Manifest Form of the Spin-Local Higher-Spin Vertex Υ^{ηη}_{ωCCC}
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Manifest Form of the Spin-Local Higher-Spin Vertex Υ ηηωCCC O.A. Gelfond , and A.V. Korybut I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,Leninsky prospect 53, 119991, Moscow, Russia Federal State Institution ”Scientific Research Institute for System Analysis of the RussianAcademy of Science”,Nakhimovsky prospect 36-1, 117218, Moscow, Russia [email protected], [email protected]
Abstract
Vasiliev system of higher-spin equations contains a free complex parameter η . Solvingthe generating system order by order one obtains physical vertices proportional to variouspowers of η and ¯ η . Recently η and ¯ η vertices in the field equations for zero-form field werewritten in the Z -dominated form implying their spin-locality by virtue of Z -dominanceLemma. Here we obtain explicit Z -independent spin-local form for the vertex Υ ηηωCCC forits ωCCC -ordered part where ω and C denote gauge one-form and field strength zero-form higher-spin fields valued in arbitrary associative algebra in which case the order ofproduct factors in ωCCC matters. ontents H + and Z -dominance lemma 6 H + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Contribution to Υ ηηωCCC modulo H + . . . . . . . . . . . . . . . . . . . . . . . . . 8 Υ ηηωCCC
116 To z -linear pre-exponentials 137 Generalised Triangle identity 148 Uniformization 169 Eliminating δ ( ρ j ) and δ ( ξ j ) . Result 1710 Final step of calculation 18
11 Conclusion 19Appendix A. B ηη B.1 d x B + W ∗ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.2 (d x B ηη + ω ∗ B ηη ) | δ ( ρ ) + W η ωC ∗ B η loc . . . . . . . . . . . . . . . . . . . . . . . 22 Appendix C. Eliminating δ ( ρ j ) and δ ( ξ j ) C.1 Terms proportional to ( p + p ) α ( p + p ) α . . . . . . . . . . . . . . . . . . . . . 24C.2 Term proportional to t α ( p α + p α ) . . . . . . . . . . . . . . . . . . . . . . . . . 25C.3 Sum of ( p + p ) α ( p + p ) α -proportional and t α ( p α + p α )–proportional terms . 26C.4 Terms proportional to δ ( ξ ) − δ ( ξ ) . . . . . . . . . . . . . . . . . . . . . . . . . 26C.5 Terms proportional to ξ δ ( ξ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Appendix D. Details of the final step of the calculation 27
D.1 ξ -independent pre-exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . 29D.2 ξ -proportional pre-exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Appendix E. Useful formulas 31 Introduction
Generating Vasiliev system of higher-spin equations [1] contains a free complex parameter η .Solving the generating system order by order one obtains vertices proportional to various powersof η and ¯ η . In the recent paper [2], η and ¯ η vertices were obtained in the sector of equationsfor zero-form fields, containing, in particular, a part of the φ vertex for the scalar field φ inthe theory. Though being seemingly Z -dependent, in [2] these vertices were written in the Z -dominated form which implies their spin-locality by virtue of Z -dominance Lemma of [3].In this paper we obtain explicit Z -independent spin-local form for the vertex Υ ηηωCCC startingfrom the Z -dominated expression of [2]. The label ωCCC refers to the ωCCC -ordered partof the vertex where ω and C denote gauge one-form and field strength zero-form higher-spin(HS) fields valued in arbitrary associative algebra in which case the order of product factors in ωCCC matters.HS gauge theory differs from usual local field theories because it contains infinite tower ofgauge fields of all spins and the number of space-time derivatives increases with the spins offields in the vertex [4, 5, 6, 7]. However one may ask for spin-locality [8, 9, 10, 11] which impliesspace-time locality in the lowest orders of perturbation theory [10]. Demanding spin-localityone actually fixes functional space for possible field redefinitions that is highly important forthe predictability of the theory.There are several ways to study the issue of (non)locality in HS gauge theory. One isreconstruction the vertices from the boundary by the holographic prescription based on theKlebanov-Polyakov conjecture [12] (see also [13], [14]). Alternatively, one can analyze verticesdirectly in the bulk starting from the generating equations of [1]. The latter approach developedin [10, 11, 2, 3, 15] is free from any holographic duality assumptions but demands careful choiceof the homotopy scheme to determine the choice of field variables compatible with spin-localityof the vertices. The issue of (non)locality of HS gauge theories was also considered in [16] and[17] with somewhat opposite conclusions.From the holographic point of view the vertex that contains φ was argued to be essentiallynon-local [18] or at least should have non-locality of very specific form presented in [19]. Onthe other hand, the holomorphic, i.e., η and antiholomorphic ¯ η vertices, where η is a complexparameter in the HS equations, were recently obtained in [2] where they were shown to bespin-local by virtue of Z -dominance lemma of [3]. The computation was done directly in thebulk starting from the non-linear HS system of [1].In this formalism HS fields are described by one-forms ω ( Y ; K | x ) and zero-forms C ( Y ; K | x )where x are space-time coordinates while Y A = ( y α , ¯ y ˙ α ) are auxiliary spinor variables. Bothdotted and undotted indices are two-component, α, ˙ α = 1 ,
2, while K = ( k, ¯ k ) are outer Kleinoperators satisfying k ∗ k = ¯ k ∗ ¯ k = 1 , { k, y α } ∗ = { k, z α } ∗ = { ¯ k, ¯ y ˙ α } ∗ = { ¯ k, ¯ z ˙ α } ∗ = { k, θ α } ∗ = { ¯ k, ¯ θ ˙ α } ∗ = 0 , (1.1)[ k, ¯ y ˙ α ] ∗ = [ k, ¯ z ˙ α ] ∗ = [¯ k, y α ] = [¯ k, z α ] ∗ = [ k, ¯ θ ˙ α ] ∗ = [¯ k, θ α ] ∗ = 0 . Schematically, non-linear HS equations in the unfolded form read asd x ω + ω ∗ ω = Υ( ω, ω, C ) + Υ( ω, ω, C, C ) + . . . , (1.2)d x C + ω ∗ C − C ∗ ω = Υ( ω, C, C ) + Υ( ω, C, C, C ) + . . . . (1.3)3s recalled in Section 2, generating equations of [1] that reproduce the form of equations(1.2) and (1.3) have a simple form as a result of doubling of spinor variables, namely ω ( Y ; K | x ) −→ W ( Z ; Y ; K | x ) , C ( Y ; K | x ) −→ B ( Z ; Y ; K | x ) . Equations (1.2) and (1.3) result from the generating equations of [1] upon order by orderreconstruction of Z -dependence (for more detail see Section 2). The final form of equations(1.2) and (1.3) turns out to be Z -independent as a consequence of consistency of the equationsof [1]. This fact may not be manifest however since the r.h.s.’s of HS equations usually has aform of the sum of Z -dependent terms.HS equations have remarkable property found in [20] that they remain consistent withthe fields W and B valued in any associative algebra. For instance W and B can belong tothe matrix algebra M at n with any n . Since in that case the components of W and B do notcommute, different orderings of the fields should be considered independently. (Mathematically,HS equations with this property correspond to A ∞ strong homotopy algebra introduced byStasheff in [21],[22],[23].) For instance, holomorphic ( i.e., ¯ η -independent) vertices in the zero-form sector can be represented in the formΥ η ( ω, C, C ) = Υ ηωCC +Υ ηCωC +Υ ηCCω , Υ ηη ( ω, C, C, C ) = Υ ηηωCCC +Υ ηηCωCC +Υ ηηCCωC +Υ ηηCCCω , . . . (1.4)where the subscripts of the vertices Υ refer to the ordering of the product factors.The vertices obtained in [2] were shown to be spin-local due to the Z -dominance Lemmaof [3] that identifies terms that must drop from the r.h.s.’s of HS equations together with the Z -dependence. Recall that spin-locality implies that the vertices are local in terms of spinorvariables for any finite subset of fields of different spins [15] (for more detail on the notionof spin-locality see [15]). Analogous vertices in the one-form sector have been shown to bespin-local earlier in [11].The main achievement of [2] consists of finding such solution of the generating systemin the third order in C that all spin-nonlocal terms containing infinite towers of derivativesin y (¯ y ) between C -fields in the(anti)holomorphic in η (¯ η ) sector do not contribute to η (¯ η )vertices by virtue of Z -dominance Lemma. Thus [2] gives spin-local expressions for the verticesΥ ηη ( ω, C, C, C ) which, however, have a form of a sum of a number of Z -dependent terms. Tomake spin-locality manifest one must remove the seeming Z-dependence from the vertex of [2].Technically, this can be done with the help of partial integrations and the Schouten identity.The aim of this paper is to show how this works in practice.Since the straightforward derivation presented in this paper is technically involved we confineourselves to the analysis of the particular vertex Υ ηηωCCC (1.4) not only to compute its manifestlyspin-local form but also to illustrate how Z -dominance Lemma works. Though this analysis canbe extended to the other cubic vertices, we hope to elaborate more efficient tools to computethem all elsewhere.The rest of the paper is organized as follows. In Section 2, the necessary background on HSequations is presented with brief recollection on the procedure of derivation of vertices fromthe generating system. Section 3 reviews the notion of the H + space as well as the justificationfor a computation modulo H + . In Section 4, we present step-by-step scheme of computationsperformed in this paper. Section 5 contains the final manifestly spin-local expression for Υ ηηωCCC vertex. In Sections 6 , 7 , 8 , 9 and 10 technical details of the steps sketched in Section 4 arepresented. In particular, in Section 7 we introduce important Generalised Triangle identity
Spin- s HS fields are encoded in two generating functions, namely, the space-time one-form ω ( y, ¯ y, x ) = d x µ ω µ ( y, ¯ y, x ) = X n,m d x µ ω µα ...α n , ˙ α ... ˙ α m ( x ) y α . . . y α n ¯ y ˙ α . . . ¯ y ˙ α m , s = 2 + m + n C ( y, ¯ y, x ) = X n,m C α ...α n , ˙ α ... ˙ α m ( x ) y α . . . y α n ¯ y ˙ α . . . ¯ y ˙ α m , s = | m − n | . (2.2)where α = 1 , α = 1 , y α and ¯ y ˙ α can be combined into an sp (4) spinor Y A = ( y α , ¯ y ˙ α ), A = 1 , ..., ω, ω, C, C, . . . ) (1.2) and Υ( ω, C, C, . . . ) (1.3) result from the generating sys-tem of [1] d x W + W ∗ W = 0 , (2.3)d x S + W ∗ S + S ∗ W = 0 , (2.4)d x B + W ∗ B − B ∗ W = 0 , (2.5) S ∗ S = i ( θ A θ A + ηB ∗ γ + ¯ ηB ∗ ¯ γ ) , (2.6) S ∗ B − B ∗ S = 0 . (2.7)Apart from space-time coordinates x , the fields W ( Z ; Y ; K | x ), S ( Z ; Y ; K | x ) and B ( Z ; Y ; K | x )depend on Y A , Z A = ( z α , ¯ z ˙ α ) and Klein operators K = ( k, ¯ k ) (1.1). W is a space-time one-form, i.e., W = dx ν W ν while S -field is a one-form in Z spinor directions θ A = ( θ α , ¯ θ ˙ α ), { θ A , θ B } = 0, i.e., S ( Z ; Y ; K ) = θ A S A ( Z ; Y ; K ) . (2.8) B is a zero-form.Star product is defined as follows( f ∗ g )( Z ; Y ; K ) = 1(2 π ) Z d U d V e iU A V A f ( Z + U, Y + U ; K ) g ( Z − V, Y + V ; K ) . (2.9)Elements γ = θ α θ α e iz α y α k and ¯ γ = ¯ θ ˙ α ¯ θ ˙ α e i ¯ z ˙ α ¯ y ˙ α ¯ k (2.10)are central because θ = 0 since θ α is a two-component anticommuting spinor.5 .2 Perturbation theory Starting with a particular solution of the form B ( Z ; Y ; K ) = 0 , S ( Z ; Y ; K ) = θ α z α + ¯ θ ˙ α ¯ z ˙ α , W ( Z ; Y ; K ) = ω ( Y ; K ) , (2.11)which indeed solves (2.3)-(2.7) provided that ω ( Y ; K ) satisfies zero-curvature condition,d ω + ω ∗ ω = 0 , (2.12)one develops perturbation theory. Starting from (2.7) one finds[ S , B ] ∗ = 0 . (2.13)From (2.9) one deduces that[ Z A , f ( Z ; Y ; K )] ∗ = − i ∂∂Z A f ( Z ; Y ; K ) . (2.14)Hence, equation (2.13) yields[ S , B ] = − iθ A ∂∂Z A B = − i d Z B = 0 = ⇒ B ( Z ; Y ; K ) = C ( Y ; K ) . (2.15)The Z -independent C -field that appears as the first-order part of B is the same that entersequations (1.2), (1.3). The perturbative procedure can be continued further leading to theequations of the form d Z Φ k +1 = J (Φ k , Φ k − , . . . ) , (2.16)where Φ k is either W , S or B field of the k -th order of perturbation theory, identified with thedegree of C -field in the corresponding expression, i.e., W = ω + W ( ω, C ) + W ( ω, C, C ) + . . . , S = S + S ( C ) + S ( C, C ) + . . . ,B = C + B ( C, C ) + B ( C, C, C ) + . . . .
