Recursion Relations in p -adic Mellin Space
PPUPT-2577
Recursion Relations in p -adic Mellin Space Christian Baadsgaard Jepsen ∗ & Sarthak Parikh † Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA Division of Physics, Mathematics and Astronomy,California Institute of Technology, Pasadena, CA 91125, USA
Abstract
In this work, we formulate a set of rules for writing down p -adic Mellin amplitudes attree-level. The rules lead to closed-form expressions for Mellin amplitudes for arbitraryscalar bulk diagrams. The prescription is recursive in nature, with two different physicalinterpretations: one as a recursion on the number of internal lines in the diagram, andthe other as reminiscent of on-shell BCFW recursion for flat-space amplitudes, especiallywhen viewed in auxiliary momentum space. The prescriptions are proven in full generality,and their close connection with Feynman rules for real Mellin amplitudes is explained. Wealso show that the integrands in the Mellin-Barnes representation of both real and p -adicMellin amplitudes, the so-called pre-amplitudes, can be constructed according to virtuallyidentical rules, and that these pre-amplitudes themselves may be re-expressed as productsof particular Mellin amplitudes with complexified conformal dimensions.December 27, 2018 ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] D ec ontents p -adic Mellin amplitudes . . . . . . . . . . . . . . 113.2 Feynman rules for real Mellin amplitudes . . . . . . . . . . . . . . . . . . . . 19 p -adic on-shell recursion relation and proof . . . . . . . . . . . . . . . . . . . 38 B.1 Outline of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.2 Details of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
C Inductive Proof of Prescription I for Mellin Amplitudes 61
C.1 The “leg adding” operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 63C.2 Integrating pre-amplitudes using “leg addition” . . . . . . . . . . . . . . . . 64C.3 Proof of the general “leg adding” operation . . . . . . . . . . . . . . . . . . . 69
D BCFW-Shifts of Auxiliary Momenta 76
D.1 Four-point exchange diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 76D.2 Five-point diagram with two internal lines . . . . . . . . . . . . . . . . . . . 831
Introduction and Summary
Mellin amplitudes share many properties with flat space scattering amplitudes and havebeen especially useful in studying holographic CFTs [1]. Until very recently, this storyhas remained restricted to the study of amplitudes in field theories defined over real-valuedspacetimes. In this paper, we continue the holographic investigation of Mellin amplitudesin the context of p -adic AdS/CFT [2, 3] initiated in Ref. [4]. These so-called p -adic Mellinamplitudes were shown to have analytic properties similar to those of traditional real Mellinamplitudes for several classes of bulk diagrams [4]. At the same time, p -adic Mellin am-plitudes were found to be significantly simpler than their real counterparts owing to theabsence of descendants [5] in the corresponding p -adic CFTs, thus providing a new tech-nically simpler arena for studying generic features of Mellin amplitudes. In this paper, weestablish systematically the analytic similarities between the real and p -adic amplitudes for all tree-level bulk diagrams involving scalars. The main questions we answer are: What isthe p -adic Mellin amplitude of an arbitrary tree-level diagram, and what recursion relationsdo the amplitudes obey? We establish three prescriptions for writing down arbitrary Mellinamplitudes, two of which are recursive, and proceed to establish the precise connections withanalogous prescriptions satisfied by corresponding real Mellin amplitudes.For real Mellin amplitudes, the final expressions, obtained via the Feynman rules forMellin amplitudes, are usually written in terms of unevaluated infinite sums over termscorresponding to the exchange of descendant fields in the intermediate channels. For scalaroperators, the amplitudes for tree-level bulk diagrams take the following schematic form,which we refer to as the series representation of the Mellin amplitudes [6, 7, 8], M ∼ ∞ (cid:88) m i ,...,m iK =0 (cid:32)(cid:89) i ∈ I s i − ∆ i − m i (cid:33) (cid:32) (cid:89) vertices V ( { m I } ) (cid:33) I = { i , . . . , i K } , (1.1)where the set I labels the internal lines of the diagram, each admitting the exchange of asingle trace operator of conformal dimension ∆ i for i ∈ I (with non-zero m i correspondingto the exchange of descendants from its conformal family), and each contact vertex in thediagram has an associated factor V which depends on the conformal dimensions of theexternal operator insertions incident at that vertex, as well as the dimensions ∆ i and theassociated integers m i of operators running along internal lines incident on the same vertex.For each internal line i ∈ I there is a “propagator factor”, which depends on a particularMandelstam-like variable s i , which itself depends on the Mellin variables. To obtain the full2ellin amplitude, one must sum over all integers m i for i ∈ I ; typically these sums are hardto evaluate analytically. This series representation makes the pole structure of the Mellinamplitude manifest — the Mandelstam variables pick up poles in the infinite sum (1.1) whenfields propagating along the internal lines go on-shell, signaling the exchange of single-traceoperators and their descendants in the intermediate channels, and the residue at each polefactorizes into a product of left and right sub-amplitudes [6, 7, 9].Alternatively, real (and p -adic) Mellin amplitudes also admit a contour representation,which we will refer to as the Mellin-Barnes integral representation. Schematically, it takesthe form M ∼ (cid:32)(cid:89) i ∈ I (cid:90) i ∞− i ∞ dc i πi f ∆ i ( c i ) (cid:33) (cid:102) M ( { c I } , { γ ij } ) I = { i , . . . , i K } . (1.2)where the weight f ∆ i ( c i ) depends on the dimension ∆ i of the internal operator exchangedalong the internal line i ∈ I as well as the spacetime dimension, and takes as argument a com-plex parameter c i , while (cid:102) M , which following Refs. [10, 11] we refer to as the “pre-amplitude”,is a function that is significantly simpler in form than the full, integrated amplitude M anddepends on the Mellin variables γ ij , the complex parameters c I , and the dimensions solelyof external operators.While in principle the series and Mellin-Barnes representations solve the problem ofobtaining real tree-level scalar correlators, they do not yield a closed-form expression for theamplitudes (except in a few simple cases such as the three- and four-point functions). Andwhile Mellin amplitudes for special kinds of one-loop diagrams were already available in theearly days of the Mellin program [1], and recent progress has been made in studying bulk-diagrams at one-loop, both in position space [12, 13] and Mellin space [10, 11], the currenttechnology at loop-level leaves more to be desired when compared with the state-of-the-art for flat-space amplitudes. The flat-space amplitudes program has benefited from manyextremely powerful tools and techniques which go beyond summing Feynman diagrams, suchas BCFW recursion [14], but these have so far remained elusive in Mellin space and serve asone of the motivations for this work. It should be mentioned, though, that methods based on general consistency and symmetry havebeen successful in a variety of settings, e.g. conformal bootstrap for higher dimensional CFTs in posi-tion space [15, 16, 17, 18, 19, 20, 21] and Mellin space [22], use of crossing symmetry to constrain 1-loopMellin amplitudes [23, 24, 25], as well as techniques utilizing superconformal Ward identities to obtain totalon-shell amplitudes without resorting to summing individual bulk diagrams in e.g. type IIB supergravity inboth position and Mellin space [26, 27], to name a few. See also Refs. [28, 29] for work on BCFW recursionrelations for bulk diagrams in momentum space . p -adic Mellin amplitudes (see Ref. [4] for a detailed introduction and section 2for a quick overview, definitions and conventions), explicit computations of the amplitudesfor tree-diagrams with up to three internal lines [4] hint at the existence of recursion relationsobeyed by the p -adic Mellin amplitudes of arbitrary tree-level bulk diagrams. In section 3,we present such a relation which is recursive in the number of internal lines in the diagram,with a proof in appendix C. The prescription consists of assigning factors to each vertex andinternal line of a given bulk diagram and then writing down the Mellin amplitude by takinga product over all these factors, but also subtracting off all possible diagrams obtained bycollapsing in the original diagram every possible subset of internal lines. The expressions forthe diagrams to be subtracted off are obtained by a recursive application of this procedure.In the end the entire Mellin amplitude is expressible in closed-form in terms of the factorsassociated with contact interaction vertices and internal lines. Explicit examples are alsoprovided, and in section 3.2 we present the Feynman rules for real Mellin amplitudes forcomparison.As pointed out above, real and p -adic Mellin amplitudes admit a Mellin-Barnes represen-tation in terms of multi-contour integrals. The reason such a representation exists is that byapplying the split representation to all the bulk-to-bulk propagators of the diagram, a Mellinamplitude can be written as a multi-dimensional contour integral over what is referred toas the pre-amplitude . In section 4 we formulate a set of rules for constructing any tree-levelpre-amplitude. Curiously, this prescription applies universally to both p -adic and real Mellinamplitudes.In the same section we also describe an interesting connection between pre-amplitudes andMellin amplitudes, which holds true for both p -adic and real Mellin amplitudes: We show thatpre-amplitudes may be obtained by taking products of diagrams which appear in the recursiveprescription for Mellin amplitudes, except with the dimensions in the diagrams assignedspecific complex values that depend on the complex variables of the contour integrals. Thisprescription is proven in section B via an inductive argument, and explicit examples areprovided in section 4.2.We emphasize that the claim of the previous paragraph essentially is that the integrand in the Mellin-Barnes integral representation of the Mellin amplitude secretly takes the samefunctional form as the integral , i.e. the full Mellin amplitude. Thus the pre-amplitude isgiven simply by a product of a particular set of full Mellin amplitudes with the dimensions In section 3 we refer to these diagrams as “undressed diagrams”, and they are essentially the same asMellin amplitudes up to simple overall factors associated with each vertex. These diagrams can be obtainedvia the prescriptions of sections 3 and 5 over p -adics and the Feynman rules [6, 7, 8] over reals.
4f certain operators set to special values. Although the series and Mellin-Barnes integralrepresentations of the real Mellin amplitudes obfuscate this property, it holds true even overthe reals as discussed in section 4.1.In section 5 we move on to present a different type of recursion relation obeyed by p -adic Mellin amplitudes, which makes the factorization property of p -adic Mellin amplitudesmanifest. This prescription allows for an arbitrary tree-level bulk diagram to be expressedin terms of lower-point bulk diagrams with shifted Mandelstam-like invariants, constructedout of the same interaction vertices as the original diagram. We prove in section 5 that thisprescription follows from the recursive prescription from section 3. We also argue that anadaptation of this prescription applies to real Mellin amplitudes and in fact gives back theFeynman rules prescription. In appendix D we illustrate with the help of explicit examples,how this prescription originates from an application of Cauchy’s residue theorem to complexdeformed Mellin amplitudes obtained via complex shifting (auxiliary) momenta, `a la on-shellBCFW recursion relations in flat space [14]. Thus this prescription is closer in form to on-shell recursive relations in flat space since it provides a decomposition of Mellin amplitudesinto products of lower-point sub-amplitudes with all external legs “on-shell”, joined togetherby “propagator” factors, reminiscent of BCFW recursion. We stress, however, that therecursive relation applies at the level of individual diagrams, and not at the level of fullamplitudes. We close with final comments and future directions in section 6. In this section we recall some basic facts and results for p -adic holography and Mellin ampli-tudes and provide relevant definitions and normalizations which will be used throughout. More details can be found in Ref. [4] (see also Refs. [2, 30]). Many results in p -adic AdS/CFTclosely resemble their real counterparts, not just in physical interpretation but also in theirmathematical formulation. To facilitate comparison and emphasize the striking similarities,we will often present a p -adic result or definition along with its real counterpart from con-ventional continuum AdS/CFT. In such cases, we will label the equations with a “ Q p n ” or an“ R n ” depending on whether they hold in p -adic AdS/CFT or real Euclidean AdS n +1 /CFT n .In the p -adic setup, the bulk geometry is given by the Bruhat–Tits tree T p n , which is aninfinite ( p n + 1)-regular graph without cycles, and the bulk scalar degrees of freedom liveon the vertices of the tree. The boundary of the tree, ∂ T p n , is the projective line over the For recent developments in p -adic holography, see Refs. [2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,41, 42]. n unramified extension of the p -adic numbers, denoted P ( Q p n ). Q p n may also bethought of as an n -dimensional vector-space over the base field Q p , thus we will refer to n as the dimension of the boundary field theory. The vertices of the tree can be labelled bybulk coordinates ( z , z ) where z , the bulk depth coordinate, is an integral power of p and z ∈ Q p n represents the boundary direction. The vertices are not labelled uniquely by thiscoordinate representation: two coordinate pairs ( z , z ) and ( z ′ , z ′ ) label the same vertex iff z = z ′ and z ′ − z ∈ z Z p n , where Z p n = { x ∈ Q p n (cid:12)(cid:12) | x | p ≤ } . | · | p denotes the ultrametric p -adic norm, | · | p : Q p n → R ≥ . On the real side, the bulk geometry is ( n + 1)-dimensionalEuclidean anti-de Sitter space. In the two formalisms, the free bulk actions are given by Q p n (cid:1) S kin = (cid:88) ⟨ ( z ,z )( w ,w ) ⟩
