Analyticity of Off-shell Green's Functions in Superstring Field Theory
aa r X i v : . [ h e p - t h ] D ec Analyticity of Off-shell Green’s Functions in SuperstringField Theory
Ritabrata Bhattacharya ∗ and Ratul Mahanta †∗ Chennai Mathematical Institute,H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, India † Harish-Chandra Research Institute, HBNI,Chhatnag Road, Jhunsi, Allahabad 211019, India
Abstract
We consider the off-shell momentum space Green’s functions in closed superstring fieldtheory. Recently in [1], the off-shell Green’s functions — after explicitly removing contributionsof massless states — have been shown to be analytic on a domain (to be called the LESdomain) in complex external momenta variables. Analyticity of off-shell Green’s functions inlocal QFTs without massless states in the primitive domain is a well-known result. Usingcomplex Lorentz transformations and Bochner’s theorem allow to extend the LES domain toa larger subset of the primitive domain. For the 2-, 3- and 4-point functions, the full primitivedomain is recovered. For the 5-point function, we are not able to obtain the full primitivedomain analytically, only a large part of it is recovered. While this problem arises also forhigher-point functions, it is expected to be only a technical issue. electronic address: [email protected], [email protected] ontents ˜ D ′ T λ
165 Conclusions 17A Convexity of T λ , T ~θλ
18B Nonconvexity of S ~θ T ~θλ
19C Path-connectedness of S ~θ T ~θλ
20D Thickening of S ~θ T ~θλ
21E The cone C (4)+12
22F Difficulty arising for C ′ (5)+12 Closed superstring field theory (SFT) which is designed to reproduce perturbative amplitudesof superstring theory is a quantum field theory with countable infinite number of fields and non-local interaction vertices, whose action is directly written in momentum space (for a detailedreview, see [2]). It contains massless states. Off-shell momentum space amputated n -pointGreen’s function in SFT is defined by usual momentum space Feynman rules [2, 3]. It can Hereafter Green’s functions will always refer to momentum space amputated Green’s functions.
1e computed by summing over connected Feynman diagrams with n amputated external legscarrying ingoing D -momenta p , . . . , p n .We consider the off-shell n -point Green’s function G ( p , . . . , p n ) in SFT after explicitlyremoving contributions coming from one or more massless internal propagators (for more de-tails, see [1]). That is, the relevant part of the perturbative expansion of the off-shell n -pointGreen’s function keeps only those Feynman diagrams which do not contain any internal linecorresponding to a massless particle. We call this part of the off-shell Green’s function asthe infrared safe part . In [1] de Lacroix, Erbin and Sen (LES) showed that the infrared safepart of the off-shell n -point Green’s function G ( p , . . . , p n ) in SFT as a function of ( n − D complex variables (taking into account the momentum conservation P na =1 p a = 0 for externalmomenta) is analytic on a domain which we call as the LES domain. At the heart of thisresult, it has been proven that each of the relevant Feynman diagrams F( p , . . . , p n ) has anintegral representation in terms of loop integrals as presented below, whenever the externalmomenta lie on the LES domain.F( p , . . . , p n ) = Z L Y r =1 d D k r (2 π ) D f ( k , . . . , k L ; p , . . . , p n ) Q I s =1 (cid:16)(cid:0) ℓ s ( { k r } ; { p a } ) (cid:1) + m s (cid:17) . (1)The above analytic function F multiplied by a factor of (2 π ) D δ ( D ) ( p + · · · + p n ) gives theusual Feynman diagram. Equation (1) represents an n -legged L -loop graph with I internallines, where k r is a loop momentum, ℓ s is the momentum of the internal line with mass m s ( = 0) , and f is a regular function whenever its arguments take finite complex values. Thefunction f contains the product of the vertex factors associated with the vertices of the graph,as well as possible momentum dependence from the numerators of the internal propagators.The momentum ℓ s of an internal line is usually a linear combination of the loop momentaand the external momenta. Due to certain non-local properties of the vertices in SFT, thegraph is manifestly UV finite as long as for each r , k r integration contour ends at ± i ∞ , and k ir , i = 1 , . . . , ( D −
1) integration contours end at ±∞ . The prescription for the loop integrationcontours has been given as follows [3]. At origin, i.e. p a = 0 ∀ a = 1 , . . . , n each loop energyintegral is to be taken along the imaginary axis from − i ∞ to i ∞ and each spatial component Hereafter off-shell Green’s functions in SFT will refer to this part of respective off-shell Green’s functionsin SFT, if not explicitly stated. This part of the off-shell Green’s functions — when all the external particlesare massless — precisely gives the vertices of the Wilsonian effective field theory of massless fields, obtainedby integrating out the massive fields in superstring theory [2, 22]. In our notation, ℓ s = − ( ℓ s ) + P D − i =1 ( ℓ is ) for each s = 1 , . . . , I .
2f loop momenta is to be taken along the real axis from −∞ to ∞ . With this, F(0,. . . ,0) hasbeen shown to be finite as all the poles of the integrand in any complex k µr plane are at finitedistance away from the loop integration contour. As we vary the external momenta from theorigin to other complex values if some of such poles approach the k µr contour, the contour hasto be bent away from those poles keeping its ends fixed at ± i ∞ for loop energies and ±∞ forloop momenta. It has been shown that there exists a path inside the LES domain connectingthe origin to any other point p ≡ ( p , . . . , p n ) of the LES domain such that when we varyexternal momenta from origin to that point p along that path, the loop integration contoursin any graph can be deformed away avoiding poles of the integrand which approach them [1].Hence the integral representation (1) for F( p , . . . , p n ) when the external momenta lie on theLES domain is well defined where the poles of the integrand are at finite distance away fromthe (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 , . . . , p n ) as a function of ( n − D complex variables (externalingoing D -momenta p , . . . , p n subject to momentum conservation) is known to be analytic ona domain called the primitive domain [4–8]. This result follows from causality constraints onthe position space Green’s functions in a local QFT and representing the momentum spaceGreen’s functions as Fourier transforms of the position space correlators . The primitive do-main contains the LES domain as a proper subset. In several complex variables, the analyticitydomain cannot be arbitrary. For example, the shape of the primitive domain allows the actualdomain of holomorphy of G ( p , . . . , p n ) to be larger than itself (e.g. [9, 10]). This can be usedto prove various analyticity properties [11–18] of the S-matrix of QFTs, since the S-matrix isdefined as the on-shell connected n -point Green’s function for n ≥