To obtain dynamical equations (1.2), (1.3) one should plug obtained solutions into equations(2.3) and (2.5). For instance, (2.5) up to the third order in C -field isd x C + [ ω, C ] ∗ = − d x B − [ W , C ] ∗ − d x B − [ W , B ] ∗ − [ W , C ] ∗ + . . . (2.17)Though the fields W , W and B , B and hence various terms that enter (2.17) are Z -dependent, equations (2.3)-(2.7) are designed in such a way that, as a consequence of theirconsistency, the sum of the terms on the r.h.s. of (2.17) is Z -independent. To see this it suf-fices to apply d Z realized as i [ S , ] ∗ to the r.h.s. of (2.17) and make sure that it gives zeroby virtue of already solved equations. For more detail we refer the reader to the review [24]. H + and Z -dominance lemma H + In this Section the definition of the space H + [2] that plays a crucial role in our computationis recollected. Function f ( z, y | θ ) of the form f ( z, y | θ ) = Z d T e i T z α y α φ ( T z, y |T θ, T ) (3.1)6elongs to the space H + if there exists such a real ε >
0, thatlim
T → T − ε φ ( w, u | θ, T ) = 0 . (3.2)Note that this definition does not demand any specific behaviour of φ at T → H +0 of [15].In the sequel we use two main types of functions that obey (3.2): φ ( T z, y |T θ, T ) = T δ T e φ ( T z, y |T θ ) , φ ( T z, y |T θ, T ) = ϑ ( T − δ ) 1 T e φ ( T z, y |T θ ) (3.3)with some δ , >
0. (Note that the second option with δ > δ . Here step-function is denoted as ϑ to distinguish it from theanticommuting variables θ .)Space H + can be represented as the direct sum H + = H +0 ⊕ H +1 ⊕ H +2 , (3.4)where φ ( w, u | θ, T ) ∈ H + p are degree- p forms in θ satisfying (3.2).All terms from H + on the r.h.s. of HS field equations must vanish by Z -dominance Lemma[3]. Following [2] this can be understood as follows. All the expressions from (2.17) have theform (3.1) and the only way to obtain Z -independent non-vanishing expression is to bring thehidden T dependence in φ ( T z, y |T θ, T ) to δ ( T ). If a function contains an additional factorof T ε or is isolated from T = 0, it cannot contribute to the Z -independent answer whichis the content of Z -dominance Lemma [3]. This just means that functions of the class H +0 cannot contribute to the Z -independent equations (1.3). Application of this fact to locality isstraightforward once this is shown that all terms containing infinite towers of higher derivativesin the vertices of interest belong to H +0 and, therefore, do not contribute to HS equations. Thisis what was in particular shown in [2]. As in [2] we use exponential form for all the expressions below where by ωCCC we assume ω ( y ω , ¯ y )¯ ∗ C ( y , ¯ y )¯ ∗ C ( y , ¯ y )¯ ∗ C ( y , ¯ y ) (3.5)with ¯ ∗ denoting star-product with respect to ¯ y . Derivatives ∂ ω and ∂ j act on auxiliary variablesas follows ∂ ωα = ∂∂ y αω , ∂ jα = ∂∂ y αj . (3.6)After all the derivatives in y ω and y j are evaluated the latter are set to zero, i.e., y ω = y j = 0 . (3.7)In this paper we use the following notation of [2]: t α := − i∂ ωα , p jα := − i∂ jα , (3.8) Z d n ρ + := Z dρ . . . dρ n ϑ ( ρ ) . . . ϑ ( ρ n ) . (3.9)7 .3 Contribution to Υ ηηωCCC modulo H + The η C vertex in the equations on the zero-forms C resulting from equations of [1] isΥ ηη ( ω, C, C, C ) = − (d x B ηη + [ ω, B ηη ] ∗ + [ W η , B η ] ∗ + [ W ηη , C ] ∗ + d x B η ) . (3.10)Recall, that, being Z -independent, Υ ηη is a sum of Z -dependent terms that makes its Z -independence implicit.As explained in Introduction, Υ ηη can be decomposed into parts with different orderings offields ω and C . In this paper we considerΥ ηηωCCC := Υ ηη ( ω, C, C, C ) (cid:12)(cid:12)(cid:12) ωCCC . (3.11)Since the terms from H + do not contribute to the physical vertex such terms can be discarded.Following [2] equality up to terms from H + referred to as weak equality is denoted as ≈ .We start with the following results of [2]: b Υ ηηωCCC ≈ Υ ηηωCCC = − (cid:16) W η ωC ∗ B η loc + W ηη ωCC ∗ C + d x B η loc (cid:12)(cid:12) ωCCC + ω ∗ B ηη + d x B ηη (cid:12)(cid:12) ωCCC (cid:17) , (3.12)where W η ωC ∗ B η loc ≈ η Z d T T Z dσ Z d ρ + δ − X i =1 ρ i ! ( z γ t γ ) (cid:2) z α y α + σz α t α (cid:3) ( ρ + ρ ) ×× exp n i T z α y α + i (1 − σ ) t α ∂ α − i ρ σρ + ρ t α p α + i ρ σρ + ρ t α p α + i T z α (cid:16) − ( ρ + ρ + σρ ) t α − ( ρ + ρ ) p α + ( ρ − ρ ) p α + ( ρ + ρ ) p α (cid:17) + iy α (cid:16) σt α − ρ ρ + ρ p α + ρ ρ + ρ p α (cid:17)o ωCCC , (3.13) W ηη ωCC ∗ C ≈ − η Z d T T Z d ρ + δ − X i =1 ρ i ! ρ ( z γ t γ ) ( ρ + ρ )( ρ + ρ ) ×× exp n i T z α y α + i T z α (cid:16) (1 − ρ ) t α − ( ρ + ρ ) p α + ( ρ + ρ ) p α + p α (cid:17) + iy α t α + ρ ρ ( ρ + ρ )( ρ + ρ ) ( iy α t α + it α p α ) + i (cid:18) (1 − ρ ) ρ ρ + ρ + ρ (cid:19) t α p α − i ρ ρ ρ + ρ t α p α o ωCCC, (3.14)d x B η loc (cid:12)(cid:12) ωCCC ≈ η Z d T Z dξ Z d ρ + δ − X i =1 ρ i ! ( z α y α ) h ( T z α − ξy α ) t α i ×× exp n i T z α y α + i (1 − ρ ) t α p α − iρ t α p α + i T z α (cid:16) − ( ρ + ρ ) t α − ρ p α + ( ρ + ρ ) p α + p α (cid:17) + iy α (cid:16) ξ ( ρ + ρ ) t α + ξρ p α − ξ ( ρ + ρ ) p α + (1 − ξ ) p α (cid:17)o ωCCC , (3.15)8 ∗ B ηη ≈ − η Z d T T Z d ρ + δ − X i =1 ρ i ! Z dξ ρ [ z α ( y α + t α )] ( ρ + ρ )( ρ + ρ ) ×× exp n i T z α y α + i T z α (cid:16) − t α − ( ρ + ρ ) p α + ( ρ − ρ ) p α + ( ρ + ρ ) p α (cid:17) + iy α t α + i (1 − ξ ) y α (cid:18) ρ ρ + ρ p α − ρ ρ + ρ p α (cid:19) + iξ y α (cid:18) ρ ρ + ρ p α − ρ ρ + ρ p α (cid:19) + i (1 − ξ ) ρ ρ + ρ t α p α − i (cid:18) (1 − ξ ) ρ ρ + ρ + ξρ ρ + ρ (cid:19) t α p α + i ξρ ρ + ρ t α p α o ωCCC, (3.16)d x B ηη (cid:12)(cid:12) ωCCC ≈ η Z d T T Z d ρ + δ − X i =1 ρ i ! Z dξ ρ ( z α y α ) ( ρ + ρ )( ρ + ρ ) ×× exp n i T z α y α + i T z α (cid:16) − ( ρ + ρ )( t α + p α ) + ( ρ − ρ ) p α + ( ρ + ρ ) p α (cid:17) + it α p α + i (1 − ξ ) y α (cid:18) ρ ρ + ρ ( t α + p α ) − ρ ρ + ρ p α (cid:19) + ξ y α (cid:18) ρ ρ + ρ p α − ρ ρ + ρ p α (cid:19) o ωCCC. (3.17)The sum of r.h.s.’s of (3.13)-(3.17) yields b Υ ηηωCCC ( Z ; Y ).Note, that all terms on the r.h.s.’s of (3.13)-(3.17) contain no p jα p iα contractions in theexponentials, hence being spin-local [2]. Thus b Υ ηηωCCC ( Z ; Y ) is also spin-local.Let us emphasize that only the full expression for Υ ηηωCCC ( Y ) (3.11) is Z -independent, while b Υ ηηωCCC ( Z ; Y ) (3.12) with discarded terms in H + is not. This does not allow one to find mani-festly Z -independent expression for Υ ηηωCCC by setting for instance Z = 0 in Eqs. (3.13)-(3.17).In this paper Z -dependence of b Υ ηηωCCC ( Z ; Y ) is eliminated modulo terms in H + by virtue ofpartial integration and the Schouten identity. As a result, b Υ ηηωCCC ( Z ; Y ) ≈ bb Υ ηηωCCC ( Y ) , where bb Υ ηηωCCC ( Y ) is manifestly spin-local and Z -independent. Since H +0 -terms do not contributeto the vertex by Z-dominance Lemma [3]Υ ηηωCCC ( Y ) = bb Υ ηηωCCC ( Y ) . Our goal is to find the manifest form of bb Υ ηηωCCC ( Y ). The calculation scheme is as follows. • I. We start from the expression Eqs. (3.13)-(3.17) for the vertex obtained in [2].9