12 ( ϕ i ( z ,z ) − ϕ i ( w ,w ) ) + (cid:88) ( z ,z ) ∈T pn m i ϕ i ( z ,z ) , R n (cid:1) S kin = (cid:90) AdS n +1 dZ (cid:20) (cid:0) ∇ ϕ i ( Z ) (cid:1) + 12 m i ϕ i ( Z ) (cid:21) , (2.1)where we have expressed the real action in embedding space coordinates such that the bulkcoordinate Z lives in an ( n + 2)-dimensional Minkowski space satisfying Z ≡ Z · Z = − ϕ i have masses m i and scaling dimensions ∆ i , related to each othervia Q p n (cid:1) m = − ζ p ( − ∆) ζ p (∆ − n ) , R n (cid:1) m = ∆(∆ − n ) , (2.2)where ζ v : C → C , which will be called the local zeta function, is a meromorphic functiondefined for v = p and v = ∞ by ζ p ( z ) = 11 − p − z , ζ ∞ ( z ) = π − z Γ (cid:16) z (cid:17) . (2.3)The bulk-to-boundary propagators for a bulk scalar of dimension ∆ are given by Q p n (cid:1) K ∆ ( z , z ; x ) = ζ p (2∆) | z | ∆ p | z , z − x | s , R n (cid:1) K ∆ ( Z, P ) = ζ ∞ (2∆)( − P · Z ) ∆ , (2.4)6here |· , ·| s denotes the supremum norm, | x, y | s ≡ sup {| x | p , | y | p } , (2.5)and the real bulk-to-boundary propagator is expressed in the embedding space formalism,with the boundary coordinate satisfying P = 0.The p -adic and real bulk-to-bulk propagators, each of which admits a split representation in terms of integrals over bulk-to-boundary propagators, are given by Q p n (cid:1) G ∆ ( z , z ; w , w ) = ζ p (2∆) p − ∆ d [ z ,z ; w ,w ] = (cid:90) iπ log p − iπ log p dc πi f ∆ ( c ) (cid:90) ∂ T pn dx K h − c ( z , z ; x ) K h + c ( w , w ; x ) , R n (cid:1) G ∆ ( Z, W ) = ζ ∞ (2∆)( Z − W )
2∆ 2 F (cid:18) ∆ , ∆ − h + 12 ; 2∆ − h + 1; − Z − W ) (cid:19) = 12 (cid:90) i ∞− i ∞ dc πi f ∆ ( c ) (cid:90) ∂ AdS dP K h − c ( Z, P ) K h + c ( W, P ) , (2.6)where we have defined h ≡ n , (2.7)with Q p n (cid:1) f ∆ ( c ) ≡ ν ∆ m − m h − c ζ p (2∆ − h ) ζ p (2 c ) ζ p ( − c ) log p , R n (cid:1) f ∆ ( c ) ≡ ν ∆ m − m h − c ζ ∞ (2∆ − h ) ζ ∞ (2 c ) ζ ∞ ( − c ) , (2.8)and Q p n (cid:1) ν ∆ ≡ p ∆ + − p ∆ − = p ∆ ζ p (2∆ − n ) , R n (cid:1) ν ∆ ≡ ∆ + − ∆ − = 2∆ − n , (2.9)and d [ z , z ; w , w ] denotes the graph distance between two nodes ( z , z ) and ( w , w ) on theBruhat–Tits tree, while ( Z − W ) is related to the chordal distance in real continuum AdS.We note that consistent with Ref. [4], we are using non-standard normalizations for thebulk-to-boundary and bulk-to-bulk propagators, so that the p -adic and real bulk-to-bulk7ropagators satisfy the equations Q p n (cid:1) (cid:0) □ z ,z + m (cid:1) G ∆ ( z , z ; w , w ) = 2 ν ∆ ζ p (2∆ − h ) δ ( z , z ; w , w ) , R n (cid:1) (cid:0) −∇ Z + m (cid:1) G ∆ ( Z, W ) = 2 ν ∆ ζ ∞ (2∆ − h ) δ ( Z − W ) . (2.10)We will consider bulk contact-interactions of the type Q p n (cid:1) (cid:88) ( z ,z ) ∈T pn N (cid:89) i =1 ϕ i ( z ,z ) , R n (cid:1) (cid:90) AdS n +1 dX N (cid:89) i =1 ϕ i ( X ) , (2.11)and study boundary correlators ⟨O ∆ ( x ) . . . O ∆ N ( x N ) ⟩ , which are functions of boundaryinsertion points x i , and are built holographically from N -point position space amplitudes A ( { x i } ). These amplitudes are given by products of bulk-to-boundary and bulk-to-bulkpropagators with bulk vertices integrated over the entire bulk, and they depend on theboundary insertion points only via the N ( N − / Mellin amplitudes are defined via multi-dimensional inverse-Mellin transforms of theposition space amplitudes, Q p n (cid:1) A ( { x i } ) = (cid:90) [ dγ ] M ( { γ ij } ) (cid:89) ≤ i The amplitude with exactly one internal line, that is the exchange amplitude,is depicted thus: M exchange = s ∆ ...... i R i L . (2.21)10he associated Mandelstam invariant s is given by s = (cid:88) ∆ i L − (cid:88) γ i L j L = (cid:88) ∆ i R − (cid:88) γ i R j R , (2.22)where the limits of the various sums have been left implicit, but it should be clear from thecontext what indices are being summed over. For example, (cid:80) ∆ i L = (cid:80) i L ∆ i L is a sum overall possible values that i L takes, and (cid:80) γ i L j L = (cid:80) i L Using the graphical representations above, we can rewrite the closed-form p -adic Mellin amplitude for the arbitrary-point exchange diagram as A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R A = (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) , (3.3)12here we have defined A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB ≡ ( − (cid:26)(cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) − (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19)(cid:27) . (3.4)It is straightforward to verify that this is simply a diagrammatic rewriting of equation (4.42)of Ref. [4] provided we identify the Mandelstam invariant s A with the expression in (2.22). Example (iv). Similar to the previous example, the p -adic Mellin amplitude for a bulkdiagram with two internal lines (but arbitrary external insertions) is given by A B ... i L ... i R ... i U = (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) ... i L ... i R ... i U A B ... i U (cid:18) A ... i L ... i R A ... i L ... i R A ... i L B ... i R ... i L ... i R AB (cid:19) C A B ... i L ... i R ... i U A B ... i L ... i R ... i U B ... i L ... i R ... i U A ... i L ... i R ... i U , (3.5)where we have defined C A B ... i L ... i R ... i U A B ... i L ... i R ... i U B ... i L ... i R ... i U A ... i L ... i R ... i U ≡ ( − (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) − A B C ... i L ... i R ... i U ... i r B ... i L ... i U A ... i R ... i U A B C ... i L ... i R ... i l ... i r ( − C A B ... i L ... i R ... i U A B ... i L ... i R ... i U B ... i L ... i R ... i U A ... i L ... i R ... i U − A B C ... i L ... i R ... i U ... i r B ... i L ... i U A ... i R ... i U A B C ... i L ... i R ... i l ... i r ( − C A B ... i L ... i R ... i U A B ... i L ... i R ... i U B ... i L ... i R ... i U A ... i L ... i R ... i U − ... i L ... i R ... i U A B ... i U . (3.6)This reproduces the amplitude given in equation (4.56) of Ref. [4] if we set the Mandelstam-like variables s A , s B to s A = (cid:88) ∆ i L − (cid:88) γ i L j L , s B = (cid:88) ∆ i R − (cid:88) γ i R j R , (3.7)13nd use (3.4) to further reduce (3.6) to an expression given entirely in terms of vertex andinternal line factors (3.1)-(3.2).The two examples above are suggestive of the recursive procedure for reconstructing p -adic Mellin amplitudes which we will now describe. In this prescription, no integration isnecessary and the basic building blocks are the bulk contact-interaction vertices and internallines, (3.1) and (3.2) respectively. This recursive procedure will be proven in the appendices. Prescription I ( p -adic Mellin amplitudes) . The Mellin amplitude for a particular bulk dia-gram is given by the product over a vertex factor (3.1) for each contact vertex in the diagram,times what we will refer to as an “undressed Mellin amplitude” (which we have depicted inred in (3.3) and (3.5)). The undressed amplitudes are constructed recursively as follows: • The undressed amplitude for a bulk diagram with no internal lines is equal to one. • The undressed amplitude for a bulk diagram with one or more internal lines is given byan overall factor of minus one raised to the number of internal lines times the followingcombination: the product over all internal line factors (3.2) associated with the bulkdiagram, minus all possible undressed diagrams associated with bulk diagrams obtainedby collapsing all possible subsets of internal lines in the original bulk diagram, witheach such subtracted term weighted by minus one raised to the number of remaining internal lines and also weighted by the vertex factor(s) (3.1) associated with any new contact vertex (or vertices) generated upon affecting such a collapse. Schematically, this prescription takes the form:(undressed diagram) = ( − (cid:40) (cid:89) (cid:0) internal line factors (cid:1) − (cid:88) (cid:104) ( − (cid:0) new vertex factor(s) (cid:1) × (cid:0) reduced undressed diagram (cid:1)(cid:105)(cid:41) . (3.8)14e make two remarks about the prescription described above: ⋆ Each vertex factor is given by the p -adic local zeta function (3.1) whose argument isgiven by the sum over the scaling dimensions of all (internal and external) operatorsincident on the vertex, minus the dimension of the boundary field theory n . ⋆ In each factor (3.2) associated with an internal line, the Mandelstam variable carried bythe line is given by a combination of Mellin variables γ ij and external scaling dimensions∆ i as defined in (2.17), with the subset S in (2.17) taken to be all the external indiceson one side of the internal line in question.As is clear from the recursive prescription for the undressed amplitudes, the recursion is onthe number of internal lines (exchanges), with each undressed amplitude expressible in termsof undressed amplitudes containing fewer internal lines (exchange channels).It will be often convenient to represent this recursive procedure diagrammatically. Todistinguish between the full Mellin amplitudes and undressed Mellin amplitudes, we will usethe following convention: ⋆ Mellin amplitudes will be represented like standard Witten diagrams, drawn in blackwith the Poincar´e disk shown explicitly. ⋆ Undressed amplitudes will be represented in red with no Poincar´e disk shown, and wewill refer to these as “undressed diagrams”.For illustrative purposes, we apply this procedure to two non-trivial diagrams, namelythe two topologically distinct types of diagrams involving three internal lines.15 xample (v). For the diagram with three internal lines arranged in a series, the recursiveprocedure leads to the following Mellin amplitude. A B C ... i L ... i R ... i l ... i r = (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) A B C ... i L ... i R ... i l ... i r A B C ... i L ... i R ... i l ... i r A B ... i l B C ... i r B C ... i L ... i R ... i l ... i r A B C ... i L ... i R ... i l ... i r A B C ... i L ... i R ... i l ... i r A B ... i l B C ... i r B C ... i L ... i R ... i l ... i r (cid:18) A ... i L ... i R A ... i L ... i R A ... i L C ... i R ... i L ... i R AB (cid:19) A B C ... i L ... i R ... i l ... i r A B C ... i L ... i R ... i l ... i r A B ... i l B C ... i r B C ... i L ... i R ... i l ... i r , (3.9)where the undressed diagram is given by A B C ... i L ... i R ... i l ... i r A B C ... i L ... i R ... i l ... i r A B ... i l B C ... i r B C ... i L ... i R ... i l ... i r =( − (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) (cid:18) C A B ... i L ... i R ... i U A B ... i L ... i R ... i U B ... i L ... i R ... i U A ... i L ... i R ... i U (cid:19) − A B C ... i L ... i R ... i l ... i r B ... i L ... i l B ... i R ... i r ( − A B C ... i L ... i R ... i l ... i r A B C ... i L ... i R ... i l ... i r A B ... i l B C ... i r B C ... i L ... i R ... i l ... i r − A B C ... i L ... i R ... i l ... i r A ... i R ... i l ... i r C ... i L ... i l ... i r A C ... i l ... i r ( − A B ... i L ... i R ... i l ... i r A C ... i L ... i R ... i l ... i r C ... i L ... i R ... i l ... i r B ... i L ... i R ... i l ... i r − A B C ... i L ... i R ... i l ... i r B ... i L ... i l B ... i R ... i r ( − A B ... i L ... i R ... i l ... i r A C ... i L ... i R ... i l ... i r C ... i L ... i R ... i l ... i r B ... i L ... i R ... i l ... i r − A B C ... i L ... i R ... i l ... i r A ... i R ... i l ... i r C ... i L ... i l ... i r A C ... i l ... i r ( − A ... i L ... i R ... i l ... i r ... i L ... i R ... i l ... i r − A B C ... i L ... i R ... i l ... i r A ... i R ... i l ... i r C ... i L ... i l ... i r A C ... i l ... i r ( − A B ... i L ... i R ... i l ... i r A C ... i L ... i R ... i l ... i r C ... i L ... i R ... i l ... i r B ... i L ... i R ... i l ... i r − A B C ... i L ... i R ... i l ... i r B ... i L ... i l B ... i R ... i r A B C ... i L ... i R ... i l ... i r B ... i L ... i l B ... i R ... i r ( − A B ... i L ... i R ... i l ... i r A C ... i L ... i R ... i l ... i r C ... i L ... i R ... i l ... i r B ... i L ... i R ... i l ... i r − A ... i L ... i R ... i l ... i r ... i L ... i R ... i l ... i r . (3.10)16t should be clear in the diagrammatic representation above which internal lines were col-lapsed in each of the terms subtracted off from the first term, the product over all internalline factors. Further, according to the prescription, the Mandelstam-like variables whichfeature in this example are given by s A = (cid:88) ∆ i L − (cid:88) γ i L j L ,s B = (cid:88) ∆ i L + (cid:88) ∆ i l − (cid:88) γ i L j L − (cid:88) γ i l j l − (cid:88) γ i L j l ,s C = (cid:88) ∆ i R − (cid:88) γ i R j R . (3.11)Note that there is no restriction on the indices in the sum, (cid:80) γ i L j l = (cid:80) i L ,j l γ i L j l , while inother sums there is one, e.g. (cid:80) γ i L j L = (cid:80) i L According to the recursive prescription, the Mellin amplitude for the stardiagram, which is a bulk diagram where three internal lines meet at a central bulk vertex,takes the form A BC ... i L ... i R ... i U ... i D = (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) A BC ... i L ... i R ... i U ... i D A BC ... i U C ... i D A BC ... i L ... i R ... i U ... i D A BC ... i U C ... i D (cid:18) A ... i L ... i R A ... i L ... i R A ... i L B ... i R ... i L ... i R AB (cid:19) A BC ... i L ... i R ... i U ... i D A BC ... i L ... i R ... i U ... i D A BC ... i U C ... i D , (3.12)17here the undressed diagram is given by A BC ... i L ... i R ... i U ... i D A BC ... i L ... i R ... i U ... i D A BC ... i U C ... i D =( − (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) (cid:18) A ... i L ... i R A ... i L ... i R A ... i L A ... i R ... i L ... i R AB (cid:19) C ... i L ... i R ... i U ... i D A ... i L ... i R ... i U ... i D B ... i L ... i R ... i U ... i D ... i L ... i R ... i U ... i D C − A BC ... i L ... i R ... i U ... i D BC ... i L ... i U A C ... i R ... i U A B ... i U ... i D ( − AC ... i L ... i R ... i U ... i D A B ... i L ... i R ... i U ... i D BC ... i L ... i R ... i U ... i D − A BC ... i L ... i R ... i U ... i D BC ... i L ... i U A C ... i R ... i U A B ... i U ... i D ( − AC ... i L ... i R ... i U ... i D A B ... i L ... i R ... i U ... i D BC ... i L ... i R ... i U ... i D − A BC ... i L ... i R ... i U ... i D BC ... i L ... i U A C ... i R ... i U A B ... i U ... i D ( − AC ... i L ... i R ... i U ... i D A B ... i L ... i R ... i U ... i D BC ... i L ... i R ... i U ... i D − A BC ... i L ... i R ... i U ... i D C ... i L ... i R ... i U A ... i R ... i U ... i D B ... i L ... i U ... i D ( − C ... i L ... i R ... i U ... i D A ... i L ... i R ... i U ... i D B ... i L ... i R ... i U ... i D ... i L ... i R ... i U ... i D − A BC ... i L ... i R ... i U ... i D C ... i L ... i R ... i U A ... i R ... i U ... i D B ... i L ... i U ... i D ( − C ... i L ... i R ... i U ... i D A ... i L ... i R ... i U ... i D B ... i L ... i R ... i U ... i D ... i L ... i R ... i U ... i D − A BC ... i L ... i R ... i U ... i D C ... i L ... i R ... i U A ... i R ... i U ... i D B ... i L ... i U ... i D ( − C ... i L ... i R ... i U ... i D A ... i L ... i R ... i U ... i D B ... i L ... i R ... i U ... i D ... i L ... i R ... i U ... i D − C ... i L ... i R ... i U ... i D A ... i L ... i R ... i U ... i D B ... i L ... i R ... i U ... i D ... i L ... i R ... i U ... i D , (3.13)with the Mandelstam-like variables given by s A = (cid:88) ∆ i L − (cid:88) γ i L j L ,s B = (cid:88) ∆ i R − (cid:88) γ i R j R ,s C = (cid:88) ∆ i D − (cid:88) γ i D j D . (3.14)18ust like in the previous example, the undressed diagram in (3.13) may recursively be reducedfurther using expressions (3.4) and (3.6), to give an amplitude expressed solely in terms ofthe internal line and contact vertex factors. It is again easily verified that this result matchesthe previously computed amplitude given in equation (4.78) of Ref. [4]. It is interesting to compare the recursive procedure describe above with the so-called Feyn-man rules for real Mellin amplitudes. Prescription II (Feynman rules for real Mellin amplitudes [6, 7, 8]) . Given a tree-level bulkdiagram, (adapted to the conventions and normalization described in section 2) the Feynmanrules are: • Label the internal lines with an index j running from 