QFT in the flat chart of de Sitter space
aa r X i v : . [ g r- q c ] D ec 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 [email protected] , [email protected] Abstract
We study the correlators for interacting quantum field theory in the flat chart of de Sitter spaceat all orders in perturbation. The correlators are calculated in the in-in formalism which are oftenapplied to the calculations in the cosmological perturbation. It is shown that these correlators arede Sitter invariant. They are compared with the correlators calculated based on the Euclidean fieldtheory. We then find that these two correlators are identical. This correspondence has been alreadyshown graph by graph but we give an alternative proof of it by direct calculation. ontents In recent years, there have been rapid progresses in the precise measurement of the observable quantitiesin cosmology, e.g., the non-Gaussianity of the fluctuations generated during inflation, which is expectedto be a powerful tool as a probe of the early universe. Along with the development of these precisemeasurements, 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 in-flationary background, in which one does not generally know how to define the interacting vacuum. Oneoften uses the iǫ prescription in cosmology to calculate the correlators perturbatively. (See, for example,Ref. [1].) In the Minkowski, this prescription is known to perturbatively give the Poincar´e invariantcorrelators 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 inflationaryera, but the physical meaning of it is not as clear as in the Minkowski case. Our main interest in thispaper is in the meaning of the iǫ prescription for interacting field theory in de Sitter space.The free scalar quantum field theory in de Sitter space is well understood [2–5], while interacting oneis a hot subject with a lot of debate [6–43]. We focus in the present paper on the problem whether the iǫ prescription for interacting theory breaks de Sitter invariance.Since de Sitter space is maximally symmetric and possesses SO (4 , iǫ prescription in the flat chart are de Sitterinvariant. Notice, for example, that in the latter the integration region for the vertices in calculatingcorrelators 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 massivescalar field. Namely, the iǫ prescription does not break de Sitter invariance for interacting massive scalarfield. Furthermore, the vacuum defined by the iǫ prescription has been shown to be equivalent to the2uclidean vacuum. The main ideas in Ref. [11] are as follows. They start from correlators defined on anEuclidean sphere and take, on the Euclidean sphere, coordinates such that when we Wick rotate the timecoordinate continues to the static chart of the Lorentzian de Sitter space. Then, after the deformation ofthe integral path of the Euclidean time, fall-off of the propagator in the large separation limit leads to theidentity of the two correlators at least on the static chart. From the analyticity of the in-in correlatorsfor their time coordinates, and the uniqueness of the analytic continuation, it is shown that the in-incorrelators 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 inde Sitter space has especially been a topic of much discussion. (See, e.g. [13, 21–23].) Our final goal isto extend the correspondence between the two vacua to those interacting massless field theory. It is alsoworth 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 wherethe propagator does not fall off in general, since the proof of the correspondence between the two vacuarelies on this decay property of the propagator at a large separation as explained above. In order toattack those theories, we take another approach. That is, we directly calculate the correlators with the iǫ prescription. We derive, along this way, the analytic Mellin-Barnes formulae for the correlators ofquantum fields in the flat chart. The resulting correlators are shown to be completely the same as theanalytic continuations of the ones considered in the Euclidean field theory in Ref. [10]. Thus we find thatthe iǫ 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 potentialto be extended to wider range of theories which include interacting massless theory such as derivativelyinteracting 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 regularizationscheme 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 N -pt correlator with single vertex. Then in Sec. 5,we extend the discussion to arbitrary graphs. We give a brief summary in Sec. 6. In this section, we briefly review free scalar quantum field theory on de Sitter space, especially in the flatchart. We also introduce Pauli-Villars regularization scheme for later use.