II. To z -linear pre-exponentials.Using partial integration and the Schouten identity we transform Eqs. (3.13)-(3.17) tothe form with z -linear pre-exponentials modulo weakly Z -independent terms. Theseexpressions are collected in Section 6, Eqs. (6.1)-(6.4). The respective cohomology termsbeing a part of the vertex Υ ωCCC are presented in Section 5 . • III. Uniformization.We observe that the r.h.s.’s of Eqs. (6.1)-(6.4) can be re-written modulo cohomology andweakly zero terms in a form of integrals R d Γ over the same integration domain I Z d Γ z α f α ( y, t, p , p , p |T , ξ i , ρ i ) E ωCCC , (4.1)where the integrand contains an overall exponential function EE = E z E, (4.2) E z := exp i n T z α ( y + P ) α o , (4.3) E := exp i n − ξ ρ (1 − ρ − ρ )(1 − ρ ) (cid:0) y + P (cid:1) α y α (4.4)+ ξ ρ (1 − ρ − ρ )(1 − ρ ) (cid:0) y + P (cid:1) α ˜ t α + ρ (1 − ρ − ρ ) ( p + p ) α y α − ρ (1 − ρ − ρ )(1 − ρ ) ρ t α y α + ρ (1 − ρ ) ( p α + p α ) t α + p α y α + p α t α o , ˜ t = ρ ρ + ρ t , (4.5) P = P + (1 − ρ ) t , (4.6) P = (1 − ρ − ρ )( p + p ) − (1 − ρ )( p + p ) , (4.7)the integral over I is denoted as Z d Γ = Z d T Z d ξ + δ − X i =1 ξ i ! Z d ρ + δ − X j =1 ρ j ! . (4.8)Eqs. (6.1)-(6.4) transformed to the form (4.1) are collected in Section 8, Eqs. (8.2)-(8.5). • IV. Elimination of δ -functions.Using partial integration and the Schouten identity we eliminate all factors of δ ( ρ i ) and δ ( ξ i ) from Eqs. (8.2)-(8.5). The result is presented in Section 9, Eqs. (9.1)-(9.4). • V. Final step.Finally, we show in Section 10 that a sum of the r.h.s.’s of Eqs. (9.2)-(9.4) equals Z -independent cohomology term up to terms in H + .By collecting all resulting Z -independent terms we finally obtain the manifest expressionfor vertex Υ ηηωCCC , being a sum of expressions (5.2)-(5.12).10 Main result Υ ηηωCCC Here the final manifestly Z -independent ωCCC contribution to the equations is presented.Vertex Υ ηηωCCC is Υ ηηωCCC = X j =1 J j (5.1)with J i given in Eqs. (5.2)-(5.12). Note that the integration regions may differ for differentterms J j in the vertex, depending on their genesis.Firstly we note that B ηη (A.10), that contains a Z -independent part, generates cohomologiesboth from ω ∗ B ηη and from d x B ηη , J = − η Z d Γ δ ( ξ ) ρ ( ρ + ρ )( ρ + ρ ) δ ( ρ ) E ωCCC, (5.2) J = η Z d Γ δ ( ξ ) ρ ( ρ + ρ )( ρ + ρ ) δ ( ρ ) E ωCCC . (5.3)Recall that E and d Γ are defined in (4.4) and (4.8), respectively. (Note, that, here and below,the integrands on the r.h.s.’s of expressions for J i are T -independent, hence the factor of R d T in d Γ equals one.)Other cohomology terms are collected from (9.2), (9.3), (9.4), (10.1), (D.4), (B.1), (B.3),(B.4) and (B.5), respectively, J = − iη Z d Γ δ ( ξ ) 1( ρ + ρ )(1 − ρ ) n ρ t α ( p + p ) α (cid:2) −→ ∂ ρ − −→ ∂ ρ (cid:3) + ρ ( p + p ) α ( p + p ) α (cid:2) −→ ∂ ρ −−→ ∂ ρ (cid:3) + ρ t α ( p + p ) α (cid:2) −→ ∂ ρ −−→ ∂ ρ (cid:3) + ρ + ρ (1 − ρ ) t α ( p + p ) α o E ωCCC , (5.4) J = iη Z d Γ δ ( ξ )1 − ρ (cid:16) − ρ (1 − ρ − ρ ) (1 − ρ ) t γ y γ − ρ (1 − ρ − ρ )(1 − ρ ) t γ y γ [ −−→ ∂ ρ + −→ ∂ ρ ] − ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ ( y + ˜ t ) γ [ −→ ∂ ρ − −→ ∂ ρ ] (cid:17) EωCCC , (5.5) J = − i η Z d Γ δ ( ξ ) h ξ ( −→ ∂ ξ − −→ ∂ ξ ) in − ρ (1 − ρ − ρ ) (1 − ρ )( ρ + ρ ) ( p α + p α ) γ t γ − ρ (1 − ρ − ρ ) (1 − ρ ) t α y α + 1( ρ + ρ )(1 − ρ )( ρ + ρ ) ( p α + p α ) t α o E ωCCC , (5.6) J = i η Z d Γ δ ( ξ ) ρ (1 − ρ − ρ )(1 − ρ ) ( ρ + ρ ) ( p + p ) γ ( t ) γ EωCCC , (5.7)11 = − η Z d Γ δ ( ξ ) ξ ρ ρ ( ρ + ρ ) (1 − ρ ) ( ρ + ρ ) ×× (cid:0) y + (1 − ρ − ρ )( p + p ) + (1 − ρ ) t (cid:1) γ (cid:0) y + ˜ t (cid:1) γ t α y α E ωCCC , (5.8) J = − η Z d Γ δ ( ρ ) (cid:16) ρ δ ( ξ ) + h iδ ( ρ ) − ( p α + p α ) t α in iδ ( ξ ) + ˜ t γ y γ o(cid:17) E ωCCC , (5.9) J = iη Z d Γ δ ( ρ ) δ ( ρ ) δ ( ξ ) exp n − iξ ( p + p + t − ρ ( p + p )) α ( y ) α − ξ ( y + p + p − ρ ( p + p )) γ ( t ) γ + (1 − ρ )( p + p ) γ y γ + p γ y γ + t β p β o ωCCC , (5.10) J = − iη Z d Γ δ ( ρ ) δ ( ξ ) δ ( ρ ) exp i n − ξ ( y + p + p + t − ρ ( p + p )) α ( y ) α + (1 − ρ )( p + p ) γ y γ + p γ y γ + t β p β o ωCCC , (5.11) J = iη Z d Γ δ ( ρ ) δ ( ρ ) y α t α exp i n ( y + P + t ) γ ( ξ t − ξ y ) γ + (1 − ρ )( p + p ) γ y γ + p γ y γ + t β p β o ωCCC . (5.12)Let us emphasize, that neither exponential function E (4.4) nor the exponentials on the r.h.s.’s of Eqs. (5.10)-(5.12) contain ∂ iα ∂ kα terms. Hence, as anticipated, all J j are spin-local.One can see that though having poles in pre-exponentials these expressions are well defined.For instance a potentially dangerous factor on the r.h.s. of (5.2) is dominated by 1 as followsfrom the inequality ρ − ( ρ + ρ )( ρ + ρ ) = − ρ ρ ≤ Q ϑ ( ρ i ) δ (1 − P ρ i ) δ ( ρ ). Analogous simple reasoning applies to the r.h.s. of (5.3).The case of (5.4)-(5.8) is a bit more tricky. By partial integrations one obtains from (5.4)-(5.6) J + J + J = iη Z d Γ δ ( ξ ) 1( ρ + ρ )(1 − ρ ) n − δ ( ρ ) t α ( p + p ) α (5.13)+[ δ ( ρ ) − δ ( ρ )] ρ ( p + p ) α ( p + p ) α + t α ( p + p ) α − δ ( ρ ) ρ t α ( p + p ) α − δ ( ρ ) ρ (1 − ρ ) t γ y γ + [ δ ( ρ ) − δ ( ρ )] ρ (1 − ρ ) ( p + p ) γ ( y + ˜ t ) γ − δ ( ξ ) (cid:16) − ρ ( ρ + ρ )( ρ + ρ ) ( p α + p α ) γ t γ − ρ ( ρ + ρ )(1 − ρ ) t α y α + 1( ρ + ρ ) ( p α + p α ) t α (cid:17)o E ωCCC .