3. These properties areto be read as the artifact of the shape of the primitive domain, irrespective of the functionalform of G ( p , . . . , p n ) which is defined on it. In particular, the derivations [10–12] of certainanalyticity properties [11,12,19,20] of the S-matrix use the information of only the LES domainas a subregion of the primitive domain.From [1] we already know that the infrared safe part of the off-shell n -point Green’s functionin SFT is analytic on the LES domain. Hence [1] basically established that in superstringtheory any possible departure from those analyticity properties of the full S-matrix that rely But the lack of a position space description of closed superstring field theory forces us to work directly inthe momentum space. For example, [10] recovered the JLD domain, [11] proved the crossing symmetry of the 2 → without massless states,analyticity properties [13–18] of the S-matrix rely on the properties of the primitive domain thatare not restricted to its LES subregion. We know that local QFTs with massless states couldpossibly deviate from these properties for their full S-matrix. However, any such departuresare entirely due to the presence of massless states, i.e. the infrared safe parts of respectiveamplitudes in local QFTs with massless states must satisfy all these analyticity properties.At this stage, it is natural to ask that whether the infrared safe part of the S-matrix of thesuperstring theory (despite having non-local vertices) satisfy the last-mentioned analyticityproperties, or not. They satisfy them, only if the relevant part of the off-shell n -point Green’sfunction in SFT can be shown to be analytic on the full primitive domain extending the LESdomain.In this paper, we aim to generalize the result of [1] by showing that the infrared safepart of the off-shell n -point Green’s function in SFT is analytic on a larger domain thanthe LES domain. As will be reviewed in section 2.2, the analyticity property of the off-shellGreen’s functions in SFT is invariant under the action of a D -dimensional complex Lorentztransformation on all the external momenta. Thus the off-shell Green’s functions in SFT arealso analytic at points that are obtained by the action of such transformations on points inthe LES domain [1]. We consider LES domain adjoining these new points. The primitivedomain essentially contains the union of a certain family of convex tube domains. We call suchtubes as the primitive tubes. Within each such primitive tube, we identify a connected tubewhich is also contained in the LES domain and its shape allows us to carry out a holomorphicextension to its convex hull inside the corresponding primitive tube, due to a classic theoremby Bochner [21]. The domain thus found may still be smaller. We explicitly work out thecases of the three-, four- and five-point functions to determine whether such convex hulls fullyobtain respective primitive tubes, or not. The extension to the primitive domain is trivialfor the two-point function. In the case of three-point function, indeed such extensions yieldall the 6 possible primitive tubes. Also for the four-point function, by such extensions allthe 32 possible primitive tubes are obtained. Whereas for the five-point function, out of 370possible primitive tubes, for 350 of them we are able to show that such extensions obtain eachof them fully. The technique that we employ for aforesaid checks seems difficult to implement4nalytically for the remaining 20 primitive tubes whose shapes are complicated. This technicaldifficulty is a feature of the higher point functions as well. However in any case, our workestablishes that based on a geometric consideration only, the LES domain is holomorphicallyextended inside all the primitive tubes. This extension does not depend on the details of theGreen’s functions. Thus with respect to all the analyticity properties of the S-matrix whichcan be obtained from this extended domain , superstring theory behaves like a standard localQFT that has massless states.We organize the paper as follows. In section 2, we briefly review certain properties of theprimitive domain and the LES domain which are useful for our purpose. In section 3, we startwith the general scheme to extend the LES domain holomorphically. We apply this schemeto 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-pointfunction. We discuss certain real limits within each of the primitive tubes in section 4. In this section we review certain properties of the two domains, namely the primitive domainand the LES domain which will be useful for our analysis. Both the domains are domains inthe complex manifold C ( n − D given by p + · · · + p n = 0. The origin of C ( n − D which is givenby p a = 0 ∀ a = 1 , . . . , n will be denoted by O . We count p a as positive if ingoing, negativeotherwise. We shall use Minkowski metric with mostly plus signature. The primitive domain D is given by D = (cid:26) p ≡ ( p , . . . , p n ) : X na =1 p a = 0 and for each I ∈ ℘ ∗ ( X )either, Im P I = 0 , (Im P I ) ≤ P I = 0 , − P I < M I (cid:27) , (2)where X = { , . . . , n } is the set of first n natural numbers. ℘ ∗ ( X ) = { I ( X, except ∅} is thecollection of all non-empty proper subsets of X . P I is defined to be equal to P a ∈ I p a . M I isthe threshold of production of any (multi-particle) state in a channel containing the external Note that for the 2-, 3- and 4-point functions, the extended domain is equal to the primitive domain. { p a , a ∈ I } , i.e. M I is the invariant threshold mass for producing any set ofintermediate states in the collision of particles carrying total momentum P I .Clearly, O ∈ D . Primitive domain is star-shaped with respect to O , i.e. the straight linesegment connecting O and any point p ∈ D which is given by (cid:8) tp : t ∈ [0 , (cid:9) lie entirelyinside D . Hence the primitive domain is path-connected as any two points p (1) , p (2) ∈ D canbe connected via the straight line segments p (1) O and Op (2) . Furthermore primitive domain issimply 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 domains denoted by {T λ , λ ∈ Λ ( n ) } [7, 9, 23, 24]. Any member T λ of this family will be calleda primitive tube. To describe this family of tubes the following definitions are needed. Weconsider the space R n − of n real variables s , . . . , s n linked by the relation s + · · · + s n = 0.We define S I = P 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 − into openconvex cones with common apex at the origin. Any such cone will be called a cell, γ λ . Withina cell each S I is of definite sign λ ( I ). Thus a cell can be written as γ λ = (cid:26) s ∈ R n − : λ ( I ) S I > ∀ I ∈ ℘ ∗ ( X ) (cid:27) , (3)where λ : ℘ ∗ ( X ) → {− , } is a sign-valued map with the following properties.