Analyticity of Off-shell Green's Functions in Superstring Field Theory

We consider the off-shell momentum space Green's functions in closed superstring field theory. Recently in \cite{LES2019}, the off-shell Green's functions -- after explicitly removing contributions of massless states -- have been shown to be analytic on a domain (to be called the LES domain) in complex external momenta variables. We extend the LES domain further to a larger domain within the primitive domain (where the analyticity of off-shell Green's functions in local QFTs without massless states is a well known result). The LES domain can be extended up to the union of certain convex tubes (i.e. primitive tubes) using Bochner's tube theorem and the fact that under complex Lorentz transformations the off-shell Green's functions retain their analyticity property. Up to the four-point function, we obtain such tubes analytically, e.g. for the four-point function there are 32 possible primitive tubes and all of them are obtained in this article. For the five-point function, out of 370 primitive tubes we are able to obtain 350 of them fully. For each of the remaining 20 tubes, it is difficult to show analytically that the application of Bochner's theorem yields the full tube. And this feature occurs for higher point functions as well.


Introduction
Closed superstring field theory (SFT) which is designed to reproduce perturbative amplitudes of superstring theory is a quantum field theory with countable infinite number of fields and nonlocal interaction vertices, whose action is directly written in momentum space (for a detailed review, see [2]). It contains massless states. Off-shell momentum space amputated n-point Green's function 1 in SFT is defined by usual momentum space Feynman rules [2,3]. It can 1 Hereafter Green's functions will always refer to momentum space amputated Green's functions.
be computed by summing over connected Feynman diagrams with n amputated external legs carrying ingoing D-momenta p 1 , . . . , p n .
In [1] de Lacroix, Erbin and Sen (LES) showed that the off-shell n-point Green's function G(p 1 , . . . , p n ) in SFT -after explicitly removing contributions where any of the internal propagator is that of a massless particle -as a function of n complex external momenta p 1 , . . . , p n is analytic on a domain which we call as the LES domain. At the heart of this result the following statement has been proven. That is, each of the relevant Feynman diagrams F(p 1 , . . . , p n ) -those which do not contain any internal line corresponding to a massless particle -in the perturbative expansion of the off-shell n-point Green's function in SFT has an integral representation in terms of loop integrals as presented below, whenever the external momenta lie on the LES domain.
(2π) D δ (D) (p 1 + · · · + p n )F(p 1 , . . . , p n ) = (2π) D δ (D) (p 1 + · · · + p n ) Equation (1) represents an n-legged L-loop graph with I internal lines, where k r is a loop momentum, ℓ s is the momentum of the internal line with mass m s ( = 0) 2 , and f is a regular function whenever its arguments take finite complex values. The function f contains the product of the vertex factors associated with the internal vertices of the graph. The momentum ℓ s of an internal line is usually a linear combination of the loop momenta and the external momenta. Due to certain non-local properties of the vertices in SFT, the graph is manifestly UV finite as long as for each r, k 0 r integration contour ends at ±i∞, and k i r , i = 1, . . . , (D − 1) integration contours end at ±∞. The prescription for the loop integration contours has been given as follows. At origin, i.e. p a = 0 ∀a = 1, . . . , n each loop energy integral is to be taken along the imaginary axis from −i∞ to i∞ and each spatial component of loop momenta is to be taken along the real axis from −∞ to ∞. With this, F(0,. . . ,0) has been shown to be finite as all the poles of the integrand in any complex k µ r plane are at finite distance away from the loop integration contour. As we vary the external momenta from the origin to other complex values if some of such poles approach the k µ r contour, the contour has to be bent away from those poles keeping its ends fixed at ±i∞ for loop energies and ±∞ for loop momenta.
It has been shown that there exists a path inside the LES domain connecting the origin to any other point of the LES domain such that when we vary external momenta from origin to that point along that path, the loop integration contours in any graph can be deformed away avoiding poles of the integrand which approach them [1]. Hence, the integral representation (1) for F(p 1 , . . . , p n ) when the external momenta lie on the LES domain is well defined where the poles of the integrand are at finite distance away from the (deformed)loop integration contours.