1 to v − and associate to eachinternal line an integer n j and a factor of L j ( s j , ∆ j , n j ) = 1 n j ! (1 + ∆ j − h ) n j (∆ j − s j ) / n j , (3.15) where s j is the Mandelstam invariant associated to internal leg j , and ( a ) n j ≡ Γ( a + n j )Γ( a ) is the Pochhammer symbol. • Label the vertices with an index j running from one to v . For a vertex j connected to L legs with scaling dimensions ∆ to ∆ L , label the internal legs by indices 1 to l andthe external legs by indices l + 1 to L , let n to n l be the integers associated with theinternal legs, and associate to the vertex j a factor of V j ( { ∆ i } , { n i } ) = 12 ζ ∞ (cid:0) L (cid:88) i =1 ∆ i − n (cid:1) × F ( l ) A (cid:32) L (cid:88) i =1 ∆ i − h ; {− n , . . . , − n l } ; { − h, . . . , l − h } ; 1 , . . . , (cid:33) , (3.16) where F ( l ) A is the Lauricella hypergeometric series [43, 44, 45, 7], defined as F ( l ) A ( g ; { a , . . . , a l } ; { b , . . . , b l } ; x , . . . , x l ) = (cid:34) l (cid:89) i =1 ∞ (cid:88) n i =0 (cid:35) ( g ) (cid:80) li =1 n i l (cid:89) i =1 ( a i ) n i ( b i ) n i x n i i n i ! . (3.17)19 The Mellin amplitude is given by taking the product of all line and vertex factors andthen summing over the integers associated with the internal legs: M ( { γ ij } , { ∆ i } ) = (cid:34) v − (cid:89) i =1 ∞ (cid:88) n i =0 (cid:35) (cid:34) v − (cid:89) j =1 L j ( s j , ∆ j , n j ) (cid:35) (cid:34) v (cid:89) j =1 V j ( { ∆ i } , { n i } ) (cid:35) . (3.18)These rules are arguably more compact and straightforward than the prescription forobtaining the p -adic Mellin amplitudes, but this apparent simplicity is deceptive and merelydue to the fact that the real Mellin amplitude is left in the form of unevaluated infinite sums,while the prescription in the p -adic case fully reduces the Mellin amplitudes to finite sums ofelementary functions. For this reason, prescriptions I and II do not highlight the similarityin the structure of real and p -adic Mellin amplitudes. For a unified perspective on real and p -adic amplitudes, it is useful to turn instead to pre-amplitudes which will be the subjectof the next section. However, we do note at this point that as in the p -adic case, we candecompose both real and p -adic Mellin amplitudes into a product of contact vertices and an“undressed diagram”: M ( { γ ij } , { ∆ i } ) = (all contact vertices ) × (undressed diagram) , (3.19)where in the real case the vertex factors are defined as • Contact vertex: ∆ ∆ ∆ ∆ ∆ f ...∆ ∆ ∆ ∆ ∆ N ... 1 ≡ ζ ∞ (cid:32) f (cid:88) i =1 ∆ i − n (cid:33) . (3.20)With this definition in place, equations (3.3), (3.5), (3.9), and (3.12) continue to apply forreal Mellin amplitudes, with the explicit expressions for the undressed diagrams over the p -adics (given by prescription I) getting replaced by infinite sums over products of Lauricellafunctions over the reals (given by prescription II). Equation (3.19) serves as the definition ofundressed diagrams over reals. We will show in the next section that the undressed diagramsover the reals and the p -adics are in fact closely related to each other via a concrete relation.20 Construction of Pre-Amplitudes Mellin amplitudes M , real and p -adic, admit a representation called the Mellin-Barnes repre-sentation in terms of a contour integral over a pre-amplitude (cid:102) M , defined in (2.20). Comparedto Mellin amplitudes, pre-amplitudes are markedly simpler: they do not depend on internalscaling dimensions, and for any given bulk diagram the corresponding pre-amplitude is sim-ply a product over factors associated with each vertex of the bulk diagram. Furthermore,pre-amplitudes of real and p -adic AdS/CFT take identical forms, and for this reason we cansimultaneously present the prescription for either type of pre-amplitude. In the real case,the prescription essentially follows from Ref. [8] after adjusting overall coefficients to accountfor our normalization conventions described in section 2. For the p -adic case, we present aninductive proof of the prescription in appendix B. Prescription III (Pre-amplitudes) . For a diagram with v vertices labelled by indices j ∈{ , . . . , v } , the pre-amplitude is constructed as follows: • Associate to each internal leg an orientation and a complex variable c i . • For a vertex j connected to L legs with scaling dimensions ∆ to ∆ L , let indices 1 to l label internal legs and let indices l + 1 to L label external leg. Futhermore, definescaling dimensions (cid:101) ∆ i as follows: if i ∈ { , . . . , l } so that i labels an internal leg,define a complex scaling dimension (cid:101) ∆ i = h ± c i , where the plus sign is chosen if theorientation of the internal leg is incoming and the minus sign is chosen if the internalleg is outgoing with respect to the vertex; if i ∈ { l + 1 , . . . , L } so that i labels an externalleg, define (cid:101) ∆ i = ∆ i . Also associate to each vertex a factor of: Q p n (cid:1) (cid:101) V j ( { s i } , { (cid:101) ∆ i } ) = ζ p (cid:0) L (cid:88) i =1 (cid:101) ∆ i − n (cid:1)(cid:20) l (cid:89) i =1 ζ p (1) | | p (cid:90) Q p dx i | x i | p | x i | (cid:101) ∆ i − si p | , x i | (cid:101) ∆ i + si − n s (cid:21) | , x , . . . , x l | n − (cid:80) Li =1 (cid:101) ∆ i s , R n (cid:1) (cid:101) V j ( { s i } , { (cid:101) ∆ i } ) =12 ζ ∞ (cid:0) L (cid:88) i =1 (cid:101) ∆ i − n (cid:1) (cid:34) l (cid:89) i =1 (cid:90) ∞ dx i | x i | | x i | (cid:101) ∆ i − si | x i | (cid:101) ∆ i + si − n (cid:35) (cid:12)(cid:12) l (cid:88) i =1 x i (cid:12)(cid:12) n − (cid:80) Li =1 (cid:101) ∆ i . (4.1) Here { s i } is the set of Mandelstam-like variables associated with the internal lines and Q p denotes the set of p -adic squares, see (A.8). The Mellin pre-amplitude is given by the product of these vertex factors: (cid:102) M ( { γ ij } , { ∆ ext, i } , { c i } ) = v (cid:89) j =1 (cid:101) V j ( { s i } , { (cid:101) ∆ i } ) . (4.2)We point out that the integral representation of the pre-amplitude vertex factor in (4.1)over the p -adics and the reals is related to undressed diagrams as defined in (3.19). Thus theseintegrals admit a recursive construction via prescription I for the p -adics and prescription IIfor the reals. This point is explained in more detail next. The reader might have noticed that the contact vertices defined in (3.1) and (3.20) appearin the prescription above for pre-amplitude vertex factors, and as we commented above theremaining factors in (4.1) can be identified with undressed diagrams. Essentially the claim isthat pre-amplitudes decompose into products of Mellin amplitudes with a special assignmentof internal dimensions, so that the Mellin-Barnes integral representation expresses any Mellinamplitude as a contour integral over simpler Mellin amplitudes.More precisely, the pre-amplitude vertex factor (cid:101) V j ( { s i } , { (cid:101) ∆ i } ) from (4.1) associated witha given vertex in a bulk diagram, with internal lines assigned orientations according toprescription III as displayed in black in (4.3), is expressible as a product of a contact vertexand an undressed diagram associated to a full Mellin amplitude, as shown: ∆ l +1 ∆ L ...... s , ∆ s l , ∆ l : (cid:101) V j ( { s i } , { (cid:101) ∆ i } ) = ∆ l +1 ∆ L ...... h + c h + c l ∆ l +1 ∆ L ...... s ,h + c h + c l ,s l h − c h − c l , (4.3)where the contact vertex factor (shown in blue in (4.3)) is given by (3.1) over p -adics and(3.20) over reals, and the undressed diagram (shown in red in (4.3)) can be explicitly con-structed over the p -adics using the recursive prescription I, while over the reals it is con-22tructed using the Feynman rule prescription II.We point out that the undressed diagram in (4.3) has l internal lines, with Mandelstaminvariants s i and propagating complex dimensions h + c i . Furthermore, the l vertices atwhich operators of dimensions h − c i are incident should formally be thought of as contactinteraction vertices, each of which has precisely one leg incident on another bulk vertex,while all other legs connect to external operator insertions.We now prove this non-trivial claim, beginning with the real case. From the Feynmanrules and definitions in section 3.2, it follows that R n (cid:1) ∆ l +1 ∆ L ...... s ,h + c h + c l ,s l h − c h − c l = (cid:34) l (cid:89) i =1 ∞ (cid:88) n i =0 (cid:35) (cid:34) l (cid:89) j =1 n j ! (1 + (cid:101) ∆ j − h ) n j ( (cid:101) ∆ j − s j ) / n j F (1) A (cid:32) (cid:101) ∆ j + h − c j − h ; {− n j } ; { c j } ; 1 (cid:33)(cid:35) × F ( l ) A (cid:32) L (cid:88) i =1 (cid:101) ∆ i − h ; {− n , . . . , − n l } ; { c , . . . , c l } ; 1 , . . . , (cid:33) . (4.4)Since (cid:101) ∆ j = h + c j for j ∈ { , . . . , l } , the Lauricella functions F (1) A in the above expressionare all equal to unity. Using the definition (3.17) for the Lauricella function as a multiplesum, changing the order of summation between this multiple sum and the sums over n i , andapplying the identity ∞ (cid:88) n =0 ( − n ) m (1 + c ) n n !( A + n ) = Γ( A + m )Γ( − c )Γ( A − c ) = ( A ) m β ∞ (2 A, − c ) , (4.5)where β ∞ ( · , · ) is related to the usual Euler Beta function and was defined in (2.24), one can23arry out the sums over the integers n i to find, R n (cid:1) ∆ l +1 ∆ L ...... s ,h + c h + c l ,s l h − c h − c l = F ( l ) A (cid:32) (cid:80) Li =1 (cid:101) ∆ i − n (cid:40) (cid:101) ∆ − s , . . . , (cid:101) ∆ l − s l (cid:41) ; { c , . . . , c l } ; 1 , . . . , (cid:33) l (cid:89) i =1 β ∞ (cid:0) (cid:101) ∆ i − s i , − c i (cid:1) = (cid:34) l (cid:89) i =1 (cid:90) ∞ dx i | x i | | x i | (cid:101) ∆ i − si | x i | (cid:101) ∆ i + si − n (cid:35) (cid:12)(cid:12) l (cid:88) i =1 x i (cid:12)(cid:12) n − (cid:80) Li =1 (cid:101) ∆ i , (4.6)where the last equality arises from a formal substitution justified via an appropriate analyticcontinuation; we refer the reader to equation (3.30) of Ref. [8]. This proves (4.3) over R n .We now turn to the p -adic case, where we need to prove the identity Q p n (cid:1) ∆ l +1 ∆ L ...... s ,h + c h + c l ,s l h − c h − c l ! = (cid:20) l (cid:89) i =1 ζ p (1) | | p (cid:90) Q p dx i | x i | p | x i | ci + h − si p | , x i | ci + si − h s (cid:21) | , x , . . . , x l | h − ∆ − (cid:80) li =1( ci + h )2 s ≡ D , (4.7)where ∆ ≡ (cid:80) Li = l +1 ∆ i and the left-hand side of the equation is obtained using prescription Iin section 3.1 and we have introduced the symbol D as a convenient shorthand for the right-hand side. As a first step, we change to variables t i ∈ Q p satisfying x i = t i . Rather thanpicking a specific root for each x i , we can include both roots in the domain of t i at the costof a factor of 2 / | | p per variable. We get that D = l (cid:89) i =1 (cid:20) ζ p (1) (cid:90) Q p dt i | t i | p | t i | h + c i − s i p | , t i | c i + s i − hs (cid:21) | , t , . . . , t l | h − ∆ − (cid:80) li =1 ( c i + h ) s . (4.8)24tarting from this expression one can verify (4.7) when l = 1: D (cid:12)(cid:12) l =1 = ζ p (1) (cid:90) Q p dt | t | p | t | h + c − s p | , t | s − ∆ s = ζ p (∆ − c − h ) − ζ p ( s − c − h )= h + c s h − c ...∆ ∆ L , (4.9)where we refer the reader to (A.12) for the second equality, and in the final equality we usedthe expression for the undressed diagram with a single internal leg, given in (3.4).Having established the base case, we now proceed to prove (4.7) inductively by showingthat (4.8) satisfies the same recursion relation as the undressed diagram in (4.7). This canbe achieved by partitioning the domain of the multi-dimensional integral in (4.8) into thefollowing sub-domains:1. the part of the domain where | t | p , | t | p , . . . , | t l | p ≤ | t | p = | t | p = · · · = | t l | p > 1, and finally3. for each proper subset { u , u , . . . , u g } ⊂ { , , . . . , l } , with complementary set labeled { u g +1 , u g +1 , . . . , u l } , the part of the domain where | t u | p = | t u | p = · · · = | t u g | p > { , | t u g +1 | p , | t u g +2 | p , . . . , | t u l | p } .For the first part of the domain, the integral in D evaluates to D (cid:12)(cid:12) sub-domain 1 = l (cid:89) i =1 (cid:20) ζ p (1) (cid:90) Q p dt i | t i | p | t i | − s i + h + c i p γ p ( t i ) (cid:21) = l (cid:89) i =1 ζ p ( − s i + h + c i ) , (4.10)where γ p is the characteristic function of Z p , defined in (A.1) and we used identity (A.5)to obtain the second equality. The contribution to D coming from the second sub-domainevaluates to D (cid:12)(cid:12) sub-domain 2 = ζ p (1) (cid:90) Q p dt | t | p | t | h − ∆ − (cid:80) li =1 ( − c i + h ) p γ p (cid:18) pt (cid:19) = − ζ p (cid:0) h − ∆ − l (cid:88) i =1 ( − c i + h ) (cid:1) . (4.11)For the last type of sub-domain, the contribution to D arising from the subset { u , ..., u g } D (cid:12)(cid:12) { u ,...,u g } partof sub-domain 3 = l (cid:89) i = g +1 (cid:34) ζ p (1) (cid:90) Q p dt u i | t u i | p | t u i | − s ui + h + c ui p | , t u i | s ui − h + c ui s (cid:35) × ζ p (1) (cid:90) Q p dt | t | p | t | h − ∆ − (cid:80) gi =1 ( h − c ui ) − (cid:80) li = g +1 ( h + c ui ) p γ p (cid:18) (1 , t u g +1 , . . . , t u l ) s pt (cid:19) . (4.12)By changing variable from t to u ≡ t/ (1 , t u g +1 , . . . , t u f ) s , the u integral factors out and canimmediately be evaluated. One obtains D (cid:12)(cid:12) { u ,...,u g } partof sub-domain 3 = − ζ p (cid:32) h − ∆ − g (cid:88) i =1 ( h − c u i ) − l (cid:88) i = g +1 ( h + c u i ) (cid:33) × l (cid:89) i = g +1 (cid:20) ζ p (1) (cid:90) Q p dt u i | t u i | p | t u i | − s ui + h + c ui p | , t u i | s ui − h + c ui s (cid:21) × | , t u g +1 , . . . , t u l | h − ∆ − (cid:80) gi =1 ( h − c ui ) − (cid:80) li = g +1 ( h + c ui ) s . (4.13)We notice that the above expression is equal to a local zeta function times an integral ofthe same form as the original integral (4.8) except with l − g instead of l internal legs andwith ∆ − (cid:80) gi =1 ( h − c u i ) substituted for ∆. Via an inductive reasoning, we assume thatthis remaining integral obeys (4.7), that is it is equal to the undressed diagram obtained bycollapsing legs indexed by u to u g in the undressed diagram on the right-hand side of (4.7),and now proceed to show that this implies the full diagram obeys (4.7) too.In section 3.1 we defined undressed diagrams as satisfying the recursion relation (3.8).From the decomposition of D into sub-domains as described above, we have in fact recoveredthe same recursive structure except we did not pick up explicit factors of ( − andthe local zeta functions in (4.10), (4.11), and (4.13) appear with minus the argument withwhich they would have appeared in the vertex factors and internal leg factors as prescribedby (3.8). That is, compared to (3.8), we find here D = (cid:89) (internal line factors w/ opposite sign arguments) − (cid:88) (new vertex factors w/ opposite sign arguments) × (reduced undressed diagram) . (4.14)However, the two decompositions (3.8) and (4.14) are in fact equivalent, by virtue of the26 -adic local zeta function identity ζ p ( x ) + ζ p ( − x ) = 1 , (4.15)using which one can show that if one flips the sign of 1) the degree of extension n , 2) allthe scaling dimensions ∆ i , and 3) all the Mandelstam invariants s i inside the arguments ofthe local zeta functions, then any undressed diagram transform into itself times a factor of( − . The equivalence of (3.8) and (4.14) follows directly from this, completingthe proof of (4.7). In light of the identities for undressed diagrams derived in the previous subsection, theprescription for real and p -adic pre-amplitudes can be recast in a diagrammatic form thatapplies to both cases at once. In the contour integrals which follow, it will be understoodthat in the real case the contours run parallel to the imaginary axis, with limits from − i ∞ to i ∞ , while in the p -adic case the contour wraps once around the complex cylinder, so thatthe limits are from − iπ log p to iπ log p . With these conventions in place, let us consider someexplicit examples for illustrative purposes. Example (vii). The exchange diagram Mellin amplitude and pre-amplitude take the fol-lowing form: s ∆ ...... i R i L = (cid:32) ∆... i L (cid:33) (cid:32) ∆ ... i R (cid:33) (cid:32) ∆ s ... i L ... i R (cid:33) = (cid:90) dc πi f ∆ ( c ) (cid:32) h + c ... i L (cid:33) (cid:32) h + cs h − c ... i L (cid:33) × (cid:32) h − c ... i R (cid:33) (cid:32) h − ch + c s ... i R (cid:33) , (4.16)where we remind the reader that f ∆ ( c ) was defined in (2.8), the vertex factors are givenby (3.1) over p -adics and (3.20) over reals, and the undressed diagrams are obtained fromprescription I over p -adics and prescription II over reals. It can be easily checked that thisreproduces the pre-amplitudes in (2.23). 