Constraining non--BPS interactions from counterterms in three loop maximal supergravity
aa r X i v : . [ h e p - t h ] O c t Constraining non–BPS interactions from countertermsin three loop maximal supergravity
Anirban Basu Harish–Chandra Research Institute, Chhatnag Road, Jhusi,Allahabad 211019, India
Abstract
The structure of one, two and three loop counterterms imposes strong constraintson several non–BPS interactions in the low momentum expansion of the three loopfour graviton amplitude in maximal supergravity. The constraints are imposed bydemanding consistency with string amplitudes. We analyze these constraints imposedon the D R interaction in 11 dimensional supergravity compactified on T . Wealso discuss partial contributions to interactions at higher orders in the momentumexpansion. email address: [email protected] Introduction
Obtaining the effective action of string theory in various backgrounds is useful from the lowenergy perspective. It is also useful in order to get a detailed quantitative understanding ofthe perturbative and non–perturbative duality symmetries of string theory. The effectiveaction which encodes S matrix elements of the theory is manifestly duality invariant . Itcontains both local and non–local terms, the later coming from integrating out masslessmodes in the loop diagrams. While calculating the effective action in arbitrary backgroundsis rather complicated, several terms have been obtained for the case of toroidal compactifi-cations preserving maximal supersymmetry [1–15]. While a class of BPS interactions havebeen understood in detail, non–BPS interactions have been hardly analyzed.Maximal supergravity has played an important role in determining these interactionsbecause of the ability to calculate multiloop amplitudes [2, 6, 20, 22–28]. In particular,the one and two loop amplitudes have yielded several interactions in the effective action.At three loops the leading interaction is the 1 / D R interaction, whose modulidependent coefficient is highly constrained. The non–BPS D R interaction is the firstsubleading interaction in the low momentum expansion of the three loop four gravitonamplitude. We perform a simple analysis to determine the constraints counterterms imposeon the moduli dependent coefficient of this interaction. To do so, we isolate the one, two andthree loop ultraviolet divergences of this three loop amplitude which have to be cancelledin the quantum theory by local counterterms. The structure of these counterterms is highlyconstrained by the structure of string theory, which uniquely fixes their renormalized values.Thus demanding the cancellation of the divergences gives us several finite contributions tothe D R interaction. This includes contributions that are inconsistent with the structureof perturbative string amplitudes at various genera. Hence these contributions must cancelcompletely in the final answer, which includes the finite supergravity contributions fromthree loops, as well as finite and regularized contributions from higher loops. Thus weobtain an intricate interplay between the cancellation of divergences and string perturbationtheory, which sheds light on the structure of quantum supergravity. We also considersome simple counterterm contributions to the D R , D R , D R and D R interactionswhich we regularize. Our analysis leads to perturbative contribution to string amplitudesat various genera. We perform the calculations in 11 dimensional supergravity on T to beconcrete, though our analysis can be generalized to arbitrary toroidal compactifications.Our analysis clearly shows the complications involved in the analysis of non–BPS in-teractions compared to their BPS counterparts. Not only are they expected to receivecontributions from all loops in supergravity, but also their contributions at at every looporder are more involved than the BPS ones. Such interactions have not been studied indetail , and are crucial in understanding the effective action beyond the first few orders inthe low momentum expansion.We begin by reviewing the various relations expressing quantities in M theory compact-ified on T in terms of the moduli of the type IIB theory [16–19]. Keeping only the scalars For self–dual theories, what we want are duality covariant equations of motion in the given background. See [24, 26, 29–33] for some analysis of non–BPS interactions. T compactification of M theory where R and R arethe dimensionless radii (in units of l ) of the two circles and dropping the 1 form gaugepotentials for simplicity, the line element in M theory is given by ds = G MN dx M dx N = G (9) µν dx µ dx ν + R l ( dx − Cdx ) + R l dx , (1.1)where x and x are dimensionless angular coordinates. The 9 dimensional metric G (9) µν = g Bµν where g Bµν is the type IIB metric in the string frame. The dimensionless volume V (inunits of 4 π l ) and the complex structure Ω of the T are related to the type IIB moduliby the relations V = R R = e φ B / r − / B , Ω = C , Ω = R R = e − φ B , (1.2)where φ B is the type IIB dilaton, and r B is the dimensionless radius of the tenth dimension(in units of l s ) in the type IIB string frame metric given by r B = 1 R √ R , (1.3)and C is the type IIB 0 form potential. This enables us to express the interactions in Mtheory on T in terms of type IIB interactions on S .The T dual type IIA theory has metric and moduli given by g Bµν = g Aµν , r B = r − A , e − φ B = r A e − φ A , C = C , (1.4)where g Aµν is the type IIA metric in the string frame, r A is the dimensionless radius of thetenth dimension (in units of l s ) in this metric, and C is the 1 form potential. Thus theresults we obtain for the type IIB theory can be easily converted to results for the type IIAtheory.Finally, the 11 dimensional Planck length is related to the string length by the relation l = e φ B / r − / B l s . (1.5)In the next section, we discuss one and two loop counterterms needed to cancel oneand two loop ultraviolet divergences in the four graviton amplitude. We then discuss thegeneral structure of three loop divergences which is the main focus of the paper. The variouscounterterm contributions to the D R interaction are discussed in detail in the followingsection, based on the structure of the three loop four graviton amplitude. Contributionsfrom loop diagrams involving both the ladder and Mercedes skeletons are considered, whichlead to several finite terms in the effective action. Consistency with string perturbationtheory imposes severe constraints on which terms can survive in the effective action. As aresult certain terms which seem to survive in the effective action based on our analysis mustvanish in the amplitude when all other contributions are taken into account. Thus stringtheory plays a decisive role in regulating the ultraviolet divergences in a way consistentwith string duality. Details of some of the calculations are mentioned in the appendices.2 Counterterms for one and two loop ultraviolet divergences
At one loop, the leading correction to the Einstein–Hilbert action is the 1 / R interaction. The R interaction is Λ divergent [2, 20, 21, 34, 35]. This leads to a term in theeffective action of the form l − Z d x p − G (9) V R (Λ l ) , (2.6)leading to the interaction l − s Z d x p − g B r − B R (Λ l ) (2.7)in the type IIB effective action which receives perturbative contribution at genus one. Thisis cancelled by a one loop counterterm with coefficient c given by4 π / l ) + c = 4 ζ (2) (2.8)leaving behind a finite remainder in the M theory effective action. The next interaction inthe low energy expansion is the D R interaction which is finite. Hence (2 .