We consider D -dimensional de Sitter space dS D with, for simplicity, unit radius. This is a hyperboloidembedded in ( D + 1)-dimensional Minkowski space with metric η ab = ( − , + , · · · , +). The embedding isspecified by η ab X a X b = 1 . (2.1)It is convenient to define the invariant distance between two points X and Y in de Sitter space by theMinkowski inner product of X and Y , which we denote as Z ( X, Y ) := η ab X a Y b , (2.2)as in Ref. [9]. For brevity, we often use alternative notation Z XY for Z ( X, Y ), Z Y for Z ( X , Y ) and soforth in the following. 3he coordinates ( η, x ) in the flat chart are related to the embedding coordinates as X = 12 (cid:18) η − η (cid:19) − || x || η , X D = − (cid:18) η + 1 η (cid:19) + || x || η , X α = − x α η , ( α = 1 , , · · · , D − , (2.3)where || x || means the norm of ( D − x . The flat chart coordinates with −∞ < η < x ∈ R D − span just a half of the whole spacetime region. In fact, the linear combination X + X D = − η , (2.4)is restricted to the positive side for negative η . The metric in the flat chart is expressed as ds = 1 η ( − dη + d x ) . (2.5)Expressed in the flat chart coordinates, the invariant distance between X and X ′ , Z ( X, X ′ ) is given by Z ( X, X ′ ) = 1 + ( η − η ′ ) − || x − x ′ || ηη ′ , (2.6)where ( η, x ) and ( η ′ , x ′ ) are the flat chart coordinates corresponding to X and X ′ , respectively. We now consider a massive free scalar QFT on de Sitter space. We focus on the Green’s function G ( X, Y )given by G ( X, Y ) = Γ ( − σ ) Γ ( σ + D − π ) D/ Γ ( D/ F (cid:18) − σ, σ + D − D Z XY (cid:19) , (2.7)which corresponds to taking Bunch-Davies vacuum [44] or Euclidean vacuum [45]. σ is related to themass of the field m by σ = − D −
12 + s(cid:18) D − (cid:19) − m . (2.8)Expressing the hypergeometric function in the Barnes representation, we have G ( X, Y ) = Z ν (cid:18) − Z XY (cid:19) ν Γ ( − ν ) ψ ( ν ) , (2.9)with ψ ( ν ) := 1(4 π ) D/ Γ (cid:20) − σ + ν, σ + D − ν, − D − ν D + σ, − D − σ (cid:21) . (2.10)Here Γ (cid:20) α , α , · · · β , β , · · · (cid:21) , stands for Γ ( α ) Γ ( α ) · · · / Γ ( β ) Γ ( β ) · · · , and the symbol R ν ( · · · ) means the Barnes integral. TheBarnes integral is an integral along a straight line, C , that traverses from − i ∞ to + i ∞ parallel to the4maginary axis with the factor 1 / πi : Z ν ( · · · ) := Z C dν πi ( · · · ) . (2.11)The integrand of the Barnes integral includes sequences of poles. For example, Γ ( z ) possesses a sequenceof poles at z = 0 , − , − , · · · . The integration path C is taken to avoid all the sequences of poles in theintegrand. In the case of the above Green’s function, C is taken to satisfymax {− Re σ − D + 1 , Re σ } < Re ν < min (cid:26) − D , (cid:27) . (2.12)This region of the integration path is called “fundamental strip,” and the poles such that are associatedwith Gamma functions like Γ ( · · · − ν ) ( Γ ( · · · + ν )) and hence such that line up on the right (left) handside of this strip are called right (left) poles. (See Fig. 1.) The symbol like R ν is used to represent theBarnes integral in this meaning in the following. Because we consider interacting theory in the present paper, we have to introduce some ultra-violetregularization scheme. We make use of the Pauli-Villars regularization. This scheme attaches somemassive propagators, G i ( X, Y ), defined in Eq. (2.9) with m replaced by the regulator mass M i , tothe original one, G ( X, Y ), so that we replace the original propagator in a graph with the regularizedpropagator G reg ( X, Y ) := G ( X, Y ) + X i C i G i ( X, Y ) . (2.13)The coefficients C i are chosen so that the regularized propagator G reg ( X, Y ) becomes finite in the coin-cidence limit Y → X , which leads to the conditions X i C i = − , X i C i M i = 0 , X i C i M i = 0 , · · · . (2.14)This regularization scheme affects the pole structure of ψ ( ν ) in (2.9), eliminating the first several rightpoles of ψ ( ν ) which are responsible for the behaviour of the Green’s function in the coincidence limit [10].The regularized Green’s function is written as G reg ( X, Y ) = Z ν (cid:18) − Z XY (cid:19) ν Γ ( − ν ) ψ reg ( ν ) , (2.15)where we assume that ψ reg ( ν ) is regularized to be analytic in the regionRe σ < Re ν < p , (2.16)with p a sufficiently large positive constant. (See Fig. 1.) In the following sections, we drop, for simplicity,the symbols such as reg on G and ψ . We now