(i) ∀ I ∈ ℘ ∗ ( X ) λ ( I ) = − λ ( X \ I ) , (ii) ∀ I, J ∈ ℘ ∗ ( X ) with I ∩ J = ∅ and λ ( I ) = λ ( J ) λ ( I ∪ J ) = λ ( I ) = λ ( J ) . (4)The first property is compatible with s + · · · + s n = 0 and the second property is compatiblewith 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 A subset of C m is called a tube if it is equal to R m + iA for some subset A of R m where m is a givennatural number. A is called the base of the tube. A subset A of R m is called a cone if any point p ∈ A = ⇒ αp ∈ A ∀ α > T λ (primitive tube) given by T λ = (cid:26) p ≡ ( p , . . . , p n ) : X na =1 p a = 0 , λ ( I )Im P I ∈ V + ∀ I ∈ ℘ ∗ ( X ) (cid:27) = R ( n − D + i C λ , C λ = (cid:26) Im p ∈ R ( n − D : λ ( I )Im P I ∈ V + ∀ I ∈ ℘ ∗ ( X ) (cid:27) , (5)where V + is the open forward lightcone in R D . C λ is the conical base of the tube T λ . Althoughthe primitive domain is non-convex the primitive tubes T λ are convex (see appendix A). Henceeach tube T λ is path-connected as the entire straight line segment p (1) p (2) connecting any twopoints p (1) , p (2) ∈ T λ is contained in the tube T λ . The LES domain is given by D ′ = (cid:26) p ≡ ( p , . . . , p n ) : ∀ a Im p µa = 0 , µ = 0 , X na =1 p a = 0 and for each I ∈ ℘ ∗ ( X )either, Im P I = 0 , (Im P I ) ≤ P I = 0 , − P I < M I (cid:27) , (6)where all the Im p a thereby Im P I are allowed to lie only on the two dimensional Lorentzianplane p − p . Clearly O ∈ D ′ and the LES domain D ′ is contained in the primitive domain D . In [1] it has been argued that the domain of holomorphy of the n -point Green’s function G ( p , . . . , p n ) in SFT is a connected region in C ( n − D containing the origin, and it is invariantunder the action of Lorentz transformations ˜Λ with complex parameters, i.e. ˜Λ is any complexmatrix satisfying ˜Λ T η ˜Λ = η for η being the Minkowski metric in R D . We call the set ofsuch matrices the complex Lorentz group, L . In general, the action of a complex Lorentztransformation ˜Λ is defined on the complex manifold C ( n − D taking a point to another pointof the same manifold given by ( p , . . . , p n ) ( ˜Λ p , . . . , ˜Λ p n ) , (7) With i = √− By a two dimensional Lorentzian plane we refer to any two dimensional plane in R D which contains the p -axis. p ˜Λ p . Note that the same ˜Λ acts on all p a .As as consequence, the result of [1] automatically generalizes to a larger domain than theLES domain D ′ , i.e. G ( p ) is analytic on the domain ˜ D ′ given by˜ D ′ = (cid:26) ˜Λ p : p ∈ D ′ , ˜Λ ∈ L (cid:27) . (8)Clearly ˜ D ′ ⊃ D ′ , since L contains the identity matrix. Hereafter we refer ˜ D ′ as the LESdomain. ˜ D ′ 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 containedin the LES domain ˜ D ′ (described in section 2.2). Any member of this family is convex and itcan be characterized by a set of D − ~θ which specifies a two dimensional Lorentzianplane p − p ~θ with ~θ = 0 specifying the p − p plane. Hence we denote a member by T ~θλ .The convex tube T ~θλ is given by T ~θλ = (cid:26) p ≡ ( p , . . . , p n ) : ∀ a Im p a ∈ p − p ~θ plane; X na =1 p a = 0 , and λ ( I )Im P I ∈ V + ∀ I ∈ ℘ ∗ ( X ) (cid:27) = R ( n − D + i C ~θλ , C ~θλ = (cid:26) (Im p , . . . , Im p n ) on manifold X na =1 Im p a = 0 such that ∀ a Im p a ∈ p − p ~θ plane and λ ( I )Im P I ∈ V + ∀ I ∈ ℘ ∗ ( X ) (cid:27) , (9)where the base C ~θλ is a subset of R ( n − D . Any such tube T ~θλ can be obtained by acting a realrotation on the tube T ~θ =0 λ . Here the axis p ~θ lies in the subspace R D − of points ( p , . . . , p D − ). It can be specified by a point on theunit sphere S D − in R D − . With a set of D − θ , . . . , θ D − ) ≡ ~θ where 0 ≤ θ , . . . , θ D − ≤ π and0 ≤ θ D − < π , the axis p ~θ can be explicitly written as p ~θ = (cos θ , sin θ cos θ , sin θ sin θ cos θ , . . . , sin θ · · · sin θ D − cos θ D − ) . S ~θ T ~θλ lies inside the primitive tube T λ as well as it is contained in the LES domain˜ D ′ . Now S ~θ T ~θλ is a tube given by R ( n − D + i ( S ~θ C ~θλ ). Although each tube T ~θλ is convex (seeappendix A) the tube S ~θ T ~θλ is non-convex (see appendix B). However the tube S ~θ T ~θλ ispath-connected (see appendix C).We apply Bochner’s tube theorem [21, 25] on the connected tube ( S ~θ T ~θλ ) . The theoremstates that any open connected tube R m + iA has a holomorphic extension to the domain R m + i Ch( A ) where Ch( A ) is the smallest convex set containing the set A , called the convexhull of A .By the above application, since S ~θ C ~θλ is non-convex we always have a holomorphic extensionof S ~θ T ~θλ to a domain given by R ( n − D + i Ch( S ~θ C ~θλ ). Furthermore C λ contains Ch( S ~θ C ~θλ ) since C λ is a convex cone containing S ~θ C ~θλ . In subsequent subsections, we deal with the explicit casesof the three-, four- and five-point functions, where up to the four-point function we obtain thatfor each C λ such extension yields the full of C λ , i.e. Ch( S ~θ C ~θλ ) = C λ , and for five-point functionwe obtain subcases in which we are able to prove this equality.For the two-point function (i.e. n = 2), T λ = S ~θ T ~θλ . That is, whenever p ∈ T λ , Im p (= − Im p ) lies on some two dimensional Lorentzian plane. Remarks
The properties of the tube S ~θ T ~θλ as a domain in several complex variables have been used hereto extend it holomorphically. From the work of [1], we know that all the relevant Feynmandiagrams (those which do not have any internal line of a massless particle) in the perturbativeexpansion of the n -point Green’s function are analytic in the common tube S ~θ T ~θλ . Hence ourextension of S ~θ T ~θλ is valid to all orders in perturbation theory.A proper application of Bochner’s tube theorem requires us to thicken the connected tube S ~θ T ~θλ in order to make it open. But the thickened tubes (as in appendix D) are not identical forall the Feynman diagrams. However the intersection of all these thickened tubes correspondingto distinct diagrams certainly contains S ~θ T ~θλ . We consider any relevant Feynman diagram.The corresponding thickened tube can be holomorphically extended to its convex hull due to S ~θ T ~θλ is non-convex when n > In [25] it has been stated as the ‘convex tube theorem’ at the end of its third section. This version of thetheorem is suitable for our purpose. This tube can be thickened in order to make it open (see appendix D). By a holomorphic extension Ω ′ of a domain Ω in C m we mean any larger domain Ω ′ containing Ω, withthe property that all the functions which are holomorphic on Ω are also holomorphic on Ω ′ . S ~θ T ~θλ ) = R ( n − D + i Ch( S ~θ C ~θλ ). Clearly, Ch( S ~θ T ~θλ ) lies in the intersection of all such extensions corresponding todistinct diagrams since Ch( S ~θ T ~θλ ) only includes convex combinations of points from S ~θ T ~θλ .Hence Ch( S ~θ T ~θλ ) is the domain where all the relevant Feynman diagrams (at all orders inperturbation theory) are analytic. 