8) is the onlyone loop counterterm (along with counterterms for interactions in the same supermultiplet)which leaves a finite remainder for terms in the effective action.At two loops [6, 22, 24], the leading interaction is the 1 / D R interaction whichhas a Λ primitive divergence. The other divergent interactions are the D R , . . . , D R interactions which have Λ , . . . , lnΛ primitive divergences respectively. Thus the primitivedivergences lead to terms in the effective action of the form l n − Z d x p − G (9) V D n R (Λ l ) − n (2.9)for 2 ≤ n ≤
5, and l Z d x p − G (9) V D R ln(Λ l ) . (2.10)Consequently, these lead to terms in the effective action of the type IIB theory given by l n − s Z d x p − g B r − n/ − B e nφ B / D n R (Λ l ) − n , (2.11)for 2 ≤ n ≤
5, and l s Z d x p − g B r − B e φ B D R ln(Λ l ) . (2.12)The primitive divergences for the D R , D R nd D R interactions are completelycancelled by two loop counterterms leaving no finite remainders as they would lead to termsinconsistent with perturbative string theory. The D R interaction has Λ divergence. Thisis cancelled by a counterterm which could leave a finite remainder determined by the genustwo D R amplitude in string theory. However this finite remainder must vanish as a3onsequence of the structure of three loop supergravity [28]. This is because a five point twoloop Λ counterterm in the same supermultiplet is needed to cancel subleading divergencesof the D R interaction at three loops, and any finite remainder would be inconsistent withperturbative string theory. For the D R interaction, the counterterm that cancels theln(Λ l ) divergence can leave a finite remainder determined by the genus three amplitudeof the D R interaction. The renormalized value of this counterterm will not be neededin our analysis.All other two loop interactions at higher orders in the momentum expansion are finite.However, for the various interactions discussed above (as well as those in the same super-multiplet) which have a primitive two loop divergence, apart from the finite contributions,there are possible one loop subdivergences. Only the Λ subdivergence yields a finite re-mainder using (2 . D R , D R and D R interactions. For the D R , D R and D R interactions, this leads to terms [6, 28] l Z d x p − G (9) V − / D R (Λ l ) E / (Ω , ¯Ω) ,l Z d x p − G (9) V − / D R (Λ l ) E / (Ω , ¯Ω) ,l Z d x p − G (9) V / D R (Λ l ) E / (Ω , ¯Ω) (2.13)in the effective action respectively. The details for the D R interaction are given inappendix A leading to ( A. l s Z d x p − g B r B D R (Λ l ) (cid:16) ζ (5) e − φ B + 83 ζ (4) e φ B + . . . (cid:17) ,l s Z d x p − g B r − B D R (Λ l ) (cid:16) ζ (3) + 4 ζ (2) e φ B + . . . (cid:17) , l s Z d x p − g B r − B e φ B D R (Λ l ) ln (cid:16) e − φ B πe − γ (cid:17) + . . . , (2.14)where we have dropped exponentially suppressed corrections. These lead to finite contri-butions using the counterterm in (2 . e − φ B . This is notthe case for the logarithmic term in the D R interaction in (2 . D R interaction in 9 dimensions has an infrared logarithmic divergence, which is capturedby (2 .
14) [22, 24]. The scale of this infrared divergent logarithm is moduli dependent inthe supergravity calculation. Thus we see how the counterterm analysis of the ultravioletdivergences gives us information about infrared divergences in the theory. These infraredeffects are also present in the finite part of the two loop D R interaction.4 The structure of three loop ultraviolet divergences
The leading interaction at three loops [23, 25, 27] is the D R interaction which has Λ primitive divergence. This is cancelled by a three loop counterterm with coefficient z whichleaves a finite remainder fixed by the genus two D R amplitude given by [28] h (Λ l ) + z = 24 ζ (4) , (3.15)where h is an irrelevant constant. The other divergent interactions are the non–BPS D R , . . . , D R interactions which have Λ , . . . , Λ primitive divergences respectively.The primitive divergences of these interactions of the form D n R (4 ≤ n ≤
10) yieldthe terms l n − Z d x p − G (9) V D n R (Λ l ) − n (3.16)in the effective action, leading to terms in the effective action l n − s Z d x p − g B r − (1+2 n/ B e nφ B / D n R (Λ l ) − n (3.17)in the type IIB theory. Consistency with perturbative string theory shows that all the prim-itive three loop divergences are cancelled by counterterms without leaving finite remainders,expect the ones for the D R and D R interactions which can receive finite contributionsdetermined by the structure of the genus three e φ B r − B and genus four e φ B r − B amplitudesrespectively. Hence we see the structure of the counterterms imposes strong constraints onthe couplings.Each of these interactions also have subdivergences which must be cancelled by oneand two loop counterterms. Among these subdivergences at one loop, only those of theform Λ can leave finite remainders using (2 . or lnΛ as determined by the structure of the one and two loopcounterterms. While the three loop lnΛ subdivergence must be cancelled by the two loop D R counterterm, the Λ subdivergence must be cancelled by a product of two Λ R counterterms.All other interactions at higher orders in the momentum expansion are finite. Our aimis to analyze the constraints imposed by counterterms on the D R interaction. D R interaction The three loop four graviton amplitude is given by [23, 25, 27] A (3)4 = (4 π ) κ (2 π ) X S h I ( a ) + I ( b ) + 12 I ( c ) + 14 I ( d ) + 2 I ( e ) + 2 I ( f ) + 4 I ( g ) + 12 I ( h ) + 2 I ( i ) i K , (4.18)5here K is the linearized approximation to R in momentum space, and 2 κ = (2 π ) l .Also S, T and U are the Mandelstam variables defined by S = − G MN ( k + k ) M ( k + k ) N , T = − G MN ( k + k ) M ( k + k ) N and U = − G MN ( k + k ) M ( k + k ) N , where G MN is the M theory metric, and the external momenta are labelled by k iM ( i = 1 , . . . ,
4) andsatisfy P i k iM = 0 and k i = 0. Also S represents the 6 independent permutations of theexternal legs marked { , , } keeping the external leg { } fixed. The external momentaare directed inwards in all the loop diagrams in figures 1 and 2. We shall use the notation σ n = S n + T n + U n (4.19)throughout. We denote the low momentum expansion of the analytic part of the amplitudeby A (3)4 = (4 π ) κ (2 π ) [ I + I + . . . ] , (4.20)where I n receives contributions at O ( σ n ), and hence we are interested in I .The primitive D R Λ divergence cancels as discussed before, and hence we need toconsider the one and two loop subdivergences only. a b d Figure 1: Three loop diagrams from the ladder skeletonIn (4 . N ( x ) are given by [25] N ( a ) = N ( b ) = N ( c ) = N ( d ) = S ,N ( e ) = N ( f ) = N ( g ) = S τ τ ,N ( h ) = (cid:16) S ( τ + τ ) + T ( τ + τ ) + ST (cid:17) + (cid:16) S ( τ + τ ) − T ( τ + τ ) (cid:17)(cid:16) τ + τ + τ + τ , (cid:17) + S ( τ τ + τ τ , ) + T ( τ τ + τ τ , ) + U ( τ τ + τ τ , ) ,N ( i ) = ( Sτ − T τ ) − τ ( S τ + T τ ) − τ ( S τ + U τ ) − τ ( T τ + U τ ) − l S T − l ST + l ST U, (4.21)6 e fg hc i
Figure 2: Three loop diagrams from the Mercedes skeletonwhere τ ij = − k i · l j ( i ≤ , j ≥ . (4.22)The various momenta l i are denoted in figure 2.In the calculations below, we drop various one and two loop counterterm contributionsthat do not leave any finite remainder as discussed above. Hence for the D R interactionwe need to consider only the O (Λ ) and O (Λ ) divergences. Also while we refer to eachcontribution mentioned in (4 .
