Quick overview

Motivation: By now the Virasoro-Shapiro amplitude for flat space tree-level tachyon scattering has been known for many decades. But determining analogous formulas for tree-level scattering on general curved backgrounds still remains a challenge.

Concrete Goal: Compute string scattering amplitudes to all orders in α' in a small AdS curvature expansion around flat space.

Scenario 1: World-sheet theory unknown.

Strategy:

1. Analytic Bootstrap: Determine the low energy expansion of the  amplitude using Regge boundedness of the corresponding correlator in the dual CFT and the CFT data of heavy operators known e.g. from integrability.

2. Observation: 	No matter what the World-sheet theory is, it should express the scattering amplitude as some integral over String World-sheets. The integrand is strongly constrained by demanding crossing symmetry and consistency with the low energy expansion.

3. Assumption: Considering all constraints, it was conjectured that the typical world sheet integrands should be expressible as some weighted sum of Single Valued  Multi PolyLogarithms (SVMPL).

4. Computation: If we can reduce the number of unknowns in the small AdS curvature expansion up to a certain order to finitely many coefficients, we can compute those by matching with the structure imposed by the decomposition of the dual correlator into conformal blocks.

Some of the papers that have applied/developed this strategy are:

Today we look at a paper, where the NLO term in the curvature expansion for the four-point scattering amplitude of Dilatons in (Subscript[AdS, 3])(xS^3)(Subscript[xM, 4]) with pure RR flux is computed using the above strategy. ((Subscript[M, 4]) is (T^4) or K3. At tree level the Dilaton is not sensitive to the difference.)

Note that a few days ago, there was also another paper on the arxiv, where a similar computation was done for graviton scattering of IIA on AdS4 x CP3. In that case, using extra input from ABJM integrability, allowed for the computation not only of the first, but also of the second curvature correction.

Scenario 2:  World-sheet theory known.

Strategy: Write down the integral and expand it. (Relevance: An important check for the SVMPL Assumption in scenario 1.)

Some details of the computation

0. Single Valued Multi Polylogarithms

1.1. MPLs

Clear[L]

L[{}]:=1
L[{0}]:=Log[z]
L[{0,0}]:=(1/2)((Log[z])^2)
L[{0,0,0}]:=(1/6)((Log[z])^3)

FullSimplify[f[z]/.DSolve[D[f[z],z]==((L[{}])/(z-1))&&f[0]==0,f,{z}][[1]]//Expand]

L[{1}]:=Log[1-z];

L[w_]:=L[w]=FullSimplify[f[z]/.DSolve[D[f[z],z]==((L[w[[2;;]]])/(z-w[[1]]))&&f[0]==0,f,{z}][[1]]//Expand,Assumptions->z>10]

L[{0,0,1}]

L[{0,1,1}]

L[{1,1,1}]

1.2. SVMPLs

Length3Q[w_]:=(Length[w]==3)
SVL[w_?Length3Q]:=L[w]+(L[Reverse[w]]/.z->zb)+L[w[[1;;2]]](L[{w[[3]]}]/.z->zb)+L[{w[[1]]}](Reverse[L[w[[2;;3]]]]/.z->zb)

SVL[{}]=1;
SVL[{0}]=Log[z zb];
SVL[{1}]=Log[(1-z)(1- zb)];
SVL[{0,0}]=(1/2)((Log[z zb])^2);

1. Flat space expansion

The expansion around flat space is the expansion around 1/λ = 0. Here, λ is defined as

Hence, we are interested in (A^((1))) in the expansion

(A^((0))) is just the flat space Virasoro Shapiro amplitude

2. World sheet integral Ansatz

The Ansatz for the world-sheet integral dictated by crossing symmetry and conformal invariance is

For (G^((0))) one gets

The (G^((1))) Ansatz dictated by the transcendentality of the coefficients in the low energy expansion is

where,  the (p^s) and (p^a) are symmetric and anti-symmetric homogenous polynomials of degree 2

ps[i_]:=cs[i,1]((S^2)+(T^2))+cs[i,2]S T;
pa[i_]:=ca[i,1]((S^2)-(T^2));

(L^a) and (L^s) are defined as

Ls[w_]:=SVL[w]+(SVL[w]/.z->1-z)+(SVL[w]/.z->zb)+(SVL[w]/.z->1-zb);
La[w_]:=SVL[w]-(SVL[w]/.z->1-z)+(SVL[w]/.z->zb)-(SVL[w]/.z->1-zb);

and the specific symmetric and anti-symmetric SVMPL chosen in the Ansatz are

wList={{0,0,0},{0,0,1},{0,1,0}};
G1=Sum[Ls[wList[[i]]]ps[i]+La[wList[[i]]]pa[i],{i,1,3}]+Zeta[3]ps[4]

In total there are 4x2 + 3 = 11 coefficients to fix.

3. Flat space expansion from holography

The full string scattering amplitude including curvature correction is given by the Borel transform of the corresponding CFT Mellin amplitude.