On the other hand in local quantum field theories without massless particles, the off-shell n-point Green's function G(p 1 , . . . , p n ) as a function of n complex external ingoing D-momenta p 1 , . . . , p n is known to be analytic on a domain called the primitive domain [4][5][6][7][8]. This result follows from causality constraints on the position space Green's functions in a local QFT and representing the momentum space Green's functions as Fourier transforms of the position space correlators 3 . The primitive domain contains the LES domain as a proper subset. In several complex variables, many a time the shape of a domain alone forces all the functions which are analytic on it to be analytic on a larger domain. The shape of the primitive domain is such that the actual domain of holomorphy of G(p 1 , . . . , p n ) is larger than the primitive domain (e.g. [9,10]). This can be used to prove various analyticity properties [11][12][13][14][15][16][17][18] of the S-matrix of those local QFTs that have no massless states, since the S-matrix is defined as the on-shell Green's function. These properties are to be read as the artifact of the shape of the primitive domain, irrespective of the functional form of G(p 1 , . . . , p n ) which is defined on it. In particular, the derivations [10][11][12] of certain analyticity properties [11,12,19,20] of the S-matrix use the information of only the LES domain as a subregion of the primitive domain.
Recall that a part of the off-shell n-point Green's function in SFT which is obtained by eliminating the contributions of massless states is analytic on the LES domain. Hence the work of [1] in effect establishes that -in superstring theory any possible departure from those analyticity properties of the S-matrix named in the above paragraph which rely only on the LES domain is solely due to the presence of massless states. This is an effect also present in those local QFTs that have massless states. Thus, with respect to the aforenamed properties [11,12,19,20], the S-matrix of the superstring theory displays the same analytic behaviour as that of a local QFT with massless particles. However, with respect to the properties of the S-matrix of local QFTs without massless states that rely on the full primitive domain (e.g. [13][14][15][16][17][18]), the analytic behaviour of the S-matrix of local theories with massless states could possibly deviate. Any such departures are entirely due to the presence of massless states, i.e. the infrared safe part must satisfy all those analyticity properties. At this stage, it is natural to ask that with respect to the last-mentioned properties whether the S-matrix of the superstring theory has an identical analytic behaviour to that of a standard local QFT with massless states, or not. They are identical on this aspect, only if the relevant part of the off-shell n-point Green's function in SFT can be shown to be analytic on the full primitive domain extending the LES domain.
In this paper, we aim to generalize the result of [1] by showing that the infrared safe part of the off-shell n-point Green's function in SFT, i.e. after eliminating the contributions of massless states is analytic on a larger domain than the LES domain staying within the primitive domain 4 . As will be reviewed in section 2.2, the analyticity property of the off-shell Green's functions in SFT is invariant under the action of a D-dimensional complex Lorentz transformation on all the external momenta. Thus the off-shell Green's functions in SFT are also analytic at points that are obtained by the action of such transformations on points in the LES domain. We consider LES domain adjoining these new points. The primitive domain essentially contains the union of a certain family of tube domains, in which we call the members as the primitive tubes. Within each such primitive tube, we identify a connected tube which is also contained in the LES domain and its shape is such that it always admits holomorphic extension inside the corresponding primitive tube possibly obtaining full of it, due to a classic theorem by Bochner [21]. We explicitly work out the cases of the three-point, four-point and five-point functions to determine whether in each case such extensions fully obtain all the possible primitive tubes or not. In the case of three-point function, indeed such extensions yield all the 6 possible primitive tubes. Also for the four-point function, all the 32 possible primitive tubes are obtained by such extensions. Whereas for the five-point function, out of 370 possible primitive tubes, for 350 of them we are able to show that such extensions obtain each of them fully. The technique that we employ for aforesaid checks seems difficult to implement analytically for the remaining 20 primitive tubes whose shapes are complicated.
However in any case, our work establishes that based on a geometric consideration only, the LES domain is holomorphically extended inside all the primitive tubes. Thus with respect to all the analyticity properties of the S-matrix which can be obtained from this extended domain 5 , superstring theory behaves like a standard local QFT that has massless states. 4 Hereafter off-shell Green's functions in SFT will refer to this part of respective off-shell Green's functions in SFT, if not explicitly stated. This part of the off-shell Green's functions -when all the external particles are massless -precisely gives the vertices of the Wilsonian effective field theory of massless fields, obtained by integrating out the massive fields in superstring theory [2,22]. 5 Note that up to the four-point function, the extended domain is equal to the primitive domain.