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 containsthe union of 6 mutually disjoint tubes denoted by {T (3) ± a , a = 1 , , } and these primitive tubesare given by T (3) ± a = (cid:26) p ∈ C D : Im p ∈ C (3) ± a (cid:27) , (10)where p = ( p , p , p ) is linked by p + p + p = 0. Their conical bases are defined by C (3)+ a = −C (3) − a = (cid:26) Im p : Im p b , Im p c ∈ V + (cid:27) , (11)where ( abc ) = permutation of (123). In order to define each of the above cones C (3)+ a ( C (3) − a ),we require a certain pair of imaginary external momenta which (or their negative) are specifiedto be in the open forward lightcone V + . For a given conical base, this in turn fixes all otherIm P I to be in specific lightcone .The cones (11) reside on the manifold Im p + Im p + Im p = 0. In order to assigncoordinates to the points in C (3)+ a , let us choose { Im p b , Im p c } as our set of basis vectors. Onthe other hand, for C (3) − a , let us choose {− Im p b , − Im p c } as our basis. With this, any of theabove cones is contained in a R D and is of the following common form C (3) = (cid:26) ~Q = ( P α , P β ) : P α , P β ∈ V + (cid:27) , (12)where any ~Q ∈ C (3) can be written as a D × ~Q = P α P β P α P β ... ... P D − α P D − β , (13) Primitive tubes are generally defined in (5). Here, an additional superscript (3) in the notations for thetubes stands for the 3-point function. For n = 3, the total number of possible P I = 2 − P r > + qP D − i =1 ( P ir ) ∀ r = α, β ensuring that both the columns belong tothe forward lightcone V + . Hence given a ~Q , the quantities P r − pP i ( P ir ) , r = α, β are apair of positive numbers. Furthermore, the two columns of ~Q in general do not lie on a sametwo dimensional Lorentzian plane.Now we consider cones C (3) ~θ containing points ~ ˜ Q where both the columns not only belongto V + but also lie on a same two dimensional Lorentzian plane p − p ~θ characterized by a ~θ . We shall show that taking points from these cones for various ~θ , a convex combination ofthem represents given ~Q in (13). This will complete the proof of Ch( S ~θ C (3) ~θ ) = C (3) , since wealready have C (3) ⊃ Ch( S ~θ C (3) ~θ ) as discussed right above the remarks in section 3.We consider two points ~ ˜ Q ∈ C (3) ~θ and ~ ˜ Q ∈ C (3) ~θ given by ~ ˜ Q = 2 P α − ǫ ǫP α P D − α , ~ ˜ Q = 2 ǫ P β − ǫ P β ... ...0 P D − β , (14)where we take any ǫ satisfying 0 < ǫ < min (cid:8) P r − qP D − i =1 ( P ir ) , r = α, β (cid:9) so that foreach r = α, β we have P r − ǫ > + qP D − i =1 ( P ir ) . Both the columns of ~ ˜ Q lie on a same twodimensional Lorentzian plane p − p ~θ where also the first column P α of ~Q in (13) lies. Similarly,both the columns of ~ ˜ Q lie on the two dimensional Lorentzian plane where P β lies. Now it iseasy to check that the following relation holds ~ ˜ Q ~ ˜ Q ~Q . (15)Equation (15) establishes that each of the 6 primitive tubes given in (10) can be obtained asholomorphic extension of a tube where the latter is contained in the LES domain ˜ D ′ . For four-point function, we have n = 4 and the maps λ ( I ) (described by equation (4)) can begiven explicitly as follows. In this case, the primitive domain D essentially contains the unionof 32 mutually disjoint tubes denoted by {T (4) ± a , T (4) ± ab , ≤ a, b ≤ , a = b } and theseprimitive tubes are given by [7, 9] T (4) ± a = (cid:26) p ∈ C D : Im p ∈ C (4) ± a (cid:27) , T (4) ± ab = (cid:26) p ∈ C D : Im p ∈ C (4) ± ab (cid:27) , (16) Here a superscript (4) in the notations for the tubes stands for the 4-point function. p = ( p , . . . , p ) is linked by p + · · · + p = 0. Their conical bases are defined by C (4)+ a = −C (4) − a = (cid:26) Im p : Im p b , Im p c , Im p d ∈ V + (cid:27) , C (4)+ ab = −C (4) − ab = (cid:26) Im p : − Im p b , Im ( p b + p c ) , Im ( p b + p d ) ∈ V + (cid:27) , (17)where ( abcd ) = permutation of (1234). Note that in order to describe each of the above coneswe require a certain set of three Im P I each of which (or its negative) is specified to be in theopen forward lightcone V + . For a given conical base, this in turn fixes all other Im P I to bein specific lightcone (see appendix E).The cones (17) reside on the manifold Im p + · · · + Im p = 0. Due to this link wecan choose any three linear combinations of Im p , . . . , Im p which are linearly independentas 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 (4)+ a , whereas we choose {− Im p b , − Im p c , − Im p d } forthe cones C (4) − a . Besides, as our basis, we choose {− Im p b , Im ( p b + p c ) , Im ( p b + p d ) } forthe cones C (4)+ ab , whereas we choose { Im p b , − Im ( p b + p c ) , − Im ( p b + p d ) } for the cones C (4) − ab .With this, any of the above cones is contained in a R D and is of the following common form C (4) = (cid:26) ~Q = ( P α , P β , P γ ) : P α , P β , P γ ∈ V + (cid:27) , (18)where any ~Q ∈ C (4) can be written as a D × ~Q = P α P β P γ P α P β P γ ... ... ... P D − α P D − β P D − γ , (19)with conditions P r > + qP D − i =1 ( P ir ) ∀ r = α, β, γ ensuring that each of the columns belongto the forward lightcone V + . Hence given a ~Q the quantities P r − pP i ( P ir ) , r = α, β, γ arethree positive numbers. Furthermore the columns of ~Q in general do not lie on a same twodimensional Lorentzian plane.Now we consider cones C (4) ~θ containing points ~ ˜ Q where all the three columns not onlybelong to V + but also lie on the same two dimensional Lorentzian plane p − p ~θ characterized For n = 4, the total number of possible P I = 2 − 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 (4)+ ab .This change of basis is a linear invertible transformation and the work of this subsection can be recast in thisnew basis (e.g., see appendix E).