The Mellin amplitude decomposes into a sum over Mack polynomials, which are Mellin transforms of conformal blocks, weighted by OPE coefficients (plus a protected part)

Assuming sufficiently well behaved convergence of the OPE, we swap the Borel transform and the above decomposition of the Mellin Amplitude to get the contributions of individual primaries to the string scattering amplitude.

We are interested in the Poles of (Subscript[A, τ,l]) in S. These have an expansion of the form

where the first few terms are

RN[0][τf_,l_]:=((16 ChebyshevT[l,-1-((2 T)/S)])/((π^3) (S^2) (-4 S τf+((τf)^3))));
RN[1][τf_,l_]:=-((8 (4 (3+2 l) S+(1+2 l) ((τf)^2)) ChebyshevT[l,1+((2 T)/S)])/( (π^3) (S^2) ((τf)^2) (((-4 S+((τf)^2)))^2)));
RN[2][τf_,l_]:=(-(2/3) l S (S+2 T) (4 S-((τf)^2)) (48 (S^2)-152 S ((τf)^2)+11 ((τf)^4)) ChebyshevT[-1+l,1+((2 T)/S)]-(2/3) (384 (l^2) S T (S+T) ((τf)^2) (4 S-((τf)^2))+T (S+T) (192 (S^3)+4080 (S^2) ((τf)^2)-668 S ((τf)^4)+5 ((τf)^6))-l (4 S-((τf)^2)) (48 (S^4)+20 (T^2) ((τf)^4)+8 (S^3) (24 T-19 ((τf)^2))-20 S T ((τf)^2) (64 T-((τf)^2))+(S^2) (192 (T^2)-1280 T ((τf)^2)+11 ((τf)^4)))) ChebyshevT[l,1+((2 T)/S)])/((π^3) (S^2) T (S+T) ((τf)^3) (((-4 S+((τf)^2)))^4));

In Addition to the expansion of the blocks, there is also an expansion of the CFT data.

4. Matching

4.1. The flat space limit

σ2=(S^2)+(T^2)+(U^2);
A0=-((σ2 Gamma[-S]Gamma[-T]Gamma[-U])/(4Gamma[S+1]Gamma[T+1]Gamma[U+1]))/.U->-S-T;
FunctionPoles[A0,S]

RN[0][τ,l]

We want poles at non negative integer S and thus τ0[δ,l]=2(Sqrt[δ]) with δ=1,2,....

δ=4;
lhalfmax=5;
BlockExpansion=(π^3)Sum[Residue[RN[0][2(Sqrt[δ]),2l]f0[δ,2l],{S,δ}]//FullSimplify,{l,0,lhalfmax}];
VirasoroShapiro=Residue[A0,{S,δ}]//FullSimplify;
Solve[CoefficientList[BlockExpansion-VirasoroShapiro,T]==0]

We see that coefficients with 2δ>l vanish. The nonvanishing coefficients match with those given in the paper.