18) as I ( x ) , the total contribution after the sum over S isperformed is referred to as I ( X ) . These are contributions to I from the loop diagrams a, b and d in figure 1. O (Λ ) counterterm contributions We first consider the contribution from the diagram a . We give some of the details of theanalysis for I ( a ) and for all the others, we simply write down the answer. For I ( a ) , in theuncompactified theory from (4 .
21) we see that the one loop subdivergences are given by I ( a ) = S Z d rr h Z d p Z d q p q ( p + q ) + Z d pp Z d qq i . (4.23)These must be regularized by including the contributions from one loop counterterms. Wenow evaluate the integrals in (4 .
23) as well as the others to follow in the background T × , . For all the cases, the 11 dimensional loop momentum p M decomposes as { p µ , l I /l } where p µ is the 9 dimensional momentum and l I ( I = 1 ,
2) is the KK momentum along T .Thus consider the integral Z d p Z d q p q ( p + q ) (4.24)in (4 .
23) in the compactified theory. We introduce a Schwinger parameter for every prop-agator. Hence the product of the propagators in the compactified theory coming from thedenominator of (4 .
24) is given by Z ∞ Y i =1 dσ i e − P j =1 σ j q j e − (cid:16) ( σ + σ + σ ) m +( σ + σ ) n + σ ( m + n ) (cid:17) /l , (4.25)where q j = { p, p, p, q, q, p + q } (4.26)and m ≡ G IJ m I m J , (4.27)where the metric of T is given by G IJ = V Ω (cid:18) | Ω | − Ω − Ω (cid:19) . (4.28)Thus compactified on T , (4 .
24) becomes1(4 π l V ) Z ∞ Y i =1 dσ i F L ( σ, λ, ρ ) Z d p Z d qe − σp − λq − ρ ( p + q ) (4.29)where F L ( σ, λ, ρ ) = X m I ,n I e − G IJ (cid:16) σm I m J + λn I n J + ρ ( m + n ) I ( m + n ) J (cid:17) /l , (4.30)and we have defined σ = σ + σ + σ , λ = σ + σ , ρ = σ . (4.31)We now define u = σ σ , u = σ + σ σ , v = σ σ , (4.33)leading to Z ∞ Y i =1 dσ i = Z ∞ dσdλdρσ λ Z du Z u du Z dv . (4.34) Thus 0 ≤ u , u , v ≤ , u ≤ u . (4.32) Z d p Z d qe − σp − λq − ρ ( p + q ) = π / ∆ − / (4.35)where ∆ = σλ + λρ + ρσ, (4.36)(4 .
24) when compactified on T gives us I = π (4 π l V ) Z ∞ dσdλdρ σ λ / F L ( σ, λ, ρ ) . (4.37)All the integrals we need can be done using the same logic and so we only give the finalanswers. Hence in the 9 dimensional theory compactified on T , to cancel this Λ divergencewe must introduce the counterterm δ (3) I ( A ) = 2 σ · c π l (4 π ) h I + I i , (4.38)where I = π π ) l (cid:16)
25 (Λ l ) + 34 π / V − / E / (Ω , ¯Ω) (cid:17) . (4.39)The counterterm involving I is depicted by a in figure 3, while the one involving I isdepicted in figure 4. In the integral I which involves the one loop integral, we haveperformed Poisson resummation to go from the KK momenta to winding momenta toperform the integrals. a b Figure 3: Planar and non–planar one loop counterterms for the ladder skeleton diagramsSimilarly from figure b , divergence cancellation gives the counterterm δ (3) I ( B ) = 2 σ · c π l (4 π ) h I + I i (4.40)in the compactified theory, where I = π (4 π l V ) Z ∞ dσdλdρ σλρ ∆ / F L ( σ, λ, ρ ) . (4.41)9igure 4: Another one loop counterterm for the ladder skeleton diagramsThis counterterm is depicted by b in figure 3.Finally from I ( d ) , we get the counterterm δ (3) I ( D ) = 2 σ · c π l (4 π ) · I (4.42)in the compactified theory.Thus in (4 .
18) to cancel the Λ divergence we get the total counterterm contribution δ (3) I ( A ) + δ (3) I ( B ) + 14 δ (3) I ( D ) = 2 σ · c π l (4 π ) h I + I i , (4.43)where I = π π l V ) Z ∞ dσdλdρ ∆ − / h ∆( σ + λ + ρ ) + 3 σλρ i F L ( σ, λ, ρ ) . (4.44)Apart from cancelling the divergence, from the term involving I in (4 . .
8) we get a finite contribution to I in the regularized theory given by I = σ (4 π ) l · π V − E / (Ω , ¯Ω) . (4.45)This leads to a term in the effective action of the form l Z d x p − G (9) V − D R E / (Ω , ¯Ω) , (4.46)which in the type IIB theory becomes l s Z d x p − g B r B (cid:16) ζ (5) e − φ B + 323 ζ (4) ζ (5) + 649 ζ (4) e φ B (cid:17) D R + . . . . (4.47)The first term is inconsistent with string perturbation theory, and hence this must cancelwhen other contributions of this kind are added. The second and third terms yield con-tributions at genus one and three respectively. Contributions of this type generalize for10igher derivative interactions as described in appendix C for the D R , D R , D R and D R interactions.In fact, the genus one contribution in (4 .
47) gives A (3)4 = (2 π l r B ) K r B ( σ l s ) π ζ (5) r B · π l r B is needed to obtain the correct normaliza-tion and is common to the multiloop amplitudes. The remaining part yields a contributionof the form π ζ (5) r B at genus one for the D R amplitude. In fact, this structure preciselyagrees with that obtained from string perturbation theory [36] upto a numerical factor.The overall factor is not expected to match as there are other contributions of this type aswell. In fact, we shall encounter one such source of contribution later on.We shall consider the finite contribution to I coming from the I term in (4 .
43) later. O (Λ ) counterterm contributions We now consider the Λ divergences. From (4 . I ( A ) = I ( B ) = I ( D ) , (4.49)leading to δ (3) I ( A ) + δ (3) I ( B ) + 14 δ (3) I ( D ) = σ (4 π ) l · π / · (cid:16) c π (cid:17) . h
27 (Λ l ) + 158 π / V − / E / (Ω , ¯Ω) i . (4.50)This counterterm is depicted in figure 5.Figure 5: Product of one loop counterterms for the ladder skeleton diagramsThus from (2 . I given by I = σ (4 π ) l · π · V − / E / (Ω , ¯Ω) . (4.51) One has to send r B → r − B to see the agreement using the pertubative equality of the type IIA and IIBamplitudes. l Z d x p − G (9) V − / D R E / (Ω , ¯Ω)= l Z d x p − G (9) V − / D R (cid:16) ζ (7)Ω / + 3215 ζ (6)Ω − / + . . . (cid:17) , (4.52)which in the type IIB theory becomes l s Z d x p − g B r B (cid:16) ζ (7) e − φ B + 3215 ζ (6) e φ B + . . . (cid:17) D R . (4.53)This leads to terms involving genus zero and genus three in the type IIB theory . However,we see that this does not yield the complete genus zero amplitude which has a differentcoefficient. This is not a contradiction as we have only considered the contributions thatarise from counterterms at three loops. We have not considered the finite contributions aswell as contributions from higher loops. In fact, it is not difficult to see that there are indeedcontributions of the kind (4 .
