Contents
###### Abstract

We study the correlators for interacting quantum field theory in the flat chart of de Sitter space at all orders in perturbation. The correlators are calculated in the in-in formalism which are often applied to the calculations in the cosmological perturbation. It is shown that these correlators are de Sitter invariant. They are compared with the correlators calculated based on the Euclidean field theory. We then find that these two correlators are identical. This correspondence has been already shown graph by graph but we give an alternative proof of it by direct calculation.

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1.1

QFT in the flat chart of de Sitter space

Yusuke Korai and Takahiro Tanaka

Yukawa Institute for Theoretical Physics, Kyoto University,

Kyoto, 606-8502, Japan

## 1 Introduction

In recent years, there have been rapid progresses in the precise measurement of the observable quantities in cosmology, e.g., the non-Gaussianity of the fluctuations generated during inflation, which is expected to be a powerful tool as a probe of the early universe. Along with the development of these precise measurements, the need arises for the accurate theoretical predictions of the corresponding quantities.

When computing the non-Gaussianity, one needs to discuss interacting quantum field theory on inflationary background, in which one does not generally know how to define the interacting vacuum. One often uses the prescription in cosmology to calculate the correlators perturbatively. (See, for example, Ref. [1].) In the Minkowski, this prescription is known to perturbatively give the Poincaré invariant correlators for interacting theory, defining interacting vacuum as the lowest energy eigen state. Indeed, this prescription also enables us to calculate the non-Gaussianity or higher correlations in the inflationary era, but the physical meaning of it is not as clear as in the Minkowski case. Our main interest in this paper is in the meaning of the prescription for interacting field theory in de Sitter space.

The free scalar quantum field theory in de Sitter space is well understood [2, 3, 4, 5], while interacting one is a hot subject with a lot of debate [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. We focus in the present paper on the problem whether the prescription for interacting theory breaks de Sitter invariance.

Since de Sitter space is maximally symmetric and possesses , de Sitter symmetry, it is strongly expected to exist a de Sitter invariant vacuum even for interacting theory. In fact, a de Sitter invariant vacuum for interacting theory is defined by constructing arbitrary correlators perturbatively at all orders using the Euclidean method [10]. While the vacuum state thus constructed are manifestly de Sitter invariant, it is not obvious whether the ones defined by the prescription in the flat chart are de Sitter invariant. Notice, for example, that in the latter the integration region for the vertices in calculating correlators are restricted to the future of the cosmological horizon, which is not de Sitter invariant.

Actually, this problem has been already resolved affirmatively in Ref. [11] for interacting massive scalar field. Namely, the prescription does not break de Sitter invariance for interacting massive scalar field. Furthermore, the vacuum defined by the prescription has been shown to be equivalent to the Euclidean vacuum. The main ideas in Ref. [11] are as follows. They start from correlators defined on an Euclidean sphere and take, on the Euclidean sphere, coordinates such that when we Wick rotate the time coordinate continues to the static chart of the Lorentzian de Sitter space. Then, after the deformation of the integral path of the Euclidean time, fall-off of the propagator in the large separation limit leads to the identity of the two correlators at least on the static chart. From the analyticity of the in-in correlators for their time coordinates, and the uniqueness of the analytic continuation, it is shown that the in-in correlators in the flat chart are identical to the analytic continuations of those on an Euclidean shpere.

Then, it is natural to ask what becomes of in massless field theory. What happens for graviton in de Sitter space has especially been a topic of much discussion. (See, e.g. [21, 22, 23, 13].) Our final goal is to extend the correspondence between the two vacua to those interacting massless field theory. It is also worth considering derivatively interacting massless scalar field, which can be a step toward graviton.

It seems difficult to extend the discussion of massive field theory above to massless field theory where the propagator does not fall off in general, since the proof of the correspondence between the two vacua relies on this decay property of the propagator at a large separation as explained above. In order to attack those theories, we take another approach. That is, we directly calculate the correlators with the prescription. We derive, along this way, the analytic Mellin-Barnes formulae for the correlators of quantum fields in the flat chart. The resulting correlators are shown to be completely the same as the analytic continuations of the ones considered in the Euclidean field theory in Ref. [10]. Thus we find that the prescription in de Sitter space gives the vacuum state corresponding to the Euclidean field theory. Although we consider only massive theory in the present paper, we believe that our proof has potential to be extended to wider range of theories which include interacting massless theory such as derivatively interacting one, since it does not employ the decay property of the propagator.