We organize the paper as follows. In section 2, we briefly review certain properties of the primitive domain and the LES domain which are useful for our purpose. In section 3, we start with the general scheme to extend the LES domain holomorphically. We apply this scheme to the case of the three-point function in subsection 3.1. In subsection 3.2 (and appendix E) we deal with the case of the four-point function, and in subsection 3.3 the case of five-point function. We discuss certain real limits within each of the primitive tubes in section 4.

Review
In this section we review certain properties of the two domains, namely the primitive domain and the LES domain which will be useful for our analysis. Both the domains are domain in the complex manifold C (n−1)D given by p 1 + · · · + p n = 0. The origin of C (n−1)D which is given by p a = 0 ∀a = 1, . . . , n will be denoted by O. We count p a as positive if ingoing, negative otherwise. We shall use Minkowski metric with mostly plus signature.

The primitive domain
The primitive domain D is given by either, Im P I = 0, (Im P I ) 2 ≤ 0 or, Im P I = 0, −P 2 where X = {1, . . . , n} is the set of first n natural numbers. ℘ * (X) = {I X, except ∅} is the collection of all non-empty proper subsets of X. P I is defined to be equal to a∈I p a . M I is the threshold of production of any (multi-particle) state in a channel containing the external states in the set {p a , a ∈ I}, i.e. M I is the invariant threshold mass for producing any set of intermediate states in the collision of particles carrying total momentum P I .
Primitive domain is star-shaped with respect to O, i.e. the straight line segment connecting O and any point p ∈ D which is given by tp : t ∈ [0, 1] lie entirely inside D. Hence the primitive domain is path-connected as any two points p (1) , p (2) ∈ D can be connected via the straight line segments p (1) O and Op (2) . Furthermore primitive domain is simply connected, i.e. any closed curve within D can be continuously shrunk to the point O, which is a property of a star-shaped domain.
Primitive domain essentially contains the union of a family of mutually disjoint tube 6 domains denoted by {T λ , λ ∈ Λ (n) } [7,9,23,24]. Any member T λ of this family will be called a primitive tube. To describe this family of tubes the following definitions are needed. We consider the space R n−1 of n real variables s 1 , . . . , s n linked by the relation s 1 + · · · + s n = 0.
We define S I = a∈I s a for each I ∈ ℘ * (X). The family of planes {S I = 0, I ∈ ℘ * (X)} (where the planes S I = 0 and S X\I = 0 are identical) divides the above space R n−1 into open convex cones 7 with common apex at the origin. Any such cone will be called a cell, γ λ . Within a cell each S I is of definite sign λ(I). Thus a cell can be written as where λ : ℘ * (X) → {−1, 1} is a sign-valued map with the following properties.
The first property is compatible with s 1 + · · · + s n = 0 and the second property is compatible with S I∪J = S I + S J whenever I ∩ J = ∅. Λ (n) denotes the collection of all possible maps λ satisfying above properties. Corresponding to each cell γ λ , now we associate an open convex tube domain T λ (primitive tube) given by 8 where V + is the open forward lightcone in R D . C λ is the conical base of the tube T λ . Although the primitive domain is non-convex the primitive tubes T λ are convex (see appendix A). Hence each tube T λ is path-connected as the entire straight line segment p (1) p (2) connecting any two points p (1) , p (2) ∈ T λ is contained in the tube T λ .

The LES domain
The LES domain is given by p a = 0 and for each I ∈ ℘ * (X) either, Im P I = 0, (Im P I ) 2 ≤ 0 or, Im P I = 0, −P 2 where all the Im p a thereby Im P I are allowed to lie only on the two dimensional Lorentzian In [1] it has been argued that the domain of holomorphy of the n-point Green's function G(p 1 , . . . , p n ) in SFT is a connected region in C (n−1)D containing the origin, and it is invariant under the action of Lorentz transformationsΛ with complex parameters, i.e.Λ is any complex matrix satisfyingΛ T ηΛ = η for η being the Minkowski metric in R D . We call the set of such matrices the complex Lorentz group, L. In general, the action of a complex Lorentz transformationΛ is defined on the complex manifold C (n−1)D taking a point to another point of the same manifold given by which we abbreviate as p →Λp. Note that the sameΛ acts on all p a .
As as consequence, the result of [1] automatically generalizes to a larger domain than the LES domain D ′ , i.e. G(p) is analytic on the domainD ′ given bỹ ClearlyD ′ ⊃ D ′ , since L contains the identity matrix. Hereafter we referD ′ as the LES domain.