12y a ~θ . We shall show that taking points from these cones for various ~θ , a convex combinationof them represents given ~Q in (19). This will complete the proof of Ch( S ~θ C (4) ~θ ) = C (4) , sincewe already have C (4) ⊃ Ch( S ~θ C (4) ~θ ) as discussed right above the remarks in section 3.We consider three points ~ ˜ Q ∈ C (4) ~θ , ~ ˜ Q ∈ C (4) ~θ and ~ ˜ Q ∈ C (4) ~θ given by ~ ˜ Q = 3 P α − ǫ ǫ/ ǫ/ P α P D − α , ~ ˜ Q = 3 ǫ/ P β − ǫ ǫ/ P β P D − β ,~ ˜ Q = 3 ǫ/ ǫ/ P γ − ǫ P γ ... ... ...0 0 P D − γ , (20)where we take any ǫ satisfying 0 < ǫ < min (cid:8) P r − qP D − i =1 ( P ir ) , r = α, β, γ (cid:9) so that foreach r = α, β, γ we have P r − ǫ > + qP D − i =1 ( P ir ) . All the three columns of ~ ˜ Q lie on a sametwo dimensional Lorentzian plane p − p ~θ where also the first column P α of ~Q in (19) lies.Similarly, all the columns of ~ ˜ Q lie on the two dimensional Lorentzian plane where P β lies, andall the columns of ~ ˜ Q lie on the two dimensional Lorentzian plane where P γ lies. Now it is easyto check that the following relation holds ~ ˜ Q ~ ˜ Q ~ ˜ Q ~Q . (21)Equation (21) establishes that each of the 32 primitive tubes given in (16) can be obtained asholomorphic extension of a tube where the latter is contained in the LES domain ˜ D ′ .13 .3 Five-point function For five-point function, we have n = 5 and in this case the primitive domain D essentiallycontains the union of 370 mutually disjoint tubes whose conical bases are given by [7] C (5)+ a = −C (5) − a = (cid:26) Im p : Im p b , Im p c , Im p d , Im p e ∈ V + (cid:27) , C (5)+ ab = −C (5) − ab = (cid:26) Im p : − Im p b , Im ( p b + p c ) , Im ( p b + p d ) , Im ( p b + p e ) ∈ V + (cid:27) , C ′ (5)+ ab = −C ′ (5) − ab = (cid:26) Im p : Im ( p a + p c ) , Im ( p a + p d ) , Im ( p a + p e ) , Im ( p b + p c ) , Im ( p b + p d ) , Im ( p b + p e ) ∈ V + (cid:27) , C (5)+ ab,c = −C (5) − ab,c = (cid:26) Im p : Im p c , − Im ( p b + p c ) , Im ( p b + p d ) , Im ( p b + p e ) ∈ V + (cid:27) , C ′ (5)+ ab,c = −C ′ (5) − ab,c = (cid:26) Im p : − Im ( p b + p c ) , Im ( p a + p c ) , Im ( p b + p d ) , Im ( p b + p e ) ∈ V + (cid:27) , C (5)+ a,bc = −C (5) − a,bc = (cid:26) Im p : Im p d , Im p e , Im ( p a + p b ) , Im ( p a + p c ) ∈ V + (cid:27) , (22)where ( abcde ) = permutation of (12345) and Im p = (Im p , . . . , Im p ) is linked by the relationIm p + · · · + Im p = 0. Due to this link we can choose any four linear combinations ofIm p , . . . , Im p which are linearly independent as our set of basis vectors, to describe a conewhich is given from the above list (22). Hence any of these cones is contained in a R D with achoice for a basis.We note that in order to describe each of the cones in (22) except the cones C ′ (5)+ ab , C ′ (5) − ab ,we require a certain set of four Im P I each of which (or its negative) is specified to be in theopen forward lightcone V + . This in turn fixes all other Im P I to be in specific lightcone . Nowwe confine ourselves to these cones which are 350 in numbers . To describe any of these coneswe choose the corresponding certain set of four Im P I as our basis (in a similar manner to thecases of the three-point and four-point functions, as demonstrated in detail in the sections 3.1, Here a superscript (5) in the notations for the cones stands for the 5-point function. And following [7] weuse a prime only to distinguish between two classes of cones having same indices ( ab ) or ( ab, c ). For n = 5, the total number of possible P I = 2 − Each of C ′ (5)+ ab and C ′ (5) − ab is symmetric under the interchange of a, b which is evident from (22). Hence theyare 20 in total. C (5) = (cid:26) ~Q = ( P α , P β , P γ , P δ ) : P α , P β , P γ , P δ ∈ V + (cid:27) , (23)where any ~Q ∈ C (5) can be written as a D × ~Q = P α P β P γ P δ P α P β P γ P δ ... ... ... ... P D − α P D − β P D − γ P D − δ , (24)with conditions P r > + qP D − i =1 ( P ir ) ∀ r = α, β, γ, δ . Given a ~Q as in (24) it can now berepresented as the following convex combination. ~ ˜ Q ~ ˜ Q ~ ˜ Q ~ ˜ Q ~Q , (25)where ~ ˜ Q r , r = 1 , . . . , ~ ˜ Q = 4 P α − ǫ ǫ/ ǫ/ ǫ/ P α P D − α , ~ ˜ Q = 4 ǫ/ P β − ǫ ǫ/ ǫ/ P β P D − β ,~ ˜ Q = 4 ǫ/ ǫ/ P γ − ǫ ǫ/
30 0 P γ P D − γ , ~ ˜ Q = 4 ǫ/ ǫ/ ǫ/ P δ − ǫ P δ ... ... ... ...0 0 0 P D − δ , (26)where we take any ǫ satisfying the condition: 0 < ǫ < min (cid:8) P r − qP D − i =1 ( P ir ) , r = α, β, γ, δ (cid:9) .Equation (25) establishes that each of the primitive tubes described by (22) except theones whose conical bases are C ′ (5)+ ab , C ′ (5) − ab can be obtained as holomorphic extension of a tubewhere the latter is contained in the LES domain ˜ D ′ .The above technique has limitations. Following difficulty arrises when we consider theremaining 20 cones which are given by C ′ (5)+ ab , C ′ (5) − ab . To describe points in C ′ (5)+ ab let us choosethe set (cid:26) Im ( p a + p c ) , Im ( p a + p d ) , Im ( p a + p e ) , Im ( p b + p c ) (cid:27)
15s our basis, and to describe points in ˜ C ′ (5) − ab let us choose the set (cid:26) − Im ( p a + p c ) , − Im ( p a + p d ) , − Im ( p a + p e ) , − Im ( p b + p c ) (cid:27) as our basis. With this any of the cones C ′ (5)+ ab , C ′ (5) − ab is of the following common form C ′ (5) = (cid:26) ~Q = ( P α , P β , P γ , P δ ) : P α , P β , P γ , P δ , ( P β + P δ − P α ) and ( P γ + P δ − P α ) ∈ V + (cid:27) . (27)Due to additional constraints on the linear combinations ( P β + P δ − P α ) and ( P γ + P δ − P α ) thetechnique which we have employed in earlier cases seems difficult to implement here analytically,in order to check the validity of Ch( S ~θ C ′ (5) ~θ ) = C ′ (5) . Here each cone C ′ (5) ~θ is to be obtainedfrom C ′ (5) 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 − p ~θ . That is, it is difficult to finda set of points ~ ˜ Q r , r = 1 , . . . , m for some m , each of which has four columns satisfying thesix conditions as stated in (27) and furthermore all the four columns lie on a two dimensionalLorentzian plane, in such a way that a convex combination of these m points produce a generalpoint ~Q in (27). As an illustration we work with one of these 20 problematic cones in appendixF (in which case, as a trial we take m = 4). T λ In section 3, we have shown that for an n -point Green’s function and given any λ from thepossible set Λ ( n ) , the tube S ~θ T ~θλ has holomorphic extension inside the primitive tube T λ wherethe former tube is contained in the LES domain ˜ D ′ .As per the equations (6) and (8), if we take the limit Im P I → , Im P I ∈ T ~θλ for a collectionof subsets { I } ⊂ ℘ ∗ ( X ), the n -point Green’s function G ( p ) in SFT is finite whenever we restricttheir real parts by − P I < M I for each I belonging to that collection { I } . Here Re P J arekept arbitrary for all J ∈ ℘ ∗ ( X ) \ { I } . In fact, for a given collection { I } by taking such limitswithin T ~θλ for any ~θ and restricting corresponding real parts, we reach to same value G ( p ) forall ~θ . Now that S ~θ T ~θλ has an unique holomorphic extension given by Ch( S ~θ T ~θλ ) ⊂ T λ , wereach to above value G ( p ) in the limit Im P I → , Im P I ∈ Ch( S ~θ T ~θλ ) with above constraintson the real parts.Evidently the family of holomorphic functions { G λ ( p ) , λ ∈ Λ ( n ) } defined on the family Here m ≤ D + 1, due to Carath´eodory’s theorem: if A is a non-empty subset of R q , then any point of theconvex hull of A is representable as a convex combination of at most q + 1 points of A . Here G λ ( p ) denotes the analytic continuation of G ( p ) defined on S ~θ T ~θλ to the domain Ch( S ~θ T ~θλ ).