52) that arise at four loops. Consider the four loop diagram a in figure 6 which has the D R interaction as the leading term in the low momentumexpansion [30, 31]. The primitive four loop D R divergence is Λ . There is a subleadingthree loop Λ divergence which is cancelled by the three loop five point counterterm b infigure 6. Thus at four loops, this counterterm yields a contribution of the form δ (4) I (4) ∼ σ ˆ zl h
27 (Λ l ) + 158 π / V − / E / (Ω , ¯Ω) i , (4.54)where the analytic part of the four loop four graviton amplitude is expanded as A (4)4 = (4 π ) κ (2 π ) h I (4) + . . . i . (4.55)Here the three loop five point counterterm which has coefficient proportional to ˆ z lies inthe same supermultiplet as the three loop primitive counterterm for the D R interactionand hence [28] ˆ z ∼ (Λ l ) + ζ (4) . (4.56)It immediately follows that we get a finite contribution of the form (4 .
52) in the effectiveaction. While there are several contributions of this kind, we see that simply the threeloop counterterm analysis shows the existence of this term in the effective action with thecorrect perturbative structure and transcendentality of the various coefficients.
These are contributions from loop diagrams c, e, f, g, h and i in figure 2. Such counterterm contributions to the D R , D R and D R interactions have been discussed inappendix C. b Figure 6: A four loop diagram and a three loop counterterm O (Λ ) counterterm contributions There are no contributions from I ( c ) .From the loop diagram diagram e , we get Λ divergent contributions which have to becancelled by one loop counterterms. We describe this case in detail, as the analysis for theother cases proceeds along the same lines.For I ( e ) , from (4 .
21) we see that the one loop subdivergences are given by I ( e ) = − S Z d rr ( J + J ) − S Z d r ( k · r )( k · r ) r J (4.57)where J = Z d pd qk · ( q + k + k ) k · qq ( q + k ) ( q + k + k ) p ( p + k ) ( p + q ) , J = Z d pd qk · ( q + k + k ) k · qq ( q + k ) ( q + k + k ) p ( p + k ) ( p − q + k + k ) , (4.58)and J = Z d pd qp ( p + k ) q ( q + k ) ( p + q + k + k ) , (4.59)and the O ( k ) terms have to be kept in I ( e ) . The divergences in the terms involving J and J in (4 .
57) are cancelled by a four point counterterm depicted by a in figure 7, while thedivergences in the term involving J is cancelled by a five point counterterm in figure 8.Thus the counterterm is given by δ (3) I ( e ) = − S · c π l (4 π ) ( J + J ) − S · ˆ c π l (4 π ) J , (4.60)13here ˆ c is the coefficient of the one loop five point counterterm which is is the samesupermultiplet as the R counterterm.The expressions for J and J have been obtained in ( B. B. J can be calculated similarly. a b Figure 7: The planar and non–planar one loop four point counterterm diagramsFigure 8: The one loop five point counterterm diagramOne can calculate all the contributions coming from all the other loop diagrams inexactly the same way as described in appendix B. However, we shall refrain from doing so,as the calculations are quite cumbersome. Instead we shall consider the constraints imposedby the symmetries of the two loop skeleton diagram on the structure of the integrals, whichwill turn out to be enough for our purposes. It is easy to see what are the one loopcounterterm diagrams that contribute at O (Λ ). Diagram f only involves a in figure 7,while g involves a in figure 7 and figure 8. Diagram h only involves figure 8, and i involvesfigure 8 and b in figure 7. Ignoring the various numerical factors, we now consider thevarious integrals that arise while calculating the various diagrams. Though this tediousexercise involves considering the contributions from each diagram separately and includingthe numerators in (4 . counterterm contribution to I is of the form δ (3) I = c π σ l (4 π ) · π (4 π l V ) Z ∞ dσdλdρF L ( σ, λ, ρ ) P ( σ, λ, ρ ) (4.61)where P ( σ, λ, ρ ) = ˆ a σ λ + ˆ a σλρ ∆ / + ˆ a σ λ + ˆ a σ λρ + ˆ a σ λ ρ ∆ / + ˆ a σ λ + ˆ a σ λ ρ + ˆ a σ λ ρ + ˆ a σ λ ρ ∆ / , (4.62)and we have not distinguished between c and ˆ c in the coefficient of (4 .
61) because bothleave a finite remainder proportional to ζ (2) using (2 . . a i have vanishing transcendentality.We now analyze the structure that arises in (4 . O (∆ − / )because there are at most two derivatives of Schwinger parameters acting on ∆ − / . Atevery fixed order in ∆, the number of Schwinger parameters is completely determined bydimensional analysis as each Schwinger parameter has dimension l . In fact, there areseveral vanishing terms in (4 . σ term vanishes in the ∆ − / term,(ii) σ , σ λ terms vanish in the ∆ − / term,(iii) σ , σ λ , σ λ , σ λρ terms vanish in the ∆ − / term.In (4 . and each term is independent. While mostof the terms that vanish do not arise at all, the vanishing of the σ λ and σ λρ terms in the∆ − / term happens from cancellations between various contributions to the integrand. Apossible σ λ term always arises only in the combination (in arbitrary dimensions D ) Z ∞ dσdλdρF L σ λ (cid:16) ∂ ∂λ + ∂ ∂ρ − ∂ ∂λ∂ρ (cid:17) ∆ − D/ = Z ∞ dσdλdρF L σ λ h D ( D + 2)4∆ D/ (cid:16) ( σ + ρ ) + ( σ + λ ) − σ + ρ )( σ + λ ) (cid:17) + D ∆ D/ i , (4.63)and thus the σ λ term vanishes. The remaining terms are already in (4 . σ λρ term happens because it always arises only in the combination Z ∞ dσdλdρF L σ λρ (cid:16) ∂ ∂λ + ∂ ∂ρ − ∂ ∂λ∂ρ (cid:17) ∆ − D/ . (4.64)Note that vanishing happens in all dimensions. The contributions of the type (4 .
63) and(4 .
64) only come from diagrams f and i respectively. The symmetry group of the two loop skeleton diagram is S , which amounts to interchanging the threeSchwinger parameters among themselves.
15e now express P ( σ, λ, ρ ) in a manifestly S invariant way. To do so, we write thevarious non–invariant expressions in (4 .