move on to the interacting theory. The interacting QFT in the flat chart of the Lorentzian deSitter space is discussed in the present and the succeeding sections. When we express the correlatorsin the wave number representation, we employ the iǫ prescription to calculate the correlators for theinteracting vacuum. This prescription regularizes the oscillatory behaviour of the Green’s functions at5 (cid:0)(cid:27) (cid:0) D (cid:0) 1 (cid:27) 1 (cid:0) D=2 0 (cid:23)(cid:0)(cid:27) (cid:0) D (cid:0) 1 (cid:27) p0 Figure 1: The left figure shows the pole structure for ψ ( ν ) which is not regularized. There are twoseries of left poles from ν = σ and ν = − σ − D −
1, and right poles from ν = 1 − D/
2. The right oneshows the pole structure for ψ reg ( ν ) which is Pauli-Villars regularized. The shaded region represents thefundamental strip in each figure.infinity in time and makes the vertex integral converge. Although what we discuss in the present paperis the position space representation of the correlators, we also employ the iǫ prescription to specify theinteracting 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. Let us consider N -pt Green’s function. The contribution to N -pt correlator at the lowest order inperturbation theory is given by V N ( X , · · · , X N ) = Z Ω dV Y G ( X , Y ) · · · G ( X N , Y ) . (3.1)In the in-in formalism with the iǫ prescription, the integration region Ω for the vertex integral is specifiedas follows.We first introduce an η -integration path P on the η -plane, independently of the spatial coordinates y , defined as a curve which starts from −∞ e − iǫ and ends at −∞ e iǫ as shown in Fig. 2. All the externalpoints are also supposed to be placed along this path. In case of the wave number representation, thisconstruction completes the definition of the in-in path on the η -plane. If we take the η -path along P , theintegral 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 representationto compute the correlators. The vertex integrals involve the spatial integration, too. As a starting point,we set the region of the vertex integral Ω to P × R D − . If we first carry out the spatial integrationbefore temporal one, the integral would diverge because we then pick up the contributions from distantspacelike region. On the other hand, if we integrate first for the time variable and then for the spatialones, the integral is convergent as we see in Sec. 4. This means that the integral over P × R D − is notwell-defined as a multiple integral.To make the integral to be well-defined as a multiple integral, we modify the integral region bydeforming the path of the η -integral P [11]. There are branching points on the η -plane, which correspondto the intersections with the light cones emanating from the external points. On the η -plane for fixed y , G ( X i , Y ) has the same structure of Riemann surface as that of (1 − Z iY ) ν i , where ν i is some complex6 (cid:17)i(cid:17)i;(cid:0) (cid:17)i;+(cid:0)1 ei(cid:15)(cid:0)1 e(cid:0)i(cid:15) Figure 2: This figure shows the η -path P which is later deformed to P y . The dots represent the timecoordinate η i of the external points and the crosses are the branching points corresponding to the lightcones emanating from the external points. The dashed lines are the branch cuts.number, and 1 − Z iY − η + η i, + )( η − η i, − )4( − η )( − η i ) , (3.2)where η i, ± := η i ± || x i − y || , ( i = 1 , · · · , N − , N ) . (3.3)Namely, the integrand has the same structure of Riemann surface as that of( − η ) − ( D + P ν i ) N Y i =1 ( − η + η i, + ) ν i ( η − η i, − ) ν i . (3.4)The time integration is unchanged even if we deform the integration contour as long as it does notcross singularities of the integrand. Thus, we deform the contour P to P y such that the maximum valueof the real part of η on P y is equal to max i { Re η i, − } + b where b is a small real positive constant. (SeeFig. 