This paper is organized as follows. In Sec. 2, we briefly review how to describe de Sitter space, especially the flat chart, and massive free scalar quantum field theory on it. Pauli-Villars regularization scheme is also introduced. Then we proceed to the interacting theory, in Sec. 3, 4 and 5. We consider, in Sec. 3 and 4, a tree graph which contributes to an -pt correlator with single vertex. Then in Sec. 5, we extend the discussion to arbitrary graphs. We give a brief summary in Sec. 6.

## 2 Preliminaries

In this section, we briefly review free scalar quantum field theory on de Sitter space, especially in the flat chart. We also introduce Pauli-Villars regularization scheme for later use.

### 2.1 de Sitter space

We consider -dimensional de Sitter space with, for simplicity, unit radius. This is a hyperboloid embedded in -dimensional Minkowski space with metric . The embedding is specified by

 ηabXaXb=1 . (2.1)

It is convenient to define the invariant distance between two points and in de Sitter space by the Minkowski inner product of and , which we denote as

 Z(X,Y):=ηabXaYb , (2.2)

as in Ref. [9]. For brevity, we often use alternative notation for , for and so forth in the following.

The coordinates in the flat chart are related to the embedding coordinates as

 X0=12(η−1η)−||%\boldmath$x$||22η ,XD=−12(η+1η)+||\boldmathx||22η ,Xα=−xαη ,(α=1,2,⋯,D−1) ,

where means the norm of -vector . The flat chart coordinates with and span just a half of the whole spacetime region. In fact, the linear combination

 X0+XD=−1η , (2.4)

is restricted to the positive side for negative . The metric in the flat chart is expressed as

 ds2=1η2(−dη2+d\boldmathx2) . (2.5)

Expressed in the flat chart coordinates, the invariant distance between and , is given by

 Z(X,X′)=1+(η−η′)2−||% \boldmathx−\boldmathx′||22ηη′ , (2.6)

where and are the flat chart coordinates corresponding to and , respectively.

### 2.2 Free QFT on de Sitter

We now consider a massive free scalar QFT on de Sitter space. We focus on the Green’s function given by

 G(X,Y)=Γ(−σ)Γ(σ+D−1)(4π)D/2Γ(D/2)2F1(−σ,σ+D−1;D2;1+ZXY2) , (2.7)

which corresponds to taking Bunch-Davies vacuum [44] or Euclidean vacuum [45]. is related to the mass of the field by

 σ=−D−12+√(D−12)2−m2 . (2.8)

Expressing the hypergeometric function in the Barnes representation, we have

 G(X,Y)=∫ν(1−ZXY2)νΓ(−ν)ψ(ν) , (2.9)

with

 ψ(ν):=1(4π)D/2Γ[−σ+ν,σ+D−1+ν,1−D2−νD2+σ,1−D2−σ] . (2.10)

Here

 Γ[α1,α2,⋯β1,β2,⋯] ,

stands for , and the symbol means the Barnes integral. The Barnes integral is an integral along a straight line, , that traverses from to parallel to the imaginary axis with the factor :

 ∫ν(⋯):=∫Cdν2πi(⋯) . (2.11)

The integrand of the Barnes integral includes sequences of poles. For example, possesses a sequence of poles at . The integration path is taken to avoid all the sequences of poles in the integrand. In the case of the above Green’s function, is taken to satisfy

 max{−Reσ−D+1,Reσ}

This region of the integration path is called “fundamental strip,” and the poles such that are associated with Gamma functions like () and hence such that line up on the right (left) hand side of this strip are called right (left) poles. (See Fig. 1.) The symbol like is used to represent the Barnes integral in this meaning in the following.

### 2.3 Pauli-Villars Regularization

Because we consider interacting theory in the present paper, we have to introduce some ultra-violet regularization scheme. We make use of the Pauli-Villars regularization. This scheme attaches some massive propagators, , defined in Eq. (2.9) with replaced by the regulator mass , to the original one, , so that we replace the original propagator in a graph with the regularized propagator

 Greg(X,Y):=G(X,Y)+∑iCiGi(X,Y) . (2.13)

The coefficients are chosen so that the regularized propagator becomes finite in the coincidence limit , which leads to the conditions

 ∑iCi=−1 ,∑iCiM2i=0 ,∑iCiM3i=0 , ⋯ . (2.14)

This regularization scheme affects the pole structure of in (2.9), eliminating the first several right poles of which are responsible for the behaviour of the Green’s function in the coincidence limit [10]. The regularized Green’s function is written as

 Greg(X,Y)=∫ν(1−ZXY2)νΓ(−ν)ψreg(ν) , (2.15)

where we assume that is regularized to be analytic in the region

 Reσ

with a sufficiently large positive constant. (See Fig. 1.) In the following sections, we drop, for simplicity, the symbols such as on and .