Using that, due to the factor of δ (1 − P ρ i ), for positive ρ i it holds ρ ( ρ + ρ )(1 − ρ ) − − ρ (1 − ( ρ + ρ ))( ρ + ρ )(1 − ρ ) ≤ , (5.14)1( ρ + ρ )(1 − ρ ) ≤ ρ + ρ )(1 − ρ − ρ ) = 1( ρ + ρ ) + 1( ρ + ρ ) , (5.15)12ne can make sure that each of the expressions with poles in the pre-exponential in Eqs. (5.7),(5.8) and (5.13) can be represented in the form of a sum of integrals with integrable pre-exponentials. For instance, the potentially dangerous factor in (5.8), by virtue of (5.14) and(5.15) satisfies ρ ρ ( ρ + ρ ) (1 − ρ ) ( ρ + ρ ) ≤ − ρ )( ρ + ρ ) + 1( ρ + ρ ) + 1( ρ + ρ ) . (5.16)Each of the terms on the r.h.s. of Eq. (5.16) is integrable, because integration is over a three-dimensional compact area P ρ i = 1 in the positive quadrant. For instance consider the firstterm. Swopping ρ ↔ ρ one has Z d ρ + δ (1 − X ρ i ) 1(1 − ρ )( ρ + ρ ) = Z d ρ + ϑ (1 − X ρ i ) 1(1 − ρ )( ρ + ρ ) = (5.17) − Z dρ Z − ρ dρ log( ρ + ρ )( ρ + ρ ) = 12 Z dρ log ( ρ ) , which is integrable.Analogously other seemingly dangerous factors can be shown to be harmless as well. z -linear pre-exponentials Step II of the calculation scheme of Section 4 is to transform Eqs. (3.13)-(3.17) to the Z -independent cohomology terms and the terms with pre-exponentials linear in z .To this end, from (A.10) one straightforwardly obtains that ω ∗ B ηη ≈ J + η Z d Γ δ ( ξ ) δ ( ρ )(1 − ρ )(1 − ρ ) " − ρ ( z α ( y α + t α ))( p β + p β )( p β + p β )+ i h(cid:16) δ ( ρ ) + δ ( ρ ) (cid:17) (1 − ρ )(1 − ρ ) − δ ( ξ ) i z α (cid:16) (1 − ρ )( p α + p α ) − (1 − ρ )( p α + p α ) (cid:17) + iz α ( p α + p α )(1 − ρ ) (cid:16) δ ( ξ ) − δ ( ξ ) (cid:17) exp n i T z α (cid:0) y α + t α +(1 − ρ )( p α + p α ) − (1 − ρ )( p α + p α ) (cid:1) + i (1 − ξ ) ρ ρ + ρ ( y α + t α )( p α + p α ) + iξ ρ ρ + ρ ( y α + t α )( p α + p α ) − i ( y α + t α ) p α o ωCCC , (6.1)where J is the cohomology term (5.2). Analogously,d x B ηη ≈ J − η Z d Γ δ ( ξ ) δ ( ρ )(1 − ρ )(1 − ρ ) " − ρ ( z α y α )( p β + t β + p β )( p β + p β )+ i h(cid:16) δ ( ρ ) + δ ( ρ ) (cid:17) (1 − ρ )(1 − ρ ) − δ ( ξ ) i z α (cid:16) (1 − ρ )( p α + t α + p α ) − (1 − ρ )( p α + p α ) (cid:17) + iz α ( p α + t α + p α )(1 − ρ ) (cid:16) δ ( ξ ) − δ ( ξ ) (cid:17) exp n i T z α (cid:0) y α +(1 − ρ )( p α + t α + p α ) − (1 − ρ )( p α + p α ) (cid:1) + i (1 − ξ ) ρ ρ + ρ y α ( p α + t α + p α ) + iξ ρ ρ + ρ y α ( p α + p α ) − iy α p α + it β p β o ωCCC (6.2)13ith J (5.3).Using the Schouten identity and partial integrations one obtains from Eqs. (3.13)-(3.15),respectively, W η ωC ∗ B η ≈ η Z d T Z dτ Z dσ Z dσ " i ( z α t α ) δ (1 − τ )+ z α ( p α + p α )1 − τ (cid:16) i (cid:0) δ ( σ ) − δ (1 − σ ) (cid:1) − (cid:2) y α + p α + p α − σ ( p α + p α ) (cid:3) t α (cid:17) exp n i T z α y α + i T z α (cid:16) τ ( p α + p α ) − ((1 − τ ) + σ τ )( p α + p α ) + (cid:0) σ + τ (1 − σ ) (cid:1) t α (cid:17) + it α p α + iσ (cid:2) y α + p α + p α − σ ( p α + p α ) (cid:3) t α − i (cid:16) σ p α − (1 − σ ) p α (cid:17) y α o ωCCC , (6.3) W ηη ωCC ∗ C ≈ − iη Z d Γ δ ( ξ ) δ ( ρ ) ( z γ t γ ) ρ + ρ h − ρ (cid:0) δ ( ρ ) + it α ( p α + p α ) (cid:1) + ξ δ ( ξ ) i ×× exp n i T z α y α + i T z α (cid:16) (1 − ρ − ρ )( p α + p α ) − (1 − ρ )( p α + p α ) + (1 − ρ ) t α (cid:17) + iy α (cid:18) ξ ρ − ρ t α + p α (cid:19) + i (cid:18) − ρ − ξ ρ ρ − ρ (cid:19) t α p α − i (1 − ξ ) ρ t α p α + i ξ ρ − ρ t α p α o ωCCC , (6.4)d x B η ≈ iη Z d Γ δ ( ξ ) δ ( ρ ) ( z α y α ) h it γ ( p γ + p γ ) + δ ( ρ ) − δ ( ρ ) i ×× exp n i T z α y α + i T z α (cid:0) (1 − ρ − ρ )( p α + p α ) − (1 − ρ )( p α + p α ) + (1 − ρ ) t α (cid:1) + i (1 − ρ ) t β p β − iρ t β p β + iξ y α (cid:16) ( ρ + ρ ) t α + ρ p α − (1 − ρ ) p α − p α (cid:17) + iy α p α o ωCCC. (6.5) Here a useful identity playing the key role in our computations is introduced.For any F ( x, y ) consider I = Z [0 , dτ Z d ξ + δ (1 − ξ − ξ − ξ ) (7.1) z γ h ( a − a ) γ δ ( ξ ) + ( a − a ) γ δ ( ξ ) + ( a − a ) γ δ ( ξ ) i F (cid:0) τ z β P β , ( − ξ a − ξ a − ξ a ) α P α (cid:1) with arbitrary τ, ξ - independent P and a i .Let G ( x, y ) be a solution to differential equation ∂∂x G ( x, y ) = ∂∂y F ( x, y ) . (7.2)14ence I = Z [0 , dτ Z d ξ + δ (1 − ξ − ξ − ξ ) (7.3)( a − a ) α ( a − a ) α −→ ∂ τ G (cid:0) τ z β P β , ( − ξ a − ξ a − ξ a ) α P α (cid:1) . Note that there is a factor of ( a − a ) α ( a − a ) α equal to the area of triangle spanned by thevectors a , a , a on the r.h.s. of (7.3).This identity is closely related to identity (3.24) of [10], that, in turn, expresses triangleidentity of [25]. Hence, (7.3) will be referred to as Generalised Triangle identity or GT identity .Note that, for appropriate G partial integration on the r.h.s. of (7.3) in τ gives z -independent (cohomology) term plus H + -term. Namely, I = − Z d ξ + δ (1 − ξ − ξ − ξ ) (7.4)( a − a ) α ( a − a ) α G (cid:0) , ( − ξ a − ξ a − ξ a ) α P α (cid:1) + Z d ξδ (1 − ξ − ξ − ξ )( a − a ) α ( a − a ) α G (cid:0) z β P β , ( − ξ a − ξ a − ξ a ) α P α (cid:1) . The second term on the r.h.s. belongs to H + if G is of the form (3.1) satisfying (3.2).To prove GT identity let us perform partial integration on the r.h.s. of (7.1) with respectto ξ i . This yields I = Z [0 , dτ Z d ξ + δ (1 − ξ − ξ − ξ ) (7.5) h z γ ( a − a ) γ P α a α + z γ ( a − a ) γ P α a α + z γ ( a − a ) γ P α a α i × ∂∂y F (cid:0) τ z α P α , − ( ξ a + ξ a + ξ a ) α P α (cid:1) . The Schouten identity yields h z γ a γ P α ( a − a ) α + z γ a γ P α ( a − a ) α + z γ a γ P α ( a − a ) α i = (7.6) h z γ P γ (cid:8) a α ( a − a ) α + a α ( a − a ) α + a α ( a − a ) α (cid:9) + z γ ( a − a ) γ P α a α + z γ ( a − a ) γ P α a α + z γ ( a − a ) γ P α a α i . One can observe that h z γ ( a − a ) γ P α a α + z γ ( a − a ) γ P α a α + z γ ( a − a ) γ P α a α i = (7.7) − h z γ a γ P α ( a − a ) α + z γ a γ P α ( a − a ) α + z γ a γ P α ( a − a ) α i , whence it follows (7.3). 