Extension of the LES domainD ′
We identify a family of tubes lying inside the primitive tube T λ which is a convex tube domain (described by equation (5) in section 2.1) such that any member of the family is also contained in the LES domainD ′ (described in section 2.2). Any member of this family is convex and it can be characterized by a set of D − 2 angles θ which specifies a two dimensional Lorentzian plane p 0 − p θ10 with θ = 0 specifying the p 0 − p 1 plane. Hence we denote a member by T θ λ . The convex tube T θ λ is given by Im p a = 0 such that ∀a Im p a ∈ p 0 − p θ plane and λ(I)Im P I ∈ V + ∀I ∈ ℘ * (X) , where the base C θ λ is a subset of R (n−1)D . Any such tube T θ λ can be obtained by acting a real rotation on the tube T θ=0 λ . Clearly θ T θ λ lies inside the primitive tube T λ as well as it is contained in the LES domaiñ D ′ . Now θ T θ λ is a tube given by R (n−1)D + i( θ C θ λ ). Although each tube T θ λ is convex (see appendix A) the tube θ T θ λ is non-convex 11 (see appendix B). However the tube θ T θ λ is path-connected (see appendix C).
We apply Bochner's tube theorem [21,25] 12 on the connected tube ( θ T θ λ ) 13 . The theorem states that any open connected tube R m + iA has a holomorphic extension 14 to the domain R m + iCh(A) where Ch(A) is the smallest convex set containing the set A, called the convex hull of A.
By the above application, since θ C θ λ is non-convex we always have a holomorphic extension of θ T θ λ to a domain given by In subsequent subsections, we deal with the explicit cases of the three-point, four-point and five-point functions, where up to the four-point function we obtain that for each C λ such extension yields the full of C λ , i.e. Ch( θ C θ λ ) = C λ , and for five-point function we obtain subcases in which we are able to prove this equality.

Remarks
The properties of the tube θ T θ λ as a domain in several complex variables have been used here to extend it holomorphically. From the work of [1], we know that all the relevant Feynman diagrams (those which do not have any internal line of a massless particle) in the perturbative expansion of the n-point Green's function are analytic in the common tube θ T θ λ . Hence our extension of θ T θ λ is valid to all orders in perturbation theory. A proper application of Bochner's tube theorem requires us to thicken the connected tube θ T θ λ in order to make it open. But the thickened tubes (as in appendix D) are not identical for all the Feynman diagrams. However the intersection of all these thickened tubes corresponding to distinct diagrams certainly contains θ T θ λ . We consider any relevant Feynman diagram. The corresponding thickened tube can be holomorphically extended to its convex hull due to Bochner's tube theorem. This extended domain contains the tube Ch( θ T θ λ ) = R (n−1)D + iCh( θ C θ λ ). Clearly, Ch( θ T θ λ ) lies in the intersection of all such extensions corresponding to distinct diagrams since Ch( θ T θ λ ) only includes convex combinations of points from θ T θ λ . Hence Ch( θ T θ λ ) is the domain where all the relevant Feynman diagrams (at all orders in perturbation theory) are analytic.

Three-point function
For three-point function, we have n = 3 and the sign-valued maps λ(I) (described by equation (4)) can be given explicitly as follows. In this case, the primitive domain D essentially contains the union of 6 mutually disjoint tubes denoted by {T ± a , a = 1, 2, 3} and this primitive tubes are given by where p = (p 1 , p 2 , p 3 ) is linked by p 1 + p 2 + p 3 = 0. Their conical bases are defined by where (abc) = permutation of (123). In order to define each of the above cones C ′+ a (C ′− a ), we require a certain pair of imaginary external momenta which (or their negative) are specified to be in the open forward lightcone V + . For a given conical base, this in turn fixes all other Im P I to be in specific lightcone 15 .
The cones (11) reside on the manifold Im p 1 + Im p 2 + Im p 3 = 0. In order to assign coordinates to the points in C ′+ a , let us choose {Im p b , Im p c } as our set of basis vectors. On the other hand, for C ′− a , let us choose {−Im p b , −Im p c } as our basis. With this, any of the above cones is contained in a R 2D and is of the following common form where any Q ∈ C ′ can be written as a D × 2 matrix given by with conditions P 0 r > + D−1 i=1 (P i r ) 2 ∀r = α, β ensuring that both the columns belong to the forward lightcone V + . Hence given a Q, the quantities P 0 r − i (P i r ) 2 , r = α, β are a pair of positive numbers. Furthermore, the two columns of Q in general do not lie on a same two dimensional Lorentzian plane. Now we consider cones C ′ θ containing points Q where both the columns not only belong to V + but also lie on a same two dimensional Lorentzian plane p 0 − p θ characterized by a θ.