16f mutually disjoint tubes { Ch( S ~θ T ~θλ ) , λ ∈ Λ ( n ) } coincide on a real domain R given by R = (cid:26) p ≡ ( p , . . . , p n ) : X na =1 p a = 0 and ∀ I ∈ ℘ ∗ ( X ) − P I < M I (cid:27) . (28) In this paper, we have shown that for any n -point Green’s function in superstring field theory,the LES domain ˜ D ′ due to its shape always admits a holomorphic extension within the primitivedomain D where the latter is basically the union of the convex primitive tubes. In the processwe have found that the LES domain ˜ D ′ contains a non-convex connected tube within eachconvex primitive tube. The former tube being non-convex allows to include all the new pointsfrom its convex hull which is the set of all convex combinations of points in that tube. Theconvex tube thus obtained is a holomorphic extension of the former non-convex tube due to aclassic 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. Wehave proved this result, in section 3.1 for the three-point function obtaining all the 6 primitivetubes, and in section 3.2 for the four-point function obtaining all the 32 primitive tubes. Theappropriate real limits within those tubes in both cases are also attained (as discussed insection 4).In section 3.3, we are able to show that for the five-point function such extension yieldsthe 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 20primitive tubes, their shape being complicated. However within all these 370 extensions insiderespective 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 ofthe S-matrix which can be obtained relying on above extended domain inside the primitivedomain, the infrared safe part of the S-matrix of superstring theory has similar behaviour tothat of a standard local QFT. Any non-analyticity of the full S-matrix of SFT is entirely due tothe presence of massless states — which is also the case for a standard local QFT. The currentapproach is perturbative (because it uses Feynman diagrams) whereas the original proof ofprimitive analyticity for local QFTs is non-perturbative. Thus superstring amplitudes mightalso have potential singularities on the primitive domain arising from non-perturbative effects.17ocal QFTs are free from those. Furthermore, in local QFTs the following estimate holds ona primitive tube T λ for each truncated cone K r ⊂ C λ ∪ { } . (cid:12)(cid:12) G (Re p + i Im p ) (cid:12)(cid:12) ≤ A (1+ k Re p k ) m k Im p k l ∀ Re p ∈ R ( n − D , Im p ∈ K r \ { } , (29)where the numbers A, m, l > K r [4] . This is guaranteed as the off-shell n -pointGreen’s function in a local QFT is equal to the Fourier-Laplace transform of some generalizedfunction (more precisely, a tempered distribution) which is a position space correlator, andthe analyticity of the off-shell Green’s function on a primitive tube follows from causalityconstraints on the position space correlators. Equation (29) is a place where superstring fieldtheory could differ since typically its non-local vertices prevent us from defining position spacecorrelators.The difficulty arising for the twenty primitive tubes in the case for the five-point functionhas been demonstrated in appendix F. This is a generic feature that arises for all higher-point functions, in the course of determining whether or not the application of Bochner’s tubetheorem yields certain primitive tubes fully. Solving this may require numerical analysis. Weleave this for future work. Acknowledgements
We would like to thank Ashoke Sen for useful discussions. We also thank Harold Erbin andAnshuman Maharana for useful comments on the draft.
A Convexity of T λ , T ~θλ The tube T λ (described by equation (5)) is convex [13]. We derive it as follows. We take any m points p (1) , . . . , p ( m ) ∈ T λ and consider the convex combination q = P mr =1 t r p ( r ) where each t r ≥ P mr =1 t r = 1. Now if q ∈ T λ then T λ is convex. A cone has been generally defined in footnote 8. Given a cone K and a positive number r , a truncatedcone is defined as (using k v k to denote the Euclidean norm of v ): K r = { v ∈ K : k v k≤ r } . (cid:12)(cid:12) z (cid:12)(cid:12) denotes the modulus of a complex number z . P na =1 q a = 0. We define Q I = P a ∈ I q a for each I ∈ ℘ ∗ ( X ). Hence we get Q I = P mr =1 t r P ( r ) I where for each r we have P ( r ) I = P a ∈ I p ( r ) a . We also have λ ( I )Im P ( r ) I ∈ V + since p (1) , . . . , p ( m ) ∈ T λ . Therefore λ ( I )Im Q I = P mr =1 t r λ ( I )Im P ( r ) I ∈ V + . Hence q ∈ T λ .This proves the convexity of T λ .Taking any m points p (1) , . . . , p ( m ) ∈ T ~θλ (described by equation (9)), now we show that theconvex combination q = P mr =1 t r p ( r ) belongs to T ~θλ . In present case, since for each r = 1 , . . . , m the Im p ( r ) a ∀ a , in turn all Im P ( r ) I lie on the two dimensional Lorentzian plane p − p ~θ , thereforewe get that the Im Q I for all I including singletons lie on the same plane p − p ~θ . Hence T ~θλ is convex following similar steps to the above case. B Nonconvexity of S ~θ T ~θλ We take a point p (1) = ( p (1)1 , . . . , p (1) n ) ∈ T ~θ =0 λ . Hence Im p (1) a ∀ a = 1 , . . . , n lie on the twodimensional Lorentzian plane p − p . Now suppose the point p (2) is obtained by acting areal rotation on the point p (1) so that Im p (2) a ∀ a = 1 , . . . , n now lie on the two dimen-sional Lorentzian plane p − p . Hence p (2) ∈ T ~θ λ where ~θ characterizes the two dimensionalLorentzian plane p − p . Now for the point q = p (1) + p (2) which is the mid-point of thestraight line segment connecting the two points p (1) , p (2) we have Im q a ∀ a = 1 , . . . , n lying onthe two dimensional Lorentzian plane p − p ~θ where the p ~θ -axis lies on the two dimensionalplane p − p making an angle 45 with the positive p -axis. Hence the point q ∈ T ~θ λ . Howeverwe can change Im p (2)1 little bit, keeping all real parts and Im p (2) a , a = 2 , . . . , n unchanged toobtain a new point ˜ p (2) such that ˜ p (2) still belongs to T ~θ λ . Consequently we get two points p (1) ∈ T ~θ =0 λ and ˜ p (2) ∈ T ~θ λ , for which the mid-point of the straight line segment connectingthem is given by ˜ q = p (1) + ˜ p (2) . Although Im ˜ q a ∀ a = 2 , . . . , n lie on the two dimensionalLorentzian plane p − p ~θ , the D -momenta Im ˜ q = p (1)1 + ˜ p (2)1 does not lie on the two di-mensional Lorentzian plane p − p ~θ anymore. Hence ˜ q / ∈ S ~θ T ~θλ . In other words, S ~θ T ~θλ isnon-convex.To see this, let us take { p , . . . , p n − } as our basis to describe points on the complex manifold p + · · · + p n = 0. A generic point p = ( p , . . . , p n − ) on this manifold can be represented byan unique D × ( n −