3.) This deformation on the η -plane is significant when the spatial coordinates of the vertex islargely separated from those of relevant external points. To the contrary, when || x i − y || is small for i that realizes the maximum among Re η i, − , the modified contour P y is almost identical to the original one P . Using this P y , we define the integration regionΩ := (cid:8) ( η, y ) | η ∈ P y , y ∈ R D − (cid:9) , (3.5)in C × R D − . The result of the integral is the same as that is obtained by integrating first for time andthen for space for the original integration region, but we emphasize that the integral over Ω is now amultiple integral. Let us return to Eq. (3.1). Inserting Eq. (2.9) into Eq. (3.1), we have V N ( X , · · · , X N ) = Z Ω dV Y Z ν · · · Z ν N " N Y i =1 Γ ( − ν i ) ψ ( ν i ) × " N Y i =1 (cid:18) − Z iY (cid:19) ν i . (3.6)7 (cid:17)i(cid:17)i;(cid:0) (cid:17)i;+(cid:0)1 ei(cid:15)(cid:0)1 e(cid:0)i(cid:15) z }| {b Figure 3: This figure represents the deformed contour P y for fixed spatial coordinate y . The originalpath P is deformed as long as it does not cross the singularities.If we can exchange the order of the integrals, R Ω dV Y and R ν · · · R ν N , we are led to calculate the followingintegral M ( ν , · · · , ν N − , ν N ) = Z Ω dV Y (cid:18) − Z Y (cid:19) ν · · · (cid:18) − Z NY (cid:19) ν N . (3.7)The first problem is to calculate this integral. This quantity is shown to have an analytic Mellin-Barnesrepresentation in Sec. 4, and hence if this exchange of the order of integration is allowed, V N can berepresented in an analytic Mellin-Barnes form. It is not trivial whether this exchange of the order of theintegration is allowed or not. This is the second problems. The same problem arises also for arbitrarygraphs as for the tree level graphs. We will extend our discussion to arbitrary graphs in Sec. 5. The goal of this section is to compute the master integral: M ( ν , · · · , ν n , ν N ) = Z Ω dV Y (cid:18) − Z Y (cid:19) ν · · · (cid:18) − Z nY (cid:19) ν n (cid:18) − Z NY (cid:19) ν N , (4.1)where we have introduced n := N − dV Y = dη d D − y ( − η ) D , (4.2)is the invariant volume, and Ω is defined in Eq. (3.5). In order to evaluate the above expression (4.1), we introduce the following generating function A ( α , · · · , α n ) := Z Ω dV Y N X i =1 α i − Z iY ! λ , (4.3)following Ref. [10], in which it was used to evaluate the master integral on an Euclidean sphere. HereRe λ < , α , · · · , α n ≥ , α N := 1 , (4.4)8 (cid:0)1 ei(cid:15)(cid:0)1 e(cid:0)i(cid:15) (cid:17)i;(cid:0) (cid:17)j;+(cid:17)i;+(cid:17)j;(cid:0) Figure 4: A figure representing the in-in path P y for the external points which lie on the real Lorentziansection and are mutually spacelike separated. The crosses represent the branching points and the dottedlines the branch cuts.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.1) to the integrand of (4.3) to obtain A ( α , · · · , α n ) = Z Ω dV Y Γ ( − λ ) Z u ( α ) u · · · Z u n ( α n ) u n × Γ [ − u , · · · , − u n , − u N ] (cid:18) − Z X (cid:19) u · · · (cid:18) − Z NX (cid:19) u N , (4.5)where u N := λ − n X i =1 u i . (4.6)[Step 2.] Next, we exchange the order of the integration, R Ω dV Y and R u · · · R u n , to have A ( α , · · · , α n ) = 1 Γ ( − λ ) Z u ( α ) u · · · Z u n ( α n ) u n × Γ [ − u , · · · , − u n , − u N ] M ( u , · · · , u n , u N ) . (4.7)Thus, the Mellin transform of A gives M .However, we have to prove that [Step 1.] and [Step 2.] are indeed possible, which is the goal ofthis subsection. In particular, [Step 2.] requires that the integral over Ω is a multiple integral. Theconvergence of the integral is rather obvious when we consider the corresponding integral over a compactEuclidean sphere, while it is not in the present case where the integration region is non-compact. In thissubsection, we assume, for a technical reason, that the time coordinates of all external points lie on thereal Lorentzian section, i.e. η i ∈ R − , y i ∈ R D − , and furthermore, that any pairs of them are mutuallyspacelike separated.Since the definition of the in-in path described in Sec. 3.1 requires the external points to lie alongthe in-in path and therefore their time coordinates are complex in general, we need some explanations ofthe in-in path for this configuration. The path is defined on the η -plane by taking the limit Im η i → P y introduced in Sec. 3.1. It seems that the path in this limit must, at least partly, lie on the η -realaxis. However, since the external points are mutually spacelike, the branch cuts, lying on the η -real axis,do not cover the whole η -real axis. Therefore, the limit can be taken without the pass P y crossing thebranch cuts, and hence the in-in path in this limit is simply a contour going from −∞ e − iǫ to −∞ e iǫ as9 (cid:17)j;+(cid:17)j;(cid:0) (cid:17)i;+(cid:17)i;(cid:0) (cid:17) Figure 5: The dot represents the time coordinate of a vertex