We shall show that taking points from these cones for various θ, a convex combination of them represents given Q in (13). This will complete the proof of Ch( θ C ′ θ ) = C ′ , since we already have C ′ ⊃ Ch( θ C ′ θ ) as discussed right above the remarks in section 3.
where we take any ǫ satisfying 0 < ǫ < min P 0 Both the columns of Q 1 lie on a same two dimensional Lorentzian plane p 0 −p θ 1 where also the first column P α of Q in (13) lies. Similarly, both the columns of Q 2 lie on the two dimensional Lorentzian plane where P β lies. Now it is easy to check that the following relation holds Equation (15) establishes that each of the 6 primitive tubes given in (10) can be obtained as holomorphic extension of a tube where the latter is contained in the LES domainD ′ .

Four-point function
For four-point function, we have n = 4 and the maps λ(I) (described by equation (4)) can be given explicitly as follows. In this case, the primitive domain D essentially contains the union of 32 mutually disjoint tubes denoted by {T ± a , T ± ab , 1 ≤ a, b ≤ 4, a = b} and this primitive tubes are given by [7,9] T ± a = p ∈ C 3D : Im p ∈ C ± a , T ± ab = p ∈ C 3D : Im p ∈ C ± ab , where p = (p 1 , . . . , p 4 ) is linked by p 1 + · · · + p 4 = 0. Their conical bases are defined by where (abcd) = permutation of (1234). Note that in order to describe each of the above cones we require a certain set of three Im P I each of which (or its negative) is specified to be in the open forward lightcone V + . For a given conical base, this in turn fixes all other Im P I to be in specific lightcone 16 (see appendix E). The cones (17) reside on the manifold Im p 1 + · · · + Im p 4 = 0. Due to this link we can choose any three linear combinations of Im p 1 , . . . , Im p 4 which are linearly independent as our set of basis vectors, to describe a given cone. In particular as our basis, we choose {Im p b , Im p c , Im p d } for the cones C + a , whereas we choose {−Im p b , −Im p c , −Im p d } for the cones C − a . Besides, as our basis, we choose {−Im p b , Im (p b + p c ), Im (p b + p d )} for the cones C + ab 17 , whereas we choose {Im p b , −Im (p b + p c ), −Im (p b + p d )} for the cones C − ab . With this, any of the above cones is contained in a R 3D and is of the following common form where any Q ∈ C can be written as a D × 3 matrix given by with conditions P 0 r > + D−1 i=1 (P i r ) 2 ∀r = α, β, γ ensuring that each of the columns belong to the forward lightcone V + . Hence given a Q the quantities P 0 r − i (P i r ) 2 , r = α, β, γ are three positive numbers. Furthermore the columns of Q in general do not lie on a same two dimensional Lorentzian plane. Now we consider cones C θ containing points Q where all the three columns not only belong to V + but also lie on the same two dimensional Lorentzian plane p 0 − p θ characterized by a θ. We shall show that taking points from these cones for various θ, a convex combination of them represents given Q in (19). This will complete the proof of Ch( θ C θ ) = C, since we already have C ⊃ Ch( θ C θ ) as discussed right above the remarks in section 3.
We consider three points Q 1 ∈ C θ 1 , Q 2 ∈ C θ 2 and Q 3 ∈ C θ 3 given by 17 Instead, one can choose {Im p b , Im p c , Im p d } as the basis to describe points in any of the cones C + ab . This change of basis is a linear invertible transformation and the work of this subsection can be recast in this new basis (e.g., see appendix E).
where we take any ǫ satisfying 0 < ǫ < min P 0 r − D−1 i=1 (P i r ) 2 , r = α, β, γ so that for each r = α, β, γ we have P 0 All the three columns of Q 1 lie on a same two dimensional Lorentzian plane p 0 − p θ 1 where also the first column P α of Q in (19) lies. Similarly, all the columns of Q 2 lie on the two dimensional Lorentzian plane where P β lies, and all the columns of Q 3 lie on the two dimensional Lorentzian plane where P γ lies. Now it is easy to check that the following relation holds Equation (21) establishes that each of the 32 primitive tubes given in (16) can be obtained as holomorphic extension of a tube where the latter is contained in the LES domainD ′ .