## 3 Interacting QFT: Single Vertex

We now move on to the interacting theory. The interacting QFT in the flat chart of the Lorentzian de Sitter space is discussed in the present and the succeeding sections. When we express the correlators in the wave number representation, we employ the prescription to calculate the correlators for the interacting vacuum. This prescription regularizes the oscillatory behaviour of the Green’s functions at infinity in time and makes the vertex integral converge. Although what we discuss in the present paper is the position space representation of the correlators, we also employ the prescription to specify the interacting vacuum.

In this section, we discuss perturbative calculations of a single vertex tree graph for the correlators. Then, we identify the problems to be solved to accomplish this calculation, which are solved in Sec. 4. In Sec. 5 the results for single vertex tree graphs are extended to arbitrary graphs.

### 3.1 Definition of the In-in Path

Let us consider -pt Green’s function. The contribution to -pt correlator at the lowest order in perturbation theory is given by

 VN(X1,⋯,XN)=∫ΩdVYG(X1,Y)⋯G(XN,Y) . (3.1)

In the in-in formalism with the prescription, the integration region for the vertex integral is specified as follows.

We first introduce an -integration path on the -plane, independently of the spatial coordinates , defined as a curve which starts from and ends at as shown in Fig. 2. All the external points are also supposed to be placed along this path. In case of the wave number representation, this construction completes the definition of the in-in path on the -plane. If we take the -path along , the integral converges with the integrand vanishing fast enough in the past.

For the purpose of the present paper, it is more convenient to use the position space representation to compute the correlators. The vertex integrals involve the spatial integration, too. As a starting point, we set the region of the vertex integral to . If we first carry out the spatial integration before temporal one, the integral would diverge because we then pick up the contributions from distant spacelike region. On the other hand, if we integrate first for the time variable and then for the spatial ones, the integral is convergent as we see in Sec. 4. This means that the integral over is not well-defined as a multiple integral.

To make the integral to be well-defined as a multiple integral, we modify the integral region by deforming the path of the -integral [11]. There are branching points on the -plane, which correspond to the intersections with the light cones emanating from the external points. On the -plane for fixed , has the same structure of Riemann surface as that of , where is some complex number, and

 1−ZiY2=(−η+ηi,+)(η−ηi,−)4(−η)(−ηi) , (3.2)

where

 ηi,±:=ηi±||\boldmathxi−% \boldmathy|| ,(i=1,⋯,N−1,N) . (3.3)

Namely, the integrand has the same structure of Riemann surface as that of

 (−η)−(D+∑νi)N∏i=1(−η+ηi,+)νi(η−ηi,−)νi . (3.4)

The time integration is unchanged even if we deform the integration contour as long as it does not cross singularities of the integrand. Thus, we deform the contour to such that the maximum value of the real part of on is equal to where is a small real positive constant. (See Fig. 3.) This deformation on the -plane is significant when the spatial coordinates of the vertex is largely separated from those of relevant external points. To the contrary, when is small for that realizes the maximum among , the modified contour is almost identical to the original one .

Using this , we define the integration region

 (3.5)

in . The result of the integral is the same as that is obtained by integrating first for time and then for space for the original integration region, but we emphasize that the integral over is now a multiple integral.

### 3.2 Problems to Be Solved in the Calculations

Let us return to Eq. (3.1). Inserting Eq. (2.9) into Eq. (3.1), we have

 VN(X1,⋯,XN)=∫ΩdVY∫ν1⋯∫νN[N∏i=1Γ(−νi)ψ(νi)]×[N∏i=1(1−ZiY2)νi] . (3.6)

If we can exchange the order of the integrals, and , we are led to calculate the following integral

 M(ν1,⋯,νN−1,νN)=∫ΩdVY(1−Z1Y2)ν1⋯(1−ZNY2)νN . (3.7)

The first problem is to calculate this integral. This quantity is shown to have an analytic Mellin-Barnes representation in Sec. 4, and hence if this exchange of the order of integration is allowed, can be represented in an analytic Mellin-Barnes form. It is not trivial whether this exchange of the order of the integration is allowed or not. This is the second problems. The same problem arises also for arbitrary graphs as for the tree level graphs. We will extend our discussion to arbitrary graphs in Sec. 5.