on P y . The remainder of the summationof the arguments of two vectors relevant to the subscript i and that relevant to the subscript j gives | arg(1 − Z iY ) − arg(1 − Z jY ) | as in (4.10).shown in Fig. 4.Proof of [Step 1.]: Note that the following inequalities hold for arbitrary Y ∈ Ω: | arg(1 − Z iY ) − arg(1 − Z jY ) | < π , ( i, j = 1 , , · · · , N ) . (4.8)In fact, arg(1 − Z iY ) is given byarg(1 − Z iY ) = arg( − η + η i, + ) + arg( η − η i, − ) − arg( − η ) − arg( − η i ) , (4.9)and then, noticing that arg( − η i ) = 0 since all the external points are on the real Lorentzian section, wehave | arg(1 − Z iY ) − arg(1 − Z jY ) | = | arg( − η + η i, + ) + arg( η − η i, − ) − arg( − η + η j, + ) − arg( η − η j, − ) | . (4.10)This quantity is less than π for any ( η, y ) ∈ Ω. (See Fig. 5.) The inequality (4.8) is the sufficient conditionthat the formula (A.1) can be applied to the integrand of Eq. (4.3). For the later purpose, we modify theintegration path P y as such that satisfies | arg(1 − Z iY ) − arg(1 − Z jY ) | < π − δ , (4.11)for any i and j with a small positive number δ . This can be achieved easily. Because | arg(1 − Z iY ) − arg(1 − Z jY ) | is close to π only in the small region surrounding the interval ( η i, − , η j, − ) or ( η i, + , η j, + ), thepath can be chosen to avoid this region.Proof of [Step 2.]: We denote the integration paths for u , · · · , u n as C , · · · , C n , respectively, anddefine C := C × · · · × C n . The sufficient condition to allow to exchange the order of the integration, R Ω dV Y and R C Q ni =1 du i / πi , is that the integral is absolutely convergent (Fubini’s theorem). In thepresent case, we should examine the following integral1 | Γ ( − λ ) | Z C n Y k =1 (cid:12)(cid:12)(cid:12)(cid:12) du k πi (cid:12)(cid:12)(cid:12)(cid:12) | ( α ) u · · · ( α n ) u n | | Γ [ − u , · · · , − u n , − u N ] |× (cid:20)Z Ω | dV Y | (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z Y (cid:19) u (cid:12)(cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z NY (cid:19) u N (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) , (4.12)10here | dV Y | = | dη | d D − y | − 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 Ω integral in the large brackets in Eq. (4.12) for fixed u , · · · , u n : (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z Y (cid:19) u (cid:12)(cid:12)(cid:12)(cid:12) · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z NY (cid:19) u N (cid:12)(cid:12)(cid:12)(cid:12) . (4.14)Notice that | (1 − Z iY ) u i | = | − Z iY | Re u i exp [ − arg(1 − Z iY )Im u i ] . (4.15)Along the integration path of u i parallel to the imaginary axis, Im u i varies while Re u i is fixed. Takinginto account that u N includes u i as given in Eq. (4.6), the part depending on Im u i in Eq. (4.14) isfactored out as exp [ { arg(1 − Z NY ) − arg(1 − Z iY ) } Im u i ] . (4.16)Since | arg(1 − Z NY ) − arg(1 − Z iY ) | is bounded as shown in Eq. (4.11), this factor is bounded from aboveby exp [( π − δ ) | Im u i | ] . Therefore, noticing that α i is real positive number, we find that[Eq. (4.12)] < | Γ ( − λ ) | (cid:12)(cid:12) ( α ) Re u · · · ( α n ) Re u n (cid:12)(cid:12) Z Ω | dV Y | (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z Y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Re u · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z NY (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Re u N × Z C n Y k =1 (cid:12)(cid:12)(cid:12)(cid:12) du k πi (cid:12)(cid:12)(cid:12)(cid:12) | Γ [ − u , · · · , − u n , − u N ] | e ( π − δ )( | Im u | + ··· + | Im u n | ) . (4.17)Since | Γ ( x + iy ) | ≈ (2 π ) / e − π | y | / | y | x − / ( | y | → + ∞ ), u k integrals in the second line in the last expres-sion are convergent. Therefore, our remaining task is to show that the volume integral Z Ω | dV Y | (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z Y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Re u · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z NY (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Re u N , (4.18)is also finite.For this purpose, we first introduce a representative point X with coordinates in the flat chart definedby ( η , x ) := N X i =1 p i ( η i , x i ) , ( p i ≥ , X p i = 1) , (4.19)and a domain D far from X in terms of the invariant distance by D := { Y | | Z Y | > Z } ∩ Ω . (4.20)Note that if we take Z to be sufficiently large, we see that | Z iY | ≥ const . × | Z Y | , ( Y ∈ D , i = 1 , , · · · , N ) . (4.21)We divide the region Ω into (Ω \ D ) and D , and evaluate each contribution to (4.18) separately.