62) in terms of S invariants. We construct thefollowing nine invariants:(i) At cubic order: p = σλρ, p = ∆( σ + λ + ρ ) . (4.65)(ii) At fifth order: p = ∆ p , p = ∆ p , p = σλρ ( σ + λ + ρ ) . (4.66)(iii) At seventh order: p = ∆ p , p = ∆ p , p = ∆ p , p = ( σλρ ) ( σ + λ + ρ ) . (4.67)Thus insider the integral in (4 . σλρ → p , σ λ → p − p , (4.68)(ii) at fifth order σ λ → p − p − p , σ λρ → p , σ λ ρ → p , (4.69)(iii) at seventh order σ λ → p − p − p p , σ λ ρ → p − p ,σ λ ρ → p − p , σ λ ρ → p . (4.70)Thus, we have that P ( σ, λ, ρ ) = a σ + λ + ρ ∆ / + a σλρ ∆ / + a σλρ ( σ + λ + ρ )∆ / + a ( σλρ ) ( σ + λ + ρ )∆ / , (4.71)where the undetermined numerical factors a i have vanishing transcendentality. Thus from(4 . δ (3) I = c π σ (4 π ) X ˆ m I , ˆ n I Z ∞ d ˆ σd ˆ λd ˆ ρe − π l G IJ (cid:16) ˆ λ ˆ m I ˆ m J +ˆ σ ˆ n I ˆ n J +ˆ ρ ( ˆ m +ˆ n ) I ( ˆ m +ˆ n ) J (cid:17) × h a (ˆ σ + ˆ λ + ˆ ρ ) ˆ∆ / + a ˆ σ ˆ λ ˆ ρ ˆ∆ / + a ˆ σ ˆ λ ˆ ρ (ˆ σ + ˆ λ + ˆ ρ )ˆ∆ / + a (ˆ σ ˆ λ ˆ ρ ) (ˆ σ + ˆ λ + ˆ ρ )ˆ∆ / i (4.72)16here we have Poisson resummed to go from momentum modes to winding modes anddefined [6, 7] ˆ ρ = ρ ∆ , ˆ σ = σ ∆ , ˆ λ = λ ∆ , (4.73)and ˆ∆ = ˆ σ ˆ ρ + ˆ σ ˆ λ + ˆ ρ ˆ λ = ∆ − . (4.74)Thus ˆ σ, ˆ λ, ˆ ρ have dimensions l − . Further defining τ = ˆ ρ ˆ ρ + ˆ λ , τ = p ˆ∆ˆ ρ + ˆ λ , V = l p ˆ∆ , (4.75)we get that δ (3) I = 6 c π σ (4 π ) l X ˆ m I , ˆ n I Z ∞ dV V Z F d ττ e − π G IJ ( ˆ m +ˆ nτ ) I ( ˆ m +ˆ n ¯ τ ) J V /τ A ( τ, ¯ τ ) (4.76)where d τ = dτ dτ and F is the fundamental domain of SL (2 , Z ) defined by F = {− ≤ τ ≤ , τ ≥ , | τ | ≥ } . (4.77)Also A is given by A ( τ, ¯ τ ) = a τ ( τ − T + 1) + a Tτ ( τ − T ) + a Tτ ( τ − T ) (cid:16) − T + ( τ − T ) (cid:17) + a T τ ( τ − T ) ( τ − T + 1) , (4.78)where T = | τ | − τ . (4.79)Thus the amplitude boils down to an integral over the moduli space of an auxiliary T parametrized by volume V and complex structure τ , where the integral of the complexstructure τ is over F . The integrand involves an SL (2 , Z ) invariant lattice factor, and anon– SL (2 , Z ) invariant function A ( τ, ¯ τ ).Now let us consider the structure of (4 . m I = ˆ n I = 0 in the lattice sum whichis of the form (Λ l ) arising from the boundary of the V integral cutoff at V ∼ (Λ l ) .Along with (2 .
8) this yields the two loop Λ primitive divergence.The subleading divergence is calculated as in appendix A, leading to∆ Ω δ (3) I div = 6 c π σ (4 π ) l Z ∞ dV V Z / − / dτ h A ∂ ˆ F L ∂τ − ˆ F L ∂ A ∂τ i(cid:12)(cid:12)(cid:12) τ =(Λ l ) /V (4.80)where ˆ F L is defined in ( A. . A → ( a + a T ) τ , ∂ A ∂τ → ( a + a T ) , (4.81)17eading to ∆ Ω δ (3) I div = 27 c π / σ (4 π ) l (cid:16) a + a (cid:17) (Λ l ) V − / E / (Ω , ¯Ω) , (4.82)and thus δ (3) I div ∼ Λ l − (cid:16) a + a (cid:17) σ V − / E / (Ω , ¯Ω) , (4.83)on using (2 . .
76) giveus that δ (3) I ∼ σ Λ + σ Λ l − V − / E / (Ω , ¯Ω) , (4.84)which cancel divergences in the three loop amplitude and leave no finite remainder. Thefinite remainder comes from the finite part of the integral in (4 .
76) which we now consider.For this purpose, it is very useful to note that A splits into a sum of functions A i eachof which satisfies Poisson equation on F . The structure of these equations is determinedrecursively [24]. We first start with the leading term in the small τ limit in (4 . a T (1 − T ) τ , (4.85)and construct a Poisson equation which has (4 .
85) as the dominant term in the small τ limit which we call A below. The leading subdominant term in this equation is O ( τ − ),which is subtracted from the O ( τ − ) terms in (4 .
78) which yields the leading term in thenext Poisson equation. This procedure proceeds recursively till all the terms are exhausted.We use the relations (cid:16) ∂T∂τ (cid:17) = 1 − T, ∂ T∂τ = 2( δ ( τ ) −
1) (4.86)repeatedly in our analysis. Thus we get that A ( τ, ¯ τ ) = X i =1 A i ( τ, ¯ τ ) (4.87)where we now discuss the structure of A i . Including the leading order term in the small τ expansion in the definition of A , we get that A = a ˆ A , (4.88)where ˆ A satisfies the Poisson equation∆ τ ˆ A = 56 ˆ A − τ δ ( τ ) − τ h τ + 1 τ i δ ( τ ) . (4.89)In (4 . A is given byˆ A = T (1 − T ) τ + T (6 − T + 45 T )13 τ + 3(1 − T + 140 T − T )143 τ + 5(11 − T + 210 T )429 τ + 5(11 − T ) τ
429 + 3 τ . (4.90)18ote that we get an O ( τ ) term which is not there in (4 . A i . At the first subleading order we get that A = − (cid:16) a + 6 a (cid:17) ˆ A (4.91)where ∆ τ ˆ A = 30 ˆ A − τ δ ( τ ) − τ h τ + 1 τ i δ ( τ ) . (4.92)Again ˆ A is given byˆ A = T (1 − T ) τ + 1 − T + 36 T − T τ + 2(4 − T + 35 T )21 τ + 4(2 − T ) τ
21 + τ . (4.93)The remaining Poisson equations for A i exactly follow the same pattern. We do notwrite them down explicitly as they are quite messy, and simply write down their generalform. The equations for A , . . . , A are given by∆ τ A = 12 A + ( τ + τ + τ − ) δ ( τ ) , ∆ τ A = 2 A + ( τ + τ ) δ ( τ ) , ∆ τ A = τ δ ( τ ) , ∆ τ A = 6 A , (4.94)where A i takes the form A ∼ T + T + T τ + 1 + T + T τ + τ (1 + T ) + τ , A ∼ T + T τ + τ (1 + T ) + τ , A ∼ τ (1 + T ) + τ , A ∼ τ , (4.95)where the various neglected coefficients are linear combinations of a i . Note that the eigen-value in every Poisson equation for A i is easily determined by the power of τ in the leadingcontribution for small τ .Now the finite part of (4 .