## 4 Computation of the Master Integral

The goal of this section is to compute the master integral:

 M(ν1,⋯,νn,νN)=∫ΩdVY(1−Z1Y2)ν1⋯(1−ZnY2)νn(1−ZNY2)νN , (4.1)

where we have introduced for convenience,

 dVY=dηdD−1y(−η)D , (4.2)

is the invariant volume, and is defined in Eq. (3.5).

### 4.1 Generating Function for the Master Integral

In order to evaluate the above expression (4.1), we introduce the following generating function

 A(α1,⋯,αn):=∫ΩdVY(N∑i=1αi1−ZiY2)λ , (4.3)

following Ref. [10], in which it was used to evaluate the master integral on an Euclidean sphere. Here

 Reλ<0 ,α1 , ⋯ , αn≥0 ,αN:=1 , (4.4)

are assumed.

In this subsection we establish the relation between the generating function and the master integral. Formally, in the same way as in the Euclidean case discussed in Ref. [10], the generating function (4.3) seems to be related to the master integral (3.7) also in the present case as follows:

[Step 1.] We first apply Eq. (A) to the integrand of (4.3) to obtain

 A(α1,⋯,αn) = ∫ΩdVY1Γ(−λ)∫u1(α1)u1⋯∫un(αn)un (4.5) ×Γ[−u1,⋯,−un,−uN](1−Z1X2)u1⋯(1−ZNX2)uN ,

where

 uN:=λ−n∑i=1ui . (4.6)

[Step 2.] Next, we exchange the order of the integration, and , to have

 A(α1,⋯,αn) = 1Γ(−λ)∫u1(α1)u1⋯∫un(αn)un (4.7) ×Γ[−u1,⋯,−un,−uN]M(u1,⋯,un,uN) .

Thus, the Mellin transform of gives .

However, we have to prove that [Step 1.] and [Step 2.] are indeed possible, which is the goal of this subsection. In particular, [Step 2.] requires that the integral over is a multiple integral. The convergence of the integral is rather obvious when we consider the corresponding integral over a compact Euclidean sphere, while it is not in the present case where the integration region is non-compact. In this subsection, we assume, for a technical reason, that the time coordinates of all external points lie on the real Lorentzian section, i.e. , and furthermore, that any pairs of them are mutually spacelike separated.

Since the definition of the in-in path described in Sec. 3.1 requires the external points to lie along the in-in path and therefore their time coordinates are complex in general, we need some explanations of the in-in path for this configuration. The path is defined on the -plane by taking the limit in introduced in Sec. 3.1. It seems that the path in this limit must, at least partly, lie on the -real axis. However, since the external points are mutually spacelike, the branch cuts, lying on the -real axis, do not cover the whole -real axis. Therefore, the limit can be taken without the pass crossing the branch cuts, and hence the in-in path in this limit is simply a contour going from to as shown in Fig. 4.

Proof of [Step 1.]: Note that the following inequalities hold for arbitrary :

 |arg(1−ZiY)−arg(1−ZjY)|<π , (i,j=1,2,⋯,N) . (4.8)

In fact, is given by

 arg(1−ZiY)=arg(−η+ηi,+)+arg(η−ηi,−)−arg(−η)−arg(−ηi) , (4.9)

and then, noticing that since all the external points are on the real Lorentzian section, we have

 |arg(1−ZiY)−arg(1−ZjY)| =|arg(−η+ηi,+)+arg(η−ηi,−)−arg(−η+ηj,+)−arg(η−ηj,−)| . (4.10)

This quantity is less than for any . (See Fig. 5.) The inequality (4.8) is the sufficient condition that the formula (A) can be applied to the integrand of Eq. (4.3). For the later purpose, we modify the integration path as such that satisfies

 |arg(1−ZiY)−arg(1−ZjY)|<π−δ , (4.11)

for any and with a small positive number . This can be achieved easily. Because is close to only in the small region surrounding the interval or , the path can be chosen to avoid this region.