(i) Integral over Ω \ D : 11e further divide Ω \ D into K defined by K := { ( η, y ) ∈ Ω \ D | || x − y || > R } , (4.22)and its complement (Ω \ D ) \ K . R is set large enough for K not to include any external points. (SeeFig. 6.)(i-a) Integral over (Ω \ D ) \ K :The region (Ω \ D ) \ K is compact but it contains the coincidence points ( η, y ) = ( η i , x i ) at which theintegrand of (4.18) diverges. Since y ≈ x i around them, the path P y is identical to the original one P ,and hence η = η i + iδη with real δη . Then, we have | − Z iY | Re u i ≈ (( δη ) + || x i − y || ) Re u i (4.23)around a point ( η i , x i ), which shows that (4.18) is finite as long as we choose the integration path of u i to satisfy Re u i > − D/ , ( i = 1 , , · · · , n, N ) , (4.24)which does not conflict with [Step 1.]. Recall that the fundamental strip of Eq. (4.5) contains the pathswith Re u i for all i being infinitesimally small negative constants.(i-b) Integral over K :We first see that, for Y ∈ K , | − Z iY | is bounded both from below and from above by positiveconstants. Recall that the η -path P y is defined by deforming P not to touch η i, − except for the case with y ≈ x i , which occurs in (Ω \ D ) \ K . Therefore, | − Z iY | does not vanish, bounded from below by someconstant c − ( > | − Z iY | is bounded from above by some constant c + . If | − Z iY | is sufficiently large, Z Y will be larger than Z . Then, by the definition of K , Y is not includedin K . Thus, we conclude that for some positive constants c ± , c − < | − Z iY | < c + , ( Y ∈ K ) . (4.25)Furthermore, one can claim that the volume of the region K is finite, i.e., Z K | dV Y | < + ∞ . (4.26)In showing this, the non-trivial point is that the region K extends to infinitely large || y || . However, theregion of the η -integral is confined to the interval η i ( y ) , − − b ′ ≤ Re η ≤ η i ( y ) , − + b , (4.27)where b is the same constant used in defining the path P y and i ( y ) is the label of the external point suchthat η i ( y ) , − > η k, − for all k = i ( y ). Here the point is that one can choose a large positive constant b ′ tobe independent of y . In fact, the invariant distance between X := ( η , x ), and the point correspondingto the above lower bound Y bdry := ( η i ( y ) , − − b ′ , y ) = ( η i ( y ) − || x i ( y ) − y || − b ′ , y ) is evaluated as | − Z ( X , Y bdry ) | = (cid:12)(cid:12)(cid:12)(cid:12) ( η − η i ( y ) + || x i ( y ) − y || + b ′ ) − || x i ( y ) − y || η ( η i ( y ) − || x i ( y ) − y || − b ′ ) (cid:12)(cid:12)(cid:12)(cid:12) & b ′ | − η | , ( || x i ( y ) − y || + b ′ → + ∞ ) . (4.28)In the last inequality we assumed || x i ( y ) − y || + b ′ → + ∞ but this should be a good approximation in12 D0 K((cid:10)nD0)nKX0
Figure 6: This figure is a schematic of how we divide the integration region Ω. There are D , K and (Ω \ D ) \ K . The dots except for X represent the external points. The dashed lines representschematically the “past light cone of X .”the region K . Therefore, if b ′ / ( |− η | ) is taken sufficiently large compared with Z , the above range of η covers the whole region of K . Thus, the volume R K | dV Y | is bounded by Z K | dV Y | < c Z || x − y || >R d D − y Z η i ( y ) , − + bη i ( y ) , − − b ′ | dη || η | D < c Z || x − y || >R d D − y || x i ( y ) − y || D < + ∞ , (4.29)where c and c are some appropriately chosen constants of O (1). In the second inequality we used | η | > | η i ( y ) , − + b | ≈ || x i ( y ) − y || . Therefore, the integral over K is proven to be finite.(ii) Integral over D :We next proceed to the integral over D . Using Eq. (4.21), one can easily bound the volume integralof our current concern from above as Z D | dV Y | (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z Y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Re u · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − Z NY (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Re u N < c × − λ Z D | dV Y | | Z Y | Re λ . (4.30)where c is a constant of O (1), and we have used the relation P N u i = λ .In order to show this integral is finite, we use Z Y as a time coordinate instead of η , which leads theintegration measure to transform as dη d D − y = − η Z Y p || x − y || + η ( Z Y − ! η d ( Z Y ) d D − y . (4.31)We substitute this into the right hand side of (4.30). Approximating Z Y − ≈ Z Y and introducing x := y/ ( − η Z Y ), we find that the integral is finite as Z D | dV Y | | Z Y | Re λ < c Z + ∞ Z dZZ − Re λ Z + ∞ dx x/ √ x (cid:18) x (1 + √ x ) (cid:19) D − < + ∞ , (4.32)where again c is a constant of O (1). 