15 useful particular case of GT identity is that with F ( x, y ) = f ( x + y ), namely Z [0 , dτ Z d ξ + δ (1 − ξ − ξ − ξ ) z γ h ( a − a ) γ δ ( ξ ) (7.8)+( a − a ) γ δ ( ξ ) + ( a − a ) γ δ ( ξ ) i f (cid:0) ( τ z − ξ a − ξ a − ξ a ) α P α (cid:1) = − Z [0 , dτ Z d ξ + δ (1 − ξ − ξ − ξ )( a − a ) α ( a − a ) α −→ ∂ τ f (cid:0) ( τ z − ξ a − ξ a − ξ a ) α P α (cid:1) . Step III of Section 4 is to uniformize the r.h.s. ’s of Eqs. (6.1)-(6.5) putting them into the form(4.1), where GT identity (7.1) plays an important role. Details of uniformization are given inAppendix B (p. 21).As a result, Eq. (3.12) yields b Υ ηηωCCC (cid:12)(cid:12)(cid:12) mod cohomology ≈ X j =1 F j (8.1)with F j presented in (8.2)-(8.5).Note that different terms of F j will be considered separately in what is follows. For thefuture convenience the underbraced terms are re-numerated, being denoted as F j,k , where j refers to F j while k refers to the respective underbraced term in the expression for F j . Forinstance, F = F , + F , + F , + F , , etc . − ω ∗ B ηη (cid:12)(cid:12)(cid:12) mod δ ( ρ )& δ ( T ) ≈ F := − η Z d Γ δ ( ξ ) δ ( ρ )(1 − ρ − ρ )(1 − ρ ) h ρ ( z β P β )( p α + p α )( p α + p α ) | {z } + iδ ( ρ )(1 − ρ − ρ )(1 − ρ )( z α P α ) | {z } + − iξ δ ( ξ )( z α P α ) | {z } + i (1 − ρ − ρ ) z α ( p α + p α ) (cid:16) δ ( ξ ) − δ ( ξ ) (cid:17)| {z } i E ωCCC , (8.2) − d x B ηη (cid:12)(cid:12)(cid:12) mod δ ( ρ )& δ ( T ) ≈ F := + η Z d Γ δ ( ξ ) δ ( ρ )(1 − ρ − ρ )(1 − ρ ) h ρ ( z β P β )( p α + p α )( p α + p α ) | {z } + ρ (1 − ρ )( z β t β ) t α ( p α + p α ) | {z } + ρ (1 − ρ )( z β t β )( p α + p α )( p α + p α ) | {z } + ρ ( z β P β ) t α ( p α + p α ) | {z } + iδ ( ρ )(1 − ρ − ρ )(1 − ρ )( z α P α ) | {z } + − iξ δ ( ξ )( z α P α ) | {z } + − iξ δ ( ξ )(1 − ρ )( z α t α ) | {z } + i (1 − ρ − ρ ) z α ( p α + p α ) (cid:16) δ ( ξ ) − δ ( ξ ) (cid:17)| {z } + i (1 − ρ − ρ ) z α t α (cid:16) δ ( ξ ) − δ ( ξ ) (cid:17)| {z } i E ωCCC , (8.3)16 d x B η − W ηη ωCC ∗ C (cid:12)(cid:12)(cid:12) mod δ ( T ) ≈ F := − η Z d Γ δ ( ρ ) δ ( ξ ) " iδ ( ρ )( z α P α ) | {z } + − i ( z α t α ) ξ δ ( ξ ) ρ + ρ | {z } + t α ( p α + p α ) z γ P γ | {z } + iδ ( ρ ) z α ( −P α ) | {z } + t γ ( p γ + p γ ) z α t α (cid:18) (1 − ρ ) − ρ ρ + ρ (cid:19)| {z } E ωCCC , (8.4) − (d x B ηη + ω ∗ B ηη ) (cid:12)(cid:12)(cid:12) δ ( ρ ) (cid:12)(cid:12)(cid:12) mod δ ( T ) − W η ωC ∗ B η loc ≈ F := − η Z d Γ δ ( ξ ) δ ( ξ ) z α ( p α + p α )( ρ + ρ )( ρ + ρ ) ×× i (cid:16) δ ( ρ ) − δ ( ρ ) (cid:17) E | {z } + iE z (cid:18) ∂∂ρ − ∂∂ρ (cid:19) E | {z } ωCCC. (8.5)Note that F , + F , = 0 , (8.6) F , + F , = 0 . (8.7)Let us emphasise that, by virtue (E.1), each F j is of the form (4.1) as expected.Note that during uniformizing procedure the vertices (5.9) -(5.12) are obtained in AppendixB (p. 21). δ ( ρ j ) and δ ( ξ j ) . Result The fourth step of Section 4 is to eliminate all δ ( ρ i ) and δ ( ξ ) and δ ( ξ ) from the pre-exponentialson the r.h.s.’s of Eqs. (8.2)-(8.5).More precisely, using partial integration, the Schouten identity and Generalised Triangleidentity (7.3), taking into account Eqs. (4.5)-(4.7) one finds that Eq. (8.1) yields (cid:0) b Υ ηηωCCC − G − G − G (cid:1)(cid:12)(cid:12) mod cohomology ≈ , (9.1)where G := J + η Z d Γ δ ( ξ ) z γ ( ( y γ + e t γ ) ρ t α ( p α + p α )(1 − ρ − ρ )(1 − ρ ) E z " ∂∂ρ − ∂∂ρ E + ( y γ + e t γ ) ρ ( p α + p α )( p α + p α )(1 − ρ − ρ )(1 − ρ ) E z " ∂∂ρ − ∂∂ρ E + ( y γ + ˜ t γ ) ρ t α ( p α + p α )(1 − ρ − ρ )(1 − ρ ) E z " ∂∂ρ − ∂∂ρ E + ( y γ + ˜ t γ ) ( ρ + ρ ) t α ( p α + p α )(1 − ρ − ρ )(1 − ρ ) E + ( y γ + ˜ t γ ) ρ t α ( p α + p α )(1 − ρ − ρ ) (1 − ρ ) E + ρ t γ ( p α + p α )( p α + p α + t α − ˜ t α )(1 − ρ − ρ )(1 − ρ )( ρ + ρ ) E ) ωCCC , (9.2)17 := J + η Z d Γ δ ( ξ )1 − ρ z α ( ρ ( y α + ˜ t α ) t γ ( y γ + P γ )(1 − ρ − ρ ) (1 − ρ ) E− ρ ρ t α ( y γ + P γ ) t γ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) E − ρ ( y α + ˜ t α ) t γ ( p γ + p γ )(1 − ρ − ρ )(1 − ρ ) E− ρ ( p α + p α )( y γ + P γ ) t γ (1 − ρ − ρ )( ρ + ρ )(1 − ρ ) E + E z ρ t γ ( y γ + P γ )( y α + ˜ t α )(1 − ρ − ρ )(1 − ρ ) " ∂∂ρ − ∂∂ρ E + E z ρ ( y α + ˜ t α )( p γ + p γ )( y γ + P γ )(1 − ρ − ρ )(1 − ρ ) " ∂∂ρ − ∂∂ρ E ) ωCCC , (9.3) G := J + η Z d Γ δ ( ξ ) ξ " ∂∂ξ − ∂∂ξ ×× z α ( ρ t α ( p γ + p γ )( y γ + ˜ t γ )(1 − ρ − ρ ) (1 − ρ )( ρ + ρ ) + − ρ t α (˜ t γ + y γ )( y γ + P γ )(1 − ρ − ρ ) (1 − ρ ) ( ρ + ρ )+ − ρ ( y α + ˜ t α )( t γ y γ )(1 − ρ − ρ ) (1 − ρ ) + ( y α + ˜ t α )( p γ + p γ ) t γ (1 − ρ − ρ )(1 − ρ ) ) E ωCCC , (9.4)with J , J and J being the cohomology terms (5.4), (5.5) and (5.6), respectively. (Details ofthe derivation are presented in Appendix C (p.24).)Note that schematically G + G + G = Z d Γ δ ( ξ ) z α g α ( y, t, p , p , p | ρ, ξ ) E ωCCC + J + J + J , (9.5)as expected . Let us stress that g α ( y, t, p , p , p | ρ, ξ ) on the r.h.s. of (9.5) is free from adistributional behaviour.
10 Final step of calculation
Here this is shown that the sum of the r.h.s.’s of Eqs. (9.2)-(9.4) gives a Z -independent coho-mology term up to terms in H + .More in detail, the expression G + G + G of the form (9.5) consists of two types ofterms with the pre-exponential of degree four and six in z, y, t, p , p , p , respectively. Thatwith degree-four pre-exponential separately equals a Z -independent cohomology term up toterms in H + . This is considered in Section 10.1. The term with degree-six pre-exponential isconsidered in Section 10.2. As a result of these calculations J (5.7) and J (5.8) are obtained.18 Consider the sum of expressions with z -dependent degree-four pre-exponential from Eqs. (9.2),(9.3) and (9.4), denoting it as S . Partial integration yields S ≈ J + η Z d Γ δ ( ξ ) h ρ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) t α z α ( p + p ) γ ( t − ˜ t ) γ (10.1)+ ρ ρ (1 − ρ − ρ )(1 − ρ ) ( ρ + ρ ) t γ z γ (cid:0) y + P (cid:1) α t α + ρ (1 − ρ − ρ )(1 − ρ ) ( ρ + ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ z α ( t ) α + ρ (1 − ρ − ρ ) (1 − ρ )( ρ + ρ ) t α z α ( p + p ) α ( y + ˜ t ) α + ρ (1 − ρ − ρ ) (1 − ρ ) ( ρ + ρ ) (cid:0) − P + ˜ t (cid:1) γ (cid:0) y + ˜ t (cid:1) γ z α t α i E ωCCC , where the cohomology term J is given in (5.7) . It is not hard to see that the integrand of theremaining term is zero by virtue of the Schouten identity. Terms of this type either appear in (9.2), (9.3) via differentiation in ρ j or in (9.4) via differen-tiation in ξ j . Denoting a sum of these terms as S we obtain S = + η Z d Γ δ ( ξ ) n ( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) h E z ( −→ ∂ ρ − −→ ∂ ρ ) E i (10.2)+ E z ρ (1 − ρ − ρ )(1 − ρ ) h ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ z α ( y + ˜ t ) α i [ −→ ∂ ρ − −→ ∂ ρ ] E (cid:17) + (cid:16)h E z i ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α [ −→ ∂ ρ − −→ ∂ ρ ] E + iξ h + n + ρ ρ (1 − ρ − ρ ) (1 − ρ ) ( ρ + ρ ) (cid:0) y + (1 − ρ − ρ )( p + p ) + (1 − ρ ) t (cid:1) γ (cid:0) y + ˜ t (cid:1) γ z α t α − ρ ρ (1 − ρ − ρ ) (1 − ρ ) (cid:0) y + ˜ t (cid:1) γ z γ t α y α + ρ (1 − ρ − ρ ) (1 − ρ ) (cid:0) y + ˜ t (cid:1) γ z γ ( p α + p α ) t α o E i × (cid:0) y + P (cid:1) α ( y + ˜ t ) α o ωCCC Recall that the integral measure dΓ(4.8) contains the factor of δ (1 − P ξ i ). Hence taking intoaccount the factor of δ ( ξ ) on the r.h.s. of (10.2) the dependence on ξ , ξ can be eliminated bythe substitution ξ → − ξ , ξ →
0. Then we consider separately the terms that contain anddo not contain ξ in the pre-exponentials. As shown in Appendix D, those with ξ -proportionalpre-exponentials give J (5.8) up to H + , while those with ξ -independent pre-exponentials givezero up to H + .