Five-point function
For five-point function, we have n = 5 and in this case the primitive domain D essentially contains the union of 370 mutually disjoint tubes whose conical bases are given by [7] C + a = −C − a = Im p : Im p b , Im p c , Im p d , Im p e ∈ V + , C + ab = −C − ab = Im p : −Im p b , Im (p b + p c ), Im (p b + p d ), Im (p b + p e ) ∈ V + , C ′+ ab = −C ′− ab = Im p : Im (p a + p c ), Im (p a + p d ), Im (p a + p e ), where (abcde) = permutation of (12345) and Im p = (Im p 1 , . . . , Im p 5 ) is linked by the relation Im p 1 + · · · + Im p 5 = 0. Due to this link we can choose any four linear combinations of Im p 1 , . . . , Im p 5 which are linearly independent as our set of basis vectors, to describe a cone which is given from the above list (22). Hence any of these cones is contained in a R 4D with a choice for a basis.
We note that in order to describe each of the cones in (22) except the conesC ′+ ab ,C ′− ab , we require a certain set of four Im P I each of which (or its negative) is specified to be in the open forward lightcone V + . This in turn fixes all other Im P I to be in specific lightcone 18 . Now we confine ourselves to these cones which are 350 in numbers 19 . To describe any of these cones we choose the corresponding certain set of four Im P I as our basis (in a similar manner to the cases of the three-point and four-point functions, as demonstrated in detail in the sections 3.1, and 3.2 respectively). With this, any of these cones is of the following common form where any Q ∈C can be written as a D × 4 matrix given by with conditions P 0 r > + D−1 i=1 (P i r ) 2 ∀r = α, β, γ, δ. Given a Q as in (24) it can now be represented as the following convex combination.
where Q r , r = 1, . . . , 4 are given by 18 For n = 5, the total number of possible P I = 2 5 − 2 = 30. 19 Each ofC ′+ ab andC ′− ab is symmetric under the interchange of a, b which is evident from (22). Hence they are 20 in total.
where we take any ǫ satisfying the condition: 0 < ǫ < min P 0 r − D−1 i=1 (P i r ) 2 , r = α, β, γ, δ . Equation (25) establishes that each of the primitive tubes described by (22) except the ones whose conical bases areC ′+ ab ,C ′− ab can be obtained as holomorphic extension of a tube where the latter is contained in the LES domainD ′ .
The above technique has limitations. Following difficulty arrises when we consider the remaining 20 cones which are given byC ′+ ab ,C ′− ab . To describe points inC ′+ ab let us choose the set Im (p a + p c ), Im (p a + p d ), Im (p a + p e ), Im (p b + p c ) as our basis, and to describe points inC ′− ab let us choose the set as our basis. With this any of the conesC ′+ ab ,C ′− ab is of the following common form Due to additional constraints on the linear combinations (P β + P δ − P α ) and (P γ + P δ − P α ) the technique which we have employed in earlier cases seems difficult to implement here analytically, in order to check the validity of Ch( θC ′ θ ) =C ′ . Here each coneC ′ θ is to be obtained fromC ′ by putting further restrictions on its points Q = (P α , P β , P γ , P δ ) so that ∀r = α, β, γ, δ P r lies on the two dimensional Lorentzian plane p 0 − p θ . That is, it is difficult to find a set of points Q r , r = 1, . . . , m for some m 20 , each of which has four columns satisfying the six conditions as stated in (27) and furthermore all the four columns lie on a two dimensional Lorentzian plane, in such a way that a convex combination of these m points produce a general point Q in (27).
As an illustration we work with one of these 20 problematic cones in appendix F (in which case, as a trial we take m = 4).

Limits within T λ
In section 3, we have shown that for an n-point Green's function and given any λ from the possible set Λ (n) , the tube θ T θ λ has holomorphic extension inside the primitive tube T λ where the former tube is contained in the LES domainD ′ .
