computed in a straightforward way following the same procedure.
The study of the p-adic Ghoshal-Kawano amplitudes requires the use of multivariate local zeta functions involving multiplicative characters and a phase factor including the noncommutative parameter θ. These are new mathematical objects. We call these objects Ghoshal-Kawano zeta functions. In sections IV and V, by using Hironaka's resolution of singularities theorem, we prove that these integrals admit meromorphic continuation as complex functions in the external momenta of the N external particles.
Four and five point amplitudes were computed explicitly in sections VI and VII, see (36) and (56). The four-point amplitude (36) coincides with the one obtained in [50]. The five-point amplitude was not obtained previously.
In section VIII we study again the amplitudes from the worldsheet view point. We compute the limit p → 1 limit for four and five point amplitudes resulting in the formulae (57) and (58), respectively. The four-point amplitude (57) agrees with the heuristic computation given by the superposition of formulae (14) and (16).
As we mentioned before, in the computation of Ghoshal-Kawano amplitudes at the treelevel, the noncommutative effect coming from the constant B-field arises only as a global phase factor because only planar diagrams are involved. In the computation of amplitudes at one-loop or multi-loops non-planar diagrams systematically arise. It would be very interesting to study the possibility of finding a non-trivial noncommutative effect as the IR/UV mixing as a result of the contribution of one-loop non-planar diagrams. Probably the multiloop analysis of the p-adic string theory studied in [56], will play an important role for the analysis of the IR/UV mixing and other interesting effects of the B-field in p-adic string theory amplitudes.
On the other hand, we think that the study of the amplitudes (1) without the ad hoc normalization x 1 = 0, x N −1 = 1, x N = ∞ may provide new insights on the effects of the B-field in p-adic string theory amplitudes. However, the study of these amplitudes is more involved than the one done here. Some of this work is in progress and will be reported elsewhere.
If f is a continuous function on K 0 , then f (σ(y)) det ∂H i ∂y j (y) p d n y, (x = H(y)).
For the proof of this theorem the reader may consult [39,Prop. 7.4.1] or [60, Section 10.1.2].

C. Some arithmetic functions
In this section we review some arithmetic functions that we shall use through this article.

Multiplicative characters
A multiplicative character (or quasi-character) of the group Q × p , · is a continuous homomorphism χ : Q × p → C × satisfying χ (xy) = χ (x) χ (y). Every multiplicative character has the form χ (x) = |x| s p χ 0 (ac(x)) , for some s ∈ C, where χ 0 is the restriction of χ to Z × p , which is a continuous multiplicative character of Z × p , · into the complex unit circle.

The Legendre symbol
For a an integer number and p a prime number, the Legendre symbol is defined as The following formulas are used in several calculations in this article: Take for x ∈ Q × p , ac(x) = x 0 + x 1 p + . . . ∈ Z × p , then x → ( x 0 p )