76) comes from the lattice sum where( ˆ m , ˆ m ) = (0 , , (ˆ n , ˆ n ) = (0 , . (4.96)To obtain this contribution, we use the relation∆ Ω X ˆ m I , ˆ n I e − π G IJ ( ˆ m +ˆ nτ ) I ( ˆ m +ˆ n ¯ τ ) J V /τ = ∆ τ X ˆ m I , ˆ n I e − π G IJ ( ˆ m +ˆ nτ ) I ( ˆ m +ˆ n ¯ τ ) J V /τ . (4.97)Thus defining δ (3) I finite = 6 c π σ (4 π ) l X i =1 I finitei (4.98)19e get that ∆ Ω I finitei = ′ X Z ∞ dV V Z F d ττ e − π G IJ ( ˆ m +ˆ nτ ) I ( ˆ m +ˆ n ¯ τ ) J V /τ ∆ τ A i (4.99)where we have integrated by parts twice, and the boundary contributions vanish using(4 . .
99) stands for the sum in (4 . A i satisfiesPoisson equation, we get an expression for the finite part of the amplitude. Let us considerthe contribution due to A to be concrete. From (4 . (cid:16) ∆ Ω − (cid:17) I finite = − a ′ X Z ∞ dV V Z ∞ dτ τ h
983 + 15( τ + τ − ) i e − π V V (cid:16) τ | ˆ m + ˆ m Ω | + τ − | ˆ n +ˆ n Ω | (cid:17) / Ω . (4.100)Using the symmetry of the integral under τ → τ − we get that (cid:16) ∆ Ω − (cid:17) I finite = − a ′ X Z ∞ dV V Z ∞ dτ τ h
983 + 15( τ + τ − ) i e − π V V (cid:16) τ | ˆ m + ˆ m Ω | + τ − | ˆ n +ˆ n Ω | (cid:17) / Ω . (4.101)Now substituting V τ = x, V τ − = y, (4.102)the integrals can easily be performed leading to (cid:16) ∆ Ω − (cid:17) I finite = − a π V − (cid:16) E / (Ω , ¯Ω) + 225 E / (Ω , ¯Ω) E / (Ω , ¯Ω) (cid:17) . (4.103)Exactly similar is the analysis for the expressions leading to equations for I finitei for i =2 , . . . ,
6. The source term involving τ δ ( τ ) in the Poisson equation leads to E / , while thesource term involving τ ( τ + τ − ) leads to E / E / . Note that the source terms of thesecond kind are actually of the form2 τ (cid:16) Aτ + Bτ (cid:17) δ ( τ ) = ( A + B ) τ (cid:16) τ + 1 τ (cid:17) δ ( τ ) + ( A − B ) τ (cid:16) τ − τ (cid:17) δ ( τ ) (4.104)where A and B are constants. The first term in (4 . τ leads to E / E / in the Poisson equation. From the general structure of the analysis of the twoloop supergravity amplitudes [24], we expect A = B , and we proceed ignoring such terms .It would be interesting to see if they vanish, and include their contributions otherwise. They produce source terms ′ X Z ∞ dV V Z ∞ dτ τ (cid:16) τ − τ (cid:17) e − π V V (cid:16) τ | ˆ m + ˆ m Ω | + τ − | ˆ n +ˆ n Ω | (cid:17) / Ω (4.105) (cid:16) ∆ Ω − λ i (cid:17) I finitei ∼ V − π (cid:16) E / (Ω , ¯Ω) + E / (Ω , ¯Ω) E / (Ω , ¯Ω) (cid:17) , (4.107)for i = 1 , , λ i = 56 , ,
12 respectively, and (cid:16) ∆ Ω − (cid:17) I finite ∼ V − π E / (Ω , ¯Ω) E / (Ω , ¯Ω) , (4.108)while contributions from A and A vanish . This conclusion is also true for the (4 . π V I finite ∼ c Ω + c Ω − + (cid:16) ζ (5) + ζ (3) ζ (7) (cid:17) Ω + ζ (2) ζ (7)Ω + ζ (4) ζ (5)Ω + ζ (3) ζ (6)Ω − + ζ (8)Ω − ,π V I finite ∼ c Ω + c Ω − + (cid:16) ζ (5) + ζ (3) ζ (7) (cid:17) Ω + ζ (2) ζ (7)Ω + ζ (4) ζ (5)Ω + ζ (3) ζ (6)Ω − + ζ (8)Ω − ,π V I finite ∼ c Ω + c Ω − + (cid:16) ζ (5) + ζ (3) ζ (7) (cid:17) Ω + ζ (2) ζ (7)Ω + ζ (4) ζ (5)Ω + ζ (3) ζ (6)Ω − + ζ (8)Ω − lnΩ ,π V I finite ∼ c Ω + c Ω − + ζ (3) ζ (7)Ω + ζ (2) ζ (7)Ω + ζ (4) ζ (5)Ω + ζ (3) ζ (6)Ω − lnΩ + ζ (8)Ω − . (4.110)Of them several are ruled out by the structure of string perturbation theory, while theremaining lead to terms in the effective action given by l s Z d x p − g B r B h ζ (4) ζ (5) + ζ (3) ζ (6) e φ B (cid:16) e − φ B ) (cid:17) + c e φ B + ζ (8) e φ B (cid:16) e − φ B ) (cid:17) + c e φ B + c e φ B + c e φ B i D R + . . . , (4.111)yielding perturbative contributions upto genus five. Note that it produces logarithmicallyinfrared divergent contributions at genus two and three. in the Poisson equation. Its leading perturbative contribution at large Ω is given by setting ˆ m = ˆ n = 0in (4 . V − Ω X ˆ m =0 , ˆ n =0 Z ∞ dV V Z ∞ dτ τ (cid:16) τ − τ (cid:17) ) e − π V (cid:16) τ ˆ m + τ − ˆ n (cid:17) (4.106)on simply rescaling V , which is inconsistent with string perturbation theory. We have that I finite ∼ E (Ω , ¯Ω) ∼ ζ (6)Ω + ζ (5)Ω − (4.109)which is inconsistent with string perturbation theory and must vanish in the whole amplitude.. Some of these terms have already arisen before in (4 . c and c respectively. To determine them, we multiply(4 . E (Ω , ¯Ω) and E (Ω , ¯Ω) respectivey and integrate over the fumdamental domainof SL (2 , Z ) Ω . This leads to [7] π V c ∼ π X k =0 k Z ∞ d Ω Ω (cid:16) µ ( | k | , / K (2 π | k | Ω )+ µ ( | k | , / µ ( | k | , / K (2 π | k | Ω ) K (2 π | k | Ω ) (cid:17) , (4.112)and π V c ∼ π X k =0 k Z ∞ d Ω Ω (cid:16) µ ( | k | , / K (2 π | k | Ω )+ µ ( | k | , / µ ( | k | , / K (2 π | k | Ω ) K (2 π | k | Ω ) (cid:17) , (4.113)where µ ( k, s ) = X m> ,m | k m s − . (4.114)Thus π V c ∼ π − ∞ X k =1 k h µ ( k, / + µ ( k, / µ ( k, / i ∼ ζ (12) ,π V c ∼ π − ∞ X k =1 k h µ ( k, / + µ ( k, / µ ( k, / i ∼ ζ (10) , (4.115)on using Ramanujan’s formula ∞ X k =1 µ ( k, s ) µ ( k, s ′ ) k r = ζ ( r ) ζ ( r + 2 s − ζ ( r + 2 s ′ − ζ ( r + 2 s + 2 s ′ − ζ (2 r + 2 s + 2 s ′ − . (4.116)This leads to genus four and five contributions of the form ζ (8) e φ B r B and ζ (10) e φ B r B respectively, on dividing by a factor of π to get the correct transcendentality. On knowingthe exact coefficients, one can determine c and c