As per the equations (6) and (8), if we take the limit Im P I → 0, Im P I ∈ T θ λ for a collection of subsets {I} ⊂ ℘ * (X), the n-point Green's function G(p) in SFT is finite whenever we restrict their real parts by −P 2 I < M 2 I for each I belonging to that collection {I}. Here Re P J are kept arbitrary for all J ∈ ℘ * (X) \ {I}. In fact, for a given collection {I} by taking such limits within T θ λ for any θ and restricting corresponding real parts, we reach to same value G(p) for all θ. Now that θ T θ λ has an unique holomorphic extension given by Ch( θ T θ λ ) ⊂ T λ , we reach to above value G(p) in the limit Im P I → 0, Im P I ∈ Ch( θ T θ λ ) with above constraints on the real parts.

Conclusions
In this paper, we have shown that for any n-point Green's function in superstring field theory, the LES domainD ′ due to its shape always admits a holomorphic extension within the primitive domain D where the latter is basically the union of the convex primitive tubes. In the process we have found that the LES domainD ′ contains a non-convex connected tube within each convex primitive tube. The former tube being non-convex allows to include all the new points from its convex hull which is the set of all convex combinations of points in that tube. The convex tube thus obtained is a holomorphic extension of the former non-convex tube due to a classic theorem by Bochner, and lies inside the corresponding primitive tube.
Up to the four-point function such extension yields the full of the primitive domain. We have proved this result, in section 3.1 for the three-point function obtaining all the 6 primitive tubes, and in section 3.2 for the four-point function obtaining all the 32 primitive tubes. The appropriate real limits within those tubes in both cases are also attained (as discussed in section 4).
In section 3.3, we are able to show that for the five-point function such extension yields the full of 350 primitive tubes out of 370 primitive tubes which are possible in this case. The technique employed in this subsection can not be applied (as it is) for the remaining 20 primitive tubes, their shape being complicated. However within all these 370 extensions inside respective primitive tubes (obtaining 350 of them fully) the appropriate real limits are attained (as discussed in section 4).
As a consequence, our result shows that with respect to all the analyticity properties of the S-matrix which can be obtained relying on above extended domain inside the primitive domain, superstring theory (more precisely, its infrared safe part) has same behaviour as that of a standard local quantum field theory. Any non-analyticity of the full S-matrix is entirely due to the presence of massless states -which is also the case for a standard local QFT.
The difficulty arising for the twenty primitive tubes in the case for the five-point function has been demonstrated in appendix F. This is a generic feature that arises for all higherpoint functions, in the course of determining whether or not the application of Bochner's tube theorem yields certain primitive tubes fully. Solving this may require numerical analysis. We leave this for future work. a ∀a = 1, . . . , n lie on the two dimensional Lorentzian plane p 0 − p 1 . Now suppose the point p (2) is obtained by acting a real rotation 22 on the point p (1) so that Im p (2) a ∀a = 1, . . . , n now lie on the two dimensional Lorentzian plane p 0 − p 2 . Hence p (2) ∈ T θ 2 λ where θ 2 characterizes the two dimensional Lorentzian plane p 0 − p 2 . Now for the point q = 1 2 p (1) + 1 2 p (2) which is the mid-point of the straight line segment connecting the two points p (1) , p (2) we have Im q a ∀a = 1, . . . , n lying on the two dimensional Lorentzian plane p 0 − p θ 3 where the p θ 3 -axis lies on the two dimensional plane p 1 − p 2 making an angle 45 0 with the positive p 2 -axis. Hence the point q ∈ T θ 3 λ . However we can change Im p (2) 1 little bit, keeping all real parts and Im p (2) a , a = 2, . . . , n unchanged to obtain a new pointp (2) such thatp (2) still belongs to T θ 2 λ 23 . Consequently we get two points p (1) ∈ T θ=0 λ andp (2) ∈ T θ 2 λ , for which the mid-point of the straight line segment connecting them is given byq = 1 2 p (1) + 1 2p (2) . Although Imq a ∀a = 2, . . . , n lie on the two dimensional 1 does not lie on the two dimensional Lorentzian plane p 0 − p θ 3 anymore. Henceq / ∈ θ T θ λ . In other words, θ T θ λ is non-convex.
To see this, let us take {p 1 , . . . , p n−1 } as our basis to describe points on the complex manifold p 1 + · · · + p n = 0. A generic point p = (p 1 , . . . , p n−1 ) on this manifold can be represented by an unique D × (n − 1) matrix where the a-th column represent the D-momenta p a . Since each T θ λ thereby θ T θ λ reside on this manifold now the imaginary parts of the points p (1) , p (2) ,p (2) , 22 In equation (7), we have defined actions of complex Lorentz transformations which include real rotations. 23 In support of this see appendix D.
q andq can be represented in terms of D × (n − 1) matrices as follows 24 .