27 xample (viii). The Mellin amplitude and pre-amplitude for the diagram with two internallines, (3.5), are: ... i U s A ∆ A s B ∆ B ...... i R i L = (cid:32) ∆ A ... i L (cid:33) ... i U ∆ A ∆ B (cid:32) ∆ B ... i R (cid:33) × ... i L ... i R ... i U ∆ A ∆ B s A s B = (cid:90) dc A πi f ∆ A ( c A ) (cid:90) dc B πi f ∆ B ( c B ) (cid:32) h − c A ... i L (cid:33) (cid:32) h − c A s A h + c A ... i L (cid:33) × ... i U h + c A h − c B ... i U h + c A h − c B s A s B h − c A h + c B × (cid:32) h + c B ... i R (cid:33) (cid:32) h + c B h − c B s B ... i R (cid:33) . (4.17)28 xample (ix). The Mellin amplitude for the diagram with three internal lines arranged inseries, (3.9) decomposes into a pre-amplitude thus: s B ∆ B ... i l ... i r s A ∆ A s C ∆ C ...... i R i L = (cid:90) dc A πi f ∆ A ( c A ) (cid:90) dc B πi f ∆ B ( c B ) (cid:90) dc C πi f ∆ B ( c C ) × (cid:32) h − c A ... i L (cid:33) (cid:32) h − c A s A h + c A ... i L (cid:33) × ... i l h + c A h − c B ... i l h + c A h − c B s A s B h − c A h + c B × ... i r h + c B h − c C ... i r h + c B h − c C s B s C h − c A h + c B × (cid:32) h + c C ... i R (cid:33) (cid:32) h + c C h − c C s C ... i R (cid:33) . (4.18) Example (x). As a final example, consider the Mellin amplitude for the diagram with threeinternal lines meeting at a center vertex, (3.12). It exhibits the following decomposition intoa contour integral over its pre-amplitude: 29 C ∆ C ... i U ... i D s A ∆ A s B ∆ B ...... i R i L = (cid:90) dc A πi f ∆ A ( c A ) (cid:90) dc B πi f ∆ B ( c B ) (cid:90) dc C πi f ∆ C ( c C ) × (cid:32) h − c A ... i L (cid:33) (cid:32) h − c A s A h + c A ... i L (cid:33) × (cid:32) h + c B ... i R (cid:33) (cid:32) h + c B h − c B s B ... i R (cid:33) × h + c C ... i U h + c A h − c B s C h + c C h − c C ... i U s A h + c A h − c A s B h − c B h + c B × h − c C ... i D s C h − c C h + c C ... i D . (4.19) In section 3 we showed how to write a Mellin amplitude as the product of vertex factorstimes a single undressed amplitude. In sections 4.1-4.2 we showed how to obtain the pre-amplitude by multiplying together one vertex factor and one undressed amplitude withcomplex dimensions for each vertex in the bulk diagram. In this subsection we explain thediagrammatic interpretation of performing contour integrals over the pre-amplitude to obtainthe full Mellin amplitude: for each integral that is carried out, two undressed diagrams mergeand become one, and after carrying out all the contour integrals, all undressed diagrams have30erged into the single undressed amplitude.As an example of how this, consider the Mellin amplitude M for the diagram withthree internal legs arranged in series as given in equation (4.18). Carrying out the c A integralgives the following: M ≡ (cid:90) dc B πi f ∆ B ( c B ) (cid:90) dc C πi f ∆ C ( c C ) × (cid:32) ∆ A ... i L (cid:33) ... i L ... i l ∆ A h − c B h + c B ... i l ∆ A h − c B × ... i r h + c B h − c C h − c B h + c C ... i r h + c B h − c C (cid:32) h + c C h − c C ... i R (cid:33) × (cid:32) h + c C ... i R (cid:33) . (4.20)Carrying out the c B integral next, one finds that M = (cid:90) dc C πi f ∆ C ( c C ) (cid:32) ∆ A ... i L (cid:33) ... i l ∆ A ∆ B × ... i L ... i l ... i r ∆ A ∆ B s A s B h − c C h + c C ... i r ∆ B h − c C × (cid:32) h + c C h − c C ... i R (cid:33) (cid:32) h + c C ... i R (cid:33) . (4.21)Finally, carrying out the c C integral yields 31 = (cid:32) ∆ A ... i L (cid:33) ... i l ∆ A ∆ B ... i r ∆ B ∆ C (cid:32) ∆ C ... i R (cid:33) × ... i L ... i l ... i r ∆ A ∆ B s A s B ∆ C ... i R . (4.22)We see that carrying out contour integrals of the form above effectively glues together two ofthe undressed diagrams defined in section 3. This holds true generally: given a preamplitude, (cid:102) M ≡ F ... i A h − c . i ... i .. DC E F ... i ... i ... i ... i B A h − cs h + c × (cid:32) h + csh − c ... i (cid:33) (cid:18) h + c ... i (cid:19) , (4.23)the following identity holds, (cid:90) dc πi f ∆ ( c ) (cid:102) M = F ... i A ∆ . i ... i .. DC E F ... i ... i ... i ... i B A ∆ s ... i (cid:18) ∆ ... i (cid:19) . (4.24)At the real place, this fact follows from the Feynman rules for Mellin amplitudes and pre-amplitudes that already exist in the literature (see Refs. [6, 7, 8]), and which in this paper wehave presented in a manner which highlights the common features that hold true over both32he reals and the p -adics. At the p -adic place, this fact requires a proof which is provided inappendix C. The recursive structure of p -adic Mellin amplitudes described in section 3.1 is of a somewhatunusual kind: the amplitude of an N -point bulk diagram is expressed in terms of amplitudesof N -point diagrams with fewer internal lines. Such a recursion relation invokes interactionvertices of higher and higher degree, that is, interaction terms involving more and morefields. A natural question to ask is whether it is possible to set up a recursive prescriptionin a form more similar to the familiar on-shell recursions in flat-space, such as the BCFWrecursion relation [14], where no new interaction vertices are introduced and a higher-pointamplitude is expressed in terms of lower point amplitudes of the same theory. In this sectionwe show that this is indeed possible, though we emphasize that the recursion we formulatein this section applies at the level of individual bulk diagrams.We start in section 5.1 by describing the factorization property of undressed diagrams andexplaining how this property allows us to reformulate the recursion relation (3.8) in a mannerthat will play as a precursor to the on-shell recursion of this section. Then, in section 5.2, wepresent what we refer to as the “on-shell” recursion relation, following up with a proof thatestablishes the equivalence between the recursive prescription I of section 3 and the on-shellprescription of this section. At the end of this section we comment on its connection with theFeynman rules for real Mellin amplitudes stated earlier as prescription II, and in appendixD, with the help of two explicit examples, we demonstrate the similarities (and differences)between this prescription and flat-space BCFW recursion. Consider an arbitrary undressed diagram: DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i . (5.1)33t follows from the recursive prescription I of section 3.1 that the sum of all the terms of theundressed amplitude proportional to the internal line factor ζ p ( s G − ∆ G ) is itself given byan undressed diagram: − ζ p ( s G − ∆ G ) DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i . (5.2)More generally, the coefficient of an internal line factor in an undressed diagram is givenby the product of the left- and right-hand (undressed) diagrams obtained by cutting thediagram across the chosen internal line, and assigning the off-shell internal leg dimensionto the newly generated external legs of the left- and right-hand undressed diagrams. Forexample, in the diagram (5.1), adding together all the terms containing the internal linefactor ζ p ( s A − ∆ A ) gives, − ζ p ( s A − ∆ A ) DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i . (5.3)The procedure generalizes straightforwardly if one wants to pick out terms containing mul-tiple propagator factors: (cid:0) terms in undressed diagram D containing propagator factors ζ p ( s − ∆ ) ...ζ p ( s f − ∆ f ) (cid:1) = ( − f (cid:34) f (cid:89) I =1 ζ p ( s I − ∆ I ) (cid:35) × (cid:32) product of undressed diagrams resulting fromremoving internal legs 1 to f from D (cid:33) . (5.4)For example, in diagram (5.1), the terms containing both factors ζ p ( s A − ∆ A ) and ζ p ( s B − ∆ B )sum to 34 − ζ p ( s A − ∆ A ) ζ p ( s B − ∆ B ) DC.. .i .. .i BB E. . .i . . .i AFA . . .i . . .i G ... i DC.. .i .. .i BB E. . .i . . .i AFA . . .i . . .i G ... i DC.. .i .. .i BB E. . .i . . .i AFA . . .i . . .i G ... i . (5.5)These observations reflect the meromorphic nature of arbitrary Mellin amplitudes, whichhave simple poles at values of Mandelstam-like invariants that put any internal leg “on-shell”. For p -adic Mellin amplitudes it is clear from the above that such poles occur inthe internal leg factor (3.2) when the Mandelstam invariant of the associated leg equals theconformal dimension of the bulk field propagating along the leg. These poles correspond tothe exchange of single-trace operators in the boundary theory. Thus p -adic Mellin amplitudeshave the right factorization properties of (AdS) scattering amplitudes.Another property of undressed diagrams that follows immediately from the recursiveprescription I is that the momentum independent terms (i.e. terms independent of any Man-delstam variable s I and thereby also independent of the Mellin variables γ ij ) are in one-to-onecorrespondence with the terms exhibiting momentum dependence. If for a given undresseddiagram one takes all the momentum-dependent terms, sets all internal leg factors ζ p ( s − ∆)to unity (which can be achieved by sending all Mandelstam variables s to infinity), andmultiplies with an overall factor of ( − × ζ p (Σ), where Σ is the sum of all external scalingdimensions minus n = 2 h , Σ ≡ (cid:88) ∆ external − h , (5.6)then one exactly recovers the momentum independent part of the undressed diagram: (cid:0) momentum independent terms (cid:1) = − ζ p (Σ) lim ∀ s →∞ (cid:0) momentum dependent terms (cid:1) . (5.7)By applying the inclusion-exclusion principle to these two properties we can derive yet an-other recursive formula for undressed diagrams, as we now explain: As discussed above, givenany diagram, we can, for each internal leg I , sum over the terms proportional to ζ p ( s I − ∆ I ),the sum of which gives us − ζ p ( s I − ∆ I ) times a smaller diagram or a product of two smallerdiagrams. Summing − ζ p ( s I − ∆ I ) times these smaller diagrams over all I , we obtain a sumcontaining all momentum-dependent terms. But in doing so, terms containing two factors35f ζ p ( s − ∆) (i.e. terms proportional to ζ p ( s I − ∆ I ) ζ p ( s J − ∆ J ) for I ̸ = J ) have been double-counted. This over-counting can be compensated for by subtracting off all terms with thetwo factors of ζ p ( s − ∆), which by the factorization property above, are also given in terms ofsmaller diagrams. But now terms containing three factors of ζ p ( s − ∆) haven’t been includedat all, so we add those terms back in. Carrying out this alternating sum until one reaches theone term that that contains the product of ζ p ( s I − ∆ I ) over all I , we get an expression for allthe momentum-dependent terms. (Note that the alternating sign of the inclusion-exclusionprinciple is cancelled by the ( − f prefactor in (5.4) so that each term comes with a minussign.) We then add in the momentum-independent terms to get an expression for the fullundressed diagram under consideration, by adding − ζ p (Σ) times the momentum-dependentterms, except with all Mandelstam variables set to infinity. Thus we arrive at the formula(undressed diagram) = − (cid:88) ℓ =1 (cid:88) all possible sets of ℓ internal legs { I ,...,I ℓ } × (cid:34) ℓ (cid:89) i =1 ζ p ( s I i − ∆ I i ) − ζ p (Σ) lim ∀ s →∞ (cid:35) × (undressed diagrams left after removing legs { I , . . . , I ℓ } ) , (5.8)where by “removing a leg” we mean cutting or factorizing the diagram across the particularleg. Concretely, writing out the first couple of terms on the right-hand side as well as thelast term, this formula says that(undressed diagram) = − (cid:88) I (cid:104) ζ p ( s I − ∆ I ) − ζ p (Σ) lim ∀ s →∞ (cid:105) × (undressed diagrams left after removing leg I ) − (cid:88) I An alternate formulation of this prescription (in auxiliary momentum space) is providedin appendix D. Before demonstrating the equivalence between prescriptions IV and I, letus consider a few examples. The derivation of (5.11) for the case of four- and five-pointdiagrams is discussed in significant detail in appendix D, using the familiar technique ofcomplexifying the amplitude and BCFW-shifting the (auxiliary) momenta. We focus hereon the final decomposition for two six-point examples to illustrate how to apply (5.11).38 xample (xii). For the six-point series diagram, the on-shell recursion (5.11) dictates ∆ A s A ∆ B s B ∆ C s C 12 436 5∆ B s B − s A + ∆ A s C − s A + ∆ A ∆ C A A s A − s C + ∆ C s B − s C + ∆ C ∆ B 12 5 C A s A − s B + ∆ B 12 6 B ∆ C s C − s B + ∆ B B A C 43 1 = − ∆ A s A ∆ B s B ∆ C s C 12 436 5∆ B s B − s A + ∆ A s C − s A + ∆ A ∆ C A A s A − s C + ∆ C s B − s C + ∆ C ∆ B 12 5 C A s A − s B + ∆ B 12 6 B ∆ C s C − s B + ∆ B B A C 43 1 β p ( s A − ∆ A , n − ∆ Σ ) ∆ A s A ∆ B s B ∆ C s C 12 436 5∆ B s B − s A + ∆ A s C − s A + ∆ A ∆ C A A s A − s C + ∆ C s B − s C + ∆ C ∆ B 12 5 C A s A − s B + ∆ B 12 6 B ∆ C s C − s B + ∆ B B A C 43 1 − ∆ A s A ∆ B s B ∆ C s C 12 436 5∆ B s B − s A + ∆ A s C − s A + ∆ A ∆ C A A s A − s C + ∆ C s B − s C + ∆ C ∆ B 12 5 C A s A − s B + ∆ B 12 6 B ∆ C s C − s B + ∆ B B A C 43 1 β p ( s B − ∆ B , n − ∆ Σ ) ∆ A s A ∆ B s B ∆ C s C 12 436 5∆ B s B − s A + ∆ A s C − s A + ∆ A ∆ C A A s A − s C + ∆ C s B − s C + ∆ C ∆ B 12 5 C A s A − s B + ∆ B 12 6 B ∆ C s C − s B + ∆ B B A C 43 1 − ∆ A s A ∆ B s B ∆ C s C 12 436 5∆ B s B − s A + ∆ A s C − s A + ∆ A ∆ C A A s A − s C + ∆ C s B − s C + ∆ C ∆ B 12 5 C A s A − s B + ∆ B 12 6 B ∆ C s C − s B + ∆ B B A C 43 1 β p ( s C − ∆ C , n − ∆ Σ ) ∆ A s A ∆ B s B ∆ C s C 12 436 5∆ B s B − s A + ∆ A s C − s A + ∆ A ∆ C A A s A − s C + ∆ C s B − s C + ∆ C ∆ B 12 5 C A s A − s B + ∆ B 12 6 B ∆ C s C − s B + ∆ B B A C 43 1 , (5.12)where we have defined ∆ Σ ≡ (cid:80) i =1 ∆ i , and the Mandelstam variables are s A ≡ ∆ + ∆ − γ , s C ≡ ∆ + ∆ − γ ,s B ≡ s A + ∆ − γ − γ = s C + ∆ − γ − γ , (5.13)where the Mellin variables γ ij satisfy (2.13) for N = 6. Further, in (5.12), we have indicatedthe Mandelstam variables associated with the internal legs on top of the lines, and in ther.h.s. in the sub-amplitudes, we have explicitly displayed the shifted Mandelstam variables. Example (xiii). Likewise, for the six-point star diagram, (5.11) gives39 A ∆ A s C s B ∆ C ∆ B C s C − s A + ∆ A s B − s A + ∆ A ∆ B 65 34 A ∆ A s A − s B + ∆ B s C − s B + ∆ B ∆ C 12 65 BC B A = − s A ∆ A s C s B ∆ C ∆ B C s C − s A + ∆ A s B − s A + ∆ A ∆ B 65 34 A ∆ A s A − s B + ∆ B s C − s B + ∆ B ∆ C 12 65 BC B A β p ( s A − ∆ A , n − ∆ Σ ) s A ∆ A s C s B ∆ C ∆ B C s C − s A + ∆ A s B − s A + ∆ A ∆ B 65 34 A ∆ A s A − s B + ∆ B s C − s B + ∆ B ∆ C 12 65 BC B A − s A ∆ A s C s B ∆ C ∆ B C s C − s A + ∆ A s B − s A + ∆ A ∆ B 65 34 A ∆ A s A − s B + ∆ B s C − s B + ∆ B ∆ C 12 65 BC B A β p ( s B − ∆ B , n − ∆ Σ ) s A ∆ A s C s B ∆ C ∆ B C s C − s A + ∆ A s B − s A + ∆ A ∆ B 65 34 A ∆ A s A − s B + ∆ B s C − s B + ∆ B ∆ C 12 65 BC B A − s A ∆ A s C s B ∆ C ∆ B C s C − s A + ∆ A s B − s A + ∆ A ∆ B 65 34 A ∆ A s A − s B + ∆ B s C − s B + ∆ B ∆ C 12 65 BC B A β p ( s C − ∆ C , n − ∆ Σ ) ∆ B s B − s C + ∆ C s A − s C + ∆ C ∆ A 34 12 C , (5.14)where this time s A ≡ ∆ + ∆ − γ , s B ≡ ∆ + ∆ − γ , s C ≡ ∆ + ∆ − γ . (5.15)Equations (5.12) and (5.14) can be reduced further by repeated application of (5.11)(see e.g. equations (D.30) and (D.52) in appendix D) to expressions which involve only (inthis case, three-point) contact interactions, whose Mellin amplitudes are given simply bythe respective vertex factors (3.1). It is straightforward to confirm that (5.12) and (5.14)agree with the Mellin amplitudes for diagrams with three internal legs computed from firstprinciples in Ref. [4] and rederived here in examples (v)-(vi) using prescription I.We now prove that prescription IV is mathematically equivalent to the recursive prescrip-tion I described in section 3. (We will later prove prescription I in appendix C.) Starting witha Mellin amplitude M given in terms of the recursive prescription I, let M ( z ) be the complexdeformation obtained by replacing each Mandelstam invariant s I with s I ( z ) ≡ s I − z . Inthat case the function, I ( z ) ≡ log p (cid:20) ζ p ( z ) − ζ p ( (cid:88) i ∆ i − n ) (cid:21) M ( z ) , (5.16)40s defined on a “complex cylinder” where the imaginary axis wraps around a circle of radius1 / log p . In other words z ∈ R × (cid:104) − π log p , π log p (cid:17) . From here on, we will use the shorthand M = M (0). Consider now the sum S ( r ) ≡ (cid:90) r + iπ log p r − iπ log p dz I ( z ) + (cid:90) − r − iπ log p − r + iπ log p dz I ( z ) , (5.17)where r is some positive number. In the limit as r tends to infinity, S ( r ) tends to zero as wewill now show. As the real part of z tends to plus or minus infinity, the integrands in (5.17)become constant, and the integrals evaluate to 2 π/ log p (which is the circumference of thecylindrical manifold) times the constant integrand. This asymptotic behavior follows fromthe fact that I ( z ) depends on z solely through the arguments of p -adic local zeta-functions,which themselves have the asymptotic behavior ζ p ( z ) → z ] → ∞ , z ] → −∞ . (5.18)We see then that when r → ∞ , the first term in (5.17) gives (cid:90) ∞ + iπ log p ∞− iπ log p dz I ( z ) = 2 π (cid:20) − ζ p ( (cid:88) i ∆ i − n ) (cid:21) M ( z = ∞ )= 2 π (cid:20) − ζ p ( (cid:88) i ∆ i − n ) (cid:21)(cid:18) the momentum-independent terms of M (cid:19) . (5.19)For the second term in (5.17), noting that the contour runs in the opposite direction, we get (cid:90) −∞− iπ log p −∞ + iπ log p dz I ( z ) = 2 π ζ p ( (cid:88) i ∆ i − n ) M ( z = −∞ ) . (5.20)We split the expression (5.20) into two parts:1. A part proportional to the momentum-independent part of M : this part exactly cancelsthe second term inside the square brackets in the second line of (5.19).2. A part proportional to the momentum-dependent part of M : since z is taken to minusinfinity, all factors of ζ p in M which carry Mandelstam variable dependence tend tounity. Using equation (5.7), this part is seen to exactly cancel with the first term insidesquare brackets in the second line of (5.19).41e see then that lim r →∞ S ( r ) = 0. But from this it follows that the sum over all residuesof I ( z ) defined in (5.16) vanishes. For we may shift the contour of the first term in (5.17)from (cid:82) r − iπ log p r + iπ log p to (cid:82) − r − iπ log p − r + iπ log p so that this term cancels with the second term, provided we add in2 πi times the sum of all the residues in the strip − r < Re[ z ] < r : × ×− r r Re[ z ]Im[ z ] × ×− r r Re[ z ]Im[ z ]1 = × ×− r r Re[ z ]Im[ z ] × ×− r r Re[ z ]Im[ z ]1 . The residue of I ( z ) at z = 0 equals M = M (0), the original un-deformed amplitude. Since M ( z ) is obtained from M by replacing in M each factor of ζ p ( s I − ∆ I ) with ζ p ( s I − ∆ I − z ),we see that the remaining residues of M ( z ) occur at z ∗ = s I − ∆ I , exactly when thecomplex-shifted internal leg goes on shell, i.e. s I ( z ∗ ) = ∆ I . Furthermore, from (5.16) weconclude that each of these residues is equal to a factor of (cid:20) ζ p ( s I − ∆ I ) − ζ p ( (cid:80) i ∆ i − n ) (cid:21) = β p ( s I − ∆ I , n − (cid:80) i ∆ i ) times the (on-shell) left and the right sub-amplitudes (evaluated at z = z ∗ ), as follows from the factorization property described in section 5.1.Thus upon equating the residue of I ( z ) at z = 0 with minus the sum of all remainingresidues, we recover the on-shell recursion formula (5.11), proving prescription IV assumingprescription I.Given the strong similarities between the structure of real and p -adic Mellin amplitudesas illuminated in this paper so far, it is natural to wonder if the on-shell recursion relationspresented in this section also have a real counterpart. It turns out (a trivial restatementof) the Feynman rules for real Mellin amplitudes (as given in prescription II) provides theclosest analog, with an important subtlety: The on-shell recursion given in prescription IVhas a particularly simple form owing to the absence of descendant fields in the p -adic setup,which is no longer the case over the reals. Thus (5.11) continues to hold for real Mellinamplitudes, provided we additionally assign to each internal leg I an integer m I , replacethe propagator factor β p ( · , · ) above by the propagator factor L ( s I , ∆ I , m I ) defined in (3.15),update the Mandelstam shift operation in (5.11) to be s J → s J − s I + ∆ I + 2 m I , and finallyinclude infinite sums over all integers m I ∈ [0 , ∞ ) in (5.11). The fact that this prescriptionreduces to the Feynman rules trivially follows from the partial fraction identity,1 D . . . D f = f (cid:88) i =1 D i (cid:34)(cid:89) j ̸ = i D j − D i (cid:35) , (5.21)42pplied to the product of factors 1 / ( s I − ∆ I − m I ) appearing in the propagator factors L ( s I , ∆ I , m I ). In this paper we completed the task, initiated in Ref. [4], of computing p -adic Mellin am-plitudes for all tree-level, arbitrary-point bulk diagrams involving scalars. This is achievedby establishing various prescriptions for constructing Mellin amplitudes which overcome theneed to perform bulk integrations altogether: • Prescription I provides an effective computational strategy, recursive on the number ofinternal lines; • Prescription IV provides an even more effective “on-shell” strategy; • Prescription III (along with (4.3)) provides a recipe for directly writing down the Mellinpre-amplitudes defined via Mellin-Barnes contour integrals in (2.20).In fact, we showed further that the pre-amplitude prescription III of section 4 applies si-multaneously to both p -adic and real Mellin pre-amplitudes, and provides a precise, unifiedrecipe for obtaining the pre-amplitudes from full Mellin amplitudes. This demonstrates theclose ties between the real and p -adic formulations of AdS/CFT.While the real and p -adic Mellin amplitudes share many common properties, such as therecursion relations mentioned above, physical factorization properties, the analytic form ofthe pre-amplitudes, and the fact that they are meromorphic functions which carry simplepoles corresponding to exchange of single-trace operators, the p -adic amplitudes are signif-icantly simpler than the real amplitudes owing to the fact that the p -adic CFTs we arestudying lack descendants [5, 30]. The simplicity of p -adic Mellin amplitudes as comparedto their real cousins is for example highlighted in the structure of the on-shell recursion re-lation IV described in section 5, which is related to, though simpler than the Feynman rulesprescription II for real Mellin amplitudes. Thus the p -adic formulation provides a promising,powerful computational tool for investigating the harder to work with real Mellin amplitudes.Part of our motivation for studying p -adic Mellin space is the hope that the simplicityand tractability of the p -adic setting may lead to new inspiration and results for real Mellinamplitudes. We should like to think of the relation between real Mellin amplitudes and thecorresponding real pre-amplitudes from section 4.1 as an instantiation of this hope; it may43e worthwhile to also investigate whether such relations exist for spinning amplitudes andloop amplitudes.Another promising avenue is the study of p -adic Mellin amplitudes involving bulk fermions [39]and amplitudes at loop level. It is tempting to note that the real Mellin loop-amplitudes,especially in the Mellin-Barnes integral form for pre-amplitudes [10, 11], are expressible interms of the local zeta function ζ ∞ , which naturally suggests analogous candidates for p -adicloop amplitudes. It would be interesting to explore this connection more systematically. Wealso wonder whether (possibly a generalization of) the recursion relations put forward in thispaper extend to loop amplitudes.In this paper we have restricted ourselves to computing Mellin amplitudes associated toindividual bulk diagrams. It remains to see whether it is possible to lift the on-shell recursionof section 5, which is applicable at the level of individual diagrams, to one which appliesto total amplitudes, i.e. the sums over constitutive diagrams of a bulk theory, and to otherbulk theories over reals. Nevertheless, the techniques of this paper should still prove usefulin extending the construction of the putative bulk dual of the free p -adic O ( N ) model, atleast at tree-level [30].On a closing note, we emphasize that at present the mounting number of remarkablesimilarities between the conventional (real) and p -adic AdS/CFT formulations are still math-ematical curiosities which remain to be fully understood. It is tempting to wonder whetherone may think of the real and p -adic formulations as “local formulations” of AdS/CFT atthe place at infinity and the finite places, respectively. This immediately leads to the ques-tion [46]: Is there an adelic formulation of AdS/CFT, i.e. a “global” formulation which sub-sumes the real and p -adic formulations in a single field-independent (field in the mathematicssense) framework? Such a unified formulation would constitute an important conceptual stepforward in understanding as yet undiscovered mathematical aspects of AdS/CFT. Acknowledgments C. B. J. and S. P. thank Steven S. Gubser, Matilde Marcolli, and Brian Trundy for usefuldiscussions and encouragement. S. P. thanks Perimeter Institute for their kind hospitalitywhile this work was in progress. The work of C. B. J. was supported in part by the De-partment of Energy under Grant No. DE-FG02-91ER40671, by the US NSF under GrantNo. PHY-1620059, and by the Simons Foundation, Grant 511167 (SSG). The work of S. P.was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter44nstitute is supported by the Government of Canada through the Department of Innovation,Science and Economic Development and by the Province of Ontario through the Ministry ofResearch, Innovation and Science. A Useful Formulae and Conventions We now recall some useful functions and formulae from Ref. [4], to which we refer for furtherdetails. The characteristic function takes a p -adic argument, and is defined to be γ p ( x ) = 2 log p πi (cid:90) ϵ + iπ p ϵ − iπ p dγ ζ p (2 γ ) | x | γp = | x | p ≤ , p -adic variables. If we let { S , ..., S N } denote afinite set of p -adic numbers and let m be an index such that | S m | p = sup {| S | p , ..., | S N | p } , thenit follows from the definition of the characteristic function that it satisfies the factorizationproperty γ p ( S ) ...γ p ( S N ) = γ p (cid:0) S m (cid:1) . (A.2)Furthermore, letting { x , ..., x N } denote another set of p -adic numbers of equal cardinality,it follows from the ultra-metricity of the p -adic norm that N (cid:89) i =1 i ̸ = m γ p (cid:0) S i ( x i − x m ) (cid:1) = (cid:89) ≤ i B.1 Outline of the proof In this section we prove that at tree-level any p -adic Mellin amplitude is given in terms ofa multiple contour integral over a pre-amplitude according to prescription III of section 4.Subsequently, in the next appendix we prove that the performing all the contour integrals46ver a pre-amplitude reproduces the Mellin amplitude given by the recursive prescription Ipresented in section 3.1.The proof for the pre-amplitude prescription is inductive. A possible inductive approachone might attempt to adopt is weak induction on the number of internal legs. However,given the pre-amplitude for an arbitrary bulk diagram, it turns out to be technically difficultto find the pre-amplitude for a diagram obtained by inserting an extra internal leg at anarbitrary position in the diagram. It is significantly more tractable to attach extra internallegs to a special kind of bulk vertex, one that only has one internal leg attached to it. Thusour inductive strategy will be to assume that the pre-amplitude prescription III is satisfiedby an arbitrary bulk diagram D L , and a bulk diagram D R built from f internal lines allconnected to the same vertex: D L = V R , D R = R R R f . (B.1)Here the grey circles each indicate an arbitrary number of vertices and internal lines thatform part of the diagram D L , and we have omitted drawing the external lines, of which anarbitrary number may be connected to any vertex. We will then consider the bulk diagram D M obtained by merging the two diagrams by fusing the two vertices labelled R in (B.1),to get D M = V R R R f , (B.2)where the merged diagram has a set of external legs which is the union of the set of ex-ternal legs for D L and D R . Using the inductive assumptions we will demonstrate that this47ulk-diagram has a Mellin amplitude given by the pre-amplitude prescription III, therebycompleting the inductive proof. B.2 Details of the proof The “left-hand diagram” D L . As a first step in the proof, we will recast the pre-amplitude for D L in a more useful form. To this end, it is useful to first consider a smallerdiagram D l , namely the one obtained by removing the vertex R and the internal line con-nected to it (i.e. collapsing the internal line joining R to V in D L ). We keep the numberand dimensions of external legs attached to vertex V in D l arbitrary for now but fix themlater. The smaller diagram takes the following form, where this time we show a subset ofthe external legs, D l = V ... i V ... i V I ... i V g ... i V J ... i V K V I V g V J V K V R + c − cV R + c + c f + c − c R − c R f − c f R + c + c f R − c R f − c f . (B.3)The vertices of this diagram are labelled V , V , . . . , V g , and the external legs are labelled bythe lower-case alphabet i or j , where we sometimes put a subscript on them to indicate whichvertex they are incident on. Specifically the label i V I runs over the external legs incidenton vertex V I , and i V runs over all labels i V , i V , . . . , i V g . We posit that the position spacepre-amplitude (cid:101) A l for D l , denoted with a tilde and defined via A ( x ) ≡ (cid:32)(cid:89) I (cid:90) dc πi f ∆ I ( c I ) (cid:33) (cid:101) A ( x ) , (B.4)48an be written in the form (cid:101) A l = L (cid:34) g − (cid:89) L =1 (cid:90) Q p dt L | t L | p (cid:35) G ( { t L } ) (cid:88) a ∈{ ,p } (cid:34)(cid:89) i V ζ p (1) | | p (cid:90) a Q p ds i V | s i V | p | s i V | (cid:80) ∆ iV p (cid:35) × (cid:89) ≤ I ≤ J ≤ g (cid:89) i VI ,i VJ γ p (cid:18) f IJ ( { t L } ) s i VI s i VJ x i VI i VJ (cid:19) (B.5)where G ( { t L } ) and f IJ ( { t L } ) are, respectively, real and p -adic valued functions whose exactform we will not need, except for the fact that the functions f IJ are valued in Q p . L and G ( { t L } ) also depend on external scaling dimensions ∆ i V , the degree of the field extension n ,and complex variables c I , but we suppress such dependencies. The fact that the (cid:101) A l has theform (B.5) can be seen inductively. For the initial step, we note that equations (4.34) and(4.48) in Ref. [4] show that the diagrams with one and two internal legs have a position spacepre-amplitude of the form (B.5). We will now assume (B.5) and show that the diagram D L (with one additional internal leg) has a pre-amplitude of the same form as well.Applying the split representation (2.6) to the internal leg connecting vertices V and R in D L , D L = (cid:90) dc πi f ∆ ( c ) (cid:90) P ( Q pn ) dx V R ...i V ... x i R h − c h + c i V I i V g i V J i V K ... ...... ... , (B.6)we find that the position space pre-amplitude (cid:101) A L is given by a boundary integral over theproduct of 1) the pre-amplitude (cid:101) A l as written in (B.5) with appropriately chosen externaldimensions incident at V , and 2) a contact amplitude associated to the vertex R . We fix thedimension of one of the external legs incident at V in D l to a complex-shifted value consistent We are allowing for the possibility of vertices V I with no external legs incident to them. Such verticeshave no associated variables s i VI as i V I runs over the empty set. But the total number of t L variables remains2 g − g is the total number of vertices, with or without incident external legs. V in D L . Label the complex-shifted dimension ∆ .We define (cid:101) L = L (cid:12)(cid:12) ∆ → h − c , (cid:101) G ( { t L } ) = G ( { t L } ) (cid:12)(cid:12) ∆ → h − c , (B.7)and rename the integration variable s associated to this external leg to t . Furthermore,we will now take the set { i V } to not include the index i = 1 associated to this leg. Forthe contact amplitude associated to the vertex R , we apply the identity (A.11). Again wemust assign a complex value to the dimension of an external leg consistent with the splitrepresentation, and we call the integration variable associated with this external leg u , whilethe other external legs (with real scaling dimensions) are labelled by an index i R . The splitrepresentation then leads to (cid:101) A L = (cid:90) Q pn dx (cid:101) L (cid:20) g − (cid:89) L =1 (cid:90) Q p dt L | t L | p (cid:21) (cid:101) G ( { t L } ) × (cid:88) a ∈{ ,p } (cid:20) (cid:89) i V ζ p (1) | | p (cid:90) a Q p ds i V | s i V | p | s i V | (cid:80) ∆ iV p (cid:21) ζ p (1) | | p (cid:90) a Q p dt | t | p | t | h − cp × (cid:20) (cid:89) ≤ I ≤ J ≤ g (cid:89) i VI ,i VJ γ p (cid:18) f IJ ( t L ) s i VI s i VJ x i VI i VJ (cid:19)(cid:21) × (cid:20) (cid:89) ≤ I ≤ g (cid:89) i VI γ p (cid:0) f I ( t L ) ts i VI ( x − x i VI ) (cid:1)(cid:21) × ζ p (cid:0) (cid:88) ∆ i R − h + c (cid:1) × (cid:88) b ∈{ ,p } (cid:20) (cid:89) i R ζ p (1) | | p (cid:90) b Q p ds i R | s i R | p | s i R | (cid:80) ∆ iR p (cid:21) ζ p (1) | | p (cid:90) b Q p du | u | p | u | h + cp × (cid:20) (cid:89) i R In order to obtain a p -adic Mellin amplitude from a pre-amplitude one must carry outa contour integral around a cylindrical manifold for each internal leg of the diagram. Insection 4.3 we proved that such contour integrals over pre-amplitudes, given according toprescription III of section 4, correctly reproduce Mellin amplitudes as defined in (2.12).What remains to be shown to demonstrate that the recursion relations of section 3.1 aretrue is that carrying out the contour integrals over the pre-amplitudes exactly reproducesthe Mellin amplitudes given by prescription I. In this appendix we prove this for arbitrarytree-level bulk diagrams.Consider first the exchange amplitude. The fact that carrying out the contour integralover its pre-amplitude yields the appropriate Mellin amplitude (see (4.16)) relies mathemat-61cally on the following identity:log p πi (cid:90) iπ log p − iπ log p dc ζ p ( A + c ) ζ p ( A − c ) ζ p ( B − c ) ζ p ( C + c ) ζ p (2 c ) ζ p ( − c ) × (cid:18) ζ p ( D − c ) + ζ p ( B + c ) − (cid:19)(cid:18) ζ p ( D + c ) + ζ p ( C − c ) − (cid:19) = 2 ζ p ( A + B ) ζ p ( A + C ) (cid:18) ζ p ( B + C ) + ζ p ( A + D ) − (cid:19) , (C.1)where A, B, C, D > 0. This identity is a rewriting of equation (4.39) in Ref. [4].For the bulk diagram with two internal lines, the fact that carrying out the contourintegral over the pre-amplitude yields the desired Mellin amplitude (see (4.17)) follows from(C.1) and the following identity:log p πi (cid:90) iπ log p − iπ log p dc ζ p ( A + c ) ζ p ( A − c ) ζ p ( B − c ) ζ p ( C + c ) ζ p (2 c ) ζ p ( − c ) (cid:18) ζ p ( D − c ) + ζ p ( B + c ) − (cid:19) × (cid:18) ζ p ( E + c ) ζ p ( D + c ) + ζ p ( C − c ) ζ p ( E − c ) + ζ p ( E + c ) ζ p ( E − c ) − ζ p ( E + c ) − ζ p ( E − c ) (cid:19) = 2 ζ p ( A + B ) ζ p ( A + C ) (cid:18) ζ p ( B + C ) ζ p ( B + E ) + ζ p ( A + E ) ζ p ( A + D )+ ζ p ( A + E ) ζ p ( B + E ) − ζ p ( A + E ) − ζ p ( B + E ) (cid:19) . (C.2)In fact in general, the fact that the pre-amplitudes given by prescription III of section 4integrate to the appropriate Mellin amplitudes given by prescription I of section 3.1 hingesmathematically on an infinite tower of increasingly convoluted contour integral identities forthe local zeta function ζ p . In the diagrammatic notation we have employed in this paper,it turns out these integral identities have a simple interpretation, expressed in the form ofidentity (4.24). In fact such a diagrammatic interpretation holds true for both real as wellas p -adic Mellin amplitudes. In this appendix we will prove identity (4.24) over the p -adics. The factors of two on the r.h.s sides of equations (C.1) and (C.2) arise non-trivially: If one computesthe contour integrals on the left-hand sides by summing over the residues, it is only the full sum of residuesthat conspire to yield two times a sum of zeta functions. This holds true for all identities in the previouslymentioned infinite tower of identities. .1 The “leg adding” operation An important part of the proof involves showing there exists a mathematical operationthat adds an extra internal line to an undressed diagram, except the operation should onlyoutput terms that do not depend on the Mandelstam variable of the added internal line.The previous statement will become clear after considering a few examples. In the simplestexample the starting point is the undressed diagram with no internal lines, which by definitionequals unity, and after the “leg adding” operation one obtains an undressed diagram in aparticular limit: (cid:88) z ∗ ∈{ ∆ , ∆ iR } Res z ∗ (cid:34) ζ p (cid:0) ∆ i L + z − n (cid:1) × (1) × ζ p (cid:0) ∆ i R − z (cid:1) ζ p (cid:0) ∆ − z (cid:1)(cid:35) = − ζ p (cid:0) ∆ + ∆ i L − n (cid:1) lim s →−∞ (cid:32) ∆ L s L z ... i L ∆ L s L . . .i ∆ R s R ... i L ... i R ∆ s ... i L ... i R zs U ∆ U s D ∆ D .. .i U .. .i D ... i R s U ∆ U s D ∆ D .. .i U .. .i D ∆ R s R (cid:33) . (C.3)Here as elsewhere, we have shortened notation by omitting a summation symbol that isimplied, eg. ∆ i R ≡ (cid:80) i R ∆ i R . The limit s → −∞ kills off the momentum dependent factor ζ p ( s − ∆) of the undressed diagram. The l.h.s. above is symmetric w.r.t. ∆ and ∆ i R , and sothe same is true for the r.h.s., although we have chosen to write it in a way that does notmake this symmetry manifest.The above example is almost too simple to be informative, so let us consider the actionof the “leg adding operation” on a diagram that has one internal line to begin with: (cid:88) z ∗ ∈{ ∆ R , ∆ iR } Res z ∗ (cid:34) ζ p (cid:0) ∆ i U + ∆ L + z − n (cid:1) ∆ L s L ... i L z. . .i U ∆ L s L . . .i U ∆ R s R ... i L ... i R s A ∆ A s B ∆ B ∆ C s C ... i L ... i R ... i U ... i D s A ∆ A s B ∆ B z ... i L ... i R ... i U ζ p (cid:0) ∆ i R − z (cid:1) ζ p (cid:0) ∆ R − z (cid:1)(cid:35) = − ζ p (cid:0) ∆ L + ∆ R + ∆ i U − n (cid:1) lim s R →−∞ ∆ L s L z ... i L . . .i U ∆ L s L . . .i U ∆ R s R ... i L ... i R s A ∆ A s B ∆ B ∆ C s C ... i L ... i R ... i U ... i D s A ∆ A s B ∆ B z ... i L ... i R ... i U . (C.4)Consider one final example, where we now attach an extra internal line onto a vertex that63lready has two internal lines incident on it: (cid:88) z ∗ ∈{ ∆ C , ∆ iD } Res z ∗ (cid:34) ζ p (cid:0) ∆ i U + ∆ A + ∆ B + z − n (cid:1) ∆ L s L z ... i L . . .i U ∆ L s L . . .i U ∆ R s R ... i L ... i R s A ∆ A s B ∆ B ∆ C s C ... i L ... i R ... i U ... i D s A ∆ A s B ∆ B z ... i L ... i R ... i U × ζ p (cid:0) ∆ i D − z (cid:1) ζ p (cid:0) ∆ C − z (cid:1)(cid:35) = − ζ p (cid:0) ∆ A + ∆ B + ∆ C + ∆ i U − n (cid:1) lim s C →−∞ s A ∆ A s B ∆ B ∆ C s C ... i L ... i R ... i U ... i D s A ∆ A s B ∆ B z ... i L ... i R ... i U . (C.5)These examples demonstrate how the operation of “adding an extra internal line” works.Assuming that such an operation exists for any undressed diagram, it is easy to provethat the type of contour integral displayed in the preceding subsection can be employed toglue together any two undressed diagrams. This proof is carried out in the next subsection.Then, in the subsection after the next, we justify the above-mentioned assumption by provinginductively that the “internal leg adding operation” can be applied to any undressed diagram,and in so doing we complete the proof of the recursive prescription I. C.2 Integrating pre-amplitudes using “leg addition” Given an arbitrary undressed diagram parametrized by the complex number c , D ≡ . i ... i .. DC E F ... i ... i ... i ... i B A h − cs h + c , (C.6)64e need to show that if it appears as part of a pre-amplitude as follows: (cid:102) M = F ... i A h − c (cid:32) h + ch − c s ... i (cid:33) (cid:32) h + c ... i (cid:33) D , (C.7)then the contour integral over c as described below effectively glues together the two un-dressed diagrams. That is, M = (cid:90) iπ log p − iπ log p dc πi f ∆ ( c ) (cid:102) M = F ... i A ∆ . i ... i .. DC E F ... i ... i ... i ... i B A ∆ s ... i (cid:32) ∆ ... i (cid:33) , (C.8)where f ∆ ( c ) is given by (2.8). We will show this inductively, by first assuming that the gluingof an extra internal line onto a diagram works for any diagram with fewer internal lines thanthe arbitrary diagram D we started with.Now, split the undressed diagram into two parts: D = D + D , where D consists of allthe terms in D that are proportional to an internal line factor, say ζ p ( s E − ∆ E ), while D consists of all the remaining terms of D . That is, D = − ζ p ( s E − ∆ E ) . i ... i .. DC E F ... i ... i ... i B A h − cs h + c , (C.9)where we used the factorization property of undressed diagrams discussed in section 5.1, and65 = lim s E →−∞ D = lim s E →−∞ . i ... i .. DC E F ... i ... i ... i ... i B A h − cs h + c = − (cid:88) z ∗ ∈{ ∆ E , ∆ i } Res z ∗ (cid:34) ζ p (cid:0) ∆ A + ∆ B + ∆ i + z − n (cid:1) . i ... i .. DC z F ... i ... i ... i B A h − cs h + c × ζ p (cid:0) ∆ i − z (cid:1) ζ p (cid:0) ∆ E − z (cid:1)(cid:35) ζ p (cid:0) ∆ A + ∆ B + ∆ E + ∆ i − n (cid:1) , (C.10)where we used the “leg adding operation” from the previous subsection to write the finalequality above. (The validity of such a leg adding operation is proven in full generality inthe next subsection.)Essentially, the point of this exercise was to re-express D and D (and thus in turn D ) in terms of a diagram with one fewer internal line than the original undressed diagram D . Having done that, we can now carry out the contour integral in (C.8) by breaking (cid:102) M into two terms, each of which, using the assumption of our inductive setup, admits an extrainternal line to be glued on.Explicitly, splitting up the integrated amplitude M = M + M into the parts com-ing from D and D , respectively, the inductive assumption allows us to straightforwardlyevaluate the two parts. The first part is given by66 ≡ (cid:90) iπ log p − iπ log p dc πi f ∆ ( c ) F ... i A h − c (cid:32) h + ch − c s ... i R (cid:33) (cid:32) h + c ... i (cid:33) D = − F ... i A ∆ ζ p ( s E − ∆ E ) . i ... i .. DC E F ... i ... i ... i B A ∆ s ... i × (cid:32) ∆ ... i (cid:33) . (C.11)And the second part is given by M ≡ (cid:90) iπ log p − iπ log p dc πi f ∆ ( c ) F ... i A h − c (cid:32) h + ch − c s ... i (cid:33) (cid:32) h + c ... i (cid:33) D , (C.12)which, using the leg adding operation, we may re-express as67 = − (cid:88) z ∗ ∈{ ∆ E , ∆ i } Res z ∗ (cid:34) ζ p (cid:0) ∆ A + ∆ B + ∆ i + z − n (cid:1) (cid:90) iπ log p − iπ log p dc π f ∆ ( c ) F ... i A h − c × (cid:32) h + ch − c s ... i (cid:33) (cid:32) h + c ... i (cid:33) × . i ... i .. DC z F ... i ... i ... i B A h − cs h + c × ζ p (cid:0) ∆ i − z (cid:1) ζ p (cid:0) ∆ E − z (cid:1)(cid:35) ζ p (cid:0) ∆ A + ∆ B + ∆ E + ∆ i − n (cid:1) . (C.13)Here we have made use of the fact that the leg adding operation commutes with the contourintegral so that we may choose to first carry out the contour integral, which is easily doneusing the inductive assumption: M = − (cid:88) z ∗ ∈{ ∆ E , ∆ i } Res z ∗ (cid:34) ζ p (cid:0) ∆ A + ∆ B + ∆ i + z − n (cid:1) × n − c n + c ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s ∆ ... i n + c ... i A n − cF ... i ∆ ... i A ∆ F ... i DC.. .i .. .i B . . .i E FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i FA . . .i . . .i s ∆ ... i DC.. .i .. .i B . . .i z FA . . .i . . .i s n − c n + cDC.. .i .. .i B z. . .i FA . . .i . . .i s ∆ ... i (cid:18) n − c n + c ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s ∆ ... i n + c ... i A n − cF ... i ∆ ... i A ∆ F ... i (cid:19) × ζ p (cid:0) ∆ i − z (cid:1) ζ p (cid:0) ∆ E − z (cid:1)(cid:35) ζ p (cid:0) ∆ A + ∆ B + ∆ E + ∆ i − n (cid:1) . (C.14)At this point we can use the leg adding operation as described in the previous subsection to68e-express M in terms of a larger diagram: M = n − c n + c ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s ∆ ... i n + c ... i A n − cF ... i ∆ ... i A ∆ F ... i lim s E →−∞ n − c n + c ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s ∆ ... i n + c ... i A n − cF ... i ∆ ... i A ∆ F ... i (cid:18) n − c n + c ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s ∆ ... i n + c ... i A n − cF ... i ∆ ... i A ∆ F ... i (cid:19) . (C.15)Comparing the final expressions for M and M , we see that up to a common pre-factor, M is given by the s E -independent terms of a diagram, and M is given by the s E -dependentterms of the same diagram. We thus conclude that their sum is equal to M = n − c n + c ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s ∆ ... i n + c ... i A n − cF ... i ∆ ... i A ∆ F ... i n − c n + c ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s ∆ ... i n + c ... i A n − cF ... i ∆ ... i A ∆ F ... i (cid:18) n − c n + c ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s n − c n + cDC.. .i .. .i B E. . .i . . .i FA . . .i . . .i s ∆ ... i n + c ... i A n − cF ... i ∆ ... i A ∆ F ... i (cid:19) , (C.16)which is precisely what we wanted to show in (C.8). C.3 Proof of the general “leg adding” operation The last task that remains to be accomplished to have a complete proof of the recursiveprescription for p -adic Mellin amplitudes is to show that the leg adding operation worksgenerally. That is, given an arbitrary undressed diagram D ≡ DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z A , (C.17)we need to show that we can obtain a diagram with an extra internal leg (more precisely, thepart of the diagram that does not depend on a particular Mandelstam variable), by takingan appropriate sum of residues of D . Concretely, we need to show that the following sum of69esidues, B ≡ (cid:88) z ∗ ∈{ ∆ E , ∆ i } Res z ∗ (cid:34) ζ p (cid:0) ∆ A + ∆ B + ∆ i + z − n (cid:1) DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z A × ζ p (cid:0) ∆ i − z (cid:1) ζ p (cid:0) ∆ E − z (cid:1)(cid:35) (C.18)is equal to the following limit, C ≡ − ζ p (cid:0) ∆ A + ∆ B + ∆ E + ∆ i − n (cid:1) lim s E →−∞ DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i . (C.19)We will show this by strong induction on the number of internal lines. The idea is to applythe formula (5.8) to the undressed diagram in equation (C.18) in order to re-cast it in termsof diagrams with fewer internal lines and then use induction to evaluate the contribution to B from these smaller diagrams. Specifically, applying the formula (5.8) to the diagram in(C.18), we may decompose B as B = − B A − B B − B C − B D − B F − B G − B AB − B AC − B AD − · · · − B F G − B ABC − B ABD − B ABF − · · · − B DF G ... − B ABCDF G , (C.20)70here B I i ...I if represents the contribution to B arising from the term in the sum in (5.8)which has l = f and I = I i , . . . , I f = I i f . For example, B A = (cid:88) z ∗ ∈{ ∆ E , ∆ i } Res z ∗ (cid:34) ζ p (cid:0) ∆ A + ∆ B + ∆ i + z − n (cid:1) ζ p (cid:0) ∆ i − z (cid:1) ζ p (cid:0) ∆ E − z (cid:1) × (cid:104) ζ p ( s A − ∆ A ) − ζ p (∆ i i i i i i , + z − n ) lim ∀ s →∞ (cid:105) DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z A × DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i (cid:35) , (C.21)where we are using the convention,∆ i ...i k ,i k +1 ...i ℓ ≡ k (cid:88) j =1 ∆ i j − ℓ (cid:88) j = k +1 ∆ i j . (C.22)so that ∆ i i i i i i , = ∆ i + ∆ i + ∆ i + ∆ i + ∆ i + ∆ i . (C.23)and summation is implied over each of the terms on the r.h.s. It may be remarked rightaway that we can decompose C in a similar manner: C = − C A − C B − C C − C D − C E − C F − C G − C AB − C AC − C AD − · · · − C F G − C ABC − C ABD − C ABE − · · · − C EF G ... − C ABCDEF G , (C.24)where, introducing the definition Σ ≡ ∆ i i i i i i i , − n , (C.25)71he term C A , for example, is given by C A = − ζ p (∆ ABEi , − n ) lim s E →−∞ (cid:2) ζ p ( s A − ∆ A ) − ζ p (Σ) lim ∀ s →∞ (cid:3) DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i . (C.26)The two limits s E → −∞ and ∀ s → ∞ do not commute, since any term containing a factorof ζ p ( s E − ∆ E ) is killed off if the s E → −∞ is taken first but not if it is taken last. We cantherefore commute the two limits only if we add back in these terms. That is, we may write C A = C A, + C A, , where C A, = − ζ p (∆ ABEi , − n ) (cid:2) ζ p ( s A − ∆ A ) − ζ p (Σ) lim ∀ s →∞ (cid:3) lim s E →−∞ DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i , (C.27) C A, = − ζ p (∆ ABEi , − n ) ζ p (Σ) lim ∀ s →∞ DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z ADC.. .i .. .i B . . .i E A DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i . (C.28)In equation (C.24) we get extra terms as compared to (C.20) because the diagram in(C.19) has an extra internal line, but we should remember that the s E → −∞ limit in(C.19) kills off the part of any term C I ...I f that contains a factor of ζ p ( s E − ∆ E ).72et us now look more closely at the contribution B A . It is straightforward to check that ζ p (∆ i − z ) ζ p (∆ i i i i i i , + z − n ) = ζ p (Σ) (cid:18) ζ p (∆ i − z ) + ζ p (∆ i i i i i i , + z − n ) − (cid:19) . (C.29)Using this, we write B A = B A, + B A, , where the first part is given by B A, = (cid:88) z ∗ ∈{ ∆ E , ∆ i } Res z ∗ (cid:34) ζ p (cid:0) ∆ A + ∆ B + ∆ i + z − n (cid:1) ζ p (cid:0) ∆ i − z (cid:1) ζ p (cid:0) ∆ E − z (cid:1) × (cid:104) ζ p ( s A − ∆ A ) − ζ p (Σ) lim ∀ s →∞ (cid:105) DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z A DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i (cid:35) , (C.30)and the second part of B A is given by B A, = − (cid:88) z ∗ ∈{ ∆ E , ∆ i } Res z ∗ (cid:34) ζ p (cid:0) ∆ A + ∆ B + ∆ i + z − n (cid:1) ζ p (cid:0) ∆ E − z (cid:1) × ζ p (Σ) (cid:18) ζ p (∆ i i i i i i , + z − n ) − (cid:19) lim ∀ s →∞ DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z A DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i (cid:35) . (C.31)Using the fact that the s → ∞ limit commutes with the sum over residues and invoking theinductive assumption, the first part is seen to equal B A, = − ζ p (∆ A + ∆ B + ∆ E + ∆ i − n ) × (cid:104) ζ p ( s A − ∆ A ) − ζ p (Σ) lim ∀ s →∞ (cid:105) lim s E →−∞ DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i . (C.32)We notice that B A, is exactly equal to C A, . And this observation does not depend on anyspecial property of B A not shared with any of the other terms on the r.h.s. of (C.20): we73ay decompose any term B I ...I f on the r.h.s. of (C.20) into two parts, called B I ...I f , and B I ...I f , , such that B I ...I f , = C I ...I f , . So we have now matched some of (C.18) with someof (C.19).What remains to be shown is that the remaining part of B is equal to the remaining partof C . The remaining part of C consists of 1) terms C I ...I f , where I i ̸ = for all i ∈ { , , . . . , f } and, 2) terms C I ...I f for which I i = E for some i ∈ { , , . . . , f } . However, note that thesum of all these terms admits a simple diagrammatic interpretation, as we now explain. Itwas pointed out in section 5.1 that for any undressed diagram, the momentum-independentterms are in one-to-one correspondence with the terms proportional to internal line factors ζ p ( s − ∆). Thus in (C.19) when we take the limit s E → −∞ , although we kill off the termsproportional to ζ p ( s E − ∆ E ), we do not kill off the momentum-independent terms whichare in one-to-one correspondence with these killed-off terms. It is exactly these surviving,unpartnered momentum-independent terms that we have yet to account for. But as theseremaining terms stand in one-to-one correspondence with the killed-off terms containing thepropagator factor ζ p ( s E − ∆), the factorization property of undressed diagrams from section5.1 allows us to express these remaining unaccounted terms of C as ζ p (cid:0) ∆ A + ∆ B + ∆ E + ∆ i − n (cid:1) ζ p (Σ) lim ∀ s →∞ DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i E FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z ADC.. .i .. .i B . . .i E A . (C.33)Let’s return to the second part of B A . We see in the expression for B A, in (C.31) thatthere is no longer a pole at z = ∆ i , so that B A, is simply equal to the residue at ∆ E . Thus,we have B A, = − ζ p (cid:0) ∆ A + ∆ B + ∆ E + ∆ i − n (cid:1) ζ p (Σ) × (cid:18) − ζ p (∆ i i i i i i , + ∆ E − n ) (cid:19) lim ∀ s →∞ DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z ADC.. .i .. .i B . . .i E A DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i . (C.34)74e can rewrite this expression in a more suggestive form as B A, = − ζ p (cid:0) ∆ A + ∆ B + ∆ E + ∆ i − n (cid:1) ζ p (Σ) × lim ∀ s →∞ (cid:18) ζ p ( s A − ∆ A ) − ζ p (∆ i i i i i i , + ∆ E − n ) (cid:19) DC.. .i .. .i B . . .i FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z FA . . .i . . .i G ... i DC.. .i .. .i B . . .i z ADC.. .i .. .i B . . .i E A × DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i G ... i DC.. .i .. .i B E. . .i . . .i FA . . .i . . .i GDC.. .i .. .i B E. . .i . . .i AFA . . .i . . .i G ... i , (C.35)where now the symbol ∀ s → ∞ also includes the limit s A → ∞ . We recognize the aboveas exactly the contribution from the l = 1 , I = A term if we apply the formula (5.8) to theundressed diagram in (C.33). Again there is nothing special about the B A term. In generalwhen we split a term B I i ...I if on the right-hand side of (C.21) into B I i ...I if , and B I i ...I if , ,the second term B I i ...I if , exactly reproduces the term with l = f , I = I i , . . . , I f = I i f onapplying the formula (5.8) to (C.33). We conclude that the sum over these contributionscoming from all the terms on the right-hand side of (C.20), that is, the sum − B A, − B B, − B C, − B D, − B F, − B G, − B AB, − B AC, − B AD, − · · · − B F G, − B ABC, − B ABD, − B ABF, − · · · − B DF G, ... − B ABCDF G, (C.36)is exactly equal to (C.33), which shows that B exactly reproduces all of C .This concludes the proof of the leg adding operation, and in turn proves the recursionprescription I from section 3. 75 BCFW-Shifts of Auxiliary Momenta In this appendix, we provide an alternative but equivalent form of the recursive prescription(5.11) from section 5, in auxiliary momentum space. The p -adic Mellin amplitude of a tree-level bulk diagram with external legs labeled { , . . . N } and auxiliary momenta (defined insection 2) labeled { k , . . . , k N } , can be written as M = − (cid:88) partitions I M ( I, − ˆ k I )( z I ) × (propagator) × M ( I c , ˆ k I )( z I ) , (D.1)where the partition of external legs, I is a subset I ⊂ { , . . . N } such that it can be obtainedby splitting the diagram into two by cutting across any chosen internal line, and I c is thecomplement of I . The hatted momenta are defined to be ˆ k I ≡ − (cid:80) a ∈ I k a ( z ), where k a ( z )are the complex-shifted external momenta which are on-shell at all values of z ∈ C , while ˆ k I goes on-shell only at z = z I .The amplitude M ( I, − ˆ k I )( z ) is one of the z -dependent sub-amplitudes obtained by split-ting the original diagram into two by cutting across the chosen internal line, and the factorof “(propagator)” is the Mellin space “propagator” β p ( − k I − ∆ I , n − (cid:80) N i =1 ∆ i ) defined in(2.24), where the unhatted momentum k I = − (cid:80) a ∈ I k a is constructed out of un-shifted ex-ternal momentum variables, and ∆ I is the conformal dimension of the bulk field propagatingalong the chosen internal line. From here on, we will sometimes refer to the on-shell recursion above as BCFW-typerecursion because of the similarity of (D.1) with the usual BCFW decomposition, but westress that we are working at the level of individual diagrams. We note that the recursionin (D.1) is identical to the recursion relation (5.11) except (5.11) is written without anydirect reference to auxiliary momenta. We now explain this recursive prescription in detailwith the help of two illustrative examples. D.1 Four-point exchange diagram We start by demonstrating the BCFW-type recursion (D.1) for the simplest case, the four-point function. The full tree-level four-point bulk (Witten) diagram for a theory with cubic Here we have made the implicit assumption that all internal lines get complex-shifted – to ensure thisis true in general, a “multi-line BCFW-shift” of the external momenta may be required. The final Mellinamplitude M will be independent of the choice of the particular BCFW-shift employed, as long as all internallines develop a z -dependence and consequently can be put on-shell at particular values of z . ∼ + + + , (D.2)where we must sum over all fields which can propagate in the intermediate channels in the s -, t - and u -channel exchange diagrams. We will be suppressing coupling constants and anysymmetry factors associated with the diagrams. These details depend on the specifics of thebulk theory, particularly the bulk Lagrangian. We now show that each individual diagramin (D.2) in Mellin space admits a BCFW-type recursion relation, with the basic buildingblocks of the recursion being the contact diagrams. Consequently, we’ll find the four-pointcontact diagram in (D.2) is already in its ‘maximally reduced’ form.Let’s focus on the s -channel exchange where a scalar field dual to a (single-trace) operatorof conformal dimension ∆ A propagates along the internal leg. It turns out the form of theMellin amplitude for the ( s -channel) four-point exchange diagram as written in example (iii)is, rather trivially, already in the desired BCFW-type on-shell recursion form. However inthis subsection, we will take the time to describe in detail the various ingredients which gointo recognizing the BCFW-type recursion structure, as they illuminate how this works withmore complicated higher-point diagrams.Define M − int∆ , ;∆ , ( s A , ∆ A ) ≡ ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 = − V ∆ A, V ∆ A, β p (cid:32) s A − ∆ A , n − (cid:88) i =1 ∆ i (cid:33) , (D.3)where the vertex factor V ∆ A, , defined using the convention (C.22), via (3.1), V (∆ ...f, ) ≡ ζ p (∆ ...f, − n ) , (D.4)is the Mellin amplitude associated with the 3-point contact diagram between operators of A similar analysis can be done for the t - and u -channel exchange diagrams. , ∆ and ∆ A . Further, we have defined s A ≡ ∆ + ∆ − γ = ∆ + ∆ − γ , (D.5)and β p is defined in (2.24). To go to the auxiliary momentum space, described in section2, we assign to each external leg of the four-point s -channel exchange diagram an auxiliarymomentum, as shown below: M − int∆ , ;∆ , ( s A , ∆ A ) = k A ∆ A k k k k ≡ M − int − k − k ; − k − k (cid:0) − k A , ∆ A (cid:1) , (D.6)where the external momenta are on-shell, − k i = ∆ i i = 1 , . . . , , (D.7)while the internal momentum is not, and momentum is conserved at each vertex. We aresuppressing the Lorentz indices on the momentum variables. To obtain a BCFW-type rela-tion for the exchange diagram, we begin by complexifying some of the external momenta insuch a way that the complex-shifted external legs remain on-shell and still satisfy momentumconservation at each vertex for all values of the complex parameter, and as a result of theshift the momentum along the internal leg gets complex-shifted as well. For instance, wemay choose the following “two-line shift” that preserves momentum conservation, k → k ( z ) ≡ k + zq , k → k ( z ) ≡ k − zq , (D.8)where z ∈ C , and q is a constrained momentum variable which will in general be complex-valued. The constraints on q arise from the following on-shell requirements, − k ( z ) = ∆ , − k ( z ) = ∆ , ∀ z ∈ C . (D.9)As desired, the momentum running through the internal leg gets complex-shifted as well, k A → k A ( z ) ≡ k ( z ) + k . (D.10)78he conditions (D.9) are met if q satisfies q = 0 q · k = 0 q · k = 0 . (D.11)Momentum conservation along with the constraints above then imply q · k = − q · k . (D.12)Owing to the fact that the auxiliary momenta k i are not null, it is in general not possible tospecify in a Lorentz covariant manner the momentum variable q satisfying (D.11). However,the explicit form for q will not be important in the following; q will enter in the calculationsvia its dot-product with various momenta and it will be sufficient for us to inquire how thesedot-products are related to each other without establishing the explicit form for any of them.The complex-shifts (D.8) transform the Mandelstam invariant s A as well, s A → s A ( z ) ≡ − ( k ( z ) + k ) = s A − zq · k = − q · k ( z − z A ) + ∆ A , (D.13)where we have defined z A ≡ s A − ∆ A q · k . (D.14)From this it is clear that the complex-shifted momentum running through the internal leg, k