f0list={f0[1,0]->(7/32),f0[4,0]->(13/6),f0[9,0]->(12930438637926492183/1379227385882214400),f0[16,0]->(30540149740456016/1123136891296875),f0[1,2]->(1/32),f0[4,2]->(7/2),f0[9,2]->(24109654015419440229/1379227385882214400),f0[16,2]->(7146080687579606224/135899563846921875),f0[2,0]->(15/(16 (Sqrt[2]))),f0[3,0]->((12411 (Sqrt[3]))/16384),f0[5,0]->((96717225 (Sqrt[5]))/67108864),f0[6,0]->((11973123 (Sqrt[(3/2)]))/3276800),f0[7,0]->((155687368062779 (Sqrt[7]))/69578470195200),f0[8,0]->((106006 (Sqrt[2]))/19845),f0[10,0]->((19876590764203625 (Sqrt[(5/2)]))/2761569482047488),f0[11,0]->((157948953934079575281847327 (Sqrt[11]))/38609539749432357027840000),f0[12,0]->((319212278817 (Sqrt[3]))/34693120000),f0[13,0]->((176583458425496963931480259921883 (Sqrt[13]))/34443910174418400709277908992000),f0[14,0]->((13379788998055473385083391 (Sqrt[(7/2)]))/1180135640051233062912000),f0[15,0]->((369551612606711045971094595184275 (Sqrt[15]))/59355899128804224336553389850624),f0[17,0]->((318016920095921418691025353151599616027421056759 (Sqrt[17]))/43068290920326274779330246049253333925888000000),f0[18,0]->(391822169375799319226175663888833199/(8179595398056230942375149568000000 (Sqrt[2]))),f0[4,4]->(65/36),f0[9,4]->(348013302709679919/24629060462182400),f0[16,4]->(3874019978551603688/81539738308153125),f0[4,6]->(1/2),f0[9,6]->(967676151727807749/98516241848729600),f0[16,6]->(3266196375876313424/81539738308153125),f0[4,8]->(1/36),f0[9,8]->(1991378786773760091/344806846470553600),f0[16,8]->(512611767068464072/16307947661630625),f0[9,10]->(1923122230596910647/689613692941107200),f0[16,10]->(9324143295359747968/407698691540765625),f0[2,2]->(1/(Sqrt[2])),f0[3,2]->((35703 (Sqrt[3]))/32768),f0[5,2]->((1491049825 (Sqrt[5]))/603979776),f0[6,2]->((1326987 (Sqrt[(3/2)]))/204800),f0[7,2]->((1129587534443179 (Sqrt[7]))/278313880780800),f0[8,2]->((325594 (Sqrt[2]))/33075),f0[10,2]->((2337162606202325 (Sqrt[(5/2)]))/172598092627968),f0[11,2]->((598520767316918019484383121 (Sqrt[11]))/77219079498864714055680000),f0[12,2]->((2128890131703 (Sqrt[3]))/121425920000),f0[13,2]->((48293798417918747099524735117241 (Sqrt[13]))/4920558596345485815611129856000),f0[14,2]->((2191626838763165406414413 (Sqrt[(7/2)]))/100579742049820999680000),f0[15,2]->((5700896445111660296502741191237675 (Sqrt[15]))/474847193030433794692427118804992),f0[17,2]->((14795866537807201285441809976333510002106769063 (Sqrt[17]))/1033638982087830594703925905182080014221312000),f0[18,2]->(9515032033640288284316122494374889/(102244942475702886779689369600000 (Sqrt[2]))),f0[9,12]->(25875765257907897/24629060462182400),f0[16,12]->(6255810680750301952/407698691540765625),f0[9,14]->(775703736696989259/2758454771764428800),f0[16,14]->(768317487206107136/81539738308153125),f0[9,16]->(60326101703732157/1379227385882214400),f0[16,16]->(142509000791324672/27179912769384375),f0[9,18]->(5559060566555523/2758454771764428800),f0[16,18]->(32226201823870976/12354505804265625),f0[2,4]->(1/(16 (Sqrt[2]))),f0[3,4]->((6237 (Sqrt[3]))/16384),f0[5,4]->((231490625 (Sqrt[5]))/150994944),f0[6,4]->((29382759 (Sqrt[(3/2)]))/6553600),f0[7,4]->((15529073351533 (Sqrt[7]))/5153960755200),f0[8,4]->((761617 (Sqrt[2]))/99225),f0[10,4]->((31070056931389825 (Sqrt[(5/2)]))/2761569482047488),f0[11,4]->((28230741698348579718113687 (Sqrt[11]))/4289948861048039669760000),f0[12,4]->((3678318050201 (Sqrt[3]))/242851840000),f0[13,4]->((123435574360751005908971889534659 (Sqrt[13]))/14351629239341000295532462080000),f0[14,4]->((1368263817079908428184945881 (Sqrt[(7/2)]))/70808138403073983774720000),f0[15,4]->((2553376537541052412443624077990075 (Sqrt[15]))/237423596515216897346213559402496),f0[17,4]->((11691370588924762607168436816668336220205940807 (Sqrt[17]))/897256060840130724569380126026111123456000000),f0[18,4]->(139485866402711990852893328548021137/(1635919079611246188475029913600000 (Sqrt[2]))),f0[16,20]->(154845122714927104/135899563846921875),f0[16,22]->(34755517792387072/81539738308153125),f0[16,24]->(2148946889670656/16307947661630625),f0[3,6]->((729 (Sqrt[3]))/32768),f0[5,6]->((86828125 (Sqrt[5]))/134217728),f0[6,6]->((192051 (Sqrt[(3/2)]))/81920),f0[7,6]->((11113991966077 (Sqrt[7]))/6184752906240),f0[8,6]->((1226 (Sqrt[2]))/245),f0[10,6]->((708367713146875 (Sqrt[(5/2)]))/86299046313984),f0[11,6]->((384956784317948202700215623 (Sqrt[11]))/77219079498864714055680000),f0[12,6]->((18647356293 (Sqrt[3]))/1576960000),f0[13,6]->((96749020698495425692803654165493 (Sqrt[13]))/14058738846701388044603228160000),f0[14,6]->((596885830599291322218101 (Sqrt[(7/2)]))/37824860258052341760000),f0[15,6]->((4242736752798830494095083563895025 (Sqrt[15]))/474847193030433794692427118804992),f0[17,6]->((71875678749229606207243733593638789936678838463 (Sqrt[17]))/6460243638048941216899536907388000088883200000),f0[18,6]->(110682306261779297041084543783137/(1503602095230924805583667200000 (Sqrt[2]))),f0[16,26]->(2600499618512896/81539738308153125),f0[16,28]->(18004502904832/3261589532326125),f0[16,30]->(233096465088512/407698691540765625),f0[16,32]->(8796093022208/407698691540765625),f0[5,8]->((91796875 (Sqrt[5]))/603979776),f0[6,8]->((2797173 (Sqrt[(3/2)]))/3276800),f0[7,8]->((6378481360853 (Sqrt[7]))/7730941132800),f0[8,8]->((263857 (Sqrt[2]))/99225),f0[10,8]->((3588894365265625 (Sqrt[(5/2)]))/690392370511872),f0[11,8]->((6451364522196238764315779 (Sqrt[11]))/1930476987471617851392000),f0[12,8]->((402678397529 (Sqrt[3]))/48570368000),f0[13,8]->((1723328663194134099136732207488863 (Sqrt[13]))/344439101744184007092779089920000),f0[14,8]->((19316716549172712446446103 (Sqrt[(7/2)]))/1634033963147861164032000),f0[15,8]->((16628264017037641655039843834375 (Sqrt[15]))/2422689760359356095369526116352),f0[17,8]->((143613877885693472861214131747346204773046615839 (Sqrt[17]))/16150609095122353042248842268470000222208000000),f0[18,8]->(3595183951500906295855815192943809/(60144083809236992223346688000000 (Sqrt[2]))),f0[5,10]->((9765625 (Sqrt[5]))/1207959552),f0[6,10]->((72171 (Sqrt[(3/2)]))/409600),f0[7,10]->((74246641872907 (Sqrt[7]))/278313880780800),f0[8,10]->((36304 (Sqrt[2]))/33075),f0[10,10]->((241838787109375 (Sqrt[(5/2)]))/86299046313984),f0[11,10]->((100653689805551704828525013 (Sqrt[11]))/51479386332576476037120000),f0[12,10]->((2241053883 (Sqrt[3]))/433664000),f0[13,10]->((6778718828237514255615266678137403 (Sqrt[13]))/2066634610465104042556674539520000),f0[14,10]->((21443140493316402563485643 (Sqrt[(7/2)]))/2655305190115274391552000),f0[15,10]->((2303852966664712599912412589359375 (Sqrt[15]))/474847193030433794692427118804992),f0[17,10]->((214197159772469024080249308358551973102525381663 (Sqrt[17]))/32301218190244706084497684536940000444416000000),f0[18,10]->(342349464841601452463991539695473/(7518010476154624027918336000000 (Sqrt[2]))),f0[6,12]->((59049 (Sqrt[(3/2)]))/6553600),f0[7,12]->((6879119738897 (Sqrt[7]))/139156940390400),f0[8,12]->((6368 (Sqrt[2]))/19845),f0[10,12]->((6878863876953125 (Sqrt[(5/2)]))/5523138964094976),f0[11,12]->((75454010670950667100975531 (Sqrt[11]))/77219079498864714055680000),f0[12,12]->((2446019613 (Sqrt[3]))/867328000),f0[13,12]->((165084707433938892536418955973669 (Sqrt[13]))/86109775436046001773194772480000),f0[14,12]->((353127795272229245322177907 (Sqrt[(7/2)]))/70808138403073983774720000),f0[15,12]->((744240989867707110096318603515625 (Sqrt[15]))/237423596515216897346213559402496),f0[17,12]->((37095518035658699053793556416135991485979296653 (Sqrt[17]))/8075304547561176521124421134235000111104000000),f0[18,12]->(7800350313156067988348244434045379/(240576335236947968893386752000000 (Sqrt[2]))),f0[7,14]->((678223072849 (Sqrt[7]))/278313880780800),f0[8,14]->((256 (Sqrt[2]))/4725),f0[10,14]->((149412841796875 (Sqrt[(5/2)]))/345196185255936),f0[11,14]->((186893846470254263857714571 (Sqrt[11]))/463314476993188284334080000),f0[12,14]->((79960644207 (Sqrt[3]))/60712960000),f0[13,14]->((1015473408770489713153602346750403 (Sqrt[13]))/1033317305232552021278337269760000),f0[14,14]->((18256662843157223541350231 (Sqrt[(7/2)]))/6638262975288185978880000),f0[15,14]->((870689855945996819472136376953125 (Sqrt[15]))/474847193030433794692427118804992),f0[17,14]->((21091392097602625260156757222157858664633992411 (Sqrt[17]))/7178048486721045796555041008208888987648000000),f0[18,14]->(219539785725311709934579685337981/(10224494247570288677968936960000 (Sqrt[2]))),f0[8,16]->((256 (Sqrt[2]))/99225),f0[10,16]->((295672607421875 (Sqrt[(5/2)]))/2761569482047488),f0[11,16]->((3019650230259965334702307 (Sqrt[11]))/23165723849659414216704000),f0[12,16]->((1603764279 (Sqrt[3]))/3153920000),f0[13,16]->((222908354400351119593920700143491 (Sqrt[13]))/516658652616276010639168634880000),f0[14,16]->((70698138997738459514202917 (Sqrt[(7/2)]))/53106103802305487831040000),f0[15,16]->((56899105454744165340802001953125 (Sqrt[15]))/59355899128804224336553389850624),f0[17,16]->((55611334437312196934072155880381831253059657953 (Sqrt[17]))/32301218190244706084497684536940000444416000000),f0[18,16]->(5374004692630647167101024114932279/(408979769902811547118757478400000 (Sqrt[2]))),f0[10,18]->((762939453125 (Sqrt[(5/2)]))/49313740750848),f0[11,18]->((4635797896899374194311509 (Sqrt[11]))/154438158997729428111360000),f0[12,18]->((1697127309 (Sqrt[3]))/11038720000),f0[13,18]->((53952758401426084988476460332249 (Sqrt[13]))/344439101744184007092779089920000),f0[14,18]->((10440753141129127900097 (Sqrt[(7/2)]))/18912430129026170880000),f0[15,18]->((208949570745513627562591552734375 (Sqrt[15]))/474847193030433794692427118804992),f0[17,18]->((1790152875159002485327953473556309182918790187 (Sqrt[17]))/1957649587287557944515011184056969723904000000),f0[18,18]->(144893332597222364552001250574769/(19662488937635170534555648000000 (Sqrt[2]))),f0[10,20]->((3814697265625 (Sqrt[(5/2)]))/5523138964094976),f0[11,20]->((939831742920786332853797 (Sqrt[11]))/231657238496594142167040000),f0[12,20]->((8051862591 (Sqrt[3]))/242851840000),f0[13,20]->((11505464793675213585069467609819 (Sqrt[13]))/258329326308138005319584317440000),f0[14,20]->((618664332623548270943723 (Sqrt[(7/2)]))/3268067926295722328064000),f0[15,20]->((3160155524181509639739990234375 (Sqrt[15]))/18263353578093607488170273800192),f0[17,20]->((28981670082523264323087933995336372805976463 (Sqrt[17]))/66738054112075839017557199456487604224000000),f0[18,20]->(15322533711635132564857554582975303/(4089797699028115471187574784000000 (Sqrt[2]))),f0[11,22]->((81402749386839761113321 (Sqrt[11]))/463314476993188284334080000),f0[12,22]->((511777683 (Sqrt[3]))/121425920000),f0[13,22]->((18680739035866441675732656623597 (Sqrt[13]))/2066634610465104042556674539520000),f0[14,22]->((10383287821041952078129 (Sqrt[(7/2)]))/204254245393482645504000),f0[15,22]->((26709731383003420886993408203125 (Sqrt[15]))/474847193030433794692427118804992),f0[17,22]->((1949185124895233946714032968465927775999582939 (Sqrt[17]))/10767072730081568694832561512313333481472000000),f0[18,22]->(12801606924018101926917980001087/(7518010476154624027918336000000 (Sqrt[2]))),f0[12,24]->((43046721 (Sqrt[3]))/242851840000),f0[13,24]->((373989730787941283299629203729 (Sqrt[13]))/344439101744184007092779089920000),f0[14,24]->((345202010200440597834259 (Sqrt[(7/2)]))/35404069201536991887360000),f0[15,24]->((1699637617184107303619384765625 (Sqrt[15]))/118711798257608448673106779701248),f0[17,24]->((208688020297852692661815625422336052923319253 (Sqrt[17]))/3230121819024470608449768453694000044441600000),f0[18,24]->(5835165879554652433576412637873/(8592011972748141746192384000000 (Sqrt[2]))),f0[13,26]->((91733330193268616658399616009 (Sqrt[13]))/2066634610465104042556674539520000),f0[14,26]->((1341068619663964900807 (Sqrt[(7/2)]))/1206956904597851996160000),f0[15,26]->((1236783980314300060272216796875 (Sqrt[15]))/474847193030433794692427118804992),f0[17,26]->((1893075985746630060921153753217001087682881 (Sqrt[17]))/99388363662291403336915952421353847521280000),f0[18,26]->(158415752522855876702566542417/(683455497832238547992576000000 (Sqrt[2]))),f0[14,28]->((9387480337647754305649 (Sqrt[(7/2)]))/212424415209221951324160000),f0[15,28]->((9583205159604549407958984375 (Sqrt[15]))/33917656645030985335173365628928),f0[17,28]->((35495071438462368988598652931321269666341213 (Sqrt[17]))/8075304547561176521124421134235000111104000000),f0[18,28]->(5714795635986492039924197439201/(87482303722526534143049728000000 (Sqrt[2]))),f0[15,30]->((5195713640749454498291015625 (Sqrt[15]))/474847193030433794692427118804992),f0[17,30]->((14428799658214237737043831013716471691934037 (Sqrt[17]))/19877672732458280667383190484270769504256000000),f0[18,30]->(866626588897278303908488615911/(60144083809236992223346688000000 (Sqrt[2]))),f0[17,32]->((40254497110927943179349807054456171205137 (Sqrt[17]))/559328453510730841290003195444848492544000000),f0[18,32]->(18616114953890247749341199093829/(8179595398056230942375149568000000 (Sqrt[2]))),f0[17,34]->((684326450885775034048946719925754910487329 (Sqrt[17]))/258409745521957648675981476295520003555328000000),f0[18,34]->(221376048770573507241315373383/(1022449424757028867796893696000000 (Sqrt[2]))),f0[18,36]->(127173474825648610542883299603/(16359190796112461884750299136000000 (Sqrt[2])))};