11 Conclusion
In this paper starting from Z -dominated expression obtained in [2] the manifestly spin-localholomorphic vertex Υ ηηωCCC in the equation (1.3) is obtained for the ωCCC ordering. Besides19valuation the expression for the vertex, our analysis provides how Z -dominance implies spin-locality.One of the main technical difficulties towards Z -independent expression was uniformization,that is bringing the exponential factors to the same form, of all contributions (3.13)-(3.17)with the least amount of new integration parameters possible. Practically, some part of theuniformization procedure heavily used the Generalized Triangle identity of Section 7 playingimportant role in our analysis.Let us stress that spin-locality of the vertices obtained in [2] follows from Z -dominanceLemma. However the evaluation the explicit spin-local vertex Υ η ωCCC achieved in this paper istechnically involved, to derive explicit form of other spin-local vertices in this and higher ordersa more elegant approach to this problem is highly desirable. Acknowledgments
We would like to thank Mikhail Vasiliev for fruitful discussions and useful comments on themanuscript. We acknowledge a partial support from the Russian Basic Research FoundationGrant No 20-02-00208. The work of OG is partially supported by the FGU FNC SRISA RAS(theme 0065-2019-0007).
Appendix A. B ηη B ηη modulo H + terms from [2] is given by B ηη ≈ − η Z d Γ δ ( ξ ) δ ( ρ ) T ρ ( z α y α ) ( ρ + ρ )( ρ + ρ ) exp (cid:0) F (cid:1) CCC , (A.1)where d Γ is defined in (4.8), F = i T z α ( y α + P α ) + i (1 − ξ ) ρ ρ + ρ y α ( p α + p α ) + iξ ρ ρ + ρ y α ( p α + p α ) − iy α p α , (A.2) P = (1 − ρ )( p + p ) − (1 − ρ )( p + p ) . (A.3)Performing partial integration with respect to T twice we obtain B ηη ≈ η Z d Γ δ ( ξ ) δ ( ρ ) ρ (1 − ρ )(1 − ρ ) h δ ( T ) + iz α P α + iz α P α (cid:16) i T z α P α (cid:17)i exp (cid:0) F (cid:1) CCC . (A.4)Noticing that ∂∂ρ F = − i T z α ( p α + p α ) − i (1 − ξ ) ρ ( ρ + ρ ) y α ( p α + p α ) , (A.5) ∂∂ρ F = = i T z α ( p α + p α ) − i ξ ρ ( ρ + ρ ) y α ( p α + p α ) (A.6)20nd performing partial integration with respect to ρ and ρ we obtain B ηη ≈ iη Z d Γ δ ( ξ ) δ ( ρ )(1 − ρ )(1 − ρ ) " − iρ δ ( T ) + z α P α (cid:0) (1 − ρ )(1 − ρ ) ( δ ( ρ ) + δ ( ρ )) − (cid:1) − i ρ z α P α (cid:18) ξ y α ( p α + p α )( ρ + ρ ) + ξ y α ( p α + p α )( ρ + ρ ) (cid:19) exp (cid:0) F (cid:1) CCC. (A.7)Observing that ∂ F ∂ξ = iρ ρ + ρ y α ( p α + p α ) − iρ ρ + ρ y α ( p α + p α ) (A.8)and using the Schouten identity z α ( p α + p α ) y β ( p β + p β ) = z α y α ( p β + p β )( p β + p β ) + z α ( p α + p α ) y β ( p β + p β ) (A.9)after partial integration with respect to ξ we obtain B ηη ≈ iη Z d Γ δ ( ξ ) δ ( ρ )(1 − ρ )(1 − ρ ) " − iρ δ ( T ) + z α ( p α + p α )(1 − ρ ) (cid:16) δ ( ξ ) − δ ( ξ ) (cid:17) + z α P α h (1 − ρ )(1 − ρ ) (cid:16) δ ( ρ )+ δ ( ρ ) (cid:17) − δ ( ξ ) ξ i + iρ z α y α ( p β + p β )( p β + p β ) exp (cid:0) F (cid:1) CCC. (A.10)The δ ( T )-proportional term gives rise to J (5.2) and J (5.3). Appendix B. Uniformization Detail
Here some details of the transformation of integrands (6.1)–(6.5) to the form (4.1) are presented.Uniformization can be easily achieved for Eqs. (6.1) and (6.2) modulo δ ( ρ )-proportionalterms. Indeed, eliminating δ ( ρ )-proportional term from the r.h.s. of (6.1), adding an inte-gration parameter ρ and a factor of δ ( ρ ), one obtains (8.2). Analogously, eliminating δ ( ρ )-proportional term from the r.h.s. (6.2), adding an integration parameter ρ , swapping ρ ↔ ρ and then adding a factor of δ ( ρ ) one obtains (8.3).To transform integrands of Eqs. (6.4) and (6.5), as well as δ ( ρ )-proportional terms of theintegrands of Eqs. (6.1) and (6.2), to the form (4.1) GT identity (7.1) is used in Sections B.1and B.2. B.1 d x B + W ∗ C Noticing that the exponential of (6.4) coincides with E at ξ = 0, while the exponential of (6.5)coincides with E (4.2) at ξ = 0, one can easily make sure, that only the δ ( ξ )-proportionalterm of (6.4) and the δ ( ρ )-proportional term of (6.5) have the desired form (4.1).Using that E (4.2) does not depend on ξ , swapping ξ ↔ ξ in the remaining part of (6.5),then swapping ξ ↔ ξ in the remaining part of (6.4), one then can apply GT identity (7.8) to21he sum of the two obtained terms . As a result, Eqs. (6.4), (6.5) yieldd x B η loc + W ηη ωCC ∗ C ≈ η Z d Γ δ ( ρ ) δ ( ξ ) h − i ( z α t α ) ρ + ρ δ ( ξ ) − i ( z α y α ) δ ( ρ ) i E ωCCC (B.1)+ η Z d Γ δ ( ρ ) h iδ ( ρ ) − t γ ( p γ + p γ ) in δ ( T ) e t α y α + δ ( ξ )( z α e t α + z α y α ) o E ωCCC , where the terms in the second row of formula (B.1) result from applying GT -identity. Rewritingthe underlined part as the result of differentiation with respect to T and performing partialintegration one obtains Eq. (8.4) plus the cohomology term J (5.9). B.2 (d x B ηη + ω ∗ B ηη ) | δ ( ρ ) + W η ωC ∗ B η loc Uniformization of the sum of δ ( ρ ) − proportional terms on the r.h.s.’s of (6.2) and (6.1) is donewith the help of GT identity (7.8) as follows. Denoting e P = y + p + p + t − ρ ( p + p ) (B.2)one can see that partial integration in T yieldsd x B ηη (cid:12)(cid:12)(cid:12)(cid:12) δ ( ρ ) ≈ − iη Z d Γ δ ( ρ ) δ ( ρ ) δ ( ξ ) h iδ ( T ) − z α y α i exp n i T z α e P α − iξ e P α y α + i (1 − ρ )( p α + p α ) y α + ip α y α + it β p β o ωCCC, (B.3) ω ∗ B ηη (cid:12)(cid:12)(cid:12)(cid:12) δ ( ρ ) ≈ iη Z d Γ δ ( ρ ) δ ( ρ ) δ ( ξ ) h iδ ( T ) − z α ( y α + t α ) i exp n i T z α e P α − iξ e P α y α + iξ e P α t α + i (1 − ρ )( p α + p α ) y α + ip α y α + it β p β o ωCCC . (B.4)The sum of (B.3) and (B.4) gives (cid:16) d x B ηη + ω ∗ B ηη (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) δ ( ρ ) ≈ iη Z d Γ δ ( ρ ) δ ( ρ ) h z γ ( − t γ − y α ) δ ( ξ ) + z γ y γ δ ( ξ ) + z γ t γ δ ( ξ ) i ×× exp n i T z α e P α − iξ e P α y α + iξ e P α t α + i (1 − ρ )( p α + p α ) y α + ip α y α + it β p β o ωCCC − iη Z d Γ δ ( ρ ) δ ( ρ )( z γ t γ ) δ ( ξ ) exp n i T z α e P α − iξ e P α y α + iξ e P α t α + i (1 − ρ )( p α + p α ) y α + ip α y α + it β p β o ωCCC + J + J (B.5)with J (5.10) and J (5.11). By virtue of GT identity (7.8) the first term weakly equals J (5.12). Finally, Eq. (B.5) yields (cid:16) d x B ηη + ω ∗ B ηη (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) δ ( ρ ) ≈ − iη Z d Γ δ ( ρ ) δ ( ρ )( z γ t γ ) δ ( ξ ) exp n i T z α e P α − iξ e P α y α + iξ e P α t α + i (1 − ρ )( p α + p α ) y α + ip α y α + it β p β o ωCCC + J + J + J . (B.6)22onsider W η ωC ∗ B η loc (3.13). This is convenient to change integration variables, moving fromthe integration over simplex to integration over square. As a result W η ωC ∗ B η loc ≈ η Z d T T Z d τ + δ (1 − τ − τ ) Z dσ Z dσ ( z α t α ) × h z α y α + σ z α t α i exp n i T z α y α + i (1 − σ ) σ t α p α + iσ σ t α p α + i (1 − σ ) t α p α + i T z α (cid:16) ( τ + τ σ ) t α + τ p α − ( τ − τ (1 − σ )) p α − ( τ + σ τ ) p α (cid:17) + iσ y α t α − i (1 − σ ) y α p α + iσ y α p α + iσ y α p α o ωCCC. (B.7)Partial integration with respect to T yields W η ωC ∗ B η loc ≈ − η Z d T Z d τ + δ (1 − τ − τ ) Z dσ Z dσ ( z α t α ) × h T z α (cid:16) τ ( p α + p α ) − ( τ + σ τ )( p α + p α ) (cid:17) − i T τ (1 − σ ) z α t α i exp( F ) ωCCC , (B.8)where F = i T z α y α + it β p β + iσ (cid:16) y α t α + ( p α + p α ) t α − σ ( p α + p α ) t α (cid:17) − i (cid:0) σ p α − (1 − σ ) p α (cid:1) y α + i T z α (cid:16) τ ( p α + p α ) − ( τ + σ τ )( p α + p α ) + ( σ + τ (1 − σ )) t α (cid:17) . (B.9)By virtue of evident formulas τ (cid:18) ∂∂τ − ∂∂τ (cid:19) F = i T z α (cid:16) τ ( p + p ) + (cid:2) ( τ + τ ) − ( τ + σ τ ) (cid:3) ( p + p ) + τ (1 − σ ) t (cid:17) α ,∂∂σ F = i T (1 − τ ) z α t α + i (cid:16) y α + p α + p α − σ ( p α + p α ) (cid:17) t α , Eq. (B.7) acquires the form W η ωC ∗ B η loc ≈ η Z d T Z d τ + δ (1 − τ − τ ) Z dσ Z dσ (cid:20) iz α t α τ (cid:18) ∂∂τ − ∂∂τ (cid:19) − z α ( p α + p α )1 − τ (cid:18) i ∂∂σ + (cid:16) y α + p α + p α − σ ( p α + p α ) (cid:17) t α (cid:19) + iz α t α (cid:21) exp( F ) ωCCC. (B.10)After partial integrations in τ , τ and σ one obtains W η ωC ∗ B η loc ≈ η Z d T Z d τ + δ (1 − τ − τ ) Z dσ Z dσ (cid:20) iz α t α δ ( τ ) (B.11)+ z α ( p α + p α )1 − τ (cid:16) i (cid:0) δ ( σ ) − δ (1 − σ ) (cid:1) − (cid:16) y α + p α + p α − σ ( p α + p α ) (cid:17) t α (cid:17) (cid:21) exp( F ) ωCCC . r.h.s. of Eq. (B.11)cancels the r.h.s. of Eq. (B.6). Performing integration with respect to τ in the remaining partof (B.11), after the following change of the integration variables Z dσ Z dτ Z dσ f ( σ , − σ , τ , σ )= Z d ρ + δ − X j =1 ρ j ! ρ + ρ )(1 − ρ − ρ ) f (cid:18) ρ − ρ − ρ , ρ − ρ − ρ , ρ + ρ , ρ ρ + ρ (cid:19) , exp( F ) (B.9) acquires the form E (4.2). As a result, the sum of Eq. (B.11) and Eq. (B.6) byvirtue Eq. (E.1) yields Eq. (8.5). Appendix C. Eliminating δ ( ρ j ) and δ ( ξ j ) To eliminate δ ( ρ j ) and δ ( ξ j ) from of the r.h.s.’s of Eqs. (8.2), (8.3) this is convenient to groupsimilar pre-exponential terms as in Sections C.1 -C.5. C.1 Terms proportional to ( p + p ) α ( p + p ) α Consider F , + F , of (8.2) and (8.3), respectively. Partial integration with respect to ρ and ρ yields F , + F , ≈ − η Z d Γ δ ( ξ ) ρ (1 − ρ − ρ )(1 − ρ ) ( p α + p α )( p α + p α ) ×× ( z γ P γ ) (cid:18) ∂∂ρ − ∂∂ρ (cid:19) E ωCCC. (C.1)By direct calculation, Eq. (C.1) gives F , + F , ≈ − η Z d Γ δ ( ξ ) ρ (1 − ρ − ρ )(1 − ρ ) ( p α + p α )( p α + p α ) × " E z (cid:18) ∂∂ρ − ∂∂ρ (cid:19) ( z γ P γ ) E + ( z γ P γ ) T ( z α t α ) E ωCCC . (C.2)By virtue of the Schouten identity z α t α ( p + p ) γ ( p + p ) γ = t α ( p + p ) α z γ ( p + p ) γ + t α ( p + p ) α ( p + p ) γ z γ (C.3)and its consequence z α t α ( p + p ) γ ( p + p ) γ E = t α ( p + p ) α " i ←− ∂∂ρ − ←− ∂∂ρ ! E z E + iE z (cid:18) ∂∂ρ − ∂∂ρ (cid:19) E + t α ( p + p ) α " i ←− ∂∂ρ − ←− ∂∂ρ ! E z E + iE z (cid:18) ∂∂ρ − ∂∂ρ (cid:19) E (C.4)24q. (C.1) yields F , + F , ≈ + η Z d Γ δ ( ξ ) ( ( z γ P γ ) ρ (1 − ρ − ρ )(1 − ρ ) ×× ( p + p ) α ( p + p ) α E z " ∂∂ρ − ∂∂ρ E + t α ( p + p ) α " δ ( ρ ) E − E z ∂∂ρ − ∂∂ρ ! E + t α ( p + p ) α " δ ( ρ ) E − E z ∂∂ρ − ∂∂ρ ! E + ρ (1 − ρ − ρ )(1 − ρ ) (cid:16) t α z α ( p + p ) γ ( p + p ) γ E (cid:17) +( z γ P γ ) − − ρ − ρ (1 − ρ − ρ )(1 − ρ ) t α ( p + p ) α E − − ρ − ρ − ρ (1 − ρ − ρ ) (1 − ρ ) t α ( p + p ) α E !) ωCCC . (C.5)One can see that δ ( ρ )- and δ ( ρ )-proportional terms on the r.h.s. of (C.5) (the underlinedones) cancel terms F , (8.3) and F , (8.4), respectively. C.2 Term proportional to t α ( p α + p α ) Consider term F , of F (8.4). By virtue of the following identity ρ ( ρ + ρ )(1 − ρ ) ( δ ( ρ ) − δ ( ρ )) = 1 (C.6) F , ≈ − η Z d Γ δ ( ξ ) ρ ( ρ + ρ )(1 − ρ ) (cid:16) δ ( ρ ) − δ ( ρ ) (cid:17)h ( p α + p α ) t α ( z γ t γ ) (cid:16) (1 − ρ ) − ρ ( ρ + ρ ) (cid:17) E i ωCCC. (C.7)Partial integrations along with the Schouten identity t α ( p α + p α )( p γ + p γ ) z γ = − t α z α ( p γ + p γ )( p γ + p γ ) + t α ( p α + p α )( p + p ) γ z γ (C.8)and realization of the underlined terms as derivative of E z along with further partial integrationyields 25 , ≈ − η Z d Γ δ ( ξ ) " ρ (1 − ρ ) (cid:16) ( p α + p α ) t α z γ t γ (cid:17) E + ρ ρ ( ρ + ρ )(1 − ρ ) (cid:16) ( p α + p α ) t α z γ t γ (cid:17) E z (cid:20) ∂∂ρ − ∂∂ρ (cid:21) E + ρ ρ ( ρ + ρ )(1 − ρ ) ( z α t α ) ×× − ( p + p ) γ ( p + p ) γ E z " ∂∂ρ − ∂∂ρ E − δ ( ρ )( p + p ) γ ( p + p ) γ E− t α (( p + p ) α ) E z " ∂∂ρ − ∂∂ρ E − δ ( ρ ) t α (( p + p ) α ) E ! + ( z α t α ) ρ (1 − ρ )( ρ + ρ ) ( p + p ) γ ( p + p ) γ E + ρ ( ρ + ρ ) t α (( p + p ) α ) E ! ωCCC. (C.9)One can see that the sum of the underlined δ ( ρ )-proportional terms cancels F , + F , of (8.3). C.3 Sum of ( p + p ) α ( p + p ) α -proportional and t α ( p α + p α ) –proportionalterms Summing up F , + F , (C.5), F , (8.4), F , (C.9) and F , + F , + F , (8.3), then performingpartial integrations and using the following simple identities(1 − ρ ) − ρ ( ρ + ρ ) = ρ ( ρ + ρ )( ρ + ρ ) , (C.10) − ρ ( ρ + ρ ) + ρ ( ρ + ρ ) ρ (1 − ρ − ρ )(1 − ρ ) = − ρ ρ ( ρ + ρ ) (1 − ρ − ρ )(1 − ρ ) , (C.11)one obtains by virtue of Eqs. (4.5)-(4.7) F , + F , + F , + F , + F , + F , + F , = G (C.12)with G (9.2). C.4 Terms proportional to δ ( ξ ) − δ ( ξ ) Consider a sum of F , (8.2) and F , (8.3). Performing partial integration with respect to ρ and ρ , then applying the Schouten identity one obtains F , + F , ≈ − η Z d Γ δ ( ξ ) " ∂∂ρ − ∂∂ρ iz α ( p α + p α )1 − ρ (cid:16) δ ( ξ ) − δ ( ξ ) (cid:17) E ωCCC == − η Z d Γ δ ( ξ ) (cid:16) δ ( ξ ) − δ ( ξ ) (cid:17)( i z γ t γ (1 − ρ ) E z " ∂∂ρ − ∂∂ρ E + (cid:16) δ ( ρ ) − δ ( ρ ) (cid:17) E ! + i z α ( p α + p α )(1 − ρ ) E z " ∂∂ρ − ∂∂ρ E ) ωCCC. (C.13)26he underlined δ ( ρ )-proportional term compensates F , of (8.3). The double underlined δ ( ρ )-proportional term vanishes due to the factor of ( δ ( ξ ) − δ ( ξ )) which after partial integra-tions in ξ and ξ produces an expression proportional to ρ .Summing up F , + F , (C.13) and F , (8.3), performing partial integrations with respectto ξ and T along with the Schouten identity one obtains F , + F , + F , ≈ G (C.14)with G (9.3). C.5 Terms proportional to ξ δ ( ξ ) Consider a sum of F , (8.2), F , (8.3) and F , (8.5). F , + F , + F , ≈ iη Z d Γ δ ( ξ ) δ ( ξ )[ δ ( ρ ) − δ ( ρ )]( ρ + ρ ) z α (cid:26) P α (1 − ρ ) − ξ ( p α + p α )( ρ + ρ ) (cid:27) E ωCCC. (C.15)Partial integration yields F , + F , + F , ≈ iη Z d Γ δ ( ξ ) δ ( ξ ) ξ ( z α t α " ρ + ρ (cid:18) E z (cid:20) ∂∂ρ − ∂∂ρ (cid:21) E + h δ ( ρ ) − δ ( ρ ) i E (cid:19) + 11 − ρ (cid:18) E z (cid:20) ∂∂ρ − ∂∂ρ (cid:21) E + h δ ( ρ ) − δ ( ρ ) i E (cid:19) + (cid:20) z α ( p α + p α ) ρ + ρ + z α ( p α + p α )1 − ρ (cid:21) E z (cid:20) ∂∂ρ − ∂∂ρ (cid:21) E ) ωCCC . (C.16)One can see that the underlined δ ( ρ )-proportional terms vanish due to the factor of δ (1 − P ρ i ) (4.8), while δ ( ρ )-proportional term compensates F , (8.3) and δ ( ρ )-proportional termcompensates F , (8.4).Summing up F , (8.3), F , (8.3), F , and F , + F , + F , (8.5), and then performingpartial integration in T one obtains by virtue of the Schouten identity F , + F , + F , + F , + F , + F , ≈ G := η Z d Γ δ ( ξ ) δ ( ξ ) ×× ( ρ ( z α t α )( p γ + p γ )( y γ + ˜ t γ )(1 − ρ − ρ ) (1 − ρ )( ρ + ρ ) + ρ h (˜ t γ + y γ )( y γ + P γ )( z α t α ) + iδ ( T ) t γ (˜ t γ − P γ ) i (1 − ρ − ρ ) (1 − ρ ) ( ρ + ρ )+ ρ (cid:2) iδ ( T ) − z γ ( y γ + ˜ t γ ) (cid:3) ( t α y α )(1 − ρ − ρ ) (1 − ρ ) + (cid:2) − iδ ( T ) + z γ ( y γ + ˜ t γ ) (cid:3) ( p α + p α ) t α (1 − ρ − ρ )(1 − ρ ) ) E ωCCC . (C.17)Since by the partial integration procedure ξ δ ( ξ ) ≡ ξ ( ∂ ξ − ∂ ξ ), (C.17) yields G (9.4).27 ppendix D. Details of the final step of the calculation By virtue of Eqs. (E.1)-(E.3), Eq. (10.2) yields S = + i η Z d Γ δ ( ξ ) (D.1) n + ( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) ξ − ρ − ρ (1 − ρ − ρ )(1 − ρ ) P α y α +( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) ξ − ρ − ρ (1 − ρ − ρ )(1 − ρ ) (cid:0) y + P (cid:1) α ˜ t α +( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α )( − ) ξ ρ (1 − ρ − ρ )(1 − ρ ) ( p α + p