A ( z ) goes on-shell at z = z A . Consequently the complexified Mellin amplitude is given by M ( z ) ≡ − V ∆ A, V ∆ A, β p (cid:32) s A ( z ) − ∆ A , n − (cid:88) i =1 ∆ i (cid:33) , (D.15)so that the original Mellin amplitude is recovered upon setting z = 0, M (0) = M − int∆ , ;∆ , ( s A , ∆ A ) . (D.16)Consider now the integral I ≡ (cid:73) C dz πi M ( z ) z , (D.17) The momentum variables k i and q are respectively, real and complex valued ( n + 1)-dimensional Lorentzvectors with a Lorentzian inner product. If ( n + 1) = 3, then only q = 0 satisfies (D.11) for arbitrary k , k .Thus we need at least ( n + 1) ≥ N to admit a non-vanishing complex-deformation zq . This requirementis consistent with the inequality mentioned in section 2. Note that the complex Mellin amplitude M ( z ) in this appendix is different from the one defined insection 5 above (5.16). C is chosen to be a circle of infinite radius centered at origin. The integrandhas simple poles at z = 0 , z = z A + 12 q · k πim log p m ∈ Z , (D.18)where the infinite sequence of poles arises from the factor of ζ p ( s A ( z ) − ∆ A ) in the β p function.We will now apply the residue theorem to obtain an expression for the original amplitudein terms of the residues from the remaining poles. It is clear that the residue at z = 0 isprecisely the original amplitude M (0). The residue at the remaining poles takes the formRes z = z A + q · k πim log p (cid:18) M ( z ) z (cid:19) = V ∆ A, V ∆ A, ( s A − ∆ A ) log p + 2 πim m ∈ Z . (D.19)Thus we conclude that I = M (0) + V ∆ A, V ∆ A, + ∞ (cid:88) m = −∞ s A − ∆ A ) log p + 2 πim . (D.20)At this point, one hopes that M ( z ) falls off sufficiently fast at large z , so that the contourintegral I = 0, i.e. the boundary term vanishes. The infinite sum in (D.20) evaluates to give + ∞ (cid:88) m = −∞ s A − ∆ A ) log p + 2 πim = ζ p ( s A − ∆ A ) − , (D.21)so the claim that I = 0 becomes M (0) ? = − V ∆ A, (cid:18) ζ p ( s A − ∆ A ) − (cid:19) V ∆ A, ? = −M ( k ( z ) , k , − k ( z ) − k ) (cid:12)(cid:12)(cid:12) z = z A (cid:18) ζ p ( s A − ∆ A ) − (cid:19) M ( k ( z ) + k , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z A , (D.22)where in the second line above we have defined the sub-amplitude M ( k a , k b , k c ) ≡ V − k a , − k b , − k c (D.23)with k a + k b + k c = 0. Consider the sub-amplitude M ( k ( z ) , k , − k ( z ) − k ) = V ∆ , ∆ ,s A ( z ) . (D.24)80t represents an “off-shell” three-point contact interaction with incoming momenta k ( z ) , k and − k ( z ) − k . At generic values of z , two of the three external legs are on-shell, while thethird one is off-shell, but the third leg goes on-shell at z = z A , and we write M ( k ( z A ) , k , − k ( z A ) − k ) = k + k ∆ A k k k k − k ( z A ) − k k ( z A ) k k ( z A ) + k k k ( z A ) 1 . (D.25)Thus if it is true that I = 0, we can interpret the second line of (D.22) as the claimthat we have rewritten the original Mellin amplitude for the s -channel four-point exchangediagram as a product of two on-shell sub-amplitudes obtained by cutting across the internalleg of the original diagram along which the field dual to an operator of dimension ∆ A propagates, times a putative “propagator” running across the internal leg given by theexpression ζ p ( s A − ∆ A ) − / not vanish, as the complex Mellin amplitude M ( z ) does not vanish at infinity. Instead, we find I = 12 V ∆ A, V ∆ A, V ∆ , (1 + p n − ∆ , )= M ( k ( z ) , k , − k ( z ) − k ) (cid:12)(cid:12)(cid:12) z = z A (cid:18) V ∆ , (1 + p n − ∆ , )2 (cid:19) M ( k ( z ) + k , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z A , (D.26)where the second line is merely a suggestive rewriting of the first line. Taking the boundaryterm into account, the corrected version of (D.22) is then given by M (0)= −M ( k ( z ) , k , − k ( z ) − k ) (cid:12)(cid:12)(cid:12) z = z A (cid:18) ζ p ( s A − ∆ A ) − (cid:19) M ( k ( z ) + k , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z A + M ( k ( z ) , k , − k ( z ) − k ) (cid:12)(cid:12)(cid:12) z = z A (cid:18) V ∆ , (1 + p n − ∆ , )2 (cid:19) M ( k ( z ) + k , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z A , (D.27)81hich simplifies to the compact expression M (0)= −M ( k ( z ) , k , − k ( z ) − k ) (cid:12)(cid:12)(cid:12) z = z A β p ( s A − ∆ A , n − ∆ , ) M ( k ( z ) + k , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z A . (D.28)The amplitude in (D.28) is, indeed, mathematically identical to the starting point (D.3),and in fact is a special case of (D.1). More importantly, we see the boundary contributiondoes not destroy the nice decomposition of the amplitude first observed in (D.22); instead itcombines nicely with the residue from the poles of the complexified Mellin amplitude in sucha way that the original amplitude is still given by a product of two on-shell sub-amplitudesobtained by cutting across the internal leg, times a propagator running across the sameinternal leg, which we refer to as the “Mellin space propagator” joining the left and rightsub-amplitudes. We comment on the appearance of non-vanishing boundary terms at theend of this appendix.Diagrammatically, we may write (D.28) as k + k ∆ A k k k k − k ( z A ) − k k ( z A ) k k ( z A ) + k k k ( z A ) 1 = − k + k ∆ A k k k k − k ( z A ) − k k ( z A ) k k ( z A ) + k k k ( z A ) 1 β p ( s A − ∆ A , n − ∆ , ) k + k ∆ A k k k k − k ( z A ) − k k ( z A ) k k ( z A ) + k k k ( z A ) 1 . (D.29)We may also equivalently depict the identity (D.28) as ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 = − ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 β p ( s A − ∆ A , n − ∆ , ) ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 , (D.30)which avoids reference to the auxiliary momentum space and any BCFW-shifts altogether,and takes the form presented in (5.11).Admittedly, the example of the four-point exchange diagram is quite special in that thesub-amplitudes turn out to be contact diagrams and thus have no Mandelstam dependence.Nevertheless, we demonstrate in the next subsection that the BCFW-type decomposition(D.1) holds, rather non-trivially, for the Mellin amplitude of the five-point diagram with twointernal exchanges. 82 A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 (a) ∆ A (b) Figure 1: (a) A five-point diagram built out of cubic interaction vertices. The Mandelstamvariables s A and s B are defined in (D.33). (b) A five-point diagram built out of cubic andquartic interaction vertices. D.2 Five-point diagram with two internal lines In a bulk theory with cubic interaction vertices, five-point diagrams of the kind shown infigure 1a, with two internal legs, are allowed. Further, if the theory contains quartic inter-action vertices as well, exchange diagrams of the kind shown in figure 1b are also possible.However, exchange diagrams like the one in figure 1b fall in the category of diagrams dis-cussed in the previous subsection, and results obtained there extend trivially to arbitraryexchange diagrams (that is, bulk diagrams with exactly one internal line).So in this subsection, we focus on the tree-level five-point diagram built solely from cubicvertices. A BCFW-type recursion can be set up in each channel individually , so just like inthe previous subsection, we will restrict attention to a particular channel, which will be theone shown in figure 1a; an identical analysis will hold in all other channels.As with the four-point exchange diagram, we start with the Mellin amplitude for thediagram in figure 1a, which is a special case of the diagram evaluated in example (iv), M − int∆ , ;∆ ;∆ , ( s A , ∆ A ; s B , ∆ B ) ≡ ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 = V ∆ A, V ∆ A B, V ∆ B, (cid:20) ζ p ( s A − ∆ A ) ζ p ( s B − ∆ B ) − V ∆ A, β p ( s A − ∆ A , n − ∆ Σ ) − V ∆ B, β p ( s B − ∆ B , n − ∆ Σ ) − V ∆ Σ (cid:21) , (D.31)83here the contact amplitudes V ∆ i ...if , were defined in (D.4), β p was defined in (2.24), andwe are using the shorthand ∆ Σ ≡ (cid:88) i =1 ∆ i = ∆ , . (D.32)Furthermore, with the help of the Mellin variable constraints (2.13) for N = 5, we havedefined s A ≡ ∆ + ∆ − γ = ∆ + ∆ + ∆ − γ − γ − γ s B ≡ ∆ + ∆ − γ = ∆ + ∆ + ∆ − γ − γ − γ . (D.33)Unlike the case of the four-point exchange diagram, it is not immediately clear that theamplitude in (D.31) admits a BCFW-type decomposition; in the rest of this appendix weshow precisely how this works out.We start by passing to the auxiliary momentum space, with the momentum assignmentsas shown: M − int∆ , ;∆ ;∆ , ( s A , ∆ A ; s B , ∆ B ) = k ∆ A ∆ B k A k B k k k k , (D.34)where, like before, the external momenta are on-shell, − k i = ∆ i i = 1 , . . . , , (D.35)and momentum is conserved at each vertex. Once again we employ a complex-shift ofmomenta; specifically we apply the two-line BCFW-shift (D.8) subject to the on-shell con-straints (D.9). The on-shell conditions lead us to constraints on q as written down in (D.11).Momentum conservation along with constraints (D.11) further implies q · ( k + k + k ) = 0 . (D.36)Looking ahead, it will be convenient to set q · k = 0 . (D.37)We are free to make this choice and still have (at least one) non-vanishing solution for thecomplex momentum variable q , as long as ( n + 1) ≥ 5, since (D.11) and (D.37) amount to 484onditions on the ( n +1)-component vector q . This proviso is consistent with the requirement( n + 1) ≥ N = 5 which ensures there are precisely N ( N − / k A → k A ( z ) ≡ k ( z ) + k , k B → k B ( z ) ≡ k + k ( z ) , (D.38)which allows the possibility of setting the internal legs on-shell at specific (non-zero) valuesof the complex parameter z . In particular, s A ( z ) ≡ − k A ( z ) = s A − zq · k = − q · k ( z − z A ) + ∆ A s B ( z ) ≡ − k B ( z ) = s B + 2 zq · k = 2 q · k ( z − z B ) + ∆ B , (D.39)where we have defined z A ≡ s A − ∆ A q · k , z B ≡ − s B − ∆ B q · k . (D.40)We conclude that k A ( z ) goes on-shell at z = z A , while k B ( z ) goes on-shell at z = z B .Equation (D.39) also suggests what the complex-shifted Mellin amplitude should look like, M ( z ) ≡ M − int∆ , ;∆ ;∆ , ( s A ( z ) , ∆ A ; s B ( z ) , ∆ B ) , (D.41)that is, we simply promote the Mandelstam variables s A and s B in (D.31) to their complex-shifted versions.Like in the previous subsection, consider now the contour integral I ≡ (cid:73) C dz πi M ( z ) z , (D.42)where the contour C is a circle of infinite radius centered at origin. It is clear from theexplicit form in (D.31) that the integrand in (D.42) has simple poles at z = 0 , z = z A + 12 q · k πim log p z = z B + 12 q · k πim log p m ∈ Z . (D.43)We will first apply the residue theorem to evaluate (D.42). The residue at z = 0 reproducesthe original amplitude M (0) which we are interested in, while the residues at the remaining85oles evaluate toRes z = z A + q · k πim log p (cid:18) M ( z ) z (cid:19) = − V ∆ A, V ∆ A B, V ∆ B, β p ( s B − ∆ B − s A + ∆ A , n − ∆ A, )( s A − ∆ A ) log p + 2 πim Res z = z B − q · k πim log p (cid:18) M ( z ) z (cid:19) = − V ∆ A, V ∆ A B, V ∆ B, β p ( s A − ∆ A − s B + ∆ B , n − ∆ B, )( s B − ∆ B ) log p + 2 πim , (D.44)for all m ∈ Z , where we made important use of the identity q · k = − q · k which followsfrom (D.36)-(D.37). Using (D.21) to sum up the residues, we obtain M (0)= −M ( k ( z ) , k , − k ( z ) − k ) (cid:12)(cid:12)(cid:12) z = z A (cid:18) ζ p ( s A − ∆ A ) − (cid:19) M ∆ B ( k ( z ) + k , k , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z A − M ∆ A ( k ( z ) , k , k , k + k ( z )) (cid:12)(cid:12)(cid:12) z = z B (cid:18) ζ p ( s B − ∆ B ) − (cid:19) M ( − k − k ( z ) , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z B + I , (D.45)where we have used (D.23) and have defined the (off-shell) sub-amplitude M ∆ ( k a , k b , k c , k d ) ≡ − V − k a , − k b , ∆ V − k c , − k d , ∆ × β p (cid:0) − ( k a + k b ) − ∆ , n + k a + k b + k c + k d (cid:1) (D.46)with k a + k b + k c + k d = 0. The sub-amplitude M ( k ( z ) , k , − k ( z ) − k ) = V ∆ , ∆ ,s A ( z ) (D.47)in (D.45) is on-shell at z = z A , i.e. all external momenta go on-shell at z = z A . Moreover,at this value of z , it precisely takes the form of an (on-shell) three-point contact Mellinamplitude. Similarly, the sub-amplitude, M ∆ B ( k ( z ) + k , k , k , k ( z )) = − V s A ( z ) , ∆ , ∆ B V ∆ , ∆ , ∆ B × β p ( s B ( z ) − ∆ B , n − ∆ − ∆ − ∆ − s A ( z )) (D.48)is on-shell at z = z A , whence s A ( z A ) = ∆ A , s B ( z A ) = − ( k + k ( z A )) = s B − s A + ∆ A ,and we recognize from the previous subsection that (D.48) takes precisely the form of an86on-shell) four-point exchange diagram (see, e.g. (D.6)): M ∆ B ( k ( z A ) + k , k , k , k ( z A )) = k k + k k + k ∆ B ∆ A k k k k − k ( z A ) − k k ( z A ) k − k − k ( z A )∆ B k k ( z A ) + k k ( z A ) k k ( z B ) + k ∆ A k ( z B ) k k k + k ( z B ) − k − k ( z B ) k k ( z B ) 1 = M − int − k − ( k ( z A )+ k ) ; − k − k ( z A ) (cid:0) − ( k + k ( z A )) , ∆ B (cid:1) = M − int∆ +∆ A ;∆ +∆ ( s B − s A + ∆ A , ∆ B ) . (D.49)A similar interpretation can be given to the sub-amplitudes in the second term on the r.h.s.of (D.45), which are to be evaluated at z = z B .We still need to take into account the boundary contribution in (D.45), denoted by I .Just like in the previous subsection, such a contribution is non-vanishing and can be directlycomputed by evaluating the contour integral at infinity in (D.42). We omit the detailsof the computation, but point out that just like in the previous subsection, the boundarycontribution combines with the sum of residues at all (non-zero) poles to give M (0)= −M ( k ( z ) , k , − k ( z ) − k ) (cid:12)(cid:12)(cid:12) z = z A β p ( s A − ∆ A , n − ∆ Σ ) M ∆ B ( k ( z ) + k , k , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z A − M ∆ A ( k ( z ) , k , k , k + k ( z )) (cid:12)(cid:12)(cid:12) z = z B β p ( s B − ∆ B , n − ∆ Σ ) M ( − k − k ( z ) , k , k ( z )) (cid:12)(cid:12)(cid:12) z = z B . (D.50)87e may write this diagrammatically as k k + k k + k ∆ B ∆ A k k k k − k ( z A ) − k k ( z A ) k − k − k ( z A )∆ B k k ( z A ) + k k ( z A ) k k ( z B ) + k ∆ A k ( z B ) k k k + k ( z B ) − k − k ( z B ) k k ( z B ) 1 = − k k + k k + k ∆ B ∆ A k k k k − k ( z A ) − k k ( z A ) k − k − k ( z A )∆ B k k ( z A ) + k k ( z A ) k k ( z B ) + k ∆ A k ( z B ) k k k + k ( z B ) − k − k ( z B ) k k ( z B ) 1 β p ( s A − ∆ A , n − ∆ Σ ) k k + k k + k ∆ B ∆ A k k k k − k ( z A ) − k k ( z A ) k − k − k ( z A )∆ B k k ( z A ) + k k ( z A ) k k ( z B ) + k ∆ A k ( z B ) k k k + k ( z B ) − k − k ( z B ) k k ( z B ) 1 − k k + k k + k ∆ B ∆ A k k k k − k ( z A ) − k k ( z A ) k − k − k ( z A )∆ B k k ( z A ) + k k ( z A ) k k ( z B ) + k ∆ A k ( z B ) k k k + k ( z B ) − k − k ( z B ) k k ( z B ) 1 β p ( s B − ∆ B , n − ∆ Σ ) k k + k k + k ∆ B ∆ A k k k k − k ( z A ) − k k ( z A ) k − k − k ( z A )∆ B k k ( z A ) + k k ( z A ) k k ( z B ) + k ∆ A k ( z B ) k k k + k ( z B ) − k − k ( z B ) k k ( z B ) 1 . (D.51)Equivalently, the result (D.50) may also be recast in the form of (5.11) as ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 = − ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 β p ( s A − ∆ A , n − ∆ Σ ) ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 − ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 β p ( s B − ∆ B , n − ∆ Σ ) ∆ A s A ∆ B s B 12 435∆ A s A − s B + ∆ B 12 5 B ∆ B s B − s A + ∆ A A A B A A s A 12 43 1 , (D.52)which avoids any reference to momentum variables. 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