13iii) Summary:We have shown in this subsection that the integral (4.12) is indeed finite when the external points X , · · · , X n and X N lie on the real Lorentzian section and are mutually in spacelike separation, as longas the integration contours for u , · · · , u n satisfy the additional conditions (4.4) and (4.24):Re λ = Re N X i u i < , Re u > − D , · · · , Re u n > − D , Re u N > − D . (4.33)Then, the order of two integrals R Ω dV Y and R C Q ni =1 du i / πi in Eq. (4.5) are exchangeable, whichimplies that the master integral M is given by the repeated Mellin transform of A . Furthermore, underthese conditions A ( α , · · · , α n ) is finite and thus from Eq. (4.7) the master integral M ( u , · · · , u n , u N )is also finite. That is, the master integral M ( u , · · · , u n , u N ) is finite when the external points are inthe real Lorentzian section and are mutually in spacelike separation, with the conditions (4.33) satisfied.The analytic expression for M is given in the succeeding subsection, where the conditions on the externalpoints are relaxed. We now proceed to compute A and hence M , to show its equivalence to the analytic continuation of theEuclidean correlators. Again in this subsection we first assume that all the external points X i lie on thereal Lorentzian section and that they are mutually in spacelike separation. After that, we show that thetime coordinates of the external points η i in the obtained expression for M can be analytically continuedto any point on the in-in path.The expression for A given in Eq. (4.3) can be transformed into A ( α , · · · , α n ) = Z R D − d D − y Z P y dη ( − η ) D − λ N X i =1 α i − V · Y ! λ , (4.34)where V · Y is an inner product of V and Y with respect to ( D + 1)-dimensional Minkowski metric and V = N X i =1 α i X i . (4.35)Notice that V + V D = N X i =1 α i − η i > , V = N X i =1 α i − η i x i . (4.36)Setting τ := V + V D η , R := ( V + V D ) x − V = N X i =1 α i − η i ( y − x i ) , (4.37) V · Y can be expressed as V · Y = − τ + R − V · V τ , (4.38)14here V · V := η ab V a V b = N X i =1 α i ! + 2 N X i 2, and τ ± := 12 n − F ± p R + J o , F := N X i =1 α i , J := 2 N X i 1, the path C can be contracted to the forward and backward paths along the negative real axis. Noticing that only15 BA (cid:24) Figure 7: The path C in the integral (4.45).the argument of ξ γ changes between these two paths, one can transform I ( α, β, γ ) as I ( α, β, γ ) = 12 πi Z ∞ ( − dx ) e − iπγ x γ ( x + A ) α ( x + B ) β + 12 πi Z + ∞ ( − dx ) e iπγ x γ · · · = 1 Γ [ − γ, γ + 1] Z + ∞ dx ( x + A ) α ( x + B ) β x γ . (4.46)Next, we expand ( x + B ) β , using Eq. (A.2), as( x + B ) β = Z µ Γ (cid:20) − β + µ, − µ − β (cid:21) x µ B β − µ , (Re β < Re µ < . (4.47)Substituting this into Eq. (4.46), we carry out x integral first to obtain I ( α, β, γ ) = Z µ Γ (cid:20) − β + µ, − µ, γ + 1 + µ, − α − γ − − µ − γ, γ + 1 , − β, − α (cid:21) A α + γ +1+ µ B β − µ . (4.48)Of course, the convergence of x integral imposes a condition Re ( α + γ + 1 + µ ) < 0. This is in factsatisfied because we can set Re µ arbitrarily close to Re β as long as Re β < Re µ ( < 0) is maintained. Ifwe change the integration variable from µ → µ − α − γ − 1, we obtain the expression (4.45).Finally, we remove the restriction Re γ > − 1. In fact, the integrand is analytic for γ and the ξ integration is uniformly convergent for γ , as long as Re ( α + β + γ + 1) < 0. Therefore, the integral isanalytic for γ , which enables us to remove the restriction Re γ > − A ( α , · · · , α n ) = − πi × − D − λ +1 Z R D − d D − R Z µ Γ (cid:20) D + λ + µ, D − λ − µ, − µ, − D + 1 − µD + λ, − λ, − λ, λ + 1 (cid:21) × − µ h F + p R + J i µ ( R + J ) − D + λ +1 − µ . (4.49)We next carry out R integration, using the formula Z R D − d D − R (cid:2) R + J (cid:3) ν/ h F + p R + J i µ = π D − Z κ Γ (cid:20) − µ + κ, − κ, − κ + ν + D − − µ, − κ + ν (cid:21) F µ − κ J ν + κ + D − , (4.50)which is valid when Re ( µ + ν + D − < 0. The idea of the proof of the above formula is not so different16rom that of the formula (4.45). One applies Eq. (A.2) to