in an analogous way . O (Λ ) counterterm contributions There are no contributions at this order from I ( c ) . From the diagrams e, f, g, h and i ,the divergence is cancelled by a Λ two loop primitive counterterm which leaves no finiteremainder as discussed before. Hence there are no finite contributions that remain. The individual contributions have a divergence as Ω → SL (2 , Z ) Ω . D R interaction at three loops coming from regularizingthe ultraviolet divergences. The various loop diagrams that contribute to this amplitudeare obtained from two types of skeleton diagrams: the ladder and the Mercedes skeletondiagrams. Focussing only on the divergences which can potentially yield a finite remainder,we see that the relevant one loop subdivergent contributions are O (Λ ) and the two loopsubdivergent contributions are O (Λ ). On regularizing these divergences, the finite contri-butions come from the finite remainder of the O (Λ ) counterterm of one loop supergravity.It is the square of these counterterms that contribute at O (Λ ).While the O (Λ ) divergent contribution does not leave any remainder for the diagramsthat arise from the Mercedes skeleton, it does leave a remainder (4 .
52) for the diagramsthat arise from the ladder skeleton, which provides a part of the complete answer. Thisinvolves the SL (2 , Z ) invariant coupling E / (Ω , ¯Ω). On the other hand, regularizing thethe O (Λ ) divergent contributions is more involved. For the diagrams that arise from theMercedes skeleton and some of those that arise from the ladder skeleton, we regularizeby including the effects of four and five point one loop counterterms. The regularizedamplitude involves performing two loop integrals, and yield Poisson equations with sourceterms of the form E / (Ω , ¯Ω) and E / (Ω , ¯Ω) E / (Ω , ¯Ω). The remaining contribution thatcomes from the diagram that arises from the ladder skeleton involves the square of oneloop integrals leading to E / (Ω , ¯Ω). These produce contributions upto genus five in stringperturbation theory.Thus from the analysis of the various counterterms we see that there are several possiblenon–vanishing contributions to the D R interaction from three loop quantum supergrav-ity. Perturbatively, they lead to contributions upto genus five. It would be interesting togeneralize the analysis to other non–BPS interactions at higher orders in the momentumexpansion at three loops and beyond. D R interaction Ignoring an irrelevant overall numerical factor, the two loop D R interaction is given by I D R = π σ l X ˆ m I ˆ n I Z ∞ dV V Z F d ττ e − π G IJ ( ˆ m +ˆ nτ ) I ( ˆ m +ˆ n ¯ τ ) J V /τ B ( τ, ¯ τ ) , (A.117)where [24] B ( τ, ¯ τ ) = 45 τ + (1 − T ) + 2(2 − T + 40 T )5 τ + 2 T (11 − T )5 τ + 32 T τ , (A.118)where T is given by (4 . m I = ˆ n I = 0 and is of the form Λ coming from the boundary of the V integral cutoff at V ∼ ( l Λ) . 23he subleading ultraviolet divergence comes from the boundary of moduli space when τ → ∞ keeping V fixed [6]. Hence we only need to isolate the boundary contribution,which is done along the lines of [6] and so we only mention the results. This subdivergentpart is given by∆ Ω I D R div = π σ l Z ∞ dV V Z / − / dτ h B ∂ ˆ F L ∂τ − ˆ F L ∂ B ∂τ i(cid:12)(cid:12)(cid:12) τ =(Λ l ) /V (A.119)where ˆ F L = X ˆ m I , ˆ n I e − π G IJ ( ˆ m +ˆ nτ ) I ( ˆ m +ˆ n ¯ τ ) J V /τ , (A.120)and we have defined ∆ Ω = 4Ω ∂ ∂ Ω ¯ ∂ Ω . (A.121)This structure comes from integrating by parts and considering only the boundary contribu-tion, as all the other contributions are finite (apart from the two loop primitive divergence).Noting that as τ → ∞ , B → τ , ∂ B ∂τ → τ , (A.122)we get that ∆ Ω I D R div = − π / σ l Λ V − / E / (Ω , ¯Ω) (A.123)leading to I D R div = 12 π / σ l Λ V − / E / (Ω , ¯Ω) , (A.124)leading to the term in the effective action of the form (2 . B Performing the various one loop counterterm integrals
After factorizing out the one loop counterterms, we need to calculate several two loopintegrals in the compactified theory. When the integrands in these integrals are expandedto the required order in the external momenta, we are left with integrands that involve twoor four powers of the continuous loop momenta along with Lorentz scalars constructed outof the loop momenta. Thus the Lorentz structure of these integrals is fixed. Hence in these D dimensional integrals, we set ( D = 9 for our case) p µ p ν → η µν p D , p µ q ν → η µν p · qD , (B.125)24nd p µ p ν p λ p ρ → D ( D + 2) ( η µν η λρ + η µλ η νρ + η µρ η νλ )( p ) ,p µ p ν p λ q ρ → D ( D + 2) ( η µν η λρ + η µλ η νρ + η µρ η νλ ) p ( p · q ) ,p µ p ν q λ q ρ → η µν η λρ ( D + 2) D ( D − (cid:16) ( D + 1) p q − p · q ) (cid:17) + η µλ η νρ + η µρ η νλ ( D + 2)( D − (cid:16) ( p · q ) − p q D (cid:17) . (B.126)We now evaluate the integrals at O ( k ) in (4 . J we have to keep the O ( k ) term.Thus compactified on T , we get that J = π S π l V ) Z ∞ dσdλdρF L h σλρ + 3 σ λρ ∂∂σ − σ λ (cid:16) ∂∂σ − ∂∂ρ (cid:17)i ∆ − / = π S π l V ) Z ∞ dσdλdρF L σλρ ∆ − / . (B.127)In these integrals there are non–trivial factors of loop momenta in the numerator whichare taken care of by replacing them with appropriate