Proof of [Step 2.]: We denote the integration paths for as , respectively, and define . The sufficient condition to allow to exchange the order of the integration, and , is that the integral is absolutely convergent (Fubini’s theorem). In the present case, we should examine the following integral

 1|Γ(−λ)|∫Cn∏k=1∣∣∣duk2πi∣∣∣|(α1)u1⋯(αn)un||Γ[−u1,⋯,−un,−uN]| ×[∫Ω|dVY|∣∣∣(1−Z1Y2)u1∣∣∣⋯∣∣∣(1−ZNY2)uN∣∣∣] , (4.12)

where

 |dVY|=|dη|dD−1y|−η|D . (4.13)

If this integral is finite, then we can justify the exchange of the order of integrals in [Step 2.].

To show this, we focus on the integrand of the integral in the large brackets in Eq. (4.1) for fixed :

 ∣∣∣(1−Z1Y2)u1∣∣∣⋯∣∣∣(1−ZNY2)uN∣∣∣ . (4.14)

Notice that

 |(1−ZiY)ui|=|1−ZiY|Reuiexp[−arg(1−ZiY)Imui] . (4.15)

Along the integration path of parallel to the imaginary axis, varies while is fixed. Taking into account that includes as given in Eq. (4.6), the part depending on in Eq. (4.14) is factored out as

 (4.16)

Since is bounded as shown in Eq. (4.11), this factor is bounded from above by Therefore, noticing that is real positive number, we find that

 [Eq. (???)] < 1|Γ(−λ)|∣∣(α1)Reu1⋯(αn)Reun∣∣∫Ω|dVY|∣∣∣(1−Z1Y2)∣∣∣Reu1⋯∣∣∣(1−ZNY2)∣∣∣ReuN (4.18) ×∫Cn∏k=1∣∣∣duk2πi∣∣∣|Γ[−u1,⋯,−un,−uN]|e(π−δ)(|Imu1|+⋯+|Imun|) .

Since , integrals in the second line in the last expression are convergent. Therefore, our remaining task is to show that the volume integral

 ∫Ω|dVY|∣∣∣(1−Z1Y2)∣∣∣Reu1⋯∣∣∣(1−ZNY2)∣∣∣ReuN, (4.19)

is also finite.

For this purpose, we first introduce a representative point with coordinates in the flat chart defined by

 (η0,\boldmathx0):=N∑i=1pi(ηi,\boldmathxi) ,(pi≥0,∑pi=1) , (4.20)

and a domain far from in terms of the invariant distance by

 D0:={Y||Z0Y|>Z0}∩Ω . (4.21)

Note that if we take to be sufficiently large, we see that

 |ZiY|≥const.×|Z0Y| ,(Y∈D0,i=1,2,⋯,N) . (4.22)

We divide the region into and , and evaluate each contribution to (4.19) separately.

(i) Integral over :

We further divide into defined by

 K:={(η,\boldmathy)∈Ω∖D0|||\boldmathx0−\boldmathy||>R} , (4.23)

and its complement . is set large enough for not to include any external points. (See Fig. 6.)

(i-a) Integral over :

The region is compact but it contains the coincidence points at which the integrand of (4.19) diverges. Since around them, the path is identical to the original one , and hence with real . Then, we have

 |1−ZiY|Reui≈((δη)2+||% \boldmathxi−\boldmathy||2)Reui (4.24)

around a point , which shows that (4.19) is finite as long as we choose the integration path of to satisfy

 Reui>−D/2 ,(i=1,2,⋯,n,N) , (4.25)

which does not conflict with [Step 1.]. Recall that the fundamental strip of Eq. (4.5) contains the paths with for all being infinitesimally small negative constants.

(i-b) Integral over :

We first see that, for , is bounded both from below and from above by positive constants. Recall that the -path is defined by deforming not to touch except for the case with , which occurs in . Therefore, does not vanish, bounded from below by some constant . It is also easy to show that is bounded from above by some constant . If is sufficiently large, will be larger than . Then, by the definition of , is not included in . Thus, we conclude that for some positive constants ,

 c−<|1−ZiY|

Furthermore, one can claim that the volume of the region is finite, i.e.,

 ∫K|dVY|<+∞ . (4.27)

In showing this, the non-trivial point is that the region extends to infinitely large . However, the region of the -integral is confined to the interval

 ηi(\boldmath{\scriptsizey}),−−b′≤Reη≤ηi(\boldmath{\scriptsizey}),−+b , (4.28)

where is the same constant used in defining the path and is the label of the external point such that for all . Here the point is that one can choose a large positive constant to be independent of . In fact, the invariant distance between , and the point corresponding to the above lower bound is evaluated as

 |1−Z(X0,Ybdry)| = ∣∣ ∣ ∣∣(η0−ηi(\boldmath{\scriptsizey})+||\boldmathxi(\boldmath{\scriptsizey})−% \boldmathy||+b′)2−||\boldmathxi(\boldmath{% \scriptsizey})−\boldmathy||22η0(ηi(% \boldmath{\scriptsizey})−||\boldmathxi(\boldmath{% \scriptsizey})−\boldmathy||−b′)∣∣ ∣ ∣∣ (4.29) ≳ b′|−η0| , (||\boldmathx% i(\boldmath{\scriptsizey})−\boldmathy||+b′→+∞) .