α ) y α − ( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) ξ ρ (1 − ρ − ρ )(1 − ρ ) ( p α + p α )˜ t α ++( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α )( − ) ρ + ρ (1 − ρ ) (( p + p )) α y α +( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α )( − ) ρ (1 − ρ ) ( t ) α y α +( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α )( − ) ρ (1 − ρ ) ( p α + p α ) t α + ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ z α ( y + ˜ t ) α ( − ) ξ ρ (1 − ρ − ρ )(1 − ρ ) t α y α + ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ z α ( y + ˜ t ) α × ( − ) ξ ρ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) (cid:0) y α + P α (cid:1) t α + ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ z α ( y + ˜ t ) α − ρ ) t α y α + ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ z α ( y + ˜ t ) α ( − ) 1(1 − ρ ) ( p α + p α ) t α ++ ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ξ ρ (1 − ρ − ρ ) ( − ( p + p )) α y α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ξ ρ (1 − ρ − ρ ) (cid:0) ( − ( p + p )) α (cid:1) ˜ t α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α × ξ ρ ρ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) t α y α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ξ − ρ ) ( p α + p α ) y α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ξ − ρ ) ( p α + p α )˜ t α ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α × ( − ) ξ ρ (1 − ρ − ρ )(1 − ρ ) ρ ( ρ + ρ ) (cid:0) y α + P α (cid:1) t α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ( − ) 1(1 − ρ ) ( p α + p α ) y α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ( − ) 1(1 − ρ ) ( p α + p α ) t α ++ ξ h + n + ρ ρ (1 − ρ − ρ ) (1 − ρ ) ( ρ + ρ ) (cid:0) y + (1 − ρ − ρ )( p + p ) + (1 − ρ ) t (cid:1) γ (cid:0) y + ˜ t (cid:1) γ z α t α − ρ ρ (1 − ρ − ρ ) (1 − ρ ) (cid:0) y + ˜ t (cid:1) γ z γ t α y α + ρ (1 − ρ − ρ ) (1 − ρ ) (cid:0) y + ˜ t (cid:1) γ z γ ( p α + p α ) t α o E i × (cid:0) y + P (cid:1) α ( y + ˜ t ) α o ωCCC . Terms from the r.h.s. of (D.1) with ξ -independent pre-exponentials are considered in SectionD.1, while those with ξ -proportional pre-exponentials are considered in Section D.2. D.1 ξ -independent pre-exponentials Here we consider only pre-exponentials, omitting for brevity integrals, integral measures etc of(D.1). By virtue of the Schouten identity taking into account that P ρ i = 1 Eq. (D.1) yields Integrand ( S ) (cid:12)(cid:12)(cid:12) mod ξ = ( y + ˜ t ) ν z ν n − ρ ( ρ + ρ )(1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) (( p + p )) α y α − ρ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) ( t ) α y α − ρ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α )( p α + p α ) t α + ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ t α y α − ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ ( p α + p α ) t α − ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ ( p α + p α ) y α − ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ ( p α + p α ) t α o == ( y + ˜ t ) ν z ν ρ (1 − ρ − ρ )(1 − ρ ) n ρ t α (( p + p ) α )( p α + p α ) t α + ( p + p ) γ y γ t α y α + (D.2) − ( p + p ) γ (1 − ρ ) t γ ( p α + p α ) t α − t γ y γ ( p α + p α ) y α − t γ (1 − ρ − ρ )( p + p ) γ ( p α + p α ) t α o = ( y + ˜ t ) ν z ν ρ (1 − ρ − ρ )(1 − ρ ) n − ρ t α (( p + p ) α )( p α + p α ) t α (D.3) − ( p + p ) γ (1 − ρ ) t γ ( p α + p α ) t α − t γ (1 − ρ − ρ )( p + p ) γ ( p α + p α ) t α o ≡ . .2 ξ -proportional pre-exponentials S (cid:12)(cid:12)(cid:12) ξ = J + i η Z d Γ δ ( ξ ) (D.4) n ( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) ξ − ρ − ρ (1 − ρ − ρ )(1 − ρ ) P α y α +( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) ξ − ρ − ρ (1 − ρ − ρ )(1 − ρ ) (cid:0) y + P (cid:1) α ˜ t α +( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α )( − ) ξ ρ (1 − ρ − ρ )(1 − ρ ) ( p α + p α ) y α +( y + ˜ t ) γ z γ ρ (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α )( − ) ξ ρ (1 − ρ − ρ )(1 − ρ ) ( p α + p α )˜ t α + ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ z α ( y + ˜ t ) α ( − ) ξ ρ (1 − ρ − ρ )(1 − ρ ) t α y α + ρ (1 − ρ − ρ )(1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ z α ( y + ˜ t ) α ( − ) ξ ρ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) × (cid:0) y α + P α (cid:1) t α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ξ ρ (1 − ρ − ρ ) ( − ( p + p )) α y α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ξ ρ (1 − ρ − ρ ) (cid:0) ( − ( p + p )) α (cid:1) ˜ t α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α × ξ ρ ρ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) t α y α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ξ − ρ ) ( p α + p α ) y α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α ξ − ρ ) ( p α + p α )˜ t α + ρ (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ z α ( y + ˜ t ) α × ( − ) ξ ρ (1 − ρ − ρ )(1 − ρ ) ρ ( ρ + ρ ) (cid:0) y α + P α (cid:1) t α + ξ hn + ρ ρ (1 − ρ − ρ ) (1 − ρ ) ( ρ + ρ ) (cid:0) y + (1 − ρ − ρ )( p + p ) + (1 − ρ ) t (cid:1) γ (cid:0) y + ˜ t (cid:1) γ n t α (cid:0) y + P (cid:1) α z σ ( y + ˜ t ) σ o − ρ ρ (1 − ρ − ρ ) (1 − ρ ) (cid:0) y + ˜ t (cid:1) γ z γ t α y α (cid:0) y + P (cid:1) σ ( y + ˜ t ) σ + ρ (1 − ρ − ρ ) (1 − ρ ) (cid:0) y + ˜ t (cid:1) γ z γ (cid:0) y + P (cid:1) σ ( y + ˜ t ) σ ( p α + p α ) t α o E io ωCCC , J is the cohomology term (5.8) . This yields S (cid:12)(cid:12)(cid:12) ξ ≈ J + i η Z d Γ δ ( ξ ) ρ (1 − ρ − ρ ) (1 − ρ ) (D.5) ξ ( y + ˜ t ) γ z γ n + (1 − ρ − ρ )(1 − ρ ) t α (( p + p ) α ) (cid:0) y + P (cid:1) α ( y + ˜ t ) α − ρ t α (( p + p ) α )( p α + p α )( y + ˜ t ) α − ρ (1 − ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ t α y α − ρ (1 − ρ )( ρ + ρ ) ( p + p ) γ (cid:0) y + (1 − ρ ) t (cid:1) γ (cid:0) y + P (cid:1) α t α − ρ (1 − ρ − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ ( p + p ) α ( y + ˜ t ) α + ρ ρ (1 − ρ )( ρ + ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ t α y α + (1 − ρ − ρ )(1 − ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ ( p α + p α )( y + ˜ t ) α − ρ ρ (1 − ρ )( ρ + ρ ) t γ (cid:0) y + (1 − ρ − ρ )( p + p ) (cid:1) γ (cid:0) y α + P α (cid:1) t α + ρ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) (cid:0) y + (1 − ρ ) t (cid:1) γ (cid:0) y + ˜ t (cid:1) γ t α (cid:0) y + P (cid:1) α + ρ (1 − ρ )( ρ + ρ ) (cid:0) ( p + p ) (cid:1) γ (cid:0) y + ˜ t (cid:1) γ t α (cid:0) y + P (cid:1) α − ρ (1 − ρ − ρ )(1 − ρ ) t α y α (cid:0) y + P (cid:1) σ ( y + ˜ t ) σ + 1(1 − ρ ) (cid:0) y + P (cid:1) σ ( y + ˜ t ) σ ( p α + p α ) t α E o ωCCC ≡ J since, using the Schouten identity, one can see that the pre-exponential of the integrand on the r.h.s. of (D.5) equals zero. Appendix E. Useful formulas
From (4.4) one has (cid:18) ∂∂ρ − ∂∂ρ (cid:19) E = i n + ξ ρ (1 − ρ − ρ )(1 − ρ ) t α y α (E.1)+ ξ ρ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) (cid:0) y α + P α (cid:1) t α + 1(1 − ρ ) ( y + p α + p α ) t α o E ∂∂ρ − ∂∂ρ (cid:19) E = i n ξ − ρ − ρ (1 − ρ − ρ )(1 − ρ ) (cid:0) y α + P α (cid:1) ( y + ˜ t ) α (E.2) − ξ ρ (1 − ρ − ρ )(1 − ρ ) ( p α + p α )( y + ˜ t ) α − ρ + ρ (1 − ρ ) (( p + p )) α y α − ρ (1 − ρ ) ( t ) α y α − ρ (1 − ρ ) ( p α + p α ) t α o E , (cid:18) ∂∂ρ − ∂∂ρ (cid:19) E = i n ξ ρ (1 − ρ − ρ ) (cid:0) ( − ( p + p )) α (cid:1) ( y + ˜ t ) α (E.3)+ ξ ρ ρ (1 − ρ − ρ )(1 − ρ )( ρ + ρ ) t α y α + ξ − ρ ) ( p α + p α )( y + ˜ t ) α − ξ ρ (1 − ρ − ρ )(1 − ρ ) ρ ( ρ + ρ ) (cid:0) y α + P α (cid:1) t α − − ρ ) ( p α + p α ) y α − − ρ ) ( p α + p α ) t α o E .
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