1) matrix where the a -th column represent the D -momenta p a . Since each T ~θλ thereby S ~θ T ~θλ reside on this manifold now the imaginary parts of the points p (1) , p (2) , ˜ p (2) , In equation (7), we have defined actions of complex Lorentz transformations which include real rotations. In support of this see appendix D. and ˜ q can be represented in terms of D × ( n −
1) matrices as follows . p (1) = p p · · · p n − p p · · · p n − · · ·
00 0 · · · · · · , p (2) = p p · · · p n − · · · p p · · · p n − · · · · · · , ˜ p (2) = p p · · · p n − · · · p p · · · p n − · · · · · · ,q = p p · · · p n − p
11 12 p · · · p n − p
11 12 p · · · p n − · · · · · · , ˜ q = p p · · · p n − p
11 12 p · · · p n − ˜ p
11 12 p · · · p n − · · · · · · . (30)Clearly all the columns of Im q lie on the two dimensional Lorentzian plane p − p ~θ wherethe p ~θ -axis lies on the two dimensional plane p − p making an angle 45 with the positive p -axis. It is also evident that the first column of Im ˜ q does not lie on the two dimensionalLorentzian plane p − p ~θ although all the other columns of Im ˜ q lie on it. C Path-connectedness of S ~θ T ~θλ We take any two points p (1) , p (2) ∈ S ~θ T ~θλ . To show that the tube S ~θ T ~θλ is path-connected itis sufficient to find a path connecting the points p (1) , p (2) staying inside the tube S ~θ T ~θλ .Let p (1) ∈ T ~θ λ and p (2) ∈ T ~θ λ for some ~θ , ~θ . Now if ~θ = ~θ (= ~θ, say) then p (1) , p (2) belong to the same tube T ~θλ . Since the tube T ~θλ is convex (see appendix A) the straight linesegment connecting p (1) , p (2) lies entirely inside T ~θλ . Now we consider the case when ~θ = ~θ . Inthis case p (1) = ( p (1)1 , . . . , p (1) n ), where all the Im p (1) a thereby Im P (1) I lie on the two dimensionalLorentzian plane p − p ~θ . We consider the point ˜ p (1) = (˜ p (1)1 , . . . , ˜ p (1) n ) ∈ C ( n − D given by ∀ a Im ˜ p (1)0 a = Im p (1)0 a ; Im ˜ p (1) ia = 0 , i = 1 , . . . , D − p (1) µa = Re p (1) µa , µ = 0 , . . . , D − , (31) For notational simplicity, we omit the prefix ‘Im’ in all the entries in the equation (30) however the entriesshould be understood as respective imaginary parts. p (1) ia , i = 1 , . . . , D − ∀ a in p (1) to obtain thepoint ˜ p (1) . As the Im p (1)0 a component remains unaltered ∀ a the point ˜ p (1) ∈ T ~θλ for all ~θ .In particular ˜ p (1) belongs to both the tubes T ~θ λ and T ~θ λ . Hence T ~θ λ being a convex tubethe straight line segment p (1) ˜ p (1) connecting p (1) , ˜ p (1) lies entirely inside T ~θ λ , and T ~θ λ beinga convex tube the straight line segment ˜ p (1) p (2) connecting ˜ p (1) , p (2) lies entirely inside T ~θ λ .Joining these two segments we get a path connecting p (1) , p (2) staying inside the tube S ~θ T ~θλ .This completes the proof.To see that ˜ p (1) ∈ T ~θλ for all ~θ , let us consider ˜ P (1) I = P a ∈ I ˜ p (1) a for an arbitrary non-emptyproper subset I of X . Therefore we have Im ˜ P (1) I = P a ∈ I Im ˜ p (1) a and it is timelike because (cid:0) Im ˜ P (1) I (cid:1) = − (cid:0)X a ∈ I Im ˜ p (1)0 a (cid:1) + X D − i =1 (cid:0)X a ∈ I Im ˜ p (1) ia (cid:1) = − (cid:0)X a ∈ I Im p (1)0 a (cid:1) < . (32)Furthermore Im ˜ P (1) I and Im P (1) I belong to the same lightcone because of the followingIm ˜ P (1)0 I = X a ∈ I Im ˜ p (1)0 a = X a ∈ I Im p (1)0 a = Im P (1)0 I = ⇒ sgn (cid:0) Im ˜ P (1)0 I (cid:1) = sgn (cid:0) Im P (1)0 I (cid:1) . (33) D Thickening of S ~θ T ~θλ As mentioned in the introduction 1, the work of [1] showed that at any point p belonging tothe LES domain D ′ all the relevant Feynman diagrams in the perturbative expansion of an n -point Green’s function in SFT are analytic. Any such Feynman diagram at the point p hasan integral representation in terms of loop integrals where the poles of the integrand are atfinite distance away from any of the loop integration contours. As per the discussion aroundequation (8), the above statement also holds for any point p belonging to the LES domain ˜ D ′ .Hence for any p ∈ S ~θ T ~θλ (thereby Im p ∈ S ~θ C ~θλ ⊂ R ( n − D ) we can allow a small open ball B Im p in R ( n − D centered at Im p such that for any point p ′ ∈ B Im p the aforementioned polesof the integrand are still at a finite distance away from the loop integration contours in a givenFeynman diagram [1]. Consequently, the same integral representation in terms of loop integralsholds at any of the new points p ′ . Therefore, by allowing such open balls for each p ∈ S ~θ T ~θλ we can make S ~θ T ~θλ open, in which the given Feynman diagram still remains analytic. In this As per the discussion in section 1, diagrams that do not have any massless internal propagator are onlyrelevant. S ~θ T ~θλ can be thickened individually for all the relevant Feynman diagrams (at all ordersin perturbation theory). E The cone C (4)+12The cone C (4)+12 taken from the list (17) can be written as C (4)+12 = (cid:26) Im p : − Im p , Im ( p + p ) , Im ( p + p ) ∈ V + (cid:27) . (34) C (4)+12 resides on the manifold Im p + · · · + Im p = 0. Here we show how specifying Im p in V − , and Im ( p + p ) and Im ( p + p ) 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 (cid:26) Im p , Im p , Im p , Im ( p + p ) , Im ( p + p ) , Im ( p + p ) , Im ( p + p + p ) (cid:27) . (35)We want to see that for any Im p inside the cone C (4)+12 in which lightcone each element ofthe above set lies. Knowing this, similar information for any other possible Im P I can bedetermined using the relation Im p + · · · + Im p = 0. Now the following information can beobtained since for any Im p ∈ C (4)+12 we have − Im p , Im ( p + p ) , Im ( p + p ) ∈ V + .Im p = − Im p + Im ( p + p ) ∈ V + , Im p = − Im p + Im ( p + p ) ∈ V + , Im ( p + p ) = Im p + Im p ∈ V + , Im ( p + p + p ) = − Im p + Im ( p + p ) + Im ( p + p ) ∈ V + . (36)Hence the signs λ ( I ) corresponding to the cone C (4)+12 is now known for any non-empty propersubset I of { , . . . , } .In section 3.2, with {− Im p , Im ( p + p ) , Im ( p + p ) } as our basis, points in the cone C (4)+12 have been described. However any point in C (4)+12 can uniquely be written in a new basisgiven by { Im p , Im p , Im p } since such a change of basis is a linear transformation L withdet( L ) = −
1. This transformation law can be stated as: any point ~Q = ( P α , P β , P γ ) written inthe basis {− Im p , Im ( p + p ) , Im ( p + p ) } can be written as L ~Q = ( − P α , P α + P β , P α + P β )in basis { Im p , Im p , Im p } . V − (= − V + ) is the open backward lightcone in R D . ~Q in the cone C (4)+12 as given in (19) in the new basis reads as L ~Q = − P α P α + P β P α + P γ − P α P α + P β P α + P γ ... ... ... − P D − α P D − α + P D − β P D − α + P D − γ , (37)with conditions P r > + qP D − i =1 ( P ir ) ∀ r = α, β, γ . And the points in (20) in the new basisread as L ~ ˜ Q = 3 − P α + ǫ P α − ǫ/ P α − ǫ/ − P α P α P α ... ... ... − P D − α P D − α P D − α , L ~ ˜ Q = 3 − ǫ/ P β − ǫ/ ǫ P β P D − β , L ~ ˜ Q = 3 − ǫ/ ǫ P γ − ǫ/