4.2. Order (λ^(-1/4))

At this order, the amplitude should vanish. Imposing this fixes the order (λ^(-1/4)) CFT data.

4.3. Order (λ^(-1/2))

At this order, we match the conformal block expansion with the world-sheet ansatz, to fix both the order (λ^(-1/2)) CFT data and the free coefficients in the Ansatz.

5. Result

f2list={f2[1,0]->(219/1024),f2[1,2]->(93/1024),f2[2,0]->-(1221/(1024 (Sqrt[2]))),f2[2,2]->(87/(64 (Sqrt[2]))),f2[2,4]->(213/(1024 (Sqrt[2]))),f2[3,0]->-((3050955 (Sqrt[3]))/524288),f2[3,2]->-((5984055 (Sqrt[3]))/1048576),f2[3,4]->-((145773 (Sqrt[3]))/524288),f2[3,6]->((38151 (Sqrt[3]))/1048576),f2[4,0]->-(833141/20736),f2[4,2]->-(1191853/20736),f2[4,4]->-(888515/41472),f2[4,6]->-(55355/20736),f2[4,8]->-(299/4608),f2[5,0]->-((8542435552735 (Sqrt[5]))/173946175488),f2[5,2]->-((13833719779295 (Sqrt[5]))/173946175488),f2[5,4]->-((1853060419375 (Sqrt[5]))/43486543872),f2[5,6]->-((4755815646875 (Sqrt[5]))/347892350976),f2[5,8]->-((329376953125 (Sqrt[5]))/173946175488),f2[5,10]->-((2693359375 (Sqrt[5]))/38654705664),f2[6,0]->-((208250553303 (Sqrt[(3/2)]))/1048576000),f2[6,2]->-((22368054537 (Sqrt[(3/2)]))/65536000),f2[6,4]->-((2277062867799 (Sqrt[(3/2)]))/10485760000),f2[6,6]->-((12865141413 (Sqrt[(3/2)]))/131072000),f2[6,8]->-((147632470533 (Sqrt[(3/2)]))/5242880000),f2[6,10]->-((101964501 (Sqrt[(3/2)]))/26214400),f2[6,12]->-((65091681 (Sqrt[(3/2)]))/419430400),f2[7,0]->-((3560268200161592143 (Sqrt[7]))/20038599416217600),f2[7,2]->-((211060636443019114717 (Sqrt[7]))/667953313873920000),f2[7,4]->-((74251416037463753977 (Sqrt[7]))/333976656936960000),f2[7,6]->-((5391162478277524753 (Sqrt[7]))/44530220924928000),f2[7,8]->-((8074005570904514059 (Sqrt[7]))/166988328468480000),f2[7,10]->-((8445040150227707453 (Sqrt[7]))/667953313873920000),f2[7,12]->-((67705998948549997 (Sqrt[7]))/40077198832435200),f2[7,14]->-((611078988636949 (Sqrt[7]))/8906044184985600),f2[8,0]->-(81704161715341/(70013160000 (Sqrt[2]))),f2[8,2]->-(148563163311277/(70013160000 (Sqrt[2]))),f2[8,4]->-(10632031232159/(6667920000 (Sqrt[2]))),f2[8,6]->-(506668900757/(518616000 (Sqrt[2]))),f2[8,8]->-(22098096093817/(46675440000 (Sqrt[2]))),f2[8,10]->-(1497642991421/(8751645000 (Sqrt[2]))),f2[8,12]->-(3686245601/(89302500 (Sqrt[2]))),f2[8,14]->-((2358383 (Sqrt[2]))/893025),f2[8,16]->-((10603 (Sqrt[2]))/99225),f2[9,0]->-(72913759957443317508079893/54065713526582804480000),f2[9,2]->-(134643287123897830242254991/54065713526582804480000),f2[9,4]->-(13244322323951517522607659/6758214190822850560000),f2[9,6]->-(1408386559520149760160081/1081314270531656089600),f2[9,8]->-(9694451967687390682380897/13516428381645701120000),f2[9,10]->-(8544307048431761577970053/27032856763291402240000),f2[9,12]->-(709163425186050348183069/6758214190822850560000),f2[9,14]->-(51966809641299653389809/2206763817411543040000),f2[9,16]->-(25392576198025516131/8827055269646172160),f2[9,18]->-(10141579493586125793/88270552696461721600),f2[10,0]->-((102806681191612804465615 (Sqrt[(5/2)]))/77942537061308301312),f2[10,2]->-((49385864534930122415 (Sqrt[(5/2)]))/20046948832641024),f2[10,4]->-((5790041119606174319765 (Sqrt[(5/2)]))/2886760631900307456),f2[10,6]->-((1148482965262962780625 (Sqrt[(5/2)]))/811901427721961472),f2[10,8]->-((5540758679914387203125 (Sqrt[(5/2)]))/6495211421775691776),f2[10,10]->-((349196876586086328125 (Sqrt[(5/2)]))/811901427721961472),f2[10,12]->-((27236845995886357421875 (Sqrt[(5/2)]))/155885074122616602624),f2[10,14]->-((174786015691650390625 (Sqrt[(5/2)]))/3247605710887845888),f2[10,16]->-((856742789306640625 (Sqrt[(5/2)]))/75745905793302528),f2[10,18]->-((87406158447265625 (Sqrt[(5/2)]))/66277667569139712),f2[10,20]->-((6098175048828125 (Sqrt[(5/2)]))/117826964567359488),f2Zeta3[1,0]->(7/16),f2Zeta3[1,2]->(1/16),f2Zeta3[2,0]->(15/(2 (Sqrt[2]))),f2Zeta3[2,2]->4 (Sqrt[2]),f2Zeta3[2,4]->(1/(2 (Sqrt[2]))),f2Zeta3[3,0]->((111699 (Sqrt[3]))/8192),f2Zeta3[3,2]->((321327 (Sqrt[3]))/16384),f2Zeta3[3,4]->((56133 (Sqrt[3]))/8192),f2Zeta3[3,6]->((6561 (Sqrt[3]))/16384),f2Zeta3[4,0]->(208/3),f2Zeta3[4,2]->112,f2Zeta3[4,4]->(520/9),f2Zeta3[4,6]->16,f2Zeta3[4,8]->(8/9),f2Zeta3[5,0]->((2417930625 (Sqrt[5]))/33554432),f2Zeta3[5,2]->((37276245625 (Sqrt[5]))/301989888),f2Zeta3[5,4]->((5787265625 (Sqrt[5]))/75497472),f2Zeta3[5,6]->((2170703125 (Sqrt[5]))/67108864),f2Zeta3[5,8]->((2294921875 (Sqrt[5]))/301989888),f2Zeta3[5,10]->((244140625 (Sqrt[5]))/603979776),f2Zeta3[6,0]->((107758107 (Sqrt[(3/2)]))/409600),f2Zeta3[6,2]->((11942883 (Sqrt[(3/2)]))/25600),f2Zeta3[6,4]->((264444831 (Sqrt[(3/2)]))/819200),f2Zeta3[6,6]->((1728459 (Sqrt[(3/2)]))/10240),f2Zeta3[6,8]->((25174557 (Sqrt[(3/2)]))/409600),f2Zeta3[6,10]->((649539 (Sqrt[(3/2)]))/51200),f2Zeta3[6,12]->((531441 (Sqrt[(3/2)]))/819200),f2Zeta3[7,0]->((7628681035076171 (Sqrt[7]))/34789235097600),f2Zeta3[7,2]->((55349789187715771 (Sqrt[7]))/139156940390400),f2Zeta3[7,4]->((760924594225117 (Sqrt[7]))/2576980377600),f2Zeta3[7,6]->((544585606337773 (Sqrt[7]))/3092376453120),f2Zeta3[7,8]->((312545586681797 (Sqrt[7]))/3865470566400),f2Zeta3[7,10]->((3638085451772443 (Sqrt[7]))/139156940390400),f2Zeta3[7,12]->((337076867205953 (Sqrt[7]))/69578470195200),f2Zeta3[7,14]->((33232930569601 (Sqrt[7]))/139156940390400),f2Zeta3[8,0]->((13568768 (Sqrt[2]))/19845),f2Zeta3[8,2]->((41676032 (Sqrt[2]))/33075),f2Zeta3[8,4]->((97486976 (Sqrt[2]))/99225),f2Zeta3[8,6]->((156928 (Sqrt[2]))/245),f2Zeta3[8,8]->((33773696 (Sqrt[2]))/99225),f2Zeta3[8,10]->((4646912 (Sqrt[2]))/33075),f2Zeta3[8,12]->((815104 (Sqrt[2]))/19845),f2Zeta3[8,14]->((32768 (Sqrt[2]))/4725),f2Zeta3[8,16]->((32768 (Sqrt[2]))/99225),f2Zeta3[9,0]->(1047365529672045866823/689613692941107200),f2Zeta3[9,2]->(1952881975248974658549/689613692941107200),f2Zeta3[9,4]->(28189077519484073439/12314530231091200),f2Zeta3[9,6]->(78381768289952427669/49258120924364800),f2Zeta3[9,8]->(161301681728674567371/172403423235276800),f2Zeta3[9,10]->(155772900678349762407/344806846470553600),f2Zeta3[9,12]->(2095936985890539657/12314530231091200),f2Zeta3[9,14]->(62832002672456129979/1379227385882214400),f2Zeta3[9,16]->(4886414238002304717/689613692941107200),f2Zeta3[9,18]->(450283905890997363/1379227385882214400),f2Zeta3[10,0]->((496914769105090625 (Sqrt[(5/2)]))/345196185255936),f2Zeta3[10,2]->((58429065155058125 (Sqrt[(5/2)]))/21574761578496),f2Zeta3[10,4]->((776751423284745625 (Sqrt[(5/2)]))/345196185255936),f2Zeta3[10,6]->((17709192828671875 (Sqrt[(5/2)]))/10787380789248),f2Zeta3[10,8]->((89722359131640625 (Sqrt[(5/2)]))/86299046313984),f2Zeta3[10,10]->((6045969677734375 (Sqrt[(5/2)]))/10787380789248),f2Zeta3[10,12]->((171971596923828125 (Sqrt[(5/2)]))/690392370511872),f2Zeta3[10,14]->((3735321044921875 (Sqrt[(5/2)]))/43149523156992),f2Zeta3[10,16]->((7391815185546875 (Sqrt[(5/2)]))/345196185255936),f2Zeta3[10,18]->((19073486328125 (Sqrt[(5/2)]))/6164217593856),f2Zeta3[10,20]->((95367431640625 (Sqrt[(5/2)]))/690392370511872),τ2[1,0]->(15/56),τ2[1,2]->(5/8),τ2[2,0]->(89/(40 (Sqrt[2]))),τ2[2,2]->(15/(8 (Sqrt[2]))),τ2[2,4]->(21/(8 (Sqrt[2]))),τ2[3,0]->((20677 (Sqrt[3]))/11032),τ2[3,2]->(168623/(31736 (Sqrt[3]))),τ2[3,4]->(3089/(616 (Sqrt[3]))),τ2[3,6]->(49/(8 (Sqrt[3]))),τ2[4,0]->(9911/1872),τ2[4,2]->(15643/3024),τ2[4,4]->(15527/3120),τ2[4,6]->(2093/432),τ2[4,8]->(89/16),τ2[5,0]->((2869115087 (Sqrt[5]))/835636824),τ2[5,2]->((4858716979 (Sqrt[5]))/1431407832),τ2[5,4]->((5889979 (Sqrt[5]))/1777848),τ2[5,6]->(19371371/(1200312 (Sqrt[5]))),τ2[5,8]->(17933/(1128 (Sqrt[5]))),τ2[5,10]->(141/(8 (Sqrt[5]))),τ2[6,0]->(90004261/(3547592 (Sqrt[6]))),τ2[6,2]->((619372 (Sqrt[6]))/147443),τ2[6,4]->((154262003 (Sqrt[(3/2)]))/18655720),τ2[6,6]->((9610 (Sqrt[6]))/2371),τ2[6,8]->(3662461/(153480 (Sqrt[6]))),τ2[6,10]->((130 (Sqrt[(2/3)]))/11),τ2[6,12]->(205/(8 (Sqrt[6]))),τ2[7,0]->(2684330302694221/(76255037418504 (Sqrt[7]))),τ2[7,2]->(32324089806855151/(922112273014840 (Sqrt[7]))),τ2[7,4]->(11532669047539/(332627262840 (Sqrt[7]))),τ2[7,6]->(55317926557/(1619440680 (Sqrt[7]))),τ2[7,8]->(4462704511/(132774360 (Sqrt[7]))),τ2[7,10]->(49745497/(1501960 (Sqrt[7]))),τ2[7,12]->(56065/(1704 (Sqrt[7]))),τ2[7,14]->(281/(8 (Sqrt[7]))),τ2[8,0]->(10387858357/(445225200 (Sqrt[2]))),τ2[8,2]->(19087991689/(820496880 (Sqrt[2]))),τ2[8,4]->(1406250193/(60929360 (Sqrt[2]))),τ2[8,6]->(345031/(15120 (Sqrt[2]))),τ2[8,8]->(3327802619/(147759920 (Sqrt[2]))),τ2[8,10]->(254097737/(11435760 (Sqrt[2]))),τ2[8,12]->(5242529/(238800 (Sqrt[2]))),τ2[8,14]->(22021/(1008 (Sqrt[2]))),τ2[8,16]->(369/(16 (Sqrt[2]))),τ2[9,0]->(296773713861252515861/14899270858516122680),τ2[9,2]->(1656534145985722154201/83342013880462262520),τ2[9,4]->(1422097266735499757/71974894303152120),τ2[9,6]->(1132295795518500979/57815737155509928),τ2[9,8]->(6776798475429569/349732181180760),τ2[9,10]->(26622542423147/1389896094440),τ2[9,12]->(275387055239/14545350120),τ2[9,14]->(686271371/36620520),τ2[9,16]->(43759/2344),τ2[9,18]->(469/24),τ2[10,0]->(3316164417810200067/(44523563311816120 (Sqrt[10]))),τ2[10,2]->((77859009324721903 (Sqrt[(5/2)]))/5235244237893208),τ2[10,4]->(5151420304355374171/(69596927526313208 (Sqrt[10]))),τ2[10,6]->(933015698438137/(12693949419592 (Sqrt[10]))),τ2[10,8]->(72093796209679/(989430569624 (Sqrt[10]))),τ2[10,10]->(500327588693/(6934001704 (Sqrt[10]))),τ2[10,12]->(2817080617479/(39446157016 (Sqrt[10]))),τ2[10,14]->(484514393/(6854344 (Sqrt[10]))),τ2[10,16]->(27150039/(387544 (Sqrt[10]))),τ2[10,18]->(3911/(56 (Sqrt[10]))),τ2[10,20]->(581/(8 (Sqrt[10])))};