In the last inequality we assumed but this should be a good approximation in the region . Therefore, if is taken sufficiently large compared with , the above range of covers the whole region of . Thus, the volume is bounded by

 ∫K|dVY|RdD−1y∫ηi(\boldmath{% \scriptsizey}),−+bηi(% \boldmath{\scriptsizey}),−−b′|dη||η|DRdD−1y||\boldmathxi(\boldmath{\scriptsizey})−\boldmath% y||D<+∞ , (4.30)

where and are some appropriately chosen constants of . In the second inequality we used . Therefore, the integral over is proven to be finite.

(ii) Integral over :

We next proceed to the integral over . Using Eq. (4.22), one can easily bound the volume integral of our current concern from above as

 (4.31)

where is a constant of , and we have used the relation .

In order to show this integral is finite, we use as a time coordinate instead of , which leads the integration measure to transform as

 dηdD−1y=⎛⎜ ⎜⎝1−η0Z0Y√||% \boldmathx0−\boldmathy||2+η20(Z20Y−1)⎞⎟ ⎟⎠η0d(Z0Y)dD−1y . (4.32)

We substitute this into the right hand side of (4.31). Approximating and introducing , we find that the integral is finite as

 ∫D0|dVY||Z0Y|Reλ

where again is a constant of .

(iii) Summary:

We have shown in this subsection that the integral (4.1) is indeed finite when the external points and lie on the real Lorentzian section and are mutually in spacelike separation, as long as the integration contours for satisfy the additional conditions (4.4) and (4.25):

 Reλ=ReN∑iui<0 , Reu1>−D2, ⋯ ,Reun>−D2,ReuN>−D2 . (4.34)

Then, the order of two integrals and in Eq. (4.5) are exchangeable, which implies that the master integral is given by the repeated Mellin transform of . Furthermore, under these conditions is finite and thus from Eq. (4.7) the master integral is also finite. That is, the master integral is finite when the external points are in the real Lorentzian section and are mutually in spacelike separation, with the conditions (4.34) satisfied. The analytic expression for is given in the succeeding subsection, where the conditions on the external points are relaxed.

### 4.2 Calculation of the Generating Function

We now proceed to compute and hence , to show its equivalence to the analytic continuation of the Euclidean correlators. Again in this subsection we first assume that all the external points lie on the real Lorentzian section and that they are mutually in spacelike separation. After that, we show that the time coordinates of the external points in the obtained expression for can be analytically continued to any point on the in-in path.

The expression for given in Eq. (4.3) can be transformed into

 A(α1,⋯,αn) = ∫RD−1dD−1y∫P\boldmath{% \scriptsizey}dη(−η)D2−λ(N∑i=1αi−V⋅Y)λ , (4.35)

where is an inner product of and with respect to -dimensional Minkowski metric and

 V=N∑i=1αiXi . (4.36)

Notice that

 V0+VD=N∑i=1αi−ηi>0 ,\boldmathV=N∑i=1αi−ηi% \boldmathxi . (4.37)

Setting

 τ:=V0+VD2η ,\boldmathR:=(V0+VD)\boldmathx−\boldmathV=N∑i=1αi−ηi(\boldmathy−\boldmathxi) , (4.38)

can be expressed as

 V⋅Y=−τ+\boldmathR2−V⋅V41τ , (4.39)

where

 V⋅V:=ηabVaVb=(N∑i=1αi)2+2N∑i

Thus, we obtain

 A(α1,⋯,αn) = ∫RD−1dD−1y∫P\boldmath{% \scriptsizey}dη(−η)D2−λ(N∑i=1αi−V⋅Y)λ (4.41) = 2−λ∫RD−1dD−1y(V0+VD2)D−1∫P′\boldmath{\scriptsizey}dτ(−τ)−D[N∑i=1αi+τ−\boldmathR