derivatives of the Schwinger parame-ters. For example, the integral − Z d pd q ( p · k )( q · k ) p q ( p + q ) (B.128)in the uncompactified theory, becomes S Z ∞ dσdλdρF L σ λ Z d pd q (cid:16) ( p + q ) − p − q (cid:17) e − σp − λq − ρ ( p + q ) = − π S Z ∞ dσdλdρF L σ λ (cid:16) ∂∂ρ − ∂∂σ − ∂∂λ (cid:17) ∆ − / (B.129)on compactifying on T .In J and J we have to keep the O ( k ) term. It is easier to write down the expressionafter performing the sum in (4 . T , J is given by X S S J = π σ π l V ) Z ∞ dσdλdρF L h σ λ ∂∂σ + σ λ ∂ ∂σ − σ λ (cid:16) ∂ ∂σ∂ρ − ∂ ∂σ∂λ − ∂ ∂σ (cid:17)i ∆ − / = − π σ π l V ) Z ∞ dσdλdρF L σ λ ( λ + ρ )∆ − / . (B.130)Expressing the result in an S symmetric way, we get that X S S J = − π σ π l V ) Z ∞ dσdλdρF L ∆ − / h ∆( σ + λ + ρ ) − σλρ i . (B.131) J can be calculated in exactly the same way, but the details are more involved and weomit the expression. 25 More interactions involving one loop counterterms
C.1 Finite contributions from one loop counterterms
We considered the contribution to the D R interaction from the counterterm in figure4 in the main text. This involves the square of one loop amplitudes which are easilyobtained leading to (4 . D R , D R , D R and D R interactions have twoloop primitive divergences of the form Λ , Λ , Λ and Λ respectively, and hence receiveone loop counterterm contributions in figure 4. We analyze these contributions below.They are obtained only from I ( a ) by expanding I ( a ) = S Z d rr Z d pp ( p + k ) ( p + k + k ) Z d qq ( q + k ) ( q + k + k ) (C.132)for r → ∞ in the compactified theory, to the relevant order in the momentum expansion.Thus compactified on T , the integral T = Z d pp ( p + k ) ( p + k + k ) (C.133)has to be expanded only upto O ( S ). This integral gives us T = π / π l V X m I Z ∞ dσσ − / e − G IJ m I m J σ/l Z dω Z ω dω e (1 − ω )( ω − ω ) σS . (C.134)The q integral in ( C. O ( S ), this equals T = π / π h l (cid:16)
25 (Λ l ) + 34 π / V − / E / (Ω , ¯Ω) (cid:17) + S l (cid:16)
23 (Λ l ) + 12 π / V − / E / (Ω , ¯Ω) (cid:17) + S l (cid:16) l ) + 1 π / V − / E / (Ω , ¯Ω) (cid:17) + S l (cid:16) − l ) + 12 π / V / E / (Ω , ¯Ω) (cid:17) + S l (cid:16) − l ) + 34 π / V / E / (Ω , ¯Ω) (cid:17)i , (C.135)where we have used Γ( s ) E s (Ω , ¯Ω) = π s − Γ(1 − s ) E − s (Ω , ¯Ω) (C.136)for s = − / , − /
2. Thus the counterterm is given by δ (3) I ( a ) = c π S l (4 π ) T , (C.137)26here we have to keep terms upto O ( S ). Using (2 . I = π π ) l h σ π l V − E / + σ π l V − E / E / + σ V − π l (cid:16) E / E / + 524 E / (cid:17) + σ V − π l (cid:16) E / E / + 56 π E / E / (cid:17) + σ V − π l (cid:16) E / + 15 π E / + 56 π E / (cid:17)i . (C.138)Dropping irrelevant numerical factors, these lead to terms in the effective action of the form(the D R interaction is given in (4 . l Z d x p − G (9) h V − E / E / D R + l V − ( E / E / + E / ) D R + l V − ( E / E / + π E / E / ) D R + l ( E / + π E / + π E / ) D R i (C.139)for the various interactions. Dropping exponentially suppressed contributions, this leads toseveral perturbative contributions in the type IIB effective action of the form l s Z d x p − g B r B h ζ (3) ζ (5) e − φ B + ζ (2) ζ (5) + ζ (3) ζ (4) e φ B + ζ (6) e φ B i D R + l s Z d x p − g B r − B h ln( e − φ B ) (cid:16) ζ (5) + ζ (4) e φ B (cid:17) + ζ (3) + ζ (2) ζ (3) e φ B + ζ (4) e φ B i D R + l s Z d x p − g B r − B h ζ (2)ln( e − φ B ) (cid:16) ζ (3) e φ B + ζ (2) e φ B (cid:17) + ζ (3) ζ (5) + ζ (2) ζ (5) e φ B + ζ (3) ζ (4) e φ B + ζ (6) e φ B i D R + l s Z d x p − g B r − B h ζ (4) e φ B ln ( e − φ B ) + ζ (2) (cid:16) ζ (3) e φ B + ζ (2) ζ (3) e φ B + ζ (4) e φ B (cid:17) + ζ (5) + ζ (4) ζ (5) e φ B + ζ (8) e φ B i D R . (C.140)This leads to several perturbative contributions from genus zero to genus five for stringamplitudes. The tree level and one loop D R terms agree with known results. The D R , D R and D R interactions have infrared divergent logarithmic terms. C.2 Finite contributions from product of one loop counterterms
Proceeding like the analysis above, we see that apart from the D R interaction, the D R , D R and D R interactions also receive finite contributions from the countertermin figure 5. The integrals involved are simple one loop integrals which receive contributions27rom I ( a ) , I ( b ) and I ( d ) to yield (the D R contribution is given in (4 . δ (3) I = 9 π π ) l (cid:16) c π (cid:17) h σ (cid:16)
25 (Λ l ) + 34 π / V − / E / (Ω , ¯Ω) (cid:17) + σ l (cid:16)
23 (Λ l ) + 12 π / V − / E / (Ω , ¯Ω) (cid:17) + σ l (cid:16) l ) + 1 π / V − / E / (Ω , ¯Ω) (cid:17)i , (C.141)leading to finite contributions given by I = π l h σ π / V − / E / + σ l π / V − / E / + σ l π / V − / E / i (C.142)on using (2 . C. l Z d x p − G (9) h V − / E / D R + l V − / E / D R + l V / E / D R i (C.143)which produces terms of the form l s Z d x p − g B r − B h ζ (5) + ζ (4) e φ B i D R + l s Z d x p − g B r − B h ζ (3) e φ B + ζ (2) e φ B i D R + l s Z d x p − g B r − B e φ B ln( e − φ B ) D R (C.144)in the type IIB effective action on dropping exponentially suppressed corrections. References [1] M. B. Green and M. Gutperle, “Effects of D-instantons,”
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