h F + p R + J i µ to obtain h F + p R + J i µ = Z κ Γ (cid:20) − µ + κ, − κ − µ (cid:21) F µ − κ ( R + J ) κ/ , (Re µ < Re κ < . (4.51)Substituting this into the left hand side of Eq. (4.50), we obtain Z R D − d D − R (cid:2) R + J (cid:3) ν/ h F + p R + J i µ = Ω D − Z κ Γ (cid:20) − µ + κ, − κ − µ (cid:21) F µ − κ J ν + κ + D − × Z + ∞ d Ξ Ξ D − (1 + Ξ) ν + κ , (4.52)where we have introduced a new integration variable Ξ := ( || R || /J ) , andΩ D − = 2 π D − Γ (cid:0) D − (cid:1) , (4.53)is the surface area of D − ν + κ + D − < κ arbitrarily close to Re µ as long as Re µ < Re κ ( < 0) ismaintained. Integration over Ξ leads to (4.50).Applying the formula (4.50) to the expression for the generating function (4.49), and replacing theintegration variables κ and µ , respectively, with w := κ − µ + λ ρ := µ + D − , (4.54)we obtain A ( α , · · · , α n ) = ( − i )2 − λ π D +12 Z w Γ (cid:20) w − λ, − w − w + D − , D + λ, − λ, − λ, λ + 1 (cid:21) × Z ρ − ρ Γ [ − λ + ρ, λ + 1 + ρ, − ρ, λ + D − − w − ρ ] . (4.55)Finally, we perform the ρ integration in the above expression for A , using the formula Z ρ − ρ Γ [ λ + 1 + ρ, − λ + ρ, − ρ, λ + a − − ρ ] = 2 λ + a − √ π Γ (cid:20) − λ, λ + 1 , a − , λ + a (cid:21) , (4.56)which can be proven as follows. If we close the ρ -path on the left hand side of (4.56) to the right, wehave Z ρ − ρ Γ [ λ + 1 + ρ, − λ + ρ, − ρ, λ + a − − ρ ]= Γ [ − λ, λ + 1 , λ + a − F (cid:18) − λ, λ + 1; 2 − λ − a ; 12 (cid:19) +2 − λ − a Γ [2 λ + a, a − , − λ − a ] F (cid:18) λ + a, a − λ + a ; 12 (cid:19) . (4.57)Now applying the following formulae, known respectively as Bailey’s summation theorem and the Gauss’17econd summation theorem [46], F (cid:18) α, − α ; γ ; 12 (cid:19) = 2 − γ √ π Γ " γ γ + α , γ +(1 − α )2 , (4.58) F (cid:18) α, β ; α + β + 12 ; 12 (cid:19) = √ π Γ (cid:20) α + β + α + , β + (cid:21) , (4.59)we obtain after simple calculations (4.56). Substituting a = D − w in (4.56), we find A ( α , · · · , α n ) = ( − i )(4 π ) D/ Z w Γ (cid:20) w − λ, − w, λ + D − wD + λ, − λ (cid:21) F λ − w (cid:18) J (cid:19) w . (4.60)Recalling the definition of F and J , (4.41), we expand F λ − w J w to be integrals with respect tothe power law indices of α i ’s using Eq. (A.1) [10]. Since the master integral M is given by the Mellintransform of the generating function A , we finally obtain M ( ν , · · · , ν N ) = ( − i ) (4 π ) D/ Γ ( D + P ν i ) [ Q Γ ( − ν i )] Z ( h ij ) Y i We would like to thank H. Kitamoto and V. Onemli for valuable comments. YK also thanks to R. Saito,K. Sugimura and K. Nakata for useful and interesting discussions. YK is supported by the Grant-in-Aidfor JSPS Fellows No. 24-4198. TT is supported by the Grand-in-Aid for Scientific Research Nos. 21111006,21244033, 24103001 and 24103006. This work was also supported by the Grant-in-Aid for the GlobalCOE programs, “The Next Generation of Physics, Spun from Universality and Emergence” from theMinistry of Education, Culture, Sports, Science and Technology of Japan. A Formula Let A , · · · , A n +1 be complex numbers satisfying | arg A i − arg A j | < π ( ∀ i, j ). Then, the followingformula is true as a repeated integral and also as a multiple integral since the integral is easily shown tobe independent of the order of the integration:( A + A + · · · + A n +1 ) λ = 1 Γ ( − λ ) Z u · · · Z u n Γ h − λ + X u i , − u , · · · , u n i ( A ) u · · · ( A n ) u n ( A n +1 ) λ − P u i . (A.1)Proof of (A.1):The basic formula is the following:( a + b ) λ = 1 Γ ( − λ ) Z µ Γ [ − λ + µ, − µ ] a µ b λ − µ , ( | arg a − arg b | < π ) . (A.2)One applies this formula (A.2) with a = A n , b = A + · · · + A n − + A n +1 , and then again apply (A.2) to( A + · · · + A n − + A n +1 ) λ − µ in the result of the previous step with a = A n − , b = A + · · · + A n − + A n +1 .Repeating the same operation, one formally reaches (A.1). The point is that the conditions | arg A − arg A n +1 | < π , | arg A − arg( A + A n +1 ) | < π , · · ·| arg A n − arg( A + · · · + A n − + A n +1 ) | < π , (A.3)are required to perform the above transformation. To allow the exchange of the order of the repeatedintegration without changing the result, we impose stronger conditions | arg A P (1) − arg A n +1 | < π , | arg A P (2) − arg( A P (1) + A n +1 ) | < π , · · ·| arg A P ( n ) − arg( A P (1) + · · · + A P ( n − + A n +1 ) | < π , (A.4)for any permutation P . 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