(29) Clearly all the columns of Im q lie on the two dimensional Lorentzian plane p 0 − p θ 3 where the p θ 3 -axis lies on the two dimensional plane p 1 − p 2 making an angle 45 0 with the positive p 2 -axis. It is also evident that the first column of Imq does not lie on the two dimensional Lorentzian plane p 0 − p θ 3 although all the other columns of Imq lie on it.
To see thatp (1) ∈ T θ λ for all θ, let us considerP a for an arbitrary non-empty proper subset I of X. Therefore we have ImP a and it is timelike because a∈I Furthermore ImP Hence for any p ∈ θ T θ λ (thereby Im p ∈ θ C θ λ ⊂ R (n−1)D ) we can allow a small open ball B Im p in R (n−1)D centered at Im p such that for any point p ′ ∈ B Im p the aforementioned poles of the integrand are still at a finite distance away from the loop integration contours in a given Feynman diagram. Consequently, the same integral representation in terms of loop integrals holds at any of the new points p ′ . Therefore, by allowing such open balls for each p ∈ θ T θ λ we can make θ T θ λ open, in which the given Feynman diagram still remains analytic. In this way, θ T θ λ can be thickened individually for all the relevant Feynman diagrams (at all orders in perturbation theory).
E The cone C +

12
The cone C + 12 taken from the list (17) can be written as C + 12 resides on the manifold Im p 1 + · · · + Im p 4 = 0. Here we show how specifying Im p 2 in V −26 , and Im (p 2 + p 3 ) and Im (p 2 + p 4 ) in V + in turn determine the sign-valued map λ(I) (described by equation (4)) uniquely. Now we consider the following set of seven Im P I Im p 2 , Im p 3 , Im p 4 , Im (p 2 + p 3 ), Im (p 2 + p 4 ), Im (p 3 + p 4 ), Im (p 2 + p 3 + p 4 ) . (34) We want to see that for any Im p inside the cone C + 12 in which lightcone each element of the above set lies. Knowing this, similar information for any other possible Im P I can be determined using the relation Im p 1 + · · · + Im p 4 = 0. Now the following information can be obtained since for any Im p ∈ C + 12 we have −Im p 2 , Im (p 2 + p 3 ), Im (p 2 + p 4 ) ∈ V + .
Hence the signs λ(I) corresponding to the cone C + 12 is now known for any non-empty proper subset I of {1, . . . , 4}.
In section 3.2, with {−Im p 2 , Im (p 2 + p 3 ), Im (p 2 + p 4 )} as our basis, points in the cone C + 12 have been described. However any point in C + 12 can uniquely be written in a new basis given by {Im p 2 , Im p 3 , Im p 4 } since such a change of basis is a linear transformation L with det(L) = −1. This transformation law can be stated as: any point Q = (P α , P β , P γ ) written in the basis {−Im p 2 , Im (p 2 +p 3 ), Im (p 2 +p 4 )} can be written as L Q = (−P α , P α +P β , P α +P β ) in basis {Im p 2 , Im p 3 , Im p 4 }.
Hence the point Q in the cone C + 12 as given in (19) in the new basis reads as with conditions P 0 r > + D−1 i=1 (P i r ) 2 ∀r = α, β, γ. And the points in (20) in the new basis read as where ǫ satisfies the condition: 0 < ǫ < min P 0 r − D−1 i=1 (P i r ) 2 , r = α, β, γ . Clearly above points L Q, L Q 1 , L Q 2 and L Q 3 written in the basis {Im p 2 , Im p 3 , Im p 4 } are consistent with (35). Furthermore each columns of L Q 1 lie on the same two dimensional Lorentzian plane where P α lies. Similarly all the columns of L Q 2 lie on the two dimensional Lorentzian plane where P β lies, and all the columns of L Q 3 lie on the two dimensional Lorentzian plane where P γ lies. Now it is easy to check that the following relation holds Which depicts nothing but the linearity of L on (21).
Now for each term in the decomposition (40), evidently all the four columns lie on a two dimensional Lorentzian plane. We need to solve for P 1 , P 2 , P 3 , P 4 by inverting the following If we find that all the P r are in V + , then the proof is done and we can say thatC ′+ 12 = Ch θC ′+, θ 12 . But subject to conditions (41), solving (45) seems to be difficult analytically.