20 0 P γ ... ... ...0 0 P D − γ , (38)where ǫ satisfies the condition: 0 < ǫ < min (cid:8) P r − qP D − i =1 ( P ir ) , r = α, β, γ (cid:9) . Clearly abovepoints L ~Q , L ~ ˜ Q , L ~ ˜ Q and L ~ ˜ Q written in the basis { Im p , Im p , Im p } are consistent with(36). Furthermore each columns of L ~ ˜ Q lie on the same two dimensional Lorentzian planewhere P α lies. Similarly all the columns of L ~ ˜ Q lie on the two dimensional Lorentzian planewhere P β lies, and all the columns of L ~ ˜ Q lie on the two dimensional Lorentzian plane where P γ lies. Now it is easy to check that the following relation holds L ~ ˜ Q L ~ ˜ Q L ~ ˜ Q L ~Q . (39)Which depicts nothing but the linearity of L on (21). F Difficulty arising for C ′ (5)+12Consider a 5-point function, i.e. n = 5. We take the following conditions which describe oneof the 20 problematic cones, C ′ (5)+12 .Im ( p + p ) , Im ( p + p ) , Im ( p + p ) , Im ( p + p ) ∈ V + , − Im ( p + p + p ) , − Im ( p + p + p ) ∈ V + . (40)23bove conditions in turn imply that Im p , Im p ∈ V + and Im p , Im p ∈ V − .In this case, we choose (cid:8) Im p , Im p , − Im p , − Im p (cid:9) as our basis to assign coordinatesto the points in the cone C ′ (5)+12 . The goal is to find P , P , P , P ∈ V + , in terms of whichwe can decompose a generic point (cid:0) Im p , Im p , − Im p , − Im p (cid:1) in the cone in a convexcombination of several points. Each of the terms in the decomposition should be in S ~θ C ′ (5)+ ,~θ .Consider the following decomposition in a sum of four terms, (cid:0) Im p , Im p , − Im p , − Im p (cid:1) = (cid:0) α P , α P , α P , α P (cid:1) + (cid:0) β P , β P , β P , β P (cid:1) + (cid:0) γ P , γ P , γ P , γ P (cid:1) + (cid:0) δ P , δ P , δ P , δ P (cid:1) , (41)where α r , β r , γ r , δ r , r = 1 , . . . , α r , α ≥ α , α ; α ≤ α + α ; α ≥ α , α ; α ≤ α + α . (42)Exactly the same conditions should hold for β r , γ r and δ r as well. Note that in case of equalitywe can stay within the cone using ǫ prescription for the p components. For example, supposewe have, α = α ; β > β ; γ > γ ; δ > δ , (43)and all other inequalities (strictly) as in (42), without loss of generality. Then we can writethe r.h.s. of (41) as (cid:0) α P + τ ¯ ǫ, α P , α P , α P (cid:1) + (cid:0) β P − τ ¯ ǫ, β P , β P , β P (cid:1) + (cid:0) γ P , γ P , γ P , γ P (cid:1) + (cid:0) δ P , δ P , δ P , δ P (cid:1) , (44)where 0 < τ < min (cid:8) β − β , β − β (cid:9) ;¯ ǫ ≡ ǫ , < ǫ < min (cid:8) P r − | ~P r | , r = 1 , . . . , (cid:9) . (45)Clearly P − τβ ¯ ǫ, P − τβ − β ¯ ǫ, P − τβ − β ¯ ǫ ∈ V + since τβ , τβ − β , τβ − β <
1. Hence for eachterm in (44), we remain inside the cone ˜ C ′ +12 . 24ow for each term in the decomposition (41), evidently all the four columns lie on a twodimensional Lorentzian plane. We need to solve for P , P , P , P by inverting the followingmatrix equation, α α α α β β β β γ γ γ γ δ δ δ δ P P P P = Im p Im p − Im p − Im p . (46)If we find that all the P r are in V + , then the proof is done and we can say that C ′ (5)+12 =Ch (cid:0) S ~θ C ′ (5)+ ,~θ (cid:1) . But subject to conditions (42), solving (46) seems to be difficult analytically. References [1] C. de Lacroix, H. Erbin and A. Sen, “Analyticity and crossing symmetry of superstringloop amplitudes,” JHEP (2019) 139 [arXiv:1810.07197].[2] C. de Lacroix, H. Erbin, S.P. Kashyap, A. Sen and M. Verma, “Closed Superstring FieldTheory and its Applications,” Int. J. Mod. Phys. A 32 (2017) 1730021 [arXiv:1703.06410].[3] R. Pius and A. Sen, “Cutkosky rules for superstring field theory,” JHEP (2016) 024[Erratum ibid. (2018) 122] [arXiv:1604.01783].[4] N.N. Bogolyubov, A.A. Logunov, A.I. Oksak and I.T. Todorov, “General principles ofquantum field theory,” Mathematical physics and applied mathematics, Kluwer, DordrechtThe Netherlands (1990).[5] O. Steinmann, “ ¨Uber den Zusammenhang zwischen Wightmanfunktionen und retardiertenKommutatoren, I,” Helv. Phys. Acta (1960) 257.[6] D. Ruelle, “Connection between wightman functions and green functions in p -space,”Nuovo Cim. (1961) 356.[7] H. Araki and N. Burgoyne, “Properties of the momentum space analytic function,” NuovoCim. (1960) 342.[8] H. Araki, “Generalized Retarded Functions and Analytic Function in Momentum Spacein Quantum Field Theory,” J. Math. Phys. (1961) 163.259] J. Bros, H. Epstein and V.J. Glaser, “Some rigorous analyticity properties of the four-pointfunction in momentum space,” Nuovo Cim. (1964) 1265.[10] J. Bros, A. Messiah and R. Stora,“ A Problem of Analytic Completion Related to theJost-Lehmann-Dyson Formula,” J. Math. Phys. (1961) 639.[11] J. Bros, H. Epstein and V. Glaser, “A proof of the crossing property for two-particleamplitudes in general quantum field theory,” Commun. Math. Phys. (1965) 240.[12] J. Bros, “Derivation of asymptotic crossing domains for multiparticle processes in ax-iomatic quantum field theory: a general approach and a complete proof for 2 → (1986) 325.[13] V.P. Pavlov, “Analytic structure of the 3 → , 277–284 (1978). https://doi.org/10.1007/BF01032423 .[14] L.M. Muzafarov and V.P. Pavlov, “Analyticity of the 3 → , 376–383 (1978). https://doi.org/10.1007/BF01039107 .[15] A.A. Logunov, B.V. Medvedev, L.M. Muzafarov, M.K. Polivanov and A.D. Sukhanov,“Analytic structure of the 3 → , 677–687(1979). https://doi.org/10.1007/BF01018717 .[16] J. Bros, “Analytic structure of Green’s functions in quantum field theory,” in: OsterwalderK. (eds) Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, vol 116.Springer, Berlin, Heidelberg (1980).[17] B.V. Medvedev, V.P. Pavlov, M.K. Polivanov and A.D. Sukhanov, “Analyticproperties of many-particle amplitudes,” Theor. Math. Phys. , 723–732 (1982). https://doi.org/10.1007/BF01018410 .[18] B.V. Medvedev, V.P. Pavlov, M.K. Polivanov and A.D. Sukhanov, “Analytic proper-ties of multiparticle production amplitudes,” Theor. Math. Phys. , 427–440 (1984). https://doi.org/10.1007/BF01018176 .[19] R. Jost and H. Lehmann, “Integral-Darstellung kausaler Kommutatoren,” Nuovo Cim. (1957) 1598. 2620] F.J. Dyson, “Integral representations of causal commutators,” Phys. Rev. (1958)1460.[21] S. Bochner, “A Theorem on Analytic Continuation of Functions in SeveralVariables,” Annals of Mathematics, Second Series, (1938), no. 1, 14–19. http://doi.org/10.2307/1968709 .[22] A. Sen, “Wilsonian Effective Action of Superstring Theory,” JHEP (2017) 108[arXiv:1609.00459].[23] M. Lassalle, “Analyticity properties implied by the many-particle structure of the n -pointfunction in general quantum field theory. I. Convolution of n -point functions associatedwith a graph,” Commun. Math. Phys. (1974), no. 3, 185–226.[24] J. Bros and M. Lassalle, “Analyticity properties and many-particle structure in generalquantum field theory. II. One-particle irreducible n -point functions,” Commun. Math.Phys.43